Shell-like Structures: Non-classical Theories and Applications

Advanced Structured Materials Volume 15 Series Editors Andreas Öchsner, Johor, Malaysia Lucas Filipe Martins da Silva,...

Author: Holm Altenbach | Victor A. Eremeyev

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Advanced Structured Materials Volume 15

Series Editors Andreas Öchsner, Johor, Malaysia Lucas Filipe Martins da Silva, Porto, Portugal Holm Altenbach, Halle, Germany

For further volumes: http://www.springer.com/series/8611

Holm Altenbach Victor A. Eremeyev Editors •

Shell-like Structures Non-classical Theories and Applications

123

Prof. Dr.-Ing.habil. Dr.h.c. Holm Altenbach Lehrstuhl Technische Mechanik Institut für Mechanik Fakultät für Maschinenbau Otto-von-Guericke-Universität Magdeburg Universitätsplatz 2 D-39106 Magdeburg Germany e-mail: [email protected]

Prof. Dr. Victor A. Eremeyev Lehrstuhl Technische Mechanik Institut für Mechanik Fakultät für Maschinenbau Otto-von-Guericke-Universität Magdeburg Universitätsplatz 2 D-39106 Magdeburg Germany and South Federal University and South Science Centre of RASci Rostov on Don Russia e-mail: [email protected]

ISSN 1869-8433 ISBN 978-3-642-21854-5 DOI 10.1007/978-3-642-21855-2

e-ISSN 1869-8441 e-ISBN 978-3-642-21855-2

Springer Heidelberg Dordrecht London New York Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Shell-like structures are used in Civil and Aero-space engineering as basic structural elements. As a model of analysis are also used such structures in other branches, e.g. Mechanical engineering, but also in new branches like Medicine and Biology. New applications are primarily related to new materials—instead of steel or concrete, now one has to analyze laminates, foams, functionally graded materials, nanofilms, biological membranes, soft tissues, etc. The new trends in applications demand the improvements of the theoretical foundations of shell theory, since new effects must be taken into account. For example, in the case of small-size shell-like structures (thin films, multiwalled nanotubes) the surface effect plays an important role in the mechanical analysis of these structural elements. Scientific meetings like conferences or colloquia are regularly organized by civil engineers, mathematicians, etc. Within the EUROMECH Colloquium 527 Shell-like structures – Non-classical Theories and Applications, which will bring together specialists from different areas, various items related to the colloquium title are discussed by international experts. The forum was addressed to scientists and researchers from industries. The focus was related to the following problems: new theories (based on two-dimensional field equations but describing non-classical effects), mathematical methods, e.g. the asymptotical analysis, new constitutive equations (for materials like sandwiches, foams, biological membranes, etc. and which can be combined with the two-dimensional shell equations), complex structures (folded, branching and/or self-intersecting shell structures, etc.) and shell-like structures on different scales (thin- and nanofilms, nanotubes, and nanoparticles) or very thin structures (similar to membranes, but with bending stiffness). In addition, coupled effects, phase transitions in shells and refined shell thermodynamics are discussed. This book contains papers submitted before the above mentioned colloquium. The reason for publication on such early stage was that a lot of the tentative colloquium participants gave a positive response on our call to publish their contribution in advance. By this way each participant is informed in some details about the forthcoming presentations at the beginning of the conference, which v

vi

Preface

stimulates a deeper scientific discussion. Finally, more than 50 papers were received and after reviewing 48 were included in this volume. It should be mentioned that some very long submissions (over 50 pages) were rejected and shorter versions was prepared for publication in this volume. The Full-length contributions will be published in the SpringerBriefs (http://www.springer.com/briefs) in 2011. This book deals with various items of the theories of shells, plates, beams, etc. In addition, traditional and new applications are presented. Its contents is split into 8 parts: • • • • • • • •

Mathematical Problems Dynamics and Stability Nonlinear Models and Coupled Fields Numerical Analysis Engineering Design Micro- and Nanomechanical Applications Biomechanics FGM and Laminated Plates and Shells

This monograph is not only based on the authors’ contributions. The publication and the conference were supported by different organizations and you can find some personal acknowledgements at the end of various papers. Here we acknowledge our main supporters: • • • • • •

Martin-Luther-Universität Halle-Wittenberg, Leucorea (Lutherstadt Wittenberg), Deutsche Forschungsgemeinschaft, Deutscher Akademischer Austauschdienst, Japan Society for the Promotion of Science (ID No. RC 21115001), Springer Publisher.

Only a few persons will be listed here: Dr. Christoph Baumann (Springer) for solving all our publishing problems just in time, Mrs. Barbara Renner for checking the language of some papers, Ms. Anna Girchenko, for writing a lot of letters, etc., Mr. Andreas Kutschke for solving technical and other problems. Last but not least we have to thank Prof. Henry Petryk (representative of the EUROMECH) and the members of the scientific committee (Profs René de Borst, Elena A. Ivanova, Reinhold Kienzler, Gennady I. Mikhasev, Wojciech Pietraszkiewicz, Paolo Podio-Guidugli) for their support. Prof. Dr. Holm Altenbach Prof. Dr. Victor A. Eremeyev

Contents

Part I Mathematical Problems 1

Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells and the Asymptotic Method of Their Solution . . . Lenser Abgar Aghalovyan

3

2

Analytical Solution for the Bending of a Plate on a Functionally Graded Layer of Complex Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Sergey Aizikovich, Andrey Vasiliev, Igor Sevostianov, Irina Trubchik, Ludmila Evich and Elena Ambalova

3

Analysis of the Deformation of Multi-layered Orthotropic Cylindrical Elastic Shells Using the Direct Approach . . . . . . . . . . . . . . . 29 Mircea Bˆırsan and Holm Altenbach

4

Asymptotic Integration of One Narrow Plate Problem . . . . . . . . . . . . . . 53 Valentina O. Finiukova and Alexander M. Stolyar

5

On Cusped Shell-like Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 George Jaiani

6

Effect of the Tangential Loads on the Bending of Elastic Plates . . . . . . 75 Kristine L. Martirosyan

7

On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Jemal Peradze

8

Mathematical Models of Micropolar Elastic Thin Shells . . . . . . . . . . . . 91 Samvel H. Sargsyan

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Contents

Part II Dynamics and Stability 9

Closed-Form Approximate Solution for the Postbuckling Behavior of Orthotropic Shallow Shells Under Axial Compression . . . . . . . . . . . . 103 Matthias Beerhorst, Michael Seibel and Christian Mittelstedt

10 Nonlinear Magnetoelastic Waves in a Plate . . . . . . . . . . . . . . . . . . . . . . . . 125 Vladimir I. Erofeev, Alexey O. Malkhanov, Aleksandr I. Zemlyanukhin and Vladimir M. Catson 11 Basic Concepts in the Stability Theory of Thin-Walled Structures . . . 135 Ardeshir Guran and Leonid P. Lebedev 12 High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Elena A. Ivanova 13 On the Reconstruction of Inhomogeneous Initial Stresses in Plates . . . 165 Rostislav D. Nedin and Alexander O. Vatulyan 14 Dynamic Response of Pre-Stressed Spatially Curved ThinWalled Beams of Open Profile Impacted by a Falling Elastic Hemispherical-Nosed Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Yury A. Rossikhin and Marina V. Shitikova 15 On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Denis Sheydakov Part III Nonlinear Models and Coupled Fields 16 On the Nonlinear Theory of Two-Phase Shells . . . . . . . . . . . . . . . . . . . . . 219 Victor A. Eremeyev and Wojciech Pietraszkiewicz 17 A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 An Danh Nguyen, Marcus Stoffel and Dieter Weichert 18 On Constitutive Restrictions in the Resultant Thermomechanics of Shells with Interstitial Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Wojciech Pietraszkiewicz 19 Free Finite Rotations in Deformation of Thin Bodies . . . . . . . . . . . . . . . 261 Leonid I. Shkutin 20 On Universal Deformations of Nonlinear Isotropic Elastic Shells . . . . 279 Leonid M. Zubov

Contents

ix

Part IV Numerical Analysis 21 Application of Genetic Algorithms to the Shape Optimization of the Nonlinearly Elastic Corrugated Membranes . . . . . . . . . . . . . . . . . . . . . . 297 Mikhail Karyakin and Taisiya Sigaeva 22 Advances in Quadrilateral Shell Elements with Drilling Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Stephan Kugler, Peter A. Fotiu and Justin Murin 23 Invariant-Based Geometrically Nonlinear Formulation of a Triangular Finite Element of Laminated Shells . . . . . . . . . . . . . . . . . . . . 329 Stanislav V. Levyakov 24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Teodoro Merlini and Marco Morandini 25 An Algorithm for the Automatisation of Pseudo Reductions of PDE Systems Arising from the Uniform-Approximation Technique . . . . . . . 377 Patrick Schneider and Reinhold Kienzler 26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Krzysztof Wi´sniewski and Ewa Turska Part V Engineering Design 27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core – Numerical Aspects and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Vyacheslav N. Burlayenko and Tomasz Sadowski 28 On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions 441 Jacek Chrósćielewski, Violetta Konopińska and Wojciech Pietraszkiewicz 29 Thermal Stress and Strain of Solar Cells in Photovoltaic Modules . . . 453 Ulrich Eitner, Sarah Kajari-Schröder, Marc Köntges and Holm Altenbach 30 Computational Models of Laminated Glass Plate under Transverse Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Ivelin V. Ivanov, Dimitar S. Velchev, Tomasz Sadowski and Marcin Kneć 31 Unbending of Curved Tube by Internal Pressure . . . . . . . . . . . . . . . . . . . 491 Alexei M. Kolesnikov

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Contents

32 Characterization of Polymeric Interlayers in Laminated Glass Beams for Photovoltaic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Stefan-H. Schulze, Matthias Pander, Konstantin Naumenko, Anna Girchenko and Holm Altenbach 33 On the Determination of Edge Reinforcement Properties for Optimum Lightweight Design of Composite Stiffeners . . . . . . . . . . . . . . 507 Philipp Weißgraeber, Christian Mittelstedt and Wilfried Becker Part VI Micro- and Nanomechanical Applications 34 Evaluation of the Mechanical Parameters of Nanotubes by Means of Nonclassical Theories of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Svetlana M. Bauer, Andrei M. Ermakov, Stanislava V. Kashtanova and Nikita F. Morozov 35 Gauss-Codazzi Equations for Thin Films and Nanotubes Containing Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Svyatoslav Derezin 36 Effective Mechanical Properties of Closed-Cell Foams Investigated with a Microstructural Model and Numerical Homogenisation . . . . . . 549 Nina-Carolin Fahlbusch and Wilfried Becker 37 What Shell Theory Fits Carbon Nanotubes? . . . . . . . . . . . . . . . . . . . . . . 561 Antonino Favata and Paolo Podio–Guidugli 38 A Variationally Consistent Derivation of Microcontinuum Theories . . 571 Johannes Meenen, Holm Altenbach, Victor Eremeyev and Konstantin Naumenko 39 Shell-Models for Multi- L ayer Carbon Nano- Particles . . . . . . . . . . . . . . 585 Melanie Todt, Franz G. Rammerstorfer, Markus A. Hartmann, Oskar Paris and Franz D. Fischer Part VII Biomechanics 40 Mechanics of Biological Membranes from Lattice Homogenization . . 605 Mohamed Assidi, Francisco Dos Reis and Jean François Ganghoffer 41 Biological and Synthetic Membranes: Modeling and Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 Rasa Kazakevi˘ci¯ut˙e-Makovska and Holger Steeb 42 Nonclassical Theories of Shells in Application to Soft Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 Eva B. Voronkova, Svetlana M. Bauer and Anders Eriksson

Contents

xi

Part VIII FGM and Laminated Plates and Shells 43 Axisymmetric Bending Analysis of Two Directional Functionally Graded Circular Plates Using Third Order Shear Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 Reza Akbari Alashti and Hossein Rahbari 44 Stability Analysis of Functionally Graded Plates Subject to Thermal Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 Mokhtar Bouazza, A. Tounsi, E. A. Adda-Bedia and A. Megueni 45 A Best Theory Diagram for Metallic and Laminated Shells . . . . . . . . . 681 Erasmo Carrera, Maria Cinefra and Marco Petrolo 46 In-Plane Strain and Stress Fields in Theories of Shearable Laminated Plates Subject to Transverse Loads . . . . . . . . . . . . . . . . . . . . 699 Giovanni Formica, Marzio Lembo and Paolo Podio-Guidugli 47 On the Use of a New Concept of Sampling Surfaces in Shell Theory . . 715 Gennady M. Kulikov and Svetlana V. Plotnikova 48 Theory of Thin Adaptive Laminated Shells Based on Magnetorheological Materials and Its Application in Problems on Vibration Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 Gennady I. Mikhasev, Marina G. Botogova and Evgeniya V. Korobko

Part I

Mathematical Problems

Chapter 1

Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells and the Asymptotic Method of Their Solution Lenser Abgar Aghalovyan

Abstract Classical and improved theories of shells consider only one class of boundary value problems - on the facial surfaces of the shell the values of the corresponding stress tensor components are given. If on the facial surfaces other conditions - displacement vector or mixed boundary conditions of theory of elasticity - are given, a nonclassical boundary value problem of the theory of shells arises, which is important in some application cases. It is proved that the Kirchhoff-Love hypotheses of the classical theory of shells are not applicable for the solution of this class of problems. It is shown that such problems can be solved by the asymptotic method of solution of singularly perturbed differential equations. Non-contradictory asymptotic orders of the stress tensor components and a displacement vector are established. The iteration processes for the determination of all sought values with the beforehand given asymptotic accuracy are built. As special cases, the solutions of static and dynamic problems, illustrating the effectiveness of the asymptotic method, are presented. Nonclassical problems related to free and forced vibrations of isotropic and anisotropic shells are considered. The amplitudes of forced vibrations are determined. For layered shells the efficiency of the method is shown. Keywords Asymptotic method · Nonclassical boundary value problems · Shell theory · Layered shell

1.1 Introduction The classical theory of plates and shells considers one class of boundary value problems only. It is assumed that on the facial surfaces of the plate or the shell the values of the corresponding stress tensor components (first boundary value problem of L. A. Aghalovyan (B) Institute of Mechanics of National Academy of Sciences of Armenia, Armenia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 1, © Springer-Verlag Berlin Heidelberg 2011

3

4

L.A. Aghalovyan

elasticity theory) are given. Improved theories of plates and shells (in the sense of Reissner, Ambartsumian or Timoshenko) are devoted to this case, too. With respect to applications, particularly, in soil mechanics or foundation engineering, seismology, geophysics, in interaction of thin and rigid bodies and others, the problems ifon the facial surfaces of the plate or the shell other conditions—displacement vector or mixed conditions of elasticity theory - are given, are important. It is proved that the Kirchhoff-Love hypotheses of the classical theory are not applicable for the solution of these classes of problems [1]. Taking into account that plates and shells are structural elements—one of their geometrical dimensions (thickness) significantly differs from the others. Hence, it is possible, to correlate the three-dimensional problem of elasticity theory to dimensionless coordinates and components of the displacement vector and to obtain a system of equations containing a small dimensionless geometrical parameter. This system of differential equations turned out to be singularly perturbed by the small parameter. For its solution the asymptotic method is found to be effective. The solution of similar equations and systems is the combination of the solutions of the internal problem (I int) and the boundary layer (Ib ) [7, 8] I = I int + Ib

(1.1)

The asymptotic theory of isotropic shells (the first boundary problem of elasticity theory) was established by Goldenveizer [6] among others and for anisotropic plates and shells, for example, in [1]. Below we show the effectiveness of the asymptotic method for the solution of static and dynamic problems of anisotropic homogeneous and multi-layered shells considering the second or the mixed boundary value problem.

1.2 Second and Mixed Static Boundary Value Problem of Anisotropic Thermoelastic Shells Let us have an anisotropic shell of thickness 2h covering the domain D = {(α, β, γ) : (α, β) ∈ D0 , −h ≤ γ ≤ h}, where D0 is the middle surface, related to the lines of curvature α, β, the rectilinear axis γ is directed perpendicularly to the middle surface. The given volume forces with the component F j (α, β, γ) and temperature effects, which are taken into account by the Duhamel-Neumann model, acting on the shell. The change of the temperature field is characterized by the function θ(α, β, γ) = T (α, β, γ) − T 0 (α, β, γ), which is considered to be known. Anisotropy is assumed to be general and is described by 21 elasticity parameters. It is required to find the solutions of the equations and correlations of the spatial problem of elasticity theory in the domain D under the boundary conditions uα (−h) = u− (α, β), uβ (−h) = v− (α, β), uγ (−h) = u−γ (α, β)

(1.2)

1 Nonclassical Spatial Boundary Value Problems of Statics and Dynamics of Shells

5

and in one of the variants of the boundary conditions at γ = h a) σαγ (h) = σ+αγ , (α, β), σγγ (h) = σ+γγ (α, β), b) uα (h) = u+ (α, β), uβ (h) = v+ (α, β), uγ (h) = w+ (α, β),

(1.3) (1.4)

c) uα (h) = u+ (α, β), uβ (h) = v+ (α, β),

(1.5)

d) σαγ (h) =

σ+αγ (α, β),

+

σγγ (h) = σ+γγ (α, β),

uγ (h) = w (α, β)

(1.6)

and under the conditions on the lateral surface of the shell, which are not specified, but they result in the boundary layer. It is considered that the functions σ+jk , u± , v± , w± belong to the class C n , ∀n. In order to decrease the calculations, we use the components of the nonsymmetric stress tensor τi j : ταα = (1 + γ/R2)σαα , τββ = (1 + γ/R1)σββ , τγγ = (1 + γ/R1)(1 + γ/R2)σγγ , ταγ = (1 + γ/R2)σαγ , τβγ = (1 + γ/R1)σβγ , (1 + γ/R1)ταβ = (1 + γ/R2)τβα

(1.7)

ταβ = (1 + γ/R2)σαβ , τβα = (1 + γ/R1)σαβ , (the symmetry condition),

where σ jk are the components of the symmetric tensor. The following equilibrium and constitutive equations for an anisotropic body, taking into account temperature deformations, in the chosen system of coordinates are valid: Equilibrium equations ∂ταγ 2ταγ 1 ∂ 1 ∂ (Bταα ) − kβ τββ + (Aτβα ) + kα ταβ + (1 + γ/R1) + + Fα∗ = 0, AB ∂α AB ∂β ∂γ R1 ∂τβγ 2τβγ 1 ∂ 1 ∂ (Bτββ ) − kα ταα + (Bταβ ) + kβ τβα + (1 + γ/R2) + + Fβ∗ = 0, AB ∂β AB ∂α ∂γ R 2 ∂τγγ ταα τββ 1 ∂ταγ 1 ∂τβγ − + + + + kβ ταγ + kα τβγ + F γ∗ = 0, ∂γ R1 R2 A ∂α B ∂β F ∗j = (1 + γ/R1)(1 + γ/R2)F j (α, β, γ), j = α, β, γ (1.8) Constitutive equations (state equations) uγ γ 1 ∂uα γ 1+ + k α uβ + = 1+ (a11 ταα + a15ταγ + a16 ταβ ) R2 A ∂α R1 R1 γ + 1+ (a12τββ + a14τβγ ) + a13τγγ + α11 δ11 (γ)θ, R2 uγ γ 1 ∂uβ γ 1+ + k β uα + = 1+ (a12 ταα + a25ταγ + a26 ταβ ) R1 B ∂β R2 R1 γ + 1+ (a22τββ + a24τβγ ) + a23τγγ + α22 δ11 (γ)θ, R2

6

L.A. Aghalovyan

∂uγ γ δ11 (γ) = 1+ (a13 ταα + a35ταγ + a36 ταβ ) ∂γ R1 γ + 1+ (a23 τββ + a34 τβγ ) + a33τγγ + α33 δ11 (γ)θ, R2 γ 1 ∂uα γ 1 ∂uβ 1+ − kβ u β + 1 + − k α uα R1 B ∂β R2 A ∂α γ = 1+ (a16 ταα + a56ταγ + a66 ταβ ) + a36τγγ R1 γ + 1+ (a26 τββ + a46 τβγ ) + α12δ11 (γ)θ, R2 ∂uα γ uα 1 γ ∂uγ δ11 (γ) − 1+ + 1+ ∂γ R2 R1 A R2 ∂α γ = 1+ (a15 ταα + a55ταγ + a56 ταβ ) + a35τγγ R1 γ + 1+ (a25 τββ + a45 τβγ ) + α13δ11 (γ)θ, R2 ∂uβ γ uβ 1 γ ∂uγ δ11 (γ) − 1+ + 1+ ∂γ R 1 R2 B R1 ∂β γ = 1+ (a14 ταα + a45ταγ + a46 ταβ ) + a34τγγ R1 γ + 1+ (a24 τββ + a44 τβγ ) + α23δ11 (γ)θ, R2 1 1 γ2 δ11 (γ) = 1 + γ + + R1 R 2 R1 R2

(1.9)

1 ∂A 1 ∂B where kα = AB ∂β , kβ = AB ∂α are geodesic curvatures, A, B are the coefficients of the first quadratic form, R1 , R2 are the principal radii of the middle (coordinate) surface curvature of the shell, ai j are the elasticity parameters, αi j are the coefficients of the thermal expansion. In order to solve the stated boundary problems, in Eqs (1.8) and (1.9) we introduce dimensionless coordinates and displacements by formulae

α = Rξ, β = Rη, γ = εRζ = hζ, uα = Ru, uβ = Rv, uγ = Rw,

(1.10)

where R is the characteristic dimension of the shell (least of radii of curvature and linear dimensions of the middle surface), ε = h/R is the small parameter. As a result we have the system, which is singularly perturbed with respect to the parameter ε. Its solution has the form of (1.1). The solution of the internal problem will be sought in the form of I int = εqI +s I (s) ,

s = 0, N

(1.11)


7

The summation by integer values from 0 to N, N is the number of approximations. The values of indexes q I should be determined in that moment, that after substitution (1.11) into the transformed, according to (1.10), equations (1.8), (1.9) and equating in each equation the coefficients with similar degrees of ε, we can get a non-contradictory system for determining desired values I (s) . Such a system was first time presented in [1] qσ = −1 for stresses,

qu = 0 for all displacements

(1.12)

Note that the asymptotics (1.11), (1.12) essentially differ from the asymptotics in the first boundary-value problem (classical theory) [6] and specific conclusion arises. The contribution of the volume forces and the temperature field will be comparable with the contribution of exterior forces, if the corresponding volumes allow the decompositions F α = ε−2+s F α(s) ,

(α, β, γ);

θ = ε−1+s θ(s) ,

s = 0, N

(1.13)

Substituting (1.11)-(1.13) into the transformed system (1.8), (1.9) and solving again the obtained system, we get: (s) (s) τ(s) αγ (ξ, η, ζ) = ταγ0 (ξ, η) + ταγ∗ (ξ, η, ζ),

(α, β)

(s) (s) τ(s) γγ (ξ, η, ζ) = τγγ0 (ξ, η) + τγγ∗ (ξ, η, ζ), (s) (s) (s) (s) τ(s) αα (ξ, η, ζ) = A13 τγγ0 + A14 τβγ0 + A15 ταγ0 + ταα∗ (ξ, η, ζ),

(α, β; 1, 2)

(s) (s) (s) (s) τ(s) αβ (ξ, η, ζ) = A63 τγγ0 + A64 τβγ0 + A65 ταγ0 + ταβ∗ (ξ, η, ζ),

(1.14)

(s) (s) (s) (s) u(s) (ξ, η, ζ) = ζ(A53 τ(s) γγ0 + A54 τβγ0 + A55 ταγ0 ) + u0 (ξ, η) + u∗ (ξ, η, ζ), (s) (s) (s) (s) v(s) (ξ, η, ζ) = ζ(A43 τ(s) γγ0 + A44 τβγ0 + A45 ταγ0 ) + v0 (ξ, η) + v∗ (ξ, η, ζ), (s) (s) (s) (s) w(s) (ξ, η, ζ) = ζ(A33 τ(s) γγ0 + A34 τβγ0 + A35 ταγ0 ) + w0 (ξ, η) + w∗ (ξ, η, ζ) (s) (s) (s) (s) (s) In Eqs (1.14) the functions τ(s) αγ0 , τβγ0 , τγγ0 , u0 , v0 , w0 , which will be uniquely determined by satisfying of the boundary conditions (1.2)-(1.6), are unknown yet. The values with star for every s are known, if the previous approximations are determined, particularly, we have ζ 1 ∂ 1 ∂ τ(s) (ξ, η, ζ) = − (Bτ(s−1) (Aτ(s−1) αγ∗ αα ) + βα ) + AB ∂η 0 AB ∂ξ ⎛ ⎞ ⎜⎜⎜ ∂τ(s−1) ⎟ αγ (s−1) (s−1) (s) (s−1) ⎟ ⎜ +R(kα ταβ − kβ τββ ) + RFα + r1 ⎜⎝ζ + 2ταγ ⎟⎟⎟⎠ + ∂ζ (s−1) (s−2) 2 +ζR(r1 + r2 )F α + ζ r1 r2 RFα dζ, (α, β; ξ, η; r1 , r2 ; A, B) (1.15) ζ (s) (s) (s) (s) u(s) (a15 τ(s) ∗ = αα∗ + a25 τββ∗ + a35 τγγ∗ + a45 τβγ∗ + a55 ταγ∗ + 0

(s) a56 τ(s) αβ∗ + α13 θ −

1 ∂w(s−1) (s−1) + r1 u(s−1) + ζ(a15r1 τ(s−1) αα + a25 r2 τββ + A ∂ξ

8

L.A. Aghalovyan

(s−1) (s−1) (s−1) (s−1) ∂u +a45r2 τ(s−1) + a r τ + a r τ + (r + r ) α θ − − 1 2 13 55 1 αγ 56 1 αβ βγ ∂ζ 1 ∂w(s−2) ∂u(s−2) r2 + r1 r2 u(s−2) ) + ζ 2 r1 r2 α13 θ(s−2) − ) dζ, A ∂ξ ∂ζ (s) (s) (s) τ(s) αα∗ (ξ, η, ζ) = B11 P1 + B12 P2 + B16 P3 ,

Q

(m)

r1 = R/R1 , r2 = R/R2

≡ 0 at m < 0 (s)

There are analogous formulae for the rest Q∗ [1]. In Eqs (1.14), (1.15) B jk = (a jl akl − a jk all )/Δ,

B jk = Bk j

Bll = (a j j akk − a2jk )/Δ,

( j k l j; j, k, l = 1, 2, 6) Alm = −a1m Bl1 − a2m Bl2 − a6m Bl6 , Amn Anm Anm = an1 A1m + an2 A2m + an6 A6m + anm , Δ=

(1.16)

(n, m = 3, 4, 5)

a11 a22 a66 + 2a12a26 a16 − a11 a226 − a22 a216 − a66 a212

The solution of the interior problem (1.11), (1.12), (1.14) permits us to satisfy all the variants of the boundary conditions (1.2), (1.3)-(1.6). Satisfying the conditions (1.2) we get (s) (s) −(s) u(s) = (1 + ζ)(A53τ(s) (ξ, η) − γγ0 + A54 τβγ0 + A55 ταγ0 ) + u (s) −u(s) ∗ (ζ = −1) + u∗ (ξ, η, ζ); (u, v, w; A5k , A4k , A3k ; k = 3, 4, 5) u−(0) = u− /R, u−(s) = 0, s 0

(1.17)

Satisfying the conditions (1.3) we get (s) +(s) (s) τ(s) = τ+(s) αγ (ζ = 1), (α, β), τγγ0 = τγγ − τγγ∗ (ζ = 1), αγ0

(1.18)

where +(1) +(s) + + τ+(0) αγ = εσαγ , ταγ = εr2 σαγ , ταγ = 0, s > 1 +(1) + + τ+(0) γγ = εσγγ , τγγ = ε(r1 + r2 )σγγ

τ+(2) γγ

= εr1 r2 σ+γγ , τ+(s) γγ

(1.19)

= 0, s > 2

Thus, the solution of the interior problem, corresponding to boundary conditions (1.2), (1.3), will be determined by Eqs (1.11), (1.12), (1.14), (1.17)-(1.19). From here it follows, that unlike the conditions of the first boundary value problem, the solution of the internal problem is uniquely determined by means of the function values, entering into the boundary conditions on the facial surfaces γ = ±h. This solution, as a rule, will not satisfy the conditions on the lateral surface of the shell. Rising misunderstanding is removed with the help of the solution for the boundary layer (Ib ). The values of the boundary layer, when removing from the lateral surface into the interior of the shell, decrease exponentially.


9

In general case the solution for the boundary layer is built and conjugated with the solution of the internal problem with the described in [1] method. Satisfying the conditions (1.4) we have τ(s) = (B55Vα(s) + B54Vβ(s) + B53Vγ(s) )/Δ, αγ0 (s) (s) (s) τ(s) βγ0 = (B45 Vα + B44Vβ + B43 Vγ )/Δ,

(1.20)

τ(s) = (B35Vα(s) + B34Vβ(s) + B33Vγ(s) )/Δ, γγ0 where 1 (s) Vα(s) = [u+(s) − u−(s) + u(s) ∗ (ζ = −1) − u∗ (ζ = 1)], 2 u±(0) = u± /R, u±(s) = 0, s 0, (α, β, γ; u, v, w), B jk = A jk All − A jl Akl , Bll = A jk Ak j − A j j Akk , B jk Bk j , j k l j, j, k, l = 3, 4, 5,

(1.21)

Δ = A33 B33 + A34 A43 + A35 B53 The solution of the internal problem, corresponding to boundary conditions (1.2), (1.4), will be determined by Eqs (1.11), (1.12), (1.14), (1.17), (1.20). In the analoguous way conditions (1.2), (1.5); (1.2), (1.6) are satisfied.

1.3 Solutions of Illustrative Problems On the base of the derived in Sect. 1.2 formulae we get solutions of some applied problems. A) Let the exterior surface of the lying horizontally orthotropic cylindrical shell be rigidly fastened, and inside pressure of the constant intensity P acts. The weight influence of the shell will be taken into account, too. Let α be the length of the generator, β is the length of the arch directing the cylinder counted from vertical (Fig. 1.1). Then A = B = 1, F α = 0,

r1 = 0,

r2 = 1,

F β = ρg sin ϕ,

kα = kβ = 0,

F γ = −ρg cos ϕ,

1 = 0, R1 ϕ = β/R

R2 = R, (1.22)

and uα (γ = h) = uβ (γ = h) = uγ (γ = h) = 0 σγγ (γ = −h) = −P, σαγ (γ = −h) = σβγ (γ = −h) = 0

(1.23) (1.24)

will be boundary conditions, where R is the radius of the middle surface of the cylinder, ρ is the density. It is considered that the temperature field does not change.

10

L.A. Aghalovyan

Fig. 1.1 Cylindrical shell

Using Eqs (1.11)-(1.19) and restricting the asymptotic expansion to O(ε2 ) exactly, which is enough for practical applications, according to (1.22) and conditions (1.23), (1.24), we get the following solution of the internal problem: σαγ = σαβ = 0,

uα = 0,

R−h h+γ (h + γ)2 P− A23 P + (h + γ)ρg cosϕ + A23 ρg cos ϕ, R+γ R+γ 2(R + γ) (h + γ)2 σβγ = (h + γ)ρg sinϕ − (2 + A23)ρg sin ϕ, 2R a12 A33 h − γ a12 a44 (h − γ)2 σαα = A13 σγγ − (2P − (3h + γ)ρg cosϕ) + ρg cos ϕ, Δ1 R + γ 2Δ1 R + γ R+γ A33 a11 h − γ σββ = A23 σγγ + (2P − (3h + γ)ρg cosϕ) R Δ1 R a44a11 h − γ γ − ρg cos ϕ + A23 (P − (h + γ)ρg cosϕ), (1.25) 2Δ1 R R 1 ρg uβ = a44 (h − γ)2ρg sin ϕ + (h − γ)(a44(1 + A23)(γ2 + 4γh + 7h2) 2 6R +A33 (γ − h)(γ + 5h)) sinϕ, 1 a44 A23 uγ = A33 (h − γ)(2P − (3h + γ)ρg cosϕ) − (h − γ)3 ρg cos ϕ 2 6R 1 − A33 (h − γ)((γ + 3h − 4A23h)P + 2A23h2 ρg cos ϕ), 2R A13 = −(a22a13 − a12a23 )/Δ1 , A23 = −(a11a23 − a12a13 )/Δ1 , σγγ = −

A33 = a13 A13 + a23 A23 + a33 ,

Δ1 = a11 a22 − a212

If the cylindrical shell has a finite length, the solution (1.25) will be just beginning with the distances from the end-wall of the cylinder, equal to the zone of extension


11

of the boundary layer. The boundary layer may be studied with the described in [1] method. B) The exterior one is free from loadings, the volume forces are absent. The shell is under the action of a variable temperature field. For the simplicity of the calculation we shall assume that the temperature field changes along the thickness of the shell linearly (θ = aγ + b). Using the same formulae and being restricted to the exactness of O(ε2 ), we get the solution: γ+h C22 [a(γ − h) + 2b], R+γ γ+h σαα = C11 (aγ + β) + A13C22 [a(γ − h) + 2b] 2(R + h) a12 γ−h − C33 [a(γ + h) + 2b], 2Δ1 R+γ γ+h σββ = C22 (aγ + b) + A23C22 [a(γ − h) + 2b] 2R a12 γ−h + C33 [a(γ + h) + 2b], (1.26) 2Δ1 R 1 uα = α13 (γ − h)[a(γ + h) + 2b], 2 1 1 uβ = α23 (γ − h)[a(γ + h) + 2b] + α23 (γ − h)[a(γ − h)(γ + 2h) + 3b(γ + 3h)], 2 6R 1 b uγ = C33 (γ − h)[a(γ + h) + 2b] + A33C22 (γ − h)(γ + 3h) 2 2R b a 2 − A23C33 (γ − h) + (A33C22 − A23C33 )(γ − h)2(γ + 2h), 2R 6R C11 = (a12 α22 − a22 α11 )/Δ1 , C22 = (a12α11 − a11α22 )/Δ1 , C33 = A13 α11 + A23 α22 + α33 σαγ = σβγ = σαβ = 0,

σγγ =

The results obtained in Sects 1.2 and 1.3 may be extended to two-layered and multilayered shells. The asymptotics (1.11), (1.12) are reserved for each layer, and the number of the layer must be attached to the solution (1.14) in the form of the index. Satisfying the boundary conditions on the facial surfaces and conditions of full contact between the layers, all the desired displacement vector components and the stresses tensor for each layer of the shell are determined [1, 2]. As illustration of the above described we present one solution for the two-layered shell. C) We have a two-layered orthotropic cylindrical shell with layers thicknesses h1 , h2 , respectively. The surface γ = h1 is rigidly fastened, and on the interior surface γ = −h2 the pressure of the constant intensity P acts. It is considered that the volume forces are absent, and the temperature field does not change. Ascribing the numbers of the layers index i, i = 1, 2 to the solution (1.14) and fulfilling all the required procedure, with exactness O(ε2 ), ε = h/R, h = max(h1, h2 ) we have the solution

12

L.A. Aghalovyan (i) (i) σ(i) αγ = σβγ = σαβ = 0,

(i) u(i) α = uβ = 0,

i = 1, 2,

R − h2 P (2) P − (A(i) 23 γ + A23 h2 ) R + γ , R+γ P (i) (i) (2) (i) σαα = −A13 (R − h2 + h2 A23 + γA23 ) R+γ σ(i) γγ = −

+

a(i) 12 Δ(i) 1

(A(i) γ − h1 A(1) ) 33 33

P , R+γ

(1.27)

(i) (2) (i) σ(i) ββ = −A23 (R − h2 − γ + h2 A23 + γA23 )

(i) P a11 (i) P + (i) (A33 γ − h1 A(1) ) , 33 R R Δ 1

u(i) γ

=

(1) (2) P −(A(i) 33 γ − h1 A33 )(R − h2 + h2 A23 ) R

1 P h1 P − (A(1) h21 − γ2 A(i) ) − A(1) (A(i) γ − h1 A(1) ) , 33 33 33 23 23 2 R R Δ(i) = a(i) a(i) − (a(i) )2 11 22 12

1.4 Non-Classical Spatial Dynamic Problems for Shells The asymptotic method is effective for the investigation of forced and free vibrations of shells, as well. Below we consider two types of forced vibrations: 1. Forced vibrations caused by the conditions at γ = −h uα (−h) = u− (α, β) exp(iΩt), uγ (−h) = w− (α, β) exp(iΩt)

uβ (−h) = v− (α, β) exp(iΩt) (1.28)

Simulating, particularly, seismic actions on the shell, where Ω is the frequency of the forcing actions. At γ = h the conditions of free or rigidly fixed surface are given: ταγ (h) = τβγ (h) = τγγ (h) = 0 (1.29) or uα (h) = uβ (h) = uγ (h) = 0

(1.30)

2. Forced vibrations of the shell with rigidly fastened exterior surface under the action of pulsing loading uα (γ = −h) = 0, uβ (γ = −h) = 0, uγ (γ = −h) = 0 σ jγ (γ = h) = σ+jγ (α, β) exp(iΩt), j = α, β, γ

(1.31) (1.32)

In order to solve the formulated dynamic problems, in the dynamic equations of three-dimensional problem of elasticity theory, obtained from Eqs (1.8), (1.9), substituting in the equations of equilibrium (1.8) Fα∗ , Fβ∗ , F γ∗ into


γ γ ∂2 u j F j = −ρ 1 + 1+ , R1 R2 ∂t2

j = α, β, γ

13

(1.33)

We again pass to dimensionless coordinates and displacements (1.10) and the solution of the transformed system is sought in the form of Qαβ = Q jk (ξ, η, ζ) exp(iΩt),

(α, β, γ), j, k = 1, 2, 3,

(1.34)

where Qα,β is any of the stresses and displacements. As a result we again obtain a singularly perturbed system, the solution of which has the form of (1.1). The solution of the internal problem is sought in the form of −1+s (s) τint τ jk (ξ, η, ζ), j, k = 1, 2, 3; s = 0, N, jk (ξ, η, ζ) = ε

(1.35)

(u (ξ, η, ζ), v , w ) = ε (u (ξ, η, ζ), v , w ) int

int

int

s

(s)

(s)

(s)

Substituting (1.35) into the according to (1.10), (1.34) transformed equations, we get a recurrent system, from where all the stresses may be expressed through the displacements [4, 5]. For determining the displacements in the case of orthotropic shells, we have the equations ∂2 u(s) (s−1) + a55 Ω2∗ u(s) = Fu (u, v; a55, a44 ), ∂ζ 2 ∂2 w(s) Δ 2 (s) + Ω w = F w(s−1) , Ω2∗ = ρh2 Ω2 , 2 Δ12 ∗ ∂ζ

(1.36)

where F u(s−1) , Fw(s−1) are known functions [5], and Δ = a11 Δ23 + a13Δ2 + a12 Δ1 , Δi j = aii a j j − a2i j , i, j = 1, 2, 3, Δ1 = a13 a23 − a33 a12 , Δ2 = a12 a23 − a22a13

(1.37)

The solution of the system (1.36) is (s)

(s)

u(s) (ξ, β, γ) = C1 (ξ, η) sin δu ζ + C2 (ξ, η) cos δu ζ + u(s) (ξ, η, ζ), v(s) (ξ, β, γ) = C3(s) (ξ, η) sin δv ζ + C4(s) (ξ, η) cos δv ζ + v(s) (ξ, η, ζ), w (ξ, β, γ) (s)

(1.38)

= C5(s) (ξ, η) sin δw ζ + C6(s) (ξ, η) cos δw ζ + w(s) (ξ, η, ζ)

where u(s) , v(s) , w(s) are special solutions of equations (1.36). In dynamic problems it is necessary to build the solution for the boundary layer as well. It is possible to use the formulae, presented in [5]. Having expressions for stresses and displacements, without any difficulties boundary conditions (1.28) (1.32) are satisfied. It is possible to show that in case of problem (1.28), (1.29) the solution will be finite, if cos 2δu 0,

cos 2δv 0,

cos2δw 0

(1.39)

14

L.A. Aghalovyan

Those values Ω, for which conditions (1.39) or the analogous conditions for the rest of the boundary conditions are not fulfilled, coincide with the values of the frequencies of free vibrations [3]. With these values of frequencies Ω of the exterior action resonance will rise.

References 1. Aghalovyan, L.A.: Asymptotic Theory of Anisotropic Plates and Shells (in Russian). Moscow, Nauka (1997) 2. Aghalovyan, L.A., Gevorgyan, R.S.: Nonclassical Boundary-value Problems of Anisotropic Layered Beams, Plates and Shells. Publishing House of the National Academy of Sciences of Armenia. Yerevan (2005) 3. Agalovyan, L.A., Gulghazaryan, L.G.: Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells. J. Appl. Math. Mech. 70, 102–115 (2006) 4. Agalovyan, L.A., Gulghazaryan, L.G.: Forced vibrations of orthotropic shells: nonclassical boundary-value problems. Int. Appl. Mech. 45(8), 105–122 (2009) 5. Aghalovyan, L.A., Gevorgyan, R.S., Ghulghazaryan, L.G.: The asymptotic solutions of 3D Dynamic problems for orthotropic cylindrical and toroidal shells. Proc. National Academy of Sciences of Armenia. Mechanics. 63(1), 6–22 (2010) 6. Gol’denveizer, A.L.: The Theory of Thin Elastic Shells (in Russian). Nauka, Moscow (1976) 7. Lomov, S.A.: Introduction in General Theory of Singular Perturbations (in Russian). Nauka, Moscow (1981) 8. Nayfeh, A.H.: Perturbations Methods. Wiley, New York (1973)

Chapter 2

Analytical Solution for the Bending of a Plate on a Functionally Graded Layer of Complex Structure Sergey Aizikovich, Andrey Vasiliev, Igor Sevostianov, Irina Trubchik, Ludmila Evich and Elena Ambalova

Abstract The problem of interaction between an asymmetrically loaded thin circular plate and a supporting elastic foundation is reduced to the solution of system of the dual integral equations for the unknown normal contact pressure and differential equation of plate bending. The supporting medium is an isotropic elastic functionally graded layer of complex structure of constant thickness lying, with or without friction, on a homogeneous elastic half-space. In designing raft foundations, prediction of differential settlements and of bending moments induced in the foundation is of great importance. It is common practice to assume that the raft foundation behaves like a thin elastic plate. Development of analytical methods of the solution of such problems for a foundation with elastic properties varying in-depth is based on the numerical construction of transforms for the kernels of integral equations of the corresponding contact problems [1]. For the construction of the approximate analytical solution of the corresponding integral equations the bilateral asymptotic method is used [2]. This method is found to be effective for rigid, as well as for flexible plates, in contrast to the method of orthogonal polynomials, collocation method, or asymptotic methods (of small or large parameter). Representation of plate deflections in the form of series by their own oscillations [3] is used under corresponding boundary conditions. It is possible to reduce the specified problems to the solution of the systems of linear algebraic equations. We provide the analysis of influence of various laws of variation of the elastic properties of the coating on the distribution of contact pressures under the circular plate; its deflection; surface settlement of the functionally-graded layer outside of plate; S. Aizikovich (B) · A. Vasiliev · I. Trubchik · L. Evich · E. Ambalova Department of Scientific Research, Don State Technical University, Gagarin sqr., 1, Rostov-on-Don, 344000, Russia e-mail: [email protected] I. Sevostianov Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 880003, USA e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 2, © Springer-Verlag Berlin Heidelberg 2011

15

16

S. Aizikovich et al.

and value of the radial and tangential moments in the thin circular plate. The developed method allows one to construct analytical solutions with prescribed accuracy and gives the opportunity to conduct multiparametric and qualitative analysis of the problem with minimal computation time. Keywords Plate · FGM foundation · Dual integral equations

Introduction For the behavior of the supporting soil, various models have been proposed and applied within the framework of linear elasticity. Well known of them are: Winkler springs, half-space continuum, and layered continuum. In this paper we discuss interaction of an axisymmetrically loaded thin circular Kirchhoff plate and functionally graded layer coupled with elastic half space. This elasticity problem models the actual problem of circular raft foundations which are used to support large cylindrical structures such as chimneys, silos, water tanks, etc. Many attempts have been made to treat them approximately using various techniques [4]– [15], since it is impossible to obtain closed form analytical solutions for interaction problem of this type. The aim of this paper is to develop an effective approximate analytical method for solution of this problem using integral transform. Present work assumes that a variation of Lame coefficients in the isotropic elastic functionallygraded layer has general form, i.e. they are arbitrarily continuous or piecewise continuous functions vertical coordinate. Friction between plate and the foundation is neglected. It is assumed that all deformations are elastic.

2.1 Interaction of a Thin Circular Plate with a Non-Homogeneous Half-Space. Statement and Formulation of the Problem An elastic layer of constant thickness H is lying, with or without friction, on a semiinfinite elastic base (Fig. 2.1).

Fig. 2.1 Axisymmetrically loaded circular plate on a functionally-graded halfspace

2 Analytical Solution of the Bending of Plates

17

A circular plate of radius R and constant thickness h governed by the KirchhoffLove thin plate theory rests on the surface z = 0 of the elastic layer. Under the action of an axisymmetrically distributed load of intensity p(r) and of the reactive normal contact pressure q(r) of the elastic layer, the plate is deflected with the deflection shape described by function w(r). Assuming no loss of contact at the interface between the plate and the elastic layer, the surface normal displacement of the elastic layer is also described by the function w(r). The surface z = 0 is free of tangential stresses while outside the contact region it is also free of normal stresses. It is required to determine the function w(r), the normal contact pressure distribution q(r) and the bending moments induced in the plate. The mathematical treatment of the stated axisymmetric interaction problem involves solution of two coupled boundary value problems, one for the bending of the thin circular plate with free-edge boundary conditions and the other for the stress and displacement fields in the foundation. Lamé coefficients Λ and M of the half-space vary with depth as follows: Λ = Λ0 (z), M = M0 (z), − H z 0 Λ = Λ1 = Λ0 (−H), M = M1 = M0 (−H), − ∞ < z < −H

(2.1)

We use static equilibrium equations of the theory of elasticity written in the displacements in the case of axially symmetric deformation for the cylindrical coordinate system. It is required to determine the distribution of contact pressure q(r ) under the plate and the shape of the deflection w(r ). According to accepted assumptions, the considered problem is reduced to solution of the following system of equations Dw(r) = p(r) − q(r), 0 r 1 ∞

(2.2)

∞ Q(α)L(αλ)J0 (αr)dα = sλw(r), 0 r 1;

0

Q(α)J0 (αr)dα = 0, r > 1 (2.3) 0

where D is differential operator of the plate bending in cylindrical coordinate system, p(r) is external loading, and q(r) stays for contact pressures under the plate. Dual integral equation (2.3) establishes the connection between contact pressures and the bending profile of the plate lying on elastic functionally-graded coating of the elastic half-space. Here 1 Q(α) =

q(ρ)J0 (αρ)ρdρ; 0

q(r) = (2π)

−1

∞ Q(α)J0 (αr)αdα,

(2.4)

0

λ = H/R is dimensionless geometrical parameter, s = ΘR3 D−1 — parameter which describes deflection rigidity of the plate, D — bending rigidity of the plate Θ = 2M(z) (Λ(z) + M(z))/ (2M(z) + Λ(z)), r = r /R. Function w(r) should satisfy traction-free boundary conditions, 2

d w ν dw

d

+ = 0, (w) (2.5)

r=1 = 0, dr dr2 r dr r=1

18


where ν is the Poisson’s ratio of the plate, and — Laplace operator (written in cylindrical coordinate system). Construction of kernel’s transform L(u) in the general case of properties variation across the functionally-graded layer is considered in [16]. When the following conditions are satisfied: min Θ(z) c1 > 0,

z∈(−∞;0]

max Θ(z) c2 < ∞, lim Θ(z) = const z→−∞

z∈(−∞;0]

(2.6)

it can be demonstrated [17], that the kernel transform has the following properties: L(u) = A + B |u| + O(|u|2 ), u → 0; L(u) = 1 + C |u|−1 + O(|u|−2 ),

A = lim Θ(0)/Θ(z), z→−∞

u → ∞;

B, C = const.

(2.7) (2.8)

2.2 Construction of the Solution We will utilize the function of bending in the form of series expansion with respect to own bendings of a circular plate with a free boundary [3] w(r) =

∞

1 wm ϕm (r),

wm =

m=0

ϕ0 (r) =

√

2,

w(ρ)ϕm (ρ)ρdρ,

(2.9)

0

ϕm (r) = Am J0 (km r) − J1 (km )I1−1 (km )I0 (km r) ,

Values Am and km for m = 0, 1, . . . , 10 are given in [3]. Taking into account linearity of problem, the reactive normal contact pressure are represented as linear combination of special solutions with the same coefficients wm , as for bending function w(r) in (2.9): q(r) =

∞

wm qm (r),

0 r 1.

(2.10)

m=0

We will determine special solution qm (r) (m = 0, 1, . . .) from integral equation (2.3) similarly to [1], [16]. We assume that function L(u) in (2.3) belongs to the class ΠN (Σ M , SN, M ), if it has the form: ⎧N ⎪ 2 2 −2 2 2 −2 −1 ⎪ ⎪ ⎪ (α + ai λ )(α + bi λ ) ≡ LN (λα) ∈ ΠN , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎨ M L(λα) = ⎪ ⎪ ck λ−1 | α | (α2 + dk2 λ−2 )−1 ≡ LΣM (λα) ∈ Σ M , ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ LN (λα) + LΣ (λα) ∈ SN, M . M


19

Here ai , bi (i = 1, . . . , N), ck , dk (k = 1, . . . , M) — certain complex constants (ai − ak )(bi − bk ) 0, k i. It has been proved [1], that provided the function L(u) possesses properties (2.7), (2.8), it admits approximation by expressions of type L(λα) = LN (λα) + LΣ∞ (λα).

(2.11)

Using (2.11), eq. (2.3) can be written in terms of the operator in the form Π N q + Σ∞ q = f.

(2.12)

Definition 2.1. Condition A will be said to hold for equation (2.12) if when one can construct a closed solution for it by following [18]. We denote it by q N = Π−1 N f.

(2.13)

Condition A means that for functions f (x) in certain class W(c, d), there exists a function q(x) in some class V(c, d) such that (2.13) holds. It follows from representation (2.13), that q N V(c, d) m(ΠN ) f W(c, d) ,

m(ΠN ) = const.

(2.14)

Let us substitute in the right part of (2.3) m-eigenfunction ϕm (r) instead of w(r). The closed approximate solution of the equation (2.3) of type (2.13) has the following form: ⎡ ⎤ N −1 ⎥⎥⎥ √ −1 ⎢⎢⎢ −1 N 0 −1 q0 (r) = 2 2π λs ⎢⎢⎣⎢LN (0) 1 − r2 + Ci Ψ (r, ai λ )⎥⎥⎦⎥ , (2.15) i=0

⎡ ⎢⎢

⎢ N qm (r) = 2π−1 Am λs ⎢⎢⎢⎢⎣L−1 N (λkm ) f (r, km ) J1 (km )I1−1 (km )L−1 N (iλkm )Ψ(r, km ) +

−

N j=0

1 ch A sh Atdt Ψ(r, a) = √ −a ; √ t2 − r 2 1 − r2 r

⎤ ⎥⎥

⎥ Cim Ψ (r, iAi λ−1 )⎥⎥⎥⎥⎦ ,

(2.16) m = 1, 2, . . . ,

1 cos ε sin tεdt f (r, ε) = √ +ε . √ t2 − r 2 1 − r2 r

(2.17)

Constants Cim are determined from the following systems of linear algebraic equations N i=1

−1 Ci0 α(ai λ−1 , bk λ−1 ) + L−1 N (0)λbk = 0,

k = 1, 2, . . . , N,

(2.18)

20

S. Aizikovich et al. N Cim α(ai λ−1 , bk λ−1 )+β(bk λ−1 , km )−J1 (km )I1−1 (km )γ(bk λ−1 , km ) = 0, i=1

k = 1, 2, . . . , N;

(2.19)

m = 1, 2, . . . ,

α(a, b) = (a sh a + b ch a)(b2 − a2 )−1 , 2 2 −1 β(μ, b) = (b cosμ − μ sin μ)L−1 N (λμ)(b + μ ) , 2 2 −1 γ(k, b) = (k sh k + b ch k)L−1 N (iλk)(b − k ) . N In turn, contact pressures qm (r) can be presented as series (2.9):

N qm (r) =

∞

1 ymj ϕ j (r),

ymj

j=0

=

N qm (ρ)ϕ j (ρ)ρdρ.

(2.20)

0

Let’s write the conditions required to present f (r) in the form (2.20). We use the following definitions: Definition 2.2. We define C k (0, 1) as the space of functions having continuous derivatives on an interval [0, 1]. Definition 2.3. We define L2 (0, 1) as the space of functions f , defined on interval 0 r 1 , measurable on this interval by Lebesgue’s norm, and such, that 1 | f (r)|2 dr < ∞, 0

where integral is taken in the Lebesgue sense. Then, the following theorem takes place [19]. Theorem 2.1. Let function f ∈ C 4 (0, 1) satisfies boundary conditions (2.5). Then representation (2.9) on [0, 1] takes place, and a series converges on [0, 1] evenly. More general expansion and completeness theorem in L2 (0, 1) also takes place [19]. Theorem 2.2. If f ∈ L2 (0, 1), than equation (2.9) is satisfied, and it is understood in the sense that 1 k lim f − fm ϕm (r) = 0, fm = f (ρ)ϕm (ρ)ρdρ. (2.21) k→∞ 2 0 m=0 L (0,1)

Besides, Parseval’s equality takes place f 2L2 (0, 1) =

∞

fm2 .

(2.22)

m=0

We assume that function p(r) can be presented in the form of series expansion

2 Analytical Solution of the Bending of Plates ∞

p(r) =

21

pm ϕm (r),

1

pm =

p(ρ)ϕm (ρ)ρdρ.

(2.23)

0

m=0

Then, substituting decompositions (2.9), (2.10), (2.23) into (2.2), we obtain infinite system of linear algebraic equations for coefficients wm : −4 wm − akm

∞

−4 w j E mj = pm km ,

m = 0, 1, 2, . . . ;

a = −1,

(2.24)

j=0

⎡

⎤

N ⎢⎢⎢ ⎥⎥⎥ m −1 −1 m E mj = 2π−1A j λs ⎢⎢⎢⎣L−1 (λk )F −J (k )I (k )L (iλk )R + Cnm S nm ⎥⎥⎥⎦, j 1 m 1 m N j j N j n=0

F mj = Rmj =

1

0

S nm =

f (ρ, k j )ϕm (ρ)ρdρ = Am [x(k j , km ) − J1 (km )I1−1 (km )X(km, k j )],

(2.26)

Ψ (ρ, k j )ϕm (ρ)ρdρ = Am [X(k j , km ) − J1 (km )I1−1 (km )x(ik j , ikm )],

(2.27)

0 1

0

(2.25)

1

Ψ(ρ, an λ−1 )ϕm (ρ)ρdρ =Am [X(anλ−1, km )−J1 (km )I1−1(km )x(an λ−1, ikm )].

Here j = 1, 2, . . . ,

(2.28) m = 0, 1, 2, . . .;

x(a, b) = b−1 (cosa sin b+a2−1 (sin (a−b)(a−b)−1 −sin(a+b)(a+b)−1 )), X(a, b) = (a2 + b2 )−1 (a sh a cos b + b sinb ch a).

(2.29)

In particular, ⎡ ⎤ N ⎢⎢⎢ ⎥⎥⎥ √ −1 −1 −1 E0m = 2 2π−1 λs ⎢⎢⎢⎣ L−1 Cnm S nm ⎥⎥⎥⎦ , N (0)Am km (sin km − J1 (km )I1 (km ) sh km ) +2 n=1

⎡ N ⎤ ⎥⎥⎥ √ −1 ⎢⎢⎢ √ 0 0 0 ⎢ E0 = 2π λs ⎢⎢⎣2+ Cn S n ⎥⎥⎥⎦ , S n0 = 2(an λ−1 sh (an λ−1 )− ch (an λ−1 )).

(2.30) (2.31)

n=1

We have

∞

−4 2 |pm km | M

m=0

∞

|pm|2 < ∞

(2.32)

m=0

at the restrictions imposed on p(r). From estimation (2.14) and condition of equilibrium of plate, 2π1

2π1 q(r)rdrdϕ =

0 0

p(r)rdrdϕ 0 0

(2.33)

22


We have ∞ m=0

−8 km

∞

|E mj |2

j=0

∞ ∞

|E mj |2 M(ΠN , Σ∞ )

m=0 j=0

∞

|pm|2 < ∞.

(2.34)

m=0

It can be solved by reduction method [20] (by replacement with system from n equations with n unknowns): −4 wm − akm

n−1

−4 w j E mj = pm km ,

m = 0, 1, 2, . . ., n − 1,

a = −1,

(2.35)

j=0

Besides, if n is sufficiently large, system (2.35) is solvable and the convergence of approximated solutions to the exact one takes place. When coefficients wm (m = 0, 1, . . ., n) are determined for the fixed value, equation (2.35) yields the contact pressure n q(r) = wm qm (r), 0 r 1, (2.36) m=0

and plate deflections w(r) =

n

wm ϕm (r),

0 r 1.

(2.37)

m=0

Practical calculations require the values of radial Mr and tangential moments Mϕ in plate. Let us write out their expressions in terms of the plate deflections: 2 D d 2 w ν dw D d w 1 dw Mr = − 2 + , Mϕ = − 2 ν 2 + . (2.38) r dr R dr2 r dr R dr Using expressions for plate deflections (2.9), we receive: Mr = Mϕ =

M D

R2

m=0

M D

R2

2 wm Am km

2 wm Am km

m=0

where Vi (km r) = Ji (km r) + Bm Ii (km r),

ν−1 V1 (km r) + V0 (km r) , km r

1−ν V1 (km r) + νV0 (km r) , km r

i = 0, 1.

(2.39)


23

2.3 Determination of the Surface Displacement Outside of the Plate Equation (2.3) has been derived as a result of the following representation of vertical displacement of the surface foundation: 1 f (r) = sλ

∞ Q(α)L(λα)J0 (αr)dα.

(2.40)

0

Below we derive analytical expression for function f (r) in (2.40), when L(α) ∈ ΠN , r > 1. For L(α) ∈ Π N function QN (α) is received in analytical form in the process of solving equation (2.3). Using formula (2.40), we receive expression for function f (r), r > 1, if L(α) ∈ ΠN we designate it as f N (r). According to (2.9), function QN (α) can be written as ∞ N QN (α) = w m Qm (α). (2.41) m=0

Corresponding expressions for functions fmN (r) are given by ⎧ ⎪ ⎪ ⎪ ⎨ fmN (r) = 2π−1 Am ⎪ G(r, km ) − BmG(r, ikm )+ ⎪ ⎪ ⎩ ⎡ ⎤⎫ N N ⎢⎢ ⎥⎥⎪ ⎪ ⎪ −1 ⎢⎢⎢ m −1 ⎥⎥⎥⎬ + Dn bn λ In ⎢⎢⎣Ψn (km )−BmΨn (ikm )+ C j γn (ia j λ )⎥⎥⎦⎪ , ⎪ ⎪ ⎭ n=1 j=1 r > 1,

m = 0, 1, 2, . . .,

2 −2 2 Ψn (a) = L−1 N (λα)γn (a), γn (a) = a sin a/(bn λ + a ),

r In = 1

exp[λ−1 bn (t − 1)] dt, √ r 2 − t2 Dn = (a2n b−2 n − 1)

Finally f N (r) =

M

1 0 N $ −b2n + a2i

i=1,in

−b2n + b2i

wm fmN (r),

r > 1,

where wm are the same magnitudes as those in (2.9).

(2.43)

cos at dt, √ r2 − t2

G(r, a) =

m=0

(2.42)

.

(2.44)

24


2.4 Numerical Examples For illustration, let us consider bending of a circular plate by uniformly distributed load of unitary intensity. The plate is impressed into a half-space; variation of the Young’s modulus with depth is described by E(z) = E 0 φ(z),

−1 z 0,

E(z) = E 0 φ(−1),

(2.45)

−∞ < z < −1,

and Poisson’s ratio of the foundation ν = 1/3, −∞ < z 0. Poisson’s ratio of plate νn = 0,15. For uniquely distributed load, the conditions of the Theorem 2.1 (p(r) √ =p= 1) are fulfilled, decomposition coefficients (2.10) are given by: p0 = p 2, pm = 1, m = 1, 2, . . .. Let’s consider the following types of non-homogeneity: 1) monotonic (power): a) increasing with depth φ1 (z) = 0,1 + z2ak ,

ak =

ln 0,1(k − 1) , 2 ln 0,5

k = 2,

b) decreasing with depth φ2 (z) = 1,1 − z2ak ,

ak =

ln(1, 1 − k/10) , 2 ln 0,5

k = 9,

2) non-monotonic φ3 (z) = 1,1 + sin(πz),

φ4 (z) = 0,1 − sin(πz).

Figures 2.2, 2.5 presents graphical diagrams for parameter χ(r) = q N (r)/q0 (r) characterizing distribution of normal contact pressure under the plate on non-homogeneous foundation in comparison with homogeneous q0 (r) (for E = E 0 φ(−1)) at various values of λ.

Fig. 2.2 Graphical diagrams of the distribution of normal contact pressures under the plate on non-homogeneous foundation in comparison with homogeneous one. a) s = 0, 1, b) s = 3

y 1.0 φ(y) 0.0 0.1 ϕ1 -1.0

6

τ0

16

8

8

4

2

5

0

16 2

4 2

τ0

0.5 a)

1

r

2

4

0

8 0.2

0.5

0.5 b)

1

r



y 1.0 φ(y) 0.0 0.1 ϕ2 -1.0


τ0

6

τ0

8

16 8 4 0.2

16

4 2

5

8

0

2 0.2

τ0

1.25

0.5

1

r

2

0

τ0

0.2

1

0.5

8

1

0

4

0.5 b)

2

8

0.5

0

1

r

0

1.5

τ0 4 16 2

1 0

2

8

0

0.5 b)

0.5

0

0.5

a)

1

r

0

1

r

1

r

τ0 4 16

1

0.2

r

2

a)

y 1.0 0.0 0.1 φ(y) -0.5 ϕ4 -1.0

1

0.2

0.5 16 4

0.5

16

0.25

0.5

4 0.5

a)

y 1.0 0.0 0.1 φ(y) -0.5 ϕ3 -1.0


25

0.2

0

0.5

8

2

0.5 b)

The digits curves correspond to the values of λ for which the calculation was conducted. Values q N (r), q0 (r) are found with formula (2.13) at N = 10, M = 10. In fig. 2.2 χ(r) corresponds to φ1 (z), in fig. 2.3 — φ2 (z), in fig. 2.4 — φ3 (z), and in fig. 2.5 — φ4 (z). One might conclude then, that, in the case of monotonously decreasing Young’s modulus described by φ2 (z) the decrease of coefficient is observed for contact pressure as compare to its value for homogeneous half-space (fig. 2.5). In such case the formulation of the problem must be modified. The contact zone of plate with foundation can be determined from condition of vanish of contact pressures on the boundary of contact. The lift-off area extends at flexural rigidity increase of plate (fig. 2.5b). It is typical for non-monotonic variations φ3 (z) and φ4 (z) that in case when φ(z) increases with depth from surface (φ3 (z)), the increase of the value χ(r), is observed. In this case, plate separation from of the foundation does not take place.

26

S. Aizikovich et al. y 1.0 φ(y) 0.0 0.1 ϕ1 -1.0

Fig. 2.6 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous half-space. a) s = 0, 1, b) s = 3

2

4

Δ(r) 1

16 8

0.5

8

0.5

0.5

0.5

0.2

0

y 1.0 φ(y) 0.0 0.1 ϕ2 -1.0

0

r

4

2 a)

0

0

2

3

8

4 2

8

r

4

2

Δ(r)

1

0

2

2

16 8

4

1

0.5

0.2 0.5

2

2

2

4

r

0

0

0.5 b)

a) Δ(r)

2

0.5 0.2

1

2 4

0

2 a)

4

r

0

r

4

r

4 0.5 8

16

1

Δ(r)

1

8

0.5 0

r

Δ(r)

8

0.2 4

4

b)

16

1

y 1.0 0.0 0.1 φ(y) -0.5 ϕ4 -1.0

16

0.5 0.2

0.2

0

r

2

2 0.5

2

4

4

a)

0

4

Δ(r)

0

Fig. 2.8 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous one. a) s = 0, 1, b) s = 3

2

b)

16

0

y 1.0 0.0 0.1 φ(y) -0.5 ϕ3 -1.0

16

0.2

Δ(r)

10 5

Fig. 2.7 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous half-space. a) s = 0, 1, b) s = 3

Fig. 2.9 Relative settlement of the surface of the inhomogeneous half-space as compared to the homogeneous one. a) s = 0, 1, b) s = 3

Δ(r)

1

0

0.2 2

16

2 b)

If φ(z) decreases with depth from surface (φ4 (z)), the decrease of the value χ(r) is observed at approach from within to boundary of plate. The form of curves χ(r) depends on the values of λ, and s. One can see from Figs 2.2–2.5 that distribution of contact pressures essentially depends on the thickness of non-homogeneous layer, type of non-homogeneity, and rigidity of the plate. Figures 2.6–2.9 illustrate the surface displacement. ⎧ ⎪ ⎪ ⎨wN (r)/w0 (r), 0 r 1, Δ(r) = ⎪ ⎪ ⎩ fN (r)/ f0 (r), r > 1,

2 Analytical Solution of the Bending of Plates y 1.0 φ(y) 0.0 0.1 ϕ1 -1.0

0,0 -0,2

27

Mr0

-0,6

2

-0,01 0 M -0,02 r -0,03

Mφ0

0.2

-1,0

-1,4 1,0 0,6

16

0,0

16 4 0.5

0,6 1,0

0

-0,04 -0,05 1,0

2

8

0.5

Mφ0

0.2

0 0,2

0,4

1,0

b)

a)

Fig. 2.10 Distribution of the bending moments. a) s = 0, 1, b) s = 3 y 1.0 φ(y) 0.0 0.1 ϕ2 -1.0

12

0.5

Mr0

0.2

2

8

0

Mφ

1,0

0

Mφ

Mr0

0,2

16

4

0,4

0,0 0,4

0 a)

0,4

1,0

-0,2 1,0 0,6

0,6 1,0

0 b)

Fig. 2.11 Distribution of the bending moments. a) s = 0, 1, b) s = 3 y 1.0 0.0 0.1 φ(y) -0.5 ϕ3 -1.0

0

Mr

1,5

0

0.2

4

8

16

0,0

0,04

0.5

-1,5 -3,0 1,0

0,08

Mφ

0 a)

0,4

0.2 4

8

0.5

-0,04 1,0

Mφ0

16

0,0

2

0,4

Mr0

0,4

1,0

2

0 0,4 b)

1,0

Fig. 2.12 Distribution of the bending moments. a) s = 0, 1, b) s = 3 y 1.0 φ(y) 0.0 0.1 -0.5 ϕ4 -1.0

8

0

Mφ

0,0

4 2

Mr

-1,5

0.5

0,4

0 a)

0,4

1,0

-0,06 1,0

0

Mφ

8

0.2

-0,04

0.2

2

16

-0,02

-3,0 1,0

0

16

1,5 Mr0 0,0

0.5

4

0,4

0 0,4 b)

1,0

Fig. 2.13 Distribution of the bending moments. a) s = 0, 1, b) s = 3

of non-homogeneous half-space of type (2.45) in comparison with homogeneous w0 (r) (under the plate and outside of it) at various values of λ and s. Figures 2.10–2.13 illustrate behavior of dimensionless radial (Mr0 = Mr /M0r ) and tangential moments (Mϕ0 = Mϕ /M0ϕ ). Values Mr (M0r ), Mϕ (M0ϕ ) are calculated with formulas (2.39) at N = 10, M = 10.

28


Acknowledgements The work is performed with the financial support of RFBR (09-08-011410a, 10-08-01296-a, 10-08-00839-a, 10-08-90025-Bel-a, 11-08-91168-GFEN-a), GK No P 1107, GK No 02.740.11.0413, GK No 02.740.11.5193, AVCP 2.1.2/10063, NASA EPSCOR Grant GR0002488; Los Alamos National Labs MOU Grant GR0002842.

References 1. Aizikovich, S.M., Alexandrov, V.M., Kalker, J.J., Krenev, L.I., Trubchik, I.S. (2002) Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties, Int. J. Solids Structures, 39 10, 2745–2772. 2. Aizikovich, S.M., Alexandrov, V.M., Trubchik, I.S. (2009) Bilateral asymptotic solution of one class of dual integral equations of the static contact problems for the foundations inhomogeneous in depth. Operator Theory: Advances and Applications, 39 3–17. 3. Tseitlin, A.I. The bending of a circular plate resting on a linearly deformable foundation. Izv. Akad. Nauk SSSR. MTT (1969)1 99–112. 4. Popov, G. Ya. Plates on a linearly elastic foundation (a survey). Prikl. Mekh. 8(3) (1971) – pp. 3–17. 5. Hooper, J. A. (1978) Foundation interaction analysis, in Developmenls in Soil Mechanics-l (Edited by R. C. Scott), Applied Science Publ., London, 149–213. 6. Selvadurai, A. P. S. (1979) Elastic Analysis of Soil-Foundation Interacrion: Developments in Geotechnical Engineering. Elsevier, Amsterdam, Vol. 17. 7. Gladwell, G.M.L. (1980) Contact Problems in the Classical Theory of Elasticity. 8. Ishkova, A. G. (1951) Bending of a circular plate on the elastic half-space under the action of a uniformly distributed axisymmetrical load. Uch. Zap. Mosk. Gos. Univ. 3, 202–225. (in Russian) 9. Palmov, V. A.(1960) The contact problem of a plate on an elastic foundation. PMM, Vol. 3, 416–422. 10. Barden, L. (1965) Contact pressures under circular slabs. Struct. Engineer, 5 (43), 153–154. 11. Brown, P. T. (1969) Numerical analyses of uniformly loaded circular rafts on deep elastic foundations. Geotechnique, Vol. 19, 399–404. 12. Brown, P. T. (1969) Numerical analyses of uniformly loaded circular rafts on elastic layers of finite depth. Geotechnique, Vol. 19, 301–306. 13. Chattopadhyay, R., Ghosh, A. (1969) Analysis of circular plates on semi-infinite elastic substrates. Indian J. Technol. 7, 312–319. 14. Selvadurai,A. P. S. (1979) The interaction between a uniformly loaded circular plate and an isotropic half-space: A variational approach. J. Strucr. Mech. 7(3), 231–246. 15. Selvadurai, A. P. S. (1980) Elastic contact between a flexible ircular plate and a transversely isotropic elastic halfspace. Int. J. Solids Struct. 16, 167–176. 16. Aizikovich, S.M. (1982) Asymptotic solutions of contact problems of the theory of elasticity for media inhomogeneous in depth. Prikl. Mat. Mekh. 46 (1), 148–158. 17. Aizikovich, S.M., Aleksandrov, V.M. (1982) On the properties of compliance functions corresponding to polylayered and continuously inhomogeneous half-space. Dokl Akad Nauk SSSR, 266 (1), 40–43 (English Transl. in Soviet. Phys. Dokl. 27). 18. Aleksandrov, V.M.(1973) The solution of a class of dual equations. Dokl. Akad. Nauk SSSR 210 (1), 55–58. 19. Coddington, E.A., Levinson, N. Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955. 20. Kantorovich, L.V., Akilov, G.P. Functional analysis, 2nd rev. ed., Nauka, Moscow, 1977.

Chapter 3

Analysis of the Deformation of Multi-layered Orthotropic Cylindrical Elastic Shells Using the Direct Approach Mircea Bˆırsan and Holm Altenbach

Abstract In this paper we analyze the deformation of cylindrical multi-layered elastic shells using the direct approach to shell theory. In this approach, the thin shelllike bodies are modeled as deformable surfaces with a triad of vectors (directors) attached to each point. This triad of directors rotates during deformation and describes the rotations of the thickness filament of the shell. We consider a general set of constitutive equations which can model orthotropic multi-layered shells. For this type of shells we investigate the equilibrium of thinwalled tubes (not necessarily circular) subjected to external body loads and to resultant forces and moments applied to the end edges. We present a general procedure to derive the analytical solution of this problem. We consider that the external body loads are given polynomials in the axial coordinate, which coefficients can be arbitrary functions of the circumferential coordinate. We illustrate our method in the case of circular cylindrical three-layered shells and obtain the solution in closed form. For isotropic shells, the solution is in agreement with classical known results. Keywords Direct approach · Multilayered shell · Reissner theory · Simple shell · Orthotropic material · Cylindrical shell

M. Bˆırsan (B) Department of Mathematics, “A.I. Cuza” University of Ias¸i, 700506 Ias¸i, Romania Faculty of Civil Engineering and Architecture, Lublin University of Technology, 20-618 Lublin, Poland e-mail: [email protected] H. Altenbach Department of Engineering Sciences, Martin-Luther-University Halle-Wittenberg, 06099 Halle (Saale), Germany Faculty of Mechanical Engineering, Otto-von-Guericke-University, 39106 Magdeburg, Germany e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 3, © Springer-Verlag Berlin Heidelberg 2011

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M. Bˆırsan and H. Altenbach

3.1 Introduction The paper is concerned with the deformation of thin elastic shells modeled by the direct approach. We employ the theory of simple shells, which was introduced by Zhilin [17,18]. In this approach, the thin shell-like bodies are modeled as deformable surfaces with a triad of rigidly rotating directors attached to each point. The model has been successfully applied in the case of plates to describe the mechanical behavior of functionally graded, viscoelastic or porous plates in [2–4]. A mathematical study of the theory of simple shells is presented in [7]. We investigate the deformation of cylindrical shells with arbitrary shape of crosssection (not necessarily circular), and made of orthotropic materials. We consider a quite general set of constitutive equations which can describe various types of composite shells, such as multi-layered shells or reinforced shells. Moreover, our results can be applied also to non-homogeneous shells, since the constitutive coefficients are assumed to be arbitrary functions of the circumferential coordinate. In this framework, we consider the following mechanical problem: find the equilibrium of cylindrical shells under the action of resultant forces and moments applied to the end edges of the cylindrical shell, and the action of given body loads (body forces and couples). We present a general analytical solution procedure for this problem. Thus, the solution for the displacement and rotation fields is expressed in terms of the exact solutions to auxiliary boundary-value problems for ordinary differential equations. The idea of this method is taken from the three-dimensional linear elasticity: in this context, the deformation of solid cylinders made from Cosserat materials has been investigated in details, see e.g. [10]. Adapting the method of Ies¸an [11], we investigate the deformation of cylindrical surfaces with directors. We derive first the equilibrium solution of our problem for the case when the body loads are independent of the axial coordinate, but they depend on the circumferential coordinate of the surface. Then, we consider the more general case when the body loads are polynomials in the axial coordinate (of degree n) and we obtain a solution of our equilibrium problem by the mathematical induction procedure with respect to the degree n. We show that our solutions are in agreement with the classical results concerning the deformation of cylindrical elastic shells. Finally, we apply this general method to the case of a three-layered circular cylindrical shell, and determine the closed-form solution for the displacement and rotation fields.

3.2 Basic Equations for Shells. General Geometry We consider a direct approach to shells, in which each material point of the surface is connected with an orthonormal triad of vectors (directors). This model has been suggested first by the Cosserat brothers [9] and it has been developed by Zhilin in [17,18], who established the constitutive equations and presented several interesting

3 Analysis of the Deformation of Multi-Layered Orthotropic Cylindrical Elastic Shells

31

applications. This approach, also called the theory of simple shells, can also account thermal effects in shells [8]. Let S denote the actual configuration of the shell at time t, and Dk (k = 1, 2, 3) designate the 3 directors connected to each point. Then, the configuration S of the shell is determined by the functions {R(x, t), Dk (x, t)} , where R is the position vector of any point with curvilinear material coordinates x = (x1 , x2 ), see Fig. 3.1. The triad

S S0

d3 d 2 (x1 , x2 )

D3

D2 D1

d1

(x1 , x2 )

r

R O

Fig. 3.1 Reference configuration S0 and actual configuration S of the shell

of directors { Dk } remains orthonormal during deformation, so that Dk · Dm = δkm (the Kronecker symbol). The rotations of the triad { Dk } describe the rotations of the thickness filament of the shell during deformation. In the reference configuration S0 (at t = 0) we denote by r(x) = R(x, 0) the position vector and by dk (x) = Dk (x, 0) the directors. Then, the rotation tensor P is defined by P(x, t) = Dk (x, t) ⊗ dk (x). Here, the usual summation convention over repeated indices is employed, and the direct tensor notation in the sense of [12]. Let P˙ denote the material time derivative of P, and ∂α P = ∂∂P xα . We introduce the angular velocity ω as the axial vector of the antisymmetric tensor P˙ · PT (the superscript T denotes the transpose). Also, let Φ α be the axial vector of the antisymmetric tensor ˙ t) = ω (x, t)×P(x, t) and ∂α P(x, t) = Φ α (x, t)×P(x, t). Through∂α P · PT . We have P(x, out the paper, the Greek indices take the values {1, 2}, while the Latin indices range over the set {1, 2, 3} (unless otherwise specified). Let n denote the unit normal to the reference surface S0 , the covariant base vectors in the tangent plane are rα (x) = ∂α r(x), and the dual basis {rα (x)} is defined by rα · rβ = δαβ . We introduce the first fundamental tensor a, the second fundamental tensor b, and the alternator tensor c given by a = rα ⊗ rα ,

b = −rα ⊗ ∂α n,

c = −a × n = −n × a.

The nonlinear theory of simple shells is presented in [5, 18]. In this paper, we confine our attention to the linear theory and present the relevant equations.

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The infinitesimal displacement vector u and the infinitesimal rotation vector ϕ are introduced by the relations u = R−r

and

ϕ˙ = ω ,

∂ αϕ = Φ α .

ϕ × 1, where 1 is the unit tensor of Then, the rotation tensor is expressed as P = 1 +ϕ second order. Let us choose the reference director d3 (x) to coincide with the unit normal to S0 , i.e. d3 = n. We may consider that ϕ (x, t) · n(x) = 0, since the (drilling) rotations about the normal n as usual do not intervene in the governing equations in the linear theory of simple shells (exceptional cases, for example, folded structures, will be not discussed here). The vector fields u and ϕ can be decomposed as u = uα rα + u3 n , ϕ = ϕα rα , which shows that we have 5 independent kinematical scalar fields (three displacements ui and two rotations ϕα ). Thus, the theory of simple shells is in this sense a Reissner-type theory for shells. ϕ , κ = rα ⊗ ∂αϕ . Then, the Let us denote by e and κ the tensors e = rα ⊗ ∂α u + a ×ϕ strain tensors for simple shells are expressed by ε=

& 1% e · a + a · eT , γ = e · n = (rα ⊗ ∂α u) · n + c · ϕ , 2 1 k = κ · a + (e · ·c)b. 2

(3.1)

Here ε is a symmetric tensor describing the extensional and in-plane shear strains of the shell, γ is a vector which accounts for the transverse shear deformation, and k is a tensor for the bending and twist strains. For any subset P0 of the reference surface S0 , we designate by ν the external unit normal to the boundary curve ∂P0 which lies in the tangent plane, and by t and m the vectors of external force and moment acting on the boundary ∂P0 . Then, the force tensor T and the moment tensor M satisfy the relations of Cauchy type t = ν · T, m = ν · M . The equations of equilibrium have the form rα · ∂α T + F = 0,

rα · ∂α M + T× + L = 0.

(3.2)

Here the vectors F and L are the external forces and moments per unit area of the surface S0 . The term T× denotes the vector invariant (or ‘Gibbsian cross’) of the tensor T, see e.g. Appendix A in [13]. γ , k) stand for the inWe present next the constitutive equations. Let U = U(εε ,γ ternal energy function, which will be assumed to be a quadratic function of its arguments. This means that “eigenstresses” presented by linear terms and nonlinear elastic behavior are ignored. For general orthotropic shells, U can be expressed as ρU =

1 1 1 ε · ·C1 · ·εε + ε · ·C2 · ·k + k · ·C3 · ·k + γ · Γ · γ , 2 2 2

(3.3)

where ρ is the mass density in the reference configuration, Ci represent stiffness tensors of order 4, and Γ is a stiffness tensor of order 2 (the transverse shear


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stiffness). These stiffness tensors satisfy certain symmetry conditions and they express the effective elastic properties and the thickness geometry of the shell. The functional form (3.3) can describe a large variety of shells, like for instance multilayered shells, as we will see later on. The stiffness tensors have been determined for various classes of shells and plates in [1–4, 18]. The constitutive equations for simple shells are 1 ∂(ρU) sym(T · a) ≡ (T · a + a · TT ) = = C1 · ·εε + C2 · ·k, 2 ∂εε ∂(ρU) ∂(ρU) N ≡ T·n = = Γ · γ, M= = (εε · ·C2 + C3 · ·k)T . ∂γγ ∂k

(3.4)

For shells of constant thickness made of non-polar materials, we have the following restrictions [18] TT · ·c + MT · ·b = 0,

M · n = 0.

(3.5)

We notice that the relation (3.5)1 gives the antisymmetric part of T · a . In view of the constitutive equations (3.4)1,2 , the force tensor T will be completely determined. The expressions of the constitutive tensors Ci and Γ are presented in [18], Sect. 3.10. Let us consider that the vectors rα are directed along the principal directions of the tensor b and the basis {rα } is orthogonal. We introduce the tensors a1 = a = r1 ⊗ r1 + r2 ⊗ r2 , a3 = c = r1 ⊗ r2 − r2 ⊗ r1 ,

a2 = r1 ⊗ r1 − r2 ⊗ r2 , a4 = r1 ⊗ r2 + r2 ⊗ r1 .

Then the fourth order tensors Ci and the second order tensor Γ are expressed by C1 = A11 a1 ⊗ a1 + A12 (a1 ⊗ a2 + a2 ⊗ a1 ) + A22a2 ⊗ a2 + A44 a4 ⊗ a4 , C2 = B13a1 ⊗ a3 + B14a1 ⊗ a4 + B23a2 ⊗ a3 + B24a2 ⊗ a4 + B41a4 ⊗ a1 + B42a4 ⊗ a2 , C3 = C11 a1 ⊗ a1 + C12 (a1 ⊗ a2 + a2 ⊗ a1 ) + C22a2 ⊗ a2 + C33 a3 ⊗ a3 + C34 (a3 ⊗ a4 + a4 ⊗ a3 ) + C44a4 ⊗ a4 , Γ = Γ1 a1 + Γ2 a2 ,

(3.6)

where the coefficients Ars , Brs , Crs and Γα are the effective stiffness moduli. As mentioned by Zhilin [18], Sect. 3.10, the formulas (3.6) have a wide range of applicability. By a suitable choice of the effective stiffness moduli, these equations can be used to describe the mechanical behavior of the following types of shells: transversal isotropic shells with constant thickness; orthotropic multi-layered shells (with symmetric or non-symmetric sequences of layers); reinforced shells; or shells with several inelastic layers. In this paper, we shall employ the relations (3.6), together with the constitutive equations (3.3)-(3.5), to investigate the deformation of orthotropic multi-layered cylindrical shells made of elastic materials.

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3.3 Cylindrical Shells with Arbitrary Cross-Section Shape. Formulation of the Problem In what follows we assume that the reference configuration S0 is a cylindrical surface (see Fig. 3.2). We refer the cylindrical surface to a Cartesian orthogonal frame Ox1 x2 x3 , such that the generators of S0 are parallel to the axis Ox3 . We denote the material surface coordinates by x1 = s, x2 = z, where the axial coordinate z is the distance to the plane x1 Ox2 , while the circumferential coordinate s is the arclength parameter along the cross-section curves z = const. Let ei be the unit vectors along the Oxi axes. Then, the reference surface S0 is characterized by the parametric equations r = r(s, z) = xα (s) eα + z e3 ,

s ∈ [0, s¯], z ∈ [0, z¯],

(3.7)

where x1 (s), x2 (s) are arbitrary functions which describe the shape of the cylinder’s cross-section. For our subsequent analysis we assume that the functions xα (s) are of class C 3 [0, s¯]. Let us designate by τ (s) the unit tangent vector to the cross-section curve, n(s) the unit normal vector to the surface, and r(s) the radius of curvature of the cross-section curve. Then, the fields introduced previously have the expressions r1 = r1 = τ = xα eα ,

r2 = r2 = e3 , n = eαβ xβ eα , c = eαβ rα ⊗ rβ , 1 a = τ ⊗ττ + e3 ⊗ e3 , b = − τ ⊗ττ, r(s) = [eαβ xα (s)xβ (s)]−1 , r(s)

(3.8)

where eαβ stands for the two-dimensional alternator (e12 = −e21 = 1, e11 = e22 = 0), and ( ) = d(ds) . Due to the identification r1 = τ , r2 = e3 we denote the components of the above tensors in the following manner u = u1τ + u2 e3 + u3 n, ϕ = −ϕ2τ + ϕ1 e3 , γ = γ1τ + γ2 e3 , ε = ε1τ ⊗ττ + ε2 e3 ⊗ e3 + ε12τ ⊗ e3 + ε21 e3 ⊗ττ, k = −δ1τ ⊗ττ + δ2 e3 ⊗ e3 + κ1τ ⊗ e3 − κ2 e3 ⊗ττ, T = T 1τ ⊗ττ + T 2 e3 ⊗ e3 + T 12τ ⊗ e3 + T 21e3 ⊗ττ + N1τ ⊗ n + N2 e3 ⊗ n, M = −M12τ ⊗ττ + M21 e3 ⊗ e3 + M1τ ⊗ e3 − M2 e3 ⊗ττ.

(3.9)

Cz¯ (s, z) e3 r Fig. 3.2 Reference configuration S0 of the cylindrical shell

O e1

τ n

e2 C0


35

In view of (3.1), (3.8) and (3.9), the components of the strain tensors are expressed by the geometrical equations 1 1 ε1 = u1,1 + u3 , ε2 = u2,2 , ε12 = ε21 = (u1,2 + u2,1 ), r 2 1 γ1 = u3,1 − u1 + ϕ1 , γ2 = u3,2 + ϕ2 , r 1 δ1 = ϕ2,1 + (u1,2 − u2,1 ), δ2 = ϕ1,2 , κ1 = ϕ1,1 , κ2 = ϕ2,2 . 2r Here we designate the partial derivatives in the usual manner: f,1 = ∂f ∂x2

∂f ∂z

∂f ∂x1

(3.10)

=

∂f ∂s

, f,2 =

= , for any function f . The equilibrium equations (3.2) can be put in the operator form P(v) = L , L = −(F i , Lα ), (3.11) ϕ ) is the field of displacements and rotations, L represents the external where v = (u,ϕ body loads acting on the shell, and the operator P = (P1 , ..., P5 ) is given by P1 (v) = T 1,1 (v) + T 21,2(v) + 1r N1 (v),

P2 (v) = T 12,1 (v) + T 2,2(v), 1 P3 (v) = N1,1 (v) + N2,2(v) − T 1 (v), r P4 (v) = M1,1 (v) + M21,2 (v) − N1(v), P5 (v) = M12,1 (v) + M2,2(v) − N2(v).

(3.12)

To write the constitutive equations in component form, we use the relations (3.3)(3.6) and (3.10)6,7 . Thus, we have T 1 =a1 ε1 + a3ε2 + b1 κ1 + b2 κ2 , T 2 = a2 ε2 + a3 ε1 + b3κ1 + b4 κ2 , 1 1 1 1 1 1 T 12 = b5 − c4 δ1 + b6 − c6 δ2 + a4 − b5 ε12 , N1 = d1 γ1 , 2 r 2 r 2 r (3.13) 1 1 1 1 1 1 T 21 = b5 + c4 δ1 + b6 + c6 δ2 + a4 + b5 ε12 , N2 = d2 γ2 , 2 r 2 r 2 r M1 = b1 ε1 + b3ε2 + c1 κ1 + c3 κ2 , M2 = b2 ε1 + b4 ε2 + c3 κ1 + c2 κ2 , M12 = b5 ε12 + c4 δ1 + c6 δ2 , M21 = b6 ε12 + c6 δ1 + c5 δ2 , where the coefficients ar (r = 1, ..., 4), b s , c s (s = 1, ..., 6) and dγ are expressed in terms of the effective elastic moduli Ars , Brs , Crs and Γα by a1 = A11 + 2A12 + A22 , a2 = A11 − 2A12 + A22 , a3 = A11 − A22 , a4 = 4A44 , b1 = −B13 − B23 + B14 + B24, b2 = −(B13 + B23 + B14 + B24), b3 = −B13 + B23 + B14 − B24, b4 = −B13 + B23 − B14 + B24, b5 = −2(B41 + B42), b6 = 2(B41 − B42), d1 = Γ1 + Γ2 , d2 = Γ1 − Γ2 , c1 = C33 − 2C34 + C44 , c2 = C33 + 2C34 + C44 , c3 = C33 − C44 , c4 = C11 + 2C12 + C22 , c5 = C11 − 2C12 + C22 , c6 = −C11 + C22 . (3.14) The general form of the constitutive equations (3.13), (3.14) is applicable for orthotropic multi-layered cylindrical shells. For each given material distribution in the layers, a suitable choice of the effective elastic moduli Ars , Brs , Crs and Γα need to

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be made. In our work, we present a general solution procedure based on the form (3.13), (3.14) for the constitutive equations. Then, we apply this method to determine the deformation of 3-layered cylindrical shells. We mention that, in the general case (3.13), (3.14), the effective elastic moduli Ars , Brs , Crs and Γα are prescribed functions of the circumferential coordinate s, so that our approach can treat also cylindrical shells with circumferential nonhomogeneity. Let us formulate the mechanical problem which will be investigated: determine the equilibrium deformation of a multi-layered cylindrical shell under the action of body loads F, L, and subjected to given resultant forces and resultant moments acting on its end edges. In what follows, we consider closed cylindrical surfaces (i.e. thin-walled tubes), but we mention that the same method can be applied also for open cylindrical surfaces. We denote by C0 the end edge of the cylindrical surface S0 characterized by z = 0, and Cz¯ the end edge corresponding to z = z¯. Then, the necessary and sufficient conditions for the equilibrium of the cylindrical shell are F da + t dl = 0, S0 C0 ∪Cz¯ (3.15) (L + r × F) da + (m + r × t) dl = 0. S0

C0 ∪Cz¯

These relations express the overall balance of forces and couples applied to the ϕ), we introduce the cylindrical shell. For any displacement and rotation field v = (u,ϕ vector functionals F (v) and M (v) given by F (v) = t(v) dl, M (v) = [m(v) + r × t(v)] dl, (3.16) C0

C0

which represent the resultant force and the resultant moment about O, respectively, acting on the end edge C0 and corresponding to the field v. Then, the global boundary conditions on the end edge C0 have the form F (v) = F 0 ,

M (v) = M 0 ,

(3.17)

where F 0 and M 0 are the prescribed resultant force and moment acting on C0 . We mention that the global boundary conditions on the opposite end edge Cz¯ can be easily derived from relations (3.15), in view of (3.16) and (3.17). We denote the components of the vectors F and M by: F = Fi ei and M = Mi ei . Since we consider closed cylindrical shells, we impose that the displacement and ' ( ϕ (s, z) satisfy the continuity conditions for s = 0, s¯: rotation field v(s, z) = u(s, z),ϕ v(0, z) = v( s¯, z),

∂v ∂v (0, z) = ( s¯, z), ∂s ∂s

∂2 v ∂2 v (0, z) = 2 ( s¯, z), 2 ∂s ∂s

∀ z ∈ [0, z¯]. (3.18)

To recapitulate, the basic equations for our shell problem are: the geometrical relations (3.10), the equilibrium equations (3.11), (3.12), the constitutive equations


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(3.13), the boundary conditions (3.16), (3.17) and the continuity conditions (3.18). As we can see, the data of our problem are the body loads F, L, the resultant force F 0 and the resultant moment M 0 acting on C0 , while the unknown is the displaceϕ). We denote this problem by Π(F F 0 ,M M0 , F, L). Let ment and rotation field v = (u,ϕ ϕ) of us designate by D(F, L) the set of all displacement and rotation fields v = (u,ϕ class C 1 (S¯ 0 ) ∩ C 2 (S0 ) which satisfy the equilibrium equations (3.11), (3.12), and the continuity conditions (3.18). We mention that the corresponding problem for solid cylinders has been extensively studied in the three-dimensional theory of elasticity (see e.g. [10] and the references given therein). In this context, if the body loads are absent (F = 0, L = F 0 ,M M 0 , 0, 0) is called the relaxed Saint-Venant problem. 0) then the problem Π(F Also, if the body loads F and L are independent of z (but they depend only on F 0 ,M M0 , F, L) is known as the Almansi-Michell problem. Moreover, if s) then Π(F F and L are polynomial functions of z (with coefficients depending on s) then F 0 ,M M 0 , F, L) is usually called the Almansi problem. Due to their importance Π(F in engineering, these problems have received considerable attention in the threedimensional elasticity theory. An interesting general method to investigate the deformation of solid cylinders has been presented in the book [10]. In our work we consider the corresponding problems for shells and adapt the method of Ies¸an [10] to study the deformation of thin-walled tubes. We remark first F 0 ,M M 0 , F, L) is not unique, due to the presence that the solution of the problem Π(F of global boundary conditions (3.17) on the end edges. However, by virtue of SaintVenant’s principle for shells, the difference between any two solutions is negligible, except possibly near the end edges. In the next sections we present a general method to construct solutions of these shell problems.

3.4 Solution Procedure An important role in our solution procedure will be played by the following result. ϕ) ∈ D(F, L) be a displacement and rotation field such that Theorem 3.1. Let v = (u,ϕ ' ∂F ∂L ( ∂v 1 ¯ 2 is of class C ( S ) ∩ C (S0 ). Then, we have ∂v 0 ∂z ∂z ∈ D ∂z , ∂z . Moreover, the resultant forces and moments acting on C0 , corresponding to the field ∂v ∂z , are given by F

∂v ∂z

=

C0

F dl,

M

∂v ∂z

= eαβ Fβ (v)eα +

C0

(L + r × F) dl.

(3.19)

Proof. From the linearity of the theory and the definition of the set D(F, L) it fol' ∂F ∂L ( lows that ∂v ∂z ∈ D ∂z , ∂z . Next, to prove the relations (3.19), we employ the equations (3.16), the equilibrium equations (3.11), (3.12), and the continuity conditions (3.18). These calculations are not detailed here. The preceding theorem has the following important consequence.

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Corollary 3.1. Assume that the external body loads F and L do not depend on the ϕ ) be a solution of the problem Π(F F 0 ,M M0 , F, L), axial coordinate z, and let v = (u,ϕ ∂v ∂v 1 ¯ 2 such that ∂z ∈ C (S0 ) ∩ C (S0 ). Then, the field ∂z is a solution of the problem G,Q Q, 0, 0), where the resultant force G and the resultant moment Q are expressed Π(G by G=

C0

F dl,

Q = eαβ Fβ0 eα +

C0

(L + r × F) dl.

(3.20)

∂L ∂v Proof. Indeed, in this case we have ∂F ∂z = 0, ∂z = 0, so that ∂z ∈ D(0, 0) by virtue of Theorem 3.1. The expressions of G and Q in (3.20) follow from the relations (3.19).

These results will be used in our subsequent developments. F 0 ,M M0 , F, L), it is useful to analyze first the In order to solve our problem Π(F solutions of this problem that depend only on s.

3.4.1 Solutions Depending Only on the Circumferential Coordinate Let us consider the equilibrium equations (3.11), (3.12), together with the continuity ϕ(s)) that depend only on s. conditions (3.18), and search for solutions v(s) = (u(s),ϕ More precisely, we consider the equations P(v(s)) = L(s)

for

s ∈ [0, s¯],

(3.21)

and the boundary conditions v (0) = v ( s¯),

v(0) = v( s¯),

(3.22)

where the body loads L(s) = −(Fi (s), Lα (s)) are assumed to depend only on s. Concerning the existence of solutions to the boundary-value problem (3.21), (3.22), we have the following result. Lemma 3.1. The boundary-value problem (3.21), (3.22) admits a solution v(s) = ϕ(s)) if and only if the external body loads F(s) and L(s) satisfy the following (u(s),ϕ conditions F dl = 0, (L + r × F) dl · e3 = 0. (3.23) C0

C0

Moreover, if the body loads F(s) and L(s) satisfy in addition the continuity conditions F(0) = F( s¯),

L(0) = L( s¯),

then the solution v(s) of the boundary-value problem (3.21), (3.22) verifies the continuity relation v (0) = v ( s¯). The proof of Lemma 3.1 is not difficult, but it necessitates lengthy calculations, and will be omitted. We only mention that the equilibrium equations (3.21) reduce to a


39

system of linear ordinary differential equations for the unknowns ui (s) and ϕα (s). Using the relations (3.12), the constitutive equations (3.13), and the boundary conditions (3.22), we are able to determine the solution of equations (3.21) in analytic form, provided the necessary and sufficient conditions (3.23) are satisfied. As a physical interpretation, the condition (3.23)1 expresses that the resultant force of the external body forces acting on C0 is zero. The condition (3.23)2 means that the axial component of the resultant moment about O of the external body moments and couples acting on C0 is vanishing. Since the problem (3.21), (3.22) is related only to the cross-section curve of the cylindrical shell and the fields do not depend on z, we call the boundary-value problem of type (3.21), (3.22) the cross-section plane problem. In view of the above results, we know how to determine the analytic solution of cross-section plane problems. In the following sections, we will have the opportunity to verify the conditions (3.23) and to solve several cross-section plane problems of the type (3.21), (3.22).

3.4.2 Basic Solutions to the Equilibrium Equations Suggested by the results concerning the deformation of the three-dimensional cylinϕ) of the equilibrium equations (in the ders [10], we search for the solutions v = (u,ϕ absence of the external body loads) such that ∂v ∂z is a rigid body displacement of the cylindrical shell. This type of solutions will be useful to construct the general F 0 ,M M0 , F, L). In other words, we look for displacement solutions of the problem Π(F ϕ(s, z)) that satisfy the equations and rotation fields v(s, z) = (u(s, z),ϕ P(v) = 0

with

v(0, z) = v( s¯, z),

v (0, z) = v ( s¯, z),

(3.24)

and

ϕ ∂u ∂ϕ ϕ∗ × r, = u∗ +ϕ = a · ϕ∗ , (3.25) ∂z ∂z where u∗ and ϕ ∗ are some arbitrary constant vectors. The relations (3.25) express ' ϕ( ∂ϕ the fact that ∂u ∂z , ∂z is a rigid body displacement and rotation field of the simple ϕ) has shell. Integrating the equations (3.25) with respect to z, we obtain that v = (u,ϕ the following form (up to a rigid displacement field) 1 u1 = − z2 (ωα xα ) + ω4 z(eαβ xα xβ ) + y1 (s), 2 u2 = z(ωα xα + ω3 ) + y2(s), 1 u3 = z2 (eαβ ωβ xα ) − ω4 z(xα xα ) + y3(s), 2 ϕ1 = ω4 z + ζ1 (s), ϕ2 = z(eαβ ωα xβ ) + ζ2 (s),

(3.26)

where ω1 , ..., ω4 are arbitrary constants, while yi (s) and ζα (s) are arbitrary functions of class C 2 [0, s¯]. We denote the components by y = y1τ + y2 e3 + y3 n , ζ = −ζ2τ + ζ1 e3

40


' ( ϕ) and by w(s) = y(s),ζζ (s) . If we regroup the terms containing ωk , the field v = (u,ϕ given by (3.26) can be written in the form v(s, z) = w(s) +

4

ωk vˆ(k) (s, z),

(3.27)

k=1

where we have denoted by vˆ(k) (s, z), k = 1, ..., 4, the fields 1 1 vˆ(α) = − z2 xατ + z xα e3 − z2 eαβ xβ n , −z eαβ xβτ , 2 ' 2 ( vˆ(3) = (z e3 , 0), vˆ(4) = z eαβ xα xβτ − z xα xα n , z e3 .

(3.28)

Imposing that v satisfy the equilibrium equation (3.24)1 from (3.27) we get P(w(s)) = −

4

ωk P(ˆv(k) (s, z)).

(3.29)

k=1

By virtue of the linearity, we separate (3.29) into 4 different boundary-value problems (k = 1, ..., 4): ' ( ' ( P w(k) (s) = −P vˆ(k) (s, z) with

w(k) (0) = w(k) ( s¯), w(k) (0) = w(k) ( s¯)

(3.30)

Since the fields v(k) are given by (3.28), it easy to show that (3.30) represents a crosssection plane problem of the type (3.21), (3.22). By a straightforward calculation we verify that the conditions (3.23) are satisfied in our case and thus, in view of Lemma 3.1, we can solve the problem (3.30) to find the unknown w(k) (s). In what follows, we consider that the field w(k) (s) have been determined in this way, for each k = 1, ..., 4. On the basis of (3.27), (3.29) and (3.30), we obtain v(s, z) =

4

ωk v(k) (s, z)

with v(k) (s, z) = w(k) (s) + vˆ (k) (s, z).

(3.31)

k=1

Let ω ˆ denote the quadruple ω ˆ = (ω1 , ω2 , ω3 , ω4 ). In order to indicate the dependence on the arbitrary constants ωk , we designate the field (3.31)1 by v{ω}, ˆ i.e. we put v{ω} ˆ =

4

ωk v(k) .

(3.32)

k=1

Since the fields vˆ(k) and w(k) are known, then the basic solutions v(k) are also known (k = 1, ..., 4). For any constants ωk , the displacement and rotation field v{ω} ˆ defined by (3.32) possesses the following important properties: i) v{ω} ˆ is a solution of the equilibrium equations with zero body loads, i.e. we have P(v{ω}) ˆ = 0;

(3.33)


41

ˆ ii) the field ∂v{∂zω} is a rigid body displacement and rotation field of the cylindrical shell; iii) the strain tensors corresponding to to the field v{ω} ˆ are independent of the axial coordinate; iv) we have Fα (v{ω}) ˆ = 0, α = 1, 2. (3.34)

We mention that these properties are analogous to the properties of the wellknown Saint-Venant’s solutions in three-dimensional elasticity. These solutions v{ω} ˆ will be employed in the next sections to express the soF 0 ,M M0 , F, L). We introduce the constants Drk , (r, k = lutions of our problem Π(F 1, ..., 4), defined by ' ( Dαk = eαβ Mβ v(k) ,

' ( D3k = −F3 v(k) ,

' ( D4k = −M3 v(k) ,

(3.35)

which are known quantities, since v(k) are known and the functionals F3 and Mi are given by (3.16). We observe that the field v{ω} ˆ does not depend on the external loads, but they depend only on the geometry of the cylindrical shell and the material properties. It can be calculated for every specific geometry and given material properties of shells. In Sects 3.5 and 3.6 we compute the field v{ω} ˆ for different types of circular cylindrical shells.

3.4.3 Cylindrical Shells Subjected to Terminal Resultant Forces and Moments F 0 ,M M0 , 0, 0), which consists in the equiLet us present a solution of the problem Π(F librium deformation of cylindrical shells under the action of resultant forces F 0 and resultant moments M 0 acting on the end edges, in the absence of body loads (F = 0, L = 0). The result is given in the next theorem. F 0 ,M M0 , 0, 0) concerning the deformation of Theorem 3.2. Consider the problem Π(F cylindrical shells due to terminal resultant forces and moments. Then, there exists a ϕ) of this problem having the form solution v = (u,ϕ z v= v{ω} ˆ dz + v{ˆη} + w(s), (3.36) 0

where ω ˆ = (ω1 , ω2 , ω3 , ω4 ) and ηˆ = (η1 , η2 , η3 , η4 ) are quadruples of constants and w(s) is a displacement and rotation field of class C 2 [0, s¯] which depends only on the circumferential coordinate s. Proof. Let us determine the constants ω, ˆ η, ˆ and the field w(s) such that the field v given by (3.36) satisfies v ∈ D(0, 0) and Fi (v) = Fi0 ,

Mi (v) = M0i .

(3.37)

42


First, we shall use the Corollary 3.1 to determine the constants ω. ˆ If we differentiate (3.36) with respect to z and take into account the property (ii) for the field v{ˆη}, then we obtain (up to a rigid body displacement field) ∂v = v{ω} ˆ . ∂z

(3.38)

On the other hand, from Corollary 3.1 we deduce that ∂v ∂z is a solution of the problem 0 Π(0, eαβ Fβ eα , 0, 0). Then, in view of (3.38) and (3.34), we have F3 (v{ω}) ˆ = 0,

Mα (v{ω}) ˆ = eαβ Fβ0 ,

M3 (v{ω}) ˆ = 0.

Using the notations (3.35), the last relations become ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ ' ( ⎜⎜⎜ Dkr ωr ⎟⎟⎟⎠ = − F10 , −F20 , 0, 0 . ⎝ r=1

(3.39)

k=1,...,4

The relations (3.39) represent a system of 4 algebraic linear equations for the constants ωk (k = 1, ..., 4). In what follows, we consider that ωk have been thus determined. Next, we apply the Lemma 3.1 to determine the field w(s). In view of (3.36), the equilibrium equations P(v) = 0 can be written as z P(w(s)) = −P v{ω} ˆ dz − P(v{ˆη}). 0

Taking into account (3.33), from the last equations we obtain the following boundary-value problem for the unknown w(s): z P(w(s)) = −P v{ω} ˆ dz with w(0) = w( s¯), w (0) = w ( s¯). (3.40) 0

') z

( ' ( ∂ We notice that ∂z P 0 v{ω}dz ˆ = P v{ω} ˆ = 0, in view of (3.33). Thus, the righthand side of (3.40)1 does not depend on z, so that (3.40) represents a cross-section plane problem of the type (3.21), (3.22). We apply the Lemma 3.1 for the crosssection plane problem (3.40) and we verify that the conditions (3.23) are satisfied in our case. Then, we can solve the problem (3.40) to obtain the field w(s). In what follows, we assume that w(s) has been determined. Finally, we compute the constants ηˆ by imposing the end edge conditions (3.37). If we denote by z

w(s, ˆ z) = w(s) +

v{ω}dz ˆ

0

for brevity, then from the conditions (3.37) we deduce F3 (v{ˆη}) = F30 − F3 (w), ˆ

Mi (v{ˆη}) = M0i − Mi (w), ˆ


or equivalently, using the notations (3.35), ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ % ' & ( ⎜⎜⎜ Dkr ηr ⎟⎟⎟⎠ = eαβ M0β − Mβ (w) ˆ , F3 (w) ˆ − F30 , M3 (w) ˆ − M03 . ⎝ r=1

43

(3.41)

k=1,...,4

Relations (3.41) form a system of 4 algebraic equations which permits the determination of the constants ηk (k = 1, ..., 4). The proof is complete. Remark. The proof of Theorem 3.2 describes the modality to construct effecF 0 ,M M0 , 0, 0). tively the solution of the form (3.36) to the problem Π(F

3.4.4 Deformation of Loaded Cylindrical Shells In this section we analyze the equilibrium of cylindrical shells when we have also F 0 ,M M0 , F, L). external body loads F and L acting on the shell, i.e the problem Π(F For the first stage, we assume that the external body loads F and L do not depend on the axial coordinate z, but they are arbitrary functions of s. In this case, the solution of our problem is given by the following result. F 0 ,M M0 , F, L) for the deformation of loaded Theorem 3.3. Consider the problem Π(F cylindrical shells in the case when the body loads F and L depend only on the ϕ) of the circumferential coordinate s. Then, this problem admits a solution v = (u,ϕ form z

v= 0

z

z

v{ω} ˆ dz dz +

0

ˆ + z w(s) + w(s), v{ˆη} dz + v{ξ} ˜

(3.42)

0

where ω, ˆ ηˆ and ξˆ are quadruples of constants and w(s), w(s) ˜ are displacement and rotation fields of class C 2 [0, s¯] which depend only on s. Proof. We follow the same idea as in the preceding proof. First, we determine the constants ω, ˆ ηˆ and the field w(s) by applying the Corollary 3.1. Indeed, by differentiating (3.42) with respect to z we find (up to a rigid body field) z ∂v = v{ω} ˆ dz + v{ˆη} + w(s). (3.43) ∂z 0 On the other hand, by virtue of Corollary 3.1, ∂v ∂z is a solution of the problem G,Q Q, 0, 0) where G and Q are given by (3.20). If we apply the Theorem 3.2 (and Π(G its proof) for the solution (3.43), we obtain that the constants ωr are given by the algebraic system ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ ' ( ⎜⎜⎜⎝ Dkr ωr ⎟⎟⎟⎠ = − G1 , −G2 , 0, 0 , (3.44) r=1

k=1,...,4

44


the field w(s) is the solution of the cross-section plane problem (3.40), while the constants ηr are determined by the equations ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ % ' & ( ⎜⎜⎜⎝ Dkr ηr ⎟⎟⎟⎠ = eαβ Qβ − Mβ (w) ˆ , F3 (w) ˆ − G3 , M3 (w) ˆ − Q3 . (3.45) r=1

k=1,...,4

Thus, we have found ω, ˆ ηˆ and w(s). Let us denote for brevity by z z z V{ω, ˆ η, ˆ w} = v{ω} ˆ dz dz + v{ˆη} dz + z w(s), 0

0

(3.46)

0

which is a known field. Then, the equilibrium equations P(v) = −(F, L) reduce to the following cross-section plane problem for the determination of the field w(s) ˜ ' ( P(w(s)) ˜ = −(F, L) − P V{ω, ˆ η, ˆ w}

with w(0) ˜ = w( ˜ s¯), w˜ (0) = w˜ ( s¯).

(3.47)

We can verify that the conditions (3.23) are satisfied in the case of the problem (3.47), so that we can compute its solution w(s). ˜ Finally, in order to determine the constants ξr (r = 1, ..., 4) we impose the end edge conditions of the form (3.37), with v given by (3.42). These conditions reduce to the following system of algebraic equations 4

% ' (& Dαr ξr = eαβ M0β − Mβ w˜ + V{ω, ˆ η, ˆ w} ,

r=1

4 r=1 4

' ( D3r ξr = F3 w˜ + V{ω, ˆ ηˆ , w} − F30 ,

(3.48)

' ( D4r ξr = M3 w˜ + V{ω, ˆ ηˆ , w} − M03 ,

r=1

which allows for the determination of the constants ξr . The proof is complete.

For the second part of this section, we consider the case when the external body loads are polynomials in the axial coordinate z of arbitrary degree n, i.e. F=

n k=0

F(k) (s) zk ,

L=

n

L(k) (s) zk ,

(3.49)

k=0

where the coefficients F(k) (s) and L(k) (s) are given functions of s. By virtue of (3.49) and the linearity of the theory, we can additively decompose F 0 ,M M0 , F, L) as the sum of the problems Π(F F 0 ,M M0 , F(0) , L(0) ) and our problem Π(F k k F 0 ,M M 0 , F(0) , L(0) ) Π(0, 0, F(k) z , L(k) z ), k = 1, ..., n. We remark that the problem Π(F has already been solved in Theorem 3.3. Let us find also a solution of the problem Π(0, 0, F(k) zk , L(k) zk ). To this aim, we employ the method of mathematical induction with respect to the degree k = 1, ..., n. In other words, we assume that we know a


45

solution of the problem Π(0, 0, F(k) zk , L(k) zk ) and we want to find a solution v to the problem Π(0, 0, F(k+1) zk+1 , L(k+1) zk+1 ). The procedure to compute the new solution v is given by the next result. Theorem 3.4. Assume that v∗ is a solution of the problem Π(0, 0, F(k+1) zk , L(k+1) zk ). ϕ) of the Then, the problem Π(0, 0, F(k+1) zk+1 , L(k+1) zk+1 ) admits a solution v = (u,ϕ form z

v = (k + 1)

v∗ dz + v{ω} ˆ + w∗ (s) ,

(3.50)

0

where ω ˆ = (ω1 , ω2 , ω3 , ω4 ), is a quadruple of constants and w(s), w∗ (s) is a field of displacement and rotation which depends only on s. Proof. Let us determine w∗ (s) and the constants ωk . Imposing that the field (3.50) satisfies the relation v ∈ D(F(k+1) zk+1 , L(k+1) zk+1 ), we derive the following boundaryvalue problem for the field w∗ (s) z 1 ∗ k+1 k+1 ∗ P(w (s)) = − (F(k+1) z , L(k+1) z ) − P v dz , (3.51) k+1 0 ∗ ∗ ∗ ∗ w (0) = w ( s¯), w (0) = w ( s¯). Since the conditions of Lemma 3.1 are satisfied, we can use the cross-section plane problem (3.51) to determine w∗ (s). Next, from the end edge boundary conditions Fi = 0, Mi = 0 we get the system of 4 algebraic equations ⎛ 4 ⎞ ⎜⎜⎜ ⎟⎟⎟ % & ⎜⎜⎝⎜ Dkr ωr ⎟⎟⎠⎟ = eβα Mβ (w), ¯ F3 (w), ¯ M3 (w) ¯ , (3.52) r=1

k=1,...,4

)z where we have denoted by w(s, ¯ z) = 0 v∗ dz + w∗ (s). From Eqs (3.52) we find the constants ωr and the proof is complete Remark. The proofs of Theorems 3.3 and 3.4 present the method to construct F 0 ,M M0 , F, L), when F and L are polynomials in z, the solution of the problem Π(F of arbitrary degree. In the general case when F and L are given (smooth enough) functions of (s, z) we can approximate them by polynomials in z of the form (3.49), F 0 ,M M0 , F, L). and then apply the solution procedure to solve the problem Π(F In the next sections we confine our attention to circular cylindrical shells and apply our solution procedure to different types of material.

3.5 Circular Cylindrical Shells Let us consider in this section circular cylindrical shells made of isotropic materials. In this simple case, we are able to obtain the solution in closed form and to compare it with the results obtained in previous approaches to shell problems.

46


The cross-section curves are circles of radius r0 , so that the parametric equations of the reference surface S0 is given by (3.7), with x1 (s) = r0 cos

s , r0

x2 (s) = r0 sin

s , r0

s ∈ [0, 2πr0 ].

(3.53)

In this case we have the relations xα = eβα

xβ , r0

xα = −

xα , r02

r(s) = r0 .

(3.54)

Let us denote by E the Young’s modulus, ν the Poisson’s ratio, μ the shear modulus of the isotropic material and by h the thickness of the shell. According to [18] Chapt. 3, in the case of isotropic shells the effective stiffness moduli introduced in (3.6) are given by Eh E h3 ν , A12 = 0, A22 = A44 = μh, B13 = , 2(1 − ν) 24r0 (1 − ν)2 E h3 E h3 E h3 B14 = − , B24 = B42 = 0, B41 = , B23 = , 24r0 (1 − ν) 48r0 (1 + ν) 24r0 (1 − ν2 ) h2 E h3 E h3 E h3 C11 = , C = , C = C = , 33 22 44 24(1 − ν) 24(1 + ν) 48r02 24(1 + ν) π2 C12 = C34 = 0, Γ1 = μ h Γ0 , Γ2 = 0 Γ0 = . 12 (3.55) 2 The coefficient Γ0 = π12 stands for the shear correction factor. If we denote by C, D, and γ the following quantities A11 =

C=

Eh , 1 − ν2

D=

E h3 , 12(1 − ν2)

γ=

1 h 2 , 48 r0

(3.56)

then in view of (3.13), (3.14) and (3.55), the constitutive equations in our case are D (κ1 + ν2 κ2 ), r0 (1 − ν) D T 2 = C(νε1 + ε2 ) − [νκ1 + (ν2 + ν − 1)κ2], r0 (1 − ν) * + D T 12 = (1 − ν) C(1 + γ)ε12 + [−(2 + γ)δ1 + γ δ2 ] , 4r0 , D T 21 = (1 − ν) C(1 − γ)ε12 + [γ δ1 + (2 − γ)δ2] , 4r0 π2 N1 = (μ h Γ0 )γ1 , N2 = (μ h Γ0 )γ2 Γ0 = , 12 D M1 = D(κ1 + νκ2 ) − (ε1 + νε2 ), r0 (1 − ν) D M2 = D(νκ1 + κ2 ) − [ν2 ε1 + (ν2 + ν − 1)ε2], r0 (1 − ν) T 1 = C(ε1 + νε2 ) −


(1 − ν)D 1 M12 = − ε12 + (1 + γ)δ1 + (1 − γ)δ2 , 2 r 0 (1 − ν)D 1 M21 = ε12 + (1 − γ)δ1 + (1 + γ)δ2 . 2 r0

47

(3.57)

F 0 ,M M0 , 0, 0), i.e. we consider Our aim is to find the solution of the problem Π(F circular cylindrical shells deformed by terminal resultant forces F 0 and moments M 0 . Let us determine first the basic solution field v{ω}. ˆ Following the procedure described in Sect. 3.4.2, we find that the displacement and rotation field v{ω} ˆ for isotropic circular cylindrical shells is given by u3 = −

u1 = 2

z2 (eαβ ωα xβ ) + ω4 r0 z, 2r 0

z + r0 F ωα xα − ν r0 ω3 , 2r0

u2 = z (ωα xα + ω3 ),

ϕ1 = ω4 z − Geαβ ωα xβ ,

ϕ2 =

where the coefficients F and G are constants specified by the relations 1−ν 1−ν 4γ 4γ ν2 Γ0 + 1 F + Γ0 + G = −ν + , 1−ν 1−ν 2 2 1−ν 4γ 1−ν 4γ ν2 Γ0 + F+ Γ0 + 4γ G = − . 2 1−ν 2 1−ν

z ωα xα , r0 (3.58)

(3.59)

Using the definitions (3.35) and (3.16) we compute next the coefficients Drk as follows 4γν2 4γ D11 = D22 = π r03 C (1 + Fν) + (G − F) + (4 − 5ν + 2ν3) , 1−ν 1−ν D33 = 2π r0 C(1 − ν2 ), D44 = π r03 C(1 − ν)(1 + 11γ + γ2), Dkr = 0 for k r. (3.60) We are now able to express the solution. According to the procedure presented in F 0 ,M M0 , 0, 0) in the form Sect. 3.4.3, we search for a solution v of the problem Π(F (3.36). Following the steps described in the proof of Theorem 3.2, we obtain the solution z3 z2 (eαβ ωα xβ ) + (eαβ ηα xβ ) + η4 r0 z, 2r0 6r0 1 u2 = z2 + Pr02 ωα xα + z (ηα xα + η3 ), 2 3 2 z z u3 = − + z r0 F ωα xα + − + r0 F ηα xα − ν r0 η3 , 6r0 2r0 2 z z ϕ1 = −zGeαβ ωα xβ − Geαβ ηα xβ + η4 z , ϕ2 = + r0 Q ωα xα + ηα xα , 2r0 r0 u1 =

(3.61)

where the constants ωα and ηk are expressed in terms of the resultants Fi0 and M0i by eαβ M0β F0 M0 F0 ω α = − α , ηα = , η3 = − 3 , η4 = − 3 . (3.62) D11 D11 D33 D44

48


The constant coefficients F and G are given by (3.59), while the constant coefficients P and Q are determined by the equations (1 + 3γ + γ2)P − 2γ(2 + γ)Q = K, 2γ(2 + γ)P − [Γ0 + 4γ(1 + γ)]Q = L,

(3.63)

with the notations . / 4γ K = 2 1 + νF + 1−ν (νG − ν2 − ν + 1) − (1 − ν)γ2G , ( 4γ ' 1−ν2 2 2 2 L = 1−ν 2 Γ0 F + 1−ν 2 G + ν F + ν + 2ν − 2 − 2(1 − ν)γ G.

(3.64)

From the above solution, we can see the geometrical significance of the overall strain measures ωα and ηk . Thus, η3 , ηα and η4 , respectively, represent measures of stretch, curvature and twist of the cylindrical shell considered as a beam, while ωα are global measures of strain appropriate to flexure. In what follows, let us consider the case of very thin shells (h r0 ), i.e. the limiting case when the parameter γ tends to zero: 1 % h &2 γ= → 0. (3.65) 48 r0 In view of (3.65), the solution (3.59)-(3.63) simplifies. Indeed, from (3.59), (3.60) and (3.63) we get D11 = π r03 E h, D33 = 2π r0 E h, D44 = 2π r03 μ h, F = −G = −ν, P = 2(1 − ν2), Q = ν(1−ν) 2 .

(3.66)

E where μ = 2(1+ν) is the shear modulus of the material. Then, the relations (3.62) between the overall strain measures ηk and the resultants F30 , M0i become

ηα =

eαβ M0β π r03 E h

,

η3 = −

F30 2π r0 E h

,

η4 = −

M03 2π r03 μh

.

(3.67)

The above relations (3.67) are in accordance with the results obtained previously for the deformation of thin cylindrical shells, presented in the classical books [16] Sect. 102, [15] Sect. 47, or in the papers [6,14]. Moreover, the displacement field ui given M0 , 0, 0) is in perfect by (3.61) for the extension-bending-torsion problem Π(F30 e3 ,M agreement with the corresponding results presented by Reissner and Tsai [14]. We can conclude that our solution is in agreement with previously known results from the literature, in the particular case of circular cylindrical thin shells made of isotropic materials. In addition, the solution procedure presented in Sect. 3.4 allows for the treatment of more complex structures, such as multi-layered orthotropic cylindrical shells (not necessarily circular). As an example, we study in the next section the deformation of a three-layered cylindrical shell.


49

3.6 Three-Layered Shells Let us present first the constitutive equations for three-layered shells with general geometry. We consider shells with a symmetric arrangement of the 3 layers (Fig. 3.3a): the two exterior layers have the same thickness h2 and they are made of the same isotropic material, while the inner layer has the thickness 2h1 and is made from a different isotropic material. Thus, the total thickness of the shell is 2h, where h = h1 + h2 .

h2 2h1 h2 (a)

(b)

Fig. 3.3 Three-layered shell: (a) General geometry; (b) Circular cylindrical shell

We denote by E 1 , ν1 and μ1 the Young’s modulus, Poisson’s ratio and shear modulus of the interior layer, respectively. The notations E2 , ν2 and μ2 will be used for the exterior layers. The constitutive equations for a three-layered isotropic shell have been presented in [18], Chapt. 5. In order to keep the calculations as simple as possible, we consider a simplified version of the constitutive equations, in which the tensor C2 = 0. More precisely, the effective stiffness moduli which appear in the constitutive equations (3.6) have the following expressions [18] E1 h1 E2 h2 A22 1−ν1 + 1−ν2 , E1 h31 E2 (h3 −h31 ) C33 = 3(1−ν1 ) + 3(1−ν2 ) ,

A11 =

C11 = C12 = C34 = 0,

= A44 =

E1 h1 1+ν1

E2 h2 + 1+ν , 2

C22 = C44 = Bkl = 0

E1 h31 3(1+ν1 )

(∀ k, l),

A12 = 0, +

E2 (h3 −h31 ) 3(1+ν2 )

(3.68)

Γ2 = 0.

For the transverse shear stiffness Γ1 , Zhilin [18] proposes the value ⎛ ⎞ 3 E 2 (h3 − h31 ) ⎟⎟⎟ λ2 λ2 ⎜⎜⎜⎜ E1 h1 ⎟⎟ , Γ1 = 2 C22 = 2 ⎜⎝ + 3(1 + ν2) ⎠ h h 3(1 + ν1) where λ is the smallest positive solution of the equation λh1 λh2 μ1 tan tan = . h h μ2

,

(3.69)

50


We notice that Γ (s) ≤ Γ1 ≤ Γ (p) ,

Γ (s) =

π2 μ1 h1 μ2 h2 , 6 4μ1 h1 + μ2 h2

Γ (p) =

π2 (μ1 h1 + μ2 h2 ). 6

The value Γ (p) corresponds to the connection in parallel, while Γ (s) represents the connection in series of the layers’ stiffness. It is useful to introduce the notations E1 h1 C1 = 2 1−ν 2 , 1

E2 h2 C2 = 2 1−ν 2 , 2

D1 =

3 2 E 1 h1 3 1−ν2 1

,

D2 =

3 3 2 E2 (h −h1 ) 3 1−ν22

.

(3.70)

In view of (3.68)–(3.70), the constitutive equations (3.13) for the case of threelayered cylindrical shells read T 1 = (C1 + C2 )ε1 + (C1 ν1 + C2 ν2 )ε2 , T 2 = (C1 ν1 + C2 ν2 )ε1 + (C1 + C2 )ε2 , T 12 = [C1 (1 − ν1) + C2 (1 − ν2 )]ε12 − 4r1 [D1 (1 − ν1 ) + D2 (1 − ν2)](δ1 + δ2), T 21 = [C1 (1 − ν1) + C2 (1 − ν2 )]ε12 + 4r1 [D1 (1 − ν1 ) + D2 (1 − ν2)](δ1 + δ2), λ2 λ2 N1 = 2h N2 = 2h 2 [D1 (1−ν1 ) + D2 (1−ν2 )]γ1 , 2 [D1 (1−ν1 ) + D2 (1−ν2 )]γ2 , M1 = (D1 +D2 )κ1 + (D1 ν1 +D2 ν2 )κ2 , M2 = (D1 ν1 +D2 ν2 )κ1 + (D1 +D2 )κ2 , M12 = M21 = 12 [D1 (1 − ν1 ) + D2 (1 − ν2 )](δ1 + δ2 ). (3.71) For the remaining of this section, we restrict our attention to circular cylindrical shells, three-layered (Fig. 3.3b). The geometry of the reference surface S0 is described by the equations (3.7), (3.53) and (3.54). We want to determine the equilibrium of this shell under the action of given body loads and arbitrary resultant forces F 0 ,M M0 , F, L). and moments applied to the end edges, i.e. to solve the problem Π(F To this aim, we apply the solution procedure presented in Sect. 3.4, and we determine first the basic solution v{ω}. ˆ For three-layered shells, we obtain the following expression for v{ω} ˆ z2 (eαβ ωα xβ ) + ω4 r0 z , u2 = z (ωα xα + ω3 ), 2r0 z2 ' ( u3 = − ωα xα + r0 Fωα xα + Hω3 , 2r0 z ϕ1 = ω4 z − G eαβ ωα xβ , ϕ2 = ω α x α , r0 u1 =

where the constants F and G are specified by the relations ( / λ2 ' λ2 . (C1 +C2 ) + 2h α ) F + 2h2 Dα (1−να ) G = −(Cα να ) , 2 Dα (1−ν 0 1 / ( λ2 . λ2 ' 1 2 Dα (1−να ) F + D1r+D + 2h 2 2 Dα (1−να ) G = r2 (Dα να ) , 2h2 0

0

(3.72)

(3.73)


51

and the constant H is given by H=−

C1 ν1 + C2 ν2 . C1 + C 2

(3.74)

The coefficients Drk in our case are computed on the basis of Eqs (3.35) and (3.16) ⎡ ⎤ ⎢ ( Dα ν α D1 + D2 ⎥⎥⎥⎥ 3 ⎢⎢⎢ ' D11 = D22 = π r0 ⎢⎣F Cα να − G 2 + (C1 + C2 ) + ⎥⎦ , r0 r02 (C1 + C2 )2 − (Cα να )2 (3.75) D33 = 2π r0 , Dkr = 0 for k r, ⎡ C1 + C2 ⎤ ⎢⎢' ( 9 ' (⎥⎥ D44 = π r03 ⎢⎢⎢⎣ Cα (1 − να) + 2 Dα (1 − να ) ⎥⎥⎥⎦ . 4r0 For the equilibrium problem that we want to solve, we take external body loads independent of the axial coordinate z. More precisely, we consider the body loads F and L of the following form F = F (a) e3 + F (p) n ,

L = 0,

(3.76)

where F (a) and F (p) are given constants. The load system (3.76) represents an uniform pressure force F (p) combined with a constant body force F (a) acting in the axial direction. F 0 ,M M0 , F, According to Theorem 3.3, we search for a solution of our problem Π(F L) in the form (3.42), where the field v{ω} ˆ is given by (3.72). Repeating the steps described in the proof of Theorem 3.3, we find after some calculations the following ϕ) form for the solution v = (u,ϕ z3 z2 (eαβ ηα xβ ) + (eαβ ξα xβ ) + ξ4 r0 z, 6r0 2r0 2 ( z ' u2 = ηα xα + η3 + z (ξα xα + ξ3 ) + 2Pr02 ηα xα , 2 r2 F (p) z3 z2 u3 = − ηα xα − ξα xα + z r0 (Fηα xα +Hη3 ) + r0 (Fξα xα +Hξ3 ) + 0 , 6r0 2r0 C1 +C2 ϕ1 = −zG (eαβ ηα xβ ) + ξ4 z − G (eαβ ξα xβ ) , z2 z ϕ2 = ηα xα + ξα xα + Q r0 ηα xα − H r0 η3 , 2r0 r0 (3.77) where the constants ηi are expressed in terms of the forces Fα0 and F (a) by the relations F0 2π r0 F (a) ηα = − α , η3 = − , (3.78) D11 D33 u1 =

and the constants ξk are given in terms of F30 , M0i and F (p) by ξα =

eαβ M0β D11

,

ξ3 = −

F30 D33

+

r0 F (p) (Cα να ) , (C1 +C2 )2 − (Cα να )2

ξ4 = −

M03 D44

.

(3.79)

52


The constants coefficients P and Q are specified by the equations Cα (1 − να ) F(Cα να ) + (C1 + C2 ) 1 + 4r02 P − Q = G + 4r02 , D (1 − ν ) Dα (1 − να ) α α ⎛ ⎞ ' ( ⎜⎜ λ2 r2 ⎟⎟ λ2 r 2 G Dα (1 + να ) − 2(D1 + D2 ) P − ⎜⎜⎜⎝1 + 2 0 ⎟⎟⎟⎠ Q = 2 0 F + . Dα (1 − να) h h

(3.80)

The solution (3.80) represents the displacement and rotation field for the deformed three-layered shell, corresponding to the given system of external loads. In the same way, we can apply the general solution procedure to solve various practical problems concerning multi-layered cylindrical shells. Acknowledgements For the first author M.B.: the research leading to these results has received funding from the European Union Seventh Framework Programme, FP7-REGPOT-2009-1 under grant agreement no:245479, and from the Polish Ministry of Science and Higher Education, grant no:1471-1/7,PR UE/2010/7; and sponsorship by the Alexander von Humboldt–Foundation.

References 1. Altenbach H (2000) An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int J Solids Struct 37: 3503-3520 2. Altenbach H, Eremeyev VA (2008) Direct approach-based analysis of plates composed of functionally graded materials. Arch Appl Mech 78: 775-794 3. Altenbach H, Eremeyev VA (2009) On the bending of viscoelastic plates made of polymer foams. Acta Mech 204: 137-154 4. Altenbach H, Eremeyev VA (2010) On the effective stiffness of plates made of hyperelastic materials with initial stresses. Int J Non-Lin Mech 45: 976-981 5. Altenbach H, Zhilin PA (2004) The theory of simple elastic shells. In: Kienzler R, Altenbach H, Ott I (eds) Theories of Shells and Plates, Lecture Notes in Applied and Computational Mechanics 16: 1-12. Springer, Berlin 6. Berdichevsky V, Armanios E, Badir A (1992) Theory of anisotropic thin-walled closed-crosssection beams. Comp Eng 2: 411-432 7. Bˆırsan M, Altenbach H (2010) A mathematical study of the linear theory for orthotropic elastic simple shells. Math Meth Appl Sci 33: 1399-1413 8. Bˆırsan M, Altenbach H (2011) On the dynamical theory of thermoelastic simple shells. ZAMM, doi: 10.1002/zamm.201000057 9. Cosserat E, Cosserat F (1909) Théorie des corps déformables. Herman et Fils, Paris 10. Ies¸an D (2009) Classical and Generalized Models of Elastic Rods. Chapman & Hall / CRC Press, Boca Raton - London - New York 11. Ies¸an D (2011) Deformation of porous Cosserat elastic bars. Int J Solids Struct 48: 573-583 12. Lurie AI (2005) Theory of Elasticity. Springer, Berlin 13. Naumenko K, Altenbach H (2007) Modeling of Creep for Structural Analysis. SpringerVerlag, Berlin 14. Reissner E, Tsai WT (1972) Pure bending, stretching, and twisting of anisotropic cylindrical shells. J Appl Mech 39: 148-154 15. Sokolnikoff IS (1956) Mathematical Theory of Elasticity. McGraw-Hill, New York 16. Timoshenko S, Goodier JN (1951) Theory of Elasticity. McGraw-Hill, New York 17. Zhilin PA (1976) Mechanics of deformable directed surfaces. Int J Solids Struct 12: 635-648 18. Zhilin PA (2006) Applied Mechanics: Foundations of Shell Theory (in Russian). Politekhn Univ Publ, Sankt Petersburg

Chapter 4

Asymptotic Integration of One Narrow Plate Problem Valentina O. Finiukova and Alexander M. Stolyar

Abstract The derivation of a narrow plate model is carried out in the paper. It has been based on the method of asymptotic integration in connection with the boundary layer method and applied to one non-linear dynamic problem. It allows to reduce the solution of the given problem to a sequence of linear or nonlinear one-dimensional initial boundary-value problems (as a result of the first iteration process) and a sequence of two-dimensional linear problems (second order process) and get the solution of a given problem with high accuracy. The small parameter is equal to the relation of the lengths of adjacent sides of the plate. It has been obtained that the main term of the asymptotic expansion satisfies the known equation of a beam theory. Some examples are calculated in order to find the limits of asymptotics obtained. Keywords Asymptotic integration · Narrow plate

4.1 Introduction In this paper we consider one singularly disturbed initial-boundary value problem, which describes the dynamic behavior of so-called narrow rectangular plates with free longitudinal sides. Both the asymptotic integration and the boundary layer method are applied to this problem. The solution is constructed as power series in parameter δ which is a ratio of adjacent sides of the plate. Coefficients of the first series are obtained as a result of the first iteration process by its direct substitution in equations and boundary (and initial) conditions. The main terms of expansions satisfy the known linear or non-linear equations of lower dimension. During the first iteration process discrepancies appear. They are compensated during the V. O. Finiukova (B) · A. M. Stolyar Southern Federal University, Milchakova, 8a, 344090 Rostov-on-Don, Russia e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 4, © Springer-Verlag Berlin Heidelberg 2011

53

54

V.O. Finiukova and A.M. Stolyar

second iteration process by boundary-layer functions which are the solutions of the corresponding generalized linear bi-harmonic equations for a semi-strip. The transfer from three-dimensional linear equations of elasticity theory to the two-dimensional one in the case of thin domains have been investigated in [1, 3, 14, 17] and in [2, 5, 12] – for non-linear problems. Non-linear problems of oscillations and dynamic stability of narrow cylindrical panel are considered in [8]. The beam type equations are derived on the base of Mindlin-Timoshenko plate equations in [6, 7]. The following problems are considered in [10]: 1. 2. 3. 4.

linear bending of the plate on non-linear elastic foundation; linear bending of orthotropic plate on linear foundation; linear oscillations of orthotropic plate on linear foundation; oscillations of orthotropic plate (von Kármán type equations [2]).

The transition from two-dimensional plate equations to one-dimensional beam equations takes place when longitudinal sides of the plate are free. Another boundary conditions are defined which allow such transition; some relationship between mechanical parameters are found which makes such transition impossible [10]. In this paper we develop the approach presented in [10] to consider the problem concerning oscillations of orthotropic rectangular plate on the linear foundation.

4.2 Oscillations of Narrow Orthotropic Rectangular Plate on Elastic Foundation. Governing Equations Let us consider the corresponding initial-boundary value problem, given in dimensionless form (here we use von Kármán plate equations [2]): c2 ∂4y w + 2c3δ2 ∂2y ∂2x w + δ4 ∂4x w + k1δ4 w + δ4∂2t w = δ2 L(w, Φ) + δ4 q(x, t),

(4.1)

c2 ∂4y Φ + 2c1 δ2 ∂2y ∂2x Φ + δ4 ∂4x Φ = −αc2 δ2 L(w, w), ∂ ∂ ∂ a [w, ∂t w]t=0 = 0, ∂ x = , ∂y = , ∂t = , δ = ∂x ∂y ∂t l

(4.2)

L(w, Φ) = ∂2x w∂2y Φ + ∂2y w∂2x Φ − 2∂ x ∂y w∂ x ∂y Φ, ∂2x Φ, ∂ x ∂y Φ, ∂2y w + ν1 δ2 ∂2x w, ∂3y w + c4 δ2 ∂y ∂2x w = 0, y=±1 w, ∂2x w, δ2 ∂2x Φ − ν2 ∂2y Φ, δ2 ∂3x Φ + c5 ∂ x ∂2y Φ =0

(4.3)

x=0, 1

(4.4) (4.5)

Besides we use known identity (written in dimensionless form as well) 1% 1 & ∂2y Φ − ν1 δ2 ∂2x Φ dx = αδ2 (∂ x w)2 dx 0

0

(4.6)

4 Asymptotic Integration of One Narrow Plate Problem

55

The introduced coefficients are connected with dimensionless values by the following relations: x1 = lx, c2 =

D2 , D1

Di =

x2 = ay, c3 =

D3 , D1

E i h3 , 12(1 − ν1ν2 )

W = wh,

q=

Ql4 , D1 h

c4 =

D3 + 2Dk , D2

τ = ct,

c2 =

c5 = 2c1 + ν2 ,

k1 =

Dk =

ρhl4 , D1

Kl4 , D1

Gh3 , 12

2c1 =

D3 = D1 ν2 + 2Dk ,

F = D1 Φ,

i = 1, 2,

E2 − 2ν2 , G

α = 6(1 − ν1ν2 ),

E1 ν2 = E 2 ν1 ,

where W and h are the displacement and the thickness of the plate, K is the foundation reaction coefficient, E i is the Young’s modulus, νi is the Poisson’s ratio, G is the shear modulus, ρ is the material density, F is the function of forces, Q(x1 , τ) is the transverse load, τ is the time coordinate. We suppose that our plate occupies the domain Ω: Ω = {(x1 , x2 ) : 0 ≤ x1 ≤ l, −a ≤ x2 ≤ a}. Equations (4.4) and (4.5) describe boundary conditions for free of forces and moments longitudinal sides of the plate (x2 = ±a) and simply supported edges x1 = 0, x1 = l. The parameter δ is a characteristic of the relative width of the plate. We suppose it to be small. It stands near several terms of Eqs (4.1), (4.2) and the boundary conditions (4.4), (4.5) (and identity (4.6)) and makes the problem (4.1)–(4.5) singularly disturbed.

4.3 First Iteration Process Following the approach of asymptotic integration method we construct solution of (4.1)–(4.5) as power series: w=

∞

w2k (x, y, t)δ2k ,

Φ=

k=0

∞

Φ2k (x, y, t)δ2k

(4.7)

k=0

Then we substitute (4.7) in (4.1)–(4.5) and gather coefficients standing near δ of the same degree. We find as a result of the first step: . / w0 = w0,0 (x, t), Φ0 = 0, w0,0 , ∂2x w0,0 = 0, w0,0 , ∂t w0,0 t=0 = 0 (4.8) x=0,1

Function w0,0 is undefined yet. Second step results in following relations: w2 = w2,0 (x, t) + w2,2 (x, t)y2 ,

w2 , ∂2x w2

x=0,1

=0

(4.9)

56


We may get from the first boundary condition in (4.9)

w2,0

x=0,1 = 0,

(4.10)

but we can not fulfill the second boundary condition in (4.9). The discrepancy will be eliminated using boundary layer functions further on. Besides we get formulas for Φ2 using (4.6) 1 d2 (t) = α 2

Φ2 = d2 (t)y , 2

1

' ( ∂ x w0,0 2 dx

0

On the third step we find the corresponding relations for definition of w4 and Φ4 , but some of them can not be solved because of discrepancies on the edges x = 0, x = 1. However, condition of compatibility of several equations, obtained on the first three steps of the first iteration process, allows to get the initial-boundary value problem for the main term of asymptotic expansion. We write it in dimension variables l% &2 E 1 h3 4 1 E1 h 2 ∂ x1 W + KW + ρh∂2τ W = ∂ x1 W ∂ x1 W dx1 + Q, 12 2 l 0 W, ∂2x1 W = 0, [W, ∂τ W]τ=0 = 0

(4.11)

x=0,l

Equation (4.11) with boundary and initial conditions describes oscillations of the beam on the linear foundation with simply supported edges. Continuing first iteration process we get the following relations for constructing series (4.7): k {w2k , Φ2k } = y2 j {w2k,2 j (x, t), Φ2k,2 j (x, t)} j=0

Functions w2k,2 j (where j 0) and Φ2k,2 j depend on the functions obtained on the previous steps of the first iteration process, functions w2k,0 satisfy the linear differential equations. For instance, one could get the following relations: Φ2,2 = d2 (t),

Φ4,0 = Φ4,4 =

&2 αν1 % 2 ∂ x w0,0 , 12

(1 − ν1ν2 )∂4x w2,0 + kw2,0 + ∂2t w2,0 − 2Φ2,2 ∂2x w2,0 1 1 +2α∂2x w0,0 w2,0 ∂2x w0,0 dx = q2 − 2Φ4,4 ∂2x w0,0 + 4 Φ4,4 dx, 0

(4.12)

0

where q2 denotes the known functions on the current step of the first iteration process.


57

4.4 Second Iteration Process In order to remove discrepancies which appear in first iteration process we use boundary layer functions. We construct the solution of the problem (4.1)–(4.5) as follows: ∞ w = δ2k {w2k (x, y, t) + u2k (ξ, y, t) + v2k (η, y, t)} , k=0

Φ=

∞

(4.13) δ {Φ2k (x, y, t) + ϕ2k (ξ, y, t) + ψ2k (η, y, t)} , 2k

k=0

where ξ = x/δ, η = (1 − x)/δ are the so-called “extended” variables, u2k , ϕ2k and v2k , ψ2k are the boundary layer functions, which are concentrated near the edges x = 0 and x = 1, respectively. After substituting (4.13) in (4.1)–(4.6) and taking into account the results of the first iteration process one can obtain the relations concerning the boundary layer functions. On the first steps of the second iteration process we get u0 = ϕ0 = 0; then it is easy to show that the boundary value problems for the functions u2k , ϕ2k and v2k , ψ2k are linear (k 0). Let us consider first non-trivial boundary value problems.

4.4.1 Solution of Generalized Bi-Harmonic Problem Function u4 satisfies the following equations c2 ∂4y u4 + 2c3 ∂2y ∂2ξ u4 + ∂4ξ u4 = 0,

u4 , ∂2ξ u4

(4.14)

= 0, ξ=1/δ ∂2y u4 + ν1 ∂2ξ u4 , ∂3y u4 + c4 ∂y ∂2ξ u4 = 0, y=±1

u4 ξ=0 = −w4 x=0 , ∂2ξ u4

ξ=0 = −∂2x w2

x=0

(4.15)

Here the variable t is a parameter. The form of the solution of the problem (4.14), (4.15) depends on relation between mechanical constants. If 2 G 1 ν2 1 = , √ E 1 2 ν1 1 + ν1 ν2 then the solution of the problem (4.14), (4.15) may be reduced to the case of isotropic plate/shell [8, 9]. If 2 G 1 ν2 1 ≺ , (4.16) √ E 1 2 ν1 1 + ν1 ν2

58


then the solution of problem (4.14), (4.15) is given by the following series1 : 0

u4 =

E k e−sk ξ F k (y) + 2 Re

∞

k=− j+1

E k e−sk ξ F k (y)

k=1

The so-called Papkovich functions Fk (y) satisfy the boundary value problems: c2 FkIV + 2c1 s2k F k + s4k Fk = 0,

[Fk + ν1 s2k F k , F k + c4 s2k F k ]y=±1 = 0,

(4.17)

where sk are the “reduced” roots of equation κ · sinhz − sin(ωz) = 0,

(4.18)

z = 2βsk , j is the number of its real roots; κ, ω and β are the constants, which are connected with mechanical parameters of the plate; one can obtain asymptotic expressions for real and imaginary parts of the complex roots of Eq. (4.18) as follows: 4πω ωπ 4k + 1 ln κ − k · − + O e 1+ω2 , 2 1 + ω2 1 + ω2 4πω π 4k + 1 ω ln κ − k Imz = · + + O e 1+ω2 2 1 + ω2 1 + ω2 Re z =

k→∞

(4.19)

Equations (4.19) are obtained in the case, when κ and ω are supposed to be fixed. Coefficients Ek are obtained using the identities [11, 15]

1% −1

& Fk Fn − ps2k s2n Fk F n dx = 0,

1% −1

k n,

& F k F¯ n − ps2k s¯2n F k F¯ n dx = 0,

p=

c1 , c2

for ∀ k, n,

(4.20)

(4.21)

where the overlined variables are the complex conjugate, as follows: Ek =

1

(F k f1 − s2k Fk f2 ) dy

−1 1 −1

(Fk )2 − s4k Fk2

(4.22) dy

In connection with the calculation of E k the problem of representation of two real functions f1 (y) and f2 (y) as series of Papkovich functions f1 (y) =

∞ k=− j+1

1

E k Fk (y),

f2 (y) =

∞

s2k E k Fk (y),

(4.23)

k=− j+1

If instead of inequality (4.16) the opposite consideration arises, the problem can be solved in a similar way.


59

has been considered [11]. Different problems of such kind are investigated in [4, 9, 13, 16] and so on including the problem of completeness of the system of Papkovich elementary solutions for bi-harmonic operator in a semi-strip. Let us give the conditions of probability of expansions (4.23) (theorem 1 was proved in [11], taking after Grinberg’s method [4]; proof of the theorem 2 is similar to the first one). Theorem 4.1. Let an even function f1 (y) be continuously differentiable and its second derivative be absolutely integrated on [0, 1]. Then the sum of the first series in (4.23) coincides with the function f1 (y) when y ∈ [0, 1) if the following relations take place 1 ν2 f1 (1) + f2 (y)dy = 0, (4.24) 0

c1 ν1 ν2 f1 (1) − 2ν1c4 f1 (1) + 2(2c3ν1 − c1 )

1

1

f1 (y) dy + c1ν1

0

f2 (y)y2 dy = 0

0

Theorem 4.2. Let an even function f2 (y) be continuously differentiable and its second derivative be absolutely integrated on [0, 1]. Let function f1 (y) meet the same requirements. Then the sum of the second series in (4.23) coincides with the function f2 (y) when y ∈ [0, 1) if the first relation (4.24) takes place. Equalities (4.24) may be called the conditions of convergence of the series (4.23). They have practical application as well. In the first iteration process we got a sequence of linear equations, which allows to obtain functions w2k,0 ; for instance function w2,0 satisfies Eq. (4.12). We have also two boundary conditions (4.10) for w2,0 . But we failed trying to get boundary values for ∂2x w2,0 . We needed the second iteration process to obtain them (functions −w4 | x=0 and −∂2x w2 | x=0 (see Eqs (4.15)) play the role of functions f1 (y) and f2 (y)). For that purpose we substitute −w4 | x=0 and −∂2x w2 | x=0 instead of f1 (y) and f2 (y) in the first relation (4.24) and taking into account the results of the first iteration process we get the missing boundary value

1 G 4 2 ∂ x w2,0 x=0 = ν1 1 + 8ν1 ∂ x w0,0

x=0 6 E1 Function v4 satisfies the boundary value problem similar to (4.14)–(4.15). Its solution allows to get the boundary value of ∂2x w2,0 on the edge x = 1.

4.4.2 One More Boundary Value Problem Function ϕ2 satisfies the following equations c2 ∂4y ϕ2 + 2c1 ∂2y ∂2ξ ϕ2 + ∂4ξ ϕ2 = 0, [Aϕ2 , Bϕ2 ]ξ=1/δ = 0,

∂2ξ ϕ2 , ∂y ∂ξ ϕ2

y=±1

= 0,

Aϕ2 |ξ=0 = 2ν2 d2 (t),

(4.25) Bϕ2 |ξ=0 = 0,

60


A = ∂2ξ − ν2 ∂2y ,

B = ∂2ξ + c5 ∂ξ ∂2y

The solution of the problem (4.25) is also constructed as a series of Papkovich functions and may be reduced to the case considered in [4].

4.5 Estimation of the Remainder of Series in the Linear Case If we investigate the linear model of an orthotropic plate [10] and consider the linear initial boundary value problem instead of the problem (4.1)–(4.5) one can estimate the remainder term of solution. Let the solution be given by the first series (4.13). We use the notation U 2n =

n

δ2k {w2k (x, y, t) + u2k (ξ, y, t) + v2k (η, y, t)}

(4.26)

k=0

and the following known definition. Definition 4.1. We call function U the asymptotic δ-approach to solution w in the domain Ω, if supΩ |w − U| → 0 when δ → 0. Besides if supΩ |w − U| = O(δn ) then we call function U the asymptotic approach to solution w in Ω with accuracy of δn order. Using this definition, one can establish the following theorem. Theorem 4.3. Let the function q(x, t), x ∈ [0, 1], has two continuous derivatives and ∂2x q(0, t) = ∂2x q(1, t) = 0. Then the function U4 is an asymptotic approach to solution w of the corresponding linear problem [10], i. e. w(x, y, t) = U 4 + O(δ6 ), when δ → 0.

4.6 Closing Remarks In this paper we use the method of asymptotic integration with boundary layer method for reducing of nonlinear plate initial boundary value problem to the sequence of initial boundary value problems of smaller dimension; the main term of asymptotic expansion satisfies the known non-linear integro-differential equation and junior terms are the solutions of linear beam oscillations problems. Boundary values, which we need for the definition of mentioned junior terms are calculated using the conditions of convergence of the series, which appear as the solutions of the corresponding boundary-layer problems. These solutions are the eliminators of discrepancies, which appear during the first iteration process. Remark 4.1. Let the longitudinal sides of the plate be not absolutely free, but the bending moment M(x1 , τ) be given there on the boundaries x2 = ±a. It means that instead of the third equality (4.4) we have the following condition:


∂2y w + ν1 δ2 ∂2x w

y=±1

= −δ2 m(x, t),

61

m(x, t) =

M(x1 , ct) 2 l D1 h

In that case one may also transit the plate problem to the problem of beam oscillations under reduced load Q(x1 , τ) + ν1 ∂2x1 M(x1 , τ). Remark 4.2. Let the third equality in (4.4) be replaced by [∂y w]y=±1 = 0. It means that free clamping of the longitudinal sides of the plate takes place. In that case one may also transit the plate problem to the problem of beam oscillations with reduced elasticity modulus. The corresponding equation is D1 ∂4x1 W + KW + ρh∂2τ W = Q(x1 , τ) Remark 4.3. Let the Young’s modulus be connected by the relation E1 /E 2 = O(δ2 ). It means that the rigidity of the material is much more in Ox2 direction than in longitudinal direction Ox1 . Asymptotic integration does not lead to any transition in that case. Remark 4.4. While carrying out the asymptotic integration of the plate equations, we need to obtain initial values of the first iteration process functions of course. It is a special part of the problem, we did not present it in this paper. We only note here that some matching conditions are necessary. For instance, if we keep the terms up to O(δ4 ) order in solution (4.13) of the problem, then the following equalities must take place q(1, 0) = ∂2x q(1, 0) = ∂t q(1, 0) = q(0, 0) = ∂2x q(0, 0) = ∂t q(0, 0) = 0 Remark 4.5. For the purpose of estimation of the boundaries of asymptotics obtained application, numerical algorithms of beams, plates and cylindrical panels oscillations and dynamical buckling have been developed on the base of finite difference method. Corresponding numerical analysis has been carried out and estimations defined for isotropic material [8, 10].

References 1. Aghalovyan, L. A.: On the boundary layer of the orthotropic plates. (in Russian). Izv. AN Arm. SSR. Mechanics. 26, 2, 27–43 (1973). 2. Ciarlet, P. G., Rabier, P.: Les equations de von Kármán. Springer-Verlag, Berlin-HeidelbergNew York (1980). 3. Gol’denweiser, A. L.: Theory of Thin Shells (in Russian). Nauka, Moscow (1976). 4. Grinberg, G. A.: On the method, suggested by P. F. Papkovich for solution of the plane problem of the theory of elasticity in rectangular domain and the problem of bending of rectangular thin plate with two fastened edges and its some generalizations (in Russian). Prikl. Mat. Mech. 17, 2, 211–228 (1953). 5. Kucherenko, V. V., Popov, V. A.: Asymptotics of solution of elasticity theory problems in thin domains (in Russian). Dokl. AN SSSR. 274, 1, 58–61 (1984). 6. Russell, D. L., White, L. W.: Formulation and Validation of Dynamical Models for Narrow Plate Motion. Applied Mathematics and Computation. 58, 103–141 (1993).

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7. Russell, D. L., White, L. W.: The lowed narrow plate model. Electronic J. Diff. Equations. 27, 1–19 (2000). 8. Srubshchik, L. S., Stolyar, A. M., Tsibulin, V. G.: Asymptotic integration of nonlinear equations of cylindrical panel vibrations. J. Appl. Math. Mech. 52, 4, 511–518 (1988). 9. Stolyar, A. M.: Asymptotic integration of the equation of vibrations of long rectangular plate (in Russian). Izv. Sev.-Kavk. Nauchn. Tsentra Vyssh. Shk., Estestv. Nauki, 4 (56), 46–50 (1986). 10. Stolyar, A. M.: Asymptotic analysis of the problems of statics and dynamics of narrow rectangular plates. (in Russian). Izv. Vyssh. Ucheb. Zaved., Sev.-Kavk. Region, Estestv. Nauki, Specvypusk Pseudo-differential Equations and Some Problems of Mathematical Physics. 107–111 (2005). 11. Stolyar, A. M.: Representation of two real functions as series of P. F. Papkovich functions in one problem of limiting transition (in Russian). Izv. Vyssh. Ucheb. Zaved., Sev.-Kavk. Region, Estestv. Nauki, Specvypusk Actual Problems of Mathematical Hydrodynamics. 203– 206 (2009). 12. Sugimoto, N.: Nonlinear theory for flexural motions of thin elastic plate. Trans. ASME. J. Appl. Mech. 48, 2, 377–390 (1981). 13. Ustinov, Yu. A., Yudovich, V. I.: On the completeness of a system of elementary solutions of the biharmonic equation in a semi-strip. J. Appl. Math. Mech. 37, 4, 665–674 (1973). 14. Ustinov, Yu. A.: Boundary problems and the problem of limiting transition from threedimensional problems of the theory of elasticity to two-dimensional problems for heterogeneous plates. Dr. habil. Thesis (in Russian). Moscow (1977). 15. Vasil’ev, V. V., Lur’e, S. A.: Plane problem of elasticity theory for orthotropic clamped strip (in Russian). Izv. AN SSSR. MTT. 5, 125–135 (1984). 16. Vorovich, I. I.: Some mathematical problems of the theory of plates and shells (in Russian). In Trans. of II All-Union Congress on Theoretical and Applied Mechanics, pp. 116–136. Moscow (1966). 17. Vorovich, I. I.: Some results and problems of the asymptotic theory of plates and shells. (in Russian). In Trans. of I All-Union school on theory and numerical methods of calculation of plates and shells, pp. 51–149. Tbilisi (1975).

Chapter 5

On Cusped Shell-like Structures George Jaiani

Abstract This paper is updated concise survey of results concerning elastic cusped shells, plates, and beams and cusped prismatic shell-fluid interaction problems. Keywords Cusped beams · Cusped plates · Cusped prismatic shells · Mathematical modeling · Linear elasticity

5.1 Introduction The present paper is an updated concise survey of investigations concerning elastic cusped (in other words, said differently, sharpened, cuspidate, cuspate, cuspidal, tapered) shell-like structures, namely, cusped (standard and prismatic) shells, plates, and beams, as well as cusped shell-like elastic body – fluid interaction problems. Under cusped shells (see, e.g., [1, 2]) we understand shells whose thickness vanishes either on a part or on the whole boundary of either the standard shell “middle” surface or prismatic shell projection. Beams are called cusped (see [3, 4]) if at least at one of the endpoints of their axis the cross-section area vanishes. Mathematically, one is lead to posing and solving, in the static case, certain boundary value problems (BVPs) for even order equations and systems of elliptic type with order degeneration, while in the dynamical case one faces initial boundary value problems (IBVPs) for even order equations and systems of hyperbolic type with order degeneration. In the study of cusped shell-like bodies, forces concentrated along cusped edges and at points may be encountered; moreover, it may happen that well-posed displacement boundary conditions (BCs) along cusped edges depends on their geometry and sharpness. At present, we can count on a sufficiently complete mathematical G. Jaiani (B) I. Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, 2 University St., 0186, Tbilisi, Georgia e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 5, © Springer-Verlag Berlin Heidelberg 2011

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theory of elastic cusped prismatic shells and beams; however, the study of cusped standard shells of general form remains topical.

5.2 Geometry of Structures Under Consideration Investigations of cusped elastic prismatic shells, actually, takes its origin from the fifties of the last century, namely, in 1955 I. Vekua [1, 2] raised the problem of investigation of elastic cusped prismatic shells, whose thickness on the prismatic shell entire boundary or on its part vanishes. Such bodies, considered as 3D ones, may occupy 3D domains with, in general, non-Lipschitz boundaries. In practice, such cusped prismatic shells, in particular, cusped plates, and cusped beams (i.e., beams whose cross-sections’ area vanishes at least at one end of the beam) are often encountered in spatial structures with partly fixed edges, e.g., stadium ceilings, aircraft wings, submarine wings, etc., in machine-tool design, as in cutting-machines, planning-machines, in astronautics, turbines, and in many other application fields of engineering. (+)

x3=h(x1,x2) x3

Fig. 5.1 Prismatic shell of a constant thickness. ∂Ω is a Lipschitz surface

x2

(-)

x3=h(x1,x2)

x1

(+)

x3=h(x1,x2) (-)

x3

Fig. 5.2 Sharp cusped prismatic shell with a semicircle projection. ∂Ω is a Lipschitz surface

x3=h(x1,x2) x2 x1

G0 = g

0

Let Ox1 x2 x3 be an anticlockwise-oriented rectangular Cartesian frame of origin O. We conditionally assume the x3 -axis vertical. The elastic body is called a prismatic shell if it is bounded above and below by, respectively, the surfaces (so called face surfaces) (+)

(−)

x3 = h (x1 , x2 ) and x3 = h (x1 , x2 ),

5 On Cusped Shell-like Structures

65 (+)

x3=h(x1,x2)

g1

g2

x3

Fig. 5.3 Cusped plate with sharp γ1 and blunt γ2 edges, γ0 = γ1 ∪ γ2 . ∂Ω is a nonLipschitz boundary

x2 (-)

x1

x3=h(x1,x2)

(+)

x3=h(x1,x2)

Fig. 5.4 Blunt cusped plate with the edge γ0 . ∂Ω is a Lipschitz boundary

G0 = g

(-)

0

x3=h(x1, x2) =0

Fig. 5.5 Cross-sections of a prismatic (left) and a standard shell with the same midsurface

laterally by a cylindrical surface Γ of generatrix parallel to the x3 -axis. By Γ0 we denote a part of Γ which degenerates in a curve. In other words, the 3D elastic prismatic shell-like body occupies a bounded region Ω with boundary ∂Ω, which is defined as: * + (−) (+) Ω := (x1 , x2 , x3 ) ∈ R3 : (x1 , x2 ) ∈ ω, h (x1 , x2 ) < x3 < h (x1 , x2 ) , (5.1) where ω := ω ∪ ∂ω is the so-called projection of the prismatic shell Ω := Ω ∪ ∂Ω (see Figs 5.1-5.4); γ = ∂ω and ∂Ω denote boundaries of ω and Ω, respectively. * (+) (−) > 0 for (x1 , x2 ) ∈ ω, 2h(x1 , x2 ) := h (x1 , x2 ) − h (x1 , x2 ) ≥ 0 for (x1 , x2 ) ∈ ∂ω is the thickness of the prismatic shell Ω at the points (x1 , x2 ) ∈ ω ¯ = ω ∪ ∂ω. max{2h} is essentially less then characteristic dimensions of ω. Let (+)

(−)

2h(x1 , x2 ) := h (x1 , x2 ) + h (x1 , x2 ).

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In the symmetric case of the prismatic shells, i.e., when

(−)

h (x1 , x2 )

(+)

= − h (x1 , x2 ), i.e., 2h(x1 , x2 ) = 0, we have to do with plates of variable thickness 2h(x1 , x2 ) and a middle-plane ω (see Figs 5.3, 5.4). Prismatic shells are called cusped if a set γ0 , consisting of (x1 , x2 ) ∈ ∂ω for which 2h(x1 , x2 ) = 0, is not empty (see Figs 5.3, 5.4, 5.11–5.18). Distinctions between the prismatic shell of a constant thickness and the standard shell of a constant thickness are shown in Fig. 5.5. The lateral boundary of the standard shell is orthogonal to the “middle surface”, while the lateral boundary of the prismatic shell is orthogonal to the prismatic shell’s projection on x3 = 0. Fig. 5.6 Cusped beam with rectangular cross-sections. ∂Ω is a Lipschitz boundary

Fig. 5.7 A cusped beam with rectangular crosssections. ∂Ω is a Lipschitz boundary

Fig. 5.9 A conical wedge. ∂Ω is a Lipschitz boundary

Fig. 5.8 A cusped circular beam. ∂Ω is a Lipschitz boundary

Fig. 5.10 A cusped circular beam. ∂Ω is a non-Lipschitz boundary

3 (−) (+) If in (5.1) ω := (x1 , x2 ) ∈ R2 : 0 < x1 < L, h2 (x1 ) < x2 < h2 (x1 ), 0 ≤ 2h2 (x1 ) := 4 (+) (−) h2 (x1 ) − h2 (x1 ) L, L = const > 0 , the prismatic shell-like body Ω will become a b

beam-like body with a cross-section of arbitrary form. In particular, let a domain Ω of R3 occupied by an elastic beam be * (−) (+) Ωb := (x1 , x2 , x3 ) ∈ R3 : 0 < x1 < L, hi (x1 ) < xi < hi (x1 ), + (+) (−) 5 2hi (x1 ) := hi − hi ≥ 0, hi ∈ C([0, L]) C 2 (]0, L[), i = 2, 3, L = const > 0

and 2h3 and 2h2 be correspondingly the thickness and the width of the beam and their maxima be essentially less then the length L of the bar; the superscript “b” means beam, see Figs 5.11–5.18 showing at the cusp the angle ϕ between tangents

5 On Cusped Shell-like Structures (+)

67

(−)

T and T , where typical longitudinal (vertical and horizontal) sections of beams are given, and Figs 5.6–5.10. If at least one of the conditions 2hi (0) = 0 and 2hi (L) = 0, i = 1, 2, is fulfilled, a beam is called the cusped one. The last class of beams consists of beams with rectangular cross-sections which may degenerate in segments or points at the beams ends, see Figs 5.6, 5.7. (+)

T

(-)

x2

T

Fig. 5.12 ϕ = π

Fig. 5.11 ϕ = 0

(+)

(+)

T

(-)

T

Fig. 5.13 ϕ =

T

(-)

x2 π 2

Fig. 5.14

(+)

π 2

T

x2

(+)

Fig. 5.17 0 < ϕ < π

x2

T

π 2

(+)

T

T

T

(-)

Fig. 5.16 0 < ϕ
0, x2 ≥ 0.

(5.5)

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Since 1972 Jaiani investigated the tension-compression problem for such cusped prismatic shells and constructed effective solutions of BVPs in integrated stresses, when ω is the half-plane, half-strip, or angle. A qualitative part of Jaiani’s abovementioned results can be summarized as follows. Statement 5.1 Let n be the inward normal of the prismatic shell projection boundary. In the case of the tension-compression (N = 0) problem on the cusped edge, where (see Figs 5.15–5.18, which can be considered as cusped prismatic shells’ cross-sections) 0 ≤ ∂h ∂n < ∞ (in the case (5.5) this means k2 ≥ 1), which will be called a sharp cusped edge, the displacement vector can not be prescribed, while on the cusped edge, where (see Figs 5.11–5.14) ∂h ∂n = + ∞ (in the case (5.5) this means k2 < 1), which will be called a blunt cusped edge, the displacement vector can be prescribed. In the case of the classical bending problem when at a plate cusped edge ∂h = O(dk−1 ) as d → 0, k = const > 0, ∂n

(5.6)

where d is the distance between an interior reference point of the plate projection and the cusped edge, the edge can not be fixed if k ≥ 1/3, but it can be fixed if 0 < k < 1/3; it can not be freely supported if k ≥ 2/3 , and it can be freely supported if 0 < k < 2/3; it can be free or arbitrarily loaded by a shear force and a bending moment if k > 0. Note that in the case (5.5), the condition (5.6) implies that d2 = x2 and k = k2 = k1 /3. In other words, when a profile, i.e., cross-section (see Fig. 5.19) of the plate in a neighborhood of a plate boundary point P (blunt cusp): (i) lies in the green zone then all three main BVPs can be correctly posed; (ii) lies in the blue zone then the edge can be either simply supported or be free (or be arbitrarily loaded); (iii) lies in the red zone then such edge can be only free (or be arbitrarily loaded). The last from the above three assertions is also valid for the sharp cusp.

Fig. 5.19 Zones of setting of different BCs at cusped edges depending on geometry of tapering (sharpening)


71

Fig. 5.20 A cusped cylindrical shell

The cylindrical bending of a cylindrical shell has been considered by Tsiskarishvili and Khomasuridze when the thickness has the following form h = h0 sinκ ϕ, h0 , κ = const > 0, where ϕ ∈] 0, ϕ0 [ is the polar angle counted from the plane Oζ x (see Fig. 5.20). Let w be a radial component of the displacement vector, i.e., deflection, which obviously characterizes bending, and v be an angular component of the displacement vector, i.e., displacement in the middle surface (cylinder) orthogonal to the element of the cylinder which (i.e., displacement v) characterizes tension-compression of cylindrical shell. They proved (as foreseen by Jaiani) that deflection w can be given on the cusped edge only when κ ∈ 0, 23 , its first derivative can be given only when κ ∈ 0, 13 , but v can be given if the cusped edge is blunt, i.e., κ ∈] 0, 1 [, as in the zero approximation (compare with Statement 1). The same authors have considered the strength problem of a uniformly loaded 3D elastic structure consisting of a rotational cylindrical shell of constant thickness and two rotational cusped conical shells with a linearly changing thickness. Jaiani has proved that Statement 1 remains valid in the case of classical bending of orthotropic cusped plates. However, for general cusped shells and general anisotropic cusped plates, corresponding analysis is yet to be done. The classical bending of plates with the flexural rigidity (5.4) in energetic and in weighted Sobolev spaces has been studied by Jaiani [5]. In the energetic space some restrictions on the lateral load has been relaxed by Devdariani. In 2004 Jaiani has studied bending of an elastic cusped Kirchhoff-Love plate on an elastic foundation with a constant compliance. In 2002 in the case of the cylindrical bending Chinchaladze has explored the well-posedness of the Keldysh type and weighted BVPs for the Kirchhoff-Love plate with two cusped edges and studied general and harmonic vibrations of such plates are studied. She has shown that the setting of BCs depends on the geometry of sharpening of plate edges, while the setting of initial conditions is independent of it. In 2009 for geometrically non-linear cusped Timoshenko plate with flexural rigidity (5.4) Chinchaladze and Jaiani proved that at a cusped edge the bending moment and intersecting force can be prescribed only if 0 ≤ k1 < 2 and 0 ≤ k1 < 1, respectively (in contrast to the linear case, when these quantities can always be

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prescribed). Applying the function-analytic method developed by Fichera, in the N = 0 approximation%for general & form cusped prismatic shells Jaiani established that at blunt cusped edge ∂h = +∞ displacement vector components can be prescribed, ∂n % & ∂h while sharp cusped edge 0 ≤ ∂n < ∞ should be freed from BCs (the Keldysh type BVP for displacements). The last result concerning sharp cusped edges is true for the N-th approximation as well. In the case (5.5) the tension-compression system of the N = 1 approximation is investigated by Devdariani, Jaiani, Kharibegashvili, and Natroshvili in 2000. The existence and uniqueness of generalized solutions of BVPs with Dirichlet (for weighted zero-moments when κ2 < 1 and for weighted first moments when κ2 < 1/3) and Keldysh type (for weighted zero-moments when κ2 ≥ 1 and for weighted first moments when κ2 ≥ 1/3) BCs is proved in weighted Sobolev spaces. In [6] the vibration tension-compression system of the N = 1 approximation is studied when 2h(x1 , x2 ) ≥ hκ xκ2 , hκ = const > 0, κ = const ≥ 0, x2 ≥ 0. The method of investigation of hierarchical models based on the idea to get Korn’s type inequality for 2D models from the 3D Korn’s inequality for non-cusped domains and then use Lax-Milgram theorem belongs to Gordeziani. This method found its complete realization in [7]. The analogous approach is developed by Schwab [8]. This idea for cusped but Lipschitz 3D domains with corresponding modifications was successfully used in [9]. In the case when cusped prismatic shell occupies a Lipschitz 3D domain; on face surfaces stress vectors, while on the noncusped edge weighted moments of displacement vector components are given, with the help of variational methods, the existence and uniqueness theorems for the corresponding 2D BVPs are proved. The above method does not allow to consider BVPs when on the cusped edge either displacements or loads are prescribed. In the case under consideration the loads are concentrated along the cusped edge. In [10] the well-posedness of BVPs for elastic cusped plates (i.e., symmetric prismatic shells) in the N-th approximation N ≥ 0 of I.Vekua’s hierarchical models. Note, that for the r-th order moments Dirichlet and Keldysh BCs are correct 1 1 when κ < 2r+1 and κ ≥ 2r+1 (r=0,1,2,...N), respectively. In the zero approximation of Vekua’s hierarchical models of prismatic shells, when the thickness has the form (5.5) Jaiani has solved the Flamant, Cerutti and Carothers’ type problems in the explicit form. The purpose of [11] is to show that internal concentrated contact interactions are at times to be expected within the framework of Vekua’s hierarchical theory of prismatic shells [1]. The results can be summarized as follows. When equilibrium problems for cusped prismatic shells subject to concentrated loads are formulated within the framework of the zero approximation, (i) in both the Flamanttype problem and Cerruti-type problem, where the external concentrated force acts in direction x1 , force-like internal concentrated interactions do occur, while they do not occur in the Cerruti-type problem where the external concentrated force acts in direction x3 ; (ii) in the Carothers-type equilibrium problem, neither forcelike nor couple-like internal concentrated interactions occur. These results parallel closely those obtained, by Podio-Guidugli. He has considered the plane versions of Flamant’s, Cerruti’s, and Carothers’ problems.


73

5.5 Cusped Beams In 1879 G. Kirchhoff constructed solutions for the harmonic eigenvibration of beams having a form either of a wedge (see Fig. 5.18) or a conical wedge (see Fig. 5.9) by means of Bessel functions. In 1916, using the above exact solution for the wedge shaped beam, Timoshenko investigated the eigenvibration when cusped beam end is free while the non-cusped end is built-in. In 1980 Uzunov numerically solved the problem of bending of the cusped circular beam (see Fig. 5.8) on an elastic foundation with a constant compliance. The blunt cusped end is free and the noncusped end is clamped. In 1990–1995 the bending vibration of homogeneous Euler-Bernoulli cone beams and beams of continuously varying rectangular cross-sections, when one side (width) of the cross-section is constant, while the other side (thickness) has the form (5.5), where x2 is the axial coordinate measured from the cusped end, were considered by Naguleswaran. In these investigations the cusped end is always free. Shavlakadze has studied the contact problem for an unbounded elastic medium composed of two welded half-planes x1 > 0 and x1 < 0 having different elastic constants and strengthened on the semi-axis x2 > 0 by a cusped thin beam (as an inclusion) with a free cusped end. The well-possedness of BVPs in weighted displacements for (0,0) and (1,0) approximations are investigated by Jaiani. Setting of BCs in the general (N3 , N2 ) approximation is considered in [3]. Kharibegashvili and Jaiani have studied well-possedness of dynamical problems in (0,0) approximation with mixed BCs at beam ends when either integrated stresses or weighted displacement components are prescribed. In [4] for cusped Euler-Bernulli beams all the reasonable BVPs are solved in explicit (integral) forms. Moreover, the existence and uniqueness of weak (generalized) solutions to vibration problems in suitably chosen weighted Sobolev spaces are established. In [12] a dynamical problem in the (0,0) approximation of elastic cusped prismatic beams is investigated when stresses are applied at the lateral surfaces and the ends of the beam. Two types of cusped ends are considered when the beam cross-section turns into either a point or a straight line segment. Correspondingly, at the cusped end either a force concentrated at the point or forces concentrated along the straight line segment are applied. The existence and uniqueness theorems in appropriate weighted Sobolev spaces are proved.

5.6 Relations of 3D, 2D, and 1D Problems Paper [13] deals with the analysis of the physical and geometrical sense of N-th (N=0,1,. . . ) order moments and weighted moments of the stress tensor and the displacement vector in the theory of cusped prismatic shells. The peculiarities of the setting of BCs at cusped edges in terms of moments and weighted moments are analyzed. The relation of such BCs to the BCs of the 3D theory of elasticity is also discussed.

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5.7 Cusped Prismatic Shell-Fluid Interaction Problems Works of Chinchaladze and Jaiani are devoted to some solid-fluid interaction problems when the solid part is an elastic cusped plate. Namely, the adequate transmission conditions on the interface for the interaction problem, when for elastic part the classical Kirchhoff-Love model is used, are studied; the bending of KirchhoffLove plates with two cusped edges under action of either incompressible ideal or incompressible viscous fluids has been considered as well, in particular, harmonic vibration is studied. In 2005-2006 by Chinchaladze, Gilbert an interaction between an elastic prismatic shell and an incompressible fluid, when in the elastic part the N = 0 approximation of Vekua’s hierarchical models for a cusped elastic prismatic shell is used, is investigated. References to all the works of the scientists mentioned in the present paper can be found in the bibliographies of papers [4, 9, 14]. An expanded version of this exploratory survey, supplemented with a large bibliography, will be published in Springer Briefs series.

References 1. Vekua, I. N. (1955). On one method of calculating of prismatic shells. Trudy Tbilis. Mat. Inst., 21, 191-259 (in Russian). 2. Vekua, I. N. (1985). Shell Theory: General Methods of Construction. Pitman, Boston. 3. Jaiani, G. (2001). On a mathematical model of bars with variable rectangular cross-sections. ZAMM Z. Angew. Math. Mech., 81 (3), 147-173. 4. Jaiani, G. V. (2002). Theory of Cusped Euler-Bernoulli Beams and Kirchoff-Love Plates. Lect. Notes TICMI, 3. 5. Mikhlin, S. G. (1970). Variational Methods in Mathematical Physics. Nauka, Moscow. 6. Jaiani, G., Schulze, B.-W. (2007). Some Degenerate Elliptic Systems and Applications to Cusped Plates. Mathematische Nachrichten, 280 (4), 388-407. 7. Avalishvili, M., Gordeziani, D. (2003). Investigation of two-dimensional models of elastic prismatic shell. Georgian Math. J., 10 (1), 17-36. 8. Schwab, C. (1996). A-posteriori modelling error estimation for hierarchic Plate Models. Numer. Math., 74, 221-259. 9. Jaiani, G., Kharibegashvili, S., Natroshvili, D., Wendland, W.L. (2004). Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary. J. Elasticity, 77 (2), 95-122. 10. Chinchaladze, N., Gilbert, R., Jaiani, G., Kharibegashvili, S., Natroshvili, D. (2008). Existence and uniqueness theorems for cusped prismatic shells in the N-th hierarchical model. Math. Methods Appl. Sci., 31(11), 1345-1367. 11. Chinchaladze, N., Jaiani, G., Maistrenko, B., Podio-Guidugli, P. (2011). Concentrated contact interactions in cuspidate prismatic shell-like bodies. Archive of Applied Mechanics, DOI: 10.1007/s00419-010-0496-6. 12. Chinchaladze, N., Gilbert, R., Jaiani, G., Kharibegashvili, S., Natroshvili, D. (2010). Cusped elastic beams under the action of stresses and concentrated forces. Appl. Anal. 89(5), 757774. 13. Jaiani, G. (2008). On Physical and Mathematical Moments and the Setting of Boundary Conditions for Cusped Prismatic Shells and Beams. Proceedings of the IUTAM Symposium on Relation of Shell, plate, Beam, and 3D Models, IUTAM Bookseries, 9, 133-146, Springer. 14. Jaiani, G.V. (1996). Elastic bodies with non-smooth boundaries–cusped plates and shells. Z. Angew. Math. Mech., 76, Suppl. 2, 117-120.

Chapter 6

Effect of the Tangential Loads on the Bending of Elastic Plates Kristine L. Martirosyan

Abstract In this work, the problem of bending of a semi-infinite plate under the action of tangential loads is considered on the base of the classical theory, theory of Reissner-Hencky-Mindlin in Vasilyev’s sense, and theory of Ambartsumyan. The comparisons between deflections, transverse shear forces and moments by firstorder and higher-order refined theories are performed. Keywords Bending · Tangential loads · Plate theories

6.1 Problem Formulation The problem of bending of plates in the presence of tangential loads is considered on the basis of the following three theories: classical theory of Kirchhoff (K) [1]; improved theory of first order - the theory of Reissner (R) [2]; improved theory of higher order - the theory of Ambartsumyan (A) [3]. Another approach of analogous problems investigation is suggested in [4]. We consider a semi-infinite plate with thickness 2h, on the front surfaces of which tangential loads are given. Rectangular Cartesian coordinate system (x, y, z) is chosen so that the middle plane of the plate coincides with the plane (Oxy). Let us assume that the rectangular plate occupies the region 0 ≤ x < ∞, 0 ≤ y ≤ b, −h ≤ z ≤ h (Fig. 6.1). It is supposed that on the front surfaces tangential loads are given: z=h:

σ33 = 0, σ31 = X + (x, y), σ32 = 0,

z = −h : σ33 = 0, σ31 = −X − (x, y), σ32 = 0

(6.1)

K. L. Martirosyan (B) National Academy of Sciences, Institute of Mechanics, Yerevan, Armenia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 6, © Springer-Verlag Berlin Heidelberg 2011

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K.L. Martirosyan

x

O

y

−X −

y

X+

z Fig. 6.1 Semi-infinite plate band under tangential loads.

Concerning the displacements, by the theories (K), (R) and (A) the following respective assumptions are accepted [5]: • Kirchhoff theory (K)

∂W , ∂x ∂W U2 = V − z , ∂y U3 = W

(6.2)

U1 = U − zθ1 , U2 = V − zθ2 , U3 = W

(6.3)

∂W z z 1 + X1 + X2 + g(z)ϕ1 , ∂x 2G 2h G ∂W 1 U2 = V − z + g(z)ϕ2 , ∂y G U3 = W

(6.4)

U1 = U − z

• Reissner theory (R)

• Ambartsumyan theory (A) U1 = U − z

Here, U, V are displacements of the median plane; W is the deflection of the plate; ϕ1 , ϕ2 , θ1 , θ2 are functions independent of the coordinate z, G is the shear modulus, and X1 = X + − X − , X2 = X + + X − are the tangential loads. The function g(z) in Eqs (6.4) is defined as z2 g(z) = z 1 − 2 (6.5) 3h

6 Effect of the Tangential Loads on the Bending of Elastic Plates

77

The equations of bending of the plate have the form • by theory (K) Δ2 W =

h ∂X1 D ∂x

(6.6)

• by theory (R) ⎧ ∂θ1 ∂θ2 ⎪ ⎪ ⎪ ΔW − − = 0, ⎪ ⎪ ⎪ ∂x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂θ1 ∂θ2 4Gh ∂W 2h ⎨ D Δθ1 + θ + + − θ1 = X1 , ⎪ ⎪ ⎪ ∂x ∂x ∂y 1 − ν ∂x 1 −ν ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂θ2 ∂θ1 4Gh ∂W ⎪ ⎪ ⎪ D Δθ2 + θ + + − θ2 = 0 ⎩ ∂y ∂y ∂x 1 − ν ∂y

(6.7)

• by theory (A) ⎧ ∂ϕ1 ∂ϕ2 3 ∂X1 ⎪ ⎪ ⎪ + =− , ⎪ ⎪ ⎪ ∂x ∂y 4 ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 8h3 ∂ ∂ϕ1 ∂ϕ2 4h 2h3 ∂2 X1 1 − ν ∂2 X1 ⎨ ∂ D ΔW − Δϕ +θ + + ϕ = + , 1 1 ⎪ ⎪ ⎪ ∂x 15 ∂x ∂x ∂y 3 3(1 − ν) ∂x2 2 ∂y2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 8h3 ∂ ∂ϕ1 ∂ϕ2 4h h3 θ ∂2 X1 ⎪ ⎪ ⎪ D ΔW − Δϕ2 + θ + + ϕ2 = ⎪ ⎩ ∂y 15 ∂y ∂x ∂y 3 3 ∂x∂y (6.8) In Eqs (6.6) to (6.8), ν is the Poisson‘s ratio, D is the bending stiffness of the plate. θ and D have the following form θ=

1+ν , 1−ν

D=

4Gh3 3(1 − ν)

The applied loads are given as it follows: X1 = τ0 sin λy,

X2 = 0

at λ =

π b

(6.9)

The edges of the plate at y = 0, b are hinged; the corresponding boundary conditions are then: • by theory (K): W = 0,

∂W 2 = 0 at ∂y2

y = 0, b

(6.10)

• by theory (R): W = 0,

θ1 = 0,

∂θ2 = 0 at ∂y

y = 0, b

(6.11)

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K.L. Martirosyan

• by theory (A): 5 ∂2 W 4 ∂ϕ2 W = 0, ϕ1 = − X1 , − =0 2 8 5G ∂y ∂y

at

y = 0, b

(6.12)

With X1 the only tangential loads applied on front surfaces (6.9). It should be noted, that in the case of X2 = 0, generalized plane stress has a trivial solution. By the theories (K), (R) and (A) equations in the particular case of one-dimensional problem under the conditions (6.10), (6.11) and (6.12) have solutions. Therefore, solutions of Eqs (6.6), (6.7) and (6.8) in the two-dimensional case by analogy with the approach of Nadai [1] we will search under the conditions • by theory (K):

lim W = 0

(6.13)

x→∞

• by theory (R):

lim W = 0,

x→∞

lim θ1 = −

x→∞

lim θ2 = 0,

τ0 , 2G(η2 + 1)

(6.14)

x→∞

where

hλ η= √ 3

• by theory (A):

lim W = 0,

x→∞

lim ϕ1 = −

x→∞

lim ϕ2 = 0,

5λ2 τ0 , 8χ2

(6.15)

x→∞

2

where χ=h

−1

( 5' 1 + ξ2 , 2

2 ξ=

2 λh 5

6.2 Solutions for Lightweight Restraint Conditions at x = 0 Let the edge x = 0 has the condition of lightweight restraint; the corresponding boundary conditions have the following form: • by the theory (K):

W = 0, ∂W = 0 at ∂x

x=0

(6.16)


79

• According to the theory (R) W = 0, θ1 = 0, ∂θ2 = 0 at ∂x

(6.17) x=0

• According to the theory (A) W = 0, ∂W 4 X1 − ϕ1 = ∂x 5G 2G ∂ϕ1 ∂ϕ2 5 ∂X1 − + = 0 at ∂y ∂x 8 ∂y

(6.18) x=0

The expressions for deflections will have the following form: • by the theory (K):

W=0

• by the theory (R): W=

(6.19)

e−λx xτ0 sin λy 2G(1 + 2η2(1 + θ))

(6.20)

e−λx xτ0 sin λy, 2G(1 + β)

(6.21)

• by the theory (A): W= where

4ξ2 1−ν From Eqs (6.20) and (6.21), it is obvious that the deflections by the theories (R) and (A) are identical when h2 λ2 1. In addition, we can calculate the estimated value for the moment M1 (0, y) and the forces N1 (0, y), N2 (x, 0). It is obtained, that the moments M1 (0, y) by the theories (R) and (A) coincide when hλ 1. The forces N1 (0, y) computed by the theories (R) and (A) coincide when h2 λ2 1. The expressions for the forces N2 (x, 0) when h2 λ2 1 are of the form β=

• by theory (R)

• by theory (A)

8ηe−λx N2 (x, 0) = −ηhτ0 e−(xα/h) − ; 1−ν

(6.22)

5ξhτ0 −xχ 4ξe−λx N2 (x, 0) = − e − ; 6 1−ν

(6.23)

When hλ 1 in corner point (x = 0, y = 0), we find that the ratio of the force N √2 (0, 0) calculated by the theory (R) to that calculated by the theory (A) is equal to 6/5.

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K.L. Martirosyan

6.3 Solutions for Straitened Sliding Contact Conditions at x = 0 At the edge x = 0 the straitened sliding contact conditions are of the form • by the theory (K)

∂W = 0, ∂x ∂W = 0, X1 (6.24) ∂y 3 ∂ W h = 3 D ∂x Thus, according to Kirchhoff’s theory for the function of the deflection, three boundary conditions are obtained instead of the required two. When satisfying the condition of self-conjugacy of the problem [6], there are two variants of boundary conditions here: either the first two conditions are satisfied for W, which would be consistent with the task of anchorage, or the first and third conditions are satisfied, which coincides with the problem for sliding contact. Ambiguity of the formulation of the problem with the boundary conditions by the theory of Kirchhoff shows evidence of the need for applying refined theories. [7]. • by the theory (R) ∂W = 0, (6.25) ∂x θ1 = θ2 = 0 • by the theory (A)

∂W 1 + X1 = 0, ∂x 10G ∂W 4 − ϕ2 = 0, ∂y 5G

(6.26)

3 ϕ1 = − X1 4 Satisfying the boundary conditions (6.25) and (6.26), we obtain the expressions for deflections: • by the theory (R): W=

e−λx (1 + λx)τ0 % & sin λy; 2Gλη 1 + η2 1 − 2η2 (1 + θ) + 2η 1 + η2 (1 + θ)

• by the theory (A): % % & % && e−λx γλx 1 + ξ2 + β 6 + γ + ξ2 + γξ2 + 5λx τ0 sin λy W= % & , 10Gλβ 1 + ξ2 ξ + γ 1 + ξ2

(6.27)

(6.28)


where

81

⎛ ⎞ ⎟⎟⎟ 4ξ2 ⎜⎜⎜⎜ ξ ⎟⎟⎠ γ= ⎜⎝1 − 1−ν 1 + ξ2

From Eqs (6.27) and (6.28), it is obvious that the deflections by the theories (R) and (A), when h2 λ2 1, have the form: • by the theory (R): √ −λx 3e (1 + λx)τ0 sin λy W= , (6.29) 4η 2Ghλ2 (1 + 1−ν ) • by the theory (A): W=

3e−λx (1 + λx)τ0 sin λy √ 4ξ 10Ghλ2 (1 + 1−ν )

(6.30)

When hλ 1, we obtain that the ratio of the deflection, calculated by the theory √ (R), to the magnitude of the deflection, calculated by the theory (A) is equal to 5/6. Calculated values for the force N1 (0, y) by the theories (R) and (A) coincide and are identically equal to zero. When hλ 1, the ratio of the moment, calculated by √ the theory (R), to the magnitude of the moment, calculated by the theory (A) is 5/6. When hλ 1 in corner point x = 0, y = 0, we obtain that the ratio of the force, N2 (0, 0), calculated by √ the theory (R), to the magnitude of cutting force, calculated by the theory (A) is 6/5.

6.4 Conclusions The bending of a semi-infinite plate under the action of tangential loads has been studied analytically using three theories: the classical theory of Kirchhoff (K), the first order shear deformation theory of Reissner (R), and the refined higher-order theory of Ambartsumyan (A). It is found that the first theory is not satisfactory in general, while the solutions of the last two theories partly coincide for particular thickness ratios. These closed-form solutions can be helpful for numerical solutions assessment.

References 1. Timoshenko S., Woinowsky-Kriger S. Theory of Plates and Shells. McGraw Hill, New York, 1985. 2. Vasilyev, V. V.. Classical theory of plates: history and modern analysis. Russian Sc. Acad. Publishers, MTT, 1998, No 3, 46-58 (in Russian) 3. Ambartsumyan S.A. Theory of Anisotropic Plates (Strength, Stability, and Vibrations). TECHNOMIC Co, Stamford, 1970. 4. Kirakosyan R.M. On the improved theory cylindrical ortotropic plates of variable thickness. Proc. NAS of Armenia, 47. N. 5-6, 1994 (in Russian).

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5. Belybekyan M.V. On the equations of the theory of plates, taking into account the transverse shear. In: Problems of the mechanics of thin deformable bodies, Yerevan. Gitutyun (Ed.), NAS RA, 2002, p.67-88, (in Russian). 6. Belubekyan V.M, Belubekyan M.V. On the boundary conditions of plate theory. Proc. NAS of Armenia, V. 52. N. 2, 1999, p.11-21, (in Russian). 7. Belubekyan M.V. A paradox of the theory of Kirchhoff’s boundary conditions. In the coll. The study of modern scientific problems in high schools. Pub. Armenia, Yerevan, 2000, v.2. p. 134-138.

Chapter 7

On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates Jemal Peradze

Abstract The boundary value problem for a Timoshenko system of differential equations describing the static behavior of a plate is considered. Two sought functions are expressed through the third one for which an integro-differential equation with the Dirichlet condition on the boundary is written. The application of the Galerkin method to the obtained problem leads to a nonlinear system of algebraic equations which is solved by iteration. The condition of iteration process convergence is established and its rate is estimated. Keywords Timoshenko equation · Galerkin method · Jacobi iteration · Convergence of the iteration method

7.1 Problem Formulation The nonlinear Timoshenko model for plates and shells is important from the theoretical and engineering standpoints. Mathematical analysis of the corresponding system of equations is connected with a big difficulty because of the complex structure inherent in these equations. I. Vorovich [7] regarded the nonlinear system of Timoshenko equations as one of the unsolved problems of the mathematical theory of plates and shells in terms of solvability and construction of approximate solution algorithms. As far as we know, presently this problem remains far from being solved. In this context, the study of simplified one-dimensional variants of the Timoshenko system will help get a better insight into the nature inherent in these models and will make it easier to proceed to the investigation of the two-dimensional case. J. Peradze (B) Tbilisi State University, 2, University Str., Tbilisi 0186, Georgia & Georgian Technical University, 77, M. Kostava Str., Tbilisi 0174, Georgia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 7, © Springer-Verlag Berlin Heidelberg 2011

83

84

J. Peradze

Here we consider the problem of convergence and accuracy of the iteration process of solution of the one-dimensional axisymmetric boundary value problem for the Timoshenko plate. All constants participating in the obtained convergence conditions and the method error estimate are expressed explicitly through the initial data of the problem, which, in the case of practical application of the method, enables us to define in advance the number of iteration steps and to calculate the upper bound of the error. Let us consider the nonlinear system of differential equations 1 % 2 & w + p(x) = 0, 2 ( Eh ' Eh 1 2 k02 w + ψ + u + w w + q(x) = 0, 2(1 + ν) 2 1 − ν2 u +

' ( h2 ψ − k02 w + ψ = 0; 6(1 − ν)

(7.1)

0 < x < 1,

with the boundary conditions u(0) = u(1) = 0,

w(0) = w(1) = 0,

ψ (0) = ψ (1) = 0.

(7.2)

Here u = u(x), w = w(x) and ψ = ψ(x) are the functions to be defined, p(x) and q(x) are the given functions, ν, E, h and k0 are the given constants, 0 < ν < 12 . System (7.1) is obtained from the system of Timoshenko equations for a shell presented in [7, p. 42], on the one hand, by throwing away the variables y and t and assuming that k x = ky = 0, and, on the other hand, by preserving the terms with cubic nonlinearity. Thus system (7.1) characterizes the static state of the plate under the action of axially symmetric load. Note that this system can also be obtained from the system of Timoshenko equations for a plate from [2, p. 24].

7.2 Reduction of the Problem Let us return to problem (7.1),(7.2). Using the first and the third equation of system (7.1) and the corresponding boundary conditions from (7.2), the functions u(x) and ψ(x) can be obtained through the function w(x) 1 x 1 x 2 w (x) dx − w (ξ) dξ + x (1 − ξ)p(ξ) dξ − (x − ξ)p(ξ) dξ, 2 0 0 0 0 x 1 σ ψ(x) = − cosh σ(x − 1) cosh σξw (ξ) dξ + coshσx cosh σ(ξ − 1)w (ξ) dξ , sinh σ 0 x (7.3) where k0 σ= 6(1 − ν) . h x u(x) = 2

1

2

7 On the Convergence of an Iteration Method in Timoshenko’s Theory of Plates

85

Using (7.3) in the second equation of system (7.1), we come the equation for the function w(x) 1 x Eh 1 − ν 2 1 1 2 k + w dx + (1 − x)p(x) dx − p(ξ) dξ w − p(x)w 2 0 2 0 1 − ν2 0 0 x 3Ek04 1 − ν − sinhσ(x − 1) cosh σξ w (ξ) dξ h sinhσ 1 + ν 0 1 + sinh σx cosh σ(ξ − 1)w (ξ) dξ + q(x) = 0, (7.4) x

to which we add the boundary condition from (7.2) w(0) = w(1) = 0.

(7.5)

After solving problem (7.4), (7.5), we substitute w(x) in (7.3) and find other unknown functions u(x) and ψ(x).

7.3 The Numerical Algorithm Let us consider a numerical algorithm of the solution of problem (7.4), (7.5). The approximation of w(x) with respect to the spatial variable is written as the finite sum wn (x) =

n 1 wni sin iπx, iπ i=1

(7.6)

where, in case we use the Galerkin method [4], the coefficients wni satisfy the nonlinear system of equations

p1i + p2 +

n 1 w2n j wni + p3i j wn j + qi = 0, i j=1 j=1

n

i = 1, 2, . . ., n.

(7.7)

Here the following notation is used p1i =

3 + (hπi) 2 1 x

1 2k02 (1−ν)

j p3i j = 8 − i qi

1

=−

0

8(1 − ν2 ) Ehπ

0 1

,

p2 = 4

1

(1 − x)p(x) dx,

0

1 1 p(ξ) dξ sin iπx sin jπx d + p(x) sin iπx cos jπx dx , iπ 0

q(x) sin iπx dx. 0

(7.8)

86

J. Peradze

Note that to obtain the i-th equation of system (7.7), in the left-hand part of (7.4) we should replace w(x) by wn (x), multiply the obtained expression by iπ1 sin iπx and, after that, using the relations * 1 0, i j, sin iπx sin jπx dx = 1 0 2 , i = j, eax eax sin bx dx = 2 2 (a sin bx − b cosbx), a +b eax eax cos bx dx = 2 2 (a cosbx + b sin bx), a +b integrate it over x from 0 to 1 and equate the result to zero. We will solve system (7.7) by using the iteration method

p1i + p2 + w2ni,k+1 +

n j=1 ji

n 1 w2n j,k wni,k+1 +p3ii wni,k+1 + p3i j wn j,k + qi = 0, (7.9) i j=1 ji

where wni,k+l is the (k + l)-th approximation of wni , l = 0, 1, and k = 0, 1, . . . , i = 1, 2, . . ., n,. This method is nothing else but the Jacobi iteration process [5]. After defining wni,k , i = 1, 2, . . ., n, we use them in formula (7.6) instead of wni and, as a result, find the approximation of the function w(x), which, when used in (7.3) gives the approximation for the functions u(x) and ψ(x). Note that each i-th equation of system (7.9) is a cubic equation for wni,k+1 . Therefore, using the Cardano formula [1], wni,k+1 can be written in the explicit form wni,k+1 = σi,1 − σi,2 , where

k = 0, 1, . . .,

1 ⎡ ⎛ 2 ⎞1 ⎤ ⎢⎢⎢ 3 ⎟ 2 ⎥⎥ 3 ⎜ r s ⎥ si ⎜ ⎟ ⎥ ⎢ σi,l = ⎢⎢⎢⎢(−1)l + ⎜⎜⎜⎝ i + i ⎟⎟⎟⎠ ⎥⎥⎥⎥ , ⎣ 2 4 27 ⎦

ri = p1i + p2 + p3ii +

n

w2n j,k ,

j=1 ji

si =

i = 1, 2, . . ., n,

l = 1, 2,

n 1 qi + p3i j wn j,k . i

(7.10)

(7.11)

(7.12)

j=1 ji

7.4 The Jacobi Matrix Let us rewrite system (7.10) in the form ' ( wni,k+1 = ϕi wn1,k , wn2,k , . . . , wnn,k

(7.13)


87

and consider the Jacobi matrix

∂ϕi J= ∂wn j,k

n .

(7.14)

i, j=1

By virtue of (7.10)–(7.13) we conclude that the diagonal elements of the matrix J are equal to zero. As to the nondiagonal elements, i j, we have for them ⎡ ⎤ ⎛ 2 ⎞− 1 ⎥⎥ ⎢ 2 ⎜⎜⎜ si ri3 ⎟⎟⎟ 2 si ⎥⎥ ∂ϕi 1 1 ⎢⎢⎢⎢ 2 l 2 ⎢⎢ p3i j + (−1) ⎜⎜⎝ + ⎟⎟⎠ =− p3i j + ri wn j,k ⎥⎥⎥⎥ . (7.15) ⎦ ∂wn j,k 6 4 27 2 9 σ2 ⎢⎣ i,l

l=1

By (7.11) ri σi,1 · σi,2 = , 3

σ3i,2 − σ3i,1

⎛ 2 ⎞1 ⎜⎜⎜ si ri3 ⎟⎟⎟ 2 σ3i,1 + σ3i,2 . ⎜⎜⎝ + ⎟⎟⎠ = 4 27 2

= si ,

(7.16)

Using these relations in (7.15) we get ∂ϕi 1 1 2 1 = − p3i j 2 ri + wn j,k si 4 . ri 2 2 ∂wn j,k 6 σi,1 − 3 + σi,2 3 σi,1 + ( 3 ) + σ4i,2

(7.17)

Taking into account the first equality of (7.16), we find that σ2i,1 −

ri 1 + σ2i,2 ≥ ri , 3 3

σ4i,1 + σ4i,2 ≥

2 2 r . 9 i

(7.18)

By (7.17) and (7.18) we write

∂ϕi

≤ φ + φ , 1i j 2i j

∂wn j,k

(7.19)

where 1 1 φ1i j = |p3i j | , 2 |ri |

φ2i j = 2|wn j,k | |si |

1 . ri2

(7.20)

Let us estimate either of φli j , l = 1, 2. Suppose that the initial data of the problem – the function p(x) and the constants ν, E, h and k0 – are such that p1i > |p2 + p3ii | .

(7.21)

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J. Peradze

As follows from (7.8), (7.21) implies the fulfillment of the relation ⎛ ⎜⎜⎜ ⎜⎜⎝

⎞−1 1

1 3 ⎟⎟⎟⎟

+ > 4 (1 − x)p(x) dx ⎟ ⎠

2k02 (1 − ν) (hiπ)2 0 1 x

4 1 −8 p(ξ) dξ sin2 iπx dx + p(x) sin 2iπx dx

, iπ 0 0 0 i = 1, 2, . . ., n.

(7.22)

In that case, by virtue of (7.20) and (7.11) we have φ1i j ≤

1 |p3i j | , 2ci

(7.23)

where ci = p1i − |p2 + p3ii | . x 1√ Further, using (7.20), (7.21), (7.12) and the fact that max c+x 2 = 2 c , x > 0, c > 0, x we come to a conclusion that ⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ ⎜⎜⎜ 12 12 ⎟⎟⎟⎟ n n ⎜ ⎟⎟ |wn j,k | |si | |wn j,k | ⎜⎜ 1 |qi | 1 ⎜⎜⎜ φ2i j = 2 ≤2 + p23il w2nl,k ⎟⎟⎟⎟ n 2 ⎟⎟⎟ |ri | ri | ci + wn j,k ⎜⎜⎜ i ci c + w2 l=1 l=1 ⎜⎜⎜ ⎟⎟⎟ i nl,k ⎜⎝ ⎟⎠ l i l i l=1 li

1 1 |qi | 1 ≤ √ + √ ci i ci 2 ci

n

p23il

12

.

(7.24)

l=1 li

7.5 The Iteration Method Convergence and Error Let us introduce the vector and matrix norms by means of the expressions and max

n

1≤ j≤n i=1

n i=1

|vi |

|mi j | for v = (vi )ni=1 and M = (mi j )ni, j=1 . Now we will formulate the main

result. Assume that the norm of the matrix J is smaller than a number Δ, 0 < Δ < 1. According to (7.14), (7.19), (7.23) and (7.24), this is equivalent to the requirement 2 n n n 1 1 1 1 |qi | max |p3i j | + 2 √ + p23i j < Δ. 2 1≤ j≤n ci 2 i=1 ci i ci 1

i=1 i j

j=1 ji

(7.25)


89

Then, by virtue of Banach’s contraction principle [3] there exists a unique solution wni , i = 1, 2, . . ., n, of system (7.7) to which the sequence of approximations wni,k of the iteration method converges as k → ∞, whereas the error method decreases with a geometrical progression rate n

Δk |wni,0 − wni,1 |, 1 − Δ i=1 n

|wni,k − wni | ≤

i=1

k = 0, 1, . . . .

7.6 An Alternative Form of Requirements To conclude, we replace conditions (7.22) and (7.25) by more rigid but easily verifiable conditions. For this, we require the fulfillment of p1i > |p2 | + |p3ii | instead of (7.21), then apply (7.8) and the following relations obtained by means of the Cauchy–Bunyakowski inequality

1

≤ 1 p , (1 − x)p(x) dx

√ 0 0 3

1 x

p(ξ) dξ sin iπx sin jπx dx

≤

0

0

1 √ p0 , i j, 2 2

√ 3 p(ξ) dξ sin2 iπx dx

≤ p0 , 4 0 0

1

≤ 1 p , i j, p(x) sin iπx cos jπx dx

2 0 0

1

1

p(x) sin iπx cos iπx dx

≤ √ p0 ,

0 2 2

1

x

) 1 2 1 where p0 = 0 p2 (x) dx . As a result, instead of requirement (7.22) we will have ⎛ ⎜⎜ c = ⎜⎜⎜⎝

⎞−1 ⎛ √ ⎞ ⎜⎜⎜ 5 1 6 ⎟⎟⎟⎟ 2 ⎟⎟⎟ ⎜ ⎟⎠ p0 > 0, + − + ⎟ ⎝√ ⎠ π k02 (1 − ν) (hπ)2 3

(7.26)

and (7.25) will be replaced by the condition 1 2c

n √ 1 1 4 2 1 + √ + n + 2(n − 1)n p0 + √ max |qi | < Δ. π i c 1≤i≤n 2 π i=1

(7.27)

It is not difficult to verify that conditions (7.22) and (7.25) as well as (7.26) and (7.27) are fulfilled for sufficiently small functions p(x) and q(x).

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J. Peradze

References 1. Birkhoff, G., Mac Lane, S.: A brief Survey of Modern Algebra. Second edition. The Macmillan Co., New York; Collier-Macmillan Ltd., London (1965) 2. Lagnese, J., Lions, J.-L.: Modelling analysis and control of thin plates. Recherches en Mathematiques Appliquees [Research in Applied Mathematics], 6. Masson, Paris (1988) 3. Lebedev, L. P., Vorovich, I. I.: Functional Analysis in Mechanics. Springer Monographs in Mathematics. Springer, New York (2003) 4. Marchuk, G. I.: Methods of Numerical Mathematics. Second edition. Transl. from the Russian by Arthur A. Brown. Applications of Mathematics, 2. Springer, New York–Berlin (1982) 5. Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. Reprint of the 1970 original. Classics in Applied Mathematics, 30. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000) 6. Vol’mir, A. S.: Nonlinear Dynamics of Plates and Shells (in Russian). Nauka, Moscow (1972) 7. Vorovich, I. I.: Nonlinear Theory of Shallow Shells. Applied Mathematical Sciences, 133. Springer, New York (1999).

Chapter 8

Mathematical Models of Micropolar Elastic Thin Shells Samvel H. Sargsyan

Abstract In the present paper on the basis of mathematical (asymptotic) confirmed hypotheses method, depending on the values of dimensionless physical parameters, mathematical models of micropolar elastic thin shells with free rotation, with constrained rotation, with “small transverse stiffness” are constructed. Transverse shear and related to them strains are completely taken into account in the suggested theories of micropolar shells. On the basis of these models specific problems of strength and dynamics of micropolar elastic thin shells, plates and bars can be studied. Keywords Micropolar thin shells · Free and constrained rotations · Asymptotic methods

8.1 Introduction One of the actual problems of the micropolar (asymmetric, moment) theory of elasticity is the construction of general applied theories of shells, plates and bars [1, 2]. The main problem of the general theory of micropolar elasticity of thin bars, plates and shells is the approximate, but adequate reduction of the three-dimensional or two-dimensional equations of the theory of micropolar elasticity to applied twodimensional or one-dimensional equations. In [6] the asymptotic theory of micropolar elastic thin shells is constructed for a thin domain. It should be noted that the construction of the models of micropolar elastic thin shells by the asymptotic theory is connected with rather difficult and unwieldy mathematical calculations. From this point of view in this paper the method of hypotheses is applied for the construction of the models of micropolar elastic thin shells. The hypotheses are formulated on the S. H. Sargsyan (B) National Academy of Sciences of the Republic of Armenia, Gyumri State Pedagogical Institute, Armenia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 8, © Springer-Verlag Berlin Heidelberg 2011

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basis of the qualitative results of the asymptotic integration of boundary problems of in the three-dimensional micropolar theory of elasticity for thin shell domain. It should be noted that the suggested models of micropolar elastic thin shells based on the asymptotic theory and on the mentioned method of hypotheses coincide. Depending on the values of the physical dimensionless parameters, the models of micropolar elastic thin shells with free rotation, with constrained rotation, with “small transverse stiffness” are established. The transverse shear and the related strains are completely taken into account.

8.2 Problem Statement An isotropic shell of constant thickness 2h is considered as a three-dimensional elastic micropolar body. We proceed from the fundamental equations of the spatial static problem of linear micropolar theory of elasticity with independent fields of displacements and rotations [4]: • Equilibrium equations ∇m σmn = 0, • Elasticity relations *

∇m μmn + enmk σmk = 0

(8.1)

σmn = (μ + α)γmn + (μ − α)γnm + λγkk δnm μmn = (γ + ε)κmn + (γ − ε)κnm + βκkk δnm

(8.2)

γmn = ∇m Vn − ekmn ωk ,

(8.3)

• Geometric relations κmn = ∇m ωn .

Here σ, ˆ μˆ are the tensors of the force and the moment stresses, γˆ , κˆ are the strain tensor and the bending-torsion tensor, V, ω are the vectors of displacement and free rotation, λ, μ, α, β, γ, ε are the elastic parameters of the micropolar material. The indices m, n, k take the values 1, 2, 3. We will use orthogonal curvilinear coordinates αk , adopted in the theory of shells. Appropriate boundary conditions will be attached to the governing equations (8.1) - (8.3) of three-dimensional micropolar theory of elasticity. On the surfaces α3 = ±h of the shell force and moment stresses are given (i = 1, 2): σ3i = ±q±i , σ33 = ±q±3 , μ3i = ±m±i , μ33 = ±m±3 ,

(8.4)

On the boundary Σ of the shell boundary conditions in force and moment stresses or displacements and rotations or in mixed form can be prescribed. It is assumed that the shell thickness is small compared with typical radii of curvature of the shell’s middle surface, and hence, we will proceed from the following

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basic concept: in the static case the general stress-strain state (SSS) of a thin threedimensional body, forming a shell, is composed of the internal SSS, covering the shell, and boundary layers, localizing near the surface of the shell edge Σ. The construction of the general applied two-dimensional theory of micropolar elastic thin shells is closely connected with the formulation of the internal problem. During the determination of both the internal and the external SSS of the shell [6], the values of the physical parameters of the shell material play an important role. From this point of view the following dimensionless physical parameters are introduced: μ , α

R2 μ , β

R2 μ , γ

R2 μ , ε

(8.5)

where R, which is a characteristic radius of curvature of the middle surface of the shell, is a scale factor.

8.3 Model of Micropolar Elastic Thin Shells with Independent Fields of Displacements and Rotations The case when the dimensionless physical parameters (8.5) take the following values μ R2 μ R2 μ R2 μ ∼ 1, ∼ 1, ∼ 1, ∼1 (8.6) α β γ ε is considered. Taking into account the qualitative results of the asymptotic solution of equations (8.1)–(8.3) with the above mentioned boundary conditions, for the case (8.6) in [6] a theory of micropolar elastic thin shells with independent rotations based on the following, rather general assumptions (hypotheses) is deduced: a) During the deformation initially straight and normal to the median surface fibers are rotated freely in space as a whole rigid body, without changing its length and without remaining perpendicular to the deformed middle surface. The hypothesis is mathematically written as follows: Vi = ui (α1 , α2 ) + α3 Ψi (α1 , α2 ), V3 = w(α1 , α2 ), ωi = Ωi (α1 , α2 ), ω3 = Ω3 (α1 , α2 ) + α3 ι(α1 , α2 )

(8.7)

Thus, normal to the middle surface displacement and tangential rotations are constant functions along the shell thickness, and the tangential displacements and normal rotation are changing by a linear law. It should be noted that from the point of view of displacements the hypothesis (8.7) are similar to Timoshenko’s kinematic hypothesis in the classical theory of elastic shells [5]. The hypothesis (8.7) is called Timoshenko’s generalized kinematic hypothesis in the micropolar theory of shells. b) In the generalized Hooke’s law (8.2) the force stress σ33 can be neglected in comparison to the force stresses σii .

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c) Using the assumption of thin-walled shells, it is assumed 1+

α3 ≈1 Ri

d) In computing the strains, bending-torsion, force and moment stresses, first for the force stresses σ3i and moment stress μ33 we will take: 0

σ3i = σ3i (α1 , α2 ),

0

μ33 = μ33 (α1 , α2 )

(8.8)

After determining of the mentioned quantities, the values of σ3i and μ33 will be finally defined by addition of summands to the corresponding values (8.8), obtained by integration of the first two or the sixth equilibrium equations of (8.1). The condition will be required for the summands that quantities averaged through the shell thickness are equal to zero. The basic system of equations and boundary conditions of the general theory of micropolar elastic thin shells with independent fields of displacements and rotations, constructed with the help of hypotheses a) - d), is the following: Equilibrium equations: 1 ∂T ii 1 ∂A j 1 ∂S ji 1 ∂Ai Ni3 + (T ii − T j j ) + + (S ji + S i j ) + = −(q+i + q−i ), Ai ∂αi Ai A j ∂αi A j ∂α j Ai A j ∂α j Ri 1 ∂Mii 1 ∂A j 1 ∂H ji 1 ∂Ai + (Mii − M j j )+ + (H ji + Hi j ) − N3i = −h(q+i − q−i ), Ai ∂αi Ai A j ∂αi A j ∂α j Ai A j ∂α j T 11 T 22 1 ∂(A2 N13 ) ∂(A1 N23 ) + − + = q+3 + q−3 , R1 R 2 A1 A2 ∂α1 ∂α2 (8.9) 1 ∂Lii 1 ∂A j 1 ∂L ji 1 ∂Ai Li3 + (Lii − L j j ) + + (L ji + Li j ) + Ai ∂αi Ai A j ∂αi A j ∂α j Ai A j ∂α j Ri + − +(−1) j (N j3 − N3 j ) = −(m + m ), i i L11 L22 1 ∂(A2 L13 ) ∂(A1 L23 ) (8.10) + − + − (S 12 − S 21 ) = (m+3 + m−3 ), R1 R2 A1 A2 ∂α1 ∂α2 1 ∂(A2 Λ13 ) ∂(A1 Λ23 ) L33 − + − (H12 − H21 ) = h(m+3 − m−3 ). A1 A2 ∂α1 ∂α2 Elasticity relations: 2Eh [Γii + νΓ j j ], S i j = 2h[(μ + α)Γi j + (μ − α)Γ ji], 1 − ν2 3 2Eh 2h3 (8.11) Mii = [Kii + νK j j ], Hi j = [(μ + α)Ki j + (μ − α)K ji ], 2 3 3(1 − ν ) Ni3 = 2h(μ + α)Γi3 + 2h(μ − α)Γ3i, N3i = 2h(μ + α)Γ3i + 2h(μ − α)Γi3,

T ii =


Lii = 2h

4γ(β + γ) 2γβ β κii + κjj + L33 , β + 2γ β + 2γ β + 2γ

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Li j = 2h[(γ + ε)κi j + (γ − ε)κ ji ],

4γε γ − ε m+i − m−i L33 = 2h[(β + 2γ)ι + β(κ11 + κ22 )], Li3 = 2h κi3 + , γ+ε γ+ε 2 + − 2h3 4γε γ − ε mi + mi Λi3 = li3 + . 3 γ+ε γ + ε 2h

(8.12)

Geometric relations: 1 ∂ui 1 ∂Ai w 1 ∂u j 1 ∂Ai + uj + , Γi j = − ui − (−1) jΩ3 , Ai ∂αi Ai A j ∂α j Ri Ai ∂αi Ai A j ∂α j 1 ∂Ψi 1 ∂Ai 1 ∂Ψ j 1 ∂Ai Kii = + Ψ j, Ki j = − Ψi − (−1) j ι, Ai ∂αi Ai A j ∂α j Ai ∂αi Ai A j ∂α j Γ3i = Ψi − (−1) j Ω j , Γi3 = −ϑi + (−1) jΩ j , (8.13) 1 ∂w ui 1 ∂Ωi 1 ∂Ai Ω3 ϑi = − + , κii = + Ω2 + , Ai ∂αi Ri Ai ∂αi Ai A j ∂α j Ri 1 ∂Ω j 1 ∂Ai 1 ∂Ω3 Ωi (8.14) κi j = − Ωi , κi3 = − , Ai ∂αi Ai A j ∂α j Ai ∂αi Ri 1 ∂ι li3 = . Ai ∂αi Γii =

Boundary conditions (for α1 = const) ∗ or u = u∗ , ∗ or w = w∗ , T 11 = T 11 N13 = N13 1 1 ∗ or u = u∗ , ∗ ∗ S 12 = S 12 M = M 2 11 2 11 or K11 = K11 , ∗ or K = K ∗ , H12 = H12 12 12 ∗ , L11 = L∗11 or κ11 = κ11 ∗ ∗ , L13 = L13 or κ13 = κ13

∗ , L12 = L∗12 or κ12 = κ12 ∗ ∗ Λ13 = Λ13 or l13 = l13

(8.15)

(8.16)

In the model (8.9)–(8.16) of micropolar elastic shells the transverse shear and the related deformations are completely taken into account. One obtains a system of 52 equations for 52 unknown functions (i, j = 1, 2, i j): ui , w, Ψi , Ωi , Ω3 , ι, ϑi , T ii , S i j , Ni3 , N3i , Mii , Hi j , Lii , Li j , L33 , Li3 , Λi3 , Γii , Γi j , Γi3 , Γ3i , Kii , Ki j , κii , κi j , κi3 , li3 . The system of differential equations (8.9)–(8.14) is of 18-th order with 9 boundary conditions (8.16) on each of the contours of the shell’s middle surface Γ. Choosing appropriately Lamé’s parameters Ai the basic relations of the statics of micropolar cylindrical shells and shells of revolution will be obtained from the general equations (8.9)–(8.14) of the shells theory. From the system of equations (8.9)–(8.14) and boundary conditions (8.15) and (8.16) of micropolar shells, when α = 0, the boundary problem of the classical theory of shells will be separated on the basis of Timoshenko’s hypothesis [5] (with some differences related to the static hypothesis d)). If in the model (8.9)–(8.16) the transverse shear is neglected: Γi3 + Γ3i = 0

or

Ψi = ϑi

(8.17)

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a model of micropolar elastic thin shells with independent rotations will be obtained, where instead of Timoshenko’s generalized kinematic hypothesis Kirchhoff-Love’s generalized kinematic hypothesis in the case of micropolar shell will be accepted (so, we take the formula (8.7) taking into account (8.17)). The system of equations of this model of micropolar shells is the following: the equilibrium equations (8.9) and (8.10); elasticity relations from (8.11) and (8.12) for quantities T ii , S i j , Mii , Hi j , Lii , Li j , L33 , Li3 , Λi3 , where formula Ni3 − N3i = 4hα(Γi3 − Γ3i )

(8.18)

must be added; geometric relations from (8.13) and (8.14) for quantities Γii , Γi j , κii , κi j , κi3 ,li3 , where formulas Kii =

1 ∂ϑi 1 ∂Ai + ϑ j, Ai ∂αi Ai A j ∂α j ϑi = −

1 ∂w ui + , Ai ∂αi Ri

Ki j =

1 ∂ϑ j 1 ∂Ai − ϑi − (−1) j ι, Ai ∂αi Ai A j ∂α j

Γi3 − Γ3i = 2[−ϑi + (−1) j Ω j ]

(8.19)

must be added. The main equations of micropolar elastic thin shells of the model with independent rotation, obtained by Kirchhoff-Love’s generalized kinematic hypotheses, must be solved together with the boundary conditions of the classical theory of elastic shells [7] and conditions (8.16). When α = 0, from the mentioned main system of equations and boundary conditions of micropolar elastic thin shells both the general equations and the boundary conditions of the classical theory of elastic shells will be separated on the basis of the Kirchhoff-Love’s hypotheses.

8.4 Model of Micropolar Elastic Thin Shells with Constrained Rotation The case when the dimensionless physical parameters (8.5) have the following values: R2 μ R2 μ R2 μ α μ, ∼ 1, ∼ 1, ∼1 (8.20) β γ ε is considered. The asymptotic analysis of the posed boundary problem [6] for system of equations (8.1)–(8.3) in the case of (8.20) shows that the asymptotic approximations of the vector of rotation ω are connected with the approximations of the vector of displacement V as in the classical theory of elasticity: 1 ω = rotV 2

(8.21)


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It means that in this case the suggested applied two-dimensional theory of micropolar shells is in the frame of micropolar theory with constrained rotation (or Cosserat pseudo continuum [3]). It should be noted that the general three-dimensional micropolar theory of elasticity with constrained rotation has some special features, for example: a) in this model the elastic body is characterized by four elastic constants λ, μ (or E, ν) and γ, ε (or l, η); b) from the six boundary conditions (8.4) remain five [3]: it is impossible to prescribe arbitrary boundary conditions for the moment stress μ33 . On the basis of the results of the asymptotic method of integration of the posed boundary problem for systems of equations (8.1)–(8.3), when the dimensionless physical parameters (8.5) take the values (8.20) [6], the following assumptions (hypotheses) can be applied for the construction of the general applied two-dimensional theory of micropolar shells with constrained rotation: 1. these are assumptions a)–d) of the previous section (but in this case the assumption d) must be referred only to the force stresses σ3i ); 2. the condition of constrained rotation (8.21) The basic system of equations of general applied two-dimensional theory of micropolar elastic thin shells with constrained rotation, in which the transverse shear and the related deformations are completely taken into account, is expressed as follows: equilibrium equations (8.9) and (8.10); elasticity relations from (8.11) and (8.12) for quantities T ii , Mii , Li j , Li3 , Λi3 , where formulas S 12 + S 21 = 4μh(Γ12 + Γ21), Ni3 + N3i = 4μh(Γi3 + Γ3i ), 3 H12 + H21 = 2h3 2μ(K12 + K21 ), Lii = 4γhκii

(8.22)

must be added; geometric relations from (8.13) and (8.14) for quantities Γii , Kii , Γi3 , Γ3i , κii , κi j , κi3 , li3 , ϑi , where formulas Γi j =

1 ∂u j 1 ∂A j 1 ∂Ψ j 1 ∂Ai − ui , Ki j = − Ψi , Ai ∂αi Ai A j ∂α j Ai ∂αi Ai A j ∂α j 1 Ωi = −(−1)i (Ψ j + ϑ j ), Ω3 = (Γ12 − Γ21 ), 2 1 ι = (K12 − K21 ) 2

(8.23)

must be added. Boundary conditions (8.15) and (8.16) should be added to the system of equations of micropolar elastic shells with constrained rotation. The system of equations of the theory of micropolar shells with constrained rotation has 18th order with nine boundary conditions (8.15) and (8.16) on each edge of the middle surface Γ. This system contains 51 equations for 51 unknown functions: T ii , Mii , S i j , Ni3 , N3i , Hi j , Lii , Li j , Li3 , Λi3 , Γii , Kii , Γi j , Ki j , Γi3 , Γ3i , κii , κi j , κi3 , li3 , ui , w, Ψi , ϑi , Ωi , Ω3 , ι. If transverse shear is neglected in this system of equations, i.e. if formulas (8.17) are valid, the model of micropolar elastic shells with constrained rotation will be

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obtained, when Kirchhoff-Love’s generalized hypothesis is valid instead of Timoshenko’s generalized kinematic hypothesis. Main equations and boundary conditions of this model of micropolar shells with constrained rotation are the followings: equilibrium equations (8.9) and (8.10), where Ni3 are neglected; elasticity relations from (8.11) and (8.12) for quantities T ii , Mii , Li j , Li3 , Λi3 , elasticity relations from (8.22) for quantities S 12 + S 21 , H12 + H21 , Lii ; geometric relations from (8.13) and (8.14) for quantities Γii , κii , κi j , κi3 , li3 , ϑi , geometric relation from (8.19) for quantity Kii , geometric relations from (8.23) for quantities Ω3 , ι and formulas Γi j =

1 ∂u j 1 ∂Ai − ui , Ai ∂αi Ai A j ∂α j

Ki j =

1 ∂ϑ j 1 ∂Ai − ϑi , Ai ∂αi Ai A j ∂α j

(8.24)

Ωi = −(−1)iϑ j . The boundary conditions for this model of micropolar elastic thin shells, when α1 = const, are the followings: ∗ T 11 = T 11 or u1 = u∗1 ,

S 12 +

H12 − L11 ∗ = S 12 or u2 = u∗2 , R2

∗ ∗ M11 + L12 = M11 or K11 = K11 ,

N13 +

1 ∂(H12 − L11 ) ∗ = N13 or w = w∗ , A2 ∂α2

(8.25)

∗ L13 = L∗13 or κ13 = κ13 ,

Λ13 = Λ∗13 or l13 = l∗13 .

8.5 Model of Micropolar Shells with “Small Transverse Stiffness” The following case for the physical dimensionless parameters (8.5) is considered: α ∼ μ,

R2 α R2 α R2 α 1, 1, 1 β γ ε

(8.26)

On the basis of asymptotic analysis of the formulated boundary problem [6] for the system of equations (8.1)–(8.3) in a thin three-dimensional domain and considering (8.26) the following asymptotically confirmed assumptions (hypotheses) can be formulated:


99

• the assumptions a)–d) of the section three; • σi j −σ ji , σi3 −σ3i can be neglected in the moment equations of equilibrium (8.1), but taking into consideration the values of physical dimensionless parameters (8.26) (α is a small quantity for the given R) these terms are kept in the physical relations (8.2). It must be noted that this model of micropolar elastic shells, which is obtained on the basis of the accepted hypotheses, is called model with ”small shear stiffness” taking into consideration the fact that the physical parameter α is a shear modulus like the classical sher modulus μ. In this model the ”moment part” of the problem is separated as an independent boundary problem. The main system of equations and boundary conditions of the theory of micropolar elastic thin shells with ”small shear stiffness” with consideration of transverse shear is the following: • the ”moment part” of the problem: equilibrium equations (8.10) without (S 12 − S 21 ), (Ni3 − N3i ), (H12 − H21 ); elasticity relations (8.12); geometric relations (8.14); boundary conditions (8.16); • the ”force part” of the problem: equilibrium equations (8.9); elasticity relations (8.11); geometric relations (8.13); boundary conditions (8.15). The governing system of equations and boundary conditions of the theory of micropolar elastic thin shells with ”small transverse stiffness” without transverse shear is the following: • the ”moment part” of the problem: equilibrium equations (8.10); elasticity relations (8.12); geometric relations (8.14); boundary conditions (8.16); • the ”force part” of the problem: equilibrium equations (8.9); elasticity relations from (8.11) (except the last one) and elasticity relation (8.18); geometric relations from (8.13), (8.14) (except the last five relations of (8.13)) and geometric relations (8.19); boundary conditions of Kirchhoff-Love’s classical theory of elastic shells and conditions (8.16). In the model of micropolar shells with ”small transverse stiffness” (with or without transverse shear) the following fact is important: when the ”moment part” of the problem has zero solution (i.e. when the above mentioned corresponding equations and boundary conditions are homogeneous) the ”force part” of the problem doesn’t coincide with the classical models of elastic shells, as in the physical relations (8.11) members with physical constant α are present.

Conclusion General applied two-dimensional theory of micropolar elastic isotropic shells is constructed, when transverse shear and related to them strains are completely taken into consideration. Depending on the values of dimensionless physical parameters, theories of micropolar elastic thin shells with free fields of displacements and rotations,

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with constrained rotation, with “small shear stiffness” are constructed. General applied two-dimensional theory of micropolar thin plates and general applied onedimensional theory of micropolar thin bars can be obtained as private cases of the theory of shells.

References 1. Altenbach, J., Altenbach, H., Eremeyev, V. A.: On generalized Cosserat-type theories of plates and shells: A short review and bibliography. Arch. Appl. Mech. 80(1), 73–92 (2010) 2. Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM 89(4), 242–256 (2009) 3. Koiter, W.T.: Couple-Stresses in the Theory of Elasticity. Proc. Koninkl. Neterland. Akad. Wetensh. Pt. I-II. 67, 17–44 (1964) 4. Palmov, V.A.: Fundamental equations of the theory of asymmetric elasticity. Applied Mathematics and Mechanics. 28(3), 496–505 (1964) 5. Pelech, B.L.: Theory of Shells with Finite Transverse Shear Stiffness (in Russian). Naukowa Dumka, Kiev (1973) 6. Sargsyan, S.H.: General theory of elastic thin shells on the basis of asymmetrical theory of elasticity (in Russian). Doklady NAS RA. 108(4), 309–319 (2008) 7. Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw Hill, New York (1959)

Part II

Dynamics and Stability

Chapter 9

Closed-Form Approximate Solution for the Postbuckling Behavior of Orthotropic Shallow Shells Under Axial Compression Matthias Beerhorst, Michael Seibel and Christian Mittelstedt

Abstract The current paper deals with a closed-form approximate solution for the postbuckling behavior of an unstiffened, singly-curved, orthotropic shell. As loading condition the case of uniform axial compression is treated. Concerning the boundary conditions all edges are supposed to be simply supported. Additionally, geometrical imperfections in form of an initial deflection of the shell can be accounted for. Choosing rather simple shape functions for the deflection a closed-form expression for the Airy stress function is obtained from the compatibility condition. As the equilibrium condition cannot be satisfied exactly the solution procedures of Galerkin as well as Ritz are employed to obtain an approximate solution. The resulting expressions from these procedures again allow for a closed-form solution of the load-deflection-relationship. After the force and the amplitude are known all other state variables such as stresses and displacements can be evaluated in a closed-form manner. Due to the rather simple formulation of the deflection shape the algorithm is limited to cases where the qualitative shape of the deflection does not change significantly. On the other hand the very high computational efficiency of the described solution procedure makes it ideally suited for use in the field of optimization and preliminary design, if the applied load does not exceed the linear buckling load too much. Keywords Postbuckling · Orthotropic shells · Closed-form solution · Analytical

M. Beerhorst (B) · M. Seibel HAW Hamburg, Berliner Tor 9, D-20099 Hamburg, Germany e-mail: [email protected],[email protected] C. Mittelstedt ELAN GmbH, Team Method and Tools, Karnapp 25, D-21079 Hamburg, Germany e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 9, © Springer-Verlag Berlin Heidelberg 2011

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9.1 Introduction Especially in those fields of engineering where minimum weight is of primary importance composite materials find an ever increasing use. In the case of thinwalled structural members stability problems occur if the structure is loaded in compression or shear. For analysis purposes structural members are often decomposed into relative simple geometries like plates and shells. Because of this the calculation of the stability behavior of plates and shells has been the subject of scientific research for over a hundred years. An important quantity in the design is the linear buckling load which describes the load at which the undeformed structure may abruptly undergo a deformation into its normal direction. However, since the linear buckling load is often significantly below the maximum load capacity of the structure, the so called post buckling behavior is also of special interest. For detailed analysis of the postbuckling behavior various iterative procedures including global approaches using Ritz- or Galerkin-methods (e.g. Shin et al. [15]), semi-discretizing approaches like the finite-strip method (e.g. Zou and Qiao [18]) or the discrete finiteelement-method (e.g. Reddy [13]) have been applied. The focus of the present work lies on the development of a solution procedure which employs a global approach in combination with only few shape functions instead of whole series in order to allow for a closed-form solution. As it has been shown lately for example by Diaconu and Weaver ( [3], [4]) who examined the case of an compressively loaded infinitely long plate as well as Mittelstedt and Schröder [12] who dealt with a plate of finite length, closed-form-type solutions offer the merit of a very high computational efficiency while delivering quite satisfying results if the structure is not loaded far beyond the linear buckling load. In the present work the approach developed in [12] is extended to the geometry of singly-curved, shallow shells (see Fig. 9.1). Among the various shell theories 0

Nx

0

Nx

Fig. 9.1 Singly curved, shallow shell under uniaxial compression

existing (Donnell [5], Sanders-Koiter [14], [8], Love [10] and Flügge [6]) the one of Donnell which is valid for thin shells with small curvature was selected for the present work. Comparisons of the different theories can, inter alia, be found in the works of Simitses et al. [16] and Jaunky and Knight [7]. Beginning with the presentation of the governing equations related to this problem non dimensional parameters are introduced in order to transform the derived

9 Approximate Solution for the Postbuckling Behavior of Orthotropic Shells

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expressions in a generic manner. With the aid of a simple function of known shape but unknown amplitude for the deflection and a function of the same shape and known amplitude for the initial deflection, the compatibility requirement yields a closed-form expression for the Airy stress function. Employing the solution procedures of Galerkin as well as Ritz, the equilibrium condition is satisfied in an integral sense resulting in approximate expressions for the load-deflection-dependency. As a closure the resulting load-deflection-curves, buckling mode shapes and average strains are compared to an approach which uses full series expansions for the deflections as well as the stress function.

9.2 Governing Equations 9.2.1 Classical Formulations The geometry and coordinate systems of the shell which is dealt with in the present paper are visualized in Fig. 9.2. The shell has the length a, mid-surface arc length b, thickness t and mid-surface radius R. Since the shell in the present work is assumed

b a x, z, y, y= R

R

Fig. 9.2 Geometry and coordinate systems

to be thin-walled and of small curvature the material law which describes the relationship between forces and moments on the one hand and strains and curvatures on the other hand may be described by using the classical laminate plate theory [9]:

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⎧ 0⎫ ⎡ ⎪ ⎢⎢⎢A11 Nx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢A12 Ny0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ Nxy ⎪ ⎬ ⎢⎢⎢⎢A16 = ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ B11 ⎪ ⎪ M 0x ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ B 0 ⎪ ⎪ ⎪ M ⎪ ⎪ y⎪ ⎢⎢ 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ M 0 ⎭ ⎢⎣ B 16 xy

A12 A16 B11 B12 A22 A26 B12 B22 A26 A66 B16 B26 B12 B16 D11 D12 B22 B26 D12 D22 B26 B66 D16 D26

⎤⎧ 0 ⎫ B16 ⎥⎥⎥ ⎪ εx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎥⎥⎥ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ B26 ⎥⎥⎥ ⎪ ε ⎪ ⎪ y ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎥⎥⎥ ⎪ ⎪ ⎪ 0 ⎪ ⎪ B66 ⎥⎥⎥ ⎪ ε ⎨ xy ⎪ ⎬ ⎥⎥⎥ ⎪ . ⎪ ⎪ ⎪ 0 ⎪ ⎪ D16 ⎥⎥⎥⎥ ⎪ κ ⎪ ⎪ x⎪ ⎪ ⎪ ⎪ ⎥⎥ ⎪ ⎪ ⎪ ⎪ ⎪ D26 ⎥⎥⎥⎥⎥ ⎪ ⎪ κy0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎦ ⎪ ⎪ ⎪ ⎩ 0 D κ ⎭ 66

(9.1)

xy

Gathering terms in sub-vectors and sub-matrices yields: ⎧ ⎫ 0⎪ ⎪ ⎪ ⎨N ⎪ ⎬ = ⎪ ⎪ ⎪ ⎩M0⎪ ⎭

⎡ ⎤⎧ ⎫ 0⎪ ⎢⎢⎢A B ⎥⎥⎥ ⎪ ⎪ ⎨ε ⎪ ⎬ ⎢⎢⎢⎣ ⎥⎥⎥⎦ ⎪ ⎪ ⎪. 0⎪ ⎩ BD κ ⎭

(9.2)

Herein a column vector is indicated by one underline and a matrix by two underlines. The partially inverted form of (9.1) reads: ⎤⎧ ⎫ ⎧ ⎫ ⎡ ⎢ 6 6 ⎪ B ⎥⎥⎥⎥ ⎪ ⎪ ⎪ ⎪N 0 ⎪ ⎪ ⎨ ε0 ⎪ ⎬ ⎢⎢⎢⎢ A ⎬ ⎥⎥⎥ ⎨ = (9.3) ⎢ ⎪ ⎪ ⎪ ⎪ T ⎢ ⎪ ⎪ ⎪ ⎪, 0 ⎦ ⎩ M 0 ⎭ ⎣−6 ⎩ 6 B D κ ⎭ with: 6 = A−1 , A

6 B = −A−1 B,

6 = D − B A−1 B. D

(9.4)

The strain distribution in the shell can be calculated from the in-plane strains and the curvatures of the reference surface according to the following kinematic relations: ε x = ε0x + zκ0x ,

εy = ε0y + zκy0 ,

ε xy = ε0xy + zκ0xy .

(9.5)

Herein the curvatures are identical to those of the linear plate theory and read: κ0x = −

∂2 w , ∂x2

κy0 = −

∂2 w , ∂y2

κ0xy = −2

∂2 w . ∂x∂y

(9.6)

In contrast to this the membrane strains have additional nonlinear components consisting of derivatives of the deflection which has been separated into a part with constant amplitude describing the initial deflection w0 and a load dependent variable part w: 2 ∂u 1 ∂w ∂w ∂w0 + + , ∂x 2 ∂x ∂x ∂x 2 ∂v 1 ∂w ∂w ∂w0 w 0 εy = + + + , ∂y 2 ∂y ∂y ∂y R ∂u ∂v ∂w ∂w ∂w ∂w0 ∂w ∂w0 ε0xy = + + + + . ∂y ∂x ∂x ∂y ∂x ∂y ∂y ∂x ε0x =

(9.7) (9.8) (9.9)


107

These equations are identical to those proposed for the flat plate with initial deflection by Marguerre [11] with an additional term wR in ε0y describing the influence of the radius. Since it is assumed that the shell is stress free in the initial state only the variable deflection w and not w0 is accounted for in this term. Furthermore the initial deflection w0 is assumed to be small in comparison to w in the later loading stages so that terms including w0 in second order are neglected. From the six equilibrium conditions concerning the forces and moments the following three equations can be derived ( [16]): ∂Nx ∂Nxy + = 0, ∂x ∂y

(9.10)

∂Nxy ∂Ny + = 0, ∂x ∂y

(9.11)

∂2 M xy ∂2 My ∂2 M x ∂2 (w + w0 ) + 2 + + N x ∂x∂y ∂x2 ∂y2 ∂x2 2 2 ∂ (w + w0 ) ∂ (w + w0) Ny + 2N xy + Ny − + p = 0. ∂x∂y R ∂y2

(9.12)

Wherein p denotes pressure acting normal to the shell surface. By introducing the Airy stress function (9.13) as given below, conditions (9.10) and (9.11) derived from equilibrium are identically fulfilled. Nx0 =

∂2 Ψ , ∂y2

Ny0 =

∂2 Ψ , ∂x2

0 Nxy =−

∂2 Ψ ∂x∂y

(9.13)

Apart from equilibrium the compatibility condition of the in-plane strains (9.14) is taken into account. 2 0 2 0 ∂2 ε0x ∂ εy ∂ ε xy + − =0 (9.14) ∂x∂y ∂y2 ∂x2 Using the partially inverted material law (9.3) in conjunction with the Airy stress function (9.13) the moments can be described as: ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w − B + B − D − D − 2D , 21 11 12 61 16 ∂x∂y ∂x∂y ∂y2 ∂x2 ∂x2 ∂y2 ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w My0 = −B12 2 − B22 2 + B62 − D12 2 − D22 2 − 2D26 , ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w M 0xy = −B16 2 − B26 2 + B66 − D16 2 − D26 2 − 2D66 . (9.15) ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y M 0x = −B11

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Inserting the above equations into the third equilibrium-equation (9.12) yields: ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ + (B11 + B22 − 2B66) 2 2 + B12 4 + (2B26 − B61) 3 4 ∂x ∂x ∂y ∂y ∂x ∂y 4 4 4 ∂ ψ ∂ w ∂ w ∂4 w (D ) + (2B16 − B62) + D + 2 + 2D + D 11 12 22 66 ∂x∂y3 ∂x4 ∂x2 ∂y2 ∂y4 2 4 4 2 2 ∂ w ∂ w ∂ ψ ∂ w ∂ w0 +4D16 3 + 4D26 2 2 − 2 + ∂x ∂y ∂x ∂y ∂y ∂x2 ∂x2 2 2 2 2 2 2 ∂ ψ ∂ w ∂ w0 ∂ ψ ∂ w ∂ w0 1 +2 + − 2 + − + p = 0. (9.16) ∂x∂y ∂x∂y ∂x∂y ∂x ∂y2 ∂y2 R B21

The membrane strains formulated according to the partially inverted material law (9.3) read: ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w − A + A − B − B − 2B , 12 11 12 16 16 ∂x∂y ∂x∂y ∂y2 ∂x2 ∂x2 ∂y2 ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w ε0y = −A12 2 − A22 2 − A26 − B21 2 − B22 2 − B26 , ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ∂2 ψ ∂2 ψ ∂2 ψ ∂2 w ∂2 w ∂2 w ε0xy = A16 2 + A26 2 − A66 − B61 2 − B62 2 − 2B66 . ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ε0x = −A11

(9.17) (9.18) (9.19)

Inserting these expressions into the compatibility-equation (9.14) results in: ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ (2A ) + A + + A − 2A 11 12 66 16 ∂x4 ∂y4 ∂x2 ∂y2 ∂x∂y3 4 4 4 ∂ ψ ∂ w ∂ w ∂4 w −A26 3 − B21 4 − B12 4 + (2B66 − B11 − B22) 2 2 ∂x ∂y ∂x ∂y ∂x ∂y ∂4 w ∂4 w + (B62 − 2B16) + (B61 − 2B26) 3 = 0. ∂x∂y3 ∂x ∂y A22

(9.20)

Another formulation of the compatibility expression is obtained by inserting the membrane strains resulting from the kinematic relations (9.5) into (9.14):

∂2 w ∂x∂y

2

∂2 w ∂2 w ∂2 w ∂2 w0 ∂2 w ∂2 w0 − − 2 ∂x2 ∂y2 ∂x2 ∂y2 ∂y ∂x2 ∂2 w ∂2 w0 1 ∂2 w +2 + = 0. ∂x∂y ∂x∂y R ∂x2 −

(9.21)


109

Now both representations of the compatibility condition are equated to achieve a relation between the Airy stress function and the deflection functions: ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ + A11 4 + (2A12 + A66) 2 2 − 2A16 − A26 3 4 3 ∂x ∂y ∂x ∂y ∂x∂y ∂x ∂y 4 4 4 ∂ w ∂ w ∂ w ∂4 w −B21 4 − B12 4 + (2B66 − B11 − B22) 2 2 + (B62 − 2B16) ∂x ∂y ∂x ∂y ∂x∂y3 2 ∂4 w ∂2 w ∂2 w ∂2 w ∂2 w ∂2 w0 + (B61 − 2B26) 3 = − 2 2− 2 ∂x∂y ∂x ∂y ∂x ∂y ∂x ∂y2 ∂2 w ∂2 w0 ∂2 w ∂2 w0 1 ∂2 w − 2 +2 + . (9.22) 2 ∂x∂y ∂x∂y R ∂x2 ∂y ∂x A22

The further attempts to solve the problem will mainly rely on finding a solution of the equilibrium condition in the form of (9.16) and the compatibility condition (9.22).

9.2.2 Non-Dimensional Quantities In order to get a more general insight into the influence of geometry and laminate parameters it is convenient to introduce non dimensional quantities (cf. [4]). The coordinates are related to the length and width of the shell as: y η= . b

x ξ= , a

(9.23)

Additionally a dimensionless radius R and pressure p are defined: R=R

7 AD , b2

p= p

a2 b 2 . 8 7 D 611 D 622 AD

Herein the following abbreviation was used: 8 4 7 611 A 622 D 611 D 622 . AD = A

(9.24)

(9.25)

Nondimensionalization of the stress function ψ and the potential of the system Π is achieved by using relations (9.26). ψ= 8

ψ 611 D 622 D

,

Π =Π

ab . 8 72 D 611 D 622 AD

(9.26)

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The displacements become: u=

ua 2

7 AD

,

v=

vb 2

7 AD

,

w=

w , 7 AD

w0 =

w0 . 7 AD

(9.27)

Further abbreviations proposed by Diaconu and Weaver [4] are: 9 9 611 622 a 4 A b2 6 B21 a 4 D αA = , αB = 2 , αD = , b A 622 7 b D 611 a AD βA =

612 + A 666 2A , 8 6 6 2 A11 A22

γA = 8 4

δA = 8 4

616 A 63 A 6 A 11 22 626 A 611 A 63 A 22

ηA = 8

μB = θB =

6 B11 , 7 AD b6 B61 a AD 7

612 A 611 A 622 A

ρB = ,

ζB =

,

,

6 B22 , 7 AD b 26 B26 a AD 7

βB =

6 B11 + 6 B22 − 2 6 B66 , 7 AD

612 + 2D 666 D βD = 8 , 6 6 D11 D22

γB =

b 26 B26 − 6 B61 , a 7 AD

γD = 8

δB =

a 26 B16 − 6 B62 , b 7 AD

δD = 8

ηB =

a2 6 B12 , 2 7 b AD

ηD = 8

ωB = ,

k1 =

6 B66 , 7 AD a2 b 2 , √ 7 D11 D22 AD

4

4

616 D 63 D 6 D 11 22 626 D 611 D 63 D 22 612 D 611 D 622 D

,

,

,

a6 B16 a6 B62 , τB = , b AD 7 b AD 7 a 2 b2 k2 = . (9.28) 72 AD

νB =

A non dimensional formulation of the equilibrium (9.16) is now obtained by using the non dimensional quantities above and multiplying the resulting expression with k1 from (9.28): αB

∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ 1 ∂4 w ∂4 w + β + η + γ + δ + + 2β B B B B D ∂ξ4 ∂ξ2 ∂η2 ∂η4 ∂ξ3 ∂η ∂ξ∂η3 α2D ∂ξ4 ∂ξ2 ∂η2 ∂4 w γ D ∂4 w ∂4 w ∂2 ψ ∂2 w ∂2 w0 + α2D 4 + 4 + 4α δ − + D D αD ∂ξ3 ∂η ∂η ∂ξ∂η3 ∂η2 ∂ξ2 ∂ξ2 ∂ 2 ψ ∂ 2 w ∂ 2 w0 ∂2 ψ ∂2 w ∂2 w0 1 +2 + − 2 + − + p = 0. (9.29) ∂ξ∂η ∂ξ∂η ∂ξ∂η ∂ξ ∂η2 ∂η2 R


111

The procedure for transforming the compatibility equation (9.22) works analogously but this time the multiplication factor is k2 from (9.28): 1 ∂4 ψ ∂4 ψ ∂4 ψ ∂4 ψ δA ∂4 ψ + α2A 4 + 2βA 2 2 − 2αAγA −2 2 4 3 αA ∂ξ3 ∂η ∂η ∂ξ ∂η ∂ξ∂η αA ∂ξ ∂4 w ∂4 w ∂4 w ∂4 w ∂4 w − η − β − δ − γ B B B B ∂ξ4 ∂η4 ∂ξ2 ∂η2 ∂ξ∂η3 ∂ξ3 ∂η 2 2 ∂ w ∂2 w ∂2 w ∂2 w ∂2 w0 = − 2 2− 2 ∂ξ∂η ∂ξ ∂η ∂ξ ∂η2 2 2 2 2 ∂ w ∂ w0 ∂ w ∂ w0 1 ∂2 w − 2 +2 + . ∂ξ∂η ∂ξ∂η R ∂ξ2 ∂η ∂ξ2 −αB

(9.30)

The relation of loads and Airy stress function is also transformed as: 0

N x π2 =

∂2 ψ , ∂η2

0

N y π2 =

∂2 ψ , ∂ξ2

0

N xy π2 = −

∂2 ψ , ∂ξ∂η

(9.31)

where: 0

Nx =

Nx0 b2 , 8 611 D 622 π2 D

0

Ny =

Ny0 a2 , 8 611 D 622 π2 D

0

N xy =

0 ab Nxy . 8 611 D 622 π2 D

(9.32)

In the special case of symmetric, orthotropic laminates the absence of many coupling terms leads to a drastic simplification of the expressions above. More precisely the non dimensional parameters γA , δA , γD , δ D and all abbreviation parameters of (9.28) with subindex B vanish. The equilibrium condition (9.29) then simplifies to: 4 1 ∂4 w ∂4 w ∂2 ψ ∂2 w ∂2 w0 2 ∂ w + 2βD 2 2 + αD 4 − 2 + ∂ξ ∂η ∂η ∂η ∂ξ2 ∂ξ2 α2D ∂ξ4 ∂ 2 ψ ∂ 2 w ∂ 2 w0 ∂2 ψ ∂2 w ∂2 w0 1 +2 + − 2 + − + p = 0, (9.33) ∂ξ∂η ∂ξ∂η ∂ξ∂η ∂ξ ∂η2 ∂η2 R and the compatibility condition reads: 2 2 4 1 ∂4 ψ ∂4 ψ ∂ w ∂2 w ∂2 w 2∂ ψ + α + 2β = − A A ∂ξ∂η ∂η4 ∂ξ2 ∂η2 ∂ξ2 ∂η2 α2A ∂ξ4 ∂2 w ∂2 w0 ∂2 w ∂2 w0 ∂2 w ∂2 w0 1 ∂2 w − 2 − + 2 + . ∂ξ∂η ∂ξ∂η R ∂ξ2 ∂ξ ∂η2 ∂η2 ∂ξ2

(9.34)

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9.2.3 In-Plane Displacements In the field of structural design not only the deflection of the shell but also the in-plane displacements are of interest. The partial derivatives of the in-plane displacements as shown below in (9.35) to (9.37) are calculated from the kinematic relationships (9.5). 2 ∂u 1 ∂w ∂w ∂w0 = ε0x − − ∂x 2 ∂x ∂x ∂x 2 ∂v 1 ∂w ∂w ∂w0 w = ε0y − − − ∂y 2 ∂y ∂y ∂y R ∂u ∂v ∂w ∂w ∂w ∂w0 ∂w ∂w0 + = ε0xy − − − ∂y ∂x ∂x ∂y ∂x ∂y ∂y ∂x

(9.35) (9.36) (9.37)

Inserting the membrane strains of (9.17) to (9.19) into (9.35) to (9.37) and performing the transformation yields: ∂u ∂2 ψ ∂2 ψ ∂w ∂w0 = α2A 2 + ηA 2 − , ∂ξ ∂ξ ∂ξ ∂η ∂ξ 2 ∂v ∂2 ψ 1 ∂2 ψ 1 ∂w ∂w ∂w0 w = ηA 2 + 2 2 − − − , ∂η 2 ∂η ∂η ∂η R ∂η αA ∂ξ ∂u ∂v ∂2 ψ ∂w ∂w ∂w ∂w0 ∂w ∂w0 + = 2(ηA − βA ) − − − , ∂η ∂ξ ∂ξ∂η ∂ξ ∂η ∂ξ ∂η ∂η ∂ξ

(9.38) (9.39) (9.40)

in case of an symmetric, orthotropic laminate. Often the overall change in length and width is of special interest, which is calculated by integration over the length and width as: 1 ∂2 ψ ∂2 ψ ∂w ∂w0 Δa = α2A 2 + ηA 2 − dξ, (9.41) ∂ξ ∂ξ ∂η ∂ξ 0 and:

⎞ 2 1 ⎛⎜ ∂w ∂w0 w ⎟⎟⎟⎟ ⎜⎜⎜ ∂2 ψ 1 ∂2 ψ 1 ∂w Δb = ⎝⎜ηA ∂η2 + 2 ∂ξ2 − 2 ∂η − ∂η ∂η − ⎟⎠ dη. αA R 0

(9.42)

If these values are related to the nondimensionalized initial length a and width b of the shell, the average strains become: εa =

Δa Δa = , a a

εb =

Δb b

=

Δb . b

(9.43)

The nondimensionalized length a and width b may be calculated from (9.27), which describes the displacements, by replacing the characters u with a and v with b.


113

9.3 Closed-Form Solution 9.3.1 Boundary Conditions The edges of the shell are assumed to be simply supported, which demands vanishing normal displacements and bending moments along the edges. w (ξ = 0, η) = 0,

M η (ξ = 0, η) = 0,

(9.44)

w (ξ = 1, η) = 0,

M η (ξ = 1, η) = 0,

(9.45)

w (ξ, η = 0) = 0,

M ξ (ξ, η = 0) = 0,

(9.46)

w (ξ, η = 1) = 0,

M ξ (ξ, η = 1) = 0.

(9.47)

Regarding the in-plane displacements no specific boundary conditions are assumed, so the edges can freely move in-plane.

9.3.2 Linear Buckling Loads For the validation of the results later on the linear buckling loads of the plate and the shell will be referred to. For brevity only the resulting expressions are cited here, details can for example be found in [17]. The linear buckling load of an orthotropic plate with symmetric lay-up is: 0

Nx =

αD m + m αD

2 + 2 (βD − 1),

and for the shell: ⎡ ⎢⎢⎢ 2 n 2 0 2 ⎢⎢⎢⎢ m 1 N x = n ⎢⎢ + 2βD + αD + 2 % ⎢⎣ n αD m (πn)4 R m

1

& % 1 2 n n αA + 2βA + m α A

(9.48)

⎤ ⎥⎥⎥ ⎥⎥⎥ &2 ⎥⎥⎥⎥ . (9.49) ⎦

Herein the numbers of halfwaves m and n which lead to the lowest buckling load are the relevant ones.

9.3.3 Compatibility Condition - Stress Function As a first step towards the solution of the postbuckling problem the compatibility condition (9.34) is used to obtain a closed-form solution of the stress function. As this partial differential equation depends on the stress function on the one hand and

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the deflection on the other hand the shape of the deflection is assumed a priori to reduce the equation to a partial differential equation of only one unknown function. Products of sine-functions which fulfill the boundary conditions given in (9.44) – (9.47) are chosen as shape functions for the variable deflection w and the initial deflection w0 : w = A sin(mπξ) sin(nπη),

(9.50)

w0 = B sin(mπξ) sin(nπη).

(9.51)

Herein A and B are the unknown and known deflection amplitudes, respectively. The number of buckling halfwaves is expressed through m in longitudinal and n in circumferential direction. Inserting the shape functions into the compatibility condition (9.34) results in: 1 ∂4 ψ ∂4 ψ ∂4 ψ m2 + α2A 4 + 2βA 2 2 = −π2 A sin(mπξ) sin(nπη) 2 4 ∂η ∂ξ ∂η αA ∂ξ R % & 4 2 2 2 +π m n A(A + 2B) cos (mπξ) + cos2 (nπη) − 1 .

(9.52)

Since the above partial differential equation is of linear nature the overall solution of the stress function can be found by superposition of a homogeneous solution and the particular solutions: ψ = ψh + ψ p1 + ψ p2 . (9.53) The homogeneous solution of the problem corresponds to the situation of a plate with zero deflection. According to [12] it can be obtained via integration while neglecting the integration constants: 0 ψh = N x π2 dη dξ. (9.54) After integration the homogeneous solution reads: 0

ψh =

N x π2 η2 . 2

(9.55)

From the right-hand side of the compatibility condition (9.52) it can be observed that the particular solution should consist of a product of sine-functions and a sum of squares of cosine-functions. So the first and second particular solution are assumed to be of the form: ψ p1 = C sin(mπξ) sin(nπη),

(9.56)

ψ p2 = D cos (mπξ) + E cos (nπη).

(9.57)

2

2

Herein the first particular solution describes the influence of the radius, whereas the second is identical to the solution for a flat plate. Subsequently inserting the


115

particular solutions into the compatibility condition, the unknowns C, D and E can be calculated by comparison of coefficients:

C= Rπ2 D=

−A m2 α2A

4

+ α2A mn 2 + 2βA n2

α2A n2 A(A + 2B) 16m2

,

E=

,

m2 A(A + 2B) . 16α2A n2

(9.58)

9.3.4 Equilibrium Condition - Load-Deflection-Relation After a closed form solution for the stress function was determined by using the compatibility condition, the relationship between the applied load and the corresponding amplitude of the deflection has to be found. This can be done be using the equilibrium condition (9.33). Because this equation does not allow for a closedform solution two approximation approaches are employed. First the minimization of the potential energy of the system as suggested by Ritz is outlined. According to Ritz the stationary value of potential energy is found when the derivations of the total potential of the system with respect to the variables vanish: ∂Π = 0. ∂Ai

(9.59)

The total potential of the system is the sum of the internal and the external potential: Π = Π i + Π e. In the present case the inner energy of the shell reads: % & 1 Πi = σx ε x + σy εy + σ xy γ xy dV. 2 V

(9.60)

(9.61)

Inserting the strains according to the kinematics (9.5) and performing integration over the thickness yields: 1 Πi = 2

a b% 0

0

& 0 0 N x0 ε0x + Ny0 ε0y + N xy γ xy + M 0x κ0x + My0 κy0 + M 0xy κ0xy dydx.

(9.62)

Now the laminate forces and moments are replaced by the relations (9.13) and (9.15) and the strains and curvatures are replaced by (9.17)–(9.19) and (9.6):

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a b

1 Πi = 2

0

0

2 2 2 2 ∂ ψ 6 612 ∂ ψ ∂ ψ A11 + 2 A ∂y2 ∂x21 ∂y2 2 2 2 2 622 ∂ ψ + A 666 ∂ ψ +A 2 ∂x1 ∂y ∂x ⎛ ⎞2 2 2 2 ⎜ ⎟ 611 ⎜⎜⎜⎜⎝ ∂ w ⎟⎟⎟⎟⎠ + 2 D 612 ∂ w ∂ w +D ∂x2 ∂y2 ∂x21 2 2 2 2 622 ∂ w + 4 D 666 ∂ w +D dydx. ∂xdx2 ∂y2

(9.63)

Equation (9.64) shows the internal energy after transformation. 1 Πi = 2

0

1

1 2 2 ∂ ψ ∂2 ψ ∂2 ψ α2A + 2η A ∂η2 ∂ξ2 ∂η2 0 2 2 2 2 1 ∂ ψ ∂ ψ + 2 + 2 (βA − ηA ) 2 ∂ξ∂η αA ∂ξ 2 1 ∂2 w ∂2 w ∂2 w + 2 + 2η D ∂ξ2 ∂η2 αD ∂ξ2 2 2 2 2 ∂ w ∂ w (β ) + α2D + 2 − η dηdξ. D D 2 ∂ξ∂η ∂η

(9.64)

The potential of the external load is identical to the external work performed by this load: 1 1 0 du Πe = − N x dηdξ. (9.65) dξ 0 0 With the derivative of the displacement u according to (9.38) the external potential becomes: 1 1 ∂2 ψ ∂2 ψ ∂w ∂w0 0 Πe = − N x α2A 2 + ηA 2 − dηdξ. (9.66) ∂ξ ∂ξ ∂η ∂ξ 0 0 In the present case the only unknown variable is the deflection amplitude A of the variable deflection w, so only condition (9.67) below has to be fulfilled to find the state of stationary potential energy. ∂Π = 0. ∂A

(9.67)

The above condition leads to an expression which contains third order terms of A 0 and linear terms of the external load N x . Now two options exist for finding the relation between load and amplitude.


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• The first one, which may be called load controlled, is to set the value of the load and then calculate the amplitude, which is the maximum real root of the polynomial of third order for A. • The second one, which will be referred as amplitude controlled, is to solve (9.67) 0 for N x . From the resulting expression the load can be directly calculated for a given amplitude. The advantage of the first method is that setting the range of the load might be more intuitive as the engineer is often interested in the behavior of the structure up to a certain load level, for example two times the linear buckling load. On the other hand the second approach can also handle situations in which the absolute value of the amplitude decreases with increasing load, for example when dealing with cylindrical shells. Another way to approximately satisfy the equilibrium condition is to apply the Galerkin-procedure. This method is based on the main idea to eliminate the weighted residuum of the differential equation in an integral sense. For the present case this leads to: 1 1 . / R w (ξ, η) W(ξ, η)dηdξ = 0, (9.68) 0

0

with the first factor being the residuum (cf. (9.34)): 4 . / 1 ∂4 w ∂4 w ∂2 ψ ∂2 w ∂2 w0 2 ∂ w R w (ξ, η) = 2 + 2βD 2 2 + αD 4 − 2 + ∂ξ ∂η ∂η ∂η ∂ξ2 ∂ξ2 αD ∂ξ4 ∂2 ψ ∂2 w ∂2 w0 ∂2 ψ ∂2 w ∂2 w0 1 +2 + − 2 + − . ∂ξ∂η ∂ξ∂η ∂ξ∂η ∂ξ ∂η2 ∂η2 R

(9.69)

The weighting function is identical to the deflection function with the amplitude set to unity: W(ξ, η) = sin(mπξ) sin(nπη). (9.70) After inserting the deflection functions (9.50) and (9.51) and the stress function (9.53) into the residuum-equation (9.69) the integrals of (9.68) can be evaluated. The result is an expression which contains third order terms of A and linear terms of 0 the external load N x . In order to obtain the load-amplitude-relationship either loadcontrolling or amplitude-controlling as described above for the Ritz-method may be employed.

9.4 Results and Discussion In order to establish a basis for estimating the accuracy and applicability of the proposed closed-form solution it is compared to a solution procedure which uses Fourier series expansions for the deflections:

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w= w0 =

M N

Amn sin (mπξ) sin (nπη),

(9.71)

Bmn sin (mπξ) sin (nπη),

(9.72)

m=1 n=1 M N m=1 n=1

as well as the particular parts of stress function: ψ p1 = ψ p2 =

2M 2N m=0 n=0 2M 2N

Cmn sin(mπξ) sin(nπη),

(9.73)

Dmn cos(mπξ) cos(nπη),

(9.74)

m=0 n=0

similar as proposed for example by [2] and [1]. Because the algorithm based on this idea is capable of analyzing unsymmetrical lay-ups it will be referenced as PBUS (Postbuckling of unsymmetric shells). Analogously the present method will be referred to as PBOS (Postbuckling of orthotropic shells). Additionally the linear buckling loads of the flat plate as well as the shallow shell as given in (9.48) and (9.49) will be used as reference. The load-deflection curves do also include the postbuckling deflection of a flat plate, which is achieved by setting the value for the radius to infinity. As in-plane coordinates for the load-deflection-history the point with the maximum deflection obtained from the eigenvalue analysis which lies nearest to the center of the shell is chosen. For the further analyses the material properties are taken from a CFRP T300/913C UD-prepreg: E11 = 93000 MPa, E22 = 7300 MPa, G12 = 4700 MPa, ν12 = 0.3. In all examples the thickness of each ply is tPly = 0.25 mm and the orientation of the eight plies in degree is [90/90/0/0] s. The numbers of buckling halfwaves m and n which are chosen for the present analysis are determined from the eigenvalue analysis according to (9.49). The amplitude B of the initial deformation is set to 10.4% of the laminate thickness. The current example deals with a shell of the length a = 500 mm, width b = 200 mm and radius R = 2000 mm. Performing the eigenvalue analysis reveals that the 0 linear buckling load is N x = −3.8 and the corresponding eigenmode consists of one halfwave in circumferential direction and four halfwaves in longitudinal direction. Carrying out a convergence study with PBUS showed that using 6 x 6 terms in this case is more than sufficient. In Fig. 9.3 the load-deflection-history is shown for the coordinates ξ = 5/8 and η = 0.5. It can be seen that the deflection of the plate increases most rapidly with increasing load and that the results of PBOS and PBUS are in excellent agreement. As expected the results of the shell show a slower increasing amplitude which indicates a stiffer structural behavior. The results of the Galerkin and Ritz approach are identical. On first glance the resulting curve from the PBUS-tool which uses a Fourier expansion of 6 x 6 terms for the deflection function


-12 -10

Nx0

-8 -6 -4

PBOS Shell PBOS Plate Plate Linear Shell Linear PBUS PBUS (Mean) PBUS Plate

-2 0

0

1

2

3

4

w+w0

5

6

7

8

Fig. 9.3 Deflection history for point at ξ = 5/8 and η = 0.5

8 6 4

w+w0

2 0

-2 -4 -6 -8

PBUS PBOS

-10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

Fig. 9.4 Deflection along ξ in the middle of the shell (η = 0.5) at N x = −12

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1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 9.5 Contour plot of initial deflection

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0 0

Fig. 9.6 Contour plot of deflection in the deep postbuckling region (N x = −12)


-0.065 -0.066

a [%]

-0.067 -0.068 PBUS PBOS

-0.069 -0.070 -0.071 -0.072 -0.073 0.0

0.2

0.4

0.6

0.8

1.0

0

Fig. 9.7 Average strain εa at N x = −6

0.06 PBUS PBOS

0.04

b [%]

0.02 0.00 -0.02 -0.04 -0.06 0.0

0

0.2

Fig. 9.8 Average strain εb at N x = −6

0.4

0.6

0.8

1.0

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0 -2

Nx0

-4 -6 -8 -10 -12 -0.20

PBUS PBOS

-0.15

-0.10

-0.05

a [%]

0.00

0.05

Fig. 9.9 Strain history of the average strain εa in the middle of the shell (η = 0.5)

might be surprising because it shows a smaller deflection and thus a stiffer behavior than the present approach. This effect can be explained by the fact that the absolute value of the buckling peaks is not the same in the PBUS-analysis which can be seen in Fig. 9.4. Unlike in the case of a flat plate, due to the influence of the radius, the absolute values of the buckling peaks pointing inward the shell and thus in negative ζ-direction are greater than the ones pointing in the outward direction. Because the present method has only one degree of freedom with respect to the amplitude of the deflection it is not able to capture this effect and despite that returns an amplitude which is more like the arithmetic mean of the absolute values of the buckling peaks of PBUS. This mean value of the deflection is also visualized in Fig. 9.3 and is in good agreement with deflection calculated with PBOS. Another aspect which comes into play is the shape of the buckling modes at different load levels. At the stage of very low load levels the mode shape is identical to the linear eigenmode and can be described by a single product of two sine-functions. A contour plot (Fig. 9.5) shows elliptical contour lines at this point. If the load is increased the crest of the buckling wave flattens which leads to mode shapes which are often described as bath tub shaped and are visualized in Fig. 9.6. A comparison of the average strains in longitudinal and transversal direction is 0 given in Figs 9.7 and 9.8, respectively . With an amount of N x = −6 the load level is roughly 1.5 times the linear buckling load which is an important design load in aircraft design, as limit load is often set to the linear buckling load and ultimate load is 150% of limit load. Figure 9.7 shows that the average strain εa is qualitatively


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similar while being constant for the present approximation and only slightly varying in case of PBUS. The relative deviation amounts to approximately 10%. For the average strain εb depicted in Fig. 9.8 the qualitatively behavior is similar again, but the deviation at the peaks is relatively high and amounts to about −30%. In Fig. 9.9 the average longitudinal strain at the middle of the shell (η = 0.5) is given in dependence of the applied load. As it could be expect from Fig. 9.7 the agreement between the two methods is satisfying, while the structural behavior simulated by PBOS is stiffer.

9.5 Conclusions In the present contribution an algorithm of high computational efficiency for the estimation of the postbuckling behavior of shallow orthotropic shells was presented. As it can be seen from the comparison with more detailed analysis methods the results are in excellent agreement for the flat plate. With decreasing radius the irregular deflection pattern of the shell is only simulated in an averaging sense by the present method. While the average strains in longitudinal direction are approximated with satisfying accuracy up to a load level of 1.5 times the linear buckling load, the strains in circumferential direction show large deviations in the peak regions of the deflection. Returning results in milliseconds while being capable of tracing the main structural behavior up to the early postbuckling region, the developed algorithm is well suited for the fields of preliminary design and optimization.

References 1. Bürmann, P.: A semi-analytical model for the post-buckling analysis of stringer- and framestiffened cylindrical panels under combined loading. Ph.D. thesis, Technische Universität Braunschweig (2006) 2. Byklum, E.: Ultimate strength analysis of stiffened steel and aluminium panels using semianalytical methods. Ph.D. thesis, NTNU Trondheim (2002) 3. Diaconu, C.G., Weaver, P.M.: Approximate solution and optimum design of compressionloaded, postbuckled laminated composite plates. AIAA Journal 43(4) (2005) 4. Diaconu, C.G., Weaver, P.M.: Postbuckling of long unsymmetrically laminated composite plates under axial compression. International Journal of Solids and Structures 43, 6978–6997 (2006) 5. Donnell, L.H.: Stability of thin-walled tubes under torsion. NACA Report 479 (1933) 6. Flügge, W.: Stresses in shells, 2nd edn. Springer, Berlin (1973) 7. Jaunky, N., Knight, N.F.: An assessment of shell theories for buckling of circular cylindrical laminated composite panels loaded in axial compression. International Journal of Solids and Structures 36, 3799–3820 (1999) 8. Koiter, W.T.: A consistent first approximation in general theory of thin elastic shells. the theory of thin elastic shells. In: Proceedings IUTAM Symposium, pp. 12–33. Delft (1959) 9. Kollar, L., Springer, G.: Mechanics of composite structures. Cambridge University Press, Cambridge (2003)

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10. Love, A.: A treatise on the mathematical theory of elasticity, 4th edn. Dover Publication, New York (1927) 11. Marguerre, K.: Die mittragende breite der gedrückten platte. Luftfahrtsforschung 14 3, 121– 128 (1937) 12. Mittelstedt, C., Schröder, K.-U.: Postbuckling of compressively loaded imperfect composite plates: closed-form approximate solutions. International Journal of Structural Stability and Dynamics 10, 761–778 (2010) 13. Reddy, J.: Mechanics of laminated composite plates and shells, 2nd edn. CRC PRESS, Boca Raton et al. (2004) 14. Sanders, J.: An improved first-approximation theory for thin shells. NASA-TR-R-24 (1959) 15. Shin, D.K., Griffin, O.H., Gürdal, Z.: Postbuckling response of laminated plates under uniaxial compression. International Journal of Non-Linear Mechanics 28, 95–115 (1993) 16. Simitses, G.J., Shaw, D., Sheinman, I.: Stability of cylindrical shells, by various nonlinear shell theories. Zeitschrift für Angewandte Mathematik und Mechanik 65, 159–166 (1985) 17. Wiedemann, J.: Leichtbau - Elemente und Konstruktion, 3rd edn. Klassiker der Technik. Springer, Berlin et al. (2007) 18. Zou, G., Qiao, P.: Higher-Order finite strip method for postbuckling analysis of imperfect composite plates. Journal of Engineering Mechanics (2002)

Chapter 10

Nonlinear Magnetoelastic Waves in a Plate Vladimir I. Erofeev, Alexey O. Malkhanov, Aleksandr I. Zemlyanukhin and Vladimir M. Catson

Abstract We consider an elastic plate made of conductive material in an external magnetic field. It is shown that the distribution of intense waves in such a system can be described by nonlinear evolution equations, combining the well-known Khokhlov–Zabolotskaya–Kuznetsov and Kodomtsev–Petviashvili model equations. The features of the propagation of two-dimensional nonlinear magnetoelastic waves are analytically and numerically analyzed. Keywords Magnetoelasticity · Elastic waves · Magnetic field · Plate

10.1 Introduction As a scientific direction magnetoelasticity formed in the mid-50th of the XXth century at the intersection of solid mechanics, electrodynamics and acoustics. The first works were initiated by the problems of geophysics—the need to describe the wave dynamics of the deep layers of the Earth taking into account its electrical conductivity and interaction with the geomagnetic field [1]. Since that time, the dynamic V. I. Erofeev (B) Nizhny Novgorod Branch of Blagonravov Mechanical Engineering Research Institute of RAS, Belinsky St. 85, 603024, Nizhny Novgorod & Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia e-mail: [email protected] A. O. Malkhanov Nizhny Novgorod Branch of Blagonravov Mechanical Engineering Research Institute RAS, Belinsky St. 85, 603024, Nizhny Novgorod, Russia e-mail: [email protected] A. I. Zemlyanukhin · V. M. Catson Saratov State Technical University, Saratov, Politekhnicheskaja St. 77, 410054 Saratov, Russia e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 10, © Springer-Verlag Berlin Heidelberg 2011

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processes in the interaction of electromagnetic and strain fields are being actively studied. This is associated with numerous physical, technical and technological applications. Among these problems is the problem of strength of structures and mechanisms operating in conditions of strong magnetic fields. The inclusion of fields of different physical nature in mechanical systems opens new possibilities for the development of engineering and technology. Magnetoelasticity effects are manifested or in strong magnetic fields created when the load can significantly affect the wave and dissipative properties of the medium [2], or in thin bodies: rods, plates, shells [3]. The infinite magnetoelastic medium is characterized by the anisotropy of the properties which are induced by the magnetic field. The magnetic field in a medium with finite conductivity leads to an additional mechanism of dissipation. These properties of magnetoelastic systems open new possibilities for practical applications. In this paper we study the propagation of nonlinear magnetoelastic waves in a plate. Particular attention is paid to the formation of two-dimensional spatially localized waves. The problem of the magnetic field effect on the localization of the strain wave considered earlier in [4] for the one-dimensional case.

10.2 Equations of Magnetoelasticity In magnetoelasticity, the influence of the magnetic field on the deformation field is described employing the Lorentz forces Fm = ρe E + j × B,

(10.1)

which are introduced into equations of motion of an elastic body ρ

∂2 u = (λ + μ) grad div u + μ Δu + Fnonlinear + Fm ∂t2

(10.2)

Here E is the intensity of the magnetic field; j is the vector of the electric current density, B is the magnetic induction vector; ρe is the volumetric density of the electric charges; u is the displacement vector; λ, μ are the Lamé constants; ρ is the density of the material; t is the time. The force Fnonlinear includes elements which result from the consideration of elastic nonlinearity. If only the quadratic nonlinearity is taken into account, then the components of the vector can be represented through the gradients of displacements as follows % &' ( F i = μ + A4 ul,kk ul,i + ui,l ul,kk + 2ul,k ui,lk % &' ( (10.3) + λ + μ + A4 + B ul,k ul,ik + ui,l uk,lk + (λ + B) ul,l ui,kk % &' ( A + (B + 2C) ul,l uk,ki + λ + μ + 4 + B ul,k ul,ki + ui,l uk,lk

10 Nonlinear Magnetoelastic Waves in a Plate

127

Here A, B, C are the Landau constants, index after comma means differentiation with respect to the corresponding coordinate, repeating indices mean summation. From the Maxwell equations, one can obtain equations for the electric and magnetic inductions (D) and (B), respectively, ∂D ∂B ∂u c2 = rotH − j, = rot ×B + ΔB, (10.4) ∂t ∂t ∂t 4πσ which should be added by the electromagnetic equations of state j = σE,

D = εe E,

B = μe H

(10.5)

These equations should be solved together with Eqs (10.1) and (10.2). Here H is intensity of the magnetic field, σ is the conductivity, εe is the permittivity and μe is the magnetic conductivity, c is the speed of light in vacuum. In magnetoelasticity, both the biasing current (∂D/∂t = 0) and the density of free electric charges (ρe = 0) are neglected. Due to this, equations of magnetoelasticity can be written as follows ∂2 u 1 (rot H × H) , = (λ + μ) grad div u + μΔu + Fnonlinear + 4π ∂t2 ∂H ∂u c2 = rot ×H + ΔH. ∂t ∂t 4πσ

ρ

(10.6)

We assume that the external constant magnetic field with the intensity H0 is transverse to the direction of wave propagation. The total magnetic field consists of its permanent value and the perturbations that result from the interaction with the strain H = H0 n + h,

(10.7)

where h is a small disturbance of the magnetic field, n is the normal vector. Further, we consider the propagation of longitudinal waves in nonlinear elastic homogeneous plate in an external magnetic field (Fig. 10.1). In this case a small perturbation and the total magnetic field vectors are as follows: % & % & h = h x , h y , h z , H = h x , h y , H0 + h z . (10.8) As is well known [3], the idea of bringing three-dimensional equations of elasticity theory to two-dimensional equations of equilibrium or dynamics of the plates is to express the quantities describing the stress-strain state at any point of the body, through the new values, which vary along the medial surface of the plate. The transition from an infinite number of degrees of freedom in the normal direction x⊥ = (x2 , x3 ) to a finite number of degrees of freedom (finite number of modes) is performed by approximating the displacement with polynomials. It is usually in the powers of the transverse coordinates, and as a small parameter serves the

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Fig. 10.1 Plate in a magnetic field

relative thickness of the plate 2kz h, where h is the half-thickness of the plate, kz is the normal component of the wave vector. If ωh < cl , (cl – propagation velocity of longitudinal elastic waves in an infinite medium, cl = (λ + 2μ)/ρ) the following approximation of the displacement is valid [6] λz u1 = u (x, y, t) , u2 = v (x, y, t) , u3 = − divV, (10.9) λ + 2μ where u, v –⎛ denote the displacement of the middle of the plate; x = x1 ; y = x2 ; ⎞ ⎜⎜⎜ u ⎟⎟⎟ z = x3 ; V = ⎜⎜⎜⎝ ⎟⎟⎟⎠. v Magnetoelasticity equations for a plate undergoing a longitudinal vibration can be written as follows 4 2 ∂2 u 2 ∂2 u % 2 2 & ∂2 v λ2 h2 ∂ u ∂4 v 2∂ u − c p 2 − cl − cτ −c − + ∂x∂y τ ∂y2 3 (λ + 2μ)2 ∂x2 ∂t2 ∂x∂y∂t2 ∂t2 ∂x 2 2 2 4 4 4 4 λ cτ h ∂ u ∂ u ∂ v ∂ v 1 ∂hz + + 2 2+ 3 + + H0 2 ∂x4 3 4πρ ∂x ⎤ ∂x ∂y ∂x∂y 3 (λ + 2μ) ⎡ ∂x ∂y 2 ⎢⎢⎢ ∂ ∂u ∂v ∂u ∂v ⎥⎥⎥ ∂u ∂2 u ∂ ∂u ∂v ⎢ ⎥⎥ = α1 + α2 ⎢⎣2 + + + + ∂x ∂x2 ∂y ∂y ∂x ∂x ∂y ∂x ∂y ∂x ⎦ ∂hy ∂h x ∂v ∂2 v ∂v ∂2 v 1 ∂hz + α3 + α4 − h + h − ; z y ∂y ∂x∂y ∂x ∂x2 4πρ ∂x ∂x ∂y 4 (10.10) 2 ∂2 v 2 ∂2 v % 2 2 & ∂2 u λ2 h 2 ∂ v ∂4 u 2∂ v − c p 2 − cl − cτ −c − + ∂y∂x τ ∂x2 3 (λ + 2μ)2 ∂y2 ∂t2 ∂y∂x∂t2 ∂t2 ∂y λ2 c2τ h2 ∂4 v ∂4 v ∂4 u ∂4 u 1 ∂hz + + + + + H0 2 ∂y4 2 ∂x2 3 ∂x 3 4πρ ∂y ⎤ ∂y ∂y ∂y∂x 3 (λ + 2μ) ⎡ 2 2 ⎢ ∂v ∂ v ∂ ∂v ∂u ⎥⎥⎥⎥ ⎢⎢⎢ ∂ ∂v ∂u ∂v ∂u = α1 + α 2 + + + + ⎢ ⎥ 2⎣ ∂y ∂y2 ∂x ∂x ∂y ∂y ∂x ∂y ∂x ∂y ⎦ ∂hy ∂h x ∂u ∂2 u ∂u ∂2 u 1 ∂hz + α3 + α4 − hz − hx − ; 2 ∂x ∂y∂x ∂y ∂y 4πρ ∂y ∂x ∂y


129

∂h x ∂ ∂u ∂v c2 ∂2 h x ∂2 h x = hy − h x + + 2 ; ∂t ∂y ∂t ∂t 4πσ ⎛∂x2 ∂y ⎞ ∂hy ∂ ∂u ∂v c2 ⎜⎜⎜ ∂2 hy ∂2 hy ⎟⎟⎟ ⎜⎝ =− hy − h x + + 2 ⎟⎠ ; 2 ∂t ∂x ∂t ∂t 4πσ ∂y ∂x ∂hz ∂ ∂u ∂ ∂v c2 ∂2 hz ∂2 hz (H0 + hz ) (H0 + hz ) =− − + + . ∂t ∂x ∂t ∂y ∂t 4πσ ∂x2 ∂y2 2 0 % &2 1 Here c p = cl 1 − c2l − 2c2τ /c4l is the velocity of the longitudinal wave extending in the plate in the absence of geometric dispersion and interaction with the magnetic field; cτ is the propagation velocity of elastic shear waves in an infinite medium; cτ = μ/ρ.

10.3 Beam of Longitudinal Waves Propagation Consider the propagation of the beam longitudinal waves along the x axis, restricting the area in which the parameters of nonlinearity, dispersion and diffraction of the same order (∼ ε). Suppose that the beam of waves is a limited, slightly divergent and close to a plane wave. Let us introduce the ”ray” coordinates: √ ξ = x − c˜ t; η = εx; χ = εy, (10.11) and new functions u = u; v =

√ √ √ εv; hz = hz ; h x = εh x ; hy = εhy .

(10.12)

The choice of variables in the form (10.11) and replacement (10.12) reflect the fact that, by virtue of nonlinearity, dispersion and diffraction divergence, the displacements and the components of the %magnetic field varies both along the beam propa√ & gation direction (∼ ε), and across ∼ ε . Substituting (10.11), (10.12) into (10.10) and holding down the only terms with order of ε not higher than the first, we get the evolutionary equation for axial (longitudinal) strain U = ∂u/∂ξ: ∂ ∂U ∂U ∂3 U ∂2 U ∂2 U + αU + β 3 − δ 2 + γ 2 = 0, ∂ξ ∂η ∂ξ ∂ξ ∂ξ ∂χ where α=

3−

% c &2 A

cl

% &2 % &2 c cτ 6λ2 − (λ+2μ) + cpl g 2 1 − cl % &2 % &2 − 1, c 2 2 cpl + ccAl

(10.13)

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λ2 h2 β= 6 (λ + 2μ)2

% c p &2 % c &2 % c &2 + cAl − cτl cl % c &2 % &2 , 2 cpl + ccAl

c2 8% & % & , 2 cp 2 8πσc2l + ccAl cl % &2

δ=

% &2 % &2 c 1 + ccA − ccτ 2 − cp l l l γ = % &2 % &2 % &2 % &2 % &2 , cp cp cA cA 2 2 cl + cl + cl − ccτl cl g=

2A + 6B + 2C , ρc2l

c˜ =

8 c2p + c2A ;

8 : here cA = H02 4πρ is the Alfven wave velocity. Equation (10.13) generalizes the well known model equations of Khokhlov– Zabolotskaya–Kuznetsov and Kadomtsev–Petviashvili, which follow from (10.13), when β = 0 and δ = 0 respectively [7]. To obtain an analytical solution “by simplest equations method” [8] we take as a simple Riccati equation Yz = −Y 2 + aY + b, (10.14) which has a solution in the following form √ 1 4b + a2 1 Y(z) = a + tanh 4b + a2 (z + C2 ) , 2 2 2

(10.15)

where a and b define the later of the over determined system of equations. Equation (10.13) is of the order of singularity, which is equal to two, so the solution of the equation will be sought in the form of: U (z) = A0 + A1 Y + A2 Y 2 + B1

Y z

Y

+ B2

Y 2 z

Y

,

(10.16)

here z = k0 ξ + k1 χ + k2η, Ai , Bi – arbitrary constants. In view of (10.14) we obtain: U (z) = (A2 + B2 ) Y 2 + (A1 − B1 − 2aB2) Y + A0 + aB1+ % & bB1 + 2abB2 B2 b2 +B2 a2 − 2b + + 2 Y Y

(10.17)

Substituting (10.17) into (10.13) and equating terms with the same degree of Yz , we obtain a system of equations, because of the cumbersome nature of which we show here only the first two equations: %

&4 1 αB22 + 6βB2 2 %

= 0,

& 14βB2 α + αB1 B2 + 2βB1 + 2aαB22 + 2δB2 b3 = 0 ...

(10.18)


131

Solution of the system is as follows: A2 = −

12β 1δ , a = − , A0 = A1 = B1 = B2 = b = 0. α 5β

Then the exact solution of Eq. (10.13) takes the form of a shock wave: 2 3 δ2 k0 ξ + k1 χ + k2 η U (ξ, χ, η) = − −1 + tanh , 25 αβ 2

(10.19)

(10.20)

5βγ k2

6 δ where is k0 = 15 βδ , k2 = − 125 + δ 1 , k1 is an arbitrary constant. The requirement β2 of positive definiteness of the velocity of the wave (k2 > 0) imposes a restriction on the k1 coefficient: 6 δ4 k12 > . (10.21) 625 γβ3 3

During the verification of the solution, (10.20) served as an initial condition in numerical simulations. Calculations were based on the pseudospectral semi-implicit scheme [9] with parameters: 256 × 64 – the dimension of the grid; Δξ = 0.25 – step length along the ξ axis, Δχ = 0.25 – step length along the χ axis, Δη = 0.003 – step length along the η axis. The evolution of the perturbation is shown in Fig. 10.2. The shock front propagates without disturbances in the positive direction of ξ, which indicates the reliability of the resulting analytical solutions. Also in the numerical experiment was considered another form of initial conditions: 2 ξ − 32 U0 (ξ, χ) = 12sech sech (χ−8) (10.22) 4 The dome of the perturbation (Fig. 10.3) is moving forward on ξ, while spreading to the sides of χ, the latter effect dominates. With the passage of “time” η is closer to the borders of the perturbation amplitude grows to a certain limit (U0 = 7.1), constantly spreading outward and moving forward, which leads to cross-structures

Fig. 10.2 Transmission of the shock wave (α = −1, β = 1, γ = 1, δ = 1)

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Fig. 10.3 Initial disturbance for the numeric experiment

Fig. 10.4 Result of the evolution of initial disturbance with dissipation (η = 5)

Fig. 10.5 Result of the evolution of initial disturbance without dissipation (η = 5)

(Fig. 10.4 and 10.5). The height of the cross in the maximum point equals to two (U 0 = 2) (Fig. 10.4). Figure 10.5 shows the evolution of the initial disturbance in the equation of Kadomtsev–Petviashvili. From a comparison of cross-structures in Fig. 10.4 and 10.5 that the evolution of the initial momentum for both equations is qualitatively the same character – the spreading of the sides and the formation of cruciform


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Fig. 10.6 Dependence of the beam amplitude on the external magnetic field intensity

Fig. 10.7 Dependence of the beam widths along axes on the external magnetic field intensity

structures. The only difference is decreasing amplitude and increasing blurring of the perturbation due to dissipation (δ 0). Let us trace the change in amplitude and width of the wave beam along each coordinate, depending on the external magnetic field. Defining the arbitrary constants as follows: √ 7 δ2 1 δ3 k1 = , k2 = (10.23) 25 γβ3 125 β2 we obtain expressions for the amplitude and width of the beam along axes: 3 δ2 10β 250β2 50 γβ3 A=− , Δξ = , Δη = , Δχ = √ (10.24) 25 αβ δ δ3 7 δ2 For condensed matter in magnetic fields with strength up to 10 Tl Alfven wave velocity is less than the propagation velocity of longitudinal waves [5], so changing the parameters investigated in the 0 ≤ cA/cl < 1. Practical interest are the same values in the interval 0 ≤ cA/cl < 0.3. Figure 10.6 shows the variation of the amplitude of the wave beam. It is seen that with increasing external magnetic field amplitude decreases. At the same time, the width of the beam along each of axes increases with increasing magnetic field (Fig. 10.7).

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10.4 Conclusion As a result, analytical studies and numerical simulations demonstrated the possibility of formation of intense space-localized magnetoelastic waves in a plate. The dependencies of wave parameters (amplitude, velocity, width) on the external magnetic field, showing that using a magnetic field can control the characteristics of localized waves. Acknowledgements The work has been supported by the Russian Foundation for Basic Researchers, grant No 09–08–00827 and No 09–08–00188.

References 1. Knopoff, L.: The interaction between elastic waves motion and a magnetic field in an electric conductor. J. Geophys. 60, 441–456 (1955) 2. Bagdasarian, G.E., Danoyan, Z.N.: Electromagnetoelastic Waves. Yerevan State University, Yerevan (2006) (in Russian) 3. Ambartsumian, S.A., Bagdasarian, G.E., Belubekyan, M.V.: Magnetoelasticity of Thin Shells and Plates. Nauka, Moscow (1977) (in Russian) 4. Erofeyev, V.I., Malkhanov, A.O.: Localized magnetoelastic waves formation. Int. Rev. of Mech. Eng. 4, 581–585 (2010) 5. Erofeyev, V.I.: Wave Processes in Solids with Microstructure. World Scientific, Singapore (2003) 6. Erofeyev, V.I., Potapov, A.I., Soldatov, I.N.: Nonlinear Waves in Elastic Bodies with Spatial Dispersion. Gorky Univ. Publ., Gorky (1986) (in Russian) 7. Dodd, R.K., Eilbek, J.C., Gibbon J.D., Morris, H.C.: Solitons and Nonlinear Wave Equations. Academic Press, London (1984) 8. Kydryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons and Fractals. 24, 1217–1231 (2005) 9. Press, W.H., Teukolsky, S.L., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. The Art of Scientific Somputing. Cambrige University Press, Cambridge (1992)

Chapter 11

Basic Concepts in the Stability Theory of Thin-Walled Structures Ardeshir Guran and Leonid P. Lebedev

Abstract Numerous specialized books and papers have been written about the subject of stability in mechanics. Most of these concentrate on methods for obtaining critical values of certain parameters and typically contain algorithms and graphs generated for describing important but very specific problems. In the present paper we take a step back and discuss the truly central notions regarding mechanical stability. Our intention is to treat the required concepts on a fairly elementary level, while simultaneously offering a bit of useful historical perspective. We also attempt to explain some common discrepancies between theoretical and practical results. Keywords Stability · Thin-wall structure · Shell · Plate

11.1 Introduction The terms “stable” and “stability” are somewhat overused. In everyday life we call various processes stable when they exhibit steady, periodic, or closely self-similar behaviors with respect to changes in certain parameters such as those of loading. The term “loss of stability” is associated with qualitative changes in processes or state behaviors. In this way, the notion of stability is similar to the mathematical notion of continuous dependence on time or some other parameter. Effects related to loss of stability are given special designations like “phase transition” in physics. Each natural science (biology, geology, chemistry) employs its own terms for loss A. Guran (B) Institute of Structronics, Julius Raab Straße 10, A-4040 Linz, Austria & Abteilung Robotik, Johannes Kepler University, Altenberger Straße 69 A-4040 Linz, Austria e-mail: [email protected] L. P. Lebedev Universidad Nacional de Colombia, Cr. 45, # 2685, Bogotá D.C., Colombia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 11, © Springer-Verlag Berlin Heidelberg 2011

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of stability with regard to its particular objects of study. The same can be said for the various branches of engineering. Because valid mathematical descriptions take the form of precise statements involving equations or inequalities, however, we must lay down more precise explanations for the terms mentioned above. No universal definition, acceptable in all circumstances, is available. We shall consider only some aspects of stability theory in mechanics. In particular, we shall confine ourselves to the theory of thin-walled structures. These structures play important roles in engineering practice. Structural stability is often assumed in civil engineering design. On the other hand, instabilities are easily produced and can severely damage a structure. Buckling under load represents a typical example of loss of stability for thin-walled structures such as plates and shells. The buckling of a compressed column is a phenomenon with which every engineering student should be familiar. It is, of course, easily replicated using an ordinary plastic or metal ruler, and was first studied mathematically by Leonhard Euler (1701–1783) some 250 years ago. Although buckling can be catastrophic when it occurs on the body of a submarine, the same phenomenon provides the operating principle for the ubiquitous thermal switch. So the requirements of engineering practice have necessitated further development of stability theory by both engineers and mathematicians. Scientific notions seldom develop in a simple or straightforward manner. Stability has been studied for many years, and we still do not fully understand effects as simple as the buckling of a spherical cap under pressure. The major stability criteria and associated mathematical apparatus remain topics of active research interest in mechanics. Since mechanics depends heavily on the tools of calculus, its manner of approach to stability considerations was inherited largely from that subject. Only so much guidance could be borrowed directly from mathematics, however; without the incorporation of a clear engineering-level understanding based on practical considerations, it would have been impossible to construct sensible tools for predicting loss of stability. Meanwhile, mathematicians can only process the information that engineers have supplied. Furthermore, this processing can occur only with the benefit of tools that were constructed throughout the long history of mathematics — much of which preceded the advent of modern engineering practice. The stimulus provided to mathematics has been considerable though. Throughout the history of science we see a wonderful symbiosis between mathematics and real life. We shall restrict our discussion to the main ideas and methods associated with stability (and instability) in one of the oldest of the natural sciences: mechanics. In particular, we will concentrate on applications within the theory of elasticity (cf., also [2, 3, 20, 30]).

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11.2 Some Historical Remarks The use of mathematical tools in mechanics is so intensive that it is possible to regard mechanics as a branch of mathematics.1 But the modern era has seen at least some divergence between the modes of investigation used in the two subjects. In particular, the development of mechanics is not tied to a strictly axiomatic approach; rather, the changing requirements of actual engineering practice continually come into play as we noted above. Here we would like to discuss those stability-related notions that have evolved within mechanics. It was from these notions that certain portions of mathematics (e.g., stability theory for differential equations, branching of solutions to nonlinear equations in Hilbert spaces) were able to draw significant developmental inspiration. The first serious advancement in the understanding of stability for elastic structures was made by Euler [9] during his work on axially compressed rods called elastica. In the framework of statics, the stability problem can be treated only via nonlinear equations. Euler considered the case for which the degree of compression present at the rod ends is increased in proportion to a parameter. Until some “critical value” of the parameter, now known as Euler’s value, is reached, the rod must remain straight. Beyond this value the rod can assume two other symmetric equilibrium shapes. The closer the loading parameter is to Euler’s value, the closer the buckled shape stays to rectilinear. Thus, at the critical value itself we observe three possible solutions; furthermore, the dependence on the loading parameter is continuous (with respect to some norm). Such a point is called a point of branching. In experiments one commonly observes a buckled shape, hence Euler reasonably supposed that his critical point represents the least axial compression above which the straight shape is unstable. The total energy for the buckled state is less than that for the straight shape. Later work by Lagrange (1736–1813) [35] and Dirichlet (1805–1859) [8] paved the way for judging stability of the buckled state under a compression that exceeds Euler’s critical value. In view of the continuous dependence of both the stress and displacement fields on the load parameter near the critical point, it makes sense to invoke the tools of calculus. Suggesting itself here is an approach based on linearization of the problem about the solution corresponding to the straight rod shape. Euler was thus able to pose a linear boundary value problem having nontrivial solutions at certain critical values of the load parameter, the smallest of which was his original critical value. In this way Euler introduced an early eigenvalue problem in the theory of ordinary differential equations. His method was subsequently extended to other, more complex problems of elastic stability, but remained substantially the same. In engineering practice Euler’s critical points, which are found for certain equilibrium states when we know the main state of a structure, are associated with loss of stability. This is confirmed by experiment: as a rule, when we increase a load well above its critical value the structure in question will fail. In abstract equations, however, branching 1

We might mention that in Russian universities it is common for divisions of mechanics to thrive within mathematics departments.

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solutions need not deviate greatly from the main branch, so it is possible that a branching point could be neglected. We may be able to calculate Euler’s critical value then, but the situation reminds us of that for the critical values of an ordinary function in calculus: we set the first derivative to zero as a necessary condition in order to identify candidates for maxima and minima, but must do additional work in order to clarify the real nature of these points. It is clear that branching tools alone can tell us nothing about the stability or instability of a structural state. In engineering practice we are interested not only in the behavior of a structure in its ideal equilibrium state, but also under random external loads that vary (slowly or rapidly) with time. It is common for designers to rely on the fact that the state of a good stable structure will deviate only slightly from equilibrium when subject to light excitation. Unfortunately the picture is very different for states that lie close to instability. So it is not surprising that a study of unstable states should be based on dynamical considerations. Classical mechanics is divided two areas — statics and dynamics — and the resulting methodological subdivision was inherited by the theory of elasticity. Euler’s linearization method pertains to statics. We shall now consider the stability problem from the dynamical point of view. One of the oldest and most interesting stability problems is that for the motion of the Earth around the Sun, taking into account the influence of the Moon and other planets. A simple approximation, and one still not fully treated, is the so-called three-body problem. So we consider the stability question for a system of three point masses moving purely under their mutual gravitational forces. We wish to learn whether the Earth’s trajectory, which has been watched closely throughout history, will remain the same forever under this model. This means we must consider semiinfinite time intervals and other abstract notions that violate our understanding of the processes by which the Sun is slowly dying. Nonetheless, this approximate model could tell us something about the Earth’s future. Lagrange proposed a definition for stability of motion, relevant in cases where small changes in external conditions cause only finite changes in the motion of a system. His notion was appropriate for certain types of motion in mechanics. The final form of definition of dynamic stability was developed by A.M. Lyapunov (1857–1918) [42, 43]. It is the stability with respect to the initial data of the motion. This type of the definition is determined by existence of resonance frequencies in elastic structures. The resonance is wellknown in bridge design, since it was quickly found that a comparatively small group of soldiers marching across a bridge could cause a collapse (whereas the bridge could survive a much larger static load). The resonances of linear and nonlinear oscillating systems are well-known. Their existence predetermined that the main definitions for stability and instability in dynamics would differ significantly from those used in statics. Let us consider why the phenomenon of resonance brought other notions into dynamic stability. A simple but instructive mechanical system consists of a point mass m, attached to a spring having stiffness coefficient k, and that oscillates under a harmonic force F = f cos ωt. The equation of motion is m x(t) ¨ + kx(t) = f cos ω0 t.


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√

If f = 0, we observe simple harmonic oscillations at frequency ω0 = k/m. It seems reasonable to consider the stability of such oscillations, using the same idea as for statics: if the system is stable, then small perturbations in the external forces should bring small perturbations to the motion. Can we apply this successfully to our present system? When ω ω0 , a small f will give rise to small perturbations in amplitude. When ω comes close to ω0 , however, these perturbations will begin to grow, and when ω = ω0 the solution will contain a term proportional to t cosω0 t. This term will grow infinitely with time, regardless of the magnitude of f . The same thing occurs with any elastic structure in dynamics: there exist resonant frequencies at which an arbitrarily small external force can excite growing oscillations. This effect can be explained in terms of energy that accumulates in the system during each cycle of the oscillation. Of course, we rarely see oscillations driven toward infinite amplitude! In fact, when a certain amplitude is reached the linear model will no longer apply; furthermore, in any real system friction will extract energy from the system. For many elastic systems, frictional forces reduce forced resonances; this is why we could use dynamics tools to study stability. If we had a rigorous understanding of friction, we might be able to consider the problem of dynamic stability in the same way as for statics (i.e., through the change of state of a system under the perturbation of forces or other system parameters). Unfortunately this is not the case: no good universal model exists to describe the myriad real effects covered under the term “friction.” Models of a very restricted nature, when coupled with the equations of elasticity, yield complex boundary value problems that were considered intractable until very recently (with the advent of digital computation). So the ideas of dynamic stability were developed around overly-idealized elastic structures, leading ultimately to a dead end. Progress could continue only through consideration of particular types of stability of motion. For the three-body problem, for example, one can consider stability of motion under small perturbations in the initial conditions. This kind of stability is called Lyapunov stability, after A.M. Lyapunov (1857–1918) who developed its principal definitions and methods. Lyapunov’s notions spread quickly throughout dynamics and are now considered the keys to understanding stability in a great many circumstances. Lyapunov stability is applicable to elastic structures under various types of loading. In many cases static stability of some states of elastic structures is confirmed within the framework of Lyapunov’s ideas, and so much of Euler’s theory of stability was eventually confirmed for many elastic structures. This was first discovered by Movchan [45, 46] for some particular elastic systems under potential forces. However, the stability of systems under non-potential loads can be studied only on a dynamical basis.

11.3 Notions and Definitions for Stability in the Theory of Thin-Walled Structures So what steps must we take in order to study stability or instability of a given body, process, or state?

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1. We must construct a model adequate for predicting loss of stability. We must establish what we mean by stability, and present our selected notion in mathematical form. 2. We must establish necessary conditions for stability and instability of a process or state of a body, and develop methods for checking these. This includes tests for stability and instability of solutions to boundary value problems. 3. We must establish sufficient conditions for stability and instability. In addition, we must provide some means for practical selection of stable and unstable processes or states. Finally, we must furnish guidelines for theoretical and experimental studies that can generate reliable results in the area. The term “loss of stability” has acquired its meaning gradually, and through several mechanisms: (a) engineering intuition and qualitative observations made on the behavior of structures; (b) the influence of existing mathematical tools for researching the behavior of structures; (c) experimental procedures that ultimately confirmed the applicability of available methods. All three steps are equally important. Typically, however, textbooks confine themselves to Steps 2 and 3 (and only that portion of the latter which can be mathematically formalized). The remainder would require the kind of “fuzzy” explanation that is normally excluded from textbooks. Let us touch on a few aspects of this. Adequate models. One example of a stability problem in mechanics is the famous problem of turbulence for the flow of a liquid through a pipe. At some speed, the highly-organized laminar flow pattern is replaced by the seemingly chaotic pattern characteristic of turbulent flow. The latter is not as efficient for transportation of the liquid, and is therefore more expensive. At this time there is no clear confirmation that the Navier–Stokes model for a viscous liquid can adequately describe turbulence or how it arises, and so it seems that Step 1 is crucial for the theory of stability of liquid motion (possibly much more so than the techniques of Step 2). The situation is much better for an elastic ruler under axial compression. Precise experiments show that Euler’s elastica model is appropriate for studying stability of a compressed straight ruler, and his linearization technique seems appropriate for finding critical loads for elastic structures. However, we can be in doubt whether the elastica model for studying stability applies to all cases involving real solids. For example, we may consider a buckled spot on a tin can produced by fingertip pressure; since the spot does not automatically disappear upon removal of the pressure, it is not so clear whether we can exploit Euler’s theory to describe this situation within the framework of elasticity theory. (The reader should understand that a simple tin can is prestressed by its mode of production; some portions are in plastic states, so the question whether elastic theory would apply is not simple.) Next, restricting ourselves to elastic structures, we note that linear models of 3-D bodies, beams, plates, and shells cannot reveal anything about stability. Indeed, if as is common in stability theory for elastic structures, we increase the load in proportion to some parameter, then the solution to a linear problem will also be proportional to this parameter. There will be no indication of any loss of stability. So we must employ nonlinear models for static studies of instability effects.


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However, various nonlinear equations can be formulated for the same real object. They differ in both their degrees of accuracy and their possibilities for predicting certain effects. We must select the simplest one that can still describe the effects we wish to study. This stage of investigation is quite informal, since we need to evaluate the possibilities of a model without the benefit of calculations. Stable and unstable states and processes. Sometimes it is easy to distinguish between stable and unstable states visually, but not to quantify this difference in mathematical terms. Such is the case with turbulent flow. Here, the only way to determine stability is to use the dynamics of the process under disturbances in some adjustable parameters. However, the independent time variable adds another dimension to the problem and this can increase the difficulties associated with numerical solution. For this reason, practitioners use static tools whenever possible. Euler’s method of finding critical points. The buckling of a ruler under axial compression gives us a good example of when we can avoid considering process dynamics. We recall what happens in this example. Until some critical value of compression is reached, the ruler maintains its straight shape; any deflection produced by a small normal force disappears as soon as the force is removed. When the critical value is exceeded, the straight state becomes unstable; any transverse force will deflect the ruler into a curved state. This change occurs dynamically and the resulting curvature can be significant. But for compression just above Euler’s critical value, the change in shape happens relatively slowly and could be regarded as quasistatic. This is why we can apply static tools in this case. Here the critical point is a point of branching of solutions, and can be found through the application of functional analytic methods. Since the picture of the loss of stability for a ruler was extended for application to some shapes of plates and shells, we should devote special attention to the location of critical points in given parameter ranges for certain main states of an object. The method developed by Euler will yield only critical points on some main branch of a solution. The next question is whether these correspond to loads at which loss of stability can occur. So we consider how to find a critical point on a branch of solutions which depends on some set of parameters. A boundary value problem for an elastic object can be written as an equation G(x, ω) = 0

(11.1)

in an unknown variable x ∈ X, where X is a Banach space. The nonlinear operator G takes values in a Banach space Z. The parameter ω, characterizing the load on the object or some other significant variable, is an element of another Banach space Ω. Its values are usually considered to be proportional to a numerical parameter λ, and we shall proceed with this understanding. For stability, we check a branch of solutions defined by the equation x = x(λ). In these terms, equation (11.1) takes the form F(x, λ) = 0. To get the equation for the branching point we introduce a new unknown y by x = x(λ) + y and try to find the values λk of λ where there arise new branches of solutions y = y(λ) to the equation F(x(λ) + y(λ), λ) = 0. In this, we assume that the solutions depend continuously on λ, and so at a critical point

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y(λk ) = 0. Linearizing the equation F(x(λ) + y(λ), λ) = 0 with respect to y at the point λ, we get F(x(λ) + y, λ) = B(x(λ), λ)y + N(x(λ), y, λ) = 0,

(11.2)

where B is a linear operator in y, and N, the nonlinear part of F, is small with respect to y near the branching point λ; that is, N(x(λ), y, λ) = o (y). The operator B(x(λ), λ)y is the Gâteaux derivative of F at the point (x(λ), λ). Consider the problem B(x(λ), λ)y = 0, (11.3) supposing that B(x(λ), λ) is a bounded linear operator with respect to y that depends on a parameter λ. This equation has a trivial solution for any λ. For an axially compressed ruler, this zero solution corresponds to the straight ruler shape. Let us find values of λ for which this equation has nontrivial solutions. For many (but not all) problems of mechanics, this is possible only for a discrete set of values for λ which are called critical values or eigenvalues. The corresponding nontrivial solutions are called eigenfunctions or eigensolutions. Usually we consider loads that increase in proportion to λ, and so there is a smallest critical value that we call Euler’s critical value. For a ruler, it can be shown that this is the branching point for solutions, and that for λ less than Euler’s value there are no nontrivial solutions y for the complete nonlinear problem. The linear branching equation cannot give us the amplitude of the branching part of the solution. So special techniques were developed to find after-critical solutions (we will not pause to consider this). We only note that for the problem of buckling of a plate having general shape and compressed in its midplane, the results mentioned above for the ruler have been generalized (Vorovich [55–57]): there is an Euler value for compression below which there are no buckled equilibrium shapes, and from this critical load we see branching of solutions. The existence of Euler’s critical load has also been established for momentless solutions for shallow shells. It turned out that for some shells, however, there exist other types of solutions under loads “less” than the critical value. When there is a stable nontrivial state for a shell that is not subjected to loads, the shell is called non-rigid (Vorovich [55–57]). Such solutions cannot be found through direct application of the methods of branching theory. Thus we know that for thin-walled structures there is a smallest critical point of the loading parameter λ from which solution branches emanate. We must still ask which of the branches correspond to real shapes of the structure, and hence which of the solutions are stable. It is clear that we cannot answer these questions by using only static tools, but need to bring in dynamic considerations. Energy method. In classical mechanics, results on stability have been established — in large part by Lagrange and Dirichlet. They demonstrated that for an equilibrium state of a system of mass points under conservative load, at which the total potential energy assumes its minimum value, the trajectories beginning from some small neighborhood of the equilibrium point with sufficiently small velocity remain in some small neighborhood of that point. Thus this equilibrium can be considered stable.


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Although elastic systems under potential loads belong to another class of systems, it was thought that the same idea could be applied. This happened because stability results were obtained for linear problems of elasticity that were quite close to those for classical mechanics. So the idea that minimum total energy corresponds to a stable equilibrium solution took on a life of its own. This permits us to avoid detailed considerations of the dynamics of various motions. When each of the solutions is stable by the energy criterion, we need to select which of them is more probable. The branch for which the total potential energy of the solutions is less is regarded as more probable. For a long time this viewpoint on stability was supported by direct experimentation with rods and some plates, and by attempts at calculating the dynamics of certain problems. The tools of the calculus of variations and functional analysis allowed us to justify this method, extending the Lagrange–Dirichlet theorem on stability of finite-dimensional systems to some problems of nonlinear elasticity, including those for plates and shells. The minimality of the total potential energy (for conservative loads) at an equilibrium point can signify stability of equilibrium, whereas its non-minimality can signify instability. There may be systems for which we do not see branching of solutions, but we can still use the energy method to find points at which loss of stability occurs. So the energy method is not a simple supplementation of Euler’s linearization method. It is worth repeating that the energy method of testing an equilibrium solution for stability is done in the static mode, but its roots lie in the dynamics of processes. Lyapunov’s definition of stability. The most sensible of the tests for stability of some system equilibrium or motion involves checking what happens with the system over time after some disturbance in the load or other variable parameter. We saw that even for a linear elastic system, because of resonance effects, we cannot consider stability with respect to disturbances in forces. This brings us to a special stability test developed by Lyapunov. Although it is not appropriate for all types of problems, the associated ideas are versatile and can be used in many circumstances. Lyapunov introduced a few main definitions to describe the stability of a solution to a system of ordinary differential equations (cf., Merkin [44]). These could be seen as defining continuity of this solution with respect to the initial data (which could be regarded as the system input) when the solution is considered as a whole on a semi-infinite time interval. Namely, he studied the stability of a solution y = y(t) ∈ IRn of a Cauchy problem y (t) = F(y(t),t),

y(t0 ) = y0

(11.4)

with a sufficiently smooth function F. The main definition states that the solution is stable if for any ε > 0 there is a δ = δ(ε) > 0 such that for any ξ that satisfies ξ < δ the solution yξ (t) to this equation with initial value yξ (t0 ) = y0 + ξ satisfies the inequality yξ (t) − y(t) < ε, where the norm is a standard norm in IRn . If this definition is not fulfilled — that is, if for some ε we cannot find a corresponding δ with the needed properties — then the solution is called unstable (in the sense of Lyapunov).

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Let us consider a solution to the equation as an element of the normed space C(IRn ; t0 , ∞) of functions that are continuous on [t0 , ∞) , and treat the problem as the result of the action of an operator from the set of initial conditions y0 ∈ IRn to the set of solutions y = y(t) ∈ IRn ; t0 , ∞). Now the definition of a stable solution in the sense of Lyapunov coincides with the way in which we define the continuity of an operator at a point in functional analysis. In addition, Lyapunov introduced the notion of asymptotic stability of a solution to the above equation. A solution y = y(t) is asymptotically stable if it is stable in the above meaning and yξ (t) − y(t) → 0 as t → ∞. In the above terms, this means that the difference yξ (t) − y(t) belongs to the subspace of C(IRn ; t0 , ∞) whose elements x(t) satisfy x(t) → 0 as t → ∞. Note that we could introduce a special requirement on the order of convergence to zero of yξ (t) − y(t) as t → ∞, and could proceed to consider various types of asymptotic stability such as exponential stability, etc. So we see that Lyapunov’s stability represents nothing more than a special notion of operator continuity. Our next issue is the practical determination of stability or instability of a solution. Here Lyapunov used two longstanding ideas from mechanics. The first one was to linearize the equation with respect to the difference x(t) = yξ (t) − y(t): that is, to consider the equation for the difference x (t) = F(y(t) + x(t),t)−F(y(t),t),

(11.5)

and linearize its right-hand side with respect to x(t). For the linearized equation Lyapunov found some conditions to insure that its solution remains small if the initial values are small. Together with the conditions under which the behavior of the solution to the linearized equation determines stability or instability of the solution of the initial nonlinear equation, these are the main parts of what is called Lyapunov’s first method for the study of stability. Lyapunov’s second method generalizes an idea contained in the proof of the Lagrange–Dirichlet stability criterion. The criterion was based on the energy conservation theorem for a system of mass points. Lyapunov proposed that we consider functions of a special type that can play the role of energy in this proof, and demonstrated how this may be done practically. Both methods turned out to be applicable to a variety of problems. An extension of Lyapunov’s notions to elastic structures presents some difficulty, because we must now deal with equations involving partial derivatives. All functions and vectors appearing in the setup for the stability problem take values in the space IRn , where, as the reader is aware, all norms are equivalent. With solutions to boundary value problems for spatial elastic structures, we encounter functions in several variables that describe the fields of displacements, strains, and stresses in the body. In this case, over a set of functions, we can introduce various norms which are not equivalent. This means we can consider different types of stability that are not equivalent when introducing different functional spaces in which we consider the continuity of solutions to corresponding problems. In the theory of shells, one of the most attractive (from the viewpoint of the justification of stability results) types of norm to use for this purpose is one defined by the quadratic portion of the elastic


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energy. Of course, other types of norms can be used for such purposes as well (see Atanacovich–Guran [3]). We have been discussing stability with respect to disturbances in the initial data. We can also consider a load disturbance, provided it has small amplitude and finite duration or finite amplitude and small duration. Such a disturbance can do work and thereby introduce additional small energy into a mechanical system. After the disturbance has ceased, its effects can be incorporated into and considered under the heading of a disturbance in the “initial data” (with respect to a new initial time). We can even consider load disturbances of infinite duration, provided that the total energy added by these is small. The three notions of stability discussed above should not be regarded as mutually conflicting. We should understand that Lyapunov’s dynamic definition of stability is the most universal. Because of the restrictions imposed by the static approach, the two other methods can show stability in the situation when in dynamics the solution is unstable. There are problems for which static approaches regularly give erroneous results, so we must clearly understand the range of applicability of each approach. Stability of Periodic Solution by Means of Poincaré map [52]. Let us consider a periodic system described by a nonlinear system of ordinary differential equations y = g(y, t),

(11.6)

where y ∈ IRm , and g(y, t) is a periodic function of time t with period T > 0, that is g(y, t) = g(y, t + T ) for any y and t. We assume that g(y, t) is a smooth function of y and a continuous function of t, ensuring existence and uniqueness of a solution with the initial condition y0 = y(t0 ) on the semi-infinite time interval t > t0 . A solution y(t) of the system (11.6) is said to be periodic if y(t) = y(t + T ) (11.7) for any instant t. It is sufficient to check condition (11.7) at an instant t = t0 . By uniqueness of the solution y(t) and periodicity of g(y, t), if y(t) = y(t0 ) = y(t0 + T ) then (11.7) holds for any t. A periodic solution can be stable, asymptotically stable, or unstable. Let us consider a map f (y), which transfers a point y = y(0) in the state space to the point f (y) = y(T ), where y(t) is a solution of the system (11.6). The map f (y) is called a Poincaré map. The Poincaré map describes the dynamics of the system over the period T . Successive application of the Poincaré map on a point y yields f k (y) = f ( f (. . . f (y) . . .)) = y(kT ), ; k times

which is the value of the solution y(t) at t = kT . The point y0 is called stationary for the map f (y) if f (y0 ) = y0 .

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The relation of the Poincaré map to the periodic system (11.6) is given by the following theorems. Theorem 1. A point y0 is stationary for the Poincaré map f (y) if and only if the solution y(t) of the system (11.6) with the initial condition y(0) = y0 is periodic with period T . The Poincaré map determines the discrete dynamical system y = f 0 (y) → f 1 (y) → . . . → f k (y) → . . .

(11.8)

where k represents discrete (integer) time. The stationary point y0 remains unchanged under the action of the Poincaré map. Points y that are close to y0 can approach or move off the stationary point under multiple action of the map f (y). Such behavior determines stability properties of the stationary point y0 for the discrete dynamical system (11.8). Definition. The stationary point y0 of the map f (y) is said to be stable if for any ε > 0 there exists δ > 0 such that f k (y) − f k (y0 ) < ε whenever y − y0 < δ for any integer k > 0. If, in addition, f k (y) − f k (y0 ) → 0 as k → ∞ then the stationary point is said to be asymptotically stable. Although the Poincaré map f (y) contains only partial information regarding the system (11.6) (using this map we cannot predict the state of the system at instant t kT ), it is sufficient to make a decision on the stability of a periodic solution to (11.6). Theorem 2. A periodic solution y(t) of the system (11.6) is stable (asymptotically stable) if and only if the stationary point y0 = y(0) of the Poincaré map f (y) is stable (asymptotically stable). Let y˜ (t) be a periodic solution of (11.6). Introducing the vector x(t) = y(t) − y˜ (t) describing the deviation from the periodic solution and using (11.6), we obtain x = y − y˜ = g(˜y(t) + x(t), t) − g(˜y(t), t)). Assuming that x is small and neglecting the higher order terms, we find x = G(t)x where G(t) =

∂g

∂y y=˜y(t)

(11.9)


147

is the m× m Jacobian matrix of the vector function g(y, t) evaluated at y = y˜ (t). As the functions g(y, t) and y˜ (t) are periodic with period T , the matrix G(t) is periodic with period T as well. The linear periodical system (11.9) represents the linearization of the system (11.6) near the periodic solution y˜ (t). Analogously, we perform the linearization of the Poincaré map y → f (y) near the stationary point y˜ 0 = y˜ (0) as x → Fx, where x = y − y˜ 0 and F=

∂ f

∂y y=˜y0

is the m × m Jacobian matrix of the mapping f (x) evaluated at the stationary point x˜ .

11.4 Why Do We See Discrepancies between Theoretical and Experimental Results on the Stability of Thin-Walled Structures? Precise experiments with buckling of spherical caps under pressure, as well with other types of shells, demonstrate good agreement with theory. But in engineering practice we see that actual critical values can differ significantly from predicted values. Why does this happen? First, conditions in practice are quite far from those of precise experimentation. What does this mean? In short, it could be said that the discrepancy in values is produced by inappropriate modeling for studying these effects. We must remember that in calculations we deal not with real objects but with their mathematical models. These can offer good descriptions in some circumstances and poor descriptions in others. Our experience with the design and real-life performance of structures is evidence of the reliability of many mathematical models. But no model is perfect under all circumstances. As a rule, the strength of a structure is described by a linear model which is crude but sufficient for the purpose. The problem of stability of a solution is more fragile. First of all, the picture of buckling for each real structure has some peculiarities that may or may not be taken into account by a model. Many results on stability of thin-walled structures are obtained by Euler’s method of linearization described above. Critical points thus found are quite sensitive to many things such as imperfections in the shape of shells, their boundary conditions, and even to the technology of production of a shell. Imperfections result in actual values significantly below the predicted ones. So much research effort is devoted to the question of how we should account for imperfections in order to obtain more realistic numerical results. Ideal models turn out to be inappropriate for finding critical loads for stability of some objects, and researchers seek more appropriate ones. Models need some robustness with respect to the critical loads they determine; small disturbances in the shape of objects, in loads, or in supports should give small deviations in critical load values. We would like to discuss why this

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remains a challenging problem for engineers and mathematicians, and what should be taken into account by a good model of a thin-walled object in a study of its stability. The points we will consider play different roles in the overall effect. The order in which we consider them does not indicate their relative importance. Imperfections. Various nonlinear models are used to describe shells in practice. In each model there is an assumed distribution of strains along the normal to the shell’s midsurface. Additional assumptions result in a boundary value problem having a relatively simple (but still cumbersome) structure. The model of a shell is an approximation to a 3-D model of the real shell; here we should understand that, as with any approximation, it takes on only some main features of the more general model. Hence it may produce results that do not agree with those produced by the more accurate model. The reasons are as follows. First, the shape of the midsurface differs a bit from its intended shape. Small deviations, if taken into account, bring new terms into the equations. This can change the critical values calculated by Euler’s method, and certain values can simply disappear. Another problem is the thickness of the shell. It enters the coefficients of the higher-order derivative terms, and therefore can have a strong influence on the critical point values. However, the imperfections we are discussing affect solutions not only in the framework of a two-dimensional shell model; more serious are their effects on the distribution of strains along the normal to the midsurface. Irregularities in the shape of a shell face frequently introduce complicated 3 − D internal stress distributions that cannot be well-approximated by a 2-D approximation. Stress concentration can lead to plasticity effects near such irregularities. It is clear that these effects can contribute to significant discrepancies between theoretical and practical values for critical loads. Stochastic studies on the influence of imperfections are important for the analysis of buckling of thin-walled structures (see Guran and Ahmadi [22], Palassopoulos [48], and Guran [11]). Prestressing and inelasticity. We have mentioned that a material can demonstrate inelastic — in particular, plastic — properties because of defects in the shape of a shell. This can happen at sufficiently high loads. It can also result from certain production technologies, which leave behind internal stresses. Existing technological methods significantly reduce the level of strain introduced by cutting, pressing, or other stages of production, but cannot eliminate such strains completely. Unfortunately we can seldom determine these strain fields without destroying the object, hence it is difficult to account for them. Since the dependence of shell deformation on load increment is not linear, we encounter significant indeterminacy in trying to find critical loads while trying to account for prestressing of the shell (Guran [12, 49]). Boundary conditions. When solving boundary value problems for shells and plates, we use ideal boundary conditions such as Dirichlet’s conditions. However, real clamping can be non-rigid. From the viewpoint of the strained state of a structure, the difference between solutions to problems with ideal and non-ideal boundary conditions may be negligible if the pliability of the support is low. But as to the critical values of the load it may be significant. In experiments with preserving membranes in the Laboratory of Shells at Rostov State University, it was found that


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even traces of oil on the clamped area of a membrane affects the critical buckling pressure quite significantly, making results on critical pressure practically random. In the same experiments with thin-walled preserving caps, it was seen that normally the buckled spot on the membrane starts to develop near the edge. At the clamped edge there are some unusual conditions because of the technology of cap production: along the bent curve we see a deep crack, and it seems the material in this area is in a plastic state. It is clear that much of this happens to other types of clamped shells and plates, and so a proper type of boundary condition can play a crucial role in finding critical loads for real objects. Temperature effects. Normally thermoeffects are not taken into account when finding critical loads. However, their influence can be crucial in some circumstances. Let us consider the problem of asymmetric buckling of a spherical thin-walled cap. During testing and in real conditions, the increase in pressure is not infinitely slow; this means that the process of loading a cap is not completely isothermal — it is intermediate between isothermal and isentropic. Any change in strain is supplemented by a small change in temperature that depends on the change in strain. This means that inside a cap whose deformation state under pressure is not homogeneous, nor in different points, nor along the normals to the midsurface, there arise heat flows because of which the strain state changes slightly with time. Thus the process of loading the cap, even if it is quite slow and can be considered as quasistatic, depends on time. Such light changes in strain should be neglected when we consider the strain state. But in the theory of stability we consider more fragile characteristics: the critical point and the subsequent development of the deformation under the pressure increase. Since for a cap, near the first critical point where there arise branches of asymmetric solutions, the symmetric solution is also stable, it is possible to imagine (as far as we know nobody has tried to study this effect) that in some regime of the load increase, this non-robust critical point (which is not a point when we consider a time-dependent problem) can be “circumvented” so that the asymmetric branches do not appear. We suppose that such behavior of a thin-walled structure is quite probable in the neighborhood of critical points that do not possess the property of robustness. This direction in stability theory deserves further study.

11.5 Conclusion Stability theory is an interesting and important field of applied mathematics, having numerous applications in the natural sciences as well as in aerospace, naval, mechanical, civil, and electrical engineering. Stability theory was always important in astronomy and celestial mechanics, and during the last decade it has been applied to the study of processes in chemistry, biology, economics, and the social sciences. In this keynote article we presented the central notions of the stability theory of mechanical systems. We also attempted to explain some common discrepancies between theoretical and experimental results in the stability of thin-walled mechanical structures. Thousands of papers have been devoted to the topics mentioned above.

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The interested reader can consult [1–7, 10, 13–21, 23–34, 47, 50, 51, 53–55, 57] for more information. Theoretical analysis of these problems requires advanced tools of functional analysis, some of which can be found in [2, 30, 36–41, 58–60]. We hope this article will assist readers not only in finding stable solutions that can provide for safe operation of structures, machines, and devices made of thin-walled elements in engineering, but also in developing stable and predictable processes in the technical, natural, and social sciences. Acknowledgements The first author is indebted to his collaborators Professors Teodor Atanakovic (University of Novi Sad), Vladimir Beletskii (Moscow State University), Iliya Blekhman (IPM, Saint Petersburg), Horst Leipholz (University of Waterloo), Raymond Plaut (Virginia Polytechnic Institute), Karl Popp (Technical University of Hannover), Richard Rand (Cornell University), Fred Rimrott (University of Toronto), John Roorda (University of Waterloo), Alexander Seyranian (Moscow State Lomonsov University), Bez Tabarrok (University of Victoria), and Peter Tovstik (Saint Petersburg State University) for their contributions to his understanding of the Stability theory of mechanical systems. This work is prepared to honor Professor Felix Leonidovich Chernousko (Director, Institute for Problems in Mechanics of the Russian Academy of Sciences in Moscow) on his birthday; may he have excellent health, a long life, much happiness, and many more years of creative work in science.

References 1. E. Abed, Y. Chou, A. Guran, and A. Tits, Nonlinear stabilization and parametric optimization in the benchmark nonlinear control design problem, Proc. IEEE American Control Conf., (Seattle, WA, 4357–4359, 1995). 2. S. S. Antman, Nonlinear Problems of Elasticity (Springer-Verlag, New York, 1995). 3. T. M. Atanakovic and A. Guran, Theory of Elasticity for Scientists and Engineers (Birkhauser, Boston, 2000). 4. T. M. Atanackovic and A. Guran, A generalized Greenhill problem with shear deformation, compressibility and imperfections, Theoretical and Applied Mechanics 25, 1–20 (1999). 5. V. V. Beletskii, “Resonance Phenomena at Rotation of Artificial and Natural Celestial Bodies”, in Satellite Dynamic, G.E.O. Giacaglia (Ed), Springer-Verlag, Berlin, 1975, pp. 191–232. 6. I. Blekhman, Selected Topics in Vibrational Mechanics (World Scientific, Singapore–New Jersey–London–Hong Kong, 2004). 7. E. Byskov, Smooth postbuckling stresses by a modified finite element method, Int. Journal for Numerical Methods in Engineering 28, 2877-2888 (1989). ¨ 8. J. P. L. Dirichlet, Uber die stabilität des Gleichgewichts, J. reine angew. Math 32, 85–88 (1846). 9. L. Euler, Methodus Inveniendi Lineas Curvas; Maximi Minimive Proprietate Gaudentes (Appendix. De Curvas Elasticas), (Marcus Michaelem, Bousquet, Lausanneand Geneva, 1744). 10. M. Farshad and A. Guran, “Adaptive Material and Structures”, in 3rd International Congress on Mechatronic, Prague, Czech Technical University in Prague, A. Guran and M. Valasek (Editors), 2004, pp. 319–374. 11. A. Guran, On the Stability of an Elastic Imperfect Column Including Axial Compressibility, Journal of Applied Mathematics and Mechanics-ZAMM (Zeitschrift fur Angewandte Mathematik und Mechanik) 72(10), 481–485 (1992). 12. A. Guran, Influence of various types of load-dependent supports on the stability of a compressible column model, Acta Mech. 97, 91–100 (1993). 13. A. Guran, “Shape control of smart structures by using fluidic actuators”, in The 3rd International IEEE Scientific Conference on Physics and Control, 2007.


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14. A. Guran, “Controlling the Chaos Using Fuzzy Estimation in a Gyrostat Satellite”, in Recent Progress in Controlling Chaos, M. A. F. Sanjuan and C. Grebogi (Eds), World Scientific, Singapore–New Jersey–London–Hong Kong, 2010. 15. A. Guran, “Buckling of a column as a benchmark nonlinear control design”, in The 3rd international IEEE scientific conference on physics and Control, J. Kurths, A. Fradkov, G. Chen (Eds), University of Potsdam, Germany, 2007. 16. A. Guran, “Ernst Mach and Peter Salcher: The development of nonlinear wave mechanics during the period 1850-1950”, in International Sysposium Peter Salcher and Ernst Mach: A Successful Teamwork, Rijeka, Croatia, 23-25 September, 2004. 17. A. Guran et al, “Oscillations, Bifurcations and Chaos in Ziegler’s Pendulum with Eccentric Load in a gravitational field”, in Proceedings of the 3rd International Conference on Complex Systems and Applications(ICCSA 2009), University of Le Havre Le Havre, Normandy, France, 2009. 18. A. Guran et al, On the Stability of a Spinning Viscoelastic Column, Mechanics Based Design of Structures and Machines 19(4), 437–455 (1991). 19. A. Guran et al, Studies in Spatial Motion of a Gyro on an Elastic Foundation, Mechanics Based Design of Structures and Machines An International Journal 21(2), 185–199 185–199 (1993). 20. A. Guran and T. M.Atanackovic, Lecture Notes on Theory of Elasticity (in Russian) (St. Petersburg University Press Petersburg, 2002). 21. A. Guran and T. M. Atanackovic, Fluid conveying pipe with shear and compressibility, Eur. J. Mech., A/Solids 17, 121–137 (1997)). 22. A. Guran and G. Ahmadi, “Postbuckling Behavior and Imperfection Sensivity of an Elastic Compressible Rod whose Flexural Resistance Changes with Load”, in Second International Conference on Application of Mathematics in Technical and Natural Sciences, Sozopol, Bulgaria, 2010 (to appear). 23. A. Guran, A. Bajaj, N. Perkins, G.DEluterio, and C. Pierre, Stability of Gyroscopic Systems (World Scientific, Singapore–New Jersey–London–Hong Kong, 1999). 24. A. Guran and M. Khoshnood, “Control of an Inverted Double Pendulum with Eccentric Fluid Load by Means of Computed Torque Method”, in Proceedings of EUROMECH Colloquium 515 Advanced Applications and Perspectives of Multibody System Dynamics, Blagoevgrad, 2010, pp. 36–37. 25. A. Guran, L. Sperling (Guest Editors), Wissenschaftliche Zeitschrift fur Grundlagen und Anwendungen der Technischen Mechanik, 24, Special issue In memoriam Friedrich P.J. Rimrott (2004). 26. A. Guran and I. G. Tadjbakhsh, “A mechanical actuator to suppress vibration”, in Proc. Smart Structures and Materials (SPIE), Smart Structures and Intelligent Systems, W. Hagood (Ed) 1917, 1993, pp. 1113–1114. 27. A. Guran and B. Yousefghahari, “Biomechanical study of spine”, in XIII Mediterranean Congress of Rheumatology; Clinical and Experimental Rheumatology 27, 726, 2009. 28. A. Guran and B. Yousefghahari, “A nonlinear Multibody System Dynamics Model to Predict the Impact Response of Human Chest during Cardiopulmonary Resuscitations”, in Proceedings of EUROMECH Colloquium 515 Advanced Applications and Perspectives of Multibody System Dynamics, Blagoevgrad, 2010, pp. 70–71. 29. A. Guran and B. Yousefghahari, “Shape-control of carbon nanotubes by means of nanofluidic actuators”, in Proc. Workshop on Biomedical applications of functionalized carbon nanotubes, Schloss Eckberg, Dresden, Germany, 2010. 30. R. J.Knops and E. W. Wilkens, Theory of Elastic Stability, Handbuch der Physik. Bd. VIa/3 (Springer-Verlag, Berlin, 1973). 31. W. T. Koiter, On the stability of elastic equilibrium, (Dissertation, Delft, 1945) in Trans. NASA, 10:833, 1967. 32. W. T. Koiter, “Elastic stability and postbuckling behavior”, in Proc. Symp. Non-Lrnear Probl., R.E. Langer, ed., Univ. of Wisconsin Press, 1963, pp. 257-275. 33. W. T. Koiter, “The non-linear buckling problem of complete sphercal shells under uniform external pressure”, Proc. Konik, Ned. Akad. Wetensch. Ser B, 72:40, 1969.

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34. W. T. Koiter, Current trends in the theory of buckling,Buckling Structures, 1-16 (1976). 35. J. L. Lagrange, Mécanique Analutique” (Courier, Paris, 1788). 36. L. P. Lebedev and M. J. Cloud, The Calculus of Variations, Optimal Control, and Functional Analysis (World Scientific, Singapore–New Jersey–London–Hong Kong, 2003). 37. L. P. Lebedev and M. J. Cloud, Tensor Analysis (World Scientific, Singapore–New Jersey– London–Hong Kong, 2003). 38. L. P. Lebedev and M.,J. Cloud, Introduction to Mathematical Elasticity (World Scientific, Singapore–New Jersey–London–Hong Kong, 2009). 39. L. P. Lebedev, M. J. Cloud, and V. A. Eremeyev, Tensor Analysis with Applications in Mechanics (World Scientific, Singapore–New Jersey–London–Hong Kong, 2010). 40. L. P. Lebedev and I. I. Vorovich, Functional Analysis in Mechanics (Springer, New York, 2002). 41. L. P. Lebedev, I. I. Vorovich, and G. M. L. Gladwell, Functional Analysis: Applications in Mechanics and Inverse Problems, (Kluwer Academic Publishers, 1996 (2/e 2002)). 42. A. M. Lyapunov, On the stability of ellipsoidal figures of equilibrium of a rotating fluid (Master’s thesis (in Russian), Sankt-Petersbourgue, Akademy of Sciences, 109 pp, 1884). 43. A. M. Lyapunov, The General Problem of Motion Stability (in Russian) (Doctoral dissertation, Kharkov, Russia, 250 pp, 1892). 44. D. R. Merkin, Introduction to Stability of Motion (Springer-Verlag, New York, 1996). 45. A. A. Movchan, On the direct Liapunov method in the stability problems for elastic systems (in Russian), Prikl. Mat. Mekh. 23(3) (1959). 46. A. A. Movchan, On stability of motion of continuous bodies. Lagrange’s theorem and its inverse (in Russian), Inzhenernyj sbornik, 29, (1960). 47. J. F. Olesen and E. Byskov, Accurate detemination of asymptotic postbuckling stresses by the finite element method, Computer and Structures 15, 157-163 (1982). 48. G. V. Palassopoulos, “Effect of stochastic imperfections on the buckling strength of imperfectionsensitive structures”, in Fifth World Congress on Computational Mechanics, Vienna, Austria, 2002. 49. R. H. Plaut and A. Guran, Buckling of plates with stiffening elastically restrained edges, Journal of Engineering Mechanics–ASCE 120(2), 408–411 (1994). 50. J. Roorda, Stability of structures with small imperfections, Journal of Engineering Mechanics– ASCE, 87-106 (1965), . 51. J. Roorda and A. H. Chilver, Frame buckling: an illustration of the perturbation technique, Int.J.Journal Non-Linear Mechanics 5, 235-246 (1970). 52. A. P. Seyranian and A. A. Mailybeav, Multiparameter Stability Theory with Mechanical Applications (World Scientific, Singapore–New Jersey–London–Hong Kong, 2003). 53. B. Tabarrok and F. P. J. Rimrott, Variational Methods and Complementary Formulations in Dynamics (Kluwer Academic Publishers, Netherland, 1994). 54. P. E. Tovstik and A. L. Smirnov, Asymptotic Methods in the Buckling Theory of Elastic Shells (World Scientific, Singapore–New Jersey–London–Hong Kong, 2001). 55. I. I. Vorovich, Some question of shell stability “in the large”, Dokl.Akad.Nauk SSSR 122, 37–40 (1958). 56. I. I. Vorovich, The problem of non-uniqueness and stability in the non-linear mechanics of continuum mechanics, in Applied Mechanics, Proc. Thirteenth Intern. Congr. Theor. Appl. Mech., (Springer, Berlin, 340–357 (1973). 57. I. I. Vorovich, Nonlinear Theory of Shallow Shells, (Springer-Verlag, New York, 1999, translation from Russian edition by Nauka, Moscow, 1989). 58. I. I. Vorovich and L. P. Lebedev, Some issues of continuum mechanics and mathematical problems in the theory of thin-walled structures, International Applied Mechanics (Prikladnaya Mekhanika) 38(4), 387–398 (2002). 59. V. A. Yakubovich, G. A. Leonov,and A. Kh. Gelig, Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities (World Scientific, Singapore–New Jersey–London– Hong Kong, 2004). 60. V. Yurkevich and A. Guran, Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities (Book Reviews), IEEE Transaction on Automatic Control 50(4), 542–543 (2005).

Chapter 12

High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory Elena A. Ivanova

Abstract The classic plate theory by Kirchhoff allows to accurately describe the processes slowly varying by time. To solve the problems of the plate vibrations in the case of the external loads quickly varying by time the Reissner’s type plate theory should be used. The Reissner’s type plate theory includes three eigenfrequency spectra: one low-frequency spectrum whose asymptotic order is O(h), and two high-frequency spectra whose asymptotic order is O(h−1 ). Solving the problems of plate vibrations under the action of the quickly varying by time loads it is necessary to take into account vibrations with eigenfrequencies from the highfrequency spectra. That is why the problem of plate free vibrations with eigenfrequencies whose asymptotic order is O(h−1 ) is interesting and practically important. In this paper asymptotic analysis of the equations of the Reissner’s type plate theory for high-frequency free vibrations is carried out and the approximate equations of plate vibrations with eigenfrequencies of the asymptotic order O(h−1 ) are proposed. Asymptotic analysis of the equations of plate free vibrations shows that behavior of the functions defining the stress-strain state of the plate for high-frequency free vibrations differs from it for low-frequency free vibrations. For high-frequency free vibrations the solution includes the functions which quickly vary along the space coordinates but which are not the boundary layer type functions. Because of that using of the exact equations of the Reissner’s type theory in numerical procedures is difficult. Approximate equations of high-frequency free vibrations of plates independent of quickly varying along the space coordinates functions are formulated in this paper. These equations describe vibrations with eigenfrequencies from the high-frequency spectra only, like the classic plate theory describes vibrations with eigenfrequencies from the low-frequency spectrum only. Keywords Reissner plate · High-frequency vibration E. A. Ivanova (B) Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bolshoy pr. V.O., 61, 199178, Saint Petersburg, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 12, © Springer-Verlag Berlin Heidelberg 2011

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12.1 Introduction It is known that the transverse shear deformations [1–4] and the inertia of rotation [5,6] must be taken into account in some problems of forced vibrations of plates, in particular, for the plate vibrations under impact and other rapidly time-varying loads, for the thick plates and for the composite laminates. [7–10]. Therefore, the solution of the problem of eigenfrequencies and modes of plate vibrations for the Reissner theory [11, 12] is of great importance. The problem of law-frequency free vibrations of the Reissner plate is studied in detail. It is well known that the solution of the problem includes the slowly varying functions of the space coordinates and the rapidly varying functions of boundary-layer type [13]. In contrast to lawfrequency free vibrations, the problem of high-frequency free vibrations of plate has been studied insufficiently. As far as we know, the problem was studied only in the paper [14] where the equations of free plate vibrations are obtained on the basis of three-dimensional elasticity by a variational-asymptotic method for various frequency spectra. In contrast to [14], we obtain the approximate equations of high-frequency free plate vibrations starting from the asymptotic analysis of the equations of the Reissner plate theory. We show that the functions that describe the stress-strain state of the plate for the law-frequency and high-frequency vibrations, have quite a different character of varying with respect to the space coordinates. For highfrequency free vibrations the solutions include rapidly varying functions as well as for low-frequency ones, but for the high-frequency case these functions are not the boundary-layer type functions. They deeply penetrate into the plate domain. In contrast to [14], our attempt revises the problem of high-frequency vibrations so as to make it more convenient for numerical implementation and to cover all possible types of boundary conditions. We also note that the given equations for high-frequency vibrations differ from the equations in [14].

12.2 Summary of the Basic Equations of Free Vibrations of Reissner’s Plate Let us consider the problem of vibrations of a plate with taking account of the inertia of rotation, and of the transverse shear deformation. The deflection w, the vector Ψ of rotation angles, the vector N of shear forces, and the moment tensor M are related to the displacements and stresses in three-dimensional elasticity as follows [15, 16]: hw = N=

h/2 −h/2

h/2

−h/2

u · n dz ,

a · τ · n dz ,

h Ψ= 3

M=

h/2 −h/2

h/2 −h/2

a · τ · az dz ,

u z dz (12.1) a = E − nn.

12 High-Frequency Free Vibrations of Plates in the Reissner’s Type Theory

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Here u and τ are are the displacement vector and the stress tensor in the threedimensional theory, h is the plate thickness, n is the vector of the unit normal to the plate plane, E is the unit tensor. The theory of plate with taking account of the inertia of rotation, and of the transverse shear deformation includes the following equations. The equations of the motion are ∇ · N + ρhP = ρhw¨ ,

∇·M−N =

1 3¨ ρh Ψ, 12

(12.2)

where P(x, y, t) is the external load, ρ is the mass density. Constitutive equations take the form . / M = D (1 − μ)æ + μ tr æ a .

N = GhΓγ ,

(12.3)

Here γ is the transverse shear deformation vector, æ is the bending–twisting tensor, D = Eh3 /[12(1 −μ2)] is the bending stiffness, E is the Young modulus, μ is Poisson’s ratio, GhΓ is the shear stiffness, Γ is the coefficient of transverse shear, G = E/[2(1+ μ)] is the shear modulus. Geometric relations are γ = ∇w + Ψ,

æ=

& 1% ∇Ψ + ∇ΨT . 2

(12.4)

The kinematic boundary conditions acquire the form w|c = w∗ ,

ν · Ψ|c = Ψν∗ ,

τ · Ψ|c = Ψτ∗ .

(12.5)

The force boundary conditions can be written as follows ν · N|c = Nν∗ ,

ν · M · ν|c = Mν∗ ,

ν · M · τ|c = Mτ∗ .

(12.6)

Here ν and τ the unit outward normal vector and the unit tangent vector to the plate contour, respectively; the vectors ν, τ, and n, are assumed to form a right-handed system; Ψν∗ and Ψτ∗ are angles of rotation about the tangent vector and the normal vector to the plate contour, respectively; Nν∗ is the lateral force, Mν∗ is bending moment, Mτ∗ is the torque. Introducing potentials Φ and F we reduce the equations of the plate theory to the more convenient form [17]: 2 3 .... ρh3 2 ¨ ¨ + ρ h Φ + ρh P = 0, D ΔΔΦ + ρhΦ − 1+ ΔΦ (12.7) 12 Γ(1 − μ) 12GΓ 12Γ ρ F − F¨ = 0. (12.8) G h2 The quantities characterizing the stress-strain state of a plate are expressed in terms of the potentials Φ and F by the formulas ΔF −

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w = −Φ +

h2 ρh2 ¨ ΔΦ − Φ, 6Γ(1 − μ) 12GΓ

Ψ = ∇Φ + ∇F × n,

ρh3 ¨ ∇Φ + GhΓ∇F × n, 12 ( 1−μ ' M = D (1 − μ)∇∇Φ + μΔΦ a + ∇∇F × n − n × ∇∇F . 2 N = D ∇ΔΦ −

(12.9)

It is known [18] that in the Reissner’s type theory of plates there are three spectra of eigenfrequencies, which satisfy the following asymptotic estimates: 9 ω(2) i

=

(1) 2 (1) ω(1) i = hω1i + h ω2i + ... 9

12GΓ + ω(2) + ... 0i ρh2

ω(3) i =

12GΓ + ω(3) + ... 0i ρh2

(12.10)

where the first spectrum in Eqs (12.10) describes the low frequency bending vibrations, whereas the second and the third spectra in (12.10) characterize the high frequency shear and bending vibrations.

12.3 Asymptotic Analysis of the Equations of Reissner’s Plate Theory For the free high-frequency vibrations, the functions Φ and F substantially differ from those for the low-frequency vibrations or static bending. The function F varies slowly with respect to the spatial coordinates and is not of boundary-layer type. This is due to the fact that the leading terms in the first and in the second components in Eq. (12.8) cancel each other. For the approximate statement of the problem, Eq. (12.8) remains the same. A very important feature of the high-frequency vibrations is that the asymptotic orders of the functions F and Φ are the same: F ∼ Φ . The penetrating potential Φ for the high-frequency vibrations has quite a different structure than in the low-frequency case; namely, along with functions, slowly varying with respect to the spatial coordinates, it includes a rapidly varying function, which was lacking in the preceding cases. Let us denote this rapidly varying function by ϕ, and retain the notation Φ for the slowly varying component as well as for the penetrating potential itself. This seems convenient, since there is an asymptotic relation ϕ ∼ h2 Φ . The function ϕ seemingly need not be taken into account since it is relatively small. However, this is not the case, and the function ϕ may exert influence on the leading terms of some characteristics of the stress-strain state, which can depend not only on the penetrating potential, but also on its derivatives of order ≤ 3. Let us proceed from Eq. (12.7) to the approximate equations for the components Φ and ϕ. We suppose that for the functions Φ, and ϕ the following asymptotic estimates hold


12GΓ F¨ = − + O(1) F, ρh2 ∂F ∂F ∼ ∼ F, ∂x ∂y

¨ = − 12GΓ + O(1) Φ , Φ ρh2

∂Φ ∂Φ ∼ ∼ Φ, ∂x ∂y

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12GΓ ϕ¨ = − + O(1) ϕ, ρh2

∂ϕ ∂ϕ 1 ∼ ∼ ϕ, ∂x ∂y h

ϕ ∼ h2 Φ ∼ h2 F.

12.4 Approximate Formulation of the Problem of High-Frequency Free Vibrations On substituting the expression for the penetrating potential Φ + ϕ into Eq. (12.7) and by retaining only the leading terms, we obtain Gh A(Φ) + D B(ϕ) = 0, % 2 & 12Γ ρ ¨ A(Φ) = Γ + ΔΦ − 2 Φ − Φ, 1−μ G h 12 % Γ(1 − μ) & B(ϕ) = Δ Δϕ + 2 1 + ϕ . 2 h

(12.11)

Note that the first equation in (12.11) contains an obvious contradiction. On the one hand, A(Φ) is a slowly varying function, since it depends on Φ, and B(ϕ) is a rapidly varying function, since it depends on ϕ. On the other hand, according to the first equation in (12.11), the functions A(Φ) and B(ϕ) are proportional. Thus, A(Φ) and B(ϕ) are slowly varying and rapidly varying simultaneously, that is, they are both zero. Hence, the first equation in (12.11) represents the two equations A(Φ) = 0,

B(ϕ) = 0.

(12.12)

The first equation in (12.12) has the form % 2 & 12Γ ρ ¨ Γ+ ΔΦ − 2 Φ − Φ = 0. 1−μ G h

(12.13)

Equation (12.13) permits us to find the leading term of the slowly varying part of the penetrating potential. It should be noted that Eq. (12.13) for Φ coincides with Eq. (12.8) for F to within a constant coefficient of the first summand, and the behavior of Φ is similar to that of F in the case of high-frequency vibrations. The second equation in (12.12) is Δz(ϕ) = 0,

z(ϕ) = Δϕ +

12 % Γ(1 − μ) & 1+ ϕ. 2 2 h

(12.14)

The solution of the equation Δz = 0 is a slowly varying function. Since z(ϕ) varies rapidly, all solution except for zero are excluded: z ≡ 0 . Then the equation for ϕ acquires the form

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12 % Γ(1 − μ) & 1 + ϕ = 0. (12.15) 2 h2 Equation (12.15) allows one to find the leading term of the rapidly varying part of the penetrating potential. Note that in contrast to the low-frequency vibrations, the rapidly varying function for the high-frequency vibrations is not of boundary-layer type, but penetrates into the entire plate domain. For the high-frequency vibrations, the characteristics of stress-strain state of the plate have the following asymptotic representations: Δϕ +

% & h2 2 ΔΦ − 1 + ϕ, 12 Γ(1 − μ) ' ( Ψ = ∇Φ + ∇F × n , N = GhΓ ∇Φ + ∇F × n , w=−

(12.16)

( 1−μ ' M = D (1 − μ)∇∇(Φ + ϕ) + μΔ(Φ + ϕ) a+ ∇∇F × n − n × ∇∇F . 2 Before proceeding to the statement of the boundary conditions, let us focus our attention on the following important property of ϕ. Consider a boundary condition depending on ϕ, for example, w|c = 0. Obviously, ϕ cannot vary rapidly on the boundary, since the other components in the boundary condition vary slowly. A similar conclusion holds for any boundary condition that depends on ϕ. In addition, ϕ is generally nonzero on the plate contour, since otherwise it would obviously be zero in the entire plate domain. The most natural conclusion follows: on the plate boundary, the function ϕ loses the property of being rapidly varying and becomes a slowly varying function along the contour. We point out that ϕ varies slowly only on the plate boundary; it rapidly varies in every direction in the interior of the domain arbitrarily close to the boundary. Thus, the tangent derivative of ϕ on the plate contour has the same asymptotic order as ϕ: ∂ϕ/∂τ|c ∼ ϕ. The kinematic boundary conditions acquire the form

2 % h 2 &

∂Φ ∂F

∂Φ ∂F

− ΔΦ − 1 + ϕ = 0, + = 0, −

= 0. 12 Γ(1 − μ) c ∂ν ∂τ c ∂τ ∂ν c (12.17) The force boundary conditions are written as follows: ∂Φ ∂F

GhΓ +

= 0, ∂ν ∂τ c

∂2 F % 1 ∂Φ ∂2 Φ & 12 % Γ(1 − μ) &

D(1 − μ) − + +D ΔΦ − 2 1 + ϕ = 0, (12.18)

c ∂ν∂τ R ∂ν ∂τ2 2 h

∂2 Φ % 1 ∂F ∂2 F & % ρh3 &

D(1 − μ) + + 2 − F¨ + GhΓ F

= 0.

c ∂ν∂τ R ∂ν ∂τ 12 The physical meaning of the boundary conditions (12.17) and (12.18) is the same as that of conditions (12.5) and (12.6), respectively. Equations (12.8), (12.13), and (12.15) supplemented with the boundary conditions (12.17) and (12.18) form an


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approximate statement of the problem of free high-frequency vibrations, which permits us to find the leading terms of the characteristics of the stress-strain state and the eigenfrequencies with a relativeerror O(h4 ). Recall that the main terms of the eigenfrequencies are equivalent to 12GΓ/(ρh2) .

12.5 Formulation of the Problem of High-Frequency Free Vibrations of the Reissner’s Plate without Taking Account of Rapidly Varying Function As was noted in the preceding, ϕ is a rapidly varying function and penetrates into the entire plate domain. Hence, the cited statement for the high-frequency vibrations is practically invalid in numerical implementations. Is it possible to revise this problem without taking account of ϕ? Before answering this question, let us note the following: the vector Ψ of rotation angles and the vector N of shearing forces, which do not depend on ϕ (see Eqs (12.16)), are two orders of magnitude larger than the deflection w and the moment tensor M respectively. Thus allows one to claim, that the stress-strain state is mainly characterized by the vector of rotation angles and the vector of shear forces, whereas the deflection and the moment tensor play a less important role. Therefore, the statement of this problem without the function ϕ is consistent in principle. Thus, let us eliminate Eq. (12.15) from the system of equations for the highfrequency vibrations. Then the order of the system with respect to the spatia] derivatives is reduced from six to four. Hence, the three boundary conditions in the original statement must be replaced by two boundary conditions independent of ϕ. One of the original three boundary conditions is the third condition in (12.17) or in (12.18). Since these conditions do not depend on ϕ, they are retained. Two other conditions are replaced by one condition according to the following rule: • the second condition in (12.17) and the first condition in (12.18) are equivalent, and if they are given simultaneously, there is no need to choose one of them (note that the function ϕ is identically zero for these boundary conditions); • if the first and the second conditions in (12.17) are given, the second condition must be retained, since it is independent of ϕ (whereas the first condition depends on ϕ); • if the first and the second conditions in (12.18) are given, the first condition must be retained, since it is independent of ϕ (whereas the second condition depends on ϕ); • since the first condition in (12.17) and the second condition in (12.13) depend on ϕ they can be replaced by the combination

∂2 F % 1 ∂Φ ∂2 Φ & % ρh3 & ¨ + GhΓ Φ D(1 − μ) − + 2 + Φ ∂ν∂τ R ∂ν ∂τ 12

= 0. c

(12.19)

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The physical meaning of the boundary conditions without the function ϕ is the following. The third condition in (12.17) means that the angle Ψτ of rotation about the normal to the plate contour is zero. The second condition in (12.17) (and the equivalent first condition in (12.18)) indicates that the angle Ψν of rotation about the tangent to the plate contour is zero. The third condition in (12.18) means that the torque Mτ is zero. Condition (12.19) indicates that the reduced bending moment Mν − GhΓw is zero.

12.6 Asymptotic and Numerical Analysis of High-Frequency Free Vibrations of Rectangular Plates Free vibrations of rectangular plates with frequencies belonging to high-frequency spectra are studied. The results predicted by the exact Reissner’s type theory are compared with those predicted by an approximate theory of high-frequency free vibrations which takes into account only functions slowly varying with respect to the spatial coordinates. It is well known that in solving some dynamical problems of plates, in particular, problems on forced vibrations under the action of impact loadings, one cannot ignore high-frequency vibrations which are associated with the inertia of rotation and the transverse shear deformation. As pointed out above, for high-frequency vibrations, the solution contains functions rapidly varying with respect to spatial coordinates and penetrating into the entire domain of the plate. The presence of such functions makes the exact equations of the Reissner theory practically unsuitable for the numerical analysis of the problems. Above an approximate statement of the problem on high-frequency free vibrations of a plate was suggested; only functions that vary slowly with respect to the spatial coordinates are taken into account. The asymptotic accuracy of this statement is O(h) compared with unity in determining eigenfrequencies and O(h4 ) in determining eigenfrequencies. This difference in the accuracy is accounted for by the fact that the leading terms of asymptotic expansions for all eigenfrequencies coincide and are known, whereas the approximate theory defines the first correcting term in the asymptotic expansions for the eigenfrequencies. Of course, the asymptotic accuracy of a theory is an important characteristic. However, to assess an asymptotic theory from the viewpoint of its practical significance, the actual accuracy of the theory is of importance rather than its asymptotic accuracy. (By the actual accuracy we mean the relative difference of the value of a quantity predicted by the approximate theory and the value of that quantity predicted by the exact theory for the given value of the small parameter.) In what follows we deal with the analysis of the actual accuracy of the approximate theory of highfrequency vibrations which was suggested above. The purpose on this research is to determine the area of applicability of the theory. The investigation is exemplified by problems having exact analytical solution, which allows us to rule out practically any errors of calculations. Now we consider rectangular plates two opposite sides of which are hinged. Let us consider a plate


161

occupying a domain −a ≤ x ≤ a, −b ≤ y ≤ b. The constrained hinged support conditions are assumed to be satisfied at the sides y = ±b: w|c = 0,

Mν |c = 0,

Ψτ |c = 0.

(12.20)

The boundary conditions at the sides x = ±a can be arbitrary. We study vibrations symmetric with respect to the axes x = 0 and y = 0. The eigenforms satisfying the differential equations (12.7) and (12.8) and the boundary conditions (12.20) at y = ±b have the form Φn (x, y) = [C1n cos (λ1n x) + C2n cos (λ2n x)] cos (μn y), Fn (x, y) = C3n sin (δn x) sin (μn y), √ √ (12.21) μn = (2n − 1)π/(2b), λ1n = An − Bn , λ2n = An + Bn , δn = ρω2n /G − 12Γ/h2 − μ2n , An = [1 + Γ(1 − μ)/2]ρω2n/(2GΓ) − μ2n , Bn = ρh/D + ([1 − Γ(1 − μ)/2]ρω2n/(2GΓ))2. By satisfying the boundary conditions at x = ±a one reduces the problem to solving a system of homogeneous algebraic equations for the coefficients C1n ,C2n ,C3n . By equating the determinant of this system to zero, one obtains an equation for determining eigenfrequencies. We considered all types of boundary conditions possible in the Reissner’s type theory and obtained the frequency equations for each of them. Let us discuss the solution of the problem according to the approximate theory of high-frequency free vibrations. It can be readily shown that the eigenforms satisfying the differential equations (12.8) and (12.13) and the constrained hinged support conditions (12.20) at y = ±b have the form Φn (x, y) = C1n cos (λ1n x) cos (μn y), Fn (x, y) = C3n sin (δn x) sin (μn y), 2 2 ρ ω0n ρ 12GΓ λ1n = − μ2n , δn = ω0n − μ2n , ω0n = ω2n − . G Γ + 2/(1 − μ) G ρh2 (12.22) The asymptotic analysis shows that the frequency equations and the eigenforms predicted by tho asymptotic theory in question follow from the exact frequency equations and the exact eigenforms within an O(h) asymptotic error for all types of boundary conditions (by exact frequency equations and exact eigenforms we mean those obtained by the Reissner’s type theory). The numerical analysis was carried out for the problem discussed above. Computations were performed for plates of dimensions a = b = 1 m and thicknesses h = 0, 1 m and h = 0, 04 m with the elastic constants E = 2, 1 · 1011 Pa, μ = 0, 25, Γ = 5/6, and ρ = 7, 951 · 103 kg/m3 . The key results can be summarized as follows. The first 10 eigenfrequencies of high-frequency spectra are found. The calculations were performed by the exact

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theory and by the approximate theory for all types of boundary conditions possible in the Reissner’s type theory. The actual errors of the approximate theory are found. The calculations were performed for all types of boundary conditions and the plate thicknesses 0, 1 m and 0, 04 m. For the case in which the free edge conditions were imposed at the sides x = ±a, more detailed investigation was carried out. The first 10 eigenfrequencies for plates of thickness 0, 2 m, 0, 3 m, 0, 4 m, and 0, 5 m and the corresponding actual errors are calculated. The eigenforms corresponding to the first 10 eigenfrequencies are determined. The calculations were performed by the exact and approximate theories for all types of boundary conditions and the plate thickness 0, 1 m. In the case of the free edge conditions, the eigenforms are found for the plate of thickness 0, 04 m as well. For the eigenforms in calculation of which the approximate theory leads to the largest errors, the graphs of the rotation angles Ψx and Ψy versus the coordinate x are constructed. Let us point out some general features characteristic of high-frequency spectra and the approximation of these spectra by the approximate theory. The frequencies of the bending spectrum are higher than those of the shear spectrum. (Calculation show that for all types of boundary conditions at x = ±a, only two frequencies of the first ten belong to the bending spectrum.) The accuracy for the frequencies belonging to the bending spectrum is, as a rule, less than that for the frequencies belonging to the shear spectrum. This is quite natural, since in the approximate theory, the equation responsible for the shear vibrations is exact, whereas the equation responsible for the bending vibrations is approximate. While the approximate theories when applied to low-frequency vibrations yield higher values for the eigenfrequencies compared with the exact ones, this is not the case for high-frequency vibrations. For high-frequency vibrations, no monotonic increase in the relative error with the mode number is observed either. Of course, lower frequencies are, on the average, predicted more accurately than higher frequencies. However, for high frequency vibrations, the situation in which a frequency with a larger mode number is predicted more accurately than many frequencies with smaller mode numbers is usual. Let us briefly dwell on the results of calculation of eigenforms. We carried out the investigation of the accuracy provided by the approximate theory for eigenforms as follows. For all types of boundary conditions, analytical expressions for the potentials F(x, y) and Φ(x, y) were obtained according to the exact and approximate theories. Then the eigenforms represented by F(x, y) were compared with those represented by Φ(x, y). As a result, it was established that most of eigenforms predicted by the approximate theory virtually coincide with those predicted by the exact Reissner’s type theory. The errors turned out to be noticeable only in the case of the free edge conditions at x = ±a. Therefore, the subsequent discussion pertains just to this type of boundary condition (h = 0, 1 m). It has been established that the eigenforms predicted by the approximate theory and by the exact theory are in quite good agreement, which allows us to conclude that the approximate theory provides high accuracy in predicting eigenforms as well. This statement is valid for the overwhelming majority of eigenforms. Nevertheless, in exceptional cases, the difference in eigenforms predicted by the exact and approximate theories can be


163

large. It should be emphasized that such cases are encountered rather rarely: among eigenforms found in the course of the present investigation (10 eigenforms were calculated for each of the eight types of boundary conditions), the exact and the approximate theories were established to disagree only in one case.

12.7 Discussion of the Physical Meaning of Obtained Results From the physical viewpoint, it seems obvious that the high-frequency vibrations are produced by the shear phenomena. The presented asymptotic estimates confirm this assertion. Indeed, for the low-frequency vibrations the vector Ψ = −∇w + γ of rotation angles is actually determined by the deflection w, and the vector γ of transverse shear deformation represents the unimportant refinements: γ ∼ hΨ in the vicinity of the boundary and γ ∼ h2 Ψ inside the domain. For the high-frequency vibrations the situation is quite opposite: the vector Ψ of rotation angles practically coincides with the vector of transverse shear deformation γ, and the deflection w adds unimportant refinements:Ψ ∼ γ, ∇w ∼ hΨ. Since the nature of the high-frequency shear and high-frequency bending vibrations is the same (in particular, this is confirmed by the coincidence of leading eigenfrequency terms in the shear and bending spectra), the equations for both vibrations naturally seem to be similar. This is just the case in the suggested statement of problem for the high-frequency vibrations: Eq. (12.13) for the bending vibrations practically coincides with Eq. (12.8) for the shear vibrations. Thus, if we assume similarity of the bending and shear vibrations, then the order of the system with respect to the space coordinates is reduced. This means that yet another function is not included. Since we already have the equations for the shear and bending vibrations, the equation for this function is likely to be the static equation. In the presented statement, Eq. (12.15) for ϕ just does not contain the time derivatives. From the physical considerations, it is obvious that this function characterizes the bending phenomena and has no importance in the problem of highfrequency vibrations. This assertion is supported by the fact that this function rapidly varies with respect to the space coordinates, as we can see from Eq. (12.15), and penetrates into the entire plate domain. Would this function be of primary importance for high-frequency vibrations, we should adopt that the plate theory does not apply to this problem. Thus, the possibility of stating the problem without rapidly varying functions seems to be natural from the physical viewpoint. The boundary conditions in the proposed statement are also worth saying a few words. First, it must be noted that there are four possible boundary conditions, and two of them are kinematic: Ψν |c = 0 (the angle of rotation about the tangent) and Ψτ |c = 0 (the angle of rotation about the normal); the other two are the force conditions: Mν∗

c = 0 (reduced bending moment) and Mτ |c = 0 (torque). Note that the meaning of the kinematic and the force conditions slightly change in proceeding from the original statement to the approximate statement in the Reissner’s type theory. For the approximate statement, the conditions Ψν |c = 0 and

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Nν |c = 0 are equivalent, since N = GhΓΨ , and the force condition Nν |c = 0 becomes kinematic. The kinematic condition w|c = 0 disappears in the approximate state ment, and the deflection appears in the condition Mν∗

c = 0 , since Mν∗ = Mν −GhΓw , that is, it becomes part of the force boundary condition. A certain symmetry for the problem of high-frequency vibrations is not restricted by the similarity of Eqs (12.8) and (12.13). The similarity also occurs for the boundary conditions. In fact, the con dition Ψν |c = 0 yields Ψτ |c = 0 , and the condition Mν∗

c = 0 yields Mτ |c = 0 by the substitution Φ → F, F → −Φ. Such symmetry also supports the cited statement (rather aesthetically than physically).

References 1. Reissner, E.: On the theory of bending of elastic plates. J. Math. Phys. 23(1944)4, 184–191 2. Reissner, E.: The effects of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12(1945)2, 69–77 3. Reissner, E.: On transverse bending of plates, including the effects of transverse shear deformation. Int. J. Solids. Struct. 11(1975)5, 569–573 4. Reissner, E.: On the theory of transverse bending of elastic plates. Int. J. Solids. Struct. 12(1976)8, 545–554 5. Mindlin, R. D.: Influence of rotatory inertia and shear on flexural motion of isotropic, elastic plates. J. Appl. Mech. 18(1951)1, 31–38 6. Shen, R. W.: Reissner-Mindlin plate theory for elastodynamics. J. Appl. Math. 3, 179–189 (2004) 7. Reissner, E.: On bending of elastic plates. Quart. Appl. Math. 51, 55–68 (1947) 8. Reissner, E.: A note on bending of plates, including the effects of transverse shearing and normal strains. ZAMP. 32(1981)6, 764–767 9. Wenbin, Yu: Mathematical construction of a Reissner-Mindlin plate theory for composite laminates. Int. J. Solids Struct. 42(2005)26, 6680–6699 10. Batista, M.: Refined Mindlin-Reissner theory of forced vibrations of shear deformable plates. Engineering Structures. 33(2011)1, 265–272 11. Mindlin, R. D., Schacknow A., Deresiewicz H.: Flexural vibrations of rectangular plates. J. Appl. Mech. 23(1956)3, 430–436 12. Shen, H. S., Yang, J., Zhang, L.: Free and forced vibrations of Reissner–Mindlin plates with free edges resting on elastic foundation. J. Sound Vibr. 244(2001)2, 299–320 13. Kuznetsov, E. N.: Edge effect in the bending inextensible plates. J. Appl. Mech. 49(1982)3, 649–651 14. Berdichevskii, V. L.: A high-frequency long-wave vibrations of plates. Dokl. AN SSSR. 235(1977)6, 1319–1322 15. Altenbach, H., Zhilin, P. A.: The general theory of elastic simple shells. Advances in Mechanics. 11(1988)4, 107–148 16. Zhilin, P. A.: On the Poison and Kirchhoff theory of plates from the modern point of view. Izv. RAN. Mekhanika Tverdogo Tela (Mechanics of Solids). (1992)3, 18–64 17. Bolotin, V. V. (ed): Vibrations in Engineering (in Russian). Vol. 1, Mashinostroenie, Moscow (1978) 18. Zhilin, P. A., Il’icheva, T. P.: Vibration spectra and modes of rectangular parallelepiped, obtained on the basis of three-dimensional elasticity and plate theory. Izv. AN SSSR. Mekhanika Tverdogo Tela (Mechanics of Solids). (1980)2, 94–103

Chapter 13

On the Reconstruction of Inhomogeneous Initial Stresses in Plates Rostislav D. Nedin and Alexander O. Vatulyan

Abstract The direct problem formulation of the steady-state vibration of a thin elastic isotropic plate with non-homogeneous prestress field is described using the variational principle of Lagrange. The solving calculations of the model direct problems were made using finite element method. The investigation of the prestress level influence on the frequency response functions of the plate points was made. The operator relation of the inverse problem is constructed, which binds together the unknown prestress components and the given displacement field functions in the frequency set. The series of computation experiments on the identification of the smooth laws of the uniaxial pretension in the rectangular plate was conducted. On the base of these experiments, the conclusion about the efficiency of the proposed scheme of solving the important inverse problems class was drawn. The preferable frequency ranges are given for that the reconstruction of the initial stresses is most efficient. Keywords Thin plates · Inhomogeneous initial stresses · Inverse problem · Identification

13.1 Introduction The solid equilibrium problems in the presence of the initial stress state have been attracted attention of researchers for a long time. Stresses that exist in the solid without an application of any external (force or thermal) actions are termed initial stresses (also prestresses, residual stresses). Such stresses are often the results of welding, toughening, heat treatment and other manufacturing processes [1,3–5,10]. R. D. Nedin (B) · A. O. Vatulyan Southern Federal University, Faculty of Mathematics, Mechanics and Computer Sciences, ul. Milchakova 8a, 344090, Rostov-on-Don, Russia e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 13, © Springer-Verlag Berlin Heidelberg 2011

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Mostly the homogeneous initial stress state model is used, at that it is enough to measure the elastic waves velocities for the estimation of such stresses [14]. At the same time the initial stress state is often essentially non-homogeneous and it depends on coordinates, especially about stress concentrators — cavities, insertions, cracks, weld seams neighborhoods. The opportunity of non-homogeneous prestress state identification can be used in extremely claiming nondestructive method in the fields of building, mechanical engineering, oil-and-gas sphere, biomechanics, modeling of manufacturing methods of the composite and functionally-gradient materials production. One of the most efficient nondestructive methods of non-homogeneous prestress state estimation is acoustical method. In the present paper the acoustical method is used for the theoretical research of the non-homogeneous prestress state determination opportunity. Even in the limits of the linearized model, the solving of the identification problem of the non-homogeneous prestress state can be realized only on basis of the solving the inverse elasticity theory problem with variable characteristics, which is non-linear and ill-posed problem. At that even for the influence estimation calculations, the elasticity theory problems with variable characteristics must be solved, and that is possible only with use of computing techniques, for example the finite element method (FEM) [16]. The principal difficulty of identification problems research is the complicated construction procedure of operator equation, which connects the unknown and given functions. And that conditioned by the variability of the differential operators coefficients and by the impossibility to construct the general presentations of corresponding elastic theory problems solutions in an explicit form [15]. In the present paper the direct problem formulation of the steady-state vibration of a thin elastic isotropic plate with non-homogeneous prestress field is described. In the case of homogeneous prestress state, such formulation was given in [12, 13]. On the basis of linearization method the inverse problem was reduced to the Fredholm integral equation of the first kind with continuous kernel, which was solved using Tikhonov’s regularization method [11, 17]. The solving calculations of the model problems were made using FEM in the package FreeFem++ [6, 22]; the series of computation experiment on the identification of the smooth laws of the uniaxial pretension in the rectangular plate was conducted; the preferable frequency ranges are given for that the reconstruction of the initial stresses is most efficient.

13.2 Formulation of Out-of-Plane Vibrations of the Thin Prestressed Plate Consider a thin elastic isotropic plate occupying the volume V with the boundary ∂V of arbitrary section outline and thickness 2h. Denote a plane section in the middle level by S . Regard ∂S = l = lu ∪ lσ ; the plate is fixed rigidly on the lateral surface part lu × [−h, h] and it is free from loads on lσ × [−h, h] (Fig. 13.1).

13 On the Reconstruction of Inhomogeneous Initial Stresses in Plates

167

Fig. 13.1 On the left: plate’s image in Cartesian coordinates. On the right: plate’s section of area S in the middle level (i.e. x3 = 0)

The stationary vibrations conditions of the plate are caused by periodic body q iωt forces with intensity 2h e , directed along the axis x3 . The linearized vibration equations are T i j , j +ρω2 ui + F i = 0, (13.1) where the components of the non-symmetric Piola tensor are [5, 10] T i j = σi j + ui,m σ0m j ,

(13.2)

the components of the symmetric tensor σi j submit to the Hooke’s law [9] σi j = λ∗ δi j uk,k + 2μ#i j ,

(13.3)

2λμ where λ∗ = λ+2μ — Lame parameter for two-dimensional stress state [11], μ — shear modulus (these coefficients are constants), σ0m j — the components of the symmetric prestress tensor, submitting the equilibrium equations:

σ0i j , j = 0.

(13.4)

Consider small deformations, regarding the Cauchy relations: & 1% ui , j +u j ,i . 2

(13.5)

T i j n j |∂V = Pi = 0.

(13.6)

#i j = Suppose no boundary loads:

In this problem let’s consider existing of 2-dimensional non-homogeneous prestess field in the plate, which is described by three components of prestress tensor σ011 , σ012 , σ022 being the functions of the coordinates x1 , x2 . We take Kirchhoff– Love hypotheses for thin plates, i.e. we neglect the stress component σ33 and all the components of the displacement vector depend on one function w = w(x1 , x2 ): u1 = −x3 w,1 ;

u2 = −x3 w,2 ;

u3 = w.

(13.7)

168


Therefore the nonzero components of the stress tensor become . / σ11 = −x3 (λ∗ + 2μ)w,11 +λ∗ w,22 , σ12 = −2μx3 w,12 , . / σ22 = −x3 λ∗ w,11 +(λ∗ + 2μ)w,22 .

(13.8) (13.9) (13.10)

13.3 Derivation of the Motion Equation and the Boundary Conditions Let us apply the variational principle of Lagrange to the general formulation of the vibration problem of 3-dimensional solid, taking into account (13.2)–(13.10) [2, 7, 8, 19]. Consider the motion equations (13.1) T i j , j +ρω2 ui + F i = 0. Multiplying these equations by displacement variations δui and integrating by volume V, we have % & T i j , j δui + ρω2 ui δui + Fi δui dV = 0, V

or

% & (T i j δui ), j −T i j (δui ), j +ρω2 ui δui + F i δui dV = 0. V

Applying the Gauss–Ostrogradski formula, we obtain: T i j n j δui d(∂V) − T i j δui , j dV + ρω2 ui δui dV + Fi δui dV = 0. ∂V

V

V

V

By lack of external forces (13.6), the first integral vanishes. Therefore − T i j δui , j dV + ρω2 ui δui dV + F i δui dV = 0. V

V

(13.11)

V

Regarding V = S × [−h, h] and using (13.7)–(13.10), let’s integrate all terms in (13.11) by x3 . For that re-arrange elements of integration in every integral. The q body force vector has nonzero component F 3 = 2h . Hence V

1 F i δui dV = 2h

V

1 qδu3dV = 2h

h

1 · dx3

−h

qδwdS =

S

qδwdS . S

(13.12)


169

The displacement variations are δu1 = −x3 δw,1 ;

δu2 = −x3 δw,2 ;

δu3 = δw.

(13.13)

The nonzero T i j expressions are T 11 = σ11 + σ011 u1 ,1 +σ012 u1 ,2 ,

(13.14)

= σ12 + σ012 u1 ,1 +σ022 u1 ,2 , = σ12 + σ011 u2 ,1 +σ012 u2 ,2 , = σ22 + σ012 u2 ,1 +σ022 u2 ,2 , = σ011 u3 ,1 +σ012 u3 ,2 , = σ012 u3 ,1 +σ022 u3 ,2 .

(13.15)

T 12 T 21 T 22 T 31 T 32

(13.16) (13.17) (13.18) (13.19)

Taking into consideration (13.13)-(13.19), let us integrate the equality (13.11) by coordinate x3 and use the Green’s formula. After simplification we have: ? 2 2 2 − h3 T˜ αβ δw,α nβ d(∂S ) + h3 T˜ αβ ,β δw,α dS + h3 ρω2 wα δw,α dS + 3 3 3 ∂S

S

+ 2h

ρω2 wδwdS − 2h

S

S

σ0αβ w,β δw,α dS +

S

qδwdS = 0, S

introducing σ˜ 11 = (λ∗ + 2μ)w,11 +λ∗ w,22 , σ˜ 12 = 2μw,12 , σ˜ 22 = λ∗ w,11 +(λ∗ + 2μ)w,22 , T˜ 11 = (λ∗ + 2μ)w,11 +λ∗ w,22 +σ011 w,11 +σ012 w,12 , T˜ 12 = 2μw,12 +σ0 w,11 +σ0 w,12 , 12

22

T˜ 21 = 2μw,12 +σ011 w,12 +σ012 w,22 , T˜ 22 = λ∗ w,11 +(λ∗ + 2μ)w,22 +σ012 w,12 +σ022 w,22 . Note, that all the function marked with tilde don’t depend on x3 . Using the constructions which are typical for the plates theory [7, 12], we finally obtain 2 − DΔ2 w + h3 (σ0mβ w,αm ),αβ −2hσ0αβ w,αβ + 3 S

2 + h3 ρω2 Δw − 2hρω2 w − q δwdS − 3

170


Fig. 13.2 Directions of the tangent normal τ and the outer normal n at an arbitrary point of the closed contour ∂S [7]

? − ∂S

∂ Gt (δw)d(∂S ) + ∂n

? ∂S

∂ 2 ∂w (Ht ) − Nt + h3 ρω2 δwd(∂S ) = 0, ∂τ 3 ∂n

(13.20)

2 2Eh3 where h3 (λ∗ + 2μ) = = D is the cylindrical rigidity of plate; 3 3(1 − ν2 ) 2 G t = h3 T˜ αβ nα nβ — bending moment on ∂S ; 3 2 Ht = h3 (T˜ 2β nβ n1 − T˜ 1β nβ n2 ) — twisting moment on ∂S ; 3 ∂(Δw) 2 3 0 Nt = −D − h (σmβ w,αm ),β nα + 2hσ0αβ nα w,β — shear force on ∂S ; ∂n 3 T˜ αβ = λ∗ δαβ w,kk +2μw,αβ +w,αm σ0mβ , ∂ ∂n

and

∂ ∂τ

α, β, k, m = 1, 2;

— normal and tagentional derivatives to ∂S relatively (Fig. 13.2).

Equating the coefficients by independent variations to zero, we get the boundary problem: 2 2 DΔ2 w + h3 (σ0mβ w,αm ),αβ −2hσ0αβ w,αβ + h3 ρω2 Δw − 2hρω2 w − q = 0, 3 3

∂w

= 0, G t |lσ = 0, w|lu = 0, (13.21) ∂n lu

∂ 2 ∂w

(Ht ) − Nt + h3 ρω2

= 0. ∂τ 3 ∂n l σ

Note, that in case of homogeneous prestress field (σ0αβ = const) and neglect of rotatory inertia 23 h3 ρω2 Δw, we obtain the equation given in [12, 13]. In case of band plate the formulation (13.21) and the formulation which given in [20] are congruent.

13.4 Solution of the Direct Problem The computations of the direct problems solutions were obtained using FEM in the package FreeFem++. The series of computations experiments on the solving of direct problem (13.21) was made for various section shapes (rectangles, ellipses,


171

Fig. 13.3 Image of the prestressed cantilever with rectangular section under uniformly distributed periodic load of intensity q. At that l is the cantilever’s length, b and h are the section dimensions. This cantilever may be considered as a band plate of hight h and section with dimensions l and b which is clamped at one side edge

rings, T-shaped regions and others), boundary conditions and types of prestress state (uniaxial and 2-dimensional). All the components of the prestress tensor σ0αβ , α, β = 1, 2 were taken as the functions of coordinates; at that the continuous functions were used (linear, quadratic, exponential, trigonometric) and the piecewise functions with several break points were used too. As a result of problem solving the displacement field w(x1 , x2 , ω) was calculated for the given vibration frequency ω. The accuracy analysis was made for the FreeFem+ solution of the problem (13.21). For that the obtained solution in case of band plate (the section shape is the elongate rectangular) with homogeneous characteristics (the material parameters, the density, the components of the prestress tensor) was compared with the analytical solution of the analogous problem of bending vibrations of the prestressed cantilever with rectangular section under uniformly distributed load (Fig. 13.3) [20]. At that, the case of uniaxial prestress state was considered, i.e. the only nonzero component of the prestress tensor is the component σ011 = const. The prestressed cantilever problem is described by the following boundary problem for the differential equation of the fourth order [20] (J(E + σ011 )w ) − (Fσ011 w ) − ρFω2 w − q = 0, w(0) = 0, w (0) = 0, w (l) = 0, (J(E + σ011 )w (l)) − (Fσ011 w (l)) = 0,

(13.22) (13.23) (13.24)

where l — rod length, J = 2bh3 /3 — moment of inertia, b — section width, 2h — section hight, F = 2bh — section area, q — uniformly distributed load value. Note, that we use slightly modified formulation of the prestressed cantilever problem — instead of girder characteristics Young modulus E we take the value E/(1 − ν2 ), which is included in the cylindrical rigidity formula (ν is the Poisson coefficient).

172


Remark. You can derive the formulation of the prestressed cantilever vibration problem (13.22)-(13.24) with homogeneous material parameters (E and ν) from the formulation of the direct vibration problem of thin prestressed plate (13.21), deduced in Sect. 13.3 . For that the special case must be considered: w = w(x1 ), σ011 (x2 ) 0, the rest components of the prestress tensor equal zero. The solution of the boundary problem (13.22)–(13.24) in dimensionless form is w(ξ) = C1 cosh βξ + C2 sinh βξ + C3 cos θξ + C4 sin θξ + where ξ = x/l ∈ [0, 1] and @ A B8 κ14 + 4(1 + τ)κ22 + κ12 β= , 2(1 + τ) σ011

@ A B8 θ=

q0 , κ24

κ14 + 4(1 + τ)κ22 − κ12 2(1 + τ)

(13.25)

,

Fl2 ρFl4 ω2 ql2 τ, κ22 = τ, q0 = , E J EJ EJ and Ci — constants revealed from the boundary conditions (13.23)–(13.24) (since the last one are not manageable, these constants are not presented). The various ratios of rod dimensions were considered for carrying out the accuracy analysis. The result for the rod with following parameters is shown below: l = 0.2 m, b = 0.013 m, h = 0.01 m, E = 1.96 · 1011 Pa, ν = 0.29, ρ = 7.8 · 103 kg/m3 , q = 300 kg/m, σ011 = 98 MPa. The frequency response function for the tag of the rod with foregoing parameters is illustrated in Fig. 13.4. ω The whole frequency range f = 2π ([ f ] = Hz) consists of few region divided by resonance frequencies f1 = 15.1 Hz, f2 = 87.3 Hz, f3 = 241.1 Hz, f4 = 471 Hz. Below the range 0 < f < f1 is termed as the first frequency range, the range f1 < f < f2 — as the second frequency range, the range f2 < f < f3 — as the third frequency range, the range f3 < f < f4 — as the fourth frequency range. τ=

,

κ12 =

Fig. 13.4 Frequency response function of the cantilever’s tag. Frequencies f1 = 15.1, f2 = 87.3, f3 = 241.1 are the bend resonances


173

The comparing results of the analytical solution wr (x1 ) for the rod with the numerical solution wp (x1 , b/2) for the plate are illustrated in Fig. 13.5. δ is given as the numerical solution error in percent δ=

|wp (x1 , b/2) − wr(x1 )| 100% max |wr (x1 )|

The comparing analysis of the solutions of two problems revealed that the relative error in the nodes of the numerical solution wp (x1 , x2 ) in comparison with the analytical solution wr (x1 ) for the frequencies from the first and the second frequency ranges is less than 0.07% for the finite elements mesh 20 × 100. This implies very close approximation scheme and allows to use FreeFem++ for solving other problems in future. The isolines graphics of the FEM-solution wp (x1 , x2 ) for the plate with parameters a = 2 m, b = 1 m, h = 0.1 m, E = 1.96 · 1011 Pa, ν = 0.29, ρ = 7.8 · 103 kg/m3, q = 300 kg/m, for different prestress states are shown in Figs 13.6–13.8 below (the axis x1 is directed vertically downward, the axis x2 is directed horizontally to the right). The results for the plate with a clamped boundary (w = ∂w ∂n = 0 at the ∂S ) in case of statics ( f = 0 Hz) are shown in Fig. 13.6. The results for the pre-stretched plate ∂w (σ011 = const) which is clamped at one side edge (w = ∂x = 0 at the boundary x1 = 1 0), for various frequency values, are illustrated in Fig. 13.7. The isolines graphics for the plate with same boundary conditions as in Fig. 13.7 in the case of nonhomogeneous uniaxial prestresses σ011 (x2 ) for the frequency value f = 0.7 Hz are given in Fig. 13.8.

13.5 Analysis of the Prestress Level Influence on the Frequency Response Functions of the Plate Points The investigation of the prestress σ011 = const influence on the frequency response functions of the rectangular plate point {x1 = a; x2 = b/2; x3 = 0} was made for the first and the second frequency ranges, for the various ratios of plate dimensions a : b : h (Fig. 13.9). The graphics of the prestresses influences on the frequency response functions for the ratio of plate dimensions a : b : h = 20 : 10 : 1 are shown in the Fig. 13.10. The investigation of the prestress level influence on the frequency response functions of the rectangular plate points was made for the various ratios of plate dimensions a : b : h. It is significant that I almost does not depend on ratio a : b; on the other hand it depends essentially on ratios h : b and h : a. The plate is thicker (i.e. the coefficient h/a or h/b is more) the I is less (i.e. the prestress level σ011 influences weaker on the frequency response functions) At the same time, with increase of these coefficients all the resonance frequencies increase too; the frequencies change

174


Fig. 13.5 The comparing results of the analytical solution wr (x1 ) (shown as firm line) for the rod with the numerical solution wp (x1 , x2 ) (shown as small squares) for the plate. Four different vibration modes are considered: a — from the first frequency range, for f = 7 Hz; b — from the second frequency range, for f = 40 Hz; c — from the third frequency range, for f = 120 Hz; d — from the fourth frequency range, for f = 300 Hz (see also Fig. 13.4)


175

is often used in the identification problem in the case of the homogeneous stress fields. Thus, the experimental results demonstrated that the discrepancy of the frequency response functions quite enough for using prestress reconstruction procedure [21].

13.6 Formulation of the Inverse Problem of the Non-homogeneous Prestress Reconstruction in the Solid In [17] the reciprocity relation was built, which allows to construct the iterative process in the prestresses identification problem. Set a problem of function σ0i j (x1 , x2 ) identification using some additional information. The most popular ways of its definition are the following two: 1) the assignment of the displacement field w inside the region for a fixed frequency; 2) the assignment of the displacement field w at

Fig. 13.6 From left to right: σ0αβ = 0; σ011 = const; σ022 = const; σ012 = const; σ0αβ = const

Fig. 13.7 From left to right: f = 0.5 Hz; f = 5 Hz; f = 16 Hz; f = 30 Hz; f = 80 Hz

176


π Fig. 13.8 From left to right: σ011 = 0 Pa; σ011 (x2 ) = 109 sin( 100 x2 ) Pa; σ011 (x2 ) = 106 (e0.05x2 − 0.5) 0 6 0.035x 0 6 0.045x 2 Pa; σ (x ) = −10 (e 2 − 100) Pa, x ∈ [0, 100] Pa; σ11 (x2 ) = −5 · 10 sin(0.04x2 )e 2 11 2

the boundary part lσ in the frequency set ωk ∈ [ω− , ω+ ]. In the limits of the first definition way the inverse problem is linear, in the limits of the second definition it’s nonlinear [18]. Consider the second way of the additional information assignment. Let’s apply the fundamental reciprocity relation to the out-of-plane vibration problem of a plate. Using the hypotheses and assumptions for plates, we have * 2 2 δσ011 h3 (w,211 +w,212 ) + 2hw,21 + 2δσ012 h3 w,12 Δw + 2hw,1 w,2 3 3 S

+δσ022

2 3 2 h (w,12 +w,222 ) + 2hw,22 3

+

dS =

q(w − f )dS ,

(13.26)

∂S

where δσ0αβ — corrections to the prestress functions σ0αβ . Consider the case of uniaxial prestretch along the axis x1 — the only one nonzero component of the prestress tensor is σ011 . For satisfaction of the equilibrium equations we regard that the function σ011 depends only on the coordinate x2 . In that case the equation (13.26) takes the form

Fig. 13.9 Image of the prestretched rectangular plate in Cartesian coordinates. At that σ011 is the only nonzero prestress tensor component. The plate is clamped

at one ∂w

edge: w| x1 =0 = ∂x1 =0 x1 =0


177

Fig. 13.10 The analysis of the prestress level influence on the frequency response functions of the plate point x1 = a; x2 = b/2; x3 = 0. At these pictures the following plate parameters are taken: a = 2 m, b = 1 m, h = 0.1 m, E = 1.96 · 1011 Pa, ν = 0.29, ρ = 7.8 · 103 kg/m3 , q = 300 kg/m. On the left figure the first frequency range (0, f1 ) is considered, on the right figure — the second one ( f1 , f2 ) ( f1 = 1.6 Hz — the first resonance frequency, f2 = 8.8 Hz — the second resonance frequency). The I value characterizes the prestress influence value in comparison with no-prestress state (σ011 = 0) in

percentage terms. This value is defined by formula I =

w(a,b/2)| 0 −w(a,b/2)|σ0 =0

σ 0

11

11 100%.

w(a,b/2)|σ0 =0

11

The influ-

ence graphics are shown using different line styles corresponding to the values of the dimensionless parameter τ = σ011 /E. Several values of τ lying in the range 10−5 ÷ 10−3 were considered

δσ011 S

2 3 2 h (w,11 +w,212 ) + 2hw,21 dS = 3

q(w − f )dS .

(13.27)

∂S

The presented equation is the Fredholm integral equation of the first kind relatively the correction δσ011 to the unknown uniaxial prestress function. Remark. In the paper [21] the nonlinear inverse problem of in-plane vibration of the plate with non-homogeneous uniaxial prestress field was formulated and solved. At that the various ways of loading at the plate’s lateral surface were considered. The Fredholm integral equation of the first kind was constructed analogously to (13.27). The scheme of the non-homogeneous uniaxial prestress function identification was proposed, which was based on the iterative process construction; on the base of computing experiments for the different laws of variation, the conclusion about the efficiency of the inverse problem solving scheme was drawn. The preferable loads and frequency ranges were given for that the reconstruction of the prestress function was the most efficient.

178


13.7 Formulation of the Inverse Problem of the Uniaxial Non-Homogeneous Prestress Reconstruction in the Rectangular Plate Now consider the inverse problem for the rectangular plate with uniaxial nonhomogeneous prestress field existing inside it (Fig. 13.9). The problem takes the following form 2 2 DΔ2 w + h3 (σ0mβ w,αm ),αβ −2hσ0αβ w,αβ + h3 ρω2 Δw − 2hρω2 w − q = 0, 3 3 ∂w w|lu = 0, |l = 0, Gt |lσ = 0, ∂n u

∂ 2 3 2 ∂w

(Ht ) − Nt + h ρω

= 0, ∂τ 3 ∂n l σ

where lu = {x1 = 0; x2 ∈ [0, b]}, lσ = {x1 = a; x2 ∈ [0, b]} ∪ {x1 ∈ [0, a]; x2 = 0} ∪ {x1 ∈ [0, a]; x2 = b}. As given parameters we use the plate’s dimensions, the material constants (the q cylindrical rigidity D and the density ρ), the body forces intensity 2h . Let’s set the 0 identification problem of the prestress function σ11 (x2 ). In the event that the additional information is the assignment of the displacement field w at the whole area S in the frequency set ωk ∈ [ω− , ω+ ], the inverse problem becomes linear and reduces to the solving of the Fredholm integral equation of the first kind (on the basis of (13.27)) b K(x2 , ω)σ011 (x2 )dx2 = F(ω), ω ∈ [ω− , ω+ ], (13.28) 0

where

a K(x2 , ω) =

2 3 2 h (w,11 +w,212 ) + 2hw,21 dx1 , 3

(13.29)

0

b a F(ω) =

q(w − f )dx1dx2 , 0

(13.30)

0

Thereby the operator relation is constructed, which binds together the desired quantity (i.e. the prestress component) and the given displacement field functions in the frequency set. Note, that solving of the presented Fredholm integral equation of the first kind (13.28) is ill-posed problem. For its solving the Tikhonov’s regularization method with the automatic regularization parameter searching was used [11].


179

13.8 Numerical Experiments on the Prestress Identification The series of computing experiments on the identification of the uniaxial nonhomogeneous prestress σ011 (x2 ) was conducted. The displacement field w at the area S was given. The various function classes were considered for the reconstruction — linear, polynomial, exponential, trigonometrical and more complicated analytical dependencies [21]. The vibration frequencies f of the applied load were chosen among the first, second and third frequency ranges of the frequency response functions. In all the experiments the prestress functions σ011 (x2 ) were chosen in such way that the fracσ0

tion max E11 (E is the Young modulus) varied in the range 10−5 ÷ 10−3 . In the majority of computations the essential deterioration of the unknown function σ011 (x2 ) identification had a place at the ends of segment x2 ∈ [0, b]. In some cases such error at the ends exceed 30%, whereas in the distance of the ends it varied up to 10%. Such accuracy deterioration of the inverse problem solution relates with the rough calculation of the second order partial derivatives of w function, which appear in the kernel formula (13.29). It is significant that the kernel does not tend to zero near the ends of the segment x2 ∈ [0, b]. In the event that the values σ011 (0) and σ011 (b) are given, the identification quality near these ends is much more high. As it turned out, for the majority of the experiments the consideration of the second frequency range gives a much more accurate identification result, than the consideration of the first or third frequency ranges. The results of uniaxial prestress function identification for the plate with parameters a = 2m, b = 1m, h = 0.1m, E = 1.96 · 1011Pa, ν = 0.29, ρ = 7.8 · 103kg/m3 are shown below. The resonance frequencies are f1 = 1.6Hz, f1 = 8.8Hz, f1 = 23.1Hz, f1 = 43Hz. The identification results in the second frequency range are illustrated in Fig. 13.11–13.13. At that the values σ011 (0) and σ011 (b) are accepted as given. The exact function σ011 (x2 ) is shown at the graphics as a firm line; the identification result — as small squares. The error δ of the inverse problem solution is given in percentage terms by the formula

σ011 (x2 ) − σ0∗

11 (x2 ) δ= % & 100%, 0∗ max |σ11 (x2 )| x2 ∈[0,b]

where σ011 (x2 ) — the identified law of variation, σ0∗ (x ) — the exact law of varia11 2 tion of prestress.

180

Fig. 13.11 Identification results. Exact law σ011 (x2 ) = 2.5 · 106 (e0.035x2 + 4) Pa, x2 ∈ [0, 100]; f− = 1Hz, f+ = 5Hz

Fig. 13.12 Identification results. Exact law σ011 (x2 ) = −106 (e0.045x2 − 100) Pa, x2 ∈ [0, 100]; f− = 3 Hz, f+ = 8 Hz

Fig. 13.13 Identification results. Exact law σ011 (x2 ) = −5 · 106 sin(0.04x2 )e0.035x2 Pa, x2 ∈ [0, 100]; f− = 2.9 Hz, f+ = 5 Hz



181

13.9 Conclusion The direct problem formulation of the steady-state vibration of a thin elastic isotropic plate with non-homogeneous prestress field is described using the variational principle of Lagrange. The solving calculations of the model direct problems were made using FEM in the package FreeFem++. The investigation of the prestress level influence on the frequency response functions of the plate points was made. It revealed that the frequency ranges near the resonances are the most efficient in the view of the identification. Within the limits of Newton method ideology of solving nonlinear inverse problems, the first kind integral Fregholm’s equation with continuous kernel with respect to correction of the prestress functions is formulated. The series of computation experiments on the identification of the smooth laws (linear, polynomial, exponential, trigonometrical and more complicated analytical dependences) of the uniaxial pretension in the rectangular plate was conducted. On the base of computing experiments, the conclusion about the efficiency of the proposed scheme of solving the important inverse problems class was drawn. The preferable frequency ranges are given for that the reconstruction of the initial stresses is most efficient. Acknowledgements The present work is done with the support of Russian Foundation of Basic Research “Development of new methods for identification of localized and dispersed heterogeneousness in solid bodies” (project code 10-01-00194-a) and Federal Target Program “Scientific and research-and-educational staff of innovation Russia” on 2009 - 2013 years (state contract P596).

References 1. Birger, I.A.: Residual Stresses (in Russian). State scientific-and-technical publishing of machine-building literature, Moscow (1963) 2. Bittencourt, M.L., Pereiray C.E.L.: Procedures for Teaching Variational Formulation and Finite Element Approximation of Mechanical Problems applied to the Kirchhoff Plate Model. Department of Mechanical Design, State University of Campinas, Campinas, Brazil (2004) 3. Deaconu, V.: Finite Element Modelling of Residual Stress – A Powerful Tool in the Aid of Structural Integrity Assessment of Welded Structures. 5th Int. Conference Structural Integrity of Welded Structures (ISCS2007), Timisora, Romania (2007) 4. Feng, Zh.: Processes and Mechanisms of Welding Residual Stress and Distortion. Woodhead Publishing Limited, Cambridge, England (2005) 5. Guz, A. N.: Elastic Waves in Compressible Materials with Initial Stresses and Nondestructive Ultrasonic Method of Two-Ply Residual Stresses Definition (in Russian). Int. Appl. Mechanics, v. 30, 1, 3–17 (1994) 6. Hecht, F., Pironneau, O., Le Hyaric A., Ohtsuka K.: FreeFem++. Version 2.17-1.. (Available via DIALOG, 1999), http://www.freefem.org/ff++ 7. Leybenzon, L. S.: Course of Elasticity Theory (in Russian). State Publishing of Technicaland-Theoretical Literature, Leningrad (1947) 8. Mikhlin, S.G.: Variational Methods in Mathematical Physics (in Russian). Nauka, Moscow (1970) 9. Novatsky, V.: Theory of Elasticity (in Russian). Mir, Moscow (1975)

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10. Robertson, R.L.: Determining Residual Stress from Boundary. Measurements: A Linearized Approach. Journal of Elasticity, vol 52. Kluwer Academic Publishers, Netherlands, 63–73 (1998) 11. Tikhonov, A.N., Arsenin, V.Ya.: Methods of Ill-Posed Problems Solving (in Russian). Nauka, Moscow (1979) 12. Timoshenko, S.P.: Course of Elasticity Theory (in Russian). Naukova Dumka, Kiev (1972) 13. Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells (in Russian). Nauka, Moscow (1966) 14. Tovstik, P.E.: Vibration and Stability of Prestressed Plate Lying on Elastic Foundation. Appl. mechanics and mathematics, v.1, 106–120 (2009) 15. Vatulyan, A.O.: Identification Problems of Inhomogeneous Properties of Solids. Bulletin of Samara State University, issue 54, 4, 93–103 (2007) 16. Vatulyan, A.O.: Inverse Problems in Mechanics of Deformable Solids (in Russian). Phizmatlit, Moscow (2007) 17. Vatulyan, A.O.: Iterative Processes in Inverse Coefficient Problems. Abstracts of XIV international conference ”Present-day problems of continuum mechanics”. South Federal University Publishing, Rostov-on-Don, Russia, 81–85 (2010) 18. Vatulyan, A.O.: Regarding Inverse Coefficient Problems in Linear Mechanics of Deformable Solids. Appl. mechanics and mathematics, 6, 911–918 (2010) 19. Vatulyan, A.O.: Regarding Variational Formulation of Inverse Coefficient Problems for Elastic Bodies. Proceedings of RAS, v. 422, 2, 182–184 (2008) 20. Vatulyan, A.O., Dudarev, V.V.: Concerning Some Problems of Non-Homogeneous Prestress State in Elastic Bodies. The publishing of Saratov University. New series. S. Mathematics. Mechanics. Computer Science, v. 9, issue 4, part 2, 25–32 (2009) 21. Vatulyan, A.O., Dudarev, V.V., Nedin, R.D., Saakyan, Ya.G.: Regarding Some Problems of Prestress Identification. Abstracts of XIV international conference ”Present-day problems of continuum mechanics”. South Federal University Publishing, Rostov-on-Don, 86–90 (2010) 22. Zhukov, M.Yu., Shiryaeva, E.V.: Using Finite Element Package FreeFem++ for Problems of Hydrodynamics, Electrophoresis and Biology. South Federal University Publishing, Rostovon-Don (2008)

Chapter 14

Dynamic Response of Pre-Stressed Spatially Curved Thin-Walled Beams of Open Profile Impacted by a Falling Elastic Hemispherical-Nosed Rod Yury A. Rossikhin and Marina V. Shitikova

Abstract The dynamic stability with respect to small perturbations, as well as the local damage of geometrically nonlinear elastic spatially curved open section beams with axial precompression have been analyzed. Transient waves, which are the surfaces of strong discontinuity and wherein the stress and strain fields experience discontinuities, are used as small perturbations, in so doing the discontinuities are considered to be of small values. Such waves are initiated during low-velocity impacts upon thin-walled beams. The theory of discontinuities and the method of ray expansions, which allow one to find the desired fields behind the fronts of the transient waves in terms of discontinuities in time-derivatives of the values to be found, are used as the methods of solution for short-time dynamic processes. The example of using the ray expansions for analyzing the impact response of spatially curved thin-walled beams of open profile is demonstrated by solving the problem about the normal impact of an elastic hemispherical-nosed rod upon an elastic arch representing itself a channel-beam curved along an arc of the circumference. The influence of the initial stresses on the dynamic fields has been investigated. Keywords Spatially curved thin-walled beam of open profile · Surface of strong discontinuity · Ray method · Impact · Shock interaction

14.1 Introduction Thin-walled beams of open section are extensively used as structural components in different structures in civil, mechanical and aeronautical engineering fields. These structures have to resist dynamic loads such as wind, traffic and earthquake Y. A. Rossikhin (B) · M. V. Shitikova Research Center of Wave Dynamics, Voronezh State University of Architecture and Civil Engineering, Voronezh 394006, Russia e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 14, © Springer-Verlag Berlin Heidelberg 2011

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loadings, so that the understanding of the dynamic behavior of the structures becomes increasingly important. The classical theory of thin-walled uniform open cross-section straight beams as well as horizontally curved ones was developed by Vlasov [30] in the early 60-s without due account for rotational inertia and transverse shear deformations. The Vlasov theory is the generalization of the Bernoulli-Navier law to the thin-walled open section beams by including the sectorial warping of the section into account by the law of sectorial areas, providing that the first derivative of the torsion angle with respect to the longitudinal axis serves as a measure of the warping of the section. However, as it has been recently shown [27] Vlasov’s equations are inappropriate for use in the problems dealing with the transient wave propagation. Many researchers have tried to modify the Vlasov theory for dynamic analysis of elastic isotropic thin-walled beams with uniform cross-section by including into consideration the rotary inertia and transverse shear deformations ([2], [4], [9], [15], [19], [20], [22]), and/or considering coupled bending–torsional vibration of axially loaded thin-walled beams ([3], [8], [17], [31]). The increasing use of curved thin-walled beams in highway bridges and aircraft has resulted in considerable effort being directed toward developing accurate methods for analyzing the dynamic behaviour of such structures ([5], [10], [13], [14], [16], [21]). Curved members in modern bridges and architectural structures continue to predominate because of emphasis on aesthetics and transportation alignment restrictions in metropolitan areas. It is well known that Timoshenko [29] in order to generalize the Bernoulli-Euler beam model has proposed the transverse shear angle to be the independent variable. This starting point was the basis for the derivation of a set of two hyperbolic differential equations describing the dynamic behaviour of a beam, resulting in the fact that two transient waves propagate in the Timoshenkobeam with finite velocities: the longitudinal wave with the velocity equal to G L = E/$, and the wave of trans verse shear with the velocity equal to GT = Kμ/$, where E and μ are the elastic moduli, $ is the density, and K is the shear coefficient which is weakly dependent on the geometry of the beam. Many of the up-to-date technical articles involve the derivation of the equations which, from the authors viewpoint, should describe the dynamic behaviour of thinwalled beams of the Timoshenko type. The critical overview of the cited above papers has been carried out recently in [27]. It has been emphasized that only inclusion into consideration of three factors at a time, namely: (1) transverse shear deformations, (2) rotary inertia, and (3) warping deformations as the independent field – could lead to the correct system of hyperbolic equations of the Timoshenko type for describing the dynamic behaviour of thin bodies. Ignoring one of the factors or its incomplete consideration immediately results in an incorrect set of governing equations. It has been shown that all papers in the field can be divided into three groups. The papers, wherein the governing set of equations is both hyperbolic and correct from the viewpoint of the physically admissible magnitudes of the velocities of the transient waves resulting from these equations, fall into the first category,

14 Dynamics of Spatially Curved Thin-Walled Beams

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i.e. the velocity of the longitudinal wave is G L = E/$, while the velocities of the three transverse shear waves, in the general case of arbitrary cross sections of thinwalled beams with open profile, depend essentially of the geometry of the beam ([7], [15], [20], [21]). There are seven independent unknowns in the displacement field in the general case if only primary warping is included into consideration [15], or with additional three generalized displacements describing the variation of the secondary warping due to non-uniform bending and torsion [7], or with additional three variables describing a “complete homogeneous deformation of the microstructure” [20]. As this takes place, different authors obtain different magnitudes for the velocities of transverse shear waves. The second category involves the articles presenting hyperbolic but incorrect equations from the above mentioned viewpoint due to ignoring at least one of the aspects required, i.e. resulting in incorrect magnitudes of the transient waves. This concerns, first of all, the velocity of the longitudinal waves which should not deviate from G L = E/ρ, nevertheless, there are some examples [19] where such a situation takes place. Secondly, in some papers one can find equations looking like hyperbolic ones ( [3], [22], [30]) but from which it is impossible to obtain the velocity, at least, of one transient wave at all. In such papers, usually six generalized displacements are independent while warping is assumed to be dependent on the derivative of the torsional rotation with respect to the beam axial coordinate [22] or is neglected in the analysis [3]. In other words, there is a hybrid of two approaches: Timoshenko’s beam theory and Vlasov’s thin-walled beam theory, some times resulting to a set of equations wherein some of them are hyperbolic, while others are not. Thirdly, not all inertia terms are included into consideration [16]. The papers providing the governing system of non-hyperbolic equations belong to the third group. In such papers, the waves of transverse shear are the diffusion waves possessing infinitely large velocities, and therefore the dynamic equations presented in [2], [4] and [17] cannot be named as the Timoshenko type equations. The simple but effective procedure for checking for the category, within which this or that paper falls in, has been proposed by Rossikhin and Shitikova [27] based on the theory of discontinuities [28] and illustrated by several examples. It has been shown that only the theories of the first group, such as the Korbut-Lazarev theory which provides the physically admissible velocities of propagation of transient waves, could be used for solving the problems dealing with transient wave propagation, while the theories belonging to the second and third group could be adopted for static problems only. However, the presence of three or four transverse shear waves, which propagate with different velocities dependent strongly on geometric characteristics of the thinwalled beam, severely limits the application of such theories in solving engineering problems. As for the experimental verification of the existence of the three shear waves in thin-walled beams of open section, then it appears to be hampered by the fact that the velocities of these waves depend on the choice of the beam’s cross section. This immediately raises the question of whether it is possible to create such a theory of dynamic behaviour of thin-walled beams, within the framework of which,

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instead of three transverse shear waves, only one shear-like wave will propagate with the velocity independent of the characteristics of the beam’s cross section, as it takes place in the case of the longitudinal-flexural-warping wave. The positive answer to this question will be given below.

14.2 Problem Formulation and Governing Equations For investigating the dynamic behaviour of thin-walled beams of open section, we shall proceed from three-dimensional equations of isotropic elasticity written in the Cartesian system of coordinates x1 , x2 , and x3 . Let us consider a certain unperturbed equilibrium of an elastic body characterized by the displacement vector u0i , stress tensor σ0i j , and the vector of volume forces Xi0 (surface forces are absent). The characteristics of the unperturbed equilibrium state satisfy the following geometrically nonlinear equations and boundary conditions on the surface [6]: {σ0jk (δik + u0i,k )}, j + Xi0 = 0 ,

(14.1)

σ0jk (δik + u0i,k )ν j = 0 ,

(14.2)

where δik is the Kronecker’s symbol, ν j are the components of the unit vector normal to the boundary surface, Latin indices take on the magnitudes 1,2,3, a Latin index after comma denotes the partial derivative with respect to the corresponding spatial coordinate x1 , x2 , x3 , and the summation is understood over the repeated indices. Let us perturb the body with some small deviations from the unperturbed equilibrium: ui , σi j , Xi = −$üi , where $ is the material density, and an overdot denotes the time-derivative. The characteristic components of the perturbed motion u˜ i , σ ˜ ij , and X˜ i take the form u˜ i = u0i + ui ,

σ ˜ i j = σ0i j + σi j ,

X˜ i = −$˙vi ,

(14.3)

where vi is the displacement velocity. The components of the perturbed state satisfy the following equations and boundary conditions: , σ ˜ ik (δik + u˜ i,k ) , j + X˜ i = 0 , ' ( σ˜ jk δik + u˜ i,k ν j = 0 .

(14.4) (14.5)

Substituting (14.3) in (14.4) and (14.5), carrying out the linearization of the resulting equations, and taking (14.1) and (14.2) into account, we find 3 % & 4 σ jk δik + u0i,k + σ0jk ui,k = $üi , (14.6) ,j 3 % & 4 σ jk δik + u0i,k + σ0jk ui,k ν j = 0 . (14.7)


187

Suppose that the unperturbed state deviates a little from the initial undeformed state. In the majority of engineering problems the given assumption is found to be valid. In this case, the terms u0i,k can be neglected, and Eqs (14.2), (14.6) and (14.7) at σ0i j = ci j , where ci j (i, j = 1, 2, 3) are certain constant values (moreover, some of them could be vanished according to the conditions of the problem under consideration), take the following form: σi j, j + σ0jl ui,l j = $˙vi , σi j ν j = 0 ,

σ0i j ν j

=0.

(14.8) (14.9)

Equations (14.8)–(14.9) should be considered together with the relationship % & σ ˙ i j = λvl,l δi j + μ vi, j + v j,i , (14.10) where λ and μ are Lamé constants. Suppose that the wave surface of strong discontinuity exists in a spatially curved thin-walled beam of open section. Then let us differentiate (14.8), (14.10) and (14.9) k times with respect to time, write them on the both sides of the wave surface, and take their difference. As a result we obtain % & . / σi j,(k+1) = λ vl,l(k) δi j + μ vi, j(k) + v j,i(k) , (14.11) . / σi j, j(k) + σ0jl eil, j(k−1) = $ vi,(k+1) , (14.12) σi j,(k) ν j = 0 , (14.13) &+ % &− . / % where ei j = vi, j , and Z,(k) = ∂k Z/∂tk − ∂k Z/∂tk . Equation (14.12) lacks the discontinuities in the external forces, since they are continuous on the wave surface of strong discontinuity. For the ease of further treatment, let us introduce two sets of coordinates: λ , τ, ξ with the unit vectors λ {λi }, τ {τi }, and ξ {ξi }, and λ, x, y with the unit vectors λ, k {ki }, and s {si }. The axes λ, τ, ξ are the natural axes for the curved axis of the beam, in so doing the λ-axis is the tangent to the beam’s axis, the τ-axis is its binormal, the ξ-axis is its main normal, s is the arc length calculated from a certain point with the coordinate s0 along the beam axis (Fig. 14.1), while the x− and y−axes are the main central axes of the beam’s normal section. The angle ϕ(s) is the angle between the x− and τ−axes and the y− and ξ−axes. Following [26], it can be shown that dsi = ki (K + τ) − κλi cos ϕ(s) , ds dki = −si (K + τ) + κλi sin ϕ(s) , ds dλi = −ki κ sin ϕ(s) + si κ cos ϕ(s) , ds

(14.14) (14.15) (14.16)

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Fig. 14.1 Scheme of a spatially curved linear elastic beam of arbitrary open cross-section

where K = dϕ/ds, while κ(s) and τ(s) are the curvature and the torsion of the beam’s axis, respectively. Note that it is precisely these three values, K(s), κ(s) and τ(s), that are of prime engineering interest in studying the spatially curved thin-walled beams of open profile. Moreover, as it will be shown in the further analysis, these values produce new features in the dynamic response of such beams as compared with straight thin-walled beams of open cross-section. Considering the conditions of compatibility ∂ σi j,(k) d[σi j,(k) ] ∂[σi j,(k) ] −1 σi j, j(k) = −G σi j,(k+1) λ j + λj + kj + s j (14.17) ds ∂x ∂y . / . / d[eil,(k) ] ∂ eil,(k) ∂ eil,(k) . / eil, j(k) = −G −1 eil,(k+1) λ j + λj + kj + s j (14.18) ds ∂x ∂y . / . / . / d vi,(k) ∂ vi,(k) ∂ vi,(k) . / . / . / eil,(k) = vi,l(k) = −G−1 vi,(k+1) λl + λl + kl + sl (14.19) ds ∂x ∂y as well as formulas (14.14)–(14.16), Eqs (14.11) and (14.12) can be rewritten as % & [σi j,(k+1) ] = −G−1 λ[vl,(k+1) ]λl δi j − G−1 μ [vi,(k+1) ]λ j + [v j,(k+1) ]λi d[vl,(k) ] ∂[vl,(k) ] ∂[vl,(k) ] +λ λl + kl + sl δ i j ds ∂x ∂y d[v j,(k) ] d[vi,(k) ] ∂[vi,(k) ] +μ λj + λi + kj ds ds ∂x ∂[v j,(k) ] ∂[v j,(k) ] ∂[vi,(k) ] + ki + sj + si , (14.20) ∂x ∂y ∂y


189

d[σi j,(k) ] ∂[σi j,(k) ] ∂[σi j,(k) ] λj + kj + sj ds ∂x ∂y d[vi,(k) ] 0 ∂[vi,(k−1) ] 0 + G−2 [vi,(k+1) ]σ0λλ − 2G−1 σλλ − 2G −1 σλx ds ∂x ∂[vi,(k−1) ] 0 d2 [vi,(k−1) ] 0 ∂ d[vi,(k−1) ] 0 − 2G−1 σλy + σ + 2 σλx λλ ∂y ∂x ds ds2 4 ∂ d[vi,(k−1) ] 0 ∂[vi,(k−1) ] 3 +2 σλy + −(K + τ)σ0λy + κσ0λλ sin ϕ ∂y ds ∂x 4 ∂[vi,(k−1) ] 3 + (K + τ)σ0λx − κσ0λλ cos ϕ ∂y & d[vi,(k−1) ] % 0 −1 + G [vi,(k) ] − κ σλx sin ϕ − σ0λy cos ϕ , (14.21) ds

$[vi,(k+1) ] = −G −1 [σi j,(k+1) ]λ j +

where σ0λλ = σ0i j λi λ j , σ0λx = σ0i j λi k j , and σ0λy = σ0i j λi s j . Analysis of the influence of the preliminary shear stresses σ0λx and σ0λy and the axial compression or tension stresses σ0λλ and their different combinations on the dynamic response of thin-walled rods of open profile is a very complicated problem which is of great importance in civil engineering and bridge construction. However, reference to Eq. (14.21) shows that the value σ0λλ is multiplied by the k + 1-order jump and by the s-derivative of the k-order, i.e. it has a strong impact both on the velocities of the surfaces of strong discontinuity and on the discontinuities, in contrast to the values σ0λx and σ0λy which are multiplied only by the k- and k − 1-order jumps, i.e. they exert weak effect only on the jumps. That is why, in the further part of the present paper, we shall neglect σ0λx and σ0λy with respect to the value σ0λλ .

14.2.1 Dynamic Response of Axially Prestressed Spatially Curved Thin-Walled Rods of Open Profile Thus, assuming further that σ0λx = σ0λy = 0 and σ0λλ 0, from (14.21) we have d[σi j,(k) ] ∂[σi j,(k) ] ∂[σi j,(k) ] ρ[vi,(k+1) ] = −G−1 [σi j,(k+1) ]λ j + λj + kj + sj ds ∂x ∂y d[vi,(k) ] d2 [vi,(k−1) ] ∂[vi,(k−1) ] + G−2 [vi,(k+1) ] − 2G −1 + +κ sin ϕ 2 ds ∂x ds ∂[vi,(k−1) ] − κ cos ϕ σ0λλ . (14.22) ∂y To satisfy Eq. (14.13), it is sufficient to put σi j,(k) ki k j = 0 , σi j,(k) si s j = 0 ,

σi j,(k) si k j = 0 .

(14.23)

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Then the boundary surface will be free from the normal and tangential stresses. Now we expand the value [vi,(k) ] entering into (14.20) and (14.22)–(14.23) in terms of three mutually orthogonal vectors λ, k, s. As a result we obtain .

/ vi,(k) = ω(k) λi + θ(k) ki + η(k) si ,

(14.24)

where ω(k) = [vi,(k) ]λi , θ(k) = [vi,(k) ]ki , and η(k) = [vi,(k) ]si . Suppose that for a thin-walled beam of open section the following velocity fields are fulfilled [30], [11]: 1y

ω(k) = ω(k) (s, s1 ) = ω0(k) (s) + ω ˜ 1x ˜ (k) (s)x(s1 ) + ψ˜ (k) ωA (s1 ) , (k) (s)y(s1 ) + ω

(14.25)

0 θ(k) = θ(k) (s, s1 ) = θ(k) (s) − ω1λ (k) (s)(y(s1 ) − ay ) ,

(14.26)

η(k) = η(k) (s, s1 ) = η0(k) (s) + ω1λ (k) (s)(x(s1 ) − a x ) ,

(14.27)

where s1 is the arc length measured along the cross-section profile from the point M0 , which corresponds to the arc length of s01 , to the point M[x(s1 ), y(s1 )], which corresponds to the arc length of s1 (Fig. 14.2).

Fig. 14.2 Scheme of the cross section of a thin-walled beam with a generic open cross-section

Gol’denveizer [11] proposed that the angles of in-plane rotation do not coincide with the first derivatives of the lateral displacement components and, analogously, warping does not coincide with the first derivative of the torsional rotation. It should be emphasized that it was precisely Gol’denveizer [11] who pioneered in combining Timoshenko’s beam theory [29] and Vlasov thin-walled beam theory [30] (note that the first edition of Vlasov’s book was published in Moscow in 1940) and who suggested to characterize the displacements of the thin-walled beam’s cross-section by seven generalized displacements. In order to refine the structure of formula (14.25) in the case of a spatially curved thin-walled beam, let us differentiate (14.24), which involves only the terms


191

0 , and η0 , with respect to s with due account for (14.14)–(14.16). As a result ω0(k) , θ(k) (k) we obtain ⎛ 0 ⎞ . / ⎟⎟⎟ d vi(k) 0 ⎜⎜⎜⎜ dω(k) 0 0 = ⎜⎝⎜ + θ(k) κ sin ϕ − η(k) κ cos ϕ⎟⎟⎠⎟ λi ds ds ⎛ 0 ⎞ ⎜⎜⎜ dθ(k) ⎟⎟⎟ 0 0 ⎜ + ⎜⎜⎝ − ω(k) κ sin ϕ + η(k) (K + τ)⎟⎟⎟⎠ ki ds ⎛ 0 ⎞ ⎜⎜⎜ dη(k) ⎟⎟⎟ 0 0 + ⎜⎜⎝⎜ + ω(k) κ cos ϕ − θ(k) (K + τ)⎟⎟⎠⎟ si . (14.28) ds

Moreover, we shall use the relationship for the discontinuity in the k-order derivative with respect to time t of the angular velocity . / 1y Ωi(k) = ω1λ ˜ 1x ˜ (k) si . (k) ki + ω (k) λi + ω

(14.29)

Differentiating (14.29) with respect to s and considering (14.14)–(14.16) yields ⎛ 1λ ⎞ ⎟⎟⎟ d[Ωi(k) ] ⎜⎜⎜⎜ dω(k) 1y 1x = ⎜⎜⎝ +ω ˜ (k) κ sin ϕ − ω ˜ (k) κ cos ϕ⎟⎟⎟⎠ λi ds ds ⎛ 1x ⎞ ⎜⎜⎜ dω ⎟⎟⎟ ˜ (k) 1y + ⎜⎜⎝⎜ − ω1λ ˜ (k) (K + τ)⎟⎟⎠⎟ ki (k) κ sin ϕ + ω ds ⎛ 1y ⎞ ⎜⎜⎜ dω ⎟⎟⎟ ˜ (k) ⎜ 1λ 1x + ⎜⎜⎜ + ω(k) κ cos ϕ − ω ˜ (k) (K + τ)⎟⎟⎟⎟ si . (14.30) ⎝ ds ⎠ If we suppose that transverse shear strains are absent, then we are led to the following relationships, which are in compliance with the Vlasov theory: ⎛ 0 ⎞ ⎜⎜⎜ dη(k) ⎟⎟⎟ 1x 0 0 ⎜ ω ˜ (k) = − ⎜⎜⎝ + ω(k) κ cos ϕ − θ(k) (K + τ)⎟⎟⎟⎠ , (14.31) ds 1y ω ˜ (k)

ψ˜ (k)

⎛ 0 ⎞ ⎜⎜⎜ dθ(k) ⎟⎟⎟ 0 0 = − ⎜⎜⎜⎝ − ω(k) κ sin ϕ + η(k) (K + τ)⎟⎟⎟⎠ , ds

(14.32)

⎛ 1λ ⎞ ⎜⎜⎜ dω(k) ⎟⎟⎟ 1y 1x ⎜ = − ⎜⎜⎝ +ω ˜ (k) κ sin ϕ − ω ˜ (k) κ cos ϕ⎟⎟⎟⎠ ds =−

dω1λ (k)

+

dη0(k)

κ sin ϕ −

0 dθ(k)

κ cos ϕ ds ds% ds & 0 + ω0(k) κ2 sin ϕ − θ(k) sin ϕ + η0(k) cos ϕ κ(K + τ).

(14.33)

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If we now substitute (14.31)–(14.33) in (14.25), then we obtain a formula for defining the k-order time-derivative in the discontinuity in the velocity of translation along the λ-axis without account for transverse shear deformations. Following Gol’denveizer [11], in order to take the transverse shear deforma0 tions into account, it is essential to substitute the derivatives dω1λ (k) /ds, dη(k) /ds, and 1y

0 /ds by the independent functions ψ , ω1x , and ω , respectively, i.e. to rewrite dθ(k) (k) (k) (k) Eqs (14.31)–(14.33) as % & 1x 0 0 ω ˜ 1x (14.34) (k) = − ω(k) + ω(k) κ cos ϕ − θ(k) (K + τ) ,

% 1y & 1y ω ˜ (k) = − ω(k) − ω0(k) κ sin ϕ + η0(k) (K + τ) ,

(14.35)

% & 1y ψ˜ (k) = − ψ(k) + ω ˜ 1x κ sin ϕ − ω ˜ κ cos ϕ (k) (k) 1y

= −ψ(k) + ω1x (k) κ sin ϕ − ω(k) κ cos ϕ % & 0 2 0 + ω(k) κ sin ϕ − θ(k) sin ϕ + η0(k) cosϕ κ(K + τ) .

(14.36)

As a result, instead of formula (14.25), we obtain 0 0 ω(k) (s, s1 ) = ω0(k) (s) − ω1x (k) (s) + ω(k) κ cos ϕ − θ(k) (K + τ) y(s1 ) 1y − ω(k) (s) − ω0(k) κ sin ϕ + η0(k) (K + τ) x(s1 ) 1y 0 2 + −ψ(k) (s) + ω1x (k) (s)κ sin ϕ − ω(k) (s)κ cos ϕ + ω(k) κ sin 2ϕ % & 0 − θ(k) sin ϕ + η0(k) cos ϕ κ(K + τ) ωA (s1 ). (14.37) Formulas (14.26) and (14.27) remain unchanged. If we put K = κ = τ = 0 in (14.37), then we are led to the case of a straight thin-walled rod of open profile. In further treatment we will use formulas (14.25)–(14.27), but in the final relationships we 1y will carry out the substitution of the values ψ˜ (k) , ω ˜ 1x ˜ (k) by their magnitudes (k) , and ω defined by formulas (14.34)–(14.36). At k = 0, the values entering in (14.26), (14.27), and (14.37) have the following 0 physical meaning: ω0(0) (s), θ(0) (s) and η0(0) (s) are the discontinuities in the velocities of translatory motion of the section as a rigid body together with the point C (the center of gravity of the cross-section) along the λ−, x− and y−axes, respectively, 1y 1λ ω1x (0) (s), ω(0) (s) and ω(0) (s) are the discontinuities in the angular velocities of the cross-section’s rotation as a rigid whole around the x−, y− and λ−axes, respectively, ψ(0) (s) is the discontinuity in the velocity of warping of the cross section, ωA (s1 ) is twice the area of the sector AM0 M, the point A is the center of bending (Fig. 14.2) with the coordinates [30] 1 1 ax = ωC (s1 )y(s1 )dF , ay = − ωC (s1 )x(s1 )dF . (14.38) Ix Iy F

F


193

Since the section is referred to the main central axes and the coordinates of the point A are defined by formulas (14.38), then I xy = Iωx = Iωy = S y = S x = 0. Note that the sectorial area, generally speaking, depends on two coordinates, namely: the initial point s01 and the terminal point s11 , i.e. ωA (s01 , s11 ). Let us choose s11 in such a way that the relationship 1 ωA (s01 , s11 ) = ωA (s01 , s1 )dF (14.39) F F

will be valid. Then, as it is shown in [30], taking the point with the coordinate s11 as the origin of the arc length measuring, we obtain S ω = ωA (s11 , s1 )dF = 0 . (14.40) F

Let us name this point as the null sectorial point [30]. Since there may exist several such points, the null sectorial point nearest to the point A can be named as the main null sectorial point, and we shall take it as the initial point of reading. Thus, using the above mentioned choice of the coordinates and the initial points of measuring, the functions 1, x(s1 ), y(s1 ), and ωA (s11 , s1 ) occur to be orthogonal. If we substitute (14.20) in the first two equations of (14.23) and consider (14.24), then we are led to the relationships dω(k−1) λ −1 ε x,(k) = εy,(k) = G ω(k) − + κη(k−1) cos ϕ − κθ(k−1) sin ϕ , (14.41) 2(λ + μ) ds where

' ( ' ( ∂ [vi,(k−1) ]ki ∂ [vi,(k−1) ]si ε x,(k) = , εy,(k) = . ∂x ∂y

Equation (14.23), after the substitution of (14.20), (14.24), (14.26), and (14.27) in it, is fulfilled unconditionally. Multiplying (14.20) successively by λi λ j , λi k j and λi s j and considering (14.14)– (14.16), after integration of the final equations over the cross-sectional area of the beam, we could ) obtain the relationships for) discontinuities in the generalized forces Nλ(k+1) = [σi j,(k+1) ]λi λ j dF, Qλx(k+1) = [σi j,(k+1) ]λi k j dF, and Qλy(k+1) = F F ) [σi j,(k+1) ]λi s j dF. In a similar way we could find the relations for discontinuities in F

the ) derivatives of the bending moments ) with respect to the x− and y−axes M x(k+1) = [σi j,(k+1) ]λi λ j ydF and My(k+1) = [σi j,(k+1) ]λi λ j xdF, in the derivatives of the F F ) bimoment B(k+1) = [σi j,(k+1) ]λi λ j ωA dF, and in the derivatives of the bendingF ) ) torsional moment M A(k+1) = [σi j,(k+1) ]λi s j (x − a x )dF − [σi j,(k+1) ]λi k j (y − ay)dF. F

F

194


Multiplying (14.22) successively by λi , ki , and si with due account for (14.14)– (14.16), then substituting (14.25)–(14.27) in the obtained equations and integrating them over the cross-sectional area of the beam, after elimination of the generalized forces we are led to the following equations: −2

G ($G −

$G21 − σ0λλ )ω0(k+1)

−1

%

$G 21 + σ0λλ

& dω0(k)

= −2G ds & % & % −1 2 2 0 − G $G 1 + 2$G 2 + 2σλλ κ a x cosϕ + ay sin ϕ ω1λ (k) 0 − η0(k) cos ϕ + θ(k) sin ϕ + F 1(k−1) ,

(14.42)

0 −1 2 0 G −2 ($G2 − $G22 − σ0λλ )(θ(k+1) + ay ω1λ (k+1) ) = −2G ($G 2 + σλλ ) * % + & % & d 0 × θ(k) + ay ω1λ + (K + τ) η0(k) − a x ω1λ (k) (k) ds % & + G−1 $G 21 + $G22 + 2σ0λλ κω0(k) sin ϕ 3 1y 4 + G−1 $G22 ω(k) − ω0(k) κ sin φ + η0(k) (K + τ) + F2(k−1) ,

(14.43)

% & −1 G−2 ($G 2 − $G22 − σ0λλ )(η0(k+1) − a x ω1λ $G22 + σ0λλ (k+1) ) = −2G * % & % &+ d 0 0 1λ × η(k) − a x ω1λ − (K + τ) θ + a ω y (k) (k) (k) ds % & − G−1 $G 21 + $G22 + 2σ0λλ κω0(k) cos ϕ 3 4 0 0 + G−1 $G22 ω1x (k) + ω(k) κ cos φ − θ(k) (K + τ) + F3(k−1) ,

(14.44)

2

0 0 − G −2 ($G2 − $G 21 − σ0λλ ) ω1x (k+1) + ω(k+1) κ cos ϕ − θ(k+1) (K + τ) % & d 0 = 2G −1 ρG 21 + σ0λλ ω1x + ω0(k) κ cos ϕ − θ(k) (K + τ) (k) ds & % + κ $G 21 + 2$G 22 + 2σ0λλ G−1 ω1λ (14.45) (k) sin ϕ + F 4(k−1) , 1y − G−2 ($G 2 − $G 21 − σ0λλ ) ω(k+1) − ω0(k+1) κ sin ϕ + η0(k+1) (K + τ) % & d 1y = 2G −1 ρG 21 + σ0λλ ω(k) − ω0(k) κ sin ϕ + η0(k) (K + τ) ds & % 2 2 + κ $G 1 + 2$G 2 + 2σ0λλ G −1 ω1λ (14.46) (k) cos ϕ + F5(k−1) ,


− G−2 ($G2 − − = −

195

1y $G21 − σ0λλ ) ψ(k+1) − ω1x (k+1) κ sin ϕ + ω(k+1) κ cos ϕ % & 0 ω0(k+1) κ2 sin 2ϕ + θ(k+1) sin ϕ + η0(k+1) cosϕ κ(K + τ) % & d 1y 2G−1 ρG21 + σ0λλ ψ(k) − ω1x (k) κ sin ϕ + ω(k) κ cos ϕ ds % & 0 ω0(k) κ2 sin 2ϕ + θ(k) sin ϕ + η0(k) cos ϕ κ(K + τ) + F 6(k−1) (14.47)

% & 0 0 G−2 ($G 2 − $G22 − σ0λλ ) IPA ω1λ (k+1) + ay Fθ(k+1) − a x Fη(k+1) % &* d % & −1 2 0 0 0 = −2G ρG2 + σλλ IPA ω1λ (k) + ay Fθ(k) − a x Fη(k) ds % &4 0 + F(K + τ) a x θ(k) + ayη0(k) % & + G−1 ρG22 ay F + κIy cos ϕ $G21 + $G 22 + 2σ0λλ 1y × ω(k) − ω0(k) κ sin ϕ + η0(k) (K + τ) % & + G−1 −$G22 a x F + κI x sin ϕ $G21 + $G 22 + 2σ0λλ (14.48) 1x 0 0 × ω(k) + ω(k) κ cos ϕ − θ(k) (K + τ) % &% & + G−1κF $G21 + $G 22 + 2σ0λλ a x cos ϕ + ay sin ϕ ω0(k) + F7(k−1) , where G 21 = E$−1 , G22 = μ$−1 , and F j(k−1) ( j = 1, 2, ..., 7) are the functions of the k − 1-order. The system of seven equations, (14.42)–(14.49), involves seven unknown 1y 0 1λ values: ω0(k) , θ(k) , η0(k) , ω1x (k) , ω(k) , ω(k) , and ψ(k) , which are defined uniquely from this set of equations. To find these values on the quasi-longitudinal shock wave and 8 on the quasi-transverse shock 8 wave, we should put in all equations G = G I = G21 + $−1 σ0λλ and G = G II =

G22 + $−1 σ0λλ , respectively.

14.2.2 Construction of the Desired Wave Fields in Terms of the Ray Series Following the previous two papers by Rossikhin and Shitikova [24] and [26] devoted to the dynamic behaviour of thin elastic bodies, where thin plates and shells have been considered in [24], and spatially curved and twisted slender rod-like solids have been studied in [26], as a method applicable for solving dynamic problems resulting in propagation of wave surfaces of strong and weak discontinuity we will use the method of ray expansions [1]. This method is one of the methods of perturbation technique, where time is used as a small parameter. Thus, knowing the discontinuities of the desired stress and velocity fields determined above within an accuracy of arbitrary constants on the two waves of strong discontinuity, quasi-longitudinal and quasi-transverse, propagating in the

196


thin-walled beam of open profile, we could construct the fields of the desired functions also with an accuracy of the arbitrary constants utilizing the ray series [1], which are the power series with variable coefficients and which allow one to construct the solution behind the wave fronts of strong discontinuity [24], [26]: k ∞ 1 α s s Z(t, s) = [Z ](s) t − H t− , k! ,(k) Gα Gα α=I,II

(14.49)

k=0

where Z is the desired value, H(t − s/Gα ) is the unit Heaviside function, and the index α = I, II labels the ordinal number of the wave propagating with the velocity G α . The arbitrary constants entering into the ray expansions are determined from the initial and boundary conditions. The example of using the ray expansions (14.49) for analyzing the impact response of spatially curved thin-walled beams of open cross-section will be demonstrated below by solving the problem about the normal impact of an elastic spherically-headed rod upon an elastic arch, representing itself a channel-beam curved along an arc of the circumference.

14.3 Impact Response of a Thin-Walled Beam of Open Profile During the past two decades foreign object impact damage to structures has received a great deal of attention, since thin-walled structures are known to be susceptible to damage resulting from accidental impact by foreign objects. Impact on aircraft structures or civil engineering structures, for instance, from dropped tools, hail, and debris thrown up from the runway, poses a problem of great concern to designers. Since the impact response is not purely a function of material’s properties and depends also on the dynamic structural behaviour of a target, it is important to have a basic understanding of the structural response and how it is affected by different parameters [25]. From this point of view, analytical models are useful as they allow systematic parametric investigation and provide a foundation for prediction of impact damage. An impact response analysis requires a good estimate of contact force throughout the impact duration. Low velocity impact problems, which also took the local indentation into account, have been solved by many authors. Reference to the stateof-the-art paper [25] shows that in most studies it was assumed that the impacted structure was free of any initial stresses. But this does not adequately reflect the real multidirectional complex loading states that the materials experience during their service life. In practice, the composite facing of a structure may be under a preload, e.g. a sandwich structure with laminate facing under bending loads, jet engine fan blades subjected to centrifugal forces. Even when stationary on the runway a composite airframe is under pre-stress. The dynamic response of straight thin-walled beams of generic open profile impacted by an elastic sphere was investigated in [23] utilyzing the technical


197

theory by Korbut and Lazarev [15]. A numerical approach to determining the transient response of straight nonrectangular bars subjected to transverse elastic impact was described in [18] using finite element method. Below in order to analyze the impact response of spatially curved thin-walled beams of open profile we shall implement the theoretical results which have already been described in Sect. 14.2, since the derived hyperbolic system of recurrent equations together with the ray expansions allow one to describe the short-time processes, in particular, the processes of shock interaction.

14.3.1 Impact of an Elastic Hemispherical-Nosed Rod against a Thin-Walled Beam of Open Profile Let us consider the normal impact of an elastic thin spherically-headed rod of circular cross-section with the radius r0 against a lateral surface of the initially stressed thin-walled beam of open section (Fig. 14.3). For simplicity, we shall suppose −1 that the thin-walled beam has the constant curvature κ = R−1 t = R , while the torsion τ = 0. Moreover, the beam possesses one axis of symmetry, along which the impactor moves.

Fig. 14.3 Scheme of shock interaction

At the moment of impact, the velocity of the impacting rod is equal to V0 , and the longitudinal shock wave begins to propagate along the rod with the velocity 8

G 0 = E 0 $−1 0 , where E 0 is its elastic modulus, and $0 is its density. Using the dynamic condition of compatibility on the front of this wave coupling the stress σ− and velocity v− fields [25] ' ( σ− = ρG0 V0 − v− , (14.50)

− the contact stress σ =σ could be defined as cont

y=0

% & σcont = ρG 0 V0 − α˙ − η0 ,

(14.51)

198


where α˙ + η0 = v−

y=0 is the normal velocity of the beam’s points within the contact domain, α is the value governing the local bearing of the target’s material during the process of its contact interaction with the impactor, and an overdot denotes the time-derivative. The interacting bodies are considered to be of rather long extent in order to ignore reflected waves, since they arrive at the place of contact after the rebounce of the impactor from the target. Formula (14.51) allows one to find the contact force % & P = πr02 $0 G 0 V0 − α˙ − η0 . (14.52) However, the contact force can be determined not only via Eq. (14.52), but using the Hertz’s law as well [12], [25] P = kα3/2 ,

(14.53)

where k is the contact stiffness coefficient depending on the geometry of colliding bodies, as well as on their elastic constants 4 √ k = E ∗ R , 3

1 1 1 = + , R r 0 Rt

1 − ν20 1 − ν2t 1 = + . E∗ E0 Et

(14.54)

Here, r0 is the radius of the rod’s rounded end (Fig. 14.3), Rt is the radius of curvature of the target in the place of impact, and E0 , ν0 and Et , νt are the elastic modulus and the Poisson’s ratio of the indenter and the target, respectively. Eliminating the force P from (14.52) and (14.53), we are led to the equation for determining the value α(t) α˙ + η0 +

k α1/2 = V0 . πρ0 G0 R

(14.55)

Since the contours of the beam’s cross sections remain rigid during the process of impact, then all sections involving by the contact domain form a layer which moves as rigid whole. Let us name it as a contact layer (Fig. 14.3). If we write the series (14.49) on the unknown moving boundary a(t) of the contact domain, and consider the effect of ‘retardation’ [25] implying in the fact that the transient waves detach from the boundary of the contact domain not immediately at the moment of impact t = 0, but after some time duration t = t∗ = r0 V0 /2G22 elapsed from the moment of impact, then in the series (14.49) we should change the values t and s by t − t∗ and a − a∗ , respectively, where s = a∗ is the location of the contact region boundary at t = t∗ , a = a∗ + a0 (t − t∗ ) + a1 (t − t∗ )2 + ..., (14.56) −1 ∗ −1 ∗ 2 t − t∗ − (a − a∗)G −1 α = (1 − a0G α )(t − t ) − a1Gα (t − t ) + ...,

and a0 , a1 , ... are yet unknown constant values, and

a∗

= r0 V0G−1 II .

(14.57)


199

With the assumptions made, the motion of the contact domain is described by one equation M η˙ 0 = 2Qλy + P , (14.58) where M = 2$Fa(t − t∗ ) and 2a(t − t∗ ) are the mass and the width of the contact spot, respectively, while the value α is connected with the value a according to the Hertz’s theory by the following relationship: α = R −1 a2 ,

(14.59)

Equation (14.53), the initial condition α˙ = V0 ,

(14.60)

which may be written at t = t∗ due the smallness of the time t∗ , as well as the boundary conditions θ0 = ω1y = ω1λ = ψ = 0 , ω1x = −˙aR−1 , ω0 = a˙

(14.61)

should be added to Eqs (14.55) and (14.58). Representing the value α in terms of a series α = α∗ + α1 (t − t∗ ) + α2 (t − t∗ )2 + α3 (t − t∗ )3 + ...,

(14.62)

where α∗ ≈ 2V0 t∗ = r0 V02 G−2 II , and α1 , α2 , α3 , ... are yet unknown constants, and substituting (14.56) and (14.62) in (14.59), we could find with due account for the initial condition (14.60) all coefficients of the series (14.56), as well as α1 = V0 . Since the solution behind the wave fronts up to the contact domain is constructed in terms of the ray series (14.49), then we should first determine the ray series coefficients for the desired values. This could be carried out by solving Eqs (14.42)– (14.49) at k = 0, 1, 2, ... putting K = τ = 0, κ = R−1 , and ϕ = χ = 0 (Fig. 14.3). Knowing the ray series coefficients, we could write the ray series for all desired values, and then substitute them in Eqs (14.56)–(14.62). Finally equating the coefficients at equal powers of t − t∗ , we could find the coefficients αk , among them, ⎛ ⎞ ⎜⎜⎜ V0 πρ0 R 1/2 α∗3/2 2G2 α∗1/2 ⎟⎟⎟ 1 k α2 = − + + ⎝⎜ ⎠⎟ < 0. 2 πρ0 R α∗1/2 G 0 ρF G 0 R 1/2 Substituting the found arbitrary constants in the ray series, we obtain the final expressions for the desired fields. Thus, for example, knowing the value α2 , it is possible to determine α(t) (14.62) and a(t) (14.56), and therefore to obtain the typical time-dependence of the contact force (14.53) within an accuracy of (t − t∗ )2 , since α2 is a negative value: 3 43/2 P(t − t∗ ) − P∗ = k V0 (t − t∗ ) + α2 (t − t∗ )2 ,

(14.63)

200


where P∗ = P|t=t∗ = kα∗3/2 . Equating to zero the expression for the contact force (14.63), we obtain the approximate formula for the duration of contact of the impacting rod with the thin-walled beam of open section. 14.3.1.1 Numerical Example As an example, let us consider the impact of a steel rod with a rounded end upon a steel arch with a constant radius of curvature and zero torsion, the cross section of which represents a channel (Fig. 14.3). The dimensionless time t˜ − t˜∗ = (t − t∗ )( 25 )2/3 V01/3 k2/3 (πr02 $G0 )−2/3 dependence of the dimensionless contact force C− P C∗ = (P − P∗ )(πr2 $G 0 V0 )−1 is presented in Fig. 14.4 for different levels of the P 0 initial axial compression σ Cλλ = σ0λλ ($G 22 )−1 . Reference to Fig. 14.4 shows that the increase in the initial axial compression results in the increase of both the maximal.

Fig. 14.4 Dimensionless time-dependence of the dimensionless contact force

Fig. 14.5 C σλλ -dependence of the dimensionless initial velocity of impact in the case when the contact stress is equal to the yield limit

The curve of the C σλλ -dependence of the dimensionless initial velocity of impact C0 = V0G−1 , resulting in the local damage of the thin-walled open-section beam in V 0 the place of contact is shown in Fig. 14.5 at the given magnitude of the dimensionless yield limit C σy = σy ($0 G20 )−1 . From Fig. 14.5 it is evident that with the increase in the initial axial compression the initial velocity of impact, which may lead to the local damage of the structure, decreases.

14.4 Conclusion Starting from the three-dimensional dynamic theory of linear elasticity and the Vlasov and Gol’denveizer theories, the dynamic theory of thin-walled beams of open section has been proposed with due account for the axial precompression. As this takes place, the derived hyperbolic system of recurrent equations together with


201

the ray expansions allow one to describe the short-time processes, in particular, the processes of shock interaction. The dynamic stability with respect to small perturbations, as well as the local damage of geometrically nonlinear elastic spatially curved open section beams with axial precompression have been analyzed. Transient waves, which are the surfaces of strong discontinuity and wherein the stress and strain fields experience discontinuities, have been used as small perturbations, in so doing the discontinuities have been considered to be of small values. Such waves could be initiated during lowvelocity impacts upon thin-walled beams. The approach proposed permits to solve analytically for the first time, to the authors’ knowledge, the problem of normal impact of a rod against an elastic thinwalled beam of open section making allowance for the beam’s translatory and rotational motions, warping, rotary inertia, shear deformation, the local bearing due to the Hertz’s theory, and the initial axial compression of the beam. The method of expansion along the rays behind the wave front is valid for short times after the wave front has passed. That is why it is well suited for solving the problems of shock interactions, since the duration of contact is very short, and thus a small number of terms in ray expansions is sufficient to achieve reasonable accuracy of the solution. The theory constructed allows one to investigate the influence of the initial stresses on the dynamic fields and to answer a set of major questions, among them: (1) What magnitude should the initial velocity of impact take on at the given axial precompression of the thin-walled open section beam in order to produce its local damage at the place of shock interaction between the target and the impactor? (2) How does the level of the initial axial compression influence the maximal contact force and the duration of contact? The graphs presented in Figs 14.4 and 14.5 are the answers to the above questions. Acknowledgements The research described in this publication was made possible in part by Grants No. 2.1.2/520 and No. 2.1.2/12275 from the Russian Ministry of High Education.

References 1. Achenbach JD and Reddy DP (1967) Note on wave propagation in linearly viscoelastic media. ZAMP 18:141–144 2. Arpaci A, Bozdag SE and Sunbuloglu E (2003) Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effects. J Sound Vibr 260:889–900 3. Banerjee JR and Williams FW (1994) Coupled bending-torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element. Int J Solids Struct 31:749–762 4. Bersin AN and Tanaka M (1997) Coupled flexural-torsional vibrations of Timoshenko beams. J Sound Vibr 207:47–59 5. Bhattacharya B (1975) Coupled vibrations of thin-walled open section curved beams. ASCE J Struct Eng 13:29–35 6. Bolotin VV (1961) Nonconservative problems of the theory of elastic stability (in Russian). Fizmatlit, Moscow 7. Capuani D, Savoia M and Laudiero F (1992) A generalization of the Timoshenko beam model for coupled vibration analysis of thin-walled beams. Earthquake Eng Struct Dyn 21:859–879

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8. Cortínez VH and Rossi RE (1998) Dynamics of shear deformable thin-walled open beams subjected to initial stresses. Rev Internac Métod Numér Cálc Diseñ Ingr 14(3):293–316 9. de Boer R and Sass HH (1975) Der Stoβ auf gerade dünnwandige Träger. Ingenieur-Archiv 44(3):177–188 10. Gendy AS and Saleeb AF (1994) Vibration analysis of coupled extensional/flexural/torsional modes of curved beams with arbitrary thin-walled sections. J Sound Vibr 174:261–274 11. Gol’denveizer AL (1949) To the theory of thin-walled beams (in Russian). Prikl Mat Mekh 13:561–596 12. Goldsmith W (1960) Impact. Arnold, London 13. Kim NI, Seo KJ and Kim MY (2003) Free vibration and spatially stability of non-symmetric thin-walled curved beams with variable curvatures. Int J Solids Struct 40:3107–3128 14. Kim NI and Kim MY (2005) Exact dynamic stiffness matrix of non-symmetric thin-walled curved beams subjected to initial axial force. J Sound Vib 284:851–878 15. Korbut BA and Lazarev VI (1974) Equations of flexural-torsional waves in thin-walled bars of open cross section. Int Appl Mech 10:640–644 16. Korbut BA and Lazareva GV (1982) Dynamic theory of thin curvilinear beams. Int Appl Mech 18:476–482 17. Li J, Shen R, Hua H and Jin X (2004) Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams Int J Mech Sciences 46:299–320 18. Lin Y, Lai WK and Lin KL (1998) A numerical approach to determining the transient response of nonrectangular bars subjected to transverse elastic impact. J Acoust Soc Am 103:1468– 1474 19. Mescherjakov VB (1968) Free vibrations of thin-walled open section beams with account for shear deformations (in Russian). Proc Moscow Inst Railway Transport Eng 260:94–102 20. Muller P (1983) Torsional-flexural waves in thin-walled open beams. J Sound Vibr 87:115– 141 21. Piovan MT, Cortínez VH and Rossi RE (2000) Out-of-plane vibrations of shear deformable continuous horizontally curved thin-walled beams. J Sound Vibr 237:101–118 22. Prokić A (2006) On fivefold coupled vibrations of Timoshenko thin-walled beams. Eng Struct 28:54–62 23. Rossikhin YuA, Shitikova MV (1999) The impact of a sphere on a Timoshenko thin-walled beam of open section with due account for middle surface extension. ASME J Pressure Vessel Tech 121:375–383 24. Rossikhin YuA, Shitikova MV (2007) The method of ray expansions for investigating transient wave processes in thin elastic plates and shells. Acta Mech 189:87–121 25. Rossikhin YuA, Shitikova MV (2007) Transient response of thin bodies subjected to impact: Wave approach. Shock Vibr Digest 39:273–309 26. Rossikhin YuA, Shitikova MV (2008) The method of ray expansions for solving boundaryvalue dynamic problems for spatially curved rods of arbitrary cross-section. Acta Mech 200:213–238 27. Rossikhin YuA, Shitikova MV (2010) The analysis of the transient dynamic response of elastic thin-walled beams of open section via the ray method. Int J Mech 4:9–21 28. Thomas TY (1961) Plastic flow and fracture in solids. Academic Press, New York 29. Timoshenko SP (1928) Vibration problems in engineering. Van Nostrand, New York 30. Vlasov VZ (1961) Thin-walled elastic beams. Published for the National Science Foundation, Washington, D.C. by the Israel Program for Scientific Translations, Jerusalem (Eng. trnsl. from the 2d Russian Ed. published in 1959 in Moscow). 31. Vörös GM (2009) On coupled bending–torsional vibrations of beams with initial loads. Mech Res Com 36:603–611

Chapter 15

On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression Denis Sheydakov

Abstract In the present paper, in the framework of a general stability theory for three-dimensional bodies the buckling analysis has been carried out for a rectangular sandwich plate subject to biaxial compression. The sandwich plate consists of a porous core, covered by a hard and stiff shell. The behavior of a coating is investigated in the framework of a classic (non-polar) continuum model, while to describe the properties of a core the Cosserat continuum model is used. Using the linearization method in the vicinity of a basic state, the neutral equilibrium equations have been derived, which describe the perturbed state of a sandwich plate. The linearized boundary-value problems have been formulated both for a case of a general sandwich plate and for a sandwich plate with identical top and bottom coatings. By solving these problems numerically for some specific materials, the critical curves and corresponding buckling modes can be found, and the stability regions can be constructed in the plane of loading parameters (relative axial compressions). Keywords Buckling analysis · Sandwich plate · Biaxial compression

15.1 Introduction The problem of equilibrium stability for deformable bodies is of major importance both from theoretical and practical point of view, because the exhaustion of loadcarrying capability and collapse of buildings and engineering structures quite often occurs due to buckling under external loads. In the case of elastic medium, the stability theory is extensively developed for classic non-polar materials. There is large number of papers on stability both for thin and thin-walled bodies in the form of D. Sheydakov (B) South Scientific Center of Russian Academy of Sciences, Chekhova Ave. 41, 344006 Rostov-on-Don, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 15, © Springer-Verlag Berlin Heidelberg 2011

203

204

D. Sheydakov

rods, plates and shells, and for massive (three-dimensional) bodies. However, due to the increasing number of new constructional materials, the problem of stability for bodies with a complex structure becomes relevant. The present research is dedicated to the buckling analysis of nonlinearly elastic plates of highly porous materials, such as metallic and polymeric foams. As a rule, these plates have a sandwich structure (porous core covered by hard and stiff shells), which is necessary for corrosion protection and optimization of mechanical properties during loading. In majority of papers the behavior of foams is studied in the framework of a classic continuum model. But it seems more reasonable to use a model of micropolar medium or Cosserat continuum model [1, 3, 4, 7, 12] for analysis of highly porous materials, because it allows describing their distinctive properties more adequately. Any particle of the Cosserat continuum has an additional inner degree of freedom a microrotation. As a result of this, a deformation in the couple-stress elasticity theory is described not by one, but by two strain tensors-the stretch tensor and wryness tensor-which greatly complicates the study of micropolar bodies.

15.2 Equilibrium of the Rectangular Sandwich Plate Subject to Biaxial Compression We consider the rectangular sandwich plate of length b1 , width b2 and thickness H. The behavior of the porous core of thickness 2a is described by the model of micropolar elastic body. The top coating of thickness h+ and bottom coating of thickness h− are made of classic non-polar materials. In the case of biaxial compression of the plate, the position of a particle in the strained state is given by the radius vectors R, R+ and R− (here and below, by subscripts ‘+’ and ‘−’ we denote the quantities related to the top and bottom coatings, respectively, without subscripts ‘+’ or ‘−’ – related to the core of the sandwich plate) [6, 13]: X1 = α1 x1 ,

0 ≤ x1 ≤ b1

X2 = α2 x2 , 0 ≤ x2 ≤ b2 ⎧ ⎪ ⎪ f+ (x3 ), a ≤ x3 ≤ a + h+ ⎪ ⎪ ⎪ ⎨ X3 = ⎪ f (x3 ), |x3 | ≤ a ⎪ ⎪ ⎪ ⎪ ⎩ f (x ), −(a + h ) ≤ x ≤ −a − 3 − 3

(15.1)

R+ = α1 x1 e1 + α2 x2 e2 + f+ (x3 )e3 ,

a ≤ x3 ≤ a + h+

R = α1 x1 e1 + α2 x2 e2 + f (x3 )e3 ,

|x3 | ≤ a

R− = α1 x1 e1 + α2 x2 e2 + f− (x3 )e3 ,

−(a + h−) ≤ x3 ≤ −a

(15.2)

15 On Stability of Elastic Rectangular Sandwich Plate Subject to Biaxial Compression

205

Here x1 , x2 , x3 are Cartesian coordinates in the reference configuration (Lagrangian coordinates), X1 , X2 , X3 are Eulerian coordinates, {e1 , e2 , e3 } are orthonormal vector basis of Cartesian coordinates, α1 and α2 are compression ratios along the axes x1 and x2 , respectively, f (x3 ) and f± (x3 ) are unknown functions, which characterize the strain in thickness direction of the sandwich plate. In addition, for |x3 | ≤ a a proper orthogonal tensor of microrotation H is given, which represents the rotation of the particle for micropolar medium and at the considered strain is equal to the unit tensor E: H = E = e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3

(15.3)

According to expressions (15.1), (15.2), the deformation gradients C and C± are (hereinafter denotes the derivative with respect to x3 ): C+ = grad R+ = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f+ e3 ⊗ e3 ,

a ≤ x3 ≤ a + h +

C = grad R = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f e3 ⊗ e3 ,

|x3 | ≤ a

C− = grad R− = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f− e3 ⊗ e3 ,

−(a + h−) ≤ x3 ≤ −a

(15.4)

where grad is gradient in the Lagrangian coordinates. It follows from relations (15.3), (15.4) that for the porous core (|x3 | ≤ a) of the plate the wryness tensor L is equal to zero [8, 9] '

(

L × E = − grad H · HT = 0

and stretch tensor Y is expressed as follows Y = C · HT = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f e3 ⊗ e3

(15.5)

According to (15.4), for the top (a ≤ x3 ≤ a + h+) and bottom (−a − h− ≤ x3 ≤ −a) coatings the expressions for stretch tensors U± and macrorotation tensors A± have the form [6]: %

U± = C± · CT±

&1 2

= α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + f± e3 ⊗ e3 A± = U−1 ± · C± = E

(15.6)

We assume that the elastic properties of the sandwich plate are described by the physically linear material, whose specific strain energy in the case of micropolar body is a quadratic form of the tensors Y − E and L [2, 5]:

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D. Sheydakov

% & 1 1 W (Y, L) = λtr2 (Y − E) + (μ + κ) tr (Y − E) · (Y − E)T + 2 2 % & 1 1 1 1 + μtr (Y − E)2 + γ1 tr2 L + γ2 tr L · LT + γ3 tr L2 2 2 2 2 μ + κ > 0,

λ + 2μ + κ > 0,

γ2 ≥ 0,

(15.7)

γ1 + γ2 + γ3 > 0

and in the case of classic non-polar medium – quadratic form of the tensors U± − E [6]: 1 W± (U± ) = λ± tr2 (U± − E) + μ± tr (U± − E)2 , μ± > 0, λ± + 2μ± > 0 (15.8) 2 Here λ, μ and λ± , μ± are Lame coefficients for the plate core and coatings, respectively, κ, γ1 , γ2 , γ3 are micropolar elastic coefficients. It follows from expressions (15.3), (15.5), and (15.7) that for the porous core the Piola-type couple stress tensor is equal to zero in the case of biaxial compression of the sandwich plate G=

% & ∂W · H = γ1 (tr L) E + γ2 L + γ3 LT · H = 0 ∂L

and Piola-type stress tensor D is D=

% % & & ∂W · H = λtr (Y − E) E + μ YT − E + (μ + κ) (Y − E) · H = ∂Y

= (λs + χ (α1 − 1)) e1 ⊗ e1 + (λs + χ (α2 − 1)) e2 ⊗ e2 + ' ' (( + λs + χ f − 1 e3 ⊗ e3 ; s = α1 + α2 + f − 3, χ = 2μ + κ

(15.9)

According to (15.6), (15.8), the expressions of Piola stress tensors D± for coatings have the form: D± =

∂W± · A± = (λ± tr (U± − E) E + 2μ± (U± − E)) · A± = ∂U±

= (λ± s± + 2μ± (α1 − 1)) e1 ⊗ e1 + (λ± s± + 2μ± (α2 − 1)) e2 ⊗ e2 + ' ' (( + λ± s± + 2μ± f± − 1 e3 ⊗ e3 ; s± = α1 + α2 + f± − 3

(15.10)

The equilibrium equations of nonlinear micropolar elasticity in the absence of mass forces and moments are written as follows [2, 13] % & divD = 0, divG + CT · D = 0 (15.11) ×

where div is the divergence in the Lagrangian coordinates. The symbol × represents the vector invariant of a second-order tensor: K× = (Kmn em ⊗ en )× = Kmn em × en


207

The equilibrium equations for classic non-polar continuum in the absence of mass forces have the form [6]: divD± = 0 (15.12) Boundary conditions e3 · D+ | x3 =a+h+ = 0, e3 · D± | x3 =±a = e3 · D| x3 =±a ,

e3 · D− | x3 =−(a+h− ) = 0

f± (±a) = f (±a) ,

f (0) = 0

(15.13)

express the absence of external loads on the faces of the sandwich plate (x3 = a + h+ , x3 = −a − h−), rigid coupling of top and bottom coatings with the core (x3 = ±a) and the absence of vertical displacement on the plane x3 = 0. By solving the boundary problem (15.11) – (15.13) while taking into account the relations (15.9), (15.10), we found the unknown functions f (x3 ) and f± (x3 ) f (x3 ) = α3 x3 ,

f+ (x3 ) = α+3 (x3 − a) + aα3, α3 = 1 +

λ(2 − α1 − α2 ) , λ+χ

f− (x3 ) = α−3 (x3 + a) − aα3 α±3 = 1 +

λ± (2 − α1 − α2 ) λ± + 2μ±

15.3 The Perturbed State Suppose that in addition to the above-described state of equilibrium for the sandwich plate, there is an infinitely close equilibrium state under the same external loads, which is determined by: 1) for the micropolar core–the radius vector R + ηv, and microrotation tensor H − ηH × ω, 2) for the coatings–the radius vectors R± + ηv± . Here η is a small parameter, v and v± are vectors of additional displacements, ω is a linear incremental rotation vector, which characterizes the small rotation of the particles for micropolar medium, measured from the initial strain state. The perturbed state of equilibrium for the micropolar medium is described by the equations [2]: divD• = 0, divG• + gradvT · D + CT · D• = 0 (15.14) ×

d

D (R + ηv, H − ηH × ω)

dη η=0

d

• G = G (R + ηv, H − ηH × ω)

dη η=0 D• =

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D. Sheydakov

where D• and G• are the linearized Piola-type stress tensor and couple stress tensor. In the case of physically linear micropolar material (15.7) for these tensors the following relations are valid: • & ∂W ∂W • % ' • ( D• = ·H+ · H = λ tr Y E + (μ + κ) Y• + μY•T · H− ∂Y ∂Y (15.15) % % & & T − λtr (Y − E) E + μ Y − E + (μ + κ) (Y − E) · H × ω

∂W G = ∂L •

•

·H+

& ∂W • % ' • ( · H = γ1 tr L E + γ2 L• + γ3 L•T · H− ∂L % & − γ1 (tr L) E + γ2 L + γ3 LT · H × ω

'

(

Y• = gradv + C × ω · HT ,

(15.16)

L• = grad ω · HT

Here Y• is the linearized stretch tensor, L• is the linearized wryness tensor. The equations of neutral equilibrium in the framework of non-polar nonlinear theory of elasticity have the form [6]:

d

div D•± = 0, D•± = D± (R± + ηv± )

(15.17) dη η=0 Representations of the linearized Piola stress tensors D•± for physically linear material (15.8) are obtained by linearization of constitutive relations (15.10) with regard to (15.6): • ' ' ( ( ∂W± ∂W± D•± = · A± + · A•± = λ± trU•± E + 2μ± U•± · A± + ∂U± ∂U± (15.18) ( −1 ' • + (λ± tr (U± − E) E + 2μ± (U± − E)) · U± · gradv± − U± · A± Here U•± are the linearized stretch tensors, which can be expressed in terms of the linearized Cauchy-Green deformation tensors: % &• (U± · U± )• = C± · CT± ⇓

'

U•± · U± + U± · U•± = gradv± · CT± + C± · gradv±

(T

Linearized boundary conditions on the faces of the sandwich plate (x3 = a + h+ , x3 = −a − h−) and on the interfaces (x3 = ±a) are written as follows:

e3 · D•+

x =a+h = 0, e3 · D•−

x =−(a+h ) = 0 + − 3 3 (15.19)

•

•

•

e3 · D± x =±a = e3 · D x =±a , v± | x3 =±a = v| x3 =±a , e3 · G x =±a = 0 3

3

3


209

We assume that at the ends of the plate (x1 = 0, b1 ; x2 = 0, b2 ) there is no friction and constant normal displacements are given. This leads to the following linearized boundary conditions: 1) for the porous core of the plate (|x3 | ≤ a):

e1 · D• · e2

x

1 =0, b1

e1 · G• · e1

x

= e1 · D• · e3

x

1 =0, b1

1 =0, b1

e2 · D · e1

x •

2 =0, b2

e2 · G• · e2

x

= e2 · ω| x1 =0, b1 = e3 · ω| x1 =0, b1 = 0 (15.20)

= e2 · D · e3

x •

2 =0, b2

= e1 · v| x1 =0, b1 = 0

2 =0, b2

= e2 · v| x2 =0, b2 = 0

= e1 · ω| x2 =0, b2 = e3 · ω| x2 =0, b2 = 0

2) for the top and bottom coatings (a ≤ x3 ≤ a + h+, −a − h− ≤ x3 ≤ −a):

e1 · D•± · e2

x =0, b = e1 · D•± · e3

x =0, b = e1 · v± |x1 =0, b1 = 0 1 1 1 1

• • e2 · D± · e1 x =0, b = e2 · D± · e3 x =0, b = e2 · v± |x2 =0, b2 = 0 2

2

2

(15.21)

2

We write the vectors of additional displacements v and v± , and incremental rotation ω in the basis of Cartesian coordinates: v = v1 e1 + v2 e2 + v3 e3 v± = v±1 e1 + v±2 e2 + v±3 e3 ω = ω 1 e1 + ω 2 e2 + ω 3 e3

(15.22)

With respect to representation (15.22), the expressions for the linearized stretch tensors Y• and U•± , and wryness tensor L• have the form: ∂v1 ∂v2 ∂v3 Y• = e1 ⊗ e1 + − α1 ω 3 e1 ⊗ e2 + + α1 ω 2 e1 ⊗ e3 + ∂x1 ∂x1 ∂x1 ∂v1 ∂v2 ∂v3 + + α2 ω 3 e2 ⊗ e1 + e2 ⊗ e2 + − α2 ω 1 e2 ⊗ e3 + (15.23) ∂x2 ∂x2 ∂x2 ∂v1 ∂v2 ∂v3 + − α3 ω 2 e3 ⊗ e1 + + α3 ω 1 e3 ⊗ e2 + e3 ⊗ e3 ∂x3 ∂x3 ∂x3 ∂v±2 ∂v±3 1 ± (e2 ⊗ e3 + e3 ⊗ e2 ) + α + α 2 3 ∂x1 α2 + α±3 ∂x3 ∂x2 ∂v±3 ∂v±2 ∂v±1 1 ± (e1 ⊗ e3 + e3 ⊗ e1 ) + + e2 ⊗ e2 + α1 + α3 ∂x2 α1 + α±3 ∂x3 ∂x1 ∂v±3 ∂v±1 ∂v±2 1 (e1 ⊗ e2 + e2 ⊗ e1 ) + e3 ⊗ e3 + α1 + α2 ∂x3 α1 + α2 ∂x2 ∂x1

U•± =

∂v±1

e1 ⊗ e1 +

(15.24)

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D. Sheydakov

∂ω 1 ∂ω 2 ∂ω 3 ∂ω 1 e1 ⊗ e1 + e1 ⊗ e2 + e1 ⊗ e3 + e2 ⊗ e1 + ∂x1 ∂x1 ∂x1 ∂x2 ∂ω 2 ∂ω 3 ∂ω 1 ∂ω 2 ∂ω 3 + e2 ⊗ e2 + e2 ⊗ e3 + e3 ⊗ e1 + e3 ⊗ e2 + e3 ⊗ e3 ∂x2 ∂x2 ∂x3 ∂x3 ∂x3 L• =

(15.25)

According to relations (15.3) – (15.6), (15.15), (15.16), (15.18), (15.22) – (15.24), the components of the linearized Piola-type stress tensor D• and couple stress tensor G• , and Piola stress tensors D•± are written as follows: e1 · D• · e1 = (λ + χ)

∂v1 ∂v2 ∂v3 +λ +λ ∂x1 ∂x2 ∂x3

∂v2 ∂v1 +μ +(λs + μ(α1 + α2 )− χ)ω3 ∂x1 ∂x2 ∂v3 ∂v1 e1 · D• · e3 = (μ + κ) +μ − (λs + μ (α1 + α3 ) − χ) ω2 ∂x1 ∂x3 ∂v1 ∂v2 e2 · D• · e1 = (μ + κ) +μ −(λs + μ(α1 + α2 ) − χ) ω3 ∂x2 ∂x1 ∂v1 ∂v2 ∂v3 e2 · D• · e2 = λ + (λ + χ) +λ ∂x1 ∂x2 ∂x3 ∂v3 ∂v2 e2 · D• · e3 = (μ + κ) +μ + (λs + μ (α2 + α3 ) − χ) ω1 ∂x2 ∂x3 ∂v1 ∂v3 e3 · D• · e1 = (μ + κ) +μ + (λs + μ (α1 + α3 ) − χ) ω2 ∂x3 ∂x1 ∂v2 ∂v3 e3 · D• · e2 = (μ + κ) +μ − (λs + μ (α2 + α3 ) − χ) ω1 ∂x3 ∂x2 ∂v1 ∂v2 ∂v3 e3 · D• · e3 = λ +λ + (λ + χ) ∂x1 ∂x2 ∂x3 e1 · D• · e2 = (μ + κ)

∂ω1 ∂ω2 ∂ω3 + γ1 + γ1 ∂x1 ∂x2 ∂x3 ∂ω ∂ω ∂ω ∂ω 2 1 1 2 e1 · G• · e2 = γ2 + γ3 , e2 · G• · e1 = γ2 + γ3 ∂x1 ∂x2 ∂x2 ∂x1 ∂ω3 ∂ω1 ∂ω1 ∂ω3 e1 · G• · e3 = γ2 + γ3 , e3 · G• · e1 = γ2 + γ3 ∂x1 ∂x3 ∂x3 ∂x1 ∂ω ∂ω ∂ω 1 2 3 e2 · G• · e2 = γ1 + (γ1 + γ2 + γ3 ) + γ1 ∂x1 ∂x2 ∂x3 ∂ω3 ∂ω2 ∂ω2 ∂ω3 e2 · G• · e3 = γ2 + γ3 , e3 · G• · e2 = γ2 + γ3 ∂x2 ∂x3 ∂x3 ∂x2 ∂ω ∂ω ∂ω 1 2 3 e3 · G• · e3 = γ1 + γ1 + (γ1 + γ2 + γ3 ) ∂x1 ∂x2 ∂x3

(15.26)

e1 · G• · e1 = (γ1 + γ2 + γ3 )

(15.27)


∂v ∂v1 ∂v + λ± 2 + λ± 3 ∂x1 ∂x2 ∂x3 ± ± ± ∂v ∂v ∂v λ s − 2μ ± ± ± 2 e1 · D•± · e2 = 2μ± 2 + − 1 ∂x1 α1 + α2 ∂x1 ∂x2 ± ± ∂v λ± s± − 2μ± ∂v3 ∂v±1 e1 · D•± · e3 = 2μ± 3 + − ∂x1 α1 + α±3 ∂x1 ∂x3 ∂v±1 λ± s± − 2μ± ∂v±1 ∂v±2 • e2 · D± · e1 = 2μ± + − ∂x2 α1 + α2 ∂x2 ∂x1 ∂v± ∂v± ∂v± e2 · D•± · e2 = λ± 1 + (λ± + 2μ± ) 2 + λ± 3 ∂x1 ∂x2 ∂x3 ± ± ± ∂v ∂v ∂v λ s − 2μ ± ± ± 3 3 • e2 · D± · e3 = 2μ± + − 2 ∂x2 α2 + α±3 ∂x2 ∂x3 ∂v± λ± s± − 2μ± ∂v±1 ∂v±3 e3 · D•± · e1 = 2μ± 1 + − ∂x3 α1 + α±3 ∂x3 ∂x1 ∂v± λ± s± − 2μ± ∂v±2 ∂v±3 e3 · D•± · e2 = 2μ± 2 + − ∂x3 α2 + α±3 ∂x3 ∂x2 ∂v± ∂v± ∂v± e3 · D•± · e3 = λ± 1 + λ± 2 + (λ± + 2μ± ) 3 ∂x1 ∂x2 ∂x3 e1 · D•± · e1 = (λ± + 2μ± )

±

±

211

±

(15.28)

Expressions (15.14), (15.17), describing the perturbed state of equilibrium for a porous sandwich plate, constitute a system of twelve partial differential equations in the twelve unknown functions v1 , v2 , v3 , v±1 , v±2 , v±3 , ω1 , ω2 , ω3 . Substitutions v1 = V1 (r) sin β1 x1 cos β2 x2 , v2 = V2 (r) cos β1 x1 sin β2 x2 , v3 = V3 (r) cosβ1 x1 cos β2 x2 ,

v±1 = V1± (r) sin β1 x1 cos β2 x2 v±2 = V2± (r) cos β1 x1 sin β2 x2 v±3 = V3± (r) cos β1 x1 cos β2 x2

ω1 = Ω1 (r) cos β1 x1 sin β2 x2 ω2 = Ω2 (r) sin β1 x1 cos β2 x2 ω3 = Ω3 (r) sin β1 x1 sin β2 x2 β1 = πm1 /b1 ,

β2 = πm2 /b2 ,

(15.29)

(15.30)

m1,2 = 1, 2, ...

lead to the separation of variables x1 , x2 in these equations and allows to satisfy the linearized boundary conditions (15.20), (15.21) at the ends of the plate. By taking into account the relations (15.4), (15.9), (15.22), (15.26)–(15.29), equations of neutral equilibrium (15.14), (15.17) are written as follows:

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D. Sheydakov

% % && (μ + κ) V1 − (λ + μ) β21 + (μ + κ) β21 + β22 V1 − − (λ + μ)β1 β2 V2 − (λ + μ) β1 V3 + B2Ω2 − B3 β2 Ω3 = 0 % % && (μ + κ) V2 − (λ + μ) β22 + (μ + κ) β21 + β22 V2 − − (λ + μ)β1 β2 V1 − (λ + μ) β2 V3 − B1Ω1 + B3 β1 Ω3 = 0 % & (λ + χ) V3 − (μ + κ) β21 + β22 V3 + (λ + μ) β1 V1 + + (λ + μ)β2 V2 + B1β2 Ω1 − B2β1 Ω2 = 0

(15.31)

% % & & 2 2 2 γ2 Ω 1 − (γ1 + γ3 ) β1 + γ2 β1 + β2 − B1 (α2 + α3 ) Ω1 − − (γ1 + γ3 )β1 β2 Ω2 + (γ1 + γ3 ) β1 Ω3 + B1V2 + B1β2 V3 = 0 % % & & 2 2 2 γ2 Ω 2 − (γ1 + γ3 ) β2 + γ2 β1 + β2 − B2 (α1 + α3 ) Ω2 − − (γ1 + γ3 )β1 β2 Ω1 + (γ1 + γ3 ) β2 Ω3 − B2V1 − B2β1 V3 = 0 % % & & 2 2 (γ1 + γ2 + γ3 ) Ω 3 − γ2 β1 + β2 − B3 (α1 + α2 ) Ω3 − − (γ1 + γ3 )β1 Ω1 − (γ1 + γ3 ) β2 Ω2 − B3 β2 V1 + B3β1 V2 = 0 %

& % % & & 2μ+ + B+2 (V1+ ) − (λ+ + 2μ+ ) β21 + 2μ+ + B+3 β22 V1+ − % & % & − λ+ − B+3 β1 β2 V2+ − λ+ − B+2 β1 (V3+ ) = 0

%

& % % & & 2μ+ + B+1 (V2+ ) − (λ+ + 2μ+ ) β22 + 2μ+ + B+3 β21 V2+ − % & % & − λ+ − B+3 β1 β2 V1+ − λ+ − B+1 β2 (V3+ ) = 0

(15.32)

%% & % & & (λ+ + 2μ+ ) (V3+ ) − 2μ+ + B+2 β21 + 2μ+ + B+1 β22 V3+ + % & % & + λ+ − B+2 β1 (V1+ ) + λ+ − B+1 β2 (V2+ ) = 0 %

& % % & & 2μ− + B−2 (V1− ) − (λ− + 2μ− ) β21 + 2μ− + B−3 β22 V1− − % & % & − λ− − B−3 β1 β2 V2− − λ− − B−2 β1 (V3− ) = 0

%

& % % & & 2μ− + B−1 (V2− ) − (λ− + 2μ− ) β22 + 2μ− + B−3 β21 V2− − % & % & − λ− − B−3 β1 β2 V1− − λ− − B−1 β2 (V3− ) = 0

(15.33)


213

%% & % & & (λ− + 2μ− ) (V3− ) − 2μ− + B−2 β21 + 2μ− + B−1 β22 V3− + % & % & + λ− − B−2 β1 (V1− ) + λ− − B−1 β2 (V2− ) = 0 Here we use the following notations B1 = λ (α1 − 3) + (λ + μ) (α2 + α3 ) − χ B2 = λ (α2 − 3) + (λ + μ) (α1 + α3 ) − χ B3 = λ (α3 − 3) + (λ + μ) (α1 + α2 ) − χ B±1 =

λ± s± − 2μ± , α2 + α±3

B±2 =

λ± s± − 2μ± , α1 + α±3

B±3 =

λ± s± − 2μ± α1 + α2

The expressions for the linearized boundary conditions (15.19) take the form: 1) for x3 = a + h+: % & % & % & % & 2μ+ + B+2 V1+ + β1 B+2 V3+ = 0, 2μ+ + B+1 V2+ + β2 B+1 V3+ = 0 (15.34) % & λ+ β1 V1+ + λ+ β2 V2+ + (λ+ + 2μ+ ) V3+ = 0 2) for x3 = a: % & % & 2μ+ + B+2 V1+ + β1 B+2 V3+ − (μ + κ) V1 + β1 μV3 − B2Ω2 = 0 % & % & 2μ+ + B+1 V2+ + β2 B+1 V3+ − (μ + κ) V2 + β2 μV3 + B1Ω1 = 0 % & λ+ β1 V1+ + λ+ β2 V2+ + (λ+ + 2μ+ ) V3+ − λβ1 V1 − −λβ2 V2 − (λ + χ) V3 = 0

γ2 Ω1 + γ3 β1 Ω3 = 0,

(15.35)

γ2 Ω2 + γ3 β2 Ω3 = 0 γ1 β1 Ω1 + γ1 β2 Ω2 − (γ1 + γ2 + γ3 ) Ω3 = 0 V1 − V1+ = 0, V2 − V2+ = 0, V3 − V3+ = 0

3) for x3 = −a: % & % & 2μ− + B−2 V1− + β1 B−2 V3− − (μ + κ) V1 + β1 μV3 − B2Ω2 = 0 % & % & 2μ− + B−1 V2− + β2 B−1 V3− − (μ + κ) V2 + β2 μV3 + B1Ω1 = 0 % & λ− β1 V1− + λ− β2 V2− + (λ− + 2μ− ) V3− − λβ1 V1 − −λβ2 V2 − (λ + χ) V3 = 0

γ2 Ω1 + γ3 β1 Ω3 = 0,

γ2 Ω2 + γ3 β2 Ω3 = 0 γ1 β1 Ω1 + γ1 β2 Ω2 − (γ1 + γ2 + γ3 ) Ω3 = 0 V1 − V1− = 0, V2 − V2− = 0, V3 − V3− = 0

(15.36)

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D. Sheydakov

4) for x3 = −a − h−: % & % & 2μ− + B−2 V1− + β1 B−2 V3− = 0,

%

& % & 2μ− + B−1 V2− + β2 B−1 V3− = 0 % & λ− β1 V1− + λ− β2 V2− + (λ− + 2μ− ) V3− = 0

(15.37)

Thus, the stability analysis of the sandwich plate is reduced to solving a linear homogeneous boundary problem (15.31) – (15.37) for a system of twelve ordinary differential equations. It is easy to show that in the case of identical top and bottom coatings (h+ = h− , λ+ = λ− , μ+ = μ− ) the boundary problem (15.31) – (15.37) has two independent sets of solutions [10]. The First set is formed by solutions for which the deflection of a plate is an odd function of x3 :

0 ≤ x3 ≤ a

a ≤ x3 ≤

H 2

⎧ ⎪ ⎪ V1 (x3 ) = V1 (−x3 ), ⎪ ⎪ ⎪ ⎨ V2 (x3 ) = V2 (−x3 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V (x ) = −V (−x ), 3 3 3 3

Ω1 (x3 ) = −Ω1 (−x3 ) Ω2 (x3 ) = −Ω2 (−x3 ) Ω3 (x3 ) = Ω3 (−x3 ) (15.38)

⎧ + ⎪ ⎪ V1 (x3 ) = V1− (−x3 ) ⎪ ⎪ ⎪ ⎨ + V2 (x3 ) = V2− (−x3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V + (x ) = −V − (−x ) 3 3 3 3

For the Second set of solutions, on the contrary, the deflection is an even function:

0 ≤ x3 ≤ a

⎧ ⎪ V1 (x3 ) = −V1 (−x3 ), ⎪ ⎪ ⎪ ⎪ ⎨ V2 (x3 ) = −V2 (−x3 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V (x ) = V (−x ), 3

a ≤ x3 ≤

H 2

3

3

3

Ω1 (x3 ) = Ω1 (−x3 ) Ω2 (x3 ) = Ω2 (−x3 ) Ω3 (x3 ) = −Ω3 (−x3 )

⎧ + ⎪ V1 (x3 ) = −V1− (−x3 ) ⎪ ⎪ ⎪ ⎪ ⎨ + V2 (x3 ) = −V2− (−x3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V + (x ) = V − (−x ) 3 3 3 3

(15.39)

Due to this property of boundary problem (15.31) – (15.37), for the study of stability it is sufficient to consider only the upper half of the sandwich plate (0 ≤ x3 ≤ H/2). The boundary conditions at x3 = 0 follows from the evenness and oddness of the unknown functions V1 , V2 , V3 , Ω1 , Ω2 , Ω3 : a) for the First set of solutions: V1 (0) = V2 (0) = V3 (0) = 0,

Ω1 (0) = Ω2 (0) = Ω3 (0) = 0

(15.40)


215

b) for the Second set of solutions: V1 (0) = V2 (0) = V3 (0) = 0,

Ω1 (0) = Ω2 (0) = Ω3 (0) = 0

(15.41)

Thus, in the case of identical top and bottom coatings, the stability analysis of the sandwich plate is reduced to solving two linear homogeneous boundary-value problems – (15.31), (15.32), (15.34), (15.35), (15.40) and (15.31), (15.32), (15.34), (15.35), (15.41) – for a system of nine ordinary differential equations.

15.4 Conclusion In the framework of bifurcation approach, we studied the stability of an elastic rectangular sandwich plate with a porous core, which is subject to biaxial compression. For the physically linear material, a system of linearized equilibrium equations was derived, which describes the behavior of the sandwich plate in a perturbed state. Using special substitutions, the linearized boundary-value problems were formulated both for a case of a general sandwich plate and for a sandwich plate with identical top and bottom coatings. In the general case, the stability analysis was reduced to solving a linear homogeneous boundary problem (15.31) – (15.37) for a system of twelve ordinary differential equations. In the case of identical top and bottom coatings, the stability analysis of the sandwich plate was reduced to solving two linear homogeneous boundary-value problems – (15.31), (15.32), (15.34), (15.35), (15.40) and (15.31), (15.32), (15.34), (15.35), (15.41) – for a system of nine ordinary differential equations. For specific porous materials and coatings these boundary-value problems can be solved numerically, using methods described in [11]. Acknowledgements This work was supported by the President of Russian Federation (grant MK6315.2010.1) and by the Russian Foundation for Basic Research (grant 11-08-01152-a).

References 1. Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann et Fils, Paris (1909) 2. Eremeyev, V.A., Zubov, L.M.: On stability of elastic bodies with couple-stresses. Mekhanika Tverdovo Tela (3), 181–190 (1994). In Russian 3. Eringen, A.C.: Microcontinuum Field Theory. I. Foundations and Solids. Springer, New York (1999) 4. Kafadar, C.B., Eringen, A.C.: Micropolar media - I. The classical theory. International Journal of Engineering Science 9, 271–305 (1971) 5. Lakes, R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Muhlhaus, H., Wiley, J. (ed.) Continuum models for materials with microstructure, pp. 1–22. New York (1995) 6. Lurie, A.I.: Non-linear Theory of Elasticity. North-Holland, Amsterdam (1990) 7. Maugin, G.A.: On the structure of the theory of polar elasticity. Philosophical Transactions of Royal Society London A 356, 1367–1395 (1998)

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D. Sheydakov

8. Nikitin, E., Zubov, L.M.: Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress. Journal of Elasticity 51, 1–22 (1998) 9. Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures 46, 774–787 (2009) 10. Sheydakov, D.N.: Stability of a rectangular plate under biaxial tension. Journal of Applied Mechanics and Technical Physics 48(4), 547–555 (2007) 11. Sheydakov, D.N.: Buckling of elastic composite rod of micropolar material subject to combined loads. In: H. Altenbach, V.I. Erofeev, G.A. Maugin (ed.) Mechanics of Generalized Continua – From Micromechanical Basics to Engineering Applications, pp. 255–271. SpringerVerlag, Berlin et al. (2011) 12. Toupin, R.A.: Theories of elasticity with couple-stress. Archives for Rational Mechanics and Analysis 17, 85–112 (1964) 13. Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997)

Part III

Nonlinear Models and Coupled Fields

Chapter 16

On the Nonlinear Theory of Two-Phase Shells Victor A. Eremeyev and Wojciech Pietraszkiewicz

Abstract We discuss the nonlinear theory of shells made of material undergoing phase transitions (PT). The interest to such thin-walled structures is motivated by applications of thin films made of martensitic materials and needs of modeling biological membranes. Here we present the resultant, two-dimensional thermodynamics of non-linear theory of shells undergoing PT. The global and local formulations of the balances of momentum, moment of momentum, energy and entropy are given. Two temperature fields on the shell base surface are introduced: the referential mean temperature and its deviation, as well as two corresponding dual fields: the referential entropy and its deviation. Additional surface heat flux and the extra heat flux vector fields appear as a result of through-the-thickness integration procedure. Within the framework of the resultant shell thermodynamics we derive the continuity conditions along the curvilinear phase interface which separates two material phases. These conditions allow us to formulate the kinetic equation describing the quasistatic motion of the interface relative to the shell base surface. The kinetic equation is expressed by the jump of the Eshelby tensor across the phase interface. In the case of thermodynamic equilibrium the variational statement of the static problem of two-phase shell is presented. Keywords Non-linear shell · Shell thermodynamics · Phase transition · Cosserat shell · Micropolar shell · Kinetic equation · Singular curve

V. A. Eremeyev (B) Martin-Luther-Universität Halle-Wittenberg, Kurt-Mothes-Str., 1, 06099 Halle (Saale), Germany and South Scientific Center of RASci & South Federal University, Milchakova St. 8a, 344090 Rostov on Don, Russia e-mail: [email protected],[email protected] W. Pietraszkiewicz Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 Gdańsk, Poland e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 16, © Springer-Verlag Berlin Heidelberg 2011

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V.A. Eremeyev and W. Pietraszkiewicz

16.1 Introduction The interest in thin-walled structures undergoing PT grows recently from their perspective applications in engineering. As examples of such structures martensitic films and biological membranes can be considered. The stress- and temperatureinduced PT are widely observed in thin structures made of superelastic shape memory alloys (SMA) and shape memory polymers, such as NiTi, NiMnGa, AgCd, AuCd, CuAlNi, polyurethane, etc, which are used in various microelectromechanical systems (MEMS). Thin plates, strips, and tubes made of SMA are used as working elements of such MEMS as micropumps, sensors, actuators, microengines etc., see e.g. [8, 48, 53] and books [5, 32, 38]. Experiments on shape memory alloys and other materials undergoing PT are often performed with thin-walled samples such as thin strips, rectangular plates or thin tubes, see [25–28, 33, 41, 52, 56] among others. Other examples of PT in thinwalled structures are tents and tunnels appearing in martensitic thin films during PT, which were discovered and investigated in [6, 8, 24, 29, 47, 48]. The major known theories of PT in deformable solids are related to three-dimensional (3D) thermoelasticity, see the books [1, 4, 5, 22] and references given therein. The first two-dimensional (2D) mechanical model of PT in thin films was proposed in [7, 30, 47], see also [5, 38]. The model consists of the Cosserat membrane with one director, but without taking into account the bending stiffness of the membrane. Alternative membrane models of PT with applications to biomembrane modelling were proposed in [2, 10, 12]. The non-linear resultant equilibrium conditions of elastic shells undergoing PT of martensitic type were formulated by Eremeyev and Pietraszkiewicz [13] within the resultant dynamically exact and kinematically unique theory of shells presented in [11, 18, 34]. These conditions were extended in [44] taking into account the line tension energy of the interface. By analogy to the 3D case, the two-phase shell was regarded in [13,44] as the Cosserat surface consisting of two material phases divided by a sufficiently smooth surface singular curve (phase interface). Existence of such a curve was confirmed by several experiments on thin-walled plates, strips, and tubes, see e.g. [25–28,33,41,52,56]. These experiments demonstrate how the macroscopic domain of the new phase forms, show its further evolution during loading and annihilation after unloading. In the case of plates and strips the new phase forms often as a few bands across the strip. In the case of tubes the new phase may appear as helical or cylindrical bands which width and shape depend on acting loads. The phase boundary between “old” and “new” phases in many cases can be interpreted as a curvilinear, coherent, sharp phase interface. The quasistatic behaviour of twophase shells has recently been analyzed in [15–17]. In this paper we discuss the resultant, two-dimensional thermomechanics of shells undergoing diffusionless, displacive phase transitions of martensitic type of the shell material. In particular, we formulate the corresponding boundary-value problem (BVP). The main attention is paid to formulation of the compatibility conditions across the phase interface and to derivation of the kinetic equation describing propagation of the phase interface during loading and unloading.

16 On the Nonlinear Theory of Two-Phase Shells

221

16.2 Kinematics of 6-Parametric Theory of Shells The kinematics of general resultant theory of shells coincides with the kinematics of 2D Cosserat continuum, see [11, 18, 34] for details. In the undeformed placement the shell is represented by the base surface M with the position vector x(θα ) and the unit normal vector n(θα ), where {θα }, α = 1, 2, are the surface curvilinear coordinates. In the deformed placement the shell is represented by the surface N = χ(M) with the position vector y = χ(x) and with the attached three directors (dα , d). The deformation of the shell is described by the relations y(x, t) = χ(x) = x + u(x, t),

dα (x, t) = Q(x, t)x,α ,

d(x, t) = Q(x, t)n(x),

(16.1)

where t is a time-like scalar parameter, χ the deformation function, u ∈ E the translation vector of M, and Q ∈ S O(3) the proper orthogonal tensor representing the workaveraged gross rotation of the shell cross sections from their undeformed shapes described by (x,α , n), where (. . .),α denotes partial differentiation with respect to θα . ˙ T ) the angular velocity vecThen υ ≡ u˙ is the translation velocity and ω ≡ ax(QQ tors, where ax(. . .) is the axial vector associated with the skew tensor (. . .), and (. ˙. .) denotes the derivative with respect to t. Within the framework of 6-parametric theory of shells considered here, the following two strain measures corresponding to the deformations (16.1) are introduced, see [11, 13, 14, 43]: E = ε α ⊗ aα ,

K = κ α ⊗ aα ,

εα = y,α − dα ,

1 κα = di × Q,α QT d i , 2

(16.2)

where (aα , n) and (di ), i = 1, 2, 3, are bases reciprocal to the base (x,α , n) and the base (dα , d), respectively. We assume that in the deformed placement the shell consists of different material phases occupying different complementary subregions separated by the curvilinear phase interface D ∈ N. For a piecewise differentiable mapping χ we can introduce on M a singular image curve C = χ−1 (D) with the position vector xC . We call a priori unknown curves D and C the phase interfaces in the deformed and reference placements, respectively. Let us note that xC and yD are kinematically independent on u and Q. This means that D and C are non-material curves, in general. For the description of motion of the surface curve C on M we introduce the phase interface velocity V ≡ x˙ C · ν , where ν ∈ T x M is the unit external normal vector to C, and ν · n = 0. Hence, y (or u), Q, and xC constitute the basic kinematic unknown variables in the theory of shells undergoing PT.

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16.3 Integral Balance Equations The resultant 2D equations of the general non-linear theory of shells can be derived exactly by direct through-the-thickness integration of 3D balance laws of linear and angular momentum as well as of the energy balance and the entropy inequality of continuum mechanics, see [11, 13, 17, 34]. In quasi-static problems discussed here the global equilibrium conditions require the total force and total torque of all loads acting upon any part P ⊂ M\C to vanish F = 0,

where F≡ M≡

f da +

P

∂P\∂M f

(16.3)

n∗ ds,

nν ds +

∂P\∂M f

(c + y × f ) da + P

M = 0,

∂P∩∂M f

(mν + y × nν ) ds +

' ∗ ( m + y × n∗ ds.

∂P∩∂M f

Here f and c are the resultant surface force and couple vector fields acting on N\D, but measured per unit area of M\C. Similarly, nν and mν are the internal contact stress and couple resultant vectors defined at an arbitrary edge ∂R of R = χ(P), while n∗ and m∗ are the external boundary resultant force and couple vectors applied along the part ∂N f of N = χ(M), respectively. The latter four vectors are measured per unit length of the corresponding undeformed edges ∂P and ∂M f , respectively. According to the Cauchy postulate, the contact vectors nν and mν can be represented through the respective internal surface stress and couple resultant tensors N and M by nν = Nνν, mν = Mνν. (16.4) The tensors N, M ∈ E ⊗ T x M defined on M\C are the resultant surface stress measures of the 1st Piola–Kirchhoff type, respectively. In the literature various descriptions of shell thermodynamics are known, see e.g. [15, 18, 20, 21, 35, 39, 40, 45, 46, 49–51, 57], where various sets of surface fields responsible for temperature were used and several formulations of the first and second laws of thermodynamics for shells were discussed. The resultant local thermomechanic energy balance and the entropy inequality for the shell can also be derived by direct through-the-thickness integration of the global 3D thermomechanic balance of energy and the entropy inequality, see [35, 49–51]. However, in the construction of 2D thermodynamic relations for shells one cannot simply transfer the notions of temperature, entropy and energy from the 3D case to their averages defined on M. For example, in the 3D shell-like body it is quite natural to associate different temperatures with its lower, upper and lateral boundary surfaces. In the resultant shell model this leads to appearance of several 2D temperature fields defined in the same point of the base surface. Each such field may require


223

its own heat conduction law. As a result, the 2D entropy inequality for shells should also be appropriately modified. Similarly, the 3D stress power density of the shell-like body takes into account also a part of stress power not expressible in terms of 2D stress resultants and stress couples, see [43], Sect. 7. In the 2D energy balance equation this may require to introduce additional surface sources of energy with their own constitutive relations, see discussion in [35, 42]. The referential form of energy balance (The 1st Law of thermomechanics) of an arbitrary part P of the shell base surface M\C can be described in analogy to the 3D energy balance, see [54, 55], by the resultant quantities [35] as ˙ + E˙ = A + Q, K

(16.5)

where K is the resultant kinetic energy, E the resultant internal energy, A the resultant mechanical power, and Q is the resultant heating. For the quasistatic process discussed here K = 0, while E, A, and Q can be represented by E≡ ρ# da, A ≡ (f · υ + c · ω ) da + (nν · υ + mν · ω ) ds P

P

∂P\∂M f

(n∗ · υ + m∗ · ω ) ds,

+ Q≡

∂P∩∂M f

ρr da + P

qν ds +

∂P\∂Mh

q∗ ds,

∂P∩∂Mh

where ρ is the resultant surface mass density in undeformed placement, ε the internal resultant surface strain energy density per unit undeformed surface mass, and r the internal surface heat supply minus heat fluxes through the upper and lower shell faces, all per unit mass of M, qν and q∗ are the surface heat fluxes through ∂P and ∂Mh , respectively. The contact heat flux qν can be represented through the surface heat flux vector q by the formula qν = q · ν . The referential form of entropy inequality (The 2nd Law of thermomechanics) of an arbitrary part P of the shell base surface M\C follows from the Clausius-Duhem inequality [54, 55], ˙ ≥ J, H (16.6) where in our case H is the resultant shell entropy and J the resultant entropy supply. For any part P ⊂ M\C these fields are defined as follows: H≡ ρη da, J ≡ ρ j da + jν ds + j∗ ds, P

P

∂P\∂Mh

∂P∩∂Mh

224


where η is the resultant internal entropy density, j the resultant entropy supply minus entropy fluxes through the upper and lower shell faces, both per unit undeformed surface mass, and jν and j∗ are the resultant entropy fluxes through the internal ∂P and external ∂Mh boundary contours, respectively. The field jν can be expressed through the referential entropy flux vector j ∈ T x M according to jν = j · ν. The relations between the resultant quantities and their 3D counterparts can be derived by use of the through-the-thickness integration procedure applied to the 3D balance equations [17]. After [39] we introduce the mean referential temperature θ(x, t) > 0 and the temperature deviation ϕ(x, t) by 1 1 1 1 1 1 1 = + , ϕ= − , (16.7) θ 2 Θ+ Θ− h Θ− Θ+ where Θ± > 0 are temperatures of the upper and lower shell faces M± taken to be equal to those prevailing in the adjoining external media, and h is the shell thickness. Unlike in the 3D entropy balance [54, 55], the resultant entropy supply j and the resultant entropy flux j take now the extended form [17], 1 j = r − ϕs, θ

1 j = q − ϕs, θ

(16.8)

where s is the resultant extra heat supply and s is the resultant extra heat flux vector.

16.4 Local Shell Equations and Constitutive Relations From the integral equilibrium equations (16.3), the energy balance equation (16.5) and the entropy inequality (16.6), after appropriate transformations follow the local Lagrangian equilibrium equations and the static boundary conditions % & Div N + f = 0, DivM + ax NFT − FNT + c = 0 in M\C, (16.9) Nνν − n∗ = 0,

Mνν − m∗ = 0 along ∂M f ,

the local resultant thermomechanic balances of energy in the referential description ρε˙ = ρr − Div q + N • E◦ + M • K◦ q · ν − q∗ = 0

in M\C,

along ∂Mh ,

(16.10)


225

and the local resultant entropy inequalities r 1 1 ρ˙η − ρ − ϕs + Div q − ϕDiv s + h · s − 2 q · g ≥ 0 in M\C, θ θ θ q q∗ ν ∗ ∗ − ϕ s − − ϕs ≥ 0 along ∂Mh , ν θ∗ θ g = Grad θ,

h = Grad ϕ,

(16.11)

g, h ∈ T x M,

of the non-linear theory of shells. Here F ≡ Grad y = y,α ⊗ aα is the surface deformation gradient, F ∈ E ⊗ T x M, Div N ≡ N ,α · aα means the surface divergence of N, (·)◦ ≡ Q dtd [QT (·)] is the co-rotational time derivative, and the scalar product of two tensors A, B ∈ E ⊗ T x M is defined by A • B ≡ tr (AT B). The fields u, Q, θ, ϕ constitute the basic thermo-kinematic independent variables of the shell boundary value problem in M\C, while the fields N, M, ε, η, χ, q, and s have to be specified by the constitutive equations. Here, as an example we discuss the constitutive equations for thermoelastic shells which take the form [17], ψ ≡ ε − θη − ϕχ = ψ(E, K, θ, ϕ), N = ρψ,E ,

M = ρψ,K ,

q = q(E, K, θ, g, ϕ, h),

η = −ψ,θ ,

χ = −ψ,ϕ ,

(16.12)

s = s(E, K, θ, g, ϕ, h),

where we have introduced the surface free energy density ψ. For thermoelastic shells the local energy balance equation (16.10) reduces to the form ρ(θη˙ + ϕχ) ˙ = ρr − Div q, (16.13) while the local entropy inequality (16.11) results in the equation − ρχ˙ + ρθs − θDiv s = cϕ,

c ≥ 0,

(16.14)

where the new constitutive function c is introduced, and the reduced dissipation inequality becomes 1 − g · q − θh · s ≥ 0. (16.15) θ Both relations (16.13) and (16.14) play the role of thermoconductivity equations in the theory of thermoelastic shells. The two equations are necessary to determine two fields: the surface mean temperature θ and the surface temperature deviation ϕ. When s = 0, the equation (16.14) contains as the special case the equation for temperature deviation established for thermoelastic beams in [50]. The simplest cases of the constitutive equations for q and s satisfying (16.15) may be taken similar to the referential Fourier law in 3D continuum mechanics: q = −c g,

s = −c⊥ h,

(16.16)

226


where c is the positive heat conductivity of the shell in tangential direction and c⊥ is the positive heat conductivity of the shell in the transverse normal direction.

16.5 Continuity Conditions Along Phase Interface and Kinetic Equation Since all fields defined on M can be discontinuous across C, the phase interface C can be considered as a surface non-material singular curve. In particular, the curvilinear phase interfaces in shells can be either coherent or incoherent in rotations, see [13]. For the coherent interface both fields y (or u) and Q are supposed to be continuous at C and the kinematic compatibility conditions along C become [[υυ]] + V[[Fνν]] = 0, ω]] + V[[ Kνν ]] = 0, [[ω

(16.17) (16.18)

where the expression [[. . .]] = (. . .)B − (. . .)A means the jump at C. The phase interface is called incoherent in rotations if only y (or u) is continuous at C but Q may be discontinous. In this case the condition (16.17) is still satisfied, but (16.18) may be violated, see [13]. Assuming [[y]] = 0 along C, from (16.3) we obtain the local Lagrangian dynamic compatibility conditions [11], [[Nνν]] = 0,

[[Mνν]] = 0,

(16.19)

which are just the local balances of forces and couples at C in the case of quasistatic deformations. Additionally, we assume that the surface temperature field θ and its deviation ϕ are continuous on the whole M, that is [[θ]] = 0,

[[ϕ]] = 0

along C.

(16.20)

The local energy balance and the entropy inequality along C corresponding to (16.5) and (16.6) are [17], V[[ρε]] + [[Nνν · υ ]] + [[Mνν · ω ]] − [[q · ν ]] = 0, .. // 1 V[[ρη]] − q · ν + ϕs · ν ≡ δ2C ≥ 0 , θ

(16.21) (16.22)

where δ2C ≥ 0 denotes the surface entropy production along C. The second thermoconductivity equation (16.14) leads to the relation along C, 1 V [[ρχ]] − [[s· ν ]] = 0 . θ

(16.23)


227

From (16.17)–(16.23) we obtain the compatibility condition in the form 3 4 θδ2C = −V [[ρψ]] − ν · NT [[Fνν]] − ν · MT [[Kνν]] along C (16.24) for the coherent phase interface, and 3 4 θδ2C = −V [[ρψ]] − ν · NT [[Fνν]]

along C

(16.25)

for the phase interface incoherent in rotations. The entropy production δ2C remains always non-negative for all thermomechanical processes. This allows us to postulate the kinetic equation, describing motion of the phase interface for all quasistatic processes, in the form V = −F (νν · [[C]]νν) ,

(16.26)

where F is the non-negative definite kinetic function depending on the jump of C at C, i.e. F(ς) ≥ 0 for ς > 0, where C = Cc ≡ ρψA − N T F − MT K for the coherent interface and C = Ci ≡ ρψA − NT F for the one incoherent in rotations, A = 1 − n ⊗ n, and 1 is the 3D unit tensor. In the nonlinear shell theory the tensors Cc and Ci play the role of the Eshelby tensors or the energy-momentum tensors. In 3D continuum mechanics Eshelby-type tensors have various applications in the configurational mechanics [4, 23, 31, 36]. In particular, in the linear theory of plates and shells the Eshelby tensor was used to formulate the 2D conservation laws and the path-independent integrals, see [31]. Following [1, 4, 16] we take the kinetic function F(ς) in the form ⎧ k(ς − ς ) 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 + ξ(ς − ς0 ) ⎪ ⎪ ⎨ F(ς) = ⎪ 0 ⎪ ⎪ ⎪ ⎪ k(ς + ς0 ) ⎪ ⎪ ⎪ ⎩ 1 − ξ(ς + ς0 )

ς ≥ ς0 , −ς0 < ς < ς0 ,

(16.27)

ς ≤ −ς0 .

Here ς0 describes the effects associated with nucleation of the new phase and action of the surface tension, see [1], ξ is a parameter describing limit value of the phase transition velocity [4], and k is a positive kinetic factor. Equation (16.26) with (16.27) can be considered as the special constitutive equation describing the motion of phase interfaces in shells.

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16.6 Variational Statement of Thermodynamic Equilibrium of Two-Phase Shell Since the contributions of Gibbs [19], the variational formulation is widely used for description of phase transitions in solids, see e.g. [3, 5, 9, 22, 37]. The weak statement of quasistatic problems of non-linear shells undergoing PT was considered by Eremeyev and Pietraskiewicz [13], where the thermodynamic continuity condition was derived and relations for the Eshelby tensor C in shells were obtained. Let us note that the variational approach requires the thermodynamic equilibrium of twophase shell to be the minimizer or the stationary point of the functional of total energy. It does not describe the evolution of C as the external loads or the temperature are changing. The evolution of phase interface can be analyzed using the kinetic equation (16.26) and the BVP presented in previous Sections. Let us consider the isothermal process, i.e. we assume that θ is constant and ϕ = 0 during loading. In this case the thermodynamic equilibrium corresponds to the local or global extremum of the functional of free energy [19, 22]. Hence, the free energy ψ plays the role of the strain energy used in [13], and we have the variational principle in the form δU = A,

U=

ρψ da,

(16.28)

M

where we use the same relation for A as in Sect. 16.3, but here υ = δu is the virtual translation vector and ω = ax (δQQT ) the virtual rotation vector. Using (16.17) and (16.18), from (16.28) follow the equilibrium equations (16.9) and the static compatibility conditions (16.19). Additionally, we obtain the thermodynamic compatibility condition [[νν · Cνν]] = 0,

(16.29)

where C = Cc for the coherent interface and C = Ci for the interface incoherent in rotations. Equation (16.29) can be used to find position of the phase interface C in the thermodynamic equilibrium state. It corresponds to the stationary solution of the kinetic equation (16.26). Assuming adiabatic behaviour, i.e. when η is constant and χ = 0, we use the variational principle (16.28) with the functional of internal energy [19, 22], U= ρε da. M

The continuity condition (16.29) was extended in [44] taking into account the phase interface energy and other line fields defined along C.


229

16.7 Conclusions We have presented the resultant, two-dimensional thermomechanics of shells undergoing diffusionless, displacive phase transitions of martensitic type of the shell material. We have used the extended thermodynamics of shells with two temperature fields, i.e. the referential surface mean temperature and the temperature deviation, as well as two dual surface entropy related fields. We have discussed the thermodynamic continuity conditions along the curvilinear phase interface for quasistatic motion and for thermodynamic equilibrium. Summarising, in the case of finite deformations the thermomechanic BVP for thermoelastic shells undergoing phase transition consists of: • the equilibrium equations (16.9)1,2 supplemented by appropriate static and kinematic boundary conditions for N, M, u, and Q, • the thermoconductivity equations (16.13) and (16.14) with appropriate boundary conditions for θ and ϕ, • the compatibility conditions (16.19), (16.20), and (16.23) along the interface C, • the kinetic equation (16.26) or the thermodynamic equilibrium condition (16.29) along C, all supplemented with the proper constitutive equations for N, M, ε, η, χ, q, and s, see [17]. The kinetic equation (16.26) is used to find position of the curvilinear interface C in its quasistatic motion, while (16.29) is used to find the equilibrium position of C. The BVP summarized above was illustrated in [17] by the 1D analytically solved example of stretching and bending of the two-phase circular plate. The somewhat simpler version of the thermomechanic shell model undergoing PT with only one surface mean temperature field was earlier developed in [15] and illustrated by the 1D analytic solution of tension and bending of two-phase tube in membrane [15] and bending [16] approximations. However, realistic 2D experimental observations on thin-walled samples presented in papers cited in Introduction can only be verified numerically by two-dimensional solutions of the BVP developed here. For this purpose one still needs to develop 2D computer codes based on the extended finite element method (XFEM) for shells with moving singular curves. Acknowledgements The first author was supported by the DFG grant No. AL 341/33-1, by the RFBR with the grant No. 09-01-00459, and by the Federal target program “Scientific and pedagogical cadre of the innovated Russia during 2009–2013” (state contract P–361), while the second author was supported by the Polish Ministry of Science and Education under grant No. N506 254237.

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References 1. Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge (2006) 2. Agrawal, A., Steigmann, D.J.: Coexistent fluid-phase equilibria in biomembranes with bending elasticity. Journal of Elasticity 93(1), 63–80 (2008) 3. Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Archive for Rational Mechanics and Analysis 100(1), 13–52 (1987) 4. Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey et al. (2008) 5. Bhattacharya, K.: Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press, Oxford (2003) 6. Bhattacharya, K., DeSimone, A., Hane, K.F., James, R.D., Palmstrøm, C.J.: Tents and tunnels on martensitic films. Materials Science and Engineering A 273(Sp. Iss. SI), 685–689 (1999) 7. Bhattacharya, K., James, R.D.: A theory of thin films of martensitic materials with applications to microactuators. Journal of the Mechanics and Physics of Solids 47(3), 531–576 (1999) 8. Bhattacharya, K., James, R.D.: The material is the machine. Science 307(5706), 53–54 (2005) 9. Bhattacharya, K., Kohn, R.V.: Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials. Archive for Rational Mechanics and Analysis 139(2), 99–180 (1997) 10. Boulbitch, A.A.: Equations of heterophase equilibrium of a biomembrane. Archive of Applied Mechanics 69(2), 83–93 (1999) 11. Chrósćielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and Dynamics of Multifold Shells: Nonlinear Theory and Finite Element Method (in Polish). Wydawnictwo IPPT PAN, Warszawa (2004) 12. Elliott, C.M., Stinner, B.: A surface phase field model for two-phase biological membranes. SIAM J. Appl. Math. 70(8), 2904–2928 (2010) 13. Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. Journal of Elasticity 74(1), 67–86 (2004) 14. Eremeyev, V.A., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. Journal of Elasticity 85(2), 125–152 (2006) 15. Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Archives of Mechanics 61(1), 41–67 (2009) 16. Eremeyev, V.A., Pietraszkiewicz, W.: On tension of a two-phase elastic tube. In: W. Pietraszkiewicz, I. Kreja (eds.) Shell Structures: Theory and Applications, Vol. 2., pp. 63–66. CRC Press, Boca Raton (2010) 17. Eremeyev, V.A., Pietraszkiewicz, W.: Thermomechanics of shells undergoing phase transition. Journal of the Mechanics and Physics of Solids DOI 10.1016/j.jmps.2011.04.005 (2011) 18. Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moscow (2008) 19. Gibbs, J.W.: On the equilibrium of heterogeneous substances. In: The Collected Works of J. Willard Gibbs, pp. 55–353. Longmans, Green & Co, New York (1928) 20. Green, A.E., Naghdi, P.M.: Non-isothermal theory of rods, plates and shells. International Journals of Solids and Structures 6(2), 635–648 (1970) 21. Green, A.E., Naghdi, P.M.: On thermal effects in the theory of shells. Proceedings of the Royal Society of London Series A 365(1721), 161–190 (1979) 22. Grinfeld, M.: Thermodynamics Methods in the Theory of Heterogeneous Systems. Longman, Harlow (1991) 23. Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. SpringerVerlag, Berlin (2000) 24. Hane, K.F.: Bulk and thin film microstructures in untwinned martensites. Journal of the Mechanics and Physics of Solids 47, 1917–1939 (1999) 25. He, Y.J., Sun, Q.P.: Effects of structural and material length scales on stress-induced martensite macro-domain patterns in tube configurations. International Journal of Solids and Structures 46(16), 3045–3060 (2009)


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26. He, Y.J., Sun, Q.P.: Scaling relationship on macroscopic helical domains in NiTi tubes. International Journal of Solids and Structures 46(24), 4242–4251 (2009) 27. He, Y.J., Sun, Q.P.: Macroscopic equilibrium domain structure and geometric compatibility in elastic phase transition of thin plates. International Journal of Mechanical Sciences 52(2), 198–211 (2010) 28. He, Y.J., Sun, Q.P.: Rate-dependent domain spacing in a stretched NiTi strip. International Journal of Solids and Structures 47(20), 2775–2783 (2010) 29. James, R.D., Hane, K.F.: Martensitic transformations and shape-memory materials. Acta Materialia 48(1), 197–222 (2000) 30. James, R.D., Rizzoni, R.: Pressurized shape memory thin films. Journal of Elasticity 59(1–3), 399–436 (2000) 31. Kienzler, R., Herrman, G.: Mechanics in Material Space with Applications to Defect and Fracture Mechanics. Springer-Verlag, Berlin (2000) 32. Lagoudas, D.C. (ed.): Shape Memory Alloys. Modeling and Engineering Applications. Springer, Berlin (2008) 33. Li, Z.Q., Sun, Q.P.: The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension. International Journal of Plasticity 18(11), 1481–1498 (2002) 34. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998) 35. Makowski, J., Pietraszkiewicz, W.: Thermomechanics of shells with singular curves. Zesz. Nauk. No 528/1487/2002, IMP PAN, Gdańsk (2002) 36. Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman Hall, London (1993) 37. Mielke, A., Theil, F., Levitas, V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Archive for Rational Mechanics and Analysis 162(2), 137–177 (2002) 38. Miyazaki, S., Fu, Y.Q., Huang, W.M. (eds.): Thin Film Shape Memory Alloys: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2009) 39. Murdoch, A.I.: On the entropy inequality for material interfaces. ZAMP 27(5), 599–605 (1976) 40. Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. The Quarterly Journal of Mechanics and Applied Mathematics 29(3), 245–274 (1976) 41. Pieczyska, E.: Activity of stress-induced martensite transformation in TiNi shape memory alloy studied by infrared technique. Journal of Modern Optics 57(18, Sp. Iss. SI), 1700–1707 (2010) 42. Pietraszkiewicz, W.: On non-linear shell thermodynamics with interstitial working. In: K. Wilmański, J. Je¸drysiak, B. Michalak (eds.) Mathematical Methods in Continuum Mechanics, Chapter 11 (in print). Politechnika Łódzka, Łód´z (2011) 43. Pietraszkiewicz, W., Chrósćielewski, J., Makowski, J.: On dynamically and kinematically exact theory of shells. In: W. Pietraszkiewicz, C. Szymczak (eds.) Shell Structures: Theory and Applications, pp. 163–167. Taylor & Francis, London (2005) 44. Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2), 150–159 (2007) 45. Rubin, M.B.: Restrictions on linear constitutive equations for a rigid heat conducting Cosserat shell. International Journal of Solids and Structures 41(24–25), 7009–7033 (2004) 46. Rubin, M.B.: Heat conduction between confocal elliptical surfaces using the theory of a Cosserat shell. International Journal of Solids and Structures 43(2), 295–306 (2006) 47. Shu, Y.C.: Heterogeneous thin films of martensitic materials. Archive of Rational Mechanics and Analysis 153(1), 39–90 (2000) 48. Shu, Y.C.: Shape-memory micropumps. Materials Transactions 43(5, Sp. Iss. SI), 1037–1044 (2002) 49. Simmonds, J.G.: The thermodynamical theory of shells: Descent from 3-dimensions without thickness expansions. In: E.K. Axelrad, F.A. Emmerling (eds.) Flexible Shells, Theory and Applications, pp. 1–11. Springer, Berlin (1984) 50. Simmonds, J.G.: A simple nonlinear thermodynamic theory of arbitrary elastic beams. Journal of Elasticity 81(1), 51–62 (2005)

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51. Simmonds, J.G.: A classical, nonlinear thermodynamic theory of elastic shells based on a single constitutive assumption. Journal of Elasticity DOI 10.1007/s10659-010-9293-2 (2011) 52. Sun, Q.P. (ed.): Mechanics of Martensitic Phase Transformation in Solids. Kluwer, Dordrecht (2002) 53. Tobushi, H., Pieczyska, E.A., Nowacki, W.K., Sakuragi, T., Sugimoto, Y.: Torsional deformation and rotary driving characteristics of SMA thin strip. Archives of Mechanics 61(3-4), 241–257 (2009) 54. Truesdell, C.: Rational Thermodynamics, 2nd edn. Springer, New York (1984) 55. Truesdell, C.A.: The Elements of Continuum Mechanics. Springer, Berlin (1966) 56. Zhang, X.H., Feng, P., He, Y.J., Yu, T.X., Sun, Q.P.: Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips. International Journal of Mechanical Sciences 52(12), 1660–1670 (2010) 57. Zhilin, P.A.: Mechanics of deformable directed surfaces. International Journals of Solids and Structures 12(9–10), 635–648 (1976)

Chapter 17

A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures An Danh Nguyen, Marcus Stoffel and Dieter Weichert

Abstract A finite element model of non-local damage viscoplasticity for dynamic analysis of thin-walled shell structures is presented. To take void nucleation and growth into account, a non-local implicit gradient formulation is employed. The free energy function includes both a non-local damage variable on the mid-surface of shell structures and a local one in shell space. Local constitutive laws considering viscoplastic behavior, isotropic hardening and isotropic ductile damage leading to softening are used. The performance of the proposed approach is demonstrated through the numerical simulation of shock-wave loaded structures. Keywords Non-local damage · Viscoplasticity · Non-linear shells · Finite element

17.1 Introduction Simulation of metallic thin-shell structures subjected to impulsive loadings plays a very important role in engineering design of many fields, i.e. mechanical and civil engineering, aerospace, crash tests, explosions. In such conditions, materials of the structures may exhibit inelastic behavior, where the impact, internal damage of microstructure can develop overtime and lead to collapse of the structure instantly. It is well-known that softening and damage effects, which are always accompanied by localization of deformation, lead to the ill-posedness of the boundary value problem and, consequently, mesh-dependence, incorrect size effect, and excessive damage localization. A possibility to overcome such the mathematical difficulty is to introduce the gradient of plastic strain or the damage parameter fields in order A. D. Nguyen (B) · M. Stoffel · D. Weichert Institute of General Mechanics, Templergraben 64, D-52062 Aachen, Germany e-mail: [email protected],[email protected] e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 17, © Springer-Verlag Berlin Heidelberg 2011

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to penalize possible sharp localization. The use of gradients in the localization of deformation and fracture allows to obtain the thickness of shear band and to simulate the effect of surface tension forces (c.f. [15–17]. Among many strain-gradient and non-local damage theories, the approach proposed by Dimitrijevic and Hackl [4] are particularly interesting for our study to investigate vibration of thin-shell structures. The approach introduces a new nonlocal damage variable to transfer the values of the inelastic variables across the element boundaries, while preserving the C0 interpolation order of displacements. Such advantages help to preserve the shell theory and to reuse existing local constitutive laws of material behavior. In this paper, based on the above approach we develop a gradient-enhanced damage model for dynamic finite element computation of viscoplastic thin-shell structures. To take void nucleation and growth into account, an enhanced free energy function, which includes both a non-local damage variable on the mid-surface of shell structures and a local one in shell space, is introduced to penalize the difference between the non-local and local damage free energy. Consequently, only the potential energy is enhanced while the standard kinetic energy is preserved under assumption that the dynamic effect due to appearance of non-local variable is ignored. The dynamic thin-shell elastic theory proposed in [7] is used to integrate the presented gradient-enhanced model into the finite element program FEAP [1] to capture finite deformation. Local constitutive laws considering viscoplastic behavior, isotropic hardening and isotropic ductile damage leading to softening in [9] are employed. The performance of the proposed formulation is illustrated through numerical simulation. The damage evolution as well as property of mesh-independence are investigated in the first example of a square plate subjected to membrane loading. Then the approach is applied to simulate a copper circular plate subjected to impulse loading. The numerical results of the latter example is validated by comparison with the experimental results obtained in [2].

17.2 Dynamic Thin Shell Model 17.2.1 Basic Kinematics of Shells The non-linear shell model proposed by [6], in which the shell is regarded as an inextensible one-director Cosserat surface, is used. An overview of the shell model is given below. Further information can be found in [5, 8].

17 A Gradient-Enhanced Damage Model for Viscoplastic Thin-Shell Structures

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17.2.1.1 Shell Configurations Reference configuration of a shell is described as D T (ξ1 , ξ2 ) B := X¯ ∈ R3 | X¯ (ξ1 , ξ2 , ξ) := X (ξ1 , ξ2 ) + ξT E X , T ) ∈ C and ξ ∈ [h− , h+ ] with (X

(17.1)

Any configuration x¯ of the shell S ⊂ R3 is assumed to be defined as D S := x¯ ∈ R3 | x¯ (ξ1 , ξ2 , ξ) := x (ξ1 , ξ2 ) + ξtt(ξ1 , ξ2 ) E with (xx, t ) ∈ C and ξ ∈ [h− , h+ ]

(17.2)

where X denotes the position vector of the reference mid-surface, x = X + u denotes the position vector of the current mid-surface, u represents displacement vector of the related mid-surface point, t denotes a unit vector field at each point of the surface, referred to as the director field, h− = −h/2; h+ = h/2 where h denotes the thickness of the shell. A mid-surface A is defined by vector x and points belongs to A are defined by ξ |ξ3 =0 = ξ1 E 1 + ξ2 E 2 . The translational velocity vector of the mid-surface point at time t > 0 is denoted as x˙ and the angular velocity of the shell-director t is denoted as ω . Since the shelldirector is inextensible, the following relation is obtained (c.f. [5]) ω = t × ˙t .

(17.3)

17.2.1.2 Motion of Shell Director Motion of the extensible-shell director may be expressed by using an orthogonal transformation t = RT .

(17.4)

ˆ through an The rotation tensor Θ is associated with a skew-symmetric tensor Θ exponential mapping Θ ˆ 1 − cosΘ Θ ˆ 2 ˆ = I + sin Θ R = exp Θ Θ+ Θ . 2 Θ Θ Θ Θ

(17.5)

E i }i=1,3 be differLet {eei }i=1,3 bestandard orthonormal vector basis in R3 and {E ent orthonormal basis for the mid-surface. The motion of the director then may be expressed in different way t = R T = R Λ 0 T¯ = Λ T¯

(17.6)

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E 1 E 2 T ] represents a transformation matrix to transform between midwhere Λ 0 = [E E i } and standard basis {eei }; Λ represents rotational matrix related to surface basis {E the current state, T¯ = T¯ i E i with T¯ i = Λ0i j .

17.2.2 Strain Measures The covariant base vectors are obtained by partial derivatives of the position vectors with respect to the convective coordinates ξi , (i = 1, 2, 3) (note that ξ3 ≡ ξ) Gi =

X ∂X ,g = ∂ξi i

∂xx , ∂ξ i

i = 1, 2, 3,

(17.7) j

whereas the contravariant base vectors are defined in a standard way by G i · G j = δi j j and gi · g j = δi , here δi denotes Kronecker symbol. Vectors A α = G α |ξ=0 , a α = g α |ξ=0 span the tangent space to the mid-surface at a point X , and at a point x , respectively. For further use, the following notations are introduced 8 G = det Gi j , A = det Aαβ = A = det Ai j , μ0 = GA , dB = μ0 dξ dM0 (17.8) Lagrangian strain tensor at a point in shell space is defined by Ei j =

& 1% gi · g j − G i · G j G i ⊗ G j 2

(17.9)

where G i and g i represent covariant basis vectors at reference and current state respectively; G i and g i represent contravariant basis vectors at reference and current state respectively. Lagrangian strain tensor at a point in shell space can be expressed through the strain measures at the mid-surface by Eαβ = #αβ + ξραβ + ξ2 tαβ 1 E 3α = Eα3 = (δα + ξρ3α ) 2 E 33 = χ

(17.10)

where #αβ , δα , χ, ραβ , ρ3α , tαβ denote the membrane strain, transverse shear, thickness-stretch, bending, couple and shear and director strain measure on the midsurface, respectively. In case of inextensible thickness, the strain measures in (17.10) can be rewritten by E (1) + ξ2 E (2) E = E (0) + ξE

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ tαβ 0 ⎥⎥⎥ ⎢⎢⎢ #αβ 1 δα ⎥⎥⎥ ⎢⎢⎢ ραβ 1 ρα3 ⎥⎥⎥ ⎢ ⎥⎥⎥⎥ 2 ⎥⎥⎥ + ξ2 ⎢⎢⎢⎢⎢ ⎥⎥⎥ = ⎢⎢⎢⎣ 1 2 ⎥⎥⎥⎦ + ξ ⎢⎢⎢⎣ 1 ⎦ ⎢ ⎢⎢⎣ ⎥⎥⎦ 2 δα 0 2 ρα3 0 0 0

(17.11)


In what follows, we set (0) (0) e := E 11 E 22 . / := # γ κ t

(0) 2E 12

(0) 2E 13

(0) 2E23

(1) E11

(1) E22

(1) 2E 12

237

t (17.12)

where (0) (0) (0) # := E 11 E22 2E 12 (0) (0) γ := 2E13 2E 23 (1) (1) (1) κ := E 11 E22 2E 12

(17.13)

17.2.3 Effective Stress Resultants For simplicity of inelastic computation, the second Piola-Kirchhoff stress resultant tensors are used. h+ n = n i j G i ⊗ G j , ni j = S i j μ¯ dξ (17.14) h− h+

m = mi j G i ⊗ G j ,

ni j =

h−

S i j ξ μ¯ dξ

(17.15)

where S i j denotes components of the second Piola-Kirchhoff stress tensor and 2 A μ¯ = ;G = det Gαβ G Spatial tensors of effective stress resultants on S0 , which are defined on the basis A α , A 3 } ≡ {A Aα , T }, are defined by {A ˜ = J¯q˜ α A α , M ˜ = J¯m N˜ = J¯n˜ αβ A α ⊗ A β , Q ˜ αβ A α ⊗ A β

(17.16)

¯

where J¯ = GG¯ . We will also use the following notation for the symmetric stress 0 resultant tensors n = nαβ A α ⊗ A β ,

q = nα3 A α ⊗ T + n3α T ⊗ A α ,

In what follows, we set r t := n11 n22 n12 . / := n q r t

n13

n23

m11

m = mαβ A α ⊗ A β (17.17)

m22

m12

t

(17.18) (17.19)

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where n := n11 n22 n12 q := n13 n23 m := m11 m22 m12

(17.20) (17.21) (17.22)

17.3 Gradient-Enhanced Damage of Shell Structures 17.3.1 Kinetic Energy The kinetic energy is defined by % & 1 Aρ0 x˙ · x˙ + Iρ0 ˙t · ˙t dA0 2 A0 1 1 1 = p·p+ π · π dA0 2 A0 Aρ0 Iρ0

Π) = K (Π

where Π := (pp , π ), the mid-surface linear momentum p is defined by p = Aρ0 x˙

Aρ0 =

h+

h−

ρ0 μ0 dξ = ρ0 h

(17.23)

and the mid-surface director momentum π is defined by π = Iρ0 ˙t

Iρ0 =

h+

h−

ρ0 ξ2 μ0 dξ = ρ0

h3 12

(17.24)

ρ0 denotes the mass density at reference configuration.

17.3.2 Potential Energy Dimitrijevic and Hackl [4] proposed a gradient-enhanced damaged model by introducing an additional non-local variable to transfer the values of the damage parameter across the element boundaries. This makes the model non-local in nature and avoids the very strong requirement of the C 1 continuous displacement field. Adapting the previous approach, we introduce here an enhanced free energy for the thin-shell structures as 2 1 2 βd ˜ Φ r e Ψ (Φ ) = : + cd ∇ϕ + ϕ − γd d dξ (17.25) 2 2 where Φ := (xx, t , ϕ). The non-local variable ϕ is defined by


ϕ=

239

h+ h−

d˜ dξ

(17.26)

where d˜ is a non-local damage parameter in the shell space. It should be noted that the proposed free energy here is different from that in [4]. First, the proposed non-local variable is introduced as a mid-surface quantity. Consequently, a local constitutive law for the non-local variable in shell space is not necessary. Second, the elastic strain energy takes the standard form. A difference between non-local and local energy is adjusted by the parameter βd representing a penalty of energy difference between non-local and local field. The degree of gradient regularization and the internal length scale is represented by the parameter cd . The final parameter γd is used as a switch between the local and non-local model for the reason of numerical computation. With such the definition of enhanced free ˜ the potential energy is written as energy Ψ, Φ Φ) dA0 − Pext (Φ Φ) P (Φ ) = Ψ˜ (Φ (17.27) A0

Φ) is the potential energy of the external loads. where Pext (Φ

17.3.3 Weak Formulation of Motion Equations By the above definitions of kinetic and potential energy, the Lagrangian differential equations of motion can be transformed to the Hamiltonian canonical equations subjected to initial conditions of configuration, momenta and non-local variable. The weak form of these equation are written as Z) ∂H(Z Φ dA0 + Φ=0 Π˙ · δΦ · δΦ (17.28) Φ ∂Φ A0 Z) ∂H(Z ˙ · δΠ Π dA0 − Π=0 Φ · δΠ (17.29) Π ∂Π A0 Φ, Π ) and H(Z Z ) := K(Π Π ) + P(Φ Φ) where Z := (Φ From (17.28) the weak form of momentum equation may be expressed as % & % & ¨ , Φ , δΦ Φ = Gine Π˙ , δΦ Φ + G stat (Φ Φ, δΦ Φ) − Gext (Φ Φ) = 0 G dyn Φ (17.30)

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where

% & % & ϕ dA0 G ine Π˙ = p˙ · δuu + π˙ · δtt + p˙ ϕ · δϕ A 0 Φ, δΦ Φ) = (nn : δ## + q : δγγ + m : δκκ) dA0 + G stat (Φ +

A0

A0

⎛ ⎜⎜ βd δϕ ⎜⎜⎝ϕ − γd

h+ h−

⎞2 ⎟⎟ d dξ⎟⎟⎠ dA0

A0

(17.31) cd ∇δϕ ∇ϕ dA0 (17.32)

Following the approach in [4], we can derive an evolution equation of nonlocal variable ϕ from (17.32). βd ϕ − γd d¯ − cd ∇2 ϕ = 0 (17.33) where d¯ =

h+

h−

d dξ

(17.34)

Equation (17.33) expresses an implicit gradient scheme for thin-shell structures.

17.4 Local Constitutive Laws To complete the model, local constitutive laws considering viscoplasticity, isotropic hardening and isotropic ductile damage leading to softening are needed. We limit ourself to investigate cases of small strains. St. Venant-Kirchhoff law of non-linear hyperelasticity and the coupled viscoplastic and damage model proposed in [9] are used. A summary of formulations of these models are given in the sections below. In more detail, readers can refer to [8, 9].

17.4.1 Hyperelasticity Law on the Mid-surface The second Piola-Kirchhoff stress tensor can be rewritten by E ] 1 + 2μE E = CE S = λtr [E % (0) & kl i jkl 2 (2) S =C E i j + ξE i(1) j + ξ Ei j , where λ =

Eν (1+ν)(1−2ν)

and μ =

E 2(1+ν)

(17.35) (17.36)

denote Lamé’s constants, and

C i jkl = H i jkl − H i j33

H 33kl H 3333

(Case of S 33 = 0).

(17.37)


241

Stress resultant components in a general case are defined by

h+

h− h+

n = ij

mi j = where

Cj =

h+ h−

h−

% & (1) 2 (2) C i jkl Ei(0) μ¯ dξ, j + ξE i j + ξ E i j

(17.38)

% (0) & (1) (2) C i jkl Ei j + ξE i j + ξ2 E i j ξ μ¯ dξ,

(17.39)

C i jkl ξ j μdξ, ¯

j = 0, 1, 2, 3, 4,

i j 33, kl 33

(17.40)

For thin-shell structures the following conditions are valid, G i = A i and G i = A i , i.e. 8

G i j = Ai j , G i j = Ai j , and μ¯ = GA = 1 and the components of the constitutive tensor become independent on ξ. Then the stress resultant components become ni j = m

αβ

h+

h−

=

h+

h−

(0) C i jkl E kl dξ,

i j 33, kl 33,

(1) 2 C αβγδ E γδ ξ dξ,

i j 33, kl 33

(17.41) (17.42)

The constitutive law of hyperelasticity on mid-surface may be expressed by r = C¯ e where

(17.43)

⎡ ⎤ ⎢⎢⎢ C 0(5×5) ⎥⎥⎥ 0 ⎥⎥⎥ C¯ (5×5) = ⎢⎢⎢⎣ 0 C 2(3×3) ⎦ Cj =

h+

h−

C i jkl ξ j μdξ, ¯

i, j, k, l = 0, 1, 2, 3, 4,

i j 33, kl 33

(17.44)

(17.45)

17.4.2 Viscoplastic Damage To take material behavior of local viscoplastic damage the constitutive laws proposed in [9] is used. A brief summary of these laws are given in Eqs (17.46)–(17.55). The overstress S ex which depends on the deviatoric stress S d and the hydrostatic in stress 13 trS is used to control the evolution of the inelastic train rate E˙ [10, 12, 13]. The evolution of the local damage variable d˙ based on the work by [12–14] takes in into account the inelastic volumetric strain rate tr E˙ as well as transient and saturation behavior. Isotropic hardening and inelastic deformation in case of undamage materials are considered by von Mises yield criterion. More details can be found in [9]

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E = E e + E in F Gn ∂S ex ˙ S ex in E˙ = E0 ∂SS Sp

H I in in d˙ = (c1 + c2 e−c3 Eν )E˙ νin + c5 (c4 − d) trE˙ F Gn 1 S ex E˙ νin = 1−d S p S ex = S eq − S F − K 1 S eq = f 1−d K˙ = b(Q − K)˙#νin S = w(d) C E 1 S d = S − m trSS 3 trSS = m S

(17.46) (17.47) (17.48) (17.49) (17.50) (17.51) (17.52) (17.53) (17.54)

(17.55)

in (d) := (1 − d)md denotes a softening function, E , E e , E in , E˙ , S ex , d, E˙ νin , where w S eq , K, f , S , S d , m denote total strain tensor, elastic strain tensor, inelastic strain tensor, over stress, damage parameter, equivalent inelastic strain, isotropic hardening stress, effective von Mises yield function, second Piola-Kirchhoff stress, deviatoric stress, unity matrix, respectively. The symbols E˙ 0 , S p , n, c1 , c2 , c3 , c4 , S F , a1 , a2 , b, Q, md denote material parameters. For finite element computation, the system of differential equations (17.47), (17.48) and (17.50) in terms of the second Piola Kirchhoff stress, isotropic hardening stress, and damage variables is solved by using the implicit backward Euler integration scheme at each Gauss point across the thickness of the shell. Then, the second Piola Kirchhoff stress as well as the local consistent tangent operator are employed to update the new stress resultants and the consistent tangent operator in the mid-surface.

17.5 Examples The presented formulation of the gradient-enhanced damage model for viscoplastic thin-shell structures is implemented in the user element subroutine compiled with FEAP [1]. The implementation is valid for thin-shell structures in general, but here two examples of plates subjected to dynamical loading are chosen to show the efficiency of the suggested approach without loosing the generality of the presented shell model. We note that all herein presented results are of preliminary nature and will have to be validated by future studies.


243

1 0.8 0.6

Displacement (mm)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Fig. 17.1 Example 1 - History of prescribed displacement 0.2 0.18

Damage parameters

0.16 0.14 0.12 0.1 0.08 0.06 d

0.04

in ν

ε

0.02 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Fig. 17.2 Case of 144 elements - Evolution of damage parameter d and equivalent inelastic strain #vin at bottom fiber of critical section

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17.5.1 Example 1 A square plate of 100mm× 100mm× 5mm, which is subjected to the prescribed horizontal displacements (Figure 17.1) at the right edge of the plate, is investigated. The material parameters given in Table 17.1 are artificial only for academic study. Two regular-structured meshing of 144 and 576 elements are considered. The evolution of the damage parameter, equivalent inelastic strain, membrane strain measure and membrane stress resultants are depicted in Figs 17.2-17.6. The response of reaction and displacement at the right edge of the plate are depicted in Figs 17.6-17.7 is quite coincident with mesh refinement. Table 17.1 Example 1 - Material parameters Parameters

Values

Units

Parameters

Values

Units

h

5

mm

κ

0.8

1

E

1.98 × 105

N mm2

ν

0.3

1

ρ

106

kg m3

Ip

0.8

1

SF

167.88

N mm2

E˙ 0

1.0

1

Sp

163.12

N mm2

np

1.0

1

Q

75

N mm2

b

175

1

a1

1.3

1

a2

1.0

1

c1

1.2

1

c2

-0.02

1

c3

60.

1

c4

40.0

1

c5

0.1

1

md

2.0

1

cd

1.0

1

β

1.0

N mm2

· mm

d0

0.01

1

γd

1.0

N mm2

· mm

I p : Loading intensity factor

17.5.2 Example 2 A copper circular plate of 138 mm diameter which is clamped on the outer-ring area of 15 mm radius length is investigated as in [2]. Due to the clamped area, the loading area is of 108 mm diameter. Alternate shock pressures are shown in Fig. 17.9 and


245

0.025 0.02

Membrane strain measure

0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 0

ε11 22 ε 12 ε 0.5

1

1.5

2

2.5

3

3.5

4

2.5

3

3.5

4

Time (s)

Fig. 17.3 Case of 144 elements - Membrane strain measures 1500

Membrane stress resultant

1000

500

0

−500

−1000

−1500 0

11

n 22 n 12 n 0.5

1

1.5

2 Time (s)

Fig. 17.4 Case of 144 elements - Membrane stress resultants

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Stress resultant n

1000

500

0

−500

−1000

11

11

ε −n ε22−n22 ε12−n12

−1500 −0.02

−0.015

−0.01

−0.005

0

0.005

Strain measure ε

0.01

0.015

0.02

0.025

0.6

0.8

1

Fig. 17.5 Case of 144 elements - (## − n ) curve 5

1.5

x 10

1

Reaction (N)

0.5

0

−0.5

−1 144 elem. 576 elem. −1.5 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Displacement (s)

Fig. 17.6 Displacement-reaction curve at the right edge of the plate


247

5

1.5

x 10

144 elem. 576 elem. 1

Reaction (N)

0.5

0

−0.5

−1

−1.5 0

0.5

1

1.5

2

Time (s)

Fig. 17.7 History of reaction at the right edge of the plate

Fig. 17.8 Example 2 - Sketch of a quarter of copper plate

2.5

3

3.5

4

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A.D. Nguyen et al. 2.5

x 10

6

2 1.5

2

Pressure (N/mm )

1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

Time (s)

Fig. 17.9 Example 2 - History of prescribed pressure −3

8

x 10

Experiment Viscoplasticity Viscoplastic damage Viscoplastic nonlocal damage

6 4

Displacement (m)

2 0 −2 −4 −6 −8 −10 −12 0

0.005

0.01

0.015

Time (s)

Fig. 17.10 Displacements at center of the copper plate

0.02

0.025


249

Table 17.2 Example 2 - Material parameters Parameters

Values

Units

Parameters

Values

Units

h

2

mm

κ

0.833

1

E

113 × 109

N mm2

ν

0.33

1

ρ

8858

kg m3

Iρ

1

1

SF

180

N mm2

E˙ 0

1.0

1

Sp

40.5

N mm2

np

8.15

1

Q

64.52

N mm2

b

153

1

a1

1.3

1

a2

1.0

1

c1

1.2

1

c2

-0.02

1

c3

60.

1

c4

40.0

1

c5

0.1

1

md

2.0

1

cd

1.0

1

β

1.0

N mm2

· mm

d0

0.01

1

γd

1.0

N mm2

· mm

I p : Loading intensity factor

material parameters of copper is given in Table 17.2. Due to axisymmetry, only a quarter of the plate as in Fig. 17.8 is considered in numerical computation. The obtained results, which are compared with other two models of viscoplasticity and damage coupled viscoplasticity, are depicted in Fig. 17.10. Vibrations are observed at the beginning of loading and loading, and dissipate quickly. This is reasonable since the energy and momentum conserving algorithm proposed in [7] is used with the values of numerical dissipation β = 0.65, γ = 0.95 and α = 0.55.

17.6 Conclusions This work proposes a gradient-enhanced damage model for viscoplastic thin-shell structures subjected to dynamically loading. The previous approach gives a possibility to remove pathological mesh dependence and with numerical difficulties that appeared in existing simulations of softening and damage phenomena. However, it should be noted that the presented numerical results are preliminary and need future systematical studies.

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References 1. FEAP-8.2.k, Finite Element Analysis Program (2008). University of California. 2. Stoffel M. (2007). An experimental method for validating mechanical models. Habilitation thesis, RWTH-Aachen. 3. Stoffel M. (2004). Evolution of plastic zones in dynamically loaded plates using different elastic-viscoplastic laws. Int. J. Sol. Struc., 41 (24-25), 6813-6830. 4. Dimitrijevic B.J., Hackl K. (2008). A method for gradient enhancement of continuum damage models. Tech. Mech., 28(1), 43-52. 5. Simo J.C., Fox D.D.(1989). On a stress resultant geometrically exact shell model. Part I. Formulation and optimal parametrization. Comput. Meth. Appl. Mech. Engng., 72, 267-304. 6. Simo J.C., Rifai M.S., Fox D.D. (1992). On stress resultant geometrically exact shell model. Part VI. Conserving algorithms for non-linear dynamics. Int. J. Numer. Meth. Engng., 34, 117-164. 7. Simo J.C., Tarnow N. (1994). A new energy and momentum conserving algorithm for the non-linear dynamics of shells. Int. J. Numer. Meth. Engng., 37, 2527-2549. 8. Brank B., Briseghella L., Tonello N., Damjanic F.B. (1998). On non-linear dynamics of shells: Implementation of energy-momentum conserving algorithm for a finite rotation shell model. Int. J. Numer. Meth. Engng, 42, 409-442. 9. Velde J., Kowalsky U., Zümendorf T., Dinkler D. (2009). 3D-FE-analysis of CT-speciemens including viscoplastic material behaviour and nonlocal damage. Comp. Mat. Sci., 46, 532-357. 10. Chaboche J.L., Rousselier G. (1983). On the Plastic and Viscoplastic Constitutive Equations– Part I: Rules Developed With Internal Variable Concept. J. Pres. Vess. Tech., 105, 153-158. 11. Chaboche J.L., Rousselier G. (1983). On the Plastic and Viscoplastic Constitutive Equations– Part II: Application of Internal Variable Concepts to the 316 Stainless Steel. J. Pres. Vess. Tech., 105, 159-164. 12. Gurson A.L. (1977). Continuum theory of ductile rupture by void nucleation and growth: Part I–Yield criteria and flow rules for porous ductile media. J. Eng. Mat. Tech., 99, 2-15. 13. Tvergaard V., Needleman A. (1984). Analysis of cup-cone fracture in a round tensile bar. Act. Metal., 32, 157-169. 14. Lemaitre J., Desmorat R., Sauzay M. (2000). Anisotropic damage law of evolution. Eur. J. Mech. A/Solids, 19(2), 187-208. 15. Nguyen A.D., Stoffel M., Weichert D. (2010). A one-dimensional dynamic analysis of straingradient viscoplasticity. Eur. J. Mech. A/Solids, 29, 1042-1050. 16. Aifantis E.C. (1992). On the role of gradients in localization of deformation and fracture. Int. J. Eng. 30, No. 30, 1279-1299. 17. Nguyen Q.S., Andrieux S. (2005). The non-local generalzed standard approach: a consistent gradient theory. Compt. Rend. Meca. 333, 139-145.

Chapter 18

On Constitutive Restrictions in the Resultant Thermomechanics of Shells with Interstitial Working Wojciech Pietraszkiewicz

Abstract We analyse the restrictions imposed by the recently derived refined, resultant 2D entropy inequality on the forms of 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. Due to the presence of additional non-classical surface fields, the entropy inequality does allow the constitutive equations to capture some kinematic and thermal longer-range spatial and temporal interactions. We also propose several forms of the 2D kinetic constitutive equations. Keywords Shell thermomechanics · Interstitial working restrictions · Kinetic constitutive equations

·

Constitutive

18.1 Introduction The resultant two-dimensional (2D) balance laws of mass, linear momentum, angular momentum, and energy as well as entropy inequality of the non-linear theory of shells were first formulated by Simmonds [1] by direct through-thethickness integration of the corresponding 3D laws of continuum thermomechanics. Pietraszkiewicz et al. [2] proved that the 3D stress power of the shell can in general be given through the 2D effective stress power expressed only by the 2D shell quantities as in [1] plus an additional stress power not expressible through 2D resultant fields. Hence, in order to balance the shell energy an additional 2D stress power, called an interstitial working after Dunn and Serrin [5] and Dunn [6], was added in [3, 4] to the 2D resultant balance of energy equivalent to one of [1]. Then appropriately refined 2D laws of shell thermomechanics of [3] can be regarded as W. Pietraszkiewicz (B) Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 Gdańsk, Poland e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 18, © Springer-Verlag Berlin Heidelberg 2011

251

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exact implications of the corresponding 3D laws of rational thermomechanics discussed by Truesdell and Toupin [7] and Truesdell and Noll [8]. In this paper we analyse the restrictions imposed by the refined, resultant 2D entropy inequality on the forms of 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. For this purpose the procedure proposed by Coleman and Noll [9] is used. It is found, in particular, that due to the presence of additional 2D fields the refined 2D entropy inequality does allow the constitutive equations to depend also on the first surface gradients of the shell strain measures as well as on the second surface gradient of the mean referential surface temperature.

18.2 Local Laws of Refined Resultant Thermomechanics of Shells Let M be a regular surface modelling the base surface of the shell in the reference (undeformed) placement. We also assume that all surface fields discussed here are smooth on any part Π ⊂ M, and that mass is not created during the thermodynamic process. Then the refined, local 2D balance laws of linear momentum, angular momentum, and energy as well as the entropy inequality of the shell derived in [3] for any Π ⊂ M read: N F T − F N T ) + ρcc = k˙ + y˙ × l , Div N + ρ f = ˙l , Div M + ax (N o o N • E + M • K + Div w ) − (ρr − Divqq) = 0 , ρε˙ − (N ˙ + N • E o + M • K o + Div w − ρθs − 1 q · g + θ Div s ≥ 0 . ˙ −ρψ − ρθη θ

(18.1)

In the mechanical balance laws (18.1)1,2 , ρ(x, t) > 0 is the resultant referential surface mass (density field), where x is a point of M and t ∈ T is time, l (x, t) and k (x, t) are the resultant surface linear momentum and angular momentum vectors per unit area of M, f (x, t) and c (x, t) are the resultant surface force and couple vectors per unit mass of M, respectively. Additionally, N (x, t) ∈ V ⊗ T x M and M (x, t) ∈ V ⊗ T x M are the respective stress resultant and stress couple tensors of the first Piola-Kirchhoff type, where V is the 3D vector space and T x M is the 2D tangent space at x ∈ M, F = Grad y ∈ V ⊗ T x M is the surface deformation gradient, where y (x, t) is the position vector of the actual (deformed) base surface, while Grad and Div are the surface gradient and divergence operators with respect to x ∈ M as defined by Gurtin and Murdoch [19], and ax (T) means the axial vector of the skew tensor T ∈ V ⊗ V, TT = −T. In the thermodynamic 2D laws (18.1)3,4 , ε(x, t), η(x, t), ψ(x, t), and r(x, t) are the resultant surface internal energy, entropy, free energy, and heat supply per unit mass of M, θ(x, t) > 0 is the surface mean referential temperature with the surface gradient g = Grad θ ∈ T x M, E (x, t) ∈ V ⊗T x M and K (x, t) ∈ V ⊗T x M are the natural stretch and bending tensors of the general six-field shell model, q (x, t) is the resultant referential heat flux vector, w (x, t) is the interstitial working flux vector, s(x, t) and s (x, t) are the respective extra surface heat supply and entropy supply vectors, ˙l is the material

18 Constitutive Restrictions in the Resultant Thermomechanics of Shells

253

time derivative of l (x, t), (.)o means the co-rotational time derivative of (.), and for A T D ). For the description of all resultant any A , D ∈ V ⊗ T x M we denote A • D = tr (A 2D fields given above in terms of the corresponding 3D fields, and for the additional relations not presented here, we refer to [3, 4, 10]. The starting point of our resultant shell thermomechanics formulated in [3] had been the classical 3D Cauchy continuum and the 3D Clausius–Duhem inequality. But our refined, resultant 2D thermomechanic shell relations (18.1) are far from being the classical ones. The kinematic structure of the resultant shell theory is that of the Cosserat [12] surface, with y (x, t) and Q (x, t) as independent kinematic field variables of the shell motion, where Q (x, t) is the proper orthogonal tensor field. The thermodynamic structure of our refined, resultant 2D laws (18.1) also does not remind that of 3D rational thermomechanics developed in [7, 8]. In particular, our 2D laws (18.1)3,4 contain the extra surface scalar field s(x, t) and divergences of the additional surface vector fields w (x, t) and s (x, t), which are not present in corresponding 3D laws of rational thermomechanics but are somewhat similar to those present in 3D extended thermodynamics, see for example [13].

18.3 Restrictions on the Form of Constitutive Equations Our exact, local, resultant 2D balance laws (18.1) are expressed through 16 surface fields, which together form the shell thermomechanic process over the domain M × T . Different groups of fields play different roles in the process. The fields y , Q , θ constitute the basic thermo-kinematic independent field variables of the initial-boundary value problem of shell thermomechanics. That only seven scalar fields can be taken as independent field variables here follows from the fact that there are only seven scalar resultant field equations (18.1)1−3 to determine them. The fields N , M , q , ε, η, w , s, s have to be specified by appropriate (material) constitutive equations and the fields l , k by appropriate kinetic constitutive equations. When all the fields above are settled, the fields f , c , r are supposed to be adjusted so as to satisfy the refined 2D balance equations (18.1)1−3 . Any such process is called an admissible thermomechanic process; it is completely determined by the evolution of deformation and temperature of the shell base surface. General requirements that the constitutive equations of our resultant shell thermomechanics must obey are analogous to those of 3D rational thermomechanics formulated by Truesdell and Toupin [7], Sect. 293. These are briefly: 1) consistency, 2) coordinate invariance, 3) well posedness, 4) material frame-indifference, 5) material symmetry, and 6) equipresence. Additional quite general requirements were proposed by Truesdell and Noll [8]: 7) determinism, 8) local action, and 9) fading memory. But in the literature one can find many other specific requirements which define particular classes of material behaviour, such as for example hyperand hypoelasticity, viscosity, incompressibility, inextensibility in some direction, slow motions, small deformations, etc. Either of these requirements can be used to define particular classes of shell constitutive behaviour as well.

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Let us introduce the referential shell stress measures N(x, t) and M(x, t) with corresponding referential shell strain measures E(x, t) and K(x, t) defined by N = QT N ,

M = QT M ,

E = QT E ,

K = QT K ,

(18.2)

so that the local 2D inequality (18.1)4 becomes ˙ + Div w − ρθs − 1 q · g + θ Div s ≥ 0 ˙ + N • E˙ + M • K − ρψ˙ − ρθη θ

in Π ⊂ M . (18.3)

If y , Q , θ are taken as the independent field variables, then from the requirement of determinism the 2D thermomechanic response described by the fields N , M , q , ε, η, w, s, s, l, k ) at the shell particle, whose position in the reference Σ = (N placement is x ∈ M, can mathematically be defined by the response functionals ∞ Σ(x, t) = Σˆ τ=0 y t (z, τ), Q t (z, τ), θt (z, τ); z, B (z) . (18.4) History φt (x, τ) ≡ φ(x, t − τ) up to the present time t of any field φ(x, t) in (18.4) is defined for all τ ∈ [0, +∞), and z ∈ M is any other place than x on M. The explicit dependence of the right-hand side of (18.4) on z and on the referential structure tensor B (z) defined in [2] indicates that thermomechanic properties at x ∈ M may depend also on properties of other shell particles in Π ⊂ M, thus allowing for material inhomogeneneity and non-uniformity in shell behaviour, in general. In the special case of (18.4), which we wish to investigate in more detail as an example, we allow the fields Σ to depend on y t , Q t , and θt only locally through the first time derivatives at τ = 0 and locally through the first surface gradients at x. Then the corresponding constitutive assumption would be ˙ , K, ˙ θ, g ; x, B (x) . Σ(x, t) = Σˆ κ E, K, E (18.5) Any function in Σˆ κ is called the response or the constitutive function and is assumed to be differentiable with respect to the fields enclosed in brackets as many times as required. The lower index κ in Σˆ κ indicates that the form of each response function depends on the choice of the reference placement.

18.3.1 Viscous Shells with Heat Conduction The shell-like body described by the 2D response functions (18.5) is usually called a viscous shell with heat conduction. The structure of its constitutive equations based on the approximate resultant 2D entropy inequality analogous to that of [1] has recently been discussed by Eremeyev and Pietraszkiewicz [10]. But in our refined 2D entropy inequality (18.3) there is the interstitial working flux vector field w (x, t) which was not present in [10]. Within 3D continuum mechanics a similar interstitial working vector field allows for modelling the material behaviour also with


255

mechanical longer-range spatial interactions at x ∈ M, see [5, 6]. Additionally, the presence of θ Div s in (18.3) suggests that also the thermal longer-range spatial interactions at x ⊂ M may be accounted for in the refined resultant shell thermomechanics. Hence, the more appropriate non-classical constitutive assumption for the viscous shell with heat conduction compatible with (18.3) is Σ(x, t) = Σˆ κ [Λ; x, B (x)] ,

˙ θ, g , G ) , Λ = (E, K, Grad E, Grad K, E˙ , K,

(18.6)

where G = Grad g = G T ∈ T x M ⊗ T x M. The thermomechanic process, in which the fields E, K, θ are constant in space and time, is usually called the equilibrium process. The equilibrium response functions of the shell stress measures N E , ME are defined by E

E

ˆ κ [Λ0 ; x, B (x)] , ME = M ˆ κ [Λ0 ; x, B (x)] , Λ0 = (E, K, 0 , 0 , 0 , 0 , θ, 0 , 0 ). NE = N (18.7) Then the dynamic, dissipative parts of the response functions of the shell stress measures are such that ˆ κD [Λ; x, B (x)] , MD = M ˆ κD [Λ; x, B (x)] , ND = N N = NE +N D , M = ME + MD .

(18.8)

According to Coleman and Noll [9], in 3D rational thermodynamics the Clausius– Duhem inequality plays the role of a restriction placed on allowable forms of the response functions. The functions must be so chosen as to satisfy the inequality in every smooth thermomechanic process compatible with the 3D balance equations. To reveal the restrictions placed on the 2D shell constitutive equations (18.6) by our refined resultant 2D entropy inequality (18.3), let us just introduce (18.6) into (18.3) to obtain ˙ + (M − ρψ,K ) • K ˙ − (ρη + ρψ,θ )θ˙ − ρψ,g ·g˙ (N − ρψ,E ) • E ˙ − ρψ,E˙ •E¨ − ρψ,K˙ •K ¨ −ρψ,Grad E •Grad E˙ − ρψ,Grad K •Grad K 1 ˙ −ρψ,G •G − ρθs − q · g + Div w + θ Div s ≥ 0 . θ

(18.9)

Any deformation–temperature path can be realised in a thermomechanic process, so that values of Λ and Λ˙ can be chosen arbitrarily. Since the left-hand side of (18.9) depends on Λ˙ linearly, in order to satisfy (18.9) the coefficients in front of time rates of variables must vanish, η = − ψ,θ , ψ,Grad E = 0 , ψ,Grad K = 0 , ψ,E˙ = 0 , ψ,K˙ = 0 , ψ,g = 0 , ψ,G = 0 .

(18.10)

These relations imply that ψ = ψˆ κ [E, K, θ; x, B (x)], and by (18.10)1 and definition ψ = ε − θη the same structure should have the response functions εˆ κ and ηˆ κ . With (18.10) the inequality (18.9) reduces to

256

W. Pietraszkiewicz

˙ + (M − ρψ,K ) • K ˙ − ρθηs − 1 q · g + Div w + θ Div s ≥ 0 . (N − ρψ,E ) • E θ

(18.11)

Please note that in (18.11), N and M still depend only on Λ, while ψ,E and ψ,K depend on E, K, θ. If the decompositions (18.7) and (18.8) are used in (18.11), the following constitutive equations for the equilibrium part of the shell stress measures are obtained: NE = ψˆ κ ,E [E, K, θ; x, B (x)] ,

ME = ψˆ κ ,K [E, K, θ; x, B (x)] .

(18.12)

Let us now apply the chain rule to the surface vector fields w and s , which allows one to expand (18.11) with (18.12) into 1 1 1 D ˙ D ˙ N • E + M • K − ρθs − q − w ,θ − s ,θ · g + w ,g + s ,g • G θ θ θ 1 G + tr w ,G + s ,G ◦ GradG θ 1 1 + tr w ,E + s ,E ◦ Grad E + tr w ,K + s ,K ◦ Grad K (18.13) θ θ 1 + tr w ,Grad E + s,Grad E Grad2 E θ 1 + tr w ,Grad K + s ,Grad K Grad2 K ≥ 0 , θ where for two 3rd-order surface tensors Γ , Δ and for two 4th-order surface tensors Φ , Ψ the 2nd-order surface tensors Γ ◦ Δ and Φ Ψ are defined such that in Cartesian components associated with T x M we have Γ ◦ Δ )αβ = Γαiλ Δiλβ , (Γ

Φ Ψ )αβ = Φαiλμ Ψiλμβ , (Φ

α, β = 1, 2.

(18.14)

Let us remind that according to [3, 5, 6] the interstitial working vector w is of entirely mechanical origins. Hence, we may additionally assume that w = wˆ κ [E, K, Grad E, Grad K; x, B(x)] , so that w ,θ = w ,g = 0 . Similarly, according to [3] the extra surface vector field s is of entirely thermal origins and we may additionally assume that . / s = sˆ κ θ, g , G ; x, B (x) , so that s,E = s,K = s,Grad E = s ,Grad K = 0. With these additional assumptions the inequality (18.13) reduces to 1 ˙ − ρθs − 1 (qq − s ,θ ) · g + 1 s ,g •G G + tr G ND • E˙ + MD • K s ,G ◦GradG θ θ θ (18.15) w w (w + tr (w , ◦Grad E) + tr , ◦Grad K) E K % & % & 2 2 + tr w ,Grad E Grad E + tr w ,Grad K Grad K ≥ 0 .


257

18.3.2 Thermoelastic Shells An important special case of viscous shells with heat conduction are thermoelastic ˙ K. ˙ In this case shells, which constitutive functions in (18.6)1 do not depend on E, the refined 2D entropy inequality takes the form (18.9), only now without terms containing ψ,E˙ and ψ,K˙ . To satisfy it we should have ψ,Grad E = 0 ,

ψ,Grad K = 0 ,

ψ,g = 0 ,

ψ,G = 0 ,

(18.16)

so that also in this case ψ = ψˆ κ [E, K, θ; x, B (x)], with the same structure of εˆ κ and ηˆ κ . As a result, the constitutive equations for N, M, and η of the thermoelastic shells are N = ρψ,E , M = ρψ,K , η = − ψ,θ . (18.17) With the constitutive equations (18.17) the reduced form of our 2D entropy inequality (18.9) becomes 1 1 1 −ρθs − q − w ,θ − s ,θ · g + w ,g + s ,g • G θ θ θ 1 G + tr w ,G + s ,G ◦ GradG θ 1 1 + tr w ,E + s,E ◦ Grad E + tr w ,K + s ,K ◦ Grad K (18.18) θ θ 1 + tr w ,Grad E + s ,Grad E Grad2 E θ 1 + tr w ,Grad K + s ,Grad K Grad2 K ≥ 0 . θ If again wˆ κ in (18.18) is assumed not to depend on θ, g and sˆ κ on E, K, Grad E, Grad K then the inequality (18.18) reduces further to 1 1 1 G + tr G −ρθs − (qq − s ,θ ) · g + s ,g •G s ,G ◦GradG θ θ θ (18.19) w, ◦Grad E) + tr (w w, ◦Grad K) + tr (w % E & K% & + tr w ,Grad E Grad2 E + tr w,Grad K Grad2 K ≥ 0 . In both cases of viscous shells with heat conduction and of thermoelastic shells the reduced 2D entropy inequalities (18.15) and (18.19) still put some constraints on allowable forms of wˆ κ and sˆ κ , detailed discussion of which we leave for future work. But the main message following from (18.15) and (18.19) is that our refined, resultant 2D entropy inequality (18.3) does allow the constitutive equations of both types of shells to depend on Grad E, Grad K, and G indeed. However, before fully revealing such an explicit dependence and its thermodynamic significance much research has still to be done.

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18.4 Kinetic Constitutive Equations In the resultant shell thermomechanics it is not apparent how the resultant linear momentum l (x, t) and angular momentum k (x, t) vector fields should be related to the ˙ Q T ) velocities of the shell as well as to 2D translational υ = y˙ and angular ω = ax (Q other kinematic and thermal surface fields. Since the fields l , k do not enter into our refined entropy inequality (18.3), it does not place any thermodynamic restrictions on their functional forms. In [3, 11] some heuristic arguments were proposed on how the 2D kinetic constitutive equations can be constructed within the classes of shell models discussed in Subsects 5.1 and 5.2. Let us remind here some of those arguments and propose some forms of the 2D kinetic constitutive equations for the general six-field theory of shells. Since l and k are of entirely mechanical origins, the 2D constitutive equations for them may be postulated in the reduced form ˙ x, B (x) , l = ˆl κ ρ, υ , ω , E, K, Grad E, Grad K, E˙ , K; (18.20) ˙ , K; ˙ x, B (x) . k = kˆ κ ρ, υ , ω , E, K, Grad E, Grad K, E In analogy to the 3D case, it is reasonable to assume that the constitutive functions ˆlκ and kˆ κ are linear functions of υ, ω, l = J 1υ + J 2ω ,

k = J 3υ + J 4ω .

(18.21)

˙ Here J A ∈ V ⊗ V, A = 1, 2, 3, 4, are tensor functions of ρ, E, K, Grad E, Grad K, E, ˙ in general. K, If the tensors J A satisfy the symmetry conditions J T1 = J 1 ,

J T2 = J 3 ,

J T4 = J 4 ,

(18.22)

we may expect existence of the 2D kinetic energy density κ(x, t) per unit area of M, which is the quadratic positive-definite function of υ , ω , κ=

1 υ · J 1υ + υ · J 2ω + ω · J 3υ + ω · J 4ω) . (υ 2

(18.23)

The function κ may serve as potential for l , k , so that (18.21) can be found from κ according to l = κ,υ , k = κ,ω . (18.24) The tensors J A may be taken to be quite complex functions of all their arguments, in general. But because of mechanical origins of l and k we may additionally assume that J A = J A (ρ, E, K). With such J A the kinetic constitutive equations (18.21) was proposed in [11]. Even simpler forms of (18.21) follow when, in analogy to the rigid-body motion, the J A are assumed in the forms


J 1 = ρ11 ,

QI 3 QT , J T2 = J 3 = ρQ

QI 4 QT , J 4 = ρQ

259

(18.25)

where 1 ∈ V ⊗V is the 3D identity tensor, while I 3 and I 4 are constant inertia tensors of the shell cross section in the reference (undeformed) placement. In this case, according to (18.24), we obtain Q I T3 Q T ω , l = ρυυ + ρQ

Q I 3 Q T υ + ρQ QI 4 QT ω . k = ρQ

(18.26)

Explicit formulas for I 3 and I 4 depend on the specific internal structure of the shell across the reference thickness. The kinetic constitutive equations (18.26) were proposed by Zhilin [14] and applied, for example, in Altenbach and Zhilin [15] and Eremeyev and Zubov [16]. In many papers on shell dynamics the inertia tensors are assumed to be I 3 = 0 , I 4 = I11, where I is the transverse geometric moment of inertia of the shell cross section. Then (18.26) reduce to the ultimate simple forms l = ρυυ ,

ω. k = ρIω

(18.27)

The kinetic constitutive equations equivalent to (18.27) were used, for example, by Zhilin [17], Simmonds [1], and Chrósćielewski et al. [18].

18.5 Conclusions The refined, resultant 2D balance of energy and entropy inequality derived in [3, 4] for the general six-field theory of shells contain three non-classical surface fields w , s, s , which play the role of somewhat similar fields appearing in 3D extended thermodynamics. We have briefly analysed restrictions imposed by our refined 2D entropy inequality on the 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. Applying the procedure of Coleman and Noll [9], it has been shown that in the both cases our refined 2D entropy inequality allows the constitutive equations to depend also on the first surface gradients of shell strain measures E, K and on the second surface gradient of the surface mean referential temperature θ. We have also provided several forms of the 2D kinetic constitutive equations obtained with the help of heuristic arguments. More definite properties of the non-classical fields w , s, s as well as reduced explicit forms of the constitutive equations, following from requirements of material frame-indifference and material symmetry for these and other types of shells, should be discussed separately. Due to the non-classical forms of the reduced inequalities (18.15) and (18.19) as well as a number of independent fields in the constitutive equations (18.6), it is natural to expect that the corresponding analyses be quite involved.

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Acknowledgements This research was supported by the Polish Ministry of Science and Education under grant No. N506 254237.

References 1. Simmonds, J.G.: The nonlinear thermodynamical theory of shells: Descent from 3-dimensions without thickness expansions. In: Axelrad, E.L., Emmerling, F.A. (eds.), Flexible Shells, Theory and Applications, pp. 1–11. Springer-Verlag, Berlin (1984). 2. Pietraszkiewicz, W., Chrósćielewski, J., Makowski J.: On dynamically and kinematically exact theory of shells. In: Pietraszkiewicz, W., Szymczak, C. (eds.), Shell Structures: Theory and Applications, pp. 163–167. Taylor & Francis, London (2005). 3. Pietraszkiewicz, W.: Refined resultant thermomechanics of shells. Int. J. Engng Sci. (2011) (submitted). 4. Pietraszkiewicz, W.: On non-linear shell thermodynamics with interstitial working. In: Wilmański, K., Je¸drysiak, J., Michalak, B. (eds.), Mathematical Methods in Continuum Mechanics. Politechnika Łódzka, Łód´z (2011) (in print). 5. Dunn, J.E., Serrin J.: On the thermodynamics of interstitial working. Arch. Rational Mech. Anal. 85 (1985) 95–133. 6. Dunn, J.E.: Interstitial working and a non-classical continuum. In: Serrin, E. (ed.), New Perspectives in Thermodynamics, pp. 187-222. Springer-Verlag, Berlin et al. (1986) 7. Truesdell, C., Toupin R.: The Classical Field Theories. In: Flügge, S. (ed.), Handbuch der Physik, Band III/1, Springer-Verlag, Berlin (1960). 8. Truesdell, C., Noll W.: The Non-Linear Field Theories of Mechanics. In: Flügge, S. (ed.), Handbuch der Physik, Band III/3, Springer-Verlag, Berlin-Heidelberg-NewYork (1965). 9. Coleman, B.D., Noll W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13 (1963), 1, 167–178. 10. Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61 (2009), 1, 125–152. 11. Chrósćielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and Dynamics of Multi-Shells: Non-Linear Theory and the Finite Element Method. Institute of Fundamental Technological Research Press (in Polish). Warsaw (2004). 12. Cosserat, E., Cosserat, F.: Théorie des Corps Deformables. Hermann et Fils, Paris (1909). 13. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics (2nd Edition). Springer, New York (1998). 14. Zhilin, P.A.: Basic equations of non-classical theory of elastic shells (in Russian). Dinamika i Prochnost’ Mashin, Trudy Leningr. Polit. In-ta 386 (1982) 29–46. 15. Altenbach, H., Zhilin, P.A.: Theory of elastic simple shells (in Russian). Advances in Mechanics 11 (1988), 107–148. 16. Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moskva (2008). 17. Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12 (1976), 9-10, 635–648. 18. Chrósćielewski, J., Makowski, J., Pietraszkiewicz, W.: Non-linear dynamics of flexible shell structures. Comp. Assisted Mech. Engng. Sci. 9 (2002) 341-357. 19. Gurtin, M. E., Murdoch, A. I.: A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal. 57 (1975), 291–323.

Chapter 19

Free Finite Rotations in Deformation of Thin Bodies Leonid I. Shkutin

Abstract The term ”thin bodies” includes shells, plates and rods. Such bodies are divided in two groups: shell-like and rod-like bodies. The first group includes shells, plates and thin-walled rods, and the second group includes beams and rods with rigid cross-sections. Two approaches to model thin bodies deformation are developed in the scientific literature: axiomatic and approximate. The axiomatic approach was developed by Bernoulli, Euler, and Cosserat brothers. The paper by Ericksen and Truesdell [1] stimulated a general interest to axiomatic models of the deformation in mechanics. The review of the relevant publications is given in the references [2–5]. This lecture is devoted to construction and application of the approximate deformation models for rod-like and shell-like bodies. Keywords Finite rotations · Rod-like structures · Shell-like structures

19.1 Separation of Local Rotations in Cauchy Continuum Let a material continuum (or a solid body) in its reference state occupies a region (volume) B with a boundary (surface) Aν . The region is parameterized by the triplets of Cartesian coordinates xK and Lagrangian local coordinates t J . Let iK is the orthonormal basis of Cartesian coordinates. The position of a point in the reference state is given by the positional vector r(t J ) measured from the origin of Cartesian coordinates. The equality g J = r, J defines the holonomic local basis g J (r) of the local coordinate system in the reference state. The upper-case Latin indices take values 1, 2, 3; the lower-case indices after comma denote derivation with respect to Lagrangian coordinates; the traditional rules of tensor analysis are used in the following. L. I. Shkutin (B) Institute of Computational Modeling, Akademgorodok, 660036 Krasnoyarsk, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 19, © Springer-Verlag Berlin Heidelberg 2011

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Displacement of an arbitrary point r of the continuum in the deformed state is measured by a vector w(r), and deformation of a point vicinity is given by the transformation r → y , g J → yJ , y = r + w , yJ = y, J . (19.1) Here y(r), yJ (r) are positional and gradient vectors of a material point in the actual state (dependence on time is not shown). For separation of local rotations accompanying the deformation, in a point of the continuum we input an inholonomic basis a J (r) which rotates like a solid body. In the reference state this basis is given by the local vectors a◦J (r), in the actual state it is given by the local orthogonal transformation a J = a◦J · Θ = Θ · a◦J ,

Θ ≡ a◦J a J ,

Θ ·Θ ≡ 1,

(19.2)

where Θ (r) and Θ (r) are the conjugate tensors-rotators. The initial local basis a◦J can differ from the holonomic basis g J . In particular, as ◦ a J an orthogonal basis can be assumed if g J is not such. It is convenient to distinguish theses bases in deformation models of thin bodies. If we take the local orthonormal bases e◦J and e J corresponding to the bases a◦J and a J , then the orthogonal transformations e◦J = i J · O◦ ,

e J = e◦J · Θ = i J · O ,

O = O◦ · Θ

(19.3)

will relate these local bases with each other and with the Cartesian basis i J . Here O◦ (a) is the given orientation tensor of the initial local basis e◦J relative to the Cartesian one. This tensor has the same components in both these bases. The components of the tensor-rotator Θ (a) coincide in the bases e◦J and e J . It is convenient to realize a rotation from the Cartesian basis i J to the local basis e J with the multiplicative tensor O = O1 · O2 · O3 , where O1 , O2 , O3 are the base rotators with orthogonal matrices O1 , O2 , O3 : ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢1 0 0 ⎥⎥⎥ ⎢⎢⎢ cos θ2 0 sin θ2 ⎥⎥⎥ ⎢⎢⎢cos θ3 − sin θ3 0⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢⎢0 cos θ1 − sin θ1 ⎥⎥⎥⎥⎥, ⎢⎢⎢⎢⎢ 0 1 0 ⎥⎥⎥⎥⎥, ⎢⎢⎢⎢⎢ sin θ3 cos θ3 0⎥⎥⎥⎥⎥. ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ 0 sin θ1 cos θ1 − sin θ2 0 cos θ2 0 0 1 Here θ1 , θ2 , θ3 are angles of the right rotations with respect to corresponding base vectors. The unknown orientation tensor O(a) has the same components OKL in the Cartesian and rotated bases : O11 = cos θ2 cos θ3 , O12 = − cos θ2 sin θ3 , O13 = sin θ2 , O21 = cos θ1 sin θ3 + sin θ1 sin θ2 cos θ3 , O22 = cos θ1 cos θ3 − sin θ1 sin θ2 sin θ3 , O23 = − sin θ1 cos θ2 , O31 = sin θ1 sin θ3 − cosθ1 sin θ2 cos θ3 , O32 = sin θ1 cos θ3 + cosθ1 sin θ2 sin θ3 , O33 = cos θ1 cos θ2 θN are unknown angles of rotation from the Cartesian basis.

(19.4)

19 Free Finite Rotations in Deformation of Thin Bodies

263

Derivation of the rotated basis eK is realized with the formula eK, J = d J × eK = d JL eL × eK = d JL eLKM e M ,

(19.5)

where d J are Darboux vectors, d JL are their components in the chosen basis, and eLKM are components of the Levi-Civita tensor. From (19.5) follow Poisson’s type formulae: d J1 = e3 · e2, J = −e2 · e3, J , d J2 = e1 · e3, J = −e3 · e1, J ,

(19.6)

d J3 = e2 · e1, J = −e1 · e2, J . Using formulae (19.4), (19.6) and the expansion eK, J = OKM, J i M = OKM, J OLM eL ,

(19.7)

we obtain the relations d J1 = −θ1, J − θ3, J sin θ2 , d J2 = −θ2, J cos θ1 + θ3, J sin θ1 cos θ2 , d J3 = −θ2, J sin θ1 − θ3, J cos θ1 cos θ2 ,

(19.8)

which express the components of Darboux vectors via the rotation angles θN . In the general case these angles are represented by sums θN = θN◦ + ϑN , where θN◦ are orientation angles of the initial local basis, and ϑN are their increments during deformation. Darboux vectors and their components form the local curvature tensor of the postrotational space. The gradient vector transformation yI = a◦I · G = GIJ a J ,

G ≡ a◦I yI = GIJ a◦I a J ,

GIJ = y, I · a J

(19.9)

introduce an asymmetric tensor-gradient G(r) with two-basis expansion: in initial and rotated bases. In the reference state this tensor takes the value G◦ = a◦I gI . An objective (asymmetric) tensor of strains W(r) is given by the formulae W ≡ G − G◦ · Θ = WIJ a◦I aJ = a◦I wI .

(19.10)

Here wI (r), WIJ (r) are vectorial and scalar components of this tensor: wI = y, I − gI · Θ = a◦I · W = WIJ a J ,

WIJ = wI · a J = GIJ − GIJ◦ .

(19.11)

Formulae (19.10), (19.11) show dependence of the strain tensor on two independent kinematic fields: displacements and rotations. The second field is given by an independent orthogonal tensor, which has three independent degrees of freedom. These rotational degrees of freedom in Cauchy continuum can be eliminated with

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the help of three scalar relations on the strain tensor. In particular, the classic conditions of its symmetry lead to the relations: WJI = WIJ .

(19.12)

The strain tensor subjected to these relations will have six independent scalar components. The variant (19.12) realizes the polar decomposition of asymmetric strain tensor (19.10). Instead of (19.12), the alternative relations can be used W12 = W13 = W23 = 0 ,

(19.13)

W21 = W31 = W32 = 0 ,

(19.14)

which eliminate a triple of tensorial components. The variants (1.12)–(1.14) give the strain tensors by symmetric and triangular matrices of components: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢W11 W12 W13 ⎥⎥⎥ ⎢⎢⎢W11 0 0 ⎥⎥⎥ ⎢⎢⎢W11 W12 W13 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢⎢W12 W22 W23 ⎥⎥⎥⎥⎥ ∼ ⎢⎢⎢⎢⎢W21 W22 0 ⎥⎥⎥⎥⎥ ∼ ⎢⎢⎢⎢⎢ 0 W22 W23 ⎥⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ W13 W23 W33 W31 W32 W33 0 0 W33 Application of the relations (19.13) and (19.14) can be justified in describing deformation of thin bodies. The strength of Cauchy’s continuum in a vicinity of point r is measured with the stress vectors σI (r) on unit coordinate areas tI = const . The equality √ sI = σI / g II (19.15) gives the Lagrangian contravariant stress vectors s I (r). Balance of the external and internal forces in an actual state of Cauchy’s continuum can be expressed with the virtual equation (f · δy − δw) dB + f ν · δydAν = 0 , (19.16) B

Aν

δw = s · δ wI = S δWIJ , S IJ = s I · a J . I

IJ

Here: δw is the virtual strain energy density per unit initial volume; δ is the corotational variation operator; S IJ are two-base components of the stress tensor; f, f ν are densities of the volume and surface forces.

19.2 A Rod Deformation Model with One-Dimensional Field of Finite Rotations A volume B occupied by a rod is usually bounded by a tubular surface A3 and two end surfaces An . A spatial local system of coordinates t J is connected with the base curve C3 of the rod so that t3 is the longitudinal parameter of the line, t1 and t2 are the transverse orthogonal coordinates. Let A is an arbitrary cross section of a rod.


265

The rod-like solid body can be described by the equation r = a + ti e◦i .

(19.17)

It expresses a 3D position vector r(t J ) via a positional vector a(t3 ) of a point of the base line and two orientation unit vectors e◦i (a), which are orthogonal to the base line and its tangential unit vector e◦3 = a, 3 . The relations gi = e◦i , g3 = e◦3 + ti e◦i, 3 , e◦3 = a, 3 (19.18) express a 3D local basis g I (r) via the orthonormal basis e◦I (a) of the base line. The rod deformation in the Cauchy model is defined with the vectors (19.1), y = r+w ,

yI = y, I = gI + w, I ,

(19.19)

where w(r) is the displacement vector field. The basis of the base line is deformed together with the rod. The local orthogonal transformation eI = e◦I · Θ ,

Θ ≡ e◦J eJ ,

Θ, i ≡ 0 ,

Θ · Θ ≡ 1,

(19.20)

gives a tensor-rotator Θ (a) and a rotating basis eI (a) with the reference value e◦I (a). Assuming that a rod remains a thin body, its actual configuration is approximated with a vector function similar to (19.17), y ≈ x + ti ei = x + ti e◦i · Θ ,

(19.21)

where x(a) is an actual position vector of a point of the base line. Approximation (19.21) gives the gradient vectors yI (19.19) of the actual state yi ≈ ei ,

y3 ≈ x, 3 + ti ei, 3 .

(19.22)

Formula (19.11) gives a nonzero strain vector corresponding to approximations (19.18) and (19.22), w3 = y, 3 − g3 · Θ ≈ u3 + ti vi3 . (19.23) The vectors of metric and torsion-bending strains at the base line are given as follows: u3 ≡ x, 3 − e◦3 · Θ = x, 3 − e3 , (19.24) vi3 ≡ ei, 3 − e◦i, 3 · Θ = ei · Ω 3 = ω 3 × ei ,

Ω 3 ≡ Θ · Θ, 3 .

(19.25)

The metric strain vector u3 (a) depends on the position vector and the tensor-rotator, while the vectors of torsion-bending strains vi3 (a) depend on the tensor-rotator, or on the axial vector ω 3 (a) generated by the skew-symmetric tensor Ω 3 . The approximation (19.21) eliminates the vectors wi which measure the strains of transversal fibres of the rod. The vector w3 measures the strains of longitudinal

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fibres. For accounting the Poisson effect, the generalized approximation of strain field is wI ≈ uI + ti viI . (19.26) Here the vectors u3 , vi3 are given by functions (19.24), (19.25), and the vectors ui , vi j are arbitrary and play the role of Lagrange multipliers in the virtual work equation. The expansions of vector-functions (19.26) are fulfilled in the rotating basis uI = u IJ e J ,

viI = viIJ e J ,

wI = wIJ e J

| ∀ JI

(19.27)

with the triangular matrices of components ⎡ ⎤ ⎢⎢⎢u11 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢u u 0 ⎥⎥⎥⎥⎥ , ⎢⎢⎢ 21 22 ⎥⎥⎥ ⎣ ⎦ u31 u32 u33

⎡ ⎤ ⎢⎢⎢vi11 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢v v ⎥ ⎢⎢⎢ i21 i22 0 ⎥⎥⎥⎥⎥ , ⎣ ⎦ vi31 vi32 vi33

⎡ ⎤ ⎢⎢⎢w11 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢w w ⎥ ⎢⎢⎢ 21 22 0 ⎥⎥⎥⎥⎥ . ⎣ ⎦ w31 w32 w33

As a result, the approximations (19.21)–(19.27) permit to realize the variant (19.13) for representation of the rod strain tensor. Derivation of rotating basis eK is fulfilled with the formula (19.5): eK, 3 = d3 × eK = d3L eLKM e M ,

(19.28)

where d3 is a Darboux vector, while d3L are its components in the chosen basis. The formulae (19.8) express these components via the rotation angles θN (from Cartesian basis to rotated one): d31 = −θ1,3 − θ3,3 sin θ2 , d32 = −θ2,3 cos θ1 + θ3,3 sin θ1 cos θ2 ,

(19.29)

d33 = −θ2,3 sin θ1 − θ3,3 cos θ1 cos θ2 . ◦ of components d When θN =θN◦ these formulae give the reference values d3K 3K . The physical components of metric (19.24) and torsion-bending (19.25) strains in the rotating basis are defined by the formulae ◦ u3J = O JK xK, 3 − e3J , vi3J = eiJK ω3K , ω3K = d3K − d3K .

(19.30)

Here xK , O JK are unknown components of the position vector and orientation tensor (19.4) in the Cartesian basis. The generalized dynamic equations of the rod are constructed by the standard method from the virtual equation (19.16) and approximations (19.21), (19.23), (19.26). As a result, we obtain the one-dimensional equations of force and moment balance t3, 3 + p = 0 , m3, 3 + x, 3 × t3 + q = 0 , (19.31)


267

with the resultant vectors of internal forces and couples t3 = σ 3 dA , m3 = ai × σ 3 ti dA , A

A

(19.32)

and with the resultant vectors p, q of external forces and couples along the center line. Representing in (19.31) t3 = T 3J e J , m3 = M3J e J , x, 3 × t3 = (e3K + u3K ) e JKL T 3L e J , we obtain the force scalar equilibrium equations T 3J, 3 + e JKL d3K T 3L + p · eJ = 0

(19.33)

and the couple scalar equilibrium equations M3J, 3 + e JKL d3K M3L + (e3K + u3K ) e JKL T 3L + q · eJ = 0 .

(19.34)

In these equations T 3J , M3J are physical components of internal forces and couples. The last terms (in form of scalar product) show that expansion of the vectorial equations is fulfilled in the rotating basis. The constitutive relations for the elastic rod are formulated by the generalized equations T 3J =

A

CJK w3K dA , M3L = eiJL

A

CJK w3K ti dA ,

(19.35)

in which the coefficients CJK can depend on strains w3L = u3L + ti vi3L (in a nonlinear manner).

19.3 A Shell Deformation Model with Two-Dimensional Field of Finite Rotations A spatial local coordinate system t J is connected with a base surface A of a shell, so that t1 and t2 are internal parameters of the surface, t3 is the transverse orthogonal coordinate, and dt3 is an element of shell transverse fibre. A volume B occupied by a shell is usually bounded by two face surfaces A1 , A2 and a boundary surface A3 , which includes boundary contour C of the base surface. Surfaces A and Am are given by the equations t3 = 0 t3 = hm , so that h1 t3 h2 (h1 and h2 are known functions of the base surface points or are constants). The volume and surface elements of the shell are given by dB = η dt3 dA ,

dA = a dt1 dt2 ,

dAm = αm dA , dA3 = η dt3 dC , η = g/a , (19.36) where g = det[g IJ ] , a = det[ai j ] are volume and surface Jacobians of the local coordinate system, αm is the metric parameter of the surface Am , dC is

268

L.I. Shkutin

the boundary contour element of the base surface. Equations (19.36) can be used for computation of parameters αm . A shell-like solid body can be given by the equation r = a + t3 e◦3 .

(19.37)

It expresses spatial position vector r(t J ) via the positional vector a(ti ) of the surface point and the unit vector e◦3 (a), which is orthogonal to the surface and its tangential unit vectors e◦i = a, i . The equations g3 = e◦3 ,

gi = a◦i + t3 b◦i ,

a◦i = a, i ,

b◦i = e◦3, i

(19.38)

express spatial local basis g I (r) via the orthonormal surface basis e◦I (a). The shell deformation in the Cauchy model is defined by the vectors (19.1): y = r+ w,

yI ≡ y, I = gI + w, I ,

(19.39)

where w(r) is the displacement vector field. The surface basis is deforming together with the shell. The local orthogonal transformation aI = a◦I · Θ ,

Θ, 3 ≡ 0 ,

Θ·Θ ≡ 1

(19.40)

relates the surface tensor-rotator Θ (a) and the rotating basis eI (a) with the reference value e◦I (a). Assuming that the shell remains a thin body, its actual configuration is approximated by the vector function similar to (19.37), y ≈ x + t3 e 3 ,

(19.41)

where x(a) is the actual position vector of the surface point, e3 is the unit vector. The linear (relative to transverse coordinate) displacement field corresponds to the function (19.41), w = y − g ≈ u + t3 v ,

u = x−a ,

v = e3 − e◦3 .

(19.42)

Approximation (19.41) gives the gradient vectors yI (19.39) of the actual state y3 = e3 ,

yi = x, i + t3 bi ,

bi ≡ e3, i .

(19.43)

Formula (19.11) gives two nonzero strain vectors related by approximations (19.38) and (19.43), wi = y, i − gi · Θ ≈ ui + t3 vi , w3 ≈ 0 . (19.44)


269

The surface vectors of metric and torsion-bending strains are given here by ui = x, i − a◦i · Θ = x, i − ai ,

x = a+u ,

vi = a3, i − b◦i · Θ = a3 · Ω i = ω i × a3 ,

Ω i ≡ Θ · Θ, i .

(19.45) (19.46)

The vectors of metric strains ui (a) depend on the position vector and tensorrotator, and the vectors of torsion-bend strains vi (a) depend on tensor-rotator, or on two axial vectors ω i (a) generated by two skew-symmetric tensors Ω i ≡ Θ · Θ, i . The approximation (19.41) eliminates the vector w3 (19.44) which measures the strains of transversal fibres of the shell. The vectors wi measure the strains of longitudinal fibres. For accounting the Poisson effect, the generalized approximation of the strain field is assumed w I ≈ u I + t3 v I . (19.47) Here the vectors ui ,vi are given by functions (19.45), (19.46), and the vectors u3 ,v3 are arbitrary and play the role of Lagrange multipliers in the virtual work equation. The expansion of vector-functions (19.47) is fulfilled in the rotated basis uI = u IJ e J , vI = vIJ e J , wI = wIJ e J with the matrix of components ⎡ ⎤ ⎢⎢⎢u11 u12 u13 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢⎢u21 u22 u23 ⎥⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ 0 0 u33

⎡ ⎤ ⎢⎢⎢v11 v12 0 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢⎢v21 v22 0 ⎥⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ 0 0 v33

| ∀ JI

(19.48)

⎡ ⎤ ⎢⎢⎢w11 w12 w13 ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢⎢w21 w22 w23 ⎥⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 w33

The approximation (19.44), in general, does not allow to fulfill condition w21 = 0 (19.9) because rotator Θ (a) is independent of coordinate t3 . But this condition is the only possible variant of elimination of one degree of rotation freedom. It is possible, for example, to require realization of condition u21 = 0 instead of w21 = 0 or condition u21 = u12 instead of w21 = w12 . The most simple variant is the elimination of one degree of freedom in the rotator. In particular, one can eliminate a basis rotation with respect to e3 (a drilling rotation). In this case the rotation field will have two degree of freedom and can be represented by two independent scalar components. Derivation of the rotated basis eK is fulfilled with the formula (19.5), eK, i = di × eK = diJ e JKL eL ,

(19.49)

where di are Darboux vectors, diJ are their components in the chosen basis. The formulae (19.8) express these components via the rotation angles θN (from Cartesian basis to rotated one): di1 = −θ1, i − θ3, i sin θ2 , di2 = −θ2, i cos θ1 + θ3, i sin θ1 cos θ2 , di3 = −θ2, i sin θ1 − θ3, i cos θ1 cos θ2 .

(19.50)

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L.I. Shkutin

◦ of components d . When θN =θN◦ these formulae give the reference values diJ iJ The physical components of metric (19.45) and torsion-bending (19.46) strains in the rotated basis are defined by the formulae −1 ◦ uiJ = A−1 i O JK x K, i − eiJ , viJ = Ai e3JK ωiK , ωiK = diK − diK .

(19.51)

Here xK , O JK are unknown components of position vector and orientation tensor ◦ are known Darboux parameters of the basis e◦ . (19.4) in the Cartesian basis, diK K The generalized dynamic equations of the shell are constructed by the standard method as a consequence of the virtual equation (19.16) and approximations (19.44)–(19.47). As a result, we obtain the two-dimensional equations of force and moment balance C i + x, i × ti +C ∇i ti + p = 0 , ∇i m q=0, (19.52) with the resultant surface vectors of internal forces and couples defined by tI =

h2

sI ηdt3 ,

mI =

h1

h2

sI ηt3 dt3 ,

C i = e3 × mi , m

(19.53)

h1

and with the surface vectors p, C q of external forces and couples (η is known geometric parameter of the shell). In components Eq. (19.52) read ∇i ti = a−1 (ati ), i , ti = A−1 i T iJ e J , a = A1 A2 , i i −1 −1 C CiJ = e3KJ MiK , C = A MiJ e J , M m = A MiK eK , m i

i

(19.54)

and we obtain the scalar formulation of shell dynamic equations: −1 (A1 A2 A−1 i T iJ ), i + e JKL diK A1 A2 Ai T iL + A1 A2 p · e J = 0 , CiJ ), i + e JKL diK A1 A2 A−1 M CiL + (A1 A2 A−1 M i

i

(19.55)

+A1 A2 e JKL (eiK + uiK )T iL + A1 A2 C q · eJ = 0 . CiJ are physical components of internal forces and couples, In these equations T iJ M the last terms (in the form of scalar product) show that expansion of the vectorial equations (19.55) is fulfilled in the rotated basis. The resulting set of differential equations (19.50), (19.51), (19.55) in the general formulation contains first order partial derivatives of five kinematic and five dynamic functions. Conditions along the shell boundary contour are formulated with the virtual equality Ci j − Q Cj ) δω j = 0 . (n◦i T iJ − PJ ) δu J + (n◦i M

(19.56)

C j are given vectors of forces and couples, δu J , δω j are virtual displaceHere PJ , Q ments and rotations in the contour points, n◦i are components of the external unit normal to the contour. The equality (19.56) allows one to correctly formulate five boundary conditions of dynamic, kinematic and mixed types.


271

Constitutive relations for the elastic shell are formulated by the generalized equations T iJ =

h2

σiJ ηi dt3 ,

h1

MiJ =

h2

σiJ ηi t3 dt3 ,

CiK = e3JK MiJ , M

(19.57)

h1

in which the stresses σiJ have to be given by linear or nonlinear functions of strains (19.48), ηi are the known functions weakly depending on the coordinate t3 .

19.4 Axisymmetric Bending Modes of Plates and Shells We consider a dome-like shell with axisymmetric base surface A. The local coordinates t J with orthonormal basis e J (t1 , t2 ) are given on the surface, so that t1 ∈ [0 , l] , t2 ∈ [0 , 2π], t3 ∈ [−h , h] are meridional, circumferential, and normal coordinates (2h is a dome thickness, l is the length of meridian). In addition, the cylindrical coordinates (x1 , t2 , x3 ) are introduced with orthonormal basis i J (t2 ) . The surface basis differs from the cylindrical one by an angle θ2 (t1 ) relative to the vector i2 . The corresponding transformation of the surface basis is realized by the rotator O2 . The dome shape in the reference state is given by parametric equations x1 = r(t1 ) , x3 = z(t1 ) , θ2 = θ(t1 ) ,

∀ t1 ∈ [0 , l] ,

(19.58)

dr/dt1 = cos θ , dz/dt1 = − sin θ , where r, z are the point coordinates in cylindrical system. We consider the axisymmetric deformation of a dome when the base surface remains the surface of revolution, so that x1 = x1 (t1 ) , x3 = x3 (t1 ) , θ2 = θ2 (t1 ) ,

∀ t1 ∈ [0 , l] ,

(19.59)

where x1 , x3 are the sought coordinates of a point in cylindrical system, θ2 is a sought rotation angle. The dome material is isotropic and linearly elastic. The full system of equations includes the kinematic equations following from (19.50) and (19.51), x1,1 = (1 + u11) cos θ2 + u13 sin θ2 , x3,1 = −(1 + u11) sin θ2 + u13 cos θ2 , v11 = (θ2 − θ),1 , v22 = r−1 (sin θ2 − sin θ) , u22 = r−1 (x2 − r) , (19.60) the dynamic equations following from (19.55), (rX1 ),1 − T 22 + r p · i1 = 0 , (rX3 ),1 + r p · i3 = 0 , (rM11 ),1 − M22 cos θ2 − r T 13 + r q · i2 = 0 , T 11 = X1 cos θ2 − X3 sin θ2 , T 13 = X1 sin θ2 + X3 cos θ2 ,

(19.61)

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L.I. Shkutin

and the generalized constitutive relations following from (19.57), u11 = (1 − ν2 )F −1 T 11 − νu22 , v11 = (1 − ν2)H −1 M11 − νv22 , u13 = γF −1 T 13 , T 22 = νT 11 + Fu22 , M22 = νM11 + Hv22 .

(19.62)

The following notations are used in this system: γ = E/G, F = 2hE, 3H = 2h3 E ; 2h is the shell thickness; E, G are modules of extensional and transverse shear; ν is Poisson’s ratio; uiJ (t1 ), vii (t1 ) are components of strains in the rotating basis; X1 (t1 ), X3 (t1 ) are components of internal forces in the cylindrical basis; T iJ (t1 ), Mii (t1 ) are components of internal forces and couples in the rotating basis; derivation is fulfilled with respect to the coordinate t1 . The resulting set of differential equations (19.60), (19.61) contains first order derivatives of three kinematic functions x1 (t1 ), x3 (t1 ), θ2 (t1 ), and three dynamic functions X1 (t1 ), X3 (t1 ), M11 (t1 ). Therefore, the values of three unknown functions have to be given on shell boundary contour.

19.4.1 Circular Plate Under Radial Compression In the cylindrical coordinates (x1 , t2 , x3 ), a circular plate is described by the parametric equalities x1 = R t1 , x3 = 0 , ∀ t1 ∈ [0 , 1] , ∀ t2 ∈ [0 , 2π] ,

(19.63)

where R , t1 are radius and radial coordinate. For the system (19.60)–(19.62), we formulate two boundary-value problems on axisymmetric deformation of the circular plate under the uniform radial compressive load of intensity P. Since the surface load is absent, one should set p = q = 0 in the system. At the pole t1 = 0, the following symmetry and regularity conditions should hold: T 11 (0) = T 22 (0) , M11 (0) = M22 (0) , X3 (0) = 0 . (19.64) At the boundary contour (t1 = l), the two variants of conditions are specified: (a) x3 (1) = 0 , X1 (1) = −P , M11 (1) = 0 , (b) x3 (1) = 0 , X1 (1) = −P , θ1 (1) = 0 . The nonlinear boundary-value problems with boundary conditions (a) and (b) were solved by the shooting method with the Mathcad software package. The branching solutions of these boundary-value problems were obtained by a variation of the load parameter P. The main results of numerical solution are presented in Figs 19.1 and 19.2 for the plate with parameters E/G = γ = 2.5 , h2 /(3R2 ) = ε = 0.025 , ν = 0.25 . The following three bifurcation values of the numerical load parameter p = P/C (C = 2Ehε) were obtained:


273

0.2

1a

w

1b

2a

2b

3a

3b

0.1

10

0

20

30

40

λ

Fig. 19.1 Bifurcation maps of plate deformation in terms of parameters λ = P/P1a and w = −u3 (0)/R, 4

3a

2a

1a

u,% 0

3 2 1

0

0.5

1.0

1.5

2.0

p

2.5

Fig. 19.2 Bifurcation maps in terms of parameters p = P/C and u = −u2 (1)/R

Problem (a): p1a 0.1085, p 2a 0.772, p 3a 1.958. Problem (b): p1b 0.3915, p 2b 1.313, p 3b 2.763. The bifurcation maps of plate deformation are shown in Fig. 19.1 in terms of parameters λ = P/P1a , w = −u3 (0)/R, and in Fig. 19.2 in terms of parameters p = P/C, u = −u2 (1)/R. Here P1a = C p1a is the classical critical load for the simply supported plate, so that λ1a = 1 for the classic value of the parameter λ. The line 0 in Fig. 19.2 is the trajectory of base (plane) strains, and lines 1a, 2a, 3a are trajectories of buckling modes. These modes are shown in [6].

19.4.2 Conical Shell Under Radial Compression In the cylindrical coordinates, a meridian of the conical shell (dome) is given by the parametric equations (19.58) of the form

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L.I. Shkutin

x1 = r = t1 cos α , x3 = z = (l − t1 ) sin α , θ2 = α , ∀ t1 ∈ [0 , l] ,

(19.65)

where l is length of the meridian, α is angle in the base of the dome. We study an axisymmetric deformation (19.60)– (19.62) of the dome under a compressive radial load P uniformly distributed along the boundary contour r = b = l cos α. The surface load is absent, so that p = q = 0 in the system (19.60)–(19.62). At the pole t1 = 0, the following symmetry and regularity conditions should hold: T 11 (0) = T 22 (0) , M11 (0) = M22 (0) , X3 (0) = 0 .

(19.66)

At the boundary point t1 = l, the conditions of simple supported movable contour are: x3 (l) = 0 , X1 (l) = −P , M11 (l) = 0 . (19.67) 3a

2a

1a

4

u,% 0

3 2 1

0

0.5

1.0

1.5

2.0

p

2.5

Fig. 19.3 Bifurcation map for the dome with angle α = π/180.

4

u,%

1

2

2

3 2 1 0

3 0.5

1.0

-1

1.5

p

2.0

Fig. 19.4 Bifurcation map for the dome with angle α = π/12.

The nonlinear boundary-value problems (19.66), (19.67) were solved by the method of shooting from the point t1 = l to the point t1 = δl, where δl is a small varying value. Some results for isotropic conical domes are shown in Figs 19.3 and 19.4 h2 /(3l2 ) = ε = 0.025 , E/G = γ = 2.5 , ν = 0.25. for the domes with parameters


275

The results are represented by diagrams in the plane (p , u) where p = P/C and u is the percentage value of boundary strain |u22(l)| (19.60) . Figure 19.3 compares the bifurcation map for the dome with angle α = π/180, with that for the plate. Solid curves 1, 2 and 3 are the first three branches (trajectories) of bending modes of the dome. They are separated by the trajectories of the plate buckling modes (dashed curves of Fig. 19.2). Figure 19.4 shows these solutions for a dome with angle α = π/12. In this case, the branches of higher modes are below the line that represents the plane modes of the plate, and even below the abscissa axis. It means that the dome can have the buckling modes with the stretched boundary contour. Bending modes corresponding to all these branches are shown in [7].

19.4.3 A Dome Axisymmetric Deformation in Thermal Cycle of Phase Transformations As the constitutive relations for a shape-memory alloy in the phase transformation cycle, we use the micromechanical relations, which are formulated in [9]: w11 = φ11 + (σ11 − νσ22 )/E , 1 2 , w13 = φ13 + σ13 /G , dφii /dq = (1 − q)λ(κ φii + sii /σ) , dφ13 /dq = (1 − q)λ(κ φ13 + σ13 /σ) , π T+ − T q = sin , T− T T+ . 2 T+ − T−

(19.68)

Here wiJ and φiJ are total and phase strains, respectively, σiJ and sii are the stresstensor and stress-deviator components, E, G and ν are elasticity moduli, κ, λ and σ are experimental parameters of thermocycle, T + and T − are the initial and final temperatures of the direct transformation, T is the current temperature, 0 q 1 is an internal state parameter defined as the volume fraction of the martensite phase. According to (19.68), growing of the phase strains is completed at q = 1. The phase transformation is treated as a quasi-static process with temperature uniformly distributed over the sample, so that the parameter q does not depend on coordinates. In the phase transformation interval, the elastic moduli in (19.68) are not constant but change from their austenite to martensite values. For the phase transformations considered, these moduli can be represented as the averaged relations E = q E− + (1− q ) E+ , ν = q ν− + (1− q ) ν+ , G = E/(2 + 2ν) , where the subscripts minus and plus refer to martensite and austenite phases, respectively. Assuming that the stresses depend on q much more weakly as compared to the phase strains, we can find an approximate solution of the differential equations (19.68):

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L.I. Shkutin

2σ11 − σ22 2σ22 − σ11 σ13 , φ22 η , φ13 η , (19.69) 3 σκ 3 σκ σκ D κ E η(q) ≡ exp 1 − (1 − q)1+λ − 1 . 1+λ This solution satisfies the following physical conditions: the phase strains are absent in the austenite phase and reach maximal values in the martensite phase. Substitution of the functions (19.69) into the first three equations (19.68) gives three-dimensional constitutive relations φ11 η

E 0 w11 η1 σ11 − η2 σ22 ,

E0 w22 η1 σ22 − η2 σ11 ,

E 0 w13 η3 σ13 ,

(19.70)

E0 2E 0 E0 E0 E0 E0 +η , η2 (q) ≡ ν +η , η3 (q) ≡ +η , E 3 σκ E 3 σκ G σκ where E 0 is a constant that has dimension of stress and which is conveniently identified as E − or E + . Equations (19.70) describe one-directional phase transformation as thermoelastic deformation with an implicit temperature dependence (via the parameter q). Integration of these equations through the shell thickness leads to the generalized constitutive relations η1 (q) ≡

C0 u11 η1 T 11 − η2 T 22 ,

C0 u22 η1 T 22 − η2 T 11 ,

l−1 H0 v11 η1 M11 − η2 M22 ,

C0 u13 η3 T 13 ,

l−1 H0 v22 η1 M22 − η2 M11 ,

(19.71)

where C0 = 2hE0 and H0 = 0 /3 are the stiffness parameters. We consider the simple supported spherical dome loaded in the austenite phase by the uniform normal pressure of intensity P. The reference shape (19.58) of the dome meridian is given by the parameters 2h3 E

r = lα−1 sin θ, z = lα−1 (cosθ − cos α) , θ = αt1 , ∀ t1 ∈ [0 , 1] , where l is length of the meridian, α is angle in the base of the dome. In the bending state the load vector p is directed along internal normal to the base surface so that p · i1 = −P sin θ2 , p · i3 = −P cos θ2 , q · i2 = 0 .

(19.72)

At the boundary point t1 = 1, the conditions of simple supported contour are specified as x1 (1) = b , x3 (1) = 0 , M11 (1) = 0 . (19.73) At the pole t1 = 0, the conditions (19.66) should hold. The boundary problem (19.66), (19.73) for the set of equations (19.60), (19.61), (19.71) and (19.72) is solved by the shooting method, for discrete values of the state parameter q ∈ [0 , 1]. The complete thermocycles, obtained by solving the directand inverse-transformation problems, are shown in Fig. 19.5a for the shell without transient buckling (α = π/36, ε = 0.025, λ = 0), and in Fig. 19.5b for the shell with


277

2 W

1

2

1

0 40

20

60

T,°C

a W 2

1

2

1

0 20

40

60

T,°C

b Fig. 19.5 Phase diagrams: a) for a shell without transient buckling (α = π/36, ε = 0.025, λ = 0), b) for a shell with transient buckling (α = π/36, ε = 0.01, λ = 1)

transient buckling (α = π/36, ε = 0.01, λ = 1) as trajectories of the maximal axial displacement (in relation to the dome height) depending on temperature. The direct transition (curves 1) was computed for the load parameter p = 0.002, much smaller than upper critical loads of the dome in the austenite and martensite phases. The inverse transition (curves 2) was computed with zero load parameter. The solid curve 1 in Fig. 19.5a shows the trajectory of direct transition with a dynamic jump at the upper critical point (T 33.3◦C, q 0.69). The solid curve 2 shows the trajectory of inverse transition with a dynamic jump at the lower critical

278

L.I. Shkutin

point (T 68.5◦C, q 0.04). These curves form the dynamic hysteresis loop with the instantaneous jumps of the dome from one equilibrium shape to another at fixed temperature. The dashed curves 1 and 2 in Fig. 19.5b form the static hysteresis loop provided that the dome deformation is directed along this loop.

Concluding Remark The papers [6–9] allow to acquaint in more detail with earlier author’s results. Acknowledgements This research was supported by the RFBR under grant No. 11-01-00053.

References 1. Ericksen, J. L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Rat. Mech. Anal. 1 (1), 295–323 (1958) 2. Shkutin, L. I.: Deformation Mechanics of Flexible Bodies. Nauka, Novosibirsk (1988) 3. Shkutin, L. I.: Generalized models of the Cosserat type for finite deformation analysis of thin bodies. J. Appl. Mech. Tech. Phys., 37 (4), 400–410 (1996) 4. Eremeyev, V. A., Zubov, L. M.: Mechanics of Elastic Shells. Nauka, Moscow (2008) 5. Pietraszkievicz, W., Eremeyev, V. A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct., 46, 774–787 (2009) 6. Shkutin, L. I.: Numerical analysis of axisymmetric buckling of plates under radial compression. J. Appl. Mech. Tech. Phys., 45 (1), 89–95 (2004) 7. Shkutin, L. I.: Numerical analysis of axisymmetric buckling of a conical shell under radial compression. J. Appl. Mech. Tech. Phys., 45 (5), 741–746 (2004) 8. Shkutin, L. I.: Analysis of axisymmetric phase strains in plates and shells. J. Appl. Mech. Tech. Phys., 48 (2), 285–291 (2007) 9. Shkutin, L. I.: Axisymmetric deformation of plates and shells with phase transformations under thermal cycling. J. Appl. Mech. Tech. Phys., 49 (2), 330–335 (2008)

Chapter 20

On Universal Deformations of Nonlinear Isotropic Elastic Shells Leonid M. Zubov

Abstract A two-dimensional nonlinear model of the elastic shell is considered. It is assumed that the shell is the deformable surface with kinematically independent fields of translations and rotations. Within this model, several sets of finite nonuniform deformations are found, which for any isotropic shell satisfy equilibrium equations without surface loads. Universal solutions are obtained for six families of deformations, which are characterized by certain fields of surface translations. Each of these families consists of several subfamilies, which differ by the rotation field. It is found out that the closed isotropic spherical shell without external loads has four different equilibrium states, and the shell remains spherical in each of these states. The general expression for the rotation field of the isotropic micropolar plate is found, using the equilibrium equations and the distorsion tensor for uniform deformations. Equilibrium of the Cosserat membrane, which has the form of minimal surface, is studied. Keywords Universal solution · Nonlinear elasticity · Cosserat shell · Micropolar shell

20.1 Basic Statements Let σ be the reference (undeformed) base surface of the shell. We refer the particle of σ to Gaussian coordinates qα , α = 1, 2, with the position vector r(q1 , q2 ) = x1 i1 + x2 i2 + x3 i3 . Here xk are the Cartesian coordinates of the point of σ, while ik is the orthonormal Cartesian frame (k = 1, 2, 3). The natural and reciprocal bases on σ are given by the following relations: L. M. Zubov (B) South Federal University, Milchakova str., 8a, 344090, Rostov on Don, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 20, © Springer-Verlag Berlin Heidelberg 2011

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rα =

∂r , ∂qα

β

rβ · rα = δα ,

rβ · n = 0 ,

(20.1)

β

where δα is the Kronecker symbol and n is the unit normal to σ. The base surface Σ of the shell after deformation is referred to the same convected coordinates qα as well, and the position of the point of Σ is given by the position vector R(q1 , q2 ) = X1 i1 + X2 i2 + X3 i3 , where Xk are the Cartesian coordinates. The boundary value problem of the nonlinear micropolar shell consists of the equilibrium equations, see [1–6], div D + f = 0 ,

divG + (FT · D)× + l = 0 ,

(20.2)

the constitutive equations D = P·H , G = K·H , ∂W(E, L) ∂W(E, L) P= , K= , ∂E ∂L

(20.3)

and the kinematic relations

1 ∂H T L = rα ⊗ · H , 2 ∂qα × ∂Φ ∂Φ Δ Δ gradΦ = rα ⊗ α , div Φ = rα · α . ∂q ∂q E = F · HT ,

(20.4)

Here W is the 2D elastic strain energy density of the shell, grad and div are the surface gradient and divergence operators with regard to points on σ, respectively, D and G are the surface stress resultant and stress couple tensors, F = gradR, and H is the microrotation tensor characterizing the rotational degrees of freedom of the shell particles. E and L are the surface stretch and wryness tensors, while f and l are the external surface force and moment vectors, respectively. A× denotes the vectorial invariant of the second-order tensor A defined by the relation A× = (Amn im in )× ≡ Amn im × in . In what follows only isotropic elastic shells are considered. This means that functions W(E, L), P(E, L), K(E, L) have the form W(QT · E · Q, εQT · L · Q) = W(E, L) ,

ε = det Q ,

P(QT · E · Q, εQT · L · Q) = QT · P(E, L) · Q ,

(20.5)

K(QT · E · Q, εQT · L · Q) = εQT · K(E, L) · Q . In (20.5), Q is any orthogonal tensor satisfying the restriction n·Q = n .

(20.6)

20 On Universal Deformations of Nonlinear Isotropic Elastic Shells

281

Equations (20.5) indicate that E and P are true second-order tensors, while L and K are second-order pseudotensors. We study finite deformations of isotropic shells without surface loads, i.e. we find solutions of (20.2) with f = l = 0. The solutions discussed below for isotropic micropolar shells form six families of non-uniform deformations. Each family is characterized by the mapping R = R(q1 , q2 ) of the surface σ into the surface Σ, which sets the translation field of the shell. Since the field of microrotations in the micropolar shell is kinematically independent upon the translation field, the proper orthogonal tensor field H(q1 , q2 ) needs to be specified for determining the shell deformed state. Therefore, each of the mentioned six families of non-uniform deformations consists of several subfamilies, which differ from each other by microrotation fields. For all subfamilies the expressions for tensors E and L are presented, which are determined with the help of (20.4). The distorsion tensor F is identical for all solutions of the family.

20.2 Families of Non-Uniform Deformations 20.2.1 Cylindrical Bending of a Rectangular Plate Let the surface σ be plain x1 = const such that n = i1 . Let also q1 = x2 , q2 = x3 . We denote by R, Φ, Z the circular cylindrical coordinates of a point on the deformed surface Σ. The cylindrical bending of the plate is specified by the equations R = R0 ,

Φ = κx2 ,

Z = αx3 ,

(20.7)

where R0 , Φ, α are constants. The tensor F corresponding to (20.7) is F = κR0 i2 ⊗ eΦ + αi3 ⊗ i3 ,

Δ

eΦ = −i1 sin Φ + i2 cos Φ .

(20.8)

Since H is the kinematically independent field, one needs to specify also the microtation field which is consistent with the mapping (20.7). Here we consider the following subfamilies for H: Subfamily 1a: H = i1 ⊗ eR + i2 ⊗ eΦ + αi3 ⊗ i3 ,

Δ

eR = i1 cosΦ + i2 sin Φ ,

E = κR0 i2 ⊗ i2 + αi3 ⊗ i3 ,

L = κi2 ⊗ i3 .

(20.9)

Subfamily 1b: H = i1 ⊗ eR − i2 ⊗ eΦ − i3 ⊗ i3 , E = − (κR0 i2 ⊗ i2 + αi3 ⊗ i3 ) ,

L = −κi2 ⊗ i3 .

(20.10)

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Subfamily 1c: H = −i1 ⊗ eR + i2 ⊗ eΦ − i3 ⊗ i3 , E = κR0 i2 ⊗ i2 − αi3 ⊗ i3 ,

L = −κi2 ⊗ i3 .

(20.11)

Subfamily 1d: H = −i1 ⊗ eR − i2 ⊗ eΦ + i3 ⊗ i3 , E = −κR0 i2 ⊗ i2 + αi3 ⊗ i3 ,

L = κi2 ⊗ i3 .

(20.12)

Microrotations in subfamilies 1a–1d differ from each other by rotations about various axes. Let us introduce components P sm = i s · P · im , Ksm = i s · K · im , (s, m = 1, 2, 3). Then Eqs (20.3), (20.4) yield P1m = K1m = 0. In Eq. (20.5) we consider Q = Q1 = 2i1 ⊗ i1 − I ,

det Q1 = 1 .

Here I is the identity tensor of the three-dimensional Euclidean space. From Eqs (20.9)–(20.12) we obtain QT1 · E · Q1 = E ,

QT1 · L · Q1 = L .

(20.13)

Equations (20.5) and (20.13) yield the relations P · Q1 = Q1 · P ,

K · Q1 = Q1 · K ,

which are equivalent to Pt1 = Kt1 = 0 ,

t = 1, 2, 3 .

(20.14)

Further, on the basis of Eqs (20.9)–(20.12) we have QT2 · E · Q2 = E ,

QT2 · L · Q2 = −L ,

Q2 = I − 2i2 ⊗ i2 ,

det Q2 = −1 , (20.15)

QT3 · E · Q3 = E ,

QT3 · L · Q3 = −L ,

Q3 = I − 2i3 ⊗ i3 ,

det Q3 = −1 .

Assuming Q = Q2 and Q = Q3 in Eq. (20.5), with the help of (20.15) in addition to (20.14) we obtain the relations P23 = P32 = 0 ,

K22 = K33 = 0 .

(20.16)


283

From Eqs (20.3), (20.8)–(20.12), (20.14), (20.15) it follows that for the uniform plate the following formulas are valid: D = D22 i2 ⊗ eΦ + D33 i3 ⊗ i3 ,

G = G23 i2 ⊗ i3 + G 32 i3 ⊗ eΦ ,

(20.17)

and the tensor components in (20.17) are constant. Considering that the tensor FT · D is symmetric in this case, we see that the moment equilibrium equation (20.2)2 with l = 0 is identically satisfied for any values of R0 , κ, α. Since Eq. (20.7) leads to ∂eΦ = −κeR , ∂x2 the first equilibrium equation of (20.2) with f = 0 results in D22 = 0 and is satisfied by choosing the constant R0 . Hence, it is proved that deformations characterized by the relations (20.7), (20.9)–(20.12) satisfy the equilibrium equations of any homogeneous isotropic plate without surface loads.

20.2.2 Straightening of a Circular Cylindrical Shell Sector We introduce the circular cylindrical coordinates r, ϕ, z of a point on the surface σ by the relations x1 = r cos ϕ , x2 = r sin ϕ , x3 = z . Let us study the circular cylindrical shell of radius r0 , and let ϕ, z be Gaussian coordinates q1 , q2 . Straightening (unbending) and stretching of the shell is set by the following transformation: X1 = A ,

X2 = ηϕ ,

X3 = αz ,

q1 = ϕ ,

q2 = z ,

(20.18)

where A, η, α are constants. Introducing the unit vectors er = i1 cos ϕ + i2 sin ϕ ,

eϕ = −i1 sin ϕ + i2 cos ϕ ,

with the help of Eqs (20.18) we obtain F=

η eϕ ⊗ i2 + αi3 ⊗ i3 . r0

(20.19)

In addition to (20.19) we introduce the following subfamilies for the microrotations: Subfamily 2a: H = er ⊗ i1 + eϕ ⊗ i2 + i3 ⊗ i3 , η 1 E = eϕ ⊗ eϕ + αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0

(20.20)

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Subfamily 2b: H = −er ⊗ i1 + eϕ ⊗ i2 − i3 ⊗ i3 , η 1 E = eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0

(20.21)

H = er ⊗ i1 − eϕ ⊗ i2 − i3 ⊗ i3 , η 1 E = − eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0

(20.22)

H = −er ⊗ i1 − eϕ ⊗ i2 + i3 ⊗ i3 , η 1 E = − eϕ ⊗ eϕ + αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0

(20.23)

Subfamily 2c:

Subfamily 2d:

As in Subsect. 20.2.1, one proves that for the uniform isotropic shell under deformations (20.18), (20.20)–(20.23) the surface stress resultant and stress couple tensors are D = D22 eϕ ⊗ i2 + D33 i3 ⊗ i3 ,

G = G23 eϕ ⊗ i3 + G32 i3 ⊗ i2 ,

(20.24)

where D22 , D33 ,G 23 ,G 32 are constants. With Eqs (20.24) and f = l = 0 both equilibrium equations (20.2) are satisfied identically for any A, η, α.

20.2.3 Bending or Eversion of a Cylindrical Shell Let us study the following family of deformations of a circular cylindrical shell with initial radius r0 : R = R0 ,

Φ = κϕ ,

Z = αz ;

q1 = ϕ ,

q2 = z ,

(20.25)

where R0 , κ, α are constants. Under such deformation a sector of the undeformed circular cylindrical shell becomes a sector of the cylindrical shell of radius R0 . Besides, a stretching or a compression along the cylinder axis occurs. The cases κ > 0, α < 0 or κ < 0, α > 0 correspond to eversion of the cylindrical shell sector (or the closed cylinder). The distorsion tensor takes the form F=

κR0 eϕ ⊗ eΦ + αi3 ⊗ i3 . r0

(20.26)

We introduce the following subfamilies of H which differ from each other again by rotations about various axes.


285

Subfamily 3a: H = er ⊗ eR + eϕ ⊗ eΦ + i3 ⊗ i3 , κR0 κ−1 E= eϕ ⊗ eϕ + αi3 ⊗ i3 , L = eϕ ⊗ i3 . r0 r0

(20.27)

H = −er ⊗ eR + eϕ ⊗ eΦ − i3 ⊗ i3 , κR0 1+κ E= eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0

(20.28)

H = er ⊗ eR − eϕ ⊗ eΦ − i3 ⊗ i3 , κR0 1+κ E=− eϕ ⊗ eϕ − αi3 ⊗ i3 , L = − eϕ ⊗ i3 . r0 r0

(20.29)

Subfamily 3b:

Subfamily 3c:

Subfamily 3d: H = −er ⊗ eR − eϕ ⊗ eΦ + i3 ⊗ i3 , κR0 κ−1 E=− eϕ ⊗ eϕ + αi3 ⊗ i3 , L = eϕ ⊗ i3 . r0 r0

(20.30)

Applying Eq. (20.5), one proves that in the case of homogeneous isotropic cylindrical shell under deformations (20.25), (20.27)–(20.30) the surface stress resultant and stress couple tensors have the following expressions with constant coefficients: D = D22 eϕ ⊗ eΦ + D33 i3 ⊗ i3 ,

G = G23 eϕ ⊗ i3 + G32 i3 ⊗ eΦ ,

(20.31)

According to (20.25), (20.26) and (20.31) we have div D = −

κ D22 eR , r0

divG = 0 ,

(FT · D)× = 0.

(20.32)

From Eqs (20.32) it follows that the moment equilibrium equation (20.2)2 with l = 0 is satisfied for any values of constants R0 , κ, α, and the force equilibrium equation with f = 0 leads to D22 = 0 and can be used to determine the constant R0 .

20.2.4 Straightening and Bending of a Cylindrical Shell Let us discuss a circular cylindrical shell which deformation is given by the relations R = R0 ,

Φ = sz ,

Z = tϕ ,

(20.33)

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L.M. Zubov

where R0 , s, t are constants. Under this deformation a cylindrical shell sector is straightened to a rectangular plate, which is bent afterwards into another cylinder sector with the axis orthogonal to that of the initial cylinder. The distorsion tensor corresponding to (20.33) takes the form F=

t eϕ ⊗ i3 + sR0 i3 ⊗ eΦ . r0

(20.34)

Consistent with (20.34) the fields H consist of the following subfamilies: Subfamily 4a: H = er ⊗ eR + eϕ ⊗ i3 − i3 ⊗ eΦ , t 1 E = eϕ ⊗ eϕ − sR0 i3 ⊗ i3 , L = si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0

(20.35)

Subfamily 4b: H = er ⊗ eR − eϕ ⊗ i3 + i3 ⊗ eΦ , t 1 E = − eϕ ⊗ eϕ + sR0 i3 ⊗ i3 , L = −si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0

(20.36)

Subfamily 4c: H = −(er ⊗ eR + eϕ ⊗ i3 + i3 ⊗ eΦ ) , t 1 E = − eϕ ⊗ eϕ − sR0 i3 ⊗ i3 , L = −si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0

(20.37)

Subfamily 4d: H = −er ⊗ eR + eϕ ⊗ i3 + i3 ⊗ eΦ , t 1 E = eϕ ⊗ eϕ + sR0 i3 ⊗ i3 , L = si3 ⊗ eϕ − eϕ ⊗ i3 . r0 r0

(20.38)

For deformations described by (20.33) and (20.35)–(20.38), the surface stress resultant and stress couple tensors of the uniform isotropic cylindrical shell have the form (20.31), which garantees that the equilibrium equations become satisfied.

20.2.5 Radially Symmetric Deformation of a Spherical Shell Let the initial state of the shell be a spherical segment of radius r0 which, in particular, may be a closed sphere. We describe position on the surface σ by the spherical coordinates r, ϕ, θ, which are related with the Cartesian coordinates by


x1 = r cos ϕ cos θ ,

x2 = r sin ϕ cos θ ,

287

x3 = r sin θ .

Let us consider longitude q1 = ϕ (0 ϕ 2π) and the geographic % the geographic & π π 2 latitude q = θ − 2 θ 2 to be the Gaussian coordinates on σ. At poles of the sphere we have θ = ± π2 . We denote spherical coordinates of the surface point in the deformed state as R, Φ, Θ. Further we use two orthonormal frames er , eϕ , eθ and eR , eΦ , eΘ , which are related with the Cartesian base by er = (i1 cos ϕ + i2 sin ϕ) cos θ + i3 sin θ , eϕ = −i1 sin ϕ + i2 cos ϕ , eθ = (−i1 cos ϕ + i2 sin ϕ) sin θ + i3 cos θ , (20.39) eR = (i1 cos Φ + i2 sin Φ) cos Θ + i3 sin Θ , eΦ = −i1 sin Φ + i2 cos Φ , eΘ = (−i1 cos Φ + i2 sin Φ) sin Θ + i3 cos Θ . We set the deformation of the sphere using the relations R = R0 = const ,

Φ=ϕ,

Θ=θ.

(20.40)

On the basis of Eqs (20.39), (20.40) we have F=

R0 (eϕ ⊗ eϕ + eθ ⊗ eθ ) . r0

The mapping (20.40) is supplemented by two subfamilies of H. Subfamily 5a: H = er ⊗ er + eϕ ⊗ eϕ + eθ ⊗ eθ = I , R0 E = (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = 0 . r0

(20.41)

(20.42)

Subfamily 5b: H = er ⊗ er − eϕ ⊗ eϕ − eθ ⊗ eθ , R0 2 E = − (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = (eθ ⊗ eϕ − eϕ ⊗ eθ ) . r0 r0

(20.43)

20.2.6 Eversion of a Spherical Segment In this case the coordinate conversion and the tensor of distortion are the following: R = R0 ,

Φ=ϕ,

Θ = −θ ,

(20.44)

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L.M. Zubov

F=

R0 (eϕ ⊗ eΦ − eθ ⊗ eΘ ) . r0

(20.45)

Two subfamilies of microrotations are given by the following relations: Subfamily 6a: H = −er ⊗ eR + eϕ ⊗ eΦ − eθ ⊗ eΘ , R0 2 E = (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = (eθ ⊗ eϕ − eϕ ⊗ eθ ) . r0 r0

(20.46)

Subfamily 6b: H = −er ⊗ eR − eϕ ⊗ eΦ + eθ ⊗ eΘ , R0 E = − (eϕ ⊗ eϕ + eθ ⊗ eθ ) , L = 0 . r0

(20.47)

Let us assume that Q = Q4 = 2er ⊗ er − I, det Q4 = 1 in Eqs (20.5). For deformations (20.40), (20.42), (20.43) and (20.44), (20.46), (20.47) the following equalities hold true: QT4 · E · Q4 = E , QT4 · L · Q4 = L . (20.48) From Eqs (20.5) and (20.48) it follows that Q4 · P = P · Q4 ,

Q4 · K = K · Q4 ,

from which we obtain eϕ · P · er = eθ · P · er = eϕ · K · er = eθ · K · er = 0 .

(20.49)

Supposing Q = Q5 = I − 2eϕ ⊗ eϕ , Q = Q6 = I − 2eθ ⊗ eθ , det Q5 = −1, detQ6 = −1 in Eqs (20.5), on the basis of (20.42), (20.43), (20.46), and (20.47) we get P = Pϕϕ eϕ ⊗ eϕ + Pθθ eθ ⊗ eθ ,

K = Kϕθ eϕ ⊗ eθ + Kθϕ eθ ⊗ eϕ .

(20.50)

Let us consider the orthogonal tensor Q7 = er ⊗ er + eϕ ⊗ eθ − eθ ⊗ eϕ ,

detQ7 = 1 .

According to Eqs (20.42), (20.43), (20.46), and (20.47) the following equalities hold true: Q7 · P = P · Q7 , Q7 · K = K · Q7 . (20.51) Using Eqs (20.50), Eqs (20.51) yield Pϕϕ = Pθθ ,

Kθϕ = −Kϕθ .

Hence, it is proved in the case of homogeneous isotropic spherical shell undergoing deformations (20.40), (20.42), (20.43), (20.44), (20.46), and (20.47), the following relations hold true:


P = a(eϕ ⊗ eϕ + eθ ⊗ eθ ) ,

K = b(eϕ ⊗ eθ − eθ ⊗ eϕ ) ,

289

(20.52)

where a and b are constants. With the help of Eqs (20.41), (20.45), and (20.52) one can check that the moment equilibrium equation (20.2) is identically satisfied with l = 0, and the force equilibrium equation (20.2) is reduced to a = 0 for f = 0 and can be used to determine the radius R0 . One can implement the eversion not only to a segment of the spherical shell but to the closed sphere as well. For example, one can perform the eversion of the whole sphere by cutting it into two halfs, everting each of them and then sticking them together along the edges. Therefore, the obtained solutions (20.40), (20.42), (20.43), and (20.44), (20.46), and (20.47) mean, in particular, that there are four different states of the closed isotropic micropolar spherical shell in the absence of external force and moment loads. The shell maintains its spherical form in each of these states.

20.3 Uniform Deformations of an Isotropic Plate Let us assume that the surface σ is a part of a plane, i.e. we study a flat shell, which is located in the plane x1 x2 and has the normal vector n = i3 . The uniform deformation of the plate is described by the relations X s = aαs xα + b s , α = 1, 2 ,

aαs = const ,

s = 1, 2, 3 ,

b s = const ,

H = const .

(20.53)

Obviously, the transformation (20.53) leaves the plate flat. Since under the uniform deformation due to Eqs (20.4), (20.53) we have L = 0 and E = const, the surface stress resultant tensor D and the stress couple tensor G are also constant in the homogeneous plate. So, the first vector equilibrium equation (20.2)1 with f = 0 is identically satisfied. Assuming Q = −I in the condition of isotropy (20.5), we get K(E, 0) = 0. Hence, G = 0 under the uniform deformation. The moment equilibrium equation (20.2)2 with l = 0 is transformed into (FT · D)× = 0 .

(20.54)

Unlike the force equilibrium equation, the moment one (20.54) is not identically satisfied under the uniform deformation but is used as the equation determining the constant field of proper orthogonal microrotation tensor H for the given distorsion tensor F. One can find the solution of nonlinear equation (20.54) if and only if a certain constitutive equation of the elastic shell is known, i.e. if the relationship P(E, 0) is specified, in general. Nevertheless, as is shown below, there is a set of solutions in

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the class of isotropic plates, which do not depend on this constitutive equation. Such solutions one can call universal for the isotropic plates. Theorem 20.1. In the case of isotropic plate, Eq. (20.54) is satisfied under such uniform deformatons, for which the surface stretch tensor E is symmetric. Proof. With the help of Eqs (20.3) and (20.4), Eq. (20.54) is transformed into (HT · ET · P · H)× = 0 . The equation is equivalent to requiring the tensor ET · P to be symmetric. Under hypothesis of the theorem, the tensor E is symmetric and the relationship P(E) satisfies Eq. (20.5) with L = 0. Since, according to definition (20.4), the tensor E satisfies the condition i3 · E = 0, its symmetry property yields E · i3 = 0. Therefore, E is the plane symmetric tensor having the spectral decomposition E = E 1 d1 ⊗ d1 + E2 d2 ⊗ d2 ,

(20.55)

where d1 , d1 are orthonormal vectors in the surface σ. In Eq. (20.5) with L = 0, assuming consequently Q = S1 , Q = S2 , Q = S3 , where S1 = I − 2d1 ⊗ d1 , S2 = I − 2d2 ⊗ d2 , S3 = 2i3 ⊗ i3 − E, and taking into account Eq. (20.55), we have the equalities P · St = St · P , t = 1, 2, 3 . (20.56) From Eq. (20.56) follow the relations dα · P · i3 = 0 ,

dα · P · dβ = 0 ,

α, β = 1, 2,

αβ,

which mean that the tensor P has the structure P = P1 d1 ⊗ d1 + P2 d2 ⊗ d2 .

(20.57)

According to Eq. (20.57), the tensor P is symmetric and coaxial with the tensor E. Hence, the tensor ET · P is symmetric as well, Q. E. D. Taking into consideration definition (20.4) of the surface stretch tensor E, the theorem just proved means that the moment equilibrium condition with l = 0 will be satisfied, if the microrotation tensor H satisfies the equation F · HT = H · FT .

(20.58)

Since we are interested only in proper orthogonal solutions of Eq. (20.58), one should add to it the following relations: H · HT = I ,

det H = 1 .

(20.59)

The shell distorsion tensor F, by definition, has the properties n·F = 0 ,

a·F 0

∀a : a· n = 0 .

(20.60)


291

It results from Eq. (20.60) that the symmetric tensor F · FT has the following spectral decomposition: F · FT = U 12 e1 ⊗ e1 + U22 e2 ⊗ e2 ,

U1 > 0, U2 > 0 ,

(20.61)

where e1 and e2 are the orthogonal unit eigenvectors located within the plane σ. The third eigenvector, relevant to zero eigenvalue of the tensor F · FT , is the normal vector n = i3 . The system of Eqs (20.58), (20.59) with the conditions (20.61) was studied earlier [7] under completely different circumstances. The solution found in [7] is nonunique and for U1 U 2 contains the following four branches: H = ±(U1−1 e1 ⊗ e1 + U2−1e2 ⊗ e2 ) · F + e3 ⊗ e3 , H = ±(U1−1 e1 ⊗ e1 − U2−1e2 ⊗ e2 ) · F − e3 ⊗ e3 , e3 = e1 × e2 ,

(20.62)

e3 = (U 1−1 e1 · F) × (U 2−1e2 · F) .

One can transform the expressions (20.62) to the explicit form % & H = J −1 F∗T ± (2J + I1 )−1/2 (I1 + J)F − F · FT · F ,

(20.63)

1 I2 = tr 2 (F · FT ) − tr (F · FT )2 , 2 & 1 % 2 ∗ 2 F = F − Ftr F + I tr F − tr F2 . 2 ∗ Here F is the tensor associated with F, [8]. Let us study the nonsingular tensor C = F + n ⊗ N in the three-dimensional Euclidian space, where N is the unit vector normal to the surface Σ of the deformed shell, and write down the polar decomposition of the tensor C, C = (F · FT )1/2 + n ⊗ n · A . (20.64) J = ± I2 ,

I1 = tr (F · FT ) ,

The proper orthogonal tensor A in Eq. (20.64) is defined by the distorsion tensor and is called the macrorotation tensor of the shell. The explicit expression is known [8] for this tensor,

% &−1/2 I2 A = 2 I2 + I1 (I1 + I2 )F − F · FT · F + F∗T .

(20.65)

Comparing Eq. (20.63) with (20.65) we see that one of solutions of the system (20.58), (20.59) coincides with the macrorotation tensor A. In the degenerate case U 1 = U 2 the solution set of the system (20.58), (20.59) becomes infinite. It is described by the formulas (20.62), where e1 , e2 are any two orthogonal unit vectors in the plane σ. If σ remains undeformed, i.e. translations Δ

are equal zero, X s = x s , we have F = g, where g = I − n ⊗ n is the first fundamental

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tensor of σ. In this case the set of rotations described by formulas (20.62) consists of the identical transformation, rotations about the normal n and about any axis orthogonal to n. These microrotations, according to (20.4), cause nontrivial values of deformation tensor E − g of the plate and, consequently, invoke stresses. Thus, the model of elastic plate of Cosserat type allows the existence of a nontrivial stress field without the metric deformation.

20.4 Equilibrium of an Isotropic Cosserat Membrane If the strain energy density W does not depend on the wryness tensor L, a micropolar shell is called a Cosserat membrane [3]. In this case, according to (20.3), K=G=0,

P = P(E) =

dW(E) . dE

(20.66)

The equilibrium equations (20.2) in the case of membrane without surface loads transform to div D = 0 ,

(20.67)

(FT · D)× = 0 .

(20.68)

We call the membrane the isotropic Cosserat membrane, if for any orthogonal tensor Q satisfying (20.6), the following condition holds true: P(QT · E · Q) = QT · P(E) · Q .

(20.69)

Bellow we study one simple solution of the equilibrium equations (20.67), (20.68) for the isotropic elastic membrane. Let us assume that particles of the membrane do not undergo translations, R = r, and the field of microrotations is set by the relation H = 2n ⊗ n − I ,

(20.70)

which describes the rotation about normal to the surface σ. The distorsion tensor and the surface stretch tensor in this case are F=g,

E = −g .

(20.71)

Assuming Q = 2n ⊗ n − I in Eq. (20.69), we get P · n = 0. Further, in Eq. (20.69) we first substitute Q = j1 ⊗ j2 − j2 ⊗ j1 + n ⊗ n , and then Q = j1 ⊗ j2 + j2 ⊗ j1 + n ⊗ n , where j1 , j2 is any orthonormal frame in the tangent plane to the surface σ. It leads to the relations


j1 · P · j2 = j2 · P · j1 = 0 ,

293

j1 · P · j1 = j2 · P · j2 .

Hence it follows that P = αg with α = const due to homogeneity of the membrane. As far as according to Eqs (20.3) and (20.71), D = −αg, FT · D = −αg, the moment equilibrium equation (20.68) is fulfilled and the force equilibrium equation (20.67) transforms to divg = 0 or (tr b)n = 0 .

(20.72)

Δ

Here b = −gradn is the second fundamental tensor [8] of the surface σ. Evidently, Eq. (20.72) is fulfilled if and only if the surface σ has the mean curvature equal zero. Such surfaces are called minimal surfaces and can be realized experimentally with the help of the soap film pulled on a wire frame. So, for the isotropic Cosserat membrane in the form of minimal surface, a nontrivial equilibrium stressed state is found, which exists without any metric deformation.

20.5 Conclusion The explicit solutions of nonlinear equilibrium equations of isotropic micropolar shells found in this work represent families of large nonuniform deformations, which are defined with the help of some parameters. These solutions are analogous to universal deformations of isotropic incompressible three-dimensional micropolar bodies [9]. The solutions obtained here differ from universal deformations of elastic shells described in [3], particularly in the fact that they are realized without action of surface moments for any values of parameters mentioned above. This property of the constructed universal solutions makes them easy-to-use for experimental finding of constitutive equations of the micropolar shells. Let us note that determination of uniform deformations of the Cosserat-type shells studied in this paper is not a trivial problem, because this leads to determination of uniform microrotation field from the nonlinear moment equilibrium equations. The equilibrium state of Cosserat membrane having the minimal surface shape just illustrates some peculiarities of the micropolar shell mathematical model. Acknowledgements This work was supported by the Russian Foundation for Basic Research (grant No 09-01-00459) and the Federal target programme “Research and Pedagogical Cadre for Innovative Russia” for 2009–2013 years (state contract No P-596).

References 1. Altenbach, H., Zhilin, P.A.: General theory of elastic simple shells (in Russian). Uspekhi Mekhaniki (Advances of Mechanics) 11 (4), 107–148 (1988) 2. Chrósćielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i dynamika powłok wielopłatowych: Nieliniowa teoria i metoda elementów skończonych. Wydawnictwo IPPT PAN, Warszawa (2004)

294

L.M. Zubov

3. Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moscow (2008) 4. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells (2nd ed.). Cambridge University Press, Cambridge (1998) 5. Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solid. Structures 12(9–10), 635–648 (1976) 6. Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997) 7. Zubov, L.M.: The representation of the displacement gradient of isotropic elastic body in terms of the Piola stress tensor. J. Appl. Math. Mech. 40(6), 1012–1019 (1976) 8. Zubov, L.M.: Methods of Nonlinear Elasticity in Shell Theory (in Russian). Izdatelstvo Rostovskogo Universiteta, Rostov-on-Don (1982) 9. Zubov, L.M.: Universal solutions for isotropic incompressible micropolar solids. Doklady Physics 55 (11), 551–555 (2010)

Part IV

Numerical Analysis

Chapter 21

Application of Genetic Algorithms to the Shape Optimization of the Nonlinearly Elastic Corrugated Membranes Mikhail Karyakin and Taisiya Sigaeva

Abstract Corrugated membranes are extremely important structural parts of a great number of devices, highly sensitive pressure sensors in particular. In engineering of corrugated shapes different factors could be taken into account as crucial. Among them – membrane’s work without buckling, buckling for the prescribed load, flatness of the membrane characteristic – dependence of the applied pressure on the liquid or gas volume, a lack of the plastic deformation and so on. In this paper an approach to different problems of optimization and design using genetic algorithm is proposed. To model elastic behavior of the membrane the nonlinear equations based on the Kirchhoff-Love hypothesis and not imposed constrains on the shallowness of the membrane’s shape are used. Several multi-parameter families of the axially symmetric ”design”, i.e. functions of the shape that describe the dependence of the middle surface prominence on the membrane radius, are considered. These designs were obtained due to special modifications of spherical dome, conical membrane and their integration. The value of the linear section length of the membrane characteristic served as an optimality criterion. The paper presents modified genetic algorithm, describes refinement mechanism of its settings and observes examples of the designs, which are optimal in the specified shape classes. Keywords Genetic algorithm theory · Shape optimization

·

Corrugated membrane

·

Nonlinear shell

M. Karyakin (B) Southern Federal University, Milchakova str., 8a, 344090, Rostov-on-Don, Russia & Southern Mathematical Institute, Vladikavkaz, Russia e-mail: [email protected] T. Sigaeva Southern Federal University, Milchakova str., 8a, 344090, Rostov-on-Don, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 21, © Springer-Verlag Berlin Heidelberg 2011

297

298

M. Karyakin and T. Sigaeva

21.1 Introduction Membrane structures are among the most common and the most efficient structural elements in nature and technology due to such properties as high resistance, minimum material, large spans, shelter function, maximum load carrying capacity, specific frequency response. The amount of deflection for a given pressure difference depends on the membrane shape (e.g. diameter and thickness), the nature of the support (e.g. strongly fixed, weakly pinned), and the membrane material properties (e.g. elastic modulus, Poisson ratio, density). One of the key problems of the membrane study is closely connected with shape optimization and deserves great attention, as it plays with the interaction between shape and structural response in order to achieve certain objectives fulfilling some constrains. It also has a wide range of application in automotive and aircraft industry, civil engineering and etc. Shape optimization problems are usually solved numerically, by using iterative methods [4]. That is, one starts with an initial guess for a shape, and then gradually evolves it, until it morphs into the optimal shape. A number of prominent specialists have been applying this technique to the shells and membranes. In [8] the optimal damping set of a two-dimensional membrane is estimated using an iterative shape optimization scheme based on level set methods and the gradient descent algorithm. Another strategy in the optimization process, based on discretization and linearization techniques, is introduced by reliable finite element formulation and used in [3] and [6] for the membrane form-finding. A more recent alternative in dealing with the shape optimization problems is the genetic algorithm (GA), developed by Holland in 1975 [7], a programming technique that mimics biological evolution as a problem-solving strategy. In [11], for example, author uses it for optimization of membrane separation modules, while in [2] it is applicated to problems of designing axisymmetric shells of minimal weight. This paper discusses implementation GA for the shape optimization of the nonlinearly elastic corrugated membranes and includes four sections. In Sect. 21.2 the theoretical approach to the membrane modeling is described. Next, modified GA can be seen in Sect. 21.3 and results of testing its effectiveness are given in Sect. 21.4. Finally, in Sect. 21.5 some conclusion are proposed.

21.2 Membrane Modeling Let the surface profile of a membrane with the thickness h and radius ρ h be the function given by z = f (r) in the cylindrical coordinate system or by points ri , zi (i = 1..N). These points can be determined, for instance, from the optical or mechanical profilometer experiments. If the profile is denoted by the points, then we can use spline approximation to retrieve it. Principal surface curvatures along the meridian and parallels respectively are defined by

21 Application of GA to the Optimization of the Corrugated Membranes

k1 = −

f (1 + f 2)

3 2

,

k2 = −

f r(1 + f 2 )

1 2

,

f =

299

∂f . ∂r

(21.1)

Nonlinear behavior of these membranes can adequately be represented by twodimensional nonlinear equations based on the Kirchhoff hypotheses [10]: 1. Each line which is initially perpendicular to the middle surface remains straight and normal to the deformed middle surface. 2. The stresses σα3 are negligibly small and are taken to be zero in the generalized Hooke‘s law, which in turn describes the relationship between the stresses T αβ and moments Mαβ with the infinitesimal strain tensor components #αβ and principal curvature tensor component καβ : T 11 = B(#11 + ν#22 ),

T 22 = B(#22 + ν#11 ),

T 12 = B(1 − ν)#12,

M11 = D(κ11 + νκ22 ),

M22 = B(κ22 + νκ11 ), M12 = B(1 − ν)κ12, 1 2 1 1 #11 = e11 + θ1 , #22 = e22 + θ22 , #12 = e12 + θ1 θ2 , 2 2 2 ∂u1 1 ∂u2 u1 e11 = C + k1 u3 , e22 = + C + k2 u3 , ∂r r ∂φ r 1 ∂u1 ∂ u2 ∂θ1 ∂ θ2 1 ∂θ1 2e12 = + rC ( ), κ11 = C , 2κ12 = rC ( ) + , r ∂φ ∂r r ∂r ∂r r r ∂φ 1 ∂θ2 C ∂u3 1 ∂u3 κ22 = + θ1 , θ1 = −C + k1 u1 , θ2 = − + k 2 u2 , r ∂φ r ∂r r ∂φ 1 Eh Eh3 C = C(r) , B , D= , (21.2) 1 2 1−ν 12(1 − ν2) (1 + f 2 ) 2 where u1 , u2 , u3 are the displacements of the membrane middle surface point along the meridian, parallel and normal respectively, while θ1 , θ2 represent angles of the normal rotation, E is the Young modulus and ν denotes the Poisson ratio; cylindrical c.s. is used. In this special case the nonlinear equilibrium equations are expressed as [5] ∂(rT 11 ) ∂T 12 S1 sin(θ1 ) − + rk1 Q1 − − rp = 0, ∂r ∂φ C C ∂(rT 12 ) ∂T 22 S2 sin(θ2 ) + + rk2 Q2 − − rp = 0, ∂r ∂φ C C ∂(rCQ1 − rS 1 ) ∂Q2 r 1 ∂S 2 cos(θ1 ) + − (k1 T 11 + k2 T 22 − − rp = 0. (21.3) ∂r ∂φ C C ∂φ C Here Q1 =

1 ∂(rM11 ) M22 1 ∂(M12 ) 1 ∂(rM12 ) M12 1 ∂(M22 ) − + , Q2 = + + , r ∂r r rC ∂φ r ∂r r rC ∂φ S 1 = T 11 θ1 + T 12 θ2 , S 2 = T 12 θ1 + T 22 θ2 ,

where Q1 , Q2 are transverse loads.

300


Boundary conditions on the membrane contour at x = 0 for the case of rigid clamping are u1 = 0, u2 = u3 = 0, θ1 = 0. (21.4) Note that equations were obtained without taking into account the shell flatness hypothesis (21.2,21.3). In addition, according author’s experience it can be claimed that for the membranes with the thickness of h/ρ < 1/50 equations based on the higher order models give almost the same result but are more complicated for a solving. To reduce this problem to the differential equations system of the first order let consider an augmented eight-dimensional vector Y = (Y1 , Y2 , ..., Y8 ) with the dimensionless components u1 u2 u3 rT 11 , Y2 = , Y3 = , Y4 = θ1 , Y5 = , h h h Bh rT 12 rM11 r(CQ1 − T 11 θ1 − T 12θ2 ) Y6 = , Y7 = , Y8 = , 2 Bh Bh Bh Y1 =

(21.5)

which in turn are proportional to the membrane middle surface displacements along the meridian, parallel and normal, angle of the normal rotation, axial and shear forces, bending moment and transverse forces. These components are continuous functions in the points, where the curvature undergoes a jump, what has an essential role in the numerical integration. In the case of axisymmetrical deformation (u2 = 0, ∂φ Yi = 0) obtained equations can be simplified and turned into the system of ordinary differential equations of first order: dY1 ν K1 + νK2 1 1 2 = − Y1 − Y3 + Y5 − Y , dx x C Cx 2C 4 Y2 ≡ 0, dY3 K1 1 = − Y1 − Y4 , dx C C dY4 ν 12 = − Y4 + Y8 , dx x Cx dY5 (1 − ν2 )C px ν K1 = Y1 − (1 − ν2)K2 Y3 + Y4 + Y5 − Y7 , dx x C x C Y6 ≡ 0, (1 − ν2 )K22 x K1 + ν2 K22 dY7 px 2 px = (1 − ν2 )K2 Y1 + Y3 + Y5 − Y + , dx C C 2C 4 C dY8 (1 − ν2 )C 1 ν 1 = Y4 + Y7 − − Y8 + Y4 Y5 . dx 12x C x C

(21.6)

Here Ki = ki h are dimensionless curvatures. Boundary conditions at point x1 = ρ/h for rigid fixing boundary are Y1 (x1 ) = 0,

Y3 (x1 ) = 0,

Y4 (x1 ) = 0.

(21.7)


301

For the numerical integration of this system let take into account the commonly used method, in which boundedness of solution at x = 0 is replaced by boundary conditions at x = x0 (x0 ≤ 1): Y5 (x0 ) = 0,

Y7 (x0 ) = 0,

Y8 (x0 ) = 0,

(21.8)

what is quite natural, since these continuous in the neighborhood of zero components are vanished at x = 0. Numerical calculating for this nonlinear two-point boundary problem (21.6, 21.7, 21.8) can effectively be done with the shooting method, in which loading parameter is a value of vertical deflection Y3 at the point x = x0 , while the set of other parameters consists of Vi , (i = 1, 2, 3): Y1 (x0 ) = V1 ,

p = V2 ,

Y4 (x0 ) = V3 ,

(21.9)

that can be determined after solving Cauchy problem using boundary conditions. This problem has the stable and durable solution, which can be found in an acceptable time even for thin shells (h/ρ < 1/300).

21.3 Genetic Algorithm The genetic algorithm [1] has been inspired by evolutionary biology and incorporates techniques such as inheritance, mutation, inverse, selection and crossover to find a better solution. In this part the scheme of the genetic algorithm processing for the case of three parameters optimization is presented. A potential solution is introduced by a “chromosome” including three “genes” or coded parameters (Fig. 21.1a). Eight bits are used to represent the type value of each gene using the reflected binary coding, also known as Gray code, a binary numerical system where two successive values differ in only one bit [9]. GA follows the basic workflow: • Initialization. By selecting N random points from the search space and converting them to the chromosomes, we get initial population of N individuals. • Crossover. At first, this operator selects two parent’s individuals and swaps their bits randomly with the small probability pc1 (Fig 21.1b). Then with the probability pc2 it is determined whether or not they are crossed using the three-stepwise crossover (the single-point crossover for each of the genes) or just copied without any modifications to the next generation. Three-stepwise crossover means the exchanging genetic material between individuals by copying parts of the genes from one parent to another in order to produce children (the single-point crossover for each of the genes) (Fig. 21.1c). After this, the population of parents is extended by children individuals.

302


• Mutation. This operator serves to introduce diversity into the children population and to look for a better solution in yet unexplored areas. An occasional bit in each gene of the chromosome has the probability pm to get inverted (Fig. 21.1d). • Inverse. The next step for the modification of the new generation is inverse, what means the exchanging parts of a chromosome before and after randomized bit with the probability pi (Fig. 21.1e). As well as mutation, this operator extends the search on the new areas of the search place. • Selection. In this part it is necessary to guarantee that the best individuals survive from generation to generation. This technique is called elitism. The parent’s and children’s generations are sorted by the fitness function. Then by the extending the selection zone from the beginning of sorted population we choose occasional individuals and if they have no analogues – we add them to the new population of N individuals. Therefore the fitter individuals have greater chance to proceed and cross, while the fact that population does not become occupied by one individual ensure the maintaining of the diversity. Finally we replace the combined population with the obtained one and repeat the process until the termination condition (maximum number of generations) is met. This algorithm can be called modified as it compares different techniques, such as three-stepwise single-point crossover, elitism strategy and truncation method in order to achieve certain results in an acceptable time.

Fig. 21.1 Genetic algorithm: (a) chromosome, (b)–(c) crossover, (d) binary mutation, (e) inverse


303

21.4 Numerical Results To demonstrate using of the proposed approach let consider two membrane form finding problems described by three parameters. For the first issue let examine membrane form of the ”double dome” type, which is sketched at the Fig. 21.2a (where for seeing imaginary the radius and height scales are different). Some of parameters describing profile and material properties for this membrane are fixed, while other ones are varying values we have to determine. This data can be clearly seen from the Fig. 21.2b. Three real-valued parameters (a, b, c) form the chromosome of 24 genes that gives appropriate accuracy. Each of the estimated parameters has its own upper and lower limits, and these intervals present the search area in the GA. This information is presented in Table 21.1. Number of chromosomes in population was variable, but we noticed that population size of 100 was good choice in all experiments. Varying of the probabilities we have obtained the optimal result using next values: pc1 = 0.3, pc2 = 0.9, pm = 0.3, pi = 0.25. Termination condition was maximum generation number, and this was set to 30. Optimization problem in this case was finding the membrane with characteristic (i.e. the dependence between the applied pressure and the change of the volume under the membrane surface) containing the highest possible flatness extensive zone. To evaluate the fitness function of each chromosome the special program modeling the compression of the membrane using information about the profile and material (a)

(b) a

c

b r

h

Fig. 21.2 Double dome: (a) 3-d membrane model, (b) membrane profile. Fixed values: E, ν, h, ρ. Parameters: a, b, c Table 21.1 Doubled dome optimization input values and results E, Pa

ν

h, mm

ρ, mm

[a1 , a2 ]

[b1 , b2 ]

[c1 , c2 ]

1.73 · 105

0.3

0.1

25

[0.04, 0.4]

[1.1, 19.1]

[0.2, 0.5]

Curve

a, mm

b, mm

c, mm

Fitness function value

1

0.1339440918

9.612588334

0.47890625

initial

2

0.0890390625

1.131196269

0.43500000

average

3

0.9143066406

5.554626929

0.41796875

best

304


parameters was developed. On the basis of the numerical solving of the boundary problem (21.6) - (21.8) this program builds the dependence of the applied pressure on the volume below the membrane and then searches the nearly linear zone with the least pitch on this ready-built curve. Nearness to the linearity means that the mean-square deflection of the function from the straight line passing through its initial and end points is not exceeding some small given value. Minimizing by GA fitness function magnitude was determined by this zone pitch-length ratio. At the Fig. 21.3a certain membranes load distribution lines (characteristics) observed in numerical experiments are shown (values of estimated parameters can be found in Table 21.1). Membrane 1 is a representative of the initial population, the second one presents typical chromosome after 10 iterations and the last one fits with the best individual from the population occurred just before meeting termination condition. Load distribution lines of the optimal membrane, plate of the same diameter and spherical dome of the same camber are compared at the Fig. 21.3b. Obtained results clearly show efficiency of the GA and possibility of the application it to the form optimization problem under given conditions. However, form implementation of the found membrane for the real devices becomes complicated due to the necessity of the preloading, since the found zone satisfying required conditions is lying in the large-scale deflections region far from the no-load state. For the next example of the using GA let consider shape optimization of the corrugated membrane already in existence (Fig. 21.4a) by fixing mechanical properties and geometric subset and varying three real-valued parameters (a, b, c) as well (Fig. 21.4b). Initial design fits with parameters values presented in the row 1 of the Table 21.2, while its characteristic is illustrated by curve 1 at the Fig. 21.5. During optimization characteristic was improved according next conditions: 1. The lead of the diagram should be nearly linear curve with the length overcoming that of the existing curve by at the least 20%. 2. Linearity of this zone must not be worse compared with existing membrane.

(a)

2

,10 Pa

(b)

2

,10 Pa

1

2 3 , mm

3

Fig. 21.3 P-V curves: (a) double dome (curves 1, 2, 3 are for an initial chromosome, for the individuals after a few iteration and for the best set of parameters respectively); (b) comparison of the spherical dome (1), plate (2) and optimal membrane’s (3) characteristics.

21 Application of GA to the Optimization of the Corrugated Membranes (a)

305

(b) a

c

b r

s h

Fig. 21.4 Truncated sinusoidal dome: (a) 3-d membrane model, (b) membrane profile. Fixed values: E, ν, h, ρ, s. Parameters: a, b, c

3. Declivity angle of this linear zone should be smaller by at the least 20% than the same angle of the membrane template. Results of the GA implementation for this problem are proposed at the Fig. 21.5 and in Table 21.2. As well as in the previous issue, curve 2 represents individual from the population of the iteration 10, while curve 3 fits with the best individual in the end of the algorithm. Optimization of the linear zone length was of about 30% (600 mm3 against 450 mm3 ); angle optimization was of about 30% (0.108 against 0.153). 2

,10 Pa 2

1 3

Fig. 21.5 P-V curves of the truncated sinusoidal domes. Curves 1, 2, 3 are for the membrane template characteristic, for the individuals after a few iteration and for the best set of parameters respectively

, mm

3

Table 21.2 Truncated sinusoidal dome optimization input values and results E, Pa

ν

h, mm

ρ, mm

s, mm

[a1 , a2 ]

[b1 , b2 ]

1.73 · 105

0.3

0.1

25

1

[2.25, 22.45] [0.2, 0.8]

Curve

a, mm

b, mm

c, mm

1

2.676803243

0.7203125000

0.09972656250

initial

2

9.745247582

0.5445312500

0.03307812500

average

3

12.45468988

0.5937500000

0.06882812250

best

[c1 , c2 ] [0.03, 0.1]

Fitness function value

306


It also must be mentioned that this research is out of the scope of the stability analysis, although this approach can be extended and for this case by means of numerical scheme proposed in [5], if the fitness function expression will be modified according some conditions, that will definitely slow the optimization program process. Stability test can be realized regardless of the GA program by analyzing only the optimal forms. Such analysis had been particularly shown that load distribution line 3 at the Fig. 21.5 has no bifurcation points. Therefore working process over the indicated load range is stable.

21.5 Conclusion Presented genetic algorithm was developed to estimate the corrugated membrane profile characteristics responding desired conditions based on the membrane compression experiment modeling results. The methodology was demonstrated with a simple model to illustrate the details of procedures involved. For the cases studied, results from simulation shows that it is possible to determine these characteristics with good accuracy. Acknowledgements This research was supported by the Federal target program “Scientific and pedagogical cadre of the innovated Russia during 2009–2013” (state contract P–361).

References 1. Back, T.: Evolutionary algorithms in heory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford Univ. Press, NY (1996) 2. Banichuk, N. V., Ivanova, S. Yu., Makeev, E. V.: Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm. Mech. Solids. 42(6), 956–964 (2007) 3. Bletzinger, K.-U.: Form Finding and Optimization of Membranes and Minimal Surfaces. Tech. Univ. of Denmark, Lyngby, (1998) 4. Delfour, M.C., Zolesio, J.P.: Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM, Philadelphia (2001) 5. Getman, I. P., Karyakin, M. I., Ustinov, U. A.: Nonlinear behavior analysis of circular membranes with arbitrary radius profile. J. Appl. Math. Mech. 74(6), 917–927 (2010) 6. Gosling, P. D., Lewis, W. J.: Form-finding of prestressed membranes using a curved quadrilateral finite element for surface definition. Comput.Structs. 61(5), 871–883 (1996) 7. Holland, J. H.: Adaptation in natural and artificial systems. Univ. of Michigan Press, Ann Arbor MI (1975) 8. Lassila T.: Optimal damping of a membrane and topological shape optimization. Struct. Multidisc. Optim. 38(1), 3438–3452 (2009) 9. Rowe, J., Whitley, D., Barbulescu, L., Watson, J. P.: Properties of Gray and binary representations. Evol. Comput. 12, 47–76 (2004) 10. Vorovich, I. I.: Nonlinear Theory of Shallow Shells. Springer, NY (1999) 11. Yuen, C.C., Aatmeeyata, Gupta, S.K. and Ray, A.K.: Multi-objective optimization of membrane separation modules using genetic algorithm. J. of Membrane Sci. 176, 177–196 (2000)

Chapter 22

Advances in Quadrilateral Shell Elements with Drilling Degrees of Freedom Stephan Kugler, Peter A. Fotiu and Justin Murin

Abstract A unique derivation of quadrilateral shell elements with six degrees of freedom at each node is presented. The theoretical and numerical formulation is based on the combination of a membrane element with drilling degrees of freedom and a shear deformable plate element. The predictive quality and the computational efficiency is improved by applying multifield variational principles in connection with suitable assumed strain fields. The resulting element formulation does not require any Gaussian quadrature since all parts of the stiffness matrix can be integrated analytically. Furthermore, the derivation is generalized to geometrical and physical nonlinearities according to a corotational updated Lagrangian description. Keywords Quadrilateral shell element · Multifield variational principles

22.1 Introduction The numerical analysis of shell structures requires stable and robust finite element formulations. The development of suitable procedures represents a challenging task in finite element research. The field of applications is widely spread and reaches from linear static analysis in structural engineering to nonlinear applications in the simulation of sheet metal manufacturing processes where geometrical and physical nonlinearities are coupled with instabilities and contact phenomena. S. Kugler (B) · P. A. Fotiu Department of Applied and Numerical Mechanics, University of Applied Sciences Wiener Neustadt, Wiener Neustadt, Austria e-mail: [email protected],[email protected] J. Murin Department of Mechanics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava, Slovak Republic e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 22, © Springer-Verlag Berlin Heidelberg 2011

307

308

S. Kugler et al. ˆ4 Φ zˆ

a) 4

ˆ1 Φ zˆ

Φ3zˆ

yˆ

b)

F

F

zˆ

F

ˆ2 Φ zˆ

F

Fig. 22.1 Drilling degrees of freedom: a) In-plane rotational degrees of freedom, b) Beam to shell connection

A general purpose shell element requires six nodal degrees of freedom (three translational and three rotational components) in order to model beam to shell connections correctly: Classical shell elements can be derived combining a membrane element and a plate element with altogether three displacements and only two rotational degrees of freedom with respect to axes in plane. Consequently, the rotational degree of freedom with respect to the shell’s normal, frequently referred to as drilling degree of freedom (Fig. 22.1a), is missing and the stiffness matrix of a structure as depicted in Fig. 22.1b is singular. Many different procedures have been proposed to introduce this missing drilling degree of freedom (e.g. [1, 2]). However, the most rigorous attempt is based on a functional that includes drilling degrees of freedom directly. Such a strategy has been done by Hughes and Brezzi [3], who used Reissner’s functional [4] including rotations as separate degrees of freedom. They showed mathematically that Reissner’s proposal can not be used for low order finite element interpolations since the LBB conditions as well as the constraint count condition for the mixed patch test are failed. Hence, they enriched Reissner’s functional by an additional term to ensure stability of low order finite element interpolations. It has to be noted that the motivation for this additive functional term arises from purely mathematical considerations and no physical interpretation is given. A different derivation of such functional suitable for the derivation of low order finite elements is given in [5,6]. Here, in contrast to [3], Cosserat’s micropolar theory of elasticity [7, 8] is used as a starting point where rotations of a point are included naturally from the outset. By neglecting couple stresses μi j within a weak form of this boundary value problem leads to the internal energy, 1 int s s Π = ci jkl εkl εi j dΩ + α εaij εaij dΩ, (22.1) 2 Ω

Ω

where Ω is the elemental volume, ci jkl represents an elasticity tensor and the constant α is an added material dependent parameter that relates to the asymmetric strains εaij . The symmetric and antimetric strain tensors εisj and εaij obey the following geometrical conditions ε sji =

& & 1% 1% ui, j + u j,i and εaji = ui, j − u j,i + ei jk ϕk . 2 2

(22.2)

22 Advances in Quadrilateral Shell Elements

309

In (22.2) the variable ui represents displacements in the i-th Cartesian direction, ϕi is the ith- component of the rotation and ei jk denotes the alternator tensor. As indicated, a large number of element formulations are proposed in literature with respect to the potential (22.1), e.g. [9]. Most of these procedures are optimized with respect to high coarse mesh accuracy as e.g. in Ibrahimbegovic et al. [10–12]. However, there are not many efficient element technologies which are well suited for explicit time integration (e.g. [13]) and advances in that field of application will be presented in this paper. Within our approach a new general purpose quadrilateral shell element is derived based on flat element geometries where membrane and plate bending properties are decoupled. If the element configuration is initially warped a rigid body projection [14, 15] is used to ensure the correct number of rigid body modes and to couple membrane and bending properties. Drilling degrees of freedom are introduced within the membrane part of the stiffness matrix in relation to functional (22.1). The plate bending element is formulated within the Mindlin-Reissner theory and, therefore, is valid for thin to moderately thick shells. It has to be pointed out that the herein proposed element formulation avoids any Gaussian quadrature since all parts of the stiffness matrix can be integrated analytically.

22.1.1 Element Configuration, Shape Functions and Rigid Body Projection Within an initially warped element configuration (dashed quadrilateral with gray nodes in Fig. 22.2) a local Cartesian coordinate system is embedded by simple vector calculus. Thereby, the local base vectors eˆ x and eˆ y enclose equal angles δ with the vectors g1 and g2 that connect the midpoints of opposing elemental sides.The nodal coordinates r¯ i with respect to the elemental Cartesian coordinate system are calculated with the aid of an auxiliary matrix T T T r¯ i = X¯ i Y¯ i Z¯i = T (ri − rm ) with T = eˆ x eˆ y eˆ z ,

projected plane element

g2

zˆ 4

yˆ

δ zg

r1

1

r1

xˆ

rˆ2 rm yg

xg

g1 warped configuration

2

Fig. 22.2 Warped and projected, plane element configuration

(22.3)

310

S. Kugler et al.

and the projected nodal coordinates1 rˆ i (black in Fig. 22.2) are defined by simply omitting the z-component of r¯ i . Therefore, the nodal coordinates of the projected plane element configuration are given with respect to the elemental Cartesian coor T dinate system by rˆ i = X¯ i Y¯ i 0 . In the following sections we discuss on the basis of these nodal coordinates rˆ i (plane element configuration) the independent derivation of a three parameter membrane element and a plate element, respectively. However, it can be shown that in arbitrarily warped element configurations some parasitic strain appears under pure rigid body motion and the corresponding element tends to be too stiff [6, 14]. Rankin and Nour-Omid [15] constructed a projection matrix that extracts the pure deformation from a given displacement field, i.e. the displacement components that are free from any rigid body motion. The projection matrix transforms an element that has non zero rigid body modes to one with invariance ˆ pro j is accordproperties under rigid body motion. The projected stiffness matrix K ingly given by ˆ pro j = Pˆ T K ˆˆ K (22.4) R PR , ˆ denotes the unprojected stiffness matrix derived from the plane element where K configuration and Pˆ R is the projection matrix reading −1 ˆPR = I(24×24) − R ˆ R ˆTR ˆ ˆT . R (22.5) ˆ is evaluated by In (22.5) the rigid body matrix R ⎡ ⎢⎢⎢ 1 0 ⎢⎢⎢ ⎡ ⎤ ⎢⎢⎢ 0 1 ˆ 1 ⎥⎥⎥ ⎢⎢⎢ R ⎢⎢⎢ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢ R ˆ 2 ⎥⎥⎥ ˆ = ⎢⎢⎢⎢ ⎥⎥⎥⎥ with R ˆ i = ⎢⎢⎢⎢ 0 0 R ⎢⎢⎢ 0 0 ⎢⎢⎢ R ˆ ⎥⎥ ⎢⎢⎢ ⎢⎢⎢⎣ 3 ⎥⎥⎥⎥⎦ ⎢⎢⎢ ˆ R4 ⎢⎢⎢⎢ 0 0 ⎣ 00

0 0 0 −riz 1 riy 0 1 0 0 0 0

⎤ riz −riy ⎥⎥⎥ ⎥⎥⎥ 0 rix ⎥⎥⎥⎥ ⎥⎥⎥ −rix 0 ⎥⎥⎥⎥ ⎥⎥ . 0 0 ⎥⎥⎥⎥⎥ ⎥⎥ 1 0 ⎥⎥⎥⎥⎥ ⎥⎦ 0 1

(22.6)

The assembly of the total stiffness matrix has to be carried out with respect to the global Cartesian base eg . Therefore, the projected stiffness matrix has to be transformed according to (22.3) which reads ⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ CT K C with T C(24,24) = diag ⎢⎢⎢⎢T, . . . , T⎥⎥⎥⎥ . ˆ pro j T K=T (22.7) ⎢⎣;⎥⎦ 8×

The derivation of isoparametric quadrilateral six parameter shell elements requires a suitable plane four-node element kinematic (see Fig. 22.3). The shape functions read 1

Note that throughout the following chapters a superposed hat indicates that the quantity refers to the elemental Cartesian coordinate system.


311

N = N1 N2 N3 N4 = Δ + bx xˆ + by yˆ + γ ξη, with

1 1 yˆ 24 yˆ 31 yˆ 42 yˆ 13 , by = xˆ42 xˆ13 xˆ24 xˆ31 , 2Ae 2Ae ⎡ ⎤T ⎢⎢⎢ Xˆ 2 yˆ 34 + Xˆ 3 yˆ 42 + Xˆ 4 yˆ 23 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎡ ⎤ ⎢⎢⎢ γ 0 ⎥⎥⎥ 1 ⎢⎢⎢⎢⎢ Xˆ 1 yˆ 43 + Xˆ 3 yˆ 14 + Xˆ 4 yˆ 31 ⎥⎥⎥⎥⎥ ⎥⎥⎥ γ= ⎢ ⎥ , Υ = ⎢⎢⎢⎣ 4Ae ⎢⎢⎢⎢⎢ Xˆ 1 yˆ 24 + Xˆ 2 yˆ 41 + Xˆ 4 yˆ 12 ⎥⎥⎥⎥⎥ 0 γ⎦ ⎢⎢⎣ ⎥⎥⎦ Xˆ 1 yˆ 32 + Xˆ 2 yˆ 13 + Xˆ 3 yˆ 21 ⎡ ⎛ 4 ⎞ ⎛ 4 ⎞ ⎤ ⎜⎜⎜ ⎟⎟⎟ ⎥⎥⎥ 1 ⎢⎢⎢⎢ ⎜⎜⎜⎜ ˆ ⎟⎟⎟⎟ Δ = ⎢⎢⎣t − ⎜⎜⎝ Xi ⎟⎟⎠ b x − ⎜⎜⎜⎝ Yˆ i ⎟⎟⎟⎠ by ⎥⎥⎥⎦ , with t= 1 1 1 1 , 4 bx =

i=1

(22.8) (22.9)

(22.10)

(22.11)

i=1

1 ( xˆ24 yˆ 31 + xˆ31 yˆ 42 ) . (22.12) 2 Note that all upper case letters refer to nodal variables i.e. Xˆ J and Yˆ J depict the nodal xˆ and yˆ coordinates for J = 1, 2, 3, 4. According to Fig. 22.3 the element kinematics of an isoparametric element is given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢ xˆ ⎥⎥⎥ ⎢⎢⎢ N 0 ⎥⎥⎥ ⎢⎢⎢ X ˆ ⎥⎥ T T ˆ = Xˆ 1 Xˆ 2 Xˆ 3 Xˆ 4 , Y ˆ = Yˆ 1 Yˆ 2 Yˆ 3 Yˆ 4 . ⎥⎥⎥⎦ ⎢⎢⎢⎣ ⎥⎥⎥⎥⎦ , X xˆ (ξξ ) = ⎢⎢⎢⎣ ⎥⎥⎥⎦ = ⎢⎢⎢⎣ ˆ yˆ 0 N Y (22.13) The Jacobian J of the elements kinematics reads ⎡ ⎤ ⎢⎢⎢ ∂ xˆ ∂ˆy ⎥⎥⎥ ∂ξ ∂ξ J = ⎢⎢⎣⎢ ∂ xˆ ∂ˆy ⎥⎥⎦⎥ . (22.14) xˆ IJ = Xˆ I − Xˆ J , yˆ IJ = Yˆ I − Yˆ J , Ae =

∂η ∂η

xˆ (ξ ) = NXˆ e

η

3

yˆ

4

3

η ξ 4

ξ xˆ

1

2

1

Fig. 22.3 Plane quadrilateral with skew angled coordinates ξ and η and node numbering

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22.2 Linear Element Formulation In this section a geometrically linear shell element with a uniform thickness h is proposed which is based on the flat element geometry discussed in Sect. 22.1.1. First we derive a three-parameter membrane element including drilling degrees of freedom (Sect. 22.2.1) and independently a plate bending element (Sect. 22.2.2). A coupling of these element properties may occur by the rigid body projection (22.4) if the element is in an arbitrarily warped configuration.

22.2.1 Membrane Element In order to introduce drilling degrees of freedom within a membrane element we refer to the lately proposed functional (22.1) (see [5, 6]) with the corresponding strain definitions (22.2). This framework can even be generalized by using Hu-Washizu multi-field variational principles, where all equations of the boundary value problem are only fulfilled in a weak sense (see e.g. [16]). By requiring certain orthogonalities between the strain displacement relations and the assumed stress fields [5, 6] an enlarged potential can be given which reads 1 s s ci jkl ε¯ kl ε¯ i j dΩ + α ε¯ aij ε¯ aij dΩ − fi ui dΩ + ti ui dΓ. (22.15) 2 Ω

Ω

Ω

Γt

In (22.15) an overbar indicates that the corresponding strain field can be assumed independently. In order to avoid any condensation procedure on element level we relate the interpolations of strain fields to nodal degrees of freedom and avoid the application of internal non-conforming degrees of freedom. Equation (22.15) has to be reduced to two-dimensional problems and the variational statement for equilibrium reads in Voigt notation, T Tˆ ˆ ˆ ˆ ˆ ˆ δε s Cε s dΩ + 2α δεa εa dΩ = δuˆ fdΩ + δuˆ T ˆtdΓ, (22.16) Ω

Ω

Ω

Γt

ˆ represents the where Γt denotes the boundary where tractions are applied and C plane stress elasticity matrix. The two dimensional strain displacement relations in the strong form are given by T & 1% εˆ s = u xˆ,xˆ uyˆ ,ˆy u x,ˆ , εˆ a = u xˆ,ˆy − uyˆ,xˆ + ϕ. ˆ ˆ y + uy, ˆ xˆ 2

(22.17)

The keystep within this derivation is to find the assumed strain interpolations εˆ s and εˆ a in two steps. Firstly, we propose them by simply using the classical bilinear shape functions for the displacements and in-plane rotations in relation to (22.17). Within the second step these corresponding results are modified for sake of accuracy and


313

efficiency. The bilinear shape functions enable the strain interpolations to be divided into a constant and a non-constant (hourglass) part denoted by 0 and H in the index and we state ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢ U xˆ ⎥⎥⎥ ˆ ⎢⎢⎢ γ 0 ⎥⎥⎥ ⎢⎢⎢ U xˆ ⎥⎥⎥ ˆ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ , εˆ s = B su0 ⎢⎢⎢⎣ ⎥⎥⎥⎦ + B suH ⎢⎢⎢⎣ (22.18) Uyˆ 0 γ ⎦ ⎣ Uyˆ ⎦ ⎛ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ ⎜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎟ ˆεa = 1 ⎜⎜⎜⎜⎜Bˆ au0 ⎢⎢⎢⎢⎢ U xˆ ⎥⎥⎥⎥⎥ + Bˆ auH ⎢⎢⎢⎢⎢ γ 0 ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢ U xˆ ⎥⎥⎥⎥⎥⎟⎟⎟⎟⎟ + Nϕ Φ ˆ, (22.19) ⎝ ⎣ ⎦ ⎣ 2 Uyˆ 0 γ ⎦ ⎣ Uyˆ ⎦⎠ T T ˆ = φ1 φ2 φ3 φ4 . The correspondwith Ui = Ui1 Ui2 U i3 U i4 for i = xˆ, yˆ and Φ ing B-matrices and the shape function for the drilling degree freedom are given by ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ b x 0 ⎥⎥⎥ ⎢⎢⎢ (ξη), xˆ −e (ξη),ˆy ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ Bˆ su0 = j0 / j ⎢⎢⎢⎢ 0 by ⎥⎥⎥⎥ , Bˆ suH = j/ j0 ⎢⎢⎢⎢ −e (ξη), xˆ (ξη),ˆy ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ by b x 0 0 ˆ Bˆ au0 = j0 / j by −b x , B auH = κ j/ j0 (ξη),ˆy − (ξη), xˆ , Nϕ = j0 / j N1 N2 N3 N4 .

(22.20)

(22.21) (22.22)

√ √ Introducing the prefactors j/ j0 and j0/ j ( j = det J and j0 = det J(ξ = η = 0), see (22.14)) renders all integrands of the stiffness matrix to polynomials which can be evaluated analytically. Since numerical integration requires at least four Gauß points, this procedure is roughly four times faster compared to classical proposals. Further, two additional scalar constants e and κ are introduced within the nonconstant B-matrices to improve the accuracy of the proposed element. Following closely the derivations of [5, 6] those constants can be identified according to the analytical representation of a higher order patch test and read e = ν , κ = 2.

(22.23)

Remark 22.1. This procedure is strongly related to selective integration where shear terms are underintegrated while the remaining terms are fully integrated. However, those procedures have at least to the author’s knowledge never been discussed for elements with drilling degrees of freedom. Consequently, the stiffness matrix of a three parameter membrane element reads ⎡ ˆ ⎢⎢⎢ K ˆ K = ⎢⎢⎢⎢⎣ 11 ˆ K 21 with

⎤ ˆ T ⎥⎥⎥ K 21 ⎥⎥ , ˆ ⎥⎥⎦ K

(22.24)

22

ˆ ˆ ˆ ˆ ˆ K11 = K suu0 + KsuuH + Kauu0 + KauuH ,

(22.25)

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ˆ =K ˆ ˆ ˆ ˆ K 21 aϕu0 + KaϕuH and K22 = Kaϕϕ ,

(22.26)

where s and a indicate that the correspondence with the symmetric and the antimetric strain fields. The analytically integrated parts of the stiffness matrix can be given explicitly, ⎡ ⎤T ⎡ ⎤ ⎢⎢⎢⎢ b x 0 ⎥⎥⎥⎥ ⎢⎢⎢⎢ b x 0 ⎥⎥⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢⎢ ⎥ ˆ ˆ ⎢⎢ 0 by ⎥⎥⎥⎥ , K suu0 = 4h j0 ⎢⎢⎢⎢ 0 by ⎥⎥⎥⎥ C (22.27) ⎢ ⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎦ by b x by b x ⎡ ⎤ % & ⎢ yˆ 213 + yˆ 224 ν ( xˆ13 yˆ 13 + xˆ24 yˆ 24 ) ⎥⎥⎥⎥ ˆ T Eh ⎢⎢⎢⎢ % & ⎥⎦⎥ Υ , KsuuH = Υ (22.28) ⎢ 6 j0 ⎣ ν ( xˆ13 yˆ 13 + xˆ24 yˆ 24 ) xˆ213 + xˆ224 T ˆ K (22.29) by −bx , auu0 = 2αh j0 by −b x ⎡ % ⎤ & ( xˆ13 yˆ 13 + xˆ24 yˆ 24 ) ⎥⎥⎥⎥ xˆ213 + xˆ224 αh ⎢⎢⎢⎢ ˆ % & ⎥⎥⎦ Υ , ⎢⎢⎣ KauuH = Υ T (22.30) 3 j0 ( xˆ13 yˆ 13 + xˆ24yˆ 24 ) yˆ 213 + yˆ 224 ⎡ ⎤ ⎢⎢⎢ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ αh ⎢⎢⎢⎢⎢ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎥⎥⎥⎥⎥ ˆ Kaϕu0 = (22.31) ⎢ ⎥, 8 ⎢⎢⎢⎢⎢ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎥⎥⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ xˆ42 xˆ13 xˆ24 xˆ31 yˆ 42 yˆ 13 yˆ 24 yˆ 31 ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ xˆ42 yˆ 42 ⎥⎥⎥ ⎢⎢⎢ 4 2 1 2 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢ xˆ yˆ ⎥⎥⎥ ⎢⎢⎢ 2 4 2 1 ⎥⎥⎥⎥⎥ αh 2αh j 31 31 0 ˆ ˆ ⎢⎢⎢ ⎥⎥⎥ Υ , K ⎢⎢⎢ ⎥⎥⎥ . KaϕuH = (22.32) aϕϕ = 3 ⎢⎢⎢⎢ xˆ24 yˆ 24 ⎥⎥⎥⎥ 9 ⎢⎢⎢⎢ 1 2 4 2 ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎣ ⎦ ⎣ ⎦ xˆ13 yˆ 13 2124

22.2.2 Plate Bending Element In this section we derive a robust and highly efficient plate bending element based on the Mindlin-Reissner theory which is valid for thin to moderately thick shells. In this derivation the bending and transverse shear energies are treated independently: While the bending performance of the element is improved according to the B-bar method [17], the transverse shear forces are interpolated separately and linked by discrete collocation constraints. The resulting formulation is similar to that of Bathe and Dvorkin [18, 19] and relies on mixed interpolations of tensorial components. The element should satisfy the following requirements: • No internal degrees of freedom should be included to achieve a computational efficient formulation. • The shear energy has to vanish in the thin plate limit.


315

The kinematic assumptions read u xˆ = zˆϕyˆ , uyˆ = −ˆzϕ xˆ , uzˆ = uzˆ ,

(22.33)

where the variable uî for î = x, ˆ yˆ , zˆ depicts the deformation in i-th direction while ϕî for î = xˆ, yˆ represents the corresponding bending angles2 . The strain definitions are given by % & # xˆ xˆ = zˆϕy,ˆ xˆ , #yˆ yˆ = −ˆzϕ xˆ,yˆ , γ xˆyˆ = zˆ ϕyˆ ,ˆy − ϕ xˆ, xˆ , γ xˆzˆ = uzˆ,xˆ + ϕyˆ , γyˆ zˆ = uzˆ,ˆy − ϕ xˆ , (22.34) % & where #î ˆj are the components of the symmetric strain tensor #î ˆj = 12 uî, ˆj + u ˆj,î and γ xˆzˆ and γyˆ zˆ are the mean shear strains according to the Mindlin theory. ˆ b denotes the plane stress elasticity tensor) We assume an isotropic material (C with the constitutive equation in bending ˆ b #ˆ b , σˆ b = C

(22.35)

T with σˆ b = σ xˆ xˆ σyˆ yˆ σ xˆyˆ ]T and #ˆ b = [ # xˆ xˆ #yˆ ˆ y # xˆ yˆ ] . The constitutive equation in shear reads 5 sˆ = Ghγˆ s , (22.36) 6 T with sˆ = [ s xˆ syˆ ]T denoting the vector of distributed shear forces (γˆ s = [ γ xˆzˆ γyˆ ˆz ] ) E and G represents the shear modulus, i.e. G = 2(1+ν) . Equilibrium requires the potential Π to be a minimum, i.e.

Πb =

1 2

Ω

Π = Πb + Π s − Πext → min, 1 σˆ Tb #ˆ b dΩ , Π s = sˆT γˆ s dA , Πext = puzˆ dA. 2 Ae

(22.37) (22.38)

Ae

Here, Ae is the area of the reference plane (mid surface) and the variable p represents the lateral load pressure acting on the plate. The functional (22.38) violates the thin plate limit (i.e. the Kirchhoff patch test) in a classical displacement based plate element. This deficiency can be resolved according to a Hu-Washizu multi field variational principle. 22.2.2.1 The Bending Part of the Functional The bending part Πb of the functional (22.37) can be improved by introducing the orthogonality % &T ˆ b dΩ = 0, #ˆ b − #ˆ b σ (22.39) Ω 2

The hat on top of the variables indicates that we refer to an elemental Cartesian base of Sect. 22.1.1 (see Fig. 22.2)

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where an overbar indicates the independent assumption of the corresponding field. The resulting potential reads T 1 Πb = #ˆ b Cb #ˆ b dΩ (22.40) 2 Ω

and gives the variational basis to assume strain fields improving the element’s performance and computational efficiency. We interpolate ⎡ ⎤ ⎡ ⎤⎡ ⎤ T ˆ ⎢⎢⎢ Φ xˆ ⎥⎥⎥ ˆ ⎢⎢⎢ γ 0 ⎥⎥⎥ ⎢⎢⎢ Φ xˆ ⎥⎥⎥ ⎥⎥⎥ + BbH ⎢⎢⎢⎣ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ , #ˆ b = # xˆ xˆ #yˆ yˆ γ xˆyˆ = Bb0 ⎢⎢⎢⎣ (22.41) Φ yˆ ⎦ 0 γ ⎦ ⎣ Φ yˆ ⎦ with 9 Bˆ b0 = zˆ

⎡ ⎤ ⎡ ⎤ 9 ⎢⎢ ν (ξη) ⎢⎢⎢ 0 bTx ⎥⎥⎥ ⎥ ,ˆy (ξη), xˆ ⎥⎥⎥ ⎢ ⎢ ⎥ ⎢ ⎥⎥⎥ ˆ ⎥ j0 ⎢⎢⎢ T j ⎢⎢⎢ ⎢⎢ −b 0 ⎥⎥⎥ , BbH = zˆ ⎢⎢ − (ξη),ˆy −ν (ξη), xˆ ⎥⎥⎥⎥⎥ , ⎥⎥⎦ ⎥⎥⎦ j ⎢⎢⎢⎣ y j0 ⎢⎢⎢⎣ −bTx bTy 0 0

(22.42)

ˆ where the modifications√of Bˆ b0 and √BbH are motivated by the considerations of Sect. 22.2.1. The prefactors j0/ j and j/ j0 again ensure that the stiffness matrix can be integrated analytically.

22.2.2.2 The Shear Part of the Functional Considering the shear part Π s of the potential (22.37), we use a modified HuWashizu formulation, where the equilibrium equations and the strain displacement relations are fulfilled in a weak sense, while the constitutive relation is satisfied in strong (differential) form. Accordingly, we obtain ˆ − 1 sˆT 6 sdA, ˆ Π s = γˆ Ts sdA (22.43) 2 5Gh A

A

where the overbar indicates an independent interpolation of the corresponding field [6]. According to the well known count condition [16] it can be shown that we have ˆ i.e. two to apply four degrees of freedom for the assumed distributed shear forces s, interpolation points for each s xˆ and syˆ , which have to be located on the elements boundary. The satisfaction of the Kirchhoff patch test can be ensured by a covariant representation of transverse shear strain, γξ3 = uzˆ,ξ + det|J| ϕη , γη3 = uzˆ,η − det|J| ϕξ ,

(22.44)


317

where the contravariant components of the rotations (ϕξ and ϕη ) are given by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢ ϕξ ⎥⎥⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢⎢⎢ ⎥⎥⎥ = J−T ⎢⎢⎢⎢⎢ ϕ xˆ ⎥⎥⎥⎥⎥ = 1 ⎢⎢⎢⎢⎢ yˆ ,η − xˆ,η ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢ ϕ xˆ ⎥⎥⎥⎥⎥ . (22.45) ⎣ ϕη ⎦ ⎣ ϕ ⎦ det |J| ⎣ −ˆy xˆ ⎦ ⎣ ϕ ⎦ yˆ ,ξ ,ξ yˆ In (22.44) γξ3 vanishes at ξ = 0 and η = ±1 while γη3 vanishes for ξ = ±1 and η = 0 if pure bending deformations are applied3. Hence, linear shape functions may be used for the interpolation, T covariant γˆ = γ γ = Ns Γ , (22.46) ξ3

s

⎡ ⎢⎢⎢ NξI 0 NξIII N s = ⎢⎢⎢⎣ 0 NηII 0 1 1 NξI = (1 − η) , NξIII = (1 + η), 2 2

η3

⎡ ⎤ ⎢⎢⎢ ΓξI ⎥⎥⎥ ⎢ ⎥⎥ ⎤ ⎢⎢⎢ ⎢⎢⎢ ΓηII ⎥⎥⎥⎥⎥ 0 ⎥⎥⎥⎥ ⎥ ⎥⎥ , Γ = ⎢⎢⎢ ⎢⎢⎢ ΓξIII ⎥⎥⎥⎥⎥ , NηIV ⎦ ⎢⎢⎢⎣ ⎥⎥⎥⎦ ΓηIV

(22.47)

1 1 NηII = (1 + ξ) , NηIV = (1 − ξ). (22.48) 2 2

covariant

Now, γˆ s is related to the nodal transverse displacements Uzˆ and the two nodal bending angles Φ xˆ and Φ yˆ by inserting the corresponding parametric location into (22.44), Γ = Auz Uzˆ + Aϕx Φ xˆ + Aϕy Φ yˆ , (22.49) where the coefficients Auz , Aϕx and Aϕy depend on the element geometry only, ⎡ ⎢⎢⎢ −1 1 0 ⎢⎢⎢ 1 ⎢⎢⎢⎢⎢ −1 0 0 Auz = ⎢⎢⎢ 2 ⎢⎢⎢ 0 0 1 ⎢⎢⎣ 0 −1 1 ⎡ ⎢⎢⎢ yˆ 12 ⎢⎢⎢ 1 ⎢⎢⎢⎢⎢ yˆ 14 Aϕx = ⎢⎢⎢ 4 ⎢⎢⎢ 0 ⎢⎢⎣ 0

⎤ 0 ⎥⎥⎥⎥ ⎥⎥ 1 ⎥⎥⎥⎥ ⎥⎥⎥ , −1 ⎥⎥⎥⎥ ⎥⎥⎦ 0

⎤ ⎡ ⎢⎢⎢ xˆ21 xˆ21 0 ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢ xˆ 0 0 0 yˆ 14 ⎥⎥⎥ ⎥⎥⎥ , Aϕy = 1 ⎢⎢⎢⎢⎢ 41 4 ⎢⎢⎢⎢ 0 0 0 yˆ 43 yˆ 43 ⎥⎥⎥⎥ ⎥⎥⎦ ⎢⎢⎣ yˆ 23 yˆ 23 0 0 xˆ32 yˆ 12 0

⎤ 0 ⎥⎥⎥⎥ ⎥⎥ 0 xˆ41 ⎥⎥⎥⎥ ⎥⎥⎥ . xˆ34 xˆ34 ⎥⎥⎥⎥ ⎥⎥⎦ xˆ32 0 0

(22.50)

According to the transformation rules between covariant and Cartesian components of a vector we finally evaluate the transverse shear strain field with respect to the elemental system ⎡ ⎤ ⎢⎢⎢ yˆ ,η −ˆy,ξ ⎥⎥⎥ covariant 1 −1 ⎢⎢⎢ ⎥⎥⎥ N s Γ . γˆ s = J γˆ s = (22.51) det|J| ⎣ − xˆ,η xˆ,ξ ⎦ 3

Here, the transverse displacements uzˆ and the bending angles ϕ xˆ and ϕyˆ are interpolated in the classical bilinear form.

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For sake of efficiency and the strong request of avoiding numerical quadrature by applying an analytical integration of transverse shear part we further modify the transformation (22.51) according to covariant ˆ covariant . γˆ s = J−1 γˆ s ≈ J−1 0 γs

(22.52)

It has to be noted that this approximation causes a violation of the constant transverse shear patch test, i.e. γ xˆzˆ = γyˆ zˆ = 1 for uzˆ = xˆ + yˆ and vanishing bending angles, in arbitrarily distorted element configurations. However, we may view the transverse shear strains as penalty functions ensuring the thin plate limit and the modifications of (22.52) will not introduce significant detoriations of performance.

22.2.2.3 The Stiffness Matrix The stiffness matrix of the quadrilateral plate element is now found from the potential T 5Gh 1 ˆT ˆ ˆ 1 Π = Πb + Πs = # b Cb # b dΩ + γˆ s γˆ dA (22.53) 2 2 6 s Ω

and its variation

δΠ b + δΠ s = Ω

T ˆ b #ˆ b dΩ + δ#ˆ b C

A

T

δγˆ s

5Gh ˆ γ dA. 6 s

(22.54)

A

Here, #ˆ b represents the assumed bending strain depending on nodal variables Φ xˆ and Φ yˆ (see (22.41) and (22.42)) while γˆ s denotes the assumed transverse shear according to (22.52). The variation of internal transverse shear energy (22.54) reads ˆ ΓT K δΠ s = δΓ sΓ ,

(22.55)

with ⎡ ⎢⎢⎢ K11 K12 K13 ⎢⎢⎢ ⎢⎢⎢ K22 K12 5Gh 5Eh ˆ = ⎢⎢⎢ −1 K NTs J−T J N dA = s s ⎢ 0 0 ⎢ 6 576 (1 + ν) j0 ⎢⎢ K11 ⎢⎢⎢ A ⎣ sym

⎤ K12 ⎥⎥⎥⎥ ⎥⎥ K24 ⎥⎥⎥⎥ ⎥⎥⎥ K12 ⎥⎥⎥⎥ ⎥⎥⎦ K22

K11 = 4 (x13 + x24 )2 + (y13 + y24)2 , K12 = 3 x224 − x213 − y213 + y224 , K13 = 2 (x13 + x24 )2 + (y13 + y24)2 , K22 = 4 (x13 − x24 )2 + (y13 − y24 )2 , K24 = 2 (x13 − x24 )2 + (y13 − y24 )2 . (22.56)


319

The variation of internal bending energy δΠ b (22.54) leads to the constant part of the bending stiffness matrix

ˆ Kb0 =

h/2

−h/2 A

⎡ ⎤T ⎡ ⎤ ⎢⎢⎢ 0 bTx ⎥⎥⎥ ⎢⎢⎢ 0 bTx ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢⎢ T ⎥⎥ h3 ⎢⎢⎢ ˆT ˆ ˆ ˆ ⎢⎢ −b 0 ⎥⎥⎥⎥ , Bb0 C Bb0 dAdˆz, = j0 ⎢⎢⎢⎢ −bTy 0 ⎥⎥⎥⎥ C y ⎢ ⎥⎥⎥ ⎥⎥⎦ ⎢⎢⎣ 3 ⎢⎢⎣ ⎦ −bTx bTy −bTx bTy

(22.57)

while the modifications of the hourglass part (22.42) allow for analytical integration since the integrand of h/2 ˆ ˆB ˆ bH dAdˆz, KbH = Bˆ TbH C (22.58) −h/2 A

is not fractional anymore, ⎡ ⎤ 2 + xˆ 2 3 ⎢⎢ ⎥⎥⎥ ( ) x ˆ −ν x ˆ y ˆ + x ˆ y ˆ Et 24 24 13 13 ⎢ ˆ 13 24 ⎥⎥⎥ Υ , ⎢⎢⎢⎣ KbH = Υ T ⎦ 72 j0 −ν ( xˆ24 yˆ 24 + xˆ13yˆ 13 ) yˆ 213 + yˆ 224

(22.59)

Hence, the total stiffness matrix of the plate element reads ⎡ ˆ ˆ ⎢⎢⎢ Kuu K uϕx ⎢⎢⎢ ˆ ˆ K = ⎢⎢⎢⎢⎢ Kϕxϕx ⎢⎢⎣ symm

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥ ϕxϕy ⎥⎥⎥ , ⎥⎥⎦ ˆ K ϕyϕy

ˆ Kuϕy ˆ K

(22.60)

with ˆ ˆ Kuu = ATuz K s Auz , and

ˆ ˆ Kuϕx = ATuz K s Aϕx ,

⎡ b ⎢ˆ ˆ = ⎢⎢⎢⎢⎢ Kϕxϕx K b ⎢⎢⎣ ˆ b Kϕyϕx ˆ ˆb ˆ Kϕxϕx = Kϕxϕx + ATϕx K s Aϕx , b ˆ ˆ ˆ A . K =K + AT K ϕyϕy

ϕyϕy

ϕy

s

ˆ ˆ Kuϕy = ATuz K s Aϕy ,

⎤ ˆ b ⎥⎥⎥ K ˆ +K ˆ ϕxϕy ⎥⎥ ⎥=K b0 bH ˆ b ⎥⎥⎦ Kϕyϕy ˆ ˆb ˆ Kϕxϕy = Kϕxϕy + ATϕx K s Aϕy ,

(22.61)

(22.62)

(22.63)

ϕy

22.3 Generalization to Geometrical Nonlinearities In this chapter we generalize the previous formulation to large deformations and large strains. Within an updated Lagrangian scheme this generalization simply amounts to an interpolation of velocities and angular velocities instead of displacements and rotations. The analytical integration of membrane and bending stiffness

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matrices requires the spin and the material tangent stiffness to be constant within an element [20]. However, this is not the case for arbitrary deformation fields. Nevertheless, these two hypotheses greatly reduce the computational effort for the calculation of the internal force vector fiint , which is related to the virtual internal power δΠ int by δΠ int = δvi fiint . A key step is the use of an elastic-plastic update only within the constant part of the symmetric membrane and bending proportions to evaluate the constant internal force vector fînt 0 discussed in Sect. 22.3.1. All remaining parts, i.e. the symmetric hourglass membrane part, the complete unsymmetric drilling rotation part, the total transverse shear part and the hourglass bending part, are assigned to the hourglass internal force vector fînt H given in Sect. 22.3.2.

22.3.1 The Constant Part of the Internal Force Vector Following the procedures of [6] the virtual internal power at the current time step tn reads4 % & n ˆ n T σˆ n dΩ, δΠint 0 = δD (22.64) 0 Ωn

ˆ n denotes the constant part of the corotational rate of deformation tenwhere D 0 sor. The corotational Cauchy stress tensor at the current time step σˆ n is evaluated according to a hypo-elastic-plastic central difference stress update. There, the constant part of the rate of deformation tensor at the time tn−1/2 = tn − 1/2Δt is calculated by ⎡ n−1/2 ⎤ ⎡ n−1/2 ⎤ ⎢⎢ ˆ ⎥⎥⎥ ⎢ˆ ⎥⎥⎥ 1 1 ⎥ n− /2 n− /2 ⎢⎢⎢ V x n−1/2 ⎢⎢⎢ ω x ⎥ ˆ ˆ ˆ D0 = B su0 ⎢⎢⎢ n−1/2 ⎥⎥⎥ + zˆBb0 ⎢⎢⎣ n−1/2 ⎥⎥⎥⎦ , (22.65) ⎣ˆ ⎦ ωˆ y Vy with

⎡ ⎤ ⎢⎢⎢ b x 0 ⎥⎥⎥

⎥⎥⎥ ⎢⎢⎢⎢ 1/2 1/2 ⎥⎥

⎢ Bˆ n− and Bˆ n− = su0 = ⎢⎢⎢ 0 by ⎥⎥⎥ b0 ⎢⎣ ⎥⎦

by bx n−1/2 t

⎡ ⎤ ⎢⎢⎢ 0 bTx ⎥⎥⎥

⎢⎢⎢ ⎥ ⎢⎢⎢ −bT 0 ⎥⎥⎥⎥⎥

. ⎢⎢⎢ y ⎥ ⎣ T T ⎥⎥⎦

−b x by n−1/2

(22.66)

t

n−1/2

n−1/2 ˆ The variables Vi and ωˆ i for i = x, y refer to the projected nodal velocities and nodal angular velocities (22.5) at tn−1/2 , respectively, while b x and by are given in (22.9). Further, we write in relation to (22.64) ⎧ h ⎫ ⎡ ⎤T ⎪ ⎪ /2 ⎪ ⎪ ⎪ ⎢⎢⎢ δV ˆ nx ⎥⎥⎥ % &T ⎪ ⎪ ⎪ ⎨ ⎬ n n n ⎢ ⎥ ˆ ˆ δΠint 0 = ⎢⎢⎣ n ⎥⎥⎦ B σ dˆ z dA+ ⎪ ⎪ su0 ⎪ ⎪ ⎪ ⎪ ˆy ⎪ ⎪ δV ⎩ ⎭ Ae

−h/2

; f

4

In this section superscripts refer to the corresponding time step.


321

⎧ h ⎫ ⎡ ⎤T ⎪ ⎪ /2 ⎪ ⎪ n ⎥⎥ % ⎪ ⎪ ⎢⎢⎢ δω & ⎪ ⎪ ˆ x ⎥⎥ n T⎨ n ⎬ ⎢ ˆ ⎢ ⎥ + ⎢⎣ n ⎥⎦ Bb0 ⎪ zˆσˆ dˆz⎪ dA, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˆy δω ⎩−h/2 ⎭ Ae ;

(22.67)

m

since the current element area is usually assumed to be constant in thickness direction. Here, the B-matrices Bˆ nsu0 and Bˆ nb0 have to be related to the current time tn . Accordingly, the constant internal force vector consisting of nodal forces (f n ) and nodal moments (mn ) is given by ⎡ ⎤ ⎢⎢⎢ f n ⎥⎥⎥ ˆn f int 0 = ⎢⎢⎢⎣ n ⎥⎥⎥⎦ , (22.68) m with

% & ˆn T f f n = Ae B su0

% &T mn = Ae Bˆ nb0 m.

and

(22.69)

The resultant forces f and the resultant moments m are evaluated using Simpson’s rule. Finally, a rigid body projection (22.4) and the coordinate transformation (22.7) has to be performed, i.e. ˆn fˆn = C TPˆ T f int 0 . (22.70) int 0

R

22.3.2 The Internal Stabilization Force Vector n The internal stabilization force vector fint H is determined by a central difference update using the analytically integrated stiffness matrices of Sect. 22.2, n−1/2 n−1/2 f˙int , H = KV

(22.71)

with a local to global coordinate transformation (22.7) in connection with a rigid body projection (22.4) ˆ K=C TT Pˆ TR KPˆ R C T. (22.72) ˆ we state For the in-plane part of K ⎡ ˆ ⎢⎢⎢ K ˆ˙f ⎢⎢⎢⎢ 11 = int H in ⎣ˆ K21 n−1/2

⎤⎡ ⎤ ˆ T ⎥⎥⎥ ⎢⎢ V ˆ n−1/2 ⎥⎥⎥ K 21 ⎥⎥ ⎢⎢⎢ ⎥⎥ 1 ⎥, ˆ ⎥⎥⎦ ⎢⎣ ω ˆ n− /2 ⎦ K 22

(22.73)

z

1/2 1/2 1/2 T ˆ n−1/2 are the nodal in-plane velocities, i.e, V ˆ n− ˆ n− where V , while ωˆ n− is V z x y 1/2 n− the drilling rotational velocity at t . The corresponding stiffness matrices read ˆ ˆ ˆ ˆ K11 = K suuH + Kauu0 + KauuH ,

(22.74)

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ˆ =K ˆ ˆ ˆ ˆ K 21 aϕu0 + KaϕuH , K22 = Kaϕϕ ,

(22.75)

which have to be found in terms of the projected nodal coordinates at tn−1/2 . The related submatrices are given in (22.28)–(22.32). ˆ For the out of plane part of K at tn−1/2 consisting of a shear and a bending part we write ⎡ ˆ ⎤⎡ ⎤ 1/2 ˆ ˆ ⎢⎢⎢ Kuu K ⎥⎥⎥ ⎢⎢ V ˆ n− ⎥⎥ K uϕx uϕy z ⎢ ⎢ ⎥⎥⎥⎥ ⎢⎢⎢ n−1/2 ⎥⎥⎥⎥ ˆ˙f n−1/2 = ⎢⎢⎢⎢⎢ ˆ ˆ (22.76) ˆ x ⎥⎥⎥⎥ , Kϕxϕx Kϕxϕy ⎥⎥⎥⎥ ⎢⎢⎢⎢ ω ⎢⎢⎢ int H Out ⎥⎦ ⎥⎥⎦ ⎢⎢⎣ n− ⎢⎣ 1/2 ⎥ ˆ symm Kϕyϕy ωˆ y where the submatrices given in (22.61)–(22.63) are evaluated at tn−1/2 . Here the ˆ ˆ bending part Kb in (22.62) only consists of the non-constant hourglass part KbH given in (22.59) because the corresponding constant part is already introduced in (22.69).

22.4 The Mass Matrix Within an efficient shell formulation the mass matrix has to be optimized for an explicit time integration. A common procedure in explicit transient dynamics is described in Hughes, Liu and Levit [21]. Their way of scaling the rotatory inertia terms will be inherited here. The scaling ensures that the critical time step is not bounded by rotary inertia terms. The consistent mass matrix is given by Mdisp = h ρNT NdA, (22.77) Ae

where N is the matrix of shape functions of the bilinear quadrilateral, ρ denotes the density of the material and h and Ae are the initial shell thickness and the initial element area, respectively. The translational masses are evaluated according to the row summing technique. The nodal rotatory inertia is computed by scaling the translational mass at the node by the factor β (see [6, 21]), i.e., * 2 + h Aelem Mrot = βMdisp , where β = max , . (22.78) 4 8

22.5 Numerical Examples In order to investigate the quality of the derivations the proposed element is tested within this section. Challenging linear and nonlinear test examples are discussed and the results in terms of the developed element technology are compared to


323

literature. More test cases can be found in Kugler [6]. Note that we usually omit units in the description of the benchmark problems, however, unit-consistency will be understood. An important issue of the proposed shell element is the proposition of suitable values for the added material constant α, that relates the skew symmetric strains to the corresponding antimetric stresses (22.1). Hughes and Brezzi [3] advise to relate this constant to the shear modulus of the material, i.e. α = μ. However, it is shown in Kugler et al. [5] that the proposed membrane element is rather insensitive to α for in-plane loadings. If a curved surface (see Fig. 22.4) is discretized with the proposed element technology, we observe that the membrane and bending stiffnesses are coupled at the inter-element boundaries. A nodal rotation that is associated to the bending strains of element two (ϕb (E2)) has a component about the axis normal to element one (E1). Therefore, the drilling rotation stiffness of E1 is multiplied by ϕ = ϕm (E1) + ϕb(E2) sin β,

(22.79)

where ϕm (E1) represents the original drilling rotation of element one (the index m indicates membrane) and β is the angle between the element normals. This observation was made by Cook [22] and was detailed in [23]. Since the bending stiffness decreases of order O(h3 ) while the membrane stiffness varies only with O(h) the large membrane stiffness of element one tends to lock the bending rotations of element two in the case of very thin shells. With increasing mesh refinement and β approaching zero this phenomenon disappears. However, when very thin smooth surfaces are discretized with coarse meshes this undesirable membrane-bending interaction influences the overall stiffness substantially. Within the present theory the coupling between drilling rotations and membrane stiffnesses can be directly manipulated by the material constant α. Consequently, we propose the usage of high values of α if the drilling stiffness is important (e.g. α = 250μ) and low values in all other cases (e.g. α = 0.005μ)5. Remark 22.2. This rarely tackled mechanism emanates directly from the six parameter formulation since only the presence of drilling stiffnesses causes that problem. Therefore, the term “membrane locking” ( [22,23]) leads to misinterpretations and,

ϕ m (E1)

ϕb (E2)sin β

ϕb (E2)

β E1

smooth curved surface

E2

Fig. 22.4 A smooth curved surface modeled with two flat shell elements 5

The drill stiffness is important under three circumstances: at beam to shell connections, if shell surfaces which has to be discretized exhibits a “true” kink or if the load is applied by concentrated couples and drill rotations are postprocessed.

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consequently, the author proposes “drill rotation locking” as a suitable term describing that mechanism.

22.5.1 Slab Supported by a Central Column As indicated, it is a main issue of this paper to derive shell elements consisting of six degrees of freedom at each node to resolve problems with beam to shell connections. To check the performance within those kind of problems a square plate 10 × 10 with a unit thickness is supported by single column of length l = 10 and a circular cross section with radius of r = 0.25. The slab is rotated by forces (load case 1) and by concentrated couples (load case two) at the vertices of the plate (see Fig. 22.5), while the beam end is fully clamped. According to St. Venant’s theory of torsion both load cases yield a tip rotation of 0.42373, where we used E = 1 · 106 for Young’s modulus and ν = 0.3 for Poisson’s ratio. Gray shaded elements denote the application of α = 250μ while for all other elements α = 0.005μ is used. The Table 22.1 Simulation results: Slab supported by a central column a) Tip rotation of the column b) Vertex rotation Vertex rotation Frey [24] Ibrahimbegovic [12] Present 0.4238

0.4237

0.4237

Forces

0.4238

Moments 0.4238

0.4237

0.4237

Moments

0.4240

Forces

results for the beam tip rotation are given in Tab. 22.1 and are compared to results from literature. Since the square plate rotates almost rigidly, we expect an almost constant drilling rotation distribution, i.e. the vertex rotation of the plate is identical compared to the beam tip rotation. All results indicate the capability of the proposed shell formulation to reproduce the exact solution in a beam to shell connection.

Fig. 22.5 Slab supported by a central column


325

22.5.2 Hemispherical Shell Within this benchmark problem a smooth hemispherical shell is loaded by opposing radial point loads alternating at 90°. Since the resulting deformation mode is mainly inextensional bending, we use this problem to evaluate the ability of the shell to handle double curved shell geometries under inextensional bending deformations correctly. Two different geometries with a thickness of h = 0.04 are given in Fig. 22.6. The first one has a hole at top and bottom (Fig. 22.6a) resulting in a mesh consisting of non warped elements. The second geometry is a full hemisphere (Fig. 22.6b) with initially warped elements where the rigid body projection (22.4) influences the results substantially. The elasticity properties are taken as E = 6.825 · 107 and ν = 0.3. The normalized radial displacements ur/u∗r (u∗r = 0.0924 at point load [25]) are given in Tab. 22.2 where α is set to α = 0.005μ. Table 22.2 Simulation results: Hemispherical shell a) With hole on top No. of elements Simo [26] SQ2 [12]

Present

8×8

1.00

1.00

1.00

16 × 16

1.01

1.01

1.01

b) Without hole on top YASE [28] MITC4 [18]

No. of elements

SRI [27]

48

0.93

0.34

192

0.98

0.84

QPH [25]

Present

0.91

0.86

0.96

0.99

0.99

0.99

The geometrically nonlinear transient problem is simulated with a density of ρ = 2.5 · 10−4 (at the initial configuration). The load is assumed to be constant with time. In order to examine the performance of the present element formulation

a)

Fig. 22.6 Hemispherical shell: a) With hole; b) Without hole

b)

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QPH

Belytschko-Tsay

0

displacement

-0.05

-0.1

-0.15

-0.2

-0.25

0

0.01

0.02

0.03

0.04

0.05

time

Fig. 22.7 Dynamic response of the hemispherical shell

according to warped element geometries the hemisphere without a hole on the top is meshed with 192 elements as shown in Fig. 22.6b. The results are compared to the Belytschko et al. [29] formulation as well as to the Belytschko and Leviathan proposals [25] (benchmark results are calculated with Ls-Dyna). The Belytschko– Lin–Tsay formulation indicates a severe amplitude decay which is atypical for explicit transient solution procedures with an elastic constitutive relation. The present formulation, however, shows an excellent over all performance.

22.5.3 Roll-Up of a Clamped Beam An initially flat shell with length L = 12, width w = 1 and height h = 1 is clamped at one end and a bending moment is applied at the other end. According to the Euler beam theory, the analytical solution for a beam rolled up to a circular arc of 0 radius r is given by 1r = M EI , where M0 is the end moment and I is the moment of inertia. Within the proposed shell theory the load can either be applied by bending moments causing an out of plane bending deformation Fig. 22.8a (load case 1), or by concentrated in-plane drilling moments causing an in-plane membrane deformation Fig. 22.8b (load case 2). We are using a regular 12x1 mesh to obtain the corresponding solutions where a Young’s modulus of E = 30 · 106 , a Poisson’s ratio ν = 0 and a density ρ = 1.2 is used. The load is applied smoothly by corresponding concentrated couples at the two end nodes M = M0/2 with M0 = 1308996.939. The deformation characteristics of both load cases are given in Fig. 22.8 for 25%, 50%, 75% and 100% of M0 , respectively. We see that both deformation fields are in excellent agreement with the analytical findings even for such a coarse meshes.


a)

327

b)

Fig. 22.8 Roll-up of a clamped beam: a) Shell bending (load case 1); b) Membrane drilling (load case 2)

22.6 Summary Within this paper a quadrilateral shell element with six degrees of freedom per node is derived. The problem is tackled by independent descriptions of the membrane and the bending properties while a coupling is only introduced for warped element geometries based on a rigid body projection scheme. The membrane part consists of drilling degrees of freedom, i.e. membrane inplane rotational degrees of freedom which are derived using a unique functional that introduces rotational degrees of freedom naturally. The bending properties are derived using the Mindlin-Reissner assumption. All parts of the shell element’s linear stiffness matrix can be evaluated analytically and no Gaussian quadrature has to be carried out at all. The proposed element fulfills all patch test conditions and gives excellent results for typical benchmark problems. Acknowledgements This work has been supported by Grant Agency VEGA No. 1/0093/10.

References 1. D. Allman, A compatible triangular element including vertex rotations for plane elasticity analysis, Computers & Structures 19 (1984) 1–8. 2. A. Cazzani, S. Atluri, Four-noded mixed finite elements, using unsymmetric stresses for linear and nonlinear analysis of membranes, Computational Mechanics 11 (1993) 229–251. 3. T. J. Hughes, F. Brezzi, On drilling degrees of freedom, Computer Methods in Applied Mechanics and Engineering 72 (1989) 105–121. 4. E. Reissner, A note on variational priciples in elasticity, International Journal of Solids and Structures 1 (1965) 93–95. 5. S. Kugler, P. Fotiu, J. Murin, A highly efficient membrane finite element with drilling degrees of freedom, Acta Mechancia 213 (2010) 323–348. 6. S. Kugler, Development of a laterally pressed quadrilateral shell element, Ph.D. thesis, STU Bratislava (2010). 7. E. Cosserat, F. Cosserat, Théorie des corps déformables, A. Hermann, Paris, 1909. 8. A. Eringen, Nonlocal continuum field theories, Springer, 2002.

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9. T. Hughes, A. Masud, I. Harari, Numerical assessment of some membrane elements with drilling degrees of freedom, Computers & Structures 55 (1995) 297–314. 10. A. Ibrahimbegovic, A novel membrane finite element with an enhanced displacement interpolation, Finite Elements in Analysis and Design 7 (1990) 167–179. 11. A. Ibrahimbegovic, R. Taylor, E. Wilson, A robust quadrilateral membrane finite element with drilling degrees of freedom, International Journal for Numerical Methods in Engineering 30 (1990) 445–457. 12. A. Ibrahimbegovic, F. Frey, Membrane quadrilateral finite elments with rotational degrees of freedom, Engineering Fracture Mechanics 43 (1992) 12–24. 13. Y. Zhu, T. Zacharia, A new one-point quadrature, quadrilateral shell element with drilling degrees of freedom, Computer Methods in Applied Mechanics and Engineering 136 (1996) 165–203. 14. T. Belytschko, I. Leviathan, Projection schemes for one-point quadrature shell elements, Computer Methods in Applied Mechanics and Engineering 115 (1994) 277–286. 15. C. Rankin, B. Nour-Omid, The use of projectors to improve finite element performance, Computers & Structures 30 (1988) 257–267. 16. O. C. Zienkiewicz, R. L. Taylor, Finite Element Method: Volume 1, The Basis (Finite Element Method ), Butterworth-Heinemann, 2000. 17. J. Simo, T. Hughes, On the variational foundations of assumed strain methods, Computer Methods in Applied Mechanics and Engineering 53 (1986) 1685–1695. 18. E. Dvorkin, K. Bathe, A continuum mechanics based four-node shell element for general nonlinear analysis, Eng. Comput. 1 (1984) 77–88. 19. K. Bathe, E. Dvorkin, A formulation of general shell elements - the use of mixed interpolation of tensorial components, International Journal for Numerical Methods in Engineering 22 (1986) 697–722. 20. Q. Zeng, A. Combescure, A new one-point quadrature, general non-linear quadrilateral shell element with physical stabilization, International Journal for Numerical Methods in Engineering 42 (1998) 1307–1338. 21. T. Hughes, W. Liu, I. Levit, ”Nonlinear dynamics finite element analysis of shells”. Nonlinear finite element analysis in Struct. Mech., Springer-Verlag Berlin, 1981. 22. R. Cook, Four-node ’flat’ shell element: Drilling degrees of freedom, membrane-bending coupling, warped geometry, and behavior, Computers & Structures 50 (1994) 549–555. 23. C. Choi, T. Lee, Efficient remedy for membrane locking of 4-node flat shell elements by nonconforming modes, Computer Methods in Applied Mechanics and Engineering 192 (2003) 1961–1971. 24. F. Frey, Shell finite elements with six degrees of freedom per node, Analytical and Computational Models for Shells (Edited by A.K. Noor, T. Belytschko, J.C. Simo), ASME (1989) 291–317. 25. T. Belytschko, I. Leviathan, Physical stabilization of the 4-node shell element with one point quadrature, Computer Methods in Applied Mechanics and Engineering 113 (1994) 321–350. 26. J. Simo, D. Fox, M. Rifai, On a stress resultant geometrically exact shell model. part ii: The linear theory; computational aspects, Computer Methods in Applied Mechanics and Engineering 73 (1989) 53–92. 27. T. Hughes, W. Liu, Nonlinear finite element analysis of shells: Part i. three dimensional shells, Computer Methods in Applied Mechanics and Engineering 26 (1981) 331–362. 28. B. Engelmann, R. Whirley, A new elasto-plastic shell formulation for dyna 3d,, Lawrence Livermore National Laboratory Report UCRL-JC-104826. 29. T. Belytschko, J. Lin, C. Tsay, Explicit algorithms for the nonlinear dynamics of shells, Computer Methods in Applied Mechanics and Engineering 42 (1984) 225–251.

Chapter 23

Invariant-Based Geometrically Nonlinear Formulation of a Triangular Finite Element of Laminated Shells Stanislav V. Levyakov

Abstract A non-standard approach is proposed to develop a simple and computationally effective triangular finite element applicable to geometrically nonlinear analysis of composite shells. The approach is based on the natural components of the stress and strain tensors and their invariants which allow one to express the strain energy of the shell in a compact form without coordinate transformations. The natural components of the tensors are referred to fibers oriented along the triangle edges. An advantage of using the natural strains is that it suffices to approximate one-dimensional functions rather than two-dimensional strain fields over the elemental area. To this end, analytical solutions of auxiliary beam bending problems are used. The validity and accuracy of the approach proposed are verified using benchmark solutions and numerical data available in the literature. Keywords Invariants · Triangular finite element · Natural strain · Composite shell · Geometrical nonlinearity

23.1 Introduction Investigation of nonlinear deformation of shells poses challenging problems to analysts and structural engineers. Except for simple geometry, loading, and boundary conditions, solution of shell problems arising in practice can be obtained only by numerical methods. Since the advent of computers, the finite element method has become a dominant numerical technique for analysis of thin-walled structures, and much effort has been spent to develop computationally effective shell finite elements. Contribution of mechanical engineers and mathematicians in this field is S. V. Levyakov (B) Department of Engineering Mathematics, Novosibirsk State Technical University, 630092 Novosibirsk, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 23, © Springer-Verlag Berlin Heidelberg 2011

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immense. Review of various approaches and techniques for constructing shell elements can be found in excellent papers by Bucalem and Bathe [6], Yang et al. [18], and Gal and Levy [8]. Modern day computers allow one to handle systems of equations with millions of unknowns and implement highly sophisticated finite-element models with large number of degrees of freedom. In the context of engineering analysis, however, simple and reliable shell finite elements are still desirable. For practical applications, there should be a sensible balance between accuracy of the numerical solution and computational cost. Much research has been focused on a triangular finite element owing to its ability to model general shell geometry and irregular boundaries. A promising approach to construct an effective finite element is to consider the element as a mechanical object and not just a domain for local approximation of unknown functions. This idea was implemented in the natural mode method proposed by J. H. Argyris [2]. The method is based on (1) decomposition of the deformation of a body into purely deformation modes and rigid-body motion and (2) introduction of the coordinates defined by the triangle edges. It was shown that the deformation modes and the triangular element stiffness are convenient to determine using three membrane strains, three bending strains, and three transverse shear strains in the direction of the triangle edges. The normal components of the tensors referred to the triangle edges are called the natural strains. It is worth noting that the idea of using natural strains corresponds to the well-known fact that the plane stress (strain) states are fully determined by three direct stresses (strains) measured in three nonparallel directions. This theoretical statement has long been applied to investigate experimentally stresses in structures using strain gauges since the direct strains are much easier to measure than shear strains. The natural-mode approach was successfully implemented in a series of facet triangular elements for solving linear and nonlinear problems of laminated shells [1–4]. However, derivation of the elements involves lengthy matrix formulas relating the stress and strain components determined in different coordinate systems. The present study is in line with the natural mode approach. It shares the idea of using natural components of strain and separating rigid-body motion from pure deformation displacements of the finite element. Distinctive features of the finiteelement formulation given below are that (a) invariants of the natural strains are used; (b) finite displacements and rotations are allowed within the element; (c) geometry of initially curved element is taken into account using nodal directors; (d) deformation of the element in the presence of arbitrarily large rotations is described using the concept of kinematic group [10]. A computational benefit of the approach is that no operations on the coordinate transformation are required since no local coordinates are introduced for the elements. The paper is a continuation of the authors previous work [11–13] and organized as follows. In Sect. 23.2, assumptions upon which the finite element is constructed are summarized. In Sect. 23.3, relations between Cartesian and natural components of the strain tensor are given in an explicit form. In Sect. 23.4, strain invariants are obtained in terms of three natural components. In Sect. 23.5, template formulas are given for calculating invariants of two-dimensional symmetric tensors. In Sects 23.6

23 Triangular Finite Element of Laminated Shells

331

and 23.7, the strain energy densities are expressed in terms of invariants for isotropic and composite materials. In Sect. 23.8, the total strain energy of a laminated shell subject to the Reissner–Mindlin assumptions is obtained in terms of natural strains. In Sect. 23.9, approximations of the natural strains within the element are considered. Sect. 23.10 contains formulas for computing the stiffness matrix, gradient, and Hessian of the element. Sect. 23.11 describes a technique for evaluating variations of the strain energy which are necessary to determine equilibrium states of shells. Finally, Sect. 23.12 demonstrates capabilities and accuracy of the finite element using linear and nonlinear problems of plates and shells.

23.2 Basic Requirements and Assumptions In the derivation of the shell finite element, we adopt the following requirements: • the element should be based on mechanistically clear assumptions; • the finite element should be applicable to linear and geometrically nonlinear analysis of composite and isotropic shells; • the finite element should be simple and inexpensive with a minimum number of degrees of freedom; • the finite element should not lock in the case of very thin structures described by the Kirchhoff–Love theory; • the finite element should be insensitive to mesh distortion. We determine configuration of a shell by geometry of its middle surface. Let a triangular element be cut from the shell by three planes normal to the middle surface. In general, the material of the shell is anisotropic and linearly elastic. In the derivation of the finite element, the following assumptions are employed: (a) strains are small compared to unity, but no restriction is imposed on the magnitude of displacements and rotations; (b) the area of the curved triangular element is equal to the area of a planar triangle whose side lengths are equal to those of the curved element; (c) each side of the curved triangular element is a planar nearly circular curve which remains planar and nearly circular after deformation; (d) the normal strains in the direction of the element edges are constant; (e) transverse shear strains are taken into account using the first order shear deformation theory which states that straight lines normal to the middle surface at the initial state remain straight but not necessarily normal to the middle surface after deformation, and these lines do not change in length during deformation. We note that assumption (b) is adopted to take into account the shell metrics. It introduces a correction to the faceted element approach where the side lengths and area of the element are smaller compared to those of the real element. Assumption (c) is used to describe finite curvature changes within the element.

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23.3 Natural Components of Strain We consider a small planar triangular element whose vertices are denoted by i, j, and k, and whose lengths of the edges opposite to the vertices are denoted by li , l j , and lk , respectively (Fig. 23.1). Position of any point within the triangle is convenient to determine by the area coordinates Li , L j , and Lk which satisfy the condition Li + L j + Lk = 1 [21]. To describe inhomogeneous properties of the material within the triangle, we introduce material Cartesian coordinates ξ1 and ξ2 . In the particular case of isotropic material, there is no need to use material coordinates. The aim of this section is to obtain explicit relations between the Cartesian components of the strain tensor S mn (m, n = 1, 2) and the natural strains S i (i = 1, 2, 3) of fibers parallel to the triangle edges. These relations are written in the form of a system of linear algebraic equations [15]: S i = αmni S mn αmni = λmi λni ,

(i = 1, 2, 3; λ1i =

ξ1k − ξ1 j , li

m, n = 1, 2), λ2i =

ξ2k − ξ2 j , li

(23.1) (23.2)

where λmi are the direction cosines of the ith edge, ξmk is the mth material coordinate of the kth vertex of the triangle, and summation is performed over m, n = 1, 2. In (23.1), (23.2), and below, subscripts i, j, and k obey the rule of cyclic permutation. Relations (23.1) can be inverted to give (summation over i = 1, 2, 3) S mn = βmni S i ,

Fig. 23.1 a) Shell finite element; b) single layer element; c) isometric triangle

(23.3)


333

l1 l2 l3 li [λmk λn j + λnk λm j − 2(λ1 j λ1k + λ2 j λ2k )δmn ] (m, n = 1, 2), (23.4) 8F 2 where F is the triangle area and δmn is Kronecker’s delta. For convenience of further derivation of the finite element, we write expressions (23.1) and (23.3) in the matrix notation βmni =

S = AS C ,

⎡ ⎢⎢⎢ l1 l2 l3 ⎢⎢⎢⎢ ⎢⎢ B= 4F 2 ⎢⎢⎢⎣ 1

S C = BS ,

(23.5)

, , S C = S 11 S 22 S 12 T , S = S1 S2 S3 T, ⎡ ⎤ ⎢⎢⎢ λ2 λ2 2λ11 λ21 ⎥⎥⎥ ⎢⎢⎢ 11 21 ⎥⎥⎥ A = ⎢⎢⎢⎢ λ212 λ222 2λ12 λ22 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ λ213 λ223 2λ13 λ23 −λ22 λ23 l1

−λ23 λ21 l2

−λ21 λ22 l3

−λ12 λ13 l1

−λ13 λ11 l2

−λ11 λ12 l3

1 2 (λ13 λ22 + λ23 λ12 )l1 2

(λ11 λ23 + λ21 λ13 )l2 12

(23.6)

(23.7) ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ , ⎥⎥⎥ ⎦

(λ12 λ21 + λ22 λ11 )l3 (23.8) where the subscript C stands for Cartesian. The transformation matrices are written in terms of αmni and βmni as ⎡ ⎤ ⎢⎢⎢ α111 α221 2α121 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ A = ⎢⎢⎢⎢ α112 α222 2α122 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ α113 α223 2α123

⎡ ⎤ ⎢⎢⎢ β111 β112 β113 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ B = ⎢⎢⎢⎢ β221 β222 β223 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ β121 β122 β123

(23.9)

Thus, we have obtained explicit relations (23.5) which allow one to readily express Cartesian components of the strain tensor in terms of natural strains and vice versa.

23.4 Invariants of Strain Tensor We focus our attention on invariants expressed in terms of quantities independent of material coordinates. The invariants written in curvilinear convective coordinates were discussed by Kuznetsov and Levyakov [13]. The classical invariants of the elasticity theory [15] are written in Cartesian coordinates as 2 IS = S 11 + S 22 , IS S = S 11 S 22 − S 12 . (23.10) Here IS and IS S are the first and second invariants of the strain tensor, respectively. Relations (23.10) can be combined with (23.3) to give invariants in the natural form. This process, however, involves cumbersome expressions. We consider a

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more straightforward technique for obtaining invariants of two-dimensional tensors in terms of natural components. According to [9], we consider the ratio of the squared area of the deformed triangle F ∗ to the squared area of the triangle in the initial undeformed state F. Given the triangle side lengths, the squared area of the triangle is determined by Heron’s formula which can be written as follows (summation over p = 1, 2, 3): F2 =

Δ , 16

Δ = (l p l p )2 − 2l2p l2p ,

(23.11)

where Δ is the discriminant of the triangle. We assume that the squared area of the deformed triangle can be determined in a similar manner(summation over p = 1, 2, 3): Δ∗ ∗2 F ∗2 = , Δ∗ = (l∗p l∗p )2 − 2l∗2 (23.12) p lp , 16 where the lengths of the triangle sides after deformation are given by 2 l∗2 p = l p (1 + 2S p)

(23.13)

In view of (23.11), (23.12), and (23.13), the ratio of the squared areas can be decomposed as follows (summation over p = 1, 2, 3): F ∗2 = 1 + 2IS + 4IS S , F2 IS = 2(aa p S p − 2a2pS p ), a p = l2p Δ−1/2 ,

IS S = (a p S p )2 − 2a2p S 2p . a = l p l p Δ−1/2 ,

(23.14) (23.15) (23.16)

where a and a p are the metric coefficients. Since the metric coefficients (23.16) are functions of side lengths of the triangle only, we call the strain invariants (23.15) the coordinate-free invariants to emphasize the fact that they do not depend on the material system of coordinates.

23.5 Templates for Invariants of Tensors Relations given for the invariants of strains given in Sect. 23.4 are valid for any symmetric two-dimensional tensor umn (m, n = 1, 2) which can be represented by the following vectors comprising Cartesian and natural components: , U C = U11 U22 U 12 T ,

, U = U1 U 2 U3 T .

(23.17)

Below, we consider formulas which we call templates since they can be applied to any symmetric two-dimensional tensor of rank two.


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Relations between Cartesian and natural components of the tensor are given by U i = αmni Umn ,

U mn = βmni U i .

(23.18)

U C = BU .

(23.19)

In matrix notation, relations (23.18) read U = AU C ,

The first and second invariants of the tensor are written in terms of Cartesian and natural components as IU = U11 + U22,

2 IUU = U 11 U22 − U12 .

(23.20)

In the natural components, expressions (23.20) become (summation over p = 1, 2, 3) IU = 2(aa p U p − 2a2p U p ),

IUU = (a p U p )2 − 2a2pU 2p .

(23.21)

Let the tensor U C be a sum of two tensors V C and W C , i.e. U C = V C + W C , where the subscript C denotes Cartesian components. In this case, relation (23.20)2 becomes IUU = IVV + IWW + 2IVW , (23.22) where invariants IVV and IWW are determined by template (23.21) in which U is replaced with V and W, respectively, and IVW is the combined invariant of two tensors. In Cartesian components, the combined invariant is given by IVW =

1 (V11 W22 + V22 W11 − 2V12W12 ). 2

(23.23)

In contrast to the definition of the combined invariant given by Kuznetsov and Levyakov [13], a factor of 1/2 is introduced in (23.23) in order that the second invariant falls out as a particular case of the combined invariant. For example, setting V C = U C and W C = U C in (23.23), we obtain the second invariant IUU given by Eq. (23.20)2 . It is worth noting that the first invariant IU can be obtained from (23.23). To this end, we set W C = U C and V C = {2 2 0}T , the latter vector representing a spherical tensor. Substituting these vectors into (23.23), we arrive at (23.20)1 . Let us express the combined invariant in terms of natural components. Substituting the expression U = V + W into (23.21)2 , we arrive at expression (23.22) in which the combined invariant is given by (summation over p, q = 1, 2, 3) IVW = (a p V p )(aq Wq ) − 2a2p(VW) p .

(23.24)

Here (VW) p = V p W p is simply a product of the components having the same subscript, i.e. no summation is performed over p. In particular case where V = {2 2 2}T

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and W = U , template (23.24) yields the expression for the first invariant IU . Setting V = U and W = U , from (23.24) one obtains the second invariant IUU . In matrix notation, templates (23.21) and (23.24) are written as IU = τ T U , where

μμ T − ρ )U U, IUU = U T (μ

μμ T − ρ )W W, IVW = V T (μ

τ = 2{a1 (a − 2a1) a2 (a − 2a2) a3 (a − 2a3)}T , ⎡ ⎤ ⎢⎢⎢ a2 0 0 ⎥⎥⎥ ⎢⎢⎢ 1 ⎥⎥⎥ μ = {a1 a2 a3 }T , ρ = 2 ⎢⎢⎢⎢ 0 a22 0 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 a23

(23.25) (23.26)

(23.27)

Consider, for example, the stress tensor σmn . Relations between the Cartesian and natural components of the stress tensor are obtained by replacing U with σ in templates (23.18). The first and second invariants can be obtained using templates (23.20) for Cartesian components and templates (23.21) for natural components of the stress tensor.

23.6 Strain Energy Density for Plane Stress We consider a linear elastic anisotropic body under plane-stress conditions. In this case, the strain energy density is written in Cartesian coordinates as follows (summation over m, n = 1, 2): 1 ΠV = σmn S mn . (23.28) 2 We write this expression in terms of invariants as ΠV =

1 (Iσ IS − 2IσS ) . 2

(23.29)

Here the combined invariant IσS is obtained using template (23.23) in which V and W are replaced with σ and S , respectively. The validity of (23.29) can be directly proved using templates (23.20) and (23.23). Thus, the energy ΠV known to be an invariant quantity is expressed in terms of the two first invariants and one combined invariant whichever material is considered. This can be regarded as a simple proof of the energy invariance with respect to the choice of coordinates. If the natural strains and stresses are given, the strain energy density (23.29) can be determined using templates (23.21) and (23.24) in which V and W are replaced with σ and S , respectively.


337

23.6.1 Isotropic Material Isotropic material is characterized by two elastic constants, Young’s modulus E and Poisson’s ratio ν. Expanding (23.29) using templates (23.20) and Hooke’s law relations, we express the strain energy in terms of the strain invariants: ΠV =

1 E (I 2 − 2(1 − ν)IS S ). 2 1 − ν2 S

(23.30)

23.6.2 Six-Constant Anisotropic Material Elastic properties of an anisotropic material are usually determined using coordinates referred to Cartesian anisotropy axes. For the strain and stress tensors represented by the vectors (23.17)1 , we obtain ⎡ ⎤ ⎢⎢⎢ d11 d12 d13 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ σC = DC S C , D C = ⎢⎢⎢⎢ (23.31) d22 d23 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ sym. d33 where D C is a matrix of elastic coefficients. For orthotropic materials, the elastic coefficients are given by d11 =

E11 , 1 − ν12ν21

d22 =

E22 , 1 − ν12ν21

d13 = d23 = 0,

d12 =

E 11 ν21 E 22 ν12 = , 1 − ν12ν21 1 − ν12 ν21

d33 = 4G 12 .

(23.32)

We write the stress-strain relations in terms of natural components. Using (23.19), from (23.31) we obtain σ = DS , D = A DC B . (23.33) Substituting the strain and stress invariants into (23.29) and using templates (23.25), we obtain the strain energy density as a bilinear form of the natural stresses and strains: 1 μμ T − ρ ), ΠV = S T H Δ σ , H Δ = τ τ T − 2(μ (23.34) 2 where H Δ is the 3 × 3 invariant symmetric matrix. It follows from (23.26), (23.27), and (23.16) that the components of the matrix H Δ depend only on the lengths of the triangle sides. Inserting (23.33) into (23.34), we express the strain energy density as a quadratic form of natural strains ΠV =

1 T S DNS , 2

D N = H Δ D,

where D N is the 3 × 3 natural matrix of elastic constants.

(23.35)

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23.7 Transverse Shear Strain Energy Density We introduce symmetric shear strain tensor Γ whose Cartesian and natural components are respectively given by [13] Γmn = γm3 γn3

(m, n = 1, 2),

Γi = γi2 .

(23.36)

Relations between Cartesian and natural components are given by template (23.18) or (23.19).The first invariant of the tensor can be obtained using templates (23.20)1 and (23.21)1 .

23.7.1 Isotropic Material The transverse shear strain energy density of an isotropic shell is written as ΠΓV =

1 2 2 G(γ13 + γ23 ), 2

(23.37)

where G = E/[2(1 + ν)] is the shear modulus. Noting that the sum in parentheses is the first invariant of the tensor Γ , we immediately obtain ΠΓV =

1 GIΓ . 2

(23.38)

23.7.2 Transversely Isotropic Material Transversely isotropic material is characterized by three engineering constants E, ν, and G = G13 = G23 , where G is the transverse shear modulus. Similar to (23.37) and (23.38), we find that 1 ΠΓV = G IΓ . (23.39) 2

23.7.3 Anisotropic Material We write the density of the transverse shear strain energy of an anisotropic material as (summation over m, n = 1, 2) ΠΓV =

1 gmn Γmn , 2

(23.40)


339

where gmn is the tensor of elastic constants. Note that templates listed in Sect. 23.5 can be applied to the tensors gmn and Γmn . Comparison of (23.40) with (23.28) suggests that expression (23.40) can be written in the form of (23.29): ΠΓV =

1 (Ig IΓ − 2IgΓ ), 2

(23.41)

where the first invariants Ig and IΓ and combined invariant IgΓ can be expressed in terms of Cartesian or natural components using templates (23.20), (23.21), (23.23), and (23.24). In the particular case of orthotropic material, we obtain gC = {G13 G 23 0}T ,

Γ C = {Γ11 Γ22 Γ12 }T ,

1 (G23 Γ11 + G 13 Γ22 ). 2 Substitution of (23.43) into (23.41) yields the well-known expression Ig = G13 + G 23 ,

IΓ = Γ11 + Γ22,

ΠΓV =

IgΓ =

1 (G 13 Γ11 + G23 Γ22 ). 2

(23.42) (23.43)

(23.44)

Using templates (23.25), we write the invariants in (23.41) in matrix form as Ig = τ T g ,

IΓ = τ T Γ ,

μμ T − ρ )Γ Γ. IgΓ = g T (μ

(23.45)

Substituting (23.45) into (23.41), we obtain the density of the transverse shear energy in matrix form 1 ΠΓV = g T H Δ Γ , (23.46) 2 where the matrix H Δ is given by (23.34)2 . The strain density (23.46) can be expressed in terms of Cartesian or natural components of the tensors gmn and Γmn .

23.8 Total Strain Energy of Laminated Shell We consider a laminated shell composed of N orthotropic layers. For each layer, we introduce material coordinates ξ1 and ξ2 measured along the principal axes of the orthotropic material. The layers are assumed to be perfectly bonded such that no slipping occurs at the interlayer boundaries. We use the Reissner–Mindlin assumptions which imply that strains are linear functions of the coordinate z in the direction normal to the middle surface of the shell, whereas the transverse shear strains are constant across the shell thickness. In this case, the natural strains are written as S = ε + zκκ ,

ε = {ε1 ε2 ε3 }T ,

κ = {κ1 κ2 κ3 }T ,

h h − ≤z≤ , 2 2

(23.47)

where εi and κi are the natural components of the tensors of membrane strains and curvature changes of the middle surface, respectively, and h is the shell thickness.

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Combining (23.35)1 , (23.46), and (23.47)1 , we obtain the strain energy of the shell in terms of the natural strain components 1 Π= (εεT D N ε + 2zκκT D N ε + z2 κ T D N κ + g T H Δ Γ ) dV, (23.48) 2 V

where V is the volume occupied by the shell.

23.9 Approximation of Natural Strains We consider a shell finite element cut by three planes normal to the shell middle surface. Three nodes i, j, and k at the element vertices lying on the shell middle surface are determined by position vectors r i , r j , and r k , respectively. Moreover, we introduce three unit vectors n i , n j , and n k normal to the middle surface of the undeformed shell (Fig. 1). The resulting set of the nodal vectors is called the kinematic group and used to describe kinematics of the element [10]. In a deformed state of the shell, these nodal vectors are denoted by r ∗m and n ∗m (m = i, j, k). Note that, according to assumption (e) in Sect. 23.2, the vectors n ∗m are unit but not necessarily normal to the deformed middle surface of the shell. Our aim is to express the natural strains in terms of the scalar products of the nodal vectors which play the role of nodal parameters.

23.9.1 Normal Strains Approximation for the normal strains εi obtained under the assumptions (a), (c), and (d) is given by [12]: e∗ sin−1 (Δχ∗i ) εi = i − 1, (23.49) li Δχ∗i where Δχ∗i

=

χ∗2k − χ∗1 j 2

,

χ∗2k

=

ψ∗2k e∗i

,

χ∗1 j

=

ψ∗1 j e∗i

,

χ2k − χ1 j ψ1 j ψ2k sin−1 (Δχi ) , χ2k = , χ1 j = , li = ei , 2 ei ei Δχi 8 ei = (rr k − r j )2 , ψ1 j = n j (rr k − r j ), ψ2k = n k (rr k − r j ), 8 e∗i = (rr ∗k − r ∗j )2 , ψ∗1 j = n ∗j (rr ∗k − r ∗j ), ψ∗2k = n ∗k (rr ∗k − r ∗j ).

Δχi =

(23.50) (23.51) (23.52) (23.53)

Natural strains calculated by formula (23.49) determine constant membrane strain state within the triangle.


341

It should be noted that relations (23.49) and (23.51)4 contain uncertainty of the form (0/0) for flat or nearly flat elements. In this case, calculations are performed using Taylor series expansions [12]

23.9.2 Curvature Changes and Transverse Shear Strains To obtain approximations of the curvature changes and transverse shear strains, we consider the bending problem of a simply supported composite beam subjected to end rotations (Fig. 23.2). The differential equations and boundary conditions governing bending of the shear deformable beam oriented along the ith side of the triangle are written as Di

% d 2 ϑi dwi & − C i ϑi + = 0, 2 dsi dsi ϑi = ϑ1 j , ϑi = ϑ2k ,

wi = 0 wi = 0

d % dwi & ϑi + = 0, dsi dsi for si = 0, for si = li .

(23.54)

(23.55)

Here Ci is the transverse shear rigidity, wi is the deflection of the beam, ϑi is the rotation, si is the coordinate measured along the ith side, and Di is the flexural rigidity of the beam. The end rotations ϑ1 j and ϑ2k are determined by the formulas ∗ ϑmn = θmn − θmn ,

θmn = sin−1 (χmn ),

∗ θmn = sin−1 (χ∗mn ).

(23.56)

The end rotations are functions of the nodal parameters of the element, which can be seen from Eqs (23.51)–(23.53) and (23.56). To determine the flexural rigidity Di , we use the coordinate transformation of the elastic tensor upon rotation of the material axes:

Fig. 23.2 Side of the shell element as a planar curve

342

S.V. Levyakov

h/2 Di =

E i (z)z2 dz,

E i (z) = d11 λ41i + d22 λ42i + (2d12 + d33 )λ21i λ22i ,

(23.57)

−h/2

where the elastic constants dmn which enter the matrix DC (see Eq. (23.31)) and λmi are the direction cosines of the ith side of the planar triangle which are calculated using the nodal material coordinates for each layer using formulas (23.2). The transverse shear rigidity which enters Eq. (23.54)1 is given by h/2 Ci = ki

gi dz,

(23.58)

−h/2

where gi is the natural component of the tensor gmn which can be calculated by template (23.18)1 and ki is the shear correction factor which takes into account nonuniform distribution of the transverse shear stress across the thickness of the beam. The shear correction factor is calculated by the following formulas [4, 17]: ⎡⎛ h/2 ⎞ h/2 ⎤ ⎟⎟⎟ ⎥⎥⎥ : ⎢⎢⎢⎢⎜⎜⎜⎜ ⎟ ⎥⎥⎥ ki = R2i ⎢⎢⎢⎢⎢⎜⎜⎜⎜⎜ Gi dz⎟⎟⎟⎟⎟ ti2 (z)G−1 dz (23.59) ⎥⎥⎥ , i ⎣⎝ ⎠ ⎦ −h/2

−h/2

h/2 Ri =

h/2 Ei (z − zNi)2 dz,

zNi =

−h/2

−h/2

⎛ h/2 ⎞−1 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ E i z dz ⎜⎜⎜ Ei z dz⎟⎟⎟⎟⎟ , ⎝ ⎠ −h/2

z ti (z) = −

E i (ζ − zNi )2 dζ.

(23.60)

−h/2

The solution of the boundary-value problem (23.54), (23.55) yields dϑi 1 ηi1 si κi = = (ϑ2k − ϑ1 j ) + (ϑ2k + ϑ1 j ) 2 − 1 , dsi li li li γi = ϑi + where ηi1 =

3 , 1 + ηi3

dwi = ηi2 (ϑ2k + ϑ1 j ), dsi

ηi2 =

1 ηi1 ηi3 , 6

ηi3 = 12

(23.61) (23.62)

Di . Ci l2i

(23.63)

The first, constant term in (23.61) describes pure bending of the beam for arbitrarily large rotations ϑ2k and ϑ1 j . The second term in (23.61) takes into account transverse shear forces which occur due to “unbalanced” end rotations ϑ2k and ϑ1 j . This antisymmetric mode of bending is taken into account in a linear approximation in view of assumption (c) in Sect. 23.2. Relations (23.62) and (23.63) imply that


343

γi → 0 as Ci → ∞. Hence, the convergence to the Kirchhoff–Love shell is achieved and shear locking is automatically eliminated. To interpolate the curvature change (23.61) into the triangular element, we recall the following relations which hold at the edge opposite to node i: Li = const,

Lj = 1 −

si , li

Lk =

si . li

(23.64)

It follows that

1 li (1 + Lk − L j ). 2 Substitution of (23.65) into (23.61) yields si =

κi =

1 ηi1 (ϑ2k − ϑ1 j ) + (ϑ2k + ϑ1 j )(Lk − L j ). li li

(23.65)

(23.66)

This expression shows that κi is constant in the direction of the median drawn from the ith vertex to the opposite side of the triangle.

23.10 Strain Energy of Triangular Shell Element Inserting (23.49), (23.61), and (23.62) into (23.48), we obtain Π=

1 T 1 1 1 ε K ε ε + (εεT K εϑ ϑ + ϑT K Tεϑε ) + ϑ T K ϑ ϑ + C TΓ Γ , 2 2 2 2 ε = {ε1 ε2 ε3 }T , ϑ = {ϑ23 ϑ12 ϑ21 ϑ13 ϑ22 ϑ11 }T , D N C dV, K ε = D N dV, K εϑ = zD V

z2C T D N C dV, V

(23.68) (23.69)

V

Kϑ =

(23.67)

⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ C TΓ = ⎜⎜⎜⎜ G T dV ⎟⎟⎟⎟ H Δ k , ⎝ ⎠

⎡ ⎤ ⎢⎢⎢ k1 0 0 ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ k = ⎢⎢⎢⎢ 0 k2 0 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 k3

(23.70)

V

(23.71)

Here the shear correction factors ki are calculated by formulas (23.59) and (23.60) and C is a 3 × 6 matrix whose nonzero entries are given by C11 = [1 + η11(L3 − L2 )]l−1 1 ,

C12 = −[1 + η11(L2 − L3 )]l−1 1 ,

C23 = [1 + η21(L1 − L3 )]l−1 2 ,

C24 = −[1 + η21(L3 − L1 )]l−1 2 ,

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S.V. Levyakov

C35 = [1 + η31(L2 − L1 )]l−1 3 ,

C36 = −[1 + η31(L1 − L2 )]l−1 3 .

(23.72)

Assuming that elastic properties are constant over the area of the element, we write (23.69) and (23.70) as ⎛ h/2 ⎞ h/2 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ D N dz⎟⎟⎟⎟ C F , Kε = F D N dz, K εϑ = F ⎜⎜⎜ zD (23.73) ⎟⎠ ⎝ −h/2

Kϑ = F

−h/2

⎛ h/2 ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ C T ⎜⎜⎜⎜⎜ z2 D N dz⎟⎟⎟⎟⎟ C dF, ⎝ ⎠ −h/2

⎛ h/2 ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ C TΓ = F ⎜⎜⎜⎜⎜ G T dz⎟⎟⎟⎟⎟ H Δ k , ⎝ ⎠

(23.74)

−h/2

⎡ ⎤ ⎢⎢⎢ l−1 −l−1 0 0 0 0 ⎥⎥⎥ 1 ⎢⎢⎢ 1 ⎥⎥⎥ −1 0 ⎥⎥⎥ . C F = ⎢⎢⎢⎢ 0 0 l−1 −l 0 ⎥⎥⎥ 2 2 ⎢⎢⎣ −1 ⎦ 0 0 0 0 l−1 −l 3 3

(23.75)

The integral over the area of the element in the expression for K ϑ is calculated exactly using three Gaussian points. Using Eq. (23.62) and relation (23.36)2 , we rewrite the last term in (23.67) as C TΓ Γ = ϑT K Γ ϑ,

(23.76)

where ⎡ ⎤ ⎢⎢⎢ CΓ1 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ M = ⎢⎢⎢⎢ 0 CΓ2 0 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ 0 0 CΓ3

KΓ = WT MW , ⎡ ⎤ ⎢⎢⎢ η12 η12 0 0 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ W = ⎢⎢⎢⎢ 0 0 η22 η22 0 0 ⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ 0 0 0 0 η32 η32

In view of (23.76), the strain energy (23.67) becomes ⎡ ⎤ ⎢⎢⎢ K ε ⎥⎥⎥ K 1 T εϑ ⎥⎥⎥ , Π = u (1) K u (1) , K = ⎢⎢⎢⎣ T 2 K εϑ K ϑ + K Γ ⎦ uT(1) = {ε1 ε2 ε3 ϑ23 ϑ12 ϑ21 ϑ13 ϑ22 ϑ11 } .

(23.77)

(23.78) (23.79)

where the components of u (1) are given by (23.49) and (23.56).

23.11 Variations of the Strain Energy and Algorithm of Solution To formulate an iterative method for determining equilibrium states of the shell, it is necessary to evaluate the first and second variations of the total energy. Since the finite-element relations considered above are highly nonlinear, straightforward


345

derivation of the variations leads to extremely cumbersome analytical expressions. An effective technique for evaluating the variations is based on the chain rule where three “levels” of varied parameters are introduced. The first level is represented by the vector (23.79) and the other two levels are represented by the vectors 3 4T u (2) = e∗1 e∗2 e∗3 ψ∗23 ψ∗12 ψ∗21 ψ∗13 ψ∗22 ψ∗11 , 3 4T u (3) = q T1 q T2 q T3 ,

3 4T q i = x∗1i x∗2i x∗3i ω1i ω2i

(i = 1, 2, 3).

(23.80) (23.81)

Variations in the vector u (3) are the degrees of freedom of the element. For each level, the first and second variations of the strain energy of the shell finite element can be written in terms of variations of the components of vectors (23.79), (23.80), and (23.81): δΠ = δuuT(m) g (m) ,

δ2 Π = δuuT(m) H (m) δuu(m) ,

(23.82)

where g (m) and H (m) are the gradient and the Hessian matrix corresponding to the mth variation level, respectively. The desired values of g = g (3) and H = H (3) are computed by the recursive formulas (summation over s): g(m+1) = u(m) g (m) ,

H (m+1) = u (m) H (m) uT (m) + g(m)s u(m)s

(m = 1, 2; s = 1, . . ., 9). (23.83) Here g(m)s are the components of the vector g(m) and u (m) and u (m)s are the matrices composed of the first and second partial derivatives of the components of the mth level with respect to the components of the (m + 1)th variation level, respectively. Formulas for calculating the matrices u (m) and u (m)s are given by Kuznetsov and Levyakov [12]. The computation process (23.83) begins for the initial values g (1) = K u (1) ,

H (1) = K .

(23.84)

Once the gradient g = g (3) and Hessian matrix H = H (3) have been calculated for individual elements, they are assembled into global vectors and matrices of the finiteelement assemblage using a standard technique [5]. To find a deformed state of the shell as a finite-element assemblage, we use the arc-length control method [19]. For a certain loading step, the Newton–Raphson system of equations has the form H p−1 δqq p + w p−1 δλ p + g p−1 = 0,

(23.85)

where H and g are total Hessian matrix and gradient of the assemblage, respectively, w is the vector composed of the derivatives of the form ∂2 W/∂qi ∂λ, W is the total potential energy of the discrete shell model, λ is the loading parameter, δqq p is the unknown vector of the increments in the generalized coordinates and the superscript p denotes the iteration number.

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23.12 Numerical Testing Below, we give results of numerical testing of the proposed finite element. The verification problems are chosen to test the capability of the finite element to adequately describe the transverse shear effects in the cases of linear and geometrically nonlinear deformations. Special attention is given to very thin plates and shells to show that the finite-element solution converges to the solutions based on the Kirchhoff– Love assumptions. The accuracy and mesh convergence of the numerical solutions are studied.

23.12.1 Linear Analysis of Orthotropic Square Plates Under Uniformly Distributed Transverse Load To study shear locking free behavior of the finite element proposed, we consider an orthotropic single-layer square plate of side length L = 0.305 m and thickness h subjected to uniformly distributed load q. The material characteristics are E 11 = 2.068 · 1010 Pa, E 22 = 0.8825 · 1010 Pa, G 12 = 0.2551 · 1010 Pa, and ν12 = 0.32. The material axes are assumed to be directed along the plate sides. Owing to symmetry of the problem, a quarter of the plate was discretized using mesh types shown in Fig. 23.3. Table 23.1 lists the dimensionless values of the central deflections of very thin and moderately thick plates. The present solution converges rapidly with the number of finite elements and agrees with the data available in the literature. For thin plate where L/h = 104 , the present solution is in excellent agreement with the value of central deflection wc = 7.492qL4/(D11 h3 ) calculated by the analytical formulas of Timoshenko and Woinowsky-Krieger [16]. For thick plate with the spanto-thickness ratio L/h = 8.696, good agreement is observed with the finite-element solution wc = 8.375qL4/(D11 h3 ) obtained by Dau et al. [7].

Fig. 23.3 Square plate and mesh types


347

Table 23.1 Central deflection of a simply supported orthotropic plate subjected to uniform loading wc D11 h3 /(qL4 )

Mesh

Mesh type

2×2

A B C

7.259 7.299 7.233

8.086 8.073 8.058

4×4

A B C

7.436 7.443 7.430

8.315 8.302 8.302

8×8

A B C

7.475 7.476 7.474

8.359 8.354 8.354

16 × 16

A B C

7.484 7.485 7.484

8.368 8.367 8.367

32 × 32

A B C

7.487 7.487 7.487

8.370 8.370 8.370

L/h

= 104

L/h = 8.696

23.12.2 Linear Analysis of Laminated Square Plates Under Uniformly Distributed Transverse Load A two-layer angle-ply square plate with clamped edges is subjected to uniformly distributed transverse load q. Each layer is made of an orthotropic material with the following characteristics [20]: E11 = 4 · 107 , E 22 = 106 , ν12 = 0.25, and G 12 = G 23 = G13 = 5 · 105. The plate dimensions are L = 20 and h = 0.02. It is worth noting that since the plate is very thin (L/h = 1000), transverse shear effects are negligible. We study the effect of fiber orientation on the laminate deflection using union-jack meshes. Due to asymmetry of deformation, the entire plate is discretized. Table 23.2 illustrates mesh convergence for the central deflections wC for various fiber angles. One can see that the convergence rate somewhat decreases as the angle of fiber orientation increases. Calculation results are compared with the results of Zhang and Kim [20] obtained by triangular element with 18 degrees of freedom.

23.12.3 Pure Bending of a Plate A cantilever plate of length l = 1 m, width b = 1 m, and thickness h = 0.001 m is loaded by bending moment M. This is a popular problem for testing the ability of nonlinear finite elements to reproduce behavior of flexible structures under finite rotations and zero membrane strains. To investigate nonlinear capabilities of the proposed finite element, we consider simple two- and eight-element models shown in Fig. 23.4. The material of the plate is assumed to be isotropic with Young’s

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Table 23.2 Central deflection of a two-layer fully clamped plate under uniform pressure for various fiber angles 1000

Fiber angle

E22 h3 wc qL4

4×4

8×8

16 × 16

32 × 32

±5◦

a) 1.1754 b) 1.242

1.0775 1.098

1.0526

1.0464

±15◦

a) 1.7617 b) 1.793

1.9493 1.950

1.9789

1.9863

±25◦

a) 1.8964 b) 2.148

2.4524 2.472

2.5531

2.5785

±35◦

a) 1.8282 b) 2.335

2.6652 2.729

2.8280

2.8696

a) 1.7879 2.7244 2.9109 2.95887 b) 2.394 2.808 Remark: a) present shell FE; b) plate element LDT18 of Zhang and Kim [20]

±45◦

Fig. 23.4 Plate under pure bending and mesh types

modulus E = 2 · 1011 Pa and Poisson’s ratio ν = 0. The load-displacement curves are compared with the exact theoretical solution in Fig. 23.5. One can see that even the two-element model with 10 degrees of freedom (mesh A in Fig. 23.4) provides accurate solution for finite bending of the plate where the tip rotation reaches approximately 57◦ for the load parameter M/D < 1, where D = Eh3 /12. For higher loads, however, nonsymmetric deformation of the two-element model becomes pronounced because of coarse mesh and divergence of the solution occurs when the tip rotation reaches 97◦ for M/D = 1.7. The numerical solution obtained by the eight-element model (mesh B in Fig. 23.4) is in excellent agreement with theoretical results for up to 360 degrees rotations where the plate is rolled up into a full circular cylinder.


349

Fig. 23.5 Moment versus tip displacements for the plate subjected to pure bending

23.12.4 Nonlinear Analysis of Laminated Square Plates Under Uniformly Distributed Transverse Load A simply supported four-layer symmetric cross-ply square plate is subjected to uniformly distributed load q. Material properties are characterized by the following ratios [20]: E11 /E 22 = 25, G12 /E22 = 0.5, G 23 /E22 = 0.2, ν12 = 0.25, and G 13 /G12 = 1, where E 11 = 172.3689 · 109 Pa. Due to symmetry, a quarter of the plate is modeled by union jack meshes. The boundary conditions for the model are such that all displacements and rotations about the vector normal to the plate edge are set equal to zero, the only nonzero rotation being about the edge. Tables 23.3 and 23.4 list the central deflections as functions of the load for thin and thick plates, respectively. The solution converges rapidly with the mesh refinement. For a coarse 2 × 2 mesh, the error in calculating the central deflection does not exceed 5%. The results are in agreement with the data obtained by LDT18 triangular element of Zhang and Kim [20] using 4 × 4 mesh but somewhat softer.

23.12.5 Snap-through of a Hinged Cylindrical Laminated Roof Figure 23.6 shows a shallow cylindrical three-layer roof subjected to a central force. Two longitudinal edges of the shell are restrained against all translations and the other two edges are free. The data are as follows [14]: R = 2540, L = 254, θ = 0.1, h = 6.35, E11 = 3300, E 22 = 1100, G 12 = 660, ν12 = 0.25, and G13 = G23 = 660. Two laminate sequences are considered: [0◦ /90◦ /0◦ ] and [90◦ /0◦/90◦ ]. A ply is of

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Table 23.3 Central defection wc /h for square simply supported [0◦ /90◦ /90◦ /0◦ ] plate under uniformly distributed load (L/h = 40) qL4 E22 h4

Zhang&Kim [20]

50

Present solution 2×2

4×4

8×8

0.2829

0.2788

0.2902

0.2933

100

0.4513

0.4443

0.4597

0.4638

150

0.5678

0.5567

0.5746

0.5790

200

0.6577

0.6430

0.6626

0.6674

250

0.7317

0.7137

0.7347

0.7398

Table 23.4 Central defection wc /h for square simply supported [0◦ /90◦ /90◦ /0◦ ] plate under uniformly distributed load (L/h = 10) qL4 E22 h4

Zhang&Kim [20]

50

Present solution 2×2

4×4

8×8

0.3317

0.3537

0.3632

0.3657

100

0.4922

0.5088

0.5198

0.5225

150

0.6002

0.6113

0.6232

0.6263

200

0.6834

0.6904

0.7030

0.7062

250

0.7521

0.7559

0.7691

0.7724

0◦ if its fibers are parallel to the longitudinal direction of the shell. All plies in the same laminate are equal in thickness. As in the most studies of this popular problem available in the literature, we consider doubly symmetric deformation. Owing to symmetry, one-quarter of the roof is modeled using an 8 × 8 union-jack mesh. Figures 23.7 and 23.8 show the load versus the deflections of two points A and B. One can see that the present solution is in excellent agreement with the benchmark solution of Sze et al. [14] shown by dark circles.

Fig. 23.6 Hinged cylindrical roof subjected to a central vertical load


351

Fig. 23.7 Load versus deflection for the cylindrical roof: laminate [90◦ /0◦ /90◦ ]

Fig. 23.8 Load versus deflection for the cylindrical roof: laminate [0◦ /90◦ /0◦ ]

23.12.6 Circular Ring Pinched by Four Loads To demonstrate the capabilities of the finite element in the presence of arbitrarily large spatial displacements and rotations, we consider a thin circular ring of radius R = 0.05 m pinched by four radial loads P. The cross section of the ring is rectangular with width b = 0.01 m and wall thickness h = 0.0005 m. The material is isotropic with Young’s modulus E = 2.0 · 1011 Pa and Poisson’s ratio ν = 0. The loaded nodes are allowed to displace only in the radial direction, which implies that rigid-body

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Fig. 23.9 Load versus radial deflection for the pinched ring

Fig. 23.10 Planar and spatial equilibrium states of the pinched ring

displacements of the ring are excluded. The ring was modeled using a union-jack mesh with two elements in the width direction and 96 elements in the circumferential direction. Figure 23.9 shows the dimensionless load versus the dimensionless deflection w/R at a loaded node. Solid curves refer to stable states of equilibrium and dashed curves to unstable. At the fundamental path, the ring assumes four-lobe symmetric configurations, which are stable only at the initial stage of loading. Upon attainment of the bifurcation point B1 , the ring buckles and becomes nearly oval but still doubly symmetric. At point B2 , new branch bifurcates from the fundamental path. This branch of unstable twisted configurations of the ring contains bifurcation points B3 and B4. At point B4 , this branch intersects a branch that refers to new planar configurations of the ring. It is interesting to note that, for P = 0, the ring is deployed into a ring of radius R/3. Some planar and spatial configurations marked in Fig. 23.9 are shown in Fig. 23.10. The spatial twisted configurations can be reproduced experimentally using a simple celluloid model compressed with the help of four threads (Fig. 23.11).


353

Fig. 23.11 Photographs of the spatial equilibrium states of the celluloid model

23.13 Concluding Remarks A nonconventional approach has been applied to develop a curved triangular finite element for linear and geometrically nonlinear analysis of shear deformable composite shells. The approach is based on invariants expressed in terms of normal stresses and strains in the direction of the element edges which are referred to as natural stresses and strains. The computational benefits of this approach are (1) concise formulation of the element with algorithmic and easily programmable formulas and (2) reduction of computations owing to the fact that no local coordinates are required. The main results of this study can be summarized as follows: • explicit relations between Cartesian and natural components of the tensors have been derived; • simple and elegant formulas have been obtained for the invariants in terms of natural components of the tensors; • the strain energy of a composite shell has been written in terms of the invariants, which is essential to formulate algorithmic technique for computing finiteelement parameters; • a curved triangular finite element with 15 degrees of freedom has been developed. Numerical testing has been performed to investigate accuracy, mesh convergence, shear locking behavior, and nonlinear capabilities of the shell element. Calculation results show that the element has excellent nonlinear bending characteristics, but its shortcoming is that membrane and transverse shear strains are constant. The finite element is shear locking free and can be used in the analysis of thin and moderately thick shells undergoing large elastic displacements and rotations. Acknowledgements The author wishes to express gratitude to Dr. V. V. Kuznetsov for useful discussions and suggestions.

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References 1. Argyris, J., Tenek, L.: Linear and geometrically nonlinear bending of isotropic and multilayered composite plates by the natural mode method. Computer Methods in Applied Mechanics and Engineering 113, 207–251 (1994) 2. Argyris, J., Tenek, L.: Natural mode method: A practicable and novel approach to the global analysis of laminated composite plates and shells. Applied Mechanics Review 49, 381–399 (1996) 3. Argyris, J., Tenek, L., Olofsson, L.: TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells. Computer Methods in Applied Mechanics and Engineering 145, 11–85 (1997) 4. Argyris, J.H., Dunne, P.C., Malejannakis, G.A., Schelkle, E.: A simple triangular facet shell element with applications to linear and non-linear equilibrium and elastic stability problems. Computer Methods in Applied Mechanics and Engineering 10, 371–403 (1977) 5. Bathe, K.J.: Finite Element Procedures. Prentice Hall, New Jersey (1996) 6. Bucalem, M.L., Bathe, K.J.: Finite element analysis of shell structures. Arch. Comput. Meth. Engng. 4(1), 3–61 (1997) 7. Dau, F., Polit, O., Touratier, M.: C1 plate and shell finite elements for geometrically nonlinear analysis of multilayered structures. Computers and Structures 84, 1264–1274 (2006) 8. Gal, E., Levy, R.: Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element. Arch. Comput. Meth. Engng. 13(3), 331–388 (2006) 9. Kuznetsov, V.V.: Shell theory based on invariants. Journal of Applied Mechanics and Technical Physics 32(5), 779–783 (1991) 10. Kuznetsov, V.V., Levyakov, S.V.: Kinematic groups and finite elements in deformable body mechanics. Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela 3, 67–82 (1994). In Russian 11. Kuznetsov, V.V., Levyakov, S.V.: Phenomenological invariant-based finite-element model for geometrically nonlinear analysis of thin shells. Computer Methods in Applied Mechanics and Engineering 196, 4952–4964 (2007) 12. Kuznetsov, V.V., Levyakov, S.V.: Geometrically nonlinear shell finite element based on the geometry of a planar curve. Finite Elements in Analysis and Design 44, 450–461 (2008) 13. Kuznetsov, V.V., Levyakov, S.V.: Phenomenological invariants and their application to geometrically nonlinear formulation of triangular finite elements of shear deformable shells. International Journal of Solids and Structures 46, 1019–1032 (2009) 14. Sze, K.Y., Liu, X.H., Lo, S.H.: Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elements in Analysis and Design 40, 1551–1569 (2004) 15. Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970) 16. Timoshenko, S., Woinowski-Krieger, S.: Theory of plates and shells, 2 edn. McGraw-Hill Inc., New York (1959) 17. Vlachoutsis, S.: Shear correction factors for plates and shells. International Journal for Numerical Methods in Engineering 33, 1537–1552 (1992) 18. Yang, H.T.Y., Saigal, S., Masud, A., Kapania, R.K.: A survey of recent shell finite elements. International Journal for Numerical Methods in Engineering 47, 101–127 (2000) 19. Yang, Y.B., Shieh, M.S.: Solution method for nonlinear problems with multiple critical points. AIAA Journal 28, 2110–2116 (1990) 20. Zhang, Y.X., Kim, K.S.: A simple displacement-based 3-node triangular element for linear and geometrically nonlinear analysis of laminated composite plates. Computer Methods in Applied Mechanics and Engineering 194, 4607–4632 (2005) 21. Zienkiewicz, O.C., Taylor, R.L.: The finite element method. Butterworth, Oxford (2000)

Chapter 24

Consistency Issues in Shell Elements for Geometrically Nonlinear Problems Teodoro Merlini and Marco Morandini

Abstract Some singular concepts and non-standard practices in the FEM solution of geometrically nonlinear shell problems are highlighted and discussed. In particular, four issues are addressed. (i) The question of the drilling rotation: a shell is essentially a non-polar medium in its tangent plane, so the drilling rotation is a redundant d.o.f. to be defined by an extra stress field, and the latter ought to hold as a primary unknown field of the surface mechanics. It is shown that a proper constitutive characterization and a sound variational formulation lead to a full micropolar setting of the shell mechanics with a true three-parametric rotation tensor. (ii) The interpolation of the orientation field on the shell surface. It is shown that an interpolation scheme firmly abiding by the rules of the SO(3) group leads naturally to frame-invariant and path-independent finite elements. (iii) The linearization of the virtual functional. Again, an approach fully complying with the special orthogonal group allows an easy and correct resolution of the mixed virtual-incremental variation variables that issue in nonlinear variational formulations involving finite rotations. (iv) The question of a good discrete representation of curved surface geometries. It is shown that a pole-based kinematics built on an integral orthogonal oriento-position field leads to a fair approximation of curved geometries and allows to build low-order finite elements that are naturally locking-free. Keywords Micropolar variational mechanics · Non-polar finite elasticity · Drilling rotations · Finite rotations and rototranslations · Multiplicative interpolation of rotations · Dual numbers and tensors · Helicoidal modeling · Nonlinear shell elements

T. Merlini (B) · M. Morandini Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 24, © Springer-Verlag Berlin Heidelberg 2011

355

356

T. Merlini and M. Morandini

24.1 Introduction Consistency is an asset for successful discrete approximations of continuum mechanics problems. An early advice for consistency is found in a short note by an eminent scientist in the field [23]: ... it may be of advantage, in the approximate solution of problems by the direct methods of the calculus of variations, to satisfy moment as well as force equilibrium equations approximately only, instead of satisfying one set exactly and the other approximately.

We find this sentence by Reissner (1965) true and topical. It raises a warning against the customary axiom that the stress tensor is symmetrical (indeed, this is true for common non-polar materials in absence of external body couples). When a finite element approximation takes this assumption implicitly, it is fulfilling exactly the equilibrium of the moments while is solving the forces equilibrium in weak form. Moreover, this consistency suggestion, advanced by Reissner in the years the first linear FEM codes were appearing, explicitly stresses the concern of treating consistently the whole equilibrium set—hence the relevant work-conjugate displacement and rotation fields—right when passing from the continuum variational mechanics to its numerically discrete approximations. Reissner himself proposed to apply such concept “to the derivation of approximate stress-strain relations for thin elastic shells”, being aware that any proposed through-the-thickness kinematics is the first discrete approximation in a solid shell element. The advice of Reissner was listened by several people that wrote important papers in the Nineties, e.g. [9, 24, 26, 27], often concerning shell mechanics. In those years, our approach to shell mechanics began by addressing first the 3D solid mechanics to get acquainted with the micropolar description. A constitutive characterization of the common non-polar medium was established and the relevant variational principles including the rotation field and a workless stress field were settled and exploited to build a first successful nonlinear solid element [13], so following Reissner’s advice. Then, in order to achieve a consistent description of discrete surface elements, a new modeling of the continuum was invented, where the particle position and orientation fields are coupled together. The geometrical representation of this so-called helicoidal modeling became affordable after discovering the dual tensor algebra [1]. The 3D variational principles were rewritten, a consistent interpolation established and a new solid element with nodal oriento-positions developed and successfully tested [15–17]. With the proper tools at hand, the computational shell mechanics was then addressed. Two levels of approximation are concerned in such an extreme geometry for a solid, a former one across the thickness [19] and a latter one on the reference surface [20]. The 3D variational mechanics developed for the non-polar medium is exploited in the former approximation, whose parameters include 2D strains and the parameters of the workless stress field. The outcome of this first step are the 2D constitutive equation and the relevant tangent map, to be used in the next step. Then, the latter approximation is accomplished using the interpolation developed in the 3D case and produces a solid shell element with four nodal oriento-positions

24 Consistency Issues in Shell Elements for Geometrically Nonlinear Problems

357

and one workless stress parameter. In either approximations we consistently exploit the helicoidal modeling [21]. The aim of this contribution is to highlight and discuss a few singular concepts and non-standard practices in the nonlinear shell element formulations, that have been elaborated along with the course of our research work in this area. They concern: in Sect. 24.2, the variational framework that yields a thorough micropolar setting of the material surface mechanics and consistently includes the drilling rotation; in Sect. 24.3, the consistent approximation of the angular part of the kinematic field on a surface element by an objective and path-independent interpolation scheme; in Sect. 24.4, the consistent linearization of the variational functionals to carry out before the discrete approximation process; in Sect. 24.5, the consistent modeling of the displacement field on curved surfaces by the proposed integral kinematics. As far as concerns the notation, boldface characters (aa, A , α , ...) are used for 1st and 2nd -order tensors, calligraphic bold capitals (A, ...) for 3rd -order tensors and blackboard bold capitals (A, ...) for 4th -order tensors. A superscript ( )S means either a symmetric tensor or the symmetric part of a tensor. The default tensor notation is index-free, but when necessary Latin indexes are used for the range 1–3 and Greek indexes for the range 1–2. The Einsteinian rule of implicit summation over repeated indexes is understood.

24.2 Thorough Micropolar Setting of the Surface Mechanics It should be noted that shells, unlike beams, are hybrid structured solids: in fact a shell, though capable of withstanding bending and torquing, is essentially a nonpolar medium in its tangent plane. So the pretense of withstanding elastically drilling rotations seems at least questionable. Rather, the drilling rotation is a redundant degree-of-freedom whose definition would entail an extra stress field. Our approach to achieve a complete and consistent micropolar setting of the 2D shell mechanics descends from the 3D solid mechanics formulated in the context of a full micropolar description.

24.2.1 Constitutive Characterization of the 3D Non-Polar Medium The characterization of the 3D non-polar medium, as proposed in [13] and restated in [17], is summarized here for the reader’s convenience. Start from considering first a hyperelastic micropolar medium, which features a strain energy w function of two strain tensor parameters [12], namely a linear strain ε and an angular strain β ; these parameters are often referred to as stretch or extension tensor and wryness or distortion tensor, respectively. The derivatives of the strain energy w(εε, β ) with respect to the strain parameters define the stress parameters Tˆ (εε , β ) = w/εε ˆ (εε, β ) = w/ββ ; such relations represent the (direct) hyperelastic constitutive and M

358


ˆ (εε, β ) equations of the micropolar medium. Inversion of functions Tˆ (εε, β ) and M ˆ ) and β (Tˆ , M ˆ ) and allows to introduce gives the inverse constitutive equations ε (Tˆ , M ˆ ) by means of a Legendre transform such the complementary-energy function v(Tˆ , M ˆ ) = εε, Tˆ + ββ, M ˆ , with , the notation of the scalar product. that w(εε , β ) + v(Tˆ , M ˆ are all working parameters for a micropolar Strains ε and β and stresses Tˆ and M medium. So we sketch: micropolar

w(εε, β ) ˆ) v(Tˆ , M

working params

ε and β ˆ Tˆ and M

Then, consider the existence of a particular micropolar medium whose strain energy w does not depend on the angular strain β . This constitutive postulate allows ˆ = w/ββ . We refer to such a medium as a pseudoto discard the couple-stress M polar medium. In the following sketch relevant to the pseudo-polar case, the linear strain and stress parameters have been split for convenience into their symmetric S and skew-symmetric parts, ε = ε S + (ax ε )× and Tˆ = Tˆ + τˆ ×,1 where ax() means the extraction of the axial vector of a tensor and τˆ denotes the axial vector of tensor Tˆ . So we write: pseudo-polar

εS

w (εε , ax ε , S β) S v (Tˆ , τˆ , )

working params

workless params

εS

ε and ax ε S Tˆ and τˆ

vanishing params

β ˆ →0 M

Finally, consider the existence of a particular pseudo-polar medium whose complementary energy v does not depend on the skew-symmetric part τˆ × of the stress Tˆ . This further constitutive postulate allows to discard vector ax ε = 12 v/τˆ and to S retain the symmetric part of the inverse constitutive equation as ε S (Tˆ ) = v S . /Tˆ

S ε S (Tˆ )

S

(εε S )

Inversion of function for the direct constitutive equation Tˆ allows to recover the strain-energy function w(εεS ) by means of a Legendre transform such S S S that w(εε S ) + v(Tˆ ) = εεS , Tˆ , and to write Tˆ (εεS ) = w/εεS . The strain energy of this medium depends on a symmetric stretch tensor (6 scalar parameters) like in classical elasticity, and this medium coincides with the common non-polar medium: non-polar

εS

w (εε , , S β) S v (Tˆ , τAˆ , )

working params

workless params

vanishing params

εS

β

ax ε → 0 ˆ →0 M

ε S Tˆ

τˆ

The foregoing constitutive characterization of the non-polar medium has important consequences in 3D continuum mechanics. With reference again to micropolar mechanics, consider the notation introduced in Table 24.1. The strain and stress 1

The notation v × denotes a skew-symmetric tensor and v is its axial vector. Tensor v × transforms w. a vector w into the cross-product vector v ×w


359

Table 24.1 Micropolar mechanical quantities and equations linear fields

angular fields

Geometry in the reference configuration: x , particle’s position vector

α , particle’s orientation tensor k a j , three angular curvature vectors

g j = x , j , three base vectors I = gj

⊗ g j,

the identity tensor

αT k a j )×) (from α T α , j = (α j k a = k a j ⊗ g , the angular curvature tensor

dxx dxx

ν , the boundary outward normal vector Geometry in the current configuration: x , particle’s position vector g j

=

x,

j,

α , particle’s orientation tensor k a j , three angular curvature vectors

three base vectors

F = g j ⊗ g j , the deformation gradient Particle’s kinematics: u = x − x , the displacement

dxx dxx

k a

αT k a j )×) (from α T α , j = (α = k a j ⊗ g j , the angular curvature tensor

Φ = α α T , the rotation

χ = F − Φ I , the linear strain

ω a = k a − Φ k a , the angular strain

Statics: f , t , the external force densities T´ , the stressa

c , m , the external couple densities ´ , the couple-stressa M

divT´ + f = 0, body linear balance T´ ν = t, boundary linear balance a

´ + 2ax(T´ F T ) + c = 0, body angular balance div M ´ ν = m , boundary angular balance M

analogs of the 1st Piola–Kirchhoff stress tensor

parameters are related to the mechanical fields of the kinematic deformation and of the stress and couple-stress by ε = ΦTχ = ΦT F − I , Tˆ = Φ T T´ ,

β = Φ T ω a = Φ T k a − k a , ˆ = ΦT M ´; M

(24.1a) (24.1b)

refer to [22] for a comprehensive analysis. Tensor Φ T F = I + ε is usually referred to as the Cosserat deformation tensor in micropolar elasticity. In the case of non-polar medium, a first noteworthy outcome follows from the ´ = ΦM ˆ : the angular condition w/ββ = 0, hence the vanishing of the couple-stress M T ´ equilibrium equations are reduced to the algebraic form 2ax(T F ) + c = 0 and to m = 0. Therefore, the non-polar medium allows for external body couples c which are balanced by the skew-symmetric part of the analogue of the Cauchy stress tensor (det F )−1 T´ F T , but it cannot be loaded by external couples m at the boundary. A second important outcome follows from the condition v/ˆτ = 0, i.e. the vanishing of the skew-symmetric part of the strain parameter ε : the compatibility condition (24.1a)1 splits into its symmetric and skew-symmetric parts and is reduced to

360


ΦT F − I )S , and an equation that forces to a condition on symmetric tensors, ε S = (Φ ΦT F ) = 0. The latter equazero the right-hand-side of the skew-symmetric part, ax(Φ tion represents a statement of the polar decomposition theorem of the deformation gradient and holds as a constitutive equation of the foregoing model of non-polar medium under micropolar description, as it constitutes the definition of the material rotation—a redundant d.o.f. in classical elasticity. A third important feature is the presence of a workless stress field, vector τˆ . Tensor Tˆ from (24.1b)1 is called the Biot stress tensor [2, 6] and is in general a non-symmetric tensor—apart from the case of isotropic media, for which a symmetrical Biot stress is expected in absence of external body couples [6]. However, it is commonly recognized that the strain energy of classical non-polar media depends on the six components of a symmetric strain tensor, so the stress parameter must be symmetrical. In the proposed characterization of the non-polar medium, just the S symmetric part Tˆ of the Biot stress works elastically, whereas its axial vector τˆ , hereafter referred to as the Biot-axial, must be retained as a workless stress field, as its contribution to the equilibrium may be significant. The primary unknown fields of the minimal formulation (the so-called one-field, or displacement formulation) of the non-polar medium are the displacement u , the parameter ϕ of the rotation (we assume the rotation vector as parameter, such that ϕ×)) and the Biot-axial τˆ . The problem is governed by the field equations Φ = exp(ϕ divT´ + f = 0,

2ax(T´ F T ) + c = 0,

ΦT F ) = 0, ax(Φ

(24.2)

and by the natural boundary conditions T´ ν = t and m = 0. The weak form of the problem follows by standard variational procedure and results in the internally constrained Principle of Virtual Work Πδ = Πint δ + Πextδ = 0, with the internal virtual work given by % & Πintδ = δ w + ˆτ×, ΦT F dV. (24.3) V

ΦT F

In (24.3) the contribution ˆτ ×, Φ F from the internal constraint adds to the strain energy. It should be noted, however, that this term descends in a natural way from the constitutive characterization of the non-polar medium and is not introduced as an appended constraint, as e.g. in [26].

24.2.2 A Linear Example When the current configuration is so close to the reference configuration that the deformation can be taken as infinitesimal, and linear stress-strain relations are assumed as well, a linear problem follows. Equations (24.2) of the so-called one-field formulation are reduced to the equations T S − curlττ + f = 0, divT

2ττ + c = 0,

ϕ = 0, curl u − 2ϕ

(24.4)


361

T S + τ ×)νν = t . For isotropic materials, the and to the natural boundary condition (T only symmetric stress tensor is given by T S = λ (tr ε S )II + 2μ ε S , with λ and μ the Lamé moduli and ε S = gradS u . As an example, consider the disk of Fig. 24.1. It is restrained along the circular boundary and is loaded with body fields of tangential force f and planar couple c . Assume a centered triad I = i j ⊗ i j as absolute reference frame so to measure the position vectors as x = x j i j . With reference to a convected system of three dimensionless coordinates ξ j in the radial (0 ≤ ξ1 ≤ 1), circumferential (0 ≤ ξ2 < 2π) and axial (−1 ≤ ξ3 ≤ 1) directions, respectively, the absolute coordinates read x1 = Rξ1 cos ξ2 , x2 = Rξ1 sin ξ2 , x3 = hξ3 and the covariant base vectors are given by g 1 = R(ii1 cos ξ2 + i 2 sin ξ2 ), g 2 = Rξ1 (−ii1 sin ξ2 + i 2 cos ξ2 ) and g 3 = h i 3 . As external loads, a uniform force field and a linearly varying couple field are considered, with densities f = f0 (−ii1 sin ξ2 + i 2 cos ξ2 ),

c = ((1 − ξ1)c0 + ξ1 c1 ) i 3 ,

(24.5)

respectively. They result in an overall couple of value 43 πR2 h(R f0 + 12 c0 + c1 ) and are balanced by a tangential reaction of density t = t (−ii1 sin ξ2 + i 2 cos ξ2 ), with t = − 13 (R f0 + 12 c0 + c1 ). Equation (24.4)2 is algebraically solved for the stress-axial τ = − 12 c. Then, the problem for the displacement u is governed by the differential equation (24.4)1 together with the natural boundary condition and the boundary constraint u = 0 at ξ1 = 1. The solution of type u = v(ξ1 ) g 2 leads to the ordinary differential problem ξ1 v,11 +3v,1 +3L = 0, where L=

with v = 0 and v,1 = −L for ξ1 = 1, R f0 − 12 (c1 − c0 ) 3μ

is a dimensionless load parameter. Function v = L(1 − ξ1 ) satisfies the equations, and the rotation field is then recovered from (24.4)3 as ϕ = 12 curluu. The final result is

Fig. 24.1 Disk under volume force and couple as external loads.

362


u = u (−ii1 sin ξ2 + i 2 cos ξ2 ),

u = LRξ1 (1 − ξ1 ),

ϕ = ϕ i3,

ϕ = L(1 − 32 ξ1 ),

τ = τii3 ,

τ=

(24.6)

− 12 ((1 − ξ1)c0 + ξ1 c1 ).

Strain and stress are recovered as T S = 2μ ε S ,

% & 1 ε S = − Lξ1 cos 2ξ2 (ii1 ⊗ i 2 + i 2 ⊗ i 1 ) + sin2ξ2 (ii2 ⊗ i 2 − i 1 ⊗ i 1 ) . 2

Plots of the solution (24.6) are shown in Fig. 24.2. In absence of body couples (left plot), the response is a tangential displacement u as expected and a rotation ϕ. The latter decreases linearly from the disk center and changes sign at ξ1 = 23 . The stress-axial τ is null everywhere. When only body couples are present, instead (right plot), the disk still deforms in general with both a tangential displacement u and a rotation ϕ but the stress-axial τ is never null. With c0 = 240 and c1 = 0 (L = 0.04) we obtain the same response as in the left plot; however τ is not null, in this case. In the particular case of uniform couple ( f0 = 0 and c1 = c0 , i.e. L = 0), displacement and rotation are null everywhere and the stress-axial is uniform. It is worth noting that we could face the same example problem from the classical displacement method, forcing the linear balance as in (24.4)1 and the angular balance by means of (24.4)2 to account for the body couples. We would obtain the same solution as in (24.6) for the unknowns u and τ, but we could never determine the rotation ϕ. However, it may be observed that such an approach would look somewhat inconsistent after considering that in a variational context the natural work-conjugate multipliers of equations (24.4)1 , (24.4)2 and (24.4)3 are virtual displacements, rotations and stress-axials, respectively. In fact, if we face the problem from a classical point of view, we are making use of just (24.4)1 and (24.4)2 whereas on) i 0i

d 0r 0d

K

i 0−Ri

K

rn)

ldi

X

rnd

l 0i

X

lii

bi l−i

dn)

−Ri

mdn)

U

mrnd

dno

di

li

U

dnd

mrn) d

−ii

dng dn 0 r

Y

dnR

ri

ia−

-

0

0 0sp

ond

−di d 0rno

i l 0i bi l−i li −Ri

iaR iab 0 l

Y

di lii

ia

ldi l

Fig. 24.2 Disk data: R = 100, h = 0.5, μ = 1000. Solution under the force alone (left plot, L = 0.04) and under the couple alone (right plot, L from 0.04 to 0.). Markers denote FE results, see Sect. 24.2.4.


363

we are retaining as unknowns the displacement and the stress-axial, work-conjugate respectively to (24.4)1 and (24.4)3 , instead. So this example stresses the consistency of the constitutive characterization of the non-polar medium sketched in Sect. 24.2.1 and of the relevant continuum mechanics based on three primary unknown fields.

24.2.3 Micropolar Surface Mechanics Shells are 2D structured solids within a 3D space. They are able to withstand outof-plane distortions by opposing couple-stresses, but if made of non-polar material they are unable, like plates, to withstand in-plane distortions. So, they behave essentially in a non-polar fashion in their tangent plane. Now, the proposed constitutive characterization of the 3D non-polar medium leads to a micropolar formulation of continuum mechanics that involves the rotation as unknown field and is able to withstand body couples. Hence, an analogous concept is expected to yield a complete micropolar formulation of the 2D shell mechanics that consistently involves a full three-parametric rotation unknown. Of course, such a formulation shall involve an extra surface-stress field in charge of defining the drilling degree-of-freedom. As usual in shell modeling, two approaches are envisaged. The direct approach tends to ignore the mechanics within the shell thickness and applies the principles of the 3D realm to formulate the intrinsic mechanics of the material surface. Formulations of this kind are often referred to as Cosserat surface models. Direct approaches to micropolar surface mechanics with unrestrained rotations are used e.g. in the early paper by Zhilin [31], endowed with a modern notation for rotational kinematics, and in the finite-element formulation by Sansour and Bednarczyk [24], based on the Kirchhoff–Love shell theory. No mention to a workless surface-stress field is found in literature except for [14], where a constitutive characterization of a nonpolar-type surface on the steps of Sect. 24.2.1 is outlined and a surface Biot-axial is consistently introduced. However, the usefulness of such an approach is questionable, as the interpretation of the surface Biot-axial itself rises new difficulties that add to the lack of constitutive laws proper to direct approaches. Alternatively, the solid shell approach is recommended. First, a 3D mechanical formulation is applied to the shell body, then the problem is reduced to a 2D model by variational means with integration over the thickness. Several solid shell approaches to micropolar surface mechanics with unrestrained rotations have appeared in literature, e.g. [7, 10, 25]. In general, solid shell approaches do not resort to a workless surface stress to define the drilling degree-of-freedom, but noteworthy exceptions are the papers by Wisniewski and Turska [27–30] and by Badur and Pietraszkiewicz [3]. In the latter paper a Kirchhoff–Love shell theory is assumed and a 2D workless stress field is introduced as Lagrange multiplier of the constraint on the symmetry of the strain. In the authors’ solid shell approach [19], the minimal formulation for non-polar materials governed by equations (24.2) in the unknowns u , ϕ and τˆ is assumed for the 3D mechanics within the thickness. A variational setting leads to the internal

364


virtual work (24.3), and the 2D shell model is obtained after integrating over the thickness the integrand in (24.3). This reduction is performed in a finite-element-like sense and implies the choice of approximate fields for both the kinematic variables (displacement and rotation) and the Biot-axial stress vector. Different shell theories can be incorporated according to the hypotheses of the approximate kinematic field, including the classical Kirchhoff–Love and Reissner–Mindlin kinematics. In [19] a new shell theory is developed and adopted (see Sect. 24.5.2) and a linear Biot-axial field across the thickness is assumed. Of course, the parameters of the throughthe-thickness Biot-axial field hold as primary unknowns of the 2D shell mechanics besides the appropriate kinematic unknown fields. The 2D model ensues straightforwardly equipped with the shell constitutive equations, which relate the 2D strains and Biot-axial parameters to the relevant work-conjugate fields in a coupled way. Thus, a thorough nonlinear micropolar formulation is established for the material surface mechanics, including drilling rotations.

24.2.4 A Nonlinear Example The example disk of Sect. 24.2.2 is analyzed by the helicoidal shell element developed in [20]. The element geometry is a curved quadrilateral surface whose approximate configuration is interpolated among the oriento-positions of four corner nodes (see Sects 24.3 and 24.5); the formulation is based on 4 × 6 nodal kinematic unknowns and a unique Biot-axial vector parameter. The disk is modeled by an axisymmetrical mesh with 8 elements in the radial direction and 32 elements in the circular direction (the elements of the internal circle degenerate into triangles). First, a De Saint Venant–Kirchhoff material with Lamé moduli λ = 1500 and μ = 1000 (i.e. E = 2600 and ν = 0.3) is used to match the linear analysis of Sect. 24.2.2. Force and couple densities are set at the nine quadrature points of each element to reproduce the loads (24.5). We ran two nonlinear computations (5 iterations to converge) with data f0 = 1.2, c0 = c1 = 0 and f0 = 0, c0 = 240, c1 = 120 respectively. The results are marked in Fig. 24.2 and agree well with the analytical ones. Then, a full nonlinear computation is addressed using a constitutive model of Neo-Hookean type, referred to as Pseudo-Hookean hyperelastic law in [18]. It descends from the strain-energy function 1 U ) = λ(ln det U )2 − 2μ ln detU U + 2μ tr (U U − I ), w(U 2 with U = I +εεS the symmetric Cosserat deformation tensor. A nearly incompressible material is chosen with moduli λ = 4999000 and μ = 1000, that means a Poisson ratio ν = 0.4999. A variable radial element size with a geometric progression of common ratio 1.2 allows to refine the mesh in the center zone. The target load is a drilling body couple linearly varying along the radius from c0 = 4800 to c1 = 0. A clipping of the deformed configuration is displayed in Fig. 24.3 and plots of the


365

Fig. 24.3 Nearly incompressible disk. Undeformed mesh (straight radial lines) and deformed mesh under drilling body couples at c0 = 4800, c1 = 0 Table 24.2 Nearly incompressible disk. Planar displacements and rotations (degrees) under drilling body couples at c0 = 4800, c1 = 0 Radial position

u

v

ϕ

0.00

0

0

72.81

6.06

-4.29

4.62

67.18

13.33

-8.33

9.94

57.65

22.06

-11.35

15.60

45.89

32.54

-12.56

20.76

32.41

45.10

-11.43

23.98

18.47

60.18

-8.05

23.23

4.33

78.28

-3.60

16.44

-8.43

100.00

0

0

-25.33

element thicknesses and Biot-axials in the radial direction are given in Fig. 24.4. Displacement and rotation values are listed in Table 24.2.

]

3

D

]15

5

−15

−

3

[ i−]]]

i

−

o !oooo2]]] ]

−15

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d]

2 −]]

Fig. 24.4 Nearly incompressible disk. Thickness and Biot-axial at c0 = 4800, c1 = 0

2

] ]

3]

K

5] o ooooosi

r]

Fig. 24.5 Nearly incompressible disk. Drill load-deflection curve

(]

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The load-rotation curve of Fig. 24.5 is drawn in 20 equal load steps with 5–6 Newton–Raphson iterations per step. This problem is not highly nonlinear and the target load can also be reached quickly in four load steps for a total of 45 iterations. The helicoidal shell element works without resorting to reduced integration nor other numerical expedients widely adopted in shell analysis like incompatible modes, assumed natural strains, enhanced assumed strains.

24.3 Consistent Approximation of the Element Kinematic Field It is common practice in finite-element technology to interpolate positions in the Euclidean vector space by looking for the weighted average position vector from the equation N W J (xx − x J ) = 0 . (24.7) J=1

Solution of (24.7) for normalized weights yields x = NJ=1 W J x J in closed form. This interpolation scheme allows to build the metric within the element in the reference configuration and the position vector x and the deformed metric in the current configuration. Thereby, in the linear vector space, the unknown displacement u = x − x can be straightforwardly interpolated from the nodal displacements as u=

N

WJ u J ,

(24.8)

J=1

leading to the isoparametric concept. All in all, the interpolation of the position field is a rather plain task. The interpolation of the orientation field, instead, is a demanding task. Orientations can be represented by the orthogonal tensors α = α j ⊗ i j that orient local orthonormal triads α j with respect to the self-reciprocal absolute reference frame I = i j ⊗ i j . Orientations belong to the special orthogonal group SO(3) and compose multiplicatively: the current orientation α is the reference orientation α rotated by tensor Φ , i.e. α = Φ α . Being an orthogonal tensor, the rotation is defined by three independent parameters; for simplicity, in the following discussion we rest on the vectorial ϕ×). parameterization defined by the exponential map Φ = exp (ϕ The simplest and straightest interpolation scheme is to interpolate the total nodal rotation vectors ϕ J just like nodal displacements are interpolated in (24.8), i.e. ϕ ϕ×) and recover the current = NJ=1 W J ϕ J , then build the rotation tensor Φ = exp (ϕ orientation as α = Φ α . It is immediate to verify that such an interpolation scheme is not frame-invariant [16]. Other approaches propose to interpolate the incremental rotation vector ϕ ∂ = N J=1 W J ϕ J∂ from the incremental nodal rotation vectors ϕ J∂ computed at the th ϕ∂ ×) α k . k configuration and recover the current orientation as α k+1 = exp(ϕ th The k configuration is either defined as the last converged configuration or the


367

configuration obtained so far during the iterative solution process. It is possible to design this type of interpolation schemes in such a way that no spurious strains are added for a rigid motion [11]. However, as first explicitly noted in [8], this type of interpolation is path-dependent. In fact, this type of interpolation naturally leads to an approximate numerical integration of the angular strain β k+1 = β k + ∂ββ (a leading strain in shell elements), for finite incremental rotation vectors ϕ ∂ . As a proof, one could design a complex load sequence in such a way to reach a final configuration coincident with the reference one, with null final stresses and strains; still the simulation will end with a final configuration that differs from the initial one [8, 11]. Of course, path-dependence can be an acceptable feature in computational mechanics, as a high precision level can be always achieved by raising the number of load steps, thus reducing the error implied in the angular strain integration. However, it is much preferable to avoid path-dependence altogether. These drawbacks, either path-dependence or frame-non-invariance, affect the above mentioned interpolation schemes mainly because they were designed keeping in mind a well-established scheme that works so charmingly when interpolating positions. One can indifferently interpolate the current positions (Eq. (24.7) applied to the current configuration) or the displacements (Eq. (24.8)) just because positions belong to a linear vector space, i.e. because u = x − x . With isoparametric elements, this fact introduces some ambiguity when defining the unknowns of the problem: the problem is often stated assuming as unknown the displacement from the reference configuration, while in the authors’ opinion the deformed configuration should be intended as the real unknown instead. This distinction is almost academic when interpolating positions, but is important when dealing with orientations: one cannot use the same interpolation scheme for the reference orientation field α, the current orientation field α and the rotation field Φ, because the interpolated orientation α will differ, in general, from Φ α [16] (Fig. 24.6). In this sense, interpolating orientations and rotations with the same scheme is inconsistent. uuJ ≡ x − x

α1

x1 u1

x1

ΦΦJ = α αT

x xJ uuJ

xxJ

x2 u2

Φ1

α1 x2

Φα

α αJ

α2

ΦΦJ

ααJ

Φ2

α2

Fig. 24.6 Interpolation of the configuration vs. interpolation of the change of configuration: displacement (left) and rotation (right). The notation a aa J means a as interpolated from the a J (J = 1, ... N).

Consistency can be achieved after removing definitely the pretense to interpolate rotations: build an interpolation scheme for either orientation fields and a posteriori recover the rotation as Φ = α α T . Just like Eq. (24.7) stems from the additive

368


composition in the linear vector space, the sought interpolation scheme should properly account for the multiplicative composition of rotations. The analogue of (24.7) for the orientation field is thus [16] N

αα TJ ) = 0 . W J log (α

(24.9)

J=1

The implicit Eq. (24.9) does not yield in general the sought weighted average orientation α in closed form, but gives a nonlinear problem to solve numerically [16]. Then, the angular curvature is obtained as well, and the local rotation Φ = α α T and the angular strain β = Φ T k a − k a are recovered by comparing the interpolated quantities in the current and in the reference configurations. This leads to the correct equivalent of an isoparametric element for rotations: the reference and the current orientation fields (but not the rotation field) are approximated with the same interpolation scheme. The proposed interpolation is frame-invariant and path-independent at the same time. Equation (24.9) averages relative orientations (just like Eq. (24.7) averages relative distances), and this fact ensures frame invariance. The mere fact of interpolating orientations instead of rotations ensures path independence.

24.4 Consistent Linearization and Discrete Approximation Despite a large attention paid, in the finite-element literature, to linearize consistently the variational principles, a careful account of the properties of the special orthogonal group still raises some consistency issues also in this field. Usually, the linearization is performed on the previously approximated principle, however let us address the linearization of a virtual functional in its continuous form. For the present purpose, we may limit ourselves to focus on just the contribution ) to the variational principle from the strain energy, namely the term Πw δ = V δw dV in Eq. (24.3). The strain energy w(εεS ) is a function of the symmetric strain parameS ter; its first and second variations define the stress parameter Tˆ (εε S ) and the elastic S S SS SS tensor Eˆ (εεS ) from δw = δεεS : w/εεS = δεεS : Tˆ and ∂ Tˆ = Eˆ : ∂εεS . In the present minimal formulation, the strain parameter is a short name for the relevant kinematic ΦT F − I )S . Therefore, the virtual functional Πw δ and its strain to match, i.e. εS ≡ (Φ (incremental) variation ∂Πw δ can be written )

S

ΦT F ) : Tˆ dV, δ(Φ ) S SS ΦT F ) : Eˆ : ∂(Φ ΦT F ) : +∂δ(Φ ΦT F ) : Tˆ dV. ∂Πw δ = V δ(Φ Πw δ =

V

(24.10)


369

A consistent development of (24.10) leads to the following expressions [13], Πw δ =

V

⎧ ⎪ ⎪ ⎨ F : ϕδ· ⎪ δF ⎪ ⎩

⎫ ⎪ ⎪ ⎬ dV, S T ⎪ ⎭ ˆ ΦT F ) ⎪ −2ax (Φ Φ Tˆ

S

(24.11a)

⎡ ⎤⎧ ⎫ SS ⎢⎢⎢ Φ Eˆ SS Φ T ⎪ ⎪ ΦEˆ Φ T : F ×T321 ⎥⎥⎥⎥ ⎪ FT ⎪ −Φ ⎨ : ∂F ⎬ ⎢ ⎥⎥⎦ ⎪ ∂Πw δ = dV F : ϕ δ · ⎢⎢⎣ δF ⎪ SS T SS T ⎪ × × ×T321 ⎩ ⎭ ˆ ˆ F : ΦE Φ F : ΦE Φ : F −F · ϕ∂ ⎪ V ⎡ ⎤⎧ ⎫ ⎥⎥⎥ ⎪ ˆ S )×T231 ⎢⎢⎢ ⎪ ⎪ Φ FT ⎪ O (Φ T ⎨ : ∂F ⎬ ⎢ ⎥ ⎢ ⎥ + F : ϕ δ · ⎢⎣ δF ⎪ S ×T132 S T S S T ⎥⎦ ⎪ ⎪ ⎪ dV ⎩ ˆ ˆ ˆ ΦT ) ΦT F ) − I tr (Φ ΦT F ) · ϕ∂ ⎭ (Φ (Φ V ⎧ ⎫ S ⎪ ⎪ ⎪ ⎪ Φ Tˆ ⎨ ⎬ + dV, (24.11b) F : ϕ ∂δ · ⎪ ∂δF ⎪ S ⎪ T ⎩ ⎭ ˆ ΦT F ) ⎪ −2ax (Φ V

A · g j ) × ⊗ g j denotes a 3rd -order tensor of skew-symmetric nature where A × = (A built from tensor A , and ( )Txyz denotes the transposition of the dyadic legs x, y and z in the order they appear. O is the 4th -order null tensor. The development leading to (24.11) is consistent in the sense that the variations of orthogonal tensors are performed respectfully of the SO(3) environment. It is worth noting that the expressions of the rotation variations used in (24.11) are parameterization-free. Independently of any parameterization, the differential of an orthogonal tensor is characterized by a differential rotation vector, and the skewsymmetric parts of subsequent differentials are characterized by further differential rotation vectors independent of the preceding ones. Independently of any parameterization, we like to denote such characteristic vectors by the symbol of the rotation vector itself with a subscript that recalls the variation in hand. Thus [15], ΦΦ T = ϕ δ ×, ∂Φ ΦΦ T = ϕ ∂ ×, ∂δΦ ΦΦ T = ϕ ∂δ × + δΦ

( 1' ϕ∂ × ϕ δ × , (24.12) ϕ δ × ϕ ∂ × +ϕ 2

with ϕ δ , ϕ ∂ and ϕ ∂δ independent differential rotation vectors. It follows that there is no means to solve directly the second differential rotation vector ϕ ∂δ for the first ones ϕ δ and ϕ ∂ . Moreover, no approximation of the unknown fields has been introduced in (24.11) yet. So Eqs (24.11) actually belong to the continuum mechanics realm and not to the finite-element technology: the contribution Πw δ + ∂Πw δ to the linearized variational principle is provided by Eqs (24.11) independently of the discrete approximation chosen to formulate a finite element. However, the nucleuses of the residual and the tangent matrix to exploit in an iterative solution process are recognizable in Eqs (24.11a) and (24.11b), respectively; the latter nucleus comes with the separate contributions from the elastic tensor and from the pre-stress state. The discrete approximation of the unknown fields is often set as a linear function of the global (nodal) variables. In (24.11) a linear interpolation is appropriate when computing the deformation gradient F that belongs to the Euclidean vector space,

370


but not for the rotation field that is subject to the SO(3) rules, as discussed in Sect. 24.3. To be used within (24.11), a nonlinear interpolation must be linearized itself in turn. If we resort to the scheme governed by Eq. (24.9), this linearization process yields the interpolation formulas ϕδ =

N

Y J · ϕ Jδ , ϕ ∂ =

J=1

N K=1

Y K · ϕ K∂ , ϕ ∂δ =

N J=1

Y J · ϕ J∂δ +

N N

Y JK : ϕ Jδ ⊗ ϕ K∂ ,

J=1 K=1

(24.13) as shown in [16] (where the curvature variation variables are also given). In (24.13) the local variation variables are linear function of the global variation variables, and it is seen that the mixed differential rotation vector ϕ ∂δ is controlled by the virtual and incremental nodal rotation vectors ϕ Jδ and ϕ K∂ and by the mixed virtualincremental nodal rotation vectors ϕ J∂δ . Application of Eqs (24.13) to Eqs (24.11), together with appropriate linear interF , ∂F F , ∂δF F , and integration over the element domain gives polation formulas for δF the contribution to the discrete linearized variational principle in the form ⎤ ⎧ ⎧ ⎫T ⎧ ⎫ ⎧ ⎫T ⎡ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ δuu J ⎪ ⎬ ⎪ ⎨ FuJ ⎪ ⎬ ⎪ ⎨ δuu J ⎪ ⎬ ⎢⎢⎢⎢ K uu JK K u ϕ JK ⎥⎥⎥⎥ ⎪ ⎨ ∂uu K ⎪ ⎬ ⎥⎥⎦ · ⎪ Πw δ + ∂Πw δ = ⎪ ·⎪ +⎪ · ⎢⎢⎣ T + ϕ J∂δ · F ϕ J , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ ⎭ ⎩ F ϕJ ⎭ ⎩ ϕ ⎭ ⎩ ⎭ K K ϕ ϕϕ JK Jδ Jδ K∂ uϕ JK (24.14) that involves the virtual and incremental variations of the nodal displacements u J . It is seen that no mixed virtual-incremental displacement variations are there, as we are linearizing Πw δ with respect to the discrete unknowns u J , not the independent F = 0.) The mixed virtual-incremental rodiscrete field δuu J . (That also means ∂δF tation vectors ϕ J∂δ are there instead, and must be solved for ϕ Jδ and ϕ J∂ before assembling the problem matrix. The resolution of ϕ J∂δ is possible at each single node, where the rotation vector ϕ J is an actually free variable. As explained in [16, 17], using the differential map ϕ J , the resolution formula of the rotation and discarding of course the term with ∂δϕ is easily obtained, ϕJ + Γ J Γ −123 : ϕ Jδ ⊗ ϕ J∂ , ϕ J∂δ = Γ J · ∂δϕ IIIJ −123

(24.15)

ϕ J ) and Γ IIIJ (ϕ ϕ J ) are the first and second differential mapping tensors2 where Γ J (ϕ ϕ J ×). Note associated with the exponential map of the total nodal rotation Φ J = exp (ϕ ϕ that the local analogue of (24.15) gives the expression for the mixed variation ∂δϕ −123 −1 = Γ ϕ ∂δ − Γ III : ϕ δ ⊗ ϕ ∂ , which is not null, of course. Using (24.15), Eq. (24.14) becomes a linear function of just the virtual and incremental variation variables, with a symmetrical contribution to the tangent matrix. It is worth noting that an attempt to arrive at (24.14) by first discretizing Eq. (24.11a) by means of Eq. (24.13) and then linearizing the result obtained, would be a quite hard and very insidious task. Also note that the foregoing discussion 2

−123

Each Γ IIIJ is a 3rd -order tensor symmetric in the rightmost dyadic legs 2 and 3.


371

applies as is to 3D variational mechanics. In the case of shell elements, the same concepts should apply first to the integration across the thickness to compute the shell constitutive equations and the relevant tangent map, then to the integration on the element surface. Note that the latter integration will involve also the angular curvatures and the relevant variation variables, a fact that in turn involves a third variation of the rotation tensor [15] to be appended to Eqs (24.12) and unavoidably complicates things further.

24.5 Consistent Modeling of the Displacement Field on Curved Surfaces Consider a low-order rectangular shell element with four corner nodes equipped with rotational d.o.f.’s. In the reference configuration the element is flat, with two vectors of the nodal triads aligned with the element sides and the third vector normal to the element plane. At a certain converged solution, the current configuration is represented by the corner nodes still lying on a plane with orientations as shown if Fig. 24.7 (left). Using the interpolation scheme of Eqs (24.7) and (24.9) with bilinear

Fig. 24.7 Shell element kinematic interpolation. Uncoupled interpolation of positions and orientations (left) vs. helicoidal interpolation (right).

weights, an internal triad is positioned and oriented as in the left picture. However, the nodal orientations made it clearly understood that the actual configuration of this piece of surface would be closer to the right picture than to the left one. Note that the interpolated orientation is identical in both pictures, whereas the interpolated position is different. In the picture on the right, the interpolated position is “driven” by the nodal orientations. Is there another consistency issue? Let us explain how we drew the picture on the right. We need dual tensors: a sort of complex tensors that obey the algebra of dual numbers [1]. A dual tensor (like a complex number) is composed as the sum of a primal part and a dual part, the latter being multiplied by the dual unit E. The dual unit is a nilpotent element such that E2 = E3 = . . . = 0 in any computations. Consider for instance the arm operator X = I + E x ×, a dual tensor made of the identity as primal part and of the cross product

372


of the position vector as dual part. X is a special dual tensor, in fact it is orthogonal, X −1 = X T . A way of describing by a single geometric entity the position and the orientation of a frame is by means of the so-called oriento-position tensor [15], a dual tensor A = X α = α + E x × α made of the orientation tensor itself as primal part and of its moment with respect to the origin as dual part. Note that A is an orthogonal dual tensor itself and represents the rototranslation of the frame from the absolute reference frame I . When a shell particle moves from its reference oriento-position A to the current oriento-position A , it undergoes a rototranslation H = A A T = X Φ X T , another orthogonal dual tensor. Thus, rototranslations compose multiplicatively just like rotations do. Therefore, an interpolated oriento-position on the element domain can be built in a very consistent way by a straightforward extension of the interpolation equation (24.9) to the dual case [16], N

A A TJ ) = 0 . W J log (A

(24.16)

J=1

Note that positions and orientations are coupled together in this interpolation scheme, so the nodal orientations do affect the interpolated position. The picture on the right of Fig. 24.7 was drawn this way.

24.5.1 Helicoidal Parameterization and Helicoidal Modeling A consistent modeling of the position field on curved surfaces (Fig. 24.7) was the motivation to further deepen the representation of motion of the particles of a deforming structured solid in discrete approximations. The rototranslation is an orthogonal dual tensor depending on six parameters. A widely used parameterization decomposes the rototranslation into two separate and independent motions, displacement and rotation; the latter is in turn parameterized mostly by means of the eigenvector of the rotation tensor (additive parameterizations like subsequent Euler angles are generally dismissed in computational mechanics). However, the rototranslation is found to be represented as the exponential map H = exp (ηη×) of a skew-symmetric dual tensor built on the dual vector η = ϕ + E $ , called the helix [5]. Hence, it is immediately seen that the helix is the natural eigenvector of the rototranslation, and as such is a favorite candidate for an appropriate integral parameterization. The helicoidal parameterization of motion is a rightful alternative to the classical additive decomposition into subsequent displacement and rotation: these motions become coupled together into a synchronous motion. Rototranslation and helicoidal parameterization inherit all the properties of the rotation and its vectorial parameterization; in particular, the analogs of Eqs (24.12) for the dual case hold true as well, and we may observe that differential helices control the tangent space of rototranslations.


373

The same parameterization may apply to the modeling of the structured solid itself. In fact, the oriento-positions of any two particles of a body in a given configuration “differ” by a rototranslation. Considering the particles belonging to an infinitesimal 3D neighborhood of a point, we may define three (generalized) curvature dual vectors k j along three coordinate lines by means of the relevant differential AT k j )×. These are the component vectors of a curvature helices, such that A T A , j = (A j dual tensor k = k j ⊗ g , which actually controls the tangent space of the orientoα grad (α αT x ), position field. In the explicit expression of the dual curvature, k = k a +Eα the primal part is the angular curvature itself (see Table 24.1) and the dual part is the co-rotational gradient of the position vector. The latter part couples intimately the spatial derivatives of the orientation with those of the position: that is, the orientation field affects the evaluation of neighboring positions. This so-called helicoidal modeling [15] is appropriate for a pole-based setting of 3D continuum mechanics. The dual mechanical quantities and statics laws of a micropolar solid are outlined in Table 24.3, where the remarkably plain structure of the equilibrium equations is in evidence.3 Note that the self-based version of the dual Table 24.3 Elements of pole-based micropolar mechanics Geometry in the reference configuration: A = α+Ex×α particle’s oriento-position tensor αT x), j AT k j )×) kj = k a j + E α (α three curvature dual vectors (from A T A , j = (A αT x ) k = k j ⊗ g j = k a + E α grad (α curvature dual tensor Geometry in the current configuration: A = α + E x × α

particle’s oriento-position tensor αT x ), j AT k j )×) k j = k a j + E α (α three curvature dual vectors (from A T A , j = (A j T α x ) curvature dual tensor k = k j ⊗ g = k a + E α grad (α

Particle’s kinematics: H = A A T = Φ + E h × Φ ω = k − Φ k = ωa + E Φ grad (Φ ΦT h)

rototranslation (hh = x − Φ x , the translation vector) dual strain tensor

Statics: b s S´ divS´ + b = 0 S´ ν = s

= f + E (cc + x × f ) m + x × t ) = t + E (m ´ + x × T´ ) = T´ + E ( M

body load dual density boundary load dual density dual stress tensor (of the 1st Piola–Kirchhoff kind) body balance boundary balance

´ , gather the angular and χ and X T S´ = T´ +E M strain and stress tensors, X T ω = ω a +Eχ linear variables of the classical micropolar mechanics (see Table 24.1). Then, the strain and stress parameters are related to the mechanical variables as in (24.1), i.e. ΦT X T ω ) and Tˆ = primal(Φ ΦT X T S´ ). In the case of hyperelastic non-polar ε = dual(Φ 3

The pole is made to coincide with the absolute origin for convenience.

374


media, the strain energy is function of just the symmetric part of the strain parameter, S ΦT X T ω )S , ax dual(Φ ΦT X T ω ) = 0, w = w(εεS ) and Tˆ = w/εεS . Hence, ε S = dual(Φ T T ´ Φ X S ) = w/εεS + τˆ ×. primal(Φ The helicoidal modeling allows to complete the interpolation scheme (24.16) as detailed in [16] and to integrate it within the formulation of finite elements [17] according to the concepts discussed in Sect. 24.3: the local rototranslation H = A A T and the kinematic dual strain ω = k − H k are recovered by comparing the interpolated quantities in the current and in the reference configurations. Note that the helicoidal interpolation proposed in [16] may be considered as the multi-coordinate version of successful multiplicative interpolations published in the Nineties for the 1D case of 2-node beam elements [4, 8]; however, it is worth stressing that the consistent design leading to Eqs (24.9) and (24.16) yields a general interpolation scheme that is ready as is for higher-order elements as well.

24.5.2 The Helicoidal Shell Element Starting from the internally constrained principle of virtual work tailored to hold the helicoidal modeling concept, a working shell element was developed in [19,20]. The proposed modeling allows a reliable representation of the position field on curved and curving surfaces, and in particular provides a faithful discrete approximation on the element domain. Moreover, in [19] the helicoidal modeling has been applied as well to the 3D mechanics across the thickness, that leads to the shell constitutive equations to be used in [20]. Note that, given the close affinity between rotations and rototranslations, all the issues concerning interpolation, linearization and discrete approximation, discussed above with reference to the rotation field alone, apply straightaway to the rototranslation field as well. So, adopting the helicoidal modeling in shell finite elements does not introduce significant complications. Rather, in a sense, it simplifies things, as the Euclidean part of motion is discarded in favor of one integral orthogonal field. The advantage of modeling helicoidally a curved surface becomes evident with the lowest-order shell elements. Lowest-order elements rely by nature on the tangent space of the underlying modeling: so, in a Euclidean context they are “flat” elements—e.g. 2D or 3D solid elements and membrane elements. The lowest-order helicoidal shell element, instead, is a truly curved element, featuring orientationdriven positions: the nodal rotations control both the internal rotations and the internal displacements. Numerical tests evidence that the element, though not equipped with widespread techniques as incompatible modes, assumed natural strains nor enhanced assumed strains, does not suffer from locking [20]. This welcome behavior can be likely ascribed to the consistent approximate kinematic field faithful to the characteristics of the underlying helicoidal modeling; presently however, an analytic proof that this element is locking-free is still lacking.


375

24.6 Conclusion Four consistency issues in the formulation of proper shell elements for geometrically nonlinear problems have been discussed. Each of the first three issues may be hopefully of interest to people resting on uncoupled position and orientation fields. Firstly, the question of the drilling rotation is addressed in Sect. 24.2 and answered to by a variationally consistent approach. Essentially, the drilling rotation is a redundant d.o.f. for a material surface that behaves in a non-polar fashion in its tangent plane, so its definition entails an extra stress field to retain as a primary unknown of the surface mechanics. A sound variational formulation based on a proper constitutive characterization allows to achieve a genuine and thorough micropolar setting of the shell mechanics, endowed with a full three-parametric rotation tensor. Secondly, the interpolation of the orientation field on the shell surface is addressed in Sect. 24.3. It is shown that an interpolation scheme consistent with the rules of the special orthogonal group is the key to arrive at a frame-invariant and path-independent finite element. Thirdly, the linearization of the virtual functional is dealt with in Sect. 24.4. Again, an approach totally respectful of the SO(3) properties leads to a consistent resolution of the mixed virtual-incremental variation variables that exist in a nonlinear variational formulation involving finite rotations. The fourth consistency issue (Sect. 24.5) concerns a new way of addressing finite element kinematics. Here, the independent position and orientation fields are coupled together into an orthogonal oriento-position integral field that inherits every features pertaining to the orientation field alone, so the discussions in Sects 24.3 and 24.4 immediately apply to this case as well. The prominent feature of this approach, which was explicitly designed for consistent kinematic approximations, is the capability it offers to build low-order shell elements that naturally do not suffer from membrane and shear locking.

References 1. Angeles, J.: The application of dual algebra to kinematic analysis. In Computational Methods in Mechanical Systems (Edited by Angeles, J. and Zakhariev, E.), Springer-Verlag, Heidelberg, Vol. 161, 1–31 (1998) 2. Atluri, S.N.: Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with application to plates and shells - I, Theory. Comput. Struct. 18, 93–116 (1984) 3. Badur, J., Pietraszkiewicz, W.: On geometrically non-linear theory of elastic shells derived from Pseudo-Cosserat continuum with constrained micro-rotations. In: W. Pietraszkiewicz (ed.) Finite rotations in structural mechanics, pp. 19–32. Springer, Berlin (1986) 4. Borri, M., Bottasso, C.: An intrinsic beam model based on a helicoidal approximation – Part I: Formulation. Int. J. Numer. Meth. Engng 37, 2267–2289 (1994) 5. Borri, M., Trainelli, L., Bottasso, C.L.: On representations and parameterizations of motion. Multibody Syst Dyn 4, 129–193 (2000) 6. Bufler, H.: The Biot stresses in nonlinear elasticity and the associated generalized variational principles. Ing. Arch. 55, 450–462 (1985)

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7. Chrósćielewski, J., Makowski, J., Stumpf, H.: Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. Int. J. Numer. Meth. Engng 35, 63–94 (1992) 8. Crisfield, M., Jelenic, G.: Objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite-element implementation. Proceedings: Mathematical, Physical and Engineering Sciences 455(1983), 1125–1147 (1999) 9. Hughes, T.J.R., Brezzi, F.: On drilling degrees of freedom. Comp. Meth. Appl. Mech. Engng 72, 105–121 (1989) 10. Ibrahimbegović, A.: Stress resultant geometrically non-linear shell theory with drilling rotations – Part I. A consistent formulation. Comp. Meth. Appl. Mech. Engng 118, 265–284 (1994) 11. Ibrahimbegović, A., Taylor, R.L.: On the role of frame-invariance in structural mechanics models at finite rotations. Comp. Meth. Appl. Mech. Engng 191(45), 5159–5176 (2002) 12. Kafadar, C.B., Eringen, A.C.: Micropolar media – I. The classical theory. Int. Jnl Engng Sci. 9, 271–305 (1971) 13. Merlini, T.: A variational formulation for finite elasticity with independent rotation and Biotaxial fields. Computat. Mech. 19, 153–168 (1997) 14. Merlini, T.: Variational formulations for the helicoidal modeling of the shell material surface. Scientific Report DIA-SR 08-06, Aracne Editrice, Roma (2008). ISBN: 978-88-548-1887-3 15. Merlini, T., Morandini, M.: The helicoidal modeling in computational finite elasticity. Part I: Variational formulation. Int. J. Solids Struct. 41, 5351–5381 (2004) 16. Merlini, T., Morandini, M.: The helicoidal modeling in computational finite elasticity. Part II: Multiplicative interpolation. Int. J. Solids Struct. 41, 5383–5409 (2004). Erratum on Int. J. Solids Struct. 42, p. 1269 (2005) 17. Merlini, T., Morandini, M.: The helicoidal modeling in computational finite elasticity. Part III: Finite element approximation for non-polar media. Int. J. Solids Struct. 42, 6475–6513 (2005) 18. Merlini, T., Morandini, M.: Helicoidal shell theory. Scientific Report DIA-SR 08-07, Aracne Editrice, Roma (2008). ISBN: 978-88-548-1889-7 19. Merlini, T., Morandini, M.: Computational shell mechanics by the helicoidal modeling. Part I: Theory. Jo. Mech. Mater. Struct. 6 (to appear) (2011) 20. Merlini, T., Morandini, M.: Computational shell mechanics by the helicoidal modeling. Part II: Shell element. J. Mech. Mater. Struct. 6 (to appear) (2011) 21. Merlini, T., Morandini, M.: The helicoidal modelling in the approximation of shell structures mechanics. In: W. Pietraszkiewicz, I. Kreja (eds.) Shell Structures: Theory and Applications, vol. 2. CRC Press, Taylor & Francis, London pp. 269-272 (2010) 22. Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46, 774–787 (2009) 23. Reissner, E.: A note on variational principles in elasticity. Int. J. Solids Struct. 1, 93–95 (1965) 24. Sansour, C., Bednarczyk, H.: The Cosserat surface as a shell model, theory and finite-element formulation. Comp. Meth. Appl. Mech. Engng 120, 1–32 (1995) 25. Sansour, C., Bufler, H.: An exact finite rotation shell theory, its mixed variational formulation, and its finite element implementation. Int. J. Numer. Meth. Engng 34, 73–115 (1992) 26. Simo, J.C., Fox, D.D., Hughes, T.J.R.: Formulation of finite elasticity with independent rotations. Comp. Meth. Appl. Mech. Engng 95, 277–288 (1992) 27. Wisniewski, K.: A shell theory with independent rotations for relaxed Biot stress and right stretch strain. Computat. Mech. 21, 101–122 (1998) 28. Wisniewski, K., Turska, E.: Kinematics of finite rotation shells with in-plane twist parameter. Comp. Meth. Appl. Mech. Engng 190, 1117–1135 (2000) 29. Wisniewski, K., Turska, E.: Warping and in-plane twist parameters in kinematics of finite rotation shells. Comp. Meth. Appl. Mech. Engng 190, 5739–5758 (2001) 30. Wisniewski, K., Turska, E.: Second-order shell kinematics implied by rotation constraintequation. Jnl Elasticity 67, 229 – 246 (2002) 31. Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976)

Chapter 25

An Algorithm for the Automatisation of Pseudo Reductions of PDE Systems Arising from the Uniform-Approximation Technique Patrick Schneider and Reinhold Kienzler

Abstract One way to develop theories for the elastic deformation of two- or one-dimensional structures (like, e.g., shells and beams) under a given load is the uniform-approximation technique (see [2] for an introduction). This technique derives lower-dimensional theories from the general three-dimensional boundary value problem of linear elasticity by the use of series-expansions. It leads to a set of power series in one or two characteristic parameters, which are truncated after a given power, defining the order of the approximating theory. Finally, a so-called pseudo reduction of the resulting PDE system in the unknown displacement coefficients is performed, as the last step of the derivation of a consistent theory. The aim is to find a main differential equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations, which express all other unknown variables in terms of the variables of the main differential equation system, so that the original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the main differential equation system. To find a valid pseudo reduction by inserting the PDEs of the original system into each other is a complicated and very time-consuming task for higher-order theories. Therefore, an structured algorithm seeking all possibilities of valid pseudo reductions (to a given number of PDEs in a given number of variables) is presented. The key idea is to reduce the problem to finding a solution of a linear equation system, by treating each product of different powers of characteristic parameters with the same variable as formally independent variables. To this end, all necessary equations, which can be build from the original PDE system, have to be identified and added to the system a-priori. Keywords Uniform-approximation technique · Pseudo reduction · Partial differential equations · Higher-order theory · Linear elasticity P. Schneider (B) · R. Kienzler Department of Production Engineering, University of Bremen, Germany e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 25, © Springer-Verlag Berlin Heidelberg 2011

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P. Schneider and R. Kienzler

25.1 Introduction 25.1.1 The Uniform-Approximation Technique One way to develop theories for the elastic deformation of two- or one-dimensional structures (disks, plates, shells, bars, beams and arcs) under a given load is the uniform-approximation technique (see [2] or [4] for an introduction). This technique derives equivalent two- or one-dimensional formulations to the general three-dimensional boundary value problem of linear elasticity by the use of seriesexpansions (see [6] for details). These formulations are sets of countably many PDEs, which are power series in one or two characteristic parameters. For example: In the case of a plate theory this characteristic parameter is the second power of the so-called plate parameter which describes the relative thickness of the plate and is assumed to be a small quantity since plates are defined as being thin. In the case of a beam, we have two parameters which are assumed to be small, since the two cross-sectional measurements are assumed to be small against the measurement in beam length direction. Because of the smallness of the characteristic parameters, an approximative theory in a finite number of PDEs in a finite number of unknown quantities, which are the series coefficients of the sought displacement field, could be generated by the truncation of this power series after a certain power. The highest power which is not neglected is called the approximation order of the derived theory.

25.1.2 The General Aim of the Pseudo-Reduction Process Since a set of many PDEs in many unknown variables is not suitable for analytical discussions, as a last step of the derivation of a consistent theory, a so-called pseudo reduction of the resulting PDE System is performed. The aim is to find a main differential equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations, which express all other unknown variables in terms of the main variables, so that the original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the main differential equation system. (See the derivation of Kirchhoff’s theory in [2] for a simple example of a pseudo reduction.) Furthermore, the reduction equations must allow us to eliminate all variables which are not main variables from all stress resultants for a complete description of the theory in terms of the main variables. More precisely: Not every variable has to be expressable independently by the reduction equations, but it is crucial that they allow us to eliminate all variables which are not main variables from all stress resultants and all non-main PDEs. This means: If unknown variables always occur in specific linear combinations of derivatives in the original PDE system and in the stress resultants, it is sufficient to express these specific forms in terms of the main variables. (See [4], Sects 3.6 and 3.7 for an

25 Automatisation of Pseudo Reductions

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example.) If we find a set of PDEs (main PDE system and reduction equations) satisfying this condition, we call them a correct pseudo reduction. The word ”pseudo” indicates that only the system of the main PDEs with the reduction equations can be equivalent to the original PDE system.

25.1.3 The Aim of the Presented Algorithm In former papers (compare [2], [3], [5] and [6]) the pseudo reduction of the PDE systems was performed by differentiating and inserting the PDEs of the original system into another. It is a complicated and very time-consuming task to find a sequence of operations that actually leads to a correct pseudo reduction. Furthermore, there was in the end no possibility to show whether the resulting pseudo reduction was the only possible one, or there are degrees of freedom. (Of course a reduction with only one main PDE of the least possible order would be the most desirable one.) Therefore we intend to find an algorithm which detects all possible pseudo reductions to a given PDE system, or shows that there are no possible pseudo reductions.

25.1.4 The Main Idea of the Presented Algorithm First of all one selects the main variables (the variables of the aimed for main differential equation system) and the PDEs which shall be used to generate the main differential equation system by elimination of the other variables. In practice there is always a first generic choice, given by the type of theory which shall be derived. Consider a plate theory for example (compare [2], and [6]). Since Kirchhoff theory is a single PDE in only one variable, which is the displacement of the mid-plane in thickness direction, say w, one would first try to develop a theory in one PDE only formulated in w. So the main variable would be w. The main differential equation would be the PDE which has the constant surface load as a right-hand side, since this is the dominating load case. In addition the triangular structure of the PDE systems, which is characteristic for PDE systems arising from the uniform-approximation technique, due to the truncation of the series (compare [2], [3], [5] and [6]), predicts this equation to be a generic choice for the main differential equation, since it seams intuitive to set the PDEs into one another from the bottom to the top of the PDE system. Or considered in another way: Since the top PDE is also the one with the most variables we will have to derive a complete set of reduction equations to eliminate all other variables and therefore we will have good chances to eliminate all variables in the stress resultants with the gained set of reduction equations. In the case of a beam theory the main PDE would also be the one with the constant surface load and the main variable would be the transverse displacement of the neutral axis. In both cases, the pseudo reduction to a single PDE in one variable turns out to be possible even for second-order theories (see [4] or [6] for the cases of a plate / the case of a

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beam will be published in a forthcoming paper). So the step of choosing the main variables and the PDEs which shall be used to generate the main differential system is in practice easier than it might look at first. After this choice, the problem of pseudo reduction is reduced to find for each main differential equation and each stress resultant a linear combination of differentials of the set of the other PDEs, so that they build the same term in nonmain variables as the corresponding equation. Then we can obtain a formulation expressed only in main variables by building the differences. If we would deal with exact linear equations (no differentials), all possibilities would be found by applying the Gauss algorithm (symbolic LU-decomposition). Our situation is more complex, since we are dealing with partial differential equations, which are furthermore truncated power series. Since all PDEs are linear in the unknown variables the differential-operators could simply be treated as factors (compare Sect. 25.3.4 below). On the other hand, the fact that the PDEs are truncated power series results in the problem that the characteristic parameters are not treatable as normal factors (see Sect. 25.3.1 below). We overcome this problems by treating each product of different powers of characteristic parameters with the same variable as a formally independent variables. Of course interpreted in these replacement variables (later called cd-variables, compare 25.3.2 below) the original PDE system does not provide enough equations to achieve a pseudo reduction. Our approach is to add ”all” (compare 25.3.5 below) equations that can be build from the equations of the original PDE system by multiplications and divisions with characteristic parameters a-priori to the system, before investigating all reduction possibilities by the application of the Gauss algorithm to this system (interpreted in cd-variables).

25.2 The Setting At first we will have a closer look on the setting of a PDE system which is generated by the uniform-approximation technique and the ”rules” arising from this setting for the reduction process.

25.2.1 The One-Dimensional Case We will deal with the case of a PDE system for deriving a one-dimensional theory. In this case we have two characteristic parameters which we will call c and d and only derivatives with respect to one coordinate, we will call x. The case of a PDE system for the derivation of a two-dimensional theory is just a simple special case of the situation treated here concerning the implementation of the algorithm, since there is only one characteristic parameter in this case, which simplifies the generation of additional equations (compare 25.3.5 below). The fact that there are derivatives with


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respect to two coordinates does not require any special treatment, besides the fact that there are two different differential factors. (compare 25.3.4 below)

25.2.2 Homogeneous Material Besides some general modeling assumptions required to develop equivalent lower-dimensional theories (see [6] for a detailed discussion of the modeling assumptions.), the materials under consideration are assumed to be linear elastic and homogeneous (but possibly non-isotropic), which means that the elements of the forth order elasticity tensor E are constant material parameters: ∀(i, j, k, l) ∈ {1, 2, 3}4 : Ei jkl = const.

25.2.3 Regularity In general we will not discuss any regularity questions, which simply means that all functions are assumed to be sufficiently smooth.

25.2.4 The Original PDE System In the one-dimensional case, the original PDE system is actually an ODE system of M ∈ N ODEs in N ∈ N sought displacement coefficients {uk : I → R| k ∈ {1, ..., N}} being scalar functions depending on one coordinate, we call x ∈ I ⊂ R. In the following, we will call this displacement coefficients the original variables. Each ODE was generated by the truncation of an infinite power series in the characteristic parameters c and d, neglecting every summand with a factor ci d j where i + j > a. As mentioned before, we call a the approximation order of the theory. Each ODE has the same specific form. For each μ ∈ M we have a n n=0 i=0

i n−i

cd

N 2

aμnikl E μnikl Dlx (uk (x)) = Kcrμ d sμ Pμ (x), x ∈ I,

(25.1)

k=1 l=0

with Pμ : I → R being a load resultant, assumed to be a given scalar function and rμ , sμ ∈ N ∪ {0} fulfilling the condition rμ + sμ ≤ a − 1. Furthermore, K ∈ R and for all (μ, n, i, k, l) in 3 4 T := (μ, n, i, k, l) ∈ {1, ..., M} × {0, ..., a}2 × {1, ..., N} × {0, 1, 2}|i ≤ n

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we have aμnikl ∈ R and E μnikl is a specific constant of the forth-order elasticity tensor E. D x denotes the differential operator with respect to x, with the convention D0x (•) = •. In addition most of the Pμ are usually neglected because they describe load cases, which are not very common in practice (compare [6] p. 52), or they are of higher order (compare Sect. 25.3.3 below). Most of the aμnikl equal 0 and the ODE system has furthermore a characteristic triangular structure: The number of the aμnikl 0 decreases for a rising μ. (See [1], pp. 72-73, [2], p. 237 and [6], pp. 53-55, for explicit examples of PDE systems of this form.)

25.2.5 Differentiability The triangular structure of the ODE system generated by the uniform-approximation approach suggest that variables could successively be eliminated, by inserting the eventually differentiated equations into each other, beginning at the bottom of the system (μ = M / see [2] for a simple example). It is clear that we have to assume that the equations keep their magnitude (concerning the characteristic parameters) during differentiation, for having a chance to reduce the system, which implies that all unknown displacement coefficients uk have to keep their magnitude during differentiation. This assumption is of course mathematically inappropriate, since we do not have any a-priori indications about the structure of the solution. Indeed this is not a pure theoretical problem. For example in second-order plate theories it is known that the rate of twist has derivatives that are of higher magnitude than the original function (compare [3]). This results in the fact that a correct pseudo reduction might not give a physically meaningful theory without further a-posteriori extensions (as discussed in [6] in detail). However, these problems are not subject of this paper, since nevertheless, a valid pseudo reduction remains an important step in the derivation of a consistent theory.

25.2.6 Multiplication with Characteristic Parameters Since all equations are power series in ci d j truncated after a certain common power i + j = a, an equation witch is multiplied with a product of certain powers of the characteristic parameters cn dm has to be truncated again after multiplication, (neglect all summands with factors cn+i d m+ j where n + m + i + j > a) to give an equation which is of the desired accuracy and therefore usable for the uniform-approximation technique.


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25.2.7 Division by Characteristic Parameters Division by a certain product of powers of characteristic parameters cn d m results in a loss of accuracy of the resulting equation. To get an equation which is appropriate to generate a consistent theory of the desired order a, the original equation has to be computed with sufficiently higher accuracy a + n + m. The generation of additional higher-order terms is not a problem, since one has closed form representations of the infinite power serieses (compare [6]).

25.3 The Algorithm In this section we will describe the algorithm for the case of the pseudo reduction of a PDE system for deriving a one-dimensional theory. The case of a PDE system for the derivation of a two-dimensional theory is just a simple special case of the situation treated here concerning the implementation of the algorithm, as already mentioned. Furthermore, we will describe the algorithm for one main PDE for linguistic simplicity. For each additional main PDE and each stress resultant the described procedure has to be performed in a complete analog manner.

25.3.1 Selection of Main Variable and Main PDE As a first step the main variables and the main PDE is selected. The result of the algorithm depends on this choice. If the result of the canonical choice (compare 25.1.4 above) is that there are no pseudo reductions possible, the next higher (concerning the degree of the corresponding basis polynomial; see [2] or [6] for details) displacement coefficients are to be declared as main variables.

25.3.2 cd-Variables As a consequence of 25.2.6 and 25.2.7, every occurrence of a product of an original variable uk with powers of characteristic parameters in an input PDE for the Gauss algorithm has to be treated as an independent variable to prevent the Gauss algorithm to perform operations that change the approximation order a of the equations. For example: One summand %appearing &in the left hand side of (25.1) could be c2 d2 3E 1111 D1x u3 (x) = 3E1111 D1x c2 d 2 u3 (x) and we would treat c2 d2 u3 (x) as a variable independent from, e.g., c2 u3 (x). Otherwise, the Gauss algorithm could, e.g., divide our example equation by d 2 , leading to an equation with missing terms of order a − 1 and a. We call the new replacement variables the cd-variables.

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Consequently (right / with regard of 25.2.6 and 25.2.7) multiplications or divisions by products of powers of characteristic parameters cn d m generate formally new equations, which might be linear independent (by means of cd-variables) of the original equations.

25.3.3 Right-Hand Sides There is no need to treat the load-resultants Pμ appearing at the right-hand sides of the equations (25.1) in an analog manner like the variables uk by introducing cd-load-resultants, because they are assumed to be given and not to be eliminated. Therefore, they do not appear in the equation system for the derivation of the linear combination which finally leads to the pseudo reduction (compare derivation of x in Sect. 25.3.6 below). The load-resultants are assumed to have a magnitude 1 concerning the common power n = i + j in ci d j themselves (in contrast to the variables uk ) which is calculable by examining a zeroth-order theory (compare [2] Sect. 4.1). This magnitude has to be considered when a truncation of the power-serieses is performed. (compare conditions for rμ and sμ and the comment on the order of Pμ in Sect. 25.2.4)

25.3.4 D-Factors Since the equations of the original PDE system are linear in the original variables uk and the load resultants Pμ and their prefactors are constant with respect to x (compare (25.1)) and Gauss algorithm only builds linear combinations of the lines of an input matrix, we could simply treat D x as a symbolic factor in our special case. Valid factors for multiplications of lines concerning the Gauss algorithm are therefore in the field 3 4 K = R ∪ {D x } ∪ Ei jkl |(i, j, k, l) ∈ {1, 2, 3}4 , (25.2) allowing the Gauss algorithm to perform integrations and differentiations of our PDEs. This coincides with the normal behavior of a computer algebra system which is able to perform symbolic LU-decompositions (e.g., Maple or Mathematica) if the E i jkl and D x are declared as symbols, allowing straight forward implementations of the presented algorithm.


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25.3.5 Generating Additional Equations Written in cd-variables and D-factors the original PDE System is merely a linear equation system with a given right-hand side. Because the Gauss algorithm can not perform valid multiplications or divisions by characteristic parameters, all necessary equations have to be generated a-priori. In most cases a pseudo reduction can be found without considering any equations, which can be build from the PDEs of the original system by divisions of characteristic parameters. Nevertheless we consider these equations for our algorithm. In a first execution of the algorithm the generation of additional equations by divisions can be omitted, if the task is only to find any valid pseudo reduction. The number of PDEs which can be build from the PDEs of the original system by divisions by products of powers of characteristic parameters cn d m is infinite, so that we can not really consider all of them. But since the aim is, finally, to generate a given linear combination of cd-variables, there are two types of these equations which are potentially not useful. The main PDEs and stress resultants only contain positive powers of c and d (compare [2], [3] and [6]). Therefore, equations with cd-variables of negative powers are potentially not useful. Furthermore there is not really a physically interpretation for these variables, because of the used seriesexpansions with only positive powers (compare [2], [3] and [6]). The second type of equations which are potentially of no use are those, which contain variables, which are not in the set of the original variables {uk |k ∈ {1, . . . , N}}, since the use of these equations for the elimination of variables would always add new variables which are also to eliminate. If we decide not to consider these two types of equations, we end up with a finite set of PDEs which can be build by divisions.

25.3.5.1 A Simple Approach for Generating All Additional Equations A simple approach for generating all additional equations would be: Make a list of all PDEs of the original system. Traverse the list and build for each PDE the new PDE which is generated by multiplication with c (with regard of 25.2.6). If it is not identical zero and not already in the list, add it to a new list. Do the same with the factor d. Then traverse the list and build for each PDE the new PDE which is generated by division by c (with regard of 25.2.7). If it does not contain negative powers of c, does not contain new variables (not in the set {uk |k ∈ {1, . . . , N}}) and is not already present, add it to the new list. Do the same with the quotient d. If the new list is not empty, append it to the list and begin again, otherwise stop. The procedure indeed always terminates. We will give a more detailed presentation of another approach in the following. Our approach calculates the equations which are to generate a-priori. This omits the testing procedure for each different equation which is build and the computation of higher-order terms for each division, which is the main reason why the approach has a higher performance, if it is implemented. Furthermore, it reveals a greater insight into the structure of the problem of generating additional equations.

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25.3.5.2 Extension by Divisions by Characteristic Parameters The PDEs which can be build by divisions by characteristic parameters among the considered set of equations can be generated by the following procedure, which is to be performed for each PDE of the original PDE system. First compute the PDE μ ∈ M with increasing approximation order b > a until the next higher approximation order b + 1 would contain terms in variables that are not in the original PDE system {uk |k ∈ {1, . . ., N}}, leading to an PDE of the form b n n=0 i=0

i n−i

cd

N 2

aμnikl E μnikl Dlx (uk (x)) = Kcrμ d sμ Pμ (x), x ∈ I.

(25.3)

k=1 l=0

Than compute the maximal excludable powers of c and d of this PDE (so that no negative powers of c and d occur after excluding). 3 ex c power(μ) := min i | (μ, n, i, k, l) ∈ T 4 with aμnikl 0 and Eμnikl 0 in (25.3) 3 ex d power(μ) := min n − i | (μ, n, i, k, l) ∈ T 4 with aμnikl 0 and Eμnikl 0 in (25.3) Traverse all products which can be build of powers of c and d up to the common power ex c power + ex d power: 3 4 c j dn− j |n = 1, ..., (ex c power + ex d power), j = 0, ..., n . If the specific factor c j d n− j is excludeable (( j ≤ ex c power) and (n − j ≤ ex d power)) and the PDE can be computed with a sufficient high accuracy to give a PDE of approximation order a after the division by the factor without adding new variables to the system (n ≤ b − a); compute the PDE and add it to the system. This gives us nearly all equations which can be build from the original PDE by divisions by powers of characteristic parameters; we only omitted to build the equations where (n > b − a). In this case it is not possible to exclude the factor without adding new variables to the system for a general factor, unless it is cn or dn . Since the maximal accuracy the PDE can be computed with, without adding new variables is b, we can avoid this by multiplying the equation with the other characteristic parameter of a power of at least a + n − b. By doing so, the algorithm so far will give us at least all ”lowest” PDEs which can be build by divisions. This means precisely all PDEs which can not be divided by a power of c or d without either containing cdvariables of negative powers or adding new variables to the system, which are not in the set of the variables of the original PDE system {uk |k ∈ {1, . . ., N}}. All ”higher” ones will be generated by the next extension step. The special case of a factor cn or d n could be treated simultaneous as table 25.1 shows. In pseudo code the procedure reads as follows.


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PDE_system:=original_PDE_system basevariables:=compute_all_variables_of(original_PDE_system) for all PDE of original_PDE_system do b:=a basevarcheck:=true while basevarcheck=true do b:=b+1 higher_order_PDE:=compute_with_order_N(PDE,b) if not(compute_all_variables_of(higher_order_PDE) 0 are scalar stretch parameters, and Q1 , Q2 are rotation tensors, each depending on one rotation angle, βα , see Fig. 26.3. Using eq. (26.13) and the drill RC equation, we obtain 1 ω ≈ (β1 + β2 ) + kπ, 2

k = 0, . . ., K,

(26.14)

398

K. Wi´sniewski and E. Turska

-b2 l2 t2

t*2 t*1

b1

t1

Fig. 26.3 Deformation of a pair of basis vectors t1 and t2

l1

for cos ω 0, λ1 c1 + λ2 c2 0 and λα ≈ 1, where sα sin βα and cα cos βα . Hence, the drilling angle ω is an average of rotations of vectors tα . This interpretation is valid also for shell, for which ω is a rotation about the director t3 and tα are the vectors tangent to the reference surface, for details see [18], p. 26.

26.4 Features of Enhanced HW Shell Elements 26.4.1 Skew Coordinates To define the assumed representations of stress and strain in the HW four-node elements, we use the skew coordinates instead of the natural ones, see [21] and [22]. This modification slightly improves accuracy of mixed elements and renders that the homogenous equilibrium equations and the compatibility condition are satisfied point-wise for elements of arbitrary shape. Natural basis at element’s center. The position vector of a four-node element in the initial configuration is approximated as follows: y(ξ, η) =

4

NI (ξ, η) yI ,

I=1

1 NI (ξ, η) (1 + ξI ξ) (1 + ηI η), 4

(26.15)

where NI are the standard bi-linear shape functions, ξ, η ∈ [−1, +1] are the natural coordinates and {ξI , ηI } are the coordinates of nodes I = 1, ..., 4. The vectors of the natural basis are defined as ∂y(ξ, η) ∂y(ξ, η) , g2 (ξ, η) , (26.16) ∂ξ ∂η

and, at the element’s center, gc1 g1

ξ,η=0 and gc2 g2

ξ,η=0 . In general, gc1 and gc2 are neither unit nor mutually orthogonal, see Fig. 26.4. g1 (ξ, η)

26 Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation

399

In the reference Cartesian basis {i1 , i2 }, the position vector is expressed as y = xi1 + yi2 . The Jacobian matrix is defined as ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ ∂x ∂x ⎥⎥⎥ ⎢⎢⎢ g1 · i1 g2 · i1 ⎥⎥⎥ ∂ξ ∂η ⎥⎥⎥ , J ⎢⎢⎢⎢⎣ ∂y ∂y ⎥⎥⎥⎥⎦ = ⎢⎢⎢⎢⎣ (26.17) ⎥⎦ g · i g · i 1 2 2 2 ∂ξ ∂η and, at the element’s center, Jc J|ξ,η=0 . h 1 c 2

g

x c(0,0)

Fig. 26.4 Natural basis at the element’s center, and the ’fictitious’ parallelogram (dotted line) obtained for y¯ = ξ gc1 + η gc2

A(1,1)

g1c

1

Skew coordinates. It is a common belief that the natural coordinates ξ, η ∈ [−1, +1] are associated with the natural basis at the element’s center {gc1 , gc2 }, see Fig. 26.4. However, this is true only for parallelograms but not for irregular (trapezoidal) shapes. The coordinates which are associated with {gc1 , gc2 } are designated as “skew” and are derived below. Consider the position vector relative to the element’s center, and express it as y¯ y − yc = xS gc1 + yS gc2 ,

(26.18)

where xS , yS are the skew coordinates. Below we express the skew coordinates in terms of the natural coordinates. First, we find the relation between the skew coordinates and the Cartesian coordinates {x, y} associated with the reference basis {i1 , i2 }. The position vector can be written in the two basis as follows: x¯ i1 + y¯ i2 = xS gc1 + yS gc2 .

(26.19)

Taking a scalar product of this equation with the vectors i1 and i2 , we obtain two equations, which can be solved for the skew coordinates, ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ xS ⎪ ⎬ −1 ⎨ x¯ ⎬ = J , (26.20) ⎪ ⎪ c ⎪ ⎪ ⎪ ⎪ ⎩ yS ⎪ ⎭ ⎩ y¯ ⎪ ⎭ where Jc is the Jacobian of eq. (26.17) at the element’s center.

400


Next, we find the relation between the skew coordinates and the natural coordinates. By approximations of eq. (26.15), we have ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x¯ ⎪ ⎬ ⎪ ⎨ x − a0 ⎪ ⎬ ⎪ ⎨ a1 ξ + a2η + a3 ξη ⎪ ⎬ ⎪ =⎪ , (26.21) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y¯ ⎪ ⎭ ⎪ ⎩ y − b0 ⎪ ⎭ ⎪ ⎩ b1 ξ + b2η + b3 ξη ⎪ ⎭ where the coefficients ai , bi are functions of the nodal positions, and a0 , b0 are coordinates of the element’s center. Using this in eq. (26.20), we obtain ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ xS ⎪ ⎬ ⎪ ⎨ ξ + A ξη ⎪ ⎬ =⎪ , (26.22) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yS ⎭ ⎩ η + B ξη ⎪ ⎭ where A and B are functions of ai , bi . Finally, the coefficients A and B can be expressed in terms of the determinant of the Jacobian J of eq. (26.17), which, in the current notation, is ⎡ ⎤ ⎢⎢⎢ a1 + a3 η a2 + a3 ξ ⎥⎥⎥ ⎥⎥⎥⎥ . J = ⎢⎢⎢⎢⎣ (26.23) ⎦ b1 + b 3 η b 2 + b2 ξ The determinant of the Jacobian matrix can be expanded as follows: det J = jc + ( j,ξ )c ξ + ( j,η )c η,

(26.24)

where jc = a1 b2 − a2 b1 , ( j,ξ )c = a1 b3 − a3 b1 , and ( j,η )c = a3 b2 − a2 b3 . It can be verified that the coefficients A and B can be expressed as A=

( j,η )c , jc

B=

( j,ξ )c , jc

(26.25)

see [21] for details. Finally, the skew coordinates are expressed in terms of the natural coordinates by Eqs (26.22) and (26.25). The skew coordinates can be easily computed and improve accuracy for the mixed (HW and HR) elements of irregular shape. Remark. For parallelograms, ( j,ξ )c = ( j,η )c = 0, so A = B = 0, and, then, the skew coordinates are equal to the natural coordinates. If we apply a formula with the natural coordinates, i.e. y¯ = ξ gc1 + η gc2 , to an irregular trapezoidal element then a ‘fictitious’ parallelogram shown in Fig. 26.4 is obtained. On the other hand, eq. (26.18), which uses the skew coordinates, reproduces the element’s shape exactly.


401

26.4.2 Assumed Stress and Strain Let take as the starting point, the shell counterpart of the functional of eq. (26.6), with the shell strains ε∗ and κ∗ instead of the 3D strain E∗ as well as the shell force and couple fields N∗ and M∗ instead of the 3D stress S∗ . We treat all the fields marked by asterisks as the assumed fields, N∗ → Na ,

M∗ → Ma ,

ε∗ → εa ,

κ ∗ → κa ,

(26.26)

which is indicated by the superscript “a”. The assumed fields are defined as follows. We define the assumed stress and strain in the natural basis at the element’s center {gck }, σa = σkl gck ⊗ gcl ,

εa = εkl gck ⊗ gcl ,

k, l = 1, 2,

(26.27)

where σkl and εkl are the contra-variant components of stress and strain, respectively. The representations of these components are assumed as polynomials of the skew coordinates xS , y s and local (elemental) parameters qi . Denote the matrices of assumed components as σξ [σkl ], and εξ [εkl ]. They are transformed to the Cartesian reference basis using the transformation rule for contravariant components, σre f = Jc σξ JTc , εre f = Jc εξ JTc , (26.28) where Jc is the Jacobian matrix J of eq. (26.17) at the element’s center. Remark. In [22], the components of strain were assumed in the co-basis {gkc }, εa = εkl gkc ⊗ glc ,

k, l = 1, 2,

(26.29)

where the co-vectors gkc are defined in the standard way by gkc · gcl = δkl . The matrix of assumed components, εξ [εkl ], was transformed to the Cartesian reference basis as follows: −1 εre f = J−T (26.30) c εξ Jc , which is the transformation rule for covariant components. For the contravariant assumed stress and the covariant assumed strain, the scalar product of them does not depend on J, σre f · εre f = tr (J σξ JT J−T εTξ J−1 ) = σξ · εξ ,

(26.31)

or, in other words, is identical regardless of the element’s shape. Then, the accuracy: (a) deteriorates for 7p or 5p strain representations, and (b) is unchanged for the 9p strain representation, see [23]. Hence, in the sequel, we use the contra-variant rule, which is suitable for all tested strain representations. The following representations were selected for the shell elements developed and tested in this paper. They all are expressed in terms of the skew coordinates xS , yS .

402


1. For the assumed stress, we use the representation, ⎡ ⎢⎢⎢ q1 + q2 yS q5 + q 6 x S + q7 y S ξ 7p: σ ⎢⎢⎢⎢⎣ sym. q3 + q4 xS

⎤ ⎥⎥⎥ ⎥⎥⎥ . ⎥⎦

(26.32)

Comparing to the well-known 5p PS representation of [11], there are two differences: the off-diagonal terms are not constant but linear in xS , yS and it is written in the skew coordinates. Another 7p representation of stress is, ⎡ ⎤ ⎢⎢⎢ q1 + q2 yS + q6 xS q5 − q7 xS − q6 yS ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ , (26.33) ⎢⎣ sym. q3 + q4 xS + q7 yS ⎦ in which the parameters q6 and q7 appear in the diagonal and off-diagonal terms. In [21], the linking transformations for several forms of this representation existing in the literature is derived. As indicate our tests for the elements with drilling rotation, the representation of eq. (26.33) is less accurate than the representation of eq. (26.32), which we use. 2. For the assumed strain, we use two representations: ⎡ ⎢⎢⎢ q1 + q2 yS + q3 xS q7 + q 8 x S + q9 y S 9p: εξ ⎢⎢⎢⎢⎣ sym. q4 + q 5 x S + q6 y S ⎡ ⎤ ⎢⎢⎢ q1 + q2 yS ⎥⎥⎥ q5 ⎢ ⎥⎥⎥ . ξ 5p: ε ⎢⎢⎢⎣ ⎥ sym. q3 + q4 xS ⎦

⎤ ⎥⎥⎥ ⎥⎥⎥ , ⎥⎦

(26.34)

(26.35)

In the 9p representation, all components are linear in xS , yS , while the 5p representation is analogous to this used in [11] for stress, but expressed in the skew coordinates. Remark. We verify the assumed representations as follows. For the assumed representations of stress, we check whether they satisfy the homogenous equilibrium equations, ∂σxy ∂σyy ∂σxx ∂σxy + = 0, + = 0, (26.36) ∂x ∂y ∂x ∂y while for the assumed representations of strain, we check whether they satisfy the compatibility condition, ∂2ε xy ∂2 ε xx ∂2 εyy + = 2 . ∂x ∂y ∂y2 ∂x2

(26.37)

We verify these equations symbolically, see [21] and [22] for details. For representations in the skew coordinates, they are satisfied point-wise, even for irregular trapezoidal element’s shapes. For the same representations but in the natural coordinates, these equations are point-wise satisfied only for parallelograms.


403

26.4.3 Enhanced Assumed Displacement Gradient (EADG) Method For the elements with drilling rotation, it is more suitable to enhance the displacement gradient than the strain, because the former affects also the drill RC. The gradient of compatible displacements uc is enhanced additively by the ˜ so the deformation gradient becomes matrix H, ˜ F ∇χ + H,

(26.38)

˜ is constructed using the transforwhere ∇χ = I + ∇uc . The enhancing matrix H mation −1 jc ˜ H J c G Jc , (26.39) j where G is the matrix of assumed enhancing modes and j det J. The EADG method is a generalization of the Incompatible Displacement method of [16] and [14], and was proposed for 2D elements in [13]. We find it beneficial for elements with the drilling rotation. For shells, the transformation (26.39) is modified, for details see [24]. The matrix G of the assumed enhancing modes corresponds to the selected 7p representation of the assumed stress, and, in the current paper, it is as follows: ⎡ ⎤ ⎢⎢⎢ 0 η q1 ⎥⎥⎥ ⎥⎥⎥⎥ . 2p, EADG2: G = ⎢⎢⎢⎢⎣ (26.40) ξ q2 0 ⎦ Besides, we also use 4p, EADG2k:

⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ 0 η (q + ζq ) 1 2 ⎥⎥⎥ , G = ⎢⎢⎢⎢⎣ ⎥⎦ ξ (q3 + ζq4) 0

(26.41)

which is a sum of the EADG2 enhancement plus an analogous enhancement depending on the coordinate ζ ∈ [−h/2, +h/2].

26.4.4 Approximation of Drilling RC For the equal-order bi-linear approximations of displacements and the drilling rotation, the drill RC is incorrectly approximated. We have already discussed this issue in [17], but here we provide an additional refinement of the method. Consider a 2D problem with drilling rotation, and denote the coefficients of the bilinear shape function of the tangent displacements and the drilling rotation as ui , vi , ωi (i = 0, . . ., 3). These coefficients are functions of nodal values of respective components.

404


The linearized form of the drill RC for a bi-unit square element is as follows: c ω + 12 (u,η − v,ξ ) = 0, and, for the equal-order bilinear approximations, it has the form 1 1 1 c ω0 + (u2 − v1 ) + ω1 + u3 ξ + ω2 − v3 η + ω3 ξη = 0. (26.42) 2 2 2 We see that the ξη-term contains only a rotational parameter but not displacement ones, so the displacement and rotational parameters are not linked, which is incorrect. For this reason, we omit this term using a linear expansion of the drill RC at the element’s center. The sampling at mid-side points of element’s edges can also be used, see [18], p. 318.

+1

-1 Mode Q2

Fig. 26.5 Spurious mode Θ2 for drilling rotation

-1

+1

Omitting of the ξη-term renders that the tangent matrix for the drill RC has one spurious zero eigenvalue, which we eliminate using the stabilization function in the penalty form, 1 Θ2 (ω1 − ω2 + ω3 − ω4 ), 4

P2 = 10−3 G V Θ22 ,

(26.43)

where V is the element volume, Θ2 is the mode shown in Fig. 26.5. This stabilization function was proposed in [9] for the Allman-type quadrilaterals, and it provides the stabilization matrix which has 1 non-zero eigenvalue. This simple stabilization is sufficient for rectangular elements, but for irregular shapes we can improve it as follows. Let us re-write the spurious mode as 1 Θ2 h ωTI , 4

(26.44)

where ωI {ω1 , ω2 , ω3 , ω4 } is the vector of nodal drilling rotations, and h {1, −1, 1, −1} is the hourglass vector. Then, instead of 14 h we can use the γ-vector, identical as this used in the gamma-stabilization of the one-integration point elements. The gamma-stabilization method stems from [8], was proposed in [6] and, afterwards, refined in numerous papers. Using the gamma-stabilization method, the mode is re-defined as follows: Θ2 γ ωTI ,

γ

1 h − (hS1 ) b1 − (hS2 ) b2 , 4

(26.45)


405

where b1 =

1 (ηS2 ) ξ − (ξS2 ) η , 4A ξ {−1, 1, 1, −1},

b2 =

1 −(ηS1 ) ξ + (ξS1 ) η , 4A

η {−1, −1, 1, 1}.

Besides, Sα are the vectors of projections of nodal relative position vectors on tα and A = 4 jc . This modification of the stabilization is relatively simple and improves performance for irregular elements.

26.4.5 Approximation of Lagrange Multiplier for Drill RC The Lagrange multiplier of the functional FRC of eq. (26.6) is assumed in the natural basis of the element center, {gck }, and transformed to the local ortho-normal basis {tck } using the transformation rule for contra-variant components ⎡ ⎤ ⎢⎢⎢ 0 T a ⎥⎥⎥ ⎢ ⎥⎥⎥ JT , T = J Lc ⎢⎢⎢⎣ ⎥ Lc a −T 0 ⎦ a

(26.46)

where the assumed representation for the Lagrange multiplier is as follows: T a (ξ, η) q15 + ξ q16 + η q17 ,

(26.47)

where J Lc is the local Jacobian at element’s center. This representation was selected in [17], as optimal for non-linear in-plane bending.

26.5 Numerical Tests In this section, we describe numerical tests of 4 enhanced HW four-node shell elements with 6 dofs/node listed in Table 26.1. They are developed from the enhanced HW functional and use the assumed representations in terms of skew coordinates, see Sect. 26.4.2. In the HW element’s indicator, the total number of elemental parameters is included. In the sequel, “parameter” is abbreviated to “p”. All the tested HW elements have an identical membrane part, with 7p Na , 9p and the 2p enhancement EADG2, which we selected as optimal for the 2D HW elements with the drilling rotation. The motivation for this form of the membrane part stems from the Cook’s membrane example, see Sect. 26.5.1. This membrane part is different than for the elements without drilling rotation! εa

406


Table 26.1 Characteristics of tested shell elements. Drill RC by PL with γ = G Element

Assumed stresses

Assumed strains

Enhancement

7p Na , 7p Ma , 4p Qa

9p εa , 9p κa , 4p εaα3

4p EADG2k

a

9p εa , 5p κa , 4p εaα3

2p EADG2

9p εa , 5p κa

2p EADG2

Enhanced HW in skew coord. HW47

a

a

HW39

7p N , 5p M , 4p Q

HW31

7p Na , 5p Ma

HW29

7p Na ,

9p εa ,

4p Qa

4p εaα3

2p EADG2

Enhanced PE EADG4

-

-

4p EADG4

EADG5A

-

-

5p EADG5A

The HW47 element has all the features included, i.e. its bending/twisting part is fully analogous to the membrane part, i.e. 7p Ma , 9p κa and the 2p EADG2 ζ-dependent enhancement. Besides, its transverse shear part is treated by the HW functional with 8 (4 stress and 4 strain) parameters. It is a very good element, which can be used for reference when the accuracy and convergence properties are considered. The other 3 elements have a reduced number of parameters and differ from the HW47 element in the bending/twisting and/or transverse shear parts. 1. The HW39 element has a simplified bending/twisting part, based on 5p Ma and 5p κa . No enhancement is used for this part. The transverse shear part is identical as in the HW47 element. 2. The HW31 element has the bending/twisting part identical as the HW39 element, but the transverse shear part is standard, i.e. derived from the (non-enhanced) potential energy. 3. The HW29 element has a standard bending/twisting part, i.e. derived from the (non-enhanced) potential energy. The transverse shear part is identical as in the HW47 element. For comparison, we use the EADG4 and EADG5A elements developed from the potential energy (PE) with the enhancement of the displacement gradient implemented as in Eqs (26.38) and (26.39), where ⎡ ⎢⎢⎢ q1 ξ ⎢ G ⎢⎢⎢⎢ ⎣ q2 ξ + q5 ξη

⎤ q3 η + q5 ξη ⎥⎥⎥⎥ ⎥⎥⎥ . ⎥⎦ q η

(26.48)

4

The underlined bi-linear mode is used only in the EADG5A element and it improves performance in the pinched hemisphere example of Sect. 26.5.2. In the EADG4 element, this mode is not used. In all (HW and EADG) elements, an identical form of the Perturbed Lagrange (PL) method is used to enforce the drill RC. It has the following features: (1) the 3-parameter Lagrange multiplier T is used, and (2) gamma-stabilization is applied to eliminate one spurious mode, see Sect. 26.4.4.


407

All elements use the Green strain, and the multipliers of additional modes are eliminated on the element’s level and updated by the scheme U2, see [18]. The 2 × 2 Gauss integration is used in all elements. For the transverse shear strains, the ANS technique of [2] is used. All tested elements have a correct number of zero eigenvalues and pass the constant strain patch tests, also with unconstrained boundary conditions for the drilling rotation, see [20].

26.5.1 Cook’s Membrane

P

44

44

A

16

This test is essential in evaluating the quality of a membrane part of a shell element, as in this test the shear deformation dominates and the elements are skew and tapered. The membrane is clamped at one end, while at the other end, the uniformly distributed tangent load P = 1 is applied, see Fig. 26.6. The data is as follows: E = 1, ν = 1/3, and the thickness h = 1. This test was proposed in [3]. Two meshes are used in computations; a coarse 2 × 2-element mesh and a fine 32 × 32-element mesh. The regularizing parameter γ = G is used.

y

Fig. 26.6 Cook’s membrane. Initial geometry and load

x 48

For the 2D+drill and 2D elements, the vertical displacement at point A is compared in Table 26.2, and the conclusions are as follows: 1. Comparing the 2D+drill elements to the 2D elements, we can evaluate the quality of implementation of the drilling RC, which should not disturb the displacements, as stipulated by eq.(26.1). We see that all the elements with the drilling rotation are slightly locked comparing to their 2D counterparts. If we used the 5p stress for the 2D+drill elements then the results would be much more locked than the results presented here for the 7p stress and the EADG2 enhancement. Note that we do not manipulate the value of the regularizing parameter γ; the

408


Table 26.2 Cook’s membrane. Linear test of 2D and 2D+drill elements Element

Number of

Displacement uy at A

characteristics

strain modes

Mesh 2 × 2

Mesh 32 × 32

HW, 2D+drill,

9

21.317

23.936

7p stress + EADG2

7

20.722

23.932

5

20.713

23.931

9

21.353

23.940

7

21.177

23.939

5

20.751

23.936

EADG4, 2D+drill

4

21.015

23.936

EADG4, 2D

4

21.050

23.940

HW, 2D, 5p stress

lower value would improve the results but we would not be able to evaluate the quality of approximations. 2. Comparing 2D+drill HW elements to the reference 2D+drill EADG4 element, i.e. with the enhanced displacement gradient element with 4 modes, we see that only the HW element using the 9p assumed strain is more accurate than the EADG4 element. For this reason, we use 9p assumed strain also to the membrane part of the HW shell elements. For the shell elements, the results of the linear analysis are presented in Table 26.3, where the vertical displacements and the drilling rotation at point A are given. All the tested HW shell elements perform identically and displacements are more accurate than by the EADG elements. Note that the drilling rotation converges from the above for both types of elements, which can be attributed to the Perturbed Lagrange method used to the drill RC. Table 26.3 Cook’s membrane. Linear test of shell HW elements. γ = G Element

Mesh 2 × 2

Mesh 32 × 32

uy

ω

uy

ω

HW, all

21.317

0.925

23.936

0.891

EADG5A

21.017

0.908

23.936

0.891

EADG4

21.015

0.908

23.936

0.891

To compare the time of generation of the tangent matrix for the tested elements (without solving the equations), we used a fine 300 × 300-element mesh, which yields 90 thousands elements and about 542 thousands unknowns. The times obtained for particular elements divided by the time for the EADG4 element are presented in Table 26.4, and they indicate that the HW elements are relatively slow. We checked that the condensation procedure used during the


409

generation of the tangent matrix consumes up to 15% of time for the HW47 element. Hence, even with a very effective condensation of the elemental parameters, the element still will be more than 2 times slower than the EADG4 element. Table 26.4 Cook’s membrane. Time of generation of tangent matrix for 300 × 300-element mesh Element Time

HW47

HW39

HW31

HW29

EADG5A

EADG4

2.56

2.08

1.90

1.88

1.06

1.00

26.5.2 Pinched Hemisphere with Hole The hemispherical shell with an 18o hole is loaded by two pairs of equal but opposite external forces, see Fig. 26.7. The shell undergoes an almost in-extensional deformation, and a membrane locking can destroy the solution. For the shown FE mesh, the elements are flat and trapezoidal. Due to the double symmetry, only a quarter of the shell is modeled, and the 16 × 16-element mesh is used. The data is as follows: E = 6.825 × 107, ν = 0.3, the thickness h = 0.01. The shell is very thin, h/R = 0.001. 18 R=10 symm. symm.

z x

y

P. P.

free

Fig. 26.7 Pinched hemisphere. a) Geometry and boundary conditions. b) Deformed shape

This example was computed using our four-node shell elements, and the nonlinear analysis was performed using the Newton method. The solution curves for the inward displacement under the force are shown in Fig. 26.8. We see that the curves for the HW47, HW39, HW31 and HW29 element almost coincide. The EADG4 element yields a locked solution but the EADG5A elements is as accurate as the HW elements. Note that the difference between the EADG4 and EADG5 elements is caused by the bi-linear terms in matrix G of eq. (26.48), because in all other aspects these elements are identical.

410

K. Wi´sniewski and E. Turska Green strain, mesh 16x16, h=0.01 5 HW47 HW39 HW31 HW29 EADG5A EADG4

4

Load

3

2

1

0 0

1

2

3

4

5

6

7

8

9

Inward displacement at force P

Fig. 26.8 Pinched hemispherical shell. Inward displacement at force

Another question is the overall effectiveness (speed and robustness) of these elements in non-linear analyzes, which depends not only on the time of a single iteration but also on the radius and rate of convergence. To illustrate the difference between the elements, we perform two simple analysis: 1. One non-linear step. In this test, we seek the maximum load for which the Newton method converges, checking every P = 0.05. The results are presented in Table 26.5, where the maximum load and the number of Newton’s iterations are given. We see that the HW elements converge much faster and in a much smaller number of iterations than the EADG elements. The worst of the HW elements is HW31. Table 26.5 Pinched hemisphere with hole. One non-linear step Element Max P to converge No. of iterations

HW47

HW39

HW31

HW29

EADG5A

EADG4

0.80

0.80

0.55

0.80

0.40

0.40

10

10

10

9

19

20

2. Several non-linear steps up to the load P = 8.8. In this test, we account for the fact that the elements converge for different load increments. Thus, we perform 11 Newton’s steps for the elements which converge for P = 0.80, 16 steps for this which converges for P = 0.55, and 22 steps for these which converge for P = 0.40. The final load is P = 8.8 for all elements. The time divided by the time for the EADG4 element and the total number of iterations to convergence are presented in Table 26.6. We see that all


411

HW elements are about two times faster than the EADG4 element, and HW29 is the fastest. Table 26.6 Pinched hemisphere with hole. Several non-linear steps up to P = 8.8 Element P No. of steps No. of iterations Time

HW47

HW39

HW31

HW29

EADG5A

EADG4

0.80

0.80

0.55

0.80

0.40

0.40

11

11

16

11

22

22

62

62

89

61

204

193

0.65

0.55

0.71

0.47

1.11

1.00

26.6 Final Remarks Four HW shell elements, with the same membrane part but differing in the bending/twisting part and/or the transverse shear part, are developed and tested. The results of numerical tests indicate that: i The comparison of the HW elements among themselves, shows that the HW elements are equally accurate but differ in efficiency; the fastest is the element with the smallest number of modes, HW29. The least efficient is HW31. ii The comparison of the EADG elements, shows that EADG5A is more accurate than EADG4, e.g. as in the pinched hemisphere example. The set of shell examples presented in [15] and [7] indicates that the HW elements outperform the EAS elements in the class of shell elements without the drilling rotation. The examples presented in the current paper for the shell elements with the drilling rotation also seem to support this conclusion. However, a broader set of linear and nonlinear tests is needed, which will be a subject of our next paper, [24]. Acknowledgements This research was partially supported by the Polish Committee for Scientific Research (KBN) under grant No. N501-290234.

References 1. Badur J., Pietraszkiewicz W.: On geometrically non-linear theory of elastic shells derived from pseudo-Cosserat continuum with constrained micro-rotations. In: Pietraszkiewicz W. (ed.) Finite Rotations in Structural Mechanics, 19–32, Springer, Berlin, 1986 2. Bathe K-J., Dvorkin E.N.: A four-node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. Int. J. Num. Meth. Engng., Vol. 21, 367–383 (1985)

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3. Cook R.D.: A plane hybrid element with rotational d.o.f. and adjustable stiffness. Int. J. Num. Meth. Engng., Vol. 24, 1499–1508 (1987) 4. Chrósćielewski J., Makowski J., Stumpf H.: Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. Int. J. Num. Meth. Engng., Vol. 35, 63–94 (1992) 5. Chrósćielewski J., Makowski J., Pietraszkiewicz W.: Statics and dynamics of multi-segmented shells. Nonlinear theory and finite element method. IFTR PAS Publisher, Warsaw, 2004 (in Polish) 6. Flanagan D.P., Belytschko T.: A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int. J. Num. Meth. Engng., Vol.17, 679–706 (1981) 7. Gruttmann F., Wagner W.: Structural analysis of composite laminates using a mixed hybrid shell element, Comput. Mech., Vol.37, 479–497 (2006) 8. Kosloff D., Frazier G.A., Treatment of hourglass pattern in low order finite element codes. Int. J. Numer. Analyt. Meths. Geomech. Vol.2, 57–72 (1978) 9. MacNeal R.H., Harder R.L.: A refined four-noded membrane element with rotational degrees of freedom. Computers & Structures. Vol. 28, No. 1, 75–84 (1988) 10. Panasz P., Wisniewski K.: Nine-node shell elements with 6 dofs/node based on two-level approximations. Finite Elements in Analysis and Design, Vol. 44, 784–796 (2008) 11. Pian T.H.H., Sumihara K.: Rational approach for assumed stress finite elements. Int. J. Num. Meth. Engng., Vol.20, 1685–1695 (1984) 12. Piltner R., Taylor R.L.: A quadrilateral mixed finite element with two enhanced strain modes. Int. J. Num. Meth. Engng., Vol.38, 1783–1808 (1995) 13. Simo J.C., Armero F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Num. Meth. Engng., Vol. 33, 1413–1449 (1992) 14. Taylor R.L., Beresford P.J., Wilson E.L.: A non-conforming element for stress analysis. Int. J. Num. Meth. Engng., Vol. 10, 1211–1220 (1976) 15. Wagner W., Gruttmann F.: A robust nonlinear mixed hybrid quadrilateral shell element. Int. J. Num. Meth. Engng., Vol.64, 635-666 (2005) 16. Wilson E.L., Taylor R.L., Doherty W.P., Ghaboussi J.: Incompatible displacement models. In: Fenves S.J., Perrone N., Robinson A.R., Schnobrich W.C. (eds.) “Numerical and Computer Methods in Finite Element Analysis”. Academic Press, New York, 43–57 (1973) 17. Wisniewski K.: Recent Improvements in formulation of mixed and mixed/enhanced shell elements. SHELL STRUCTURES: THEORY AND APPLICATIONS. Proceedings of 9th Conference SSTA’2009, Jurata, October 14-16, 2009. (W. Pietraszkiewicz, I. Kreja, Eds) General Lecture, p.35-44, ISBN 978-0-415-54883-0, Taylor & Francis, 2010 18. Wisniewski K.: Finite Rotation Shells. Basic Equations and Finite Elements for Reissner Kinematics. CIMNE-Springer, 2010 19. Wisniewski K., Turska E.: Kinematics of finite rotation shells with in-plane twist parameter. Comput. Methods Appl. Mech. Engng., Vol. 190, No. 8-10, 1117–1135 (2000) 20. Wisniewski K., Turska E.: Enhanced Allman quadrilateral for finite drilling rotations. Comput. Methods Appl. Mech. Engng., Vol. 195, No. 44-47, 6086–6109 (2006) 21. Wisniewski K., Turska E.: Improved four-node Hellinger-Reissner elements based on skew coordinates. Int. J. Num. Meth. Engng., Vol.76, 798–836 (2008) 22. Wisniewski K., Turska E.: Improved four-node Hu-Washizu elements based on skew coordinates. Computers & Structures, Vol.87, 407–424 (2009) 23. Wisniewski K., Wagner W., Turska E., Gruttmann F.: Four-node Hu-Washizu elements based on skew coordinates and contravariant assumed strain. Computers & Structures, Vol.88, 1278–1284 (2010) 24. Wisniewski K., Turska E.: Four-node Hu-Washizu shell elements with drilling rotation. In preparation, 2011

Part V

Engineering Design

Chapter 27

Dynamic Analysis of Debonded Sandwich Plates with Flexible Core – Numerical Aspects and Simulation Vyacheslav N. Burlayenko and Tomasz Sadowski

Abstract Although significant work has been done in modeling sandwich panels, models for debonded sandwich plates with a flexible core, especially the vibration analysis, are at their infancy, and it will be the main focus of this paper. This study deals with a finite element (FE) analysis of vibrations of flexible core sandwich plates that are weakened by damage embedded along the face sheet-to-core interface. The FE model developed is based on a refined general-purpose sandwich panel theory, where the first order shear deformation theory and assumptions of the 3-D elasticity theory are used for modeling the face sheets and the core, respectively. The FE mesh contains continuum shell elements for each of the face sheet layers and 3-D brick elements for the core. The comparison of the FE predictions to those known experimental and analytical results allow us to estimate the accuracy of the FE model developed, as well as to find the influence of the geometrical nonlinearity of the flexible core in vibrating and the contact nonlinearity caused by debonding on dynamics of sandwich plates. Keywords Sandwich plate · Debonding · Vibration · Finite element method

27.1 Introduction Composite sandwich panels are increasingly being utilized as load-carrying components in aircraft, aerospace and marine structures. A standard sandwich panel is commonly composed of two strong outer faces (face sheets) and a low-strength inner V. N. Burlayenko (B) Lublin University of Technology, 40 Nadbystrzycka str., Lublin, 20-618, Poland and National Technical University KhPI, 21 Frunze str., 61002, Ukraine e-mail: [email protected] T. Sadowski Lublin University of Technology, 40 Nadbystrzycka str., Lublin, 20-618, Poland e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 27, © Springer-Verlag Berlin Heidelberg 2011

415

416

V.N. Burlayenko and T. Sadowski

core. In a continuing effort for attaining higher stiffness and strength to weight ratios, the constituent sandwich materials have evolved from metallic face sheets and aluminum honeycomb core to composite materials for the face sheets, and non-metallic honeycombs and plastic foams for the core [1]. Thereby, advanced sandwich-type constructions imply the presence of a thick orthotropic core with bonded anisotropic composite laminated face sheets, where the ratio of Young’s moduli of the face sheets to the core lies between 500 and 1000. According with the sandwich structural concept, the top and bottom face sheets carry the in-plane and bending loads and are interacting through the core. The core should be strong enough in order to keep the desired distance between the face sheets and to prevent their sliding with respect to each other [2]. This ability of the core is achieved by its transverse shear strength. However, the novel sandwich panels made of the soft cores are flexible in the through-the-thickness direction. This flexibility conduces to the core compressibility and, as a result, to the change in the height of the core, Fig. 27.1. Such a transverse compression may arise due to a locally distributed load on the face sheets or unequal displacements of the face sheets from bending. Therefore, a normal transverse stress will appear apart from transverse stresses in the core. So, the plane section of the core takes on a nonlinear pattern, and a sandwich structure manifests a nonlinear response to a variety of load conditions. Thereby, the classical sandwich panel theories are insufficient to capture existing nonlinearities. One of the topics, playing a major role in the design of flat and curved composite sandwich-type panels is their vibration behavior. A good understanding of the vibration behavior of such structures is of crucial importance toward a reliable prediction of their dynamic response to time-dependent external excitations, prevention of the occurrence of the resonance, and for optimal design from the vibrational point of view [3]. Furthermore, during the service life sandwich panels often encounter low-velocity impacts, such as tool drop, runway stones, etc. Even though a visual examination of the impacted surface may reveal very little damage, known as barely visible impact damage, significant damage might exist between the face sheet and the core [4]. Manufacturing defects such as an incomplete bonding of the constituent layers or the air entrapping into the resin reach layer during the curing process may also lead to debonding existence as well. This type of damage invokes a

h1

h1/2 facesheet 1

hc/2

core undeformed section facesheet 2

hc/2

h2

h2/2

j01

u01 z1 ,w01

u1

w1

deformed section

x, u0c

zc ,w0c

uc

u02 z2 ,w02

u2 j02

wс

w2

Fig. 27.1 Cross section deformation pattern of a sandwich plate with a flexible core

27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core

417

substantial reduction of the compressive and bending strengths of the sandwich construction [5]. To ensure the structural integrity after the local striking or with the local faults, the ability of the sandwich structure to retain the load carrying capacity in the presence of the imperfections must be examined and quantified. In general, to analyze sandwich structures with a perfect interface, one of two approaches are commonly used, that is either the equivalent single layer (ESL) approach or the layer-wise (LW) one. Theories based on ESL are a straightforward extension on a sandwich panel regarded as a 2-D equivalent single layer of either the Kirchhoff’s classical theory (CPT) or the Reissner-Mindlin’s first-order shear deformation theory (FSDT) or those higher order theories (HSDT), assuming the displacements in form of continuously differentiable function of the thickness coordinate [6]. However, though ESL models are able to predict well the global dynamic behavior of thin and moderately thick sandwich panels, they still cannot account for discontinuities in displacements field and stresses at the face sheet-to-core interface. In this case the analysis of sandwich panels requires the use of layer-wise theories, where each layer is considered separately. By assuming a unique displacement field for each layer and fulfilling the interface kinematic continuity conditions, a sandwich panel is analyzed [7]. Recently, special theories, such as the high-order sandwich panel theory (HSAPT) [8] and the mixed layer-wise theory [9] have been developed. Herewith, in these theories, the formulation uses either CPT or FSDT, or HSDT for the face sheets and a 3-D elasticity theory or equivalent one for the core. Different such theories are required to model any kind of flat, cylindrical and spherical sandwich panels, because of using equilibrium approach [10]. Nevertheless, as expected, analytical modeling of sandwich-type panels with discontinues such that debonding is much more intricate than that of perfectly bonded sandwich structures. In contrast to the case of perfect sandwich panels, for which the assumptions are postulated for the structure as a whole, in the case of a sandwich panel with an imperfect interface such assumptions have to impose separately on perfect and debonded its parts and involve both compatibility conditions between those parts and contact conditions associated with local interactions in the debonding zone. Moreover, the dynamic analysis of sandwich panels with a flexible core and containing debonding is complicated by the presence of various types of nonlinearities, some of them are: (i) asymmetry with respect to the mid-surface of the panel inducing global bending-stretching coupling; (ii) contact-impact problem, which induces a local structural behavior along with a global one, and (iii) stress concentration in the vicinity of the discontinuous promoting the debonding growth. Thus, the actual dynamic behavior of such panels can be examined only by nonlinear analysis techniques. A comprehensive summary on modeling techniques used to perform a dynamic analysis of debonded (or delaminated in the case of laminated structures) composite beams, plates and shells is done by Della and Shu [11]. In line with the authors’ comments, from a mathematical point of view, analytical solutions are possible only for a limited class of idealized dynamic problems such 1-D sandwich beams or 2-D plates with through-the-width debonding. Moreover, the nonlinear effects are usually neglected because of linear models applied in most of those studies. There

418


have been only few papers studying dynamics of debonded beams including the nonlinearity effects, e.g. [12]. The complexity of the problem have progressively led to the predominance of numerical models based mainly on the finite element method (FEM). Nonlinear FE dynamic analyses of laminated beams have been accomplished, e.g. in [13, 14], where beam elements and 2-D plate elements were utilized, respectively. An analytical approach and FE model of a three-layered sandwich beam accounting for intermittent contact in the modal analysis have been developed in [15]. A further contribution to the nonlinear dynamic behavior of sandwich plates with imperfections can be found in [16], where the case of sandwich plates containing post-impact zone was considered. In this study, a FE modeling approach is used within the commercial package ABAQUS, to develop models of both intact and damaged by debonding sandwich plates with a flexible core and simulate their dynamic responses. The models proposed utilize orthotropic shell/plate elements for the face sheets and continuum either isotropic or anisotropic solid elements for the core. The presence of the transverse deformation and general material orthotropy, coupled with the contact-impact interaction of detached surfaces, makes the dynamic FE analysis rather computationally complex. In this regard, the use of the explicit integration rule provides a benefit for solving the dynamic problem in a computationally cost effective manner. Thereby, to simulate dynamic responses of sandwich plates involving contact-impact conditions in the detached segments of the debonding zone, the ABAQUS/Explicit code [17] is exploited.

27.2 FE Model Developments In fact, in sandwich panels with a flexible core stresses have an inherent threedimensional nature, consequently, from the point of view of accuracy, usually the best approach in terms of FE modeling is to utilize three-dimensional (3-D) elements for each sandwich layer (faces and core). However, this approach will lead to severe problems of aspect ratio, and require an extremely refined mesh and, as a result huge computational efforts. In practice such models are limited only to small and particular tasks [18]. An alternative to such an approach is the employment of finite elements based on partially 3-D models, where the core is represented by solid elements and the face sheets by plate or shell elements [19]. In this case reasonable computational cost is combined with the fidelity of strain-stress states associated with HSAPT that provides a good trade-off between the level of accuracy and computational efficiency. Thus, 3-D models were developed utilizing shell elements, employing the Reissner-Mindlin’s hypotheses in conjunction with the laminated plate theory on the face sheets and solid brick elements, based on the 3-D elasticity theory on the core, Fig. 27.2.


Nx

Nxy

Nxy

Ny

Nxy+Nxy,ydy

top face

s 23

419

Nx+Nx,xdx Ny+Ny,ydy Nxy+Nxy,xdx

f

sf22 tf12 sc22 tc12

tf12 t 12 c

s

f s 33 s 13 f

s 33 c

f 11

s 13 c

Mxy

s 23 c

Mx My

Mxy

My +My,ydy

Mxy + Mxy,xdx

sc22

Mxy + Mxy,ydy Mx +Mx,xdx

core

(a)

(b)

Fig. 27.2 Details of the stress state at the face sheet-to-core interface: (a) core and face sheet stresses; (b) force and moment resultants for face sheet

27.2.1 Face Sheet FE Model In general, composite sandwich plates studied in this paper are made of two composite laminated face sheets and a soft core. The faces are considered to be arbitrary thickness and consist of an elastic orthotropic material. For the face sheets it was utilized 8-node continuum shell reduced integrated SC8R finite elements. These elements are positioned on the upper and lower core sides and are directly connected to the core through their share nodes. Fig. 27.3 shows the node numbering, integration point and faces of the used finite element. The SC8R continuum shell elements discretize an entire three-dimensional body, unlike conventional shells which discretize a reference surface. These elements have displacement degrees of freedom u f = (u1 , v1 , w1 , . . . , u8 , v8 , w8 )T only, use linear interpolation, and allow mechanical loading for static and dynamic procedures. In what follows, the usual definitions of the FEM are throughout used. The displacement vector u at any point of the face sheet as a shell-like structure may be expressed uf =

8

f

f

Ni [I]ui ,

(27.1)

i=1 f

ui are the displacement vector corresponding to the ith node of the SC8R element, f Ni are the shape functions associated with node i, and [I] is the 3×3 identity matrix. For simplicity herein and further such equations may be written as Fig. 27.3 Node numbers, faces and an integration point of a SC8R element

face 2 face 6

4 5

face 5

8

7 3

x1

6 1 face 1

2 face 3

face 4

420


u f = [N] f u f

(27.2)

From a modeling point of view continuum shell elements look like threedimensional continuum solids, but their kinematic and constitutive behavior is similar to conventional shell elements. It is important to notice that the SC8R elements are general-purpose elements that can provide solutions to both thin and thick shell problems. Thereby, the hypotheses of Reissner-Mindlin’s shell theory, including the effects of transverse shear deformation are assigned to the element’s kinematics as the shell thickness increases, and the Kirchhoff’s assumptions are fulfilled within the elements as the thickness decreases, herewith the transverse shear deformation tends to be very small. Moreover, these elements account for finite membrane strains, arbitrary large rotation, and allow for a thickness change, making them suitable for nonlinear geometric analysis. In doing so, computation of the change in thickness (is only valid in geometrically nonlinear analysis) is based on the element nodal displacements, which in turn are computed from an effective elastic modulus defined at the beginning of an analysis [17]. If the displacement vector u is known at all points within the elements, the strain vector ε at any point can be calculated in terms of nodal variables as follows ε f = [B] f u f , (27.3) where [B] f are the differential operators associated with the strain-displacement relationships mentioned earlier. In general, each of the orthotropic face sheets is of composite laminate with an arbitrary lay-up. When the material and reference coordinate systems coincide, the constitutive relationship for the mth face sheet orthotropic lamina is represented by σ (m) = [Q](m)ε (m) ,

(27.4)

where σi j and εi j are the components of the stress and strain vectors, respectively, and Qi j represents the stiffness matrix with nine independent material constants and depend on the material properties and fiber orientation of the mth lamina. In accordance with the FSDT, the five components of the strain vector are taken into account such as the in-plane normal and shearing strains (ε1 , ε2 and γ12 ) varying linearly, and the transverse shearing strains ( γ13 and γ23 ) being constant trough the thickness of the laminate. This assumption necessitates the use of shear correction factors. The transverse shear stiffness is computed by matching the shear response for the shell to that of a three-dimensional solid for the case of bending about one axis and assuming a parabolic variation of transverse shear stress in shell each layer. Then the shear strain energy, expressed in terms of section resultants and strains, is equated to the strain energy of this distribution of transverse shear stresses [17]. Moreover, the stress in the thickness direction may not be zero in the continuum shell element and may cause additional strain beyond that due to Poisson’s effect. This stress is computed by penalizing the effective thickness strain with a constant ’thickness modulus’ [17]. The shell elements use bending strain measures that are approximations to those of Koiter-Sanders shell theory, i.e. the displacement field normal to the shell surface


421

does not produce any bending moments. The section force and moment resultants per unit length in the normal basis directions in a given laminate section of thickness h can be defined on this basis as (N1 , N2 , N12 ) = (M1 , M2 , M12 ) =

+ h2

− h2 + h2 − h2

(V1 , V2 ) =

(σ1 , σ2 , τ12 )dz

(σ1 , σ2 , τ12 )zdz

+ h2 − h2

(27.5)

(τ13 , τ23 )dz

This leads to the constitutive relationships related the resultants and the generalized strain vectors ⎤⎧ f ⎫ ⎧ f ⎫ ⎡ ⎪ ⎢⎢⎢ A B 0 ⎥⎥⎥ ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎥⎥ ⎪ ⎪ f⎬ ⎪ ⎢⎢ ⎪ pf ⎪ ⎪ ⎨ ⎨ ⎬ ⎢ ⎥ ⎢ ⎥ = (27.6) M B D 0 ε ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪, ⎢ ⎥ b⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Vf ⎪ ⎭ ⎢⎣ 0 0 H ⎥⎦ ⎪ ⎩εf ⎪ ⎭ s

where ε p , ε b and ε s are the in-plane, bending and transverse shear strain vectors and Ai j , Bi j , Di j and Hi j are the components of the in-plane, bending-in-plane, bending and transverse shear stiffness matrices of this section and are given by %

& A i j , B i j , Di j = Hi j =

+ h2

−h +2 h 2 − h2

% & 2 Qm i j 1, z, z dz,

(i, j = 1, 2, 3) (27.7)

Qm αβ ki k j dz,

(i, j = 1, 2, α, β = i + 4, j + 4)

Here the k1 , k2 parameters are the shear correction coefficients as defined above. If there are n laminae in the lay-up, we can rewrite the above equations as a summation of integrals over the n laminae. The material coefficients will then take the form Ai j =

n

Qm i j (hm − hm−1 )

m=1 n

% & 1 2 2 Qm i j hm − hm−1 2 m=1 n & 1 m% 3 Di j = Qi j hm − h3m−1 3 m=1 n Hi j = Qm αβ (hm − hm−1 ) ki k j , Bi j =

(27.8)

m=1

where the hm and hm−1 in these equations indicate that the mth lamina is bounded by surfaces z = hm and z = hm−1 . In the matrix notations the constitutive relationships (27.6) can be written as

422


σ f = [E] f ε f ,

(27.9)

where σ f and ε f are generalized stress and strain vectors, respectively, and [E] f is the matrix of the elastic material constants associated with equations defined above. It should be mentioned that in the case of laminated structures, continuum shell elements can be also stacked to provide more refined through-the-thickness responses of transverse shear stresses and forces [17].

27.2.2 Core FE Model Because, continuum shell elements can be connected directly to first-order continuum solids without any kinematic transition, 8-node isoparametric linear solid ’brick’ elements with incompatible mode C3D8I are chosen for core modeling. In addition to the standard displacement degrees of freedom (three translations in each a node), incompatible deformation modes are added internally to the elements. The primary effect of these modes is to eliminate the parasitic shear stresses that cause the response of the regular first-order displacement elements to be too stiff in bending [17]. The incompatible mode elements use full integration and, thus, have no hourglass modes. An effective (equivalent) core material is considered as a homogeneous continuum either isotropic for plastic foams or orthotropic for non-metallic honeycomb structures. In the latter, the material constants of the equivalent material, related to the cell geometry and actual parent material properties of the cells were calculated by using either the analytical homogenization formulae [20] or numerical FEM techniques, e.g. [21]. Although brick elements provide accurate results just for the cases when transverse shear effects are predominant and when the normal stress cannot be ignored, they have a deficiency in modeling the transverse shear stress through the thickness. It should be acknowledged in the context of the stress/displacement solution that the transverse shear stresses in solid elements usually do not vanish at the free surfaces of the structure and are usually discontinuous at layer interfaces. Isoparametric interpolation adopted for the elements C3D8I is defined in terms of the isoparametric element coordinates (g, h, r) shown in Fig. 27.4. These are material coordinates. They each span the range -1 to +1 in an element. For instance, the interpolation function defined the translation in the first direction is as follows [17] u = 18 (1 − g)(1 − h)(1 − r)u1 + 18 (1 + g)(1 − h)(1 − r)u2 + 18 (1 + g)(1 + h)(1 − r)u3 + 18 (1 − g)(1 + h)(1 − r)u4 + 18 (1 − g)(1 − h)(1 + r)u5 + 18 (1 + g)(1 − h)(1 + r)u6 + 18 (1 + g)(1 + h)(1 + r)u7 + 18 (1 − g)(1 + h)(1 + r)u8,

(27.10)

27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core face 2

423

face 5

8

7

face 6 r 5 h

3 4 x

3

4

x

face 4

g 4

x

1

1

2

3

x

Z

2 1

2

face 1

Y X

face 3

Fig. 27.4 Node ordering, face numbering and integration points on each the faces for brick elements

where ui , i = 1, . . . , 8 are nodal translations in the first direction. Analogously, one can write the interpolation functions for the translations in other directions (v, w). In general, the translations through the interpolation functions and nodal variables can be expressed as follows uc = [N]c uc (27.11) Here the matrix [N]c includes the isoparametric shape functions Ni associated with node i and uc is the vector of the nodal displacements. In the core all the six components of the strain vector are taken into account. The strain-displacement equations allowing for finite strains and rotations in largedisplacement analysis in conjunction with the displacement fields are applied to calculate these components as ε c = [B]c uc , (27.12) where [B]c is the corresponding differential operator. The stress-strain relationships based on the 3-D elasticity theory give the components of the stress vector as follows σ c = [E]cε c ,

(27.13)

where [E]c is the matrix of the elastic material constants associated with either an isotropic or orthotropic homogeneous material.

27.2.3 General 3-D FE Model The dimensions of the elements for the 2-D continuum shell and 3-D brick elements are identical on the panel plane directions. For the case of perfect bonding the links between them are provided by means of share nodes, since both the solid elements and shell elements have only the translation degree-of-freedom (DOF). Thus, the deformation compatibility between the faces and the core elements is fulfilled. In the case of the debonding presence an actual small gap is introduced

424


Fig. 27.5 3-D modeling of a sandwich plate with a circular debonding zone

in a certain zone between the face sheet and core finite elements. Because it was assumed that debonding may be placed arbitrarily within the face-to-core interface, the finite elements of the general 3-D FE mesh were generated by partition of the total model onto several parts, which were connected with each other through share nodes, Fig. 27.5. An additional adjustment was not used. The general mesh contained three different zones meshed by applying different mesh methods and mesh densities such as a fine mesh for the debonding zone, the next zone surrounding debonding with decreased mesh density, and coarse mesh that was introduced to minimize a CPU time in calculations, Fig. 27.5. No artificial adjustment of either the material or geometrical properties was made at the debonding region to ensure as close as possible a physically real case.

27.3 Aspects of FE Modeling A finite element model based on the aforementioned assumptions is derived to analyze dynamics of sandwich plates. In order to be consequent, a brief presentation of the main steps of this finite element procedure is described in the following subsections.

27.3.1 FE Equations of Motion A solution of the problem, to which solid mechanics is applied, implies finding displacements, deformations, stresses, forces, and other variables in a solid body, which have to obey the equilibrium requirement for both force and moment at all times over any arbitrary volume of the body. The FEM is based on approximating this equilibrium requirement by replacing it with a weaker requirement, that equilibrium must be maintained in an average sense over a finite number of divisions of the volume of the body [22]. This equilibrium statement is usually derived in the form of the virtual work statement for later reduction to the approximate form of equilibrium used in a finite element model. For a finite element as a body at a time instance t, occupying in the current configuration a volume V with the surface S, bounding this volume and subjected at any


425

point on S to the force t per unit of current area and at any point within V to the body force f per unit of current volume the equilibrium statement, in the form of the virtual work principle, can be written as σ · ·δεεdV = tT · δudS + fT · δudV, (27.14) V

S

V

where t, f, and Cauchy stress matrix σ are an equilibrium set ∂ t = n · σ, · σ + f = 0, σ = σ T , ∂x T ∂δu and δεε = 12 ∂δu + is the virtual strain, and δu is an arbitrary vector-valued ∂x ∂x admissible function, compatible with all kinematic constraints. To discuss the dynamic behavior, the d’Alembert force is introduced into the overall equilibrium equation. Then, body force at a point f, can be written as an externally prescribed body force F, and a d’Alembert force, i.e. f = F − ρu¨ where ρ is the current density of the material at this point and u¨ is the acceleration filed of the point. The virtual work equation (27.14) can be rewritten as T T σ · ·δεεdV = t · δudS + F · δudV − ρu¨ T · δudV (27.15) V

S

V

V

In the general case, the finite element interpolator can be written as u = [N] u,

(27.16)

where [N] are interpolation functions that depend on some material coordinate system and u are nodal variables of the finite element. The virtual field δu must be compatible with all kinematic constraints. Introducing the above interpolations the displacement will have a certain spatial variation, then, δu must also have the same spatial form δu = [N] δu

(27.17)

Thereby, the continuum variational statement is approximated by a variation over the finite set of nodal variables δu. Then, the virtual material strain associated with δu is δεε = [B] δu (27.18) Following the well-known FEM procedure the equilibrium equation, related to as the equations of motion for each a finite element will be approximated as )

. e/ . / M u¨ + Ke u = Fe ,

(27.19)

where [M ] = V ρ0 [N] [N] dV0 is the mass matrix of the element, 0 ) [Ke ] = V [B]T [E] [B] dV0 is the stiffness matrix of the element, [E] is the elasticity e

0

T

426


matrix in stress-strain relationships, and V0 is the element volume in its reference state. Including into the equations of motion the Rayleigh damping, defined through a damping matrix such that it is a linear combination of the mass and the stiffness matrices, the finite element equations of motion one can read as follows . e/ . / . / M u¨ + Ce u˙ + Ke u = Fe

(27.20)

Thereafter the assembling matrix procedure, usual in the FEM is performed and the final discretized equations of motion are the following ¨ + [C] U(t) ˙ + [K] U(t) = F(t), [M] U(t)

(27.21)

where [M], [C] and [K] are the global mass, damping and stiffness matrices obtained ˙ and U(t) ¨ are the global vectors of unknown by the assembly procedure, U(t), U(t) nodal displacements, velocities and accelerations, respectively, F(t) is the global vector of nodal external forces. The boundary conditions are imposed to the initial boundary (its position is known), giving the statement of the boundary problem at the given initial conditions of the equations of motion consistently closed. Thus, the position of any point of the body, including the boundary will be known once the system of equations is solved.

27.3.2 Contact Model In general, the vibration of a structure containing detached fragments is accompanied by intermittent contact between them. The dynamic response of the structure depends on such a contact which is a function of both space and time domains. Therefore, the extent of the contact region and the contact pressure arising between the contactable interfaces should be also determined as part of the solution. To address the contact-impact problem, the contact analysis between two deformable bodies has to be incorporated into the general FE formulation. For a body with a contactable interface Γ, the two surfaces called Γ + and Γ − are assumed, Fig. 27.6. In fact, the interaction between them is associated with the appropriate constraints. The contact constraints add the following Kuhn-Tucker conditions: the surfaces cannot interpenetrate, and the contact traction (contact pressure) must satisfy momentum conservation on the interface [23]. Using finite element discretization, one of two contacting surfaces is defined as the slave and another one is called as the master. The non-penetration conditions, usually represented by the normal component of displacement and stress, are given a gap (or ‘overclosure’) function g N (x) defining the length of the orthogonal projection of a typical slave node s on the master surface: ' ( g N (x, t) = n · x s − x ,

(27.22)

27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core Fig. 27.6 FE contact discretization

slave surface s G-

n x

xs

427

contact force

gN

master surface

G+

s penetration

where n is the outward unit vector orthogonal to the master segment, while x s and x are the vectors identifying the current positions of the salve node s and the closest nodes on the master segment, respectively. The expression has to be checked for all candidate contact nodes s, which are elements of the finite set Γ − . The normal contact pressure acting on the node s is tN = tc · n, where tc is a contact traction vector. For gN ≤ 0 the constraint equation for a node s becomes active, that is x s ∈ Γ + ∩ Γ − and tN ≥ 0, otherwise the constraint is inactive, i.e. x s ∈ Γ − and the surfaces are separated. The relationship between the gap function and contact pressure is referred to as a constitutive contact model. Analogously, the tangential contact constraints can be enforced, where a tangential gap function, called as a slip function gT (x, t) and a tangential traction tT = tc · (I − n ⊗ n) can be related by the classical Coulomb friction law [24]. The enforcement of the contact constraints is achieved by appending the additional term into the virtual work principle (27.15). The virtual work done by the contact forces including the contribution from both the normal and tangential contact responses can be written as δWc = δ (tN g N + tT gT )dΓ (27.23) Γ

Herewith, the integral over the contact surface Γ consists of two surface integrals over Γ + and Γ − . The gap functions g N and gT are functions of the displacement vector u(x,t), and following the FE approximation (27.16), they are functions of the nodal variables u. The contact traction components can be expressed to be proportional to surface penetrations (gap functions) in the penalty method, or treated as additional unknowns in the Lagrange multipliers method. These two methods can also be combined in the augmented Lagrangian formulation. Thereby, the discrete weak form of the dynamic problem involving contact-impact conditions in terms of the global nodal displacements has been formed as follows ¨ + [C] U(t) ˙ + [K] U(t) = F(t) + FN (t) + FT (t), [M] U(t)

(27.24)

where F N (t) and FT (t) are the generalized global normal and shear contact force vectors, respectively, generated so that eliminate all having penetrations between surfaces of the detached segments during oscillations, Fig. 27.6. It should be noticed that the contact forces do not depend on the shape functions [N], but instead on the approximation of the surface, that is the gap functions approximation.

428


27.3.3 Integration Rule The most computationally efficient way for solving the discrete equations of motion (27.24) is the use of the explicit integration rule together with diagonal lumped element mass matrices. The explicit central difference integration operator used can be written in the following form ˙ (i+ 2 ) = U ˙ (i− 2 ) + Δt U 1

1

(i+1) + Δt(i)

2

¨ (i) , U

˙ U (i+1) = U(i) + Δt(i+1) U

(i+ 12 )

, (27.25)

% & % & where the superscript (i) refers to the increment number and i − 12 and i + 12 refer to midincrement values. The explicit procedure requires no iterations and no tangent stiffness matrix, because the accelerations at the beginning of the each increment can be calculated quite simple by the inversion of the lumped mass matrix diagonalized in advance as follows % & ¨ (i) = [M]−1 F(i) − I(i) , U (27.26) where F(i) is the vector of external nodal forces, and I(i) is the general vector of internal nodal forces. In the case of dynamic contact existing along with global dynamic motion, the general vector of internal nodal forces will include contact forces as well, when the surfaces come into contact at a defined time increment (i). The explicit procedure integrates the equation (27.24) through time by using many small time increments. The time increment used in the analysis must be smaller than the stability limit of the central-difference operator applied. An approximation to the stability limit is defined by the smallest transit time of a dilatational wave across any of the elements in the mesh as the following [17] le Δt = min , (27.27) cd where le is the characteristic element dimension and cd is dilatational wave speed of the material.

27.3.4 FE Analysis Using the FE code ABAQUS static bending, free vibration, dynamic transient and dynamic steady-state analysis of sandwich plates containing the debonding zone along the face sheet-to-core interface were carried out. The static and free vibration analysis were run by using ABAQUS/Standard with a linear model for debonded sandwich plates developed in [25]. The other two analysis exploited the nonlinear finite element model based on the abovementioned modeling approach and were run by utilizing ABAQUS/Explicit. It is important to notice that the nonlinear FE model does not assume the advance of the debonding zone during oscillations.

27 Dynamic Analysis of Debonded Sandwich Plates with Flexible Core Fig. 27.7 Graphical interpretation of the pressureoverclosure relationship used

429 Contact pressure

tN = 0 for gN = 0 for

gN < 0 tN > 0

(open), and (closed)

Any pressure possible when in contact

No pressure when no contact

Clearance

To provide during the dynamic forced analysis the contact-impact behavior of the face sheet and the core within the debonding zone of the vibrating sandwich plate, the interaction option was activated in the FE code [17]. The contactable parts were simulated by using their surface-to-surface discretization in terms of slave and master surfaces. Such a contact formulation enforces contact conditions in an average sense over the regions nearby slave nodes rather than only at an individual slave node. Thereby, surface-to-surface contact will provide more accurate contact stress resultants, which are being used to form with greater precision the right hand side of the equations of motion (27.24). The relative motion of the interacting surfaces in the contact simulation was described with finite sliding kinematic assumptions that are the most general case, and which allow any arbitrary motion of the surfaces involving their separation, sliding and rotation. The constitutive behavior of the surfaces coming into contact in the normal direction was assumed to be governed by the ’hard contact’ model, whereas, for the sake of reducing the computational cost and because the sliding in the contacting surfaces is small, frictionless conditions were accepted between them in the tangential direction. It is worthwhile to notice that ‘hard contact’ implies that the interacting surfaces transmit no contact pressure unless the nodes of the slave surface contact the master surface and no penetration is allowed at each constraint location, Fig. 27.7. Thus, the hard contact model will minimize a measure of overclosure of nodes of the slave surface into the master surface at the constraint locations in the best way. To resolve in the dynamic analysis the normal contact constraints imposed by the physical pressure-overclosure relationship corresponding to the hard contact model applied, the penalty constraint enforcement algorithm is used. This method was chosen because it does not increase the cost of the analysis compared with the Lagrange multipliers algorithm.

27.4 Numerical Results The developed finite element model has been employed for the analysis of both perfect and debonded sandwich plates with a flexible core. Both static and dynamic loading cases have been considered. The numerical results, obtained with ABAQUS

430


were compared to known analytical or numerical solutions whenever it was possible. Some new results for soft-cored sandwich plates are below presented as well.

27.4.1 Validation In order to demonstrate the accuracy and performance of the developed FE model, several examples of undamaged sandwich composite plates subjected to static and dynamic loading have been analyzed using the package ABAQUS and the results obtained have been compared with the results published in the literature. Example 27.1. A symmetric simply supported square sandwich plate with planar dimensions a × b = 200 × 200 mm2 subjected to bi-directional sinusoidal transverse load on its top surface with the intensity of 100 Nmm−2 has been considered. The 3-D elasticity solution obtained by Pagano [26] has been used for comparison of the FEM results. The total thickness of the plate is h = 50 mm, the thickness of each face sheet is one tenth of the total thickness. The performance of several different models of discretization was assessed. A mesh-sensitivity analysis was performed for all models. A suitable mesh was established once a plateau with a deviation less than 2% in the calculated values of deflections and stresses was found. As many as 80 elements along the length and 10 elements across the width were used. One SC8R element per ply across the thickness of the face sheet with seven cross-section integration points and eight C3D8I elements across the thickness of the core were utilized, as it is important to keep these first-order solid elements cubic in shape. The materials of the elements were modeled as orthotropic solids with constants listed in Table 27.1. The comparative results of the analysis are shown in Table 27.2, where the following normalization coefficients were utilized: f

(σ ¯ 11 , σ ¯ 22 , σ¯ 12 ) =

100E 2 h3 h2 h (σ ) (¯ ) (τ ) , σ , σ , τ , τ ¯ = , τ , w ¯ = w 11 22 12 13 23 13 23 q0 a q 0 a2 q0 a4

The stresses were calculated at the element integration points, but the values of stresses in the output file were determined at the nodes by extrapolating the stresses from the integration points [17]. As can be seen from Table 27.2, the results of the present ABAQUS model are in a good agreement with the analytical results. Fig. 27.8 showing through-the-thickness variation of the maximum normalized deflection has also presented to pay attention the necessity of using the layer-wise model for the analysis of the sandwich plate with a low strength core material. One can see the differences in the value of deflections between the top and bottom face sheets that means the change of the core height. As well the through-the-thickness % & a b variation of normalized transverse shear stresses calculated at the points 12 , 2,z % & b and a2 , 12 , z , respectively, have been presented in Figs 27.9a and 27.9b to have an


431

Table 27.1 Material properties of sandwich plates Example Ref. Components Face sheet

1

E1 , GPa

E2 = E3 , GPa

G 12 = G 13 , GPa G23 , GPa

ν12 = ν13 ν23

172.4

6.89

3.45

0.25

0.25

[26] Core E1 = E2

E3

G13 = G 23

G 12

ν12

ν13 = ν23

0.276

3.45

0.414

0.1104

0.25

0.1

Face sheet, ρ = 1627 kgm

2

1.378

−3

E1 , GPa

E2 = E3 , GPa

G 12 = G 13 , GPa G23 , GPa

ν12 = ν13 ν23

131.0

10.34

6.895

0.22

6.205

0.49

[9] Core, ρ = 97 kgm−3 E1 = E2

E3

G13 = G 23

G 12

ν12

ν13 = ν23

6.89 · 10−3

6.89 · 10−3

3.45 · 10−3

3.45 · 10−3

0.0

0.0

Face sheet, ρ = 4400 kgm−3

3

E1 = E2 = E3 , GPa

G 12 = G 13 = G23 , GPa

ν12 = ν13 = ν23

36.0

14.0

0.3

E1 = E2 = E3 , GPa

G 12 = G 13 = G23 , GPa

ν12 = ν13 = ν23

50.0 · 10−3

21.0 · 10−3

0.29

[15] Core, ρ = 52 kgm

−3

Table 27.2 Deflection and stresses in square sandwich plate under bi-directional sinusoidal load % & % & % & % & Source σ¯ 11 a2 , b2 , z σ ¯ 22 a2 , b2 , z τ¯ 13 0, b2 , 0 τ¯ 23 a2 , 0, 0 τ¯ 12 (0, 0, z) w¯ + h2

− h2

+ h2c

− h2c

+ h2

− h2

+ h2

− h2

[26] 1.556 -1.512 -0.233 0.196 0.259 -0.253

0.239

0.107

-0.144 0.148 7.425

FEA 1.268 -1.302 -0.255 0.219 0.233 -0.282

0.237

0.113

-0.137 0.140 7.367

insight into the strength of sandwich plates and demonstrate the interaction between the strong face sheet and the low density core. Example 27.2. For further validation, the free vibration analysis of sandwich panel with a flexible core is carried out using the ABAQUS software. A 20 mm thick, fivelayer (0/90/core/0/90) simply supported square of 200 mm by 200 mm sandwich plate is herein analyzed. The plate was made of composite anisotropic face sheets and an isotropic core, where the ratio of the face sheet thickness to the core thickness was 10. Material properties of the sandwich plate are given in Table 27.1. In a FE mesh one SC8R element per each ply across the thickness of the face sheet with five cross-section integration points and five C3D8I elements across the thickness of the core were used. The planar dimensions of the plate were discretized by 50 elements along each edge.

432


Fig. 27.8 Through-thethickness variation of the maximum normalized deflection

0.5 FEM

Thickness

0.25

Pagano

0

-0.25

-0.5 6.8

6.9

7.0

7.1

7.2

7.3

7.4

7.5

0.5

0.5

0.25

0.25

Thickness

Thickness

Normalized deflection

0

-0.25

-0.25

-0.5

-0.5 0

(a)

0

0.05

0.1

0.15

0.2

0.25

Normalized shear stress in direction 1

0

(b)

0.03

0.06

0.09

0.12

Normalized shear stress in direction 2

Fig. 27.9 Through-the-thickness variation of the normalized transverse shear stresses

The computed with ABAQUS results are compared with those predictions obtained 2 in [9]. Natural frequencies have been normalized by using the relation a4 ρ f Ω=ω , where h is the total plate thickness and ω is natural cyclic frequency. 2 f h E2

The comparative results of the normalized natural frequencies are presented in Table 27.3. A good agreement between the results one can be observed. To make evident the effect of ’soft’ core on the considered kinematics of the sandwich plate, the frequency spectrum containing fifty its natural frequencies have been analyzed. The results showed that high frequencies of the plate exhibit symmetric out-of-plane movements of the face sheets, so-called pumping modes, relating to the changes of the plate height. Several of such modes are presented in Fig. 27.10, where the core was excluded from the contour plot for the sake of clearness. Therefore, the linear assumptions for modeling sandwich plates with a flexible core are

Table 27.3 Comparison of non-dimensional natural frequencies for the intact sandwich plates Source

(1,1)

(1,2)

(1,3)

(2,2)

(2,3)

(3,3)

[9]

1.8480

3.2196

5.2234

4.2894

6.0942

7.6721

FEA

1.7952

3.1445

5.2235

4.1985

5.9945

7.5653


433

suitable only for describing low vibration modes, and the current model is capable to capture such new nonclassical effects.

(a)

(b)

(c)

(d)

Fig. 27.10 Contour plots of the pumping modes

Example 27.3. The free vibration analysis of a sandwich beam with a damaged region was considered in [15], where natural frequencies and relevant mode shapes were analytically calculated. That model was developed with ABAQUS to make a comparison between the predictions and assess the applicability of the FE model developed for taking into account an interfacial damage. A 300 mm long sandwich beam cored by a foam with a rectangular cross-section of 20 mm by 21 mm, containing damage of a 20 mm long at the middle span of the beam was simulated by 60 elements along the beam length and eight elements through its thickness. One element per thickness of each the face sheet and six elements across the core thickness were used. The thickness of the upper face sheet was of 0.5 mm and the lower one was 1 mm. Material properties of the sandwich beam were the same as described in [15] and they are listed in Table 27.1. The comparative values of the six first natural frequencies are shown in Table 27.4. As one can be seen, the close results are demonstrated. Table 27.4 Comparison of natural frequencies (Hz) between the intact and debonded sandwich plates Intact

Debonded

Mode

[15]

FEA

[15]

FEA

1

289.3

293.46

288.98

293.07

2

683.3

707.09

388.32

433.67

3

1096.9

1106.7

1093.2

1093.2

4

1151.6

1495.8

1146.9

1132.0

5

1778.2

1818.7

1771.3

1769.9

6

1895.3

1918.1

1842.2

2080.2

434


Table 27.5 Material properties of the sandwich plate Source

Components Face sheet, ρ = 1650 kgm−3

[27]

E1 = E3 , GPa

E2 , GPa

G12 = G23 , GPa G13 , GPa

ν12

ν13 = ν23

3.48

19.3

1.65

0.05

0.25

7.7

Core, ρ = 52 kgm−3 E1 = E2 = E3 , GPa

G12 = G13 = G 23 , GPa

ν12 = ν13 = ν23

92.1 · 10−3

27.3 · 10−3

0.42

27.4.2 Free Vibration Analysis A free vibration of a sandwich plate containing a debonding zone of the circular shape between the upper face sheet and the core is further studied. The plate was simply supported with planar dimensions of 270 mm by 180 mm and was made of a 50 mm thick foam core and a 2.4 mm thick each of the face sheets. The debonding zone’s diameter, d was of 30 mm. The material properties of the constituent layers of the plate are showed in Table 27.5. Pursuing the purpose, to assess the influence of the debonding presence on the modal parameters of the sandwich plate, the free vibration analysis of the same intact sandwich plate was performed too. The natural frequencies obtained with ABAQUS for these two plates are presented in Table 27.6. The comparative results demonstrate that debonding, firstly, changes the order of frequencies in the spectrum and, primarily, leads to reduction of magnitudes of the natural frequencies for bending modes in the low part of the frequency spectrum. Secondly, it does not almost change the natural frequencies associated with in-plane modes. Finally, debonding significantly affects the high frequencies making their values even higher than those for the intact plate, so-called thickening phenomenon appears. The above results imply that the debonding region exerts specific effects on the relevant mode shapes. Several numerically calculated mode shapes are shown in Fig. 27.11. For the better presentation the core was excluded from the contour plots.

Table 27.6 Comparison of natural frequencies (Hz) between the intact and debonded sandwich plates Bending INTa DBDb Bending INT modes modes

a

DBD In-plane INT modes

DBD Pumping INT modes

DBD

(1,1)

1078.5 995.4

(3,3)

3424.4 3461.6

1

1772.1 1771.7

1

4012.3 4035.3

(1,2)

1938.5 1818.6

(4,3)

3844.9 3895.8

2

2637.8 2637.4

–

–

–

(2,1)

1600.5 1567.2

(2,4)

3907.9 3946.7

3

3517.5 3517.1

–

–

–

(2,2)

2274.9 2267.1

(3,4)

4183.9 4220.1

–

–

–

–

–

b

is the intact sandwich plate; is the debonded sandwich plate

–


(a)

(b)

435

(c) (e)

(d)

(f)

Fig. 27.11 Contour plots of the free vibration modes of the debonded sandwich plate

One can see that debonding is visualized as a local separation between the constitutive layers. Herewith, these local interactions can be seen in all mode shapes, but they in certain modes are larger than in others. Hence, the effect of debonding on the sandwich plate’s free response will be more pronounced for such modes. Moreover, the mode shapes in the frequency spectrum will include both only a local deformation within the detached zone of the face sheet (Figs 27.11c and 27.11d) and mixed modes, which are a combination of debonding zone deformations along with the globally deformed pattern of the sandwich plate (Figs 27.11e and 27.11f). Thereby, one may conclude that the permutation in the order of modes, inversion of the mode shapes and thickening phenomenon are a result of complex local interaction phenomenon within the sandwich plate, induced by the debonding region.

27.4.3 Forced Vibration Analysis The flexibility of the low strength core affects the overall structural behavior and may lead to stress concentration in the vicinity of the discontinuities and support regions, in terms of shear and transverse normal stresses at the face sheet-to-core. To provide a deeper insight into dynamics of plates sandwiched by a flexible core and containing debonding, their general dynamic response is examined. Herewith, contact-impact conditions for describing interactions of surfaces coming into contact during oscillations are taken into account. The forced vibration analysis of sandwich plate with a flexible core is carried out using the explicit solver of the ABAQUS commercial software, as it was described in Sect. 27.3.3. Simplifying assumptions were formulated and adopted for calculations as the following • Elastic deformation was present. • Debonding was assumed exists before vibration started and was constant during oscillations. The transient response of the debonded sandwich plate, the same as in previous Sect. 27.4.2 was studied. The simply supported rectangular sandwich plate was

436


(a)

(b)

(c)

(d)

(e)

(f)

Fig. 27.12 Contour plots of deformed shapes of the debonded sandwich plate subjected to the impulse force: (a) t = 1 ms; (b) t = 3 ms; (c) t = 5 ms; (d) t = 7 ms; (e) t = 9 ms; (f) t = 10 ms

excited by an impulse force at the central point of the bottom face sheet to simulate a hammer hit. The duration of the applied force was much shorter than the analysis time, i.e. ⎧ ⎪ ⎪ ⎨ F 0 , 0 ≤ t ≤ t0 F(t) = ⎪ ⎪ ⎩ 0, t > t0 with t0 = 1 ms and F 0 = 1 kN. Fig. 27.12 presents the deformed forms of the sandwich plate during its transient movement at the different discrete time moments for first 10 ms. For the sake of clearness half the plate is presented. It can be clearly seen the intermittent contact of the detached surfaces in the vibrating sandwich plate. The comparison of results calculated for both the debonded sandwich plate and the same intact plate gives a way to assess an influence of debonding on the sandwich plate’s transient response. The finite element results such as displacements, strains and stresses can be computed at different points as time histories. As one can see from Fig. 27.13 the displacement time histories, calculated at the central points A and B are quite different for the intact (INT) and debonded (DBD) plates. In the case of the debonded plate the displacements of both the face sheet (DBD-f) and core (DBD-c) at the point A are shown, Fig. 27.13a. It is important to notice that the local interaction in the detached segments, located between the upper face sheet and the core of the plate changes significantly the transient response of the sandwich plate at its lower points, Fig. 27.13b. Such different deformation patterns in the top and bottom face sheets are expected due to the compressibility of the soft core. While the flexibility of the low strength core affects the overall structural behavior, it also leads to stress concentrations in the vicinity of the discontinuity. Fig. 27.14 demonstrates the contour plot of the Mises stress at the face sheet-to-core interface separately for the core and the face sheet at such a time moment (t = 3 ms), when it is maximum. One can see that the maximum value of the stress has been attained, as expected, on the boundary points of the debonding zone. As mentioned above, the degraded interface will generate higher in-plane stresses in the face sheet and transverse shear and normal stresses in the core, which may


437

make the damaged face sheet-to-core interface vulnerable to debonding propagation. For analyzing the stress states in the interface with% a time, variations of the & b hc stresses at a boundary point of the debonding zone, C a+d , , calculated for 2 2 2 both the intact and debonded plates are compared in Fig. 27.15. It can be seen that the transient stress time histories of the intact and debonded plates are obviously different. Besides, the amplitudes of the stress variations for the debonded sandwich plate are significantly larger than those for the intact one. Thus, the assumption of debonding not propagation during oscillations is doubtful and may be used for the sake of model simplification as preliminary strength information. Although this assertion should be considered in more details involving the crack mechanics approaches and it may be an objective of our future investigations.

5

6 INT DBD-f DBD-c

INT

Deflection, mm

Deflection, mm

4 2 0 -2 -4

0

-2.5

-5 0

(a)

DBD

2.5

0.002

0.004

0.006

0.008

0.01

Time, s

0

0.002

0.004

Fig. 27.13 Transient displacement time histories at the point: (a) A

0.006

0.008

0.01

Time, s

(b)

%

a b hc 2, 2,+ 2

(a) C

(b)

Fig. 27.14 Contour plot of Mises stress within: (a) the core; (b) the face sheet

&

; (b) B

%

a b h 2, 2,−2

&

438


40

INT DBD

4

Stress t23, MPa

20

Stress t13, MPa

8

INT DBD

0

-20 -40 -60

0 -4 -8

-12 0

0.002

0.004

0.006

0.008

0.01

0

Time, s

(a)

0.002

0.004

0.006

0.008

0.01

0.004

0.006

0.008

0.01

0.008

0.01

Time, s

(b)

15 INT

Stress s11, MPa

10

Stress s33, MPa

INT DBD

80

DBD

5

0

40

0

-40

-5

-80 0

0.002

0.004

0.006

0.008

0.01

0

0.002

Time, s

Time, s

(c)

(d) 9

90

DBD

6

Stress t12, MPa

Stress s22, MPa

INT

INT DBD

60 30 0

-30

3 0 -3 -6

-60 0

0.002

0.004

0.006

Time, s

(e)

0.008

0.01

0

0.002

0.004

0.006

Time, s

(f)

Fig. 27.15 Transient stress time histories of the: (a) transverse shear stress in direction 1; (b)transverse shear stress in direction 2; (c) transverse normal stress; (d) in-plane normal stress in direction 1; (e) in-plane normal stress in direction 2; (f) in-plane shear stress

27.5 Conclusions In the paper is explored a way of predicting the dynamic behavior of debonded sandwich plates with flexible core using free vibration analysis and dynamic transient analysis, and is examined the influence of the soft core and debonding on modal characteristics and transient response of the sandwich plates. The FE model


439

involving continuum shell elements for each of the face sheet layers and 3-D brick elements for the core is developed within the ABAQUS code. The free vibration analysis is performed with an implicit version of ABAQUS whereas the dynamic transient analysis is carried out by using its explicit version. Contact conditions, which are available in ABAQUS/Explicit only, are applied in detached segments of the debonded sandwich plates to avoid a physical unreal penetration of the surfaces coming into contact during oscillations and to simulate as close as possible the dynamic behavior of the sandwich plates in order to gain a better understanding of their performance. From results obtained in the paper the following conclusions can be drawn: • The core flexibility in the through-the-thickness direction leads to highly nonlinear responses of sandwich plates under both static and dynamic loading. Unequal deflections of the top and bottom face sheets and jumps of the transverse shear stresses at the face sheet-to-core interfaces were observed in the case of plate bending. The appearance of the pumping modes in the frequency spectrum resulted from the free vibration analysis. • Natural frequencies of a debonded sandwich plate usually decreases, compared to those for the same intact plate. The associated mode shapes along with global deformed forms contain local deformation modes induced by debonding and such local deformations are highly dependent on the mode number. As well, the high natural frequencies and mode shapes are more disturbed by the debonding presence. • Taking into account contact interactions in detached fragments for modeling debonded sandwich plates is very important to properly describe their dynamic forced response. The neglect of contact-impact conditions in predicting of their dynamic behavior will lead to the incorrect results. • The intermittent contact in detached segments of the debonding region significantly changes the dynamic transient response of the sandwich plate. Finite element results calculated have shown completely different deformed shapes of the debonded plate in comparison with those for the same intact sandwich plate. • The flexible core produces the high stress level at the boundary points of the debonding region such that may promote debonging to the propagation. Although this conclusion demands of a further study. Acknowledgements The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013), FP7 - REGPOT - 2009 - 1, under grant agreement No: 245479 and the support by Polish Ministry of Science and Higher Education, Grant No 1471-1/7PR UE/2010/7, is also acknowledged.

References 1. Librescu L, Hause T (2000) Recent developments in the modeling and behavior of advanced sandwich constructions: a survey. Compos Struct 48:1–17 2. Vinson JR (2001) Sandwich structures. Appl Mech Rev 54:201–214

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3. Hause T, Librescu L (2006) Flexural free vibration of sandwich flat panels with laminated anisotropic face sheets. J Sound Vib 297:823–841 4. Bernard ML, Lagace PA (1987) Impact resistance of composite plates. In: Proceedings of the American Society for Composites. 2nd Technical Conference 167–176 5. Akay M, Hanna R (1990) A comparison of honeycomb-core and foam-core carbonfibre/epoxy sandwich panels. Composites 21:325–231 6. Altenbach H (1998) Theories for laminated and sandwich plates: a review. Int Applied Mech 34:243–252 7. Ferreira AJM (2005) Analysis of composite plates using a layerwise shear deformation theory and multiquadrics discretization. Mech Adv Mater Struct 12:99–112 8. Frostig Y, Thomsen OT (2004) High-order free vibration of sandwich panels with a flexible core. Int J Solids Struct 41:1697–1724 9. Rao MK, Desai YM (2004) Analytical solutions for vibrations of laminated and sandwich plates using mixed theory. Compos Struct 63:361–373 10. Frostig Y, Thomsen OT, Vinson JR (2004) High-order bending analysis of unidirectional curved soft sandwich panels with disbonds and slipping layers, J Sandwich Struct Mater 6:167–194 11. Della CN, Shu D (2007) Vibration of delaminated composite laminates: a review. Applied Mechanics Reviews 60:1–20 12. Lu X, Lestari W, Hanagud S (2001) Nonlinear vibrations of a delaminated beam. J Vib Control 7:803–831 13. Ju F, Lee HP, Lee KH (1994) Dynamic response of delaminated composite beams with intermittent contact in delaminated segments. Compos Eng 4:1211–1224. 14. Kwon YW, Lannamann DL (2002) Dynamic numerical modeling and simulation of interfacial cracks in sandwich structures for damage detection. J Sandwich Struct Mater 4:175–199 15. Schwarts-Givli H, Rabinovitch O, Frostig Y (2008) Free vibrations of delaminated unidirectional sandwich panels with a transversely flexible core and general boundary conditions - a high-order approach. J Sandwich Struct Mater 10:99–131 16. Burlayenko VN, Sadowski T (2011) A numerical study of the dynamic response of sandwich plates initially damaged by low velocity impact. Comput Mater Sci doi:10.1016/j.commatsci.2011.01.009 17. ABAQUS Version 6.9-1 EF User’s Manual (2009) Dassault Systems Simulia Corp., Providence, RI, USA 18. Burton WS, Noor AK (1996) Assessment of continuum models for sandwich panel honeycomb cores. Comput Meth Appl Mech145:341–360 19. Nabarrete A, De Almeida SFM, Hansen JS (2003) Sandwich plate vibration analysis: three layer quasi three-dimensional finite element model. AIAA J 41:1547–1555 20. Gibson LJ, Ashby MF (1988) Cellular solids: structure and properties. Oxford, Pergamon Press 21. Burlayenko VN, Sadowski T (2010) Effective elastic properties of foam-filled honeycomb cores of sandwich panels. Compos Struct 92:2890–2900 22. Zienkiewicz OC (1977) The finite element method. 3rd edn. McGraw-Hill, London 23. Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia 24. Laursen TA, Simo JC (1993) A continuum-based finite element formulation for the implicit solution of multi-body large deformation frictional contact problems. Int J Numer Meth Eng 36:3451–3485 25. Burlayenko VN, Sadowski T (2010) Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates. Int J Non-Linear Mech 45:959–968 26. Pagano NJ (1970) Exact solutions for rectangular bi-directional composites and sandwich plates. J Compos Mater 4:20–35 27. Shipsha A, Hallström S, Zenkert D (2003) Failure mechanisms and modelling of impact damage in sandwich beams - a 2D approach: part I - experimental investigation. J Sandwich Struct Mater 5:7–31

Chapter 28

On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions ´ Jacek Chrósćielewski, Violetta Konopinska and Wojciech Pietraszkiewicz

Abstract The non-linear equilibrium conditions for irregular thin shells are formulated from the appropriate form of the principle of virtual displacements. 2D constitutive relations of elasto-plastic behaviour of thin shells are established by dividing the shell into n layers and then integrating the corresponding 3D constitutive relations throughout all layers at each step of non-linear incremental solution by FEM. As example, deformation and stress states in the casing of pressure measuring devise are calculated taking into account deformability of the junctions. Keywords Thin shell · Plasticity · Deformable junction · Finite element method · Casing

28.1 Introduction In the recent paper by Chrósćielewski et al. [1] we have proposed a methodology of the non-linear elasto-plastic analysis of thin shells with deformable junctions. The regular parts of the shell have been modelled by dividing the shell into n layers assumed to be in the plane stress state. The 3D incremental constitutive equations of each layer are described by the generalized elasto-plastic law of PrandtlReuss for small strains, with the associated flow rule and plasticity condition of Huber-Mises-Hencky with linear combination of isotropic and kinematic hardening. J. Chrósćielewski (B) · V. Konopińska Gdańsk University of Technology, Department of Structural Mechanics and Bridge Strctures, ul. G. Narutowicza 11/12, 80-952 Gdańsk, Poland e-mail: [email protected],[email protected] W. Pietraszkiewicz Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, ul. Gen. J. Fiszera 14, 80-952 Gdańsk, Poland e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 28, © Springer-Verlag Berlin Heidelberg 2011

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The 2D incremental constitutive equations for the shell stress resultants and stress couples are established by direct through-the-thickness integration throughout all layers of 3D relations mentioned above. The irregular shell with junctions are modelled according to Makowski et al. [2, 3] by a union of regular surface elements joined together along curvilinear surface edges, along which appropriate forms of 1D strain energy densities have been proposed. The methodology has been illustrated by numerical results of deformation and stress states in the casing of pressure measuring devise having two circular junctions between axisymmetric parts of different thickness. Only the limit cases of stiff and simply supported junctions have been analysed in [1]. In this report we first briefly remind basic non-linear relations of thin irregular shells with deformable junctions analysed within the elasto-plastic range of deformation. Then we present supplementary to [1] numerical results of deformation and stress states in the casing, where some intermediate cases of elastic junction behaviour defined by prescribed stiffness parameters are discussed.

28.2 Notation and Basic Relations According to [2, 3], the consistent field equations and jump conditions of thin irregular shell structures can be derived using two postulates. In the kinematic one, the deformation of the irregular shell is assumed to be determined by stretching and bending of the irregular surface-like material continuum being a union of regular smooth surface elements M (k) , k = 1, 2, ..., K, joined together along spatial curvilinear surface edges ∂M (k) , which in the reference configuration are together denoted as M and Γ, respectively. Then the equilibrium conditions are required to be derivable from the principle of virtual displacements (PVD) involving only dynamic fields associated with the assumed kinematics of M. Such PVD is postulated in the form G ≡ Gi − G e − GΓ = 0 ,

(28.1)

where G i means the internal virtual work, G e is the external virtual work, and G Γ accounts for an additional virtual work of forces and couples acting only along all singular curves modelling the shell junctions. Let u denote symbolically the translation field u inside all M (k) ∈ M, the translation and rotation-like fields (u, ϕ) and (uΓ , ϕΓ ) along regular parts of ∂M (k) and Γ, respectively, the translation ui at each corner Pi ∈ Γ, and the translation ub at each corner Pb ∈ ∂M. Then after complex transformations given in detail in [3] we obtain

28 On Elasto-Plastic Analysis of Thin Shells with Deformable Junctions

G(u; δu) = − +

∂M f

(Divs T + l) · δuda M\Γ

, (pν − p∗ + k) · δu + (h − h∗) δϕ ds

3 4 [f · δu]b − f ∗b · δub

+

Pb∈∂ M¯ f

+

443

∂Md

,

(pν + k) · δu + h δϕ ds + [f · δu]b

(28.2)

Pb∈∂ M¯ d

, + ([pν + k] − f Γ ) · δuΓ + ([h] − hΓ) δϕΓ ds Γ , + [f ]i − f i · δui = 0 . Pi ∈Γ¯

In (28.2), the compound tensor field T is defined in [3] through the surface symmetric stress resultant and stress couple tensors N, M, the compound vector l, pν , p∗ , k and scalar h, h∗ fields are defined in [3] through the external surface force p and ¯ (k) but measured per moment h resultant vectors, acting on each deformed surface M (k) unit area of M , as well as through the external boundary force t∗ and moment ¯f resultant h∗ vectors, prescribed along regular parts of the deformed boundary ∂ M ∗ but measured per unit length of ∂M f . Additionally, f b are the external concentrated forces prescribed at each singular point Pb ∈ ∂ M¯ f , [...] means the jump of (...) along each regular part of Γ, [...]i is the jump of (...) at each singular point of Γ, [...]b means the jump of (...) at each singular point of ∂M, while δ is the symbol of variation and Divs is the surface divergence operator. For any kinematically admissible virtual displacement δu the fields δu and δϕ identically vanish along ∂Md , so that the fourth row of (28.2) identically vanishes as well. Then the transformed PVD requires the equilibrium equations, the static boundary and corner conditions as well as the corresponding jump conditions along Γ to be satisfied. In such formulation the kinematic relations, the material and junction characterisation by the constitutive relations as well as the kinematic boundary conditions should additionally be specified. The whole set of shell relations constitute the highly non-linear boundary value problem (BVP) in terms of translations and their surface gradients as the only independent field variables. This complex BVP can effectively be solved only by numerical methods applying some incremental-iterative solution procedure. The procedure is usually based on approximation of the non-linear BVP by series of linearised BVPs. For the Lagrangian non-linear theory of thin, regular elastic shells (without junctions) such solution procedure was worked out in [4], where the general structure of incremental shell equations and corresponding buckling shell equations were explicitly derived. However, in case of highly non-linear irregular elasto-plastic shell problems it is more efficient to apply the numerical incremental-iterative procedure directly to the incremental variational functional (28.2), not to the field equations following from it.

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Let us briefly recall some statements of [4] and extend them to elasto-plastic shell problems with deformable junctions. Notice that components of the external loads in M, along ∂M f and along Γ may be specified entirely independently, in general, by now 18 dimensionless parameters λ p ∈ Λ ⊂ R18 . Then the non-linear BVP for a thin irregular shell generated by (28.2) can be presented symbolically as F(u, λ p ) = 0, where the non-linear continuously differentiable operator F is defined on the product space C(M, E 3 ) × R18 with values in the Banach space, where C(M, E 3 ) is a set of all components of u and its surface gradients up to the 4th order. In engineering applications all external loads are usually specified by a single common parameter λ ∈ Λ ⊂ R. In this case the solution u(λ) of the BVP form a one-dimensional submanifold in C(M, E 3 ) usually called the equilibrium path. u(λ) is called the weak solution if G (u(λ); δu(λ)) = 0 for all kinematically admissible δu(λ). For finding the weak solution u(λ) one usually applies the Newton-Kantorovich method [5] based on successive approximations to the exact solution at some λm+1 through solving a series of linearised BVPs following from linearisation of G (u(λ); δu(λ)) = 0 about a λm close to λm+1 , % & % & G u(i) ; δu + ΔG u(i) ; Δu(i+1) , δu = 0 . (28.3) m+1 m+1 m+1 The first term in (28.3) represents the value of G at the approximation u(i) m+1 . Since this approximation may not belong to the equilibrium path, the first term in (28.3) does not vanish, in general, and allows one to calculate the unbalanced force vector at the configuration corresponding to u(i) m+1 . The second term in (28.3) linear with regard to Δu(i+1) allows one to calculate the tangent stiffness matrix at u(i) of the m+1 m+1 non-linear BVP. If um+1 corresponds to the regular solution point then the successive approximations u(i+1) m+1 established by this method converge to um+1 with velocity of geometrical progression, provided that the initial approximation u(0) m+1 is sufficiently close to um+1 . The incremental shell relations following from (28.3) are valid for unrestricted translations, rotations, strains and/or bendings of the shell irregular base surface, arbitrary configuration-dependent external static loading, an arbitrary combination of work-conjugate boundary conditions, and arbitrary incremental constitutive relations of the shell and the junctions.

28.3 Constitutive Elasto-Plastic Modelling in Thin Shells Analysis in the elasto-plastic range of deformation of thin irregular shells with deformable junctions can be performed by the finite element method with C 1 elements worked out by Nolte and Chroscielewski [6], which has been extended here to account deformability of the shell junctions. The shell is first divided into n layers and the plane stress state is assumed within each layer. The 3D incremental constitutive equations of each layer are described by the generalized elasto-plastic law of Prandtl-Reuss for small strains, with the associated flow rule and plasticity condition


445

of Huber-Mises-Hencky (HMH) with linear combination of isotropic and kinematic hardening. In particular, in our constitutive shell model we apply the following relations: 1. Additive decomposition of differential increment of the Green strain tensor e into elastic and plastic parts, de = deE + deP . (28.4) 2. The overstress tensor, Σ = s−α ,

Σ = s − α ,

Σ)I , Σ = Σ − 13 tr(Σ

(28.5)

where s is the 2nd Piola-Kirchhoff stress tensor, s = s − 13 tr (s) I is its deviatoric part, α is the corresponding back stress tensor, and I is the identity tensor of the 3D vector space. 3. The Huber-Mises-Hencky (HMH) yield condition, 8 ∂f Σ, ε¯ P ) = σ¯ − σY (ε¯ P ) = 0 , σ f (Σ ¯ = 32 Σ : Σ , ≡r, (28.6) Σ ∂Σ where σ¯ is the HMH effective stress and σY is the yield stress in uniaxial tension. 4. The associated plastic flow rule and evolution equation, r : C E : de de = (dλ) r , dλ = , H + r : CE : r α = (1 − β)H deP , dσY = βH d ε¯ P , dα P

d ε¯ = P

8

2 3

deP : deP ,

(28.7)

where CE is the 4th -order tensor of elastic moduli, ε¯ P is the accumulated effective plastic strain, H is the strain hardening parameter, and β ∈ [0, 1] is the material parameter determining proportion between isotropic and kinematic hardening. 5. The incremental constitutive relation of the elasto-plastic continuum, ds = C EP : de ,

(28.8)

where C EP is the instantaneous tangent 4th -order tensor of the elasto-plastic material behaviour given by CEP = CE −

(CE : r) ⊗ (r : CE ) . H + r : CE : r

(28.9)

In this approach, as the hardening function we can also take the multi-segment approximation of experimental curves following from material tests in tension, if necessary. The stress increments corresponding to the strain increments are calculated from velocity relations using the Euler method of forward integration with correction following from the plasticity condition. Then the incremental constitutive equations for the shell stress resultants and stress couples are established by direct through-the-thickness integration throughout all layers of 3D relations mentioned above. All matrix relations for the finite element are calculated numerically

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using 3-point Gauss integration within the element, and up to n = 10 integration points across the shell thickness are applied.

28.4 Deformation and Stress States in Axisymmetric Casing The axisymmetric casing of measuring device being a part of a pressure installation consists of three regular thin shells of revolution: the circular cylindrical part of thickness hc , length H and diameter D, and two toroidal parts of thickness h s , inner boundary diameter dz and radius r. These dimensions are related by D = dz + 2r, see Fig. 28.1.

hs

dw

hs

hs r

dz

D

r dz

H

r

sym.

A

sym.

A

dz

hc r A-A

Fig. 28.1 The axisymmetric casing: geometry

The toroidal parts are connected with the cylindrical one by welding, while at the upper and lower ends the toroidal parts are connected with rigid parts by welding or screw joints. Thus, in this example we have different technological inaccuracies at the junctions associated with welding (or screw joints) and with change of thickness. The force P(a) acting at the inner and lower toroidal boundaries, see Fig. 28.2, comes from pressure difference applied to rigid parts of the casing and is calculated according to & 1 % (28.10) P(a) = qπ dz2 − dw2 . 4


hs

dw /2

s

w(s)

dz /2

r

H/2

hc sym.

f

u(s)

r (b)

[jG (b)]

H/2 (c)

X

sym.

sym.

(a)

sym.

sym.

[jG (a)]

P(a)

sym.

dz /2

hs

hs

Y

447

Fig. 28.2 The axisymmetric casing: scheme of analysis

In the numerical analysis of this example we use the program MINIMOD [6, 7] with the axisymmetric two-node RING element based on the theory of thin shells with finite rotations proposed in [8]. In this element two translation components u, w are approximated using the Hermite interpolation with C 1 interelement continuity of the form ⎛ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎞ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u˜ ⎪ ⎬ ⎜⎜⎜⎜ 0 ⎪ ⎨ uk ⎪ ⎬ ⎨ u, sk ⎪ ⎬⎟⎟⎟⎟ 1 ⎜ ⎟, = H + H (28.11) ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ sk ⎪ ⎪ ⎪ ⎝ k⎪ ⎩ w˜ ⎪ ⎭ ⎩ wk ⎪ ⎭ ⎩ w, sk ⎪ ⎭⎟⎠ k=1 1 where Hk0 and H sk are shape functions in the Hermite interpolation. In the analysis the following numerical data have been used: h s = 1 mm, H = 50 mm, D = 100 mm, dz = 10 mm, dw = 5 mm, r = 45 mm. Within the elastic range of deformation we take E = 210 GPa and ν = 0, 3. The plastic range of deformation is characterized by the initial plasticity limit σ0 = σY (0) = 450 MPa, the mixed isotropic-kinematic hardening is described by the parameter β = 1/2 and the tangent modulus is taken as ET = 0, 001 × E. The linear constitutive relation

hΓ = c [ϕΓ ] , c = cre f · γ , cre f = M0Γ · 2πrΓ , 1 M0Γ = σ0 hΓ , γ ∈ [0, +∞) 4

(28.12)

governs deformability of the junctions. Two values of the cylinder thickness hc have been analysed with different values of the stiffness parameter c prescribed at the junctions.

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1) hc = 2 mm The numerical results for two extreme cases c = 0 and c = ∞ of the junctions (a) and (b) have been given in [1].

Fig. 28.3 Vertical translation of the point (a) as function of pressure and the junction stiffness

In Fig. 28.3 we present how the vertical translation w(a) depends on the external pressure q. With growing q the w(a) grows initially almost linearly up to about 6 MPa. Then for q ∈ (6 − 8) MPa the plastic material behaviour makes the measurements of pressure less and less accurate. Above q = 8 MPa the graphs w(a) = f (q) become non-linear with clearly pronounced limit points for q above which the devise becomes damaged. The maximum value of the limit point corresponds to c(a) = ∞, and its minimum value to c(a) = 0. The values of c(b) have no noticeable influence on these results. In Fig. 28.4 we show how the angle of rotation ϕ changes along s/L ∈ [0, 1] , L = 1/4 (H + πr). When both junctions at (a) and (b) are stiff, i.e. c(a) = c(b) = ∞, the relative rotation at these points becomes zero and the graph has no jumps, see [1]. For both simply supported junctions, i.e. c(a) = c(b) = 0, as well as for the finite stiffness of c(a) and c(b) there are two jumps of the graph at the junctions, but this effect is pronounced only locally. In Fig. 28.5 it is shown how the bending couple M 1 along the casing meridian for q = 8 MPa is distributed. It is seen that for c(a) = c(b) = 0 there are zero values of the couple at both junctions. The stiffness parameters of either junction has only local effect, and for all non-zero stiffness values the couple between the toroidal and cylindrical parts of the casing is very mall.


449

Fig. 28.4 Angle of rotation along the casing meridian for q = 8 MPa

Fig. 28.5 Bending couple along the casing meridian for q = 8 MPa

2) hc = 1 mm Similar numerical simulations as above performed for the cylindrical thickness hc = 1 mm are given in Figs 28.6–28.8. Although the cylinder thickness here is only half of that discussed in the case 1), the overall deformability of the casing is almost the same as above.

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Fig. 28.6 Vertical translation of the point (a) as function of pressure and the junction stiffness

Fig. 28.7 Angle of rotation along the casing meridian for q = 8 MPa

From the results of 1) and 2) the following behaviour of the casing can be observed: • The overall axisymmetric carrying capacity of the casing depends primarily on plastification of its toroidal part near the junction (a). • Disturbances of deformation and stress states due to the junctions (a) and (b) are local without noticeable influence on each other. • In case 2), the shell rotations on both sides of (b) are almost the same and practically independent on the junction stiffness. However, for different thicknesses discussed in case 1), these rotations become distinct and their difference depends on the junction stiffness.


451

Fig. 28.8 Bending couple along the casing for q = 8 MPa

28.5 Conclusions We have presented a refined formulation of the principle of virtual displacements for modelling of thin irregular elasto-plastic shells with elastic junctions. The 2D constitutive relation in the interior shell domain have been established by dividing the shell into n layers and then integrating the corresponding 3D constitutive relations throughout all layers at each step of non-linear incremental solution by the FEM. We have applied the C 1 axisymmetric finite element of [6] and calculated numerically the deformation and stress states in the casing of pressure measuring devise with two circular junctions. As compared with [1], several additional values of the junction stiffness and two values of thickness of the cylindrical part have been taken into account. The influence of junction stiffness on the results have been analysed. Acknowledgements The research was supported by the Polish Ministry of Science and Education under grant No N 506 254237.

References 1. Chrósćielewski, J., Konopińska, V., Pietraszkiewicz, W.: On modelling and non-linear elastoplastic analysis of thin shells with deformable junctions. ZAMM 91, 6, (2011) (in print) 2. Makowski, J., Pietraszkiewicz, W., Stumpf, H.: On the general form of jump conditions for thin irregular shells. Arch. Mech. 50, 2, 483–495 (1998) 3. Makowski, J., Pietraszkiewicz, W., Stumpf, H.: Jump conditions in the non-linear theory of thin irregular shells. J. Elasticity 54, 1, 1–26 (1999) 4. Pietraszkiewicz, W.: Explicit Lagrangian incremental and buckling equations for the nonlinear theory of thin shells. Int. J. Non-Linear Mech. 28, 2, 209–220 (1993)

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5. Krasnosel’skii, M.A., Vainikko, G.M., Zabrejko, P.P., Rutitskii, Ya.B., Stetsenko, V.Ya.: Approximate Solutions of Operator Equations. Wolters-Nordhoff Publ., Groningen (1972) 6. Nolte, L.-P., Chroscielewski, J.: Large rotation elastic-plastic analysis of flexible shells. In: Taylor, C. et al. (eds.), Numerical Methods for Non-Linear Problems, Vol. 3, pp. 391–404. Pineridge Press, Swansea (1986) 7. Chrósćielewski, J., Branicki, Cz.: MINIMOD – Pakiet podprogramów wspomagaja¸cy badanie zagadnień nieliniowych. W: Mater. IX Konf. ,,Metody Komputerowe w Mechanice”, tom 1, str. 131-138. Kraków-Rytro (1989) 8. Pietraszkiewicz, W., Szwabowicz, M.L.: Entirely Lagrangian nonlinear theory of thin shells. Arch. Mech. 33 , 273–288 (1981)

Chapter 29

Thermal Stress and Strain of Solar Cells in Photovoltaic Modules Ulrich Eitner, Sarah Kajari-Schröder, Marc Köntges and Holm Altenbach

Abstract The long-term stability of photovoltaic (PV) modules is largely influenced by the module’s ability to withstand thermal cycling between -40◦ C and 85◦ C. Due to different coefficients of thermal expansion (CTE) of the different module materials the change in temperature creates stresses. We quantify these thermomechanical stresses by performing a Finite-Element-analysis (FEA) of a 60 cell module during thermal cycling. We therefore start by the experimental characterization of each material layer. Experiments performed with laminated samples are used to validate the computational model. We find that taking into account the viscoelasticity of the encapsulation layers gives the best agreement with experiments. The FEA of the complete module shows that the solar cells are under high compressive stress of up to 76 MPa as they are sandwiched between the stiff front glass and the strongly contracting plastic back sheet. The non-symmetrical structure of the 5.55 mm thick module with glass being the thickest component (4 mm) leads to bending during the thermal cycle. Keywords Thermal Stress · Solar cell · Photovoltaic

29.1 Introduction Photovoltaic (PV) modules are sold with warranties of 25 years. Withstanding outdoor exposure in different climates for such a long time requires a high reliability U. Eitner (B) · S. Kajari-Schröder · M. Köntges Institute for Solar Energy Research Hamelin (ISFH), Am Ohrberg 1, D-31860 Emmerthal, Germany e-mail: [email protected],[email protected],[email protected] H. Altenbach Martin-Luther-Universität Halle-Wittenberg, Kurt-Mothes-Str., 1, 06120, Halle (Saale), Germany e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 29, © Springer-Verlag Berlin Heidelberg 2011

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of the PV module design. The state-of-the-art technique to test the module’s stability is the IEC 61215 standard [1] where accelerated aging tests are defined. Among these tests is the thermal cycling procedure where thermal loads from day-and-night cycles and from seasonal temperature changes are imitated. Modules with the standard design are commercially deployed since the 1980s and this standard concept has by now proven that it meets the reliability demands [17]. However, as further cost reductions are necessary the use of different designs [12] and alternative materials [13, 14] have to be carefully investigated before being industrially implemented. In the case of thermomechanical stability it is no longer sufficient to check the electrical properties of the module after accelerated testing as defined in the standard but the creation of stresses and strains within all material layers must be quantified and well understood. This work represents the completion of a number of preliminary experimental and modeling investigations performed at the ISFH aiming at the determination of the thermomechanical stress and strain in a PV module. In 2008, we presented the FEM-simulation of a laminated string [5] using only linear elastic material models. However, experiments were required to assess the quality of the simulation. In 2009, we successfully transferred the method of digital image correlation to the field of PV module technology in order to measure the deformation of laminated solar cells contactless [7]. We determined the gap between two solar cells to deform by approximately 0.5 μm/◦ C. An analysis of the accuracy in displacement of this digital image correlation method led to ±1.14 μm [8]. The same method was then used by Meier [16] to inspect copper ribbons in PV modules. In 2010, we discussed the non-linear mechanical properties of the encapsulant EVA (ethylene-vinyl acetate) such as a strong temperature-dependence, relaxation and creep behavior [6]. In this contribution we review the material properties of the different module materials and put special emphasis on the encapsulant. We construct a viscoelastic material model for EVA and use it along with linear elastic models for the other module materials to simulate the gap experiment in [8]. Furthermore, we compare the EVA-models of linear elasticity and of temperature-dependent linear elasticity to the experimental data. The most consistent model with the experiment is then chosen for a FEM-simulation of a complete module in order to answer the central questions of thermomechanics of PV modules: How high are the stresses in the cells? Is the position of a solar cell in the module relevant for the stress in the cell?

29.2 Photovoltaic Modules A photovoltaic module is defined as a collection of individual solar cells integrated into a package for environmental protection [11, 17]. The solar cells convert the sunlight into electrical power so that their electrical interconnection in series results in an augmentation of electrical current. Depending on the module size and the solar cell type the maximum power of a standard module is between 50 and 315 Wp [11].

29 Thermal Stress and Strain of Solar Cells in Photovoltaic Modules

455

85

Temperature T [◦ C]

min. time max. 100◦ C/h

10 min

25

min. time 10 min -40 0

Fig. 29.1 Layer structure of a standard PV module

1

2

3

4

5

6

Time t [h]

Fig. 29.2 One cycle of the temperature cycling test according to IEC 61215

29.2.1 Structure A standard PV module consists of a glass superstrate, one layer of a transparent encapsulation sheet, the interconnected solar cells, a second layer of the encapsulation sheet and a plastic back sheet as shown in Fig. 29.1. This composite structure is created in a lamination step during module manufacturing. During lamination the lay-up of the different material layers is set under vaccum conditions and heated up to 150◦C. The polymeric encapsulation sheets begin to soften and flow around the interconnected solar cells. The encapsulant then cures or soldifies, adheres to the glass and the back sheet to create a solid composite structure with embedded solar cells.

29.2.2 Thermal Cycling Today PV modules are sold with warranties on 80% power after 25 years of outdoor operation. Studies on the reliability and failures observed in field [21] that began with the commercialization of terrestrial photovoltaics in the 1980s have led to a number of accelerated aging tests [17] summarized in the international standard IEC 61215 [1, 2]. Among these tests is the thermal cycling procedure where a module is subjected to a temperature cycle between -40◦ C and 85◦C as shown in Fig. 29.2. It is designed to imitate alternating day-and-night conditions and seasonal temperature changes in different climatic regions.

456 50 Change of cell gap width Δv [μm]

Fig. 29.3 Experimentally determined displacement of the gap between two solar cells in a laminate with three cells in a row. The dashed line represents a linear thermal expansion of 0.5 μm/◦ C

U. Eitner et al.

40 30 20 10 0 -10 -20 -30

slope 0.5μm/◦ C -40

-20

0

20

40

60

80

100


29.3 Experimental In order to assess the thermal deformation of solar cells in laminated module-like structures experimentally we successfully applied the digital image correlation technique with a stereo camera system [7] to the field of PV module technology. The details and accuracy of this method are outlined in [8]. The tested laminate consists of a 4 mm thick float glass plate, a transparent 100 μm thick back sheet, two layers of 0.45 mm thick EVA-based encapsulants and three 125 × 125 mm2 monocrystalline cells in a row with a distance of 2 mm to each other. The cells are not interconnected. Figure 29.3 shows the measured displacement of the gap between two solar cells. The experiment begins at room temperature therefore setting the point of zero displacement to 23◦ C. The dashed line corresponds to a linear displacement of the gap of 0.5 μm/◦ C. The experimental results are used to validate Finite-Elementsimulations of PV modules that are discussed in Sect. 29.5.1.

29.4 Material Models In order to calculate the thermomechanical behavior of a PV laminate with FiniteElement-simulations we first have to characterize every material component individually.

29.4.1 Silicon In this paper we consider the monocrystalline silicon solar cells as thin continua of bulk silicon. Silicon exhibits a cubic symmetry so that the linear elastic properties are expressed by C11 , C12 and C44 in the stiffness matrix C,


⎡ ⎢⎢⎢ C11 ⎢⎢⎢ ⎢⎢⎢ C12 ⎢⎢⎢⎢ ⎢⎢⎢ C12 C = ⎢⎢⎢⎢ ⎢⎢⎢⎢ 0 ⎢⎢⎢ ⎢⎢⎢ 0 ⎢⎢⎣ 0

C12 C12 0

0

C11 C12 0

0

C12 C11 0

0

0

0 C44 0

0

0

0 C44

0

0

0

0

⎤ 0 ⎥⎥⎥ ⎥⎥⎥ 0 ⎥⎥⎥⎥ ⎥⎥⎥ 0 ⎥⎥⎥⎥ ⎥⎥ . 0 ⎥⎥⎥⎥⎥ ⎥⎥ 0 ⎥⎥⎥⎥⎥ ⎥⎦ C44

457

(29.1)

These values are well documented in the literature, for example C11 = 164.8 GPa, C12 = 63.5 GPa and C44 = 79.0 GPa in [10]. Monocrystalline solar wafers are oriented so that the surface normal points in the crystallographic < 100 >-direction and the edges of the wafer in the < 010 >- and < 001 >-direction. Therefore, the coordinate system of C must match the crystallographic coordinate system, i.e. x=< 100 >, y=< 010 > and z=< 001 >. The coefficient of thermal expansion is temperature dependent [15, 18].

29.4.2 Glass The glass superstrate provides mechanical rigidity of the PV module. In this investigation we use soda-lime glass which is manufactured with the float glass process. Like silicon, its mechanical properties are well documented in the literature [3]. It is isotropic linear elastic with the Young’s modulus E = 73 GPa, Poisson’s ratio ν = 0.23 and the density ρ = 2.5 g/cm3 . We use a CTE of α = 8 × 10−6 1/K.

29.4.3 Back Sheet The back sheet is a multilayered polymeric sheet. It consists of a PVF(polyvinyl fluoride)-core and PET (polyethylene terephthalate)-layers. The combination of these layers provides protection of the solar cells and the encapsulant from external influences such as humidity, atmospheric exposure, aggressive substances and scratches. We determine the mechanical properties of the isovolta icosolar 2442 sheet by tensile tests at different temperatures. The measured Young’s modulus is 4.0 GPa at -35◦C, 3.6 GPa at 20◦C and 2.0 GPa at 80◦ C. The Poisson’s ratio is determined as 0.29. In this study we ignore the temperature-dependence of the Young’s modulus and use the value of E = 3.5 GPa.

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Storage modulus G [N/mm2 ]

103

102

101

100

-60 -40 -20

0

20 40 60 80 100


Fig. 29.4 Dynamic mechanical analysis of EVA at 1 Hz: elastic part G of the shear modulus

-35◦ C 102 -28◦ C -18◦ C 1◦ C 20◦ C

101

41◦ C 49◦ C 60◦ C

100 139◦ C 10−1 100

119◦ C

102

104

100◦ C

80◦ C 106

Time t [s]

Fig. 29.5 Tensile relaxation moduli for laminated EVA at different constant temperatures

29.4.4 Encapsulant The encapsulant which dominates the photovoltaic market since the 1980s is EVA [14]. It is a thermoplastic material that cures at 150◦ C where the polymeric chains crosslink so that a polymer network forms. Thereby it adheres to the glass, the back sheet and the solar cells. After this lamination step it is an elastomer and exhibits the typical properties such as low stiffness above the glass transition, high stiffness below the glass transition as well as relaxation and creep behavior. We investigate the mechanical properties by a dynamic mechanical analysis(DMA) and a series of relaxation and creep experiments at isothermal temperatures. Figure 29.4 shows the result of the DMA that is performed at the German Institute for Rubber Technology in Hanover by cyclic torsional loading of a fully cured stip specimen. From this curve we determine the glass transition temperature to be -35◦ C according to the definition given in [4]. The relaxation and creep experiments are conducted in tension where we either apply a step in elongation or in force and monitor the resulting material response in decreasing force or in increasing strain, respectively. The experimental results for the relaxation tests at different isothermal temperatures are shown in Fig. 29.5. From these experiments we construct three different material models for the encapsulant by increasing the level of detail found in the experiments and thus increasing the complexity of the material model.

29.4.4.1 Linear Elasticity In a linear elastic model the mechanical properties are expressed by two material constants. For the EVA we have to choose one value for the Young’s modulus E from Fig. 29.4 (after interconversion from shear to tensile data) and one value for the Poisson’s ratio ν. As E is obviously not constant over temperature we use the


Shift factor α−20 [-]

100 creep TTS

10−10 WLF-fit 10−20

0

50

100

150


Fig. 29.6 Values of α−20 from timetemperature-superposition of EVA and WLF-fit

Shear relaxation modulus G−20 [N/mm2 ]

103

relaxation TTS

-50

459

creep data 102 relaxation data 101 Maxwell model 100

for gap simulation

10−1 10−10 10−5 100 105 1010 1015 1020 1025 1030 Reduced time tred [s]

Fig. 29.7 Mastercurve for EVA and Generalized Maxwell model with 25 arms

highest (E = 2.1 GPa) and lowest value (E = 6.5 MPa) to construct two linear elastic models that represent extrema. Poisson’s ratio is in both cases set to 0.4.

29.4.4.2 Temperature-Dependent Linear Elasticity The temperature-dependent linear elastic model takes into account the temperaturedependence of the Young’s modulus as found in the DMA experiment in Fig. 29.4. E is thus a function of the temperature while we keep Poisson’s ratio constant at ν = 0.4.

29.4.4.3 Viscoelasticity Relaxation and creep implies that the mechanical properties change with time. The corresponding material theory is viscoelasticity. The viscoelastic constitutive equation takes the form of an integral expression over time. For a one-dimensional problem it is t σ(t) = E(t − u) ε˙ (u) du, (29.2) 0

where the function E(t) is the relaxation modulus. It can be determined by a relaxation test as shown in Fig. 29.5 where it holds E(t) =

σ(t) , ε0

with ε0 being the constant strain applied in a step at t = 0.

(29.3)

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The time-temperature-superposition allows to include the temperature dependence in the viscoelastic material description. The underlying assumption is that the relaxation curve at a higher temperature corresponds to the curve at advanced times. For many polymers it is therefore possible to create a mastercurve of overlapping relaxation curves by shifting them along the logarithmic time axis. It implies that E(t, T 1 ) = E(t/α(T 1 ), T ref ), (29.4) where α is the shift factor that describes how far the relaxation curve at T 1 has to be shifted to correspond to the curve at T ref . We use the experimental relaxation curves from Fig. 29.5 and the experimental creep curves to create a mastercurve. The corresponding shift factors α are plotted in Fig. 29.6. We apply the WilliamsLandel-Ferry (WLF) equation [20] to write α as a function of T , log10 αT ref (T ) =

−C1 (T − T ref ) , C2 + T − T ref

(29.5)

and determine the parameters C1 and C2 from a fit as shown in Fig. 29.6 with T ref = −20◦ C. It has to be mentioned that we interconvert the tensile data to shear data after the shifting. This step is not discussed in detail here. The mastercurve for the shear data that we obtain by using the WLF-formula for α is shown in Fig. 29.7 where the dark lines are the relaxation curves and the light curves are the interconverted creep curves. So far, we managed to create a viscoelastic mastercurve which includes the time- and temperature-dependence. We now have to write this mastercurve as a function of the reduced time to obtain a mechanical model for EVA. We choose the generalized Maxwell model [19] to parametrize the mastercurve G(t) by G(t) = G 0 +

t G i exp − , τi i=1

n

(29.6)

treating the G i as fit parameters while we set τi = 10i with i ∈ {−3, −2, . . ., 22}. This rheological model corresponds to an arrangement of maxwell arms in parallel, whereas each maxwell arm consists of one spring and one dashpot in series. For most polymers the viscoelastic effects are much stronger in the deviatoric than in the volumetric components [9]. Making the volumetric components independent of time, the three-dimensional viscoelastic constitutive equation for isotropic materials becomes t σ(t) = 2 G(t − u) ε˙ (u) du + K tr(ε(t)), (29.7) 0

with a constant bulk modulus K and with tr(·) being the trace of a tensor, i.e. the sum of the diagonal elements. Including arbitrary temperature profiles T (t) in the above equation, we obtain t σ(t, T (t)) = 2 G(ξ(t) − ξ(u), T ref ) ε˙ (u) du + K tr(ε(t)), (29.8) 0


with

t

ξ(t) = 0

1 du. αT ref (T (u))

461

(29.9)

Eq. 29.8 represents the viscoelastic constitutive equation used in the subsequent FEM-simulations together with the generalized Maxwell model and the WLFequation.

29.5 Finite-Element-Simulations Finite-Element-simulations are performed to calculate the stress and strain of solar cells in modules. As the composite structure of the PV module is created at lamination temperature of 150◦ C we assume a stress free inital state at this elevated temperature. The simulation model which describes the thermally induced deformation of the module has thus to start at 150◦ C. In this model each material is mechanically characterized by the material models discussed in Sect. 29.4. For EVA three alternative material descriptions have been presented: linear elasticity, temperaturedependent linear elasticity and viscoelasticity. We decide which model to use in the simulation of the complete module by comparing the experimentally determined gap displacement in Fig. 29.3 with the three simulations of the very same experiment.

29.5.1 Evaluation of Material Models The experiment that is described in Sect. 29.3 is reproduced with a two-dimensional plane-stress model. The linear elastic and the temperature-dependent linear elastic simulation are performed with Comsol Multiphysics using triangular elements and quadratic shape functions. We cool the laminate with the three solar cells from 150◦ C down to -40◦ C by parametrizing the temperature in steps of 10◦ C. The viscoelastic model is simulated with Abaqus using rectangular elements with quadratic shape functions. Due to the time dependence of the model, the complete time history has to be taken into account which we realize in three subsequent analysis steps, see Fig. 29.8. First we simulate the cool down after lamination which takes about 12 min. Then, the laminate is stored at room temperature for 24 h. Finally, the timetemperature-curve of the experiment is retraced beginning at room temperature. The simulated gap displacement is taken as the difference in calculated displacement of opposing cell edges. In the simulations the temperature of zero deformation is 150◦ C. In the original experiment, the reference temperature is 23◦ C. In order to compare simulation and experiment, we shift the simulation curves along the y-axis until the gap displacement at room temperature is zero. The result is shown in Figure 29.9. The linear elastic simulations give straight lines in the gap displacement plots whose slopes depend on the mechanical stiffness of the EVA. The gap displacement in the temperature-dependent linear elastic model increases with decreasing temperatures

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160 140

Change of cell gap width Δv [μm]


100 measure

80 60 store

40 20 0 -20

T -dep. lin. el.

40 20 0

lin. el.

-20 -40 lin. el.

-60

-40 -60

viscoelastic

60

cool

120

0

400

800

1200

1600

Time t [min]

Fig. 29.8 Full temperature history of measured laminate: ’cool’ is the cooling stage after lamination, ’store’ the time of 24 h between lamination and experiment and ’measure’ the temperature profile of the gap experiment

-80

-40

-20

0

20

40

60

80

100


Fig. 29.9 Comparison of the gap displacement data to the three simulations with linear elastic, temperature-dependent linear elastic and viscoelastic model for EVA

below room temperature. This effect strongly deviates from the experimental observation. The viscoelastic simulation data agree quite well with the measured data. At temperatures above 20◦ C the simulation curve reproduces the measurement data particularly well while at cold temperatures the simulated and experimental data exhibit a very similar shape of a decreasing slope. To conclude, the viscoelastic model gives by far the most consistent results with the experiment so that the full viscoelasticity of EVA is used in the simulation of a complete module.

29.5.2 Photovoltaic Module In the subsequent sections the preliminary steps are taken to set up the basis for the numerical investigation of the PV module. We use the viscoelastic model to describe the material behavior of EVA and linear elastic models for silicon, the glass and the solar cells as summarized in Table 29.1. In the three-dimensional simulation model we subject an unframed module to one temperature cycle of the IEC 61215 test and inspect the stresses and strains in the different module materials. The initial stress free state is again set to lamination temperature so that prior to the temperature cycle the module cools down to room temperature and is then held at 21◦C for 24 h. The complete temperature profile is shown in Fig. 29.10. The solar cells in the module are not interconnected. The FEM-simulation is carried out with the software package Abaqus using three-dimensional brick elements with linear shape functions.


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Table 29.1 Material properties used for the 3D-simulation of a PV module Thickness Density El. modulus Poisson’s ratio d

ρ

[μm]

[g/cm3 ]

E [GPa]

CTE

ν

α

[-]

[10−6 K1 ]

Float glass

4000

2.5

73

0.23

8

Back sheet

350

2.52

3.5

0.29

50.4

EVA

500

0.96

viscoelastic model

270

Solar cells

200

2.329

stiffness matrix (Eq. 29.1)

T -dep.

160 140

cool


120 100

thermal

80

cycle

60 store

40 20 0 -20 -40 -60

0

400

800

1200

1600

2000

Time t [min]

Fig. 29.10 Temperature profile for the module simulation

Fig. 29.11 Simulation model of a 60 cell module with symmetric boundary conditions

29.5.2.1 Geometry and Boundary Conditions The module contains 60 solar cells grouped in 6 lines made of 10 cells. The cell size is 125 mm × 125 mm pseudo-square. The gap between two cells is 2 mm and the outer cells have a distance of 20 mm to the module edges. The material layers have constant initial thicknesses over the module length and width as given in Table 29.1. In the layer of the solar cells the space between the cells is filled with EVA. The symmetry of the module is exploited by modeling only the lower left hand quarter of the complete module and by applying symmetric boundary conditions to the top and right hand faces as shown in Fig. 29.11. The mid-point of the module is fixed while all other surfaces and edges may freely deform.

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Fig. 29.12 First principal stress σI at the outer surface of the glass at -40◦ C

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Fig. 29.13 First principal stress σI in the midplane of the back sheet at -40◦ C

Fig. 29.14 First principal strain εI at -40◦ C: The structure is scaled in z-direction and the glass on top is hidden, the front surface is a yz-cutting plane through the centers of the solar cells in the left column of module

29.5.2.2 Simulation Results When the module cools down from the initial stress free state at 150◦ C the material assembly contracts. At -40◦ C we find the stresses at the outer surface of the glass to be tensile with the first principal stress σI between 12 MPa in the module center and 0 MPa in the corner of the module. The back sheet on the other outer face is in high tension. Over large regions the first principal stress σI reaches values of up to 41 MPa (Fig. 29.13). The embedded solar cells are under high compressive stress


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which is visualized in Fig. 29.15 by the third principal stress σIII ranging from 74 MPa to -16 MPa. Inspecting the different components of the stress tensor reveals that the in-plane stresses are one to two orders higher than the out-of-plane stress components which are thus not shown here. Figures 29.16 (σ11 ) and 29.17 (σ22 ) illustrate the compressive character of the normal stresses σ11 (-76 MPa to -4 MPa) and σ22 (-73 MPa to -4 MPa). In Fig. 29.18 we find the shear stress σ12 in the cells to range from -18 MPa to 17 MPa. In the mid-planes of the EVA layers the absolute values of the stresses are very low. However, large strains are present in the EVA at -40◦ C as is shown in Fig. 29.14 where the first principal strain εI is shown. The highest values are found in the layer between the cells and the back sheet exceeding 18% tensile strain. The strain in the EVA between the glass and the solar cells is significantly lower, reaching 7% strain. In each of the mentioned figures the local influence of the solar cells is visible. The maximum stress in the glass arises at the outer surface under the gap between the cells whereas under the cells that are close the module edges the stress is close to zero (Fig. 29.12). In the back sheet, the tensile stress increases when moving to the center of the module (Fig. 29.13). Additionally this increase is superimposed by a stress increase in the back sheet towards the center of each solar cell. In the solar cells themselves the stress distribution is similar for all cells as it increases towards the center of each cell but the stress amplitudes decrease when moving from the module edges to the module center (Figs 29.15-29.18). The position of a solar cell in the module is thus relevant for the level of stress in the cell. For the other temperatures 25◦ C and 85◦ C the distributions of stresses and strains in the material layers exhibit similar characteristics but are significantly lower in their absolute values. For theses temperatures the difference to the stress free state at 150◦ C is still negative but smaller than to the state of -40◦ C so that the compressive character of the cell stresses remains as well as the tensile character of the stresses in the back sheet.

29.5.2.3 Discussion The glass, which makes 72% of the module thickness and which has a high mechanical stiffness of 73 GPa, dominates the contraction of the module. Thereby it forces the laminated layers to follow the pure thermal shrinkage of the glass. The back sheet on the rear side can thus not contract as much as its high CTE (αbacksheet αglass ) demands, thereby leading to the large tensile stress. We discuss this fact in more detail and keep in mind that the total strain ε is continuous in the complete module structure due to continuous displacements u while the stress σ is not. The absolute value of the pure thermal strain |εth | is higher than the total strain |ε| in the back sheet by which it actually contracts. Both strains are negative, which leads to a positive mechanical strain εmech = ε − εth . It is the mechanical strain that determines the stress, i.e. σ = E εmech , so that we obtain tensile stress σ > 0 in the linear elastic back sheet. The tensile stress in the back sheet becomes lower at the module edges because the back sheet is in these regions only held by 1.2 mm

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Fig. 29.15 Third principal stress σIII in the mid-plane of the solar cells at -40◦ C

Fig. 29.16 Normal stress in x-direction σ11 in the mid-plane of the solar cells at -40◦ C

Fig. 29.17 Normal stress in y-direction σ22 in the mid-plane of the solar cells at -40◦ C

Fig. 29.18 Shear stress in the xy-plane σ12 in the mid-plane of the solar cells at -40◦ C


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thick EVA to the glass. There, the large strain of the compliant EVA allows a larger contraction of the back sheet and thus less mechanical strain. The tensile character of the back sheet sets the glass under slight bending. Therefore, the maximum stress in the glass appears at the outer surface. Furthermore, the tensile stress in the glass is extremal under the cell gap because there the tensile strain of the back sheet and the closing of the gap have a very local effect on the glass. The solar cells contract much less than the back sheet and, from a glass persective, they thereby buffer the higher compressive strains that originate from the back sheet. In between the cells this buffer layer is interrupted so that the compressive strains reach the inner surface of the glass. Due to the bending these compressive strains lead to tensile strains at the outer glass surface. The solar cells exhibit the lowest CTE of the module materials. The cells contract thus less than the glass on top and less than the back sheet under the cells. The cells are forced to shrinks more than the pure thermal strain (|ε| > |εth |) which leads to negative mechanical strain. As silicon is a stiff material we obtain high compressive stress in the solar cells. The EVA takes the function of a compliant buffer layer. The consequence is the large strain in the EVA layers.

29.6 Conclusion The discussion of the different material models for the encapsulant EVA shows that the most consistent results with experiments are attained if EVA is modeled as a viscoelastic material while the glass, the back sheet and the silicon are described by linear elasticity. Then, the slightly curved shape of the gap displacement data in [8] is reproduced as shown in Fig. 29.9. The three-dimensional FEM-simulation of a 60 cell module reveals the deformation mechanism in PV modules. The module contracts when it cools down from a stress-free lamination temperature. This deformation mode is maintained during a subsequent thermal cycle. The glass makes 70% of the module thickness and dominates the deformation which follows the thermal contraction according to the CTE of glass. The back sheet has a higher CTE than glass, can therefore not contract as much and is consequently under tensile stress. The higher contraction of the back sheet causes the complete layered structure to bend in the direction of the back sheet. The cells in between these two layers exhibit the lowest CTE and are thus under compression with up to -76 MPa stress at -40◦ C. The cells at the edges are more stressed than interior cells. The EVA is very compliant and exhibits large strains of up to 18% strain which illustrates its function as a buffer layer. Acknowledgements The authors thank Prof. R. Brendel for his guidance on the experimental parts and for his support of this work. Parts of this work were funded by the state of Lower-Saxony.

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References 1. Iec 61215:2005, crystalline silicon terrestrial photovoltaic (pv) modules - design qualification and type approval. international electrochemical commission, 2005. 2. Althaus, J.: Quality assurance for pv modules: experience from type approval testing. Photovoltaics International 3, 120–127 (2009) 3. Brueckner, R.: Materials Science and Technology: A Comprehensive Treatment, Vol. 9, Glasses and Amorphous Materials, chap. Mechanical properties of Glasses. Wiley-VCH (1991). 4. Ehrenstein, G.: Polymer-Werkstoffe. Struktur - Eigenschaften - Anwendung. Hanser Fachbuch (1999). 5. Eitner, U., Altermatt, P.P., Köntges, M., Meyer, R., Brendel, R.: A modeling approach to the optimization of interconnects for back contact cells by thermomechanical simulations of photovoltaic modules. In: Proceedings of the 23rd European Photovoltaic Solar Energy Conference, pp. 2815–2817. Valencia (2008) 6. Eitner, U., Kajari-Schröder, S., Köntges, M., Brendel, R.: Non-linear mechanical properties of ethylene-vinyl acetate (eva) and its relevance to thermomechanics of photovoltaic modules. In: Proceedings of the 25th European Photovoltaic Solar Energy Conference, pp. 4366–4368. Valencia (2010) 7. Eitner, U., Köntges, M., Brendel, R.: Measuring thermomechanical displacements of solar cells in laminates using digital image correlation. In: Proceedings of the 34th IEEE PVSC, pp. 1280–1284. Philadelphia (2009) 8. Eitner, U., Köntges, M., Brendel, R.: Use of digital image correlation technique to determine thermomechanical deformations in photovoltaic laminates: Measurements and accuracy. Solar Energy Materials and Solar Cells 94(8), 1346–1351 (2010). 9. Ferry, J.D.: Viscoelastic Properties of Polymers. Wiley (1980). 10. Greenwood, J.C.: Silicon in mechanical sensors. Journal of Physics E: Scientific Instruments 21(12), 1114–1128 (1988). 11. Häberlin, H.: Photovoltaik: Strom aus Sonnenlicht für Verbundnetz und Inselanlagen. Electrosuisse (2010) 12. de Jong, P.: Achievements and challenges in crystalline silicon back-contact module technology. Photovoltaics International 7, 138–144 (2010) 13. Kempe, M.: Design criteria for photovoltaic back-sheet and front-sheet materials. Photovoltaics International 2, 100–104 (2008) 14. Kempe, M.: Evaluation of encapsulant materials for pv applications. Photovoltaics International 9, 170–176 (2010) 15. Lyon, K.G., Salinger, G.L., Swenson, C.A., White, G.K.: Linear thermal expansion measurements on silicon from 6 to 340 k. Journal of Applied Physics 48(3), 865–868 (1977). 16. Meier, R., Kraemer, F., Schindler, S., Bagdahn, S.W.J.: Thermal and mechanical induced loading on cell interconnectors in crystalline photovoltaic modules. In: Proceedings of the 25th European Photovoltaic Solar Energy Conference, pp. 3740–3744. Valencia (2010) 17. Osterwald, C.R., McMahon, T.J.: History of accelerated and qualification testing of terrestrial photovoltaic modules: A literature review. Progress in Photovoltaics: Research and Applications 17, 11–33 (2009). 18. Roberts, R.B.: Thermal expansion reference data: silicon 300-850 k. Journal of Physics D: Applied Physics 14(10), L163–L166 (1981). 19. Tschoegl, N.W.: The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction. Springer (1989). 20. Williams, M.L., Landel, R.F., Ferry, J.D.: The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical Society 77, 3701–3707 (1955). 21. Wohlgemuth, J., Petersen, R.: Reliability of eva modules. In: Proceedings of the 23rd IEEE PVSC, pp. 1090–1094 (1993).

Chapter 30

Computational Models of Laminated Glass Plate under Transverse Static Loading Ivelin V. Ivanov, Dimitar S. Velchev, Tomasz Sadowski and Marcin Kneć

Abstract Laminated glass with Polyvinyl Butyral (PVB) interlayer became a popular safety glass for aircraft windows, architectural and automotive glazing applications. The very soft interlayer, bonding the glass plates, however, has negligible normal stress in transverse loading and it resists mainly by shear stress. The classical laminate theory obeying the principle of the straight normals remaining straight is not valid for laminated glass. Conventional Finite Elements (FE) are used to model the laminated glass in cylindrical bending to investigate the problem. Based on the assumption that the glass layers of a laminated glass plate obey Kirchoff’s classical plate theory and the PVB-interlayer transfer load by shear stress only, the differential equations of a Triplex Laminated Glass (TLG) plate are derived and a special TLG plate FE is elaborated. For each of its nodes, the element has one transverse translational, three rotational, and two additional degrees of freedom representing the slippage between the glass layers. All computational models are compared with experimental tests of a laminated glass strip in cylindrical bending. Keywords Laminate · Transverse load · Computational models

30.1 Introduction Laminated glass is widely spread as material for aircraft windows and architectural glazing of contemporary buildings. It is also used for windshields in automotive I. V. Ivanov (B) · D. S. Velchev Department of Engineering Mechanics, University of Ruse, “Studentska” 8, 7017 Ruse, Bulgaria e-mail: [email protected],[email protected] M. Kneć · T. Sadowski Department of Solid Mechanics, Lublin University of Technology, “Nadbystrzycka” 40, 20-618 Lublin, Poland e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 30, © Springer-Verlag Berlin Heidelberg 2011

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industry. Laminated glass, in most of the cases, consists of only two glass plates bonded together by a polyvinyl butyral (PVB) interlayer. The PVB-material is a rubber like elastomer [3] that keeps the shards of broken glass plates in the frame of the glass unit after the failure and makes laminated glass safety. Another advantage of laminated glass is its thermal insulation property, which is important for the architectural glazing. Despite of the advantages, however, the efficient application of laminated glass in the complex structures is limited because of the difficulties in their strength calculations at the stage of their design. The PVB-material has elastic modulus which is thousands of times less than the elastic modulus of glass. The great mismatch of the elastic moduli leads to layer interaction in the bending of laminated glasses which does not obey the principle of straight normals or pseudo normals in the theory of plates. The modelling of laminated glasses is sophisticated mainly because of the complex nonlinear behaviour of the PVB-material, which is highly viscoelastic with great temperature dependency [8]. The other nonlinearity, that could be included in the laminated glass behavior, is the geometrical nonlinearity of thin plates under transverse loadings. Since the complex behaviour of laminated glass is not very well modelled in the practice, its strength is considered as intermediate of two limiting cases: the case of layered glass — two glass plates without any interlayer between them, and the case of monolithic glass — one glass plate with thickness equal to the total thickness of the glass plates [9]. The former case determines the lower boundary of the strength, while the latter — the upper boundary. Vallabhan et al. [10] developed a mathematical model of laminated glass based on the minimization of the total potential energy in which the bending and membrane strain energy of the glass plates as well as the shear strain energy of the PVB-interlayer are included. The assumptions for the glass plates correspond to von Karman’s nonlinear theory of plates. The interaction of the glass plates is provided by the shear of the linearly elastic PVB-interlayer which depends on their bending. The finite in-plane strain accounting for large rotations of the normals is included in the model and therefore membrane stress could appear at large deflection of the laminate as a consequence of the geometrical nonlinearity in dependence on the boundary conditions. Five complicated differential equations are obtained and iteratively solved. Two interpolation parameters should be optimised in order to obtain a stable solution. As¸ik [1] developed an algorithm for implicit integration of the equations and their unconditionally stable solution. Norville et al. [7] developed a simple multilayer beam model of laminated glass. The shear force transferred by the PVB-interlayer is determined by a coefficient which has to be identified experimentally. The effective section modulus of the beam could be calculated in dependence on the coefficient which allows the maximum bending stress to be obtained. This is a simple model which gives an opportunity the strength of laminated glass to be approximately estimated at the stage of their design. Duser et al. [4] utilized the Finite Element (FE) method to model laminated glasses under transverse loadings. They used 3-D solid elements to model the layers and their interaction. This approach requires a lot of finite elements and

30 Computational Models of Laminated Glass Plate under Transverse Static Loading

471

therefore expensive computations. The PVB-material is modeled as linear viscoelastic and a nonlinear solution is performed. A statistical model based on twoparameter Weibull’s law of distribution is developed for the glass breakage and strength determination of the glass plates. As¸ik and Tezcan [2] published a mathematical model of laminated glass beams which is based on nonlinear strain-displacement relationship. The model is derived by variational principle from the membrane and bending strain energy of the glass layers and the shear strain energy of the PVB-interlayer. The nonlinearity of the differential equations of the model disappears if the laminated glass beam is simply supported and an analytical solution for this case is derived. The model is used to investigate the linear and nonlinear behavior of symmetric triplex glass beams in comparison with the laminated glass plate behavior. Ivanov [5] derived a simple but reasonable analytical model of laminated glass working as plane beam that can be used for structure optimization. The model is linear and it reveals the advantage of laminated glass if proper optimization of the plate thickness is done. The aim of this work is to develop efficient computational model for laminated glass calculations with complex geometry under transverse loading.

30.2 Experiments Laminated glass is provided for experimental investigation. The laminate has total thickness of 6.13 mm or the layers of glass/PVB/glass are 2.875/0.38/2.875 mm thick. The elastic properties of glass are assumed to be as follows: the Young’s modulus is E = 72 GPa and the Poisson’s ratio is ν = 0.23. The total mass density of the laminate is measured as ρ = 2.418 g/cm3 . In order to characterize the PVBmaterial, compressive shear tests are carried out.

30.2.1 Compressive Shear Test The PVB-material is almost incompressible and it can withstand large strain. When the glass layers of laminated glass are shattered, the integrity of the panel is still kept by the PVB-interlayer which is extremely strained at the places of glass fracture. A hyper-elastic material model is necessary to describe interlayer behavior, when dynamic loading with fracture is considered, but for quasi-static loading, linear behavior with the initial elastic moduli is enough. The approach used here to find the initial shear modulus is the same as the described in [6] by compressive shear test. The hyper-elastic constitutive laws of rubber-like materials are based on approximation of strain energy potential. There are several forms of the strain energy and they give different stability and approximation of material behavior for different

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types of loadings. The second order polynomial form of strain energy potential has constants which can be determined by simple shear test. The stress-strain relationship in simple shear is described by the polynomial τ = G 0 γ + G γ3 ,

(30.1)

where G 0 is the initial shear modulus of the PVB-material. The compressive shear test is depicted in Fig. 30.1. The tested laminated glass sample is between two wedges under 45-degree angle with respect to the direction of the loading force F and the lower wedge is on rollers. The PVB-interlayer with thickness h is practically incompressible and therefore in simple shear. The test is carried out on MTS-testing machine recording the force F and the vertical displacement u.

Fig. 30.1 Compressive shear test

The samples have nominal dimensions of 20 × 50 mm or the area of the samples is A = 1000 mm2 . The thickness of the interlayer is measured as h = 0.38 mm and the thickness of the glass layers is 2.875 mm. The shear stress and strain are determined by simple geometrical considerations as follows: √ F u 2 τ= √ , γ= . (30.2) h A 2 Three samples at 1 mm/min cross-head speed have been tested. The experimental data in range of γ = 0–2.5 are given in Fig. 30.2. The fitting-curve procedure in MATLAB®software gives the value of the material constant G 0 = 0.5173 MPa. The volumetric elasticity constants can be zero if fully incompressible material is assumed. In the case of confined material (as in the case of laminated glass PVB-interlayer) hybrid finite elements should be used, in order to avoid the volumetric locking of FE model in implicit FE code. The hybrid elements are


473

3

2.5

Experimental Curve fitting

τ (MPa)

2

1.5

1

0.5

0

0

0.5

1

1.5 γ (−)

2

2.5

3

Fig. 30.2 Compressive shear test data

computationally inefficient and when the volumetric locking should be avoided by conventional elements, the Poisson’s ratio, ν, should be lower than 0.5 and initial Young’s modulus should be calculated as E0 = 2(1 + ν)G0.

30.2.2 Cylindrical Bending Test In order to investigate the behaviour of laminated glass under transverse loading, a strip of laminated glass has been cut and strain gauges are glued at several places on its upper and lower surfaces. The strip was supported and loaded by the weight, W, of masses as it is shown in Fig. 30.3. The total mass, m, of weights was 4, 8, and 12 kg. The weight is applied through two thin bars in order to be spread over the width of the strip. The central deflection of laminated glass strip was also measured. The results are given in Table 30.1.

Table 30.1 Experimental results for cylindrical bending of laminated glass strip mass

deflection

gauge #1

gauge #2

gauge #3

gauge #4

gauge #5

gauge #6

kg

mm

×10−6

×10−6

×10−6

×10−6

×10−6

×10−6

4

1.70

−68

120

−72

76

−54

58

8

2.76

−132

185

−121

127

−94

96

12

4.82

−205

254

−171

178

−133

133

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Fig. 30.3 Experimental setup for cylindrical bending of laminated glass strip

The viscosity of the PVB-material is not accounted for in the computational models which follow but the measured data creep and the viscosity is clearly observed. The data measured in the experiment are average of the data read in loading and unloading. The data measured by gauge #1 and even those measured by gauge #2 are maybe strongly influenced by the way of applying the load because they have greater difference in absolute value, which is in contrast to the other pairs of gauges, and because the gauges are placed very close to the bar through which the load is applied.

30.3 Computational Models The problem of laminated glass under transverse loading was investigated by using the FE method. Two computational models were prepared in the commercial code ABAQUS™to analyse the cylindrical bending of the laminated glass strip. All FE models here represent only half of the laminated glass strip in experimental tests with boundary conditions providing the symmetry.

30.3.1 3-D Brick Element Model The laminated glass strip was modelled by 3-D brick elements with reduced integration in ABAQUS™commercial software for solution with ABAQUS/Standard. Each glass layer is modelled by five plies of elements and the PVB-interlayer is modelled by one ply of the same type of elements. The size of the elements in the plane of the


475

strip is 1.5 mm, which is approximately 4 times larger than the interlayer thickness and thus it is the largest size ratio of the elements. The degree of discretisation can be justified by the deformed model shown in Fig. 30.4.

Fig. 30.4 Deformation of 3-D brick element model

X Y Z

Step: Step−2 Increment 3: Step Time =

3.000

Deformed Var: U Deformation Scale Factor: +1.000e+01

The discretisation of the half of the laminated glass strip gives 1 177 056 degrees of freedom for the model and approximately so many unknowns. The linear solution of the problem for four steps of loading on a computer using parallelisation technique took 2 hours and 10 minutes. The size of the problem could not allow the model to be used for nonlinear solution. The deformed state of the 3-D brick element model in Fig. 30.4 shows the shear in the PVB-interlayer and the straight normals of the glass layers remaining straight. The result is achieved by giving the PVB-material properties E0 = 1.5312 MPa and ν = 0.48, providing the shear modulus to be G0 = 0.5173 as it is determined from the experiments. If the Young’s modulus is setup to be E 0 = 1.5416 and Poisson’s ratio ν = 0.49, providing the same shear modulus, the volumetric locking appears in the model and the straight normals of the whole laminated glass strip remain straight without any shear in the PVB-interlayer and small deflection of the strip. In such a case, hybrid 3-D brick elements should be used for modelling the PVB-interlayer, which will take longer time for calculations because the computational efficiency of that elements is low. The results of the calculations with the 3-D brick element model of the laminated glass strip were compared with the experimental ones. The relative deviation from the experimental results is given in Table 30.2. The strain on both surfaces of the

Table 30.2 Relative results for linear solutiona 3-D brick element model mass

deflection

gauge #1 & #2

gauge #3 & #4

gauge #5 & #6

kg

%

%

%

%

4

+26.3

−3.3

+0.3

+3.0

8

+37.7

+2.3

+4.7

+5.2

12

+13.2

+1.8

+6.4

+6.4

a

The positive strain is compared with the average of the experimental absolute strain values for both sides of the strip.

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I.V. Ivanov et al.

strip is the same in absolute value for linear solution. The positive values of the surface strain is compared with the average of the absolute values of the strain on the upper surface and the lower surface measured in the experiments at the same place along the strip. The strains of the linear solution match fairly well to the experimental ones, but the deflection is greater and it seems that the model is soft.

30.3.2 Shell Element Model Searching efficient solution of the laminated glass problem, special finite elements like continuum shell elements (according to the terminology of ABAQUS™software) and cohesive finite elements attracted our attention. The shell elements, including the continuum shell elements, obey the principle of straight normals remaining straight after the deformation. Cohesive elements are purposed to model thin layers with constitutive law determined by the traction separation behavior. The continuum shell elements and the cohesive finite elements are compatible and could be stacked each over the other. The half of the laminated glass strip is modeled by two plies of continuum shell elements, one for each glass layer, and one ply of cohesive elements for the PVBinterlayer. The model can be seen in Fig. 30.5. The size of the elements now does not depend on their thickness, so accepting preferable size of 5 mm, the model has only 36 408 degrees of freedom. It is so efficient that it takes only 20 seconds to solve the problem for 13 steps of loading on the same computer as for the linear solution of 3-D brick element model. The cohesive elements have no volumetric locking when the Poisson’s ratio is close to the value of 0.5 and the value of 0.49 is accepted here. The results of the linear solution of the shell element model are given in Table 30.3. The linear solution gives the same absolute values of strain on the upper and lower

U, U3 +5.480e+00 +4.958e+00 +4.437e+00 +3.915e+00 +3.393e+00 +2.872e+00 +2.350e+00 +1.828e+00 +1.306e+00 +7.845e−01 +2.628e−01 −2.589e−01 −7.807e−01

X Y Z

Step: Step−2 Increment 12: Step Time = 12.00 Primary Var: U, U3 Deformed Var: U Deformation Scale Factor: +1.000e+01

Fig. 30.5 Continuum shell element model with cohesive element interlayer


477

Table 30.3 Relative results for linear solutionb of shell element model mass

deflection

gauge #1 & #2

gauge #3 & #4

gauge #5 & #6

kg

%

%

%

%

4

+23.9

+10.5

+13.9

+15.4

8

+35.1

+17.4

+19.2

+18.9

12

+11.3

+16.6

+21.2

+19.9

b

The positive strain is compared with the average of the experimental absolute strain values for both sides of the strip.

surface for the same places along the strip, so the positive one only are compared with the average experimental strains. The model is quite soft with high deflection and high values of the strains. Since the shell element model is very efficient, a nonlinear solution is examined here. The nonlinearity is only geometrical nonlinearity that appears when high rotations of flexible structures are involved in the calculations. Many authors consider geometrical nonlinearity of laminated glass panels very important. Really, the architectural glazing could involve large area panels of laminated glass which are very flexible. The results of such nonlinear solution for the same shell element model of the half of the laminated glass strip are given in Table 30.4. The results are compared with the experimental results and the relative values are given. The model is again very soft with large strains and deflection compared with the experimental ones. The results show that there is no great difference between the linear and the nonlinear solution for the shell element model. The deflection is not so big in order to have large rotations, the strip is not constrained in membrane direction, so the geometrical nonlinearity is not rendered here. Also the strain of the PVB-interlayer is quite lower than 1% and any material nonlinearity caused by the hyperelasticity of the interlayer could not be observed. Table 30.4 Relative results of nonlinear solution for shell element model mass

deflection

gauge #1

gauge #2

gauge #3

gauge #4

gauge #5

gauge #6

kg

%

%

%

%

%

%

%

4

+23.9

+53.7

−14.1

+18.1

+11.2

+21.6

+11.4

8

+35.1

+42.3

−0.3

+23.7

+15.6

+21.7

+16.2

12

+11.1

+32.4

+3.8

+25.4

+17.2

+21.4

+18.4

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30.4 Triplex Laminated Glass (TLG) Plate Element A special plate finite element is developed here to model three layered laminated glass (triplex) panels. It is based on the assumption that the glass layers of laminated glass obey the Kirchoff’s plate theory, where plane stress with linear distribution over the thickness is assumed, and the PVB-interlayer is in pure shear.

30.4.1 Stress and Strain in Laminated Glass Plate The derivation of the stiffness matrix is based on the stationary principle of the total potential energy in equilibrium. A infinitesimal piece of laminated glass is considered which is shown in Fig. 30.6. The displacements of point P from glass layer number i, i = 1, 2, which is on distance ξi from the middle plane, moved to position P , are considered using the projections of the infinitesimal glass layer of the laminated glass element onto xzand yz-planes. The displacements are denoted u, v, and w as it is shown in Fig. 30.7. They can be determined as:

ui = uoi − ξi w,x

(30.3) vi = voi − ξi w,y

wi = w(x, y) where w,y = −w,x =

∂w ∂y − ∂w ∂x

= θx

= θ

y

are the rotations of the plate normal. The first derivatives of the displacements are the strains in glass layers:

Fig. 30.6 Infinitesimal element of laminated glass

(30.4)


479

Fig. 30.7 Displacements in glass layer

ε xi = ε xoi − ξi w,xx

εyi = εyoi − ξi w,yy

γ = −2ξ w xyi

(30.5)

i ,xy

Applying Hooke’s law, the plane stress components are derived:

E

σ xi = 1−ν 2 (ε xi + νεyi )

E σyi = 1−ν2 (νε xi + εyi )

E

τxyi = 2(1+ν) γ xyi

(30.6)

The normal resultant forces for each glass layer can be found by integration of stresses through the thickness of the layer: +h i /2

n xi =

σ xi dξi = −hi /2 +h i /2

nyi =

σyi dξi = −hi /2

Ehi (ε xoi + νεyoi ) 1 − ν2

(30.7)

Ehi (νε xoi + εyoi ) 1 − ν2

(30.8)

The normal resultant forces of both glass layers should be equal and opposite because of the equilibrium of the forces and the lack of membrane loading. They

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I.V. Ivanov et al.

should create a couple of forces which gives additional bending moment to the resistance in bending as it is shown in Fig. 30.8. Then the normal resultants can be denoted as:

Fig. 30.8 Bending moments and layer interaction

n x ≡ n x2 = −n x1

ny ≡ ny2 = −ny1

(30.9)

Applying Eqs (30.7) and (30.8) for both glass layers, i = 1, 2, four equations are obtained which are solved for the membrane strains of the glass layers and using the notation (30.9) the result is written as: n −νn ε xo1 = − xEh1 y

n −νn εyo1 = − yEh1 x

(30.10) n −νn ε xo2 = xEh2 y

n −νn εyo2 = yEh x 2

The equations above give also the relationships − h1 ε xo1 = h2 ε xo2 ,

−h1 εyo1 = h2 εyo2 .

(30.11)

The PVB-interlayer is assumed to have only pure shear strain and stress as it is shown in Fig. 30.9. The shear strain in both planes of projection can be derived from the membrane strains, the rotations of both glass layers, and their relative displacements or slippages: γ x ≡ γzx0 = ϕy − θy =

uo2 + h22 w,x − uo1 + h21 w,x 1 ' ( + w,x = δu + hC w,x , h0 h0

(30.12)

where δu = uo2 − uo1 is the slippage between the glass layers in x-direction and hC =

2h0 + h1 + h2 2

(30.13)

is the distance between the glass layers midsurfaces as it is shown in Fig. 30.8.


481

Fig. 30.9 Strain and stress in PVB-interlayer

In a similar way, the shear strain in the other plane is determined as & 1% γy ≡ γyz0 = δv + hC w,y , h0

(30.14)

where δv = vo2 − vo1 is the slippage between the glass layers in y-direction. The slippage between the glass layers is very important to explain the mechanism of glass layer interaction through the PVB-interlayer. Also, the derivatives of the slippage can represent very important relationship between the membrane straining of the glass layers: h 1 + h2 (n x − νny ) , Eh1 h2 h1 + h 2 = εyo2 − εyo1 = (ny − νn x ) . Eh1h2

δu,x = ε xo2 − ε xo1 =

(30.15)

δv,y

(30.16)

The shear stress in the PVB-interlayer can be found easily applying the Hooke’s law for shear: τx ≡ τzx0 = G0 γ x ,

τy ≡ τyz0 = G 0 γy .

(30.17)

30.4.2 Potential Strain Energy of Laminated Glass Plate The potential strain energy of laminated glass element will be presented in vector and matrix form in order to obtain then the stiffness matrix of the developed finite

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element. The potential strain energy of the PVB-interlayer depends only on the pure shear stress and strain. 1 1 Πγ = gT tdV0 = gT Cγ gdV0 , (30.18) 2 V0 2 V0 where g = [γ x γy ]T , t = [τx τy ]T , ⎡ ⎤ ⎢⎢⎢ G 0 0 ⎥⎥⎥ ⎢ ⎥⎥⎥ . Cγ = ⎢⎢⎣ 0 G0 ⎦

(30.19) (30.20)

The volume of integration is V0 , V0 = h0 Ω, where Ω is the area of the plate. The strain vector g can be expressed as a sum of two other vectors ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ γ x ⎥⎥⎥ 1 ⎢⎢⎢ δu + hC w,x ⎥⎥⎥ 1 h ⎥⎥⎦⎥ = uδ + C uθ . g = ⎢⎢⎣⎢ ⎥⎥⎦⎥ = ⎢⎢⎣⎢ (30.21) h h h0 0 δv + hC w,y 0 γy Then the potential strain energy can be constructed in another way: 1 Πγ = eT Dγ eγ dΩ , 2 Ω γ

(30.22)

where eγ = [δu δv w,x ⎡ ⎢⎢⎢ 1 0 ⎢⎢⎢ G0 ⎢⎢⎢⎢⎢ 0 1 Dγ = ⎢ h0 ⎢⎢⎢⎢ hC 0 ⎢⎢⎢ ⎣ 0 hC

w,y ]T , hC 0 hC2 0

⎤ 0 ⎥⎥⎥⎥ ⎥⎥ hC ⎥⎥⎥⎥ ⎥⎥⎥ . 0 ⎥⎥⎥⎥ ⎥⎥⎦ hC2

(30.23)

(30.24)

The derivatives of displacement functions can be expressed by a differential operator: eγ = Lγ u , (30.25) where

⎡ ⎤ ⎢⎢⎢ 1 0 0 ⎥⎥⎥ ⎥⎥ ⎢⎢⎢⎢ ⎢⎢⎢ 0 1 0 ⎥⎥⎥⎥⎥ ⎥⎥ , Lγ = ⎢⎢⎢⎢ ⎢⎢⎢ 0 0 ∂ x ⎥⎥⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ 0 0 ∂y

(30.26)

u = [δu δv w]T .

(30.27)

The potential strain energy of the PVB-interlayer is determined by three displacement functions of two variables — the coordinates x and y, which are the slippage of the glass layers δu and δv , as well as the deflection w.


483

The potential strain energy of each glass layer i, i = 1, 2, can be derived from the plane stress field in bending as follows: 1 1 T Πg i = e si dVi = eT Cei dVi , (30.28) 2 Vi i 2 Vi i where ei = [ε xi εyi γ xyi ]T ,

(30.29)

si = [σ xi σyi τxyi ] , ⎡ ⎤ ⎢⎢⎢ 1 ν 0 ⎥⎥⎥ ⎥⎥ E ⎢⎢⎢⎢ ⎢⎢⎢ ν 1 0 ⎥⎥⎥⎥⎥ , C= 2 ⎥⎥⎦ 1 − ν ⎢⎢⎣ 0 0 1−ν 2

(30.30)

T

(30.31)

and the volume of the glass layer, Vi , is Vi = hi Ω. The strain in glass layers can be represented in vector form ⎡ ⎤ ⎢⎢⎢ ε xoi − ξi w,xx ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ei = ⎢⎢⎢⎢ εyoi − ξi w,yy ⎥⎥⎥⎥ = eoi + ξi z , ⎢⎣⎢ ⎥⎦⎥ 0 − 2ξiw,xy

(30.32)

eoi = [ε xoi εyoi 0]T ,

(30.33)

z = [−w,xx − w,yy − 2w,xy ]T .

(30.34)

where

Substituting (30.32) into (30.28), one obtains 1 Πg i = (eoi + ξi z)T C(eoi + ξi z)dVi = Πoi + Πκi , 2 Vi where

1 Πoi = hi 2 Πκi =

(30.35)

1 h3i 2 12

Ω

eToi Ceoi dΩ ,

(30.36)

zT CzdΩ .

(30.37)

Ω

Using the Eqs (30.10), (30.15), and (30.16), the potential energy Eq. (30.36) can be obtained as: 11 Πoi = eT Cδ eδ dΩ , (30.38) 2 hi Ω δ where eδ = [δu,x δv,y ]T , ⎡ h21 h22 ⎢⎢⎢ 1 E ⎢⎢⎢ Cδ = 1 − ν2 (h1 + h2 )2 ⎣ ν

⎤ ν ⎥⎥⎥⎥ ⎥⎥ . 1⎦

(30.39) (30.40)

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Now, the potential strain energy of both glass layers can be summed: 1 Πg = Πo1 + Πo2 + Πκ1 + Πκ2 = eT Dg eg dΩ , 2 Ω g where

eg = [δu,x δv,y − w,xx − w,yy − 2w,xy ]T , ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ h1 h2 ⎢⎢⎢ 1 ν ⎥⎥⎥ ⎥⎥⎥ ⎢ ⎥ ⎥⎥⎥ ⎥⎥⎦ 0 ⎢⎢⎢⎢ h1 +h2 ⎢⎢⎣ ⎥⎥⎥ ⎢⎢⎢ ν1 ⎡ ⎤ ⎥⎥⎥⎥ E ⎢⎢⎢⎢ ⎢⎢⎢ 1 ν 0 ⎥⎥⎥ ⎥⎥⎥ . Dg = ⎢ ⎥⎥⎥ ⎥⎥⎥ 1 − ν2 ⎢⎢⎢⎢⎢ h31 +h32 ⎢⎢⎢⎢ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎢ 0 ν 1 0 ⎢ 12 ⎢⎢ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎣ ⎢⎣ ⎦⎦ 0 0 1−ν 2

(30.41) (30.42)

(30.43)

The vector of pseudo strain eg can be represented as differential operator applied on the displacement functions u: eg = Lg u , where

⎡ ⎢⎢⎢ ∂ x ⎢⎢⎢ ⎢⎢⎢ 0 ⎢⎢⎢ Lg = ⎢⎢⎢⎢⎢ 0 ⎢⎢⎢ ⎢⎢⎢ 0 ⎢⎢⎣ 0

0

0

∂y

0

0 −∂ x2 0 −∂y2 0 −2∂ xy

(30.44) ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ . ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦

(30.45)

The potential strain energy of the laminated glass plate is the sum of the potential shear strain energy of the PVB-interlayer and the potential strain energy of the glass layers: Π = Πγ + Πg . (30.46)

30.4.3 Stiffness Matrix Derivation The derived potential energy expressions show that they depend on three functions of displacements which totally describe the equilibrium state in the triplex laminated glass. A rectangular four-node plate finite element with six degrees of freedom per node as it is shown in Fig. 30.10 is constructed in the global coordinate system xyz. The vector of the element nodal displacements consists of four nodal vectors with the displacements of each node I, I = 1, 2, 3, 4. d = [dT1 dT2 dT3 dT4 ]T

(30.47)

The nodal displacement vectors have six displacements each: two relative displacements defining the slippage between the glass layers, δu , δv , the deflection, w, and three rotations, θx , θy , ψ:


485

Fig. 30.10 TLG rectangular plate finite element

d I = [δuI δvI wI θ xI θyI ψI ]T ,

I = 1, . . . , 4 .

(30.48)

The displacement functions can be approximated by shape functions multiplied by the nodal displacements, which are the unknown parameters: u = Nd .

(30.49)

The matrix of shape functions consists of four sub-matrices — one for each node I: N = [N1 N2 N3 N4 ] .

(30.50)

The nodal sub-matrices of shape functions have six shape functions depending on the nodal coordinates — one shape function for each degree of freedom: ⎡ ⎤ ⎢⎢⎢ N1I 0 0 0 0 0 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ N I = ⎢⎢⎢⎢ 0 N2I 0 0 0 0 ⎥⎥⎥⎥ , ⎢⎢⎣ ⎥⎥⎦ 0 0 N3I N4I N5I N6I

I = 1, . . . , 4 .

(30.51)

All shape functions are functions of two variables — the coordinates x and y. The first two functions are equivalent: N2I ≡ N1I .

(30.52)

They describe the slippage between the glass layers and they can be constructed as bi-linear shape functions from Lagrangian polynomials [11]. The remain shape functions describe the displacements in Kirchoff’s plate. They are C1 continuity shape function constructed by Hermitian polynomials [11]. The basic plate element used here is Bogner-Fox-Schmit (BFS) rectangular plate element which is well described in many sources like [11]. The basic plate element could be also any plate element. Here, we added two degrees of freedom and shape functions describing them. The construction of shape functions here is considered trivial and it is omitted. Once the shape functions are chosen, the stiffness matrix, K, can be defined from the potential strain energy, Π, by its partial derivatives with respect to the nodal displacements:

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∂Π = Kd . (30.53) ∂d First of all, the strain-displacement matrix should be defined. Starting with the shear strain in the PVB-interlayer, the strain vector can be expressed: eγ = Lγ u = Lγ Nd = Bγ d .

(30.54)

This defines the strain-displacement matrix: Bγ = Lγ N . Then the potential shear strain energy in the PVB-interlayer becomes: 1 T 1 Πγ = d BTγ Dγ Bγ dΩ d = dT Kγ d , 2 2 ω

(30.55)

(30.56)

where Kγ =

ω

BTγ Dγ Bγ dω

(30.57)

is the wanted stiffness matrix corresponding to the shear strain energy. The glass-layer strain vector is expressed by the shape functions: eg = Lg u = Lg Nd = Bg d

(30.58)

and defines the strain-displacement matrix: Bg = Lg N .

(30.59)

Using the strain-displacement matrix above, the glass strain potential energy becomes: 1 1 Πg = dT BTg Dg Bg dΩ d = dT Kg d (30.60) 2 2 ω which defines another part of the stiffness matrix of the element: Kg = BTg Dg Bg dΩ . ω

(30.61)

Finally, the stiffness matrix of TLG plate finite element is the sum of both stiffness matrices: K = Kγ + Kg . (30.62) Additional, element forces like surface pressure p, which is shown in Fig. 30.10, can be determined as a vector of nodal forces. The potential energy of the pressure p is: Πp = −

ω

wpdΩ .

(30.63)


487

The vector of traction forces is defined as: q = [0 0 p]T .

(30.64)

The potential energy can be defined for the traction forces: Π p = − uT qdΩ = −dT NT qdΩ ,

(30.65)

and the equivalent nodal forces are obtained: f= NT qdΩ

(30.66)

ω

ω

ω

as partial derivatives of the potential energy f = −∂Π p /∂d.

30.4.4 TLG Plate Element Validation The half of the laminated glass strip is modelled by TLG rectangular plate elements. The deformed model can be seen in Fig. 30.11. The size of the elements is 5 × 10 mm. The number of nodes is only 1 460 and the number of the degrees of freedom is 8 760. The TGL element is not prepared for nonlinear solution, but it is not necessary as it was shown in previous sections. The results of the TLG plate element solutions under different loadings are given in Table 30.5.

Fig. 30.11 The deformed TLG element model with dimensions and displacements in meters for m = 12 kg loading

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Table 30.5 Relative results for linear solutionc of TGL plate element model mass

deflection

gauge #1 & #2

gauge #3 & #4

gauge #5 & #6

kg

%

%

%

%

4

−1.5

−12.9

−13.2

−14.8

8

+21.0

+1.7

+3.1

0.0

12

+3.8

+7.0

+9.2

+6.4

c

The positive strain is compared with the average of the experimental absolute strain values for both sides of the strip. Load m = 4 kg 140 3−D brick shell linear shell nonlinear upper shell nonlinear lower TLG plate experimental upper experimental lower

120

strain (x10−6)

100 80 60 40 20 0 −20 0

50

100

250 200 150 distance (mm)

300

350

400

Fig. 30.12 Strain along the strip loaded by m = 4 kg

The results of the solution show as close as the 3-D brick element model solution in strain but closer to the experimental data solution in deflection. The TLG plate element model is maybe the best model among all computational models examined here. The comparison of the models can be better seen if the distribution of strain along the strip is shown as diagram. In Fig. 30.12 the distribution of the strain for the loading with m = 4 kg is shown, while on Fig. 30.13 and Fig. 30.14 the distributions for m = 8 kg and m = 12 kg are given, respectively. All they are compared with the experimental measurements at the points of strain gauges. The strain distribution is based on distance measurement starting from the central point of the strip toward its end. The diagrams show that the 3-D brick element model and the TLG plate element model are very similar in strains and quite better than the shell element model compared with the experimental data. The TLG plate element model is better in deflection than all others as well as in the computational efficiency. The TLG plate


489

Load m = 8 kg 250 3−D brick shell linear shell nonlinear upper shell nonlinear lower TLG plate experimental upper experimental lower

200

strain (x10−6)

150

100

50

0

−50

0

50

100


300

350

400

Fig. 30.13 Strain along the strip loaded by m = 8 kg Load m = 12 kg 350 3−D brick shell linear shell nonlinear upper shell nonlinear lower TLG plate experimental upper experimental lower

300

strain (x10−6)

250 200 150 100 50 0 −50

0

50

100


300

350

400

Fig. 30.14 Strain along the strip loaded by m = 12 kg

element could be developed on different basis in order to have general geometry and membrane forces. Even in this case, the maximum number of the degrees of freedom per node of the plate discretisation is 8, while for the shell element model, it is 12 or 1.5 times more. So, the computational efficiency and the accuracy of the TLG plate element are promising.

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30.5 Conclusions The laminated glass is very difficult to be modelled by finite elements. The conventional finite elements are not computationally efficient or have no sufficient accuracy. The analysis shows that the nonlinearity in static and quasi-static loading is negligible. The developed triplex laminated glass plate element is very computationally efficient and it has quite good accuracy compared with the other computational models considered here. The element is based on the Kirchoff’s plate theory for the glass layers and on the assumption that the PVB-interlayer undergoes pure shear. The element can be developed on the basis of isoparametric quad shell elements combined with membrane degrees of freedom instead of rectangular plate element. This could allow the element to be applied for complex geometry and for geometrical nonlinearity. Acknowledgements The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013), FP7 - REGPOT - 2009 - 1, under grant agreement No:245479. The support by Polish Ministry of Science and Higher Education, Grant No 1471-1/7.PR UE/2010/7, is also acknowledged as well as the support by National Science Fund of Bulgarian Ministry of Education and Science, grand agreement No DDVU 02/052-20.12.2010.

References 1. As¸ik M Z (2003) Laminated glass plates: revealing of nonlinear behavior. Computers and Structures 81:2659–2671 2. As¸ik M Z, Tezcan S (2005) A mathematical model for the behavior of laminated glass beams. Computers and Structures 83:1742–1753 3. Dhaliwal A K, Hay J N (2002) The characterization of polyvinyl butyral by thermal analysis. Thermochimica Acta 391:245–255 4. Duser A V, Jagota A, Bennison S J, (1999) Analysis of glass/polyvinyl butyral laminates subjected to uniform pressure. Journal of Engineering Mechanics 125:435–442 5. Ivanov I V (2006) Analysis modelling and optimization of laminated glasses as plane beam. Int J Solids Struct 43:6887–6907 6. Jagota A, Bennison S J, Smith C A (2000) Analysis of a compressive shear test for adhesion between elastomeric polymers and rigid substrates. International Journal of Fracture 104:105–130 7. Norville H S, King K W, Swofford J L (1998) Behavior and strength of laminated glass. Journal of Engineering Mechanics 124:46–53 8. Sobek W, Kutterer M, Messmer R (2000) Untersuchungen zum Schubverbund bei Verbundsicherheitsglas — Ermittlung des zeit- und temperaturabhängigen Schubmoduls von PVB. Bauingenieur 75:41–47 9. Vallabhan C V G, Minor J E, Nagalla S R (1987) Stress in layered glass units and monolithic glass plates. Journal of Structural Engineering 113:36–43 10. Vallabhan C V G, Das Y C, Magdi M, As¸ik M, Bailey J R (1993) Analysis of laminated glass units. Journal of Structural Engineering 119:1572–1585 11. Zienkiewicz O C, Taylor R L (2000) The Finite Flement Fethod (fifth ed) Vol. 2: Solid Mechanics. Butterworth-Heinemann, Oxford

Chapter 31

Unbending of Curved Tube by Internal Pressure Alexei M. Kolesnikov

Abstract In this work the effect of the unbending of a curved tube under a uniform normal pressure is investigated. The problem is considered within the framework of the nonlinear membrane theory. It is shown that the inflation of a curved tube is the special case of pure bending. The tube with a circular cross section made of a Mooney-Rivlin material is studied numerically. The dependencies between the curvature of the centerline of deformed curved tube and the internal pressure are obtained. It is found that there are the maximum pressures for the considered materials. Keywords Nonlinear elasticity · Membrane · Curved tube

31.1 Introduction A membrane that is a sector of a torus is called a curved tube. The problem of pure bending of pressurized curved tubes is considered within the framework of the nonlinear shell theory by Libai and Simmonds [5] and by Zubov [7]. In [5, 7] the approach is proposed to solve the problem of pure bending. The approach allows us to decompose the deformation into two parts: an in-plane deformation of meridional cross section, plus a rigid rotation of each of these meridional planes about some axis by linearly varying angles. In this case the equilibrium equations are reduced to the ordinary differential equations. This approach is used to solve the pure bending problem of a straight tube (cylindrical membrane) [4, 7]. The change of curvature is the result of the application of the bending moments to a straight tube. The feature of curved tube is the unbending A. M. Kolesnikov (B) Southern Federal University, ul. Milchakova 8a, Rostov-on-Don, 344090, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 31, © Springer-Verlag Berlin Heidelberg 2011

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under internal pressure in the absence of bending moments. This phenomenon is investigated in the presented work. In Sect. 31.2 the problem of pure bending is considered. Using the semi-inverse method the system of the ordinary differential equations are derived. It is shown that the inflation of a curved tube is the special case of a pure bending. In Sect. 31.3 we consider the inflation of a straight tube. The aim of this investigation is to produce curves of dimensionless pressure versus dimensionless curvature of the tube. The results are presented for the tube of a circular cross section made of a Mooney–Rivlin material in Sect. 31.4. The inflation of curved tube is compared with the inflation of straight tube. It is obtained that there are the maximum pressures for the considered materials. The maximum pressures for straight and curved tubes are closely.

31.2 Pure Bending Deformation Denote by o the surface of membrane in the reference configuration. The position of a point on o is given by r = χ1 (q1 )i1 + χ2 (q1 )e2 ,

q1 ∈ [q11 ; q12 ],

q2 ∈ [q21 ; q22 ],

(31.1)

e2 = i2 sin βq + i3 cos βq . 2

2

Here q1 and q2 are the Gaussian surface coordinates, the basis {ik } (k = 1, 2, 3) is the Cartesian basis. We assume that the cross section (q2 = const) is a closed and it is given by the functions χ1 (q1 ) and χ2 (q1 ). The parameter β is the curvature of the centerline of the torus sector. It is called the initial curvature of curved tube. We assume that the deformed membrane is a sector of torus. Denote by O the surface of deformed membrane. The position of a point on O is given by R = Χ1 (q1 )i1 + Χ2 (q1 )E2 ,

E2 = i2 sin Βq2 + i3 cosΒq2 .

(31.2)

Here the deformed cross section is given by the unknown function Χ1 (q1 ) and Χ2 (q1 ). The unknown parameter Β is the curvature of the deformed centerline. It is called the curvature of deformed curved tube. The covariant components of the metric tensors of o and O are independent of the Gaussian coordinate q2 and they form the diagonal matrices [4] g11 = χ1 2 + χ2 2 , 2

2

G11 = Χ1 + Χ2 , B11 =

Χ1 Χ2 − Χ1 Χ2 , 8 2 2 Χ1 + Χ2

B12 = 0,

g12 = 0, G12 = 0,

g22 = β2 χ22 ,

(31.3)

Χ22 ,

(31.4)

G22 = Β

2

Β Χ2 Χ1 B22 = 8 . 2 2 Χ1 + Χ2 2

(31.5)

31 Unbending of Curved Tube by Internal Pressure

493

Here the prime denotes differentiation with respect to q1 , gαγ (α, γ = 1, 2) are the covariant components of the first metric tensor of o, G αγ are the covariant components of the first metric tensor of O and Bαγ are the covariant components of the second metric tensor of O. If the thickness h of the membrane is a constant and the conditions (31.3)–(31.5) are satisfied then the deformation is called the one-dimensional deformation. In this case the equilibrium equations reduce to the system of ordinary equilibrium equations [4]. The conditions (31.3)–(31.5) put some restrictions on an external surface load. It is independent of q2 and its component is equal zero along the coordinate lines q2 . We assume that the external surface load is constant normal pressure p. The equilibrium equations may be reduced to the form [4] % & dL11 + L11 2Γ111 + Γ221 + L22 Γ122 = 0, 1 dq

(31.6)

L11 B11 + L22 B22 + p = 0.

(31.7)

Here Lαγ are the components of the Cauchy stress resultant tensor [4, 6]. We introduce the principal stretches λ1 (q1 ), λ2 (q1 ) and the function ψ(q1 ) 9 9 G11 G22 Χ2 (q1 ) 1 1 λ1 (q ) = , λ2 (q ) = , tan ψ(q1 ) = 1 . (31.8) g11 g22 Χ1 (q ) Denote by W the strain energy density of the membrane. The strain energy density can be expressed as function of the principal stretches for incompressible isotropic elastic material: W = W(λ1 , λ2 ). Therefore, using (31.4) and (31.8), the constitutive relations may be written in the form [4] L11 =

h g11 λ21 λ2

∂W , ∂λ1

L22 =

h g22 λ1 λ22

∂W , ∂λ2

L12 = L21 = 0.

(31.9)

Finally, using (31.8) and the constitutive relations (31.9), the equilibrium equations (31.6) and (31.7) reduce to the following system 2 ∂2 W ∂W ∂2 W g11 λ − − λ Β sin ψ + 1 2 1 ∂λ ∂λ ∂λ g22 2 1 2 ∂λ1 ∂W ∂2 W g22 + − λ2 = 0, (31.10) ∂λ1 ∂λ1 ∂λ2 2g22 2 g11 g22 λ2 − Β λ1 sin ψ + λ2 = 0, (31.11) g22 2g22 2 ∂W g11 p√ ψ −Β λ1 cos ψ − g22 λ1 λ2 = 0, (31.12) ∂λ1 g22 h √ Χ1 = g11 λ1 cos ψ, (31.13) √ Χ2 = g11 λ1 sin ψ. (31.14)

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If the parameter Β is fixed then the Eqs (31.10)–(31.14) and periodicity conditions are the boundary-value problem to determine the unknown functions Χ1 (q1 ) and Χ2 (q1 ). A solution can be satisfied integral edge conditions at the edge of torus sector (q2 = q21 and q2 = q22 ) [5, 7]. It can be shown that the resultant force F and the resultant moment M are independent of the cross section, F = 0 and M = M1 (q1 )i1 [4]. Denote by Y2C the coordinate of the center mass of surface which is bounded by the membrane at cross section. Then we have [4] M1 =

q12

q11

G11G 22 L22 (Y2C − Χ2 ) dq1 .

The considered deformation is the pure bending deformation. If the membrane is loaded by the uniformly normal pressure then M1 = 0. From this condition we may determine the curvature Β of the deformed torus sector. Thus, the problem of the unbending a curved tube by an internal pressure reduces to the boundary-value problem with the parameter Β. The boundary-value problem is solved numerically using a shooting method together with a Runge-Kutta integration process. The parameter Β is determined by the condition M1 = 0 using shooting method.

31.3 Cylindrical Membrane We now consider the limiting case when the membrane is a cylinder (straight tube) with a closed cross section in the reference configuration . The external load is the uniformly normal pressure p. We use the integral edge conditions F = 0 and M = 0 A closed cross section of the undeformed cylindrical membrane may be given as a circle of radius r0 [2]. If a cylindrical membrane is subjected by the uniformly distributed pressure only then the deformed membrane is the cylinder with a circle cross section of radius R0 [1, 2] R0 =

L11G11 . p

(31.15)

It follows from symmetry that the consition M = 0 is satisfied. The resultant force at the edge is represented as the sum of the pressure p and the stress L22 . Hence we have R20 πp − L22G22 2R0 π = 0. (31.16) Using the constitutive relations (31.9) and λ1 =

R0 , r0

λ2 = γ,


495

the Eqs (31.15) and (31.16) may be rewritten in the form p=

r0 h ∂W , λ1 λ2 ∂λ1

p=

2r0 h ∂W . λ21 ∂λ2

(31.17)

Here the unknown parameter γ determines the change of the cylinder length. If we shall give the strain energy density W and the pressure p then the unknown geometric parameters R0 and γ of the deformed cylindrical membrane are determined from the Eqs (31.17).

31.4 Circular Cross Section Let the cross section of undeformed curved tube is the circle of radius r0 χ1 (q1 ) = r0 sin q1 , χ2 (q1 ) = β−1 − r0 cos q1 ,

π q1 ∈ [0; 2π], q2 ∈ 0; . 2β

In this work we consider the three initial curvature β = 0.05, 0.1, 0.2. Consider a Mooney–Rivlin material. The strain energy density may be written in the form ⎡ ⎛ ⎞ ⎛ ⎞⎤ ⎜⎜⎜ 2 ⎟⎟⎟ ⎜⎜⎜ 1 ⎟⎟⎥⎥ μ ⎢⎢⎢⎢ 1 1 2 2 2 ⎜ ⎟ ⎜ (1 (1 W = ⎢⎣ + ν) ⎜⎝λ1 + λ2 + 2 2 − 3⎟⎠ + − ν) ⎜⎝ 2 + 2 + λ1 λ2 − 3⎟⎟⎟⎠⎥⎥⎥⎦ . 4 λ1 λ2 λ1 λ2 We assume that r0 = 1,

h = 0.001,

μ = 1.

The numeric results are obtained for ν = 1 and ν = 0.5. In case ν = 1 a Mooney– Rivlin material is also called a Neo–Hookean material. We introduce the dimensionless parameters p∗ =

pr0 , μh

Β∗ =

Β . β

The dependencies between the curvature and the pressure are shown in Fig. 31.1. The X-axis is the dimensionless curvature of the deformed tube Β∗ . The Y-axis is the dimensionless pressure p∗ . The solid lines correspond to the Neo–Hookean material (ν = 1). The dashed lines correspond to the Mooney–Rivlin material (ν = 0.5). For small strains (Β∗ > 0.85) the influence of the material parameter ν on curves Β∗ − p∗ is very small. The initial curvature β has a small influence to the curves Β∗ − p∗ . It is found that there is a maximum pressure for each considered material. The maximum pressures p∗max and the corresponding curvatures are presented in Table 31.1. The limiting case of the inflation of a straight tube has a maximum pressure. It is presented for β = 0 in Table 31.1. The value of p∗max slightly decreases when the initial curvature β increases.

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A.M. Kolesnikov 1

p∗

0.8

0.6

0.4

Fig. 31.1 Curvature vs. pressure. The solid lines correspond to the Neo– Hookean material (ν = 1). The dashed lines correspond to the Mooney–Rivlin material (ν = 0.5)

0.2

0

0.6

0.7

0.8

0.9

1

B∗

Table 31.1 Maximum pressure ν

1.0

0.5

β

Β∗

p∗max

0

–

0.750

0.05

0.681

0.749

0.1

0.683

0.749

0.2

0.692

0.745

0

–

0.894

0.05

0.634

0.893

0.1

0.633

0.892

0.2

0.630

0.888

The planforms of the deformed curved tubes for the undeformed curvatures β = 0.05, β = 0.1 and β = 0.2 are shown in Fig. 31.2. The gray dotted lines are the initial configuration. The black solid lines correspond to the Neo–Hookean material (ν = 1). The black dashed lines correspond to the Mooney–Rivlin material (ν = 0.5). The deformed curvatures are Β∗ = 0.9, Β∗ = 0.8 and Β∗ = 0.7. For small strain the influence of the material parameter ν on the shape of a deformed tube is very small. The cross sections of the deformed curved tubes are shown in Fig. 31.3. The gray dotted line is the initial configuration. The left side corresponds to the Neo– Hookean material (ν = 1). The right side corresponds to the Mooney–Rivlin material (ν = 0.5). The black solid lines correspond to the initial curvature β = 0.05. The gray dashed lines correspond to β = 0.2. The influence of the initail curvature to the cross section is insignificant for the Neo–Hookean material. The influence of the initail curvature to the cross section is small for the Mooney–Rivlin material. The cross sections differ for the different materials under the high pressures.


497

a)

b)

Fig. 31.2 Plane forms of the deformed curved tubes for Β∗ = 0.9, Β∗ = 0.8, Β∗ = 0.7: (a) β = 0.05; (b) β = 0.1; (c) β = 0.2. The gray dotted lines are the initial configuration. The black solid lines correspond to ν = 1. The black dashed lines correspond to ν = 0.5

c)

2

1

Fig. 31.3 Cross sections of the deformed curved tubes. The gray dotted line is the initial configuration. The left side corresponds to ν = 1. The right side corresponds to ν = 0.5 The black solid lines correspond to β = 0.05. The gray dashed lines correspond to β = 0.2

−2

−1

0

−1

−2

1

2

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A.M. Kolesnikov

31.5 Conclusions In this work we study the inflation of membrane which is a sector of torus (curved tube). The problem is considered within the framework of the nonlinear membrane theory. The inflation of curved tube is a special case of pure bending deformation. The governing equations are derived for the tube with arbitrary cross section made of incompressible elastic material. We also consider the inflation of cylindrical membrane (straight tube) to compare with the inflation of curved tube. The numerical results are presented for the circular cross section and the MooneyRivlin material. We obtain that the influence of initial curvature of tube is small to the dependence deformed curvature versus pressure. It is found that there are the maximum pressure for the considered materials. The material parameter ν has a significant influence on the maximum pressure. But for small strains the influence of the material parameter ν is very small. These influence grows when the internal pressure increases and the curvature of the deformed tube decreases. The initial curvature has a small influence on the relation pressure – deformed curvature. If the initial curvature increases then the maximum pressure slightly decreases. It is found that the maximum pressure for straight tube is slightly higher one in curved tube. For lower pressures the shapes of the deformed tubes are closely for the different materials. The difference in the shapes increases with the internal pressure. Acknowledgements This research was supported by the President of the Russian Federation (grant MK-439.2011.1).

References 1. Kabric, S.A., et. al.: General Nonlinear Theory of Elastic Shells (in Russian). Saint Petersburg Univ. Press., Saint Petersburg (2002) 2. Kolesnikov, A.M.: Compression of nonlinear elastic membranes between rigid surfaces. In: Kreja, I., Pietraszkiewicz, W. (eds.) Shell Structures: Theory and Applications, Vol. 2, pp. 71-74. Taylor & Francis Group, London (2010) 3. Kolesnikov, A.M.: Straightening of hyperelastic curved tube by internal pressure (in Russian). Proceeding of the XIV conferrence ¡¡Modern problems of continuum mechanics¿¿, vol. 1, pp. 182-186. Rostov-on-Don, (2010) 4. Kolesnikov, A.M., Zubov, L.M.: Large bending deformations of a cylindrical membrane with internal pressure. Z. Angew. Math. Mech. (2009) doi: 10.1002/zamm.200800182 5. Libai, A., Simmonds, J.G.: Nonlinear Theory of Elastic Shells. Cambridge Univ. Press., Cambridge (1998) 6. Zubov, L.M.: The Methods of Nonlinear Elasticity in Shell Theory (in Russian). Rostov Univ. Press., Rostov-on-Don (1982) 7. Zubov, L.M.: Semi-inverse solution in non-linear theory of elastic shells. Archives of Mechanics. 53, (45), 599–610 (2001)

Chapter 32

Characterization of Polymeric Interlayers in Laminated Glass Beams for Photovoltaic Applications Stefan-H. Schulze, Matthias Pander, Konstantin Naumenko, Anna Girchenko and Holm Altenbach

Abstract In this paper we present a mathematical model to describe the loaddeflection behavior of laminated glass beams under three-point bending. On the basis of first order deformation shear theory and a layerwise beam formulation the maximum deflection is computed. Results of this model are compared with bending tests on different laminated glass beams. Keywords Laminated Glass Beam · Photovoltaic Application · First order shear deformation beam theory · Layerwise beam theory

32.1 Introduction Typical solar modules consist of a transparent front glass, a thin polymeric interlayer and a thin polymeric back sheet. Especially thin film modules or crystalline cell modules used in building integrated applications are often made of an additional glass back side. One purpose for this back glass is to protect the photoelectric semiconductor against environmental stresses, such as moisture. Additionally, modules in laminated glass design exhibit a higher stiffness and can withstand higher mechanical loads if compared to modules with thin polymer back sheets. Besides distributed snow and wind loads during lifetime, these modules have to withstand distributed load tests during certification procedure. Thus, proper mechanical design S.-H. Schulze (B) · M. Pander Fraunhofer-Center for Silicon-Photovoltaics CSP Walter-Hülse-Str. 1, D-06120 Halle, Germany e-mail: [email protected],[email protected] K. Naumenko · A. Girchenko · H. Altenbach Martin-Luther-University Halle-Wittenberg, Center of Engineering Sciences, D-06099 Halle, Germany e-mail: [email protected],[email protected] e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 32, © Springer-Verlag Berlin Heidelberg 2011

499

500

S.-H. Schulze et al.

Fig. 32.1 Geometry and loading of the beam

of module interlayer, dimensions and clamping position is necessary. To evaluate the influence of polymer interlayer properties on mechanical behavior of laminated glass, beam-like structure can be used as a model. To analyze the behaviour of laminated glass beams and plates a structural mechanics model is required. The reliability of solar modules is directly connected to the functioning of solar cells that are made of brittle and thin silicon wafers. Mechanical stresses are applied to solar cells depending on the stiffness of the interlayer when a solar module is bended. Materials like ethylene-vinylacetate (EVA), polyvinylbutyral (PVB) or thermoplastic silicon elastomer (TPSE) are some important polymers that are used for embedding solar cells [3]. Becuase of the strongly different stiffness of these encapsulates, a suitable mathematical model to describe the amount of bending and shearing to the deflection of laminated glass beams under load is needed. The basic feature of the laminated glass beams considered is the small thickness and the relatively low shear stiffness of the PVB core layer if compared to the glass skin layers. In this paper we discuss the applicability of three models to the analysis of laminated glass beams. We begin with the first order shear deformation beam theory and present formulae for the maximum deflection and for the averaged bending and shear stiffness of the beam. Then we extend the model according to the layerwise beam theory. Here the behaviour of layers is characterized by independent equations. The results for the maximum deflection of the beam and the transverse shear deformation of the polymeric core layer are given in a closed analytical form. To illustrate the applicability of the beam models we perform a three-dimensional finite element analysis by the use of solid elements. Finally the load-deflection curve is compared with the experimental data of three-point-bending tests.

32.1.1 First Order Shear Deformation Theory Let x be the axial coordinate that describes the position of the cross sections with the origin in the central point of the beam, Fig. 32.1. In what follows we apply the

32 Characterization of Polymeric Interlayers for Photovoltaic Applications

501

geometrically-linear beam theory by assuming infinitesimal cross section rotations. From the equilibrium conditions for the part of the beam with the length a + x the bending moment M(x) and the shear force Q(x) can be computed as follows M(x) =

F F (a − x), Q(x) = − , 0 ≤ x ≤ a 2 2

(32.1)

The constitutive equations for the stress resultants are M(x) = Bϕ ,

Q(x) = Γ(w + ϕ),

(. . .) =

d (. . .), dx

(32.2)

where B is the bending stiffness, Γ is the shear stiffness, ϕ(x) is the cross section rotation and w(x) is the deflection. With Eqs (32.1), Eqs (32.2) can be integrated providing the cross section rotation and the deflection. Applying the boundary condition w(a) = 0 and the symmetry condition ϕ(0) = 0 the following results can be obtained F ϕ(x) = x(2a − x), 4B F F (32.3) w(x) = (a − x)(2a2 + 2xa − x2) + (a − x), 12B 2Γ Fa3 Fa wmax = w(0) = + 6B 2Γ The maximum deflection is The first term ion Eq. (32.3)3 is the deflection according to the Bernoulli-Euler beam theory, while the second term is the contribution due to transverse shear deformation. The key step in the use of Eqs (32.3) is to compute the bending and the shear stiffness of the beam from the properties of the constituents. To designate the properties of the core and skin layers let us apply the subscripts c and s, respectively. Let E i be the Young’s modulus, Gi the shear modulus, hi the thickness of the layer i, i = c, s, α = hc /h and μ = Gc /Gs . To compute the averaged stiffness we apply the approach proposed in [1], where a sandwich plate subjected to different elementary loads on two parallel boundaries is considered. Closed form solutions to this problem are derived according to the first order shear deformation plate theory and based on the three-dimensional theory of elasticity. Comparison and averaging of results provides expressions for the bending and the transverse shear stiffness of the plate. The results can also be applied to beams. In our notation and for α 1 and μ 1 the formulae presented in [1] can be given as follows B = Es

bh3 , 12

Γ=

G c bh 3α(1 − α)

(32.4)

32.1.2 Layerwise Beam Theory For very low values of the shear modulus Gc and/or the thickness of the core layer, the first order shear deformation beam (plate) theory may fail to describe the

502


Fig. 32.2 Free-body diagrams for layers with a legth dx: (a) stress resultants and interaction forces; (b) replacement of shear forces by normal forces and bending moments

deflection accurately. Indeed, if the shear modulus Gc approaches to zero for a fixed value of hc , then according to Eq. (32.4) the shear stiffness of the beam approaches to zero and the second term in Eq (32.3)3 tends to infinity. On the other hand, if G c is assumed constant and hc tends to zero, then it follows from Eq. (32.4) that the shear stiffness of the beam tends to infinity. In what follows we us discuss a layerwise beam theory that includes the transverse shear deformation of the core layer as an independent degree of freedom. Figure 32.2 (a) illustrates free-body diagrams for three layers with the infinitesimal length dx. The mechanical interactions between cross sections of the layers are characterized by normal forces Ni , shear forces Qi and bending moments Mi , i = 1, 2, 3. The interactions between layers are described by the forces distributed along the axial coordinate. They include normal forces q12 and q23 as well as shear forces τ12 and τ23 . According the hypothesis of the beam theory, beam cross sections remain plane during the deformation, i.e. they behave like rigid planes. Therefore, according to the principles of rigid body statics, the shear forces τ12 and τ23 can be replaced by the equivalent normal forces and bending moments distributed over the axis x, Fig. 32.2 (b). From the balance of forces and moments applied to each layer the following equilibrium conditions can be derived N1 + τ12 = 0,

Q1 + q12 = 0,

N2 + τ23 − τ12 = 0,

Q2 + q23 − q12 = 0,

hs = 0, 2 hs M3 − Q3 + τ23 = 0 2

M1 − Q1 + τ12

N3 − τ23 = 0,

Q3 − q23 = 0, hc M2 − Q2 + (τ12 + τ23 ) = 0, 2

(32.5)


503

For the axial displacements ui , deflections wi and rotations ϕi of the layers cross sections i, we assume the following relations u1 + ϕ1

hc hs = u2 − ϕ 2 , 2 2

u3 − ϕ3

hc hs = u2 + ϕ2 , 2 2

w1 = w2 = w3 = w

(32.6)

The kinematical constraints (32.6) imply that the layers are rigidly connected on interfaces. Sliding between the layers or delamination effects are not considered. The constitutive equations for the forces and moments can be formulated as follows Ni = Ei bhi ui ,

Qi = κGi bhi (w + ϕi ),

Mi = E i

bh3i 12

ϕi ,

(32.7)

where Ei is the Young’s modulus, Gi is the shear modulus, hi is the thickness of the layer i and κ is the shear correction factor. Because of constraints (32.6), the interaction forces τi j and qi j are not defined by the constitutive equations. They can be computed from equilibrium conditions. The general solution of linear algebraic and differential equations is straightforward and can be given in a vector-matrix form. For laminated plates the corresponding solution technique is presented in [2]. The aim of our analysis is to derive a comprehensible formula for the deflection of the beam that can be used to evaluate three point bending tests. To this end let us make additional assumptions. Because of small thickness and low stiffness of the core layer, the bending resistance of the beam is primarily determined by the skin layers. Therefore let us assume that M2 = 0 and we achieve Es bhs h2s M= ψ + (hs + hc )hΔ , (32.8) 2 3 ϕ1 + ϕ3 u3 − u 1 M = M1 + M2 , ψ = , Δ= 2 h With hc = αh and hs = (1 − α)h/2, Eq (32.8) reads M=

Es bh3 Ψ , 12

Ψ=

& 1% (1 − α)3ψ + 6(1 − α2)Δ 4

(32.9)

For very low values of μ → 0 we observe that (1 − α)/μ → ∞. Because the value of the shear force Q is limited, we should set w + ψ → 0. Let us assume that the transverse shear deformation of the beam is primarily determined by the core layer and set w + ψ = 0 (32.10) Based on Eqs (32.5) - (32.10) the following Eqs can be derived M=

(1 + α) 2 Δ − β2 Δ = − β Ψ, 2(1 − α3) 9 2 2μκ 1 − α3 β= 2 (1 − α) h 1 + νs α

Es bh3 Ψ , 12

(32.11)

504


For the beam configuration shown in Fig. 32.1, Eqs (32.11) are solved in a closed analytical form. The shear strain of the core layer γc = w + ϕc can be computed as follows ⎧ sinh β(l − a) − sinhβl ⎪ ⎪ ⎪ sinh βx + cosh βx − 1, 0 ≤ x ≤ a, ⎪ ⎪ 4G c κbh ⎨ cosh βl γc (x) = ⎪ (32.12) ⎪ (1 − coshβa) coshβ(l − x) ⎪ 3F ⎪ ⎪ , a<x≤l ⎩ cosh βl The maximum deflection can be computed as Fa3 Fa + 6B 2ΓL ; ; A B F sinh β(l − a) − sinhβl + (1 − cosβa) − sinhβa 2Γ L β cosh βl ;

wmax =

(32.13)

C

With κ = π2 /12 and assuming α2 α the coefficients β and Γ L can be given as follows 2 2π μ π2 1 1 β= , ΓL = Gc bh (32.14) 2 9 3 α (1 − α) h 6(1 + νs)α

32.2 Discussion For further discussion of the approach presented above, an interlayer thickness of 0.4mm was assumed. The shear modulus of the polymeric interlayer was set to G c =1.5MPa. In Fig. 32.3 the resulting maximum polymer shear strain at the interface glass/polymer at a load of F=35N was compared with results of finite element calculations. The strain is plotted along the beam length and the results show a higher strain for Eq. (32.12) compared to numerical solutions. To analyze the contribution of Bernoulli and Timoshenko parts in Eq. (32.13) to the maximum deflection of laminated glass beams, the equation was normalized (Fig. 32.4). For low values of Gc , parts B and C are dominant whereat part C dominates till G c =1.5MPa. With increasing modulus of the core layer, Bernoulli and Timoshenko parts dominate. This shows that the approach presented here is mainly describing the deflection of laminated glass with a low modulus core layer. To examine the validity of Eq. (32.13), three-point bending tests have been performed. Glass beams have been manufactured using a vacuum laminator and different encapsulation materials. Polymers were polyvinylbutyral PVB (E=1.5MPa), chemically cross-linked ethylenvinylacetate EVA (E=6.8MPa) and physically cross linked thermoplastic silicon elastomer TPSE (E=25.5MPa). As a result of different lamination temperatures, polymer viscosities and foil thicknesses, core layer thickness varies from hc =0.4mm for PVB, hc =0.12mm for EVA to hc =0.093mm for TPSE. In Fig. (32.5)


Fig. 32.3 Calculated shear strain after Eq. (32.12) and 3D FEM of soft interlayer at glass/polymer interface along beam length

505

Fig. 32.4 Normalized parts in Eq. (32.13)

Fig. 32.5 Measured and after Eq. (32.13) calculated maximum deflection of glass beams with different interlayer materials

Load-deflection curves for the measurements and the calculated deflection from Eq. (32.13) are shown. While good agreement can be achieved for glass beams with relatively stiff EVA and TPSE interlayer, deflection for glass beams with soft PVB interlayer are overestimated. One explanation for this behavior are the distinct loaddependent mechanical properties of amorphous PVB. In contrast, cross linked EVA and TPSE do not show strong viscoelasticity. Thus, the next steps for proper model estimation and discussion is to describe the viscoelastic behavior of soft interlayers such as amorphous PVB used for encapsulation of solar cells.

506


References 1. Altenbach, H.: An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. International Journal of Solids and Structures 37, 3503–3520 (2000) 2. Mittelstedt, C., Becker, W.: Reddy’s layerwise laminate plate theory for the computation of elastic fields in the vicinity of straight free laminate edges. Materials Science and Engineering A 498, 76–80 (2008) 3. Schulze, S.H., Pander, M., Mueller, S., Ehrich, C., Ebert, M.: Influence of vacuum lamination process on laminate properties simulation and test results. In: Proc. of the 24th European Photovoltaic Solar Energy Conference, 21-25 September 2009, Hamburg, Germany, pp. 3367– 3372 (2009)

Chapter 33

On the Determination of Edge Reinforcement Properties for Optimum Lightweight Design of Composite Stiffeners Philipp Weißgraeber, Christian Mittelstedt and Wilfried Becker

Abstract In this work the determination of the properties of an edge reinforcement of a composite plate stiffener is treated. A minimum stiffness criterion for the edge reinforcement on the basis of a closed-form buckling analysis of a composite plate with edge reinforcement and elastic clamping is given. The minimum stiffness criterion is given in explicit form and in a fully dimensionless representation. A composite stiffener designed by this criterion will exhibit a local buckling mode of the web, rather than a global simultaneous buckling of both the web and the edge reinforcement. The determination of an optimum lightweight design is discussed. In an example the criterion is applied to the dimensioning of a stiffener design. Keywords Stability · Composite stiffeners · Lightweight design

33.1 Introduction In view of the rising demand for lighter structures thin-walled composite structures are increasingly used in many fields of engineering. Typically these structures have to be stiffened to increase the critical buckling load [8]. Such structural situations can be found in many applications in aerospace engineering, where they are used in the construction of fuselage or wing sections, for example. To make use of the full lightweight potential of these structures the assessment of the buckling behavior has to cover both the stiffened shell and the stiffeners themselves. The latter are often constructed with a cross-section design that has an edge reinforcement. Typical cross section designs of open-profiled stringers are shown in Fig. 33.1. In P. Weißgraeber (B) · W. Becker Fachgebiet Strukturmechanik, TU Darmstadt, Hochschulstraße 1, D-64289 Darmstadt, Germany e-mail: [email protected] C. Mittelstedt ELAN GmbH, Team Method and Tools, Karnapp 25, 21079 Hamburg, Germany H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 33, © Springer-Verlag Berlin Heidelberg 2011

507

508

P. Weißgraeber et al.

Fig. 33.1 Open-profiled cross section designs of stiffeners

the following only open-profiled stringer designs with negligible torsional stiffness (GJ = 0) will be examined. The buckling behavior of the stiffeners is majorly influenced by the dimensions of the reinforcing flange. Depending on the extensional and especially the bending stiffness of the edge reinforcement two buckling modes of the stiffener are possible: a local buckling mode with almost zero deflection of the edge reinforcement and a global buckling mode in which the edge reinforcement follows the buckling mode of the web plate, as shown in Fig. 33.2. In this paper a buckling analysis of the reinforced stiffener that allows for the determination of a minimum stiffness criterion [7] is used. The analysis incorporates the effect of the bonding of the stiffener to the skin field in form of an elastic clamping and is performed in a fully dimensionless representation. The minimum stiffness criterion is discussed and its relevance for lightweight design of stiffeners is shown. The given analysis is restricted to the linear buckling problem. Load bearing capacities beyond buckling initiation are not taken into account. For pre-design and dimensioning processes closed-form analytical or semi-analytical are of great interest as they allow to quickly perform parameter studies or optimizations. Purely numerical analyses are very expensive and cannot replace explicit criteria. The derived criterion is applied to the dimensioning process of a composite stiffener with a I-profiled cross section design. The result is compared to that of a minimum stiffness criterion, that neglects the clamping effect of the bonding of the stiffener to the skin field [3]. A comparative numerical calculation is performed and shows the accuracy as well as the efficiency of the presented procedure. y

Fig. 33.2 Possible buckling modes of the stiffener

y

Edge reinforcement

Edge reinforcement

Web plate

Web plate

z

z

Local buckling mode

Global buckling mode

33 Edge Reinforcement Properties of Composite Stiffeners

509

33.2 Analysis The stiffener is modelled as a symmetrically laminated, orthotropic, rectangular 0 , that plate with length a, width b, thickness d under uniform compressive load N11 is simply supported on three edges, see Fig. 33.3. One of the unloaded edges is rotationally restrained with a rotational stiffness k and the other is reinforced by a flange with a bending stiffness EI, a torsional stiffness GJ and an extensional stiffness EA. For the description of the plate’s behavior the Classical Laminated Plate Theory (CLPT) [2, 5] is employed and for the flange the Euler-Bernoulli beam theory is used. In the analysis a transformation of the system into a dimensionless representation is performed using classic generic quantities [1,8]. The dimensionless quantities are shown in Table 33.1. The partial differential equation that describes the buckling behavior of an orthotropic plate [6] and the boundary conditions (BCs) of the problem in dimensionless representation read: Differential buckling equation α¯ −2

4¯ 2¯ ∂4 u¯ 3 ∂4 u¯ 3 0 2∂ u 3 3 2∂ u + 2η + α ¯ + N π = 0, 11 ∂ξ14 ∂ξ12 ∂ξ22 ∂ξ24 ∂ξ12

(33.1)

BCs at the loaded edges ξ1 = 0, ξ1 = 1: u3 (ξ1 = 0) = u3 (ξ1 = 1) = 0,

M11 (ξ1 = 0) = M11 (ξ1 = 1) = 0,

(33.2)

BCs at the reinforced edge ξ2 = 1 (Equilibrium of force and moment):

x2

Flange: EA, EI, GJ

F

b

0 N11

F

0 N11

Elastic clamping: kϕ

Fig. 33.3 The modeling of the stiffener

x1

a

Table 33.1 Generic quantities used in the dimensionless representation. Use is made of the stiffness components Di j of the CLPT and the in-plane modulus of the plate in 1-direction E p1 [2, 5] Coordinates Aspect ratio Buckling load Plate deflection Orthotropy parameter

x1 ξ2 = xb2 a8, 22 α¯ = ab 4 D D11 0 b2 0 N11 N 11 = π2 √D D 11 22 u¯ 3 = √u3 ab 66 η = D√12D+2D 11 D22

ξ1 =

Gener. Poissons’ ratio Elastic clamping stiffness

D12 D12 +2D66 kϕ = Dkb22

ε=

Bending stiffness

γ=

Extensional stiffness

δ=

EI D11 b EA E p1 db

510


Fig. 33.4 Exact solution for the buckling curve of an elastically clamped plate with edge reinforcement. m is the number of half waves in longitudinal direction, whereas the lowest buckling load with respect to m is the relevant. (Chosen values: η = 1.5, ε = .2, k¯ = 5, γ = 20, δ = .2)

2 ¯

γ ∂4 u¯ 3

∂3 u¯ 3

0 2 ∂ u 3 2

+ N 11 π δ

− α¯

2

3

α¯ 2 ∂ξ14

∂ξ ∂ξ 1 2 ξ =1 ξ =1 ξ 2

2

∂3 u¯ 3

= 0, (33.3) + (ε − 2)η 2

ξ =1 ∂ξ ∂ξ 2 1 2 =1

2 ∂2 u¯ 3

= 0. (33.4) ∂ξ12

ξ =1 2

BCs at the elastically clamped edge ξ2 = 0 (Vanishing deflection and moment equilibrium):

∂2 u¯ 3

∂u¯ 3

u¯ 3 (ξ1 , ξ2 = 0) = 0, − k = 0. (33.5) ∂ξ2 ∂ξ2 2 ξ2 =0

ξ2 =0

The correspondingly given boundary value problem is solved using a Lévy-type solution that satisfies the boundary conditions at the loaded edges, for details see [7]. The boundary conditions at the elastically clamped plate edge and at the reinforced edge are used to obtain the characteristic equation. This equation cannot be solved algebraically and has to be solved numerically. Yet the solutions are still of very high accuracy. When use is made of mathematical toolboxes the transcendental equation can be solved very quickly which allows for the creation of generic buckling diagrams. The examination of the buckling behavior approves that there are two buckling modes of the plate, as discussed beforehand. A typical result of the buckling analysis is given 0 over α), in Fig. 33.4, that shows the typical buckling curve (N11 ¯ that is subjected to a significant drop of the critical load when the change of the buckling modes occurs. It can be clearly seen, that for a lightweight design it is of major importance to ensure local buckling of the web plate. The critical aspect ratio at which the change of buckling modes occurs is essentially controlled by the bending stiffness of the reinforcing flange. The dependence of the buckling load on the flange’s bending stiffness is shown in Fig. 33.5. To find the bending stiffness needed to ensure local buckling, the buckling load of the given plate with reinforced edge is compared sim which is simply supported at all edges to the buckling load of a similar plate N¯ 11 and elastically clamped at one unloaded edge. With the aim of deriving an explicit criterion for the so-called minimum stiffness a Rayleigh-Ritz solution for the buckling load of the similar plate is utilized [4].


511

Fig. 33.5 The dependence of the buckling load on the bending stiffness of the flange. For low stiffnesses a global buckling mode with a single half wave occurs. With increasing bending stiffness a local buckling mode with three half waves in longitudinal direction occurs

Fig. 33.6 Rayleigh-Ritz solution to the buckling load of the plate with simple support on all edges according to Eq. 33.6. (Chosen values: η = 1.5, k¯ = 5)

For the explicit minimum stiffness criterion the buckling curve is separated in two intervals: from zero to the first minimum the buckling curve for m = 1 is valid and for higher aspect ratios a constant value, namely the minimum buckling load is taken: ⎧ 2 2 ⎪ 432 51+13k+k π2 ηα¯ 2 +4536 24+11k+k α¯ 4 ⎪ ⎪ 1 ⎪ ⎪ + ⎪ 2 ⎪ ⎨ α¯ 2 1116+285k+19k π4 α¯ 2 sim N¯ 11 =⎪ √ ⎪ √ ⎪ 2 ⎪ ⎪ 12(51+k(13+k))η 14 24+11k+k 36 ⎪ ⎪ + ⎩ π2 √ 2 2 1116+285k+19k 1116+285k+19k 9 2 4 1116 + 285k+ 19k 1 with αmin = π . ¯ + k) ¯ 4536(3 + k)(8

, for α¯ ≤ α1min (33.6) , for α¯ >

α1min (33.7)

An example of a resulting buckling curve is shown on Fig. 33.6. Furthermore the demand of equal buckling load is relaxed by introducing a scaling factor ρ [3, 7] 0 ! sim N 11 = ρN¯ 11 .

(33.8)

512


The scaling factor ρ should be chosen in the the range of 0.95 ρ 0.98. In [3] it is shown that for ρ = 0.98 the criterion coincides with the Windenburg criterion in the case of square isotropic plates without elastic clamping [9]. The condition (33.8) can be solved explicitly for the bending stiffness γ yielding the minimum stiffness criterion: B sim γ = f (ρ, N¯ 11 , m, k, η, ε, δ, α) ¯ = A

(33.9)

with A =A1 cosh(k1 mπ) sin(k2 mπ) + A2 cos(k2 mπ) sinh(k1 mπ) + A3 sin(k2 mπ) sinh(k1 mπ), B =B1 + cosh(k1 mπ) [B2 cos(k2 mπ) + B3 sin(k2 mπ)] + sinh(k1 mπ) [B4 cos(k2 mπ) + B5 sin(k2 mπ)] , % & A1 = k12 + k22 m2 πkk1 , % & A2 = − k12 + k22 m2 πkk2 , % &2 A3 = k12 + k22 m3 π2 , %% & % & & B1 = − kk1 k2 m k14 + k24 α¯ 4 + 2 −k12 + k22 α¯ 2 η − 2(−2 + ε)εη2 , % & B2 = − 2kk1 k2 k12 k22 α¯ 4 + (k12 − k22 )α¯ 2 η + (−2 + ε)εη2 , % & % % & sim 2 B3 =k1 k12 + k22 π kρN¯ 11 α¯ δ + k12m2 α¯ 2 k22 α¯ 2 + εη % & + m2 (−2 + ε)η k22 α¯ 2 + εη , % & % % & sim 2 B4 = − k2 k12 + k22 π kρN¯ 11 α¯ δ + k12 m2 α¯ 2 k22 α¯ 2 − (−2 + ε)η % && +m2 εη −k22 α¯ 2 + (−2 + ε)η , % &2 % sim 2 2 B5 =m k12 + k22 ρN¯ 11 π α¯ δ + mk k12 (k1 − k2 )k22 (k1 + k2 )α¯ 4 % &2 + α¯ 2 −4k12 k22 + k12 + k22 ε η + (k1 − k2 )(k1 + k2 )(−2 + ε)εη2

(33.10) (33.11)

In this criterion the only unknown is the number of half waves m ∈ N , which must be chosen so as to maximize the resulting minimum stiffness. In Fig. 33.7 an example for a resulting minimum stiffness diagram is shown. It is remarkable that the minimum stiffness does not rise infinitely with increasing aspect ratio but converges to a limit value. This behavior can only be observed at presence of an elastic clamping. If the elastic clamping is neglected respectively set to zero the minimum stiffness will not tend to an asymptotic value [3, 7].


513

Fig. 33.7 A minimum stiffness curve of an elastically clamped plate with edge reinforcement. The dashed line shows the minimum stiffness if the elastic clamping is neglected. In this case no convergence to a limit value can be observed. (Chosen values: η = 1.5, ε = .2, k¯ = 5, δ = .2, ρ = 0.95)

As the extensional stiffness is generally not independent of the bending stiffness its correct numerical value has to found iteratively, if no algebraic dependence can be found.

33.3 Determination of Optimum Lightweight Design For thin-walled structures under compressive or shear loading stability failure is often the relevant design limit. Hence the objective function ψ of an optimization of such a structure is the ratio of critical load of the structural component to its weight respectively its cross-sectional area: ψ=

Fc . A

(33.12)

In the present case the critical load of the stiffener with edge reinforcement can be given as 0

Fc = (1 + δ) · N11 ·

π2 · D11 · D22 , b

(33.13)

whereas the factor (1 + δ) takes into account the load carriage portion of the reinforcing flange. Let us consider a stiffener with an edge reinforcement of rectangular shape with height h and thickness t f (e.g. second from left in fig. 33.1). It is presumed that the flange thickness is constant and does not change with varying flange heights. In this case the dependencies of the bending stiffness γ, the extensional stiffness δ and the flange’s cross-sectional area A f to the flange height h are γ=

Et f · h3 ; 12D11b

δ=

Et f · h; E p1 db

A f = t f · h.

(33.14)

514


Fig. 33.8 The dependence of the critical load to the flange’s height. The dashed line shows the critical load of the reinforced plate and the critical load of the complete stiffener is dotted

In an example we will now consider a composite stiffener made of carbon fibre reinforced plastic (CFRP). The web plate of the stiffener has a [0°/90°/90°/0] slayup and the flange’s layup is [0°/90°] s . All layers are of equal thickness dl = 0.125 mm. The elastic properties of the CFRP-laminate are: E 11 = 138000MPa, E 22 = 8960 MPa, G12 = 7100 MPa and ν12 = 0.3. The length of the stiffener is a = 360mm, the width is b = 40 mm. The height of the flange h is subject to the following considerations. The clamping stiffness representing the connection to the grounding plate is kϕ = 300 N. All needed quantities can now be calculated: η = 0.23; k¯ = 2.33;

ε = 0.16;

α¯ = 8.28

γ = 1.07 · 10−2

h3 mm3

;

δ=

h ; 80mm

h A = (40 + )mm2 . 2

It is now possible to calculate the critical load of the reinforced plate and of the whole stiffener in dependence on the height of the flange h. The results are shown in Fig. 33.8. The critical load of the reinforced plate (dashed) rises until reaching a plateau, cf. Fig. 33.5. If the load of the complete stiffener is considered (dotted line), it is observed that the load is always increasing for higher flange’s heights. Instead of reaching a plateau the curve has a kink towards a reduced slope. The resulting dependence is almost linear, as δ ∼ h. The behavior of the introduced objective function ψ is shown in Fig. 33.9. Here again the dashed line represents the reinforced plate and the dotted line stands for the complete stiffener. If only the plate’s lightweight optimality is considered the optimum flange height can clearly be seen. In the case of the complete stiffener the dependence does not show a maximum, for higher flange heights the objective function is constant. This is a general result for stiffener designs with equal membrane modulus of both the plate and the flange. Because then the ratio of the linear function (1 + δ) and the cross-section area are A is: (1 + δ) 1 = = const A bd

(33.15)


515

Fig. 33.9 The objective function ψ = Fc /A over the flange height h. When considering only the plate a clear maximum can be observed. The complete stiffener’s objective function reaches a plateau for higher flange heights

(a) Flange height: h = 11.4 mm

(b) Flange height: h = 11.6 mm Fig. 33.10 Buckling modes in the finite element calculation

According to the given objective function all flange heights exceeding a minimum value are of equal lightweight optimality. But if we consider the stability of the reinforcing flange it is found that the smallest flange height h, that gives the maximum value of the objective function, represents the optimum lightweight design. With increasing flange height h the tendency of local flange buckling is increased. The stability of the flange has to be assessed separately [4]. When the minimum stiffness criterion is applied to the present example the following result is found: γmin = 16.3,

hmin = 11.5 mm.

(33.16)

This result coincides with the optimum found considering the objective function ψ, as discussed above. The minimum stiffness criterion as presented above leads to an optimum lightweight design of stiffeners. If the elastic clamping is neglected the minimum stiffness criterion yields much larger values γmin = 77.1314, hmin = 19.3 mm. (33.17) This happen as the minimum stiffness curve does not converge to a limit value, cf. 33.7. A numerical analysis was conducted to verify the results. A displacementcontrolled finite element analysis with S4-elements for the web plate and B33elements for the beam was used. The characteristic element was 1 mm. The result showed excellent agreement with the present analysis’ results, the error was less

516


than 0.7%. In Fig. 33.10 the buckling modes of the reinforced plate with flange heights of h = 11.4 mm and h = 11.6 mm are shown.

33.4 Conclusion In this work a solution for the buckling behavior of a symmetrically laminated, orthotropic plate with edge reinforcement and elastic clamping was recalled, that leads to an explicit minimum stiffness criterion. The explicit minimum stiffness criterion was utilized to discuss the optimum lightweight design of stiffeners. In an example the determination of an optimum lightweight design of a composite stiffeners was performed. The results showed to agree very well with numerical computations.

References 1. Barbre, R.: Beulspannungen von Rechteckplatten mit Längssteifen bei gleichmäßiger Druckbeanspruchung. Der Bauingenieur (25/26), 268–273 (1936) 2. Jones, R.: Mechanics of composite materials. McGraw-Hill Book Company (1975) 3. Mittelstedt, C., Schagerl, M.: A composite view on Windenburg’s problem: Buckling and minimum stiffness requirements of compressively loaded orthotropic plates with edge reinforcements. International Journal of Mechanical Sciences 52(3), 471–484 (2010) 4. Qiao, P., Shan, L.: Explicit local buckling analysis and design of fiber-reinforced plastic composite structural shapes. Composite Structures 70, 468–483 (2005) 5. Reddy, J.: Mechanics of laminated composite plates and shells: theory and analysis, 2. edn. CRC press (2004) 6. Turvey, G., Marshall, I.: Buckling and postbuckling of composite plates. Springer (1995) 7. Weißgraeber, P., Mittelstedt, C., Becker, W.: Buckling of composite panels: A criterion for optimum stiffener design. Aerospace Science and Technology (accepted manuscript) 8. Wiedemann, J.: Leichtbau I. Elemente und Konstruktion. Springer, 3. Auflage, (2006) 9. Windenburg, D.F.: The elastic stability of tee stiffeners. In: ’Proceeding of Fifth Congr. Appl. Mech., Cambridge, USA’ (1939)

Part VI

Micro- and Nanomechanical Applications

Chapter 34

Evaluation of the Mechanical Parameters of Nanotubes by Means of Nonclassical Theories of Shells Svetlana M. Bauer, Andrei M. Ermakov, Stanislava V. Kashtanova and Nikita F. Morozov

Abstract In [3] the stiffness of bridges and cantilevers made of natural chrysotile asbestos nanotubes has been studied by means of scanning probe microscopy. The stiffness is defined as a ratio of the value of the local load (applied to the tube) to the value of the displacement. Nanotubes with different fillers are analyzed. Experiments show that the stiffness of the tube depends on the materials for filling. The tubes with water are softer and the tubes filled with mercury are more rigid than tubes without filling materials. It was shown in [3] that the classical theory of bending can not explain the experimental results, but the experimental results well agree with the Timoshenko-Reissner theory (at least qualitatively), when the interlaminar shear modulus of elasticity changes for different filling materials. When additional factors such as lamination of structure and cylindrical anisotropy are taken into account the theory of Rodionova-Titaev-Chernykh (RTC) permits to obtain much more reliable results. In this work the authors also applied another nonclassical shell theory, namely the shell theory of Paliy-Spiro (PS) developed for shells with moderate thickness. The comparison of nonclassical shell theories (RTCh and PS) with experimental data and FEM calculations are presented. Keywords Theory of anisotropic shells · Shell theory of Paliy-Spiro · Shell theory of Rodionova-Titaev-Chernykh

34.1 Introduction Scientists have recently been actively discussing the possibility of application methods of classical mechanics to nanoobjects. It is known [1, 2] that the mechanical S. M. Bauer (B) · A. M. Ermakov · S. V. Kashtanova · N. F. Morozov St. Petersburg State University, St. Petersburg, Russia e-mail: [email protected],[email protected],[email protected] e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 34, © Springer-Verlag Berlin Heidelberg 2011

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characteristics corresponding to nano-size structure elements such as beams and plates can differ from mechanical characteristics corresponding to structures of the same material, which have ”normal” geometrical sizes. Besides size effects there is a possibility of appearance of the anisotropy of nanoobjects. In [3], [4] results of experiments, which examined mechanical properties of nanotubes made of natural chrysolite asbestos are discussed. The diameter of nanotubes is approximately equal to 32 nm, the internal diameter is 5 nm. The inner cavity of a tube was filled with water, mercury or tellurium under pressure. The rigidity of nanotubes was measured with the use of scanning probing microscopy. The rigidity was understood as the ratio between applied strain and the value of bridge deflection formed by the nanotube, which blocked an opening in porous bottom layer. The conditions of the experiment are described in detail in [3]. The experiments showed that a tube filled with water is substantially softer than a “dry” tube, i.e. a tube without any filler. Tubes filled with mercury are slightly more rigid than “dry” tubes. In [3] experimental data and the results of modeling have compared. The simplest classical models of isotropic beams and non-classical transversal isotropic models have considered. The Timoshenko-Reissner (TR) theory it is used for analytical estimation of nanotubes deflection as beams since layered structure of asbestos nanotubes allows to consider it as a transversal isotropic one. Each layer can preserve its properties but the shear modulus for the plane normal to the surface of isotropy G can change depending on the filler. The fact that a tube filled with water can be softer than a “dry” tube can be explained by decreasing of the shear modulus of the cross-section. It is known [5] that for isotropic shells and plates the TR theory asymptotically inconsistent refines the deflection. But for structures, which are made of transversal isotropic material “in case when material stiffness in tangential directions is much larger than its stiffness in the transversal direction” the TR theory makes the BernoulliKirchhoff-Love theory more precise and gives next asymptotical approximation of the three-dimensional theory [5]. The bodies “with moderately small transverse shear rigidity” are thin-walled bodies for which small parameter g = G /E (where E is the Young’s modulus in the tangential direction, G is the shear modulus for plane normal to the surface of isotropy) satisfies expression μ2 g 1 (μ = h/R, h is the thickness, R the radius of the shell). In [4] the problem of nanotubes deformation his solved applying the Rodionova-Titaev-Chernykh (RTCh) theory of anisotropic shells which permits to take into account the transverse shear and the layered structure of asbestos and cylindrical anisotropy as well. In this work the deformation of a multilayered tube under locally applied load (Fig. 34.1) is studied with the use of the theory of anisotropic shells of moderate thickness, which is developed in [7]. It is made a comparison of the results obtained with the use of the RTCh theory, the Paliy-Spiro theory (PS) and FEM calculations based on ANSYS code.

34 Evaluation of the Mechanical Parameters of Nanotubes

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34.2 Problem Definition Let α and β be cylindrical coordinates on the shell surface, α is the polar angle, β is the coordinate along the tube’s generatrix, h(i) is the thicknesses, R(i) is the radii of middle surfaces of the shell layers, and L is the tube’s length. For definition of the coefficients we will use the symbol A(i) j . Low the index j denotes the curvilinear coordinate corresponding to the value A, upper index i is the denotes the layer it corresponds, i.e. if i = 1 then A corresponds to the first inner layer, if i = N then A corresponds to the last outer layer. E1(i) , E 2(i) , E 3(i) are moduli of elasticity in the tangential and the normal directions, ν(i) are the Poisson’s ratios. jk The author studies the stress-strain state of a multilayered tube under locally applied load with the use of the improved iterative theory of anisotropic shells of Rodionova-Titaeva-Chernykh [6] and the theory of Paliy-Spiro suggested in the monograph [7]. The improved iterative RTCh theory is based on the following hypotheses: 1. the transverse tangential and normal stresses are distributed on shell’s thickness according to quadratic and cubic laws respectively; 2. the tangential and normal components of the displacement vector are distributed on the shell thickness according to quadratic and cubic laws respectively. This theory allows taking into account turns of fibers, their deviation and the change of their length. Functions which describe the displacements of the shell’s layer u1 (α, β, z), u2 (α, β, z), u3 (α, β, z) according to the RTCh theory are suggested to be found as series of Legendre polynomials P0 , P1 , P2 , P3 with aspect to the normal coordinate z ∈ − h2 , h2

Pz b a

Fig. 34.1 Geometrical model of the tube

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u1 (α, β, z) = u(α, β)P0 (z) + γ1 (α, β)P1 (z) + θ1 (α, β)P2 (z) + ϕ1 (α, β)P3 (z), u2 (α, β, z) = v(α, β)P0 (z) + γ2 (α, β)P1 (z) + θ2 (α, β)P2 (z)

(34.1)

+ ϕ2 (α, β)P3 (z), u3 (α, β, z) = w(α, β)P0 (z) + γ3 (α, β)P1 (z) + θ3 (α, β)P2 (z),

P0 (z) = 1,

P1 (z) =

2z , h

P2 (z) =

6z2 1 − , h2 2

P3 (z) =

20z3 3z − , h h3

(34.2)

u, v, w are the components of the displacement vector for points of the middle surface of the shell, γ3 and θ3 characterize the change in length of the normal to this surface, γ1 and γ2 - characterize the angles of rotation of the normal in planes (α, z), (β, z) respectively. θ1 and ϕ1 , describing normal curvature of the fiber in the plane (α, z), θ2 and ϕ2 describing the normal curvature in the plane (β, z) before deformation they were perpendicular to the shell middle surface. The Paliy-Spiro shell theory [7] assumes the following hypotheses: 1. Straight fibers of the shell, which are perpendicular to its middle surface before deformation, remain also straight after deformation. 2. Cosine of the slope angle of these fibers to the middle surface of the deformed shell is equal to the averaged angle of transverse shear. The mathematical formulation of these hypotheses gives the following equalities: u1 (α, β, z) = u(α, β) + φ(α, β)z,

u2 (α, β, z) = v(α, β) + ψ(α, β)z,

u3 (α, β, z) = w(α, β) + F(α, β, z), (34.3) φ(α, β) = γ1 (α, β) + φ0 (α, β), φ0 (α, β) = −

ψ(α, β) = γ2 (α, β) + ψ0 (α, β),

1 ∂w(α, β) 1 ∂w(α, β) + k1 u(α, β), ψ0 (α, β) = − + k2 v(α, β), A1 ∂α A2 ∂α

where φ and ψ are the angles of the normal’s rotation in planes (α, z), (β, z); φ0 , ψ0 , γ1 and γ2 — are angles of normal’s rotation to the median surface and angles of displacement in the same planes. The function F(α, β, z) characterizes the change of the length of the normal to the median surface. The Lamé’s coefficients and curvature coefficients, which determine the geometry of the cylindrical shell, have the following form (i) A(i) 1 =R ,

A(i) 2 = 1,

k1(i) =

1 , R(i)

k2(i) = 0

(34.4)


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Let us introduce the following dimensionless variables: A(i) h (i) ˜ R = 1(i) , h˜ (i) = (i) , R A2 3 4 (i) (i) ˜ (i) ˜ (i) u˜ (i) , v˜(i) , w˜ (i) , γ˜ 1,2,3 , θ˜1,2,3 , ϕ˜ (i) 1,2 , φ,0 , ψ,0 4 13 (i) (i) = u(i) , v(i) , w(i) , γ1,2,3 , θ1,2,3 , ϕ(i) , φ(i) , ψ(i) , 1,2 ,0 ,0 h E 2,3 G13,12,23 E˜ 2,3 = , G˜ 13,12,23 = , E1 E1 (i) (i) Pin1,2,3 Pout1,2,3 (i) (i) ˜ ˜ Pin1,2,3 = , Pout1,2,3 = , E1 E 14 3 (i) (i) (i) 3 (i) 4 T 0,1,2 , Q1,2 , m1,2,3 (i) T˜ 0,1,2 , Q˜ (i) , m ˜ = , 1,2 1,2,3 R(i) E1 3 (i) 4 3 (i) 4 3 (i) 4 M0,1,2 3 (i) 4 q1,2,3 M˜ 0,1,2 = (i) , q˜ 1,2,3 = , E1 R E1 h

(34.5)

where Pout x , Pin x are the values of the pressure on the internal and external shell surfaces. For simplicity let us introduce the following parameters: E 11 =

1 , 1 − ν12ν21 E˜ 3

E 12 =

E˜ 2 , 1 − ν12ν21

E22 =

ν12 , 1 − ν12ν21

ν31 + ν21 ν32 ν32 + ν21 ν31 , μ2 = , 1 − ν12ν21 1 − ν12ν21 3 3 K11 = −E11 h˜ (i) , K12 = E 22 h˜ (i) , K21 = E11 h˜ (i) μ1 , K22 = E22 h˜ (i) μ2 , 2 2 (34.6) (i) (i) h˜ h˜ K13 = E 11 (μ2 + 2ν12 μ1 ), K23 = E11 (ν12 μ1 + 2μ2 ), 2 2 h˜ (i) ˜ h˜ (i) h˜ (i) ˜ (i) h˜ (i) (i) m ˜ (i) = Pout 1 + + Pin 1 − , (x = 1, 2, 3), x x x 2 2 2 2 h˜ (i) h˜ (i) (i) (i) (i) ˜ ˜ q˜ x = Pout x 1 + − Pin x 1 − , (x = 1, 2, 3) 2 2 Ez =

1 − ν13μ1 − ν23 μ2

,

μ1 =

34.3 Correlations of the Rodionova-Titaev-Chernykh and the Paliy-Spiro Shell Theory Shell deformations of the theories under consideration are expressed through the components of displacements with the use of the equations presented in Table 34.1 Deformation components those are different for the theories are underlined. Below we present the relations for moments, strains and deformation components for the RTCh theory, which are converted for the case of a cylindrical shell. Substituting

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Table 34.1 Deformation components in RTCh and PS theories Rodionova-Titaev-Chernykh theory (i) ∂˜v(i) ˜ (i) ∂u˜ ε˜ (i) ˜ (i) ), ε˜ (i) 1 = h ( ∂α(i) + w 2 = ∂β(i) , ∂γ˜ 1(i) ∂˜γ2(i) ˜ (i) η˜ (i) ˜ 3(i) ), η˜ (i) 1 = h ( ∂α(i) + γ 2 = ∂β(i) , (i) ˜ (i) ∂w˜ − h˜ (i) u˜ (i) + 2˜γ(i) , ε˜ (i) 13 = h 1 (i) ∂α ∂w˜ (i) (i) (i) ε˜ 23 = (i) + 2˜γ2 , ∂β (i) ∂˜u(i) ˜ (i) ∂˜v , ω ω ˜ (i) = h ˜ (i) = (i) , 1 2 (i) ∂α ∂β ∂γ˜ 2(i) ∂˜γ1(i) (i) (i) (i) τ˜ 1 = h˜ , τ˜ 2 = (i) , ∂α(i) ∂β

Paliy-Spiro theory (i) ∂˜v(i) ˜ (i) ∂˜u ε˜ (i) ˜ (i) ), ε˜ (i) 1 = h ( ∂α(i) + w 2 = ∂β(i) , ˜ (i) ∂ψ˜ (i) ˜ (i) ∂φ η˜ (i) η˜ (i) 1 = h ( ∂α(i) ), 2 = ∂β(i) ,

τ˜ (i) = τ˜ (i) ˜ (i) ˜ (i) = ω ˜ (i) ˜ (i) 1 +τ 2 ,ω 1 +ω 2

τ˜ (i) = τ˜ (i) ˜ (i) ˜ (i) = ω ˜ (i) ˜ (i) 1 +τ 2 ,ω 1 +ω 2

ε˜ (i) 13 = 0, ε˜ (i) = 0, 23 ∂˜v(i) ∂˜u(i) , ω ˜ (i) = (i) , 2 (i) ∂α ∂β ˜ (i) ∂ ψ ∂ φ˜ (i) (i) (i) (i) τ˜ 1 = h˜ , τ˜ 2 = (i) , (i) ∂α ∂β ˜ (i) ω ˜ (i) 1 =h

the given in Table 34.1 dependencies in (34.7), we can receive an equation, which characterizes their connection to components of the displacement. T˜ 1(i) = E 11 h˜ (i) ε(i) + E 12h˜ (i) ε(i) + μ(i) T˜ (i) , 1 2 1 0 T˜ 2(i) = E 12 h˜ (i) ε(i) + E 22h˜ (i) ε(i) + μ(i) T˜ (i) , 1 2 2 0 M˜ 1(i) =

(i) (i) (i) ˜ (i) h˜ (i) 6 (E 11 η1 + E 12 η2 ) + μ1 M0 ,

h˜ (i) (i) (i) ˜ (i) M˜ 2(i) = (E12 η(i) 1 + E22 η2 ) + μ2 M0 , 6 (34.7)

(i) (i) T˜ 12 = T˜ 21 = G˜ (i) h˜ (i) τ˜ (i) , 12 (i) ˜ (i) = 1 G˜ (i) h˜ (i) ω M˜ 12 =M ˜ (i) , 21 6 12

Q˜ (i) = 1

Q˜ (i) = 2

5h˜ (i)G˜ (i) 13 6 5h˜ (i)G˜ (i) 23 6

ε(i) + 13

ε(i) + 23

m ˜ (i) 1 6 m ˜ (i) 2 6

− (h˜ (i))2

− h˜ (i)

G˜ (i) ∂θ3(i) 13 6 ∂α(i)

G˜ (i) ∂θ3(i) 23 6 ∂β(i)

,

,


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⎛ ⎞ ⎟⎟ ˜ (i) 2 ⎜⎜ ∂q˜ (i) ∂q˜ (i) (i) (i) (h ) ⎜ 1 2 ⎟ (i) ⎜ ⎟⎟⎟ − h˜ (i) M ˜ ˜ ˜ (i) , ⎜⎜⎝ T0 = m ˜3 + + R 1 12 ∂α(i) ∂β(i) ⎠ ⎛ (i) ⎞ (i) ˜ ∂m ˜ ⎟⎟ h˜ (i) (i) (h˜ (i) )2 (i) h˜ (i) ⎜⎜⎜⎜ ∂m ⎜⎜⎝ 1 + R˜ (i) 2 ⎟⎟⎟⎟⎠ − M0(i) = q˜ 3 + T˜ (i) 10 60 ∂α 60 1 ∂β(i) Let us substitute the following relations (34.8) for the six components of the displacements in Eqs (34.7). Thus we reduce them to dependence on the five main components of the displacement u, v, w, γ1 , γ2 : (i) (i) m1 (i) − Q˜ 1 h˜ (i) ∂θ˜3 = − , 12G13 10 ∂α(i) 10h˜ (i)G˜ (i) 13 (i) (i) q˜ (i) m2 (i) − Q˜ (i) h˜ (i) ∂γ˜ h˜ (i) ∂θ˜3 2 θ˜2(i) = 2 − (i) 3(i) , ϕ˜ (i) = − , 2 12G23 6R˜ ∂β R˜ (i) ∂β(i) 10h˜ (i)G˜ (i) 23 (i) ˜ (i) 1 % & (i) & M 1 T˜ 0 1 % (i) 1 (i) (i) (i) 0 γ˜ 3 = − μ1 ε1 + μ2 ε2 , θ˜3 = − μ1 η(i) + μ2 η(i) 1 2 (i) (i) (i) (i) ˜ ˜ 2 6 2h E˜ z h E˜ z

θ˜1(i) =

q˜ (i) 1

(i) h˜ (i) ∂γ˜ 3 − , 6 ∂α(i)

ϕ˜ (i) 1

(34.8)

The same transformation for PS theory gives: (i) (i) (i) T˜ 1 = E11 h˜ (i) ε1 + E 12 h˜ (i) ε2 i h˜ (i) (i) (i) q3 ˜ (i) + ((K11 − K12 )η˜ (i) − K η ˜ ) + μ h , 13 1 2 1 2 12

˜ (i) (i) T˜ 2(i) = E12 h˜ (i) ε(i) 1 + E 22 h ε2 i (i) h˜ (i) (i) q3 ˜ (i) + ((K21 − K22 )η˜ (i) − K η ˜ ) + μ h , 23 2 1 2 2 12 ˜ (i) 2 (i) (i) (i) (h ) (i) ˜ (i) ω T˜ 12 = G˜ (i) h ˜ + ω ˜ − τ ˜ , 12 1 2 12 1 (i) (h˜ (i) )2 (i) ˜ (i) ω T˜ 21 = G˜ (i) ˜ (i) ˜ (i) ˜2 , 12 h 1 +ω 2 + 12 τ ˜ (i) ˜ (i) = h (E 11 η(i) + E 12 η(i) M 1 1 2 6 (i) +(K11 − K12 )ε˜ (i) ˜ (i) 1 − K23 ε 2 ) + μ1

˜ (i) ˜ (i) = h (E 12 η(i) + E 22 η(i) M 2 1 2 6 (i) +(K21 − K22 )ε˜ (i) ˜ (i) 2 − K13 ε 1 ) + μ2

(34.9)

qi3 8

qi3 8

h˜ (i) ,

h˜ (i) ,

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& ˜ (i) % ˜ (i) = G˜ (i) h τ˜ (i) + τ˜ (i) − (h˜ (i) )2 ω M ˜ (i) 12 12 12 1 2 1 , & ˜ (i) % ˜ (i) = G˜ (i) h τ˜ (i) + τ˜ (i) + (h˜ (i) )2 ω M ˜ (i) , 21 12 12 1 2 2 ˜ (i) ˜ (i) (i) Q˜ (i) 1 = G 13 h γ1 ,

σ33 =

˜ (i) ˜ (i) (i) Q˜ (i) 2 = G 23 h γ2 ,

% &% & % &% & ˜ 3 (i) 1 + h˜ (i) 0.5 + z(i) − Pin ˜ 3 (i) 1 − h˜ (i) 0.5 − z(i) Pout 2 2 h h 1 + Rz(i)

,

z

σ33 dz − (μ1 ε1 + μ2 ε2 )z Ez 0 0 1 2 ε1 z μ1 η1 z3 − μ1 η1 − (i) + μ2 η2 + . 2 R R(i) 3 By using Table 34.1 we obtain moments and strains as functions of the displacements u, v, w, γ1 , γ2 . By substituting these relations into equilibrium equation of the cylindrical shell (i)

˜ β, z) = F(α,

∂T˜ 1(i) ∂α(i)

+ R˜ (i)

(i) ∂T˜ 12

∂α(i) ∂ Q˜ (i) 1 ∂α(i)

(i) ∂T˜ 21

∂β(i)

+ R˜ (i)

+ R˜ (i)

+ Q˜ (i) + q˜ (i) = 0, 1 1

∂T˜ 2(i) ∂β(i)

∂ Q˜ (i) 2 ∂β(i)

+ q˜ (i) = 0, 2

− T˜ 1(i) + q˜ (i) = 0, 3

(34.10)

⎛ ⎞ ˜ (i) ˜ (i) ⎟⎟⎟ ∂M 1 ⎜⎜⎜⎜ ∂ M 1 21 (i) ⎟⎟⎟ − Q˜ (i) + m ⎜⎜ + R˜ ˜ (i) = 0, 1 1 ∂β(i) ⎠ h˜ (i) ⎝ ∂α(i) ⎛ ⎞ ˜ (i) ˜ (i) ⎟⎟⎟ ∂M 1 ⎜⎜⎜⎜ ∂ M 12 2 (i) ⎟⎟⎟ − Q˜ (i) + m ⎜ + R˜ ˜ (i) 2 2 =0 ˜h(i) ⎜⎝ ∂α(i) ∂β(i) ⎠ one can get a system of five partial differential equations with five unknown functions for both theories. For the Rodionova-Titaev-Chernykh theory this system of equations is 14th order, and for the Paliy-Spiro theory is of 10th order. Substituting


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corresponding components of the deformation in Eqs (34.1)-(34.3) one can get all components of the stress-strain state of the shells under consideration.

34.4 Numerical Method For solving the system of Eqs (34.10) in displacements we will use the following series: u(i) (α, β) =

∞ ∞

v(i) (α, β) =

∞ ∞

w(i) (α, β) =

∞ ∞

γ1(i) (α, β) = γ2(i) (α, β) =

n=0

n=0

n=0

(i) ¯ m=0 unm sin[nα] sin[mβ], (i) ¯ m=0 vnm cos[nα] cos[mβ], (i) ¯ m=0 wnm cos[nα] sin[mβ],

m ¯ = (πm)/L,

(34.11)

∞ ∞ n=0

(i) ¯ m=0 γ1nm sin[nα] sin[mβ],

∞ ∞ n=0

(i) ¯ m=0 γ2nm cos[nα] cos[mβ]

These formulas take into account the symmetry of the shell deformation with respect to plane α = 0 and provide zero displacements u, γ1 and w while β = 0, L. Expressions for v, γ2 do not satisfy zero boundary conditions but when deformations do not concern boundary of the area these displacements are small. External and internal forces, which act on the shell surface can be represented as a product of sectional forces expanded in series. Let X1(i+1) , X2(i+1) , X3(i+1) be components of the pressure on the external surface of the ith shell, and X1(i) , X2(i) , X3(i) pressure on its internal surface. Then expressions for the load and moments become: m ˜ (i) 1 (α, β) =

∞ ∞ ˜ (i) h n=0 m=0

2

X1(i+1) nm

h˜ (i) 1+ 2

h˜ (i) +X1(i) 1 − sin[nα] sin[mβ], ¯ nm 2

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∞ ∞ ' ( h˜ (i) (i+1) q˜ (i) α, β = X1 1 + nm 1 2 n=0 m=0 h˜ (i) −X1(i) 1 − sin[nα] sin[mβ], ¯ nm 2 ' ( m ˜ (i) 2 α, β =

∞ ∞ ˜ (i) h n=0 m=0

2

X1(i+1) nm

h˜ (i) 1+ 2

(34.12)

h˜ (i) +X1(i) 1 − cos[nα] cos[mβ], ¯ nm 2 ∞ ∞ ' ( h˜ (i) (i+1) q˜ (i) α, β = X1 1 + nm 2 2 n=0 m=0 h˜ (i) −X1(i) 1 − cos[nα] cos[mβ], ¯ nm 2 ( (i) ' m ˜ 3 α, β

=

∞ ∞ ˜ (i) h n=0 m=0

+X1(i) nm ' ( q(i) 3 α, β =

2

h˜ (i) 1− 2

∞ ∞

cos[nα] sin[mβ], ¯

X1(i+1) nm

h˜ (i) 1− 2

h˜ (i) 1+ 2

n=0 m=0

−X1(i) nm

(i+1) X1nm

h˜ (i) 1+ 2

cos[nα] sin[mβ] ¯

Index i = 1 corresponds to the internal, and i = N + 1 to the external surface of the tube, which consists of N layers. Following [6], we accept the condition of rigid support layers: u˜ (i) (α, β, h/2) = u˜ (i+1) (α, β, −h/2), 1 1 u˜ (i) ˜ (i+1) (α, β, −h/2), 2 (α, β, h/2) = u 2 u˜ (i) (α, β, h/2) 3

u˜ (i+1) (α, β, −h/2) 3

(34.13)

= The load localized in a small rectangular area can be represented in the form of a product of the Fourier series of two loading functions in cross-section and longitudinal section:


⎛ ⎞ ∞ ⎜⎜⎜ C 2 L nπ nπ ⎟⎟⎟⎟ ⎜ Pa[α] = P ⎜⎜⎝ + sin( C) cos( α)⎟⎟⎠ L L n=0 nπ L L Pressure in longitudinal section of the tube is equal ⎛ ∞ ⎞ ⎜⎜⎜ 4 L mπ mπ mπ ⎟⎟⎟⎟ ⎜ Pb[β] = P ⎜⎜⎝ sin( C)sin( Lv ) sin( β)⎟⎟⎠ , L mπ L L L

529

(34.14)

(34.15)

m=0

where Lv is the center of load application, 2C the size of load application area, P the pressure in the area. Pressure area is described by function of the product of the series: Pd[α, β] = Pa[α]Pb[β] (34.16) The load is applied to the outer surface of the tube: X1(N+1) = 0, nm

X2(N+1) = 0, nm

X3(N+1) = Pd[α, β] nm

(34.17)

Pressure on the internal surface of the tube is absent: X1(1) nm = 0,

X2(1) nm = 0,

X3(1) nm = 0

(34.18)

Substituting dependencies (34.11),(34.12) and (34.16) in the system (34.10) and in conditions (34.13) we obtain a system of 8N − 3 linear algebraic equations for 5N deformation components and 3N − 3 forces of interaction between layers of (i) (i) (i) (i) (i) (i) (i) (i) shells. Each of the obtained coefficients unm , vnm , wnm , γ1nm , γ2nm , X1 , X2 , X3 is a member of the Fourier series of strain and loading functions. For realization of the aforementioned numerical method we developed a program based on code Mathematica 7.0.

34.5 Numerical Results In [4] the deformation of a nanotube is considered with the following parameters: thickness of each of the 100 layers h = 0, 135 nm, inner tube radius R = 2.5 nm, outer R = 16 nm, length of the tube L = 500 nm. For values of the modulus of elasticity of the shell E1,2,3 = 1.75∗1011 Pa, and relatively small value of the shear modulus G13 = G12 = G23 = 2.3 ∗ 107 Pa. Values of the tube deflection under action of a locally applied load calculated with the use of the RTCh theory were close to the experimental data. Poisson’s ratios ν12 = ν21 = ν32 = ν31 = ν23 = ν13 = 0.3. The Table 1 shows the values of deflection of the described tube, obtained by RTCh theory and theory of PS. Lv is a coordinate of the force application on the outer surface of the shell. Calculation of the functions of displacement was done with external force Fv = 10 nN. The area of applied load is [50 ∗ 44.7] nm2 . For comparison we present below the values of deflections of a transversal-isotropic

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S.M. Bauer et al.

Table 34.2 Deflection of the multi-layer nanotube Lv , nm

250

PS, nm

48.14 46.35 44.13 41.03 37.09 26.76 20.47 13.49 5.93

204.5 181.8 159

136.3 90.9

68.18 45.45 22.72

RTCh, nm 48.35 46.55 44.32 41.21 37.24 26.87 20.55 13.55 5.96 Ansys, nm 46.13 44.48 42.39 39.5

35.80 26.23 21.11 14.77 7.73

Table 34.3 Deflection of a single-layer shell with constant outer radius h/R

1/15

1/12

1/9

1/6

1/5

1/4

1/3

PS

628.76

501.25

373.58

245.49

203.61

161.12

120.17

RTCh

630.3

503.3

375.8

248.22

206.58

164.4

123.81

Ansys

616

485.3

360.4

232.2

191.6

150,6

95.9

tube, which were received in code Ansys 11, where 3-dimensional 20 knots element Solid 186 was used. Let us compare results obtained by the three-dimensional theory that is used in code Ansys 11 with results obtained by the aforementioned non-classical theories of shells. We consider a single-layer cylindrical shell with constant outer radius and gradually increase the thickness of the shell (and consequently reduce a radius of the middle surface of the shell). The following table lists the values of deflections at the center of considered shells. Table 34.3 gives the values of the deflections of shells. Other values that characterize the stress-strain state of shells are also similar. With increasing the relative thickness of the shell, values of deflection obtained with the PS theory are closer to the values obtained with the finite element method.

References 1. Miller, R.E., Shenoy, V.B.: Size dependent elastic properties of nano-sized structural elements. Nanotechnology, 11(3) 139–147 (2000) 2. Krivtsov, A.M., Morozov, N.F.: Anomalies in mechanical characteristics of nanometer-size objects. Doklady Physics, 46(11) 825–827 (2001) 3. Ankudinov, A.V., Bauer, S.M., Kashtanova, S.V., Morozov, N.F., Nyapshaev, I.A.: A study of mechanical stiffness of a solitary asbestos nanotubes (in Russian). Bulletin of Higher Educational Institutions. The North Caucasus Region. Series Natural Sciences. 7-9 (2009). 4. Ankudinov, A.V., Bauer, S.M., Ermakov, A.M., Kashtanova, S.V., Morozov, N.F.: On mechanical properties of asbestos nanotubes (in Russian). XIV International Conference Modern Problems of the Mechanics of Solid Media., Azov, Russia, 2010. Conference Proceedings, Vol. 1, Pp. 35-38. 5. Tovstik, P.E.: On the asymptotic character of the approximate models of beams, plates and shells (in Russian). Vestn. S. Peterburg. Univ., Mat. Mekh. Astron., 49, 49-54 (2007). 6. Rodionova, V.A., Titaev, B.F., Chernykh, K.F.: Applied Theory of Anisotropic Plates and Shells (in Russian). St. Peterburg University Press, St. Petersburg (1996). 7. Paliy, O.M., Spiro, V.E.: Anisotropic Shells in Shipbuilding. Theory and Analysis (in Russian). Sudostroenie, Leningrad (1977)

Chapter 35

Gauss-Codazzi Equations for Thin Films and Nanotubes Containing Defects Svyatoslav Derezin

Abstract In the present work we develop the theory of linear defects (dislocations and disclinations) in thin films and nanotubes. Particular attention is given to the geometrical side of the theory, namely, to the Gauss-Codazzi equations of strain compatibility (incompatibility) that determine in general the presence in the body of residual or eigen-strains. We obtain equivalent coordinate-free and covariant formulations of these equations. In addition some special cases of anholonomic isometric transformations of a plane into a surface with defects representing quasi-plastic or incoherent bending are considered. An elegant analogy is drawn with the equations describing steady motions of an ideal incompressible fluid with prescribed vorticity. Keywords Gauss-Codazzi equations · Dislocations · Disclinations · Thin films · Nanotubes

35.1 Introduction Recently, carbon nanostructures have attracted great attention because of their unique physical properties that are directly related to their exotic geometry. Carbon is known for its ability to assume extended two-dimensional sheet surfaces. The in-sheet bonding is extremely strong, the Young’s modulus of graphene is one of the highest values known for any material. Therefore these sheets are rather stable both as isolated objects, and also when curved into cylindrical geometries (nanotubes) or quasispherical geometries (fullerenes). In [1, 2] it was proposed that an appropriate shell theory is needed for the rigorous continuum modeling of carbon nanotubes containing defective structures. It was found both experimentally and theoretically [3], that graphene is always corrugated and covered by ripples that may S. Derezin (B) Southern Federal University, Milchakova str. 8a, 344090 Rostov on Don, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 35, © Springer-Verlag Berlin Heidelberg 2011

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S. Derezin

have intrinsic character or be induced by a roughness of substrate. Understanding the behavior of defects in various slender bodies like thin films is also critical for the improvement of the reliability of many micro-electro-mechanical systems [4]. A natural way that leads to emergence of local curvature in the hexagonal graphene lattice is substitution of some of the hexagons by pentagons (positive disclinations) or heptagons (negative disclinations). These types of topological defects was observed in all the previously existing graphene structures.

Fig. 35.1 Two different nanotubes: A) An ideal armchair nanotube B) A zigzag nanotube with two opposite disclinations

The curvature of the graphene surface in the presence of a disclination in the continuum limit has a delta-function singularity at the position of the defective ring. An edge dislocation (unfortunately, only in the linear approximation) can be formed by joining a pentagon and a heptagon through a line which will represent the Burgers vector of the dislocation (Fig. 35.1). Although a dislocation phenomenon appeared from the discrete structure of the crystal lattice, a continuum theory proved to be very effective in the modeling of imperfections in thin films [5]. It allows, for example, to smooth out singularities from isolated defects that make the elastic strain energy density non-integrable. Between the film and the substrate, stress usually arise from incompatibilities, or misfit, caused by different properties of materials. In this work we focus mainly on the geometrical side of such theory and treat simultaneously all the objects at hand as two-dimensional surfaces owing very nontrivial geometry in the sense of Riemann and Cartan. We suppose that stresses are purely elastic that means completely recoverable from strains, i.e. the only stored elastic energy depending on the two fundamental forms of the deformed shell is present. Hence, we drop the terms responsible for the self-energy of nucleated defects and the interaction energy between the applied stress and dislocation density.

35 Gauss-Codazzi Equations for Thin Films and Nanotubes Containing Defects

533

Historically, one of the first authors who studied dislocations in shell-like structures was Chernykh [6], he developed the linear theory of elastic shells with isolated defects. Some ideas of nonlinear theory of dislocations and disclinations in elastic shells in the framework of Cosserat media were presented in the work of Povstenko [7]. The paper is organized as follows. In section 35.2 we present general statements of the nonlinear shell theory aiming at a coordinate-free form of compatibility conditions. In section 35.3 we deal with the nonlinear theory of isolated and continuously distributed dislocations and disclinations in shells. Using the methods described earlier in [8, 10] it is shown that the presence of edge dislocations leads to the existence of Cartan’s torsion on the midsurface and the presence of screw dislocations leads to a nonsymmetrical second fundamental form with an additional degree of freedom being interpreted as the density of screw dislocations. In section 35.4 we deduce for the shell containing defects the full nonlinear system of the Gauss-Codazzi equations describing how internal defective geometry of the shell turns out to be compatible with the external geometry of the bulk space. In the case of inextensional isometric (generally anholonomic) deformations, i.e. when the metrics on the surface is Euclidean and there are no edge dislocations, the GaussCodazzi equations have very remarkable structure. It is possible draw an analogy with the two-dimensional steady flows of an ideal incompressible fluid with prescribed vorticity. In section 35.5 we discuss some general similarities of the developed theory that concern equations of the string theory in theoretical physics.

35.2 Equations of the Nonlinear Shell Theory In this section we will follow the lines particularly presented in the works of Zubov [9–11]. In the nonlinear theory we distinguish as usual between the reference and the actual states of the shell. We will employ the so-called direct approach to shells considering the shell as a two-dimensional material continuum, i.e. as a material surface possessing certain properties. The process of deformation is schematically depicted in Fig. 35.2. Here σ and Σ are the surfaces that correspond to the reference and deformed configurations of the shell respectively. The position of a point of σ is defined by the radius-vector r(q1 , q2 ), qα (α = 1, 2) being the Gaussian curvilinear coordinates on σ. Here and next the Greek indices will always take the values {1, 2} and the summation convention with respect to repeated indices will be adopted. Considering the coordinates qα to be the Lagrangian coordinates of the material surface, let us specify the point position on the surface Σ by the radius-vector R(q1 , q2 ), that is the position in the deformed configuration of the material point whose position was r(q1 , q2 ) in the reference configuration. Then u = R − r is the displacement vector. For the sake of compactness all basic equations are presented in two columns: the left one corresponds to the reference state, the right one to the deformed state. With each particle we associate the natural main

534

S. Derezin

Fig. 35.2 Deformation of a surface

rα =

∂r , ∂qα

∂R , ∂qα

(35.1)

Rα · Rβ = δαβ ,

(35.2)

R1 × R2 , |R1 × R2 |

(35.3)

Rα =

and reciprocal vector bases rα · rβ = δαβ , the unit normal vectors n=

r1 × r2 , |r1 × r2 |

N=

and the coefficients of the first fundamental form (surface metric tensor) gαβ = rα · rβ , g

αβ

α

β

= r ·r ,

Gαβ = Rα · Rβ . G

αβ

α

β

= R ·R .

(35.4) (35.5)

In the same way enter the Gauss-Weingarten formulae ∂rβ ν = γαβ rν + bαβ n, ∂qα ∂rβ β ·β = −γαν rν + bα n, ∂qα ∂n ·β = −bαβ rβ = −bα rβ , ∂qα

∂Rβ ν = Γαβ Rν + Bαβ N. ∂qα ∂Rβ β ·β = −Γαν Rν + Bα N. ∂qα ∂N ·β = −Bαβ Rβ = −Bα Rβ . ∂qα

(35.6) (35.7) (35.8)

ν , Γ ν are the connection coefficients; b , B Here, γαβ αβ αβ are the components αβ of the second fundamental form. In what follows we are going to proceed to


535

non-Riemannian geometry, so the order of indices is important. We try to keep notation of Lurie [13], where the first index always indicates the derivative. But in ν ν this section, in the case of defect-free geometry, we have conventionally γαβ = γβα , ν ν ν ν Γαβ = Γβα , and γαβ , Γαβ represent the coefficients of the Levi-Civita connection (Christoffel symbols). The coefficients of the second fundamental form could be 3 , B 3 viewed as bαβ = γαβ αβ = Γαβ , so their symmetry holds as well bαβ = bβα , Bαβ = Bβα . The distortion tensor in the case of shells has the form [9] C = F + n ⊗ N,

F = ∇ ⊗ R = rα ⊗ Rα .

(35.9)

Here ∇ = rα ∂/∂qα is the gradient operator on the surface σ. Let us pose the problem of finding the displacement vector u or equivalently the position vector R supposing that the reference configuration and therefore the position vector r are given. The solution can be derived via the usual line integration R = dr · F + R0 , (35.10) where R0 is some initial value of R. Prescribing R0 we exclude arbitrary rigid translations. The integral in (35.10) doesn’t depend on the curve of integration in a simply connected domain iff the following compatibility conditions are fulfilled ∇ · (e · F) = 0.

(35.11)

Here, e = −E × n = −n × E is Ricci’s permutation tensor on the surface σ, E is the 3D identity tensor. Compatibility conditions (35.11) are sometimes called the first order compatibil´ Cartan of ity conditions. Geometrically they demand that the torsion tensor of Elie the connection in the deformed configuration should vanish. Next we proceed to the second order compatibility conditions and construct a system of Pfaffian type for the distortion tensor ∂C = Π α · C, ∂qα μ μ ·μ ·μ Π α = (Γαβ − Γαβ )rβ ⊗ rμ + (Bαβ − bαβ )rβ ⊗ n − (Bα − bα )n ⊗ rμ .

(35.12) (35.13)

Here Π α are projections onto the surface of the well-known affine deformation tensor Π = rα Π α [13, 27]. The integrability conditions for (35.12) according to the Frobenius theorem [25] are Πβ Π α ∂Π ∂Π − + Π α · Π β − Π β · Π α = 0. (35.14) ∂qα ∂qβ Geometrically they postulate that the Riemann-Christoffel curvature tensor of the actual configuration should be identically zero.

536

S. Derezin

If conditions (35.14) are satisfied, (35.12) admits the following solution in terms of path-ordered exponential ∧ C=

(E + dr · Π ) · C0 ,

(35.15)

where C0 is an initial value that should forbid arbitrary rotations. It is well-known that the second order compatibility equations (35.14) are completely equivalent to the Gauss-Codazzi equations of the theory of surfaces. But, of course, the practical use of (35.14) will lead to formidable computations. The idea to simplify it applying a procedure based on the polar decomposition of the deformation gradient is due to Shield [15]. Later on the coordinate approach of Shield was generalized by Vallée [16] who did it in a coordinate-free form using for this purpose the specific tensor analysis developed mainly by R. Valid. Vallée together with Fortuné applied also the same ideas in the case of shell theory [17]. We mention here that the similar problem was basically considered among others in the works of Zubov, Pietraszkiewicz, Szwabowicz, and Vallée [11, 22–24]. Here we follow mainly the line presented in [11]. We shall consider a variant of the polar decomposition in shell theory proposed in [9] C = U · A + n ⊗ N, U2 = G, N = n · A. (35.16) Here, U is a positive definite (left) stretch tensor, A = A−T is a properly orthogonal rotation tensor. In fact A as well as C and F are two-point tensors because they use the vectorial bases defined in both configurations (actually, in two different positions of the same particle — before and after deformation). Tangential strains will be measured by the tensor G that has the structure (remember, that F = U · A) G = F · FT = Gαβ rα ⊗ rβ = G αβ rα ⊗ rβ .

(35.17)

Bending strains will be measured by the tensor P = Bαβ rα ⊗ rβ . Now the problem consists in reformulating the second order compatibility conditions (35.14) in terms of U and P. We use the first order compatibility conditions (35.11) that read as ∇ · (e · U · A) = 0. (35.18) Since A is orthogonal, we can differentiate the relationship A · AT = E with respect to qα and get ∂A T · A = −E × Lα , (35.19) ∂qα where Lα (α = 1, 2) are some vector fields on σ. In terms of Lα the Frobenius integrability conditions (35.14) could be rewritten in a very elegant vectorial form [11, 12] ∂Lα ∂Lβ − = Lα × Lβ . ∂qβ ∂qα

(35.20)


537

This form was widely used by Signorini (see, for example, [21]). According to the dissertation of Léonard-Fortuné [18] such vectorial form of compatibility conditions was already known to Darboux [19]. Employing Lα we construct a second order tensor L = rα ⊗ Lα , which may be called as a “Shield-like” tensor, because it has in principle the meaning of the third order tensor, introduced by Shield [15] for the case of 3D deformations. As it was shown in [11] this is possible to find a direct representation of L via U and P making use the following considerations. The first order compatibility condition could be rewritten in such a way ∇ · (e · U) + rα · (n × U × Lα ) = 0.

(35.21)

The further identity that involves two cross-products may be proved by straightforward computations rα · (n × U × Lα ) = −(det U)U−1 · L · n + [trUtrL − tr(U · L)]n,

(35.22)

provided that for a plane tensor M (which stands for a tensor belonging to the tangent space of σ) we have M∗ = −e · MT · e = (trM)g − M, (35.23) where M∗ is the adjoint tensor [14]. For a symmetric nondegenerate tensor U it follows then U∗ = −e · U · e = (det U)U−1 .

(35.24)

1 Here and next det U = (tr2 U − trU2 ) will be always understood in 2D sense. 2 Subsequently we find out the projection of L onto the normal n and denote it by h L · n = h = (det U)−1 U · ∇ · (e · U).

(35.25)

The Gauss-Weingarten formulae yields that Bαβ = −Rα ·

∂N . ∂qβ

(35.26)

All that enables us to get the relation for P P = −Rα ·

∂N α β r ⊗ r = −(∇⊗ N)· FT = −[∇⊗ (n· A)] · AT · U = (b + L · e)· U. (35.27) ∂qβ

And finally

L · g = M = −(P · U−1 − b) · e.

(35.28)

Here, g = E − n ⊗ n is a plane tensor which acts as identity tensor in the tangent plane, M is a plane tensor as well, b = bαβ rα ⊗ rβ is the second fundamental tensor of the surface in the reference configuration.

538

S. Derezin

The “Shield-like” tensor L is expressed through U and P in the following way (provided that b is certainly given) L = L · g + (L · n) ⊗ n = M + h ⊗ n.

(35.29)

Taking into account the well-known representation for the adjoint tensor [11, 12, 14] 1 1 L∗ = (Lα × Lβ ) ⊗ (rα × rβ ) = L2 − LtrL + (tr2 L − trL2 )E, (35.30) 2 2 the compatibility conditions (35.20) may be given the form ∇ · (e · L) + L∗ · n = 0.

(35.31)

For the tensor L, owing the structure of (35.29), the further useful formula is deduced L∗ = −(M∗ · h) ⊗ n + (det M)n ⊗ n, (35.32) that allows computation of L∗ in a straightforward way. For M∗ the relation (35.23) should be employed. Combining (35.31) and (35.32) we come to the invariant representation of the second order compatibility conditions

where

ˆ − e · h ⊗ n) + h · K ˆ · e = (det K ˆ )n, ∇ · (K

(35.33)

ˆ = −e · M = e · (P · U−1 − b) · e. K

(35.34)

Projecting (35.33) onto the unit normal n we obtain the Gauss equation and projecting (35.33) onto the tangent plane we obtain the Codazzi equations. Here we stress once more that the validity of the corresponding Gauss-Codazzi equations for gαβ and bαβ in the reference configuration of the shell was assumed. In what follows, aiming at the study of geometry of defects, we will need also the Gauss-Codazzi equations written in the traditional covariant form which as usual may be derived changing the order in the second order mixed derivatives for Rα , N and after that equating the coefficients. The Gauss equation looks like Rναβκ = −(Bνβ Bακ − Bαβ Bνκ ),

μ

Rναβκ = Rναβ· G μκ ,

(35.35)

μ

where Rναβ· are the components of the Riemann-Christoffel curvature tensor μ

μ

Rναβ· =

∂Γαβ ∂qν

μ

−

∂Γνβ ∂qα

μ

μ

λ λ + Γαβ Γνλ − Γνβ Γαλ .

(35.36)


539

By the standard symmetry arguments one may show that on the surface the Riemann-Christoffel curvature tensor has only one independent component, namely R1212 . Then the Gauss equation has the form R1212 = −(B11 B22 − B212).

(35.37)

Turning to the Codazzi equations we employ for the sake of geometrical clarity the components Rναβ· 3 of the Riemann-Christoffel curvature tensor in some enveloping Euclidean space. They depend of course not only on intrinsic geometry of the surface, but also on the coefficients of the second fundamental form. Rναβ· 3 =

∂Bαβ ∂q

ν

−

∂Bνβ ∂q

α

μ

μ

+ Γαβ Bνμ − Γνβ Bαμ = 0.

(35.38)

If we introduce the important notion of covariant derivative in the actual configuration compatible with the Gauss-Weingarten formulae (35.6)-(35.8) of a tensor field Aκλ··· αβ··· defined on the surface Σ, i.e. that is valid as well for nonsymmetric connections [27] ∇ν Aκλ··· αβ··· =

∂Aκλ··· αβ··· ∂qν

μ

μ

μλ···

κμ···

κλ··· κ λ − Γνα Aκλ··· μβ··· − Γνβ Aαμ··· − · · · + Γνμ Aαβ··· + Γνμ Aαβ··· + . . . , (35.39)

then the Codazzi equations may be written in a shorter form [22, 37] ∇ν Bαβ = ∇α Bνβ .

(35.40)

Now again, having in mind the symmetries of the curvature tensor, we present the only two independent equations ⎧ ⎪ ⎪ ⎨ ∇2 B11 = ∇1 B12 , (35.41) ⎪ ⎪ ⎩ ∇2 B12 = ∇1 B22 . ν and nonsymmetric coefficients B Note, that for nonsymmetric connections Γαβ αβ the transition from (35.38) to (35.40) or (35.41) is not true. The geometrical relations presented above should be certainly accomplished by statical considerations. As was already mentioned in the Introduction it is reasonable to suggest that the strain energy W of the deformed shell depends only on G αβ and Bαβ . Doing so, we arrive at the Kirchhoff-Love shell model [10]. For that model the equilibrium equations have the form

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S. Derezin β

β

∇α (ναβ − μαδ Bδ ) − Bδ ∇α μαδ + F β = 0, ∇α ∇β μαβ + Bαβ (ναβ − Bαδ μδβ ) + F = 0, 9 9 G αβ ∂W G αβ ∂W χ ν =2 , −χ μ = , g ∂G αβ g ∂Bαβ ⎧ ⎪ ⎪ ⎨ 1, α = β, 2 2 G = G 11G22 − G12 , g = g11 g22 − g12, χ = ⎪ ⎪ ⎩ 2, α β.

(35.42) (35.43) (35.44) (35.45)

Here, ναβ are the components of the tensor of membrane stresses, μαβ are the components of the tensor of bending couples, F β , F are the surface densities of body forces.

35.3 Dislocations and Disclinations in Shell-Like Structures Let us consider the case when the surfaces σ and Σ turn to be multiply-connected. Then the solution in the form of line integral R = dr · F + R0 (35.46) is still valid, but now, because of the non-trivial topology (Fig. 35.3), R has to be a multivalued function. This property implies in general the presence in the body of translational dislocations, each of which is characterized by the Burgers vector ? b N = dr · F (N = 1, 2, ..., N0). (35.47) γN

Fig. 35.3 Line integration on the multi-connected reference surface


541

Here, γN is a simple closed contour around the axis of the Nth dislocation. The total Burgers vector of a discrete set of N0 dislocations is given by B=

N0 ?

? dr · F =

N=1 γ N

dr · F,

(35.48)

γ0

where γ0 is some contour enclosing the lines of all N0 dislocations. Following [8], we transition from the discrete set of dislocations to their continuous distribution, regarding the domain σ to be simply connected and the tensor field F continuously differentiable. Then, transforming the integral in (35.48) according to Stokes’ theorem, we get ? B=

dr · F =

∇ · (e · F) dσ

γ0

(35.49)

σ0

Here, σ0 is the surface spanned over γ0 . Relationship (35.49) allows one to introduce the vectorial density of continuously distributed dislocations α as a vector field, whose flux through any arbitrary surface yields the total Burgers vector for dislocations intersecting this surface: ∇ · (e · F) = α .

(35.50)

Projecting α onto the tangent plane and the normal N it is possible to decompose it into two parts α = αν Rν + αN. (35.51) Here, αν (ν = 1, 2) are the densities of edge dislocations, α is the density of screw dislocations. Let us briefly run through disclinations — the most important defects in carbon structures because of their hexagonal lattice. In the multi-connected domain σ the solution to equations (35.12) is multivalued as well. This property can be removed by cutting the surface along some curves, which turn the multi-connected surface into a simply connected one. One may prove that the values C± of the distortion tensor on the opposite borders of the Nth cut are related by the jump C+ = C− · Φ N

where ΦN =

C−1 0 ·

(N = 1, 2, ..., N0),

(35.52)

(E + dr · Π ) · C0

(35.53)

?∧ γN

is a curvilinear multiplicative integral [10]. In view of continuity of the quantities G αβ and Bαβ , and from the polar decomposition of the surface distortion tensor (35.16), we obtain that the tensors Φ N are proper orthogonal Φ N · Φ TN = E,

Φ N = 1. detΦ

(35.54)

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S. Derezin

The transition to continuous distribution of disclinations is hampered by the fact of noncommutativity of tensors Φ N . In the framework of nonlinear plane strain, however, the finite rotations commute, and it was done in the paper [8]. Nevertheless, the possible application of the noncommutative Stokes’ theorem for producttype integrals (35.53) invented by Schlesinger [26] may help to resolve rigorously this problem. In the case of linear theory such transition is of course straightforward.

35.4 Gauss-Codazzi Equations in Presence of Defects Let’s analyze which consequences for the Gauss-Codazzi equations brings the presence of dislocations. The first order compatibility conditions (35.11) should be replaced by relation (35.50) transforming it into the so-called incompatibility conditions. Using the Gauss-Weingarten formulae (35.6)–(35.8) we compute ν ∇ · (e · F) = eαβ Γαβ Rν + eαβ Bαβ N,

where αβ

e

εαβ = √ , g

⎛ ⎞ ⎜⎜⎜ 0 1⎟⎟⎟ ⎟⎟⎟ . εαβ = ⎜⎜⎜⎝ −1 0⎠

(35.55)

(35.56)

It means that from now on the deformed configuration represents a ν and B non-Riemannian space (or a Riemann-Cartan manifold) with Γαβ αβ being not symmetric in the lower indices. The decomposition (35.51) leads to ν αν = eαβ Γαβ ,

α = eαβ Bαβ ,

(35.57)

or more concretely 1 1 1 α1 = √ (Γ12 − Γ21 ), g

1 2 2 α2 = √ (Γ12 − Γ21 ), g

1 α = √ (B12 − B21). g

(35.58)

So the lack of symmetry is caused by the presence of dislocations. Later on we will discuss this topic more precisely in terms of the torsion tensor introduced by ´ Cartan. Elie Now we proceed to a modification of the second order compatibility conditions (35.33) brought by dislocations. Instead of (35.18), (35.21), (35.25) one should use ∇ · (e · U · A) = α ,

(35.59)

∇ · (e · U) + rα · (n × U × Lα ) = α · AT ,

(35.60)

−1

h = (det U) [U · ∇ · (e · U) − α 0 ].

(35.61)

It is not difficult to prove that all the other equations remain in principle unaltered. This means that equation (35.33) together with (35.34) serve as the second order


543

compatibility (incompatibility) conditions for the shell with defects as well. Here, α 0 = α · FT represents the densities of edge dislocations only. The density of screw dislocations is hidden in the structure of tensor P = Bαβ rα ⊗ rβ being not symmetric due to the lack of symmetry in Bαβ . For the case of 3D nonlinear elasto-plasticity with dislocations such type of equations was presented in the work [20]. In the case of nonlinear plane strain, i.e. when bαβ = Bαβ = 0 we again refer the reader to the paper [8]. In order to clarify the geometrical picture we employ once more the traditional ν may be first decomposed into symmetrical covariant approach. The connection Γαβ and antisymmetrical parts ν ν ··ν Γαβ = Γ¯ αβ + S αβ ,

1 ν ν ν ν Γ¯ αβ = Γ(αβ) = (Γαβ + Γβα ), 2

1 ν ··ν ν ν S αβ = Γ[αβ] = (Γαβ − Γβα ), 2

(35.62) (35.63)

··ν ´ Cartan. In virtue of (35.57) it is completely where S αβ is the torsion tensor of Elie determined by the density of edge dislocations

1 ··ν S αβ = eαβ αν . 2

(35.64)

A straightforward application of such decomposition is not easy because the symν depends on the dislocation density as well. That’s why we introduce metric part Γ¯ αβ the more advanced decomposition ν ν ··ν Γαβ = Hαβ + T αβ , ν Hαβ

1 νμ ∂Gβμ ∂Gαμ ∂G αβ = G + − , 2 ∂qα ∂qμ ∂qβ

··ν ··ν ·ν ν T αβ = S αβ − S β·α + S ·αβ .

(35.65) (35.66)

ν are the coefficients of the Levi-Civita connection, T ··ν is the so-called Here, Hαβ αβ contortion tensor. A completely analogous procedure to separate the second fundamental form leads to

Bαβ = Kαβ + Qαβ , 1 Kαβ = B(αβ) = (Bαβ + Bβα ), 2

1 Qαβ = B[αβ] = (Bαβ − Bβα ), 2

(35.67) (35.68)

where the antisymmetric part Qαβ due to (35.57) is completely determined by the density of screw dislocations 1 Qαβ = eαβ α. (35.69) 2 From the Gauss-Weingarten formulae (35.6)–(35.8) it follows that

544

S. Derezin

∇μG αβ =

∂G αβ ν ν − Γμα Gνβ − Γμβ G αν = 0. ∂qμ

(35.70)

This means that Ricci’s lemma which is valid in the context of Riemannian geometry still holds in our situation for the deformed surface with distributed defects. So it gives us possibility to interpret the actual configuration of the shell as the metrically connected space V2 [27]. In contrast to 3D theory where some limitations are imposed on the dislocation density the Riemann-Cartan curvature tensor of V2 has the same number of symmetries as the Riemann-Christoffel tensor. Using decomposition (35.65) one may write μ μ μ Rναβ· = Kναβ· + Aναβ· , (35.71) μ

μ

∂Hαβ

μ

∂Hνβ

μ

μ

λ λ + Hαβ Hνλ − Hνβ Hαλ ,

(35.72)

μ ··μ ˆ α T ··μ + T ··λ T ··μ − T ··λ T ··μ . Aναβ· = ∇ˆ ν T αβ − ∇ αβ νλ νβ αλ νβ

(35.73)

Kναβ· =

∂qν

−

∂qα

μ

ν Here Kναβ· is the curvature tensor of the Levi-Civita connection Hαβ depending on μ the metrics Gαβ solely via (35.66), whereas Aναβ· depends on the dislocation denˆ ν is calculated with respect to the connection sity as well. The covariant derivative ∇ ν according to (35.39) (with Γ ν being replaced by H ν ). Hαβ αβ αβ Therefore the structure of the Gauss equation (35.37) remains basically unaltered with the minor exclusion that B12 B21.

R1212 = −(B11 B22 − B12 B21).

(35.74)

For the Codazzi equations the situation is a little bit different, since as was already mentioned the transition from (35.38) to the short form (35.39) is no more valid for ν and B . Thus we should substitute representations the case of nonsymmetric Γαβ αβ (35.65) and (35.67) directly into (35.38) and get Rναβ· 3 = Kναβ· 3 + Aναβ· 3 = 0, Kναβ· 3 =

∂Kαβ ∂q

ν

−

∂Kνβ ∂q

α

μ

(35.75) μ

+ Hαβ Kνμ − Hνβ Kαμ ,

ˆ α Qνβ + T ··μ (Kνμ + Qνμ ) − T ··μ (Kαμ + Qαμ ). Aναβ· 3 = ∇ˆ ν Qαβ − ∇ αβ νβ

(35.76) (35.77)

Let us consider a special case of isometric (generally anholonomic) transformations, i.e. when there are no strains at all. Such deformation may be called as the quasi-plastic [28] or incoherent [29] bending. We suppose also that in the initial state the shell is flat (bαβ = 0), then globally defined Cartesian coordinates (x1 , x2 ) may be introduced and gαβ = δαβ , G αβ = δαβ , (35.78) ν γαβ = 0,

ν Hαβ = 0.

(35.79)


545

The quantities responsible for bending will be a = a0 + a3 n,

a0 = a1 i1 + a2i2 ,

(35.80)

where a1 , a2 are the densities of edge dislocations; a3 is the density of screw dislocations. Note, that the equilibrium equations (35.43), (35.44) with no body forces applied are satisfied identically, so the Gauss-Codazzi equations serve as the only equations that would determine the state of the shell. The remaining nontrivial field is the finite rotation field A. One could find it resolving, for example, the first order incompatibility condition (35.59). Now the Gauss-Codazzi system has the following structure ⎧ ⎪ ⎪ ⎨ det K = ∇ · e · a0 − a23 , (35.81) ⎪ ⎪ ⎩ ∇ · (e · K · e) + a0 · e · K = a3 a0 − e · ∇a3 . The interesting nonlinear effect is observed — the densities of dislocations appear in a product form. ⎧ ⎪ ⎪ 2 = ' ∂a2 − ∂a1 ( − a2 , ⎪ ⎪ K11 K22 − K12 ⎪ 3 ⎪ ⎪ ∂x1 ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂K ∂K ∂a3 ⎨ 12 22 − + a1 K12 − a2 K11 = a1 a3 − , (35.82) ⎪ ⎪ ⎪ ∂x2 ∂x1 ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂K12 ∂K11 ∂a3 ⎪ ⎪ ⎪ ⎪ ⎩ ∂x − ∂x + a1 K22 − a2 K12 = a2 a3 + ∂x . 1

2

1

Suppose furthermore that edge dislocations are absent and put a = a3 . Then the system (35.82) reduces to ⎧ ⎪ 2 ⎪ K11 K22 − K12 = −a2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂K12 − ∂K22 = − ∂a , ⎨ (35.83) ⎪ ∂x2 ∂x1 ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂K ∂K ∂a ⎪ 12 11 ⎪ ⎪ ⎪ ⎩ ∂x1 − ∂x2 = ∂x1 . Or in invariant form

⎧ ⎪ ⎪ ⎨ det K = −a2 , ⎪ ⎪ ⎩ ∇ · (e · K · e) = −e · ∇a.

(35.84)

The Codazzi equations admit a straightforward integration K = −ae + ∇V.

(35.85)

Here the vector function V = V1 i1 + V2 i2 , as we will see later, plays the role analogous to the velocity vector in fluid mechanics. The functions V1 and V2 are not

546

S. Derezin

independent, since due to the symmetry of K = KT they must satisfy ∇·e·V =

∂V2 ∂x1

−

∂V1 ∂x2

= −2a.

(35.86)

Inserting representation (35.85) into the Gauss equation we get ∂V1 ∂V2 ∂x1 ∂x2

−

∂V1 ∂V2 ∂x2 ∂x1

= J(V1 , V2 ) = 0,

(35.87)

where J(V1 , V2 ) is the Jacobian of V1 and V2 . Therefore we see that the Gauss equation in addition to the constraint (35.86) implies a functional dependence between V1 and V2 . Equation (35.87) has a very closed relationship to the well-known MongeAmpère equation. Indeed, if we make a substitution V1 =

∂ψ ∂x2

,

V2 = −

∂ψ ∂x1

,

(35.88)

with ψ being analogous to the stream function, and the screw dislocation density a playing the role of vorticity function [30], then the Gauss-Codazzi equations may be given the form ⎧ ⎪ ⎪ [ψ, ψ] = 0, ⎨ (35.89) ⎪ ⎪ ⎩ Δψ = −2a. ∂2 ψ ∂2 ψ % ∂2 ψ &2 − is the Monge-Ampère operator. ∂x1 ∂x2 ∂x21 ∂x22 System (35.89) contains the homogeneneuos Monge-Ampère equation defining the Gaussian curvature of the stream function surface [31] and the Poisson equation defining the mean curvature of that surface. Hence, from the point of view of differential geometry (35.89) defines a developable surface. In fluid mechanics there exists an example of a flow connected with such kind of surfaces. This is the parallel shear flow with prescribed vorticity [31]. Integration procedure of (35.84) has lead us to the transformations (35.85), (35.88) specifying a new surface that could be easily visualized. This is not true for the surface described by the initial system (35.84), since it defines a surface owing nontrivial Riemann-Cartan geometry and located in general in a non-Riemannian enveloping space. According to the algebraic arguments of Kleinert [34] the usual 3D Euclidean space is large enough to envelop a two-dimensional non-Riemannian manifold, but the standard theorem of non-Riemannian geometry [35] says that if the torsion tensor vanishes in the whole bulk space it should immediately vanish on each subspace. Thus, the problem of direct visualizing the surfaces defined by (35.81) or (35.84) is still open. It is known [37] that in the Riemannian case this is possible to reduce the GaussCodazzi equations to a single nonlinear Liouville equation that may be coupled (in the presence of some more extra-dimensions) with a system of two nonlinear Here [ψ, ψ] =


547

equations of the Lund-Regge type. In the non-Riemannian case a lot of work has been done in [32,33], where it was shown that the Gauss-Codazzi system of general type may be reformulated as a completely integrable system.

35.5 Conclusions It has been pointed by many authors [3, 36] that equations describing defects in solid continua show striking similarity with those of the string theory [37, 38]. The geometrical approach to the quantum theory of relativistic string is based on the description of a world surface by differential forms that satisfy integrability conditions in the form of strongly nonlinear PDEs due to Gauss, Codazzi and Ricci. Therefore this means that particular structure of the curved 2D crystal lattice sets a nice field to study and unify different concepts from elasticity, topology and cosmology. For example, for a nanotube with an equal number of pentagon and heptagon defects that keep the samples flat in average it is possible to use the metrics from the solution of the cosmic string equations [3]. Acknowledgements The author is indebted to Prof. Leonid M. Zubov for stimulating discussions and valuable comments. This work was supported by the Russian Foundation for Basic Research (grant N 09-01-00459) and the Federal target programme “Research and Pedagogical Cadre for Innovative Russia” for 2009-2013 years (state contract N P596).

References 1. Kolesnikova, A.L., Romanov, A.E.: A disclination-based approach to describing the structure of fullerenes. Physics of the Solid State 40(6), 1075–1077 (1998) 2. Romanov, A.E., Sheinerman, A.G.: Energy of deformed and defective carbon clusters. Physics of the Solid State 42 (8), 1569–1574 (2000) 3. Vozmediano, M.A.H., Katsnelson, M.I., Guinea, F.: Gauge fields in graphene. Physics Reports 496, 109-148 (2010) 4. Oswald, J., Gracie, R., Khare, R., Belytschko, T.: An extended finite element method for dislocations in complex geometries: Thin films and nanotubes. Comput. Methods Appl. Mech. Engrg. 198, 1872–1886 (2009) 5. Freund, L.B., Suresh, S.: Thin Film Materials: Stress, Defect Formation and Surface Evolution. Cambridge University Press, Cambridge (2003) 6. Chernykh, K.F.: Relation between dislocations and concentrared loadings in the theory of shells. PMM U.S.S.R. 49(6), 359–371 (1959) 7. Povstenko, Yu.Z.: Continuous theory of dislocations and disclinations in a two-dimensional medium. PMM U.S.S.R. 49(6), 782–786 (1985) 8. Derezin, S.V., Zubov, L.M.: Equations of a nonlinear elastic medium with continuously distributed dislocations and disclinations. Dokl. Phys. 44(6), 391–394 (1999) 9. Zubov, L.M.: Methods of Nonlinear Elasticity in Shell Theory (in Russian). Izd-vo Rost. Un-ta, Rostov on Don (1982) 10. Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997)

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11. Zubov, L.M.: A new form of compatibility conditions in the nonlinear theory of elastic shells (in Russian). Izv. Vyssh. Uchebn. Zaved., Sev.-Kav. Region., Est. nauki. N 3, 64–65 (2000) 12. Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moscow (2008) 13. Lurie, A.I.: Nonlinear Theory of Elasticity. North-Holland, Amsterdam (1990) 14. Podio-Guidugli, P.: A primer in elasticity. Journal of Elasticity 58, 1–104 (2000) 15. Shield, R.T.: The rotation associated with large strains. SIAM J. Appl. Math. 25(3), 483–491 (1973) 16. Vallée, C.: Compatibility equations for large deformations. Int. J. Engng Sci. 30(12), 1753– 1757 (1992) 17. Vallée, C., Fortuné, D.: Compatibility equations in shell theory. Int. J. Engng Sci. 34(5), 495– 499 (1996) 18. Léonard-Fortuné, D.: Conditions de compatibilité en mécanique des solides - Méthode de Darboux. Thése, L’Université de Poitiers (2008) 19. Darboux, G.: Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal. In 4 vol., Paris, Gauthier-Villars (1887-1896) 20. Cleja-Tigoiu, S., Fortuné, D., Vallée, C.: Torsion equation in anisotropic elasto-plastic materials with continuously distributed dislocations. Math. Mech. Sol. 13, 667–689 (2008) 21. Signorini, A.: Trasformazioni termoelastiche finite. Ann. Mat. Pura Appl. 22, 33–143 (1943) 22. Szwabowicz, M.L., Pietraszkiewicz, W.: Determination of the deformed position of a thin shell from surface strains and height function. Int. J. Non-Lin. Mech. 39, 1251–1263 (2004) 23. Pietraszkiewicz, W., Vallée, C.: A method of shell theory in determination of the surface from components of its two fundamental forms. Z. Angew. Math. Mech. 87, 603–615 (2007) 24. Pietraszkiewicz, W., Szwabowicz, M.L., Vallée, C.: Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition. Int. J. Non-Lin. Mech. 43, 579–587 (2008) 25. Narasimhan, R.: Analysis on Real and Complex Manifolds. North-Holland, Amsterdam (1968) 26. Schlesinger, L.: Parallelverschiebung und Krümmungstensor. Math. Ann. 99, 413–434 (1928) 27. Norden, A.P.: Affinely Connected Spaces (in Russian). Nauka, Moscow (1976) 28. de Wit, R.: A view of the relation between the continuum theory of lattice defects and nonEuclidean geometry in the linear approximation. Int. J. Engng Sci. 19(12), 1475–1506 (1981) 29. Srolovitz, D.J., Safran, S.A., Tenne, R.: Elastic equilibriun of curved thin films. Phys. Rev. E 49(6), 5260–5270 (1994) 30. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002) 31. Yamasaki, K., Yajima, T., Iwayama, T.: Differential geometric structures of stream functions: incompressible two-dimensional flow and curvatures. J. Phys. A: Math. Theor. 44, 155501 (2011) 32. Savel’ev, M.V.: Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms. Theor. Math. Phys. 60(1), 638–647 (1984) 33. Gabeskiriya, M.A.: Lax type representation for embeddings of manifolds in non-Riemannian enveloping spaces. Theor. Math. Phys. 65(2), 1088–1091 (1985) 34. Kleinert, H.: Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation. World Scientific, Singapore (2008) 35. Eisenhart, L.P.: Non-Riemannian Geometry. AMS, New York (1927) 36. Katanaev, M.O.: Polynomial Hamiltonian form of general relativity. Theor. Math. Phys. 148(3), 1264–1294 (2006) 37. Barbashov, B.M., Nesterenko, V.V.: Introduction to Relativistic String Theory. World Scientific, Singapore (1990) 38. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. In 2 vol. Cambridge Uiversity Press, Cambridge (1986)

Chapter 36

Effective Mechanical Properties of Closed-Cell Foams Investigated with a Microstructural Model and Numerical Homogenisation Nina-Carolin Fahlbusch and Wilfried Becker

Abstract With their cost-effective production and advantageous properties (e.g. low density, low thermal conductivity, high specific stiffness) foams are an attractive material for an increasing number of applications. In this work the mechanical behaviour of closed-cell foams is analysed numerically. Besides the better understanding of the mechanisms of deformation and failure, the identification of the components of the effective elasticity tensor is the major aim of this study. Since the mechanical properties depend on the cellular microstructure, a representative volume element (RVE) of that microstructure is investigated. In an idealized manner a tetrakaidecahedral foam microstructure is considered and implemented in a finite element routine with periodical boundary conditions and a strain-energy based homogenisation concept is utilized. In this concept it is assumed that a macroscopically equivalent deformation state leads to the same strain energy in a representative volume element as in a homogenous “effective” medium with yet unknown properties [8]. The effect of imperfections, such as curved cell walls and geometry irregularities, on the effective mechanical properties is investigated and microbuckling instabilities of the cell-walls are discussed. The results are compared with literature and experimental data. Keywords Closed-cell foam · Homogenisation · Imperfections

36.1 Introduction Foams are well suited for a wide variety of applications and fields such as aerospace engineering, automotive industry and lightweight construction. Accordingly it is necessary to understand the mechanical behaviour of the cellular materials. A N.-C. Fahlbusch (B) · W. Becker Fachgebiet Strukturmechanik, TU Darmstadt, Hochschulstraße 1, D-64289 Darmstadt, Germany e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 36, © Springer-Verlag Berlin Heidelberg 2011

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N.-C. Fahlbusch and W. Becker

better understanding also allows for enhanced material design. The focus of this work is to identify the basic mechanisms of deformation and to assess the influence of geometry imperfections, which are curved cell walls and a disorder in the cell geometry. Therefore a finite element (FE) model was implemented. With a corresponding effort in principle it is possible to generate a 3D solid model by FE methods on the basis of volume image analysis [15]. These models reproduce exactly the observed cellular microstructure and enable an analysis of the effective behaviour of a foam. The influence of the imperfections, however, is very difficult to control and this method is numerically very expensive. To simplify the model, irregularities in the geometry are studied with a periodically repeating unit cell as an idealized foam structure. A literature review shows that several space-filling unit cells are available, e.g. the geometry developed by Weare and Phelan and by Kelvin [4]. Following many examples in literature (e.g. [7, 10, 11]), this study is based on the tetrakaidecahedron, a polyhedron existing of eight hexagonal and six square faces (Fig. 36.1). This geometry is easy to handle and has a relatively little anisotropy w.r.t. elastic modulus. The stiffness anisotropy is less than 10% for a foam with a relative density less than 0.2 [11]. In the past open-cell foams have been investigated thoroughly (e.g. [2, 12]) and the influence of many imperfections on the elastic behaviour of closed-cell foams has been discussed. Among them are Grenstedt’s study about the influence of cell wall waviness and cell wall thickness variations on elastic stiffness and research about the interaction between imperfections [5–7]. Simone and Gibson also researched the effects of solid distribution between the cell faces and edges [11]. The present work deals with the material ROHACELL® of the Evonik Röhm GmbH, Germany. It is a high stiffness foam based on polymethacrylimide (PMI) chemistry and has a fine, homogeneous and isotropic, 100% closed-cell structure [3]. Evonik specifies ROHACELL® as a structural foam with cell walls of constant thickness and no material concentrations at the nodes. In the following the influence of curved cell walls, specifically of curved single square and hexagonal faces and eight hexagonal faces, and geometry irregularities is studied. This paper is arranged as follows: Sect. 36.2 is devoted to the concept of the finite element analysis. In Sect.36.3 the results of the simulations are summarized.

Fig. 36.1 Idealized foam structure consisting of tetrakaidekahedrons

36 Effective Mechanical Properties of Closed-Cell Foams

551

The effects of the imperfections on the elastic stiffness are presented, as well as the results of a linear buckling analysis. The paper concludes in Sect. 36.4, where a summary is given.

36.2 Finite Element Analysis The representative volume element (RVE) was formed by one tetrakaidecahedral cell and parts of the square faces of adjacent cells (Fig. 36.2). While three square faces were modelled, the opposite square faces were left out, so that a periodic repetition of the unit cell is possible. In Table 36.1 the effective properties of ROHACELL® 51WF and 200WF specified by Evonik are listed [3]. Since no material data for Poisson’s ratio measured by Evonik are available, the results of experimental tests performed by Kraatz are used as reference data for Poisson’s ratio [9]. Kraatz quantified Poisson’s ratio as 0.32 for R51WF and 0.38 for R200WF in case of tension. Wang et al. specify 0.375 as a reference value for Poisson’s ratio for both foams and allude to the difficulties to measure Poisson’s ratio of foam materials [14].

8

4

7 3

1 Fig. 36.2 Representative volume element consisting of one tetrakaidecahedral cell and parts of the square faces of adjacent cells

6 z

y x

2

Table 36.1 Properties ROHACELL® WF measured by Evonik Röhm GmbH [3] Properties

R51WF

R200WF

Standard

Density

52 kg/m3

205 kg/m3

ISO845

Young’s modulus

75 MPa

350 MPa

ISO527-2

Shear modulus

24 MPa

150 MPa

DIN 53294

552


Whereas the elastic moduli are used for the validation of the results, the density is utilised to calculate the face thickness. All faces are assumed to have the same thickness. The edge length l of the tetrakaidecahedral geometry was identified on the basis of micro-CT-images. Following former investigations the relative density is assumed to be a significant parameter [4]. Consequently with these two data sets and the assumption of the same relative density of the real foam and the FE-model, the cell face thickness is calculated, namely for R51WF a thickness of 7.2 μm and for R200WF of 22.26 μm. Simulations carried out with models of constant relative density and variable edge length, as well as variable face thickness, show only a small influence of the edge length to the effective properties. The finite element analyses were performed with the FE software ABAQUS, while the MATLAB (MathWorks, Natick, Massachusetts) package was used to generate the input files. The output data were analysed using the ABAQUS Python development environment. As elements 4-node doubly curved general-purpose shell elements from the ABAQUS element library (called S4) were selected (cp. [1]). Only at the intersection of the considered tetrakaidecahedron with the adjacent cells 3-node triangular general-purpose shell elements (called S3) were used. On the basis of refinement tests of the mesh up to 755 000 degrees of freedom a suited mesh was chosen. The material model was isotropic. The matrix material (polymethacrylimide) was assumed to have an elastic modulus of E = 6480.0 MPa, a Poisson’s ratio of ν = 0.3 and a density of ρ = 1400.0 kg/m3 [9]. The periodic boundary conditions were defined using multi-point constraints. Therefore nodes at the corners were introduced. Four of the nodes were declared as master nodes (node 1, 2, 4, 5 in Fig. 36.2), while the others were constrained by the boundary conditions. For the simulation of a tension and shear test only the displacements at the master nodes had to be defined (cp. [2]).

36.3 Results In the following section the results calculated by the idealized model without any irregularity act as reference data. The finite element analysis gives a Young’s modulus for ROHACELL® 51WF of E0 = 78.9 MPa, a shear modulus of G0 = 32.5 MPa and a Poisson’s ratio of ν0 = 0.33. For R200WF the determined values are E0 = 316.3 MPa, G0 = 136.3 MPa and ν0 = 0.32. These results are in the same order as the experimental data (Table 36.1). Possible reasons for differences between the real foam and the FE analysis are the presumed material data for the matrix material, for example Wang et al. assume different data for the matrix material [14]. On the other hand it can be due to material concentrations at the cell edges, the so-called Plateau borders. Micro-CT-images presented by Kraatz show slight material concentrations [9]. Simone and Gibson display the dependence of the effective foam properties on the fraction of solid in the Plateau borders subjected to the relative density [11].


553

According to the imperfections and the simulated applied strain deformation, different positions of the imperfections are possible in the performed modelling. For example in the case of uniaxial strain a curved square face is conceivable to be in the orthogonal or tangential plane to the deformation axis. Consequently for every imperfection six simulations are carried out. Three of them are based on the prescribed normal strains εx , εy , εz and the others on the given shear strains γ xy , γ xz and γyz . In the following the averages of the elastic moduli resulting from the six simulations are presented.

36.3.1 Imperfections 36.3.1.1 Curved Cell Walls To analyse the effect of a wavy cell wall on the effective mechanical properties, first the geometry of a curved cell wall has to be determined. In order to do this, the cell wall is separated from the RVE. Then a fictitious pressure is applied to the cell wall, while keeping the edge nodes clamped. Afterwards the resultant displacement field is used to generate the curved cell wall. To compare the data for different materials the displacement normal to the face of the middle node of the curved cell wall is normalized to the edge length (un /l), also the elastic properties are normalized to the properties of the reference model (E/E0 , G/G 0 and ν/ν0 ). Fig. 36.3 shows the effect of one curved hexagonal face on the elastic stiffness as a function of the normalized displacement of the middle node and the relative density. It is obvious, that this imperfection has a larger influence to Young’s modulus of ROHACELL® 51WF than of 200WF. This is also valid for the shear modulus G. The effect of eight curved hexagonal faces on the elastic properties is depicted in Fig. 36.4. A significant reduction of the stiffness is observed. That means Young’s modulus of ROHACELL® 51WF decreases by 30.6% and the imperfection causes a reduction of the shear modulus G of 46.5% for un /l=0.25. The elastic moduli of the rigid foam R200WF are reduced by less than 12.7%. The influence of one curved square face is represented in Fig. 36.5. Whereas a decrease of at the most 7.2% in Young’s modulus for ROHACELL® 51WF and un /l = 0.25 is observed, the shear modulus is reduced by only less than 0.4%. For the studied imperfections the material R51WF is significantly more sensitive to imperfections than ROHACELL® 200WF. Fig. 36.3-36.5 also show diagrams, that reveal the effect of the imperfections on Poisson’s ratio. According to the experimental procedure ν is defined by values generated by the simulation of a uniaxial tension test, that are the components C11 and 12 C12 of the elasticity tensor (ν = C C+C ). For example, Fig. 36.3(d) shows the effect 11 12 of the imperfection on the elastic components in case of a uniaxial deformation. The different sensitivity of the components to the curvature causes the behaviour of Poisson’s ratio. As expected, the tension stress in the direction of the deformation

554


(a)

(b)

(c)

(d)

Fig. 36.3 Sensitivity of (a) Young’s modulus, (b) the shear modulus, (c) Poisson’s ratio and (d) the elastic components to one curved hexagonal face. The elastic properties are normalized by values determined by the reference model without any irregularity ()0

axis seems to be sustained by all “perfect” faces, whereas the compression stress leads to especially increased deformation at the curved cell wall.

36.3.1.2 Random Geometrical Irregularity As a second kind of imperfection, a geometrical irregularity of one corner node of the tetrakaidecahedron, was investigated. Therefore a random generator was implemented to create a random displacement of one corner node in a volume of 10% of the edge length. To guarantee the option of repetition of the unit cell, the opposite node was moved equally.


(a)

555

(b)

(c) Fig. 36.4 Sensitivity of (a) Young’s modulus, (b) the shear modulus and (c) Poisson’s ratio to eight curved hexagonal faces. The elastic properties are normalized by values determined by the reference model without any irregularity ()0

Of course, a single simulation does not allow a generalised conclusion. Consequently the average of 100 FE analyses is calculated and analysed. Then for example Young’s modulus is defined as 1 (k) E¯ = E , n n

k=1

where n is the number of simulations and E (k) the elastic modulus of the kth simulation. The obtained average of Young’s modulus is reduced by 1.9% in case of foam R51WF and by 0.2% in case of R200WF. Whereas the shear modulus of R51WF decreases by 1.8%, the imperfection causes a reduction of the shear modulus of

556


(a)

(b)

(c) Fig. 36.5 Sensitivity of (a) Young’s modulus, (b) the shear modulus and (c) Poisson’s ratio to one curved square face. The elastic properties are normalized by values determined by the reference model without any irregularity ()0

0.1% for R200WF. The results show, that the stiffness is not particularly sensitive to this imperfection. Results of single simulations lead to 4.0% decreased elastic properties. The average of Poisson’s ratio is almost not affected by this imperfection. That means the average is less than 0.1% reduced. As seen above, ROHACELL® 51WF shows a higher sensitivity in regard of defects.

36.3.2 Linear Buckling Analysis In order to assess the effective strength of the foams, some stability analyses have been conducted. To estimate the critical load, a linear buckling analysis was


557

Fig. 36.6 Enlarged representative volume element for the linear buckling analysis

considered. In addition to the eigenvalues, the eigenvectors, which represent the buckling mode shapes, are provided. Uni-, bi- and triaxial compression tests and a shear test were carried out. Demiray’s investigation of open-cell foams shows buckling modes related to several cells [2]. Accordingly, a model consisting of eight above-described RVEs was produced (Fig. 36.6) and the linear buckling analysis was performed. A similar model was generated by Grenstedt et al. [6, 7]. Its volume equates the volume of sixteen uniform-sized tetrakaidecahedral cells on a regular lattice. The mode shapes show no difference between the cells and the mode shape of every cell is the same as the mode shape of the one-cell model. Consequently it can be assumed that no buckling modes related to several cells exist and the investigation can be reduced to the one-cell model. In Fig. 36.7 some examples of buckling mode shapes are shown. In the case of compression there are no significant differences between the buckling modes of ROHACELL® 51WF and 200WF. Buckling occurs at the hexagonal faces. Four

(a)

(b)

(c)

Fig. 36.7 Buckling mode shapes: (a) R200WF, biaxial compression test (b) R200WF, shear test (c) R51WF, shear test

faces buckle inwards and the others outwards. Buckling of the square faces can be first determined with higher eigenvalues. The lowest critical macroscopic strains εi j are given in Table 36.2.

558


Table 36.2 Critical strain Simulated test uniaxial compression ε11 = ε

krit

biaxial compression ε11 = ε22 = ε

krit

triaxial compression ε11 = ε22 = ε33 = ε shear test γ12 = γ21 = γ

krit

krit

R51WF

R200WF

2.64 · 10−3

3.52 · 10−2

1.33 · 10−3

1.77 · 10−2

−3

0.89 · 10

1.19 · 10−2

17.8 · 10−3

20.12 · 10−2

(a)

(b)

Fig. 36.8 Postbuckling analysis for R51WF, uniaxial compression

In case of shear loading, the buckling mode shape depends on the input data of the foam. For the PMI foam R200WF, the foam with the higher relative density, a buckling mode with a single half wave at four hexagonal faces can be observed, whereas for the structural foam R51WF all hexagonal faces buckle exhibiting a buckling mode with two half waves. As expected, buckling occurs first at the faces, and not at the edges for all simulations (cp. [13]).

36.3.3 Postbuckling Analysis To get more information about the postbuckling behaviour of the foam, the pathfollowing method introduced by Riks was used. This procedure is useful for geometrically nonlinear static problems [1]. The investigations were performed for uni-, bi- and triaxial compression tests and the shear test. The considered configurations were the reference model without any irregularities, the models with applied imperfections discussed in Sect. 36.3.1, and the buckling shapes, which were provided in Sect. 36.3.2 (cp. [2]). Some of the stress-strain curves are plotted in


559

Fig. 36.8. The imperfections and the overlaid buckling shape do not have a significant effect on the elastic properties in the case of small deformations, which means only a slight nonlinearity can be observed. In case of larger deformations, the complexity of the structure becomes evident. Load minima and maxima, as well as snapback effects, can be observed depending on the considered imperfection, especially in case of ROHACELL® 51WF. These effects have to be seen in connection with different buckling shapes.

36.4 Conclusion The behaviour of a RVE of a closed-cell foam has been investigated by a FE-model. In doing so, the terakaidekahedron was adopted as unit cell and periodic boundary conditions were defined. Parameter studies were carried out and have shown the effect of imperfections on the elastic properties of the structural foam. The investigation was based on the material data of the foam ROHACELL® of the Evonik Röhm GmbH, Germany. It became obvious that ROHACELL® 51WF, with the smaller relative density, is more sensitive to the presence of imperfections than the foam 200WF. The linear buckling analysis and the path following method gave information about the buckling and postbuckling behaviour of the PMI foams.

References 1. ABAQUS Documentation, version 6.9 (2009). 2. Demiray, S.: Zur nichtlinearen Homogenisierung und mesoskopischen Simulation von Festkörperschwämmen. Dissertation, Technische Universität Darmstadt, Fachbereich Maschinenbau, Shaker Verlag, Aachen (2007). 3. Evonik Röhm GmbH, Germany http://www.rohacell.com/product/rohacell/en/products-services/ rohacell-wf/pages/default.aspx and subordinate documents. Cited 25 Jan 2011. 4. Gibson, L.J., Ashby, M.F.: Cellular Solids: Structure and properties, 2nd edition. Cambridge University Press, Cambridge (1997). 5. Grenstedt, J. L.: Influence of wavy imperfections in cell walls on elastic stiffness of cellular solids. Journal of the Mechanics and Physics of Solids 46 (1), 29-50 (1998). 6. Grensted,t J. L.: On interactions between imperfections in cellular solids. Journal of materials science 40, 5853-5857 (2005). 7. Grenstedt, J. L., Bassinet, F.: Influence of cell wall thickness variations on elastic stiffness of closed-cell cellular solids. International Journal of Mechanical Sciences 42, 1327-1338 (2000). 8. Hohe, J., Becker, W.: A probabilistic approach to the numerical homogenization of irregular solid foams in the finite strain regime. International Journal of Solids and Structures 42, 35493569 (2005). 9. Kraatz, A.: Anwendung der Invariantentheorie zur Berechnung des dreidimensionalen Versagens- und Kriechverhaltens von geschlossenzelligen Schaumstoffen unter Einbeziehung der Mikrostruktur. Dissertation, Martin-Luther-Universität Halle-Wittenberg, MathematischNaturwissenschaftlich-Technische Fakultät (2007).

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10. Mills, N. J.: Deformation mechanisms and the yield surface of low-density, closed-cell polymer foams. Journal of Materials Science 45, 5831-5843 (2010). 11. Simone, A.E., Gibson, L.J.: Effects of solid distribution on the stiffness and strength of metallic foams. Acta Materialia 46 (6), 2139-2150 (1998). 12. Ströhla, S., Winter, W., Kuhn, G.: Einfluss von Strukturstörungen auf die elastisch-plastischen Materialparameter zellularer Werkstoffe. In: Huber O., Bicker M. (Hrsg.): 2. Landshuter Leichtbau-Colloquium. Landshut: Leichtbau-Cluster, Fachhochschule Landshut, Tagungsband, 137-148 (2005). 13. Sue, J.: Effect of Microstructure of Closed Cell Foam on Strength and Effective Stiffness. Dissertation, Texas A&M University (2006). 14. Wang, J., Wang, H., Chen, X.: Experimental and numerical study of the elastic properties of PMI foams. Journal of materials science 45, 2688-2695 (2010). 15. Youssef, S., Maire, E., Gaertner, R.: Finite element modelling of the actual structure of cellular materials determined by X-ray tomography. Acta Materialia 53, 719-730 (2005).

Chapter 37

What Shell Theory Fits Carbon Nanotubes? Antonino Favata and Paolo Podio–Guidugli

Abstract We discuss what linearly elastic shell model would best capture the peculiarities of the mechanical response of carbon nanotubes, be they single- or multiwall. We argue that, at the macroscopic scale, carbon nanotubes should be modeled as orthotropic cylindrical shells. An abridged presentation of the basic ingredients of such a shell theory is given. Keywords Shell theory · Carbon nanotube · Orthotropic shell

37.1 Introduction When carbon nanotubes (CNTs) are employed as nanodevice components, they are regarded as elastic beam-like or shell-like objects and their mechanical response is characterized in terms of an as-small-as-possible number of stiffness and inertia parameters. To define and evaluate these parameters in terms of the relevant physical interaction forces and the resultant geometry is the common goal of all modelers; a way to achieve it is to to describe the mechanical behavior of CNTs by a bottomup method, which bridges between three different scales: the microscopic scale of molecular mechanics; a mesoscopic scale, at which concepts from discrete structure mechanics apply; and the macroscopic scale of continuous structure mechanics (see [1, 8–10, 13, 14], and the literature cited therein). As to what model type to pick at the macroscopic scale, whether beam-like or shell-like, a conclusive evidence is still wanted, for a number of reasons. One of these reasons, perhaps not the least important, was the lack of a fit shell theory. In this paper, we discuss what linearly elastic shell model would best capture, in our opinion, the peculiarities of A. Favata (B) · P. Podio–Guidugli Dipartimento di Ingegneria Civile, Università di Roma Tor Vergata, via Politecnico 1, 00133 Roma, Italy e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 37, © Springer-Verlag Berlin Heidelberg 2011

561

562

A. Favata and P. Podio–Guidugli

Fig. 37.1 Graphene and its zigzag and armchair roll-up axes

the mechanical response of carbon nanotubes, be they single- or multi-wall; we also describe the salient feature of such a theory, taking freely from our forthcoming study [5]. CNTs are large macromolecules, composed exclusively of carbon atoms. In imagination, SW(Single-Wall)CNTs can be obtained by rolling up into a cylindrical shape a graphene, that is, a monolayer flat sheet of graphite. There are many ways to do so, accounted for by a geometrical object, the chiral vector: ch = na1 + ma2 (Fig. 37.1). Having fixed a1 and a2 , the pairs of integers (n, m) specifies the chirality of a CNT; (n, 0)– and (n, n)–nanotubes are called, respectively, zigzag and armchair (Fig. 37.2). Chirality has an influence on the mechanical response, e.g., on the effective Young and shear moduli or on the buckling strain [2,3,6,12,15]; for MW(Multi-Wall)CNTs, the chirality of the inner walls is hard to inspect.

Fig. 37.2 The armchair-like [zigzag-like] axis of a zigzag [armchair] single-wall CNT

37 What Shell Theory Fits Carbon Nanotubes?

563

In the vast majority of papers where CNTs are regarded macroscopically as shells, textbook theories induced from classic three-dimensional isotropic elasticity are used; consequently, the shell response is characterized in terms of two elastic moduli. On the basis of the mesoscopic modeling of armchair and zigzag CNTs proposed in [1], we contend that it would be more appropriate to view all CNTs, be they single- or multi-wall and whatever their chirality, as orthotropic cylindrical shells whose mid-surface has a tangent plane coinciding with the orthotropy plane, so that the shell geometry agrees point-wise with the geometry intrinsic to the chosen type of material response (see Fig. 37.3, where the three little cylinders suggest what probes one should cut out of the shell body in order to determine its material moduli). The elastic moduli of a general orthotropic shell are nine, a number variously reduced by reducing the generality of admissible deformations. Two such reductions are considered in [5], the one drastic the other less so. A drastic reduction to four moduli is achieved by stipulating that the internal kinematical constraint typical of the Kirchhoff-Love theory of shells prevails – an assumption compatible with orthotropy and leading to a theory seemingly applicable to SWCNTs [2,3]. Stipulation of a relaxed Kirchhoff-Love constraint allowing for thickness distention leads to a theory with seven material moduli, a theory hopefully fit for MWCNTs, where thickness changes and other deformational consequences of the inter-wall forces of van der Waals type should not be ignored [4]. While many moduli enhance a theory’s potential applicability, the more they are the larger the experiment or simulation burden needed to determine them. Luckily, conspicuous simplifications are achieved for axisymmetric equilibrium problems like torsion and axial traction, the ones for which experimental tests are especially

Fig. 37.3 The geometrical elements relevant to describe the material response of CNTs modeled as shells

564


easy to set up, as well as for uniform pressure and rim flexure; remarkably, for all these problems explicit analytical solutions can be produced [5].

37.2 An Abridged Presentation of the Shell Theory We Propose The starting point of our method of deduction of a shell theory is a special version of the three-dimensional Principle of Virtual Powers (PVP). The speciality we refer to is triple: 1. the three-dimensional body we consider is shell-shaped, i.e., is a tubular neighborhood G(S, E) of constant thickness 2E of given a model surface S; for definiteness, we here take S to be a right circular cylinder (Fig. 37.4); a point p ∈ G(S, E) has position vector p := p − o = x − o + ζn(x),

x ∈ S, ζ ∈ I := (−ε, +ε),

with respect to an origin o; |ζ| is the distance of p from x, the point where the straight line through p perpendicular to S intersects S itself; 2. admissible body parts are tubular neighborhoods of thickness 2E of open subsets of S; 3. admissible virtual velocities are consistent with the representation (37.1) for admissible motions of G(S, E). With the use of such a special version of the PVP, we obtain two-dimensional balance equations in terms of two-dimensional stress measures, both at interior and boundary points of S.

Fig. 37.4 A portion of the model surface of a right cylindrical shell


565

37.2.1 Kinematics With reference to Fig. 37.4, admissible motions have the form: ζ u(x1 , ϑ, ζ) = c1 ⊗ c1 + 1 + n (ϑ) ⊗ n (ϑ) a(x1 , ϑ) ρo ' ( + w(x1 , ϑ) + ζγ n(ϑ) − ζ w,1 (x1 , ϑ)c1 + ρ−1 w, (x , ϑ)n (ϑ) . 2 1 o

(37.1)

(Here and henceforth we denote by (·),1 and (·),2 differentiation with respect to x1 and ϑ; when a field depends only on the latter variable, we prefer to write (·) instead of (·),2 .) With the use of the linear strain measure 1 E(u) = sym∇u := (∇u + ∇uT ), 2

(37.2)

it is not difficult to see that (37.1) implies that G(S, E) can change its thickness – a feature that makes the present shell theory not standard – but that material fibers orthogonal to the referential mid-surface S must remain orthogonal to it after any admissible deformation; we call this kinematic limitation the unshearability constraint. Note that the thickness-strain field has constant value: E·n⊗n ≡ γ

over G(S, E).

(37.3)

37.2.2 Balance Equations Given a material body occupying a three-dimensional open and bounded region Ω, and given a velocity field v over Ω, the internal power expenditure over a part Π of Ω associated with v is: S · ∇v Π

where S denotes the stress field in Ω; the external power expenditure over Π is: do · v + co · v, Π

∂Π

where (do , co ) denote, respectively, the distance force for unit volume and the contact force per unit area exerted on Π by its own complement with respect to Ω and by the environment of the latter; a virtual velocity field is a smooth vector field, whose support is a part, consistent with the prescriptions that define the class of admissible motions; finally, the Principle of Virtual Powers is the stipulation that S · ∇v = do · v + co · v (37.4) Π

Π

for all parts and all virtual velocity fields.

∂Π

566


Our special PVP is formulated as follows (see [5] for details). We let Ω ≡ G(S, E), consider a part collection of the type specified above and virtual velocities coherent with (37.1), of the following form: (0)

v = v1 c1 + v2 n + v3 n,

(1)

v = −v3 ,1 c1 + ρ−1 o (v2 − v3 ,2 )n + v4 n,

(37.5)

with the scalar fields vi = vi (x1 , ϑ) (i = 1, 2, 3) compactly supported in S and with v4 a constant. The internal power becomes: (0) (1) (1)( ' S · ∇v = F · ∇ v + M · ∇ v + f (3) · v , Π

P

where P is a part of S and the force tensor F, the moment tensor M, and the shear vector f (3) are the two-dimensional stress measures. As to the external power, we obtain: (0) (1) (0) (1) do · v + co · v = (qo · v + ro · v ) + (lo · v + mo · v ), Π

∂Π

P

∂P

where qo , ro are, respectively, the distance force and distance couple per unit area, and lo , mo are the contact force and contact couple per unit length. Exploiting the declared quantifications, we deduce from (37.4) the balance equations and the boundary conditions of our theory of cylindrical shells with variable thickness. The point-wise balances are: F ,1 + ρ−1 o F ,2 + qo = 0, −1 −1 −1 (F + ρ−1 o M ),1 + ρo (F + ρo M ),2 + qo + ρo ro = 0, 1 M ,11 + ρ−1 o (M + M ),12 + 2 M ,22 + (37.6) ρo

+l −l

0

−1 − ρ−1 o F + qo + ro ,1 + ρo ro ,2 = 0, 2π % & ρ−1 o M + F − ro dx1 dϑ = 0,

where F and M are the physical components of the force and moment tensors and qo and ro are the physical components of the distance force and couple. Moreover, at a boundary point belonging to a directrix of the cylinder,1 the admissible boundary conditions must consist of a list of mutually exclusive assignments of the one or the other element of the following five power-conjugate pairs: (F , a), (F + ρ−1 o M , a), (F − ρ−1 o M ,2 , w), (M , w,1 ), (M , γ).

1

(37.7)

We concentrate on this case, because it is the only one of importance when dealing with the boundary-value problems of torsion, traction, pressure, and rim flexure.


567

The unusual integral balance (37.6)4 may be in the need of some comments: it stems from testing the equality of internal and external powers by way of virtual velocity fields of the form: (1)

(1)

v=ζ v,

v = v4 n,

(37.8)

with v4 an arbitrary real number that must be taken constant over the whole of Ω, because it is meant to be a variation of the constant thickness strain E = γ (cf. (37.3)).

37.2.3 Constitutive Equations 37.2.3.1 General Orthotropic Response Let C denote the fourth-order tensor of elasticity, a linear transformation of the space of symmetric tensors, that specifies the stress response to deformations in the parent three-dimensional theory we are going to select. An orthogonal tensor Q is a symmetry transformation for the linearly elastic material described by C if it so happens that QC[E]QT = C[QEQT ] for all symmetric tensors E; the symmetry group of C is the collection of all such Q. Given a subgroup G of the rotation group, one seeks a representation formula for all elasticity tensors C such that GC ⊃ G, i.e., for all elasticity tensors sharing a given symmetry group. A linearly elastic material is called orthotropic when its stress response is insensitive to a rotation of π about a given axis c, i.e., when the symmetry group of its elasticity tensor includes that rotation. Let {ei (i = 1, 2, 3)} be an orthonormal basis of vectors. Consider the following orthonormal basis for the linear space of all symmetric tensors: 1 Vα = √ (eα ⊗ e3 + e3 ⊗ eα ) (α = 1, 2), 2 Wα = eα ⊗ eα (α not summed),

V3 = e3 ⊗ e3 ,

1 W3 = √ (e1 ⊗ e2 + e2 ⊗ e1 ). 2

(37.9)

With the use of this basis, any orthotropic elasticity tensor can be written in the following form: C = C1111 W1 ⊗ W1 + C2222 W2 ⊗ W2 + C3333 V3 ⊗ V3 + + C1212W3 ⊗ W3 + C3131V1 ⊗ V1 + C2323 V2 ⊗ V2 + + C1122(W1 ⊗ W2 + W2 ⊗ W1 ) + C1133(W1 ⊗ V3 + V3 ⊗ W1 )+ + C2233(W2 ⊗ V3 + V3 ⊗ W2 )

(37.10)

568


(cf. [7]); the orthotropic material class is then parameterized by the 9 elastic moduli C1111 , . . . , C2233 .

37.2.3.2 The Case of MWCNTs As a direct consequence of the fact that the shell model we are after incorporates the unshearability constraint, the appropriate three-dimensional response is captured by an elasticity tensor somewhat simpler than (37.10). Just as in [5], we make the shell geometry agree with the geometry intrinsic to the material response, in the sense that, at any fixed point x ∈ S, we identify e3 with n(x). With this identification, the internal constraint E(u)n · v = 0

for all v such that v · n = 0,

(37.11)

can be read as the requirement that all admissible strains be orthogonal to the following subspace of Sym: D⊥ := span(Vα , α = 1, 2).

(37.12)

Accordingly, the space Sym is split into the direct sum of two orthogonal subspaces: Sym = D ⊕ D⊥ ,

(37.13)

and the stress is split into reactive and active parts: S = S(R) + S(A) , with

S(R) = ψ(R) α Vα ,

ψ(R) α ∈ R,

S(A) = C C[E],

(37.14) C : D → D; C

(37.15)

here the coefficients ψ(R) α are constitutively unspecified, and the constraint space D implicitly defined by (37.12) and (37.13) can be identified as D = span(V3 ; Wi , i = 1, 2, 3).

(37.16)

On applying a general result proved in [11], a representation for the desired elasticity tensor C C can be deduced from the one given for C in (37.10): C C = PD C|D ,

PD := I − Vα ⊗ Vα .

where PD denotes the orthogonal projector of Sym on D. One finds the 7-parameter representation C C = C1111 W1 ⊗ W1 + C2222 W2 ⊗ W2 + C3333 V3 ⊗ V3 + C1212W3 ⊗ W3 + + C1122 (W1 ⊗ W2 + W2 ⊗ W1 ) + C1133(W1 ⊗ V3 + V3 ⊗ W1 )+ + C2233 (W2 ⊗ V3 + V3 ⊗ W2 ).

(37.17)


569

As is customary, we assume that C C is positive-definite, i.e., that E·C C[E] > 0 for all E ∈ D \ {0}.

(37.18)

37.2.3.3 The Case of SWCNTs In this case, it seems appropriate to exclude thickness changes; accordingly, the internal constraint (37.11) is reinforced a` la Kirchhoff–Love: E(u)n = 0,

(37.19)

and the procedure detailed in the previous subsection yields the elasticity tensor: 6 C = C1111 W1 ⊗ W1 + C2222 W2 ⊗ W2 + C1212 W3 ⊗ W3 + C1122 (W1 ⊗ W2 + W2 ⊗ W1 ), (37.20) 6 = span(Wi , i = 1, 2, 3) into itself. a linear transformation of the constraint space D

37.2.4 Field Equations The governing equations of our theory are arrived at by the following procedure: (i) expressions for the force and moment tensors F and M and the shear vector f (3) are obtained in terms of strain components, by the use of the constitutive equations (37.17) or (37.20); (ii) expressions for strain components in terms of displacement components are obtained, by the use of (37.2) and (37.1); (iii) the expressions resulting from substituting the expressions obtained in step (ii) into those obtained in step (i) are inserted into the balances (37.6). We exemplify this procedure in the case of the axisymmetric torsion problem under end torques, where (37.6)2 , the only relevant balance, takes the following simple form: (F + ρ−1 (37.21) o M ),1 = 0. Now, by definition, ζ F = 1+ S dζ ρo I

and

M = I

ζ 1+ ζS dζ , ρo

where S := S · n ⊗ c1 ; moreover, in view of (37.14), (37.15) and either (37.17) or (37.20), S = S (A) := 2C1212 E (here C1212 = G, the only relevant shear modulus); in addition, it follows from definition (37.2) and the form taken by (37.1) in axisymmetric situations that

570


E :=

1 ζ ∇u · (c1 ⊗ n + n ⊗ c1 ) = 1 + a,1 ; 2 ρo

all in all, after substitution and integration, ε2 F + ρ−1 M = 2ε 1 + G a ; o ρ2o but then (37.21) turns into the differential relation one would expect on recalling Saint-Venant’s torsion problem, namely, a,11 = 0

in (−l, +l).

References 1. C. Bajaj, A. Favata, P. Podio-Guidugli, On a Scale-Bridging Mechanical Model of Single-Wall Carbon Nanotubes, (2011) Forthcoming. 2. G.X. Cao, X. Chen, The effects of chirality and boundary conditions on the mechanical properties of single-walled carbon nanotubes, Intern. Journ. Sol. Struc., 44, 5447-5465 (2007). 3. T.C. Chang, Torsional behavior of chiral singlewalled carbon nanotubes is loading direction dependent, Appl. Phys. Lett., 90, 201910 (2007). 4. A. Di Carlo, A. Favata, P. Podio–Guidugli, Modeling Multi-Wall Carbon Nanotubes as Elastic Multi-Shells, 2011 (Forthcoming). 5. A. Favata, P. Podio-Guidugli, A new CNT-Oriented shell theory, submitted (2011). 6. R.V. Goldstein, V.A. Gorodtsov, A.V. Chentsov, S.V. Starikov, V.V. Stegailov, G.E. Norman, To description of mechanical properties of nanotubes. Tube wall thickness problem. Size effect, Russian Academy of Sciences, A.Yu. Ishlinsky Institute for Problems in Mechanics, Preprint 937 (2010). 7. M.E. Gurtin, The Linear Theory of Elasticity. Pp. 1-295 of Handbuch der Physik VIa/2, Springer (1972). 8. C. Li, T.-W. Chou, A structural mechanics approach for the analysis of carbon nanotubes, Int. J. Solids Struct., 40:2487 (2003). 9. G.M. Odegard, Modeling of Carbon Nanotube/Polymer Composites. In Nanoengineering of Structural, Functional and Smart Materials, eds. M.J. Schulz, A. Kelkar, M.J. Sundaresan, CRC Press (2005). 10. G.M. Odegard, Equivalent-Continuum Modeling of Nanostructured Materials. In Handbook of Theoretical and Computational Nanotechnology, eds. M. Rieth and W. Schommers, American Scientific Publishers (2006). 11. P. Podio–Guidugli, M. Vianello, The representation problem of constrained linear elasticity. J. Elasticity, 28 (1992) 271-276. 12. C.Q. Ru, Chirality-Dependent Mechanical Behavior of Carbon Nanotubes Based on an Anisotropic Elastic Shell Model, Math. Mech. Sol., 14, 88-101 (2009). 13. L. Shen, J. Li, Transversely isotropic properties of single-walled carbon nanotube, Phys. Rev. B 69:045414 (2004). 14. H. Wan and F. Delale, A structural mechanics approach for predicting the mechanical properties of carbon nanotubes. Meccanica 45: 4351 (2010). 15. Y.Y. Zhang, V.B. Tan. C.M. Wang Effect of chirality on buckling behavior of singlewalled carbon nanotubes, Jour. Appl. Phys., 100, 074304 (2006).

Chapter 38

A Variationally Consistent Derivation of Microcontinuum Theories Johannes Meenen, Holm Altenbach, Victor Eremeyev and Konstantin Naumenko

Abstract Micromorphic and micropolar continuum theories can be derived from the classical equations of continuum mechanics by introducing a local averaging technique into a mixed variational principle. This approach delivers the geometrically nonlinear strain measures, the equilibrium equations and the constitutive relations of the microcontinuum theory in a natural, variationally consistent way. It also allows to evaluate the residuum of the locally averaged theory with respect to the underlying classical continuum mechanics, and shows that the microcontinuum can be understood as a special kind of a representative volume element. Keywords Microcontinuum · Variational principle

38.1 Introduction In the last years, several authors have adressed the question how the stresses, strain measures and constitutive equations of a nonlinear Cosserat theory could be derived. An axiomatic derivation of microcontinuum field theories has been presented by Eringen [1]. In an earlier work of Suhubi and Eringen [5], a microcontinuum theory is developed by integrating the balance equations with a weighting function over a small microvolume, and a variational approach has been used by Mindlin [2]. An overview of the strain measures proposed by several authors can be found in [3]. J. Meenen (B) Aachen, Germany e-mail: [email protected] H. Altenbach · V. Eremeyev · K. Naumenko Martin-Luther-University Halle-Wittenberg, Germany e-mail: [email protected],[email protected] e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 38, © Springer-Verlag Berlin Heidelberg 2011

571

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In [1,2,5], two different length scales and their corresponding coordinate systems are used: a global scale and a microscale. The final governing equations are formulated in the global scale, whereas the microscale is used to introduce rotations (or local deformations in the case of micromorphic theories) as additional independent variables on the global scale. In all these theories, it is assumed that the displacement field in the microelement can be expanded into a Taylor or McLaurin’s series and that the microelement is so small that only the constant and the linear terms must be considered. This assumption can be avoided if the microcontinuum theory is derived from a weak form of the classical equations of continuum mechanics. For this purpose, a special test function is used which vanishes outside a fixed, finite radius. It can be shown that the weak form can then be transformed into a set of equations which is formally similar to a special case of a Cosserat theory. The variables of this microcontinuum theory can be calculated from the corresponding variables of the underlying continuum theory by an averaging procedure over a particle of finite size. In this context, a particle is a spherical region of the continuum, and it is assumed that the quantities averaged over the particle’s volume are representative for the quantities of the basic theory in the neighborhood of the particle’s center. The derivation is somewhat similar to the filtering operation used in large-eddy simulation (LES) of turbulent flows [4]. In LES, the convective terms lead to an unclosed set of filtered equations. In a Lagrangian framework, a closure problem can be avoided by averaging over a linear set of basic equations: linear in the sense that the weak form does not possess any products of the involved quantities. This is the reason for using a weak form in terms of the first Piola-Kirchhoff stress tensor and the deformation gradient, and avoiding quantities such as f.e. the right or left Cauchy-Green strain tensor or the second Piola-Kirchhoff stress tensor. It will however turn out that an approximation must be introduced as soon as the constitutive laws are evaluated.

38.2 Basic Equations The basic equations consist of the equilibrium equations in the reference configuration, expressed in terms of the first Piola-Kirchhoff stress tensor ∂PiI j ∂ai

+ ρ0 f j = 0 ,

(38.1)

the definition of the deformation gradient Fi j −

∂xi =0, ∂a j

(38.2)

38 A Variationally Consistent Derivation of Microcontinuum Theories

573

and the constitutive equations for a hyperelastic material in terms of the first PiolaKirchhoff stress tensor and the strain energy density function PiI j −

∂U(Fkl ) =0. ∂F i j

(38.3)

In these equations, ai and xi are the coordinates of a material point in the reference and in the current configuration, PiI j is the first Piola-Kirchhoff stress tensor and F i j is the deformation gradient. ρ0 and f j are the density and the distributed body force in the reference configuration, and U is the strain energy density function. Einstein’s summation convention is applied, and round parentheses such as in f (x) are exclusively used to indicate a functional dependency (i.e. that f is a function of x). These basic equations can be transformed into a weak form IE (PiI j ; vEj ) + ID(F i j ; viDj ) + IC (PiI j , F i j ; vCij ) = 0 with ⎤ ⎡⎢ ∂P I ⎥⎥⎥ ⎢⎢⎢ i j ⎢⎢⎣ IE = + ρ0 f j ⎥⎥⎥⎦ vEj dV ∂ai

(38.4)

ak ∈V

ID =

Fi j −

∂xi D v dV = 0 ∂a j i j

(38.5)

PiI j −

∂U(F kl ) C vi j dV ∂F i j

(38.6)

ak ∈V

IC = ak ∈V

and a set of suitable test functions vEj , viDj and viEj .

38.3 Equilibrium Equations For the local averaging of the equilibrium equations (38.4), a test function of the type n m wn,m j ak sk φ(ak − sk ) n

is used. In all what follows, sk are the coordinates of the center of the currently considered particle in the reference configuration. The averaging function φ is zero outside the particle radius r p : ⎧ ⎪ ⎪ ⎨ 0 : |ak − sk | < r p φ(ak − sk ) = ⎪ ⎪ ⎩ = 0 : |ak − sk | ≥ r p

(38.7)

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For the integrals of the averaging functions, the following abbreviations will be used φ(0) = φ(ak − sk )dV , (38.8) ak ∈V

φ(1) r =

φ(ak − sk )[ar − sr ]dV ,

(38.9)

ak ∈V

φ(2) rt

=

φ(ak − sk )[ar − sr ][at − st ]dV .

(38.10)

ak ∈V

Without loss of generality, the local averaging function can be normalized so that φ(0) = 1. If only isotropic functions are considered φ(ak − sk ) = φ(r)

with

r = |ak − sk | ,

(38.11)

(2) (2) it is straightforward to show that φ(1) r = 0 and φrt = φ δrt .

38.3.1 Micromorphic Theory A micromorphic theory can be derived by replacing the test function vEj of Eq. (38.4) with E,0 E,1 vE,M (38.12) j (ak , sk ) = v j + v jk ak φ(ak − sk ) . Partial integration delivers ⎤ ⎡⎢ ∂PI ⎥⎥⎥ ∂v E,M ⎢⎢⎢ i j j E,M E,M I ⎥ ⎢⎢⎣ + ρ0 f j ⎥⎥⎦ v j dV = −Pi j + ρ0 f j v j dV + ni PiI j vE,M j dA ∂ai ∂ai

ak ∈V

ak ∈V

ak ∈∂V

To postpone the treatment of boundary conditions, it is assumed that the volume is infinite. In this case, the boundary integral vanishes. Using ∂vE,M j ∂ai

= vE,0 j

∂φ ∂φ ∂φ ∂φ E,1 E,1 + vE,1 a + vE,1 δ φ = −vE,0 j ∂s − v jk ak ∂s + v jk δki φ (38.13) jk k ∂a jk ki ∂ai i i i


575

and keeping in mind that the derivative of PiI j with respect to si is zero leads to ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎬ E,0 I E,M I IE (Pi j ; v j ) = ⎪ P φdV + ρ f φdV vj ⎪ j i j ⎪ ⎪ ⎪ ⎪ ∂s ⎪ ⎪ i ⎩ ⎭ ak ∈V

ak ∈V

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎬ E,1 I I +⎪ P a φdV − δ P φdV + ρ f a φdV v jk ⎪ k ki j k ij ij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂si ⎭ ak ∈V

ak ∈V

(38.14)

ak ∈V

This leads to the equilibrium equations of the locally averaged theory ∂P(0) ij ∂si ∂P(1) i jk ∂si

+ F (0) j = 0

− P(0) + F (1) =0 kj jk

(38.15)

with the locally averaged stress and body force resultants (0) Pi j = PiI j φ(ak − sk )dV V

P(1) = i jk

PiI j ak φ(ak − sk )dV V

F (0) j

=

ρ f j φ(ak − sk )dV V

F (1) = jk

ρ f j ak φ(ak − sk )dV

(38.16)

V

These equilibrium equations are similar to the equilibrium equations of other micromorphic theories, with some exceptions. It should be noted that for small deformations, the averaged stress tensor will be symmetric, and that the ”lever arm” of P(1) is counted with respect to its length in the reference configuration. i jk Refined micromorphic theories can be easily derived by introducing higher order expansions in ak for the test function.

38.3.2 Micropolar Theory A micropolar theory can be derived as a special case with the test function E,0 E,1 vE,P φ(ak − sk ) (38.17) j (ak , sk ) = v j + ε jkr ak v¯ r

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It is straightforward to show that the weak form (38.4) transforms to ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎬ E,0 I IE (PiI j ; vE,P ) = P φdV + ρ f φdV vj ⎪ ⎪ j i j ⎪ ⎪ j ⎪ ⎪ ∂s ⎪ ⎪ ⎩ i ⎭ ak ∈V

ak ∈V

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎬ E,1 I I +⎪ P ε a φdV − ε P φdV + ρ f ε a φdV v¯r (38.18) ⎪ jkr k jkr j jkr k i j k j ⎪ ⎪ ⎪ ⎪ ∂s ⎪ ⎪ i ⎩ ⎭ ak ∈V

ak ∈V

ak ∈V

which leads to the equilibrium equations ∂P(0) ij ∂si

(0)

+Fj = 0

∂ P¯ (1) ir + ε jkr P(0) + F¯ r(1) = 0 kj ∂si

(38.19)

with the locally averaged stress and body force resultants P¯ (1) = PiI j ε jkr ak φ(ak − sk )dV ir V

F¯ r(1)

=

ρ f j ε jkr ak φ(ak − sk )dV

(38.20)

V

38.3.3 Uncoupled Micromorphic Theory A micromorphic theory with uncoupled equilibrium equations can be derived with the test function E,0 E,1 vE,R (a , s ) = v + v ˆ [a − s ] φ(ak − sk ) (38.21) k k k k j j jk Using ∂ I ∂φ Pi j [ak − sk ]φ = PiI j [ak − sk ] − PiI j δki φ ∂si ∂si the weak form becomes

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎬ E,0 I E,R I IE (Pi j ; v j ) = ⎪ P φdV + ρ f φdV vj ⎪ j i j ⎪ ⎪ ⎪ ⎪ ∂s ⎪ ⎪ i ⎩ ⎭ ak ∈V

ak ∈V

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ⎨ ⎬ E,1 I +⎪ P [a − s ]φdV + ρ f [a − s ]φdV vˆ jk ⎪ k k j k k ij ⎪ ⎪ ⎪ ⎪ ∂s ⎪ ⎪ i ⎩ ⎭ ak ∈V

ak ∈V

(38.22)


577

and the corresponding equilibrium equations are ∂P(0) ij ∂si ∂ Pˆ (1) i jk

∂si

(0)

+ Fj = 0 + Fˆ (1) =0 jk

(38.23)

with the locally averaged stress and body force resultants (1) ˆ Pi jk = PiI j [ak − sk ]φ(ak − sk )dV = P(1) − P(0) i j sk i jk V

Fˆ (1) jk

ρ f j [ak − sk ]φ(ak − sk )dV = F (1) − F (0) j sk jk

=

(38.24)

V

38.3.4 Evaluation of the Residuum The residuum of these theories with respect to the classical continuum theory can be evaluated most easily for the uncoupled micromorphic theory. Subtraction of * + * + ∂ (0) ∂ ˆ (1) ˆ (1) E,1 (0) ˆ (1) E,R (0) E,0 IE (Pi j , Pi jk ; v j ) = P + Fj vj + P + F jk vˆ jk (38.25) ∂si i j ∂si i jk from (38.22) leads to ⎧ ⎡ ⎤⎫ ⎪ ⎢⎢⎢ ⎥⎥⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎢⎢⎢ ⎬ E,0 (0) ˆ (1) E,R (0) ⎥⎥⎥⎪ I IE (PiI j ; vE,R ) − I (P , P ; v ) = P φdV − P vj ⎢ ⎥ ⎪ E ij ⎢ ⎥ i j ⎪ ⎪ j i j i jk j ⎢⎣ ⎥⎦⎪ ⎪ ⎪ ∂s ⎪ ⎪ i ⎩ ⎭ ak ∈V

⎧ ⎡ ⎤⎫ ⎪ ⎢⎢⎢ ⎥⎥⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ⎢⎢⎢ ⎥⎥⎥⎪ ⎨ ⎬ E,1 (1) I ˆ +⎪ P [a − s ]φdV − P vˆ jk (38.26) ⎢ ⎥ ⎪ k k ⎢ ⎥ i j ⎪ ⎪ i jk ⎢ ⎥ ⎪ ⎪ ∂s ⎣ ⎦ ⎪ ⎪ i ⎩ ⎭ ak ∈V

If the first Piola-Kirchhoff tensor is expanded into (2) PiI j = p(0) ˆ (1) i j (sk ) + p i jr (sk )[ar − sr ] + pi j (sk , ak )

(38.27)

in the neighbourhood of sk , the values of p(0) ˆ (1) i j and p i jr can be identified if it is assumed that the integrals (2) pi j φdV and p(2) i j [ak − sk ]φdV ak ∈V

ak ∈V

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vanish. Introducing these assumptions into the definition of the stress resultants (38.24) allows to identify p(0) ˆ (1) i j and p i jr . If the averaging function is normalized and isotropic, it can be shown that the first Piola-Kirchhoff stress tensor can be developed into PiI j = P(0) ij +

1 ˆ (1) P [ar − sr ] + p(2) ij φ(2) i jr

with the projection conditions for the residuum p(2) ij p(2) i j φdV = 0 ak ∈V

p(2) i j [ak − sk ]φdV

=0

(38.28)

ak ∈V

Introducing this expansion into (38.26) leads to ∂ p(2) i j φdV = 0 ∂si ak ∈V ∂ p(2) i j [ak − sk ]φdV = 0 ∂si

(38.29)

ak ∈V

which confirms that the expansion of PiI j is consistent with the averaging procedure. Because of (38.24), the micromorphic theory has got the same residuum and can be approximated by 1 1 (0) I Pi j = Pi j 1 − (2) sr [ar − sr ] + (2) P(1) (38.30) i jr [ar − sr ] φ φ The residuum of the micropolar theory can be evaluated in a comparable way. Subtraction of * + * + ¯ (1) ; vE,P ) = ∂ P(0) + F (0) vE,0 + ∂ P¯ (1) + ε jkr P(0) + Fˆ r(1) v¯rE,1 (38.31) IE (P(0) , P ij ir j j j kj ∂si i j ∂si ir from (38.18) leads to ⎧ ⎡ ⎤⎫ ⎪ ⎢⎢⎢ ⎥⎥⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎢⎢⎢ ⎬ E,0 (0) ⎥⎥⎥⎪ I Pi j φdV − Pi j ⎥⎥⎥⎪ vj ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ∂s ⎣ ⎦ ⎪ ⎪ ⎩ i ⎭ ak ∈V

⎧ ⎡ ⎤ ⎡ ⎤⎫ ⎪ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ ⎨ ∂ ⎢⎢⎢ ⎬ (1) (1) ⎥⎥⎥ (0) ⎥⎥⎥⎪ I I ¯ +⎪ ε P a φdV − P − ε δ P φdV − P v¯r ⎢ ⎥ ⎢ ⎥ jkr k jkr ki ⎢ ⎥ ⎢ ⎥ i j i j ⎪ ⎪ ir i j ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦⎪ ⎪ ⎪ ∂s ⎪ ⎪ i ⎩ ⎭ ak ∈V

ak ∈V


579

With the help of ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ∂ ∂ ⎢⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (0) (0) (0) ⎥ ⎢ ⎥ ⎢ I I I ⎥ ⎢ ⎥ ⎢ δki ⎢⎢⎢ Pi j φdV − Pi j ⎥⎥⎥ = Pi j φdV − Pi j ⎥⎥⎥ − sk Pi j φdV − Pi j ⎥⎥⎥⎥⎥ ⎢ sk ⎢ ∂si ⎢⎢⎣ ⎣ ⎦ ∂si ⎢⎢⎣ ⎦ ⎦ ak ∈V

ak ∈V

ak ∈V

this can be transformed into ⎧ ⎡ ⎤⎫ ⎪ ⎢⎢⎢ ⎥⎥⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎢⎢⎢ ⎬ E,0 (0) ⎥⎥⎥⎪ (1) I P φdV − P v + ε s v ¯ ⎢ ⎥ ⎪ ⎪ jkr k r ⎢ ⎥ i j ⎪ i j ⎥⎪ j ⎢ ⎪ ⎦⎪ ⎪ ⎪ ⎩ ∂si ⎣ ⎭ ak ∈V

⎧ ⎡ ⎤⎫ ⎪ ⎢⎢⎢ ⎥⎥⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎥⎥⎪ ⎨ ∂ ⎢⎢⎢ ⎬ (1) (1) (0) I ¯ +⎪ ε P [a − s ] φdV − P − P ε s v¯r ⎢ ⎥ jkr k k jkr k ⎢ ⎥ i j ⎪ ⎪ ir i j ⎢⎣ ⎥⎦⎪ ⎪ ⎪ ∂s ⎪ ⎪ i ⎩ ⎭ ak ∈V

The series expansion (2) PiI j = p(0) ¯ (1) ij + p is ε st j [at − st ] + pi j

and the projection condition for the residuum p(2) ij p(2) i j φdV = 0 ak ∈V

p(2) i j ε jkr [ak − sk ]φdV = 0

(38.32)

ak ∈V

leads for normalized and isotropic averaging functions to the approximation PiI j = P(0) ij −

1 ¯ (1) Pir − P(0) ε s εrt j [at − st ] + p(2) jkr k ij ij 2

and the residuum of the weak form is ∂ ∂si ∂ ∂si

p(2) i j φdV = 0 ak ∈V

p(2) i j ε jkr [ak − sk ]φdV = 0 ak ∈V

(38.33)

(38.34)

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38.4 Nonlinear Strain Measures A nonlinear strain measure for the micromorphic theory can be derived from the weak form with the test function D,1 viD,M = viD,0 (38.35) j j + vi jk ak φ(ak − sk ) Following the same procedure used for deriving the equilibrium equations gives the definitions of the locally averaged deformation gradients F i(0) j = F i(1) jk = and the resultants

∂x(0) i ∂s j ∂x(1) ik ∂s j

− x(0) i δk j

(38.36)

Fi(0) j =

Fi j φ(ak − sk )dV V

Fi(1) jk

=

Fi j ak φ(ak − sk )dV V

x(0) i =

xi φ(ak − sk )dV V

x(1) ik

=

xi ak φ(ak − sk )dV

(38.37)

V

For the micropolar theory, the test function D,0 D,1 viD,P j = vi j + εikr ak v¯r j φ(ak − sk )

(38.38)

can be used. This leads to ∂ x¯(1) F¯ r(1)j = r + εr jk x(0) k ∂s j with the resultants F¯ r(1)j =

(38.39)

Fi j εikr ak φdV = εikr Fi(1) jk

V

x¯(1) r

xi εikr ak φdV = εikr x(1) ik

= V

(38.40)


In the case of the uncoupled micromorphic theory, the test function is D,0 D,1 viD,R = v + v ˆ [a − s ] φ(ak − sk ) k k j ij i jk

581

(38.41)

which leads to Fˆ i(1) = jk with the resultants

∂ xˆ(1) ik ∂s j

(38.42)

Fˆ i(1) = jk

Fi j [ak − sk ]φ(ak − sk )dV V

(1) xîk

=

xi [ak − sk ]φ(ak − sk )dV

(38.43)

V

38.4.1 Evaluation of the Residuum The residuum of the displacement and strain measures can be evaluated with an approach similar to the procedure used for the stress tensor. It is assumed that the displacement and the deformation gradient in the neighbourhood of sk can be expressed by (1) (2) xi = y(0) i (sk ) + yit (sk )[at − st ] + yi (sk , ak ) (1) (2) F i j = fi(0) j (sk ) + fi jt (sk )[at − st ] + fi j (sk , ak )

With the conditions

(38.44)

y(2) i φdV = 0 and

ak ∈V

y(2) i [ak − sk ]φdV = 0 ak ∈V

fi(2) j φdV = 0 and ak ∈V

fi(2) j [ak − sk ]φdV = 0

(38.45)

ak ∈V

these expansions can be transformed into 1 (1) xˆ [ak − sk ] + y(2) i φ(2) ik 1 ˆ (1) (2) Fi j = F i(0) j + φ(2) F i jk [ak − sk ] + fi j xi = x(0) i +

(38.46)

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(0) ˆ (1) (0) (1) D,R Inserting this expansion into ID (F i j , xi ; vîD,R ) − I (F , F , x , x ˆ ; v ˆ ) delivers D j ij ij i jk i ik ∂ ∂s j ∂ ∂s j

y(2) i φ(ak − sk )dV = 0

V

y(2) i [ak − sk ]φ(ak − sk )dV

=0

(38.47)

V

The micromorphic theory has the same residuum with the expansion 1 (1) (0) x − xi sk [ak − sk ] + y(2) i φ(2) ik 1 (1) (0) (2) F i j = Fi(0) j + (2) F i jk − F i j sk [ak − sk ] + fi j φ xi = x(0) i +

(38.48)

In the micropolar case, the expansion is 1 (1) (0) x ¯ − ε x s εist [at − st ] + y(2) lks k s i l 2φ(2) 1 (0) (2) ¯ (1) Fi j = F i(0) j + 2φ(2) F s j − εlks F l j sk εits [at − st ] + fi j xi = x(0) i −

with the conditions

(38.49)

y(2) i φdV = 0 and

ak ∈V

y(2) i εikr [ak − sk ]φdV = 0 ak ∈V

fi(2) j φdV = 0 and ak ∈V

fi(2) j εikr [ak − sk ]φdV ak ∈V

The residuum of the weak form delivers ∂ ∂ y(2) φdV = 0 and y(2) i i εikr [ak − sk ]φdV = 0 ∂s j ∂s j ak ∈V

(38.50)

(38.51)

ak ∈V

38.5 Constitutive Equations For the derivation of the constitutive equation for the micromorphic theory, the test function C,0 C,1 vC,M (38.52) i j = vi j + vi jk ak φ(ak − sk )


583

is inserted into IC (Pi j ; vCij ). This leads to the relation P(0) ij = ak ∈V

P(1) = i jk

ak ∈V

∂U φdV ∂Fi j

∂U ak φdV ∂F i j

In the case of the micropolar theory, the test function C,0 C,1 vC,P i j = vi j + ε jkr ak v¯ri φ(ak − sk ) leads to

P¯ (1) rj = ak ∈V

∂U ε jkr ak φdV ∂Fi j

For the uncoupled micromorphic theory, the test function C,0 C,1 vC,R i j = vi j + vˆ i jk [ak − sk ] φ(ak − sk ) leads to

Pˆ (1) = i jk ak ∈V

∂U [ak − sk ]φdV ∂F i j

(38.53)

(38.54)

(38.55)

(38.56)

(38.57)

A exact value for these integrals cannot be calculated because the deformation gradient in the particle is only approximately known. In the most general case, these integrals can be evaluated by a numerical quadrature procedure such as a Gauss integration, if the series expansion for Fi j is inserted.

38.6 Summary Micromorphic and micropolar continuum theories can be derived from a weak form of the classical continuum theory with a test function which vanishes outside a fixed radius. In contrast to other derivations which assume that the microelement is small, this approach is valid also for a particle of arbitrary finite size, and it does not require to introduce two different length scales. The weak form also allows to evaluate the residuum between the classical continuum theory and the micromorphic theory. It can be shown that the resultants of the micromorphic theory can be interpreted as local averages of the corresponding variables of the classical continuum theory.

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References 1. Eringen, A.C.: Microcontinuum field theories: foundations and solids. Springer Verlag, New York Berlin Heidelberg (1999) 2. Mindlin, R.D.: Micro-Structure in Linear Elasticity. Arch. Rat. Mech. Anal. 11 385–414 (1962) 3. Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures 46 774–787 (2009) 4. Pope, S.: Turbulent Flows. Cambridge University Press, Cambridge (2000) 5. Suhubi, E.S., Eringen, A.C.: Nonlinear theory of simple micro-elastic solids – I. International Journal of Engineering Science 2 189–203 (1964)

Chapter 39

Shell-Models for Multi-Layer Carbon Nano-Particles Melanie Todt, Franz G. Rammerstorfer, Markus A. Hartmann, Oskar Paris and Franz D. Fischer

Abstract In many cases continuum mechanics has proved to be an appropriate method for investigating the mechanical behavior of carbon nanostructures reducing the computational requirements compared to atomistic methods significantly. The main modeling issues arising in continuum mechanics modeling of multi-layer carbon nanostructures are briefly discussed. These issues involve the continuum modeling of (i) the atomic layers, (ii) the covalent interlayer bonds, (iii) the van der Waals interactions, and (iv) the excess surface energy due to curvature. Continuum mechanics methods in conjunction with the finite element method are applied to investigate the compressive behavior of carbon crystallites and a possible growth limit of carbon onions. Keywords Nanomechanics · Layered structures · Stability · Computational modeling

39.1 Introduction Extensive experimental work has already been carried out to investigate the mechanical, electrical, optical, and chemical properties of carbon nanotubes [19, 26], carbon onions [1,45], graphite/graphene [10,20,43], and the dependency of the compressive behavior of carbon fibers on their nanostructure [4, 16, 21]. Additionally to the experimental work computational methods play an important role in gaining a M. Todt (B) · F. G. Rammerstorfer Vienna University of Technology, Vienna, Austria e-mail: [email protected],[email protected] M. A. Hartmann · O. Paris · F. D. Fischer Montanuniversität Leoben, Leoben, Austria e-mail: [email protected],[email protected] e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 39, © Springer-Verlag Berlin Heidelberg 2011

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better understanding of the properties and the behavior of carbon nanostructures. An overview about computational methods used for studying carbon nanostructures is given in [25], where the focus lies on carbon nanotubes, and in [14]. The computational methods can be subdivided in three categories: (i) methods based on atomistic approaches, (ii) methods based on continuum mechanics approaches, and (iii) multiscale methods. Computer methods based on atomistic approaches such as ab initio, molecular dynamics, and Monte Carlo simulations give detailed information about the behavior of nanostructures on the atomic level. However, these methods require extensive computational resources and are, therefore, not applicable for investigating large multi-layer nanostructures like, e.g., carbon onions. Additionally in ab initio calculations the problem of a correct treatment of van der Waals interactions between atoms in adjacent layers occurs [7]. Continuum methods are computationally less expensive than atomistic methods and, thus, make it possible to investigate the mechanical behavior of large nanostructures. In continuum mechanics modeling of nanostructures the deformations are assumed to be smooth, and the influence of defects and local atomic configurations are smeared out. Thus, deformation states, where the local behavior of the atomic structure plays an important role, cannot be captured correctly with continuum methods. Multi-scale methods [13, 14] take advantage of both, atomistic and continuum mechanics approaches and, therefore, can be used to investigate large nanostructures, where also local atomic configurations are of importance. The main issue of multi-scale methods lies on the smooth bridging of the atomistic and continuum length scales. Continuum mechanics methods can, thus, be used exclusively or within a multiscale approach to investigate the mechanical behavior of carbon nanostructures. Pure continuum mechanics models, e.g., have successfully been applied to study the mechanical behavior of carbon nanotubes [23, 41, 44] or the compressive behavior of carbon crystallites [32]. A multi-scale simulation of the fracture evolution of a carbon nanotube containing a vacany defect is presented in [13]. The focus herein is on pure continuum modeling of carbon nanostructures, and the paper is outlined as follows. In Sect. 39.2 the main issues involved in the continuum modeling of carbon nanostructures are briefly discussed. In Sects 39.3 and 39.4 some of the methods described in Sect. 39.2 are applied to investigate the compressive behavior of carbon crystallites and the growth limit of carbon onions, respectively.

39.2 Continuum Mechanics Modeling of Carbon Nanostructures Continuum mechanics modeling of multi-layer carbon nanostructures involves four major modeling issues, namely • the modeling of the atomic layers,

39 Shell-Models for Multi-Layer Carbon Nano-Particles

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• the modeling of covalent interlayer bonds, • a correct treatment of the van der Waals interactions between the layers, • the modeling of the intrinsic excess surface energy in curved carbon nanostructures. These issues will be discussed briefly within this section. Altough some configurations of covalent interlayer bonds originate from defects in the atomic layers [31], the influence of defects on the layer properties, the van der Waals interactions, and the excess surface energy is not considered here. To study the effect of defects, multi-scale methods [13, 14] bridging continuum mechanics and atomistic descriptions seem to be more appropriate than a pure continuum approach.

39.2.1 Layer Properties Different continuum models such as the truss/frame model [12] or the shell model [23, 41] exist for modeling the mechanical properties of the atomic layers, where only the shell-model will be discussed here. The atoms forming a graphene layer or the walls of carbon nanotubes are arranged in a hexagonal lattice. Thus, the material behavior of the layers is assumed to be isotropic. Nanoindentation experiments on monolayer graphene, see, e.g., [11] have shown that the material behavior can be considered as elastic until breaking of the layers. Thus, the layers can be approximated as thin shells using only three independent elastic parameters: membrane stiffness C = Eh, (39.1) bending stiffness D=

Eh3 , 12(1 − ν2)

(39.2)

and Poisson’s ratio ν. Using Eqs (39.1) and (39.2) the effective elastic modulus E and the effective thickness h of the shells can be estimated for given values of C, D, and ν. Using continuum shells for modeling the atomic layers was first proposed by Yakobson et al. [41]. Later investigations on the strain energy of carbon nanotubes [44] have shown that modeling the atomic layers as thin shells is consistent with atomistic modeling techniques. Since that time a lot of experiments, atomistic simulations, and analytical investigations have been carried out to estimate the elastic parameters describing the layer properties, see, Table 39.1. The values presented in Table 39.1, vary significantly and there is still an ongoing discussion about the thickness of the layers and the meaning of the layer thickness. The values given for E and h should, therefore, be seen as fictitious values which represent the membrane stiffness C and bending stiffness D, in combination with ν.

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Table 39.1 Estimated values for C, D, E, h and ν by various investigators using different methods Investigators Yakobson et al. [41]

C [J/m2 ] D [nNnm] E [GPa] h [nm] ν [-] Method 360

0.136

5500

0.066 0.19 molecular dynamics

377.4

0.183

5100

0.074 0.24 electronic band theory

(357.7)

(0.176)

graphene

-

-

1110

0.341

nanotube r = 3.44 nm

-

-

940

0.341 0.047

nanotube r = 3.69 nm

-

-

1060

0.341 0.125

Zhang et al. [42]

159

-

405

0.3351

Pantano et al. [23]

363

0.176

4840

0.075 0.191 finite element method

-

-

2540

0.134 0.32 molecular mechanics

graphene2

∼470

∼0.22

-

-

-

nanotubes2

∼480

∼0.225

-

-

-

-

0.133

-

-

-

Zhou et al. [44] van Lier et al. [36]

Sears et al. [29]

ab initio -

-

Wu et al. [40]

atomistic-based shell theory

Lu et al. [17]

1 2

nanoscale continuum theory

analytically 1st gen. Brenner pot.

-

0.110

-

-

-

2nd gen. Brenner pot.

-

0.225

-

-

-

incl. dihedral angle effect

Values are assumed in the cited papers. Values obtained for the unstrained nanostructures.

In Sect. 39.4 the mechanical behavior of carbon onions is investigated. The atoms forming the carbon onion layers are not only arranged in hexagonal but also in pentagonal rings. Thus, it must be checked if the values given in Table 39.1 are also appropriate for describing the mechanical behavior of carbon onion layers. For this purpose the collapse pressure p∗ and the volumetric stiffness K of a C60 fullerene, with mean radius R(0) = 0.35139 nm are estimated using Monte Carlo simulations and a continuum shell model. Details on the Monte Carlo simulations can be found in the Appendix. The volumetric stiffness K is inversely proportional to the compressibility of the system and defined as K =−

p V/V0

(39.3)

where V = 4πR3 /3 is the volume of the C60 fullerene subjected to an external pressure p and V0 = 4π(R(0) )3 /3 is the volume of the unloaded fullerene. Using continuum mechanics the collapse pressure p∗ can be estimated by the buckling pressure of a perfect thin-walled sphere [24] as E 2h2 p∗ = . (0) 2 3(1 − ν2) (R )

(39.4)


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Table 39.2 Estimated values for the collapse pressure p∗ and the compressive modulus K of a C60 fullerene using Monte Carlo (MC) simulations (see Appendix) and continuum shell theory (CST) Method MC

E [GPa] h [nm] ν -

-

-

p∗ [GPa] K [GPa] 247.79

901

CST S1 [23]

4840

0.075 0.19 254.08

853.42

CST S2 [41]

5500

0.066 0.19 223.59

853.42

CST S3 [36]

1100

0.340 0.19 1211.23 879.51

The current radius of the fullerene is calculated as pA0 (39.5) kS as the elastic radial displacement, kS = 8πEh/(1 − ν) as the elastic membrane stiffness of the layers in terms of a ”force”-displacement relation, and A0 = 4π(R(0) )2 as the surface area of the unloaded fullerene. The quantities p∗ and K are calculated using different sets of parameters, S1, S2, and S3, of E, h, and ν according to [23], [41], and [36], respectively. The results of the Monte Carlo and the continuum mechanics calculations are summarized in in Table 39.2. p∗ and K obtained for parameter sets S1 and S2 are in good agreement with the results of the Monte Carlo simulations. Parameter set S3 gives a good approximation of K but strongly overestimates p∗ . Hence, parameter set S1 or S2 should be used to describe the properties of carbon onion layers. R = R(0) − ΔR

with

ΔR =

39.2.2 Covalent Interlayer Bonding Defects in the atomic layers lead to the formation of covalent interlayer bonds between neighboring layers. In [31] different types of covalent interlayer bonds occurring in crystalline graphite are obtained using ab initio calculations. The formation of the interlayer bonds is induced either by vacancy defects or by interstitial atoms where the type of interlayer bonding depends on the defect type and the kind of layer stacking. The interlayer bonds locally reduce the interlayer distance from 0.335 nm (undisturbed graphite) to values ranging from 0.138 nm to 0.258 nm, depending on the interlayer bonding type. In [32] the interlayer bonds are modeled as trusses, having no bending and torsional stiffness and with their axial stiffness being much higher than the membrane stiffness of the layers. This simplified model can only capture effects resulting from a change in the distance of the bonded atoms in a qualitative nature. Effects resulting from changes in the inter-atomic angles cannot be captured, leading to a zero interlayer shear stiffness contribution of the bonds. Although further improvements in the continuum mechanics description of covalent interlayer bonds are necessary, the simplified model can already give some basic insight on how the location and amount of interlayer bonds influences the mechanical behavior of carbon nanostructures [32].

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39.2.3 Van der Waals Interactions In multi-layer carbon nanostructures the van der Waals (vdW) interactions between atoms on adjacent layers have to be considered as well. The vdW interactions between atoms are weak non-covalent atom–atom interactions and can be described using, e.g, the 6-12 Lennard Jones (LJ) potential 12 6 σ σ V = 4ε − , (39.6) d d with σ and ε being the LJ parameters and d being the atom–atom distance. The interaction force between two atoms is the first derivative of the potential V with respect to d reading 12 24ε σ 6 σ F= −2 . (39.7) d d d For use within a continuum mechanics model the vdW interactions have to be formulated in a form applicable within the continuum mechanics framework. In the following, appropriate formulations of the vdW interactions in form of a pressure/distance (planar structures) and a pressure/radius (curved structures) relation are briefly summarized.

39.2.3.1 Planar Multi-Layer Carbon Nanostructures In [10] a pressure/distance relation derived for graphite is given as C33 αeq 10 αeq 4 p= − , 6 α α

(39.8)

where C33 = 36.5 GPa is the elastic interlayer stiffness of graphite [43] and αeq is the equilibrium interlayer distance. The vdW bonding is assumed to be massless and its (small) shear stiffness is neglected. Although this relation does not include any curvature effects, it has been used to describe the vdW interactions in multi-walled carbon nanotubes, see, e.g., [23]. As argued in [23] results obtained with Eq. (39.8) are only valid if the radii of the nanotube walls are large enough, i.e. the curvature effects become negligible.

39.2.3.2 Curved Multi-Layer Carbon Nanostructures For curved carbon nanostructures with small layer radii, however, the curvature effect, which is – besides others – caused by different atom numbers in adjacent layers, has to be considered. In some papers the curvature effect has been taken into account by assuming that the vdW pressure on opposing faces of adjacent layers is inversely proportional to


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their surface area, leading to p1 R1 = −p2 R2 for carbon nanotubes [27] and p1 R21 = −p2 R22 for carbon onions. The quantities p1 and p2 are the vdW pressure on the inner and outer layer, respectively, and R1 ,R2 are the corresponding layer radii. In [18] and [33] more detailed continuum vdW models are derived for carbon nanotubes and carbon onions, respectively, without the above assumptions. The vdW pressures on opposing faces of two adjacent layers are evaluated by summation of all vdW interaction forces between two atoms on the different layers. Since only the radial components of the atom-atom vdW interaction forces contribute to the pressures, the other components are not considered. For deriving a continuum vdW model the discrete sum is replaced by surface integrals employing the atom area density ρ∞ leading to ⎡ 11 5 ⎤ ⎥⎥ R2 ⎢⎢⎢⎢ σ R2 − R1 σ R 2 − R1 p1 = C0 ⎢⎣231 E 13 − E 11 − 160 E 7 − E5 ⎥⎥⎥⎦ , R1 R1 + R2 R2 + R1 R1 + R2 R 2 + R1 ⎡ 11 5 ⎤ ⎥⎥ R1 ⎢⎢⎢⎢ σ R2 − R 1 σ R2 − R1 p2 = C0 ⎢⎣231 E13 + E 11 − 160 E 7 + E 5 ⎥⎥⎥⎦ R2 R1 + R2 R2 + R 1 R1 + R2 R2 + R1 (39.9) with 3π C0 = εσρ2∞ (39.10) 32 for carbon nanotubes [18] and ⎡ ⎤ ⎢⎢⎢ 2 E 1 σ11 1 5 E 2 σ5 ⎥⎥⎥⎥ ⎢ p1 = C1 ⎢⎣ 2 − (39.11) ⎥⎦ , R1 (R2 − R21 )11 (R22 − R21 )5 ⎡ ⎤ ⎢⎢⎢ 5 E 3 σ5 1 2 E 4 σ11 ⎥⎥⎥⎥ ⎢ p2 = C1 ⎢⎣ 2 − (39.12) ⎥⎦ , R2 (R2 − R21 )5 (R22 − R21 )11 with E 1 = 15R81 + 220R61R22 + 594R41R42 + 396R21R62 + 55R82 , E2 = E3 = E4 =

3R21 + 5R22 , 3R22 + 5R21 , 15R82 + 220R62R21 + 594R42R41 + 396R22R61 + 55R81

and C1 =

32π ε σ ρ2∞ (R(0) )2 (R(0) )2 , 1 2 5

(39.13) (39.14) (39.15)

,

(39.16)

(39.17)

for carbon onions [33]. In Eqs (39.9) E 13 , E 11 , E7 , E 5 are circumferential integrals involving R1 , R2 , and the circumferential angle θ, for more details see [18]. The pressure/radius relations (Eqs (39.9) and (39.11), (39.12)) are depending not only on the interlayer distance R2 − R1 but also on the layer radii R1 , R2 . Additionally, Eqs (39.11) and (39.12) depend on the radii R(0) and R(0) defining the initial configuration of the carbon 1 2

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onion layers without the vdW interactions being active. The pressures on opposing faces of adjacent layers, p1 , p2 , evaluated with these relations are not equal and do not satisfy the relations p1 R1 = −p2 R2 and p1 R21 = −p2 R22 for carbon nanotubes and carbon onions, respectively. The buckling pressure of multi-walled carbon nanotubes obtained under the assumption of p1 R1 = −p2 R2 for the vdW pressures is substantially higher than the buckling pressure calculated with the more advanced vdW model according to Eqs (39.9), see [18]. In [33] it is shown that the radial displacements of individual carbon onion layers obtained with the vdW model according to Eqs (39.11) and (39.12) are closer to results of Monte Carlo simulations than those obtained with simplified models. The above results show that correct continuum formulations are of great importance for studying the mechanical behavior of curved multi-layer carbon nanostructures. The presented pressure/radius relations (Eqs (39.9) and (39.11), (39.12)) are such correct descriptions for the homogenized behavior of the vdW interactions between adjacent layers. Local changes in the vdW interactions due to defects or non-hexagonal rings in the layers can not be captured with these relations.

39.2.4 Surface Energy Closed carbon nanostructures such as fullerenes, carbon onions, single-, and multiwalled carbon nanotubes possess an intrinsic curvature induced excess surface energy E S . As reasonably good approximation the excess surface energy can be assumed to depend on the radius R according to a power law [9] of the form E S ∝ Rβ .

(39.18)

The obtained values of β range from −1.43 to −2.51 (energy per area dependency) for fullerenes depending on the fullerene size and the model type used, for details see [9]. Thus, the surface energy becomes more important for decreasing size of the objects. For carbon nanotubes β = −2.04 (energy per area dependency) is found. The intrinsic surface energy E S is related to surface stress σ ¯ S in terms of a small strain setting [6] as ∂E S σ ¯ S = ES + S . (39.19) ∂# Here # S is the surface strain and E S , σ ¯ S have the dimension of force per unit length. S S Since the dependency of E on # is generally weak and not known in case of curved carbon nanoparticles σ ¯ S ≈ E S is assumed. The effect of the intrinsic surface stress can be applied to the continuum model of carbon nanotubes or carbon fullerenes as inwards oriented pressure pS =

σ¯ S R

(nanotubes),

pS =

2σ ¯S R

(fullerenes),

(39.20)


593

where pS vanishes fast with order |β| + 1 ≈ 3 with increasing layer radius and thus, is only of relevance for small carbon nanoparticles.

39.3 Carbon Crystallites The mechanical behavior of a classical material used in lightweight structures, namely carbon fiber reinforced plastics (CFRP), under compression or bending strongly depends on the compressive behavior of the individual carbon fibers. The compressive behavior of carbon fibers has been experimentally investigated in many articles [4, 8, 16, 21], where also a dependency of the compressive behavior on the fiber’s nanostructure is observed. The nanostructure of carbon fibers is formed by stacks of imperfect graphene layers, so called carbon crystallites [16, 22], see Fig. 39.1. In fibers subjected to compressive loading or compressive stresses due to bending buckling of these crystallites can be observed [16]. A continuum mechanics study of the compressive behavior of carbon crystallites is presented in [32] confirming observations made in [16] and with the obtained initial effective elastic modulus being in good agreement with experimental values [15]. Thus, the crystallite model and the results presented in [32] can serve as a basis for estimating the compressive behavior of carbon fibers as briefly summarized in Sects 39.3.1 and 39.3.2, respectively.

39.3.1 Shell Model The carbon crystallite assumed in [32] consists of eight graphene layers which is typical for polyacrylonitrile (PAN) based fibers. Each graphene layer is formed by carbon atoms arranged in a hexagonal lattice and oriented parallel to the fiber axis, see Fig. 39.1. Carbon atoms of neighboring layers interact via vdW forces. An equilibrium distance d002 = 0.34 nm between the layers is assumed, which is in good agreement with values given in [15, 21]. A layer size of La = 4 nm is assumed being typical for PAN based fibers, see, e.g., values given in [4, 21]. Defects in the layers and unsaturated dangling bonds at the edges can lead to the formation of

Fig. 39.1 Nanostructure of PAN-based carbon fibers. With kind permission from Springer Science+Business Media: J. Mater. Sci. 45, 2011, 6845-6848, Todt et al., Fig. 1

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covalent interlayer bonds [31, 37]. Interlayer bonds in the interior of the crystallite and at the edges are assumed to be similar reducing the interlayer distance locally to dc = 0.258 nm, which corresponds to the distance reduction due to a fourfold interstitial atom as presented in [31]. The finite element method is used to investigate the compressive behavior of the crystallites and the influence of the location and amount of covalent interlayer bonds. The layers are modeled using linear shell elements with elastic modulus E = 5.5 TPa, Poisson’s ratio ν = 0.19, and layer thickness h = 0.066 nm. The values are taken form [41] and, as discussed in Sect. 39.2.1, have to be seen as virtual ones, leading to a good representation of the membrane stiffness as well as of the bending stiffness of a single graphene layer [17, 36]. The vdW interactions between the layers are modeled using the pressure-distance relation given in Eq. (39.8). The covalent interlayer bonds are represented by single truss elements with their tensile stiffness being much higher than the membrane stiffness of the layers. To investigate the influence of the location of the covalent interlayer bonds on the compressive behavior of the fibers the bonds are either distributed randomly in the interior of the crystallite or along its edges. For a parametric study the amount of interlayer bonds is varied form 1% to 5% of interlayer-bonded atoms. For further modeling details see [32].

39.3.2 Results In a first step the length of the interlayer bonds is reduced form 0.34 nm to 0.258 nm by applying a virtual negative temperature change ΔT to the bonds. The deformations introduced by the shrinkage of the interlayer bonds strongly depend on the location of the bonds, see Fig. 39.2. Covalent interlayer bonds distributed randomly in the interior the crystallite lead only to local dimples in the layers (Fig. 39.2(a)), and a small misalignment (< 1 deg) of the crystallite with respect to the fiber axis. Fig. 39.2 Deformation states after the initial step represented by contourlines of the z-displacements of one layer on the outside of the crystallites for crystallites with interlayer bonds in the interior of the crystallite (a) and with interlayer bonds distributed only at the edges (b). With kind permission from Springer Science+Business Media: J. Mater. Sci. 45, 2011, 6845-6848, Todt et al., Fig. 3


595

Interlayer bonds distributed only at the edges cause an overall bulging of the crystallite (Fig. 39.2(b)) and a misalignment of about 4 deg. Thus, not the amount of interlayer bonds but their location determines the character of the deformation of the unloaded crystallite. The difference between the deformation states as depicted in Fig. 39.2 can be explained by a supporting effect of interlayer bonds in the interior of the crystallite preventing the formation of a bulging area. As external loading an axial compression is applied to the crystallites via prescribed displacements in y-direction on the upper edges of the crystallite with appropriate boundary conditions on the opposite edges. The prescribed y-displacements are the same for all edges, being a good approximation of the deformations also for bended fibers. Since the fiber diameter (≈1 μm to 10 μm) is usually much larger than the crystallite’s dimensions perpendicular to the fiber axis (≈5 nm), the strain can assumed to be constant along the width of the crystallite. The effective secant modulus E s is evaluated from the load displacement response of the crystallite using F y Es A = , (39.21) uy l where F y is the reaction force in y-direction corresponding to the prescribed displacements, A = La × Lc is the initial cross section and l = La is the initial height of the crystallite. The compressive behavior, expressed as the dependency of E s on the compressive strain εy , strongly depends on the deformation state after the initial bond shrinkage and thus, on the location of the interlayer bonds, see Fig. 39.3. The secant modulus E s of crystallites with interlayer bonds randomly distributed in the interior of the crystallite remains almost constant until buckling occurs, for which the critical strain and, thus, the further development of E s depend on the amount of interlayer bonds. After buckling of the crystallite Es decreases significantly. The obtained initial secant moduli range from 1105 GPA to 1158 GPA, being in good agreement with experimental values of 1140 ± 10 GPA [15]. For crystallites with the interlayer bonds distributed only at the edges no constant region of Es can be

1250 1000 Es [GPa]

Fig. 39.3 Secant modulus Es evaluated for different amounts of covalent interlayer bonds ranging from 1% to 5% of bonded atoms. The results obtained for the interlayer bonds distributed in the interior of the crystallite and at the edges only are drawn with continous lines and dashed lines, respectively. With kind permission from Springer Science+Business Media: J. Mater. Sci. 45, 2011, 6845-6848, Todt et al., Fig. 5

750 500 250 0 0

0.0025

εy [-]

0.005

0.0075

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observed, which is a consequence of the bulging introduced by the formation of the interlayer bonds. For this case Es is not significantly influenced by the amount of interlayer bonds, and the initial value of Es is much lower than the experimental values [15]. The obtained results show that interlayer bonds distributed in the interior of the crystallite have a stabilizing effect, whereas interlayer bonds distributed only at the edges de-stabilize the crystallite. Furthermore, it can be concluded that it is more likely that the formation of covalent interlayer bonds occurs not only at the edges but also in the interior of the crystallite.

39.4 Carbon Onions Carbon onions consist of a number of concentric layers of closed covalently bonded carbon nanostructures and are almost spherical in their shape. Carbon onions have first been produced by intense electron irradiation of polyhedral multi-layer carbon nanoparticles [35]. Since the discovery of carbon onions extensive experimental work has been carried out to investigate their formation, structure [35, 45], and their ability to act as nanoscopic pressure cells for the growth of nanodiamonds [1]. Carbon onions can be produced, e.g., by high temperature annealing of diamond nanoparticles [34] or the high pressure transformation of single-crystal graphite [2]. One common feature of these techniques is that the size of the produced carbon onions seems to be limited. A precursor and irradiation dose dependent growth limit has also been reported in [45]. Continuum mechanics in conjunction with the finite element method is used to investigate whether the occurrence of a structural instability limits the size of carbon onions. The hypothesis of a growth limit resulting from a structural instability is based on the idea that no further layer can be added after the size of the onion has reached a critical layer number. The continuum shell model used is described in Sect. 39.4.1, and the results are presented in Sect. 39.4.2.

39.4.1 Shell Model An axi-symmetric model of an N-layer carbon onion, see Fig. 39.4, is used for the computational stability analysis. Axi-symmetry is – under the assumption of perfectly spherical onions – justified as long as only pre-buckling states are considered. For a thin-walled spherical shell axi-symmetric buckling modes have the same physical relevance as non-axi-symmetric ones, due to the high multiplicity of the relevant eigenvalue [5]. This principal feature can be assumed to be valid for coupled multi-layer configurations, too, justifying the assumption of an axi-symmetric onion model also for buckling analysis.


597 i+1

z Ri+1

i

i-1 Ri

R

Ri-1

Fig. 39.4 Axi-symmetric model of a carbon onion consisting of three layers (N = 3)

vdW(i-1)i

A(i-1)i

vdWi(i+1)

Ai(i+1)

The individual layers are modeled using axi-symmetric three-noded shell elements with elastic modulus E = 4.84 TPa, Poisson’s ratio ν = 0.19 and a layer thickness h = 0.075 nm. The values for E, ν, h are taken from [23] and are appropriate values for modeling carbon onion layers, see Sect. 39.2.1. The radius of an onion layer can be estimated as (0) 0.103374ni − 0.424548, (39.22) R(0) i =a with a(0) = 0.142 nm being the carbon-carbon bondlength in an undeformed layer and ni is the number of atoms forming the layer [38]. According to [30] the number of atoms forming the layer can be calculated as ni = 60 i2 assuming that each onion layer is an icosahedral fullerene. The curvature induced surface energy E S [9] is accounted for by applying an inwards oriented external pressure pSi , see Sect. 39.2.4. A combination of Eqs (39.18) and (39.20) results in β−1 pSi = 2K1 (R(0) , i )

(39.23)

K1 = 0.36 nN nm/(nm)0.17 is estimated from Fig. 7 in [9] and for β an average value of -1.83 is used (see Sect. 39.2.4). The vdW interactions between adjacent layers are – as a rough simplification – modeled as linear elastic bedding. The stiffness C of the springs in terms of a pressure-distance relation is derived by linearizing Eq. (39.8) about the equilibrium interlayer distance αeq = 0.3415 nm pvdW = CΔα,

(39.24)

with C = C33 /αeq = 106.9 GPa/nm and Δα = αi, j − αeq where αi, j is the current distance between two adjacent layers i and j.

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39.4.2 Growth Process and Results The innermost layer of the carbon onion is assumed to be a C60 fullerene. For simulating the growth process each new layer N is concentrically located onto the current onion consisting of N − 1 layers. pSN is applied onto layer N and kept constant during the growth process. Since the new layer with an initial radius R(0) N does not exactly (0) fit the actual onion (RN − RN−1 αeq ), the vdW interactions have to be considered according to Eq. (39.24). pSN in combination with the vdW interactions leads to either compressive or tensile membrane forces in the new layer and also to a change in the membrane forces in the layers underneath. Every time when a further layer N is added, a buckling eigenvalue analysis is performed. The eigenvalue problem is formulated as j j (K + λ N ΔK )φ = 0∼. (39.25) ≈ N−1 ≈N N ∼

K is the tangent stiffness matrix of the N-layered onion including the effects ≈ N−1 of the surface stress and the vdW forces acting in the state of the onion with N − 1 layers. ΔK represents the contribution of the vdW interactions between layer N − 1 ≈N j

and layer N, and the surface stress of layer N. The quantities λN are the eigenvalues. The smallest eigenvalue λ1N is the factor by which the contribution to the tangent stiffness matrix of the loading state due to the N-th layer has to be multiplied in order to bring the onion with N layers to an unstable state. This means, if λ1N < 1.0, then layer N cannot be added to the carbon onion without surpassing a stability limit. The vector φ 1N is the eigenvector corresponding to λ1N , characterizing the buckling ∼ mode of the N-layered onion. Figure 39.5(a) shows the lowest, i.e., the relevant buckling eigenvalue λ1N as it depends on the number of layers N forming the carbon onion. λ1N > 1 means that the N-layered onion is in stable equilibrium. The curvature induced surface energy contributions pSi acting on the individual layers (i = 1...N) influence λ1N only for onions consisting of less than six layers and has no significant influence on the growth limit. The negative peak for N = 2 and without pSi is due to relatively large tensile membrane forces in the innermost layer i = 1 and rather low compressive membrane forces in layer i = 2 and has no physical relevance. λ1N approaches 1.0 when the onion has grown to a critical layer number N = Ncrit = 23, whether or not pSi is considered, see inset in Fig. 39.5(a). The buckling mode of the critical configuration, see Fig. 39.5(b), shows that merely the outer layers buckle, whereas the innermost layers remain perfectly spherical. The occurrence of the structural instability is mainly due to the vdW interactions, whereas the the influence of pSi is negligible. However, in reality carbon onions with a much higher number than Ncrit = 23 are found. The reason for this distinction lies in the rough simplification involved in the vdW interaction model used in this demonstration example. The linearized vdW model is only valid in the immediate vicinity of αeq . For interlayer distances larger


(b) without surface stress with surface stress

20

z

1

lowest eigenvalue λN

(a) 40

599

0

1.5 1.25 1

λN

-20 -40

R

1.0 0.75

-60 -80

0

5

0.5 20 21 22 23 24 25 N 15 10 20 25 30 number of layers N

Fig. 39.5 (a) Lowest eigenvalue λ1N depending on the number of layers N forming the carbon onion, where the i-th layer of the N-layered onion consists of n = 60 i2 atoms. A detail of the stability limit can be found in the inset. (b) Buckling mode of a carbon onion with N = 24 corresponding to the lowest eigenvalue λ1N

than αeq the linearized model behaves much stiffer than the nonlinear vdW model according to Eq. (39.8), leading especially in the outermost layers to a strong overestimation of the compressive membrane forces. However, the compressive membrane forces in the outer layers are responsible for the occurrence of a structural instability and, thus, their overestimation leads to an instability for Ncrit being lower than the actual value. Further investigations will be carried out with a vdW model according to Eqs (39.11) and (39.12). Nevertheless, the current analysis shows that, a structural instability may occur during the growth of carbon onions, as a consequence of the emerging membrane forces in the layers due to accommodation of missfitting onion layers.

39.5 Conclusion The results obtained for carbon crystallites and carbon onions show that continuum shell-models can be successfully applied to study the mechanical behavior of such kinds of carbon nanostructures. Compared to atomistic modeling techniques the computational costs of continuum mechanics methods are rather low. The local influence of defects or non-hexagonal rings on the layer properties and on the van der Waals interactions can not be captured with the simplified models presented here and further improvements of the continuum description of the interlayer bonds are required. Nevertheless, the simplified models can serve as a basis for further investigations concerning the mechanical behavior of carbon nanostructures.

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Appendix: Some Remarks on Monte Carlo Simulations As summarized in [9,33] in standard Monte Carlo rules a displacement of an individual atom of an atomic system is performed with the probability p = exp(−ΔE/kBT ) [3]. ΔE denotes the difference in the total energy of the system before and after the displacement, T is the absolute temperature, and kB is the Boltzmann constant. The total energy of the system, e.g., the C60 fullerene, can be approximated as the sum EF = E nS1 + E nB2 + EnT3 (39.26) n1

n2

n3

of two-atom bondstretching, three-atom bond bending and four-atom bond torsion contributions E nS1 , EnB2 , and E nT3 , respectively. n1 , n2 , and n3 run over all covalently bonded pairs, all bond angles, and all torsion angles, respectively. The term of bondstretching between two covalently bonded atoms i and j is described with the Morse potential, E S (ri j ) = E0 ({1 − exp[β(ri j − r0 )]}2 − 1). (39.27) with ri j being the actual bond length and r0 being the equilibrium bond length. E 0 is the bond energy and β−1 is the width of the potential. The bond-bending term is described with a harmonic potential, 1 E B (θi jk ) = kθ (cosθi jk − cosθ0 )2 , 2

(39.28)

where θi jk is the angle between the i − j and j − k bonds, θ0 is the equilibrium bond angle, and kθ is the bending force constant. The torsion contribution can be given by [28] 1 E T (φi jkl ) = kφ (1 − cos2φi jkl ), (39.29) 2 where φi jkl is the torsion angle and kφ is the torsion force constant. The parameters E 0 = 6.1322 eV, β = 1.8502 A−1, r0 = 1.4322 A−1 , kθ = 10 eV, θ0 = 120◦ , kφ = 0.35 eV used in the Monte Carlo simulation are taken from [9]. For simulating the collapse load of a C60 fullerene a force F is applied to each atom with its line of action going through the origin of the fullerene. F is increased stepwise until collapse of the fullerene can be observed at F ∗ = 40 eV/nm. The corresponding critical pressure can be obtained as p∗ =

n F∗ , 4π(R(0) )2

(39.30)

where n = 60 is the number of atoms and R(0) is the radius of the fullerene. R(0) is defined as R(0) = |x j − X| = 0.355 nm, (39.31) where x j is the position vector of the jth atom, X is the position vector of the center of mass of the fullerene, and angular brackets denote averaging over all atoms. With


601

Eqs (39.30) and (39.31) the collapse pressure of a C60 fullerene is obtained as p∗ = 247.79 GPa.

References 1. Banhart, F., Ajayan, P.M.: Carbon onions as nanoscopic pressure cells for diamond formation. Nature 382, 433–435 (1996) 2. Blank, V.D., Denisov, V.N., Kirichenko, A.N., Kulnitskiy, B.A., Yu Martushov, S., Mavrin, B.N., Perezhogin, I.A.: High pressure transformation of single-crystal graphite to form molecular carbon onions. Nanotechnology 18, 345601-1–4 (2007) 3. Binder, K.: Applications of Monte Carlo methods in statistical physics. Rep Prog Phys 60, 487–559 (1997) 4. Dobb, M.G., Guo, H., Johnson, D.J., Park, C.R.: Structure-compressional property relations in carbon fibres. Carbon 33, 1553–1559 (1995) 5. Drmota, A. Scheidl, R., Troger, H.: On the imperfection sensitivity of complete shells. Comput Mech 2, 63–74 (1987) 6. Fischer, F.D., Waitz, T., Vollath, D., Simha, N.K.: On the role of surface energy and surface stress in phase-transforming nanoparticles. Prog Mater Sci 53, 481–527 (2008) 7. Grimme, S.: Accurate description of van der Waals complexes by density functional theory including empirical conditions. Comput Chem 25, 1463–1473 (2004) 8. Hawthrone, H.M.: On non-Hookean behavior of carbon fibers in bending. J Mater Sci 28, 2531–2535 (1998) 9. Holec, D., Hartmann, M.A., Fischer, F.D., Rammerstorfer, F.G., Mayrhofer, P.H., Paris, O.: Curvature-induced excess surface energy of fullerenes: Density functional theory and Monte Carlo simulations. Phys Rev B 81, 235403-1–10 (2010) 10. Kelly, B.T.: The physics of graphite. pp 70–80. Applied Science Publishers, London (1981) 11. Lee, C., Wei, X., Kysar, J.W., Hone, J.: Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008) 12. Li, C., Chou, T.: A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40, 2487–2499 (2003) 13. Liew, K.M., Sun, Y.Z.: Computational Modelling and Simulation of Carbon Nanotubes. In: Topping, B.H.V., Adam, J.M., Pallarés, F.J., Bru, R., Romeo, M.L. (eds.) Developments and Applications in Engineering Computational Technology. Saxe-Coburg Publications, Stirlingshire, Scotland (2010) 14. Liu, W.K., Karpov, E.G., Zhang, S., Park, H.S.: An introduction to computational nanomechanics and materials. Comput Methods Appl Mech Engrg 193, 1529–1578 (2004) 15. Loidl, D., Peterlik, H., Müller, M., Riekel, C., Paris, O.: Elastic moduli of nanocrystallites in carbon fibers measured by in-situ X-ray microbeam diffraction. Carbon 41, 563–570 (2003) 16. Loidl, D., Paris, O., Burghammer, M., Riekel, C., Peterlik, H.: Direct Observation of nanocrystalline buckling in carbon fibers under bending load. Phys Rev Lett 95, 25501-1– 4 (2005) 17. Lu, Q., Arroyo, M., Huang, R.: Elastic bending modulus of monolayer graphene. J Phy D 42, 102002-1–6 (2009) 18. Lu, W.B., Wu, J., Xiao, J., Hwang, K.C., Fu, S.Y., Huang, Y.: Continuum modeling of van der Waals interactions between carbon nanotubes. Appl Phys Lett 94, 101917-1–3 (2009) 19. Kuzumaki, T., Hayashi, T., Ichinose, H., Miyazama, K., Ito, K., Ishida, Y.: In-situ observed deformation of carbon nanotubes. Phil Mag A 77, 1461–1469 (1998) 20. Meyer, J.C., Geim, A.K., Katsnelson, M.I., Novoselov, K.S., Booth, T.J., Roth, S.: The structure of suspended graphene sheets. Nature 446, 60–63 (2007) 21. Oya, N., Johnson D.J.: Longitudinal compressive behaviour and microstructure of PAN-based carbon fibres. Carbon 39, 635–645 (2001)

602

M. Todt et al.

22. Paris, O., Peterlik, H.: The structure of carbon fibres. In: Eichhorn, S., Hearle J.W.S., Jaffe, M., Kikutani, T. (eds.) Handbook of textile fibre structure Vol. 2. Woodhead Publishing Ltd, Abington (2009) 23. Pantano, A., Parks, D.M., Boyce, M.C.: Mechanics of deformation of single- and multi-wall carbon nanotubes. J Mech Phys Solids, 52, 789–821 (2004) 24. Pflüger, A.: Stabilitätsprobleme der Elastostatik. p 514. Springer-Verlag, Berlin/Heidelberg (1975) 25. Raffii-Tabar, H.: Computational physics of carbon nanotubes. Cambridge University Press, Cambridge (2008) 26. Qi, H.J., Teo, K.B.K., Lau, K.K.S., Boyce, M.J., Milne, W.I., Robertson, J., Gleason, K.K.:Determination of mechanical properties of carbon nanotubes and vertically aligned carbon nanotube forests using nanoindentation. J Mech Phys Solids 51, 2213–2237 (2003) 27. Ru, C.Q.: Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube. J Appl Phys 87, 7227–7231 (2000) 28. Sancho-Garc´ıa, J.C., Pérez-Jiménez, A.J., Pérez-Jordá, J.M., Moscardó, F.: High level ab initio calculations of the torsional potential of glyoxal. Chem Phys Lett 342,452–460 (2001) 29. Sears, A., Batra, R.C.: Macro-mechanics properties of carbon nanotubes: Molecular mechanics simulations. Phys Rev B 69, 235406-1–10 (2004) 30. Tang, A.C., Huang, F.Q.: Stability rules of icosahedral (Ih or I) fullerenes. Chem Phys Lett 247, 494–501, (1995) 31. Telling, R.H., Ewels, C.P., El-Barbary, A.A., Heggie, M.I.: Wigner defects bridge the graphite gap. Nat Mater 2, 333–337 (2003) 32. Todt, M., Rammerstofer, F.G., Paris, O., Fischer, F.D.: Nanomechanical studies of the compressive behavior of carbon fibers. J Mater Sci 45, 6845–6848 (2010) 33. Todt, M., Rammerstofer, F.G., Fischer, F.D., Mayrhofer, P.H., Holec, D., Hartmann, M.A.: Continuum modeling of van der Waals interactions between carbon onion layers. Carbon 49, 1620–1627 (2011) 34. Tomita, S., Burian, A., Dore, J.C., LeBolloch, D., Fujii, M. Hayashi, S.: Diamond nanoparticles to carbon onions transformation: X-ray diffraction studies. Carbon 40, 1469–1474 (2002) 35. Ugarte D.: Curling and closure of graphitic networks under electron-beam irradiation. Nature 359, 707–709 (1992) 36. Van Lier, G., Van Alsenoy, C., Van Doren, V., Geerlings, P.:Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene. Chem Phy Lett 326, 181–185 (2000) 37. Vollath D.: Nanomaterials. Wiley, Weinheim (2008) 38. Voytekhovsky, Y.L.: A formula to estimate the size of a fullerene. Acta Crystallogr A 59, 193–194 (2003) 39. Walther, J.H., Jaffe, R., Halicioglu, T., Koumoutsakos, P.: Carbon nanotubes in water: Structural characteristics and energetics. J Phys Chem B 105,9980–9887 (2001) 40. Wu, J., Peng, J., Hwang, K.C., Song, J., Huang, Y.: The intrinsic stiffness of single-wall carbon nanotubes. Mech Res Commun 35, 2–9 (2008) 41. Yakobson, B. I. and Brabec, C. J. and Bernholc, J.:Nanomechanics of carbon tubes: Instabilities beyond linear response. Phys Rev Lett 76, 2511–2514 (1996) 42. Zhang, P., Huang, Y., Geubelle, P.H., Klein, P.A., Hwang, K.C.: The elastic modulus of singlewall carbon nanotubes: a continuum analysis incorporating interatomic potentials. Int J Solids Struct 39, 3893–3906 (2002) 43. Zhao, Y.X., Spain, I.L.: X-ray diffraction data for graphite to 20 GPa. Phys Rev B 40, 993–997 (1989) 44. Zhou X., Zhou, J.J., Ou-Yang, Z.C.: Strain energy and Young’s modulus of single-wall carbon nanotubes calculated from electronic energy-band theory. Phys Rev B 62, 13692–13696 (2000) 45. Zwanger, M.S., Banhart, F., Seeger, A.: Formation and decay of spherical concentric-shell carbon clusters. J Cryst Growth 163, 445–454 (1996)

Part VII

Biomechanics

Chapter 40

Mechanics of Biological Membranes from Lattice Homogenization Mohamed Assidi, Francisco Dos Reis and Jean François Ganghoffer

Abstract The goal of this chapter is to set up a novel methodology for the calculation of the effective mechanical properties of biological membranes viewed as repetitive networks of elastic filaments, basing on the discrete asymptotic homogenization method. We will show that for some lattice configurations, internal structure mechanisms at the unit cell scale lead to additional flexional effects at the continuum scale, accounted for by an internal length associated to a micropolar behavior. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes for a given network topology, as closed form expressions of the geometrical and mechanical micro-parameters. A new approach, based on general beam equations, is proposed to tackle the non-linear constitutive behavior of the network, accounting for large strains and large rotations. Thereby, a perturbed equilibrium problem is set up at the unit cell level, solved by the Newton-Raphson method. This localization problem interacts with the homogenization procedure allowing the construction of the Cauchy and couple stress tensors, both steps leading to an update the network geometry and constitutive behaviour. A classification of lattices with respect to the choice of the equivalent continuum model is proposed: the Cauchy continuum and a micropolar continuum are adopted as two possible effective medium, for a given beam model. The relative ratio of the characteristic length of the micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. Calculation of the equivalent mechanical properties of the peptodoglycan membrane illustrates the proposed methodology.

M. Assidi (B) ENSEM-INPL, 2 Avenue de la Forêt de Haye, 54500 Vandoeuvre-les-Nancy, France e-mail: [email protected] F. D. Reis · J. F. Ganghoffer LEMTA - ENSEM, 2, Avenue de la Forêt de Haye, BP 160, 54054 Vandoeuvre CEDEX, France e-mail: [email protected] e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 40, © Springer-Verlag Berlin Heidelberg 2011

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40.1 Introduction One of the essential aims of bioengineering today is to answer the basic question of how individual cells and/or cell networks, viewed as a continuum architectural biomaterials, interact with each other or behave under different loading, including combined thermal, chemical and mechanical loadings in a general situation. The somewhat complex topology of the discrete microstructure of biomembranes poses a serious problem when setting up a mechanical model in view of the determination of their effective mechanical behavior. Severe deformation occurs within highly localized regions of the material where the size scale of the deformation regions is of the same order as that of the microstructure. Furthermore, the experimental determination of the mechanical properties of biological membranes is not an easy task, as adequate experimental set up have to be devised, and the membrane anisotropy and large deformability have to be accounted for. The principal objective of this work is the derivation of the effective mechanical properties of biomembranes, considered as planar networks of connected filaments, obeying entropic elasticity. An important aim is to enhance our understanding of how the architecture (topology) of those networks impacts the overall mechanical properties, and to develop appropriate and accurate micromechanical models which can be used in a predictive manner. In this context, the asymptotic homogenized technique shall be herewith involved to derive the expressions (in closed form in the linear framework) of the equivalent moduli of biological membrane, versus the geometrical and mechanical micro-parameters. The derivation of the equivalent mechanical properties of cellular biological structures is further interesting in order to understand the somewhat peculiar observed behaviour (anisotropy, negative Poisson’s ratio [1, 2]) and to possibly evaluate the load bearing capacity of the membrane. Especially, expressions of the effective properties would allow relating the continuum to the discrete level, to understand the very microscopic origin of the mechanical behaviour of the cell-wall, and to assess the effect of the membrane topology. As stated in Lim et al. [3], mechanical models for cells are derived using either the micro/nanostructural approach or the continuum one. Although providing less insight into detailed molecular mechanical events, the continuum approach is easier and more straightforward to use in computing the mechanical properties of the cell and its response under biomechanical loading. Moreover, the established continuum mechanical model can provide details on the distribution of stress and strains induced in the cell and can be integrated in finite element simulations at the scale of the whole cell. However, the knowledge of the continuum behaviour of a membrane is challenging, as it is generally highly anisotropic due to unequal length and properties of the threads within the molecular network; furthermore, biological membranes are prone to large distensions and one should ideally consider nonlinear effects. Hence, micromechanical approaches are needed in order to bridge the scales and to provide a physically based constitutive law at a continuum scale, whereby the equivalent continuum properties emerge from the nano-structural parameters related to both the geometry and mechanics of the network.

40 Mechanics of Biological Membranes from Lattice Homogenization

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This chapter is organized as follows: filaments are modeled as undulated beam susceptible of large motion from their initial configuration. Accordingly, we first set up general beam models accounting for large displacements and large rotations (Sect. 40.2). The discrete homogenization method is exposed in Sect. 40.3, leading to a general micropolar continuum model. The non-linear equilibrium problem of a lattice is written in Sect. 40.4, which is solved by a Newton-Raphson iterative procedure. As an application, the equivalent mechanical properties of the peptidoglycan network are derived in Sect. 40.5, adopting first a linear framework, and extending next to the nonlinear situation described in Sect. 40.4. A classification of the obtained equivalent continuum as either a Cauchy or micropolar effective medium is provided for this membrane (Sect. 40.5), according to the relative value of the characteristic flexural length versus the network beam length. The effective moduli and Poisson’s coefficients of this orthotropic membranes are represented versus the configuration angle of this network. The nonlinear response is illustrated by the evolution of the Cauchy stress versus a suitable strain. Finally, some perspectives of future work are given in Sect. 40.6.

40.2 Beam Equations in the Geometrically Nonlinear Framework Biomembranes are viewed as networks of connected filaments (Fig.40.1), each being modeled as an undulated beam undergoing large displacements and rotations. We denote B0 the reference configuration of the beam under consideration. We also introduce the non-linear map, which takes every material particle at the position X to its new position x in the current configuration B.

Fig. 40.1 Biomembrane viewed as a network of connected filaments

The kinematics of the curved beam is firstly established, in order to derive expressions of the resultant and moments for a given beam element. Torsion and warping are not taken into account, which restricts the present formulation to a certain class of problems. The beam strains are defined and the relation to the Green-Lagrangian strains shall be shown; we introduce for that purpose an orthogonal basis system Ai with local coordinates {ξ1 , ξ2 , ξ3 }. The axis of the beam is initially along the vector

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Ai , with an arc-length parameter s = ξ1 ∈ [0, L] of the special curve. L is the curvilinear beam length. Thus, the cross-section of the beam lies in the plane sustained by the basis vectors {A2 , A3 }. Note that ai is defined in the current configuration and a1 is not the tangent vector of the deformed reference curve. Let define the basis vectors Ai and ai , which result from the following orthogonal transformations of the fixed cartesian vectors ei , as Ai = R0 (s)ei

(40.1)

ai = R(s, t)ei

in which R0 , R ∈ S O, the group of orthogonal transformations. It appears that R may be parameterized in terms of its associated axial vector ωI with ω = |ωI |, according to the Rodrigues rotation formula R = 1+

sin (ωI ) 1 − cos(ωI ) 2 Ω+ Ω ω ω2

(40.2)

with Ω = skew (ω). The position vectors of the undeformed and deformed cross-sections are given from the following kinematics assumption: X (s, ξ2 , ξ3 ) = X0 (s) + ξ2 A2 (s) + ξ3 A3 (s) x (ξ2 , ξ3 , s, t) = x0 (s, t) + ξ2 a2 (s, t) + ξ3 a3 (s, t)

(40.3)

where a1 is assumed to be piecewise constant. Based on the kinematic assumption (40.3), the tangent vectors G i = X,i and gi = x,i are derived as G1 = X0 + ξ2 A2 + ξ3 A3

g1 = x0 + ξ2 a2 + ξ3 a3

G 2 = A2

g2 = a2

G 3 = A3

g3 = a3

(40.4)

where (, ) denotes the customary symbol for differentiation with respect to the arclength. The derivative of the orthogonal basis systems are expressed using the vector products Ai = θ0 × Ai and ai = θ × ai , where θ and θ0 denote the so-called axial vectors. The pullback of the covariant basis vectors by the rotation operators R0 = Ai ⊗ ei and R = ai ⊗ ei yields F1 := RT0 G 1 = ε0 + κ0 × d F2 := F3 := where d = ξ2 e2 + ξ3 e3 .

RT0 G 2 RT0 G 3

= e2 = e3

f1 := RT g1 = ε + κ × d f2 := RT g2 = e2 f3

:= RT g

3

= e3

(40.5)


609

The strain and curvature vectors of the current configuration are successively defined as ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ x .a1 ⎥⎥⎥ ⎢⎢⎢ a .a3 ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ 2 ⎥⎥⎥ ε := RT x0 = ⎢⎢⎢⎢ x0 .a2 ⎥⎥⎥⎥ , κ := RT θ = ⎢⎢⎢⎢ a3 .a1 ⎥⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ x0 .a3 a1 .a2

(40.6)

The corresponding strains and curvature in the reference configuration read ε0 := RT0 X0 and κ0 := RT θ0 respectively. The Green-Lagrangian tensor E = E i j G i ⊗ G j , with the dual basis vectors Gi therein, with Gi .G j = δi j , is elaborated in component form as follows ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ E 11 ⎥⎥⎥ ⎢⎢⎢ 1 (g11 − G11 ) ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ 2 ⎥⎥⎥ E = ⎢⎢⎢⎢ 2E12 ⎥⎥⎥⎥ = ⎢⎢⎢⎢ (g12 − G12 ) ⎥⎥⎥⎥ ⎢⎣⎢ ⎥⎦⎥ ⎢⎣⎢ ⎥⎥⎦ (g13 − G13 ) 2E13

(40.7)

with the metric coefficients Gi j = G i .G j = Gi .R0 RT0 G j = F i .F j and gi j = gi .g j = gi .RRT g j = fi . f j in the reference and current configuration respectively. The Green-Lagrangian strain tensor is work conjugate to the second PiolaKirchhoff stress tensor S i j . Within the present beam theory, the components S 22 , S 33 , S 23 are neglected. By considering an isotropic constitutive behaviour, one can calculate the resultants of forces and moments as follows: N= S 11 ds Mx = (ξ2 S 13 − ξ3 S 12 ) ds Γ Γ Ty = S 12 ds My = ξ3 S 11 ds (40.8) Γ Γ Tz = S 13 ds Mz = −ξ2 S 11 ds Γ

Γ

After lengthy calculations and by considering only the first order expansion of the Taylor series of the trigonometrical functions, the forces and moments resultants can be written as follows: ⎛ ⎞ &⎟⎟ % & ⎜⎜⎜ 1 % 2 2 & ω23 1 % 2 2 2 2 N = E s A ⎜⎜⎝u + u + v + − ε01 + ε02 + ε03 ⎟⎟⎟⎠ + E s Iz ω2 3 − κ03 2 2 2 T y = GA (|ω3 | (1 + u) + v − ε02) T z = −GAε03 M x = 0,

My = 0 % ' & ( Mz = −E s Iz ω3 1 + u − |ω3 |v + ε01 κ03

(40.9)

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Those expressions are written in a general case and exhibit a non-linear elastic behaviour. In order to apply the asymptotic homogenization method, some simplifications are made. We consider that all beams are rectilinear in their reference state, and suppose that no initial deformations or initial curvatures are present, implying that ε01 , ε02 , ε03 , κ03 are nil. From the continuous beam equations (40.9), we deduce the discrete relations by replacing the primed variable by their finite differences, in order to transform the initial continuous problem into a more convenient system of discrete expressions in view of the asymptotic homogenization method. Thus, Eqs (40.9) can be rewritten as ⎛ ⎛% ⎞ & % & ⎞ b⊥ 2 ⎟ ⎜⎜⎜ ⎜⎜⎜ ΔU · eb 2 2⎟ b ⎟ ⎟ ΔU · e ⎟ ψ 1⎜ ⎜ ΔU · e ⎟⎟⎟⎟ + c ⎟⎟⎟⎟⎟ N = E s A ⎜⎜⎜⎜ b + ⎜⎜⎜⎜ + ⎟ b 2 2 ⎝ l ⎠ 2 ⎟⎠ 2 ⎝ (l ) l 2 Δψ +E s Iz b &⎞ % &⎞ ⎛ ⎛l % (40.10) ⎜⎜⎜ ⎜⎜⎜ ΔU.eb ⎟⎟⎟ ΔU · eb⊥ ⎟⎟⎟ ⎜ ⎜ ⎟ ⎟ ⎟⎟ + ⎟⎠⎟ T y = GA ⎜⎝⎜ψc ⎜⎝⎜1 + lb ⎠ lb % & % &⎞ ⎛ ⎛ ⎞ ⎜⎜⎜ Δψ ⎜⎜⎜ ΔU · eb ⎟⎟⎟ ΔU · eb⊥ ⎟⎟⎟ ⎟⎟⎟ − ψc ⎟⎟⎟ Mz = −E s Iz ⎜⎜⎜⎝ b ⎜⎜⎜⎝1 + ⎠ ⎠ l lb lb In those expressions, A is the beam section, lb the beam length, Iz the quadratic moment of the considered beam, eb and eb⊥ are respectively the unit vector director of the beam and the transverse vector. E s and G are the Young and shear moduli of the considered beam respectively. ψc is the rotation of the central node of the beam (40.2). Observe that the obtained expression of the moment in (40.10) is an extension to higher order terms of the moment given in [4].

40.3 Discrete Homogenization of Networks The discrete homogenization method is a mathematical technique to derive the equivalent continuous medium behaviour of a repetitive discrete structure made of identified elementary cells. This technique is inspired from the homogenization of periodic media developed thirties years ago by Sanchez, 1980, Bakhvalov and Panasenko, 1984, Panasenko 1983 and more recently applied by warren and Byskhov [4] and Mourad and Caillerie [5]. It has been also combined with the energy method by Pradel and Sab and applied to discrete homogenization. More details about the method can be found in [6]. The discrete homogenization method consists in assuming asymptotic series expansions of both the node displacements, tension and external forces as successive powers of a small parameter labelled #, defined as the ratio of a characteristic length of the basic cell to a characteristic length of the lattice structure. Those expansions are then inserted into the equilibrium equation, conveniently expressed in weak form. The balance equation of the nodes, the


611

e2λ=Y

2

force-displacement relations and the moment-rotation relations of the beams are developed by inserting those series expansions and by using Taylor’s expansion of finite differences. The discrete sums are finally converted in the limit of a continuous density of beams into Riemann integrals, thereby highlighting continuous stress and strain measures. The calculations have been completed for a quite general truss and the results give a general and closed form expression of elastic properties in the linear framework. The method has given rise to implementation into a dedicated software, and can handle complicated membranes with an arbitrary topology. For more technical details related to the asymptotic homogenized method, one can refer to the non-exhaustive list of publications [4, 7–9]. Various analyses dealing with the impact of irregularity of the unit cell have been performed by many authors; they prove that the irregularity and natural variation of the unit cell have a small impact on the effective properties, not exceeding 9% in the case of cellular material [10,11] and 4% in the case of biological membranes. Thus, biological membranes can be assumed quasi repetitive periodical networks of threads representing the molecular chains within a unit cell, with a topology of the unit cell specific to each type of biological membrane. In this section, we present a simple model based on [12] work to get the expression of the force and moments resultants. The expressions of resultants are, here, restricted to small deformations from the initial shape. The motion of each node from the unit cell is represented by a displacement u in the x-direction, v in the y-direction and a rotation φ in z-direction. Thus, the motion of the point O/E is defined by the triad of vectors (u(O/E(b)), v(O/E(b)), φ(O/E(b))), the motion of the extremity point by the latter triad plus a small variation. A representative beam is shown in Fig.40.2, along with the forces and moments acting at its extremities.

e 2 =j

b be T= T

λ =Y 1

e1

e1 =i

cartesian coordinates

curvilinear coordinates

de

O(b)

Μο ))

b u(O(

am formed be

φo b To

b e e und

form

b N

φe

E(b)

Me b)) u(E(

C(b) b e

b Tt

ed b

eam

Mo and Me worn by e 3 =eλ 3

Fig. 40.2 Kinematic and static parameters of a lattice beam

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40.3.1 Simplified Beam Model In view of deriving a simplified beam model from the previous general one, for the purpose of an easier treatment and numerical implementation, we neglect the nonlinear part of the strains (and the coupling term between displacement and rotation) in Eq. (40.10). The normal and transverse efforts, and the moment at the beam extremities can accordingly be successively expressed versus the kinematical nodal variables as, see [6, 13]: EsA % bε b & ΔU · e lb & 12EsIz % b b⊥ & lb % O(b)ε ε E(b)ε Tt = ΔU · e − ε φ +φ 2 (lb )3 b 2 & 1 % & 12EsIz 2 (l ) % O(b)ε E(b)ε bε b⊥ M O(b)ε = ε 2φ + φ − ε ΔU · e 6 2 (lb )3 & 1 % & 12EsIz 2 (lb )2 % O(b)ε E(b)ε E(b)ε bε b⊥ M = ε φ + 2φ − ε ΔU · e 6 2 (lb )3 Nε =

(40.11)

M O(b)ε and M E(b)ε are the moment at the origin and the end positions of a generic beam respectively. Details related to the asymptotic expansion of the kinematical variables can be found in [6]. Let notice that it is necessary to retain second-order terms in the series expansions in order to take into account micropolar effects, as stated in [4].

40.3.2 Stress and Couple Stress Vectors In this section, we expose the main steps of the asymptotic homogenization technique considering micropolar effects; the equilibrium equations for the resultant and moment to be next written do not rely on a specific beam model, and one can thus consider either the general beam model (40.10), or its simplified version, (40.15). The equilibrium of forces for the whole lattice writes in virtual power form and after asymptotic developments as ' ( T εb vε (O(b)) − vε(E(b)) = 0 (40.12) νi ∈Z2 b∈BR

with v(.) a virtual velocity field choosing to vanish on the edges. The vector of effort T b decomposes into a normal and a transverse contribution as T b = N b eb + T tb eb⊥

(40.13)


613

with the normal and transverse efforts previously given in Eq. (40.15). The discrete equilibrium of moments expresses in virtual power form as the following sum over all lattice nodes

% & M O(b) · w(O(b)) + M E(b) · w(E(b)) + lb eb ∧ T εb · w(C(b)) = 0

(40.14)

νi ∈Z2 b∈BR

The director vector eb (unit length, Fig.40.2) and the beam lengths remain fixed under the adopted small strain framework. The discrete asymptotic technique requires the development of all variables as Taylor series vs. the small parameter ε, namely the beam length lεb , the beam width tεb , the lattice thickness eεb , the displacement uεn and the rotation at the lattices nodes φεn (they constitute the kinematic variables). The Bernoulli beam model is considered in this work. From the results of [9], one can express the beam length as follows lb = lb0 + εlb1 + ε2 lb2 + · · · + ε p lbp

(40.15)

The truss under consideration is completely defined by the positions of the nodes and their connectivity. Each beam links two nodes and is oriented so that it has an ˜ and an end node E(b). ˜ Although we can choose the origin node origin node O(b) as part of the reference cell, this is not necessarily the case for the end node, which nevertheless belongs to the next neighbouring cell. Moreover, we associate to each ˜ (Fig.40.2). Each extremity node has beam a node defined as its centre denoted C(b) two displacements in the two principal directions and one rotation in the plane (i, j). The beam kinematic parameters together with the efforts and moments are shown on Fig.40.2. The asymptotic development of the virtual velocity and rotation rate are next expressed: for any virtual velocity field vε (λ), a Taylor series development leads to vε (O(b)) − vε(E(b)) = vε (λε + εδib) − vε(λε ) ∂v(λε ) ib =ε δ ∂λi

(40.16)

wherein we have parameterized any point within the surface element representative of the membrane by curvilinear coordinates λi (they may be conceived as Lagrangian coordinates of the material points); this allows treating membranes which exhibit a local curvature in their reference state. The rotation rate is similarly expanded taking into account the central node of the beam wO(b)ε

= w(λ) & 1% w = wO(b)ε + wE(b)ε % & 2 ∂w(λ) i E(b)ε i w λ + εδ = w(λ) + ε δ ∂λi C(b)ε

(40.17)

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In the forthcoming development and for simplicity raisons, we assume all beams to have length l and uniform thickness t such that the ratio η = tl 1. Thus, we can simplify the expression of the bending and stretching stiffnesses by defining b the slenderness parameter η = tlb (which has a finite value). Inserting Eq. (40.11) and (40.16) into (40.12) and considering the aforementioned simplifications, one obtains after some developments and ordering following the successive powers of ε the equilibrium of forces as % & ε2 E s η ΔU1 · eb eb ε % & 1 ER (b) b⊥ ∂v (λ ) i R (b) + E s η3 ΔU 1 · eb⊥ − E s η3 lb φO + φ e δ 0 0 i 2 ∂λ (40.18) % & % + ε3 Esη ΔU2 · eb · eb + E s η3 ΔU2 · eb⊥ ε 1 ER (b) ∂φ0 ib b⊥ ∂v (λ ) i R (b) − E s η3 lb φO + φ + δ e · δ =0 1 1 i i 2 ∂λ ∂λ with ΔU 1 the first order difference of the displacement obtained versus ε as % & ER (b) OR (b) ∂u0 ib bε R (b) ΔU = ε u1 − u1 + iδ + ε2 u2ER (b) − uO (40.19) 2 ∂λ ; ;ΔU1 ΔU 2

The previous discrete equation is transformed into a continuous Riemann integral on the (surface) domain Ω when the ε tends to zero: for any enough small % parameter & regular function , the quantity ε2 g ενi can be interpreted as the Riemann sum νi ∈Z2 ) of an integral over Ω, Ω g (λ)dλ when ε → 0. Thus, the equilibrium equation in translation becomes after homogenization ∂v Si · i dλ = 0 (40.20) ∂λ Ω

evidencing a stress vector Si , which splits into a first and a second order contribution, viz Si = Si1 + εSi2 , with Eq. (40.21). Si1 =

% & E s η ΔU 1 · eb eb

b∈B R

% & 1 ER (b) R (b) + E s η3 ΔU1 · eb⊥ − E s η3 lb φO + φ eb⊥ 0 0 2 % & Si2 = E s η ΔU2 · eb eb b∈B 1 ER (b) ∂φ0 ib R (b) + E s η3 ΔU2 · eb⊥ − E s η3 lb φO + φ + δ eb⊥ 1 1 2 ∂λi

(40.21)


615

Similarly to previous developments, the moment equilibrium (40.14) is homogenized, inserting the asymptotic expansion (40.17) of the virtual rotation rate. After simplifications and passing to the limit ε → 0 in the discrete sum, the moment equilibrium equation after homogenization takes the form ∂w μi · i dλ = 0 (40.22) ∂λ Ω with the couple stress vector μi also identified on two orders, viz μi = μi1 + εμi2 , with

& (lb )3 % ER (b) R (b) φ0 − φO δib 0 12 b∈B b 3 ER (b) OR (b) ∂φ0 ib ib 3 (l ) i μ2 = Esη φ1 − φ1 + iδ δ 12 ∂λ b∈B μi1 =

E s η3

(40.23)

The very expression of the stress and couple stress vectors Si and μi in (40.20) and (40.22) is dependent on the unit cell topology and on the mechanical properties of the beam network. Observe that the homogenized equilibrium equations (40.20) % & and (40.22) involve virtual velocities (v, w), respectively equivalent to the rates u, ˙ φ˙ .

40.3.3 Stress Tensor and Micromoment Tensor In a second step, the equilibrium equations of the equivalent micropolar continuum are written in virtual power form, in order to highlight the stress tensor σ and the tensor of micromoment m as dyadic products of the evidenced stress and couple stress vectors with the gradient of the position vector with respect to the curvilinear coordinates. The following transformations from the Cartesian to the curvilinear coordinates λi are expressed, with R the position vector of any material point: ∂v ∂R = %xv · ; ∂λi ∂λi

∂w ∂R = %xw · ∂λi ∂λi

(40.24)

leading to the following equilibrium equation of the equivalent micropolar continuum % % & ∂v & ∂w i gσ · eλ · dλ + gm · eiλ · dλ = 0 Ω ; ∂λi Ω ; ∂λi (40.25) μi Si i ∂v i ∂w → S · i dλ + μ · i dλ = 0 ∂λ ∂λ Ω Ω with g the Jacobean of the transformation from cartesian to curvilinear coordinates.

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Hence, the last equation shows that the homogenized network obeys the same equilibrium equation as the equivalent continuum micropolar, with the stress and couple stress tensor identified to the following dyadic products: 1 ∂R σ = Si ⊗ i ; g ∂λ

1 ∂R m = μi ⊗ i g ∂λ

(40.26)

Using the symmetry properties of the lattice, we can simplify the expressions of the vectors of effort and couple stress. The general form of the constitutive equations of linear micropolar elasticity relating the stress and couple stress tensors to the strain and curvature tensors is as follows {σ} = [A] {ε} + [B]{κ} {m} = [C] {ε} + [D] {κ}

(40.27)

This form of the continuum constitutive law can presently be identified from the expressions of the homogenized stress and couple stress tensors together with the expressions of Si and μi : 1 ∂R 1 ∂R σ = S1 ⊗ i + ε S2 ⊗ i g g ∂λ ∂λ ; ; [A]{ε}

[B]{κ}

[C]{ε}

[D]{κ}

1 ∂R 1 ∂R m = μ1 ⊗ i + ε μ2 ⊗ i g g ∂λ ∂λ ; ;

(40.28)

We restrict ourselves to centro-symmetric unit cells in this work, which entails the vanishing of the coupling matrices [B] and [C], as shown in [14]. The expression of the stress and couple stress vectors substantially simplifies as follows

% & E s η ΔU1 · eb eb b∈BR % & ER (b) 3 b⊥ 1 3 b OR (b) + E s η ΔU1 · e − E s η l φ0 + φ0 eb⊥ 2 % b∈BR & b b⊥ ib = N1b eb + T t1 e δ

Si = Si1 =

% &3 lb ∂φ0 ib ib 3 R (b) μi = μi2 = Esη φ1ER (b) − φO + δ δ 1 12 ∂λi b∈B 1% & = M2ER (b) − M2OR (b) δib 2 b∈B b∈B

R

(40.29)


617

b , and M n , respectively, the first order normal and transverse effort and with N1b , T t1 2 the second order moment, obtained when expanding the expressions (40.11) versus the small parameter ε . Those expressions still involve the unknown displacements un1 , un2 and rotations φn0 , φn1 , which are determined for all nodes using the equilibrium equations (40.12) and (40.14). The early works of Euler and Bernoulli relative to beam mechanics suggest the novel idea to consider the displacement and rotations as independent quantities, as well as efforts and couple stresses. The idea of couple stress was explored, two centuries ago, by MacCullagh, 1839, Lord Kelvin, 1882-1890, Voigt, 1887 and was followed later by the Cosserat brothers. In the so-called micropolar or (Cosserat) theory, two degrees of freedom are present, namely the displacement and the microrotation fields (ui ,φi ). An alternative to the simple continuum model is the micropolar material developed by the pioneering authors Eringen and Suhubi (1964). Although the concept of a micropolar solid was fully developed one century ago by the Cosserat brothers (1909), it remained a purely conceptual object until man was able to produce and characterize solids (either artificial of natural) exhibiting such scale effects. Later on, theories incorporating couple stresses were developed using the full capabilities of modern continuum mechanics, [15]. As a consequence of the micropolar behavior, a size effect appears in torsion and bending, but not in tension; the Poisson’s ratio is unaffected. Note further that the use of general curvilinear coordinates allows to consider lattices having a curvature in their reference configuration, so that the homogenized beam network is in fact a thin anisotropic shell. The extension of the homogenization method to the nonlinear framework is next considered.

40.4 Non-linear Problem As biological membranes undergo thermal fluctuations, they are prone to large configurational changes. As already underlined, this modification is likely to appear due to signalling events leading to an adaptation of the membrane shape to environmental changes. We presently do not describe the chain of events leading to such a modification, but limit ourselves to modeling the net mechanical effect of such configurational changes. We shall accordingly extend the linear framework so far adopted and consider the impact of a variation of the lattice geometry on the equivalent moduli, and hence on the effective membrane behavior. We accordingly write down the non-linear equilibrium problem associated to the large perturbations of the network; the non-linearity is due to the beam directors and beam lengths changing with the applied loading. Forthcoming developements are in principle quite general in the sense they can be applied to a general beam model (40.10). In order to set the stage, let consider the equilibrium equations (40.12) and 0 (40.14). For every beam b ∈ B, one can write Bb0 = R ER (b) − ROR (b) + ∂R δ jb , ∂λ j lb0 = Bb0 and eb0 =

Bb0 . lb0

For a given λ and for each value of

∂R0 (λ) jb δ , ∂λ j

j = 1, 2, 3,

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one determine the quantities Bb0 , eb0 and calculate from specific beam model, N b0 , T b0 , M b0 , b ∈ B, allowing to calculate the stress and couple stress vectors Si0 and μi0 . In other terms, for an imposed deformation gradient at the continuum level G j , for each beam b ∈ B, one insert the quantities Bb0 = R ER (b) − ROR (b) + G j δ jb , b lb = Bb and eb = Blb into the considered beam equations (40.10) or (40.11). By differentiating the equilibrium equations (40.12) and (40.14), an incremantal form of the equilibrium is written. We introduce for this purpose a projection operator A = [I − eb ⊗ eb ], C = [I − 12 eb ⊗ eb ] and we define a rescaled resultant as Cb0 = Nbb0 , b ∈ B; the perturbation of the force equilibrium equation, (40.11), writes N l 0

%

% & & Cb0 (k) Bb(k) + N Cb0 (k) δBb(k) Δv + Cb0 (k) Bb(k) Δv δN N % b (k) & b∈B % b (k) & ∀v ∈ R3 , b∈B b (k) + δT t 0 eb⊥(k) + T t 0 δeb⊥(k) Δv + T t 0 eb⊥(k) Δv = 0 b∈B

b∈B

(40.30) with Δv. Similarly, the perturbation of the moment equilibrium (40.13) delivers

% & b (k) δM E(b)0 (k) wE(b) + δM O(b)0 (k) wO(b) + δ B ∧ T t 0 eb⊥(k) wC(b) + b∈B % b∈B% & & ∀v ∈ R3 , b (k) M E(b)0 (k) wE(b) + M O(b)0 (k) wO(b) + B ∧ T t 0 eb⊥(k) wC(b) = 0 b∈B

b∈B

(40.31) In the two previous perturbed equations, the variation of the geometry is accounted for by the perturbed quantities, successivly the beam director 1 CAδB lb &% &−1 1 % = b 2 Bb(k) ⊗ Bb(k) eb0 ⊥(k) ⊗ Bb(k) CAδB (l )

δeb = δeb⊥

(40.32) (40.33)

and length δlb = δB · eb + B · δeb = δB · eb + B ·

1 CAδB lb

(40.34) B [I + CA] · δB lb Those variations induce in turn the following perturbations of the normal effort =

Cb0 (k) Bb(k) dN b(k) [I + CA] · δB Bb(k) dlb lb & Cb0 (k) % b(k) 1 dN = b B ⊗ Bb(k) [I + CA] δBb(k) b l dl

Cb0 (k) Bb(k) = δN Cb0 (k) Bb(k) δN

(40.35)


619

and of the transverse effort δT t

b0 (k) b⊥(k)

e

=

& 1 dT t b0 (k) % b⊥(k) e ⊗ Bb(k) [I + CA] δBb(k) b b l dl

(40.36)

By inserting Eqs (40.32-40.36) into the equilibrium relation (40.30), one obtains the following perturbed equilibrium equation in translation: ⎞ ⎛⎜⎜ 1 d N & Cb0 (k) % b(k) ⎟⎟⎟ b(k) b(k) b0 (k) ⎟ ⎜⎜⎝ C B ⊗ B [I + CA] + N ⎠ δB Δv b b l dl b∈B % & Cb0 (k) Bb(k) δBb(k) Δv + N b∈B ⎛ ⎜⎜ 1 dT t b0 (k) % & ⎜⎜⎜ + eb⊥(k) ⊗ Bb(k) [I + CA] (40.37) ⎝ lb dlb b∈B ⎞ &% &−1 ⎟⎟ T t b0 (k) % + b 2 Bb(k) ⊗ Bb(k) eb0 ⊥(k) ⊗ Bb(k) CA⎟⎟⎟⎠ δBb(k) Δv (l ) % b (k) & + T t 0 eb⊥(k) Δv = 0 b∈B

One can remark it is easier to write the perturbed moment equilibrium by multiplying Eq. (40.31) by the vector Bb(k) , similarly to (40.37): δM E(b)0 (k) Bb(k) wE(b) + δM O(b)0 (k) Bb(k) wO(b) b∈B % & b (k) + δ B ∧ T t 0 eb⊥(k) Bb(k) wC(b) b∈B % & (40.38) + M E(b)0 (k) Bb(k) wE(b) + M O(b)0 (k) Bb(k) wO(b) b∈B % & b (k) + Bb(k) ∧ T t 0 eb⊥(k) Bb(k) wC(b) = 0 b∈B

In the moment equilibrium equation (40.31), one further uses the following incremental expressions

δM E/O(b)0 (k) Bb(k) =

& dM E/O(b)0 (k) % b(k) B ⊗ Bb(k) [I + CA] δBb(k) b dl

% & b (k) δ Bb(k) ∧ T t 0 eb⊥(k) Bb(k) = F 1 + F2 + F3 with

% & b (k) F 1 = δBb(k) ∧ T t 0 eb⊥(k) Bb(k) % b (k) & = T t 0 eb⊥(k) ∧ Bb(k) δBb(k)

(40.39) (40.40)

(40.41)

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% & b (k) F 2 = Bb(k) ∧ δT t 0 eb⊥(k) Bb(k) ⎛ ⎞ b (k) ⎟⎟ % & 1 dT t 0 ⎜⎜⎜⎜⎜ b(k) b(k) ⎟ = b ∧ B ⎟⎟⎟⎟⎠ eb⊥(k) ∧ Bb(k) [I + CA] δBb(k) = 0 ⎜⎜⎝B ; b l dl

(40.42)

0

% & F 3 = Bb(k) ∧ T t b0 (k) δeb⊥(k) Bb(k) ⎛ ⎞ ⎜⎜ ⎟⎟ % & b0 (k) ⎜ t ⎟⎟ eb⊥(k) ∧ Bb(k)CA δBb(k) = 0 b(k) b(k) ⎟ = − (lb1)2 dTdlb ⎜⎜⎜⎜B ∧ B ⎝;⎟⎟⎠

(40.43)

0

The moment equilibrium equation expresses finally as 1 dM E(b)0 (k) % & Bb(k) ⊗ Bb(k) [I + CA] δBb(k) wE(b) b b l dl b∈B % b (k) & t 0 + T eb⊥(k) ∧ Bb(k) δBb(k) wC(b) b∈B

1 dM O(b)0 (k) % & + Bb(k) ⊗ Bb(k) [I + CA] δBb(k) wO(b) b b l dl b∈B % & b (k) b(k) + B ∧ T t 0 eb⊥(k) Bb(k) wC(b) b∈B % & + M E(b)0 (k) Bb(k) wE(b) + M O(b)0 (k) Bb(k) wO(b) = 0

(40.44)

b∈B

Equations (40.37) and (40.44) provide the perturbed problem to be solved by an iterative Newton-Raphson technique. Let notify that this problem has a solution up to within a translation (and a rotation), thereby one has to fix at least one node of the structure to prevent the tangent stifness matrix from becoming singular. The solution of the localization problem at the unit cell scale, defined as the set of variables Bb(k+1) = Bb(k) + δBb(k) of the system formed by Eqs (40.37) and (40.44), provides the non-linear constitutive law of the unit cell. It can be considered as the (nonlinear) localization problem at the unit cell level, and has to be combined with the homogenization step of the network, realized in Sect. 40.3, providing the expression of the Cauchy stress and couple stress versus the actualized network topology at each loading increment. The main steps of the algorithm are summarized in Fig. 40.3. The so obtained constitutive law is akin to a micropolar hyperelastic anisotropic continuum (the state of anisotropy is continuously evolving according to the variation of the unit cell topology); it has in general to be numerically determined.


621

Fig. 40.3 Two-steps localization homogenization procedure for the construction of the effective non-linear constitutive law

40.5 Application: Equivalent Properties of the Peptidoglycan Cell Wall The peptidoglycan network is built from two non-equivalent chains, as shown on the face view of a Sect. (Fig. 40.4) (from [16]): sugar rings run in one direction and peptide strings form transverse links. It is believed that this molecular anisotropic organization is dictated by design principle of the cell, such that the stiffer chains act as reinforcement in the direction that shall sustain the maximal stress. Let note that the knowledge of the mechanical properties of peptidoglycan is of importance for understanding bacterial growth and form.

(a)

(b)

Fig. 40.4 Peptidoglycan network a) Dimensions of network chains (peptides and sugar rings) [16], b) Face view of the section of the peptidoglycan network and definition of the geometrical model

The peptide and glycan chains are modelled as regular beams with a regular circular cross section with radii respectively equal to 0.5nm and 1nm, [17]; the angle θ is chosen as a descriptor of the topology of the glycan network (Fig. 40.4b); it varies in-between 5 and 20 degrees, according to the dimensions of the molecular chains of the unit cell shown in Fig. 40.4a. The Young moduli are elaborated from the persistence length according to the relation ξ p = KEIB Tz [17], wherein E, Iz , T, KB are the tensile modulus, the quadratic moment (dependent on the beam cross section), the absolute temperature and Boltzmann constant respectively. The peptide

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chain is endowed with the classical properties of entropic springs, given that its end-to-end length in the network l pep = ree = 1.3nm is less than its contour length Lc = 4.2nm. The persistence length of the peptide string can be obtained from the 2 = 2ξ simplified relation ree p,peptide Lc , (the bracket denotes the average value) giving ξ p,peptide = 0.2nm. The glycan chains are comparatively much stiffer, with ξ p,glycane 10nm [18] and lgly = ree = 2nm. The average Young moduli of the glycan and peptide chains are then obtained at low temperature (T = 273K), respectively as Ygly = 4.799 × 107 J/m3 and Y pep = 1.53 × 107 J/m3 . The whole lattice is generated from the repetition of the unit cell shown in Fig. 40.4 thanks to two periodicity vectors defined in the Cartesian basis. After calculations, we extract the homogenized moduli of the hexagonal lattice from the equivalent stiffness matrix, expressed versus the geometrical and mechanical lattice parameters. Four mechanical parameters characterize the derived effective planar micropolar model: two tensile moduli E 1 ∗, E2 ∗, one shear modulus G∗, one Poisson’s ratio ν12 or ν21 , as the structure is shown to be orthotropic for all values of θ (the effective stiffness matrix is symmetrical), and two additional (micropolar) moduli γ∗ and κ. Those properties are next expressed versus the Young modulus, the slenderness ratio η and the configuration variable θ: η3 lgly Ygly cos (θ) && (% 1 − cos2 (θ) + η2 cos2 (θ) l pep + lgly sin (θ) % & η3 Ygly Y pep l pep + lgly sin (θ) % E2 ∗ = ' (& cos (θ) 2Ygly η2 l pep + lgly Ygly cos2 (θ) + η2 lgly Y pep 1 − cos2 (θ) % % & & G∗ = η3 Ygly Y pep l2gly l pep + lgly sin (θ) cos (θ) 2η2 l2gly Y pep sin (θ) l pep E 1 ∗ = %'

+2Ygly cos2 (θ) l3pep + l2pep Y pep lgly cos2 (θ) + η2 l3gly Y pep + l2pep η2 lgly Y pep −1 −l2pep η2 lgly Y pep cos2 (θ) % & lgly η2 − 1 sin (θ) cos(θ) & ν12 = ' (% 1 − η2 cos2 (θ) − cos2 (θ) l pep + lgly cos (θ) % & % & η2 − 1 sin (θ) l pep + lgly sin (θ) ν21 = 2 2 2 2 2η Ygly l pep + lgly Y pep cos (θ) + lgly η Y pep cos (θ) 2 2 2 2 2 κ = l pep cos (θ) + l pep η sin (θ) + η lgly sin (θ) cos (θ) % +η2 lgly sin3 (θ) η3 lgly 2 Y pep Ygly 2 cos2 (θ) l pep 3 Ygly & −1 +η2 lgly 3 Y pep + 2Ygly η2 sin2 (θ) l pep 3 cos (θ) 3 3 1 cos (θ) Ygly η lgly γ∗ = 12 l pep + lgly sin (θ)

(40.45)

We plot the evolution of the homogenized elastic properties versus the angle θ in Fig. 40.6, with a range of values chosen in the interval [5◦ − 30◦]. The equivalent tensile modulus along the x direction exhibits a maximum for a square topology (θ is nil) and shows strong variations around this maximum (it is an even function


(a)

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(b)

Fig. 40.5 a) Evolution of effective tensile moduli versus the geometrical parameter θ for the peptidoglycan. b) Shear modulus

of θ). The shear modulus increases continuously in the selected range of variation of θ. The calculated effective properties appear to be very close to the experimental values and to those calculated by other approaches. In [17], the shear modulus is about G∗ = 3.6 · 106 J/m2 , close to the homogenized value G∗ = 2 − 3 · 106 J/m2 . Moreover, the peptidoglycan has two different effective Young moduli in the two principal directions, respectively E1 ∗ = 1 − 3.2 · 107 J/m3 and E 2 ∗ = 3 − 5 · 106 J/m3 . The homogenized value is further close to measurements [17], which give E∗ = 2 −3 · 107 J/m3 . A (symbolic) calculation based on the effective properties shows that the network is orthotropic (viewed as a continuum), which sets a relation between the two tensile moduli and the two Poisson’s ratio. The contraction coefficient ν21 increases monotonously versus θ, since the networks contracts more when the beams align in the y direction. Poisson’s ratio ν12 shows a complex evolution pictured in Fig.40.6, increasing through a maximum (the transverse contraction is maximal for an angle around 20 degrees) and decreasing thereafter. Considering a fixed geometry (the angle θ is fixed) and eliminating the slenderness ratio η viewed as a paramter, the scaling behavior of the equivalent elastic moduli versus the effective density is obtained, showing a nonlinear monotonous increase (Fig.40.7). In order to assess the relevance of a micropolar continuum model, a micropolar characteristic length in bending lchar is evaluated versus the beam length, the angle θ and the slenderness ratio η in Eq. (40.46). The characteristic length is defined from the coefficients of the stiffness matrix (considering γ∗ an orthotropic behavior) in terms of the general definition l2char = 2(2μ∗+κ∗) , hence choosing here the x-direction l2xchar =

1 b 2 η2 − 3 sin (θ) + η2 sin (θ) + 3 (l ) 48 1 + sin (θ)

(40.46)

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(a)

(b)

Fig. 40.6 Equivalent Poisson’s ratio versus the geometrical parameter θ for peptidoglycan network

(a)

(b)

Fig. 40.7 Variation of the elastic moduli versus the density for different values of parameters θ = (5◦ , 10◦ , 15◦ , 20◦ ) and different slenderness ratio η

Two possible choices of the equivalent continuum have been considered, namely the Cauchy continuum and the micropolar continuum (the first one being a subcase of the latter), including additional rotational degrees of freedom compared to the former continuum model. The choice of the more appropriate continuum seems difficult a priori, since the equivalent properties may show unusual values as highlighted by previous results. Hence, we rather elaborate an a posteriori criterion for the choice of the equivalent continuum, relying on the comparison of the characteristic length of the micropolar continuum lchar to the beam length lb and to a macroscopic length at the scale of the lattice L: • when lchar ≤ lb or lchar L, micropolar effects are unnoticeable at the macroscopic scale, and one may adopt a Cauchy continuum;


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• when lb ≤ lchar ≤ L, or when lchar ≈ L, micropolar effects have an impact at the macroscopic scale and it is justified to use a micropolar effective continuum. The ratio of the characteristic micropolar length l xchar (equation (40.46)) to the characteristic unit cell size 2lb cos (θ) is plotted for the peptidoglycan network, with the previously adopted mechanical parameters (Fig.40.8).

Fig. 40.8 Evolution of the characteristic length of the peptidoglycan network versus θ

For classical configurations of the network (with positive θ), previous ratio is less than unity, showing that micropolar effects may be neglected (they act at a length scale lower than the unit cell size). For strongly re-entrant hexagonal lattices corresponding to angles θ ≤ −35◦, the ratio is larger than unity, thereby meaning that the microrotation effect becomes noticeable at a macroscopic scale, hence a Cosserat medium is appropriate for such networks according to previous criterion. A negative θ (corresponding to a so-called re-entrant lattice) entails a negative Poisson’s ratio (Fig.40.6b) - although not related to micropolar effects - with values becoming as negative as −5 when θ reaches − π3 . The mechanism responsible for the negative Poisson’s ratio is the deployment of the re-entrant beams in the direction transverse to the applied loading. In addition to the persistence length used to describe the correlation of chain segments orientations at the molecular scale, an additional length at the microscopic scale is introduced, the micropolar length, describing flexion effects at the scale of the unit cell. The present study shows that micropolar effects naturally emerge as an outcome of the homogenization procedure, and the micropolar length may have an impact at the scale of the whole membrane for certain topologies of the filament network. Biological materials such as bone were already reported to exhibit size effects amenable to a Cosserat model [19], slender specimens having a higher apparent

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stiffness than thick ones. We presently show that biological networks may also exhibit such size effects and provide a methodology based on multiscale modeling to quantify those effects. The rationale for introducing micropolar effects in biological networks is that the filaments in the network are generally convoluted (the end-toend distance may be much smaller than the contour length); hence, bending and twisting due to local rotations may be the dominant deformation modes of those networks. The established non-linear scheme is next applied to evaluate the stress-strain response of the peptidoglycan membrane.

6

3

x 10

σ11 (Pa) 2.5

2

1.5

1

ε11 0.5 0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Fig. 40.9 Stress strain response of the peptidoglycan network

Accordingly, the tensile behavior of this network in the x-direction is simulated (at the nano-level) by increasing the deformation gradient Gi applied at the vertical right edge of the unit cell. The non-linear response shows a geometrical hardening (the slope is increasing) in the x-direction traducing the progressive orientation of the threads into the x-direction when the unit cell length is increased (Fig. 40.9). A kinematic locking will arise for a certain strain value (about 15%).

40.6 Conclusion and Discussion Biomembranes have here been treated as single layers, which is a strong approximation as to the in-plane shear behaviour (the in-plane stretch is much less influenced by this assumption). Biological membranes consist indeed in two leaflets endowed with a relative sliding mobility. This is however a common viewpoint in the literature, since the molecular based models treat the spectrin network and lipid


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membrane (which rather has a fluid like behavior) in a globalized manner, [11]. The interest of the developed homogenization technique lies in its flexibility and capability to handle any planar lattice, with the molecular chains assimilated to nonlinear beams. In the elastic case, and for small perturbations, the effective properties are derived as closed-form expressions of the geometry of the molecular chains (characterized by the persistence length, the quadratic moment) and mechanical properties (Young modulus). The persistence length is the geometrical parameter at the nanoscale that may be used as an input to account for the entropic elasticity of the network chains. The beam length in the network model may be assimilated to the average end-to-end distance of a given chain, which is known to fluctuate with temperature and depends upon the persistence length [17]. This effect is only indirectly incorporated into the model, by changing the beam length according to temperature and the estimated value of the persistence length. The role of fluctuations of the chain segments are however not considered in the present model. The role of the fluctuations of the geometry and properties (the Young modulus is linked to the persistence length, itself affected by the temperature) over long distances may be assessed from a sensitivity analysis using the closed form expressions of the effective moduli. Going even further along this idea, one may even evaluate the nanoscopic parameters of a given membrane in a kind of inverse approach from measurements performed at a more macroscopic scale, using the expressions of the moduli to make such an adjustment. The further interest of the homogenization method is that it delivers the full compliance (or rigidity) matrix, reflecting the sometimes complex and evolving anisotropy of the so-built equivalent continuum (the material symmetry group of the discrete lattice is included in the material symmetry group of the homogenized membrane [14]). Such complex constitutive laws may then further be used in multiscale simulations of membranes submitted to given loadings, [20]. The two considered effective media, of Cauchy or Cosserat type, provide the same tensile moduli. The Cosserat media provides additional micropolar moduli, and an associated characteristic length, the importance of which deciding upon the scale at which micropolar effects are noticeable. The peptidoglycan network leads for regular configurations (for low temperatures) to a classical Cauchy continuum; the re-entrant hexagonal network of the peptidoglycan shows a characteristic length larger than the size of the unit cell, hence micropolar effects have in this case an impact at the scale of the whole network, and have to be accounted for in the homogenized constitutive law. Due to micropolar effects, the shear modulus decreases μ∗ = μ − κ/2 and becomes even negative. Hence, biological networks will deform more easily in shear and flexion/torsion, especially in the large deformation regime. The Cosserat (micropolar) continuum model is therefore relevant for modeling membranes in specific configurations for which such exotic effects (negative contraction) become pronounced. Although the likelihood of such configurations may be low according to the induced energy cost and the resulting low probability (expressed by the factor exp (−ΔE/KBT )), they may locally be present due to thermal fluctuations. Although lattices have been considered planar, the present homogenization technique allows modelling lattice shells (membranes endowed with a curvature in

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their natural state), provided suitable curvilinear coordinates are adopted. Micromechanical analyses of biological networks based on those complex constitutive laws pave the way to FE based simulations over whole cells submitted to loadings reflecting the effect of their biological environment, or in order to analyse the deformation of biological membranes and cells under a controlled loading.

References 1. Boey, S.K., Boal, D.H., Discher, D.E.: Simulations of the erythrocyte cytoskeleton at large deformation. Biophysical Journal 75, 1573–1583 (1998) 2. Boal, D., Seifert, U., Shillcock, J.C.: Negative Poisson ratio for two-dimensional networks under tension. Physical Rewiew E 48, 4274–4283 (1993) 3. Lim, C.T., Zhou, E.H., Quek, S.T.: Mechanical models for living cells-a review. Journal of Biomechanics. 39, 195–216 (2006) 4. Warren, W.E., Byskov, E.: Three-fold symmetry restrictions on two-dimensional micropolar material. European Journal of Mechanics A/Solid 21 779–792 (2002) 5. Mourad, M., Caillerie, D., Raoult, A.:, in Computational Fluid and Solid Mechanics, ed. by Klaus-Jürgen Bathe. Proceeding, vol 2 (Elsevier, Boston, Etats-Unis, 2003), p. 1779–1781 6. Dos Reis, F., Ganghoffer, J-F.: Discrete homogenization of architectured materials: implementation of the method in a simulation tool for the systematic prediction of their effective elastic properties. Technische Mechanik 30, 85–109 (2010) 7. Pradel, F., Sab, K.: Homogenization of discrete media. Compte rendu de l’Académie des sciences II(b), 699–704 (1998) 8. Tollenaere, H., Caillerie, D.: Continuous modeling of lattice structures by homogenization. Advances in Engineering Software 29, 699–705 (1998) 9. Mourad, A.: Description topologique de l’architecture fibreuse et modélisation mécanique du myocarde. Institut National Polytechnique de Grenoble (2003) 10. Alkhader, M., Vural, M.: Mechanical response of cellular solids: Role of cellular topology and microstructural irregularity. International Journal of Engineering Science 46, 1035–1051 (2008) 11. Dao, M., Li, J., Suresh, S.: Molecularly based analysis of deformation of spectrin network and human erythrocyte. Material Sciences and Engineering C 26, 1232–1244 (2006) 12. Askar A., Cakmak A. S.: A structural model of a micropolar continuum. International Journal of Engineering Sciences. 6, 583–589 (1968) 13. Dos Reis, F.: Homogénéisation automatique de milieux discret périodique. Applications aux mousses polymère et aux milieux auxétiques. Institut National Polytechnique de Lorraine (2010) 14. Trovalusci, P., Masiani, R.: Material symmetries of micropolar continua equivalent to lattices. International Journal of Solids and Structures 36, 2091–2108 (1999) 15. Mindlin, R.D.: Stress functions for Cosserat continuum. International Journal of Solids and Structures 1, 265–271 (1965) 16. Koch, A.L., Woeste, S.: Elasticity of the sacculus of Escherichia coli. Journal of Bacteriology 174, 327–341 (1992) 17. Boal, D.H. 2002. Mechanics of the Cell. The Press Syndicate of the University of CAMBRIDGE: CAMBRIDGE University Press. 18. Stokke, T., Brant, D.A.: The reliability of wormlike polysaccharide chain dimension estimated from electron micrographs. Biopolymers 30, 1161–1181 (1990) 19. Lakes, R.: On the torsional properties of single osteons. Journal of Biomechanics 28, 1409– 1410 (1995) 20. Feyel, F., Chaboche, J-L.: FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Computer Methods in Applied Mechanics and Engineering 183, 309–330 (2000)

Chapter 41

Biological and Synthetic Membranes: Modeling and Experimental Methodology ¯ e-Makovska and Holger Steeb Rasa Kazakevi˘ciut˙

Abstract This work is concerned with the problems of constitutive modeling and experimental testing of perfectly flexible membranes and membranes with bending stiffness. The elastic response of such membranes is the main topic of the work, but the phenomenon of stress softening of elastomeric membranes is also shortly discussed. Special attention is devoted to the methodology of determining response functions and involved material parameters in the respective constitutive models. It is shown that the non-linear response of isotropic perfectly flexible membranes may be deduced from the inflation test provided that the complete meridian profiles of an inflated membrane are measured at different pressure levels. For linear and semi-linear constitutive models, a efficient methodology for the identifying possible types of anisotropy is presented for both the extensional and bending responses of the membrane. This methodology enables to determine the complete set of material constants for each type of anisotropy. Keywords Membranes · Constitutive models · Experimental methodology · Inflation test · Anisotropy

41.1 Introduction Theories of two-dimensional continua have long been used to model diverse phenomena ranging from bioelasticity and fluid capillarity to elasticity of biological and synthetic membranes, and mechanics of certain structural networks. Although general theories for such continua have long been available [1, 23], the specific experimentally-determined constitutive relations for many membranes are R. Kazakevi˘ci¯ut˙e-Makovska (B) · H. Steeb Mechanics-Continuum Mechanics, Ruhr-University Bochum, D-44780 Bochum, Germany e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 41, © Springer-Verlag Berlin Heidelberg 2011

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not available or not sufficiently tested. Also, the performance of many biological membranes under specific loading conditions is not well-understood [2, 3, 7, 20]. There are different methodologies to determine constitutive relations in continuum mechanics of three-dimensional bodies and specific forms of the relevant response functions can be determined directly from selected well-designed (homogenous) tests [25]. As a rule, these methods are based on specific experimental programs and certain specimen geometries. In case of synthetic materials, samples may be fabricated in any required geometry. In contrast, the size and the shape of biological and even certain synthetic membranes are dictated by nature or their intended use. In effect, the choice of specimen geometries and the possible testing methods are very limited [6, 8]. Therefore, there is a need to identify classes of experiments that allow for a direct determination of constitutive relations while preserving the natural geometry of such membranes. In this work, it is discussed how specific forms of response functions for biological and synthetic membranes can be identified by direct and indirect measuring deformations of the membrane together with its curvature geometry. The presented approach forms a basis for the development of phenomenological constitutive models. Experiments with membranes on a macroscopic level yield the material properties which characterize the membrane as a continuum material. However, in biological membranes the material can be considered as a continuum only in the two-dimensions describing the membrane surface; the membrane retains its molecular character in the thickness dimension [7]. Consequently, it is considered the twodimensional models for membranes without any reference to the three-dimensional theories of continua. The most general problem considered in this paper is that of a flexible biological membrane having an extensional as well as a flexural stiffness and undergoing possibly inelastic deformations due to a evolution of its physical structure at the microlevel. An appropriate model rich enough to account for these effects may be built by combining the well-developed mechanical bending theory for thin shells [1, 22, 23] and theoretical concepts of micro-forces and micro-stresses such as that used in modeling of flexible membranes with voids [12] or thin polymeric films with microstructure [21]. Here it is not presented all details of such theory referring instead to the above cited papers. In Sect. 41.2, it is considered the elastic constitutive laws for membranes having no or only negligible bending stiffness and the problem of determination of the relevant response functions for isotropic membranes undergoing arbitrarily large deformations (Subsect. 41.2.2). The important problem of identification of the different types of anisotropy in membrane materials and the associated problem of determining the complete set of material constants for every type of anisotropy are discussed in Subsect. 41.2.3. Next the theory is extended to membranes with bending stiffness (Sect. 41.3). It is shown that the same approach may be applied to the analysis of membranes having extensional and bending stiffness, but only for limited classes of constitutive models. Finally, in Sect. 41.4, it is shortly discussed certain aspects of the inelastic behavior of elastomeric membranes.

41 Biological and Synthetic Membranes: Modeling and Experimental Methodology

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41.2 Elasticity of Perfectly Flexible Membranes Perfectly flexible membranes have no or only a negligible bending stiffness and hence they cannot have a transverse shear stiffness. Such membranes can only carry the extensional internal forces tangent to their deformed shapes which adopt themselves to the external loading. The governing equations for perfectly flexible membranes follow from the general theory of shells [1, 22, 23] by discarding bending effects.

41.2.1 Equilibrium Equations and Constitutive Laws The equilibrium equation for perfectly flexible membranes takes the particularly simple form [16, 22, 23] Div N + p = 0 ,

N = N αβ α α ⊗ αβ ,

(41.1)

where N is the surface stress tensor and Div denotes the surface divergence operator in the deformed configuration of the membrane. Moreover, if the membrane is elastic, it can store energy by changing its metrical properties. In effect, the mechanical response of perfectly flexible elastic membranes is determined by the strain energy density Φ (measured per unit area of the reference) which is a function of the 2 × 2 (tangential) surface deformation gradient F . The frame-indifference principle requires that Φ depends on F only through the surface right Cauchy-Green deforC ) and mation tensor C = F T F . Thus, the strain energy may be expressed as Φ = Φ (C the classical constitutive equations take the form N = j−1 F S F T ,

C ) = ∂ E Φ (E E) . S = 2 ∂C Φ (C

(41.2)

F > 0, S is the symmetric (second P − K type) surface stress tensor Here j = detF C − 1 ) denotes the classical membrane Green-Lagrange strain tensor. and E = 12 (C Further specification of the constitutive equations may be obtained from the analysis of deformation modes of the membranes. In general, changes in the overall shape or conformation of the membrane can be viewed conceptually as the superposition of local deformations of imaginary differential elements. If the membrane is thin, then the local deformation of each material element can be represented by three independent modes of deformation: • area dilatation or condensation without changing shape or curvature of the element, • extension of the element without change in surface area or curvature, and • changes in element curvature without change in element area or aspect ratio. These geometric changes are quantified by the fractional change in the element area α, the in-plane extension ratio η (at constant surface density), and changes in principal curvatures, κ1 and κ2 , of the element. Furthermore, the maximum shear is

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−η | represented by the Eulerian shear strain γ = |η4(1+α) . Hence, a extension at constant area creates pure shear deformations. The two quantities α and η are associated exclusively with the in-plane deformation of membrane and they may be determined in terms of the principal strain invariants i1 = tr C and i2 = det C through the relations % & i1 = η−1 1 + η2 α, i2 = α2 . (41.3) 2

In the case of perfectly flexible membranes, changes in the local values of curvature do not affect its strain energy. The other two deformation modes may effectively be analyzed by a Flory type multiplicative decomposition of the surface deformation 1 gradient F = F¯ J , where J = j 2 1 is the spherical tensor and F¯ denotes the unimoduT lar tensor, detF¯ = 1. By implication, C¯ = F¯ F¯ = j−1 C , and the strain energy Φ may be considered as a function of the tensor C¯ describing deformation of the membrane without area changes and j which is the measure of the latter: % & C ) = Φ C¯ , j . Φ = Φ (C (41.4) Consequently, numerous representations of the constitutive equation (41.4) may be derived with a view of experimental identification of the relevant response functions. For each kind of material symmetry of the membrane, the strain energy Φ depends on C only through a set of functionally independent invariants. In the case of isotropic membranes, Φ is a function of the two principal invariants of C . This tensor is positive definite and symmetric, and thus it possesses two real, positive roots which are the squares of the in-plane principal stretches λ1 , λ2 : i1 = tr C = λ21 + λ22 ,

i2 = det C = λ21 λ22 .

(41.5)

In effect, the strain energy Φ of isotropic membranes must be a symmetric function of the principal stretches, Φ = (i1 , i2 ) = φ (λ1 , λ2 ). It follows that the corresponding constitutive equations take the form S = 2 ((Φ1 + i1 Φ2 ) 1 − Φ2C ) , ∂Φ (i1 , i2 ) Φα = Φα (i1 , i2 ) ≡ , α = 1, 2 ∂iα

(41.6a) (41.6b)

and the main constitutive problem is to develop an appropriate experimental methodology enabling determination of the specific response functions Φα (i1 , i2 ), α = 1, 2, from the well-designed set of tests. This general constitutive equation (41.6a) also preserves its form for membranes considered as a three-dimensional thin body made of transversely isotropic material [11].


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41.2.2 Experimentations with Isotropic Membranes The most convenient tests in finite elasticity of membranes are the uniaxial stretching of a narrow strip and the biaxial stretching of a rectangular sheet. When the membrane can be cut or fabricated in the requisite geometry, these preferred tests can performed and thereby material parameters can be determined. However, there are many situations wherein one has little or no control over the geometry of specimens. Examples include tests on biological membranes whose geometry is dictated by nature as well as nondestructive evaluations of membrane material in its natural environment. Hence, there is a need for different methods of characterizing of the membrane behavior. A number of researchers have investigated the problem of the axisymmetric membrane inflation and employed it to identify the elastic properties of elastomeric materials and biological soft tissues [8, 18, 24]. Using this method, the principal stretches are easy to measure because the principal directions of stresses and strains in axisymmetric membranes are aligned with the meridian and circumferential directions. Moreover, the principal stresses depend only on the deformed shape independently of the membrane material and can be expressed in terms of the internal pressure as well as in terms of the principal curvatures of membrane which can be measured in the experiment. This experimental methodology is based on the following theoretical basis. We denote by T 1 and T 2 the principal stress resultants per unit length of the deformed membrane (i.e. the proper values of the symmetric stress tensor N ) and let p be the excess of internal pressure over external pressure. Then from the equilibrium equation (41.1) specified for axisymmetric deformations of membranes, we may derive the important relations p p κ1 T1 = , T2 = 1− , (41.7) 2κ2 κ2 2κ2 where the two principal curvatures κ1 and κ2 are given by κ1 = cos ϕ

dϕ , dr

κ2 =

sin ϕ . r

(41.8)

Here r denotes the radial coordinate, and ϕ is the angle between the tangent to the meridian of axisymmetric membrane and the axis of rotation. Furthermore, the constitutive equation (41.6a) gives % & % & 2 2 T 1 = 2λ1 λ−1 T 2 = 2λ−1 (41.9) 2 Φ1 + λ2 Φ2 , 2 λ1 Φ1 + λ1 Φ2 , where Φα (i1 , i2 ), α = 1, 2, represent the two response functions determined from inflation test. The standard inflation test involves measuring only the equal stresses T ≡ T 1 = T 2 and stretches λ ≡ λ1 = λ2 at the apex of the inflated membrane. In this case, the equilibrium equation (41.7) and the constitutive relation (41.6a) reduce to the form

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T=

% & pR = 2 Φ1 + λ2 Φ2 2

(41.10)

with R as the radius of curvature at the apex. We have carried out such test for NR/S BR blend rubber (purchased from Jäger, Germany) and the representative data are shown in Fig. 41.1. This test is also used to determine the mechanical properties of many biomembranes including the determination of the shape and growth of aneurysms ([6, 20]) as well as in description of cell division [17]. However, it must be noted that Eq. (41.10) is strictly valid only for spherical geometry.

Fig. 41.1 Standard inflation test: Plots of measured equi-biaxial stresses v.s. equi-biaxial strains at the apex of the membrane (results for three samples)

While the standard inflation test provides valuable data on the mechanical behavior of tested membranes, the measurement of stresses and stretches only at the apex do not allow to determine both response functions Φα (i1 , i2 ). In fact, the state of strains in the inflated membrane varies from the biaxial one at the apex to the strip tension at the clamped boundary, where there is no deformation in the circumferential direction, λ2 = 1. Thus, measuring both curvatures and stretches along the whole meridian of the inflated membrane (see Fig. 41.2) provides data for the behavior of membrane material under different deformation states. From these data both response functions Φα (i1 , i2 ) and hence the strain energy function Φ = Φ (i1 , i2 ) may be determined with a reasonable accuracy. By fitting the response functions to the experimental data of stresses and stretches, the elastic parameters will be identified for the specific constitutive models. It may be further observed that if the bulk membrane material is incompressible, then the condition λ2 = 1 at the clamping of the membrane corresponds to a pure shear deformation. The possibility of deducing the membrane mechanical properties from the complete measured meridian profiles at different pressure levels is one of our main current research. This approach will improve the classical analysis of aneurysms, cell division, and many other problems in


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Fig. 41.2 Complete inflation test: Plots of measured complete profiles of the deformed membrane at different pressure levels

the mechanics of biomembranes. The axisymmetric membrane inflation test is direct and non-destructive. However, it is not applicable to membranes of general shape. The primary limitation of the membrane inflation test is the inability to separately control the two principal strains or stresses. For example, an independent control of the orthogonal extensions during in-plane biaxial tests allows one to perform tests wherein one coordinate invariant measure of the finite deformation is maintained constant while the other varies, and vice versa.

41.2.3 Quantification of Anisotropic Responses The anisotropic behavior of many membranes results from their oriented microstructure. Soft tissues, textiles and coated architectural fabrics are typical examples. Within phenomenological approach, the membrane is treated as a 2D continuum and the macroscopic constitutive equations are formulated by purely mathematical considerations without explicit reference to a underlying microstructure. In particular, it may be proved [19, 26] that the strain energy function of the membrane regularly reinforced with two families of mechanically equivalent fibers, whose orientations are specified by unit vectors α 1 and α 2 , has the following general form E ) = Φ (i1 , i2 , νN ) , Φ = Φ (E

(41.11)

where E is the strain tensor introduced in Sect. 41.2.1, and i1 ≡ 1 · E ,

1 i2 = 1 · E 2 , 2

νN =

1 XN · E2 N

(41.12)

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are the three functionally independent invariants. Furthermore, it has been shown in [26] that the anisotropy tensor X N may be expressed in the following general form X N = G N−1 (C N A 1 − S N A 2 ) ,

A 1 = α 1 ⊗ α1 ,

A2 = α2 ⊗ α2 ,

(41.13)

where C N ≡ cos (2 Nθ + (N − 1) F),

S N ≡ sin (2 Nθ + (N − 1) F) .

(41.14)

The constants G and F are defined as in [26] and θ is the fibre direction. The four special types of anisotropy corresponding to N = 1, 2, 3, 4 are illustrated in Fig. 41.3. The constitutive equations for every kind of anisotropy may be derived from the

Fig. 41.3 Anisotropic membranes regularly reinforced with two or more families of mechanically equivalent fibers

strain energy (41.11) by the application of the general relation (41.2). However, the basic problem is to determine the type of anisotropy and the relevant material parameters for a given membrane material. This requires a suitable methodology of the experimental testing and data analysis. The simplest and yet not trivial case is that of linearly elastic membranes for which the strain energy function has the quadratic form of the strain tensor E 1 1 E ) = E · CE E = C αβλκ E αβ Eλκ . Φ = Φ (E (41.15) 2 2 In the most general case, the elasticity tensor C possesses six independent components and the problem reduces to their determination for each type of anisotropy. Let K denote the inverse of the elasticity tensor C. Then, the elastic modulus E ν and the Poisson ratio ν (νν, ν¯ ) in any direction specified by the unit vector ν (¯ν determines the orthogonal direction) are given by the following relations


1 = (νν ⊗ ν ) · K [νν ⊗ ν ] , Eν

ν (νν, ν¯ ) = −

(νν ⊗ ν ) · K [νν ⊗ ν ] . (¯ν ⊗ ν¯ ) · K [¯ν ⊗ ν¯ ]

637

(41.16)

Equations (41.16) provide a theoretical basis for the quantification of anisotropy and the determination of independent materials constants for the linearly elastic membranes. This follows from the theorem (cf. [19]): For the determination of all six components of the elasticity tensor C for the membrane of no or unknown symmetry, it is necessary to carry out at least three tensile tests on the specimens whose axes are neither parallel nor orthogonal to each other. The three such tests give the three elastic moduli and the three Poisson ratios. Moreover, Eqs (41.16) constitute a system of six algebraic equations for six independent components of the tensor K and hence of the elasticity tensor C. Conversely, if the type of anisotropy and hence the elasticity tensor C are known, Eqs (41.16) may be used to determine the variations of the elastic modulus with direction. This is shown in Fig. 41.4 for four types of anisotropy illustrated in Fig. 41.3. For any membrane material, we may carry out tension tests for a sufficient number of directions and next the determined elastic moduli may be represented graphically as in Fig. 41.4 from which the kind of anisotropy may be concluded. This methodology of quantify the anisotropic response of linearly

Fig. 41.4 Variation of the elasticity modulus with direction for membranes with specific anisotropy (shown in Fig. 41.3)

elastic membranes has the natural extension to at least special non-linear constitutive models with the semi-linear strain energy of the following general form (cf. [14]) E ) = w (E E ) + F (u (E E )) , Φ = Φ (E

(41.17)

638


E ) and u (E E ) are given by quadratic forms where the functions w (E 1 E ) = E · BE E, w (E 2

1 E ) = E · AE E. u (E 2

(41.18)

Here F is a monotonic increasing function of u with the property F (0) = 0, i.e. it vanishes in the reference configuration of the membrane. With the special assumption F (u) = exp u, the strain energy (41.17) reduces to the classical Fung form [8] which every since dominated in the constitutive modeling of anisotropic soft tissues. For some other choices of the function F (u), the same strain energy (41.17) may be used to describe the non-linear behavior of anisotropic architectural fabrics of the type shown in Fig. 41.5. In general, the constitutive model based on the strain

Fig. 41.5 Typical stress-strain data for coated fabrics: Tension tests in different directions (PVCcoated fabric [5])

energy function (41.17) involves twelve independent elastic constants which may be determined by the procedure presented above taking into account that tangential elasticity tensor C of the form E ) ≡ ∂2E Φ (E E ) = B + F (u)A + F (u)AE E ⊗ AE E, C (E

(41.19)

where the prime stands for the derivative with respect to u. Applying Eqs (41.16) E ) of the tangential elasticity tensor C (41.19), we may dewith the inverse K (E termine E (νν) and ν (νν, ν¯ ) at any point of the stress-strain relation for different directions. Twelve such values suffice to determine the twelve independent elastic constants appearing in the strain energy function (41.17). However, this procedure is not unique because different choices of points on the stress-strain curve leads to more or less different values of elastic constants.


639

41.3 Membranes with Bending Stiffness The theory of finite deformation of perfectly flexible elastic membranes is based on the assumption that only surface stresses and inertia must be considered. To allow for a bending resistance, surface couples must be also taken into account. There are two basic approaches to develop such a theory [1, 23] involving the concepts of oriented or Cosserat surfaces and second-grade material surfaces, respectively. Within these approaches, the familiar Kirchhoff-Love shell may be interpreted as Cosserat surface with the director constrained to coincide with the unit normal vector giving the orientation of material surface. This theory has been extensively studied in the literature [22, 23] and it is used in what follows.

41.3.1 Elastic Response Functions The elastic response of membranes with% the extensional and flexural stiffness is & determined by the strain energy Φ = Φ ααβ , kαβ considered as a function of the metric and curvature tensors of the deformed membrane. For solid membranes, this strain energy may be expressed in coordinate invariant form as [22] C ,G G) , Φ = Φ (C

(41.20)

where the surface deformation tensors C and G are defined by C = F T F = ααβ e α ⊗ e β ,

G = F T K F = kαβ e α ⊗ e β .

(41.21)

Here K denotes the curvature tensor of the membrane in the deformed configuration, F is the surface deformation gradient with respect to the reference configuration of membrane, and e α are the natural base vectors in that configuration. The corresponding stress-deformation relations for the stress and couple tensors, N and M , follow from the general rule [22] C ,G G )) F T , N = 2 j−1 F (∂C Φ (C

C ,G G )) F T . M = j−1 F (∂G Φ (C

(41.22)

The constitutive equations for membranes with special anisotropy types have been discussed in the literature (e.g. [10]), but the general methodology to determine the specific form of response functions and the involved material parameters is not known. In the following section, we shall consider only linearly elastic fabric membranes with an arbitrary extensional and flexural anisotropy in their responses.

640


41.3.2 Linearly Elastic Anisotropic Membranes Coated fabrics used as a covering material in tension structures have a bending stiffness much lower than the tensile stiffness. Therefore, the bending stiffness is usually neglected in the design of tensile structures, but it has to be taken into account if it is analyzed the possible wrinkling states typical for such structures. Investigations into the bending rigidity of fabrics started in 1930s when Peirce introduced the concept of bending stiffness for these materials and developed the Cantilever Method for measurement this stiffness. The Peirce Cantilever Test is performed on the commercially available Shirley Stiffness Tester and it is still considered as the standard method to measure the bending stiffness of fabrics [4]. Various theoretical formula for the bending stiffness in any direction have been proposed in literature assuming that the values in the orthogonal (warp and fill) directions are known. However, the derived theoretical results are often not very satisfactory. This is illustrated in Fig. 41.6 for two types of fabrics. The general constitutive

Fig. 41.6 Variation of the bending modulus with direction for anisotropic membranes with flexural stiffness: Comparison of data with classical models and results of this paper

equations for linearly elastic membranes follow from Eqs (41.22) with the strain energy function assumed in the form E ,G G) = Φ (E

& 1% E + G · DG G + E · BG G + G · BT E , E · CE 2

(41.23)


641

where C, D and B are fourth order elasticity tensors. The resulting stress-strain relations for the extensional and bending responses of the membrane are coupled. However, the numerical results obtained for the coupled and uncoupled constitutive laws presented in Fig. 41.7 and Fig. 41.8 show that it suffices to consider the uncoupled form of the law by neglecting the terms containing the tensor B. Under this assumption, the tensional and bending elasticity tensors, C and D, have at most twelve independent material constants which must be determined from suitable experimental tests.

Fig. 41.7 Comparison of the measured deflected shape of a fabric strip with the numerical solution (FEM) obtained for the coupled and uncoupled constitutive laws

Under the above assumptions, the constitutive equations derived from the strain energy (41.23) are uncoupled so that the tensile and bending properties of membranes may essentially be studied separately. The tensile behavior of membranes has been discussed in Sect. 41.2. For the case of uncoupled constitutive equations, the same approach may be applied to study of bending behavior of linearly elastic membranes. Accordingly, we consider the cylindrical bending of the membrane in any direction specified by the unit vector ν . The bending stiffness B (νν ) and the coupling coefficient μ (νν, ν¯ ) for the chosen direction ν are given by the formula analogous to (41.16), namely 1 = (νν ⊗ ν ) · G [νν ⊗ ν ] , B (νν)

μ(νν, ν¯ ) = −

(νν ⊗ ν ) · G [νν ⊗ ν ] . (¯ν ⊗ ν¯ ) · G [¯ν ⊗ ν¯ ]

(41.24)

Here G denotes the inverse of the bending elasticity tensor D. Having determined B (νν) and μ(νν , ν¯ ) for the three directions which are neither parallel nor orthogonal to each other, the system of equations (41.24)) may be used to compute the six independent components of the tensor G and hence the bending elasticity tensor D. Moreover, μ (νν, ν¯ ) is an analogue to the Poisson’s ratio giving rise to the

642


Fig. 41.8 Comparison of the measured and computed (FEM) load-deflection curves for coupled and uncoupled constitutive laws

anticlastic curvature. This procedure has been used to determine the elastic constants for cylindrical bending in the single direction, that were used in the finite element computation of the cylindrical bending of a coated fabric for which the data were known. A comparison of the data and the numerical solutions shows a very good coincidence of the results (Fig. 41.7 and Fig. 41.8). However, the determination of all six constants needed to fully model the bending behavior of membranes requires to measure the change in curvature not only in the direction of applied moment but also in the perpendicular direction. This is very difficult to realize in experiment. Moreover, this method is not applicable to biomembranes such as red blood cells. Generally, the behavior of red blood cell membranes is modeled within the concept of fluid membranes with the curvature elasticity. The appropriate constitutive equations have been recently reconsidered in a number of important publications, e.g. [3], [7], [9]. However, the experimental methodology for the determination of such constitutive laws is still under development.

41.4 Inelastic Effects in Membranes Real membranes, both natural and synthetic, display one or another kind of inelastic behavior under various loading conditions. For example, elastomeric membranes and membrane type soft tissues show a characteristic stress softening when subject to cyclic loading. As a rule, this phenomenon is accompanied by hysteresis and permanent set. All these inelastic effects may be illustrated in a simple problem.


643

41.4.1 Spherical Inflation of Balloons It has long been known that the inflation of toy balloons may provide a good illustration of the peculiar behavior of elastomeric membranes. Over years, this problem has been studied by many authors under the assumption that the deformation of the balloon is completely reversible. However, a typical inelastic effect observed in the inflation and subsequent deflation of balloons is that of a stress-softening (the Mullins effect) accompanied by a hysteresis and residual strains (the permanent set). These phenomena are also observed in the inflation of certain biological membranes [9, 13, 15]. Therefore, understanding the physical nature and properties of stress softening phenomenon and its mathematical modeling is of importance not only in science of elastomers but also in physiology of natural organs. In fact, the study of the membrane inflation has been largely stimulated by the research in physiology (see [15] for details). Recently, this problem has been considered within the concept of pseudo-elasticity originally proposed by Ogden and Roxburgh and further developed by Dorfmann and Ogden (see [15] for the reference to original works). However, neither of these two studies presents a comparison of the theoretical results with available experimental data. In addition, the analysis of their results presented in [15] shows that the main physical characteristics of the stress softening phenomenon are not properly represented and they appear to be at marked variance with the experimental data (Fig. 41.9). The primary inflation curve of the virgin balloon is very accurate described by the classical rubber elasticity such as the Gent model. However, the Dorfmann-Ogden theory with and without residual strains fail to represent the dominant characteristic of stress softening during unloading.

Fig. 41.9 Inflation of a spherical balloon: The comparison of the experimental data with model predictions (see [15] for details)

644


It may be surprised to note that many other 3D models of this phenomenon proposed over the years in literature also fail to correctly describe the simple inelastic effect observed during inflation of elastomeric balloons. One of the reasons of this fact is the lack of a reliable methodology to quantify the character of stress softening observed in elastomers and soft tissues. This point was discussed in literature within the 3D theories. Here we present the most important results of this problem directly related to the 2D theory of membranes.

41.4.2 Stress Softening of Membranes The simplest theory taking into account the inelastic behavior of membranes with and without flexural stiffness is obtained by disregarding the gradient effects which leads to the specification of the energy potential as a function of the deformation tensors C and G , and a microvariable α C ,G G , α) Φ = Φ (C

(41.25)

together with the constitutive law for microforces which yields the evolution law for the microvariable α of the following general form (details of such an approach may be found in [12, 21]) C ,G G , α) . α˙ = β (C (41.26) In this case, the resulting constitutive relations are slightly more complex due to presence of the bending stiffness. The thermodynamically admissible constitutive equations for the stress and couple tensors, N and M , may now be derived from the dissipation inequality as shown in [15, 21], but details will not be presented here. The crucial point is the specification of dependency of the energy potential (41.25) on the microvariable α. An inappropriate choice of such dependency may lead to the prediction of the monotonic stress softening behavior in order to satisfy the dissipation inequality [15] and conflicting results (Fig. 41.9). Indeed, the analysis of data for the elastomeric membranes shows that they display the characteristic S -shaped stress softening behavior in contrast to the monotonic behavior predicted by the theoretical models. To remove these discrepancies, a new methodology is needed for the analysis of the stress softening behavior of membranes with and without bending stiffness. This will serve as a guideline for the formulation of the energetically compatible constitutive equations which correctly capture the most important features of this phenomenon.

41.5 Discussions This paper presents the first results of the current work in progress on the constitutive modeling and the experimental testing of membranes, both with and without bending


645

stiffness taken into account. The considered models are formulated within a purely two-dimensional approach without any reference to the possible three-dimensional theories. This is motivated by the observation made in the relevant literature that many natural and synthetic membranes may be treated as a material continuum only in two-dimensions preserving their discrete structure in the thickness direction. Within this approach, the main concern is the methodology of identification of the two-dimensional constitutive equations for different classes of membranes. We have shown that for isotropic perfectly flexible membranes undergoing elastic deformations the relevant response functions may be deduced from the axi-symmetric inflation test provided that the complete shape of the membrane is measured at different pressure levels. This test is not destructive and therefore it applies not only to synthetic membranes but also to biomembranes in their native form. We have discussed the experimental technique and the theoretical methodology designated to determine the complete set of material parameters for membranes with no or unknown symmetries in their response. This methodology applies to the extensional and bending responses of membranes within linear and semi-linear elastic models. The possibility of extending these techniques and methods to quantify the mechanical behavior of inelastic membranes is under current investigations.

References 1. Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells. A short review and bibliography. Arch. Appl. Mech. 80, 73–92 (2010) 2. Bridgens, B.N., Gosling, P.D., Birchall, M.J.S.: Membrane material behaviour: Concepts, practice and developments. Struct. Eng. 82, 28–33 (1978) 3. Brown, F.L.H.: Elastic modeling of biomembranes and lipid bilayers. Ann. Rep. Phys. Chem. 59, 685–712 (2008) 4. Chapman, B.M.: The high-curvature creasing behaviour of fabrics. J. Textile Institute 64, 250–262 (2007) 5. Chen, S., Ding, X., Yi, H.: On the anisotropic tensile behaviors of flexible polyvinyl chloridecoated fabrics. Textile Res. J. 77, 369–374 (2007) 6. Doyle, B.J., Corbett, T.J., Cloonan, A.J., O’Donnell, M.R., Walsh, M.T., Vorp, D.A., McGloughlin, T.M.: Experimental modelling of aortic aneurysms: novel applications of silicone rubbers. Med. Eng. Phys. 31, 1002–1012 (2009) 7. Evans, E.A., Hochmuth, R.: Mechanochemical properties of membranes. Current Topics Mem. Transp. 10, 1–64 (1978) 8. Hsu, F.P.K., Schwab, C., Rigamonti, D., Humphrey, J.D.: Identification of response functions from axisymmetric membrane inflation tests: implications for biomechanics. Int. J. Solids Struct. 31, 3375–3386 (1994) 9. Humphrey, J.D.: Computer methods in membrane biomechanics. Comput. Meth. Biomech. Biomed. Engng. 1, 171–210 (1998) 10. Indelicato, G., Albano, A.: Symmetry properties of the elastic energy of a woven fabric with bending and twisting resistance. J. Elast. 94, 33–54 (2009) 11. Kazakevi˘ci¯ut˙e-Makovska, R.: Non-linear response functions for transversely isotropic elastic membranes. Civil Eng. 7(5), 345–351 (2001) 12. Kazakevi˘ci¯ut˙e-Makovska, R.: Structure of constitutive relations for isotropic elastic membranes with voids. Civil Eng. 7(1), 23–28 (2001)

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13. Kazakevi˘ci¯ut˙e-Makovska, R.: Damage-induced stress-softening effects in elastomeric and biological membranes. J. Civil Eng. Manag. 8, 68–72 (2002) 14. Kazakevi˘ci¯ut˙e-Makovska, R.: Modelling of fabric structures in civil engineering. In: Proc. Simulations in Urban Engineering pp. 145–148 (2004) 15. Kazakevi˘ci¯ut˙e-Makovska, R.: On pseudo-elastic models for stress softening in elastomeric balloons. Comp. Mat. Continua 15, 27–44 (2010) 16. Kazakevi˘ci¯ut˙e-Makovska, R.: Statics of hybrid structures composed of interacting membranes and strings. Int. J. Non-Linear Mech. 45, 186–192 (2010) 17. Lardner, T.J.: Elastic models of cytokinesis. In: Biomechanics of Cell Division, N. Akkas (Ed.), Plenum Press, New York and London pp. 247–279 (1987) 18. Mott, P.H., Roland, C.M., Hassan, S.E.: Strains in an inflated rubber sheet. Rubber Chem. Tech. 76, 326–333 (2003) 19. Rychlewski, J.: Two-dimensional Hookes tensors isotropic decomposition, effective symmetry criteria. Appl. Mech. 48, 325–345 (1996) 20. Shah, A.D., Harris, J.L., Kyrlacou, S.K., Humphrey, J.D.: Further roles of geometry and properties in the mechanics of saccular aneurysms. Comp. Meth. Biomech. Biomed. Eng. 11, 109–121 (1997) 21. Steeb, H., Diebels, S.: Modeling thin films applying an extended continuum theory based on a scalar-valued order parameter. Part I: Isothermal case. Int. J. Solids Struct. 41, 5071–5085 (2004) 22. Steigmann, D.J.: Fluid films with curvature elasticity. Arch. Rat. Mech. Anal. 150, 127–152 (1999) 23. Steigmann, D.J.: On the relationship between the Cosserat and Kirchhoff-love theories of elastic shells. Math. Mech. Solids 4, 275–288 (1999) 24. Treloar, L.R.G.: Strains in an inflated rubber sheet and the mechanism of bursting. Tran. Inst. Rubber Ind. 19, 201–212 (1944) 25. Treloar, L.R.G.: The physics of rubber elasticity (3rd ed.). Claredon Press, Oxford (2005) 26. Zheng, Q.S., Betten, J., Spencer, A.J.M.: The formulation of constitutive equations for fibrereinforced composites in plane problems: Part I. Arch. App. Mech. 62, 530–543 (1992)

Chapter 42

Nonclassical Theories of Shells in Application to Soft Biological Tissues Eva B. Voronkova, Svetlana M. Bauer and Anders Eriksson

Abstract Two non-classical theories for orthotropic plates of moderate thicknesses are discussed. In these theories both deformations, rotation and bending of the fibers and their elongations in the direction of the thickness of the shell are taken into account. The stress-strain state of a circular plate modeling the Lamina Cribrosa in the human eye is studied by means of these theories. Numerical results for displacements and stresses found with the presented theories are compared with those obtained with FEM. Keywords Soft biological tissues · Shell theory · Orthotropy

42.1 Introduction Shell-like structures are of very frequent occurrence in biological systems, either as body organs or their parts. Therefore, shell theory-based models have been used in recent years as tools to help better predict surgeries, to describe development of diseases [3]. On the other hand, the specific structural features of biological tissues (non-homogeneity, anisotropy) should be taken into account if we want to predict correct response of the organ or the tissue to a particular mechanical loading. Due to transverse shear and thickness changes neglect classical shell and plate theory

E. B. Voronkova (B) · A. Eriksson Department of Mechanics, Royal Institution of Technology KTH, Osquars Backe 18, Stockhom, 10044 Sweden e-mail: [email protected],[email protected] Svetlana M. Bauer Department of Theoretical and Applied Mechanics, St. Petersburg State University, Universitetskii pr. 28, St. Petergof, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 42, © Springer-Verlag Berlin Heidelberg 2011

647

648

E.B. Voronkova et al.

can over- or underestimate the results and the error is raised with increasing of the thickness-to-span ratio or the anisotropy degree. Starting from the basic works by Timoshenko [9], Reissner [7], Ambartsumyan [2], numerous of, so name, non-classical, refined and high-order theories of orthotropic plates and shells have been formulated during the last few years. Some of these theories are reviewed in [1], [6]. Asymptotic correctness of the Timoshenko-Reissner theory for thin shells with small shear modulus is discussed in [10]. In the present paper two refined theories for orthotropic plates of moderate thicknesses worked out by Paliy-Spiro (PS) [5] and by Rodinova-Titaev-Chernykh (RTC) [8], are discussed. The PS and RTC theories based on two different principals: hypotheses of stressed and deformation states are used for the deduction from 3-dimensional system of equations to a 2-dimensional one in the PS theory; the RTC theory authors apply mathematical approaches to develop a set of 2-dimensional equations. Namely, the stress-strain state of a circular plate modeling the Lamina Cribrosa in the human eye is studied by means of these theories. Results found with the RTC and PS theories are compared with those obtained with FEM to demonstrate the reliability of the presented theories.

42.2 Problem Formulation Let us consider a circular transversal isotropic plate with the constant thickness h, the radius R in the cylindrical coordinate system (r, α, z). Here r – variable radius, α – circumferential coordinate in the plate surface (z = 0), the z–axis is normal to this surface. The lower and upper surfaces of the plate are loaded by the uniform pressures: σzz = −p±

f or

z = ±h/2 .

(42.1)

The edge r = R of the plate is supposed to be completely fixed. Assuming the r − α plane to be the plane of the isotropy at every point, the Hooke’s law is σrr = Err err + E rα eαα + E rz ezz , σαα = E rα err + E rr eαα + E rz ezz ,

(42.2)

σzz = E rz err + E rz eαα + E zz ezz , σrα = Gerα , σrz = G rz erα , σrα = Grz erα , with the stresses σ and the strains e. The coefficients in (42.2) depend on five independent elastic parameters for the transversally isotropic material: the moduli of elasticity in the plane and transversal directions (E r and Ez ); Poisson’s ratios for the shortening in the plane of isotropy under the tension in this plane (ν) and for the shortening in the plane of isotropy under the tension in the direction orthogonal to the plane (ν ); and the shear modulus in rz-plane – G rz :

42 Nonclassical Theories of Shells in Application to Soft Biological Tissues

E rr =

E r (E z − E r ν2 ) , (1 + ν)(E r (1 − ν) − 2E zν2 )

Erz =

E r E z ν , E ( 1 − ν) − 2E rν2

E zz =

Erα =

649

E r (Ez ν + E z ν2 ) , (1 + ν)(E r (1 − ν) − 2Erν2 )

Ez2 (1 − ν) , Ez (1 − ν) − 2Er ν2

G=

Er . 2(1 + ν)

We also introduce the following notations ν∗ =

Er ν , Ez 1 − ν

q = p − − p+ ,

h m = − (p− + p+ ) . 2

Due to symmetry the distribution of the stresses does not depend on the circumferential coordinate α.

42.3 The Paliy-Spiro Theory The theory of anisotropic shells with moderate thickness was proposed by Paliy and Spiro in their monograph [5]. The basic kinematic assumptions of the PS theory are • a rectilinear element normal to the middle surface of a shell remains rectilinear after the deformation; • cosine of the slope angle of a rectilinear element normal to the middle surface is assumed to be equal to the mean transverse shear angle. Expressed in mathematical terms, these hypotheses give urPS = u(r) + zϑ ,

uzPS = w(r) + F(r, z) ,

ϑ = ϑ0 (r) + γr ,

(42.3)

where urPS , uzPS are displacements in the r and z directions; u, w are the midplane displacements; ϑ are the rotations of rectilinear fibers about α-axes; ϑ0 , γr are the rotation and shear angels of normals to the midplane in (r , z) plane; F(r, z) is a function, equal to zero at z = 0. As the plate deforms axisymmetrically, assumptions (42.3) yield the following strain-displacement relations PS err =

du d2 w −z 2 , dr dr

PS eαα =

u z dw − , r r dr

PS ezz =

∂F , ∂z

PS erz = γr . (42.4)

According to [5], a linear law is proposed for normal stress distribution along the plate thickness m z PS σzz = +q . (42.5) h h Using Eqs (42.2), (42.4), one obtains 2 2 z σzz du u z d w 1 dw F(r, z) = dz − zν∗ + + + . (42.6) dr r 2 dr2 r dr 0 Ezz

650


The constitutive equations for stress resultants, bending moments and shear force are 2 2 E r h du u d w ν dw ∗ PS ∗ h T rPS = + ν + ν m , M = −D + + ν q , r 2 r r dr 8 1 − ν2 dr dr 2 Er h u du 1 dw d2 w ∗ PS ∗ h T αPS = + ν + ν m , M = −D + ν + ν q , (42.7) α dr r dr 8 1 − ν2 r dr2 E r h3 QrPS = G rz hγ1 , D = . 12(1 − ν2) Substituting Eqs (42.7) into the system of equilibrium equations of the circular plate r

dQrPS + QrPS = −rq , dr

d(rT rPS ) − T αPS = 0 , dr

d(rMrPS ) − MαPS = rQrPS , (42.8) dr

all sought variables can be obtained with the boundary conditions, in case of a fixed edge, u = w = γr = 0 on r = R.

42.4 The Rodionova-Titaev-Chernykh Theory The refined theory of anisotropic shells worked out by Rodionova, Titaev and Chernykh [8] based on following static and kinematic assumptions • static: the transverse tangential and normal stresses are distributed along the shell thickness according to the quadratic and cubic laws, respectively; • kinematic: along the shell thickness the tangential and normal components of the displacement vector have the polynomial distributions of the third and the second powers, respectively. The authors used expansions in the Legendre polynomials of the thickness coordinate to reduced an initial three-dimensional problem. Following [8], for the displacements in the radial (uRTC ) and transversal (uRTC ) directions we have r z uRTC = u∗ (r)P0 (z) + γ1∗ (r)P1 (z) + θ1∗ (r)P2 (z) + ϕ∗1 (r)P3 (z) , r uRTC = w∗ (r)P0 (z) + γ3∗ (r)P1 (z) + θ3∗ (r)P2 (z) . z

(42.9)

Here Pi , (i = 0 . . . 3) are the Legendre polynomials P0 (z) = 1,

P1 (z) =

2z , h

P2 (z) =

6z2 1 − , h2 2

P3 (z) =

20z3 3z − . h h3

The displacement components in (42.9) have simple physical or geometrical interpretations. A tangential and a normal displacements of a point on the midplane are (u∗ − θ1∗ /2) and (w∗ − θ3∗ /2), respectively. Coefficients γ3∗ , θ3∗ characterize the alteration of the lengths of normals to the midplane. The rotation angel and curvature

42 Nonclassical Theories of Shells in Application to Soft Biological Tissues

651

of the fibers, normal to the midplane before the deformation, in (r , z) plane can be described by (2γ1∗ /h − 3ϕ∗1 /h) and by 6θ1∗ /h2 , 20ϕ∗1 /h3 , respectively. Representing the strains and stresses as the linear combinations of the Legendre’s polynomials and using the above hypotheses, one can write the stress-strain relations in the form 2 & & Er h % ∗ D% ∗ ∗ ∗ RTC ∗ ∗ h ε + νε + ν m , M = κ + νκ + ν q , r 2 2 2 1 10 1 − ν2 1 2 & & Er h % ∗ D% ∗ ∗ ∗ RTC ∗ ∗ h = ε + νε + ν m , M = κ + νκ + ν q , α 2 1 2 1 2 10 1 − ν2 dθ∗ Grz h E r h2 = 5ε∗13 − 3 , D = , 6 dr 12(1 − ν2 )

T rRTC = T αRTC QRTC r

(42.10)

where the strain components are ε∗1 =

du∗ , dr

ε∗2 =

u∗ , r

κ∗1 =

dγ1∗ , dr

κ∗2 =

γ1∗ , r

ε∗13 =

dw∗ 2 ∗ + γ . dr h 1

(42.11)

The normal and shear stresses can be written as following [8] σRTC zz (z) =

m 3z z3 + q − 2q 3 , h 2h h

σRTC rz (z) =

QRTC QRTC r P0 (z) − r P2 (z) . (42.12) h h

The problem is governed by equilibrium equations (similar to (42.8)) together with relations (42.10) and expressions for displacement components ( m ν∗ h ' ∗ ∗ ( hq ν∗ h ' ∗ − ε1 + ε2 , θ3∗ = − κ + κ∗2 , 2Ezz 2 10Ezz 6 1 ∗ ∗ h dγ3 h dθ3 θ1∗ = − , ϕ∗1 = − 6 dr 10 dr

γ3∗ =

(42.13)

and the boundary conditions on clamped outer edge of the plate u∗ = γ1∗ = w∗ = 0. In addition, at the center of the plate, the requirement of finite stresses for both theories must meet.

42.5 Results 42.5.1 Relationships for the Deflections and Stresses Let us denote rˆ = r/R ,

zˆ = z/h ,

μ = h/R ,

uˆ i = ui E r /(hq) , i = 1, 3

(42.14)

and then omit the symbol ˆ. Considering a circular plate with the clamped edge, we obtain the deflections of the plate as

652


uPS 3 uRTC 3

% & ⎡ ⎤ ∗ 2 ⎢ &⎥ m Er 3 ν 1−ν % PS 2 ⎢⎢⎢⎢ Er 2 ⎥⎥⎥⎥ = u30 + z + z ⎢⎣ − 1 − 2r ⎥⎦ , hq Ezz 2Ezz 4 μ2 & m Er 3 ν∗ (1 − ν2 ) % 2 3E r = uRTC + z + z − 1 − 2r2 , 30 ∗ 2 hq E 5Ezz 4 μ

(42.15)

where uPS , uRTC are normalized transverse displacements of the midplane 30 30 uPS 30 uRTC 30

% & &2 E r 1 − r 2 3(1 − ν2 ) % = 1 − r2 + , 16μ4 4Grz μ2 % & & 3E r 1 − r2 3(1 − ν2 ) % 2 2 = 1−r + + 16μ4 10G rz μ2 & (1 − ν2 )ν∗ % Er + 1 − 6r2 − . 2 20E zz 80μ

(42.16)

Both theories give parabolic transverse displacements in normal coordinate z. The terms of the second and third order in (42.15) are almost the same but solutions for the midplane’s normal deflection differ significantly. The RTC theory gives two additional summands in the solution of middle surface compared with that determined from the PS theory (42.16). The normal stresses according to the PS and the RTC theories are given by σPS zz /q =

m +z , qh

m 3z + − 2z3 . qh 2

(42.17)

r 3 − − 6z2 . 2μ 2

(42.18)

σRTC zz /q =

For shear stresses we obtain σPS rz /q = −

r , 2μ

σRTC rz /q =

It is important to note that material elastic constants affect neither normal nor shear stresses in (42.17, 42.18).

42.5.2 Numerical Examples For numerical examples the plate properties are taken from literature as material properties of the Lamina Cribrosa (LC) of the human eye [4]: the in-plane modulus and the Poisson’s ratio are assumed to be E r =1 MPa and ν = 0.45. For the transverse modulus and the shear modulus in r − z plane we suppose Ez = Er /n and G rz = G/n, where n is a measure of the degree of anisotropy. The Poisson ration ν is selected so that the stain energy of the plate material be positive. Calculations are performed for different n. To verify the presented theories the FEM software ABAQUS (Simula, Inc., USA) is used to calculate transverse displacement, normal and shear stresses of the plate

42 Nonclassical Theories of Shells in Application to Soft Biological Tissues Fig. 42.1 Distribution of normal stresses in a circular plate with h/R = 0.3. Dotted line with disk markers refers to the PS theory, dashed line with square markers — to the RTC theory, lines without markers correspond to FEM simulation (solid line — Er /Ez = 1, dashed — Er /Ez = 10, dot-dashed — Er /Ez = 50)

zh 12

100

13

0.2

16 13

Σrz q 0.0 0.5

80

60

40

0.4

0.6

a

1.5

0.6

0.2

0.4

1.0

0.8

Σzz q

rR 1.0

Er hq

500

400

200

0.8

0.6

300

0.8

2.0

100

20 0.2

1.0

u30

12

Er

0.4

hq

16

Fig. 42.2 Distribution of shear stresses in a circular plate at h/R = 0.3. Dotted line with disk markers refers to the PS theory, dashed line with square markers — to the RTC theory, solid line — FEM simulation at Er /Ez = 1, dot-dashed line refers to FEM simulation at Er /Ez = 50 u30

653

1.0

rR

0.2

0.4

0.6

0.8

1.0

rR

b

Fig. 42.3 The normalize midplane displacement at Er /Ez = 10 (a) and Er /Ez = 50 (b). Dotted line with disk markers corresponds to the PS theory results, dashed line with square markers — the RTC theory, solid line — results obtained by FEM

under consideration. An axisymmetric eight-node element CGAX8 is employed for the analysis. The normalized normal stresses distribution along the plate thickness given by the PS and the RTC theories compared with those obtained by FEM are plotted in Fig. 42.1. One can see that Er /E z ratio changes slightly the calculation results and the results for both theories are in good agreement with those found by FEM even for strong anisotropy. The RTC theory and FEM simulation give very close results for shear stress for isotropic plate or for a plate with relatively small anisotropy degree n (Fig. 42.2). But if the transverse modulus is two order smaller than in-plane modulus

654


value, the PS theory provides better agreement with the FEM results. The normalize midplane deflection according to the presented theories and those obtained by FEM are plotted in Fig. 42.3. As can be seen, with the increasing of anisotropy degree the PS theory fits better FEM results than the RTC theory.

42.6 Conclusions Two non-classical theories for orthotropic plates of moderate thicknesses are discussed in the paper. The exact bending solution of transversally isotropic circular plate using the PS and the RTC theory are presented. To verify the presented theories the results are compared with those obtained by FEM. The comparison shows that for the transverse displacement both theories give acceptable fit to FEM results. For normal stresses the RTC theory is slightly better than the PS-theory when compared with FEM results. For isotropic plates the shear stresses obtained by the RTC theory are in better agreement with FEM results than ones obtained according to the PS theory, but for highly anisotropy plate the PStheory predicts the shear stresses more accurately. It is noticeable that, in general case, the RTC theory has the 14th order while the PS theory is of the 10th order, which makes a calculation procedure simpler. Acknowledgements The research was supported in part by SI-sponsored Visby program from KTH and by RFBR grants 09-01-00140a, 10-01-00244a.

References 1. Altenbach, H.: Theories for laminated and sandwich plates. An overview. Mech Compos Mater 34, 333–349 (1998) 2. Ambartsumyan, S.A.: General theory of anisotropic shells. Nauka, Moscow (1974) (in Russian) 3. Bauer, S.M.: Mechanical Models of the Development of Glaucoma. In: Guran, A., Smirnov, A.L., Steigmann, D.J., Vaillancourt, R. (eds.) Advances in Mechanics of Solids In Memory of Prof E M Haseganu, pp. 153–178, World Scientific Publishing, Singapore (2006) 4. Edwards, M.E., Good T.A.: Use of a mathematical model to estimate stress and strain during elevated pressure induced lamina cribrosa deformation. Curr Eye Res 23, 215–225 (2001) 5. Paliy, O.M., Spiro, V.E.: Anosotrotic Shells in Shipbuildings. Theory and Analysis. Sudostroenie, Leningrad (1977) (in Russian) 6. Reddy, J.N., Wang, C.M.: An overview of the relationships between of the classical and shear deformation plate theories. Composites Science and Technology 60, 2327–2335 (2000) 7. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12, 69–72 (1945) 8. Rodionova, V.A., Titaev, V.F., Chernykh, K.F.: Applied Theory of Anisotropic Plares and Shells. St. Petersburg University Press, St.Petersburg (1996) (in Russian) 9. Timoshenko, S.P: On the transverse vibrations of bars of uniform cross-section. Philos Mag 169, 125–131 (1922) 10. Tovstik, P.E., Tovstik, T.P.: On the 2D models of plates and shells including the transversal shear. Z. Angew. Math. Mech. 87, 160–171 (2007)

Part VIII

FGM and Laminated Plates and Shells

Chapter 43

Axisymmetric Bending Analysis of Two Directional Functionally Graded Circular Plates Using Third Order Shear Deformation Theory Reza Akbari Alashti and Hossein Rahbari

Abstract Bending analysis of a functionally graded circular plate with clamped and simply supported boundary conditions is carried out. Material properties of the plate are assumed to be functionally graded in two directions, namely in the radial direction and through the thickness of the plate, obeying exponential distribution laws. Poisson’s ratio is assumed to be constant throughout the plate. Governing equations of a circular plate with material properties varying only through the thickness of the plate, based on the third order shear deformation theory are employed. In order to carry out the bending analysis of a two directional functionally graded circular plate, the plate is divided into number of annular plates with material properties graded only through the thickness. Material properties of each annular plate are defined by the corresponding exponential grading rule and the radial position of its center, hence these properties are assumed to be constant for each annular plate but different from its adjacent plates. Imposing the overall boundary conditions and compatibility conditions of adjacent annular plates, the governing equations of all annular plates are obtained and solved simultaneously. The analytical results are compared with the results obtained by the first order shear deformation theory and finite element method which are found to be in good agreement. Keywords Functionally graded materials · Axisymmetric bending · Circular plates

43.1 Introduction Functionally graded materials (FGMs) are new kind of composites with material properties varying continuously in some spatial directions. Featuring gradual transition in material composition and properties, these materials are designed to meet R. A. Alashti (B) · H. Rahbari Babol University of Technology, Babol, Iran e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 43, © Springer-Verlag Berlin Heidelberg 2011

657

658

R.A. Alashti and H. Rahbari

specific performance requirements. Due to the smooth variation of material properties, FGMs offer many advantages over laminated composites including improved fatigue resistance and reduction of thermal, residual and inter laminar stresses. Hence, FGMs are finding applications in many industrial fields such as aerospace and power generation industries. Circular plates made of FGMs are often employed as parts of engineering structures. However, studies on functionally graded circular plates are rare in comparison with those available on functionally graded rectangular plates [1–3]. Sureh and Mortensen [4] provided an excellent introduction to the fundamental theory of FGMs. Reddy et al. [5] have studied axisymmetric bending and stretching of functionally graded circular and annular plates using the first-order shear deformation Mindlin plate theory. Ma and Wang [6] presented relationships between axisymmetric bending and buckling solutions of functionally graded circular plates based on the third order shear deformation plate theory and the classical plate theory (CPT). Wang et al. [7] investigated the analytical solutions for axisymmetric bending of functionally graded circular and annular plates. Noiser and Fallah [8] obtained the analytical solution of the axisymmetric behavior of FGM circular plates with clamped and simply supported boundary conditions under mechanical and thermal loadings. Saidi et al. [9] carried out axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third order shear deformation plate theory(UTST). Zhong and Shang [10], carried out three dimensional analysis of a simply supported functionally gradient piezoelectric rectangular plate. Few researches have also been carried out on two directional functionally graded plates. Guojun and Zheng [11], presented the axisymmetric bending of two directional functionally graded circular and annular plates. They also carried out dynamic analysis of multi directional functionally graded annular plates [12]. In above mentioned researches, material properties are assumed to have a smooth variation usually through the thickness only. However, there are practical occasions which require grading of the materials in two or three directions. Hence, it is necessary to develop appropriate methods to investigate the mechanical responses of multi directional functionally graded structures. In this paper, the deflection of a circular plate graded through the thickness based on the UTST is calculated in terms of the deflection of a corresponding homogeneous circular plate obeying the CPT. Next, the plate is divided into number of annular plates with material properties defined by the corresponding exponential grading rule and the radial position of its center. By increasing the number of divisions, a two directional functionally graded circular plate is approximated.

43.2 Formulation of the Problem A two directional functionally graded circular plate of total thickness h, radius b and subjected to axisymmetric transverse load q is considered. The r-coordinate is assumed to be radially outward from the center, the z-coordinate along the thickness

43 Axisymmetric Bending of 2-D FGM Circular Plates Using UTST

659

and the θ-coordinate is taken along the circumference of the plate. It is assumed that the grading of the material, applied loads, and boundary conditions are axisymmetric so that the displacement uθ is identically zero and (ur , uz ) are functions of r and z. Material properties of the plate are assumed to be functions of r and z i.e. E = E(r, z) and ν = ν(r, z). According to the classical plate theory, displacement fields are defined as Ur (r, z) = u(r) − z

dwk , dr

Uz (r, z) = wk (r),

(43.1)

where u and wk are the radial displacement and transverse deflection of the midplane (i.e., z = 0 ), respectively. Displacement fields are defined on the basis of the Kirchhoff plate theory, in that both transverse shear and transverse normal effects are ignored, i.e. deformation is assumed to be entirely due to bending and in-plane stretching. The unconstrained third-order shear deformation plate theory is based on the following representation of the displacement field across the plate thickness Ur (r, z) = u(r) + zφ(r) + z3ψ(r),

Uz (r, z) = w(r),

(43.2)

where Ur and U z are the displacements along the radius and through the thickness respectively and φ and ψ denote the rotation and higher-order rotation of the transverse normal in the plane of θ = const., respectively. Linear strain components associated with the displacement field of Eq. (43.2) are ∂U r du dφ dψ = + z + z3 , ∂r dr dr dr Ur u φ ψ #θθ = = + z + z3 , r r r r ∂U r ∂U z dw γrz = + = φ+ + 3z2 ψ ∂z ∂r dr #rr =

(43.3)

The Hooke’s law for the plate is defined as σrr =

E (#rr + ν#θθ ) , 1 − ν2

σθθ =

E (#θθ + ν#rr ), 1 − ν2

σrθ =

E #rθ 1+ν

(43.4)

Equilibrium equations are obtained, using the principle of virtual displacements d (rNrr ) − Nθθ = 0, dr d (rPrr ) − Pθθ = 3rRr , dr

d (rMrr ) − Mθθ = rQr , dr d − (rQr ) = rq dr

(43.5) (43.6)

and in case of the classical plate theory, equilibrium equations are derived as d % k& k rNrr − Nθθ = 0, dr

d % k& k rMrr − Mθθ = rQkr , dr

−

d % k& rQr = rq, dr

(43.7)

660


where superscript k refers to quantities corresponding to the Kirchhoff plate theory, Nii , Mii , Pii , (i = r, θ) represent forces, moments and higher-order moments, respectively and Qr , Rr are shear force and higher-order shear force respectively which are defined by the following expressions (Nrr , Nθθ ) = (Mrr , Mθθ ) = (Prr , Pθθ ) = (Qr , Rr )

=

h/2

−h/2 h/2 −h/2 h/2 −h/2 h/2 −h/2

(σrr , σθθ ) dz, (σrr , σθθ ) zdz, (43.8) (σrr , σθθ ) z dz, 3

% & σrθ 1, z2 dz

k and M k , i.e. moment resultants corresponding to the classical plate theory and Mrr θθ are defined as 2 k 2 k d w ν dwk d w 1 dwk k k Mrr = −D + , M = −D ν + , (43.9) θθ r dr r dr dr2 dr2

where D=

Eh3 ' ( 12 1 − ν2

Using Eqs (43.3), (43.4), and (43.8), one can obtain the constitutive relations as du ν dφ ν dψ ν Nrr = A11 + u + B11 + φ + E11 + ψ , (43.10) dr r dr r dr r du u dφ φ dψ ψ Nθθ = A11 ν + + B11 ν + + E11 ν + , (43.11) dr r dr r dr r du ν dφ ν dψ ν Mrr = B11 + u + D11 + φ + F11 + ψ , (43.12) dr r dr r dr r du u dφ φ dψ ψ Mθθ = B11 ν + + D11 ν + + F11 ν + , (43.13) dr r dr r dr r du ν dφ ν dψ ν Prr = E 11 + u + F 11 + φ + H11 + ψ , (43.14) dr r dr r dr r du u dφ φ dψ ψ Pθθ = E 11 ν + + F 11 ν + + H11 ν + (43.15) dr r dr r dr r


dw + φ + 3D33ψ, dr dw Rr = D33 + φ + 3F33ψ, dr

661

Qr = A33

(43.16) (43.17)

where A11 , B11, D11 , E 11 , F 11 , H11 , A33 , D33 , F33 , are the plate stiffness coefficients defined as h/2 & E(r, z) % 2 3 4 6 A11 , B11, D11 , E 11 , F 11 , H11 = 1, z, z , z , z , z dz, 2 −h/2 1 − ν h/2 E(r, z) % 2 4 & A33 , D33 , F 33 = 1, z , z dz −h/2 2(1 + ν) where E and ν are the modulus of elasticity and the Poisson’s ratio respectively. In order to proceed with the problem, it is assumed that the plate is functionally graded in the z-direction and therefore the modulus of elasticity becomes only a function of z, E = E(z). Introducing Eqs (43.10)-(43.11) into Eq. (43.7) and twice integrating with respect to r yields u=−

B11 E11 c1 c2 φ− ψ+ r+ , A11 A11 2 r

(43.18)

where c1 and c2 are integration constants. Substituting this equation into Eqs (43.10)-(43.13), eliminates u and leads to equations which are defined in terms of φ and ψ: dφ ν dψ ν B11 B11 Mrr = α1 + φ + α2 + ψ + ν2 c1 − 2 ν1 c2 , (43.19) dr r dr r 2 r dφ φ dψ ψ B11 B11 Mθθ = α1 ν + + α2 ν + + ν2 c1 + 2 ν1 c2 , (43.20) dr r dr r 2 r dφ ν dψ ν E11 E11 Prr = α2 + φ + α3 + ψ + ν2 c1 − 2 ν1 c2 , (43.21) dr r dr r 2 r dφ φ dψ ψ E11 E 11 Pθθ = α2 ν + + α3 ν + + ν2 c1 + 2 ν1 c2 (43.22) dr r dr r 2 r where α1 = D11 −

B211 E2 B11 E11 , α2 = F11 − , α3 = H11 − 11 , A11 A11 A11

ν1 = 1 − ν, ν2 = 1 + ν

In what follows, the moment sum M k based on the CPT, the moment sum M and the higher-order moment sum P based on the UTST are presented in terms of Mik , Mi , Pi , (i = rr, θθ) respectively

662


Mk =

k + Mk Mrr θθ , ν2

M=

Mrr + Mθθ ν2

P=

Prr + Pθθ ν2

(43.23)

Using Eqs (43.9), (43.19)-(43.22) and (43.23), the following relations are obtained. r

dM k d % k& k = rMrr − Mθθ , dr dr dM d r = (rMrr ) − Mθθ , dr dr dP d r = (rPrr ) − Pθθ , dr dr

(43.24) (43.25) (43.26)

and Eqs (43.9) and (43.23) yield to the following expression. M k = −D∇2 wk

(43.27)

where ∇2 is the two-dimensional Laplacian operator in polar coordinates that is defined for axisymmetric conditions as ∇2 (•) =

d2 (•) 1 d(•) + r dr dr2

(43.28)

Substituting Eqs (43.19)-(43.22) into the second and third Eq. (43.23) and rearranging dφ φ dψ ψ M = α1 + + α2 + + B11c1 , (43.29) dr r dr r dφ φ dψ ψ P = α2 + + α3 + + E 11 c1 (43.30) dr r dr r Now, comparing second Eq. (43.7) and first (43.6) with Eqs (43.24) and (43.26) respectively, one can simply conclude Qkr =

dM k , dr

r

dP = 3rRr dr

(43.31)

Comparing second Eq. (43.6) and third (43.7) and integrating with respect to r, we have rQr = rQkr + c3 , (43.32) where c3 is integration constant. Now, rearranging Eqs (43.6), (43.25), first (43.31) and (43.32) and integrating with respect to r, lead to the following expression that establishes relations between the moment sums of the classical plate and the unconstrained third order shear deformation plate theories M = M k + c3 ln r + c4 ,

(43.33)


663

where c4 is integration constant. Substituting Eq. (43.27) into above equation yields the moment sum in terms of derivatives of wk 2 k d w 1 dwk M = −D + + c3 ln r + c4 (43.34) r dr dr2 By setting equal the moment sums of Eqs (43.29) and (43.34), integrating and rearranging, φ is defined as φ=−

α2 D dwk B11r r(2 lnr − 1) r 1 ψ− − c1 + c3 + c4 + c5 , α1 α1 dr 2α1 4α1 2α1 rα1

where c5 is integration constant. Derivation of Eq. (43.30) results 2 2 dP d φ 1 dφ φ d ψ 1 dψ ψ = α2 + − + α + − 3 dr dr2 r dr r2 dr2 r dr r2

(43.35)

(43.36)

Using Eqs (43.16) and (43.17) we obtain: Rr =

D33 αˆ 2 Qr + ψ A33 3

(43.37)

and using second Eqs (43.31), (43.32), (43.35), (43.36) and (43.37), the following expression is found 2 d ψ 1 dψ ψ k c3 αˆ 1 + − − α ˆ ψ = α ˆ Q + (43.38) 2 3 r r dr2 r dr r2 where α2 αˆ 1 = α3 − 2 , α1

⎛ ⎞ ⎜⎜⎜ D233 ⎟⎟⎟ ⎜ ⎟⎟ , αˆ 2 = 9 ⎜⎝F 33 − A33 ⎠

αˆ 3 =

3D33 α2 − A33 α1

Equation (43.38) can be rearranged as d2 ψ 1 dψ 1 2 k c3 + − + β ψ = β Q + , 1 r r dr2 r dr r2 where β2 =

αˆ 2 , αˆ 1

β1 =

αˆ 3 , αˆ 1

(43.39)

1 Qkr = − qr 2

A general solution for Eq. (43.39) is available and given as ψ = A1 I1 (βr) + A2 K1 (βr) + A3 r +

A4 , r

(43.40)

664


where A1 and A2 are constants, I1 and K1 are the first order modified Bessel function of the first and the second kinds, respectively and A3 and A4 are defined as A3 =

qαˆ 3 , 2αˆ 2

A4 = −

αˆ 3 c3 αˆ 2

Finally, ψ is obtained in terms of derivatives of wk and w, which are deflection fields based on the CPT for homogeneous plates and UTST for FGM plates, respectively. This definition leads to the expression that establishes a relation between UTST and CPT deflections. For this purpose, at first Eq. (43.16) is rearranged as φ=

Qr dw 3D33 − − ψ A33 dr A33

(43.41)

Next, equating the right hand sides of Eqs (43.35) and (43.41) and rearranging, we obtain Qr 1 dw D dwk B11r ψ= − + + c1 αˆ 3 A33 αˆ 3 dr α1 αˆ 3 dr 2α1 αˆ 3 (43.42) r(2 lnr − 1) r c5 − c3 − c4 − 4α1 αˆ 3 2α1 αˆ 3 rα1 αˆ 3 Integrating above equation with respect to r results in the deflection of a FGM plate based on the UTST, in terms of the deflection of a homogeneous plate based on the CPT, hence w=

0 lnr 1 D k M k B11 r2 r2 w + + c1 + − (ln r − 1) c3 α1 A33 4α1 A33 4α1 2 r ln r − c4 − c5 + c6 − αˆ 3 ψdr, 4α1 α1

(43.43)

where c6 is integration constant. Now, in order to study the bending of a two directional functionally graded plate, the plate is divided into number of annular plates. Material properties of each annular plate are defined by substituting the coordinates of the mid point in the grading law of the plate in the radial direction. Hence, these properties are assumed to be constant radially at each annular plate but different from the adjacent plate. Generally Poisson’s ratio ν, varies in a small range; hence, for simplicity it is assumed to be constant throughout the plate. Defining the modulus of elasticity of the plate as z r E(r, z) = E 0 exp λ1 + λ2 (43.44) h b Then the grading of each annular plate both in the radial and through the thickness directions is assumed to be in the following form z ai + ai+1 rin = ai , rout = ai+1 ⇒ E(r, z) = E 0 exp λ1 + λ2 , (43.45) h 2b


665

where E0 is the elastic modulus of the bottom of the plate (z = −h/2), λ1 and λ2 are the grading indices through the thickness and the radius of the plate, respectively. Compatibility conditions of adjacent annular plates are defined as at wi−1 = wi ,

ui−1 = ui ,

φi−1 = φi ,

Mrr,i−1 = Mrr,i ,

r = ai , i = 2, 3, · · · , n

ψi−1 = ψi ,

Nrr,i−1 = Nrr,i ,

Prr,i−1 = Prr,i ,

(43.46)

Qr,i−1 = Qr,i

In order to employ compatibility conditions of Eq. (43.46), the stress resultants of the UTST using Eqs (43.10), (43.18), (43.19), (43.21), (43.27) and (43.35) are obtained as A11 ν2 A11 ν1 c1 − 2 c2 , 2 r B11ν1 ν2 ν1 k Mrr = Mrr − 2 c2 + ωc3 + c4 − 2 c5 , 2 r r dψ ν α α4 ν2 E 11 ν1 2 k Prr = αˆ 1 + ψ + Mrr + c1 − 2 c2 dr r α1 2 r α2 ω α2 ν2 α2 ν1 + c3 + c4 − 2 c5 , α1 2α1 r α1 Nrr =

where ω=

ν ν2 1 + 2 ln r , 4 ν1

α4 = E 11 −

(43.47)

α2 B11 α1

43.3 Results and Discussion The boundary condition at the center of the plate is defined as at r = 0;

u = 0, φ = 0, ψ = 0, Qr = 0,

dw dwk = 0, Qkr = 0, =0 dr dr

(43.48)

and the boundary conditions at r = b, considering clamped or simply supported conditions at the outer radius of the plate are: • Clamped circular plate At r = b;

u = 0,

φ = 0,

ψ = 0,

w = 0,

dwk =0 dr

(43.49)

w=0

(43.50)

• Simply supported circular plate At r = b;

u = 0,

Mrr = 0,

Prr = 0,

The plate is assumed to be made of material with following properties ν = 0.3,

E 0 = 380 × 109 Pa,

λ1 = 1,

λ2 = 1

666


0 -0.155

-0.02

-0.156

Max. Deflection*10

3

-0.04 -0.157 -0.06

W*10 3

-0.158 -0.159

-0.08 -0.1

-0.16

UTST FST CPT

-0.12

-0.161

-0.14

-0.162 25

50

75

-0.16

100

0

0.25

0.5

No. of Div.

0.75

1

r/b

Fig. 43.1 Variation of wmax with number of divisions

Fig. 43.2 Variation of w of a clamped circular plate for different plate theories

The plate is subjected to a constant distributed transverse loading of 1 MPa. In order to carry out the convergence study of the method, the maximum deflection of a clamped circular plate with unit radius and thickness to radius ratio of h/b = 0.1 is considered. Variation of maximum deflection of the plate with number of divisions of the circular plate is shown in Fig. 43.1. It is clearly observed from the figure that the solution is rapidly converging. Considering the computational timing and the expected accuracy of the results, number of divisions equal to 50 is found to be sufficiently accurate for the calculation purposes. Next, variations of deflection of the circular plate according to different plate theories i.e. CPT , the first order shear deformation theory (FST) and UTST with clamped and simply supported boundary conditions are calculated and the results are presented in Figs 43.2 and 43.3 respectively.

0

-0.1

W*10

3

-0.2

-0.3

-0.4

-0.5

Fig. 43.3 Variation of w of a simply supported circular plate for different plate theories

UTST FST CPT

-0.6

0

0.25

0.5

r/b

0.75

1


667

0

0

-0.1 -0.2

-0.05

-0.3 -0.4

W*10 3

W*10 3

-0.1

-0.15

λ2=0 λ2=0.5 λ2=1 λ2=2 λ2=5

-0.2

-0.25

-0.3

0

0.25

0.5

0.75

-0.5 -0.6 -0.7

λ2=0 λ2=0.5 λ2=1 λ2=2 λ2=5

-0.8 -0.9 -1 -1.1 1

0

0.25

0.5

0.75

1

r/b

r/b

Fig. 43.4 Variation of w with λ2 (λ1 = 1), clamped circular plate

Fig. 43.5 Variation of w with λ2 (λ1 = 1), simply supported circular plate

In Figs 43.4 and 43.5, variation of lateral deflection with constant λ1 = 1 and varying λ2 from 0 to 5 for clamped and simply supported boundary conditions are carried out. It is observed that the deflection decreases as λ2 increases. It is also observed from the figures that two directional functionally graded circular plates are stiffer than those graded through the thickness graded plates. It is obvious that UTST and FST theories have excellent agreement with each other while the CPT underestimates the deflection, representing a stiffer plate. A finite element analysis of the axisymmetric bending of the circular plate is carried out. The circular plate is modeled and meshed using 2400 SOLID95 elements. Material properties of each element are defined by substituting the position of the center of the element into Eq. (43.44). The results obtained by the present method are compared with the results of the finite element analysis for clamped and simply supported boundary conditions, as shown in Table 43.1. Table 43.1 Comparsion of the w × 103 of two directional FGM plate Clamped

Simply Supported

r/b

Present

ANSYS

Error

Present

ANSYS

Error

0

-0.15429

-0.15375

0.35%

-0.69408

-0.69267

0.20%

0.2

-0.13935

-0.13887

0.34%

-0.65244

-0.65166

0.11%

0.4

-0.10196

-0.10152

0.43%

-0.54070

-0.54023

0.08%

0.6

-0.05608

-0.05573

0.61%

-0.38043

-0.38011

0.08%

0.8

-0.01716

-0.01700

0.94%

-0.19347

-0.19321

0.13%

668


43.4 Conclusions In this paper, axisymmetric bending analysis of an unconstrained third-order shear deformable, two directional functionally graded circular plate is carried out. The governing equations of the plate are solved using a convenient method by representing the results in terms of corresponding results of a plate obeying the CPT. The results obtained are compared with corresponding results of the CPT, the FST and the finite element method. It is observed that the finite element calculations are in very good agreements with the results of the FST and the UTST. It is also observed that the CPT underestimates the maximum deflection values, hence indicating a stiffer plate.

References 1. Chi S H, Chung Y L (2006) Mechanical behavior of functionally graded material plates under transverse load-part I: analysis. Int J Solids Struct 43:3657–3674 2. Chi S H, Chung Y L (2006) Mechanical behavior of functionally graded material plates under transverse load-part II: numerical results. Int J Solids Struct 43:3675–3691 3. Nguyen T K, Sab K, Bonnet G (2008) First-order shear deformation plate models for functionally graded materials. Comp Struct 83:25–36 4. Sureh A, Mortesen A (1998) Fundamentals of functionally graded materials. IOM Communications Limmited, London 5. Reddy J N, Wang C M, Kitipornachai S (1999) Axisymmetric bending of functionally graded circular and annular plates. Euro J of Mech A/Solids 18:185–199 6. Ma L S, Wang T J (2004) Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory. Int J of Solids and Struct 41:85–101 7. Wang T J, Ma L S, Shi Z F (2004) Analytical solutions for axisymmetric bending of functionally graded circular and annular plates. Acta Mech 36:348–353 8. Noiser A, Fallah F (2009) Reformation of Mindlin-Reissner governing equations of functionally graded circular plates. Acta Mech 198:209–233 9. Saidi A R, Rasouli A, Sahraee S (2009) Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third-order shear deformation plate theory. Comp Struct 89:110–119 10. Zhong Z, Shang E T (2003) Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate. Int J of Solids and Struct 40:5335–5352 11. Guojun N, Zheng Z (2007) Axisymmetric bending of two directional functionally graded circular and annular plates. Acta Mech 20:289–295 12. Guojun N, Zheng Z (2010) Dynamic analysis of multi-directional functionally graded annular plates. Appl Math Model 34:608–616

Chapter 44

Stability Analysis of Functionally Graded Plates Subject to Thermal Loads Mokhtar Bouazza, A. Tounsi, E. A. Adda-Bedia and A. Megueni

Abstract Stability analysis of functionally graded ceramic–metal plates under thermal loads is presented using the first order shear deformation theory. The effective material properties of the functionally graded plates are assumed to vary through their thickness direction according to the power-law distribution of the volume fractions of the constituents. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. The derived equilibrium and buckling equations are then solved analytically for a plate with simply supported boundary conditions. Numerical examples cover the effects of the gradient index, plate aspect ratio, side-to-thickness ratio and loading type on the critical buckling for FGM plates. Keywords Buckling · Functionally graded plate · First order shear deformation theory · Thermal loading

44.1 Introduction Functionally graded materials (FGMs) have gained wide application in a variety of industries due to their distinctive material properties that vary continuously and smoothly through certain dimensions. Compared with common composites, FGMs M. Bouazza (B) Department of Civil Engineering, University of Bechar, Bechar 08000 & Laboratory of Materials and Hydrology (LMH), University of Sidi Bel Abbes, Sidi Bel Abbes, Algeria e-mail: [email protected] A. Tounsi · E. A. Adda-Bedia Laboratory of Materials and Hydrology (LMH), University of Sidi Bel Abbes, 2200 Sidi Bel Abbes, Algeria A. Megueni Department of Mechanical Engineering, University of Sidi Bel Abbes, 2200 Sidi Bel Abbes, Algeria H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 44, © Springer-Verlag Berlin Heidelberg 2011

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avoid the inter-laminar stress gaps that are caused by mismatches in the properties of two different materials, and can be adjusted appropriately according to practical requirements. The research on FGMs mainly focuses on thermal stress analysis, the determination of static and dynamic responses, and vibration analysis. Obata and Noda [1] studied the thermal stresses in a hollow circular cylinder and a hollow sphere of an FGM, and Fukui et al. [2] examined the stresses and strains in a functionally graded thick-walled tube under uniform thermal loading. Reddy and Chin [3] presented a finite element formulation for the analysis of the dynamic thermoelastic response of functionally graded cylinders and plates that employs the first-order shear deformation plate theory to account for the transverse shear strains and the rotations. A generic static and dynamic finite element formulation was proposed by Liew et al. [4] for the modeling and control of piezoelectric shell laminates under coupled displacement and certain temperatures and electric potential fields. Pelletier and Vel [5] provided an analytical solution for the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads, and Yang and Shen [6] investigated the free vibration and dynamic instability of functionally graded cylindrical panels subjected to combined static and periodic axial forces in a thermal environment. In addition to the aforementioned linear analysis (small strains), the nonlinear response of FGMs has also attracted research interest. Praveen and Reddy [7] conducted a geometrically nonlinear transient analysis of FGM plates under thermal and mechanical loading, and Park and Kim [8] carried out a thermal postbuckling and vibration analysis of FGM plates based on the first-order shear deformation plate theory. Reddy [9] proposed a theoretical formulation for FGM plates using the third-order shear deformation plate theory, and developed a corresponding finite element model that accounts for thermomechanical coupling, time dependency, and von Kàrmàn-type geometric non-linearity. Yang et al. [10] presented a formulation for the thermo-mechanical post-buckling analysis of FGM shell panels based on the classical shell theory with von Kàrmàn-Donnell-type nonlinearity, and Hosseini Kordkheili and Naghdabadi [11] derived a finite element formulation for the geometrically nonlinear thermoelastic analysis of FGM plates and shells using the updated Lagrangian approach. Arciniega and Reddy [12] presented a tensor-based finite element formulation for the large deformation analysis of FGM shells, and Woo and Merguid [13] reported an analytical solution for the coupled large deflection of FGM plates and shallow shells under a mechanical load and in a temperature field. The present authors [14] analyzed thermal buckling of sigmoid functionally graded plates using first order shear deformation theory. In that study, the material properties of functionally graded plates were assumed to vary continuously through the thickness of the plates, according to the two power law distribution in terms of the volume fractions of constituents. In this study, the thermal buckling analysis of power-law FGM (P-FGM) are investigated by using first order shear deformation theory. The von Kàrmàn’s nonlinear strain-displacement relation is used to account for buckling due to thermal load. Material properties are varied continuously in the thickness direction according to a

44 Stability Analysis of Functionally Graded Plates Subject to Thermal Loads

671

simple power law distribution. The thermal buckling behaviors under uniform, linear and sinusoidal temperature rise across the thickness are analyzed. In addition, the effects of temperature field, volume fraction distributions, and system geometric parameters are investigated.

44.2 Functionally Graded Materials Consider a case when FGM plate is made up of a mixture of ceramic and metal as show in Fig. 44.1. The material properties vary continuously across the thickness

Fig. 44.1 Typical FGM rectangular plate

according to the following equations, which are the same as the equations proposed by Reddy et al. and Praveen et al. [3,7,9] E (z) = E m + Ecm Vf (z) ,

Ecm = E c − E m ,

α (z) = αm + αcm Vf (z) ,

αcm = αc − αm ,

(44.1)

ν (z) = ν0 , where subscripts m and c refer to properties of metal and ceramics, respectively, and Vf (z) is volume fraction of the constituents which can mostly be defined by powerlaw functions [15,16]. For power-law FGM, volume fraction function is expressed as z 1 k Vf = + (44.2) h 2

44.3 Stability Equations Assume that u, v, w denote the displacements of the neutral plane of the plate in x, y, z directions respectively; φ x , φy denote the rotations of the normals to the plate

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midplane. According to the first order shear deformation theory, the strains of the plate can be expressed [17-19] # x = u,x + zφ x,x , #y = v,y + zφy,y ,

% & γ xy = u,y + v,x + z φ x,y + φy,x ,

(44.3)

γ xz = φ x + w,x , γyz = φy + w,y The forces and moments per unit length of the plate expressed in terms of the stress components through the thickness are h/2 Ni j =

h/2 σi j dz,

Mi j =

−h/2

h/2 σi j zdz,

−h/2

Qi j =

τi j dz

(44.4)

−h/2

The nonlinear equations of equilibrium according to von Kàrmàn’s theory are given by Nx,xx + 2N xy,xy + Ny,yy = 0, M x,xx + 2M xy,xy + My,yy − Q x,x − Qy,y = 0,

(44.5)

Q x,x + Qy,y + q + Nx w,xx + Ny w,yy + 2N xy w,xy = 0 Using Eqs (44.1), (44.3) and (44.4), and assuming that the temperature variation is either linear with respect to x− and y− directions, or constant, the equilibrium Eq. (44.5) may be reduced to a set of one equation as & 2 (1 + ν) 2 % % N x w,xx + Ny w,yy + 2N xy w,xy + q E%1 & & E 1 1 − ν2 % − N x w,xx + Ny w,yy + 2N xy w,xy + q = 0, 2 E1 E3 − E2

%4 w +

(44.6)

where (E 1 , E 2 , E 3 ) =

h/2 %

& 1, z, z2 E (z) dz,

(44.7)

−h/2

h/2 (Φ, θ) =

(1, z) E (z) α (z) T (x, y, z) dz.

(44.8)

−h/2

To establish the stability equations, the critical equilibrium method is used. Assuming that the state of stable equilibrium of a general plate under thermal load may be designated by w0 . The displacement of the neighboring state is w0 + w1 ,


673

where w1 is an arbitrarily small increment of displacement. Substituting w0 + w1 into Eq. (44.6) and subtracting the original equation, results in the following stability equation & 2 (1 + ν) 2 % 0 0 % N x w1,xx + Ny0 w1,yy + 2Nxy w1,xy + q E%1 & & E1 1 − ν2 % 0 − N x0 w1,xx + Ny0 w1,yy + 2Nxy w1,xy + q = 0, 2 E1 E3 − E2

%4 w1 +

(44.9)

0 where N x0 , Ny0 , N xy refer to the pre-buckling force resultants. To determine the buckling temperature difference ΔT cr , the pre-buckling thermal forces should be found firstly. Solving the membrane form of equilibrium equations, gives the pre-buckling force resultants

N x0 = −

Φ , 1−ν

Ny0 = −

Φ , 1−ν

0 Nxy = 0.

Substituting Eqs (44.10) into Eq. (44.9), one obtains % & E 1 1 − ν2 Φ 2 (1 + ν) Φ 4 4 % w1 − % w1 + %2 w1 = 0. E1 1 − ν E1 E3 − E22 1 − ν

(44.10)

(44.11)

The simply supported boundary condition is defined as w1 = 0,

M x1 = 0,

φy1 = 0

on

x = 0, a

w1 = 0,

My1 = 0,

φ x1 = 0

on y = 0, b

(44.12)

The following approximate solution is seen to satisfy both the governing equation and the boundary conditions mπx mπy w1 = c sin sin , (44.13) a b where m, n are number of half waves in the x and y directions, respectively, and c is a constant coefficient. Substituting Eq. (44.13) into Eq. (44.11), and substituting for the thermal parameter from Eq. (8), yields: % & % & E 1 E 3 − E 22 (1 − ν)π2 m2 + n2 B2a E 1 Φ= % & % & (44.14) ' ( 2 (1 + ν) E 1 E3 − E 22 π2 m2 + n2 B2a + E12 a2 1 − ν2

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44.4 Buckling Analysis In this section, the thermal buckling behaviors of fully simply supported rectangular metal-ceramic plates under thermal environment are analyzed. The thermal load is assumed to be uniform, linear and sinusoidal temperature rise through the thickness direction. The reference temperature is assumed to be 5 ◦ C. The effects of volume fraction index and geometric parameter a/h are investigated in each case. The combination of materials consists of aluminum and alumina.The coefficients of Young’s modulus, conductivity, and thermal expansion for alumina are E c = 380 GPa, Kc = 10.4 W/mK, αc = 7.4 × 106 (1/◦ C), and for aluminum are E m = 70 GPa, Km = 204 W/mK, αm = 23 × 106 (1/◦ C), respectively. Poisson’s ratio is chosen as ν = 0.3.

44.4.1 Uniform Temperature Rises The plate initial temperature is assumed to be T i . The temperature is uniformly raised to a final value T f in which the plate buckles. The temperature change is ΔT = T f −T i . Using Eqs (44.14), (44.1), and (44.8), the buckling temperature change is obtained as Φ = PΔT, (44.15) where h/2 P=

E (z) α (z) dz.

(44.16)

−h/2

The critical temperature difference is obtained for the values of m, n that make the preceding expression a minimum. Apparently, the critical temperature difference for m = n = 1 is obtained by using minimization methods.

44.4.2 Linear Temperature Rise The temperature field under linear temperature rise through the thickness is assumed as ΔT h T (z) = z + + Tm, (44.17) h 2 where z is the coordinate variable in the thickness direction which measured from the middle plane of the plate. T m is the metal temperature and ΔT is the temperature difference between ceramic surface and metal surface, i.e., ΔT = T c − T m . For this loading case, the thermal parameter can be expressed as


Φ = PT m + XΔT, where

h/2 X=

E (z) α (z) −h/2

From Eq. (44.18) one has ΔT =

675

(44.18)

z 1 + dz. h 2

Φ − PT m . X

(44.19)

(44.20)

44.4.3 Sinusoidal Temperature Rise The temperature field under sinusoidal temperature rise across the thickness is assumed as 0 1 πz π T (z) = ΔT 1 − cos + + Tm, (44.21) 2h 4 where T (z) is the temperature gradient. From Eqs (44.21) and Eq. (44.8) can be expressed as follows: Φ = P (T m + ΔT ) − YΔT, (44.22) where

h/2 Y=

E (z) α (z) cos

−h/2

From Eq. (44.22) one has ΔT =

πz 2h

Φ − PT m P−Y

+

π dz. 4

(44.23)

(44.24)

44.5 Numerical Results and Discussion 44.5.1 Comparisons In order to prove the validity of the present formulation, results were obtained for isotropic plates and compared with the existing ones in the literature. The critical temperatures of simply supported, isotropic square plates subjected to constant and linearly varying temperature distributions obtained using first order shear deformation theory are verified against the energy method based results of Gowda and Pandalai [21] and solution of Kari et al [20] based on finite element method using semiloof element in Table 1. Both results are in excellent agreement.

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Table 44.1 Critical temperature for isotropic square plates subjected to different forms of temperature distribution Temperature distribution Analytical [21] FEM [20] Present Uniform temperature rise

63.27

63.33

63.237

Linear temperature rise

126.54

126.00

126.474

Fig. 44.2 Critical temperature of simply supported isotropic plates (a/h = 100)

Fig. 44.3 Young’s modulus variation associated with different exponent indexes for a P-FGM plate

In addition, the buckling loads for simply supported, isotropic plates under uniform temperature rise are calculated and compared in Fig. 44.2 with finite element results obtained by Kari et al [20]. It can be seen that, for most cases the present results agree well with existing results.


677

44.5.2 Buckling Analysis of FGM Plates The variation of Young’s modulus in the thickness direction of the P-FGM plate is depicted in Fig. 44.3, which shows that the Young’s modulus changes rapidly near the lowest surface, and increases quickly near the top surface. Firstly, the critical buckling temperature are calculated for functionally graded plates with different volume fraction exponent under uniform temperature rise, linear and sinusoidal temperature distribution across the thickness and are plotted in Figs 44.4 and 44.5. The plate aspect ratio is set as (a/b = 1). These two figures show that the critical buckling temperature or temperature difference increases as the relative thickness h/a increases, whatever the gradient index k is. However, the

Fig. 44.4 Critical buckling temperature rise of functionally graded square plate vs h/a (k = 0)

Fig. 44.5 Critical buckling temperature rise of a functionally graded square plate vs h/a (k = 2)

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Fig. 44.6 Critical buckling temperature rise of a functionally graded rectangular plate vs a/b (k = 0)

Fig. 44.7 Critical buckling temperature rise of a functionally graded rectangular plate vs a/b (k = 1)

critical temperature gradient under sinusoidal temperature rise is higher than that under linear and uniform temperature rise. Figures 44.6 and 44.7 show the variation trend of critical temperature difference with respect to the plate aspect ratio a/b for different values of material gradient index k. The relative thickness of the plate is set as h/a = 0.2. It is observed that with increasing the plate aspect ratio a/b from 1 to 10, the critical buckling temperature difference also increases steadily, whatever the material gradient index k is. Comparing Figs 44.4 and 44.6 with Figs 44.5 and 44.7, the responses are very similar; however, the critical temperature difference demonstrates a decreasing trend with increasing gradient index.


679

44.6 Conclusions Thermal buckling analysis of fully simply supported rectangular FGM plates under thermal environment are investigated by using first order shear deformation theory. The von Kàrmàn’s nonlinear strain-displacement relation is used to account for buckling due to thermal load. The thermal load is assumed to be uniform, linear and sinusoidal temperature rise through the thickness direction. Based on the numerical results, the following conclusions are reached: 1. The critical buckling temperature for functionally graded rectangular plates are generally lower than the corresponding values for homogeneous plates. It is very important to check the strength of the functionally graded plate due to thermal buckling, although it has many advantages as a heat resistant material. 2. Geometric parameter h/a is increased, the critical temperature gradient increases rapidly. 3. Volume fraction index k is increased, the critical temperature gradient decreases. This is because as volume fraction index is increased, the contained quantity of ceramic decreases. 4. The critical temperature under sinusoidal temperature rise has the highest value in three cases, and that under linear temperature rise is higher than that under uniform temperature rise.

References 1. Obata, Y., Noda, N.: Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded material. Journal of Thermal Stresses. 17, 471–487 (1994) 2. Fukui, Y., Yamanaka, N., Wakashima, K.: The stresses and strains in a thick-walled tube for functionally graded material under uniform thermal loading. JSME International Journal Series A: Solid Mechanics and Material Engineering. 36, 156-162 (1993) 3. Reddy, J.N., Chin, C.D.: Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses. 21, 593–626 (1998) 4. Liew, K.M., He, X.Q., Ng, T.Y., Kitipornchai, S.: Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches. International Journal for Numerical Methods in Engineering. 55, 653–668 (2002) 5. Pelletier, J.L., Vel, S.S.: An exact solution for steady-state thermoelastic response of functionally graded orthotropic cylindrical shells. Int. J. Solids Struct. 43, 1131–1158 (2006) 6. Yang, J., Shen, H.S.: Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. Journal of Sound and Vibration. 261, 871–893 (2003) 7. Praveen, G.N., Reddy, J.N.: Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35, 4457–4476 (1998) 8. Park, J.S., Kim, J.H.: Thermal postbuckling and vibration analyses of functionally graded plates. Journal of Sound and Vibration. 289, 77–93 (2006) 9. Reddy, J.N.: Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering. 47, 663–684 (2000) 10. Yang, J., Liew, K.M., Wu, Y.F., Kitipornchai, S.: Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties. Int. J. Solids Struct. 43, 307–324 (2006)

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11. Hosseini Kordkheili, S.A., Naghdabadi, R.: Geometrically nonlinear thermoelastic analysis of functionally graded shells using finite element method. International Journal for Numerical Methods in Engineering. 72, 964–986 (2007) 12. Arciniega, R.A., Reddy, J.N.: Large deformation analysis of functionally graded shells. Int. J. Solids Struct. 44, 2036–2052 (2007) 13. Woo, J., Merguid, S.A.: Nonlinear analysis of functionally graded plates and shallow shells. Int. J. Solids Struct. 38, 7409–7421 (2001) 14. Bouazza, M., Tounsi, A., Adda-Bedia, E.A., Megueni, A.: Thermal buckling of sigmoid functionally graded plates using first order shear deformation theory, MAMERN09: 3rd International Conference on Approximation Methods and Numerical Modeling in Environment and Natural Resources Pau. France, June 8-11 (2009) 15. Chi, S.H., Chung, Y.L.: Mechanical behavior of functionally graded material plates under transverse load-Part I: Analysis. Int. J. Solids Struct. 43, 3657–3674 (2006) 16. Chi, S.H., Chung, Y.L.: Mechanical behavior of functionally graded material plates under transverse load- Part II: Numerical results. Int. J. Solids Struct. 43, 3675–3691 (2006) 17. Mindlin, R. D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18, 31–38 (1951) 18. Reissner, E.: On the theory of bending of elastic plates. J. Math. Phy. 23, 184–191 (1944) 19. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, 69–77 (1945) 20. Kari, R. Thangaratnam, Palaninathan, Ramachandran, J.: Thermal buckling of composite laminated plates. Computers & Structures. 32 (5), 1117–1124 (1989) 21. Gowda, R.M.S., Padalai, K.A.V.: Thermal buckling of orthotropic plates. In Studies in Structural Mechanics (Edited by K. A. V. Padalai). IIT, Madras, 9–44 (1970)

Chapter 45

A Best Theory Diagram for Metallic and Laminated Shells Erasmo Carrera, Maria Cinefra and Marco Petrolo

Abstract In this work, refinements of classical theories are proposed in order to analyze isotropic, orthotropic and laminated plates and shells. Higher order theories have been implemented according to the Carrera Unified Formulation (CUF) and, for a given problem, the effectiveness of each employed generalized displacement variable has been established, varying the thickness ratio, the orthotropic ratio and the stacking sequence of the lay-out. A number of theories have therefore been constructed imposing a given error with respect to the available ’best solution’. The results have been restricted to the problems for which closed-form solutions are available. These show that the terms that have to be used according to a given error vary from problem to problem, but they also vary when the variable that has to be evaluated (displacement, stress components) is changed. Keywords Refined classical theories · Laminated shells and plates · Unified formulation

45.1 Introduction Laminated structures such as traditional composite panels are frequently found in aerospace vehicle applications. High transverse shear and normal deformability as E. Carrera (B) · M. Cinefra Department of Aeronautic and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy e-mail: [email protected],[email protected] M. Petrolo Department of Aeronautic and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy, and Institut Jean Le Rond d’Alembert, UMR7190 CNRS, Paris06, Case 162, Tour 55-65, 4, Place Jussieu, 75252 Paris, France e-mail: [email protected],[email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 45, © Springer-Verlag Berlin Heidelberg 2011

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well as discontinuity of physical properties make the use of structural models difficult. Accurate stress and strain fields evaluation demand the development of ad hoc theories for the analysis of these structures. Most known theories were originated from the intuition of some structural analysis pioneers. Among these, Kirchhoff [25], Love [29], Reissner [38], Mindlin [30], Vlasov [42], Koiter [26] and Naghdi [31], and others. In most cases, these ‘axiomatic’ intuitions lead to a simplified kinematics of the true three-dimensional deformation state of the considered structure: the section remains plane, the thickness deformation can be discarded, shear strains are negligible, etc... For a complete review of this topic, including laminated composite structures, the readers can refer to the many available survey articles on plates and shells [28]- [36]. As an alternative to the axiomatic approach, approximated theories have been introduced employing ’asymptotic-type’ expansions of unknown variables over the thickness. The order of magnitude of significant terms is evaluated referring to a geometrical parameter (thickness-to-length in the case of plates and shells). The asymptotic approach furnishes consistent approximations. This means that all the retained terms are those which have the same order of magnitude as the introduced perturbation parameter when the latter vanishes. Articles on the application of asymptotic methods to shell structures can be found in Cicala [17], Fettahlioglu and Steele [19], Berdichevsky [1, 2], Widera and coauthors [43, 44], and Spencer et alii [40], as well as in the monographes by Cicala [18] and Gol’denweizer [20]. Both the axiomatic and the asymptotic methods have historically been motivated by the need to work with simplified theories that are capable of leading to simple formulas and equations which can be solved by hand calculation. Up to five decades ago, in fact, it was quite prohibitive to solve problems with many unknowns (more than 5, 6); nowadays, this limitation no longer holds. Of course, the formulation of more complicated problems would be difficult without the introduction of appropriate techniques that are suitable for computer implementations. The approach herein discussed makes use of such a suitable condensed notation technique that was introduced by the first author during the last decade and it is referred to as the Carrera Unified Formulation, CUF, for beams, plates and shell structures [3,4,6–8,10]. Governing equations are given in terms of a few ’fundamental nuclei’ whose form does not depend on either the order of the introduced approximations or on the choices made for the base functions in the thickness direction. In short, CUF makes it possible to implement those terms which had been neglected by the above cited pioneers. In order to obtain more general conclusions and to draw general guidelines and recommendations in building bidimensional theories for metallic and composite plates and shells, it would be of great interest to evaluate the effectiveness of each refined theory term. This has been done in the present paper. In CUF, in fact, the role of each displacement variable in the solution is investigated by measuring the loss of accuracy due to its being neglected. A term is considered ineffective, i.e. negligible, if it does not affect the accuracy of the solution with respect to a reference 3D solution. Reduced kinematics models, based on a set of retained displacement variables, are then obtained for each considered configuration. Full and reduced models are then compared in order to highlight the

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sensitivity of a kinematics model to variations in the structural problem. This method can somehow be considered as a mixed axiomatic/asymptotic approach since it furnishes asymptotic-like results, starting from a preliminary axiomatic choice of the base functions. A companion investigation, related to plates, has been proposed in [14, 15]. The present work deals with problems with only displacement variables formulated using the Principle of Virtual Displacements (PVD). The results have been restricted to simply supported orthotropic plates and shells, subjected to harmonic distributions of transverse pressure for which closed-form solutions are available. Among the theories contained in the CUF, only the Equivalent Single Layer (ESL) models are considered, in which a laminate plate/shell is reduced to a single lamina of equivalent characteristics. The effectiveness of each displacement variable has been established, varying the thickness ratio, the orthotropic ratio and the stacking sequence of the lay-out. For a given problem, the best theory has been constructed imposing a given error with respect to the available best results.

45.2 Carrera Unified Formulation The main feature of the Carrera Unified Formulation (CUF) is the unified manner in which a large variety of beam/plate/shell structures are modeled. Details of CUF can be found in the already mentioned papers [3, 4, 6, 7, 10]. According to this formulation, the governing equations are written in terms of a few fundamental nuclei which do not formally depend on both the order of the expansion N and the approximating functions, in the thickness direction. In the case of ESL approach, which is herein considered, the displacement field is modeled in the following manner: u = F τ uτ ,

τ = 1, 2, ...., N

(45.1)

where Fτ are functions of z. uτ is the displacements vector and N stands for the order of the expansion. According to Einstein’s notation, the repeated subscript τ indicates summation. In this work, Taylor polynomials are used for the expansion: F τ = zτ−1 ,

τ = 1, 2, ...., N

(45.2)

N is assumed to be as high as 4. Therefore the displacement field is: u = u1 + z u2 + z2 u3 + z3 u4 + z4 u5 v = v1 + z v2 + z2 v3 + z3 v4 + z4 v5 w = w1 + z w2

+ z2

w3

+ z3

w4

+ z4

(45.3) w5

The Reissner-Mindlin model [30,38] (also known as First Order Shear Deformation Theory, FSDT, in the case of laminates) for the plate and the Naghdi model [31] for the shell, can be obtained acting on the Fτ expansion. Two conditions have

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to be imposed. 1) First-order approximation kinematics field, 2) the displacement component w has to be constant above the cross-section, i.e. w2 = 0. The resultant displacement model is: u = u 1 + z u2 (45.4) v = v1 + z v2 w = w1 The Kirchhoff-type approximation [25] (also known as Classical Laminate Theory, CLT) for plate and the Koiter approximation [26] for the shell, can also be obtained using a penalty technique for the shear correction factor.

45.2.1 Governing Differential Equations In this work, the Principle of Virtual Displacements (PVD) is used to obtain the governing equations and boundary conditions. In the general case of multi-layered plates/shells subjected to mechanical loads, the governing equations are: T

δuks :

k k Kkτs uu uτ = Puτ

Kkτs uu

(45.5) Pkuτ

where T indicates the transpose and k the layer. and are the fundamental nuclei for the stiffness and load terms, respectively, and they are assembled through the depicted indexes, τ and s, which consider the order of the expansion in z for the displacements. The corresponding Neumann-type boundary conditions are: k kτs k ¯τ , Π kτs d uτ = Π d u

(45.6)

where Π kτs d is the fundamental nucleus for the boundary conditions and the over-line indicates an assigned condition. For the explicit form of fundamental nuclei for the Navier-type closed-form solution and more details about the constitutive equations and geometrical relations for laminated plates and shells in the framework of CUF, one can refer to [6].

45.3 Method to Build the Best Plate/Shell Theories Significant advantages are offered by refined plate/shell theories in terms of accuracy of the solution, but a higher computational effort is necessary because of the presence of a larger number of displacement variables. This work is an effort to understand the convenience of using a fully refined model rather than a reduced one. The effectiveness of each term, as well as the terms that have to be retained in the formulation, are investigated as follows.


685

1. The problem data are fixed (geometry, boundary conditions, loadings, materials and layer lay-outs). 2. A set of output variables is chosen (maximum displacements, stress/displacement component at a given point, etc.). 3. A theory is fixed, that is, the terms that have to be considered in the expansion of u, v, and w are established. 4. A reference solution is used to establish the accuracy (the N = 4 case is assumed as the best-reference solution since it offers an excellent agreement with the 3D solutions). 5. The effectiveness of each term is numerically established measuring the error produced compared to the reference solution. 6. Any term which does not give any contribution to the mechanical response is not considered as effective in the kinematics model. 7. The most suitable kinematics model is then detected for a given structural lay-out. A graphical notation has been introduced to make the representation of the obtained results more readable. This consists of a table with three lines for the displacement components and a number of columns that depends on the number of displacement variables which are used in the expansion. All 15 terms of the expansion are reported in Table 45.1. The table is referred to the fourth-order model, N = 4, expressed in Table 45.1 Locations of the displacement variables within the tables layout N=0 N=1 N=2 N =3 N=4 u1

u2 z u3 z2 u4 z3 u5 z4

v1

v2 z

w1

w2 z w3 z2 w4 z3 w5 z4

v3 z2 v4 z3 v5 z4

Eq.(45.3). White and black triangles are used to denote the inactive and active terms respectively, as in Table 45.2. Table 45.3 shows the case in which the parabolic term of the expansion of the in-plane displacement v is discarded. The elimination of a Table 45.2 Symbols to indicate the status of a displacement variable Active term Inactive term

term, as well as the evaluation of its effectiveness in the analysis, can be obtained by exploiting a penalty technique. The corresponding results are compared with those given by a full fourth-order model using the percentage variations δw and δσ . These parameters are defined according to the following formulas:

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Table 45.3 Symbolic representation of the reduced kinematics model with v3 deactivated

δw =

w wN=4

× 100, δσzz =

σzz σzzN=4

× 100,

(45.7)

Where subscript ’N = 4’ denotes the values that correspond to the plate/shell theory given by Eq.(45.3). Parameters related to other stress or displacement values could also be introduced ( δu , δσxx , etc.). It is important to notice that a displacement variable of the expansion can be considered non effective with respect to a specific output component (displacement or stress) when, if neglected (removed from the formulation), it does not introduce any changes in the results according to a fixed accuracy. The accuracy is here fixed to be as 0.05 %, that is, a term is considered negligible if the error caused by its absence in the kinematics model is lower than 0.05 %. When conducting the analysis of each displacement variable, a reduced kinematics model, if any exists, is established which is equivalent to a fourth-order expansion. The numerical investigation has considered either plates and shells with different lay-outs: isotropic, orthotropic and cross-ply composite plates and shells. Furthermore, the effects on the definition of the reduced model of the following geometrical/mechanical parameters have been evaluated: length-to-thickness ratio a/h (for shells R/h), orthotropic ratio E L /ET and ply sequence.

45.4 Results and Discussion In the following discussion, either plate and shell structures with different geometries and lay-outs are considered. For each case, it is taken for granted that the solution obtained with the theory of Eq.(45.3) is very close to the 3D solution, as demonstrated in many first-author’s works among which [5,7,12,13,15]. Therefore, the fourth-order model has been chosen as the reference solution for the present analysis.

45.4.1 Plates A simply supported plate has been considered. A bi-sinusoidal transverse distributed load is applied to the top surface: pz = p¯ z sin(

mπx nπy ) sin( ) a b

(45.8)


687

with a = 0.1 [m], b is assumed equal to a. p¯ z is the applied load amplitude, p¯ z = 1 [kPa], and m, n are the wave number in the two in-plane, plate directions. Attention has been restricted to the case m = n = 1. w, σ xx and σzz are computed at [a/2, b/2, h/2], while σ xz is computed at [0, b/2, 0]. An isotropic plate has been considered first. Young’s modulus, E, is equal to 73 [GPa]. Poisson’s ratio, ν, is equal to 0.34. Four different length-to-thickness ratios, a/h, are considered: 100, 10, 5 and 2, that is, thin, moderately thick, thick and very thick plates, respectively. The results of the effectiveness of each displacement variable are given in Table 45.4. A thin plate geometry has been considered (a/h = 100). The percentage variation, δ, introduced neglecting each displacement variable, is evaluated for w, σ xx , σxz and σzz . The following comments can be made. 1. The constant term w1 , the linear terms, u2 and v2 and the parabolic term w3 are the most important ones to detect w, σxx and σ xz . 2. Accurate evaluations of σzz require the use of w1 , ..., w5 variables. For the sake of brevity, the tables referring to different a/h values are not reported here, but they can be found in [15]. A rather quite comprehensive analysis is instead given in Table 45.5, which considers different plate geometries. The sets of effective terms are reported, that is, the plate models required to detect the fourth-order solution are shown. The last column gives the expansion terms needed to ’exactly’ detect the whole considered outputs ’exactly’. Me states the number of terms (i. e. computational costs) of the theory necessary to meet the fourth-order accuracy requirements. The required terms are again those corresponding to the black triangles. Some remarks can be made. 1. As a/h decreases, the theories become more computationally expensive (Me increases). 2. Different choices of displacement variables are required to obtain exact different outputs. 3. All 15 terms are necessary for very thick plate geometries. Table 45.5 is, as in the paper title, an attempt to offer both guidelines and recommendations for building the best plate/shell theories for the considered problems. Orthotropic plates have been considered to assess the accuracy of the plate theory vs. orthotropic ratio, E L /E T . It is a well known fact that orthotropic plates, such as laminated composite structures, exhibit larger shear deformations than metallic structures made of isotropic materials. The analysis of such plates is therefore of particular interest for the present investigation. Young’s modulus along the transverse direction, E T , is assumed as high as 1 [GPa]. Different orthotropic ratios, E L /ET , are assumed: 5, 25 and 100, where E L stands for Young’s modulus along the longitudinal direction. Young’s thickness modulus, Ez , is assumed equal to ET . The shear moduli are assumed as high as 0.39 [GPa]. Poisson’s ratios, νLT and νLz , are equal to 0.25, and a/h is assumed equal to 10. The sets of effective variables for different orthotropic ratios are summarized in Table 45.6.

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Table 45.4 Influence of each displacement variable of a fourth order model on the solution. a/h= 100. Isotropic plate [15] δw [%]

δσxx [%]

δσxz [%]

δσzz [%]

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

1.3 × 10−5 1.2 × 10−2 1.3 × 10−2

81.3

0.2

0.2

299.7

100.0

0.2

0.2

0.2

100.0

100.0

100.0

100.0

139.1

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

94.6

74.2

101.8

−8.1 × 104

100.0

100.0

72.3

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

127.6


689

Table 45.5 Comparison of the sets of effective terms for isotropic plates with different a/h [15] w

σ xx

σ xz

σzz

COMBINED

a/h = 100 Me = 4

Me = 4

Me = 5

Me = 4

Me = 7

a/h = 10 Me = 6 Me = 10 Me = 6

Me = 9

Me = 13 '

a/h = 5 Me = 9 Me = 11 Me = 7

Me = 9

Me = 13

a/h = 2 Me = 13 Me = 14 Me = 7 Me = 13

Me = 15

Table 45.6 Comparison of the sets of effective terms for orthotropic plates with different E L /ET [15] w

σ xx

σ xz

σzz

COMBINED

Me = 7 Me = 10 Me = 6

Me = 9

Me = 13

E L /ET = 5

E L /ET = 25 Me = 6 Me = 10 Me = 4

Me = 9

Me = 13

E L /ET = 100 Me = 5

Me = 6

Me = 3

Me = 7

Me = 11

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The obtained results suggest the following comments. 1. The conclusions made for the isotropic plate are confirmed for the orthotropic plate. 2. The plate theory needed to furnish an accurate description of several outputs tends to have lower Me for larger E L /ET values (as shown in the last column of Table 45.6); 3. The orthotropic ratio, E L /E T , plays a similar role as the length-to-thickness ratio, a/h; these two parameters are the most significant in evaluating the accuracy of a given plate theory. Composite plates have been analyzed to assess the plate theory accuracy vs. stacking sequence. Attention is restricted to higher order theories, as in Eq. (45.3). The authors are aware that laminated structures require more adequate descriptions, such as those given by the so-called zig-zag theories as well as a layer-wise description. Such analysis are herein omitted but could be the subject of future investigations. The readers are addressed to the already mentioned review papers as well as to the historical review on zig-zag theories in [11]. A three-layer composite plate has been analyzed. E L is equal to 40 [GPa]. ET and E z are equal to 1 [GPa]. νLT and νLz are equal to 0.5 and 0.6, respectively. Each layer is 0.001 [m] thick. Three stacking sequences are considered: two symmetrical (0◦ ) and (0◦ /90◦ /0◦ ), and one asymmetrical (0◦ /0◦ /90◦). Table 45.7 shows the plate model for each stacking sequence and output variable as well as for the combined evaluation of w, σxx , σxz and σzz to obtain a fourth-order model accuracy. Table 45.7 Comparison of the sets of effective terms for composite plates with different stacking sequences [15] w

σ xx

σ xz

σzz

COMBINED

◦

0 Me = 5

Me = 5

Me = 4

Me = 4

Me = 7

0◦ /90◦ /0◦ Me = 5

Me = 5

Me = 4

Me = 7

Me = 7

0◦ /0◦ /90◦ Me = 9 Me = 12 Me = 10 Me = 14

Me = 14


691

The table highlights the following main aspects related to the choice of the plate theory. 1. The stacking sequence influences the construction of adequate plate models to a great extent; it plays a similar role as the geometry and the orthotropic ratio; 2. An asymmetric lamination sequence requires a considerably higher number of displacement variables than a symmetric one.

45.4.2 Shells An isotropic shell has been considered first. The geometry of the shell is described in Fig. 45.1. Referring to Ren’s work [39], a simply supported shell and a sinusoidal

Fig. 45.1 Geometry of shell

distribution of transverse pressure applied at the top surface have been considered (cylindrical bending problem): nπβ (45.9) pz = p¯ z sin b where β is the curvilinear coordinate. The attention has been restricted to the case n = 1. Rβ is assumed equal to 10 [m] and the dimension b = π3 Rβ . The amplitude of the applied load is p¯ z = 1 [kPa]. Young’s modulus, E, is equal to 73 [GPa] and Poisson’s ratio, ν, is equal to 0.34. Four different thickness ratios, Rβ /h, are considered: 100, 50, 10 and 4. The displacement w and the stresses σyy and σzz are computed at [a/2, b/2, h/2], while σyz is computed at [a/2, 0, 0]. For the sake of brevity, the study of the effectiveness of each displacement variable is not here reported, but in Table 45.8 the sets of effective terms required to

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Table 45.8 Comparison of the sets of effective terms for isotropic shells with different Rβ /h w

σyy

σyz

σzz

Me = 4

Me = 6

Me = 5

Me = 9

Me = 9

COMBINED

Rβ /h = 100

Rβ /h = 50 Me = 4

Me = 6

Me = 5 Me = 10

Me = 10

Rβ /h = 10 Me = 6

Me = 8

Me = 5 Me = 10

Me = 10

Rβ /h = 4 Me = 6

Me = 9

Me = 8 Me = 10

Me = 10

detect the fourth-order solution are reported for each thickness ratio Rβ /h. The conclusions made for the isotropic plate are valid also for the shell: as the thickness ratio decreases, the theories become more computationally expensive (Me increases). Moreover, as in the plate w3 is important to detect w, σyy and σyz and all the terms of w expansion are necessary for the exact evaluation of σzz . In this particular case, the terms of the expansion of u are non influential because a cylindrical bending problem has been considered. One can note that the constant term of the in plane displacement v is more important in the shell than in the plate because the shell has a membranal deformation even when it is very thin. This fact is due to the curvature. Orthotropic shells have been considered to assess the accuracy of the shell theory vs. orthotropic ratio, E L /ET . The geometry of the shell is cylindrical, with a/Rβ = 4, nπβ and the loading is internal sinusoidal pressure pz = p¯ z sin( mπα a ) sin( b ), with m = 1 and n = 8. Young’s modulus along the transverse direction, ET , is assumed as high as 1 [GPa]. Different orthotropic ratios, E L /E T , are assumed: 5, 25 and 100, where E L stands for Young’s modulus along the longitudinal direction. The shear moduli G LT and GT T are assumed 0.5E T and 0.2E T , respectively, and Poisson’s ratios, νLT and νT T , are equal to 0.25. Rβ /h is assumed equal to 100. The displacement w and the stresses σyy and σzz are computed at [a/2, b/2, −h/2], while σyz is computed at [a/2, 0, 0]. The 3D solution for this problem is given by Varadan and Bhaskar in [41].


693

A single-layered (90◦) shell has been analyzed (the lamination angle is measured with respect to the longitudinal axis). The sets of effective variables for different orthotropic ratios are summarized in Table 45.9. Table 45.9 Comparison of the sets of effective terms for orthotropic shells with different E L /ET w

σyy

σyz

σzz

Me = 7

Me = 8

Me = 7 Me = 10

COMBINED

E L /ET = 5 Me = 11

E L /ET = 25 Me = 8

Me = 8

Me = 7 Me = 10

Me = 10

E L /ET = 100 Me = 7

Me = 7

Me = 7 Me = 10

Me = 11

The obtained results confirm the conclusions made for the plate. The orthotropic ratio plays a similar role as the thickness ratio in evaluating the accuracy of a given theory and the sets of effective terms differ to a great extent with changing of E L /ET . Finally, composite shells have been analyzed to assess the shell theory accuracy vs. stacking sequence. In addition to the shell considered above, a three-layered symmetrical (90◦ /0◦ /90◦) shell and a two-layered asymmetric (90◦ /0◦ ) shell have been considered. In all the cases, the layers are of equal thickness. The orthotropic ratio E L /E T is taken equal to 25 and the thickness ratio Rβ /h = 100. Table 45.10 shows the shell model for each stacking sequence in order to obtain a fourth-order model accuracy. As in the plate case, the table highlights that the stacking sequence influences the construction of adequate models to a great extent and an asymmetric lamination sequence requires a higher number of displacement variables than a symmetric one. In general, from the analysis of the shell geometry, one can note that more and different displacement variables are effective in the evaluation of the different outputs, compared with the plate, if analogous problems are considered. This fact is due to the curvature that appear in the strain-displacement relations and couples membranal and bending behaviors of the shell.

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Table 45.10 Comparison of the sets of effective terms for composite shells with different stacking sequences w

σyy

σyz

σzz

COMBINED

90◦ Me = 8

Me = 8

Me = 7 Me = 10

Me = 10

90◦ /0◦ /90◦ Me = 8

Me = 9

Me = 6 Me = 10

Me = 10

Me = 12

90◦ /0◦ Me = 6

Me = 9 Me = 10 Me = 11

45.4.3 The Best Theory Diagram The approach here presented has proved its validity in constructing: 1. reduced models equivalent to a full higher-order theory; 2. reduced models able to fulfil a given accuracy input. The construction of these models has highlighted that Unified Formulation allows us, for a given problem, to obtain a diagram that in terms of accuracy (input) gives an answer to the following fundamental questions: • what is the ’minimum’ number of the terms, Nmin , to be used in a plate/shell model? • Which are the terms to be retained, that is, which are the generalized displacement variables to be used as degrees od freedom? To the best of the authors’ knowledge, there are no other available methods that can provide this kind of results. The present method of analysis is able to create plots like the one in Fig. 45.2 that gives the number of terms as function of the permitted error. This plot can be defined as the Best Theory Diagram BTD since it allows us to edit an arbitrary given theory in order to have a lower amount of terms for a given error (vertical shift, ΔN ) or, to increase the accuracy keeping the computational cost constant (horizontal shift, Δerror ). Most times, the plot presented appears as an hyperbole. CUF makes the computation of such a plot possible. Note that the diagram has the following properties: • it changes by changing problems (thickness ratio, orthotropic ratio, stacking sequence, etc.);


st Be

r eo Th

Number of Terms

Error

y

695

Arbitrary Theory

N

Di

ag ram Error

Fig. 45.2 An example of Best Theory Diagram (BTD) [16]

Fig. 45.3 Accuracy of all the possible combinations of plate models in computing w for the simplysupported plate loaded by a bi-sinusoidal load (each ’+’ indicates a different plate model) [16]

• it changes by changing output variable (displacement/stress components, or a combination of these). The validity of the BTD is tested by computing the accuracy of all the plate/shell models obtainable as a combination of the 15 terms of the fourth-order theory. The results are reported in Fig. 45.3 in the case of a simply-supported plate loaded by a bi-sinusoidal load; the transversal displacement w is considered as output variable. The BTD perfectly matches the lower boundaries of the region where all the models lie. This confirms that the BTD represents the best theory (i.e. the least cumbersome) for a given problem. The BTD permits the evaluation of any existing plate/shell model, as in the previous sections. The distance from the BTD of a given known model represents a guideline to recommend any other theory. More complex analyses will be conducted using the Finite Element Method in future work to investigate the effects of loadings, boundary conditions, etc. Some of these analyses

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have been already studied in [14, 16] where the attention was restricted to plates. Future investigations could consider the shells.

45.5 Conclusions The effectiveness of each displacement variable of higher order plate/shell theories has been investigated in this paper. The Carrera Unified Formulation (CUF) has been used for the systematic implementation of refined models. Navier-type closed-form solutions have been adopted for the analysis. Isotropic, orthotropic and composite plates and shells have been considered. The role of each displacement variable has been described in terms of displacement and stress components, referring to a fourth-order model solution. The contribution of each term to the accuracy of the solution has been evaluated, introducing the so-called mixed axiomatic/asymptotic method, which is able to recognize the effectiveness of each displacement variable of an arbitrary refined theory. It can be stated that the choice of the model which suits the accuracy requirements for a given problem is dominated by the length-to-thickness ratio, the orthotropic ratio and the lamination sequence. It has also been found that each displacement/stress component would require its own model to obtain exact results. Moreover, the number of retained terms is very closely related to the geometrical/mechanical configuration of the considered problem. In particular, the shell configurations require more displacement variables than the plates because the curvature introduces coupling effects. Remarkable benefits, in terms of total amount of problem variables, are obtained for thin structures or for symmetrical laminations. Finally, the use of full models is mandatory when a complete set of results is needed. CUF has shown to be well able to deal with a method that could be stated as a mixed assiomatic/asymptotic structural analysis of different structures. Two main benefits can be obtained. 1. It permits the accuracy of each problem variable to be evaluated by comparing the results with more detailed analyses (also provided by CUF); no mathematical/variational techniques are needed as in the case of asymptotic-type analyses. 2. It offers the possibility of considering the accuracy of the results as an input, while the output is represented by the set of displacement variables which are able to fulfill the requirement. From this analysis it is possible to draw a curve, the Best Theory Diagram BTD, that allows us to edit an arbitrary given theory in order to have a lower amount of terms for a given error or to increase the accuracy keeping the computational cost constant. Acknowledgements The financial support from the Regione Piemonte projects STEPS and MICROCOST is gratefully acknowledged.


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References 1. Berdichevsky, V. L.: Variational-asymptotic method of shell theory construction. PMM Vol. 43, 664–667 (1979). 2. Berdichevsky, V. L., Misyura, V.: Effect of accuracy loss in classical shell theory. Journal of Applied Mechanics Vol. 59, 217–223 (1992). 3. Carrera, E.: A class of two-dimensional theories for anisotropic multilayered plates analysis. Atti della accademia delle scienze di Torino. Classe di scienze fisiche matematiche e naturali Vol. 19–20, 1–39 (1995). 4. Carrera, E.: Evaluation of layer-wise mixed theories for laminated plates analysis. AIAA Journal Vol. 26, 830–839 (1998). 5. Carrera, E.: A study of transverse normal stress effect on vibration of multilayered plates and shells. Journal of Sound and Vibration Vol. 225, No. 5, 803–829 (1999). 6. Carrera, E.: Multilayered shell theories that account for a layer-wise mixed description. Part I: Governing equations. AIAA Journal Vol. 37, 1107–1116 (1999). 7. Carrera, E.: Multilayered shell theories that account for a layer-wise mixed description. Part II: Numerical evaluations. AIAA Journal Vol. 37, 1117–1124 (1999). 8. Carrera, E.: Developments, ideas and evaluations based upon the Reissner’s mixed variational theorem in the modeling of multilayered plates and shells. Applied Mechanics Reviews Vol. 54, 301–329 (2001). 9. Carrera, E.: Theories and finite elements for multilayered plates and shells. Archives of Computational Methods in Engineering Vol. 9, No. 2, 87–140 (2002). 10. Carrera, E.: Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering Vol. 10, No. 3, 216–296 (2003). 11. Carrera, E.: Historical review of zig-zag theories for multilayered plates and shells. Applied Mechanics Reviews Vol. 56, 287–308 (2003). 12. Carrera, E., Brischetto, S.: Analysis of thickness locking in classical, refined and mixed theories for layered shells. Composite Structures Vol. 85, No. 1, 83–90 (2008). 13. Carrera, E., Giunta, G., Brischetto, S.: Hierarchical closed form solutions for plates bent by localized transverse loadings: Journal of Zhejiang University SCIENCE B Vol. 8, 1026–1037 (2007). 14. Carrera, E., Miglioretti, F., Petrolo, M.: Accuracy of refined finite elements for laminated plate analysis. Composite Structures Vol. 93, 1311–1327 (2011). 15. Carrera, E., Petrolo, M.: Guidelines and recommendations to construct refinements of classical theories for metallic and composite plates. AIAA Journal Vol. 48, No. 12, 2852–2866 (2010). 16. Carrera, E., Petrolo, M., Miglioretti, F.: Guidelines and recommendations on the use of higherorder finite elements for bending analysis of plates. International Journal for Computational Methods in Engineering Science and Mechanics, in press. 17. Cicala, P.: Sulla teoria elastica della parete sottile. Giornale del Genio Civile Vol. 4, 6 and 9 (1959). 18. Cicala, P.: Systematic approximation approach to linear shell theory. Levrotto e Bella, Torino (1965). 19. Fettahlioglu, O. A., Steele, C. R.: Asymptotic solutions for orthotropic non-homogeneous shells of revolution. ASME J. Appl. Mech. Vol. 44, 753–758 (1974). 20. Gol’denweizer, A. L.: Theory of thin elastic shells. International Series of Monograph in Aeronautics and Astronautics, Pergamon Press, New York (1961). 21. Grigolyuk, E. I., Kulikov, G. M.: General directions of the development of theory of shells. Mekhanica Kompozitnykh Materialov Vol. 24, 287–298 (1988). 22. Kapania, K.: A review on the analysis of laminated shells. ASME J. Pressure Vessel Technol. Vol. 111, No. 2, 88–96 (1989). 23. Kapania, K., Raciti, S.: Recent advances in analysis of laminated beams and plates, part I: Shear effects and buckling. AIAA Journal Vol. 27, No. 7, 923–935 (1989).

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24. Kapania, K., Raciti, S.: Recent advances in analysis of laminated beams and plates, part II: Vibrations and wave propagation. AIAA Journal Vol. 27, No. 7, 935–946 (1989). 25. Kirchhoff, G.: Uber das gleichgewicht und die bewegung einer elastischen scheibe. J. Angew. Math. Vol. 40, 51–88 (1850). 26. Koiter, W. T.: On the foundations of the linear theory of thin elastic shell. Proc. Kon. Nederl. Akad. Wetensch. Vol. 73, 169–195 (1970). 27. Librescu, L.: Elasto-statics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures. Noordhoff Int, Leyden, Netherland (1976). 28. Librescu, L., Reddy, J. N.: A critical review and generalization of transverse shear deformable anisotropic plates, euromech colloquium 219, kassel. Refined Dynamical Theories of Beams, Plates and Shells and Their Applications September 1986, 32–43, I Elishakoff and Irretier (eds), Springer Verlag, Berlin (1986). 29. Love, A. E. H.: The Mathematical Theory of Elasticity. Fourth ed., Cambridge Univ Press (1927). 30. Mindlin, R. D.: Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. ASME J. Appl. Mech. Vol. 18, 1031–1036 (1950). 31. Naghdi, P. M.: The theory of shells and plates. Handbuch der Phisik Vol. 6, 425–640 (1972). 32. Noor, A. K., Burton, W. S.: Assessment of shear deformation theories for multilayered composite plates. Appl. Mech. Rev. Vol. 42, No. 1, 1–18 (1989). 33. Noor, A. K., Burton, W. S.: Assessment of computational models for multilayered composite shells. Appl. Mech. Rev. Vol. 43, No. 4, 67–97 (1989). 34. Qatu, M. S.: Recent research advances in the dynamic behavior of shells. Part 1: laminated composite shells. Applied Mechanics Reviews Vol. 55, No. 4, 325–350 (2002). 35. Qatu, M. S.: Recent research advances in the dynamic behavior of shells. Part 2: homogenous shells. Applied Mechanics Reviews Vol. 55, No. 5, 415–434 (2002). 36. Reddy, J. N.: Mechanics of laminated composite plates and shells. Theory and Analysis. Second ed., CRC Press (2004). 37. Reddy, J. N., Robbins, D. H.: Theories and computational models for composite laminates. Appl. Mech. Rev. Vol. 47, No. 6, 147–165 (1994). 38. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. ASME J. Appl. Mech. Vol. 12, 69–76 (1945). 39. Ren, J. G.: Exact solutions for laminated cylindrical shells in cylindrical bending. Composites Science and Technology Vol. 29, 169–187 (1987). 40. Spencer, A. J. M., Watson, P., Rogers, T. G.: Stress analysis of laminated circular cylindrical shells. Recent Developments in Composite Materials Structures. Presented at the Winter Annual meeting of ASME, Dallas, Nov. 1990, AD 19, AMD 113, ASME, New York (1990). 41. Varadan, T. K., Bhaskar., K.: Bending of laminated orthotropic cylindrical shells – an elasticity approach. Composite Structures Vol. 17, 141–156 (1991). 42. Vlasov, B. F.: On the equations of bending of plates. Dokla Ak Nauk Azerbeijanskoi-SSR Vol. 3, 955–979 (1957). 43. Widera, D. E. O., Fan, H.: On the derivation of a refined theory for non-homogeneous anisotropic shells of revolution. ASME J. Appl. Mech. Vol. 110, 102–105 (1988). 44. Widera, D. E. O., Logan, L.: Refined theories for nonhomogeneous anisotropic cylindrical shells: Part I-derivation. Journal of the Engineering Mechanics Division Vol. 106, No. 6, 1053–1074 (1980).

Chapter 46

In-Plane Strain and Stress Fields in Theories of Shearable Laminated Plates Subject to Transverse Loads Giovanni Formica, Marzio Lembo and Paolo Podio-Guidugli

Abstract The predictions of a new theory of orthotropic laminated plates are compared with those of two other theories equally based on a Reissner-Mindlin Ansatz for the displacement field, either layer by layer [5] or for the whole plate [4]. A wellknown merit of such an Ansatz is to allow and account for transverse shearings. What we are after here is to determine how well in-plane strain and stress fields are described. For definiteness, we consider circular plates that are axi-symmetrically loaded, whose layers are made of transversely isotropic materials and are symmetrically located with respect to the midplane of the plate. The new theory allows for an explicit analytic solution of this problem, as the simpler of the two theories considered for comparison does, but shows an accuracy closer to the other more complex theory, whose governing equations we solve numerically; as a benchmark, we use a numerical solution of the corresponding three-dimensional equilibrium problem; the results of our comparison are summarized graphically in the final section. Keywords Orthotropic laminated plate · Reissner-Mindlin plate

46.1 Introduction We present a theory of linearly elastic laminated plates based on the following assumptions: (i) layers are symmetrically located with respect to the plate’s midplane and made of a material whose response in any plane parallel to the midplane is G. Formica (B) · M. Lembo Dipartimento di Strutture, Università di Roma Tre, Via Corrado Segre 6, 00146 Roma, Italy e-mail: [email protected],[email protected] P. Podio-Guidugli Dipartimento di Ingegneria Civile, Università di Roma Tor Vergata, Viale del Politecnico 1, 00133 Roma, Italy e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 46, © Springer-Verlag Berlin Heidelberg 2011

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orthotropic; (ii) each layer undergoes only deformations of the type admitted in the Reissner-Mindlin theory of single-layered plates [1, 2]; (iii) the displacement and traction vectors are both continuous across the planes separating adjacent layers. We deduce the governing equations of our theory from a two-dimensional Principle of Virtual Power that follows by mere thickness integration from the general threedimensional PVP when the admissible displacements agree with the assumptions of continuity across the interfaces, and the class of virtual velocities is chosen consistent with the R-M kinematics. For transversely isotropic circular plates subject to transverse loads, we obtain exact and explicit analytical solutions, that we compare with the predictions of two other plate models, due to Yang, Norris and Stavsky [4] and Seide [5], both based on the R-M kinematics; as a common benchmark, we make use of a numerical solution of the associated three-dimensional problem. Accuracy in evaluation of transverse shear stresses is an issue of central importance when a new theory of laminated plates is proposed or a comparison among different theories is set up. In fact, failure of multilayered composite panels is most often due to such stresses. We do not attempt to give here an account of the large literature on the subject; rather, we refer the reader to a book by Reddy [8] and two papers by Bert [6] and Noor and Malik [7], where various theories are compared. From those papers we see that it is common to employ with reasonably good results a two-step analysis, consisting firstly in finding the in-plane stresses by means of a two-dimensional theory and, secondly, in using the three-dimensional equilibrium equations to evaluate transverse stresses. Our present contribution aims to assess the accuracy in the description of in-plane strain and stress fields achievable by the use of some first-order theories, ours included, of laminated plates susceptible of transverse shear deformations. The two theories we consider for comparison with ours have different features and different degrees of complexity: The theory of Yang, Norris and Stavsky assumes for a laminated plate the same displacement field as that of a single-layered R-M plate, and describes the equilibrium of a circular axi-symmetric plate by means of a system of three equations, whatever the number of layers is; the theory of Seide considers each layer as a R-M plate, imposes continuity of displacements and equilibrium of tractions on the planes separating two of such sub-plates, and describes the equilibrium of a circular axi-symmetric plate formed of n layers by means of a system of n + 2 equations. The theory we present is similar to that of Yang, Norris and Stavsky in the fact that needs only three equations to describe the equilibrium of a circular axi-symmetric plate, but, at least in the cases considered in the comparison, exhibits an accuracy notably closer to that of the Seide’s theory. Our paper is organized as follows. In Sect. 46.2 we introduce our first-order model of a shearable laminated plate. To ease our developments, the presentation concerns a circular, three-layer plate, subject to axi-symmetric transverse loads. In Sect. 46.3 we give analytical solutions to the flexure problem for such a plate, both when the load is uniform and when it can be given a series representation in terms of Bessel functions. In Sect. 46.4 we offer a brief presentation of the two plate models whose predictions we compare graphically with ours and comment in Sect. 46.5.

46 In-Plane Strain and Stress Fields in Theories of Shearable Laminated Plates

701

46.2 A Model of Shearable Multilayered Plate In this section we present our model of shearable laminated plate. We assume that the layers of the plate are made of orthotropic materials and, for the purpose of giving explicit analytic solutions of the equilibrium problem, we refer to the case of a circular, midplane symmetric plate, under axi-symmetric loads; it is an easy matter to extend the model to plates of general shape, whose layers are arranged in an arbitrary way, and that are subject to general loads. We denote by C the cylindrical three-dimensional body of which the plate theory gives a two-dimensional model, and denote by 2h the height of C, by S its midsection, and by R the radius of S. To describe the deformations of C, we introduce a cylindrical coordinate system (r, θ, z) with origin at the center of S, and with z-axis coinciding with the axis of C. We denote by (eer , e θ , e z ) the unit vectors associated with the coordinate system. We consider a plate composed of three layers and label as (1) the quantities pertaining to the central layer having height 2ξh, with ξ < 1, and lying between the planes z = z−(1) = −ξh and z = z+(1) = ξh; moreover, we label as (2l) and (2s) the quantities pertaining to the external layers, lying the former between the planes z = z−(2l) = −h and z = z+(2l) = −ξh, and the latter between the planes z = z−(2s) = ξh and z = z+(2s) = h. We assume that in each layer the admissible deformations are of the type occurring in the Reissner-Mindlin theory; taking into account the symmetry with respect to the z-axis, we find that the displacement field in each of the three layers has the form u (k) = u (k) (r, θ, z) = (u(k) (r) + zψ(k) (r))eer (θ) + w(k) (r)eez , (46.1) where the index k can be 2l, 1, or 2s. This form of the displacement field corresponds to the kinematical assumptions (internal constraints) that, along the z-direction, extensions vanish and transverse shear strains are uniform. In terms of the components of the strain tensor E , these restrictions require that, in each layer, (k) E zz = 0,

(k) Ezr,z = 0,

(46.2)

where a comma followed by the symbol of a variable denotes differentiation with respect to that variable. Thus, considering also the effects of symmetry, in each layer (k) (k) (k) the not identically null components of the strain are Err , Eθθ , and E rz . In a plate whose layers are made of orthotropic materials, the components of the stress S that correspond to these strains are (k)

(k)

(k)

(k)

(k

(k)

(k)

(k)

(k)

(k)

S rr = Crrrr E rr + Crrθθ Eθθ , S θθ = Cθθrr E rr + Cθθθθ Eθθ , (k) S rz

=

(k) 2C(k) rzrz E rz ,

(46.3)

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where C(k) ···· are the components of the elasticity tensor of the layer k that, in terms of the Young’s moduli and Poisson’s ratios for the directions r and θ, and shear modulus for the directions r and z, have the expressions C(k) rrrr =

C(k) rrθθ =

(k)

Er(k) 1 − ν(k) ν(k) r/θ θ/r

E r(k) ν(k) r/θ (k) 1 − ν(k) r/θ νθ/r

=

C(k) θθθθ =

,

E θ(k) ν(k) θ/r (k) 1 − ν(k) r/θ νθ/r

Eθ

1 − ν(k) ν(k) r/θ θ/r

= C(k) θθrr ,

,

(k) C(k) rzrz = G ,

(ν(k) denotes the ratio of lateral contraction along θ-direction to longitudinal extenθ/r sion along r-direction for the layer k). If the layers are composed of transversely (k) (k) (k) isotropic materials, we have E θ(k) = E r(k) , ν(k) r/θ = νθ/r , and thus Cθθθθ = Crrrr . Since the layers are mutually bonded and their detachment or sliding is prevented, the displacement field is taken to be continuous in any deformation of the plate. In addition, Eqs (46.1) show that the derivatives of the displacement are continuous within each layer. It follows that (cf., e.g., [3], Sect.175), across the interfaces between layers, jumps may occur only in derivatives of the displacement along (k) (k) (k) the normal direction z; hence, the strains Err , E θθ are continuous while E rz may suffer jumps across the interfaces. As a consequence of continuity of the displacement, we find that u(2l) = u(1) − ξh(ψ(1) − ψ(2l) ) ,

u(2s) = u(1) + ξh(ψ(1) − ψ(2s) )

w(2l) = w(1) = w(2s) = w .

(46.4)

Moreover, when we refer to C the usual assumption that three-dimensional equilibrium equations in integral form hold for arbitrary volumes also if they contain singular surfaces, we have that (cf., e.g., [3], Sect.193) across the interfaces between the layers the traction vector is continuous. This additional condition yields that in each layer the displacement field can be written as % & u (k) = u + (H (k) + zχ(k) )(w,r + ψ) − zw,r e r + weez , (46.5) where u and ψ are the functions previously denoted as u(1) and ψ(1) , (1) χ(1) = C(1) rzrz /Crzrz = 1 ,

H (1) = 0 ,

(2) χ(2) = χ(2l) = χ(2s) = C(1) rzrz /Crzrz ,

H (2) = H (2l) = H (2s) = sgn(z)ξh(χ(1) − χ(2) ) .

We notice that, while Eqs (46.2) and (46.4) have a kinematical nature, the final form (46.5) of the displacement is influenced also by the equilibrium conditions at the interfaces, a circumstance that reflects into the dependence of u (k) on the shear moduli of the layers.


703

Accordingly to Eqs (46.5), in deformations of the plate the non-null components of the strain have the expressions % & (k) E rr = (u(k) + zψ(k) ),r = u,r + H (k) + zχ(k) (w,r + ψ),r − zw,rr , % &1 1 1 1 (k) E θθ = (u(k) + zψ(k) ) = u + H (k) + zχ(k) (w,r + ψ) − z w,r , (46.6) r r r r % & ' ( 1 1 (k) E rz = w(k) + ψ(k) = χ(k) w,r + ψ . 2 ,r 2 The not identically zero stress resultants are the membrane forces Nrr and Nθθ , the shear Qr , and the bending moments Mrr and Mθθ ; they are defined by z+(k) {Nrr , Nθθ , Qr , Mrr , Mθθ } = {S rr , S θθ , S rz , S rr z, S θθ z} dz . (46.7) k

z− (k)

Introduction in these definitions of the constitutive equations (46.3) with the strains (46.6), taking into account that the external layers are composed of the same material, yields the following expressions of the stress resultants in terms of u, w, and ψ, u u ' ( Nrr = a1 u,r + a2 , Nθθ = a2 u,r + a3 , Qr = b w,r + ψ , (46.8) r r 1 1 Mrr = d1 (w,r + ψ),r + d2 (w,r + ψ) − d3w,rr − d4 w,r , r r 1 1 Mθθ = d2 (w,r + ψ),r + d5 (w,r + ψ) − d4 w,rr − d6 w,r , r r where the coefficients a1 , a2 , a3 , b, d1 , d2 , d3 , d4 , d5 , d6 are defined by z+(k) 3 4 (k) (k) {a1 , a2 , a3 } = C(k) rrrr , Crrθθ , Cθθθθ dz = k

z− (k)

k

z− (k)

(46.9) (46.10)

3 4 3 (2) (2) 4 (1) (1) (2) = 2hξ C(1) rrrr , Crrθθ , Cθθθθ + 2h(1 − ξ) Crrrr , Crrθθ , Cθθθθ , z+(k) (1) b= C(k) rzrz dz = 2hCrzrz ,

{d1 , d2 , d5 } =

k

z+ (k) % z− (k)

&3 4 (k) (k) H (k) + zχ(k) C(k) rrrr , Crrθθ , Cθθθθ zdz =

2h3 ξ3 3 (1) (1) (1) 4 Crrrr , Crrθθ , Cθθθθ 3 &3 4 h3 % (2) (2) + 3ξ(1 − ξ2 ) + χ(2) (2 − 3ξ + ξ3 ) C(2) rrrr , Crrθθ , Cθθθθ , 3 z+(k) 3 4 (k) (k) 2 {d3 , d4 , d6 } = C(k) rrrr , Crrθθ , Cθθθθ z dz = =

k

=

2h3 3

z− (k)

%

3 4 3 (2) (2) (2) 4& (1) (1) 3 ξ3 C(1) rrrr , Crrθθ , Cθθθθ + (1 − ξ ) Crrrr , Crrθθ , Cθθθθ .

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The equilibrium equations of the plate and the associated boundary conditions are obtained by integration over the thickness of the three-dimensional Principle of Virtual Power, S · ∇vv dv = b · v dv + t · v da , C

C

∂C

where b are the loads per unit volume of C, and t are the loads per unit area of the boundary ∂C of C. The virtual velocities are assumed of the form v (r, θ, z) = (v(r) + zχ(r))eer (θ) + η(r) ez , (i.e., for each layer, v(k) = v, χ(k) = χ, η(k) = η) that is consistent with the kinematical restrictions expressed by Eqs (46.2) and (46.4). After integration over the thickness, use of divergence theorem, and localization, the following plate equilibrium equations in terms of stress resultants are obtained 1 Nrr,r + (Nrr − Nθθ ) + qr = 0 , r 1 Qr,r + Qr + qz = 0 , r 1 Mrr,r + (Mrr − Mθθ ) − Qr + dr = 0 , r

in S ,

(46.11)

where qr , qz are the are loads per unit area of S in the radial and transverse directions, and dr is the couple per unit area of S in the radial direction. On the boundary ∂S of S, Eqs (46.11) are accompanied by essential conditions that assign the values of the functions u, w and ψ, or natural conditions that assign the values of the resultants Nrr , Qr and Mrr , respectively. Equations (46.11) can be deduced also by integration over the thickness of the components along the radial and transverse directions of equilibrium equation DivSS + b = 0, and of the component along the radial direction of equation z(DivSS + b ) = 0, making use of the conditions S n = t on the end faces of C (i.e., the cross-sections corresponding to the coordinates z = ±h). Substitution of the stress resultants (46.8)-(46.10) into Eqs (46.11) yields the equilibrium equations in terms of the functions u, w, and ψ: ' ( 1 a1 ru,r ,r − a3 u(r) + rqr = 0 , r ' ( b r(w,r + ψ) ,r = −qz r , 1 1 d1 (r(w,r + ψ),r ),r − d3 (rw,rr ),r − d5 (w,r + ψ) + d6 w,r = br(w,r + ψ) . r r

(46.12)

These equations show that, as a consequence of the symmetry of the plate with respect to its midplane, the membrane equation (46.12)1 corresponding to in-plane deformations is separate by the flexural equations (46.12)2,3 corresponding to transverse deformations.


705

46.3 Equilibrium Solutions We now give the solution of the flexural equations (46.12)2,3 for the equilibrium problem of a plate whose layers are composed of transversely isotropic materials (an hypothesis which implies that d5 = d1 and d6 = d3 ) and that is acted upon by a load qz that either is uniform or can be expressed in a series expansion as qz = qz (r) =

∞

Am J0 (κm r/R) ,

(46.13)

m=1

where J0 are Bessel functions of the first kind and order 0, and κm are the positive roots of the equation J0 (κ) = 0. We assume that the plate is simply supported and subject to a null distribution of moments on ∂S, so that the boundary conditions associated to Eqs (46.12)2,3 are w(R) and Mrr (R) = 0. To find the solutions of the problem, we first transform the system (46.12)2,3 in order to obtain an equation containing the unknown w only. To this end, we observe that, for d5 = d1 , d6 = d3 , and dr = 0, Eq. (46.12)3 can be written ( ( 1' 1' d1 r(w,r + ψ) ,r − d3 rw,r ,r = b(w,r + ψ) , (46.14) r r ,r ,r while integration of Eq. (46.12)2 , taking into account that Qr (0) = 0, yields the expression of w,r + ψ in terms of the load qz , r 1 w,r + ψ = − qz ρdρ . (46.15) br 0 We introduce this expression into both the members of Eq. (46.14) and arrive at the equation r 1' ( 1 d1 rw,r ,r = qz ρdρ + qz,r , (46.16) r d3 r 0 bd3 ,r for the function w. The general solution of the homogeneous form of this equation is w(r) = c1 + c2 r2 + c3 ln r . Here we see that the integration constant c3 must be zero in order that the displacement be finite at r = 0, while the values of the integration constants c1 and c2 are obtained by imposing that the general solution of nonhomogeneous equation satisfy the conditions w(R) = 0 and Mrr (R) = 0. For qz uniform, Eq. (46.16) becomes 1' ( qz r rw,r ,r = , r 2d 3 ,r

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and has the particular solution qz r 4 . 64d When qz is given by the series (46.13), Eq. (46.16) becomes ∞ ( 1' Am d1 κm R rw,r ,r = + J1 (κm r/R) , r d bR κm ,r m=1 3 w(r) =

with J1 the Bessel functions of the first kind and order 1; this equation has the particular solution 3 ∞ Am d1 κm R R w(r) = + J0 (κm r/R) . d bR κ κ m m m=1 3 Once one has the general solution w(r) of Eq. (46.16), the function ψ(r) is given by Eq. (46.15), which yields ψ(r) = −2c2 r −

(8d3 + Br2 )r qz , 16bd3

for qz uniform, and ψ(r) = −2c2 r −

∞

Am

m=1

R2 d1 κm R R + − J1 (κm r/R) , 2 bR κm bκm d3 κm

for qz given by the series (46.13).

46.4 Two Other Models We give here a brief account of two other theories of laminated plates having as a starting point the Reissner-Mindlin kinematics, with a view towards comparing their predictions with those of the theory discussed in the previous sections. The theory of Yang, Norris, and Stavsky [4] is, to our knowledge, the first example of extension to laminate plates of the Reissner-Mindlin theory. It assumes that in a multilayered plate the displacement field is of the same form as that occurring in a singlelayered plate. Thus, for a laminated plate of the type here considered, the displacement field is u (r, θ, z) = (u(r) + zψ(r))eer (θ) + w(r)eez . In the whole plate, the non-null strain components are E rr = u,r + zψ,r ,

1 E θθ = (u + zψ) , r

1 Erz = (ψ + w,r ) . 2

(46.17)


707

The equilibrium of the plate is governed by equations that in terms of resultants of stress have the form (46.11) and are accompanied by boundary conditions of the same type. Using the definition of stress resultants (46.7), the constitutive equations (46.3), and the strains (46.17), the equilibrium equations are expressed in terms of the functions u, w, and ψ. The membrane equation is the same as (46.12)1 ; for dr = 0, the flexural equations are ' ( 6 b r(w,r + ψ) ,r = −qz r , 1 d61 (rψ,r ),r − d65 ψ = 6 br(w,r + ψ) , r

(46.18)

where 6 b=

k

3 4 d61 , d65 =

k

=

z+ (k) z− (k)

% (1) & (2) C(k) rzrz dz = 2h ξCrzrz + (1 − ξ)Crzrz ,

z+ (k) 3 z− (k)

4 (k) C(k) rrrr , Cθθθθ zdz =

3 4& 2h3 % 3 3 (1) (1) 4 (2) ξ Crrrr , Cθθθθ + (1 − ξ3 ) C(2) rrrr , Cθθθθ . 3

When the layers are made of transversely isotropic materials, in the previous equations we have d61 = d65 . In the theory of Seide [5], each layer is regarded as a Reissner-Mindlin plate, and continuity of displacements and equilibrium of tractions is assumed at the planes separating adjacent layers; in a n−layer plate, the displacement in the whole plate is determined by (2n + 3) functions. In a circular plate susceptible of axi-symmetric deformation, the displacement is determined by (n + 2) functions. In the circular plate here considered, the displacement is defined by five functions and can be given the form of Eqs (46.1) with the restrictions (46.4). On adopting this representation, the unknown functions are: u, w, ψ, ψ(2l) , and ψ(2s) . The system of equations that determines these unknowns is obtained as follows: for each layer, the three equilibrium equations in the ReissnerMindlin theory that are not identically satisfied are written; for null volume forces, these equations are: & (k)− (k)+ 1 % (k) (k) Nrr − Nθθ + tr + tr = 0 , r 1 (k) (k)− (k)+ Q(k) + tz = 0 , r,r + Qr + tz r % & − 1 (k) (k) (k) (k) (k)+ Mrr,r + Mrr − Mθθ − Q(k) − tr(k) ) = 0 , r + 2h (tr r (k) Nrr,r +

(46.19)

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+

(k) (k) where 2h(k) is the height of the layer k, and t(·) , t(·) are the tractions acting on its end faces. The stress resultants in the layer k are defined by

3

(k) (k) Nrr , Nθθ , Q(k) r

3

4

4 (k) (k) Mrr , Mθθ =

=

z+ (k)

z− (k)

z+ (k) z− (k)

{S rr , S θθ , S rz , } dz .

{S rr , S θθ } (z − o(k) )dz ,

with o(k) = (z−(k) + z+(k) )/2. From Eqs (46.19)1,3 one has + 2h(k) tr(k)

(k)− 2h(k) tr

=

(k) Q(k) r − Mrr,r −

=

(k) Mrr,r

+

(k) (k) Mrr − Mθθ

r

(k) (k) Mrr − Mθθ

r

⎛ ⎞ (k) (k) ⎟ ⎜⎜⎜ N − N ⎟⎟⎟ rr (k) θθ ⎟⎟⎠ , − h(k) ⎜⎜⎝⎜Nrr,r + r ⎛ ⎜

⎜⎜ (k) (k) − Qr − h(k) ⎜⎜⎜⎝Nrr,r +

⎞ (k) (k) ⎟ Nrr − Nθθ ⎟⎟⎟ ⎟⎟⎠ . r

Then, the equations of adjacent layers are combined by using the condition that the mutual tractions between adjacent layers are in equilibrium; moreover, the traction acting on the end faces of the plate are set equal to the applied surface loads. In this way, four equations are obtained (one equation for each interface and one equation for each end face). By using the same equilibrium conditions, a fifth equation is deduced from Eqs (46.19)2 of all the layers. The boundary conditions on ∂S consist in an assignment of transverse displacement or of shear, and in an assignment of radial displacements both at interfaces and at the end faces, or of certain combinations (k) (k) of membrane forces Nrr and moments Mrr of the layers adjacent the interfaces and the end faces, combinations that are deduced by using the Principle of Virtual Power. After having solved the plate equilibrium problem, transverse stresses are calculated in each layer from the three-dimensional equilibrium equations.

46.5 Comparison We consider a simply supported circular plate of radius R and total thickness 2h, consisting of three layers; the inner layer has thickness ξ(2h), the outer layers have the same thickness. The outer layers are made of the same transversely isotropic material, with elastic moduli E (2) = 1.7 × 105 N/mm2 ,

ν(2) = 0.25 ,

G (2) = 0.5 × E (2) ;

the material of the inner layer, also transversely isotropic, is 25 times softer, in the sense that 1 (2) E (1) = E , ν(1) = 0.25 , G(1) = 0.5 × E (1) . 25 r


709

2.0

(a) 1.5

wW

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Ξ

(b)

0.50

wW

0.45

0.40 20

40

60

80

100

R h Fig. 46.1 (Uniform load, simply supported plate) Maximum displacement vs. (a) thickness ratio of the inner layer; (b) diameter-to-thickness ratio

We use the following graphic conventions: solid line ≡ three-dimensional solution, obtained by COMSOL Multiphysics ( ≡ our model ' ≡ Yang, Norris, & Stavsky [4]

≡ Seide [5]

46.5.1 Uniform Load Let W := 10−2

q z R4 , E (1) h3

710

G. Formica et al.

0.5

wW

0.4

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

rR 1.0

zh

0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

1000 Err 1.0

zh

0.5

Fig. 46.2 (Uniform load, simply supported plate) From top: Displacement vs. distance from center; radial strain and radial stress at r/R = 0.5

0.0 0.5

1.0 0.04

0.02

0.00

0.02

0.04

Srr E1

where qz is the applied load per unit area (we take qz /E (1) = 1/680), and let w/W denote the dimensionless transverse deflection at the center. We plot the dependence of the center deflection w/W on the thickness ratio ξ (Fig. 46.1a) and on R/h, the diameter scaled by the total thickness (Fig. 46.1b). Finally, for ξ = 1/3 and R/h = 10, we plot the deflection versus r/R, and for a transverse fiber at r/R = 0.5, we plot the scaled radial stress S rr and the radial strain E rr versus the scaled transverse coordinate z/h (Fig. 46.2). We omit the diagrams for the azimuthal stress and strain S θθ and Eθθ , that turn out to be qualitatively identical.

46 In-Plane Strain and Stress Fields in Theories of Shearable Laminated Plates 0.35

0.30

wW

0.25 0.20 0.15 0.10 0.05 0.00

711

0.0

0.2

0.4

0.6

0.8

1.0

rR 1.0

zh

0.5 0.0 0.5 1.0

30

20

10

0

10

20

30

1000 Err 1.0

zh

0.5

0.0 0.5

Fig. 46.3 (Bessel load, m = 1; simply supported plate) From top: Displacement vs. distance from center; radial strain and radial stress at r/R = 0.5

1.0

1.0

0.5

0.0

Srr E

0.5

1.0

1

All plate theories, ours as well as those of [4] and [5], imply for all curves in question linear or piece-wise linear shape, more or less close to the corresponding piece-wise smooth curves relative to the three-dimensional problem.

712

G. Formica et al. 0.020

0.015

wW

0.010 0.005

0.000

0.005

0.010 0.0

0.2

0.4

0.6

0.8

1.0

rR 1.0

zh

0.5

0.0 0.5 1.0

5

0

5

1000 Err 1.0

0.5

zh


0.0

0.5 1.0 0.2

0.1

0.0

0.1

0.2

1

Srr E

46.5.2 Bessel Loads In this subsection, we collect information about the displacement, stress and strain fields induced by the application of the first three Bessel loads (cf. (46.13)) (see, respectively, Figs 46.3, 46.4 and 46.5; in all cases, Am = 500 N/mm2 ). All plots, except for those of strain and stress from Yang, Norris & Stavsky theory, agree qualitatively with the benchmark three-dimensional solution, but the quantitative predictions of the various theories differ, sometimes markedly.

46 In-Plane Strain and Stress Fields in Theories of Shearable Laminated Plates 0.008 0.006

wW

0.004 0.002

713

0.000

0.002 0.0

0.2

0.4

0.6

0.8

1.0

rR 1.0

0.5

zh

0.0

0.5

1.0

4

2

0

2

4

1000 Err 1.0

zh

0.5


0.0 0.5 1.0

0.10

0.05

0.00

0.05

0.10

Srr E1

46.5.3 Concluding Remarks In the case of uniform load, all three plate theories depict displacements correctly from a qualitative point of view, with different degrees of accuracy: Seide’s theory is the most accurate, so much so that it almost reproduces the three-dimensional solution; Yang, Norris & Stavsky’s is the least accurate; ours lays in between, but closer to Seide’s. As to in-plane strains and stresses, their distributions over a transverse fiber at r/R = 0.5 are qualitatively good for all models, with the same precision hierarchy as for displacements.

714

G. Formica et al.

For Bessel loads, strong differences in response are manifested, especially for m = 2 and m = 3. The model of Yang, Norris & Stavsky not only is again the least accurate in its predictions, but also does not reproduce the sign changes of strains and stresses along the thickness. The best approximation to the three-dimensional solution is given by the Seide’s model; ours is not quite as good, but close. These results should be considered in the context of a two-step analysis for transverse stress evaluation, where accuracy in predicting in-plane stresses affects evaluation of the target transverse stresses via the three-dimensional equilibrium equations. All in all, it would appear that the model of laminated plates we propose, whose governing equations have a simple form at times susceptible of a direct analytic solution, is definitely better than Yang, Norris & Stavsky’s and, moreover, furnishes a cheap alternative to the use of the more complex theory by Seide.

References 1. Reissner E., 1945, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., 12, pp. 69-77. 2. Mindlin R.D., 1951, Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates, J. Appl. Mech., 18, pp. 31-38. 3. Truesdell C., Toupin R., 1960, The Classical Field Theories, in Handbuch der Physik, edited by S. Flügge, Springer-Verlag, Berlin. 4. Yang P.C., Norris C.H., Stavsky Y., 1966, Elastic wave propagation in heterogeneous plates, Int. J. Solids Structures, 2, pp.665-684. 5. Seide P., 1980, An improved approximate theory for bending of laminated plates, Mechanics Today, 5, pp. 451-466. 6. Bert C.W., 1984, A critical evaluation of new plate theories applied to laminated composite, Composite Structures, 2, pp.329-347. 7. Noor A.K., Malik M., 2000, An assessment of five modeling approaches for thermomechanical stress analysis of laminated composite panels, Computational Mechanics, 25, pp. 45-58. 8. Reddy J.N., 2004, Mechanics of laminated composite plates and shells: theory and analysis, 2nd ed., CRC Press, Boca Raton, Florida.

Chapter 47

On the Use of a New Concept of Sampling Surfaces in Shell Theory Gennady M. Kulikov and Svetlana V. Plotnikova

Abstract This paper focuses on the higher-order shell theory, which permits the use of 3D constitutive equations. It is based on the new concept of sampling surfaces (S-surfaces) inside the shell body. According to this concept, we introduce N not equally located S-surfaces parallel to the middle surface and choose displacements of these surfaces as fundamental shell unknowns. Such choice allows us to represent the higher-order shell formulation in a compact form and to derive straindisplacement equations, which are invariant under all rigid-body shell motions. Keywords Higher-order shell theory · Sampling surface

47.1 Introduction It is well-known that a conventional way for developing the higher-order shell theories accounting for thickness stretching is to utilize either quadratic or cubic series expansions in the thickness coordinate and to choose as unknowns the generalized displacements of the middle surface [1, 2]. In the present paper, we propose a new concept of S-surfaces inside the shell body. As S-surfaces Ω1 , Ω2 , ..., ΩN , we choose outer surfaces and any inner surfaces inside the shell body and introduce displacement vectors u1 , u2 , ..., uN of these surfaces as shell unknowns. Such choice of displacements with the consequent use of Lagrange polynomials of degree N − 1 in the thickness direction permits one to derive strain-displacement equations, which precisely represent rigid-body shell motions in a convected curvilinear coordinate system. The latter is straightforward for development of the exact geometry (EG) solid-shell element formulation. The term ”EG” reflects the fact that the parametrization of the reference surface is known and, therefore, the coefficients of the first and G. M. Kulikov (B) · S. V. Plotnikova Tambov State Technical University, Sovetskaya Street, 106, Tambov 392000, Russia e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 47, © Springer-Verlag Berlin Heidelberg 2011

715

716

G.M. Kulikov and S.V. Plotnikova

second fundamental forms of the reference surface can be taken exactly at each element node. It should be mentioned that in recent works [3–5], the higher-order shell theories with three and four equally located S-surfaces have been developed. Herein, a general case with N not equally located S-surfaces is studied.

47.2 Kinematic Description of Undeformed Shell Consider a thick shell of the thickness h. Let the midsurface Ω be described by orthogonal curvilinear coordinates θ1 and θ2 , which are referred to the lines of principal curvatures of its surface. The coordinate θ3 is oriented along the unit vector e3 normal to the reference surface Ω. Introduce the following notations: r = r(θ1 , θ2 ) is the position vector of any point of the midsurface; aα are the base vectors of the midsurface defined as aα = r,α = Aα eα , (47.1) where eα are the orthonormal base vectors; Aα are the coefficients of the first fundamental form; R = r + θ3 e3 is the position vector of any point in the shell body; RI = r + θ3I e3 are the position vectors of S-surfaces; θ3I are the transverse coordinates of S-surfaces such that θ31 = −h/2 and θ3N = h/2; gi are the base vectors in the shell body given by gα = R,α = Aα cα eα ,

g3 = R,3 = e3 ,

(47.2)

where cα = 1 +kα θ3 are the components of the shifter tensor; kα are the principal curvatures of the midsurface; gαI are the base vectors of S-surfaces (Fig. 47.1) defined as

Fig. 47.1 Geometry of the shell I gαI = R,α = Aα cαI eα ,

g3I = e3 ,

where cαI = 1 + kα θ3I are the components of the shifter tensor at S-surfaces.

(47.3)

47 On the Use of a New Concept of Sampling Surfaces in Shell Theory

717

Here and in the following developments, (. . .),i stands for the partial derivatives with respect to coordinates θi ; Greek tensorial indices α, β range from 1 to 2; Latin tensorial indices i, j, k, m range from 1 to 3; indices I, J identify the belonging of any quantity to the S-surfaces and take values 1, 2, ..., N.

47.3 Kinematic Description of Deformed Shell A position vector of the deformed shell is written as ¯ = R + u, R

(47.4)

where u is the displacement vector, which is always measured in accordance with the total Lagrangian formulation from the initial configuration to the current configuration directly. In particular, the position vectors of S-surfaces are ¯ I = RI + uI , R

uI = u(θ3I ),

(47.5)

where uI (θ1 , θ2 ) are the displacement vectors of S-surfaces. The base vectors in the current shell configuration are defined as ¯ ,i = gi + u,i . g¯ i = R

(47.6)

In particular, the base vectors of S-surfaces of the deformed shell (Fig. 47.2) are I I ¯ ,α g¯ αI = R = gαI + u,α ,

g¯ 3I = g¯ 3 (θ3I ) = e3 + β I ,

β I = u,3 (θ3I ),

(47.7) (47.8)

where β I (θ1 , θ2 ) are the values of the derivative of the displacement vector with respect to coordinate θ3 at S-surfaces. The Green-Lagrange strain tensor can be expressed as 2εi j =

1 (¯gi · g¯ j − gi · g j ), Ai A j c i c j

(47.9)

where A3 = 1 and c3 = 1. In particular, the Green-Lagrange strain components at S-surfaces are 1 2εiI j = 2εi j (θ3I ) = (¯g I · g¯ I − g I · g I ). (47.10) Ai A j ciI cIj i j i j Substituting (47.3) and (47.7) into the strain-displacement relationships (47.10) and discarding non-linear terms, one derives I 2εαβ =

1 I 1 I u,α · eβ + u · eα , I Aα cα Aβ cβI , β

718


Fig. 47.2 Initial and current configurations of the shell

I 2εα3 = β I · eα +

1 I u,α · e3 , Aα cαI

I ε33 = β I · e3 .

(47.11)

Next, we represent displacement vectors uI and β I in the reference surface frame ei as follows: uI = uiI ei , β I = βiI ei . (47.12) i

i

Using (47.12) and presentations for the derivatives of unit vectors ei with respect to orthogonal curvilinear coordinates 1 eα,α = −Bα eβ − kα e3 , Aα 1 e3,α = kα eα , Aα one obtains

Bα =

1 eβ,α = Bα eα , Aα

1 Aα,β Aα Aβ

(β α),

(47.13)

1 I I u = λiα ei , Aα ,α i

(47.14)

where I λαα =

1 I u + Bα uβI + kα u3I , Aα α,α I λ3α =

I λβα =

1 I u − Bα uαI Aα β,α

1 I u − kα uαI . Aα 3,α

( β α), (47.15)


719

Substituting (47.12) and (47.14) in strain-displacement relationships (47.11), we arrive at the index notations of these relationships I 2εαβ =

1 I 1 I λαβ + I λβα , I cβ cα

I 2εα3 = βαI +

1 I λ , cαI 3α

I ε33 = β3I .

(47.16)

Remark 1. The strain components (47.16) are objective, i.e., they represent precisely all rigid-body shell motions in any convected curvilinear coordinate system. It can be verified following a technique developed in [4, 5].

47.4 Displacement and Strain Approximations in Thickness Direction Up to this moment, no assumptions concerning displacements and strains fields have been made. We start now with the first fundamental assumption of the proposed higher-order shell theory. Let us assume that the displacement field is approximated in the thickness direction according to the following law: ui = LI uiI , (47.17) I

where LI (θ3 ) are the Lagrange polynomials of degree N − 1 expressed as LI =

$ θ3 − θ J

3

JI

θ3I − θ3J

such that LI (θ3J ) = 1 for J = I and LI (θ3J ) = 0 forJ I. The use of (47.8), (47.12) and (47.17) yields βiI = M J (θ3I )uiJ ,

(47.18)

(47.19)

J I are the derivatives of the Lagrange polynomials. Thus, the key funcwhere M I = L,3 tions βiI of the proposed higher-order shell theory are represented as a linear combination of displacements uiI . The following step consists in a choice of correct approximation of strains through the thickness of the shell. It is apparent that the best solution of the problem is to choose the strain distribution, which is similar to the displacement distribution (47.10), that is, εi j = LI εiI j . (47.20) I

720


47.5 Variational Equation Substituting strains (47.20) into the principal of the virtual work and introducing stress resultants h/2 HiIj = σi j LI c1 c2 dθ3 , (47.21) −h/2

we arrive at the following variational equation: ⎤ ⎡⎢⎢ % &⎥⎥ ⎢⎢⎢ I I N N + N 1 1 − 1 ⎥⎥⎥ Hi j δεi j − c1 c2 pi δui − c1 c2 pi δui ⎥⎥⎦ A1 A2 dθ1 dθ2 = δWΣ , ⎢⎢⎣ Ω

I

i, j

i

(47.22) where p−i , p+i are the surface loads acting on the bottom and top surfaces; WΣ is the work done by external loads applied to the boundary surface Σ. For simplicity, we restrict ourselves to the case of linear elastic materials. The natural choice for constitutive equations is the generalized Hook’s law: σi j = Ci jkm εkm . (47.23) k,m

Inserting stresses (47.23) in (47.21) and taking into account strain approximation (47.20), one finds J HiIj = DiIJjkm εkm , (47.24) J k,m

where

h/2 DiIJjkm

= Ci jkm

LI L J c1 c2 dθ3 .

(47.25)

−h/2

Remark 2. Recalling that cα are the polynomials of degree one, whereas LI and L J are the polynomials of degree N − 1, one can carry out the exact integration in (47.25) by using the n-point Gaussian quadrature rule with n = N + 1.

47.6 Finite Element Formulation Variational equation (47.22) in conjunction with (47.24) and (47.25) is the basis for developing the EG four-node solid-shell element. The abbreviation EG is explained in Introduction. The finite element formulation is based on the simple and efficient interpolation of shells via curved EG four-node solid-shell elements uiI = Nr uirI , (47.26) r


721

where Nr (ξ1 , ξ2 ) are the bilinear shape functions of the element; uirI are the displacements of S-surfaces at element nodes; ξα = (θα − cα ) /(α are the normalized curvilinear coordinates (Fig. 47.3); cα are the coordinates of the element center; 2(α are the lengths of the element; the index r runs from 1 to 4 and denotes the number of nodes.

Fig. 47.3 Biunit square in (ξ1 , ξ2 )-space mapped into the EG four-node shell element in (x1 , x2 , x3 )-space

To implement the analytical integration throughout the element [6], we employ the assumed interpolation of strain components εiI j = Nr εiI jr , εiI jr = εiI j (P˜ r ), (47.27) r

where P˜ r are the element nodes in (ξ1 , ξ2 )-space. The main idea of such approach can be traced back to the ANS method [7]. It is important to note that herein we treat the term “ANS” in a broader sense. In the proposed EG solid-shell element formulation, all strain components are assumed to vary bilinearly inside the biunit square. This implies that instead of expected non-linear interpolation of strains throughout the element the more suitable bilinear ANS interpolation is utilized. Introducing a displacement vector of the shell element T U = UT1 UT2 UT3 UT4 ,

722


N N N T Ur = u11r u12r u13r u21r u22r u23r ...u1r u2r u3r

(47.28)

and utilizing a standard finite element technique, one arrives at the element equilibrium equations KU = F, (47.29) where K is the element stiffness matrix; F is the force vector.

47.7 Numerical Examples The performance of the higher-order shell theory and EG four-node solid-shell element formulation developed is evaluated by using several exact solutions of the 3D elasticity theory extracted from the literature.

47.7.1 Square Plate Under Sinusoidal Loading Consider first a simply supported square plate (Fig. 47.4) subjected to the sinusoidally distributed pressure load p−3 = −p0 sin πθa1 sin πθb2 . The mechanical and geometrical parameters are taken as follows: E = 107 , ν = 0.3 and a = b = 1. To compare the results derived with an exact solution [8], the following dimensionless variables are introduced: U3 = 100Eh3u3 (a/2, a/2, z)/p0 a4 , S 12 = 10h2 σ12 (0, 0, z)/p0 a2 ,

S 11 = 10h2 σ11 (a/2, a/2, z)/p0 a2 , S 13 = 10hσ13 (0, a/2, z)/p0 a,

S 33 = σ33 (a/2, a/2, z)/p0 ,

Fig. 47.4 Simply supported square plate

z = θ3 /h.


723

Owing to symmetry of the problem, only one quarter of the plate is modeled by the 64 ×64 mesh of EG four-node solid-shell elements. The data listed in Tables 47.1 and 47.2 show that the S-surfaces concept developed permits one to find the numerically exact solutions even for very thick plates. Fig. 47.5 presents the distribution of stresses in the thickness direction in the case of using seven equally located Ssurfaces for different values of the slenderness ratio a/h. These results demonstrate the high potential of the proposed higher-order shell theory. This is due to the fact that the boundary conditions for transverse stresses S 13 and S 33 on the bottom and top surfaces are satisfied properly.

Fig. 47.5 Distribution of stresses S 11 , S 12 , S 13 and S 33 through the thickness of the plate for N = 7

724


Table 47.1 Results for a thick square plate with a/h = 2 Variant

U 3 (0)

S 11 (−0.5)

S 12 (−0.5)

S 13 (0)

S 33 (−0.5)

N=3

5.610

-2.683

0.830

1.596

-1.066

N=5

6.042

-3.027

1.045

2.306

-1.013

N=7

6.046

-3.013

1.045

2.276

-1.000

N=9

6.046

-3.013

1.045

2.277

-1.000

Exact [8]

6.047

-3.014

1.046

2.277

-1.000

Table 47.2 Results for thick and thin square plates with five equally located S-surfaces N=5 a/h

U 3 (0)

Exact solution [8] S 11 (−0.5)

S 12 (−0.5)

S 13 (0)

U 3 (0)

S 11 (−0.5)

S 12 (−0.5)

S 13 (0)

4

3.663

-2.174

1.026

2.369

3.663

-2.175

1.027

2.362

10

2.942

-2.004

1.056

2.384

2.942

-2.004

1.056

2.383

100

2.804

-1.975

1.063

2.387

2.804

-1.976

1.064

2.387

47.7.2 Cylindrical Shell Under Sinusoidal Loading Next, we study a simply supported cylindrical shell with L/R = 4 subjected to the sinusoidal loading p−3 = −p0 sin πθL1 cos4θ2 , where θ1 and θ2 are the longitudinal and circumferential coordinates of the midsurface; L and R are the length and radius of the shell. The shell is made of the unidirectional composite with the fibers oriented in the circumferential direction. The mechanical parameters are taken as follows:

Fig. 47.6 Simply supported cylindrical shell


725

Table 47.3 Results for a thick cylindrical shell with R/h = 2 Variant

U3 (0)

S 11 (0.5)

S 22 (0.5)

S 12 (−0.5)

S 13 (0)

S 23 (0)

S 33 (0)

N=3

6.693

1.151

1.433

-0.962

0.993

-1.674

-0.4216

N=5

7.248

0.936

4.410

-1.582

1.508

-2.123

-0.3762

N=7

7.466

1.201

5.061

-1.729

1.495

-1.981

-0.3649

N=9

7.497

1.353

5.162

-1.755

1.497

-2.063

-0.3755

Exact [9] 7.503

1.332

5.163

-1.761

1.504

-2.056

-0.37

Table 47.4 Results for thick and thin shells with seven equally located S-surfaces N=7 R/h

Exact solution [9]

U3 (0)

S 22 (0.5)

S 13 (0)

S 23 (0)

U 3 (0)

S 22 (0.5)

4

2.782

4.854

0.9863

10

0.9188 4.048

0.5199

100

0.5169 3.840

0.3927

-3.856

S 13 (0)

S 23 (0)

-2.970

2.783

4.859

0.987

-2.990

-3.665

0.9189 4.051

0.520

-3.669

0.5170 3.843

0.393

-3.859

Fig. 47.7 Distribution of transverse normal stresses S 33 through the thickness of the shell for N = 7

E L = 25ET, G LT = 0.5E T, G TT = 0.2E T, ET = 106 , νLT = νTT = 0.25. Here, subscripts L and T refer to the fiber and transverse directions of the ply. To compare the derived results with the exact solution [9] the following dimensionless variables are utilized: S 11 = 100h2 σ11 (L/2, 0, z)/p0 R2 , S 12 = 100h2 σ12 (0, π/8, z)/p0 R2 ,

S 22 = 10h2σ22 (L/2, 0, z)/p0 R2 , S 13 = 100hσ13(0, 0, z)/p0 R,

726


S 23 = 10hσ23 (L/2, π/8, z)/p0 R,

S 33 = σ33 (L/2, 0, z)/p0 ,

U3 = 10E L h3 u3 (L/2, 0, z)/p0 R4 ,

z = θ3 /h.

Due to symmetry of the problem, only one sixteenth of the shell (Fig. 47.6) is discretized with the 32 × 128 mesh of EG four-node solid-shell elements. The data listed in Tables 47.3 and 47.4 demonstrate again the high potential of the shell theory developed. Additionally, Fig. 47.7 presents the distribution of transverse normal stresses in the thickness direction in the case of using seven equally located S-surfaces for different values of the slenderness ratio R/h. It is seen that boundary conditions on the outer surfaces are satisfied correctly.

47.8 Conclusions A simple and efficient concept of S-surfaces inside the shell body has been proposed. This concept permits the use of 3D constitutive equations and leads for the sufficient number of S-surfaces to the numerically exact solutions of 3D elasticity problems for thick and thin shells. Acknowledgements This work was supported by Russian Ministry of Education and Science (Grant No 2.1.1/10003) and Russian Foundation for Basic Research (Grant No 08–01–0373).

References 1. Noor AK, Burton WS (1990) Assessment of computational models for multilayered composite shells. Appl Mech Rev 43:67–97 2. Carrera E (2002) Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comp Meth Eng 9:1–60 3. Kulikov GM (2001) Refined global approximation theory of multilayered plates and shells. J Eng Mech 127:119–125 4. Kulikov GM, Plotnikova SV (2008) Finite rotation geometrically exact four-node solid-shell element with seven displacement degrees of freedom. Comp Model Eng Sci 28:15–38 5. Kulikov GM, Carrera E (2008) Finite deformation higher-order shell models and rigid-body motions. Int J Solids Struct 45:3153-3172 6. Kulikov GM, Plotnikova SV (2006) Geometrically exact assumed stress-strain multilayered solid-shell elements based on the 3D analytical integration. Comp Struct 84:1275-1287 7. Bathe KJ, Dvorkin EN (1986) A formulation of general shell elements - the use of mixed interpolation of tensorial components. Int J Num Meth Eng 22:697-722 8. Vlasov BF (1957) On the bending of rectangular thick plate. Trans Moscow State Univ 2:2531 (in Russian) 9. Varadan TK, Bhaskar K (1991) Bending of laminated orthotropic cylindrical shells – an elasticity approach. Comp Struct 17:141-156

Chapter 48

Theory of Thin Adaptive Laminated Shells Based on Magnetorheological Materials and Its Application in Problems on Vibration Suppression Gennady I. Mikhasev, Marina G. Botogova and Evgeniya V. Korobko

Abstract A laminated cylindrical shell like a sandwich formed by embedding the magnetorheological elastomers (MRE) in between elastic layers is the subject of this investigation. Physical properties of the magnetorheological (MR) layers are assumed to be functions of the magnetic field induction. Based on both the assumptions of the generalized kinematic hypothesis of Timoshenko for a whole sandwich and the experimental data for the MRE, a system of differential equations with complex time-dependent variable coefficients describing motion of the MR adaptive shell is derived. As an example, free and forced vibrations of the MR adaptive three-layered cylinder are studied at different fixed magnetic field levels. In addition, applying the asymptotic approach, the adaptive capabilities of the MRE to suppress the running localized vibrations in the MR shell under the effect of the time-dependent magnetic field are analyzed. Keywords Thin laminated shells · Adaptive physical properties · Magnetorheological elastomers · Vibration suppression

48.1 Introduction Thin multi-layered cylindrical shells have a wide range of applications in many engineering structures, such as airborne/spaceborne vehicles, underwater objects, cars, etc ( [9], [18]). The vibroprotection of thin-walled structures experiencing an G. I. Mikhasev (B) · M. G. Botogova Belarusian State University, Minsk, Belarus e-mail: [email protected] E. V. Korobko A. V. Lykov Heat and Mass Transfer Institute of National Academy of Sciences of Belarus, Minsk, Belarus e-mail: [email protected] H. Altenbach and V.A. Eremeyev (eds.), Shell-like Structures, Advanced Structured Materials 15, DOI: 10.1007/978-3-642-21855-2 48, © Springer-Verlag Berlin Heidelberg 2011

727

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G.I. Mikhasev et al.

external vibrational load is a subject of great practical interest for mechanical engineers which design and model similar structures. The appearance of the group of new composite materials with active and adaptive properties, called smart materials, opens new possibilities for solving these problems. Some of them are electrorheological (ER) and magnetorheological (MR) composites and, particularly, magnetorheological elastomers (MRE). They belong to the group of active materials which physical properties such as viscosity and shear modulus can vary when subjected to different electric or magnetic field levels ( [8], [3]). Most papers on vibration suppression of laminated thin-walled structures have been done for the case when the interlayer smart materials are ER or MR fluids ( [17], [19], [21], [23]). However, ER and MR fluids experience essential lack. They do not keep their geometrical shape at a low electric or magnetic field level that leads to some technological problems at designing and running the solid-fluid-solid sandwich structures. It is solid smart materials such as the MRE that are mostly applicable in the vibration control of adaptive structures [4]. The MRE are magnetizable particles molded in either rubbery polymers or deformed inorganic polymer matrices. The optimum weight/density ratio of magnetic particles, carrier viscous liquid and polymer determine shear modulus, viscosity and response time being the integral characteristics of a smart material [3]. Although rather many papers are dedicated to studying the properties of magnetorheological media and elastomers (see, e.g., [3], [10], [20], [21], [22]), there are few researches on the dynamic calculation of the adaptive sandwich beams, plates and shells containing MRE as the interlayer viscoelastic filler. It is explained by the fact that the response of composite construction containing MRE significantly depends on the ratio of time scale of controlling signal providing time reaction of MRE and dynamic characteristics of the controlled construction ( [2], [14]). That is why most authors studied the adaptive sandwich structures when the applied magnetic field was a stationary one indepent of time. It is also necessary to indicate the lack of universal and satisfactory theories for thin adaptive sandwich shells. In the most general terms, the known theories proceed from the order of shell equations depending on the number of layers of a shell (see, e.g., in references [1], [7]) and are rather sophisticated for practical application. In this study, based on the assumptions of the generalized kinematic hypothesis of Timoshenko for a whole sandwich, the governing equations with complex timedependent variable coefficients describing the dynamics of the MR adaptive cylindrical shell are derived. The reduced constitutive equations for a whole sandwich are taken in the complex form, the complex shear modulus for MRE being determined experimentally at different levels of the induced magnetic field. To simplify governing equations the reduced complex Young’s and shear moduli are introduced for a whole sandwich. As opposed to the theory developed by Grigoliuk and Kulikov [5], the geometrical parameters of a shell as well as physical characteristics of layers composing the sandwich are supposed to be functions of curvilinear coordinates. In addition, complex moduli for the adaptive interlayer MR elastomers are taken in the form of functions of the time-dependent induction of the applied magnetic field.

48 Thin Adaptive Laminated Shells Based on Magnetorheological Materials

729

The principle of damping vibrations is discussed in the paper on the example of a three-layered adaptive MR cylinder. Free and forced vibrations of the shell are studied when the stationary magnetic field is induced. Variations in the reduced stiffness and damping properties of a sandwich are achieved in response to different applied magnetic field levels. The specific goal defined herein is to demonstrate the possibility of asymptotic approach developed in reference [16] to predict soft suppression of localized vibrations running over the adaptive MR shell surface under the effect of the nonstationary magnetic field.

48.2 Sandwich Structure and Properties of Damping Layers Consider a thin cylindrical shell like a sandwich (see Fig. 48.1) consisting of N transversely isotropic layers characterized by thickness hk , density ρk , Young’s modulus E k , and Poisson’s ratio νk where k = 1, 2, . . ., N, and N is an odd number. Let the layers with the odd numbers (including load-bearing ones) be made of elastic material, and the layers having the even numbers be fabricated from a viscoelastic magnetorheological (MR) material whose rheological properties depend on the intensity of a magnetic field. The middle surface of any fixed layer is taken as the original surface. The co-ordinate system α1 , α2 is illustrated in Fig. 48.1, where α1 , α2 are the axial and circumferential coordinates, respectively. In the general case the shell may be non-circular with the radius of curvature R2 (α2 ) and not closed in the circumferential direction. The shell is assumed to be bounded by the two not necessary plane edges L1 (α2 ) ≤ α1 ≤ L2 (α2 ). If every layer is made of elastic and homogeneous material, the parameters E k as well as the shear moduli G k are real constants for any k. When the sandwich is formed by embedding magnetorheological materials in between elastic layers, some of these parameters corresponding to the viscoelastic lamina with adaptive rheological properties are assumed to be the complex functions

Fig. 48.1 A laminated cylindrical shell and the curvilinear coordinate system

730


E k = E k (α1 , α2 , t) + iE k (α1 , α2 , t), G k = Gk (α1 , α2 , t) + iG k (α1 , α2 , t)

(48.1)

√ of the curvilinear coordinates α1 , α2 and time t, where i = −1. The presupposed dependence of parameters (48.1) on α1 , α2 and t is introduced here to assume inhomogeneity of the external nonstationary magnetic field. From all the variety of smart materials we consider here magnetorheological elastomers (MRE), the elastic properties of which change depending on the value of the magnetic field applied. They consist of magnetic particles in a deformed polymer matrix. The possibility to control viscoelastic properties of MRE in a wide range allows us to use them as a damping layer in smart shell-like structures. A natural inorganic polymer (bentonite clay, the size of laminar particles is 1 - 10 μm) in synthetic oil was used as a matrix for the MRE, and the particles of carbonyl iron (the particle size is about 20 μm) as a filler. The matrix for the MR elastomer was prepared by thorough rubbing the polymer in the surfactant-added oil. Carbonyl iron particles were introduced (about 30 wt. %) into the prepared matrix. The dependence of components of the complex shear moduli GMR and GMR on the induction of the magnetic field B has been determined by means of the rheometer ”Physica MCR 301” by Anton Paar. The dependencies of components of the complex shear modulus on the induction of the magnetic field (at the frequency of external impact 10 Hz) are shown in Fig. 48.2. The curves obtained demonstrate the nonlinear dependence of the shear modulus on the induction B of the magnetic field of high intensity. Only at B < 200 mT (in the pre-yield regime) our results correlate with the supposition in reference [22] about the linear dependence of the imaginary shear modulus of the MR material on the magnetic field induction. For small deformations, the majority of known elastomers demonstrate elastic properties with the modulus of elasticity E MR varying in the range of 1, 6 to 35 MPa and Poisson’s ratio ν MR not exceeding 0, 4 [20]; in this case the formula G MR = E MR /[2(1 + ν MR )]

(48.2)

3500

1400 1200

3000 G’’

2000

800

G’’, kPa

G’, kPa

G’

400

1000

0 0

200

400

600

800

1000

B, mT

Fig. 48.2 Dependence of the real and imaginary parts of the MRE shear modulus on the magnetic field induction


731

for calculating the shear modulus of isotropic material is valid. However, many of the MR elastomers (including the MRE with the properties given in Fig. 48.2), possessing isotropy in the absence of magnetic field, show directional properties at a high level of a magnetic field. For a thick sandwich this property has an essential effect on the modes for which the amplitudes of the tangential and normal displacements of the shell have the same order; the thinner the MR layer is, the less the anisotropic property affects the vibration forms of the whole sandwich. In this paper flexural vibrations of the thin laminated shell will be studied. Then every internal MR layer may be modelled as the isotropic viscoelastic medium. It may be seen from Fig. 48.2 that for the MRE under consideration the following linear approximations GMR = 4, 500 + 14, 978B(t),

E MR = 13, 230 + 45, 040B(t),

G MR = 17, 000 + 3, 680B(t),

E MR = 50, 000 + 10, 920B(t)

(48.3)

are valid in the pre-yield regime at B < 200 mT, correlations (48.3) for E MR and E MR having been found from Eq. (48.2) at ν MR = 0, 4.

48.3 Governing Equations The variant of a theory of thin laminated shells with adaptive physical properties proposed here is completely based on the theory of Grigoliuk and Kulikov ( [5]). The key idea of this theory is to introduce a unique kinematic hypothesis for the whole packet of a sandwich. Although this approach is more common than the others depending on the number of layers (see, e.g., papers [1], [7]), it leads to the system of partial differential equations which does not depend on the number of layers. The comparative analysis of the results obtained on the basis of this approach and the finite-element simulation ( [13], [15]) have shown that an accuracy of Grigoliuk and Kulikov’s governing equations is satisfactory if the shell is sufficiently thin and its vibrations occur with minor sizes of deflections or wave lengh.

48.3.1 Basic Hypotheses It is necessary to introduce some additional notations. Let δk be the distance between the original surface and the upper bound of the kth layer, ui and w the tangential and normal displacements of the original surface points respectively, u(k) i the tangential displacements of points of the kth layer, σi3 the transverse shear stresses, Θi the angles of rotation of the normal n about the vector ei (see Fig.48.1). Here i = 1, 2; k = 1, 2, . . ., N.

732


The following hypothesis of the laminated shell theory stated by Grigoliuk and Kulikov [5] are assumed here: i The distribution law of the transverse tangent stresses across the thickness of the kth layer is assumed to be of the form (k) σi3 = f0 (z)μ(0) i (α1 , α2 ) + fk (z)μi (α1 , α2 ) ,

(48.4)

where f0 (z), fk (z) are the continuous functions introduced as follows f0 (z) =

1 (z − δ0 )(δN − z), h2

fk (z) =

1 (z − δk−1 )(δk − z). h2k

(48.5)

ii Normal stresses acting on the area elements parallel to the original one are negligible with respect to the other components of the stress tensor. iii A deflection w does not depend on the coordinate z. iv The tangential displacements are distributed across thickness of the layer packet according to the generalised kinematic hypothesis of Timoshenko: u(k) i (α1 , α2 , z) = ui (α1 , α2 ) + zΘi (α1 , α2 ) + g(z)ψi(α1 , α2 ) where g(z) =

)z

(48.6)

f0 (x)dx.

0 (k) The functions μ(0) i , μi , Ψi can be found in paper [5]. It should be noted here (k) that μ(0) i , μi , ψi depend on the elements of the matrix characterizing the transverse shifted pliability of the kth layer. Hypothesis ( 48.6) permits to describe the non-linear dependence of the tangential displacements on the z coordinate; at g ≡ 0 it turns into Timoshenko’s hypothesis.

48.3.2 Strain-Displacement Relations Vibrations are assumed to be accompanied by the formation of a large number of waves so that the shell may be considered to be shallow within the limits of one semi-wave. Then, taking into account the hypotheses accepted above, the straindisplacement relations will be the following [5]: u(k) i = ui − zw, i + g(z)ψi , εi j = ei j + z κi j ψi j , ε13 = f0 (z)ψi ,

(48.7)

where 1 ei j = (ui, j + u j, i ) + κi j w, 2 1 ψi j = (ψi, j + ψ j, i ), κi j = −w, i j , i, j = 2. 2

(48.8)


733

Here the differentiation with respect to the coordinate αi is designated by the subscript i following the comma.

48.3.3 Constitutive Equations Following reference [5], the constitutive equations are assumed to be of the form σi j =

Ek Ξεi j , 1 − ν2k

(48.9)

where Ξεi j = (1 − ν)εi j + νδi j (ε11 + ε22 ),

(48.10)

δi j is Kronecker symbol (δii = 1; δi j = 0, i j), and ⎛ N ⎞−1 N E k hk νk ⎜⎜⎜⎜ Ek hk ⎟⎟⎟⎟ ⎜⎜ ⎟⎟ ν= 1 − ν2k ⎝ k=1 1 − ν2k ⎠ k=1

(48.11)

is the reduced Poisson’s ratio. In (48.10), (48.11), the parameters Ek , Gk are real constants for all elastic layers and the complex functions (48.1) when the layers are made of the MRE. Next, one can introduce the reduced modulus 1 − ν2 Ek hk , h = hk h k=1 1 − ν2k k=1 N

E=

N

and the dimensionless stiff characteristics ⎛ N ⎞−1 E k hk ⎜⎜⎜⎜ E k hk ⎟⎟⎟⎟ ⎜⎜ ⎟⎟ . γk = 1 − ν2k ⎝ k=1 1 − ν2k ⎠

(48.12)

(48.13)

of the kth layer. Then, from Eqs ( 48.10)– ( 48.13), one gets E k hk Eh = γk . 2 1 − νk 1 − ν2

(48.14)

It may be seen from Eqs (48.11), (48.12) that the reduced modulus E and Poisson’s ratio ν are also functions of the curvilinear coordinates and time. Taking into account Eqs ( 48.10)–( 48.14), it may be concluded that for the influence of the damping layers on the amplitude vibrations to be tangible, the order of the imaginary part of the dimensionless stiff characteristic (48.13) of the MR layers should coincide with the maximum order of parameters |γk | for elastic layers.

734


48.3.4 Stresses and Moments The specific membrane forces and the bending moments (stress resultants) in the sandwich are expressed in the standard way as δ

Ti j =

N k

δ

σi j dz,

Mi j =

k=1δ k−1

N k

z σi j dz.

(48.15)

k=1δ k−1

In addition to the classical forces and moments of the isotropic one-layered shell theory, the generalized specific forces and moments are introduced as follows [5] δ

Qi j =

N k

δ

f0 (z)σi3 dz,

Li j =

k=1δ k−1

N k

g(z)σi j dz.

(48.16)

k=1δ k−1

The introduction of the generalized forces and moments is caused by the presence of the additional degrees of freedom corresponding to the transverse shears in the shell. Taking Eqs (48.9), (48.10), (48.15), (48.16) into consideration, one gets the following equations Eh Eh2 Ξe + (c13 Ξκi j + c12 Ξψi j ), i j 1 − ν2 2(1 − ν2 ) 1 Eh2 Mi j = hc13 T i j + (η3 Ξκi j + η2 Ξψi j ), 2 2(1 − ν2) 1 Eh2 Li j = hc12 T i j + (η2 Ξκi j + η1 Ξψi j ), 2 2(1 − ν2)

Ti j =

(48.17)

where 1 2 h π1k = 12

δk

1 2 h π2k = 12

g (z) dz, 2

δk−1

1 2 h π3k = 12

δk g(z) dz,

N

(ζk−1 + ζk )γk ,

k=1

δk−1

η1 =

c13 =

N

ξk−1 π1k γk − 3c212 ,

k=1 N η3 = 4 (ξk2 + 3ζk−1 ζk )γk − 3c213 , k=1

c12 =

δk z g(z) dz, δk−1

N

ξk−1 π3k γk ,

k=1

η2 =

N

ξk−1 π2k γk − 3c12c13 ,

k=1

hξk = hk ,

hζn = δn (n = 0, k). (48.18)


735

Following reference [5], the so-called generalized displacements and strains are introduced as 1 1 ui = uˆ i − hc13 w, i − hc12 ψi , 2 2 1 1 ei j = eˆ i j − hc13 κ, i − hc12ψi j . 2 2

(48.19)

Let us also consider the following transformations ˆ i j = Mi j − 1 hc13 T i j , M 2

1 Lˆ i j = Li j − hc12 T i j . 2

(48.20)

Then, one obtains the following equations Eh3 (η3 Ξκi j + η2 Ξψi j ), 12(1 − ν2) Eh3 Lˆ i j = (η2 Ξκi j + η1 Ξψi j ). 12(1 − ν2 )

ˆ ij = M

(48.21)

for the reduced moments. Equations (48.16) for the generalized transverse stresses may be rewritten as Qi = G ψi , (48.22) where G=

N

k=1 N k=1

λ2

λk − λko kk

λk −

λ2ko λkk

+

G −1 k

δk λkk =

2 N λ2

k0

k=1

λkk

Gk ,

(48.23)

δk f02 (z)dz,

δk−1

λkn =

fk (z) fn (z)dz δk−1

and G(α1 , α2 , t) being the reduced shear modulus of the sandwich.

48.3.5 Equations of Motion in Stress Terms Let the shell be under the normal load qn (α1 , α2 , t). The shell edges can also experience external stresses and moments. Equations of the shell motion can be derived by using Hamilton’s principle t2 (δU − δA − δT )dt, t1

(48.24)

736


where δU and δT are the variations of the strain and kinetic energies respectively, and δA is the variation of the external boundary and the surface force energies. Expressions for the strain and kinetic energies written in terms of the specific stresses, reduced moments and generalized strains are very cumbersome and so omitted here (see reference [5]). Performing the ordinary procedure of calculating the variations in (48.24), one can derive the equations written in terms of the specific stresses and reduced moments and the boundary conditions as well. It should be noticed that these equations are the same for both the composite elastic shells and viscoelastic ones with the MR layers. When omitting the non-linear terms and the inertia forces corresponding to the tangential displacements, the equations of motion in the stress terms may be written as follows T 1i, 1 + T 2i, 2 = 0, Lˆ 1i, 1 + Lˆ 2i, 2 = 0, ˆ 11, 11 + 2 M ˆ 12, 12 + M ˆ 22, 22 − M

i = 1, 2,

N ∂2

1 T 22 − 2 R2 (α2 ) ∂t

(48.25) (48.26)

ρk hk w = −qn (α1 , α2 , t), (48.27)

k=1

and the boundary conditions are ˆ 11 = Lˆ 11 = ψ2 = 0, w=M

(48.28)

w = w, 1 = ψ1 = ψ2 = 0

(48.29)

for the joint supported and clamped edges α1 = L1 (α2 ), L2 (α2 ) respectively. The other variants of the boundary conditions, including nonhomogeneous conditions for the boundary membrane stresses and moments, are listed in reference [5].

48.3.6 Governing Equations in Terms of Displacement and Stress Functions In this subsection the governing equations for the thin composite laminated cylindrical shells with adaptive physical properties are derived, the approach developed in reference [5] and based on the introduction of force and displacement functions being utilized. Not complicated transformations of Eqs (48.8), (48.19) yield the strain compatibility condition eˆ 11, 22 − 2ê12, 12 + eˆ 22, 11 = R−1 (48.30) 2 w, 11 . Let us introduce the force function Φ so that the membrane stresses are the following T i j = δ i j Δ Φ − Φ, i j ,

(48.31)


737

where Δ = ∂2 /∂α21 + ∂2 /∂α22 is the Laplace operator in the curvilinear coordinates α1 , α2 . Then Eq. (48.25) will be satisfied identically. Expressing the generalized strains eˆ i j by means of F and inserting them into (48.30), one obtains the equation ⎡⎢⎢ 1 ν ⎤⎥⎥ 1 1−ν 1 ⎢ ⎥ ⎢ ⎥ ΔΔ Φ + − Φ, 12 = w, 11 . (48.32) ⎣ ⎦ Φ, ii + 2 Eh Eh Eh Eh R (α , j j 2 2) , ii , 12 i, j=1,2;i j The functions ψ1 , ψ2 can be represented in the form ψ1 = a, 1 + φ, 2 ,

ψ2 = a, 2 − φ, 1 ,

(48.33)

where a(α1 , α2 , t) and φ(α1 , α2 , t) are unknown functions. Substituting Eq (48.33) into (48.26), with Eqs (48.8), (48.21), (48.21) in mind, produces the following equations ,

D[η2 (−w, 11 − νw, 22 ) + η1 (a, 11 + φ, 21 + ν(a, 22 − φ, 12 ))] , 1 * + (48.34) 1−ν + D −η2 (1 − ν)w, 12 + η1 (φ, 22 − φ, 11 + 2φ, 12) = G(a, 1 + φ, 2 ), 2 ,2 * + 1−ν D −η2 (1 − ν)w, 12 + η1 (φ, 22 − φ, 11 + 2φ, 12) 2 ,1 , + D[−η2 (w, 22 + νw, 11 ) + η1 (a, 22 − φ, 12 + ν(a, 11 + φ, 21 ))] , 2 = G(a, 2 − φ, 1 ), (48.35) where Eh3 D= (48.36) 12(1 − ν2 ) is the reduced flexural rigidity of the sandwich. Finally, substituting (48.21) into the last equation (48.27), and taking into account (48.8), (48.33), one obtains the following equation ,

D[η3 (w,11 − νw,22 ) + +η2(a,11 + νa,22 )] ,11 + , + 2(1 + ν)D(η3w,12 + η2 a,12 ) ,12 +

, ∂2 w + D[η3 (w,22 − νw,11 ) + +η2 (a,22 + νa,11 )] ,22 − ρh 2 − ∂t 1 − F,11 − [(1 − ν)Dη2φ,12 ],11 + [(1 − ν)Dη2φ,12 ],22 − R2 (α2 ) −[(1 − ν)Dη2(φ,22 − φ,11 )],11 = 0.

(48.37)

Equations (48.32), (48.34), (48.35) and (48.37) can be the governing ones when modelling free or forced vibrations of the laminated shell with the variable and adaptive physical properties. They are simplified if E, ν, η j do not depend on the curvilinear coordinates α1 , α2 .

738


Let E, ν, η j be the functions only of time t. Following [5], one can introduce the displacement function χ by formulas h2 η 2 h2 w = 1 − Δ χ, a = − Δχ, (48.38) β η1 β where β=

12(1 − ν2 ) G Ehη1

(48.39)

is the shear parameter. Then Eqs (48.32), (48.34), (48.35), (48.37) will be reduced to the following equations (which are similar to v. Kármán’s ones) Eh3 η3 θh2 1 ∂2 Φ ∂2 w 1 − Δ Δ2 χ + + ρh 2 = qn (α1 , α2 , t), 2 2 β R2 (α2 ) ∂α1 12(1 − ν ) ∂t 2 Eh ∂ w h2 2 Δ Φ− = 0, w = 1 − Δ χ, (48.40) R2 (α2 ) ∂α21 β and the equation 1 − ν h2 Δφ = φ 2 β

(48.41)

with respect to the shear force φ, where θ = 1−

η22 . η1 η3

(48.42)

The boundary conditions (48.28), (48.29) can be rewritten in terms of the displacement and stress functions. So, for the joint supported edges they will be as follows [5]: χ = Δχ = Δ2 χ = Φ = ΔΦ = 0 at α1 = L1 (α2 ), L2 (α2 ).

(48.43)

It should be emphasized that Eqs (48.40), (48.41) have been derived by Grigoliuk and Kulikov [5] for the case of elastic laminated cylindrical shells with constant parameters E, ν, θ, β, ηk . Remark 48.1. Let us introduce a natural small parameter h∗ =

h , R

(48.44)

characterizing the shell thinness, where R is the characteristic size of the shell. Let y(α j , t) be any of the functions E(α j , t), ν(α j , t), ηk (α j , t) satisfying the following condition ∂y y∼R at h∗ → 0. (48.45) ∂α j


739

It is also assumed that R R

∂ (w, Φ, a) ∼ (w, Φ, a) at ∂α1

h∗ → 0,

at

h∗ → 0.

∂ (w, Φ, a) ∼ h−1/4 (w, Φ, a) ∗ ∂α2

(48.46)

Then one can prove that the solution of Eqs (48.40), (48.41) will satisfy the governing equations (48.32), (48.34), (48.35), (48.37) up to the values of the order h1/4 ∗ . Condition (48.45) means small variability of the functions E, ν, η j in the axial and circumferential directions and holds if the magnetic field induction B varies slowly versus the curvilinear coordinates α1 , α2 . On the contrary, the asymptotic correlation (48.46) means a high variability of unknown functions in the circumferential directions and corresponds to the low-frequency vibrations of the medium-length cylindrical shell [16]. So, the foregoing Remark 48.1 means that the degenerate equations (48.40), (48.41) may be applied to examine the low-frequency vibrations of thin medium-length laminated shells with variable physical parameters. Exactly these equations will be used in the section to follow for constructing the approximate (asymptotic) solutions describing the running localized vibrations in a three-layered cylindrical MR adaptive shell under the effect of a non-stationary magnetic field.

48.4 Free and Forced Vibrations under Stationary Magnetic Field The joint-supported circular three-layered cylindrical shell of the constant radius R, length L and with constant physical parameters will be considered in what follows. Let the external surfaces be elastic and not susceptible to the magnetic field, and the internal layer be made of the MRE described in Sect. 48.2. It is of interest to study the influence of the stationary magnetic field on free and forced vibrations of the shell.

48.4.1 Free Vibrations In the case of free vibrations, qn = 0. Then the suitable solution of Eqs (48.40), satisfying the boundary conditions (48.43), will be given by the following expressions: πnα1 mα2 sin , L R πnα1 mα2 Φ = ΦA exp(iΩt) sin sin , L R χ = χA exp (iΩt) sin

(48.47)

where n and m are the wave numbers in the axial and circumferential directions, respectively, and Ω = ω + iα is an unknown complex eigenfrequency. Substituting Eqs (48.47) into Eqs (48.40) yields the relation for the complex natural frequency

740

G.I. Mikhasev et al. B, mT 50

0

150

100

2

200

1

100

Ki

200

150 2

-0,5

-0,001

250 1

-1,0

3

-0,002

50

0

-1,5 3

-2,0

-0,003 (a)

-2,5

(b)

Fig. 48.3 Parameters Ki versus the magnetic induction B for various values of a thickness h2 : (a) 1 – h2 = 5mm, 2 – h2 = 6mm, 3 – h2 = 10mm; (b)1 – h2 = 10mm, 2 – h2 = 20mm, 3 – h2 = 30mm

Ω2nm = where μ4 =

R4 n 4 E 4 4 2 1 + KθΔnm μ π Δ + , nm 1 + KΔnm L4 Δ2nm ρR2

ε4 η 3 , 1 − ν2

Δnm =

n 2 R 2 m2 + , L2 π2

K=

π2 h2 . βR2

(48.48)

(48.49)

Equation (48.48) can be used to predict the passive vibration suppression by means of a choosing both the optimal thickness of the MR layer and the intensity of the stationary magnetic field. In (48.48), the principal adaptive parameters are the reduced complex magnitudes E = Er + iE i , K = Kr + iKi which depend on the magnetic field induction. The influence of these parameters on the effectiveness of the vibration suppression is different and depends strongly upon the correlation of the geometrical and physical parameters of layers composing the sandwich. In the case of passive suppression of low-frequency vibrations of the medium-length thin cylinder (here n, m are small, and R/L ∼ 1), the influence of the parameter K is negligibly small, but on the other hand, the reduced modulus E will be the determinative parameter. If the problem on damping the high-frequency vibrations is stated, then the influence of the parameter K increases. To analyze the influence of the stationary magnetic field on both the adaptive parameters and the natural frequencies, one should apply to the numerical calculations. Example 48.1. Let the external supporting surface of the shell be made of the ABS plastic SD-0170 with parameters E 1 = E3 = 1, 5 × 109 Pa, ν1 = ν3 = 0, 4, ρ1 = ρ3 = 1, 04 × 103 kg/m3 and the internal viscoelastic layer be the MRE with parameters ν2 = 0, 42, ρ2 = 2, 650 ×103kg/m3 and moduli E 2 , E 2 ,G 2 ,G 2 specified in Sect. 48.2.


741

40

3

Kr

30 20 2

10

Fig. 48.4 Parameters Kr versus the magnetic induction B for various values of a thickness h2 : 1 – h2 = 10mm, 2 – h2 = 20mm, 3 – h2 = 30mm

1 0

50

100

150

200

B, mT

The numerical computations of the parameters Kr , Ki versus the induction B of the magnetic field at L = 1m, R = 0, 5m, h1 = h3 = 0, 5mm and for various values of thickness h2 were performed. Fig. 48.3(a) shows the dependence of Ki upon the induction B for very thin shells with h2 = 5; 6; 10mm. The similar graphs of the function Ki (B) for a thicker shell (at h2 = 10; 20; 30mm) are given in Fig. 48.3(b). When comparing Fig. 48.3(a) and Fig. 48.3(b), it may be concluded that for very thin shells the dissipative parameter Ki displays weak dependence upon the magnetic field induction B, but the lager the thickness h2 of the MR layer is, the higher the damping property of the sandwich becomes. Calculations for the shells with thicknesses h2 = 10; 20; 30mm of the MR layer presented in Fig. 48.4 show that the parameter Kr decreases when the induction B increases. Figure 48.5 demonstrates the dependence of the eigenfrequency ω and the damping ratio α upon both the parameter B and the thickness h2 of the adaptive layer at n = 1, m = 6. One can see that the increase in the magnetic field intensity involves the increase in the natural frequencies and corresponding damping ratios. And decreasing the natural frequencies caused by increasing the thickness h2 is the specific property of a thin cylindrical shell [16].

48.4.2 Forced Vibrations Let the three-layered MR shell be under the normal vibrational load qn = F(α1 , α2 ) exp(iΩe t),

(48.50)

where Ωe is the frequency of the external periodic force, and F(α1 , α2 ) is its intensity non-uniformly distributed over the shell surface.

742


Fig. 48.5 Natural frequencies ω (a) and damping ratios α (b) vesus magnetic induction B for various values of a thickness h2 of the adaptive layer: 1 – h2 = 10mm; 2 – h2 = 20mm; 3 – h2 = 30mm

The solution of Eqs (48.40) with the boundary conditions (48.43) can be written by use of the expansion theorem as χ=

∞ ∞

sin

n=1 m=1

mα2 πnα1 sin xnm (t), L R

Φ=

∞ ∞ n=1 m=1

sin

πnα1 mα2 sin fnm (t). (48.51) L R

In the above equations, xnm (t) are unknown time dependent generalized coordinates. The substitution of Eqs (48.50) and (48.51) into (48.40) results in the relation fnm (t) =

EhR(h2 L2 π2 m2 n2 + h2 π4 n4 + R2 L2 β2 π2 n2 ) xnm (t) β(L2 m2 + R2 π2 n2 )

(48.52)

and the following equation ∞ ∞ n=1 m=1

[ x¨nm + Ω2nm xnm (t)] sin

πnα1 mα2 F(α1 , α2 ) sin = exp(iΩe t). L R Anm

(48.53)

In Eqs (48.53), Ωnm is the complex natural frequency defined by (48.48), and h 2 π 2 n 2 m2 Anm = ρh 1 + + 2 . (48.54) β L2 R In what follows, low-frequency vibrations of the medium-length cylindrical shell (for which R/l ∼ 1) will be considered. Then Anm ≈ ρh is the real magnitude independent of the magnetic field induction.


743

Multiplying Eq (48.53) by sin(πn1α1 /L) sin(m1 α2 /R), integrating over the shell surface, and taking into account the properties of orthogonality of modes (48.47), one obtains the following equation x¨nm + Ω2nm xnm = F nm exp(iΩe t),

(48.55)

where 2 Fnm = πRLAnm

L 2πR πnα1 mα2 F(α1 , α2 ) sin sin dα1 dα2 . L R 0

(48.56)

0

Here F nm exp(iΩe t) is the generalized force associated with the generalized coordinate xnm (t). The partial solution of Eq. (48.55) is xnm (t) =

Fnm 2 Ωnm − Ω2e

exp(iΩe t).

(48.57)

Then the general solution of Eq. (48.40) is as follows ∞ ∞ χ= Cnm exp(iΩnm t) + n=1 m=1

F nm πnα1 mα2 exp(iΩe t) sin sin , L R Ω2nm − Ω2e

(48.58)

where Cnm are arbitrary complex constants, which can be found from the initial conditions. Based on Eq. (48.58), the vibration response of the shell to the external periodic load (48.50) is ready to be predicted. Since the complex natural frequency Ωnm depends upon the adaptive complex moduli E2 (B), G 2 (B) of the MR layer, Eq. (48.58) can be also easily used to predict the vibration suppression capabilities of the MR shell for different values of the magnetic induction B. It should be noticed that Eq. (48.58) is deducted if the moduli E 2 (B), G2 (B) are constant values independent of time. At the same time the response of the magnetorheological medium to the external magnetic pulse depends significantly on the ratio of the time scale of controlling signal and the reaction of the medium itself [14]. So, when applying the magnetic field signal, the time of reaction of the MRE is about 10−3 − 10−2 sec. At the smooth change of the magnetic field induction the time of reaction can vary substantially. In any case, the obtained here Eqs (48.48), (48.58) do not reveal the reaction of the shell in the time interval comparable with the time of the MRE reaction and could be considered only at setting a certain stationary regime for the viscoelastic characteristics of the MRE. Therefore, (48.48), (48.58) could be used for solving the problem of vibrations damping only on low frequencies, when the frequency of both natural and exciting vibrations is not comparable with the speed of the MRE reaction to the magnetic field. The abrupt impact of the magnetic field is a kind of ”parametric blow” for a mechanical system and can excite additional high frequency modes that require further investigations [14]. The soft suppression of low-frequency vibrations of adaptive MR shells at a smooth change of the magnetic field induction will be studied in the section to follow.

744


48.5 Soft Suppression of Running Low-Frequency Vibrations in MR Adaptive Shells Now reverting to the problem stated in the general case in Sect. 48.2, let us consider the laminated MR non-circular cylindrical shell when L1 (α2 ), L2 (α2 ), R2 (α2 ) are variables. In addition, the nonhomogeneous magnetic induction B(α1, α2 , t) is assumed to be a function of time t, so that the adaptive parameters Ek ,G k depend also on α1 , α2 and t. The problem is to demonstrate the capabilities of soft suppression of non-stationary localized vibrations running in the circumferential direction under the time-dependent magnetic field. Let Y be one of the unknown functions appearing in Eqs (48.40). To evaluate the field-of-use restriction of the asymptotic solutions of Eqs (48.40) in the form of running localized vibrations [16], it is necessary to introduce indices of variation of the functions Y and B in time t as ∂Y Y ∼ ωc h−ς ∗ Y ∂t

at

h∗ → 0,

(48.59)

∂B B ∼ ωc h−ς ∗ B at ∂t

h∗ → 0,

(48.60)

where ωc = E r /(ρR2 ) is the characteristic frequency, E r = ReE is the real part of the reduced complex modulus E at B = 0. It is supposed that the functions B and Y satisfy (48.45) and (48.46) respectively, and ςY = −1/2, ςB ≤ −3/4. The last relations for the indices mean that the rate of change of the magnetic induction B is less than both the velocity and frequency of running vibrations [16]. Then, for analyzing the short waves running in the circumferential direction the homogeneous governing equations (48.40) written in the dimensionless form may be used ( [15], [16]): ∂ 2 Φ∗ ∂2 + ε2 2 [χ∗ − ε2 κ(ϕ, τ)Δχ∗ ] = 0, 2 ∂s ∂τ 2 ∂ ε4 Δ2 Φ∗ − g(ϕ, τ)k(ϕ) 2 [χ∗ − ε2 κ(ϕ, τ)Δχ∗ ] = 0. (48.61) ∂s

ε4 d(ϕ, τ)[1 − ε3υ(ϕ, τ)Δ]Δ2 χ∗ + k(ϕ)

Here Δ=

∂2 ∂2 + 2, 2 ∂s ∂ϕ k(ϕ) =

d(ϕ, τ) = R , R2 (ϕ)

E(ϕ, τ)(1 − ν20 )η3 (ϕ, τ) E 0 [1 − ν2(ϕ, τ)]

α1 = Rs, α2 = Rϕ,

,

g(ϕ, τ) =

t = ε−3 tc τ,

E(ϕ, τ) , E0

tc = ω−1 c ,

h2∗ h2 = , Φ = EhR2 Φ∗ , χ = ε−4 Rχ∗ , 12(1 − ν20) 12R2 (1 − ν20 ) K(ϕ, τ) K(ϕ, τ)θ(ϕ, τ) κ(ϕ, τ) = 2 2 ∼ 1, υ(ϕ, τ) = ∼ 1 at ε → 0, ε π ε3 π 2 ε8 =

(48.62)


745

where χ∗ , Φ∗ are the dimensionless displacement and stress functions, ε is a small parameter, τ is dimensionless time and tc is the characteristic time. The last equations for κ and υ imply that the variable parameters K, θ defined by (48.42) and (48.49) are assumed to be small [15]. These assumptions hold for a thin shell and those materials which are considered here as components of the sandwich. For instance, for one-layered isotropic shells [5], θ = 1/85. Following the idea stated in references [12], [16], the solution of Eqs (48.60) are constructed in the form of the series ∞

χ∗ =

χm (s, ϕ, τ, ε),

Φ∗ =

m=1

∞

Φm (s, ϕ, τ, ε),

(48.63)

m=1

where χm , Φm are the required functions localized in a neighborhood of the generatrix ϕ = qm (τ). The pair χm , Φm is called the mth wave packet (WP) with the center at ϕ = qm (τ). For the mth WP it is assumed [16] χm =

∞

3 4 χm j (s, ιm , τ) × exp iε−1 S m (ιm, τ, ε) ,

j=0

Φm =

∞

3 4 Φm j (s, ιm , τ) × exp iε−1 S m (ιm, τ, ε) ,

j=0

Sm =

1 Ωm (τ)dτ + ε1/2qm (τ)ιm + εbm (τ)ι2m , 2

(48.64)

where ιm = ε−1/2 [ϕ − qm (τ)], χm j (s, ιm , τ), Φm j (s, ιm , τ) are polynomials in ιm with coefficients being functions of the longitudinal coordinate s and time τ, |Ωm (τ)| is the momentary frequency of vibrations of the shell in a neighborhood of the WP center, pm (τ) is the wave (real) number determining the variability of waves in the circumferential direction, and the function bm (τ), where Imbm > 0 for any τ > 0, characterizes the width of the mth WP. Functions (48.64) approximate perturbations which may be generated in the shell by some transient forces applied along a generatrix. The wave packets like those (48.64) may also appear [11] as a result of the parametric excitation of the cylindrical shell having variable geometrical parameters (e.g., a curvature, a thickness or a generatrix length). The algorithm of seeking all unknown functions appearing in (48.64) may be found in references [12], [16] and is not a subject of this research. It should be only noticed that substituting Eqs (48.64) into (48.61) and the corresponding boundary conditions (48.43) produces the sequence of the one-dimensional boundary-value problems at the moving generator ϕ = qm (τ) which is the center of the mth traveling WP. So, in the zeroth order approximation ( j = 0), one obtains χm0 = Pm0 (ιm , τ)Zm (s),

Zm (s) = sin

πm{s − s1 [qm (τ)]} l[qm (τ)]

(48.65)

746


and the relation Ωm (τ) = q˙ m (τ)pm (τ) ∓ Hˆ m [pm (τ), qm(τ), τ],

(48.66)

where 9 Hˆ m (pm , qm , τ) =

π4 m4 k2 (qm )g(qm, τ) d(qm , τ)p4m + 4 4 pm l [qm (τ)] 1 + κ(qm, τ)p2m

(48.67)

is the complex Hamilton function associated with the mth WP, Pm0 (ιm, τ) is an unknown polynomial in ιm , l(qm) = s2 (qm ) − s1 (qm) = R−1 [L2 (qm) − L1 (qm )], and the dot (·) denotes differentiation with respect to τ. The sign (∓) in (48.66) indicates the availability of two branches (positive and negative) of solutions corresponding to the m± th WP running in the opposite directions. To avoid inconvenience the subscript m is omitted in what follows. The real and imaginary parts of the adaptive complex functions appearing in (48.67) are introduced as d1 = Red(q, τ), d2 = Imd(q, τ), g1 = Reg(q, τ), g2 = Img(q, τ), κ1 = Reκ(q, τ), κ2 = Imκ(q, τ),

(48.68)

where the notations Re f and Im f denote the real and imaginary parts of a function f . Then, separating the real and imaginary parts in (48.66) yields the relations for the momentary frequency ω = ReΩ and the damping ratio α = ImΩ of running vibrations: hˆ i (p, q, τ) ω = qp ˙ ± H(p, q, τ), α = − . (48.69) 2H(p, q, τ) Here 9 8 1 ˆ 2 2 ˆ ˆ H= hr + h r + hi , 2 π4 m4 k2 πg1 p4 (1 + p2 κ1 )d1 + p6κ2 d2 hˆ r (p, q, τ) = + , l 4 p4 (1 + p2κ1 )2 + p4 κ22 π4 m4 k2 πg2 p4 (1 + p2κ1 )d2 − p6 κ2 d1 hˆ i (p, q, τ) = + . l4 p4 (1 + p2κ1 )2 + p4 κ22

(48.70)

The compatibility condition for the non-homogeneous problem arising in the first order approximation ( j = 1) with respect to χm1 implies the Hamiltonian system q˙ = H p ,

p˙ = −Hq ,

(48.71)

where the subscripts p, q denote differentiation with respect to the corresponding variables p, q.


747

Finally, from the second order approximation ( j = 2), one obtains the Riccati equation b˙ + H pp b2 + 2H pq b + Hqq = 0. (48.72) and the amplitude equation r0

∂ 2 P0 ∂P0 ∂P0 + r1 ι + r2 + r3 P0 = 0. 2 ∂ι ∂τ ∂ι

(48.73)

In Eq. (48.73), r j (τ)( j = 0, 1, 2, 3) are functions expressed in terms of the Hamiltonian Hˆ and its derivatives over p and q; they are not written out here in the explicit form because of its awkwardness and may be found in references [12], [16]. Note only that, as opposed to the similar equation derived earlier for the isotropic elastic shells [16], now the coefficients r j (τ) depend upon the complex parts id2 (τ), ig2 (τ), iκ2 (τ) (see Eqs (48.68)) of the adaptive complex functions. The solution of Eq. (48.73) may be expressed by means of the Hermite polynomials [12]: Pm0 (ι, τ) = T m (τ)Hm [$(τ)ι], (48.74) where

) exp − (r1 /r2 )dτ $(τ) = 8) ) , 2 (r0 /r2 ) exp −2 (r1 /r2 )dτ dτ 3 ) ) 4m/2 4 (r0 /r2 ) exp −2 (r1 /r2 )dτ dτ T m (τ) = ) . exp (r3 /r2 )dτ

(48.75)

Here Hm (x) is the Hermit polynomial of the mth degree. The Hamilton function (48.67) and Eqs (48.71)–(48.73) are governing ones and may be used for predicting both the shell response to the localized perturbations and soft suppression of the running vibrations under the time-dependent magnetic field. To demonstrate in detail the effect of suppressing the running WP when the growing magnetic field is induced, it is necessary to apply to an example. Example 48.2. Let us consider the circular three-layered cylindrical shell with a constant length generatrix, the other geometrical and physical properties being specified in Example 48.1. Then k = 1, l are constant magnitudes, and d = g = g1 (τ) + ig2 (τ) is the dimensionless reduced complex modulus being the smooth function of time τ. Generally speaking, the influence of the reduced shear parameter κ on the dynamics of a very thin shell is inefficient [15]. Therefore, to simplify calculations κ is set equal to 0 in what follows.

748


Then the solutions of the Hamilton equations (48.71) and the Riccati equation (48.72) are τ b0 ˆ )dτ , b(τ) = p(τ) = p0 , q(τ) = Θ p (p0 ) G(τ , )τ ˆ 1 + b Θ (p ) G(τ )dτ 0 0 pp 0 2 Θ(p) =

0

π 4 m4 l4

√ 2 8 ˆ = 2 g1 (τ) + g2 (τ) + g2(τ), (48.76) G(τ) 1 2 2

+ p4 ,

where p0 , b0 are constant parameters evaluated by the initial conditions for the mth WP at τ = 0. Inserting Eqs (48.74) into (48.69) results in the relations for the momentary frequency and decrement of the running mth WP: ˆ ω(τ) = (p0 + 1)Θ p(p0 )G(τ),

α(τ) = −

g2 (τ)Θ(p0) . ˆ 2G(τ)

(48.77)

It may be seen from Eqs (48.3), (48.12), (48.76), (48.77) that if the magnetic induction B(τ) is the steadily increasing function of dimensionless time τ, then the momentary frequency ω also increases. The numerical calculations of the damping ratio α(τ) and the parameter b(τ) were performed at the linearly growing magmatic induction B = 20τ for thicknesses h2 = 5mm and h2 = 10mm of the MR layer. The external layers of the thickness h1 = h2 = 0, 5mm were assumed to be made of the ABS plastic SD-0170. Figures 48.6 shows that a smooth increase of the magnetic induction B(τ) implies the appreciable suppression of the running localized vibrations, this suppression being accompanied by rapid spreading waves. Moreover, the larger the thickness h2 of the adaptive layer is, the higher the efficiency of the vibration damping becomes. 2,0

(a)

(b)

1,5

a

Im b

0,4

1,0

2

0,2 0,5

1

1 2

0

3

6

0

9

3

6

9

t Fig. 48.6 Damping ratios α (a) and parameters Imb (b) versus dimensionless time τ: 1 – h2 = 5mm, 2 – h2 = 10mm


749

48.6 Conclusions The nonhomogeneous multilayer non-circular cylindrical shell with the viscoelastic adaptive magnetorheological elastomers sandwiched between elastic layers was considered. Based on the unique kinematics hypothesis for the whole sandwich and the constitutive equations which took into account the dependence of the complex moduli on the externally applied magnetic field, the governing equations with timedependent complex coefficients for the MR adaptive laminated shell were derived. In the case when all geometrical and physical parameters (including the magnetic induction) are constant, the solutions of the governing equations describing both free and forced vibrations have been constructed in the explicit form. From the model predictions, variations in natural frequencies, vibration amplitudes and loss factors were observed when the MR adaptive shell was subjected to different stationary magnetic field levels. Additionally, the localized vibrations running in the circumferential direction in the non-circular cylindrical MR shell with variable geometrical and physical parameters under the effect of the time dependent magnetic field were studied by using the asymptotic approach. The observed vibration suppression capabilities of the MR adaptive shells are summarized as follows: 1. Theoretical and numerical studies of the three-layered MR adaptive shells being under the effect of the stationary magnetic field revealed that the application of higher magnetic field strength on the shell resulted in shifts in eigenfrequencies toward higher frequencies and decreased the vibration amplitudes. The efficiency of the vibration suppression depends strongly on the correlation of the geometrical and physical parameters of all layers composing the adaptive sandwich. However, in any case, the increase in the thickness of the adaptive MR layer raises the efficiency of the vibration suppression. 2. The dynamic characteristics of the running localized vibrations in the MR adaptive cylindrical shell are largely influenced by the external time-dependent magnetic field. So, the smooth increase of the magnetic field induction results in the increase of the momentary frequencies and the damping ratio of the running packets of destructive bending waves, the effect of suppression being accompanied by quick spreading the damped waves over the shell surface.

References 1. Bolotin, V.V., Novichkov, Yu, N.: Mechanics of Multilayer Constructions. Mashinostroenie, Moscow (1980) 2. Choi, Y.T., Werely, N.M.: Assesment of time response characteristics of electrorheological and magnetorheological dampers. Proc. SPIE. 4331, 92–102 (2001) 3. Ginder, G.M. Rheology controlled by magnetic fields. Encyclopedia of Applied Physics. 16, 487–503 (1996) 4. Ginder, J.M., Schlotter, W.F., Nichols, M.E.: Magnetorheological elastomers in tunable vibration absorbers. Proc. SPIE. 3985, 418–424 (2001)

750


5. Grigoliuk, E.I., Kulikov, G.M.: Multilayer Reinforced Shells: Calculation of Pneumatic Tires. Mashinostroenie, Moscow (1988) 6. Hasheminejad, S. M., Maleki, M.: Free vibration and forced harmonic response of an electrorheological fluid-filled sandwich plate. Smart Mater. Struct. (2009) doi: 10.1088/09641726/18/5/055013. 7. Hsu, T.-M., Wang J. T.-S.: A theory of laminated cylindrical shells consisting of layers of orthotropic laminae. AIAA Journal. 8(12), 2141–2146 (1970) 8. Jolly, M.R., Bender, J.W., Carlson, D,J.: Properties and applications of comerciaal magnetorheological fluids. J. Intell. Mater. Syst. Struct. 10, 5–13 (1999) 9. Korjakin, A., Rikards, R., Chate, A, Altenbach, H.: Analysis of free damped vibrations of laminated composite conical shells. Composite Structures. 41, 39–47 (1998) 10. Korobko, E. V., Zhurauski, M. A., Novikova, Z. A., Kuzmin, V. A.: Rheological properties of magnetoelectrorheological fluids with complex disperse phase. Journal of Physics: Conference Series. 149, 12–65 (2009) 11. Mikhasev, G.I.: Free and parametric vibrations of cylindrical shells under static and periodic axial loads. Technische Mechanik. 17, 209–216 (1997) 12. Mikhasev, G.I.: Localized families of bending waves in a thin medium-length cylindrical shell under pressure. J. Sound Vib. 253(4), 833–857 (2002) 13. Mikhasev, G., Korchevskaya, E., Gabbert. U., Marinkovic, D.: Local buckling, stationary and non-stationary vibrations of thin composite laminated shells having the weakest spots. In: Loughlan, J. (ed.) Proceedings of the Fourth International Conference on Thin-Walled Structures, pp. 769-776. Loughbourough, UK (2004) 14. Mikhasev, G.I., Korobko, E.V., Novikova, Z.A.: On suppression of vibrations of three-layered beam containing magnitorheological composite. Mechanics of Machines, Mechanisms and Materials. 4(13), 49–53 (2010) 15. Mikhasev, G., Seeger, F., Gabbert, U.: Comparison of analytical and numerical methods for the analysis of vibration of composite shell structures. In: Kasper, R. (ed.) Entwicklungsmethoden und Entwicklunsprozesse im Maschinenbau: 5 Magdeburger Maschinenbau-Tage, pp. 175-183. Logos-Verl, Berlin (2001) 16. Mikhasev, G.I., Tovstik, P.E.: Localized Vibrations and Waves in Thin Shells. Asymptotic Methods. FIZMATLIT, Moscow (2009) 17. Park, D.W., Choi, S.B., Jung, S.B.: Shape control of an electrorheological fluid based smart plate. Proc. SPIE. 3329, 824–835 (1998) 18. Qatu, M.S.: Recent research advances on the dynamic analysis of composite shells. Composite Structures. 93(1), 14–31 (2010) 19. Shaw, J.: Hybrid control of cantilevered ER sandwich beam for vibration suppression. J. Intell. Mater. Syst. Struct. 11, 26–31 (2000) 20. White, J.L.. Choi, D.D.: Polyolefins: Processing, Structure, Development, and Properties. Carl Hanser Verlag, Munich (2005) 21. Yalcintas, M., Dai, H.: Magnetoreological and electroreological materials in adaptive structures and their perfomance comparision. J. Smart Mater. Struct. 8, 560–573 (1999) 22. Yalcintas, M., Dai, H.: Vibration suppresion capabilities of magnetorheological materials based adaptive structures. Smart. Mater. Struct. 13, 1–11 (2004) 23. Yeh, J.-Y.: Vibration and damping analysis of orthotropic cylindrical shells with electrorheological core layer. Aerospace Science and Technology. (2008) doi:10.1016/j.jsv.2008.02.012.

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