6, Engineering structures: nonclinical analysis optimal design and identification

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Author: Adnan Ibrahimbegovic | Bostjan Brank

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ec cover (i).qxd

09/08/2005

10:13

Page 1

ISBN 1-84544-163-X

ISSN 0264-4401

Volume 22 Number 5/6 2005

Engineering Computations International journal for computer-aided engineering and software Engineering structures: nonlinear analysis, optimal design and identification Guest Editors: Professor Adnan Ibrahimbegovic´ and Professor Boštjan Brank

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Engineering Computations International Journal for Computer-Aided Engineering and Software

ISSN 0264-4401 Volume 22 Number 5/6 2005

Engineering structures: nonlinear analysis, optimal design and identification Guest Editors Professor Adnan Ibrahimbegovic´ and Professor Bosˇtjan Brank

Access this journal online _________________________

483

Editorial advisory board __________________________

484

Editorial _________________________________________

485

Identification strategy in the presence of corrupted measurements O. Allix, P. Feissel and H.M. Nguyen_______________________________

487

Constrained finite rotations in dynamics of shells and Newmark implicit time-stepping schemes Bosˇtjan Brank, Said Mamouri and Adnan Ibrahimbegovic´ __________________________________________

505

Saint-Venant multi-surface plasticity model in strain space and in stress resultants J-B. Colliat, A. Ibrahimbegovic´ and L. Davenne_______________________

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536

CONTENTS

CONTENTS continued

Prediction of crack pattern distribution in reinforced concrete by coupling a strong discontinuity model of concrete cracking and a bond-slip of reinforcement model Norberto Dominguez, Delphine Brancherie, Luc Davenne and Adnan Ibrahimbegovic´ __________________________________________

558

On numerical implementation of a coupled rate dependent damage-plasticity constitutive model for concrete in application to high-rate dynamics Guillaume Herve´, Fabrice Gatuingt and Adnan Ibrahimbegovic´ _________

583

Shape optimization of two-phase inelastic material with microstructure Adnan Ibrahimbegovic´, Igor Gresˇovnik, Damijan Markovicˇ, Sergiy Melnyk and Tomazˇ Rodicˇ __________________________________

605

Parameterization based shape optimization: theory and practical implementation aspects Marko Kegl ___________________________________________________

646

Multi-scale modeling of heterogeneous structures with inelastic constitutive behaviour: part I – physical and mathematical aspects Damijan Markovic, Rainer Niekamp, Adnan Ibrahimbegovic´, Hermann G. Matthies and Robert L. Taylor _________________________

664

Some aspects of 2D and/or 3D numerical modelling of reinforced and prestressed concrete structures Pavao Marovic´, Zˇeljana Nikolic´ and Mirela Galic´ _____________________

684

Coupling FEM and BEM for computationally efficient solutions of multi-physics and multi-domain problems Boris Sˇtok and Nikolaj Mole _____________________________________

711

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EC 22,5/6

EDITORIAL ADVISORY BOARD J.E. Akin Department of Mechanical Engineering and Materials Science, Rice University, Texas, USA S.N. Atluri Computational Modeling Center, Georgia Institute of Technology, Georgia, USA

484

Z.P. Bazˇant Professor of Civil Engineering, Northwestern University, Illinois, USA T. Belytschko Department of Civil Engineering, Northwestern University, Illinois, USA P.G. Bergan A.S. Veritas Research, Hovik, Norway P. Bettess University of Durham, UK J.C. Bruch Jr Department of Mechanical and Environmental Engineering, University of California, California, USA C.L. Chenot Centre de Mise en forme des Mate´riaux, Valbonne, France Y.K. Cheung Department of Civil Engineering, University of Hong Kong, Hong Kong R. de Borst Department of Civil Engineering, Delft University of Technology, Delft, The Netherlands C.S. Desai Department of Civil Engineering & Engineering Mechanics, The University of Arizona, Arizona, USA R.H. Dodds Jr Department of Civil Engineering, University of Illinois, at UrbanaChampaign, Illinois, USA E.N. Dvorkin Center for Industrial Research, Buenos Aires, Argentina C.A. Felippa University of Colorado, Colorado, USA M. Geradin Universite´ de Lie`ge, Lie`ge, Belgium J.O. Hallquist Livermore Software Technology Corporation, California, USA T.J.R. Hughes Division of Mechanics and Computations, Stanford University, California, USA A. Ibrahimbegovic Ecole Normale Superieure de Cachan, Cachan cedex, France A.R. Ingraffea Cornell University, Ithaca, New York, USA D.W. Kelly The University of New South Wales, New South Wales, Australia B.H. Kro¨plin Universita¨t Stuttgart, Stuttgart, Germany P. Ladeveze Laboratoire de Mećanique et Technologie, Cachan, France

Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 22 No. 5/6, 2005 p. 484 # Emerald Group Publishing Limited 0264-4401

J. Mackerle Department of Mechanical Engineering, Linkoping Institute of Technology, Linkoping, Sweden H. Mang Technical University Wien, Vienna, Austria Z. Mroz Institute of Fundamental Technological Research, Warsaw, Poland

J.C. Nagtegaal Hibbit, Karlsson & Sorenson Inc., Rhode Island, USA A.K. Noor Old Dominion University, NASA Langley Research Centre, Hampton, Virginia, USA J.T. Oden Engineering Mechanics, The University of Texas at Austin, Austin, Texas, USA R. Ohayon Chair of Mechanics, Conservatoire National des Arts et Me´tiers, Paris, France E. Onãte Universidad Politećnica de Catalunã, Barcelona, Spain M. Ortiz Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California, USA D. Peric Department of Civil Engineering, University of Wales, Swansea, UK P. Pinsky Department of Civil Engineering, Stanford University, California, USA C.V. Ramakrishnan Indian Institute of Technology, New Delhi, India E. Ramm Universita¨t Stuttgart, Stuttgart, Germany F.G. Rammerstorfer Universita¨t Wien, Vienna Austria J.N. Reddy Department of Mechanical Engineering, Texas A&M University, Texas, USA B. Schrefler Instituto di Scienza e Tecnica delle Costuzioni, Dell Universita` degli Studi di Padova, Padova, Italy S.W. Sloan Department of Civil, Surveying and Environmental Engineering, University of Newcastle, New South Wales, Australia I.M. Smith University of Manchester, Manchester, UK E. Stein Institut fu¨r Baumechanik and Numeru¨rische Mechanik, Universita¨t Hannover, Hannover, Germany R.L. Taylor Department of Civil Engineering, University of California, California, USA B.H.V. Topping Civil & Offshore Engineering, Heriot-Watt University, Edinburgh, UK L.T. Watson Departments of Computer Science and Mathematics, Virginia Polytechnic Institute & State University, Virginia, USA N.P. Weatherill Department of Civil Engineering, University College of Swansea, Swansea, UK K. Willam Department of Civil Engineering, University of Colorado, Colorado, USA J.R. Williams Department of Civil Engineering, Massachusetts Institute of Technology, Massachusetts, USA E.L. Wilson Department of Civil Engineering, University of Colorado, Colorado, USA P. Wriggers Institut fu¨r Mechanik, Darmstadt, Germany G. Yagawa Department of Nuclear Engineering, University of Tokyo, Tokyo, Japan

Editorial Engineering structures: nonlinear analysis, optimal design and identification This special issue of International Journal for Engineering Computations contains extended versions of the papers first presented by several invited speakers at the NATO-sponsored advanced research workshop on “Multi-physics and multi-scale computer models in non-linear analysis and optimal design of engineering structures under extreme conditions”, which was held in Bled, Slovenia, from 13 to 17 June 2004. It also features a couple of papers contributed by young researchers, who all made short presentations of their research works at that meeting. The collected papers seek to present the most recent research achievements on this currently very active research topic, and are therefore considered to be more suitable for a scientific journal than for a book-like presentation of keynote lectures, which was already published (Ibrahimbegovic and Brank, 2004). We are grateful to our colleague (and one of the keynote lecturers at the NATO-ARW in Bled) Professor D.R.J. Owen, the Editor of International Journal for Engineering Computations, for kindly providing us an opportunity to gather all those contributions in the present special issue. The kind of questions, which were raised herein and discussed thoroughly, are rather diverse, just as one can expect for the research field as rich and as active as the one of engineering structures. When attempting a rough classification of the published papers that could help to guide potential readers, we can recognize the contributions on nonlinear structural analysis, on optimal design of structures and microstructures, as well as on the structural parameters identification. With such a wide scope of papers’ contents, rather than looking for some “logical” order which was likely to be somewhat biased by our own personal preference, we choose to simply present the papers in alphabetical order of first authors’ names. Therefore, we start the issue with the paper by Allix and co-workers on devising a computational strategy for the identification of structural properties obtained from dynamic testing in the presence of corrupted measurements, followed by the paper by Brank and collaborators on dynamics of shell structures at finite rotations and implicit time-stepping schemes; next is the paper by Colliat et al. on development of constitutive model in either strain space of the space of stress resultants with applications to limit load analysis of brittle cellular structures under high temperatures, followed by the paper of Dominguez et al. on modeling the inelastic behavior of structures build of the most classical of all composite materials – reinforced concrete, where one can take into account all the potential failure modes related to either concrete, steel or bond-slip; the paper of Herve´ et al. extends further this line of developments by considering yet more elaborate model for inelastic constitutive behavior of concrete in high rate dynamics; the paper of Ibrahimbegovic and co-workers provides a strategy for optimal design of microstructures with a particular emphasis on two-phase material and phase interface shape optimization;

Editorial

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Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 22 No. 5/6, 2005 pp. 485-486 q Emerald Group Publishing Limited 0264-4401

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the paper of Kegl also considers shape optimization problem, but directly at the structural level; the paper of Markovic and collaborators provides a computational procedure for multi-scale analysis of inelastic behavior of heterogeneous structures and its parallel computer implementation; the paper of Marovic and co-workers presents numerical models for reinforced and prestressed concrete structures with special emphasis on handling presterssing tendons and reinforcing bars; finally, the paper of Stok and Mole draws on a fruitful combination of the finite element and the boundary element methods to deal with modeling of multi-physics problem of engineering structure with its surrounding environment. Rather than elaborating some more on the main points brought forward in each paper, we invite the readers of the Journal to do their own explorations. With a wide variety of topics being addressed in this special issue, we do hope that our goal will be achieved in ensuring that each reader finds at least one of the topics of a particular interest. We would like to thank all the authors of the special issue for their contributions to this goal. Adnan Ibrahimbegovic´ and Bostjan Brank Guest Editors Reference Ibrahimbegovic, A. and Brank, B. (2004), “Multi-physics and multi-scale computer models in nonlinear analysis and optimal design of engineering structures under extreme conditions” Proceedings NATO-ARW, Bled, Slovenia, 12-17 June, IOS Press.

The Emerald Research Register for this journal is available at www.emeraldinsight.com/researchregister

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0264-4401.htm

Identification strategy in the presence of corrupted measurements O. Allix, P. Feissel and H.M. Nguyen Laboratoire de Mećanique et Technologie, ENS Cachan, Cachan, France

Identification strategy

487 Received November 2004 Revised December 2004 Accepted December 2004

Abstract Purpose – To propose and develop an identification method of material parameters from dynamics test in the presence of extensively corrupted measurements. Design/methodology/approach – The method we propose, which is based on the use of the error in constitutive relation for identification problems in the framework of transient dynamics, leads to nonstandard wave propagation problems. For solving this numerical difficulty, we used the transition matrix method for short-duration tests and the combined Riccati constant/transition matrix approach for long-duration tests. Findings – A numerical strategy adapted to the problem. Results obtained appears to be insensitive to perturbation of measurements up to a very high level of perturbation. Research limitations/implications – Only simple case of elastic bar have been treated so far. Originality/value – Without any a priori information on the level of perturbation, this method is robust with respect to the perturbation. A coupling of two resolution methods allows to deal with problem of arbitrary duration. Keywords Dynamics, Identification, Measurement Paper type Research paper

1. Introduction This work was undertaken in the context of the identification of a dynamic damage model for the prediction of the rupture of composite crash absorbers. The identification tests which were conducted made use of a system derived from a split Hopkinson bar apparatus. The data obtained from those tests, i.e. the velocities and the forces at the ends of the specimen, were non-reproducible and highly scattered. Therefore, we had to seek an identification strategy which would be as insensitive as possible to such scattering. In this case, contrary to other experimental situations, there is no a priori information available on the nature or on the level of the perturbation, which is why we designate these uncertainties as corruption. This also explains why we were a priori unable to rely on a Tikhonov regularization approach (Tikhonov and Arsenin, 1977; Andrieux and Voldoire, 1995) or on a Kalman filter approach (Kalman, 1960; Corigliano and Mariani, 2003). The method proposed here was developed specifically to deal with this context of highly scattered measurement results. It is based on the concept of modified error in constitutive relation (Ladeve`ze, 1998). This concept has been proven to be effective in the case of vibrations. Its principle consists in splitting experimental as well as theoretical quantities into two groups: the reliable quantities and the less reliable quantities. Then, the less reliable conditions are relaxed and verified globally, at best, throughout the identification strategy. This approach was extended to transient dynamics in Allix et al. (2003) and Feissel (2003) and appears to be very robust from an

Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 22 No. 5/6, 2005 pp. 487-504 q Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644400510602989

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identification point of view, but it leads to numerical difficulties. Therefore, this paper focuses on the numerical strategy. In the first section, the identification framework is described and an example which is used throughout the paper to illustrate the problem and the different methods is presented. Then, the proposed strategy is described. The associated problems lead to the resolution of a set of coupled direct and retrograde wave propagation problems. A first iterative technique is developed, which turns out to be effective only for short-duration tests. This is due to the presence of exponentially increasing functions of time in the theoretical solution which leads to very poorly conditioned systems if one uses the classical Newmark integration scheme. This is why, in our second attempt, we combine the previous approach with an algebraic Ricatti technique in order to be able to deal with dynamic tests of arbitrary duration as proposed in Formosa et al. (2002). 2. Identification framework in the case of highly scattered boundary conditions The objective of this study is the identification of model parameters from dynamic tests in the presence of highly scattered measurements. There are two problems to deal with: (1) heterogeneity and the dynamics of the test; and (2) the handling of corrupted measurements. In order to deal with the first problem, we propose to adopt an inverse variational approach. This approach consists of two steps. In the first step, one confronts the experimental data with the model and its parameters. This is usually done through a calculation. In the second step, due to the results of the previous calculation, one evaluates a cost function which characterizes the quality of the model parameters which must be identified. Therefore, in order to find the solution to the identification problem, one must minimize this cost function with respect to the model parameters. There have been many attempts to take perturbed measurements into account. Most often, these methods require an a priori knowledge of the level of perturbation. For example, this is the case of the Tikhonov regularization methods (Tikhonov and Arsenin, 1977; Andrieux and Voldoire, 1995) or the Kalman filter techniques (Kalman, 1960; Corigliano and Mariani, 2003). In the present context, there is no a priori information on the level of perturbation. This is the reason why we had to seek a specific method. In order to describe the identification strategy and the technique used for its resolution, we will simplify the identification problem to that of homogeneous elastic properties of a bar subjected to dynamic loading. This choice lends itself to a simple treatment, but the method can be applied to more complicated cases, such as field measurements. In this context, the measurements concern, on the whole time interval [0, T ], the displacement and force boundary conditions at both ends of the bar (Figure 1):

Figure 1.

. .

measured displacements: u~ 0 ðtÞ; u~ L ðtÞ where t [ ½0; T; measured forces: F~ 0 ðtÞ; F~ L ðtÞ.

The model relies on the dynamic equilibrium: ru€ 2 s;x ¼ 0 and the constitutive relation: s ¼ Eu;x ; where E is the unknown quantity of the problem (Figure 1). Then, the calculation which confronts the measurements to the model for a given Young’s modulus must be defined based on these problem data (boundary conditions, equilibrium equation and constitutive relation). One can note that the exact verification of all these data would correspond to an ill-posed problem. Therefore, some of the data must be released in defining this calculation. In order to test the method, we use a numerical example. A preliminary calculation consists in simulating the case shown in Figure 2 of a rod clamped at one end and subjected to a half-sine force at the other. The corresponding displacements are shown in Figure 3. Then, the forces and displacements at both ends obtained from this simulation are used to generate a set of measurements for the identification problem. In order to test the robustness of the method, a perturbation in terms of both displacements and forces can be added to the boundary conditions. The case shown in Figure 3 corresponds to a perturbation resulting from the application of a uniform white noise with a standard deviation of 12 percent with respect to the perturbed quantities. The figure represents the displacement part of the measurements alone.


489

3. Formulation of the proposed identification strategy In order to present an inverse approach, one must define the two steps of the identification strategy. The proposed method relies on the guiding principles of

Figure 2. The numerical test

Figure 3. Exact and perturbed boundary conditions (using uniform white noise, standard deviation 12 percent)

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the modified error in constitutive relation (Ladeve`ze, 1998), which were developed mainly in the context of model updating techniques for vibration problems (Deraemaeker et al., 2002). The experimental and theoretical quantities are divided into two groups: the reliable quantities and less reliable quantities. In the proposed method, the verification of the properties which are considered to be reliable is enforced throughout the identification process, whereas the uncertain quantities are taken into account by minimizing a modified constitutive relation error (Chouaki et al., 1998). Considering the case of 1D elasticity, let us divide the quantities into two groups as shown in Table I. 3.1 Formulation The basic problem we must solve, which corresponds to the first step of the identification strategy, is reformulated by introducing a functional which is the sum of a modeling error term and an experimental error term: Find the field u; s; ud ; f d ; minimizing: J ðs; u; ud ; f d Þ ¼

Z

T 0

1 2

Z

L

E

21

0

L L ~ :ðs 2 Eu;x Þ dx þ d f ð f d ; fd Þ 0 þ jd u ðud ; u~ d Þj0 dt 2

under the constraints: u [ UAd ðud Þ;

s [ DAd ðf d ; uÞ

ð1Þ

L

where the following notations are used: for any f: j f j0 ¼ f ð0Þ þ f ðLÞ; ud and fd are the displacement and force boundary conditions defined at x ¼ 0 and x ¼ L; UAd ðud Þ ¼ {u [ H 1 ðVÞ=u ¼ ud for x [ {0; L}; and ujt¼0 ¼ u0 ; u_ jt¼0 ¼ u_ 0 }; a field such as u is called KA (kinematically admissible); and DAd ðf d ; uÞ ¼ {s [ H div ðVÞ=sn ¼ f d for x [ {0; L}; r · u€ 2 divs ¼ 0 for x [ {0; L}}; a field such as s is called DA (dynamically admissible); in addition, u and s are linked by the equilibrium conditions. In the previous expression of the error, df represents a distance between the end measurements f~d and the corresponding boundary conditions fd deduced from the formulation. A typical choice, which will be used in the following development, is: B df ð f d ; f~d Þ ¼ ð f d 2 f~d Þ2 2 du is defined in the same manner:

Table I. The reliable and unreliable quantities in the case of 1D elasticity

Reliable

Unreliable

Equilibrium ru€ 2 s;x ¼ 0 Initial conditions uðx; 0Þ ¼ u0 u_ ðx; 0Þ ¼ u_ 0

Constitutive relation s ¼ E:u;x Boundary conditions u~ d and f~d

du ðud ; u~ d Þ ¼


A ðud 2 u~ d Þ2 2

A and B are two parameters which allow, depending on the relative confidence in those terms, to balance the two terms defining the error, namely the one in constitutive relation and the one associated with the measurements. A first choice, that also allows the functional to be dimensionaly correct, is to divide each term by its order of magnitude. This choice appears to be relevant with respect to the identification results. Then, the cost function needed in the identification step of the strategy uses the same functional as in the basic problem, evaluated at the fields which are solutions of the basic problem. Therefore, the identification problem becomes: min J ðsðEÞ; uðEÞ; ud ðEÞ; f d ðEÞ; EÞ

ð2Þ

E

where the functional J is defined in equation (1). The following notation is also used: gðEÞ ¼ J ðs ðEÞ; uðEÞ; ud ðEÞ; f d ðEÞ; EÞ

ð3Þ

where (s (E), u(E), ud(E), fd(E)), are solutions of equation (1): ðs ðEÞ; uðEÞ; ud ðEÞ; f d ðEÞÞ ¼ Arg min J ðs; u; ud ; f d ; EÞ

ð4Þ

u[UAd ðud Þ s[DAd ðf d ;uÞ ud ; f d

Therefore, the problem can be stated as: J ðs; u; ud ; f d ; EÞ

min

ð5Þ

E;ud ;f d u[UAd ðud Þ s[DAd ðf d ;uÞ

Let us mention that an alternative choice for this cost function is currently being studied. In this alternative, the identification is based on the modeling error term alone and the distance to the measurements is introduced only in the basic problem to regularize the boundary conditions (Feissel, 2003). Since the present paper focuses on the resolution strategies, this point will not be developed any further. In practice, the minimization is carried out in two steps. First, the functional to be minimized is evaluated for a given Hooke’s tensor E, leading to the determination of the unknown boundary conditions, which depend implicitly on E. Then, the minimization with respect to the value of E is performed. 3.2 The basic problem: the equation to be solved Each of the constraints of the basic problem is relaxed using a Lagrange multiplier, which leads to the following formulation: Z T L Lðs; u; ud ; f d ; u* ; l; m; EÞ ¼ J ðs; u; ud ; f d ; EÞ 2 jðu 2 ud Þlj0 dt 0

TZ

Z

L

ðru€ 2 s;x Þu* dx dt 2

2 0

0

ð6Þ

Z

T 0

L jðsn 2 f d Þmj0 dt

491

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In order to obtain the system that the solution must satisfy, the stationarity of L is enforced:

dL ¼ 0

492

ð7Þ

where dL has the following expression: Z T Z L dL ¼ ds · E 21 s 2 E u;x þu;x* dx þ jds · nu* 2 ds · nmjL0 dt 0

0

T Z

Z

L

ðduÞ;x ðs 2 E · u;x Þ þ d€uru* dx þ

2 0

0 TZ L

Z 2 0

þ

Z

L jdu · lj0

dt

ð8Þ

du* ðru€ 2 s;x Þ dx dt

0

L df d m þ ›d f þ dud l þ ›d u dt ›f ›u

T

d

0

Z

T

2 0

d

0

L jdmðsn 2 f d Þ þ dlðu 2 ud Þj0 dt

One deduces the strong form of the associated equations by taking equation (8) into account in equation (7), and by integrating the term du¨ by parts with respect to t: 8 ru€ 2 E u;xx þu;*xx ¼ 0 for x [0; L½; > > > > > > ru€ * 2 E · u;*xx ¼ 0 > > > > > < for x [ {0; L}; E u;x þu;*x n 2 B1 u* ¼ f~d where nð0Þ ¼ 21 and nðLÞ ¼ 1 > u 2 AE u;x* ¼ u~ d > > > > > > and uðx; 0Þ ¼ u0 ; u_ ðx; 0Þ ¼ u_ 0 > > > > : u* ðx; TÞ ¼ 0; u_ * ðx; TÞ ¼ 0 ð9Þ The system to be solved is a coupled direct-retrograde wave propagation problem where u must satisfy initial conditions and u* final conditions. Let us note: X ¼ ðs; u; ud ; fd ; u* ; l; mÞ; then the function L in equation (6) becomes: LðXðEÞ; EÞ ¼ gðEÞ It is interesting to note that the use of the same functional in the definition of the basic problem and in the identification step enables one to evaluate the cost function and its gradient using the solution fields with no additional calculation: DgðEÞ · q ¼

›L ›L ›X ›L ›L qþ q¼ q because of equation ð8Þ : ¼0 ›E ›X ›E ›E ›X

ð10Þ

1 ) DgðEÞq ¼ 2 2

Z

TZ L 0

0

ðE · u;x 2 sÞðE · u;x þ sÞ dx dt q E2

ð11Þ

4. Finite element formulation of the basic problem The presence of initial and final conditions poses a serious difficulty when solving the basic problem. This difficulty is related solely to the time discretization, even if in dynamics time and space are coupled. Regarding the space discretization, a classical finite element approach is used. In the case of the one-dimensional problem of a bar of length L uniformly discretized into Ne elements of length le: ( uðxÞ ¼ ½FðxÞU u* ðxÞ ¼ ½FðxÞU * where [F(x)] is the matrix which collects the finite element functions and U (respectively, U*) the vector of nodal unknowns associated with u (respectively, u*). Let us also introduce the matrix [B(x)] such that: 1ðuðxÞÞ ¼ ½BðxÞU The corresponding weak formulation (12) follows: 8 s ¼ E · ½BðxÞðU þ U * Þ > > > > Z L > > > > ðdU * T ½FðxÞT r½FðxÞU€ þ dU * T ½BðxÞT E½BðxÞðU þ U * ÞÞ dx > > > 0 > < 2dU * T F~ d 2 B1 PU * ¼ 0 > > RL > > T T T T € > > 0 ðdU ½FðxÞ r½FðxÞ · U* þ dU ½BðxÞ E½BðxÞU * Þ dx > > > > L > > : 2 A · dU T PU 2 U~ d 0 ¼ 0 U, U* satisfying the initial and final conditions: ( _ U ð0Þ ¼ U 0 Uð0Þ ¼ U_ 0 _ ðTÞ ¼ 0 U * ðTÞ ¼ 0 U* with the following notations: 2 6 6 6 6 U~ d ¼ 6 6 6 6 4

u~ d j0

3

2

f~d j0

3

7 6 7 6 0 7 7 7 6 7 6 . 7 7 6 7 and F~ d ¼ 6 .. 7 7 7 7 6 7 7 6 7 0 5 6 0 7 5 4 u~ d jL f~d jL 0 .. .

and P is the operator which extracts the value of U at x ¼ 0 and x ¼ L :

ð12Þ


493

2

EC 22,5/6

6 6 6 6 PU ¼ 6 6 6 6 4

494

U1

3

7 7 7 7 7 7 7 0 7 5 U Ne 0 .. .

Let us denote: ½M ¼

Z

L

½FðxÞT r½FðxÞ dx

and

0

½K ¼

Z

L

½BðxÞT E½BðxÞ dx 0

The system to be solved takes the following matrix expression: "

U€ € U*

"

# þ

M 21 K 2AM 21 P

#" # U M 2 1 K þ B1 P M 21 K

U*

2 ¼4

M 21 F~ d 2AM 21 PU~ d

3 5

5. Resolution of the basic problem in time As previously stated, the main numerical difficulty when solving the basic problem comes from the final conditions that U* must fulfill. Various approaches were studied in order to deal with this problem. The first family of methods contains the methods which are global in time. One constructs a global system over space and time, which enables one to take into account both the initial and the final conditions. This can be done by assembling over time the relations among the quantities at the various time steps using a time integration scheme, e.g. a Newmark scheme. Another possibility is to use a finite element formulation over time as in Constantinescu et al. (2003). These methods were applied to our problem, but their application to 3D problems would lead to the resolution of large linear systems. Furthermore, their extension to the nonlinear case is not straightforward and leads to highly nonlocal problems in time due to the time dependency of nonlinear constitutive laws. The second family of methods contains the methods which remain local in time. These, after some transformation of the system, enables one to use an incremental scheme. Three methods were tested. The first method consists in considering the system as a coupled system between U and U*. Then, the direct and retrograde equations are solved iteratively one after the other. However, such a method diverges because of the presence in the coupled problem of exponentially increasing functions of time in both the direct and the retrograde directions, which must be linked to a resonance phenomenon because the two coupled equations have the same eigenvalues. The other two methods are presented below and are, respectively, based on the transition matrix and on the algebraic Riccati equation associated with the system.

5.1 A transition matrix method In order to determine the value of the final conditions given the initial conditions, one introduces the following matrix: 2 3 2 3 M 21 K M 21 K þ B1 P M 21 F~ d 6 7 6 7 ½M a ¼ 24 5; ½S a ¼ 4 Q 5 Q 2AM 21 U~ d 2AM 21 M 21 K 2 and

½X ¼ 4

U U*

3 5

Therefore, the system can be written as: X€ ¼ M a X þ S a with the initial and final conditions: 8 < U ð0Þ ¼ U 0 ; : U * ðTÞ ¼ 0; where

" ½X ¼

_ Uð0Þ ¼ U_ 0 _ ðTÞ ¼ 0 U*

U

ð13Þ

ð14Þ

#

U*

ð15Þ

The above system, whose unknown is X, is solved using a time integration scheme. Then, the transition matrix which connects the initial and final values of X is deduced. The final step is the partial inversion of this matrix in order to deduce the initial values satisfied by U* from the final values. For example, let us consider an explicit integration scheme. The following recurrence holds: " # " # X nþ1 Xn ð16Þ ¼ Ka _ þ Rn X_ nþ1 Xn with 8 2 3 2 > Dt · Id Id þ Dt2 M a > > > 6 7

> K a ¼ 4 Dt > 5 2 > Dt 2 Dt 2 > 2M þ M M Id þ a a > a 2 2 2 > < 2 3 Dt 2 > > 2 S an > > 6 7

> > R ¼ > > n 4 Dt Id þ Dt 2 M a S an þ Dt S an þ1 5 > 2 2 2 > : Using the notation

ð17Þ


495

"

EC 22,5/6

Tn ¼

# Xn ; X_ n

one obtains:

496

T n ¼ K an21 T 1 þ

n X

K i21 a Rn2i ;

ð18Þ

i¼1

by assuming that: R0¼ 0 Then, with: M ¼ K Na t 21

and RT ¼

Nt X

K i21 a RN t 2i

ð19Þ

i¼1

The following relation between the initial and final values holds: 3 2 3 2 U Nt U1 6 _ 7 " 7 " #6 # 6 UN t 7 RT 1 M 11 M 12 6 U_ 1 7 7 6 7 6 6 * 7¼ ; 7þ 6 6 U Nt 7 RT 2 M 21 M 22 6 U *1 7 7 6 5 4 4 _* 5 UN t U_ *1

ð20Þ

where " M¼

M 11

M 12

M 21

M 22

#

It follows that: 2

3 02 * 3 1 " # U Nt U1 U 1* 4 5 ¼ M 2i @ 4 5 2 M 21 2 RT 2 A 22 U_ * U* U_ * 1

Nt

ð21Þ

1

_ at t ¼ 0: This relation enables one to determine U* and U* One advantage of this method is that it can be extended to the nonlinear case since it can be viewed as an iterative method based on the gradient of the initial conditions with respect to the final conditions. However, this formulation has a drawback which is again due to the presence of exponentially increasing functions of time in the theoretical solutions. Consequently, the associated numerical problem is very poorly conditioned, as shown in Figure 4. Furthermore, a small error in the assumption of the initial conditions would lead to a large error at the final step and, therefore, to erroneous solution fields in the case of a large time domain. Therefore, the method is not satisfactory when the time domain is large.


497

Figure 4. Conditioning of M22 depending on Nt for an explicit and implicit integration scheme

5.2 The Riccati approachs Thus, in order to deal with tests of arbitrary duration, we sought another method, namely the Ricatti method (Lions, 1968; Anderson and Moore, 1989), which is classically used in control theory for slightly different types of problems. The first step consists in formulating the problem (equation 1) in the finite element form. Z T 2 b 2 1 a ðU 2 V ÞT KðU 2 V Þ þ Pu U 2 U~ d þ Pf F 2 F~ d dt min 2 2 2 U ;V ;F 0 € M UþKV ¼F

where the notations M, K, U, F, Pu U, Pf F are defined in Section 4, and V is such that: s ¼ E½BðxÞV : Then, the problem is transformed into a first-order problem: Z T 1 T 1 ~ T ~ e Re þ ðUd 2 CqÞ QðUd 2 CqÞ dt min ð22Þ q;e 2 2 0 q_ ¼AqþGeþBF~ d

with the following notations: " # " # 8 V 2U U > > > e¼ q¼ _ ; ; > > PF 2 F~ d > U > > > > " # " # > > 0 Id 0 < A¼ ; B¼ ; 2M 21 K 0 M 21 PT > > > > " # " # > > K 0 > 0 0 > > > G¼ ; R¼ ; > > 0 B · Id 2 2M 21 K M 21 PT :

u ¼ ½F~ d ; C¼ P

0 ;

Q ¼ A · Id 2

EC 22,5/6

This minimization under constraints is carried out through the introduction of a Lagrange multiplier L. Z T

1 T 1 e Re þ 1 T Q1 2 Lð2_q þ Aq þ Ge þ BuÞ dt Lðq; e; LÞ ¼ 2 2 0

498

The stationarity of the Lagrangian function LðdL ¼ 0Þ leads to a differential system in time and in space which connects the expected fields and the multiplier L. Owing to the equilibrium constraint, the formulation yields final conditions on the Lagrange multiplier. Therefore, one must solve the following coupled problem: 8 e ¼ R 21 G T L > > < q_ ¼ Aq þ GR 21 G T L þ BF~ d ð23Þ > > T T T :L _ ¼ C QCq 2 A L 2 C QU~ d with qð0Þ ¼ q0 with q0 given LðT f Þ ¼ 0 where T f is the final time The idea behind the Riccati method is to introduce two new unknowns, K and d, in order to uncouple the direct and adjoint problems by setting LðtÞ ¼ 2KðtÞ · qðtÞ þ dðtÞ: The problem becomes: 8 q_ 2 ½A þ GR 21 G T Kq 2 GR 21 G T d 2 BF~ d ¼ 0 > > < d_ þ ½KGR 21 G T þ A T d þ KBF~ d þ C T QU~ d ¼ 0 ð24Þ > > : K_ þ KA þ A T K 2 KGR 21 G T K þ C T QC ¼ 0 with

8 qð0Þ ¼ 0 > > < dðTÞ ¼ 0 > > : KðTÞ ¼ 0

The equation (24) is obtained by solving successively the third equation, called the Riccati equation, yielding K, the second one yielding d, and finally the first one yielding q. One can then deduce L from the other solution fields. The first step is therefore to solve the Riccati equation. Two methods are presented here, depending on whether or not the Riccati matrix is assumed to be constant. 5.2.1 Method based on the Riccati differential equation. The Riccati differential equation, that has to be solved, is the following: (_ K þ KA þ A T K 2 KGR 21 G T K þ C T QC ¼ 0 ð25Þ KðTÞ ¼ 0 The majority of methods to solve this equation (Anderson and Moore, 1989) is based on the eigenvalues of the Hamiltonian matrix defined from equation (23):

" H¼

A

GR 21 G T

C T QC

2A T

#


In our problem, this matrix cannot be diagonalized. Thus one applied a numerical method which is based on a homographic scheme as proposed in (Dubois and Saidi, 2000). First, one has to write the equation (25) on the retrograde time: t ¼ T 2 t 8 > < dK 2 KA 2 A T K þ KGR 21 G T K 2 C T QC ¼ 0 dt ð26Þ > : Kð0Þ ¼ 0 Then, one introduces a strictly positive real number m, in such a way that the real matrix ½mI 2 A 2 A T is definite and positive. One introduces too a definite positive matrix M ¼ ð1=2ÞmI 2 A: The matrix A is decomposed into two parts: A ¼ A þ þ A 2 with, ( þ 1 A ¼ 2 mI A2 ¼ M Then one defines a numerical scheme for equation (26): 1 1 ðK iþ1 2 K i Þ þ ðK i GR 21 G T K iþ1 þ K iþ1 GR 21 G T X i Þ þ ðM T K iþ1 þ K iþ1 M Þ Dt 2 T ð27Þ ¼ mK i þ C QC One obtains Lyapunov equation with K iþ1 as unknown: S Tiþ1 K iþ1 þ K iþ1 S iþ1 ¼ Y iþ1

ð28Þ

with, 8 1 Dt > < S iþ1 ¼ I þ GR 21 G T K i þ DtM 2 2 > : Y iþ1 ¼ K i þ mDtK i þ DtC T QC Equation (28) is solved with the function lyap in Matlab. Once the Riccati differential equation is solved, one can deduce d and q from equation (24) using a time scheme. The obtained solution differs from the exact one only because of the use of a time scheme. As a matter of fact, the estimated error for E ¼ E 0 with exact measurements is about 3 £ 102 5, which can be considered negligible with respect to the mean value of the error on the range ½E 0 =2; 2E 0 (Figure 7). The theoretical value of the error being equal to 0 in that case, this result can be considered as numericaly satisfactory. The main drawback of this method is its numerical cost, because the Lyapunov equation (28) has to be solved at each time step. Therefore, it is used here in order to estimate the quality of the following method to solve the basic problem.

499

EC 22,5/6

5.2.2 Methods based on the algebraic Riccati equation. In order to avoid the drawback of the solving of the Riccati differential equation, a standard hypothesis is to neglect the transient part of it. One has therefore to solve the stationary part of it, the so-called algebraic Riccati equation, obtained by assuming K_ ¼ 0 in equation (25). KA þ A T K 2 KGR 21 G T K þ C T QC ¼ 0

500

ð29Þ

Such an assumption relies on the fact that the solution of equaiton (25) converges to the solution of equation (29) over the time, when the time period is long enough. Equation (29) can be solved by means of the linear quadratic regulator (LQR) method (Anderson and Moore, 1989). This approach yields erroneous results for times close to the final time T. The time interval, where the constant Riccati matrix assumption is not valid, is estimated by comparing the solution by this approach to the one by the approach based on the Riccati differential equation. In order to do that, we introduce the distance between the two solutions: Z L ex 2 U ri 2 U const dx ð30Þ erðtÞ ¼ ri 0

U ex ri

is the field of the approach based on the Riccati differential equation; and where the field of the constant Riccati matrix approach. U const ri Let us first consider the solving of the basic problem in the simple case of an elastic bar with unperturbed measurements and an initial guess of the elastic modulus E ¼ 2E 0 : The whole time interval considered involves ten forward-backward waves, which leads to a diverging solution when one uses the transition matrix method alone. Figure 5 shows, as expected, that the erroneous results of the constant Riccati matrix method are localized near the final time step. We should mention that similar results can be obtained for much longer durations.

Figure 5. Error of the constant Riccati matrix approach and the combined one by comparing to the one based on the Riccati differential equation

5.2.3 Combined Riccati/transition matrix method. In order to correct the error resulting from the hypothesis of a constant Riccati matrix, it is possible to use the matrix transition approach defined in the previous section. In fact, the error is concentrated near the final time and the Ricatti approach needs to be corrected only over a short period of time ½T 2 t; T: In the previous example, it turns out that the method needs to be corrected over a period which corresponds roughly to one round trip of the wave in the bar. The efficiency of this coupling will be pointed out in the following section on the illustration of the method.


501

6. Illustration 6.1 Description of the method proposed by Rota The first method tested was inspired by Rota (1994). It consists of using the experimental data to define two auxiliary problems: one with prescribed displacements and the other with prescribed forces. These are well-posed problems which can be solved numerically. They yield solution fields uKA and uDA (KA: kinematically admissible, DA: dynamically admissible) which depend on the Young’s modulus E chosen. Then, a distance between the two simulations based on their solution fields is defined and used to quantify the consistency of E with respect to the boundary conditions. This distance is equal to zero (i.e. perfect consistency) if the two calculations yield the same solution fields. The identification is performed by minimizing this distance with respect to E. Among other possible distances, one can choose the distance based on the constitutive relation (31): Z TZ L ðsDA 2 sKA Þð1DA 2 1KA Þ dx dt ð31Þ eðuKA ; uDA Þ ¼ 0

0

where s is the stress and 1 the strain. Figure 6 shows the cost function which needs to be minimized with respect to the relative Young’s modulus (E/E0) for perturbations consisting of high-frequency sines of various magnitudes. The conclusions are the same for other types of perturbations. In the absence of perturbation, the minimum is obtained for E ¼ E 0 and the method

Figure 6. Cost function at various perturbation levels – Rota’s method

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502

leads to a proper identification. However, for large perturbations, the curve has no minimum in the interval considered, which would lead to highly erroneous identification results. This is due to the fact that the two calculations are very different, not because of the material’s parameters, but because of the perturbations on the boundary conditions which introduce artificial inconsistencies between the Young’s modulus and the boundary conditions. This method can be found to fail starting at a level of perturbation lower than 10 percent. 6.2 Illustration of the proposed method In order to illustrate its robustness, the proposed method was applied to the numerical example presented in Section 2 and compared to the previous method, which is a reference in terms of well-controlled experiments. The identification curves for short- and long-duration loading are plotted in Figure 7. Our method appears to be valid for perturbations up to 40 percent. In order to demonstrate the advantage of the combined Riccati approach, we present an example of intermediate-duration loading in Figure 8. For T ¼ 3t0 (t0 being the time of a round trip of the wave in the bar), the identification curves of the transition matrix approach diverge (Figure 8(a)), while those of the constant Riccati approach alone are not accurate enough because the duration is too short (Figure 8(b)). Conversely, the combined Riccati-transition matrix approach leads to sensible results (Figure 8(c)). In comparison with Rota’s method, one can show that moderate perturbations have very little effect on the identification curves. Furthermore, our method yields the correct identification of the Young’s modulus even in the case of large perturbations. The method appears to be very robust with respect to perturbations on the measurements. 7. Conclusion The objective of this paper was to find a solution to the problem of the identification of material parameters from dynamic tests with measurement uncertainties. The method

Figure 7. Cost function of the proposed method at various perturbation levels


503

Figure 8. Identification curves with different approaches for T ¼ 3 t0 loading duration

we propose is based on the use of the error in constitutive relation for identification problems in the framework of transient dynamics. By adhering strictly to the basic idea which consists of partitioning quantities into a reliable set and an unreliable set, we ended up with a nonclassical dynamic problem. The main drawback of this approach is that it leads to nonstandard wave propagation problems whose resolution requires specific methods. A first attempt at solving these numerical problems was based on what is known as the matrix transition method. Unfortunately, this method appears satisfactory only for short-duration tests. Therefore, in this paper, we used a transition matrix method for short-duration problems only because of the presence of an exponentially increasing function of time in the theoretical solution which causes this solution to diverge for large time domains. A method based on the algebraic Riccati equation and applicable to long-duration tests was proposed and developed as in Formosa et al. (2002). By coupling the two methods into what we called the combined Riccati/transition matrix approach, we were able to address loading cases of arbitrary duration. This method worked well on the example we treated.

EC 22,5/6

When solving the nonstandard wave propagation problems, our approach has the great advantage of being robust with respect to measurement uncertainties, at least in the examples which we considered. The next steps will consist of adapting this method first to the case of damage, then to more complex structures.

504

References Allix, O., Feissel, P. and The´venet, P. (2003), “A delay-damage meso modeling of laminates under dynamic loading: basic aspects and identification issues”, Computers and Structures, Vol. 81 No. 12, pp. 1177-91. Anderson, B.D.O. and Moore, J.B. (1989), Optimal Control: Linear Quadratic Methods, Prentice-Hall, New York, NY. Andrieux, S. and Voldoire, F. (1995), “Stress identification in steam generator tubes from profile measurements”, Nuclear Engineering and Design, Vol. 158 Nos 2/3, pp. 417-27. Chouaki, A., Ladeve`ze, P. and Proslier, L. (1998), “Updating structural dynamics models with emphasis on the damping properties”, AIAA, Vol. 36 No. 6, pp. 1094-9. Constantinescu, A., Ivaldi, D. and Stolz, C. (2003), “Identification du chargement thermique transitoire par controˆle optimal”, actes du sixie`me colloque national en calcul des structures, Vol. 1, pp. 185-92. Corigliano, A. and Mariani, S. (2003), “The extended Kalman filter for model identification in impact dynamics”, in Onãte, E. and Owen, D.R.J. (Eds), Complas VII, Barcelona. Deraemaeker, A., Ladeve`ze, P. and Leconte, Ph. (2002), “Reduced bases for model updating in structural dynamics based on constitutive relation error”, Computer Methods in Applied Mechanics and Engineering, Vol. 191 Nos 21-22, pp. 2427-44. Dubois, F. and Saidi, A. (2000), “Unconditionnaly stable scheme for Riccati equation”, ESAIM: Proceedings, 8, pp. 39-52. Feissel, P. (2003), “Vers une strate´gie d’identification en dynamique rapide pour des donneés incertaines”, Thesis, LMT-ENS Cachan, Cachan. Formosa, F., Abou-Kandil, H. and Reynier, M. (2002), “Updating of smart structures models using piezoelectric materials”, paper presented at 3rd World Conference on Structural Control IASC, April, p. 2002. Kalman, R.E. (1960), “A new approach to linear filtering and prediction problems”, Trans. ASME J. Basic Engineering, Vol. 82, pp. 35-45. Ladeve`ze, P. (1998), “A modelling error estimator for dynamical structural model updating”, Advances in Adaptive Computational Methods in Mechanics. Lions, J.L. (1968), Controˆle optimal de syste`mes gouverne´s par des e´quations aux de´riveés partielles, DUNOD, Paris. Rota, L. (1994), “An inverse approach for identification of dynamic constitutive equations”, paper presented at the International Symposium on Inverse Problems, A.A. Balkema, Rotterdam. Tikhonov, A.N. and Arsenin, V.Y. (1977), Solutions of Ill-Posed Problems, Winston and Sons, Washington, DC. Further reading Barthe, D., Ladeve`ze, P. and Deraemaeker, A. (2003), “Validation of dynamics models with uncertainties based on the constitutive relation error”, in Onãte, E. and Owen, D.R.J. (Eds), Complas VII, Barcelona.



Constrained finite rotations in dynamics of shells and Newmark implicit time-stepping schemes Bosˇtjan Brank Faculty of Civil and Geodetic Engineering, University of Ljubljana, Ljubljana, Slovenia

Said Mamouri

Constrained finite rotations in dynamics 505 Received August 2004 Revised January 2005 Accepted January 2005

1900 Route des cretes les collines de Sophia Bat E1-BP 152, 06903 Sophia Antipolis, France, and

Adnan Ibrahimbegovic´ Ecole Normale Supe´rieure de Cachan, Cachan, France Abstract Purpose – Aims to address the issues pertaining to dynamics of constrained finite rotations as a follow-up from previous considerations in statics. Design/methodology/approach – A conceptual approach is taken. Findings – In this work the corresponding version of the Newmark time-stepping schemes for the dynamics of smooth shells employing constrained finite rotations is developed. Different possibilities to choose the constrained rotation parameters are discussed, with the special attention given to the preferred choice of the incremental rotation vector. Originality/value – The pertinent details of consistent linearization, rotation updates and illustrative numerical simulations are supplied. Keywords Shell structures, Dynamics, Numerical analysis Paper type Research paper

1. Introduction In this work, we address the issues pertaining to dynamics of constrained finite rotations as a follow-up from the previous considerations in statics (Ibrahimbegovic´ et al., 2001). The present considerations are of direct interest for nonlinear dynamics of smooth shells where, according to the classical shell theory (Naghdi, 1972), one can eliminate the rotation component around the shell-director and retain only two rotation parameters. Rotations in classical shell models are therefore unrestricted in size, but constrained in the space in the direction of the shell-director. There exists a number of possibilities as a choice for the rotation parameters of this kind (Bu¨chter and Ramm, 1992; Betsch et al., 1998; Brank and Ibrahimbegovic´, 2001) and related works. Among them, we believe, the prominent role is played by the incremental rotation vector (Ibrahimbegovic´, 1997a) as the most suitable parameter for the standard incremental solution strategy as well as for the construction of the time-stepping schemes. The latter is examined in more detail in This work was supported by the Slovenia-France research collaboration PROTEUS and by the Ministry of Education, Science and Sport of Slovenia.


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506

this work in the context of Newmark implicit time-stepping schemes. A number of illustrative examples show a very satisfying performance of the proposed schemes. For the discretization in space a four-noded isoparametric shell finite element with continuum-consistent interpolations is used. This kind of interpolations allow one to treat shell finite rotations exclusively at the element nodes. The outline of the paper is as follows. In Section 2 we recall governing equations for the dynamics of stress resultant geometrically exact shells. Equations are given in terms of the shell-director vector and no particular rotation parameters are yet associated with its rotation. In Section 3 we discuss different possibilities which can be chosen for the constrained rotation parameters, and relate those parameters with the shell-director. We focus on the vector-like rotation parameters associated with the rotation vector and on iterative rotation parameters associated with the exponential mapping formula. In Section 4 we introduce constrained incremental rotation vector, which is further examined in more detail in Section 5 where we develop implicit Newmark time integration schemes for classical smooth shells with constrained rotations. The consistent linearization aspects are addressed in Section 6 and some linearized matrices are provided in Appendix. Three numerical examples are presented in Section 7. 2. Geometrically exact shell model; dynamic formulation 2.1 Basic kinematic relations In this work, we consider a shell as a single director Cosserat surface (Naghdi, 1972, Simo and Fox, 1989; Ibrahimbegovic´, 1997b). This is a two-dimensional surface (typically chosen as the shell mid-surface) with a so-called director vector attributed to each point of the surface. The position vector for a particular point in a shell deformed configuration is assumed to be defined by the following expression

w ðj 1 ; j 2 ; tÞ þ ztðj 1 ; j 2 ; tÞ;

ðj 1 ; j 2 Þ [ A;

z [ F :¼ {h 2 ; h þ }

ð1Þ

where A defines the domain of the mid-surface parametrization and h þ 2 h 2 is the thickness of the shell. In equation (1), j 1 and j 2 are convected curvilinear coordinates and z is through the thickness coordinate. Parameter t defines time with the interval of interest defined as t [ ½t0 ¼ 0; T: It is assumed that the director vector t remains a unit vector in any deformed configuration, i.e. ktk ¼ 1 ð2Þ It follows from equation (1) that all deformed configurations of the shell are completely determined by pairs (w , t). In other words, the configuration space, denoted by C, is then defined by C :¼ {ðw ; tÞ : A ! R3 £ S 2 jw j›Aw ¼ w; t j›At ¼ t} 2

ð3Þ

where S is a unit sphere (a space of all vectors of unit length), while ›Aw and ›At are parts of the boundary where the displacement and the director field are specified, respectively. At each point of the mid-surface in the deformed configuration (at time t. t0) we define the convected frame as ð4Þ {t 1 ; t 2 ; t 3 } :¼ {w ;1 ; w ;2 ; t} where

› ð+Þ: ›j a It is considered that the vector basis in equation (4) is obtained by mapping of the frame constructed in the shell reference configuration at time t ¼ t 0 {g 1 ; g 2 ; g 3 } :¼ {w 0;1 ; w 0;2 ; t 0 } ð5Þ where ðw0 ; t 0 Þ [ C0 define the initial positions of the mid-surface and the director field, respectively. Without loss of generality we choose the set of normal coordinates by assuming that the director vector is initially orthogonal to the shell mid-surface, i.e. ð6Þ ðg 1 £ g 2 Þ £ g 3 ¼ 0 The above concepts are shown in Figure 1. Remark 1. The shell model, completely equivalent to the Cosserat surface model, can be derived from the 3d continuum by employing standard assumptions on the distribution of the displacement field in the shell body and by approximating the terms describing the shell geometry (Bu¨chter and Ramm, 1992; Bas¸ar and Ding, 1990; Brank et al., 1997, 1998). ð+Þ;a ;

Constrained finite rotations in dynamics 507

2.2 Strain measures We can define the relative deformation gradient at w0 as a linear map F : T w0 C ! T w C; which is mapping vector fields defined on the reference mid-surface onto the vector fields defined on the current mid-surface. F is given as F ¼ ti ^ gi ¼ ta ^ ga þ t3 ^ g3 ð7Þ a b where g are the dual base vectors defined through the relationship g a · g ¼ dba ; where dba is the Kronecker symbol. We note that due to the choice of normal coordinates g 3¼ g3. By making use of the relative deformation gradient, the relative Lagrangian strain measures for the shell may be defined as 1 E m;s ¼ ½F T F 2 1 2

ð8Þ

Figure 1. Base vectors in reference and current configurations

EC 22,5/6

where 1 is a unit tensor relative to the reference configuration. It follows from equationsd (7) and (8) that the components of the strain tensor E m,s may be written as

508

1 1ab ¼ ðw ;a · w ;b 2 w 0;a · w 0;b Þ 2

ð9Þ

21a3 ¼ ga ¼ w ;a · t 2 w 0;a · t 0

ð10Þ

where 1ab and ga are the classical expressions for the membrane and the shear strains (Naghdi, 1972). The Lagrangian strain measures for the bending strains can be developed by making use of the director gradient. By defining tensor G ¼ t ;a ^ g a we may write Eb ¼ FTG 2 B where B ¼ ðg a · g;b Þg a ^ g b is the curvature tensor of the shell surface at the reference configuration. The components of the strain tensor E b are then

kab ¼ w ;a · t ;b 2 w 0;a · t 0;b

ð11Þ

which are the classical expressions for the shell bending strains (Naghdi, 1972). 2.3 Constitutive relations With the strain measures (9)-(11) we can define the strain energy function by the following expression Fð1ab ; ga ; kab ; · Þ

ð12Þ

where an empty slot in equation (12) indicates that such a strain energy function should also depend upon the first and the second fundamental forms of the mid-surface. The effective stress resultants can be obtained as the corresponding partial derivatives of the strain energy, i.e. n ab ¼

›F ; ›1ab

qa ¼

›F ; ›g a

m ab ¼

›F ›kab

ð13Þ

The simplest properly invariant constitutive relations for shells are obtained by postulating small strains and isotropic shells, neglecting variation of metrics through the shell thickness and assuming a quadratic form of the strain energy n ab ¼

Eh H abgd 1gd ; 1 2 n2

m ab ¼

Eh 3 H abgd kgd ; 12ð1 2 n 2 Þ

q a ¼ kGhg ab gb ð14Þ

where E is Young’s modulus, n the Poisson ratio, k the shear correction factor, n ab, q a and m ab are effective stress resultants and couples, g ab ¼ g a · g b , and 1 H abgd ¼ ng ab g gd þ ð1 2 nÞðg ag g bd þ g ad g bg Þ 2

ð15Þ

Remark 2. In order to simplify finite element implementation one usually introduces local Cartesian coordinates at the numerical integration points which simplify tensor (15), since for orthonormal frames g ab ¼ d ab :

2.4 Kinetic energy Kinetic energy for a shell is given by the expression Z 1 _ þ I r t_ · t_ dA ½Ar w_ · w K¼ 2 A

ð16Þ

where Ar and Ir are the surface mass density and the rotation inertia of the shell-director, respectively, at the initial configuration. From the three-dimensional point of view Ar and Ir may be interpreted as the zeroth and the second moment of the mass density about the mid-surface (Brank et al., 1998), given by Ar ¼

Z

hþ

rm dz < rh

ð17Þ

h2

Ir ¼

Z

hþ

rz 2 m dz < r h2

h3 12

ð18Þ

pffiffiffiffiffi pffiffiffi Here r is a three-dimensional mass density. Geometric term m ¼ g * = g is defined pffiffiffi with g* (3d Jacobian determinant of the map w 0þ zt0) and g ¼ det ½g a · g b ; g ¼ kg 1 £ g 2 k; which is the measure of the mid-surface at the reference configuration. For thin shells it may be assumed that m ¼ 1: Remark 3. We note in passing that the moment of inertia relative to the shell mid-surface is simply a scalar value, which is certainly easier to deal with than the inertia tensor which is found in 3d beam dynamics (Ibrahimbegovic´ and Al Mikdad, 1998; Al Mikdad and Ibrahimbegovic´, 1997). 2.5 Strong and weak forms of balance equations Two-dimensional momentum balance equations for geometrically exact shells take the following form (Simo and Fox, 1989) 1 pffiffiffi € ¼ pffiffiffi an a ;a þn Arw a

ð19Þ

1 pffiffiffi I rðt £ t€Þ ¼ pffiffiffi am a ;a þw ;a £ n a þ m a

ð20Þ

where n a are stress resultants (the components of n a are n ab and n a3¼ q a), m a are a ab a3 are the applied stress couples (the components of m p ffiffiare ffi m and m ¼ 0Þ; n and m forces and couples, respectively, a ¼ kt 1 £ t 2 k is the surface Jacobian at the deformed configuration, a ¼ det½t a · t b : In equation (20) Ar and I r are the surface mass density and the rotation inertia of the shell-director, respectively, at the deformed configuration. In following D’Alambert principle, we can derive the corresponding weak form of the balance equations by introducing the “inertia” forces, multiplying the equations by test functions dw and dt and making use of the integration by parts (Hughes, 1987). The weak form of the equations of motion may then be written with respect to the reference configuration as


EC 22,5/6

510

dPðw ; t; dw ; dtÞ ¼

Z

€ · dw þ I r t€ · dtÞ dA þ dPstat ¼ dPdyn þ dPstat ¼ 0 ð21Þ ðAr w A

dPstat is the weak form of static equilibrium equations Z 1 n ab ðdw ;a · w ;b þ w ;a · dw ;b Þ þ q a ðdw ;a · t þ w ;a · dtÞ dPstat ¼ 2 A Z þ m ab ðdw ;a · t ;b þ w ;a · dt ;b Þ dA 2 dPext

ð22Þ

A

where dPext is the virtual work of the applied forces, A is mid-surface area at the initial configuration, while n ab, q a and m ab are effective stress resultants and couples. dw is an arbitrary test function which represents virtual displacements of the mid-surface, and dt is the test function for shell-director vector, which must satisfy the orthogonality condition dt · t ¼ 0

ð23Þ

emerging from the shell-director incompressibility assumption (2). In solving the finite element approximation of equation (21) by the Newton incremental-iterative method, one makes use of the linearized form given as Z 1 ›2 F 1 ðdw ;a · w ;b þ w ;a · dw ;b Þ ðDw ;g · w ;d þ w ;g ·Dw ;d Þ: Lin½dPð· Þ ¼ ½dPð ·Þ þ ›1ab ›1gd 2 A 2 þðdw ;a · t þ w ;a ·dtÞ

›2 F ðDw ;b ·t þ w ;b ·DtÞ ›ga ›gb

›2 F þðdw ;a · t ;b þ w ;a ·dt ;b Þ ðDw ;g · t ;d þ w ;g ·Dt ;d Þ dA ›kab ›kgd Z 1 þ n ab ðdw ;a ·Dw ;b þ Dw ;a ·dw ;b Þ þ q a ðdw ;a · Dt þ Dw ;a ·dtÞ 2 A þm

þ

ab

Z

a

ðdw ;a ·Dt ;b þ Dw ;a ·dt ;b Þ þ q ðw ;a ·DdtÞ þ m

ab

ð24Þ

ðw ;a ·Ddt ;b Þ dA

€ · dw þ I r Dt€ ·dt þ I r t€ ·DdtÞdA ¼ 0 ðAr Dw A

where Dw is the incremental displacement vector and Dt incremental director vector, constrained by Dt · t ¼ 0. Note, that Ddw is zero, while Ddt is generally not. The integrals given in equation (24) provide basis for computing material, geometric and mass part of symmetric tangent operator. For discussion on symmetry of tangent operator for shells with finite rotations see Suetake et al. (2001).

2.6 Finite element interpolations The spatial discretization of the problem is performed by an isoparametric finite element approximation. The following interpolations, referred to as the continuum-consistent, are used to approximate the shell geometry at any time t [ ½0; T

w ðj 1 ; j 2 ; tÞ ¼

nen X

N a ðj 1 ; j 2 Þw a ðtÞ;

tðj 1 ; j 2 ; tÞ ¼

a¼1

nen X

N a ðj 1 ; j 2 Þt a ðtÞ

ð25Þ

a¼1

where N a ðj 1 ; j 2 Þ are the corresponding shape functions for a shell element with nen nodes, ðj 1 ; j 2 Þ [ ½21; 1 £ ½21; 1 and ð+Þa are the corresponding nodal values. The virtual and incremental quantities at any time t [ ½0; T are interpolated in the same manner dw ð j 1 ; j 2 Þ ¼

nen X

N a ðj 1 ; j 2 Þdw a ;

dtðj 1 ; j 2 Þ ¼

a¼1

Dw ðj 1 ; j 2 Þ ¼

nen X

1

2

Ddtðj ; j Þ ¼

N a ðj 1 ; j 2 Þdt a

ð26Þ

a¼1

N a ðj 1 ; j 2 ÞDw a ;

Dtðj 1 ; j 2 Þ ¼

a¼1 nen X

nen X

nen X

N a ðj 1 ; j 2 ÞDt a

a¼1

ð27Þ 1

2

N a ðj ; j ÞDdt a

a¼1

The derivatives of the above interpolated functions with respect to time t, or coordinates j 1 and j 2 may be easily obtained. The interpolation of the transverse shear fields is based on the assumed strain method as suggested by Dvorkin and Bathe (1984); the variational justification of that method may be found in Simo and Hughes (1986). The assumed strain interpolations can be directly expressed by judiciously chosen displacement and rotation parameters, so that without lost of generality we can in the following present the problem as the displacement-based. Remark 4. In the present formulation we interpolate the components of the shell director vector. Discussion on more elaborate interpolation – where rotation parameters are directly interpolated – may be found in Bas¸ar and Kintzel (2001) for shells and (Zupan and Saje, 2001) for beams. 3. Shell director motion in terms of constrained finite rotation parameters and their time derivatives 3.1 Shell-director position Let us consider a motion of the director vector attached to a particular point of the shell mid-surface. Since the director vector is of unit length (equation (2)) its position at any time t [ ½t 0 ; T may be given by the finite rotation of the base vector e ; e 3 ¼ 0; 0; 1T (see Figure 1), thus having t ¼ Le;

t0 ¼ L0e

ð28Þ

where t0 is the reference position of the director vector at the initial time t ¼ t 0 and L 0 ¼ L jt¼t0 :


EC 22,5/6

In equation (28) the director vector position is described by an orthogonal tensor L which is an element of SO(3) group SOð3Þ ¼ {L : R3 ! R3 jLL T ¼ I; det L ¼ 1}

512

ð29Þ

The principal difficulty introduced by describing the shell-director position by the rotation tensor L is due to the fact that SO(3) is not a linear space. Therefore the issues of theoretical formulation, consistent linearization and update procedure become more complex. One can simplify the problem by exploiting the relations between the rotation tensor and its eigenvector. Namely, any rotation tensor L is associated with a skew-symmetric tensor Q through an exponential mapping L ¼ exp½Q and any Q further possesses its axial vector q, referred to as the rotation vector, such that Qb¼ q £ b, ;b [ R3 : Given the rotation vector q, one can reconstruct the corresponding orthogonal tensor L by using either the exponential mapping or the Rodrigues formula; see Simo and Fox (1989) and references therein ~ qÞ ¼ exp ½Q L ¼ Lð ~ qÞ ¼ cos qI þ L ¼ Lð

ð30Þ

sin q 1 2 cos q Qþ q^q q q2

ð31Þ

~ aÞ denotes rotation defined by rotation vector a and expressed either in form Here Lð (30) or (31). It will be clear from the context which form is used. L may be viewed as a composition of two orthogonal tensors, one taking us from the fixed global basis to the local basis in the reference configuration and another taking us further to the current configuration. One may rewrite the equation of shell-director motion, (equation (28)) as ~ qÞe t ¼ Le ¼ L 0 Lð

ð32Þ

~ uÞL 0 e ¼ Lð ~ uÞt 0 t ¼ Le ¼ Lð

ð33Þ

We refer to q as the material rotation vector, while u represents its spatial counterpart[1]; motivation for this notation is explained in Ibrahimbegovic´ et al. (1995). Their mutual relationship follows from equations (32) and (33) as ~ uÞL 0 ) u ¼ L 0 q ~ qÞ ¼ LT Lð Lð 0

ð34Þ

~ uÞ ¼ L 0 Lð ~ qÞLT ) q ¼ LT u Lð 0 0

ð35Þ

~ qÞ and Lð ~ uÞ rotate e and t0, respectively, without drilling Let us assume that Lð ~ uÞ are constrained by requiring that the ~ qÞ and Lð rotation. In other words, rotations Lð rotation vector component along the rotated vector plays no role in the theory. Rotation vector is therefore perpendicular to both initial and rotated vector Equation ð32Þ ) q · e ¼ 0; Equation ð33Þ ) u · t 0 ¼ 0;

q · LT0 t ¼ 0

ð36Þ

u·t ¼ 0

ð37Þ

From equations (28) and (34) it follows immediately that the constraints (36)1 and (37)1 are equivalent, as well as (36)2 and (37)2. It also follows from (36), that the material

rotational vector always lies in the plane defined by fixed vectors e1 and e2, or, in interpretation of (37)1, in the plane tangential to the mid-surface at the reference configuration.

Constrained finite rotations in dynamics

3.2 Shell-director velocity Velocity of the shell-director vector t at any time t [ ½t 0 ; T may be formally obtained as _t ¼ d t t ¼ d ½L t e ð38Þ dt t¼0 dtt¼0

513

We will derive explicit expressions of the shell-director velocity by defining a one parameter family of shell-director vectors t ! t t in four different ways, depending on the definition of a one parameter family of constrained rotations t ! L t . 3.2.1 Multiplicative update of constrained rotations. Let us multiply L from the right _ to obtain hand side with an orthogonal tensor exp ½tC _ t t ¼ L t e ¼ L exp ½t Ce

ð39Þ

By analogy, multiplication of L from the left hand side with an orthogonal tensor _ gives exp ½tW _ Le ¼ exp ½t Wt _ t t ¼ L t e ¼ exp ½tW

ð40Þ

_ are skew-symmetric tensors defining material and spatial angular _ and W where C velocities of the shell-director motion, respectively. From equations (39) and (40) we can express time derivative of L as _ ¼ d _ _ ¼ WL L L t ¼ LC ð41Þ dt t¼0 _ _ and W Mutual relationship between C _ _ ¼ L T WL; C

_ ¼ LCL _ T W

ð42Þ

further leads to the corresponding relationship between their axial vectors (Ibrahimbegovic´ et al., 1995) _ c_ ¼ L T w;

_ ¼ Lc_ w

ð43Þ

By using equation (38) we obtain from equations (39) and (40) the following _ expressions for the shell-director velocity in terms of c_ and w t_ ¼ Lðc_ £ eÞ

ð44Þ

_ £ Le ¼ w _ £t t_ ¼ w

ð45Þ

Exploiting the analogy between equations (32) and (39), we may conclude (equation (36)) that the axial vector c_ is constrained as c_ · e ¼ 0 ð46Þ while, in the same manner, we may observe from equations (33) and (40) that the axial _ is constrained as vector w

EC 22,5/6

_ ·t ¼ 0 w

ð47Þ

3.2.2 Additive update of constrained rotations. The third and the fourth possibility for the construction of tt in equation (38) exploit the additive update of rotational parameters. By using the material rotation vector q we have ~ q þ t q_ Þe t t ¼ L t e ¼ L 0 Lð

514

ð48Þ

while with its spatial counterpart u we obtain ~ u þ tuÞL ~ u þ t u_Þt 0 _ 0 e ¼ Lð t t ¼ Lð

ð49Þ

Time derivative of equations (48) and (49) leads to the expressions of the shell-director velocity in terms of the rotational vectors q and u and their velocities: t_ ¼ L 0 A q q_

ð50Þ

t_ ¼ A u u_

ð51Þ

sin q q cos q 2 sin q A ¼ 2 ðe ^ q þ EÞ þ ðq £ eÞ ^ q q q3

ð52Þ

with q

where q ¼ kqk and Eb 5 e £ b; ;b [ R3 , and sin u u cos u 2 sin u ðt 0 ^ u þ T 0 Þ þ ð u £ t Þ ^ u Au ¼ 2 0 u u3

ð53Þ

where u ¼ kuk and T 0 b ¼ t 0 £ b; ;b [ R3 . A q and A u are obtained by using equation (31) in (48) and (49), and by observing that ðq ^ qÞe ¼ 0 and ðu ^ uÞt 0 ¼ 0 which follows from equations (36) and (37). Some further details of the derivation of the above tensors may be found in Brank et al. (1997) and Ibrahimbegovic´ et al. (2001). _ and u_ comes from the time derivative of equations (34) The relation between q and (35)

u_ ¼ L 0 q_ ;

q_ ¼ LT0 u_

ð54Þ

while by using equations (50), (51) and (54) we may obtain A u ¼ L 0 A q LT0

ð55Þ

It also follows trivially from equations (36) and (37) that

q_ · e ¼ 0 , u_ · t 0 ¼ 0

ð56Þ

u_ · t ¼ 0;

ð57Þ

u · t_ ¼ 0

Further developments and commutative diagrams providing relations between the _ q_ and u_) are presented in Brank and parameters introduced above (c_ , w, Ibrahimbegovic´ (2001).

Remark 5. By multiplying t_ with t, we have t_ · t ¼ 0; the condition which follows from the incompressibility assumption (2) of the shell-director vector. It can be shown that the right hand sides of equations (44), (45), (50) and (51) satisfy this condition. 3.3 Shell-director acceleration Having concluded that we have four different possibilities to express velocity of the shell-director by using either exponential mapping formula or Rodrigues formula, we proceed by deriving expressions for the shell-director acceleration. By taking the time derivative of equations (44), (45), (50) and (51) we have _ c_ £ eÞ þ Lðc £ eÞ t€ ¼ Lð €

ð58Þ

_ £ t_ þ w € £t t€ ¼ w

ð59Þ

€ t€ ¼ L 0 ½Y q q_ þ A q q

ð60Þ

t€ ¼ Y u u_ þ A u u€

ð61Þ

where Y q is a tensor function of q and q_ obtained by the time derivative of equation (52) Yq ¼ 2 þ

q cos q 2 sin q sin q ðq · q_ Þðe ^ q þ EÞ 2 e ^ q_ 3 q q sin qð3 2 q 2 Þ 2 3q cos q ðq · q_ Þðq £ eÞ ^ q q5

ð62Þ

q cos q 2 sin q _ ½ðq £ eÞ ^ q þ ðq £ eÞ ^ q_ q3 and Y u is a tensor function of u and u_ obtained by the time derivative of equation (53) þ

Yu ¼ 2

u cos u 2 sin u sin u ðu · u_Þðt 0 ^ u þ T 0 Þ 2 t 0 ^ u_ 3 u u

þ

sin uð3 2 u 2 Þ 2 3u cos u ðu · u_Þðu £ t 0 Þ ^ u u5

þ

u cos u 2 sin u _ ½ðu £ t 0 Þ ^ u þ ðu £ t 0 Þ ^ u_ u3

ð63Þ

Some details on the derivation of tensors Y q and Y u may be found in Brank et al. _ 5 LC _ (1997). Expressions (58) and (59) can be further elaborated. By noting that L (equation (41)) we obtain from equation (58) t€ ¼ L c_ £ ðc_ £ eÞ þ c€ £ e ð64Þ ¼ L c_ ðc_ · eÞ 2 eðc_ · c_ Þ þ c€ £ e ¼ Lð2ec_ 2 þ c€ £ eÞ


EC 22,5/6

_ 3 t in expression (59) and applying similar procedure where c_ ¼ kc_ k. By using t_ 5 w as in equation (64) we have € £t _ £ ðw _ £ tÞ þ w € £ t ¼ 2tw_ 2 þ w t€ ¼ w

516

ð65Þ

_ where w_ ¼ kwk: Note that the absolute values of material and spatial angular _ velocities are equal, i.e. c_ ¼ w: Let us now check which constraints are acting on the acceleration of rotational parameters. Taking the time derivatives of equation (42) we obtain _L _ T; € ¼ LL € TþL W

€ þL _ TL _ € ¼ LTL C

ð66Þ

€ and C € are both skew-symmetric, It can be verified that angular acceleration tensors W and that their mutual relationship can be written as € ¼ LCL € T; W

€ € ¼ L T WL C

ð67Þ

We can also establish the mutual relationship of their axial vectors as € € ¼ Lc; w

€ ¼ LTw € c

ð68Þ

It follows trivially from equation (46) that

c€ · e ¼ 0

ð69Þ

€ ·t þ w _ · t_ ¼ 0, from which it With the time differentiation of equation (47) we have w _ · t_ ¼ w _ · ðw _ £ tÞ ¼ 0 that further follows by using w € ·t ¼ 0 w

ð70Þ

It also follows from equations (56) and (57) that

q€ · e ¼ 0 , u€ · t 0 ¼ 0 u€ · t ¼ 0;

u_ · t_ ¼ 0;

u · t€ 5 0

ð71Þ ð72Þ

3.4 Two versus three rotational parameters In the coordinate representation of the rotation vector (which can be either material or _ angular spatial object), angular velocity of the shell-director motion ðc_ ; wÞ, € velocity of the rotation vector ðq_ ; u_Þ acceleration of the shell-director motion ðc€ ; wÞ, and acceleration of the rotation vector ðq€ ; u€Þ, one can exploit the constraints presented in the above sections. However, we can only exploit constraints if known vectors (i.e. fixed base vectors or vectors associated with the reference configuration) appear in relations with the rotational parameters and their time derivatives. In other words, the constraints can be exploited to reduce the coordinate representation only with the material objects. The material rotational objects can be presented by two components, while on the other hand, their spatial counterparts have to be expressed by all three components which are not mutually independent.

The coordinate representation of the tensors defined above in Section 3 are therefore either ð3 £ 3Þ matrices for tensors with spatial objects or ð2 £ 3Þ; ð3 £ 2Þ and ð2 £ 2Þ matrices for tensors with material objects. Similarly, the coordinate representation of spatial vectors is of ð3 £ 1Þ form, while of material vectors is of ð2 £ 1Þ form. Material and spatial objects associated with the rotation, velocity and acceleration of the shell-director vector are summarized in Table I.


4. Constrained incremental rotation vector It has been noted (Betsch et al., 1998) and (Ibrahimbegovic´ et al., 2001) that an attractive parametrization of constrained finite rotation with the total rotation vector (q or u) will exhibit the singularity problem whenever its norm reaches a multiple of p. For overcoming this deficiency we introduce in this section an incremental rotation vector which, moreover, is fully consistent with the standard incremental solution scheme for nonlinear problems. Since it maintains additive iterative rotational updates it is also very suitable for optimization problems (Kegl, 2000; Ibrahimbegovic´ and Knopf-Lenoir, 2001). 4.1 Incremental rotation updates The evolution of configuration space variables is obtained by a step-by-step integration scheme. The time interval of interest is partitioned into the number of time steps: 0 , t 1 , · · · , tn , tnþ1 , · · · , T: At the typical time, tn, the values of translational and rotational motion components are denoted as

w n ¼ w ðtn Þ;

t n ¼ tðt n Þ

ð73Þ

where tn is defined via orthogonal tensor Ln¼ L(tn) through relation tn¼ Lne. Let us now substitute total rotation vectors q (material version) and u (spatial version) by the corresponding incremental rotation vectors qnþ 1 and unþ 1, which are reset to zero at the beginning of each solution increment. Without going through a detailed proof we can show that the relations for the position, velocity and acceleration of the shell-director vector given in Section 3 also hold for the corresponding incremental rotation vector, simply by making the following substitutions

q; q_ ; q€ ! qnþ1 ; q_ nþ1 ; q€ nþ1 u; u_ ; u€ ! unþ1 ; u_nþ1 ; u€nþ1 L 0 ; L; t 0 ; t ! L n ; L nþ1 ; t n ; t nþ1 q

q

A ;Y !

ð74Þ

Aqnþ1 ; Yqnþ1

A u ; Y u ! Aunþ1 ; Yunþ1 Material (2 comp.) Constraint Spatial (3 Comp.) Constraint Total rotation vector Angular velocity Angular acceleration Velocity of the total rotation vector Acceleration of the total rotation vector

q _ c € c q_ q€

(36) (46) (69) (56) (71)

u _ w € w u_ u€

(37) (47) (70) (57) (72)

Table I. Parameters of the shell-director motion, their time derivatives and related constraints

EC 22,5/6

In the context of step-by-step integration scheme, the new value of displacement vector at time tnþ 1 is obtained trivially as

w nþ1 ¼ w n þ u nþ1

518

ð75Þ

where u nþ1 ; Dw nþ1 are incremental displacements of the mid-surface point. Obtaining tnþ 1 can be more complicated. Namely, to update the shell-director vector, one first needs to update the orthogonal matrix Ln. Its incremental update can be carried out in terms of the incremental rotation vector as ~ nþ1 Þ ~ unþ1 ÞL n ¼ L n Lðq L nþ1 ¼ Lð

ð76Þ

Considering that Ln is an orthogonal tensor, one may obtain the following relations from equation (76) ~ nþ1 ÞLT ; ~ unþ1 Þ ¼ L n Lðq Lð n

~ unþ1 ÞL n ~ nþ1 Þ ¼ LT Lð Lðq n

ð77Þ

Furthermore, taking into account that a skew-symmetric tensor and the corresponding orthogonal tensor obtained by its exponentiation share the same eigenvectors (Ibrahimbegovic´ et al., 1995) it follows that

unþ1 ¼ L n qnþ1 ;

qnþ1 ¼ LTn unþ1

ð78Þ

Having updated the orthogonal matrix Ln by using equation (76), we may proceed with the evaluation of the shell-director at time tnþ 1. According to equation (76) we have two possibilities in terms of the incremental rotation vector ~ qnþ1 Þe t nþ1 ¼ L n Lð

ð79Þ

~ unþ1 ÞL n e 5 Lð ~ unþ1 Þt n t nþ1 ¼ Lð

ð80Þ

By exploiting similarities of equations (32) and (33) with equations (79) and (80), respectively, we can conclude that qnþ 1 and unþ 1 are subjected to the following constraints (equations (36) and (37))

qnþ1 · e ¼ 0; unþ1 · t n ¼ 0;

qnþ1 · LTn t nþ1 ¼ 0

ð81Þ

unþ1 · t nþ1 ¼ 0

ð82Þ

where, again, constraints (81)1 and (82)1 are equivalent. By using equation (74), the constraints (56) and (71) now turn to be

q_ nþ1 · e ¼ 0;

q€ nþ1 · e ¼ 0

ð83Þ

while the constraints associated with the time derivatives of unþ1 follow from equation (82). 4.2 Iterative rotation updates When an implicit time-stepping scheme is used, the final values of the state variables at time increment [tn, tnþ 1] are established by an iterative procedure carried over the increment. To that end, let superscript (i ) denote the iteration counter.

ð84Þ


is the (i)th contribution to the incremental displacement field. where The iterative values of the shell-director vector are obtained through the corresponding iterative updates of the orthogonal tensor, Lnþ 1, since

519

At each iteration the incremental displacement update is performed in a standard additive fashion as ðiÞ ðiÞ uðiþ1Þ nþ1 ¼ unþ1 þ Dunþ1

DuðiÞ nþ1

ðiþ1Þ tðiþ1Þ nþ1 ¼ Lnþ1 e

ð85Þ

In the iterative update of finite rotations we can choose between the spatial and the material representations. By making use of the material formðiÞ of the incremental ðiÞ ðiÞ _ rotation vector qðiÞ nþ1 and its iterative increment Dqnþ1 ¼ Dt qnþ1 we have

~ qðiÞ þ DqðiÞ L ¼ L Lðiþ1Þ ð86Þ n nþ1 nþ1 nþ1 The rotation update can also be performed with the spatial rotation parameters as

~ uðiÞ þ DuðiÞ L n ¼ L ð87Þ Lðiþ1Þ nþ1 nþ1 nþ1 Comparing equation (86) and (87) the following relations may be obtained ðiÞ DuðiÞ nþ1 ¼ L n Dqnþ1

ð88Þ

Noting that the material incremental rotation vector at iteration (iþ 1) is qðiÞ nþ1 þ ; it follows from equations (81) and (86) that DqðiÞ nþ1 DqðiÞ nþ1 · e ¼ 0

ð89Þ

Similarly, we can conclude from equation (82) and (87) that DuðiÞ nþ1 · t n ¼ 0;

ðiþ1Þ DuðiÞ nþ1 · tnþ1 ¼ 0

ð90Þ

Relations (89) and (90)1 are equivalent. 5. Implicit time integration schemes for constrained rotations 5.1 Description of the problem In the computation dynamics, besides computation of displacements and rotations, one also needs to obtain velocities and accelerations at the chosen instant in the time interval of interest. We use the Newmark family of algorithms for that end; for energy conserving algorithms (Simo and Tarnow, 1994; Brank et al. 1998; Briseghella et al., 2001). Standard implementation is used for computing the translational motion components, and necessary modifications are proposed for computing the components related to the constrained rotation of the shell-director. Considering the typical time interval between tn and tnþ 1 the algorithmic problem _ n ; and acceleration, can be described as: given at time tn displacement, w n, velocity, w w¨n, of translational motion of the shell mid-surface, and shell-director, t n , its constrained rotation, Ln, velocity, t_ n ; and acceleration, t€ n , find such values of w and t at time tnþ 1 that

EC 22,5/6

520

dPdyn jnþ1 þ dPstat jnþ1 ¼ 0

ð91Þ

and update velocities and accelerations of displacements and shell director by using the corresponding Newmark approximations. The update for displacements, constrained rotation tensors and shell-director vectors, which we need when solving equation (91) iteratively by Newton solution procedure, was discussed in the previous section. In this section, we address the remaining ingredients of the problem, namely the update of velocities and accelerations. 5.2 Newmark scheme for displacements For a non-linear dynamics problem with translational degrees of freedom only, the standard implementation of the Newmark algorithm can be used. We compute the velocities and accelerations at time tnþ 1 with _ nþ1 ¼ w

g b2g ðb 2 0:5gÞh _nþ _n u nþ1 þ w w bh b b

ð92Þ

1 1 0:5 2 b €n u nþ1 2 w w_ n 2 2 b bh bh

ð93Þ

€ nþ1 ¼ w

where b and g are free Newmark parameters and h ¼ t nþ1 2 t n is a typical time step. Typical choice for b ¼ 1=4 and g ¼ 1=2 leads to the scheme of second-order accuracy. 5.3 Newmark scheme for constrained finite rotations; material form It was noted in Simo and Vu-Quoc (1988), Ibrahimbegovic´ and Al Mikdad (1998) and Iura and Atluri (1988) in their work on beams that Newmark approximations for angular velocity and acceleration can directly be applied only in the material representation as

g b2g _ ðb 2 0:5gÞh € c_ nþ1 ¼ qnþ1 þ cn þ cn bh b b

ð94Þ

1 1 0:5 2 b € c€ nþ1 ¼ 2 qnþ1 2 c_ n 2 cn b bh bh

ð95Þ

where qnþ 1 is material incremental rotation vector which is zero at tn, while c_ n and c€ n are material angular velocity and material angular acceleration at tn. These approximations make sense geometrically also for shells, since all vectors in equations (94) and (95) are constrained by lying in the R2 plane perpendicular to the fixed base vector e, (equations (81), (46) and (69)), or, in another interpretation, in the plane tangential to the mid-surface at the reference configuration. Shell-director velocity follows from equation (44) t_ nþ1 ¼ L nþ1 ðc_ nþ1 £ eÞ and the shell-director acceleration from equation (64)

2 t€ nþ1 ¼ L nþ1 2ec_ þ c€ nþ1 £ e nþ1

ð96Þ

ð97Þ

~ qnþ1 Þ: By inserting equations (92), (93), (96) and (97) into the where L nþ1 ¼ L n Lð weak form of the balance equation (91), we obtain a system of non-linear equations in incremental displacements unþ 1 and incremental material rotation vector qnþ 1. 5.4 Newmark scheme for constrained finite rotations; spatial form If we multiply expressions (94) and (95) from the left hand side by Ln, and use material-spatial transformations for the rotational objects, we obtain

g b2g ðb 2 0:5gÞh _nþ €n L n c_ nþ1 ¼ unþ1 þ w w bh b b

ð98Þ

1 1 0:5 2 b €n _n2 L n c€ nþ1 ¼ 2 unþ1 2 w w b bh bh

ð99Þ

If we further multiply the left hand side of equations (98) and (99) by identity ~ T ðunþ1 ÞLð ~ unþ1 Þ; we end up with the spatial form of the Newmark approximations L for finite rotations, (Ibrahimbegovic´ and Al Mikdad, 1998) ðb 2 0:5gÞh ~ unþ1 Þ g unþ1 þ b 2 g w _ nþ1 ¼ Lð _nþ €n w w bh b b

€ nþ1 w

1 1 0:5 2 b ~ €n _n2 w ¼ Lðunþ1 Þ unþ1 2 w b bh 2 bh

ð100Þ

ð101Þ

_ and w € at any time instant are constrained by equations (82), (47) where unþ 1, and w and (70), respectively. However, as already mentioned in Section 3.4, those constraints cannot be exploited to reduce the coordinate representation of the spatial form of angular velocity and acceleration. Shell-director velocity at tnþ 1 follows from equation (45) _ nþ1 £ t nþ1 t_ nþ1 ¼ w

ð102Þ

and acceleration of the shell-director vector follows from equation (65) € nþ1 £ t nþ1 2 t nþ1 w_ 2nþ1 t€ nþ1 ¼ w

ð103Þ

Their more elaborate forms are ðb 2 0:5gÞh ~ unþ1 Þ g unþ1 £ t nþ1 þ b 2 g w _ n £ t nþ1 þ € n £ t nþ1 ð104Þ t_ nþ1 ¼ Lð w bh b b and


EC 22,5/6

0:5 2 b ~ unþ1 Þ 1 unþ1 £ t nþ1 2 1 w € _ £ t 2 £ t t€ nþ1 ¼ Lð w n n nþ1 nþ1 b bh 2 bh

g2 ðb 2 gÞ2 ðb 2 0:5gÞ2 2 _ n ·w € n ·w _nþ €n w unþ1 · unþ1 þ h w 2 2 2 b h b b2 ðb 2 gÞg ðb 2 0:5gÞg _nþ €n 2 2t nþ1 unþ1 · w unþ1 · w b 2h b2 ðb 2 gÞðb 2 0:5gÞh _ € w þ · w n n b2

2 t nþ1

522

ð105Þ

The equivalence of equations (96) and (102) and of equations (97) and (103) follows immediately by using the transformation rules between the material and the spatial rotational objects. When inserting equations (92), (93), (102) and (103) into the weak form of the balance equation (91), we obtain a system of non-linear equations in incremental displacements unþ 1 and incremental spatial rotation vector unþ 1. 5.5 Newmark scheme in terms of the shell-director vector Another possibility to obtain the shell-director velocity and acceleration at time tnþ 1 is to use Newmark approximations directly in terms of the shell-director vector time derivatives

g b2g_ b 2 0:5g € ðt nþ1 2 t n Þ þ ht n tn þ t_ nþ1 ¼ bh b b

ð106Þ

1 1 _ 0:5 2 b € t€ nþ1 ¼ 2 ðt nþ1 2 t n Þ 2 tn tn 2 b bh bh

ð107Þ

In order to compare this scheme with the one of the previous section, we insert equations (102) and (103) into equations (104) and (105), respectively. We obtain ~ unþ1 Þ g unþ1 £ t n þ b 2 g t_ n þ ðb 2 0:5gÞh ðt€ n þ t n w_ 2 Þ _t nþ1 ¼ Lð ð108Þ n bh b b ~ unþ1 Þ g unþ1 £ t n 2 1 t_ n 2 0:5 2 b ðt€ n þ t n w_ 2 Þ €t nþ1 ¼ Lð n bh b bh 2 g ðb 2 gÞ2 2 ðb 2 0:5gÞ2 h 2 2 _ € 2 t nþ1 2 2 u2nþ1 þ w w þ n n b h b2 b2 ð109Þ ðb 2 gÞg ðb 2 0:5gÞg _nþ €n 2 2t nþ1 unþ1 · w unþ1 · w b 2h b2 ðb 2 gÞðb 2 0:5gÞh €n _ n·w þ w b2

Some similarities may be noticed between expressions (106) and (108) as well as between expressions (107) and (109); however, further more detailed comparison is not obvious. 6. Linearization aspects 6.1 Linearization of the shell-director motion Let us recall that the dynamic part of the weak form of balance equations at time tnþ 1 is Z dPdyn;nþ1 ¼ ½Ar w€ nþ1 · dw þ I r t€ nþ1 · dt dA ð110Þ A

where the test function dt has to satisfy the algorithmic form of equation (23) dt · t nþ1 ¼ 0

ð111Þ

By exploiting analogy with equations (50) and (51) we may write dt in terms of the incremental rotation vectors (introduced in Section 4 and used for the time discretization in Section 5) as

dt ¼ L n Aqnþ1 dq dt ¼ Aunþ1 du

ð112Þ

where Aqnþ1 and Aunþ1 defined in equations (52) and (53), are now functions of the incremental rotation vectors qnþ 1 and unþ 1, respectively. Equation (112) satisfy condition (111). Linearization of dt with respect to intrinsic rotational variables at time tnþ 1, which are qnþ 1 and unþ 1, is not zero. It can be shown that the following forms can be obtained from equation (112) h q i ~ Ddt · b ¼ dq Y nþ1 Dqnþ1 ð113Þ h u i ~ Ddt · b ¼ du Ynþ1 Dunþ1 ~ q and Y ~ u are again functions of qnþ 1 and unþ 1, respectively. where b [ R3 and Y nþ1 nþ1 ~ q is Full form of Y nþ1 ~ q ¼ q cos q 2 sin q {qnþ1 ^ qnþ1 ½ðL n eÞ · b þ I ½ðL n ðqnþ1 £ eÞÞ · b} Y nþ1 q3 2

sin q I½ðL n eÞ · b q

sin qð3 2 q 2 Þ 2 3q cos q þ {qnþ1 ^ qnþ1 ½ðL n ðqnþ1 £ eÞÞ · b} q5 þ

q cos q 2 sin q {½A þ A T ððL n e 1 Þ · bÞ þ ½B þ B T ððL n e 2 Þ · bÞ} q3

where q ¼ kqnþ1 k and matrices A, B are defined as

ð114Þ


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524

2 A¼4

0

3

0

q1nþ1 q2nþ1

"

5;

B¼

2q1nþ1 2q2nþ1 0

0

# ;

T qnþ1 ¼ q1nþ1 ; q2nþ1 ð115Þ

~ u is given in an analogous way. Some further details may be found in Full form of Y nþ1 Brank et al. (1997). Finally, we note that the linearization of the shell-director at time tnþ 1, namely Dtnþ 1, may be expressed in terms of incremental rotational parameters as (equation (112)) Dt nþ1 ¼ L n Aqnþ1 Dqnþ1

Dt nþ1 ¼ Aunþ1 Dunþ1

ð116Þ

6.2 Linearization of the dynamic part of the weak form of balance equations € nþ1 with The linearization of dPdyn at time tnþ 1 may be obtained by linearization of w respect to unþ 1, by linearization of t€ nþ1 with respect to intrinsic rotational variables at time tnþ 1 and by exploiting one of the relation (113). We can write linearized form of dPdyn as Z DdPdyn;nþ1 ¼ ðAr Dw€ nþ1 · dw þ I r Dt€ nþ1 · dt þ I r t€ nþ1 · DdtÞ dA ð117Þ A

where linearization of the translational part is given trivially d 1 € nþ1 ðw n þ 1Du nþ1 Þ ¼ 2 Du nþ1 € nþ1 ¼ ½w Dw d1 1¼0 bh

ð118Þ

Expressions for Ddt are given in the previous section, while Dt€ nþ1 is derived below. Following equation (97) it may be written in terms of material axial velocity and acceleration as h i 2 Dt€ nþ1 ¼ L n 2Dqnþ1 £ eDc_ nþ1 þ Dqnþ1 £ ðDc€ nþ1 £ eÞ ð119Þ _ € _ þ L nþ1 2 2ðe ^ cnþ1 ÞDcnþ1 þ Dcnþ1 £ e where

g 1 Dqnþ1 ; Dc€ nþ1 ¼ 2 Dqnþ1 ð120Þ bh bh We have expressed Dt€ nþ1 with the linearized form of the material incremental rotation vector Dqnþ 1. When working with spatial objects, we can exploit equation (103) to obtain Dc_ nþ1 ¼

€ nþ1 £ t nþ1 þ w € nþ1 £ Dt nþ1 2 Dt nþ1 w_ 2nþ1 2 t nþ1 2w_ nþ1 Dw_ nþ1 Dt€ nþ1 ¼ Dw ð121Þ € nþ1 £ t nþ1 þ w € nþ1 £ Dt nþ1 2 Dt nþ1 w_ 2nþ1 2 2ðt nþ1 ^ w _ nþ1 ÞDw _ nþ1 ¼ Dw where

_ nþ1 ¼ Dw

g ~ Lðunþ1 ÞDunþ1 ; bh

€ nþ1 ¼ Dw

1 ~ Lðunþ1 ÞDunþ1 bh 2

ð122Þ

~ unþ1 ÞL n ; expression (122) By exploiting relations Dunþ1 ¼ L n Dqnþ1 and L nþ1 ¼ Lð may be also given in terms of the incremental material rotation vector as _ nþ1 ¼ Dw

g L nþ1 Dqnþ1 ; bh

€ nþ1 ¼ Dw

1 L nþ1 Dqnþ1 bh 2

ð123Þ

We may therefore express Dt€ nþ1 with Dunþ 1 or Dqnþ 1. The equivalence of equations (119) and (121) can be shown by some simple manipulations. 7. Numerical simulations 7.1 Rotational parameters and procedures used In this section we present results obtained in numerical simulations. All the computations are carried out by a research version of the computer program FEAP, developed by Professor R. L. Taylor at UC Berkeley (Zienkiewicz and Taylor, 1989). A four-noded isoparametric shell finite element with assumed strain interpolations for transverse strains (Brank et al., 1995, for details) is used to that end. Among the above discussed possible parametrizations of the shell-director motion, we have chosen the formulation based on the incremental material rotation vector. Note that within the interpolation presented in Section 2.7, all possible parametrizations of constrained finite rotations should produce the same results, however they could differ in convergence characteristics and in the range of the allowable shell-director rotation where the solution can be obtained. Comparison of results for different rotation parameters for static loading may be found in Betsch et al. (1998) and Ibrahimbegovic´ et al. (2001). Among the discussed Newmark time-stepping schemes for constrained finite rotations two different Newmark time-stepping schemes were coded. Algorithmic approximation of velocity and acceleration of the shell-director in time is obtained either with the spatial representation procedure described in Section 5.4 (named “version 1”), or with an alternative simpler procedure in terms of the shell-director vector, which is defined in Section 5.5 (named “version 2”). For both versions of Newmark time-stepping schemes the mass matrices are presented in the Appendix. 7.2 Example 1: motion of a beam-like plate The beam of length L ¼ 10; width B ¼ 1 and thickness h ¼ 0:5 is initially lying in rest in the XY plane (see Figure 2). Its material characteristics are: Young’s modulus E ¼ 21; 000; Poisson’s ratio n ¼ 0:2 and mass density r ¼ 1: Mid-surface mass density and inertia term with respect to the mid-surface are Ar ¼ hr ¼ 0:5 and I r ¼ rh 3 =12 ¼ 0:0104; respectively. The beam is subjected to the external forces and moments which are applied at one end of the beam. Force vector is directed 458 from the mid-surface and has the following components with respect to X, Y, Z coordinate system: f ¼ {0; 214:14; 14:14}T : Applied vector of moments is m ¼ { 2 20:00; 0:20; 0}; where its two non-zero components act in the direction of 2X and þ Y axis, respectively. The load is multiplied by a time function defined as: f ðtÞ ¼ t for t [ ½0; 1; f ðtÞ ¼ 2t þ 2 for t [ ½1; 2 and f ðtÞ ¼ 0 for t . 2:


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526 Figure 2. Beam-like plate: geometry and loading data

The response of the beam is calculated up to 4 s with the time step Dt ¼ h ¼ 0:01 s: Displacements and velocities of the point A which at t0 occupies position (0, L, 0) are shown in Figures 3 and 4 where “vertical”, “horizontal” and “out of plane” denote the direction of X, Y and Z coordinate, respectively. A sequence of deformed configurations during the initial 2 s (when the load is applied) is shown in Figure 5, while a sequence of deformed configurations between 2 and 3.6 s (when the beam is moving freely) is shown in Figure 6. The interval between two subsequent plotted configurations is 0.2 s. 7.3 Example 2: motion of a short cylinder This example was considered by Simo and Tarnow (1994) and Brank et al. (1998). Geometry of the short cylinder is defined by radius R ¼ 7:5; height H ¼ 3 and thickness h ¼ 0:02: The material characteristics are: Young’s modulus E ¼ 2 £ 108 ; Poisson’s ratio n ¼ 0:5 and mass density r ¼ 1: Mid-surface mass density and inertia

Figure 3. Beam-like plate: displacement time histories for a point initially lying at (0, L, 0)


Figure 4. Beam-like plate: velocity time histories for a point initially lying at (0, L, 0)

Figure 5. Beam-like plate: deformed shapes; 0 # t # 2 s (plot after every 0.2 s)

Figure 6. Beam-like plate: deformed shapes; 2 # t # 3:6 s (plot after every 0.2 s)

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528

term with respect to the mid-surface are Ar ¼ hr ¼ 0:02 and I r ¼ rh 3 =12 ¼ 6:667 £ 1027 ; respectively. Loading conditions are shown in Figure 7 and Table II. At time t ¼ 0 the initial conditions are prescribed to be zero. The shell is subjected to impulsive loads acting from 0 until 1 s with the peak reached at 0.5 s. Response is calculated up to 10 s with the time step equal to Dt ¼ h ¼ 0:05 s: Evaluation of displacements and velocities of a point which is at t ¼ 0 located at (0, 2 R, 0) is shown in Figures 8 and 9, respectively. A sequence of deformed configurations is shown in Figure 10: at 0.25, 0.75, 1.25 s (first row); 1.75, 2.25, 2.75 s (second row); 3, 3.25, 3.5 s (third row) and 3.75 s.

Figure 7. Short cylinder: geometry

Table II. Short cylinder: loading data

Figure 8. Short cylinder: displacement time histories for a point initially lying at (0, 2 R, 0)

a Nodal loads Time, t p(t)

0 [0, 21, 2 1]T p(t) 0.0 0.0

p/2 [1, 1, 1]T p(t) 0.5 5.0

p [1, 1, 1]T p(t) 1.0 0.0

3p/2 [0, 21, 2 1]T p(t)


Figure 9. Short cylinder: velocity time histories for a point initially lying at (0, 2R, 0)

Figure 10. Short cylinder: deformed configurations

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530

Figure 11. Spherical cup: geometrical and material data

Figure 12. Spherical cup: displacement time histories of the loaded ring

7.4 Example 3: dynamic snap-trough of a spherical cup The geometry of the spherical cup and material characteristics are defined in Figure 11. At the bottom of the cap the displacements in the Z (i.e. vertical) direction are restricted to be zero. The mesh is composed of 32 £ 8 finite elements. Force f ðtÞ ¼ 1:5 pðtÞ; where pðtÞ ¼ t for t [ ½0; 1 and pðtÞ ¼ 1 for t . 1; is applied at each of the 32 nodes around the top hole in 2 Z direction. Response is traced up to 3 s by using the time step Dt ¼ h ¼ 0:01 s: Two different time-integration schemes are used to calculate this example; the first one is described in Section 5.4 and marked as “version 1” in Figure 12, while the second one is described in Section 5.5 and marked as “version 2”. Figure 12 shows displacement in the 2 Z direction of the nodes around the upper hole with respect to the time. It can be seen that the scheme which approximates the rotational objects themselves is much more stable than the scheme which interpolates the time derivatives of the shell-director. The divergence for “version 2” occurs at approximately 4 s, while “version 1” is perfectly stable up to 10 s. Up to the divergence of “version 2”, both schemes give exactly the same results

(see Table III for the convergence characteristics). In Figure 13 a sequence of deformed configurations is presented at 0.05, 0.7, 0.9 s (first row); 1.1, 1.3, 1.5 s (second row); 1.7, 1.9, 2.1 s (third row); 2.3, 2.5, 2.7 s (fourth row) and 2.9 s. 8. Conclusions A detailed development of the parametrization of constrained finite rotations for dynamics of shells is presented. We recognized that the incremental rotation vector parameterization is the most suitable choice for handling the dynamics of shells in the sense of: . being able to avoid singularity problems; . maintaining simple additive iterative updates for rotations; and . being able to construct the “displacement-like” Newmark schemes for rotational degrees of freedom.

Iter. no. 1 2 3

Version 1 Residual norm Energy norm 1.78 £ 101 1.20 £ 102 2 2.95 £ 102 8

3.97 £ 102 1 1.33 £ 102 7 7.13 £ 102 19

Version 2 Residual norm Energy norm 1.78 £ 101 1.20 £ 102 2 2.95 £ 102 8

3.97 £ 102 1 1.33 £ 102 7 7.12 £ 102 19


Table III. Spherical cup. Convergence characteristics at t ¼ 0:12 s

Figure 13. Spherical cup: deformed configurations

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Note 1. In this section q and u represent the total material and the total spatial rotation vector, respectively, which are measured from the reference configuration, while in Section 4 the corresponding incremental rotation vectors, which are reset at the begining of each solution increment, will be introduced.

532 References Al Mikdad, M. and Ibrahimbegovic´, A. (1997), “Dynamique et sche´mes d’inte´gration pour mode`les de poutres geóme´tricament exact”, Revue europeenne des elements finis, Vol. 6, pp. 471-502. Bas¸ar, Y. and Ding, Y. (1990), “Theory and finite element formulation for shell structures undergoing finite rotations”, in Voyiadjis, G.Z. and Karamanlidis, D. (Eds), Advances in the Theory of Plates and Shells, Elsevier, Amsterdam, pp. 3-26. Bas¸ar, Y. and Kintzel, O. (2001), “Finite rotations and large strains in finite element shell analysis”, Comput. Meth. Eng. Sci., Vol. 4, pp. 217-30. Betsch, P., Menzel, A. and Stein, E. (1998), “On the parametrization of finite rotations in computational mechanics. A classification of concepts with application to smooth shells”, Comput. Meth. Appl. Mech. Eng., Vol. 155, pp. 273-305. Brank, B. and Ibrahimbegovic´, A. (2001), “On the relation between different parametrizations of finite rotations for shells”, Eng. Comput., Vol. 18, pp. 950-73. Brank, B., Peric´, D. and Damjanic´, F.B. (1995), “On implementation of a non-linear four-node shell finite element for thin multilayered elastic shells”, Comput. Mech., Vol. 16, pp. 341-59. Brank, B., Peric´, D. and Damjanic´, F.B. (1997), “On large deformations of thin elasto-plastic shells: implementation of a finite rotation model for quadrilateral shell element”, Int. J. Numer. Meth. Eng., Vol. 40, pp. 689-726. Brank, B., Briseghella, L., Tonello, N. and 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. Eng., Vol. 42, pp. 409-42. Briseghella, L., Majorana, C. and Pavan, P. (2001), “A conservative time integration scheme for dynamics of elasto-damaged thin shells”, Comput. Meth. Eng. Sci., Vol. 4, pp. 273-86. Bu¨chter, N. and Ramm, E. (1992), “Shell theory versus degeneration – a comparison in large rotation finite element analysis”, Int. J. Numer. Meth. Eng., Vol. 34, pp. 39-59. Dvorkin, E.N. and Bathe, K.J. (1984), “A continuum mechanics based four-node shell element for general nonlinear analysis”, Eng. Comput., Vol. 1, pp. 77-84. Hughes, T.J.R. (1987), Finite Element Method: Linear Static and Dynamic Analysis, Prentice-Hall, Englewood Cliffs, NJ. Ibrahimbegovic´, A., Frey, F. and Kozˇar, I. (1995), “Computational aspects of vector-like parametrization of three-dimensional finite rotations”, Int. J. Numer. Meth. Eng., Vol. 38, pp. 3653-73. Ibrahimbegovic´, A. (1997a), “On the choice of finite rotation parameters”, Comput. Methods Appl. Mech. Eng., Vol. 149, pp. 49-71. Ibrahimbegovic´, A. (1997b), “Stress resultant geometrically exact shell theory for finite rotations and its Fe implementation”, ASME Appl. Mech. Review, Vol. 50, pp. 199-226. Ibrahimbegovic´, A. and Al Mikdad, M. (1998), “Finite rotations in dynamics of beams and implicit time-stepping schemes”, Int. J. Numer. Meth. Eng., Vol. 41, pp. 781-814.

Ibrahimbegovic´, A. and Knopf-Lenoir, C. (2001), “Shape optimization of elastic structural systems undergoing large rotations: simultaneous solution procedure”, Comput. Meth. Eng. Sci., Vol. 4, pp. 337-44. Ibrahimbegovic´, A., Brank, B. and Courtois, P. (2001), “Stress resultant geometrically exact form of classical shell model and vector-like parametrization of constrained finite rotations”, Int. J. Numer. Meth. Eng., Vol. 52, pp. 1235-52. Iura, M. and Atluri, S.N. (1988), “Dynamic analysis of finitely stretched and rotated three-dimensional space-curved beams”, Computers and Structures, Vol. 29, pp. 875-89. Kegl, M. (2000), “Shape optimal design of structures: an efficient shape representation concept”, Int. J. Numer. Meth. Eng., Vol. 49, pp. 1571-88. Naghdi, P.M. (1972), “The theory of shells and plates”, in Flu¨gge, S. (Ed.), Encyclopedia of Physics, Springer-Verlag, Berlin. Simo, J.C. and Fox, D.D. (1989), “On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization”, Comput. Methods Appl. Mech. Eng., Vol. 72, pp. 267-304. Simo, J.C. and Hughes, T.J.R. (1986), “On the variational foundations of assumed strain methods”, J. Appl. Mech., Vol. 53, pp. 51-4. Simo, J.C. and Vu-Quoc, L. (1988), “On the dynamics in space of rods undergoing large motions: a geometrically exact approach”, Comp. Meth. Appl. Mech. Eng., Vol. 66, pp. 125-61. Simo, J.C. and Tarnow, N. (1994), “A new energy and momentum conserving algorithm for the non-linear dynamics of shells”, Int. J. Numer. Meth. Eng., Vol. 37, pp. 2527-49. Suetake, Y., Iura, M. and Atluri, S.N. (2001), “Variational formulation and symmetric tangent operator for shells with finite rotation field”, Comput. Meth. Eng. Sci., Vol. 4, pp. 329-36. Zienkiewicz, O.C. and Taylor, R.L. (1989), The Finite Element Method: Basic Formulation and Linear Problem, McGraw-Hill, New York, NY. Zupan, D. and Saje, M. (2001), “A new finite element formulation of three-dimensional beam theory based on interpolation of curvature”, Comput. Meth. Eng. Sci., Vol. 4, pp. 301-18. Further reading Simo, J.C., Fox, D.D. and Rifai, M.S. (1990), “On a stress resultant geometrically exact shell model. Part III: the computational aspects of the nonlinear theory”, Comput. Meth. Appl. Mech. Eng., Vol. 79, pp. 21-70. Simo, J.C., Rifai, M.S. and Fox, D.D. (1992), “On stress resultant geometrically exact shell model. Part VI. Conserving algorithms for non-linear dynamics”, Int. J. Numer. Meth. Eng., Vol. 34, pp. 117-64. Appendix. Mass matrix Mass matrix follows from the linearization of the dynamic part of the weak form of balance equation (117). Incremental material rotation vector and spatial Newmark scheme for rotations Let us first derive mass matrix for “version 1” of time-interpolation of constrained rotations. _ nþ1 and Dw € nþ1 with relations (123), By using equation (121) for Dt€ nþ1 ; and expressing Dw we obtain 21 2g € nþ1 2 w_ 2 I L n Aq Dqnþ1 ð124Þ _ Dt€ nþ1 ¼ ½t T 2 ^ w L þ W nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 bh 2 bh


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534

€ nþ1 are skew-symmetric matrices, i.e. T nþ1 ¼ t nþ1 £ b and where Tnþ 1 and W € nþ1 ¼ w € nþ1 £ b, b [ R3 . By defining variation, dt, and linearization of variation, Ddt, of W the shell-director vector in terms of incremental material rotation vector, see equations (112) and (113), and by using continuum-consistent interpolations of Section 2.7, we obtain the sub-matrix MIJ of the finite element mass matrix of the following form 3 2 A 0 3£2 N I N J bhr2 I 7 6 q;I T T 7 6 7 6 LIn £ N I N J I r Anþ1 7 6 7 6 21 2 g J J J 7 6 0 2£3 J _ t T 2 ^ w L þ ð125Þ M IJ ¼ 6 nþ1 nþ1 nþ1 nþ1 7 2 bh bh 7 6

7 6 2 7 6 J q;J €J _J I Ln A þ W 2 w 7 6 nþ1 nþ1 nþ1 5 4 ~ I I dJ N IY r I

;J ˜ I is a (2 £ 2) where I is a (3 £ 3) unit matrix, Aqnþ1 is a (3 £ 2) matrix, dJI is Kronecker delta and Y matrix defined as (equation (114))

0 2 B 6 6 ~ I ¼ q cos q 2sin q B Y B26 3 @ 4 q 2

q1;I nþ1

2

2;I q1;I nþ1 qnþ1

2;I q1;I nþ1 qnþ1

q2;I nþ1

3

7 T 7 I €I t t :

2 7 5 n nþ1

3

2 1 1 n oT sin q I 1;I 4 t€nþ1 A 2 þ4 5LIn q2;I nþ1 ;2qnþ1 ;0 q 0 01 10

3 0 5 tI T t€I n nþ1 1

2 3

2 q1;I q1;I q2;I nþ1 nþ1 nþ1 7 n oT sin qð32 q 2 Þ23q cos q 6 I 6 7 I 2;I 1;I t€nþ1 þ q ;2 q ;0 L 6 7 n nþ1 nþ1

2 5 4 1;I 2;I q5 qnþ1 qnþ1 q2;I nþ1 02 þ

0

q1;I nþ1

ð126Þ

3

6 7 I q cos q 2sin q B T €I B6 7 @4 1;I 2;I 2 5Ln {1;0;0} tnþ1 q3 qnþ1 2 qnþ1

1 2 3

2 2 q1;I q2;I nþ1 nþ1 C 6 7 I 7L {0;1;0}T t€I C 26 nþ1 A 4 5 n 2;I qnþ1 0 n oT 2;I 1;I 2;I I : where q ¼ kqInþ1 k; and q1;I nþ1 and qnþ1 are two non-zero components of qnþ1 ¼ qnþ1 ; qnþ1 Incremental material rotation vector and Newmark scheme in terms of shell-director For “version 2” of time-interpolation of the shell-director motion we exploit expression (107). By its linearization with respect to the shell-director at time tnþ 1, we obtain simple expression for Dt€ nþ1 ; namely

Dt€ nþ1 ¼


1 Dt nþ1 ; bh 2

which can be further elaborated by using equation (116) to obtain Dt€ nþ1 ¼

1 L n Aqnþ1 Dqnþ1 bh 2

ð127Þ

Expressing variation, dt, and linearization of variation, Ddt, of the shell-director vector with equations (112) and (113), respectively, and by using continuum-consistent interpolations of Section 2.7, we obtain the following form of the sub-matrix of mass matrix 2 3 A N I N J bhr2 I 0 3£2 6 7 7 h iT h iT ð128Þ M IJ ¼ 6 4 q;I J 5 I J q;J 1 I J I ~I 0 2£3 N N I A L L A þ N I d Y r r 2 n n nþ1 nþ1 I bh which is of simpler form that the one presented in equation (125) for “version 1”.

535



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Saint-Venant multi-surface plasticity model in strain space and in stress resultants

536

J-B. Colliat, A. Ibrahimbegovic´ and L. Davenne

Received September 2004 Revised December 2004 Accepted January 2005

Ecole Normale Supe´rieure de Cachan Laboratoire de Mećanique et Technologie (LMT), Wilson, Cachan, France Abstract Purpose – To present a new constitutive model for capturing inelastic behavior of brittle materials. Design/methodology/approach – The multi-surface plasticity theory is employed to describe the damage-induced mechanisms. An original feature in that respect concerns the multi-surface criterion which limits the principle values of elastic strains, which is equivalent to Saint-Venant plasticity model. The latter allows to represent the damage both in tension and in compression. Findings – Provides a quite realistic description of cracking phenomena in brittle materials, with a very few parameters, leading to a very useful tool for analyzing practical engineering problems. Originality/value – The model is recast in terms of stress resultants and employed within a flat shell elements in order to provide a very efficient tool for analysis of cellular structures. Moreover, a detailed description of the numerical implementation is given. Keywords Modelling, Plasticity, Physical properties of materials, Stress (materials) Paper type Literature review


1. Introduction With the present increase of the computational power, one can seek to provide a more realistic representation of the physical nature of damage mechanisms. The multiscale analysis (Markovic and Ibrahimbegovic, 2004) thus becomes a very interesting option for combining the large structural model with a more detailed (small) representation. In Ibrahimbegovic et al. (2005) we have described how to build a multiscale model for cellular structures, such as masonry wall built of hollow bricks, submitted to sustained duration of high temperature. The flat shell elements are used for modeling different panel barriers which allows to construct a detailed representation of each brick. With the model of this kind, we get a possibility of refined representation of local damage mechanisms in each block, but the number of degrees of freedom increases very quickly. The key challenge is thus to provide a very robust implementation of the local constitutive model. One such model is proposed in this work, within the framework of multi-surface plasticity, with the goal to describe simple fracture mechanism under maximum stress value, similar to Rankine criterion (Pearce and Bicánic´, 1997). The original feature of the model is to relate damage to maximum value of elastic strain, such as for St-Venant plasticity model (Saint-Venant De, 1855). The latter allows to take into account both This work was supported by the French Ministry of Research and CTTB. This support is gratefully acknowledged.

cracking in tension, perpendicular to loading, and in compression, parallel to the loading direction, as observed in simple traction and compression tests. This difference is also accounted for when further describing the softening phenomena, namely, both the fracture energy in tension and in compression can be introduced in the list of model parameters and thus provide a more reliable description of fracture mechanisms. A careful consideration of the implementation details ensures a very robust performance of the proposed model in numerical computations. The outline of the paper is as follows. In Section 2 we provide the main ingredients of the proposed multi-surface plasticity model developed within the standard thermodynamics framework. A special care in that sense is dedicated to elaborating loading/unloading conditions, which remain quite subtle for a multi-surface plasticity model. In Section 3, a generalization of this multi-surface plasticity model applicable in the framework of plates and shells, which is cast directly in terms of stress resultants is provided. A detailed description of the numerical implementation is given in Section 4. In Section 5, numerical results for several numerical simulations carried out to illustrate different model features are given. Concluding remarks are stated in Section 6. 2. Continuum formulation In this section we first present the mechanical part of the model within the two-dimensional thermodynamics framework. The main novelty concerns the choice of elastic domain, which is written in the context of multi-surface plasticity. We then show the results for a couple of simple numerical examples illustrating the most important features of this model, before turning to its implementation in terms of stress resultants. 2.1 Thermodynamics framework In order to retrieve the most important feature of brittle materials like clay, that is to say a different behavior in tension and compression cases, the two-dimensional model shown here is set as the Saint-Venant model. The latter is very much alike the Rankine model, but with the key difference to define the elastic limit with respect to the principal mechanical strains, which corresponds to the experimentally observed behavior of such materials. From the thermodynamics point of view, the free energy is written, 1 c ð1; 1p ; jÞ ¼ ð1 2 1p Þ : C : ð1 2 1p Þ þ Hðj; 1 2 1p Þ 2 ~

ð1Þ

where 1e ¼ 1 2 1p is the elastic strain defined in accordance with the classical decomposition of the total strain excluding thermal effects and j the hardening/softening variable for the isotropic case. In the equation (1), a coupling between the strain/stress state and hardening/softening process is introduced in the expression of the corresponding energy Hðj; 1 2 1p Þ: This explicit expression for hardening potentials will be given according to the softening law in (10), which allows to enter explicitly the desired value of fracture energy either in tension or compression. According to this expression for strain energy, the state laws are:

s¼

›c ›H ¼ C : ð1 2 1p Þ þ p ›ð1 2 1 Þ ~ ›ð1 2 1p Þ

ð2aÞ

Saint-Venant multi-surface plasticity model 537

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q¼2

›c ¼ 2kðj; 1 2 1p Þ ! q_ ¼ 2D · j_ 2 U : ð1_ 2 1_ p Þ ›j ~

ð2bÞ

where D ¼ ›k=›j is the classical hardening modulus and U ¼ ›k=1 2 1p ~ is a third-order tensor. Invoking the principle of maximum plastic dissipation (Lubliner, 1990), Dp ¼ s : 1_ p þ q · j_ $ 0

MaxjFi ¼0 {Dp } , Maxjg_ i Minjs {Lp } ,

8 m ›L p ›D p X ›Fi > > ¼2 þ g_ i ¼0 > > > › s › s ›s < i¼1 m > ›L p ›D p X ›Fi > > ¼ 2 þ ¼0 g_ i > > ›q ›q : ›q i¼1

ð3Þ

ð4Þ

where, Lp ðs; q; g_ i Þ ¼ 2Dp ðs; qÞ þ

m X

g_ i Fi ðs; qÞ

ð5Þ

i¼1

one can further obtain the flow rules, 1_ p ¼

m X

g_ i

›Fi with Fi ¼ 0 ›s

ð6aÞ

g_ i

›F i with ›q

ð6bÞ

i¼1

j_ ¼

m X i¼1

Fi ¼ 0

where the elastic domain is defined by a set of m $ 1 yield functions, Fi ðs; qÞ , 0 intersecting in a possibly non-smooth fashion. 2.2 Elastic domain We note with respect to the last result that in the present two-dimensional case, the elastic domain is defined by two independent surfaces Fi ð1; 1p ; jÞ; 1 # i # 2 which are defined directly in the space of state variables 1; 1p and j: An alternative representation can be given considering the spectral decomposition of strain tensor 1e ¼

II X

1ek nk ^nk with

1eI $ 1eII ;

k¼I

where the two surfaces are expressed as: G e e F1 ð1 ; jÞ :¼ 2 K þ 1I 2 1y 2 kðjÞ # 0 3

ð7aÞ

G e F2 ð1e ; jÞ :¼ 2 K þ 1II 2 1y 2 kðjÞ # 0 3

ð7bÞ

The plastic deformation occurs with respect to the criterion which is expressed in principal mechanical strains. This criterion is usually referred to as Saint-Venant criterion (Saint-Venant De, 1855). By defining the criterion in strain space, we can provide the main advantage of this model in its ability to reproduce failure for both tension and compression stress states. One can also recover the standard format of the yield criterion in the stress space, which is more efficient in numerical implementation. Namely, by employing the state laws in equation (2), we write the two surfaces in the stress space Fi ðs; qÞ ¼ Fi ð1; 1p ; jÞ according to, F1 ðs; qÞ :¼

K þ 4G=3 K 2 2G=3 sI 2 sII 2 ðsy 2 qÞ # 0 2G 2G

F2 ðs; qÞ :¼ 2

K 2 2G=3 K þ 4G=3 sI þ sII 2 ðsy 2 qÞ # 0 2G 2G


ð8aÞ

ð8bÞ

In Figure 1 we show the principal axis representation of this two-dimensional criterion in the case of perfect plasticity. The two surfaces are therefore, simply represented by straight lines. We should note that F1 $ F2 ; so that the second surface can never be the only one active (Figure 1). In pure tension mode, the limit of the elastic domain is, F1 ðsI Þ :¼

2G sI 2 sy # 0 K þ 4G 3

ð9Þ

and in pure compression: F1 ðsII Þ :¼ 2

2G sII 2 sy # 0 K 2 2G 3

ð10Þ

Figure 1. Elastic domain in principle stress space – 2D case

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The unsymmetrical feature of such material is so reproduced with a very few number of parameters (only three in perfect plasticity case). The only loading case which is not limited is the bi-axial compression along the two axes such as produced by hydrostatic pressure. 2.3 Hardening law In this approach, the hardening is supposed to be isotropic, which allows us to use only a single variable with: j ¼ j1; and the associated flux q ¼ q1: The latter leads to the second state law in a simplified form: q ¼ 2kðj; 1e Þ

ð11Þ

Actually, since the compressive and the tension failure mechanisms are reproduced according to the same fracture mode driven by the principal tensile strains, the second flow rule (6) has been chosen in order to take into account the influence of stress/strain state on the post-peak behavior. The hardening law can then be written: sy j q ¼ 2kðj; 1e Þ ¼ sy 1 2 exp 2 Gf ð1e Þ

ð12Þ

We indicate in equation (12) that the fracture energy is supposed to change continuously from a specified value in tension Gt to compression Gc (Figure 2) according to: Gf ¼

Gc þ Gt Gc 2 G t 2 tan hðbtr½1e Þ 2 2

ð13Þ

where b is a parameter to be chosen to set a more or less rapid transition. From the physical point of view, this assumption on imposing different values for Gt and Gc is related to the meaning of the fracture energy, as the amount of energy needed to create a crack per square meter. Namely, even if the two failure modes in tension and compression stress states are captured with the same failure mechanism pertaining to the principal tensile strain, the number of cracks created in those two cases is completely different. This is shown in Figure 3 representing the crack pattern in simple tension and the one in simple compression test, leading to quite different dissipated energy.

Figure 2. Influence of strain state on fracture energy – 2D case

2.4 Loading/unloading conditions and geometrical interpretation Considering multi-surface plasticity case defined by a set of admissible surfaces {Fi ¼ 0}; it has been shown by Simo et al. (1988) that the loading conditions in strain space will take the form: given any 1; _ if ;i;

›F i : C : 1_ , 0 then ›s ~

1_ p ¼ 0;

q_ ¼ 0

ð14aÞ

if ’i;

›Fi : C : 1_ . 0 then ›s ~

1_ p – 0;

q_ – 0

ð14bÞ


According to the usual terminology of computational plasticity in application to multi-surface plasticity context, a surface is said to be active in contributing to internal variable evolution if g_ i . 0: Consequently, given any 1; _ we can show that:

g_ i . 0 t

›Fi : C : 1_ . 0 ›s ~

1#i#m

ð15Þ

In other words, for a particular surface i, ›Fi =›s : C : 1_ . 0: does not necessarily ~ could finally have an active imply that the surface remains active. In the opposite, one surface i for which initial guess was not showing it, i.e. ›Fi =›s : C : 1_ , 0: Consequently, the key point of the methodology in multi-surface~ plasticity context consists in determining the set of all active surfaces (Hofstetter et al., 1993) for CAP model and Pearce and Bicánic´ (1997) for Rankine plasticity criterion) and then solving all the plastic admissibility constraint equations simultaneously. In order to illustrate more clearly this main point related to the model proposed herein, we consider in detail the loading condition (14), the case of perfect plasticity. In strain space, we first choose the curvilinear system basis defined by, gi ¼

›F i ›s

i [ ½1; . . . ; 2

ð16Þ

equipped with the scalar product defined by C: Clearly this basis {g i } is not an orthogonal one according to this scalar product,~ thus leading to distinguish between covariant and contravariant components and introduce the metric tensor gij written as: g ij ¼ kgi ; gj l ¼ gi : C : gj

~

ð17Þ

Figure 3. Failure modes in both compression and tension

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According to the first flow rule (6a), by assuming a plastic step departing from the yield surface which would not change the elastic deformation for the case of perfect plasticity, X 1_ ¼ 1_ p :¼ g_ i g i ð18Þ i

542

showing that the plastic multipliers g_ i can be seen as the contravariant components of 1_ in the basis {g i }: With respect to this interpretation, we denote as, 8 9 < = X ›Fi g_ i . 0; ;i Gþ ¼ 1_ ¼ 1_ p [ S; 1_ p ¼ g_ i ð19Þ : ; ›s i the cone for which all plastic multipliers are strictly positive. On the other hand, the consistency condition can be written as, X X ›F i _ i ¼ ›Fi : s_ ¼ ›Fi : C : 1_ 2 g_ j gj : C : g i ¼ : C : 1_ 2 g_ j gij ¼ 0 F ›s ›s ~ › s ~ ~ j j

ð20Þ

or X ›Fi : C : 1_ ¼ g_ j g ij ¼ g_i ›s ~ j

ð21Þ

According to this last result, the ›s Fi : C : 1_ can be interpreted as the covariant ~ we also denote the cone Mþ as: components of 1_ in the basis {gi }: Consequently, ›Fi 3 þ M ¼ 1_ [ S j;i; : C : 1_ . 0 ð22Þ ›s ~ We note in passing that the complementary cone M 2 implies the elastic loading, which can be written as: ›Fi M 2 ¼ 1_ [ S3 j;i; : C : 1_ # 0 ð23Þ ›s ~ According to these definitions, it should be noted that the result g_ i . 0 t ›Fi =›s : C : 1_ . 0 can be geometrically interpreted as the non-matching of M þ and Gþ ; see ~ Figure 4 for the general case and Figure 5 for the present model. The particular feature of the proposed model which deserves a special care in handling concerns the fact that the second surface F2 could be active, while ›F2 =›s : C : 1_ , 0: ~ can be written as: More precisely, the basis vector of the proposed model, g1 ¼

›F1 K þ 4G=3 K 2 2G=3 ¼ n1 ^ n1 2 n2 ^ n2 2G 2G ›s

ð24aÞ

g2 ¼

›F 1 K 2 2G=3 K þ 4G=3 ¼2 n1 ^ n1 þ n2 ^ n2 2G 2G ›s

ð24bÞ

Saint-Venant multi-surface plasticity model 543 Figure 4. Loading conditions in stress space – 2D case

Figure 5. Developed model loading conditions in stress space – 2D case

One can see from Figure 5 that the cone Gþ contains the cone M þ . Consequently, all the strain states residing in an area denoted by (A) in Figure 5 would lead to g_ 2 . 0; even if ›s F2 : C : 1_ # 0: For that reason, when considering the numerical ~ implementation, three different cases have to be taken into account. In either the first or the second one, the trial elastic state is directly furnishing the set of active surfaces (Figure 5). For the last case, where the trial state belongs to the area denoted by (A), both surfaces are active and we have to compute the corresponding values of both plastic multipliers. 3. Stress resultant multi-surface plasticity model The objective here is to show how the two-dimensional case of the multi-surface model can be incorporated into a flat shell element which proved to be very useful in constructing the models for cellular structures (Ibrahimbegovic et al., 2005). The shell element we use in this work combines the discrete Kirchhoff quadrilateral plate with a membrane element including the drilling degrees of freedom (Ibrahimbegovic et al., 1990, 1993). We denote here as eðx1 ; x2 Þ the in-plane membrane strain tensor and kðx1 ; x2 Þ the curvature tensor which can be written in a condensed form of a set of generalized strains as

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e

x¼

!

k

:

Normal membrane forces tensor N and bending moment tensor M are also written in a condensed form as N

p¼

M

! :

We can then write the constitutive tensor for stress resultant form as, 2

hC :

6 ~ J¼4 0: ~

3

0: h3 12 C

~

:

7 5;

C ¼ ðK 2 2G=3Þ¼1 ^ ¼1 þ 2GI

~

ð25Þ

where h is the shell thickness. The last is used to recast in the stress resultant form the standard thermodynamical framework with the free energy, 1 c ðx; xp ; jÞ ¼ ðx 2 xp Þ · J ðx 2 xp Þ þ Hðj; x 2 xp Þ 2 ~

ð26Þ

which provides the state laws given as,

p¼

q¼2

›c ›H ¼ J · ðx 2 xp Þ þ ›ðx 2 xp Þ › ð x 2 xp Þ ~

›c ¼ 2k j; x 2 xp ! q_ ¼ 2D · j_ 2 U : x_ 2 x_ p ›j ~

ð27aÞ

ð27bÞ

where D ¼ ›k=›j and U ¼ ›k=x 2 xp have the same meaning as in equation (2b). In ~ resultants, the second term in equation (18) is neglected. the computation of stress By assuming that the same constitutive equations remain valid in the case of inelastic process, we can obtain the plastic dissipation as: Dp ¼ pðx_ 2 x_ p Þ þ q · j_ $ 0

ð28Þ

The elastic domain can be defined directly in terms of stress resultants by a set of m $ 1 yield functions Fi ðp; qÞ; intersecting in a possibly non-smooth fashion. With such a model ingredient, the flow rules can be written by invoking the principle of maximum plastic dissipation as:

MaxjFi ¼0 {Dp } , Maxjg_ i Minjs {Lp } ,

8 m ›L p ›D p X ›F i > > ¼2 þ g_ i ¼0 > > > › p › p ›p < i¼1 m > ›L p ›D p X ›F i > > ¼ 2 þ ¼0 g_ i > > ›q ›q : ›q i¼1

ð29Þ


where Lp ðp; q; g_ i Þ ¼ 2Dp ðp; qÞ þ

m X

g_ i Fi ðp; qÞ

ð30Þ

i¼1

and then leading to the flow rules:

x_ p ¼

m X

g_ i

›Fi ›p

ð31aÞ

g_ i

›Fi ›q

ð31bÞ

i¼1

j_ ¼

m X i¼1

The computation of the plastic multipliers in (31) above is carried out according to the loading/unloading and consistency conditions. The different results presented in the 2D context take the same form in the generalized stress and strain formulation. 3.1 Elastic domain The elastic domain is here defined by four independent yield surfaces Fi ðx; x p ; jÞ; 1 # i # 4; which are directly defined in generalized strain space x; x p and j: We denote h 1þ ¼ e þ k 2

and

h 12 ¼ e 2 k 2

as the local strain tensors at the top and bottom surfaces of plates and their spectral decompositions as: 1þ=2 ¼

2 X

þ=2 þ=2 þ=2 nk ^nk :

1k

k¼1

By generalizing the presented 2D plasticity model to shells, we assume that the stress resultant plasticity starts as soon as the stress at either upper or lower face reaches the given elastic limit, leading to four different yield surfaces: G þ F1 ¼ 2 K þ ð32aÞ 1I 2 1y 2 kðjþ Þ 3

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G þ F2 ¼ 2 K þ 1II 2 1y 2 kðjþ Þ 3

ð32bÞ

G 2 F3 ¼ 2 K þ 1I 2 1y 2 kðj2 Þ 3

ð32cÞ

G 2 F4 ¼ 2 K þ 1II 2 1y 2 kðj2 Þ 3

ð32dÞ

With the use of state laws in equation (27) and the normalized values of stress resultants 0 1 ^ N ^ ¼ N=h; M ^ ¼ 6M=h2 ; p^ ¼ @ A; N ^ M we can write the four surfaces in stress resultants space Fi ðp^; qÞ ¼ Fi ðx; x p ; j Þ F1 ¼

K þ 4G=3 ^ ^ þM ^ 2 K 2 2G=3 N ^ 2ðp 2 qþ Þ N þ M y 2G 2G I II

F2 ¼ 2

F3 ¼

K 2 2G=3 ^ ^ þM ^ þ K þ 4G=3 N ^ 2ðp 2 q þ Þ N þ M y 2G 2G I II

K þ 4G=3 ^ ^ 2M ^ 2 K 2 2G=3 N ^ 2ðp 2 q 2 Þ N 2 M y 2G 2G I II

F4 ¼ 2

K 2 2G=3 ^ ^ 2M ^ þ K þ 4G=3 N ^ 2ðp 2 q 2 Þ N 2 M y 2G 2G I II

ð33aÞ

ð33bÞ

ð33cÞ

ð33dÞ

in which j†jI =II are the two principal values of the second-order symmetric tensor. The graphical illustration of this kind of criterion is shown in Figure 6.

Figure 6. Multi-surface strain based plasticity criterion in stress resultants

4. Discrete formulation In this section, we recast this non-linear evolution problem in a discrete pseudo-time setting produced by a one-step scheme. The latter amounts to taking into account the values of evolution variables at time tn xn ; xnp and jn, and computing the generalized strain increment Dxnþ1 as well as applying an implicit backward-Euler integration scheme to obtain the corresponding values of internal variables. For the best iterative value of strain increment, the computation of this kind starts by assuming the elastic trial state:

xe;trial ¼ xen þ Dxnþ1 nþ1

ð34aÞ

xp;trial ¼ xpn nþ1

ð34bÞ

jtrial ¼ jn nþ1

ð34cÞ

which further leads to the corresponding values of stress resultants:

ptrial nþ1 ¼ pn þ J · Dxnþ1

ð35aÞ

e;trial qtrial ¼ 2k j ; x nþ1 n nþ1

ð35bÞ

~

If we finally find that all yield functions remain negative, trial trial Ftrial i;nþ1 ¼ Fi pnþ1 ; qnþ1 # 0

1#i#4

ð36Þ

the trial stress state is admissible and it is accepted as final. The step is thus elastic. In the opposite, we have to compute the non-zero values of plastic multipliers in order to recover the admissibility of stress. As discussed above, the key problem consists in determining the final set of active i surfaces (with l i . 0) according to the fact that Ftrial i;nþ1 . 0 t l . 0: In that sense, for the presented model, it should be emphasized that the second (respectively fourth) surface could be active even if Ftrial 2=4;nþ1 # 0: Considering an active set of surfaces, the consistency condition leads to the non-linear system: Fi;nþ1 ðp; qÞ ¼ 0

ð37Þ

m X i ›F i pnþ1 ¼ ptrial 2 J l nþ1 ~ i¼1 ›p nþ1

ð38aÞ

qnþ1 ¼ 2k jnþ1 ; x e;trial nþ1

ð38bÞ

with the relations:


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with plastically admissible stress resultants and the plastic multipliers l i as unknowns. Contrary to a single surface plasticity model for plates (Ibrahimbegovic and Frey, 1993), the present case cannot be reduced to a single equation. We thus carry out an iterative solution for the complete system; the latter employs the following linearized form: h i ›Fi ðkÞ ðkþ1Þ ðkÞ ðp; qÞ ¼ Fi;nþ1 ðp; qÞ þ Dlðkþ1Þ ¼ 0 ð39Þ L Fi;nþ1 ›l nþ1

›F i ›F i › p ›F i › q ¼ þ ›l j ›p ›l j ›q ›l j m ›F i X ›Fj ›Fi ¼2 J þ ›p ~ j¼1 ›p ›q

›xe ›j 2D · j 2 U : j ¼ 2g ij ~ ›l ~ ›l

ð40Þ

The matrix of the linearized system, with the components ! ›F i ›F i ›Fj ›Fi ›Fj g ij ¼ ; j2 U þ D ›p ~ ›q ~ ›p ›q ›q remains positive-definite, which provides the guarantee for the solution. Once the computation of the internal variables and plastically admissible stress resultants is completed, we turn to the solution of the global set of equilibrium equations. The key ingredient in that sense pertains to the elastoplastic consistent tangent modulus, which can be written as, 0 1 X › F i * A dl i ð41Þ dp ¼ Jðdx 2 dxp Þ ¼ J @dx 2 ›p ~ ~ i where

21 2 X * 21 i › Fi J ¼ J þ d l i ›p2 ~ ~ is the algorithmic modulus. The latter is different from the continuum modulus (Simo and Taylor, 1985) due to the rotation of principal directions. The system of equations dFi ¼ 0 provides the result: dg i ¼

X j

g ij

›Fj * J dx ›p ~

ð42Þ

which finally leads to, * X ij * ›F i * ›F j ep ep g J dp ¼ J dx; J ¼ J 2 ^ J £ ›p ~ ~ ›p ~ i;j

ð43Þ

5. Numerical examples In this section, we present several numerical examples, in order to illustrate the predictive capabilities of the presented model. All the computations are carried out by using the finite element program FEAP (Zienkiewicz and Taylor, 2000). 5.1 Illustrative example with simple test case Considering pure tension and compression tests, the key point is to show the influence of the evolution of the fracture energy according to the stress state (see equation (10)). Figure 7 shows the uniaxial response of the proposed model for both cases; The first one corresponds to the “uncoupled” case for which fracture energy is assumed to remain constant for all possible stress states. Then, the second is a “coupled” case where the fracture energy is assumed to evolve according to expression (11). It is clearly shown in Figure 7 that the uncoupled case leads to a non-realistic behavior according to the fact that the amount of dissipated energy in both pure tension and compression is the same. Contrary to this situation, the coupled case allows to retrieve a physically-based uniaxial behavior by modifying the fracture energy. We reiterate on the point that this ingredient is crucial in order to obtain a representative behavior of brittle material.


5.2 Engineering application – chimney under fire In this example we present an industrial application dealing with the fire resistance of a chimney. The latter is made of an assembly of hollow clay blocks (Plate 1). The inner part of these blocks is submitted to the flow of hot gas (up to 1,0008C in 10 mn). This heating drives to the failure of the blocks and the possibility for the gas to flow outside from the chimney. According to the industrial point of view, the key point is to evaluate the time before failure. In order to compare the numerical results with the corresponding experimental data, a test has been driven on a chimney made of three hollow blocks. Thermocouples and strain gauges have been put on the block placed in middle position. The former are

Figure 7. Uniaxial stress/strain response

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Plate 1. Clay block and chimney

providing temperatures through the thickness of the block and so furnish some information concerning temperature gradient. Strain gauges have been placed on the outer face only in order to estimate stresses evolution with respect to time. By taking into account the symmetry, the numerical model for this example is made of 32 flat shell elements only. Comparing to classical solid three-dimensional modeling, this application is thus clearly showing the main advantage of using shell elements by drastically reducing the size of the discrete problem. According to the loading case, a coupled thermomechanical analysis has been driven using these flat shell elements incorporating the mechanical model presented here. Concerning the heat transfer problem, a special discrete form of the transfer equation has been developed, allowing to take into account for both average temperatures and through-the-thickness gradients (Colliat et al., 2004). This model is representing both conduction and radiative heat transfers which are leading to a nonlinear problem (Figure 8). The latter takes

Figure 8. Clay block – boundary conditions and mesh

place inside each cell and it can be taken into account by classical eight nodes solid elements (Colliat et al., 2004). The boundary condition are chosen as a convective heat transfer between the hot gas and the inner face of the block. We chose the convection coefficient in order to reproduce the temperature evolution on this face observed during the test (see Figure 9). The parameters used to drive the numerical analysis are shown in Table I. For this example, we assumed that all these parameters remain constant with respect to temperature evolution. Figure 9 shows the comparison between the numerical analysis and the experimental data. The first part deals with temperatures evolutions in time. The four curves are representing the surface temperature from the inner face to the outer face of the block. As already mentioned, the convection coefficient has been chosen in


Figure 9. Clay block – temperature and horizontal stress evolutions

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Table I. Clay block – parameters

Plate 2. Clay block – cracks pattern and failure mechanism

order to reproduce the inner face temperature evolution. According to this point, it is shown that the outer face temperature is not accurately predicted. The main reason for this is the choice of one shell element only in the heat flow direction. Nevertheless, the main objective of this computation is not to provide a detailed prediction of the temperature field (a three-dimensional analysis with solid element would be more adequate) and the accuracy obtained is sufficient for the mechanical analysis. Concerning the mechanical behavior, the numerical analysis leads to failure which starts in the inner part of the corner of the block, and propagates to the outer part. This result is completely in accordance with the experimental behavior for which the crack pattern is almost vertical in two opposite corners. Plate 2 shows the main cracks in this region after the test. Figure 9 also shows the horizontal stress evolution on the outer face of the block with respect to time. In order to compare with the experimental data provided by the strain gauge, the computed stress corresponds to the same point. It is clearly shown that the proposed model is able to reproduce with a good accuracy the mechanical behavior of this structure. The only discrepancy occurs in the first part of the heating process, where the numerical approach is overestimating the stress. This is mainly due to the overestimation of the temperature at the outer face (see Figure 9(a)). The time to

Mass density Heat capacity Thermal conductivity Thermal expansion coefficient Young’s modulus Poisson ratio sy Gt Gc

1.870 836 J kg2 1 K2 1 0,45 W m2 1 K2 1 7 £ 102 6 K2 1 12 GPa 0,2 11 MPa 80 J m2 2 10 Gt

reach the failure is well reproduced, around 10 min for both cases. This finding is of crucial concern for the industrial application. 5.3 Thermomechanical analysis of hollow brick wall In this example, we consider a thermomechanical coupling in the cellular units placed with a brick wall. We assume that the geometry and the loading allow for exploiting the periodicity conditions. This implies that the analysis can be carried out on a single cellular unit isolated from the whole structure at the level of interface with neighboring units, by applying the corresponding boundary conditions which assure periodicity. More precisely, for the typical unit assembly in a brick wall (Figure 10) with only partial overlapping of successive layers, the same periodicity conditions are enforced only over half of the brick. Therefore, the domain which is retained in the analysis corresponds to one typical unit of the size 570 £ 200 £ 200 mm3 : This domain also includes a half of the vertical and horizontal joints with the thickness equal to 10 mm. The chosen finite element mesh (Figure 11) consists of three vertical layers of flat shell element which brings the total number of these element to 384 for the entire brick. Another subtlety of the model is the choice which is made for representing the interface joints. This one is the model by elastic solid elements covering the cells of the brick placed only at the top. However, the latter does not introduce any non-symmetry in the problem, considering the periodicity in the boundary conditions. Both mechanical and thermal loading is applied in this case. The mechanical loading is supposed to represent the dead load on the brick chosen as a compressive loading of 1.3 MPa, which is introduced directly at the level of each element as the initial compressive loading in the bricks, remaining constant afterwards. The thermal loading is then applied, in terms of the uniform temperature field applied only at the brick facet exposed to fire. The time evolution of this temperature field is given as

uðtÞ ¼ u0 þ 345 log ð8t þ 1Þ


ð44Þ

where u0 is the initial temperature and t the time in minutes.

Figure 10. Unit assembly in a brick wall and periodic BC

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The mechanical and thermal properties of the brick material are chosen as given in Table II. The properties of the interface are given in Table III. First, the results are presented in terms of temperature field. Figure 12(a) shows the evolution of the temperature in three different cells (Figure 13 for locations). The experimental results are provided by thermocouples inside the cells. Therefore, we compare these values with the temperatures of the two surfaces on both sides of each cell obtained by the finite element analysis. The comparison shows we are able to capture the temperature evolutions even far from the exposed face of the wall. This result is confirmed by Figure 12(b) which shows a temperature profile 48 min after the beginning of heating. The key point in order to obtain such good result is the introduction of radiative exchanges in the heat transfer model. On the mechanical point of view, Figure 14(a) shows the evolution of the sum of vertical reactions at selected nodes. Each curve corresponds to a line of nodes parallel to the exposed face of the wall and positive values are for compression (with prestressed initial value due to mechanical constant loading). Figure 14(b) shows the comparison on the horizontal displacement of wall built with ten rows of bricks.

Table II. Mechanical and thermal properties of the brick

Density Heat capacity Conductivity (parallel to flakes) Conductivity (perpendicular to flakes) Thermal expansion coefficient at uref Young’s modulus Poisson ratio sy at uref Fracture energy

1.870 836 J kg2 1 K2 1 0.55 W m2 1 K2 1 0.35 W m2 1 K2 1 7 £ 102 6 K2 1 12 GPa 0.2 14.5 MPa 80 J m2 2

Table III. Mechanical and thermal properties of the interface

Density Heat capacity Conductivity Thermal expansion coefficient Young’s modulus Poisson ratio

2.100 950 J kg2 1 K2 1 1.15 W m2 1 K2 1 1 £ 102 5 K2 1 15 GPa 0.25


Figure 12. Brick – temperature evolutions and profile after 48 mn

Figure 13. Brick – cells facets locations

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Figure 14. Brick – total vertical reactions for the first lines and horizontal displacement

This bending is due to the temperature gradient through the wall. We show that the stiffness provided by the analysis is quite correct even if the displacement are slightly overestimated by the absence of mechanical boundary conditions. Global analysis on entire wall (e.g. without periodic boundary conditions) have been made in order to improve this result. 6. Conclusions The multi-surface model proposed herein finds its place quite naturally within the multiscale approach for modeling the cellular structures (Ibrahimbegovic et al., 2005). The model is used for the computations at the finest scale to explain the physical

nature of damage mechanisms. For that reason with a large number of computations to be performed at the finest scale, the numerical implementation of the model is carried out with a great care in order to ensure the most robust model performance. The original feature of the model concerns the plasticity criterion proposed in strain space, with respect to the principal values of elastic strain tensor. The latter allows to represent the cracking phenomena both in tension and in lateral straining under compression. We thus provide quite a realistic description of cracking phenomena in brittle materials, with a very few parameters. As such, the model can be applied to other situation where the brittle rupture is a dominant failure mode. References Colliat, J-B., Ibrahimbegovic´, A. and Davenne, L. (2004), “Heat conduction and radiative heat exchange in cellular structures using flat shell elements”, Communications in Numerical Methods in Engineering (in press). Crisfield, M. (1995), “Stress resultant plasticity for shells”, Proceedings of COMPLAS 5, Barcelona. Hofstetter, G., Simo, J.C. and Taylor, R.L. (1993), “A modified cap model: closest point solution algorithms”, Computers & Structures, Vol. 46 No. 2. Ibrahimbegovic, A. and Frey, F. (1993), “An efficient implementation of stress resultant plasticity in analysis of reissnermindlin plates”, Int. J. Numer. Meth., Vol. 36, pp. 303-20. Ibrahimbegovic, A., Colliat, J-B. and Davenne, L. (2005), “Thermomechanical coupling in folded plates and non-smooth shells”, Comput. Meth. Appl. Mech. Eng.(in press). Ibrahimbegovic, A., Frey, F. and Rebora, B. (1993), “A unified approach for modeling complex structural systems: FE with rotational degrees of freedom”, European J. Finite Elem., Vol. 2, pp. 257-86 (in French). Ibrahimbegovic, A., Taylor, R.L. and Wilson, E.L. (1990), “A robust membrane quadrilateral element with drilling degrees of freedom”, Int. J. Numer. Meth., Vol. 30, pp. 445-7. Lubliner, J. (1990), Plasticity Theory, Maxwell Macmillan International Editions, London. Markovic, D. and Ibrahimbegovic, A. (2004), “On micro-macro interface conditions for micro-scale based FEM for inelastic behavior of heterogeneous materials”, Comput. Meth. Appl. Mech. Eng., Vol. 193, pp. 5503-23. Pearce, C.J. and Bicánic´, N. (1997), “On multi-surfaces plasticity and Rankine model”, in Owen, D.J.R., Onate, E. and Hinton, E. (Eds), Proceedings of CIMNE, Barcelona, Spain. Saint-Venant De, A.J.C.B. (1855), “De la torsion des prismes”, Me´moire de l’Acade´mie des Sciences, Imprimerie Impe´riale, in French. Simo, J.C. and Taylor, R.L. (1985), “Consistent tangent operators for rate-independent elastoplasticity”, Comput. Meth. Appl. Mech. Eng., p. 48. Simo, J.C., Kennedy, J. and Govindjee, S. (1988), “Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms”, Int. J. Numer. Meth. Eng., p. 26. Zienkiewicz, O.C. and Taylor, R.L. (2000), Finite Element Method, 5th ed., Vols 1, 2 and 3, Butterworth, London.



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558 Received September 2004 Accepted January 2005


Prediction of crack pattern distribution in reinforced concrete by coupling a strong discontinuity model of concrete cracking and a bond-slip of reinforcement model Norberto Dominguez Laboratoire de Mećanique et Technologie (LMT), Ecole Normale Superieure de Cachan, Cachan, France

Delphine Brancherie Laboratoire Roberval, Universite´ de Technologie de Compie`gne, Compie`gne, France, and

Luc Davenne and Adnan Ibrahimbegovic´ Laboratoire de Mećanique et Technologie (LMT), Ecole Normale Superieure de Cachan, Cachan, France Abstract


Purpose – To provide a reinforced concrete model including bonding coupled to a classical continuum damage model of concrete, capable of predicting numerically the crack pattern distribution in a RC structure, subjected to traction forces. Design/methodology/approach – A new coupling between bonding model and an alternative model for concrete cracking is proposed and analyzed. For concrete, proposes a damage-like material model capable of combining two types of dissipative mechanisms: diffuse volume dissipation and localized surface dissipation. Findings – One of the most important contributions is the capacity of predicting maximal and minimal spacing of macro-cracks, even if the exact location of cracks remains undetermined. Another contribution is to reiterate on the insufficiency of the local damage model of concrete to handle this class of problems; much in the same manner as for localization problem which accompany strain-softening behavior. Practical implications – Bonding becomes very important to evaluate both the integrity and durability of a RC structure, or in particular to a reliable prediction of crack spacing and opening, and it should be integrated in future analysis of RC. Originality/value – Shows that introduction of the influence of concrete heterogeneities in numerical analysis can directly affect the configuration of the crack pattern distribution. Use of a strong discontinuity approach provides additional cracking information like opening of macro-cracks. Keywords Concretes, Modeling, Stress (materials), Numerical analysis Paper type Research paper

This work has been gratefully supported by Electricite´ de France (EDF) and CONACYT Me´xico.

1. Introduction Concrete and steel are two materials whose behaviors are quite complex and very different in traction (weakness of concrete) and in compression (weakness of slender steel elements). The fact that they can combine within the reinforced concrete to eliminate jointly the weakness of each component has long been recognized and widely used in civil engineering constructions. However, it is only a more recent design requirement to provide not only the guaranties on structural integrity but also structural durability related to given crack patterns, which imposes a fresh start in modeling of reinforced concrete. Therefore, the RC models assuming concrete and steel are perfectly attached one to another without considering how the real transfer of forces between them is and how they affect the performance of the whole structure are often no longer acceptable. In structures of reinforced concrete, bonding is a crucial phenomenon of local interaction between steel rebar and concrete, whose complexity depends not only on material characteristics and specific geometry of surfaces in contact but also on loading conditions as well as the structural configuration of the global problem. There are considerable difficulties to identify locally the parameters controlling the phenomena as well as the problematic of modeling the interface to be integrated in a structural analysis problem where description of steel bar bonding is supposed to remain perfect. In reality, the highly heterogeneous geometry and/or material parameters variation of concrete render the development of a local bond model quite delicate. In a typically reinforced concrete structure analysis the finite element model is constructed with a mesh where truss, triangular and quadrilateral solids may be used. Each element deals with the specific behavior of a material (either concrete or steel), and a crude model of bonding is sometimes represented by spring connectors; the latter cannot account for mesh sensibility on constitutive models, macro-crack effects and degradation of bonding. In this work we explore an alternative finite element approach, which is developed recently in order to avoid localization problems produced by concrete cracking which is referred to as “strong discontinuity approach” (Jira´sek and Zimmermann, 2001; Simo et al., 1993; Wells and Sluys, 2001). The key point of such an approach is to concentrate all the dissipation produced in a localization zone on a surface of discontinuity of the displacement field. The vast majority of works developed in this framework consider that the only source of dissipation in structures is due to the apparition and development of localization zones. This approach is convenient for the description of the rupture behavior of thin structures. However, for massive structures, such considerations are not sufficient: bulk dissipation produced in so-called “process zones”, which is due to the formation and development of micro-cracks preceding the formation of macro-cracks cannot be neglected. Reinforced concrete structures are in general massive structures, therefore we propose here in a damage-like material model capable of combining two types of dissipative mechanisms: (1) A diffuse volume dissipation due to the development in a quasi homogeneous way of micro-cracks. This diffuse mechanism is represented by a dissipative classical continuum damage model. (2) A localized surface dissipation produced by the apparition and development of localization zones or macro-cracks in the case treated here of concrete damage material. This localized dissipative mechanism is taken into account by

Crack pattern distribution

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building an adequate discrete type model linking traction on the surface of discontinuity and the displacement jump. This model is constructed within the framework of interface thermodynamics. The theoretical formulation of this kind of concrete model within the thermodynamics framework is presented in the foregoing, as well as the corresponding finite element formulation. The same thermodynamics framework is used for development of the bond-slip model proposed herein. The model is capable of accounting for physical phenomena such as cracking, frictional sliding and their coupling; the numerical implementation of the bond-slip model is provided within the framework of a zero thickness interface element, which can deal with plane strain and axisymmetric problems. The outline of the paper is as follows. In Section 2, we present the details of the concrete model, both in terms of its theoretical formulation and its numerical implementation. The salient features of the bond-slip model are described in Section 3. In Section 4, we provide the results of numerical simulations, which illustrate the behavior of the proposed model in terms of crack prediction. Finally, closing remarks are stated in Section 5. 2. Description of the strong discontinuity model for concrete cracking The main purpose of this section is to present the theoretical formulation as well as the finite element implementation of the “strong discontinuity approach” for modeling the cracking of concrete. The presented approach is capable of taking into account two types of dissipative mechanisms in order to better reproduce the behavior of massive structures: a bulk dissipation characterized by the development of micro-cracks, which is taken into account by the introduction of a continuum damage model and a surface dissipation taking place at the level of the localization zones in terms of the macro-cracks, responsible for the rupture of the structure. Two mechanisms are then forced to operate in a coupled manner potentially in each element (Ibrahimbegovic and Brancherie, 2003). We first present the theoretical formulation for both: the bulk damage model and the discontinuity damage model, as well as their modifications induced by the introduction of a displacement discontinuity. The key points of the finite element implementation are then presented with particular developments dedicated to the necessary modifications of the solution strategy (Ibrahimbegovic et al., 1998; Ibrahimbegovic and Markovic, 2003) due to the introduction of a displacement discontinuity. 2.1 Theoretical formulation As mentioned previously, the key point of the method is the introduction of a surface of displacement discontinuity on which are concentrated all localized dissipative mechanisms due to the apparition and development of localization zones. We will develop here the modification induced by this displacement discontinuity and present more precisely how to build each model associated to each dissipative mechanism. 2.1.1 Kinematics of the displacement discontinuity. We consider here a domain V split into two sub-domains Vþ and V2 by a surface of discontinuity, denoted as Gs (Figure 1).

The surface of discontinuity Gs is characterized at each point by a normal vector denoted n and a tangential vector denoted m. The discontinuous displacement field can then be written as ¼

~ uðx; tÞ ¼ uðx; tÞ þ uðtÞðHGs ðxÞ 2 wðxÞÞ


ð1Þ

as HGs ðxÞ is the Heaviside function being 1 in Vþ and 0 in V2 and w(x) is at least C 0 function, with its boundary values defined according to ( 0 if x [ ›V > V2 wðxÞ ¼ 1 if x [ ›V > Vþ

561

~ It follows therefore that uðx; tÞ has the same boundary value as the total displacement field u(x,t). Such an expression for the displacement field gives a strain field written as ¼

¼

~ 1ðx; tÞ ¼ 7s uðx; tÞ ¼ 7s uðx; tÞ þ GðxÞuðtÞ þ ðuðtÞ^nÞs dGs ðxÞ

ð2Þ

The strain field appears then to be decomposed into a regular part and a singular part, the latter accompanying the Dirac-delta function dGs ðxÞ: When considering a damage model, for which the state equation can be written as 1¼D:s the compliance D must be decomposed into a regular and a singular part: ¼ D ¼ D þ DdGs ðxÞ: The latter ensures that the stress field remains regular. This decomposition implies that all internal variables must be decomposed in the same manner, into a regular and a singular part. Each part of the decomposition is associated to a damage dissipative mechanism, with the regular part to the continuum damage model and the singular one to the discrete damage model describing the localization zone. We present subsequently the chosen formulation of the two damage models, the continuum one, associated to the bulk material, and the discrete one, associated to the localization zone. 2.1.2 Continuum damage model. In order to take into account the apparition and development of micro-cracks with a quasi homogeneous distribution, we develop an isotropic damage model. The model is constructed in a similar way as plasticity models by defining the admissible stress domain in terms of the damage function. In this case,

Figure 1. Slip line Gs separating the domain into Vþ and V2

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the internal variables are the damage compliance denoted as D (with an initial value denoted as D e for the initial undamaged compliance) and the variable associated to hardening denoted as j: The evolution equations are obtained by considering the principle of maximum damage dissipation and introducing the Lagrange multiplier g_: The main ingredients of the construction of the damage model are summarized in Table I. 2.1.3 Discrete damage model. It is proposed to take into account the dissipative behavior of localization zones, defined as macro-cracks for damage materials. It is constructed in a very similar way as the continuum model just presented. In order to control separately the normal and tangential components of the traction on the surface of discontinuity, the elastic domain is defined by a multi-surface criterion where g_1 and g_2 denote the Lagrange multipliers associated, respectively, to each damage surface defining the elastic domain. Each surface is coupled to the other via the traction-like ¼ internal variables of the model are the compliance variable q: The ¼ ¼ of the interface denoted as Q and the variable associated to softening denoted j: In Table II we summarize the main ingredients of the construction of the model. 2.2 Finite element implementation The main difficulty in dealing with the FE implementation of the proposed concrete cracking model pertains to accounting for displacement discontinuities which cannot be done by standard isoparametric interpolations. In the following, we present the choice of the suitable finite element interpolations and the modifications introduced in the resolution of the problem.

Helmholtz free energy Damage function

21

c ð1; D; jÞ ¼ 12 1 : D : 1 þ JðjÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðs; q Þ ¼ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} s : D e : s 2 p1ffiffiEffi ðsf 2 q Þ kskD e

Table I. Main ingredients of continuum damage model

Evolution equations

Helmholtz free energy

c ðu; Q; jÞ ¼ 12 u : Q

Dissipation

Damage function Constitutive equations Dissipation Table II. Main ingredients of the discrete damage model

21

: 1 and q ¼ 2 ddj JðjÞ _ : s þ q j_ Dðs; q Þ ¼ 12 s : D e _ D ¼ g_ D and j_ ¼ g_ p1ffiffiffi

Constitutive equations

Evolution equations

s¼D

kskD e

E

¼ ¼ ¼ ¼ ¼

¼

¼ 21

¼

¼ 2

¼

: u þ JðjÞ ¼

¼ f1 ðtGs ; qÞ ¼ t Gs :n 2 ðs 2 qÞ f¼ ¼ ¼ ¼ ¼ f2 ðtGs ; qÞ ¼ tGs :m 2 ðss 2 ¼sss qÞ ¼ 21

f

¼

¼

¼ ¼

: u and q ¼ 2 d¼ JðjÞ dj ¼ ¼ _ t þ ¼qj_ Dðt Gs ; qÞ ¼ 12 t Gs :Q Gs _ ¼ g_1 1 n ^ n þ g_2 1 m ^ m Q t Gs :n jt Gs :mj _ ¼ g_ þ ¼¼ss g_ j and 1 sf 2 t Gs ¼ Q

The finite element interpolation is chosen to take into account a displacement discontinuity, by considering the incompatible mode methods (Ibrahimbegovic and Wilson, 1991). More precisely, we choose the finite element interpolation according to ¼

u h ðx; tÞ ¼ NðxÞdðtÞ þ uðtÞMðxÞ


ð3Þ

N(x) is the classical shape function associated to the considered triangular element, d(t) denotes the nodal displacement and M(x) is a discontinuous function of the type presented in Figure 2. It is important to note that N(x)d(t) has the same nodal value as u h(x,t). With such an approximation, the finite element interpolation of the strain field can be written

563

¼

ð4Þ 1 h ðx; tÞ ¼ BðxÞdðtÞ þ uðtÞG r ðxÞ s where G r ðxÞ ¼ LMðxÞ with L the matrix associated to the operator 7 . The finite element interpolation of the virtual strain field can be constructed with the same scheme as ¼ d1 h ðx; tÞ ¼ BðxÞddðtÞ þ duðtÞG v ðxÞ ¼ ddðtÞ and duðtÞ denote, respectively, the virtual displacement field and virtual displacement jump field. Gv(x) is a modified incompatible mode function constructed from the function Gr(x) so as to guarantee the satisfaction of the patch-test. We note that in general Gv(x) is different from the function Gr(x). It has to be noted that, as M(x) is a discontinuous function, the functions Gr(x) and Gv(x) can be decomposed into a regular and a singular part as ¼

¼

G v ðxÞ ¼ Gv ðxÞ þ Gv ðxÞdGs and G r ðxÞ ¼ Gr ðxÞ þ Gr ðxÞdGs

ð5Þ

2.2.1 Operator split solution procedure. With those interpolations for real and virtual strain fields using the incompatible mode, the discretized problem can be written as 8 Z Nel > int;e ext;e int;e > > A f ðtÞ 2 f ðtÞ ¼ 0 with f ðtÞ ¼ B T ðxÞs dVe > < e¼1 e V Z Z Z ¼T T T > e e e > Gv ðxÞs dV ¼ Gv ðxÞs dV þ Gv ðxÞt Gs dGs ¼ 0; ; e [ 1; Neli > h ðtÞ ¼ > e e : V V Gs ð6Þ

Figure 2. Function M(x) for a three-node element

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The equilibrium of the structure is written as a system of two equations. The first one is the set of global equilibrium equations, which is classically written in the finite element method. The second one is a local equilibrium equation written in each localized element. Independently this equation can be interpreted as the weak form of the traction continuity condition along the surface of discontinuity. The consistent linearization of system (6) leads to the set of equilibrium equations, which can be written for time step n þ 1 and iteration (i) 8 i o Nel h Nel n > ¼ ðiÞ ext;e int;eðiÞ eðiÞ ðiÞ eðiÞ > > < A Knþ1 Ddnþ1 þ Frnþ1 Dunþ1 ¼ A f nþ1 ðtÞ 2 f nþ1 ðtÞ e¼1 e¼1 ð7Þ h i h i > ¼ ðiÞ eðiÞ ðiÞ eðiÞ eðiÞ eðiÞ > > FeðiÞ : vnþ1 þ Kdnþ1 Dd nþ1 þ Hnþ1 þ Kanþ1 Dunþ1 ¼ 2hnþ1 where KeðiÞ nþ1 FeðiÞ rnþ1

Z

¼

B V

¼

T

e

Z

B Ve

KeðiÞ dnþ1 ¼

Z

e ðxÞCedðiÞ nþ1 BðxÞdV ;

T

e ðxÞCedðiÞ nþ1 Gr ðxÞ dV ;

¼T

Gs

Gv ðxÞ

HeðiÞ nþ1

¼

T

e

V

FeðiÞ vnþ1

›t G s dGs ; KeðiÞ anþ1 ¼ ›d

Z

Z

¼

Z Ve

¼T

Gs

e Gv ðxÞCedðiÞ nþ1 Gr ðxÞ dV e B T ðxÞCedðiÞ nþ1 Gv ðxÞ dV

Gv ðxÞ

ð8Þ

›t Gs ¼ dGs ›u

At that stage there are a couple of possibilities to solve the above set of equilibrium equations. The first possibility consists in solving simultaneously at global level the two equations. The second possibility, which is chosen herein, consists in taking advantage of the fact that the second equation is written locally on each localized element. Then, this second equation is solved at the element level for a given value of the displacement field increment Dd ðiÞ nþ1 : This allows determining the value of the ¼ displacement jump increment DuðiÞ nþ1 : Then by static condensation at the element level, the system of equations in (7) is reduced to a single equation which takes the classical form in the finite element method as Nel

^ eðiÞ Dd ðiÞ ¼ 0 with AK nþ1 nþ1

e¼1

^ eðiÞ K nþ1

h i21 h i eðiÞ eðiÞ eðiÞ eðiÞ ¼ KeðiÞ FeðiÞ nþ1 2 Frnþ1 Hnþ1 þ Kanþ1 vnþ1 þ Kdnþ1

ð9Þ

3. Bond-slip model: constitutive equations and finite element formulation The concept of “reinforced concrete” is only possible due to the existence of bonding, which is a transfer zone of forces and stresses between concrete and the steel bars submerged in it. The global bond-slip relationship is one way of measuring experimentally this phenomenon of interaction between surfaces, but is not enough to represent what happens locally in the vicinity of steel bars surface and the concrete in contact. In this section, we develop the constitutive relations for bond slip capable of accounting for physical phenomena such as cracking and frictional sliding as well as for their coupling. To that end, we first present the thermodynamic framework and

then the numerical implementation, which fits within the framework of the finite element approximation, which is finally presented in the last paragraph. 3.1 Thermodynamic formulation The constitutive relations which relate the stress tensor to the strain tensor should include: cracking for an excessive tangential stress snt ¼ sT ; inelastic strain due to sliding 1sT ; hysteretic behavior due to friction and coupling between tangential and normal stress snn ¼ sN in sliding phase; other stress and strain components are of no interest. The interface element is activated if and only if a relative displacement exists between the two bodies in contact. The main ingredients of the proposed bond-slip model can be obtained by appealing to thermodynamics considerations (Dominguez et al., 2003) which leads to Table III. Owing to the simplicity of damage surface expressed in the strain space, the evolution of the damage variable can be integrated explicitly to obtain


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8 sffiffiffiffiffiffi "rffiffiffiffi #Bd1 9( ) < = Y0 2 pffiffiffiffiffiffi pffiffiffiffiffiffi 1 Yd 2 Y0 exp Ad1 d ¼12 : ; 1 þ Ad2 hY d 2 Y 2 iBþd2 G Yd where Ad1, Bd1, Ad2, Bd2 are material parameters. The latter is a generalized version of the model in Ragueneau et al. (2004), which is capable of accounting for large sliding in the final phase of pull-out test. The evolution equations of the sliding strain component are integrated by an implicit scheme. Among different numerical integration schemes which may be used, Euler’s backward scheme is implemented in the classical form of so-called “return mapping” algorithm (Ortiz and Simo, 1986), which ensures the convergence for any step size. Helmholtz free energy Damage function Constitutive equations

Dissipation

rc ¼ 12 ½ 1N E1N þ ð1 2 dÞ1T G1T þ 1T 2 1sT Gd 1T 2 1sT þ ga 2 þ H ðzÞ s fs sN ; sT ; X ¼ sT 2 X 2 13 sN # 0 fd ðY d ; Z Þ ¼ Y d 2 ðY 0 þ Z Þ # 0 sN ¼ E1N ; s sT ¼ Gð1 2 dÞ1T þ Gd 1T 2 1T ; s s sT ¼ Gd 1T 2 1T Y ¼ Y d þ Y s ¼ 12 1T G1 T þ 1T 2 1sT G 1T 2 1sT ; 8 X ¼ ga; < Z 1; si10T , 1iT # 12T 0 Z ¼ H ðzÞ ¼ : Z 1 ·Z 2 ; si1iT $ 12T 1 2

qffiffiffi hpffiffiffiffiffiffi i2 Y 0 þ A1d1 G2 ln ð1 þ zÞ 11T0 T h i 2z Z 2 ¼ Y 2 þ A1d2 1þz Z1 ¼

Evolution equations

_ ›fd d_ ¼ l_d ››Yfdd ; z_¼ 2 ld ›Z ; 1_sT ¼ l_s sign ssT ; and a_ ¼ 2l_s sign ssT þ 32 aX

Table III. Main ingredients of bond-slip model

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3.2 Finite element implementation We construct a zero thickness interface element from a degenerated version of standard quadrilateral element, capable to account for both tangential and normal stresses, calculated from corresponding normal and tangential strains, even for large displacements. The reference domain of a straight-edge quadrilateral element is defined by the locations of its four nodal points xea ; a ¼ 1; . . . ; 4: Only two sets of coordinates among four nodes must be given, the other two nodes have the same coordinate, which defines a quadrilateral element with zero thickness (Figure 3). 3.2.1 The “hpen” parameter. We follow the idea of interpenetration used by Ibrahimbegovic and Wilson (1992) in a contact problem’s resolution, in order to avoid the potential in stability drawbacks when dealing with contact forces. In other words, we assume that concrete in contact with steel surface has a zone of roughness, which could be compressed or crushed, so it is possible to assume a small penetration between surfaces (Figure 4). In order to maintain continuity between 2D continuum elements (for steel and concrete) and interface contact element, we introduce a geometrical parameter called “hpen”, which allows defining both normal and tangential strains for the interface element even in the interface of zero thickness. One can thus state that the “hpen” parameter has a physical value that corresponds to the maximal penetration corresponding to the thickness of compressed-pulverized concrete. From Figure 4 it is possible to deduce the following relationship hpen ¼ tA0 þ g 0 þ tB0

Figure 3. Degeneration of a four-node quadrilateral element in a one-dimensional element and local coordinates

Figure 4. Initial configuration, normal displacement and penetration for crushing of asperities

ð10Þ

If we assume that at the moment of contact, g 0 ¼ 0; and that body B is rigid, which implies that t B0 ¼ 0; the deformable thickness parameter becomes hpen ¼ tA0

ð11Þ

The “hpen” parameter can further be introduced into the geometric properties of the degenerated contact interface, which allows that normal and tangential strains be expressed as 1n ¼

uAn uA ¼ An hpen t 0

and

1t ¼

uAt uA ¼ At hpen t 0

ð12Þ

with un and ut being the normal and tangential displacements of a point at the concrete surface with respect to the steel surface. Because the interface thickness is zero, no strains in other directions are considered for calculations. These strain components are further used to comport the corresponding stress components. The parameter “hpen” also plays a crucial role in computing strain-displacement matrix and simplifying the computation of shape function derivatives. Namely for a typical four-node quadrilateral element, with the shape functions expressed as 1 N a ðj; hÞ ¼ ð1 þ ja jÞð1 þ ha hÞ 4

ð13Þ

where a ¼ 1; 2; 3; 4 are the element nodes; j,h are natural coordinates and ja,ha the values of natural coordinates, we need to build the strain-displacement matrix B 2 3 0 N a;x 6 7 0 N a;y 7 ð14Þ B¼6 4 5 N a;y N a;x In seeking that B-matrix be constructed from derivatives of shape functions by using the nodal coordinates, we ought to correct the coordinates of two nodes being the same; to that end, we first compute x,j, y,h, x,h, y,j, by introducing the parameter “hpen” in the calculations as follows 9 ye0 ¼ ye1 2 ye2 þ ye3 þ hpen 2 ye4 þ hpen =4 > > = e e e e e y ;j ¼ 2y1 þ y2 þ y3 þ hpen 2 y4 þ hpen =4 þ y0 h ð15Þ > > y e ;h ¼ 2ye 2 ye þ ye þ hpen þ ye þ hpen =4 þ y0 j ; 1

2

3

4

j ¼ det X;j ¼ x;j y;h 2x;h y;j

ð16Þ

We can then compute by standard transformation N a;x ¼

N a;j y;h 2N a;h y;j j


N a;y ¼ 2

N a;j x;h 2N a;h x;j j

ð17Þ

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The internal force vector for the joint element is finally computed by using the classical Gaussian quadrature, with two integration points for tangential direction and only one for normal direction. 4. Numerical examples In this section, we present the results of numerical simulation of several standard traction tie tests, which are used in the study of cracking of structural members of reinforced concrete subjected to traction, where bonding has an important influence in resistance as well as in the crack pattern distribution. The latter are surrounded by concrete specimens, which are typically of cylindrical or prismatic shape: the steel bar – either smooth or with ribs – is placed in the center and subjected to traction force at both ends. This traction force is transferred from the steel bar to the concrete through the interface bond. Other than characterizing the bond, one of the most important goals for a test of this kind is to provide the crack pattern distribution in the concrete body, which accompanies a gradual force transition from steel to concrete. When a reinforced concrete tie is submitted to increasing traction loads, depending on geometry as well as of material heterogeneity, a macro-crack will be created in the section where the maximal resistance to rupture is reached, unloading immediately the concrete around the crack. Because of this discontinuity, normal stresses are redistributed along the bar, producing a new crack as soon as another section reaches again the maximal resistance to rupture, separated by a distance called “length of transference” (Jaccoud, 1987). Following this process, different cracks can be created to stop only when the force is not able to damage concrete (Figure 5). Crack spacing is determined by the maximal and minimal length of transference in the tie. The effectiveness of introducing a bonding model in calculations of reinforced concrete structures was initially analyzed in our study by the simulation of long concrete tie (length of 150 cm); this tie was discretized with standard quadrilateral elements in an axisymmetric formulation: one without interface elements, and the other with non-zero interface elements. The concrete is modeled with the Mazars local damage model (Mazars, 1986). Qualitatively, it is possible to verify in Figure 6, the advantage of modeling interface in reinforced concrete analysis: for a same imposed displacement (3 mm), a discretization without joint elements is not able to reproduce a realistic crack pattern distribution as good as a discretization with joint elements. If we compare the global response in both, the different jumps in force that should appear for each fracture in concrete are clearly visible in the case of the mesh with interface elements. In the following section, we analyze this phenomenon by presenting, more particularly, the results for simulations of two series of tests: the “short tie” used in Clement’s test (Clement, 1987) and the “long tie” studied in Daoud’s test (Daoud, 2003). All the simulations were carried out by the Finite Element code FEAP developed by Taylor (2004). 4.1 Analytical study of Clement’s tie The experimental results on pull-out tests are obtained by Clement (1987) on prismatic specimen with 10 £ 10 £ 68 cm; with a high resistance steel bar of diameter equal to 10 mm placed in the middle. The standard strength concrete is used to construct each specimen, with compressive strength f c ¼ 32 MPa and tensile strength f t ¼ 2:8 MPa;


569

Figure 5. Evolution of the cracking pattern distribution on a reinforced concrete tie

measured at 28 days. The chosen cement content is 365 kg/m; water/cement ratio is 0.53, and the maximal size of aggregate was 10 mm. Seven specimens were placed horizontally in a test bench with monotonic displacement-controlled (deformation rate was 3:7 £ 1025 mm=s). The test stopped as soon as the first macro crack appeared in the specimen, which was located between 21 and 31.5 cm from edge. The experimental results for the force applied at the moment of fracture and the corresponding displacement are given in Table IV. 4.1.1 Coupling of the bond element with Mazars damage model. The simulations were carried out considering perfect adherence and the interface elements bonding the steel bar and the concrete tie. It was achieved by using quadrilateral elements in an axisymmetric formulation for all materials. In order to verify the effectiveness of the bonding model, different sizes of interface finite element were discretized; the different mesh characteristics as well as the material parameters used in these simulations are reported, respectively, in Tables V and VI (Figure 7). Figure 8 gives the load-displacement curves obtained considering meshes cle_s_068 and cle_d_068, using or not bonding elements. Compared to the experimental results, we can note that the simulation considering interface elements gives better results

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Figure 6. Qualitative comparison between two kinds of discretization for a tie of length ¼ 150 cm: without and with interface elements, using “Mazars” and “bond/slip” local constitutive models

Test Table IV. Experimental results from Clement tie test

T3 T4 T5

fc 28 days (MPa)

ft 28 days (MPa)

Applied displacement at rupture (mm)

Frupt (KN)

29.15 28.00 33.42

2.76 3.02 2.58

0.37 0.44 0.42

21.4 23.1 24.2

either in term of the limit load corresponding to the apparition of the macro-crack but also in term of the displacement corresponding to rupture. Even if, in a first instance, the results obtained from coupling the interface model to Mazars local model for damage concrete seem to be satisfactory (Figure 8), showing in particular the influence of bonding in the global response of the structure, further analysis with different element size are carried out in order to quantify the sensibility of the global response as regards the mesh discretization. The results obtained for three different mesh refinement along the bar (17, 68 and 204 elements) are given in Figure 9. We can note that the results in terms of limit load and imposed displacement at rupture are dependent on the mesh discretization which is typical of local softening models where strain localization plays an important role. In our case, we can make two important remarks. First, modeling with different element size affects directly the crack pattern distribution in the tie, modifying the position of the macro-cracks. Second, we have observed that the use of a refined mesh diminishes the advantages of modeling interface: the finer the mesh is the closer the

Mesh code

Number of nodes

Total Number of standard QUAD4 elements

Number of joint elements

Characteristic size for concrete finite element (mm)

cle_s_017 cle_d_017 cle_s_018 cle_d_018 cle_s_068 cle_d_068 cle_s_069 cle_d_069 cle_s_136 cle_d_136 cle_s_170 cle_d_170 cle_s_204 cle_d_204

124 160 130 168 657 795 666 806 1,269 1,543 3,345 3,687 3,991 4,401

74 74 78 78 496 496 503 503 972 972 2,950 2,950 3,528 3,528

– 17 – 18 – 68 – 69 – 136 – 170 – 204

40 £ 15 40 £ 15 37.7 £ 15 37.7 £ 15 10 £ 9 10 £ 9 9.85 £ 9 9.85 £ 9 5£5 5£5 4£3 4£3 3.33 £ 3 3.33 £ 3

Material Steel

Denomination

Young’s modulus Poisson’s ratio Concrete Young’s modulus Poisson’s ratio (parameters for Mazars model) Elastic limit Compression parameter A Traction parameter A Compression parameter B Traction parameter B Shear effects corrector Bond Maximal penetration Young’s modulus Tangent stiffness modulus Poisson’s ratio Threshold Brittleness damage transition Threshold for damage by friction Brittleness damage by friction Kinematics law Nonlinear hardening Lateral pressure coefficient

Parameter Es n Ec n 10 AC AT BC BT b hpen Eb Gb n 10t A0d 1ft Afd g a c


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Table V. Mesh information for simulations with Mazars law

Value 210,000 MPa 0.3 31,000 MPa 0.2 1.0 £ 102 4 1.9 0.95 1,300 21,000 1.06 0.80 mm 31,000 MPa 15,000 MPa 0.2 5.0 £ 102 4 2.34 0.36 1.2 £ 102 3 MPa2 1 10.0 MPa 1.8 MPa2 1 1.0

results are to those obtained without bond elements, probably due to the higher influence of the concrete local model in the final response (Figure 10). For these reasons, we decided to explore a new coupling between the bonding model and a damage model constructed in the framework of the strong discontinuity approach which has been designed in order to tackle mesh dependency for problems involving softening materials.

Table VI. Material parameters (coupling Mazars and bonding models)

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Figure 7. “Mazars” and “bonding” local constitutive models: (a) normal stress-strain curve for concrete behavior; (b) tangential stress-slip curve for bonding behavior

4.1.2 Coupling of the bond element with a strong discontinuity approach. The Clement’s tie has also been tested considering a model coupling the bonding model presented here above and a strong discontinuity model for concrete cracking. This strong discontinuity model has also been implemented in the Finite Element Code FEAP and modified to deal with axisymmetric problems. The material parameters used for the simulation are reported in Table VII. In order to check mesh objectivity, two different mesh refinements were tested: a mesh composed of 68 elements along the bar and a mesh composed of 136 elements. Results in terms of the load-displacement diagram are given in Figure 11. We can note that the different discretization give quite the same global response till the apparition of the first macro-cracks. The results obtained considering a 68 element mesh using interface elements are compared to the results obtained considering Mazars damage model. Those diagrams are given in terms of load displacement curve in Figure 12. There is still work in progress on the coupling between bonding elements and the strong discontinuity


573

Figure 8. Comparison of experimental and numerical results, using meshes cle_s_068 and cle_d_068: force-displacement curve

Figure 9. Comparison between numerical results, varying the element size of meshes: as mesh becomes refined, local model of concrete is more sensible to element size effects, affecting the global response of the tie

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574 Figure 10. Ratio between numerical and experimental load, showing the influence of element size in global response of a structure (mesh sensibility and numerical localization problems)

Material

Denomination

Parameter

Steel

Young’s modulus Poisson’s ratio Young’s modulus Poisson ratio Elastic limit Hardening modulus Limit stress Softening modulus Maximal penetration Young’s modulus Tangent stiffness modulus Poisson’s ratio Threshold Brittleness damage transition Threshold for damage by friction Brittleness damage by friction Kinematics law Nonlinear hardening Lateral pressure coefficient

Es n Ec n sf K ¼ sf b hpen Eb Gb n 10t A0d 1ft Afd g a c

Concrete continuum model

Discrete model Bond

Table VII. Material parameters (coupling strong discontinuity approach and bonding models)

Value 210,000 MPa 0.3 31,000 MPa 0.2 2.2 MPa 3,000 MPa 2.3 MPa 10 MPa/m 0.80 mm 31,000 MPa 15,000 MPa 0.2 5.0 £ 102 4 2.34 0.36 1.2 £ 102 3 MPa2 1 10.0 MPa 1.8 MPa2 1 1.0

approach for modeling the behavior of concrete but from the first numerical results we can appreciate better correspondence between the experimental global response and the response obtained by coupling strong discontinuity and bonding elements than by coupling local damage model and bonding elements: great jumps corresponding to the apparition of the macro-cracks for local damage model are avoided by considering the strong discontinuity model. The limit load and the imposed displacement corresponding to rupture are in good agreement with those of the experiment in


575

Figure 11. Load-displacement considering two different meshes (coupling strong discontinuities and bond elements)

particular experiment C. The crack pattern obtained for both approach (coupling strong discontinuity/ bond element and coupling Mazars damage model/bond element) are given in Figures 13 and 14. We can note that the simulation carried out with both approach predict quite the same crack pattern: a central crack and two symmetric cracks on each part, which is in good agreement with the experimental observations. The advantage of using a strong discontinuity approach for reproducing the behavior of the concrete is to provide information not only on the position and the spacing of crack but also on the opening of macro-cracks. That information is relevant for predicting the durability of reinforced concrete structures in their environment. It has to be noticed that localization in material such heterogeneous as concrete is, in general, dependent on the heterogeneities of the material properties. In order to quantify the sensibility of the prediction of the limit load as well as the crack pattern, we have carried out the same simulation as previously but considering a ¼ non-homogenous material. The limit stress sf at which discontinuities are introduced has been turned into a random variable of mean value 2.3 MPa. Several computations have been carried out considering different deviations and a random set ¼ of material properties (here sf Þ: The results in terms of the load displacement curves and the cracks pattern are given, respectively in Figures 15 and 16. The analysis of these figures allows concluding that, as expected, the crack pattern is highly dependent on the heterogeneity of the material properties of the concrete. In all cases, the global response in terms of load displacement curve is quite similar. A first interpretation seems to indicate that the heterogeneous properties of such material as concrete do not affect the global response of the tie but as local importance in the determination of the

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Figure 12. Damage variable contour in Clement’s tie using the strong discontinuities model for concrete crack

Figure 13. Damage variable contour for Clement’s tie using Mazars local damage model

position of the macro-cracks. This is in good agreement with experimental results for which the crack pattern is dependent on the specimen but the global response relies quite independent of the heterogeneous characteristic of the concrete. 4.2 Analytical study of Daoud’s tie Recently, similar tests with reinforced concrete ties have been done by Daoud (2003) with specimens of 100 cm of length, and the crack pattern distribution was more


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Figure 14. Comparison of numerical crack pattern distribution for different kind of coupling: bonding-Mazars coupling and bonding-discontinuities coupling

Figure 15. Comparison of crack pattern obtained for different random set of limit stress considering different deviations

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Figure 16. Load displacement curve considering random set of limit stress with different deviations and a homogeneous concrete

interesting, because in each specimen no less than four macro-cracks were produced and they were perfectly observables (Figure 17). The minimal and maximal spacing of crack is provided in his work (Table VIII), so we decided to mesh the specimen in order to check the capacity of the bonding model to reproduce the crack pattern distribution. The geometrical and material properties were extracted from Daoud’s thesis and the coupling between Mazars local model for damage in concrete and the steel-concrete bonding model was used again. However, the most important point in this new simulation was the introduction of the concrete heterogeneity in calculations, in order to check the effects in the global response. For this, we assigned a random variable for each concrete element which affects directly the limit of elastic strain used in the Mazars law, and thus, the internal traction resistance of concrete: different deviation of the limit of elastic strain (5, 10 and 20 percent) were chosen to verify the concrete heterogeneity influence in the tie global response. 4.2.1 Analysis of numerical results. As we mentioned, the introduction of concrete heterogeneity by implementing a random variable has affected immediately the crack pattern distribution in the tie (Figure 18). In the case of typical discretizations where the concrete properties are considered homogeneous, spacing of cracks is very uniform, with some little variations in the edges of the bar. When a small variation of concrete resistance is introduced (deviation of 5 percent), the configuration of the crack pattern distribution is closer to reality: even if the exact location of macro-cracks remains undetermined, the maximal and minimal spacing of cracks, as well as the number of cracks for a same displacement, do not vary a lot. As long as the variation of concrete resistance increases, the minimal spacing becomes smaller, and the maximal spacing grows, showing how heterogeneity can produce weaker zones in the concrete body. A comparison of experimental and numerical spacing of cracks is shown in Table IX.


579 Figure 17. Specimens of 100 cm of length tested experimentally by Daoud (2003): different diameters of rebars (12, 16 and 20 mm) as well as two type of concrete were used in tests

Test BAP-12-HA BAP-16-HA BAP-20-HA BV-12-HA BV-16-HA BV-20-HA BAP-12-smooth BAP-16-smooth BAP-20-smooth

Diameter (mm)

Minimal spacing (cm)

Maximal spacing (cm)

12 16 20 12 16 20 12 16 20

14 12.3 10 12.5 11.5 10 14 15.5 14.5

19 18.7 21 21 17 17 19 21 20

5. Conclusions Our works shows that the interface finite elements allow modeling of a fundamental characteristic of reinforced concrete with local interaction between steel bars and concrete. The latter is of crucial importance to evaluation of both the integrity and durability of a reinforced concrete structure, or in particular to a reliable prediction of crack spacing and opening. Moreover, one of the most important contributions of our work is the capacity of predicting maximal and minimal spacing of macro-cracks, even if the exact location of cracks remains undetermined. Namely, by the introduction of the influence of concrete heterogeneities in numerical analysis, it is possible to show that even a small variation in concrete tensile fracture resistance can directly affect the configuration of the crack pattern distribution. However; the peak loading needed to introduce the cracking of concrete in a pull-out test is not as sensitive to the same statistical variation of concrete heterogeneities. Another contribution of this work is to reiterate on the insufficiency of the local damage model of concrete to handle this class of problems; much in the same manner as for localization problem which accompany strain-softening behavior. Effectively,

Table VIII. Maximal and minimal spacing of cracks from Daoud’s tie tests

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Figure 18. Comparison of numerical crack pattern distribution for different kinds of concrete heterogeneity, and corresponding maximal and minimal spacing for each simulation for an imposed displacement of 3 mm (rebar diameter ¼ 12 mm)

Reference of test

Table IX. Comparison of maximal and minimal spacing of cracks

Experimental Max/min spacing (cm)

BAP-12-HA Smax Smin BAP-16-HA Smax Smin BAP-20-HA Smax Smin

19 14 18.7 12.3 21 10

Numerical simulation spacing (cm) Homogeneous Deviation of Deviation of Deviation of concrete 5 percent 10 percent 20 percent 14.7 13.2 16.2 14.7 13.2 8.8

19.1 13.2 17.6 14.7 19.1 7.4

22.0 13.2 20.5 13.2 16.2 7.4

19.1 10.3 22.0 10.3 13.2 8.8

one needs even more a sufficiently reliable model for predicting the failure and post-peak response of concrete in dealing with the prediction of ultimate load state of a reinforced concrete structure. References Clement, J.L. (1987), “Interface acier-be´ton et comportement des structures en be´ton arme´ – caracterisation – mode´lisation”, PhD thesis, University of Paris VI, Paris. Daoud, A. (2003), “Etude expe´rimentale de la liaison entre l’acier et le be´ton auto-plaçant – contribution a` la mode´lisation nume´rique de l’interface”, PhD thesis, INSA de Toulouse, Toulouse. Dominguez, N., Ragueneau, F., Ibrahimbegovic, A., Michel-Ponnelle, S. and Ghavamian, S. (2003), “Bond-slip in reinforced concrete structures, constitutive modelling and finite element implementation”, paper presented at Euro-C Conference on Computational Modeling of Concrete Structures, St Johann im Pongau.

Ibrahimbegovic, A. and Brancherie, D. (2003), “Combined hardening and softening constitutive model of plasticity: precursor to shear slip line failure”, Computational Mechanics, Vol. 31, pp. 88-100. Ibrahimbegovic, A. and Markovic, D. (2003), “Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures”, Comp. Meth. Appl. Mech. Eng, Vol. 192, pp. 3089-107. Ibrahimbegovic, A. and Wilson, E.L. (1991), “A modified method of incompatible modes”, Communications in Applied Numerical Methods, Vol. 7, pp. 187-94. Ibrahimbegovic, A. and Wilson, E.L. (1992), “Unified computational model for static and dynamic frictional contact analysis”, International Journal for Numerical Methods in Engineering, Vol. 34, pp. 233-47. Ibrahimbegovic, A., Gharzeddine, F. and Chorfi, L. (1998), “Classical plasticity and viscoplasticity models reformulated: theorical basis and numerical implementation”, International Journal for Numerical Methods in Engineering, Vol. 42, pp. 1499-535. Jaccoud, J.P. (1987), “Armature minimale pour le controˆle de la fissuration des structures en be´ton”, PhD thesis, EPF Lausanne, Lausanne,. Jira´sek, M. and Zimmermann, T. (2001), “Embedded crack model: II. Combination with smeared cracks”, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 1291-305. Mazars, J. (1986), “A description of micro- and macroscale damage of concrete structures”, Journal of Engineering Fracture Mechanics, Vol. 25 Nos 5/6, pp. 729-37. Ortiz, M. and Simo, J.C. (1986), “An analysis of a new class of integration algorithms for elastoplastic constitutive relations”, International Journal for Numerical Methods in Engineering, Vol. 23, pp. 353-66. Ragueneau, F., Dominguez, N. and Ibrahimbegovic, A. (2004), “Thermodynamic-based interface model for cohesive brittle materials: application to bond-slip in RC structures”, Computer Methods in Applied Mechanics and Engineering, Vol. 5, pp. 607-25. Simo, J.C., Oliver, J. and Armero, F. (1993), “An analysis of strong discontinuity induced by strain softening solutions in rate-independent solids”, Journal of Computational Mechanics, Vol. 12, pp. 277-96. Taylor, R.L. (2004), FEAP – A Finite Element Analysis Program, Version 7.5 User Manual, available at: www.ce.berkeley.edu/ , rlt/feap/ Wells, G.N. and Sluys, L.J. (2001), “A new method for modelling cohesive cracks using finite elements”, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 2667-82. Further reading Cox, J.V. and Hermann, L.R. (1998), “Development of a plasticity bond model for steel reinforcement”, Mechanics of Cohesive-Frictional Materials, Vol. 3, pp. 155-80. De´sir, J-M., Romdhane, M.R.B., Ulm, F-J. and Fairbairn, E.M.R. (1999), “Steel-concrete interface: revisiting constitutive and numerical model”, Computers & Structures, Vol. 71, pp. 489-503. Eligehausen, R., Popov, E.P. and Bertero, V.V. (1983), “Local bond stress-slip relationships of deformed bars under generalized excitations”, Report no. UCB/EERC-83/23 of the National Science Foundation, University of California, Berkely, FA. Hughes, T.J.R. (1987), The Finite Element Method, Prentice-Hall International Editions, Englewood Cliffs, NJ.


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Lemaitre, J. and Chaboche, J.L. (1990), Mechanics of Solids Materials, Cambridge University Press, Cambridge. Ragueneau, F., La Borderie, Ch. and Mazars, J. (2000), “Damage model for concrete like materials coupling cracking and friction, contribution towards structural damping: first uniaxial application”, Mechanics of Cohesive Frictional Materials, Vol. 5, pp. 607-25. Romdhane, B. and Ulm, F.J. (2002), “Computational mechanics of the steel-concrete interface”, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 26, pp. 99-120. Salari, M.R. and Spacone, E. (2001), “Finite element formulations of one-dimensional elements with bond-slip”, Engineering Structures, Vol. 23, pp. 815-26.



On numerical implementation of a coupled rate dependent damage-plasticity constitutive model for concrete in application to high-rate dynamics

On numerical implementation

583 Received October 2004 Accepted January 2005

Guillaume Herve´, Fabrice Gatuingt and Adnan Ibrahimbegovic´ Ecole Normale Supe´rieure de Cachan, LMT-Cachan, Cachan, France Abstract Purpose – To provide an efficient and robust constitutive equations for concrete ion application to high rate dynamics. Design/methodology/approach – Develops an explicit-implicit integration scheme for a concrete model. This robust integration scheme ensures computational efficiency. Comparison between simulations of impact of equivalent aircraft engine projectiles and the tests carried out in Sandia laboratory also demonstrate its efficiency. Findings – Shows that modeling transient high rate dynamic behavior of concrete is very important to take into account for design concrete structures in the cases of dynamic loading conditions, such as an impact on the structure. Originality/value – Proposes an original integration scheme for a coupled rate dependent damage plasticity model. Also provides a detailed consideration of the numerical stability of this kind of scheme for rate-dependent damage model. Keywords Concretes, Plasticity, Stress (materials) Paper type Research paper

1. Introduction Transient high rate dynamic behavior of concrete is a very important to take into account for design concrete structures in the case of dynamic loading conditions, such as an impact on the structure. This impact loading can be due to explosion, mind blast or an accidental collision of cars, trains or airplanes with the structure. In particular for structures that involve public safety, they have to be design to resist not only the static loading but also the dynamic loading produced extreme conditions. In order to perform a 3D non-linear analysis with a wide variety of damage mechanism in concrete we need a model sufficiently robust and capable of providing a reliable representations for different loading path. But we also need to have a numerical implementation of this model firmly under control in order to ensure a robust computation. In fact, the complexity of any practically interesting model is such that a fully implicit schemes are excluded. In other words, for the numerical simulations of fast transient dynamic, such as an impact on concrete structures, one uses an explicit time integration scheme. The computer program architecture can thus be simplified accordingly, in order to provide a solution for very complex industrial application; with the code as LS-DYNA as one of the most prominent examples, which is also used herein.


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The outline of the paper is as follows. In the next section, we briefly review the salient feature on the coupled damage plasticity model for high rate dynamics of concrete, adapted to our cases of interest. Then the pertinent details of the chosen scheme for the numerical implementation is presented. In Section 4, we briefly discuss the proposed procedure for model parameters identification. Then the stability of the numerical scheme for a rate dependent damage model is discussed in Section 5. Section 6 provides several illustrative examples to show the model performance in representing different inelastic modes of damage in dynamics. Closing remarks and conclusions are given in Section 6. 2. Coupled rate dependant damage-plasticity constitutive model for concrete This model is first proposed by Gatuingt (Gatuingt and Pijaudier-cabot, 2002) to simulate explosion in contact or impact of hard projectiles at velocity less than 350 m/s. For this kind of loading three failure mechanisms have to be described. First one which is observed under the impact is a decrease of the material porosity. This phenomenon is represented with homogenization technics by considering concrete as a matrix (cement paste and aggregates) with pores. To model the penetration of a hard striker introducing a large deviatoric strain, a plastic model based on modified Gurson’s yield function (to take into account porosity evolution) is used. The third and final mechanism is supplied to handle the case where the compressive wave can reflect on a free surface producing a traction state of stress which is represented with a damage model. These mechanical effects are combined in the relationships which relate the stresses to the elastic strains: 1 e e e sij ¼ ð1 2 DÞ K1kk dij þ 2G 1ij 2 1kk dij ð1Þ 3 where the shear G and bulk moduli K of the coupled model are defined by Mori-Tanaka’s expressions: K¼

4K M GM ð1 2 f * Þ ; 4GM þ 3K M f *

G¼

GM ð1 2 f * Þ 6K M þ 12GM f* 1þ 9K M þ 8GM

ð2Þ

with KM and GM, respectively, the bulk and shear moduli of the matrix material without pores. In the case of a smooth impact (such as impact by cars or airplanes), we can consider that the porosity will not decrease enough under the projectile to induce a variation of the concrete moduli. For the study of these kind of problems we will consider in this paper that K and G remain constant. The phenomenon of microcracking, in uniaxial tension and compression, is captured with a rate dependent damage model (Dube et al., 1996). Accounting for rate effects is necessary in order to represent the type of response one finds in dynamic experiments, mostly dynamic tensile tests (Klepaczko and Brara, 2001). In addition, the added benefit of the rate dependency is to preserve the well-posedness of the equations of motion when strain softening occurs (Needleman, 1988; Sluys, 1992). The extensive program of experiments carried out within the French research network GEO showed that there was a marked dependence between the loading rate and the curve relating

the volumetric strain to the hydrostatic stress (Gatuingt, 1999). The latter is thus captured by implementing a viscoplasticity model. Within the classical framework of small strain cinematics, we use the basic assumption of additive strain decomposition: 1_ij ¼ 1_eij þ 1_vp ij

ð3Þ 1_vp ij

where 1_ij is the total strain rate, 1_eij , the elastic one and the viscoplastic strain rate. The viscoplastic strain rates are obtained following Perzyna’s approach: _ 1_vp ij ¼ l

›F NT ›sij

ð4Þ

The viscoplastic multiplier l_ is defined with the power law which also takes into account the porosity f*: f* F NT nvp _ l¼ ð5Þ ð1 2 f * Þ mvp where mvp and nvp are material parameters. The porosity evolution is controlled by the irreversible volumetric strain only according to: df * ¼ kð1 2 f * Þf * d1pkk

ð6Þ

where parameter k is introduced in equation (6) in order to be able to calibrate the velocity with which voids are closed. In equation above, FNT is the modified Gurson’s yield function proposed by Needleman and Tvergaard (1984): 3J 2 I1 F NT ðsij ; sM ; f * Þ U 2 þ 2q1 f * cosh q2 ð7Þ 2 ð1 þ ðq3 f * Þ2 Þ ¼ 0 2sM sM where I 1 ¼ TrðsÞ; the first invariant of stress tensor, J 2 ¼ kdevðsÞk; the second invariant of the deviatoric part, sM is the stress in concrete matrix without voids and q1, q2, q3 are scalars parameters. The model of this kind treats the concrete as a porous material. This porosity has a great importance on the material behavior when the hydrostatic stress contribution is not negligible. Indeed, this model improves upon the Drucker and Prager (1952) yield surface which is often used for concrete with consequences that the material remains elastic for triaxial compression, which is in contradiction with the compaction experimentally observed (Burlion et al., 2001). The main interest in the modified Gurson’s yield function is to be closed for a hydrostatic state of stress and to provide a kind of CAP model (Gatuingt and Pijaudier-Cabot, 2002, 2003; Gurson, 1977; Ibrahimbegovic et al., 2003). The constitutive response in tension and compression is controlled by the damage evolution law governed by a rate dependent model given by: 0D D ð1=bÞ E1nD 1~ e 2 1D0 2 a1 12D A ð8Þ D_ ¼ @ mD with the damage criterion of Mazars (1986)


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f ¼ 1~ e 2 k

ð9Þ

where, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X þ 2 1~ ¼ 1ei i e

586

is the elastic equivalent strain for quasi-brittle materials (Mazars, 1986). In equation (8) above mD, nD are material parameters which control the rate effect, a, b are material parameters which govern the growth of damage in quasistatic tension and compression and 1D0 is the initial value of damage threshold. Figure 1 shows the coupled plasticity-damage model response for the hydrostatic stress state for a loading/unloading cycle. We can see that the model captures both strain hardening effect with concrete compaction and the irreversible plastic residual strain upon unloading. 3. Numerical implementation Given the complexity of the constitutive relations and the main application domain which pertains to concrete in fast transient dynamics, the model has been implemented in the explicit finite element code LS-DYNA (Hallquist, 1995). The objective of the numerical implementation of the model is to be able to calculate the new state of stress at time t þ Dt knowing the increment of strain D1 and the state of strain at time t. In this computational process, the evolution of the damage and viscoplastic strain are totally decoupled in order to preserve the computational efficiency. We refer to Figure 2

Figure 1. Hydrostatic response of the model used


587

Figure 2. Numerical integration scheme

for the flow-chart of the implemented computational procedure, which provides the summary of all different cases. In a first time, we assume that all the strain increment are elastic and we compute the equivalent elastic strain: 1eij ðt þ DtÞ ¼ 1ij ðt þ DtÞ 2 1vp ij ðtÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 P e 1~ e ðt þ DtÞ ¼ 1i ðt þ DtÞ i

ð10Þ

Furthermore, we assume that all internal variables remain fixed with their rates _ equal to zero, so that we can compute the corresponding stress rate in ðf_ * ; s_ M and DÞ the elastic predictor: e 1_ij ðt þ DtÞ ¼ 1_eij ðt þ DtÞ þ 1_vp ij ðt þ DtÞ ¼ 1_ij ðt þ DtÞ 1 s_ ¼ ð1 2 DðtÞÞ K 1_kk ðt þ DtÞdij þ 2G 1_ij ðt þ DtÞ 2 1_kk ðt þ DtÞdij 3 vp 1_ ðt þ DtÞ ¼ 0; f_ * ðt þ DtÞ ¼ 0; s_ M ðt þ DtÞ ¼ 0

ð11Þ

ij

In order to verify if the correction is needed to obtain the real state of stress, we test the positivity of the elastic equivalent strain predicted. If this strain is negative, we deal with a loading path mainly in triaxial compression and the damage variable D will never evolve. We are thus concerned with either the case 1 (elasticity) or with the case 3 (only plasticity) as shown in Figure 2. In case 2, we only have a damage evolution with no plasticity. For the “coupled” case 4, we have evolutions of the damage variable

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and of the plastic strain. In this case, we first compute the new state of damage and we use it to compute the plastic correction without any subsequent iteration. For the plastic correction, we use the return mapping algorithm (Ortiz and Simo, 1986) for the plastic part of the model assuming that the damage D is known at the state t þ Dt: The latter is computed first from the evolution of the damage during the incremental process, by using an explicit scheme due to the formulation of the damage growth (equation (8)). We can then write: vp e 1_ij ðt þ DtÞ ¼ 1_eij ðt þ DtÞ þ 1_vp ij ðt þ DtÞ ¼ 0 ) 1_ij ðt þ DtÞ ¼ 21_ij ðt þ DtÞ 1 vp vp vp _ ›F NT 1_vp s_ ij ¼ 2ð1 2 Dðt þ DtÞÞ K 1_kk dij þ 2G 1_ij 2 1_kk dij ij ¼ l 3 ›sij

›F NT _ f_ * ¼ kð1 2 f * Þf * 1_vp kk ¼ lkð1 2 f * Þf * ›skk

s_ M ¼ l_

ð12Þ

Et ›F NT sij ð1 2 f * ÞsM ›sij

Substituting equation (12)3 into equation (12)2 we obtain: ›F NT ›F NT 1 ›F NT _ s_ ij ¼ 2lð1 2 Dðt þ DtÞÞ K dij þ 2G 2 dij 3 ›skk ›skk ›sij

ð13Þ

with l_ ¼ ½f * =ð1 2 f * Þ½F NT =mvp (see equation (2)) for linear viscoplasticity ðnvp ¼ 1Þ and where plastic loading ðF NT . 0Þ is implicitly assumed. The rate of change of the yield function during the relaxation process can be written as follows (Figure 3):

›F NT ›F NT _ ›F NT ·f* þ : s_ ij þ · s_ M F_ NT ¼ ›sij ›f * › sM

ð14Þ

Substituting s_ ij ; f_ * and s_ M with their evolutions in equation (12), defined as a function of the plastic multiplier l_; we obtain an ordinary differential equation to solve:

Figure 3. Principle of the return mapping

F NT f* ›F NT ›F NT ›F NT 1 ›F NT F_ NT ¼ 2 : ð1 2 DÞ K dij þ 2G 2 dij 3 ›skk mvp 1 2 f * ›sij ›skk ›sij ð15Þ ›F NT ›F NT ›F NT Et ›F NT þ · kð1 2 f * Þf * þ · sij ›f * ›skk ›sM ð1 2 f * ÞsM ›sij This kind of problem has been solved by Ortiz and Simo (1986) introducing the instantaneous relaxation time t: ›F NT ›F NT ›F NT 1 ›F NT t ¼ mvp 1 2 f * : ð1 2 DÞ K dij þ 2G 2 dij f* 3 ›skk ›s ›skk ›sij ð16Þ ›F NT ›F NT ›F NT Et ›F NT þ · kð1 2 f * Þf * þ · sij ›f * ›skk ›sM ð1 2 f * ÞsM ›sij equation (15) reduces to: F NT 2t F_ NT ¼ 2 ) ln ðF NT Þ ¼ t t

ð17Þ

The following algorithm is then applied: the elastic predictor is computed. The return path can then reach a suitably updated yield surface by means of a sequence of straight segment Dit (Figure 3), which is the instantaneous relaxation time of the linear differential equation. " !# ( i 1 2 f * ðt i Þ ›F NT ›F iNT ›F iNT 1 ›F iNT i D t ¼ mvp : ð1 2 DÞ K dij þ 2G 2 dij f * ðt i Þ 3 ›skk ›sij ›skk ›sij 2

›F iNT ›F i · kð1 2 f * ðt i ÞÞf * ðt i Þ NT ›f * ›skk

›F i E t ðt i Þ ›F iNT 2 NT · : sij ðt i Þ ›sM ð1 2 f * ðt i ÞÞsM ðt i Þ ›sij

ð18Þ )

To update the variable sij, f* and sM during the return mapping iterative process, we use:

sij ðt iþ1 Þ ¼ sij ðt i Þ þ s_ ij ðt i ÞDi t sM ðt iþ1 Þ ¼ sM ðt i Þ þ s_ M ðt i ÞDi t _ ðt i ÞDi t f * ðt iþ1 Þ ¼ f * ðt i Þ þ f*

ð19Þ

We can thus compute F iþ1 NT as well as the new values of the internal variables. The total relaxation time is obtained with the following equation:


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t iþ1 ¼ t i þ Di t · log

F iNT F iþ1 NT

ð20Þ

We obtain the exact plastic correction for F NT ¼ 0 which is obtained when the total relaxing time is equal to the real time increment Dt (Ortiz and Simo, 1986). During this internal variable computation and in particular for the elastic prediction, we can obtain numerical problem to compute the modified Gurson’s yield function (see equation (7)). This is due to the high numerical values obtained with the cosh function when the term q2 I 1 =2sM becomes large enough. This phenomenon is shown in Figure 4. We follow Mahnken (1999), who proposed to modify the expression of the hyperbolic cosine with the power series development around a critical point. We choose for critical value q2 I 1 =2sM ¼ X c ¼ 30 and we obtain the expression: q2 I 1 # X c : F NT cosh q2 I 1 2 s 2 sM M " 2 # ð21Þ q2 I 1 q I 1 q I 2 1 2 1 2s . X c : F NT coshðX c ÞþsinhðX c Þ 2s 2X c þ 2 coshðX c Þ 2s 2X c M M M This method ensures continuity under the critical point and limits the growth up of this term when X . X c .

Figure 4. Modification of the cosh function to prevent numerical overflow

Figure 5 shows the response of the model for a hydrostatic compression (where only the plastic criterion is activated) for different strain increment but with the same strain rate. We can notice that the model response is now totally independent of the strain increment. This is due to the implicit integration scheme used for the plasticity. 4. Parameters identification The main goal of this section is to give a systematic procedure in order to calibrate and choose the model parameters. We saw in Section 2 that the constitutive equations are governing with two inelastic threshold functions. The first one is based on Mazars’s work and concerns the damage evolution. The second one is the modified Gurson’s yield function for the evolution of the plastic strain. It is important to note that, the Mazars’s threshold function is activated in traction before the Gurson’s yield function, means that only the damage constitutive law is used, refer Figure 6. The representation of the yield functions was generated with MatSGen software developed at LMT-Cachan by François (2004). So if we want to calibrate the model damage parameters we have to perform a traction test in statics and in dynamics. In addition, for a triaxial compression load path, only the Gurson’s yield function is activated which means that we will use only the viscoplastic constitutive law. To calibrate the model parameters for the viscoplastic strain evolution, we have to perform the hydrostatic compression test in statics and in dynamics. On the other hand, if we are in a load path in simple compression with low confinement, we activate the two thresholds and we will have a coupled response of the constitutive equation. Nevertheless, if we choose appropriate parameters, we can obtain


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Figure 5. Hydrostatic response for different strain increment

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Figure 6. Needleman-Tvergaard and Mazars yield function in stress space

a simple compression depending only on the damage law which will permit to calibrate the damage parameters in compression. In Table I, we present a summary of the parameter values for MB50. 5. Stability of the numerical implementation 5.1 Rate dependant damage model An homogeneous material element at a macroscopic point of view deforms in a homogeneous manner if a homogeneous stress is applied at its boundaries. But when the strain becomes larger, due to the loading, for example, concentration can occur in Plasticity parameters

Table I. Model parameters for MB50

E0 n q1 q2 q3 sM0 n k mvp nvp

Values

Units

Damage parameters

Values

Units

5.5£ 1010 0.2 1.5 0.8 1 60 15 45 1.1£102 2 1

Pa – – – – MPa – – – –

E0 n 1D 0 at bt nDt mD t ac nD c mDc

3.5£ 1010 0.2 1£ 102 4 20,000 1.6 5 0.5£ 102 4 3,000 20 0.5£ 102 3

Pa – – – – – – – – –

one element which lead to localization of strain over more or less extended area and the deformation of the considered element ceased to be homogeneous. A good way for describing localization in terms of continuum theory is the strain rate discontinuity (Rudnicki and Rice, 1975; Rice, 1976; Rice and Rudnicki, 1980). The localization implies a non uniqueness in the incremental response of a homogeneous, homogeneously strained body and also implies a vanishing speed of acceleration waves (Hadamar, 1903; Hill, 1962) The equation of the damage model is:

s ¼ ð1 2 DÞE 1

ð22Þ

with the evolution law of the damage governed by a rate dependent model (refer equation (8)) considering nD ¼ 1: This is an evolution problem which is solved by a time integration scheme. This latter provides a discretized solution of this evolution problem, which can be written as: div snþ1 ¼ 0

snþ1 ¼ ð1 2 Dnþ1 ÞE1nþ1 ¼ Gð1nþ1 ; Dnþ1 Þ DD ¼ Dnþ1 2 Dn ¼ Dt D_ nþ1 ¼ Fð1nþ1 ; Dnþ1 Þ

ð23Þ

This is a non-linear problem which can be solved by the Newton scheme. We have to ensure that this remains a well-posed problem by verifying the positiveness of the consistent tangent operator. Therefore, we linearize the non-linear evolution laws:

dsnþ1 ¼

›G ›G : d1nþ1 þ dDnþ1 ›1nþ1 ›Dnþ1

dDnþ1 ¼

›F ›F : d1nþ1 þ dDnþ1 ›1nþ1 ›Dnþ1

ð24Þ

With the equation (24)2 we have:

dDnþ1

›F : d1nþ1 ›1 ¼ nþ1 ›F 12 ›Dnþ1

ð25Þ

Using this expression of dDnþ 1 in equation (24)1 we can write:

dsnþ1 ¼ H : d1nþ1

ð26Þ

›G ›F ^ ›G ›D ›1nþ1 H¼ þ nþ1 ›F ›1nþ1 12 ›Dnþ1

ð27Þ

with

the tangent modulus of the non-linear evolution law. In our case, we have:


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›G ¼ ð1 2 Dnþ1 ÞE ›1nþ1 ›G ¼ 2E1nþ1 ›Dnþ1

594 ›F ›Dnþ1

›F Dt k1nþ1 lþ ¼ ›1nþ1 mD 1~nþ1 ð1=bÞ Dnþ1 1 Dt a 1 2 Dnþ1 Dt ¼ 2a ¼2 · mD bDnþ1 ð1 2 Dnþ1 Þ mD

ð28Þ

which leads to: k1nþ1 lþ 1~nþ1 H ¼ ð1 2 Dnþ1 ÞE þ ð29Þ Dt 1þa mD In order to solve the problem, we have to verify the equilibrium for the linearized problem: 2E1nþ1 ^ mDtD

div dsnþ1 ¼ 0

ð30Þ

which is a well-posed problem under the condition: det nH n ¼ 0

ð31Þ

With the tangent modulus obtained in equation (29) we obtain from the last expression: 2 3 6 det6 4ð1 2 Dnþ1 ÞnEn 2

7 Dt nðE1nþ1 ^ k1nþ1 lþ Þn7 5¼0 Dt mD · 1~nþ1 · 1 þ a mD

this is verified for (Bigoni and Hueckel, 1991):

mD · 1~nþ1 · 1 þ a mDtD ¼ ðk1nþ1 lþ nÞ{nEnð1 2 Dnþ1 Þ}21 ðE1nþ1 Þ Dt

ð32Þ

ð33Þ

5.2 Application to 1D bar in traction For a 1D bar in traction, we can obtain a simple form of the tangent modulus: Dt=mD ð34Þ Dt 1þa mD For this simple case, the stability condition is H ¼ 0: With this condition, we can obtain a stability time step Dt: H ¼ ð1 2 Dnþ1 ÞE 2 E1nþ1

Dt ¼

md 1nþ1 =ð1 2 Dnþ1 Þ 2 a

ð35Þ

Figure 7 shows the reduction of the critical time step with the damage increase. This curve has been obtain for model parameters identify for concrete MB50, with the values of parameters given in Table I. We can observe that for these values, the critical time step is big enough to ensure stability of the numerical scheme even at the end of the damage evolution.


595

6. Numerical simulations 6.1 Patch test In order to see if the model was implemented correctly in the FE code LS-DYNA, we decided to test two different meshes (one regular and one distorted; see Figure 8), under a homogeneous stress field – the classical path test (refer Zienkiewicz and Taylor, 2000) carried out in dynamics. First we carried out traction tests on a regular mesh cube and a distorted mesh one in order to check if the damage phenomena exhibit mesh dependency, refer Figure 9. The stress vs strain diagram clearly indicates no mesh dependency as shown in Figure 10, with two curves which remain in very good accordance. Then we carried out the same kind of tests for an hydrostatic state of stress as shown in Figure 11, with the same purpose to verify the good accordance of the model response for different meshes. Figure 12 showing the hydrostatic pressure vs

Figure 7. Critical time step vs damage

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Figure 9. Patch test in traction

voluminal strain curves for regular and distorted mesh indeed confirms good accordance in both cases. 6.2 Reinforced concrete slab Another test of the model capabilities was to carry out a large scale computation. As it would be dedicated to simulations of soft impact phenomena and especially aircraft impact, we carried out the computation of impact between 747 class Boeing like aircraft and a reinforced concrete slab. The slab was a parallelepiped of 80 £ 80 m large and 80 cm thick. The concrete within was a classical one of 25 MPa ultimate compressive stress. The aircraft impacted the slab following a 208 angle, at 252 m/s this means 900 kph the ultimate speed of this kind of aircraft in such a configuration, as it is shown in Figure 13. The finite element model of the aircraft was constructed by a mesh of Belytschko-Lin-Tsay shell elements, the slab is constituted of under-integrated (one Gauss point) brick elements. The automatic contact surface to surface LS-DYNA option for contact was used in this simulation. The reinforcement was represented with beams and a perfect contact between concrete and steel bar was assumed, by merging nodes between beams elements and 3D elements. Reinforcement ratio was 0.4 percent in each direction.


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Figure 10. Traction response for the two different meshes

Figure 11. Patch test in hydrostatic compression

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Figure 12. Hydrostatic response for the two different meshes

Figure 13. Configuration of plane and slab before impact

The results presented in Figure 14 show the evolutions of the damage on the impacted face and the opposite one. When damage reaches the maximum value of 1, the concrete is locally fully destroyed. Thus, we can conclude in this case that a huge crater has occurred on the top face of the slab, yet we can assume that the scabbing on the rear face is limited to a 15 £ 15 m area containing nearly seven disseminated scabbed discs. According to the 0.25 damage value on the half of the opposite face, we can assume that half of rear face is considerably weakened with the presence of several cracks.


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Figure 14. Damage evolution during the impact

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Figure 15. The six different missiles used in Sandia tests

Figure 16. The three different kind of impacted slabs in Sandia tests

6.3 Sandia’s Laboratory tests simulations In order to check the relevance of the constitutive relations we carried out simulations of impact tests performed in Sandia Laboratory by Sugano et al. (1993a, b). Following the test program, we simulate impact of an aircraft engine equivalent missiles, considering several sizes. In particular, we carried out simulations of LED, MED and SER missiles which means, respectively, Large size Equivalent and Deformable missile, Medium size Equivalent and Deformable missile and Small size Equivalent Rigid missile. The details for the selected three missiles are shown in Figure 15. The missiles impacted three slabs, with a particular choice depending on the kind of missiles. The slabs are shown in Figure 16, from left to right with the two slabs placed in a box have been used for small size and large size missile tests. Among several impact tests, carried out by Sugano et al., we chose to simulate impact of those exposed in Table II. In our simulations missiles where represented by using the finite element models constructed with Belytschko-Lin-Tsay shell elements, except for SER missile, whose

mesh is built with 3D elements. The concrete part of the slab was represented with under-integrated 3D elements, the reinforcement was represented with truss-bars. In each case reinforcement ratio was 0.4 percent in each direction. The qualitative results of numerical simulations, the same as those obtained in the tests, are shown in Table III; we note that penetration means that the missile penetrated the slab without having gone through it, perforation means that missile went through, scabbing means that the impact generated a scab on the rear face of the slab. All computations where stopped when the velocity of the missile stopped decreasing, this means when the missile finally got stuck in the concrete, or when concrete did not bring any resistance to its penetration, since the slab was totally damaged. The damage fringes of the simulations at the end of the computations are exposed in Figures 17-19, respectively, for impact tests S10, S28 and L5. In Figure 17, we can observe that the damage is not enough continuous to let the missile goes through, the damage fringe on the rear face exposes that a scabbing is occurring. In Figure 18, the damage fringe is continuous through the slab thickness under the impacted zone, thus we can conclude that a perforation has occurred and a scabbing too. In Figure 19, we see that the damage is localized in the impact area, a large undamaged volume separates damage on the rear face and damage on the front face. We can here conclude on a penetration with maybe some cracks or a small scab. No.

Missile type

S10 S28 L5

SER SED LED

Missile velocity (m/s)

Slab thickness (m)

Reinf. ratio

Slab type

141 196 214

0.15 0.06 1.60

0.4 0.4 0.4

Small #1 Small #1 Large #3

No.

Perforation

Scabbing

Penetration

S10 S28 L5

No Yes No

Yes Yes Some cracks

Yes Yes Yes


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Table II. Simulated tests characteristics

Table III. Tests results for each impact

Figure 17. Simulations of S10 test – Top, bottom and side views

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Figure 18. Simulations of S28 test – Top, bottom and side views

Figure 19. Simulations of L5 test – Top, bottom and side views

Considering these observations and Table III, we can conclude that obtained a very good accordance with the tests. The latter also confirms the validation of the constitutive equations that had been developed and also their implementation. 7. Conclusions We have presented a very careful consideration of the numerical implementation aspects for the constitutive model of coupled damage-plasticity, within the framework of explicit-implicit scheme used by computer code LS-DYNA. The proposed integration scheme can be considered as optimal one for the given problem, in which we combine an explicit solution of global equations of motion, with the implicit solution of local evolution equations, which ensures both the computational efficiency and admissible values of stress field during the damage computation. The model is especially suited for representing different damage mechanisms which develop in concrete under high rate loading, both in tension and compression. The former pertains to tension cut-off damage criterion, whereas the latter is represented by the plasticity model with Gurson-like criterion, with a particular feature of accounting for hardening due to concrete compaction. Several illustrative numerical examples are presented in order to show a very satisfying performance of the proposed model. The most prominent is the example showing very good comparison with the experimental results obtained in Sandia’s laboratory for impact problems on reinforced concrete slabs.

References Bigoni, D. and Hueckel, T. (1991), “Uniqueness and localization – I. Associative and non-associative elastoplasticity”, Int. J. Solids Struct., Vol. 28, pp. 197-213. Burlion, N., Pijaudier-Cabot, G. and Dahan, N. (2001), “Experimental analysis of compaction of concrete and mortar”, Int. J. Numer. Anal. Method. Geomech., Vol. 25, pp. 1467-86. Drucker, D.C. and Prager, W. (1952), “Soil mechanics and plastic analysis of limit design”, Quart. Appl. Math., p. 10. Dube, J.F., Pijaudier-Cabot, G. and La Borderie, C. (1996), “A rate dependent damage model for concrete in dynamics”, J. Eng. Mech. ASCE, Vol. 122, pp. 939-47. François, M. (2004), “MatSGen: user guide”, Note interne du LMT-Cachan. Gatuingt, F. (1999), “Pre´vision de la rupture des ouvrages en be´ton sollicite´s en dynamique rapide”, The`se de doctorat de l’ENS de Cachan, Cachan. Gatuingt, F. and Pijaudier-Cabot, G. (2002), “Coupled damage and plasticity modelling in transient dynamic analysis of concrete”, Int. J. Numer. Anal. Method. Geomech., Vol. 26, pp. 1-24. Gatuingt, F. and Pijaudier-Cabot, G. (2003), “Gurson’s plasticity coupled to damage as a CAP Model for concrete compaction in dynamics”, Constitutive Modelling of Geomaterials, CRC Press, Boca Raton, FL, pp. 12-24. Gurson, A. (1977), “Continuum theory of ductile rupture by void nucleation and growth: part I – yield criteria and flow rules for porous ductile media”, J. Eng. Mater. Tech., Vol. 99, pp. 2-15. Ibrahimbegovic, A., Markovic, D. and Gatuingt, F. (2003), “Constitutive model of coupled damage-plasticity and its finite element implementation”, Eur. J. Finite Elem., Vol. 12, pp. 381-405. Hadamar, J. (1903), “Lecons sur la Propagation des Ondes et les Equations de l’Hydrodynamique”, Hermann, Paris. Hallquist, J.O. (1995), “LS-DYNA 3D – Theoretical manual”, Livermore Software Technology Corporation, Livermore, CA. Hill, R. (1962), “Acceleration waves in solids”, J. Mech. Phys. Solids, Vol. 10, pp. 1-16. Klepaczko, J.R. and Brara, A. (2001), “An experimental method for dynamic tensile testing of concrete by spalling”, Int. J. Impact Eng., Vol. 25, pp. 387-409. Mahnken, R. (1999), “Aspects on the finite-element implementation of the Gurson model including parameter identification”, Int. J. Plasticity, Vol. 15, pp. 1111-37. Mazars, J. (1986), “A description of micro and macroscale damage of concrete structures”, J. Eng. Fract. Mech., Vol. 25, pp. 729-37. Needleman, A. (1988), “Material rate dependence and mesh sensitivity in localization problems”, Comput. Method. Appl. Mech. Eng., Vol. 67 No. 1, pp. 69-85. Needleman, A. and Tvergaard, V. (1984), “An analysis of ductile rupture in notched bars”, J. Phys. Mech. Solids, Vol. 32, pp. 461-90. Ortiz, M. and Simo, J.C. (1986), “An analysis of a new class of integration algoriths for elastoplastic constitutive relations”, Int. J. Numer. Method. Eng., Vol. 23, pp. 353-66. Rice, J.R. (1976), “The localization of plastic deformation”, Theoretical and Applied Mechanics, North-Holland, Amsterdam. Rice, J.R. and Rudnicki, J.W. (1980), “A note on somes features of the theory of localization of deformations”, Int. J. Solids Struct., Vol. 16, p. 597.


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Rudnicki, J.W. and Rice, J.R. (1975), “Conditions for the localization of deformations in pressure-sensitive dilatant materials”, J. Mech. Phys. Solids, Vol. 23, p. 371. Sluys, L.J. (1992), “Wave propagation, localisation and dispersion in softening solids”, Doct. Dissertation, Delft University of Technology, Delft,. Sugano, T., Tsubota, H., Kasai, Y., Koshika, N., Ohnuma, H., von Riensemann, W.A., Bickel, D.C. and Parks, M.B. (1993a), “Local damage to reinforced concrete structures caused by impact of aircraft engine missiles – part 1. Test program, method and results”, Nucl. Eng. Des., Vol. 140, pp. 387-405. Sugano, T., Tsubota, H., Kasai, Y., Koshika, N., Ohnuma, H., von Riensemann, W.A., Bickel, D.C. and Parks, M.B. (1993b), “Local damage to reinforced concrete structures caused by impact of aircraft engine missiles – part 2. Evaluation of test results”, Nucl. Eng. Des., Vol. 140, pp. 407-23. Zienkiewicz, O.C. and Taylor, R.L. (2000), “The finite element method – volume 1: the basis”, Butterworth and Heinemann, Elsevier Science, Barking. Further reading Burlion, N., Gatuingt, F., Pijaudier-Cabot, G. and Daudeville, L. (2000), “Compaction and tensile damage in concrete: constitutive modelling and application to dynamics”, Comput. Method. Appl. Mech. Eng., Vol. 183, pp. 291-308.



Shape optimization of two-phase inelastic material with microstructure Adnan Ibrahimbegovic´ Ecole Normale Supe´rieure de Cachan, LMT-Cachan, Cachan, France

Igor Gresˇovnik

Shape optimization

605 Received October 2004 Accepted January 2005

Ecole Normale Supe´rieure de Cachan, LMT-Cachan, Cachan, France C3M – Center for Computational Continuum Mechanics, Ljubljana, Slovenia

Damijan Markovicˇ Ecole Normale Supe´rieure de Cachan, LMT-Cachan, Cachan, France Naravoslovnotehnisˇka fakulteta, Univerza v Ljubljani, Ljubljana, Slovenia

Sergiy Melnyk Ecole Normale Supe´rieure de Cachan, LMT-Cachan, Cachan, France, and

Tomazˇ Rodicˇ Naravoslovnotehnisˇka fakulteta, Univerza v Ljubljani, Ljubljana, Slovenia C3M – Center for Computational Continuum Mechanics, Ljubljana, Slovenia Abstract Purpose – Proposes a methodology for dealing with the problem of designing a material microstructure the best suitable for a given goal. Design/methodology/approach – The chosen model problem for the design is a two-phase material, with one phase related to plasticity and another to damage. The design problem is set in terms of shape optimization of the interface between two phases. The solution procedure proposed herein is compatible with the multi-scale interpretation of the inelastic mechanisms characterizing the chosen two-phase material and it is thus capable of providing the optimal form of the material microstructure. The original approach based upon a simultaneous/sequential solution procedure for the coupled mechanics-optimization problem is proposed. Findings – Several numerical examples show a very satisfying performance of the proposed methodology. The latter can easily be adapted to other choices of design variables. Originality/value – Confirms that one can thus achieve the optimal design of the nonlinear behavior of a given two-phase material with respect to the goal specified by a cost function, by computing the optimal form of the shape interface between the phases. Keywords Optimization techniques, Structures, Modelling Paper type Research paper

The authors gratefully acknowledge the financial supports by the European Commission, through a Marie Curie Fellowship (contract number HPMF-CT-2002-02130), the French Ministry of Research and the Slovene Ministry of Education, Science and Sport. The work of AI was supported by Alexander von Humboldt Foundation.


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1. Introduction Ever increasing demands to achieve more economical design of a given material result in the need to exploit and analyze its inelastic non-linear behavior. The latter can be formally placed under control by micro-macro models (Ibrahimbegovic and Markovic, 2003), where one goes down to micro-scale in order to obtain a more reliable interpretation of the mechanisms governing the inelastic behavior. This kind of approach opens not only numerous possibilities to obtain a better description of the inelastic behavior of a material than the classical phenomenological models, but also allows one to consider very fine details of the material microstructure and design the one which is the most suitable for a given goal. A number of possibilities can be easily imagined in that respect, from designing a material which will reduce as much as possible the damage in the given zone, thus increasing the durability of the structure, all the way to designing a material which will maximize the damage in a given zone, where it is important to concentrate energy dissipation in a structure. The design procedure is called upon to guide and accomplish this task. The desired goal is set in terms of the objective cost function (Kleiber et al., 1997), dependent upon the design variables, which can be either geometric or mechanic parameters of the material and its microstructure. The former case, which is of main interest for the work described herein, is more demanding in terms of the solution procedure requirements. Therefore, a novel approach is sought herein with respect to the classical methods (Tortorelli and Michaleris, 1994; Tsay and Arora, 1990) where one separates the optimization problem from the mechanics problem and reduces the communication between the two to the sensitivity computations. The solution procedure for coupled optimization-mechanics problem relies on the method of Lagrange multipliers to bring the two problem ingredients on the same level, which allows much greater flexibility in subsequent solution steps. Another important contribution regarding the solution procedure concerns the phase interface shape representation, which allows a very efficient computation. The details of the solution procedure are presented for the chosen model problem of two-phase material, with one phase as plasticity and another phase as damage. However, the proposed procedure can easily be adapted to other cases of practical interest. The outline of the paper is as follows. In the next section we briefly review the micro-macro representation of the chosen model problem of two-phase material. A number of practical materials, such as porous metals or concrete, belong to this category. In Section 3 we present the solution procedure for the coupled optimization-mechanics problem. In Section 4 we describe the details of the microstructure representation and the interface shape parameterization. Several numerical examples are presented in Section 5 in order to illustrate a very satisfying performance of the proposed approach. Concluding remarks are stated in Section 6. We also supply an Appendix to explain the details of the proposed approach more clearly in the simple 1D setting. 2. Model problem: micro-macro model of two-phase material To fix the ideas on multi-scale modeling of inelastic behavior of materials proposed herein, we consider a model problem presented in Figure 1, which is very much representative of standard three-point bending tests, very often used for brittle materials. When trying to interpret the test results in the range of inelastic analysis

and identify more easily the dissipative mechanisms, this would lead us to take into account the details of the material microstructure, and to carry out micro-macro inelastic analysis. 2.1 Micro-macro modeling approach at coupled scales In targeting the most general application domain, we consider the problem in multiscale analysis of inelastic behavior for the case where the scales remain strongly coupled, imposing the constant communication between the scales. More precisely, in order to more easily identify a particular failure mechanism and its evolution, one is constantly obliged to go down to micro-scale before advancing the computations at the macro scale. The micro-scale can be as small as 1 mm for metals or as large as 1 cm for concrete, so that “micro-scale” terminology should only be interpreted in the relative sense as being much smaller than the macroscale characterizing the structural dimension. It can happen, such as for large aggregate concrete material, that the ratio between macroscale and micro-scale is not big enough in order to justify the classical scale separation hypothesis, which would allow the computation at the micro-scale to be carried out in advance. Two-scale finite element model can be constructed for this kind of problem as shown in Figure 1, where the finite element representation is provided for each scale. Without loss of generality we assume that all the internal variables are defined only at the micro-scale. The latter implies that the state variables can be written as: displacements at the macroscale, u M, displacements at the micro-scale, u m and the internal variables governing the evolution of inelastic dissipation at the micro-scale, collectively denoted as v. The particular model problem we consider herein pertains to a two-phase material, with the first phase or the matrix represented by a plasticity model and the second phase, represented by a damage model. The set of internal variables therefore consists of plastic strain 1 p, hardening variable for plastic phase j p, damage compliance D and damage hardening variable j d. In order to specify the evolution equations of these internal variables we choose the deviatoric plasticity model of the matrix and a simple damage criterion proportional to the spherical stress for the second phase, with a vanishing value of fracture stress in the case when we model inclusions (Ibrahimbegovic et al., 2003). The irreversible nature of the evolution of these internal variables obliges us to carry out an incremental solution procedure, by using one-step time-integration scheme. For a typical time step of one such scheme between ti and tiþ 1 we can write the central problem of multiscale analysis as follows:

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Figure 1. Micro-macro model of the three-point bending test with macroscale finite element mesh composed of a number of micro-scale finite elements with the exact finite element representation of the material microstructure

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Central problem of multiscale analysis Given the state variables at time t i ; m p p d u i ¼ uM i ; ui ; v i ¼ ð1 ; j ; D; j Þ and Dt ¼ t iþ1 2 t i :

608

Find the corresponding values at time t iþ1 ; m u iþ1 ¼ uM iþ1 ; uiþ1 ; v iþ1 ;

ð1Þ

such that the weak form of equilibrium equation is satisfied at both scales m G uM iþ1 ; uiþ1 ; v iþ1 ; w ¼ 0 and internal variables evolutions are supplied over time step v iþ1 ¼ v i þ Dt½g_iþ1 ›F=›v iþ1 ; Fbeing the yield/damage criterion, g the plastic/damage multiplier (Ibrahimbegovic et al., 2003) and w the weighting function. In the above definition we refer to our recent works (Ibrahimbegovic and Markovic, 2003; Markovic et al., n.d.) for different forms of setting up and solving the equilibrium equations depending upon the chosen scale coupling for either displacement or stress based interface and different microstructure representations. The crucial point in solving these equations pertains to intrinsically different nature of state variables and the displacement field; namely, the weak form features the spatial displacement derivatives and therefore requires the displacement continuity over the boundaries of the micro-scale elements ui ¼ N a ðhj Þuai :

ð2Þ

This choice results in a large coupled set of equilibrium equations to be solved at the level of each macroscale element, defined as the corresponding assembly of micro-scale elements. On the other hand, no derivatives appear on the internal variables and, for that reason, only independent element-wise values can be used; one typically employs the Gauss quadrature point values, so that the interpolations can formally be defined as 1 p ¼ dðh 2 h a Þ1ap

D ¼ dðh 2 h a ÞDa

j p ¼ dðh 2 h a Þjap

j d ¼ dðh 2 h a Þjad

1_ p ¼ dðh 2 h a Þ1_ap

D_ ¼ dðh 2 h a ÞD_ a

p j_ p ¼ dðh 2 h a Þj_a

d j_ d ¼ dðh 2 h a Þj_a

g_ p ¼ dðh 2 h a Þg_ ap

g_ d ¼ dðh 2 h a Þg_ ad ;

ð3Þ

where d(·) denotes the Dirac delta function and h a are abscissas of the chosen Gauss quadrature rule. The central problem is thus transformed in a very large set of independent equations, namely those for internal variable evolution written independently for each Gauss point of micro-scale elements, along with a smaller set of equilibrium equations for each micro-scale element. All the micro-scale elements contributions are assembled and solved for at the level of a single macroscale element, before solving global set of equilibrium equations, which is obtained by the standard finite element assembly procedure of macroscale elements contributions. The iterative analysis of this kind is driven either by imposed displacement (for displacement based coupling, see Ibrahimbegovic and Markovic, 2003) or imposed stress (for stress based coupling, see Markovic et al., n.d.). Once this analysis has converged we can carry on with a next iterative step at the macroscale. Therefore, the macroscale analysis amounts to formally the same procedure as the standard, single scale finite element analysis, with the only difference related to the manner in which we compute the stiffness matrix and residual vector of these elements, which are available only once the micro-scale computation is carried out. As shown by Markovic et al. (2004), the micro-macro solution procedure just described is ideally suited for parallel computations, which allows solving problems with a very large number of unknowns. 2.2 Microstructure representation: exact versus structured mesh representation The computational framework presented in the previous section relies on the finite element representation of the microstructure in order to explain the failure mechanism. Among a very large number of different possibilities we chose herein a model problem of two-phase material where a plasticity model can describe the inelastic behavior of one phase and the inelastic behavior of the other phase can be represented by a damage model. One can find a number of real materials whose inelastic behavior can be described by a two-phase model of this kind, from the porous metals with damage phase with a vanishing value of damage stress representing the voids, to concrete material where the cement paste behavior is described by a plasticity model and the aggregate behavior is described by a damage model. Moreover, for the chosen model problem we select a simple microstructure shown in Figure 2(a), where the damage phase occupies a region of circular shape surrounded by the plastic phase spreading to the boundaries of the square cell corresponding to the representative volume element. A slight modification of this microstructure is also considered where the damage phase would occupy a domain of the elliptic shape centered within the square periodic cell, several such cells forming a single macroscale element. We will first consider the finite element representation of this microstructure, which is referred to as “exact”, in the sense that the finite element mesh is exactly adjusted to the domains occupied by each phase, so that every micro-scale finite element corresponds to a sub-domain occupied by only one phase. Any particular micro-scale finite element will thus contain a homogeneous domain, so that the computations can be carried out in completely standard manner. We can also consider different convenient representations of such a microstructure, which is constructed by using a structured finite element mesh shown in Figure 2(b), where each finite element is of a rectangular shape and the same size, and therefore one micro-scale finite element domain can be shared between both phases. The standard

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Figure 2. Finite element representations of the two-phase material microstructure

computational procedure for each micro-scale finite element can no longer be used to guarantee the sufficient result accuracy and one has to consider different enhancements. Three different possibilities exist, such as: (1) Gauss numerical point (GNP) filtering (Wriggers and Zohdi, 2001); (2) incompatible mode representation (Ibrahimbegovic and Markovic, 2003); and (3) stress based representation (Markovic et al., n.d.). The accuracy of these structured mesh representations can be brought to the level the exact microstructure representation, only with the last two. 3. Solution procedure of coupled analysis and optimisation 3.1 Lagrange multiplier method basis for coupled solution procedure The classical optimization procedure, pertaining to the design of engineering structures, can be extended to the presented class of problems in order to provide the optimal design of a composite material. The notion of desired, optimal performance is more precisely specified in mathematics language in terms of the cost or objective function. This cost function is specified in terms of so-called design variables, which are used to define either geometric and/or material properties of the structure in its initial configuration (Kleiber et al., 1997). Some examples of cost functions for structures involve weight, strength or amplitude of the stress field. Any of these criteria can be exploited in designing the optimal behavior of a particular composite material, but one can also devise a number of new, less frequently used choices for the cost function that would specify better the desired inelastic behavior of a given material. One such example is related to a very important issue of material durability, where one would seek the composite material arrangement that would limit inelastic behavior to a minimum. Another example of this kind concerns the materials used in vibration isolation system in order to reduce the structural damage in structures, where one would seek to maximize the damage in the isolation layers. For either of these cases a very suitable choice for the cost function

is the inelastic dissipation. For a two-phase model material we consider herein, the total dissipation is the sum of the plastic and damage dissipation, which can be expressed according to (Ibrahimbegovic et al., 2003) 1 D_ d þ D_ p ¼ sD_ s þ q d j_ d þ s1_ p þ q p j_ p : 2

ð4Þ

where the first two terms express the damage and the last two the plastic dissipation at each point with the known values of the internal variables and their evolution. In equation (4), s is the stress tensor, whereas q d and q p are stress-like variables that describe hardening effects for damage and plasticity phase, respectively. The design problem in the classical sense can then be interpreted as the constrained minimization of cost function j(·) in terms of design variables p, which can be written as follows: ð5Þ p ¼ p* ; minGð...Þ¼0 jðp; ·Þ ; where the constraints pertain to the weak form of the equations governing the equilibrium at both macro- and micro-scale, as well as the evolution equations of the internal variables, as specified in the previous section. The cost function can be defined as minimizing the dissipation (when we want to ensure durability) or minimizing the negative of dissipation (when we want to ensure concentration of inelastic behavior in an isolation device) in a given region of the structure throughout the loading time history, which can be written as Z TZ ðD_ p þ D_ d ÞdV dt ð6Þ jðp; uðpÞ; vðpÞÞ ¼ 0

V

Rather than adopting the classical formulation of the optimization problem (Kleiber et al., 1997), we follow the previous work by Ibrahimbegovic and Knopf-Lenoir (2003) or Ibrahimbegovic et al. (2004) to make use of the Lagrange multiplier technique to eliminate the constraint in equation (5). In this process we bring the mechanics equations at the same level as the cost function. In other words, the state variables to perform mechanics analysis are also brought to the same level as the design variables and, contrary to the statement in equation (6), the state variables can now be considered as independent from the design variables. The latter implies that the solution procedure can be carried out in any chosen order, and it can thus become quite different from the classical optimization computation. The coupled analysis and optimization problem of this kind can be formulated by introducing the Lagrangian functional mind;u;V maxl Lðp; u; V ; lÞ ¼ J ðp; u; V Þ þ Gðp; u; V ; lÞ;

Shape optimization

ð7Þ

where l is the set of the Lagrange multipliers enforcing different constraints in micro-macro mechanics model, containing both the local multipliers l v for internal variables and global ones l eq for displacement field. The second term on the right hand side in equation (7) takes the same form as the weak form of the governing equilibrium and evolution equations, with the corresponding Lagrange multiplier replacing the variations of the state variables.

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The Kuhn-Tucker optimality conditions corresponding to the Lagrangian functional in equation (7) can be written as follows: Variation with respect to the Lagrange multipliers returns the corresponding macroscale and micro-scale equilibrium and internal variable evolution equations of the problem:

›L ¼G¼0 ›l

612

ð8Þ

Variation with respect to the internal variables provides a set of equations to solve at each Gauss point:

›L ›J ›G ¼ þ lV ¼0 ›V ›V ›V

ð9Þ

Variation with respect to the displacement field features the tangent operator:

›L ›J ›G ¼ þ l eq ¼0 ›u ›u ›u

ð10Þ

Variation with respect to the design variables will typically couple all the variables and lead to the most elaborate equation:

›L ›J ›G ¼ þl ¼0 ›p ›p ›p

ð11Þ

One can further simplify these equations in view of providing their discrete approximation. In particular, the choice of discrete approximation of the Lagrange multipliers is equivalent to the discrete approximations chosen for the variations of corresponding state variables, which is being replaced by the corresponding Lagrange multipliers in the Lagrangian functional in equation (7). For example, the inter-element continuity requirement at the micro-scale is imposed on the Lagrange multipliers for equilibrium equations with

leq i ¼

2 X

N b ðhi Þleq i;b :

ð12Þ

b¼1

Similarly, the discrete approximations of the Lagrange multipliers for internal variables are picked up so as to reduce their contributions to Gauss quadrature points only with p

p

d

d

F l F ¼ dðh 2 h a ÞlF ¼ dðh 2 h a ÞlF a ; l a ;

l D ¼ dðh 2 h a ÞlDa ;

p

p

l j ¼ dðh 2 h a Þlja ;

p

l 1 ¼ dðh 2 h a Þl1a d

p

d

lja ¼ dðh 2 h a Þlja;n

ð13Þ

The Lagrangian functional in equation (7) can then be written to indicate explicitly all the independent variables _ j_ d ; g_ d ; lÞ: L ¼ Lðp; 1ðuÞ; 1 p ; j p ; D; j d ; 1_ p ; j_ p ; g_ p ; D;

ð14Þ

This number of the independent variables can further be reduced by taking into account the finite difference approximations for time derivatives of all the internal variables, which is in accordance with the backward Euler time integration of the corresponding evolution equations on internal variables in the central problem in equation (1) 1_ap;n ¼ p j_a;n ¼

1ap;n 2 1ap;n21 t n 2 t n21

jap;n 2 jap;n21 tn 2 t n21

Da;n 2 Da;n21 D_ a;n ¼ tn 2 t n21 d j_a;n ¼

jad;n 2 jad;n21 t n 2 t n21

Shape optimization

613 ð15Þ

By making use of these finite difference approximations we can reduce the number of Kuhn-Tucker optimality conditions by four; more precisely, by employing the chain rule in order to express the variations with respect to internal variable rates with respect to the variations of the internal variables, the result in equation (9) can be restated as ›L 1 ›L ›L 1 ›L p þ þ ¼ 0; d1 dD ¼ 0 ›1 p Dt n ›1 p ›D Dtn ›D_ p ð16Þ ›L 1 ›L ›L 1 ›L p d þ j ¼ 0; þ j ¼ 0 d d ›j p Dtn ›j_ p ›j d Dtn ›j_ d Having thus reduced the number of unknowns and their domain of definition to a minimum needed, the solution procedure can be started. The preferred order we choose is to first solve for the all internal variable increments at all Gauss points from equation (16) above to obtain DV d and DV p ; this kind of computation also implies solving for the stress plastic and/or damage admissibility conditions Fp ¼ 0 and Fd ¼ 0: Solving equation (8) for micro-scale and macroscale displacement incremental fields Du is carried out next, by using the multiscale solution procedure, as described in the previous section. The optimization loop starts by solving from equation (10) for all Lagrange d p multipliers Dl eq ; Dl V and Dl V : The latter reduces to a linear problem, as the consequence of the choice of the dual formulation in equation (7). The last solution concerns the new increment of the design variables Dp computed from equation (11). For clarity, we provide in the Appendix all the details of this solution procedure for a 1D case of the optimal design of composite truss-bar with one part built of plastic and the rest of damage material. 3.2 Software architecture for coupling of analysis and optimization The entire solution procedure is naturally divided into two parts. The inner part consists of solution of the mechanical problem and calculation of the objective and constraint functions for given values of the design parameters, and the outer part consists of solving for optimal design variables by using the solution of the inner problem. The multi-scale solution procedure for the mechanical problem has been implemented in the finite element environment “FEAP” (Taylor, 2004). The procedure has been parallelized in such a way that the problems at the

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microscopic level (each corresponding to one macro element) are solved simultaneously over a heterogeneous network of computers (Markovic et al., 2004). This speeds up the numerical solution of the mechanical problem enough that performing optimization on top of it becomes feasible. Solution of outer part was performed by the optimization program “Inverse” (Gresˇovnik, 2000; Rodicˇ and Gresovnik, 1998; Gresˇovnik, n.d.). This program has been designed for linking optimization algorithms and other tools through a suitable interface with numerical analysis environments. It is built around interpreting language that enables flexible and transparent access to the implemented functionality for setting up the solution schemes for specific problems. The concept has been confirmed on a large variety of problems, including many in the field of metal forming (Gresˇovnik, 2000; Gresˇovnik and Rodicˇ, 1999, 2003) where numerical analyses involve highly non-linear and path dependent material behavior, large deformation, multi-body contact interaction and consequently large number of degrees of freedom. “Inverse” carries out the optimization algorithm that solves the outer problem, controls the solution of the inner mechanical problem and takes care of connection between these two parts. Prior to calculation of the objective and constraint functions, input for mechanical analysis is prepared according to the current values of design parameters. In the phase interface optimization problem we would like to solve herein, the latter corresponds to generation of the finite element mesh that is used for each macro element. The mesh for a single cell or a single macroscale element is generated first on the basis of a template mesh corresponding to a circular inclusion, by transforming node co-ordinates as described in detail in the next section. Subsequently, we assume the periodic microstructure, which allows us to combine several periodic cells in order to obtain the complete macroscale finite element mesh. The latter is stored to a file in the prescribed format where it can be accessed by the FEAP micro-macro analysis. This procedure is graphically illustrated in Figure 3. In the optimization phase, the calculation of the response functions includes solution of the mechanical problems and integration of the relevant quantities, which is performed in “FEAP”. These results are passed to “Inverse” through arguments of the analysis procedure, which was prepared in “FEAP” for the complete calculation of the response functions of the optimization problem. Although the interfaces with simulation codes usually involve more sophisticated control over program flow and internal data (Gresˇovnik, 2000), this kind of interfacing appeared the quickest way to solve our problem, largely due to openness and extensibility of “FEAP”. Main advantages of linking “Inverse” with “FEAP” and using it for optimization are more transparent definition of the problem, simple and systematic application of modifications to the original problem, and accessibility of already incorporated utilities. These include different optimization algorithms, tabulating utilities, automatic recording of algorithmic progress and other actions, debugging utilities, automatic numerical differentiation, and an useful bypass utility for avoiding memory heaping problems that may be difficult to avoid when a stand-alone numerical analysis software is arranged for iterative execution.

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Figure 3. Instances of the finite element mesh at different values of design parameters

4. Shape optimisation of microstructure 4.1 Parameterization of the shape of inclusions In order to reduce the number of design variables, we assume periodic microstructure of the material, where the material geometry can thus be described at the micro-scale level of single periodic cell. For the model problem of two-phase material studied herein, the typical periodic cell microstructure can be defined by three-parameter representation shown in Figure 4. By assuming the typical dimension of a period of microstructure or the size of a “cell” to be equal to d, we can start from the reference shape of the phase interface (or an inclusion contour) as a circle of radius r in ðfÞ ¼ r 0 ¼ d=2: At a given set of design parameters p the contour shape will be defined by r p ðf; pÞ;

ð17Þ

where ðr; fÞ are polar co-ordinates with the origin of the co-ordinate system positioned in the center of a periodic cell and rp denotes the distance of the contour from the origin (Figure 4). For the purpose of shape parameterization we will transform some pre-constructed reference mesh corresponding to the reference shape of inclusions according to parameter values. At any value of parameters p all nodes of the reference mesh will be mapped by a parameter dependent map defined over the domain of the periodic cell in

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such a way that the contour of the reference inclusion shape is mapped into the contour defined by equation (17), the boundary of a periodic cell is invariant and other points are mapped C 0 continuously with respect to co-ordinates: ; p ; x i ðpÞ ¼ F~ xð0Þ i

616

ð18Þ

where i is a node index and xð0Þ i are the reference coordinates of the node. We will conveniently define the map in polar coordinates: ð0Þ ; f ; p : ðr i ðpÞ; fi ðpÞÞ ¼ F r ð0Þ i i

ð19Þ

In particular, we define F as 8 r p ðf; pÞ > > ; f r ; r , r in ðfÞ > > < r in ðfÞ ; ð20Þ Fðr; f; pÞ ¼ r ext ðfÞ 2 r p ðf; pÞ > > > r ext ðfÞ 2 ðr ext ðfÞ 2 rÞ ; f ; r $ r in ðfÞ > : r ext ðfÞ 2 r in ðfÞ where r in ðfÞ defines the initial boundary of the inclusion, r in ðfÞ ¼ r 0 ¼ d=2; and r ext ðfÞ defines the border of a periodic cell:

Figure 4. A three-parameter description of the geometry of the phase interface or an inclusion in a periodic cell (note that the angle a is chosen as negative for more clear representation)

ð21Þ

8 d p 3p 5p 7p > > ;0 # f , _ #f, _ # f , 2p > > 2 cosðfÞ 4 4 4 4 > > < d p 3p 5p 7p r ext ðfÞ ¼ ; #f, _ #f, > > 2 sinð f Þ 4 4 4 4 > > > > : periodic ð0; 2pÞ; f , 0 _ f $ 2p

Shape optimization ð22Þ

617

By equation (19), mesh nodes of the reference mesh are moved in radial direction. Within the inclusion domain points are contracted or stretched from the cell center towards new inclusion boundary defined by r p ðf; pÞ; while in the matrix domain points are stretched from the cell boundary towards inclusion boundary (Figure 4). r p ðf; pÞ is defined for ½0 # f # 2p and we require that it satisfies the following conditions: 0 , r p ðf; pÞ , r ext ðfÞ

;p

r p ð0; pÞ ¼ r p ð2p; pÞ

›r p ›r p ð0; pÞ ¼ ð2p; pÞ ; p ›f ›f

ð23Þ

In the examples presented in the following section we will restrict parameterization to elliptical shapes of inclusions. This is achieved by a three-parameter function vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u ; ð24Þ r p ðf; a; b; aÞ ¼ u 2 tcos ðf 2 aÞ sin2 ðf 2 aÞ þ a2 b2 which defines an ellipse with half-axes lengths a and b and orientation specified by angle a between the main axis and the x coordinate axis (Figure 4). In order to satisfy the first condition in (22), we ought to impose the restrictions on parameter values 0 , jaj , d=2 ^ 0 , jbj , d=2

ð25Þ

The second and third conditions in equation (23) are already satisfied by the chosen parameterization. The choice of elliptical parameterization for our study was made for several practical reasons. Possible shapes of inclusions defined in this way are simple and corresponding structures would be easy to manufacture. At the same time the variability of achievable shapes is sufficient to significantly affect the microscopic stress state and thus the overall structural response, and therefore ensure quite a significant role for optimization. Small number of design parameters allows of a better insight into the optimization process. The number of design parameters has also a critical impact on the computational cost and provides a means of filtering off high frequency oscillations in the designed shapes. This is favorable for many shape optimization applications where, in the presence of discretization and round-off errors, some means of regularization must be introduced in order to eliminate the tendency towards erroneous oscillatory solutions (Ba¨ngtsson et al., 2003).

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Parameterization defined by equation (24) is not unique in the sense that different sets of parameters result in the same curve. We can see, for example, that r p ðf; a; b; a þ kpÞ ; r p ðf; a; b; aÞ ; k [ {ZI þ < ZI 2 }; r p ðf; b; a; a þ p=2Þ ; r p ðf; a; b; aÞ;

618

ð26Þ

r p ðf; 2a; b; aÞ ; r p ðf; a; b; a þ pÞ Unless some regularization is applied, this will inevitably lead to non-uniqueness of optimal solutions. A possible solution to this problem is to restrict the admitted set of design parameters to some set S , R3 in such a way that any pair of distinct parameter vectors within the admitted domain defines distinct shape of the inclusion, i.e. ;p 1 [ R3 ;

;p 2 [ R3 ;

p 1 [ S ^ p 2 [ S ^ p 1 – p 1 ) ’f [ ½0; 2pÞ; ð27Þ

r p ðf; p 1 Þ – r p ðf; p 2 Þ: A possible choice for S is S ¼ {ða; b; aÞ; 0 , a , d=2 ^ 0 # b , a ^ 0 # a , p}:

ð28Þ

In practice we do not need to bother about multiple solutions that essentially represent the same shapes. If the optimization algorithm converges to a solution that does not satisfy equation (28), we can simply transform parameter values by using identities like equation (26) in order to obtain the basic form of the solution. One can even argue that this is a better approach than to explicitly impose constraints on parameters. This can be illustrated on the hypothetical situation described below: Let the minimized function f ðpÞ be of the form f ðpÞ ¼ f s ða; bÞ þ f a ðaÞ;

ð29Þ 2

where f s is a continuous and bounded function on R and f a is continuous and bounded function on R; periodic with a period 2p. Let in addition f s attain a unique local minimum at ða* ; b* Þ; f a a strict local minimum at a* and a strict local maximum at a þ ; 0 , a þ , a* , 2p; and let a* and a þ also be global extremes of f a : ða* ; b* ; a* Þ is then a global minimum of f. Function f a is illustrated in Figure 5. Let us restrict the set of admissible design parameters to S ¼ {ða; b; aÞ; 0 # a , 2p};

Figure 5. A function fa(a) as described in equation (29)

ð30Þ

such that ða* ; b* ; a* Þ [ S: We can see that for any point a 1 , a þ domain S does not contain any descent path connecting the points ða* ; b* ; a 1 Þ and ða* ; b* ; a* Þ: This means that if we use a descent interior point minimization algorithm with a starting point ða* ; b* ; a 1 Þ; it will not likely converge to ða* ; b* ; a* Þ: In the given situation the algorithm would converge to ða* ; b* ; 0Þ: If we do not constraint the admissible range of parameters, there will exist a descent path connecting ða* ; b* ; a 1 Þ and ða* ; b* ; a* 2 2pÞ, and the algorithm will converge to ða* ; b* ; a* 2 2pÞ, with the same value of the minimized function f as at ða* ; b* ; a* Þ. In our case, periodicity of f(p) in p3 ¼ a follows from the fact that points ða; b; aÞ and ða; b; a þ 2pÞ of the design space represent identical designs for any a, b and a, but the assumption (29) does not generally hold. A more elaborated derivation would show that much less restrictive conditions for the minimized function and a starting guess can be defined entailing similar arguments for not restricting the parameter range in order to achieve unique parameterization. 4.2 Derivatives of the initial position of nodes with respect to shape parameters By taking into account the transform function (20), the derivatives with respect to parameters can be written as follows: 8 r ›r p ðf; pÞ > > ; 0 ; r , r in ðfÞ > > ›Fðr; f; pÞ ›ðF r ; F f Þ < r in ðfÞ ›pi ¼ ¼ r ext ðfÞ 2 r ›r p ðf; pÞ > ›pi ›p i > > ; 0 ; r $ r in ðfÞ > : r ext ðfÞ 2 r in ðfÞ ›pi

ð31Þ

In equation (31), ðr; fÞ refer to polar coordinates of nodes in the referential mesh. If r p ðf; pÞ is defined by (23), then its derivatives with respect to parameters (the half-axes of the ellipse and the angle between the first half-axis and the x direction) are as follows:

›r p ðf; pÞ ›r p ðf; a; b; aÞ ¼ ›p1 ›a vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 u cos ðf 2 aÞ2 u 1 u ¼ tcos ðf 2 aÞ2 sin ðf 2 aÞ2 a3 þ a2 b2

ð32Þ

›r p ðf; pÞ ›r p ðf; a; b; aÞ ¼ ›p2 ›b vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 u sin ðf 2 aÞ2 u 1 u ¼ 2 3 tcos ðf 2 aÞ b sin ðf 2 aÞ2 þ a2 b2

ð33Þ

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›r p ðf; pÞ ›r p ðf; a; b; aÞ ¼ ›p1 ›a cos ða 2 fÞsin ða 2 fÞ cos ða 2 fÞsin ða 2 fÞ ¼ 2 a2 b2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 u 1 u u tcos ðf 2 aÞ2 sin ðf 2 aÞ2 þ a2 b2

ð34Þ

At each design iteration we reconstruct the mesh for subsequent mechanics analysis by applying the transform in equation (20) to all nodes of the given reference mesh, which implies that all the nodal co-ordinates in the mesh are readily available. Another possible approach is to just use the parametric definition of the boundary between the inclusion and the matrix and generate the mesh upon this boundary. In order to calculate the derivatives of the nodal positions with respect to the shape parameters, we must first transform the positions of nodes produced by the mesh generation procedure to the referential co-ordinates in which the transform F is defined[1]. For this we must construct the inverse map F 2 1 defined in such a way that F 21 ðFðr; f; pÞ; pÞ ¼ ðr; fÞ: For the shape defined by transform in equation (20) we have 8 r in ðfÞ > > ~ ; f ; r~ , r p ðf; pÞ r > > > r p ðf; pÞ > > > < 0r~ðr ðfÞ 2 r ðfÞÞ þ r ðfÞr ðfÞ 1 ext ext in in 21 ~ ; ð35Þ F ð~r; f; pÞ ¼ B C 2r ð f Þr ð f ; pÞ > ext p > C; r~ $ r p ðf; pÞ >B ; f > @ A > r ext ðfÞ 2 r p ðf; pÞ > > > : 4.2.1 Transformation to Cartesian coordinates. Since the calculation will be performed in Cartesian coordinates, we need to transform F and its derivatives. The polar and Cartesian coordinates are related by 8 arctg y=x; x . 0 > > > > > p þ arctg y=x; x , 0 > > > > > > > 3 p > > ; x¼0^y,0 > > 2 > > > : 0; x ¼ y ¼ 0

4.3 Imposing geometric constraints by intermediate transforms By application of transforms like equation (20) a bad input mesh for finite element calculation can be obtained. Geometrically infeasible situation occurs when parts of

the inclusion boundary exceeds the cell boundary. Unfavorable mesh can be produced even if the geometric layout is physically permissible, because the transform can distort individual elements in such a way that angles between adjacent element edges are larger than p radians (Figure 6). Two corrective mechanisms were used in order to prevent excessive mesh distortion, both of which utilize additional intermediate maps. The first correction is performed by application of an intermediate map directly to shape parameters in order to keep them in a given range where mesh distortion is not so severe. The second correction is performed by additionally transforming the co-ordinates of the inclusion boundary in such a way that it cannot exceed the boundary of the cell. In the case where the first correction would not prevent the inclusion boundary of extending out of the cell, the second correction would produce non-elliptical shapes, thus modifying the intended set of attainable shapes. The role of the second correction is twofold: first it prevents the breakdown of the optimization algorithm when infeasible guesses are generated, and second it can produce instructive results worth of further analysis when the optimal point lies in the extreme portions of parameter space where the correction becomes effective. The corrections are performed by transforming the parameters or co-ordinates by monotonous functions with a limited range. For example, lower and upper bound on an individual parameter are enforced by setting pi ðp~ i Þ ¼ f 1 ðp~ 1 ; l min ; l max ; dÞ;

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ð37Þ

where p~ i is an optimization parameter, pi is the transformed value of this parameter actually used in the shape transform formulae, fl is the family of functions used for enforcing the bounds on parameter range, and constant parameters of the family lmin, lmax and d specify the lower, the upper bound and the transition interval, respectively. f 1 ðp~ i ; . . .Þ must be monotonously increasing function of p~ i ; with f 1 ðpi ; l min ; l max ; dÞ [ ½l min ; l max

ð38Þ

Since the optimization algorithm operates with parameters p~ rather than p, we must apply the chain rule in order to calculate the derivatives of the initial co-ordinates of mesh nodes with respect to the parameters. Instead of Fðr; f; pÞ; the actual co-ordinate transform is

Figure 6. Undesired distortion of a mesh element by the shape transform

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ð39Þ

Considering equations (31) and (37), we have:

622

~ f; pÞ df 1 ðp~ i ; l imin ; l imax ; di Þ ~ f; pðpÞÞ ~ dFðr; ›Fðr; ¼ : ›p i dp~ i dp~ i

ð40Þ

Geometrically feasible bounds on the co-ordinates of the transformed mesh were achieved by enforcing suitable bounds on the inclusion boundary r p ðf; pÞ: As in equation (37), this was accomplished by application of an additional function with a limited range to the inclusion boundary determined by equation (24). We will write ~ ¼ f 1 ðr p ðf; pðpÞÞ; ~ r pmin ðfÞ; r pmax ðfÞ; dðfÞÞ: rp ðf; pðpÞÞ

ð41Þ

~ in the formulae (20), (31) and (35), Accordingly, we must replace r p ðf; pÞ with r~p ðf; pÞ ~ ›p~ i : The last derivative is obtained as and ›r p ðf; pÞ=›pi with ›rp ðf; pÞ= ~ ~ ›r p dpi ›rp ðf;pðpÞÞ ›rp ðf;pðpÞÞ ¼ ›p~ i ›r p ›pi ›p~ i

›f 1 ðr p ðf;pÞ;r pmin ðfÞ;r pmax ðfÞ;d p ðfÞÞ

¼

›r p

ð42Þ

›r p

df 1 ðp~ i ;l imin ;l imax ;di Þ :

dp~ i f;p~ ›pi f;p~

The same family of functions was used for imposing bounds on rp as for imposing limits on parameter range, and ~ ¼ ðp1 ðp~ 1 Þ; p2 ðp~ 2 Þ; . . .Þ; pðpÞ i.e. each transformed pi depends only on one corresponding parameter p~ i : We allow the parameters of the family to be dependent on f. In this way, we can for example set the upper bound for inclusion boundary to be a square slightly smaller than the periodic cell, i.e. defined by a formula similar to equation (22), except with a smaller d. 4.4 Transforms used for limiting parameter range and inclusion boundary As mentioned before, functions fl must be monotonously increasing functions of the first argument for any parameter of the family. It must be continuously differential with respect to the first argument. Furthermore, we design functions in such a way that min þ d , x , max 2 d ) f 1 ðx; min; max; dÞ ¼ x:

ð43Þ

The functions are conveniently defined by gluing together monotonous segments that are adequately bound and satisfy the continuity conditions at the endpoints. The following form with continuous second derivatives has been chosen in the particular case:

8 min; x # min 2 d > > > > > > min þ fc1ðx 2 ðmin 2 dÞ; dÞ; min 2 d , x # min > > > > > > > min þ fc2ðx 2 ðmin þ dÞ; dÞ; min , x , min þ d > < f 1 ðx; min; max; dÞ ¼ x; min þ d # x # max 2 d > > > > max 2 fc2ððmax 2 dÞ 2 x; dÞ; max 2 d , x , max > > > > > max 2 fc1ððmax þ dÞ 2 x; dÞ; max # x , max þ d > > > > > : max; max þ d # x

Shape optimization ð44Þ

623

The corresponding derivative and inverse formulas are 8 0; x # min 2 d > > > > > derfc1ðx 2 ðmin 2 dÞ; dÞ; min 2 d , x # min > > > > > > > > derfc2ðx 2 ðmin þ dÞ; dÞ; min , x , min þ d df 1 ðx; min; max; dÞ < 1; min þ d # x # max 2 d ¼ > dx > > derfc2ððmax 2 dÞ 2 x; dÞ; max 2 d , x , max > > > > > > derfc1ððmax þ dÞ 2 x; dÞ; max # x , max þ d > > > > : 0; max þ d # x

ð45Þ

and 8 min2d;y # min > > > > > > min2d þinvfc1ðy2min;dÞ; min , y # minþd=6 > > > > > > minþd þinvfc2ðy2min;dÞ; minþd=6 , y , minþd > > < 21 f 1 ðy;min;max;dÞ ¼ y; minþd # y # max2d ; ð46Þ > > > max2d 2invfc2ðmax2y;dÞ; max2d , y , max2d=6 > > > > > > maxþd 2invfc1ðmax2y;dÞ; max2d=6 # y , max > > > > > : maxþd; max # y where the family of inverse functions f 21 is defined by the formula l f 21 1 ðf ðx; min; max; dÞ; min; max; dÞ ¼ x:

ð47Þ

The functions f1, its first and second derivatives and its inverse are shown in Figure 7. Auxiliary functions used in the definition of fl are defined as follows:

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Figure 7. The function used for limiting parameter range with its first and second derivative and inverse

fc1ðx; dÞ ¼

x3 ; 6d 2

fc2ðx; dÞ ¼ d þ x 2 derfc1ðx; dÞ ¼

x3 6d 2

dfc1ðx; dÞ x2 ¼ 2 dx 2d

ð48Þ

dfc2ðx; dÞ x2 ¼12 2 dx 2d pffiffiffiffiffiffiffiffiffiffi invfc1ðy; dÞ ¼ 3 6d 2 y derfc2ðx; dÞ ¼

invfc21ðy; dÞ ¼ x; fc2ðx; dÞ ¼ y ^ x [ ½2d; 0 The value of invfc21 is obtained by application of the analytical formula for zeros of a third order polynomial and choosing the real solution that lies in the appropriate interval ½2d; 0: There is always exactly one such value for the interval y [ ½d=6; d where invfc2 is evaluated, since fc2 is strictly monotonous on ½2d; 0 with derivative lying in ½1=2; 1:

5. Numerical examples Several numerical examples are chosen and solved in order to demonstrate the applicability of the proposed design approach. While application to design of a material with far more complex microstructure can be foreseen for many practical situations, the main goal of the presented numerical experiments was to validate the solution scheme for the chosen model material with two-phase microstructure. In particular, feasibility of combining multi-scale numerical models, featuring elasto-plastic and damage material phases at the smaller scales, with efficient gradient-based techniques for constrained optimization was investigated. We also wanted to draw some attention to problems previously experienced in the area of material forming (Gresˇovnik, 2000; Gresˇovnik and Rodicˇ, 2003), such as the presence of substantial noise in the numerical response that can badly affect the performance of classical optimization algorithms. A structural element under a given loading (prescribed displacements) was considered, as depicted in Figure 8. The element is composed of a matrix containing periodically distributed inclusions of a different material. The material properties of the matrix are described by the von Mises elasto-plastic model, using the following yield function, Fp, rffiffiffi 2 p p ðsy þ q p Þ; ð49Þ F ðs; q Þ ¼ kDevðsÞk 2 3

Shape optimization

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where DevðsÞ ¼ s 2 1=3 TrðsÞI is the deviatoric part of the stress s, TrðsÞ is its trace and I the 3 £ 3 identity matrix. In equation (49) sy represents the initial yield stress and q p the plastic hardening function, defined as q p ðj p Þ ¼ ðs p 1 2 sy Þð1 2 e 2b

p p

j

Þ;

ð50Þ

where j p is the hardening variable, s1p the plastic saturation stress and b p the plastic saturation exponent. In our analyses the matrix material parameters take the following values: sy ¼ 1:0 108 ; sp1 ¼ 5:0 108 ; b p ¼ 1; 000: On the other hand, the behavior of inclusions is described by the damage model introduced in Ibrahimbegovic et al. (2003) and very similar to the classical plasticity model described above. It is based on the fracture criterion function, Fd, Fd ðs; q d Þ ¼ TrðsÞ 2 ðsf þ q d Þ;

ð51Þ

where sf represents the initial fracture stress and q d the damage hardening function, defined as

Figure 8. Studied heterogeneous structure with periodic microstructure

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Figure 9. Finite element mesh of the periodic cell at the solutions: (a) when maximizing total plastic dissipation; and (b) when maximizing work of external forces

q d ðj d Þ ¼ ðs1d 2 sf Þð1 2 e 2b

d d

j

Þ;

ð52Þ

with j d being the damage hardening variable and sd1 ; b d are damage saturation stress and damage saturation exponent, respectively. In our analyses the inclusion material parameters take the following values: sf ¼ 0:33 £ 108 ; sd1 ¼ 0:66 £ 108 ; b d ¼ 10: The corresponding linear and isotropic elastic properties of each phase are K p ¼ 1:0 £ 1010 (matrix bulk modulus), G p ¼ 1:0 £ 1010 (matrix shear coefficient), K d ¼ 1:2 £ 1010 (inclusion bulk modulus) and G d ¼ 1:5 £ 1010 (inclusion shear coefficient). We optimized the shape of inclusions with respect to different criteria regarding the overall response of the element. Elliptical shapes of inclusions were considered using the parameterization described in Section 4. The numerical model was described in Section 2 and the solution scheme in Sections 3 and 4. Preliminary testing of the method was performed on problems with trivial solutions that can be guessed in advance. Namely, when the objective function pertains to maximizing the plastic dissipation, the optimal design which follows is the one where the inclusion size shrinks to zero and the matrix material occupies the whole domain. On the contrary, when we look for the design at which the work of external forces is maximal, the optimal solution implies that the inclusion material would occupy the whole space. Figure 9 shows the solutions obtained for both of these cases represented by the finite element mesh of the periodic cell. The applied parameterization is not capable of representing these extreme situations and the algorithm therefore converges to the representative designs with minimum and maximum inclusion volumes, respectively. The half-axes of the inclusion were not formally constrained in this case. Instead, the a priori constraints were imposed on the inclusion boundary by application of additional transforms on the boundary radius r p ðf; pÞ as described in the previous section (see equation (41)). More precisely, r pmin ðfÞ defining the boundary of the smallest possible inclusion was chosen as a circle with a very small radius and r pmax ðfÞ was chosen to be a square with the side a bit smaller than width the periodic cell. The latter was chosen intentionally in order to make the effect of the transform visually more apparent. The problem of excessive mesh distortion for each of these two designs is clearly visible in Figure 9(b). In spite of such drastic situation regarding the finite element

mesh distortion, no problems were experienced with convergence of the optimization procedure to the expected result. However, one should take into account that the results obtained by the finite element analysis are rather sensitive to mesh distortion and therefore a control on the mesh grading and mesh regularity should be incorporated accordingly when applying the approach described herein. More precisely, with the proposed solution scheme and flexible software architecture, it is easy to calculate and manipulate numerical indicators of mesh regularity within the optimization procedure. Such indicators can either be used just to provide information on when the results should no longer be too much trusted, or to actively control the optimization procedure, e.g. by using the penalty terms, in order to always force the design problem solution and subsequent mesh distortion within the range which can be considered as acceptable with respect to the mechanics simulation results remaining of sufficient accuracy. In the presented examples, we used the Pian-Sumihara elements (Pian and Sumihara, 1984) in the microscopic finite element model, which are known to be rather insensitive to shape irregularity. Different approaches can be imagined to deal with situations where excessive mesh distortion would prevent the determination of optimal solution providing the equivalent accuracy of result that is otherwise possible with respect to the accuracy of the numerical simulation of mechanics problem. Although a more detailed exploration is beyond the scope of the present work, we would like to mention two possibilities that we deem convenient for practical applications. The first possibility is to apply automatic mesh generation in order to generate the mesh for the micro level, and thus apply the shape transform described in the previous section only to the geometrical definition of the inclusion boundary. Positions of the internal mesh nodes would be in this case produced without explicit application of the shape transforms. However, we would still need to consider the explicit definition of the transforms to provide the consistent sensitivity fields over the interior of the matrix or inclusion material in the case of sensitivity analysis. In this case, the positions of generated mesh nodes should be mapped to the reference domain by the inverse transform in order to calculate derivative terms. The only additional implementation difficulty that we are able to foresee is in interfacing and manipulation of the geometry definition of the boundary and related automatic mesh generation. Sufficient tools to implement such an approach are already available in any commercial simulation environment. One must, however, anticipate that such an approach would reduce the efficiency of the optimization procedure because of the addition of non-smoothness to the numerically calculated relation between the design parameters and the objective and constraint functions. Transforming the same reference mesh over several analyses, and re-meshing only (a very few times) when the mesh becomes too distorted, should be able to alleviate these problems. According to our experience, such approach is well suited already at least for the chosen example problems. In all demonstrated cases, a single re-meshing is sufficient mesh quality, and furthermore even manual intervention would be quite adequate. Another possibility to deal with the problem of mesh distortion is to use a fixed regular mesh over the complete domain of the periodic cell. In this case, material composition and representation of the interface between the matrix and inclusion material would be dealt with at the level of an individual element rather than on the element interface level. This kind of structured mesh representation of

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Figure 10. Solution path of the Nelder-Mead simplex method in the space of shape parameters. Edges and apices of the subsequent simplexes are shown, together with their centers (red points)

the microstructure, as proposed in Ibrahimbegovic and Markovic (2003), would provide an important advantage of mesh regularity and prevent any possibility of mesh distortion and resulting ill-conditioning problem. However, the number of design variables would increase considerably with respect to the exact finite element representation employed herein, since any micro-scale element would become a potential candidate for harboring an interface between two phases and the corresponding design variables describing it. In the present case with a regular, elliptic shape of phase interface, which can be described with only three design variables, the structured mesh approach is very unlikely to be more efficient. However, when considering the best shape representation of the phase interface in a more general case with a multi-phase composite material, the latter approach should not be discarded a priori. The two above mentioned example problems of interface shape optimization were first solved by the Nelder-Mead simplex method. Rather than a single point, this method maintains a set of nþ 1 points where the objective function is evaluated through iterations, where n is the number of parameters. An instance of the solution path in the parameter space is shown in Figures 10 and 11, for the example with solution shown in Figure 9(b). Convergence of the value of the objective function and distance from the optimum is shown in Figure 12.

Shape optimization

629 Figure 11. Solution path defined by centers of simplexes shown in the previous figure. Projection to the sub-space of ellipse half-axes is shown on the right-hand side

Figure 12. Convergence of the simplex method: (a) distance from the optimum; and (b) value of the objective function

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Figure 13. Dependence of minus total plastic dissipation on the inclination angle of the inclusion. Half-axes were fixed at a ¼ 0.3 and b ¼ 0.6

These calculations typically converged with the precision of 102 3 taking in between 50 and 100 iterations (which corresponds to about 100-200 evaluations of the objective function). Different starting points were set in order to check that the algorithm converges to the same point. For the two example problems, the solution was not unique in terms of the optimal angle of the “elliptical” inclusion, since all angles resulted in the same symmetric shape due to the imposed parameter transform. In terms of the finite element mesh, various solutions agreed almost exactly. Non-uniqueness in terms of calculated optimal parameters is very unlikely to be observed in complex cases. Typical computation times for a single analysis run, carried out in parallel on eight 2.4 GHz Pentium 4 Linux workstations, ranged between 5 and 8 min. Bookkeeping time of the optimization algorithm is negligible for the proposed solution scheme; therefore, the main opportunity for improving the efficiency was seen in reduction of the number of required analyses by application of a more efficient optimization algorithm. The sequential quadratic programming (SQP, Fletcher, 1996; Lawrence and Tits, 1996) was the method of choice for its known performance in solving non-linear constrained problems. Our main concerns related to the application of this method were related to potentially noisy calculated response, especially because we calculated the derivatives of the objective and constraint functions numerically by the forward difference method. We therefore investigated behavior of the calculated response in this respect by preliminary parametric studies. Some results of these studies are shown in Figures 13-18. We examined angular dependency of the calculated response with fixed size of half-axes of the ellipse (Figures 13 and 14). These tests indicated that the total response of the loaded structure significantly depends on the orientation of the inclusions, rather than mainly on the inclusion volume. Further computations are made to get some indication on the level of noise in computed response, which is obtained by tabulating the response in the parameter space. The goal is to find out at what range of parameter change the contribution of

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631

Figure 14. Dependence on the inclination angle of the inclusion of the derivative of minus plastic dissipation with respect to: (a) the larger half-axis; and (b) the inclination angle

numerical noise is qualified as being considerable with respect to the local trend in the response. This information was crucial for the choice of step for numerical differentiation for objective function derivative computation. Since little can be concluded in advance, the chosen approach is simply to tabulate the response along individual parameters over different interval lengths, with sufficient number of points along each interval to be able to visualize the effect of noise and distinguish it from the general trend. Because of well-scaled design parameters, tabulating along directions such that all parameters change equally should give as useful first indications as tabulating along individual parameters. For better efficiency, we also replaced several uniform samplings on increasingly shorter lines by a sampling with intervals of geometrically increasing length. For the first indication of the effect of numerical noise, we performed the tabulation of calculated response starting at parameters p 1 ¼ ½0:3; 0:6; 0:1T and ending at p 2 ¼ ½0:29; 0:61; 0:11T : We sampled in 30 points, by a rather conservative factor of interval length growth of 1.2. This implies the length of the first sampling interval being about 0.1 percent and the length of the last sampling interval being about 17 percent of the whole interval. Sampling by factors of interval growth of two or more is usually adequate for this task and much more efficient, but a bit less comfortable for visualization. Figure 15 shows the variation of minus plastic dissipation along a given direction between the two points. No effect of random noise

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Figure 15. Variation of minus plastic dissipation along the straight line between the points p1¼ [0.3,0.6,0.1]T and p2¼[0.29,0.61,0.11]T in the space of shape parameters a, b and a

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Figure 16. Numerical derivative of minus plastic dissipation with respect to the first half-axis sampled in the same points as the results shown in earlier

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Figure 17. Numerical derivative of minus plastic dissipation with respect to the inclination angle

Figure 18. Convergence of the objective function for the problem whose solution is shown in Figure 9(b)

patterns can be perceived on the plots, indicating that it should be appropriate to use numerically calculated derivatives in the optimization procedure. We have fixed the length of the interval for numeric differentiation at 0.001 for all three parameters and verified the suitability of this choice by tabulating the numerical derivatives obtained by forward differentiation. Results are shown in Figures 16 and 17 for derivatives with respect to the first axis and the inclination angle, respectively. Sampling was performed in the same way as described above for the results in Figure 15. Results for derivatives with respect to the second half-axis are similar as those for the first half-axis shown in Figure 16. We can see that the amplitude of what appears to be the contribution of non-smooth response is less than 1/10 of the difference in the derivative with respect to the first axis between the points p1 and p2. This is encouraging, especially if we take into account that the variation of the first parameter is opposite to the variation of the second one, effectively compensating the terms dependent just on the size of the inclusion. Differentiation with respect to the inclination angle is more critical, obviously because the sensitivity with respect to this parameter is about two orders of magnitudes smaller (see Figure 17) than the sensitivity with respect with the first two parameters. We have therefore chosen to increase the step length for numerical differentiation with respect to this parameter to 0.01. We applied the so-called feasible sequential quadratic programming, a variant of the SQP algorithm developed by Tits et al. (Panier and Tits, 1993; Lawrence and Tits, 1996;

Bonnans et al., 1992; Lawrence et al., 1995), with forward difference numerical differentiation to solve a series of problems with different objectives and constraints. For unconstraint problems the method is reduced to the Broyden-FletcherGoldfarb-Shanno (BFGS) non-linear minimization algorithm with line search (Fletcher, 1996; Dennis and Schnabel, 1996). It was applied to the same problem whose solution by the simplex method was described above, i.e. maximization of the work of external forces (Figure 9(b)). The solution of this problem was typically obtained in three to four iterations (Figure 12), which took less than 50 numerical analysis runs in total. Four analyses were run for each point of the parameter space to calculate the objective function and its gradient. In the considered example with simple geometry and loading, various measures of overall performance such as work of external forces or plastic dissipation exhibit monotonous dependency on the volumetric ratio of the two phases and therefore size of inclusions, at any fixed shape and orientation of inclusions. Designs of microstructure obtained by minimization or maximization of such measures will correspond to designs with maximal or minimal volume of one phase, such as those shown in figure. We regard here only those geometric layouts that are elements of the design space defined by a chosen parameterization, and the solutions with minimal or maximal volume of a given space do not necessarily coincide with layouts where one phase vanishes. It is interesting to consider problems where we look for minimum of a given criterion, while a set of feasible designs is constrained with another criteria. We deal with such examples below. The choice of criteria was not motivated by any particular application, and the primary purpose of the examples is to examine the proposed solution scheme. We first look for the design that results in maximal plastic dissipation, with a prescribed upper limit on the accumulated damage. For chosen materials, both damage and plastic dissipation grow with the size of inclusion and are relatively more sensitive to size than orientation. We can expect that the constraint will be active in the optimum while exact orientation and half-axes will be adjusted according to local dependency of both damage and plastic dissipation, thus hard to be guessed or determined by manual parametric studies. A priori bounds on the half-axes were imposed implicitly by transformation of parameters (equations (37), (38), (44) and (48)). Lower bound of 0.15 and upper limit of 0.9 were imposed for both half-axes. We also wanted to avoid excessively elongated, oblong inclusion shapes and therefore, we applied additional parameter transform of this type that constrained achievable ratios between half-axes to at most 2.5. It turned that both explicit constraint on the allowed amount of damage and the imposed bound on half-axes ratio determined the optimal shape, which is shown in Figure 19. Optimal parameters were ½0:254; 0:95; 3:205T ; which corresponds to geometric parameters a < 0:634; b < 0:254 and a < 1:635: These values were, in addition to the bound imposing transform, obtained by swapping half-axes and rotating by p=2 clockwise, in order to provide a unique notation. The FSQP algorithm converged to this solution in up to ten iterations from different starting points, with 15-20 evaluations of the objective and constraint functions and their gradients. Since gradients were calculated numerically, each evaluation took five complete numerical analyses of the structure and the actual computational cost was up to 80 analyses. In this case, the optimum obtained in subsequent experiments was

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unique within the prescribed tolerance. Figure 20 shows an instance of the convergence path of the method, while variation of the distance from the optimum and values of the objective and constraint function is shown in Figure 21. In Figures 22 and 23 the results for the same solution procedure are represented, but this time all evaluations are shown, including those performed in the line searches. Two other problems were considered: maximization of plastic dissipation with constraint on maximum volume (Figure 24(a)) and minimization of plastic dissipation with constraint on the minimum work of external forces (Figure 24(b)). Similar performance of the solution scheme could be observed as in the previous case. 6. Conclusion The methodology for solving the coupled problems in nonlinear mechanics and optimal design of a heterogeneous material was developed and illustrated on a simple example of a model material with two phases, one with behavior described by plasticity and another by damage. We discussed in detail the shape optimization

Figure 19. Shape of inclusions that maximizes plastic dissipation at the constraint imposed on the damage

Figure 20. Solution path of the FSQP method to the solution from figure

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Figure 21. Course of: (a) distance from the optimum; (b) the objective function; and (c) the constraint through iterations shown in the figure

problem for the internal interface between two phases. The desired optimization goal is defined in terms of the objective and constraint functions, which are related to the micro-scale response of the material under consideration. Furthermore, we considered a periodic microstructure of the material, where each periodic cell consists of a plastic matrix phase into which a single inclusion represented by a damage model is

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Figure 22. Solution path from figure, with all evaluation points shown

incorporated. Such geometric layout is in accordance with practicability of production for artificial materials. We chose elliptical shapes of inclusions. With such a choice, as well as the periodicity of the microstructure and the exact finite element representation of each sub-domain occupied by a single phase within each cell, we manage to reduce drastically the number of design variables to only three. In other words, at each iterative stage of the shape design procedure, we can recover the position of all nodes in the mesh of micro-scale elements from the given current values of the half-axes and the inclination angle of the elliptic interface. Several important benefits are related to chosen interface parameterization. First, the form of the shape transform and number of design parameters restrict the range of achievable shapes, which can be useful for avoiding too complex shapes and regularizing the shape optimization problem. Since the shape transform is defined continuously over the whole domain of the periodic cell, derivatives of the nodal positions are readily available and can be used as input for sensitivity analysis. Finally, the finite element mesh depends smoothly on the design parameters and therefore, the calculated response is less noisy. We were therefore able to ensure the superior performance of the gradient-based sequential programming optimization algorithm over the non-gradient simplex method, even without using analytic result for sensitivity analysis of the multi-scale model, but only numerical differentiation to compute the gradients. The calculated response was smooth enough to achieve stable convergence of the gradient-based optimization algorithm. On the other hand, the level of noise was high enough that the choice of the right step size for numerical differentiation turned critical. These findings indicate that further effort in the implementation of the sensitivity analysis of the mechanical model would be worthy. One can imagine a number of possible applications of the presented approach, anywhere from designing the material microstructure which will reduce as much as possible the damage in a given zone, thus increasing the durability of the structure, to designing the material microstructure which will maximize the damage in a given

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Figure 23. Course of: (a) distance from the optimum; (b) the objective function; and (c) the constraint through evaluations shown in the figure

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Figure 24. Optimal shapes of inclusions calculated for: (a) maximization of plastic dissipation with constraint on maximum volume; and (b) minimization of plastic dissipation with constraint on the minimum work of external forces

zone, where it is important to concentrate energy dissipation in a structure. The chosen design goals in engineering applications would not only depend on the purpose for which a particular structural element would be used but also on the way how this element is integrated in the whole structure, its interaction with other parts of the structure or external media, and the range of possible loading conditions. The latter may gives rise to additional complexities, such as taking into account multiple loading conditions or simultaneous optimization of internal phase boundary and external shape of the element. We envisage that the solution scheme could be extended in a straightforward way to comply with such requirements. Minor extensions would be needed and also be possible in order to allow for non-periodic microstructure with continuous variation of its shape over macroscopic domain. One of the most crucial components of the presented approach is parameterization of the interface between the material phases. In the case of a single inclusion incorporated in the surrounded matrix material, the applied parameterization can easily be extended to cover more complex shapes. This may require more robust approach for dealing with mesh distortion, in particular automatic generation of mesh upon the mesh independent definition of the inclusion boundary. Beside the technical difficulties this would inevitably lead to more noisy response functions, and one should consider application of more optimization techniques that are more robust in the presence of noise. In order to allow even more general microstructure of the representative volume, some other technique of defining the interface boundary should be considered, such as the level set method (Wang et al., 2003). Such situation would arise, e.g. when one should account for multiple inclusions of irregular shapes and without defined location within the representative volume. In this case, use of a fixed regular mesh over the complete domain of the micro problem may be worth of consideration, such that material composition and representation of the interface between the matrix and inclusion material would be dealt with at the level of an individual element rather than on the element interface level (Ibrahimbegovic and Markovic, 2003). As the final challenge for the future work in this domain, we see the optimal design of the non-deterministic material structure, where only some probabilistic parameters of the phase distribution can be adjusted.

Note 1. Note that F defines a spatial map of the whole periodic cell consistently with the transform of the boundary between the two phases. References Ba¨ngtsson, E., Noreland, D. and Berggren, M. (2003), “Shape optimization of an acoustic horn”, Comput. Methods. Appl. Mech. Eng., Vol. 192 Nos 11/12, pp. 1533-71. Bonnans, J.F., Panier, E.R., Tits, A.L. and Zhou, J.L. (1992), “Avoiding the Maratos effect by means of a nonmonotone line search. II: inequality constrained problems – feasible iterates”, SIAM J. Numer. Anal., Vol. 29, pp. 1187-202. Dennis, J.E. Jr and Schnabel, R.B. (1996), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, PA. Fletcher, R. (1996), Practical Methods of Optimization, 2nd ed., Wiley, New York, NY. Gresˇovnik, I. (n.d.), “Quick introduction to optimization shell inverse”, available at: www.c3m.si/ inverse/doc/ other/ Gresˇovnik, I. (2000), “A general purpose computational shell for solving inverse and optimisation problems – applications to metal forming processes”, PhD thesis, University of Wales Swansea, available at: www.c3m.si/inverse/doc/phd Gresˇovnik, I. and Rodicˇ, T. (1999), “A general-purpose shell for solving inverse and optimisation problems in material forming”, in COVAS and Antonio, J. (Eds), Proc. 2nd ESAFORM Conf. on Material Forming, Guimaraes, Portugal, pp. 497-500. Gresˇovnik, I. and Rodicˇ, T. (2003), “An integral approach to optimization of forming technology”, Proceedings the 5th World Congress of Structural and Multidiscliplinary Optimization, Lido di Jesolo, Italy. Ibrahimbegovic, A. and Knopf-Lenoir, C. (2003), “Shape optimisation of elastic structural systems undergoing large rotations: simultaneous solution procedure”, Computer Model. Eng. Sci., Vol. 4, pp. 337-44. Ibrahimbegovic, A. and Markovic, D. (2003), “Strong coupling methods in multiphase and multiscale modeling of inelastic behavior of heterogeneous structures”, Comput. Meth. Appl. Mech. Eng., Vol. 192, pp. 3089-107. Ibrahimbegovic, A., Markovic, D. and Gatuingt, F. (2003), “Constitutive model of coupled damage-plasticity and its finite element implementation”, Europ. J. Finite Element, Vol. 12, pp. 381-405. Ibrahimbegovic, A., Knopf-Lenoir, C., Kucerova, A. and Villon, P. (2004), “Optimal design and optimal control of elastic structures undergoing finite rotations and deformations”, Int. J. Numer. Meth. Eng.(in press). Kleiber, M., Autmer, H., Hoen, T.D. and Kowalezyk, P. (1997), Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations, Wiley, New York, NY. Lawrence, C.T. and Tits, A.L. (1996), “Nonlinear equality constraints in feasible sequential quadratic programming”, Optimization Methods and Software, Vol. 6, pp. 265-82. Lawrence, C.T., Zhou, J.L. and Tits, A.L. (1995), User’s Guide for CFSQP Version 2.3: A C Code for Solving (Large Scale) Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints, Institute for Systems Research, College Park, MD. Markovic, D., Ibrahimbegovic, A., Niekamp, R. and Matthies, H. (2004), “Parallelized algorithm for multi-scale modeling of hetergeneous structures with inelastic constitutive behavior”, Proceedings NATO-ARW, Bled, Slovenia, June.

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Markovic, D., Niekamp, R., Ibrahimbegovic, A., Matthies, H. and Taylor, R.L. (n.d.), “Multi-scale modeling of heterogeneous structures with inelastic constitutive behavior. Part I: Physical and mathematical aspects”, Eng. Computing, Vol. 22 Nos 5/6. Melnyk, S. (2004), “Shape optimization of inclusion in a heterogeneous material”, MSc thesis, ENS-Cachan (in French). Panier, E. and Tits, A.L. (1993), “On combining feasibility, descent and superlinear convergence in inequality constrained optimization”, Mathematical Programming, Vol. 59, pp. 261-76. Pian, T.H.H. and Sumihara, K. (1984), “Rational approach for assumed stress finite elements”, Int. J. Numer. Meth. Eng., Vol. 20, pp. 1638-85. ˇ Rodic, T. and Gresovnik, I. (1998), “A computer system for solving inverse and optimization problems”, Eng. Computer, Vol. 15 No. 7, pp. 893-907. Taylor, R.L. (2004), FEAP: Finite Element Analysis Program – User’s Manual, available at: www. uc.ce.edu/rtl Tortorelli, D.A. and Michaleris, P. (1994), “Design sensitivity analysis: overview and review”, Inverse Prob. Eng., Vol. 1, pp. 71-105. Tsay, J.J. and Arora, J.S. (1990), “Nonlinear structural design sensitivity analysis for path dependent problems. Part i and ii”, Comput. Meth. Appl. Mech. Eng., Vol. 81, pp. 183-228. Wang, M.Y., Wang, X. and Guo, D. (2003), “A level set method for structural topology optimization”, Comput. Methods. Appl. Mech. Eng., Vol. 192, pp. 227-46. Wriggers, P. and Zohdi, T.I. (2001), “Computational testing of new materials”, Proceedings ECCM 2001, Crackow, Poland.

Appendix. Coupled non-linear mechanics-optimisation problem in 1D setting In order to clarify the ideas presented in the main body of the paper, we provide in this Appendix a more detailed presentation of the coupled nonlinear mechanics-optimization problem in a simple 1D setting. In order to have the same ingredients as in the original problem, we chose the simplest possible case where one macroscale truss-bar element consists of two micro-scale bar elements, one with constitutive behaviour described by plastic and another by damage model (see Figure A1). The behavior of the plastic component is described by the classical hardening plasticity model (Ibrahimbegovic et al., 2003) with three fundamental ingredients of the additive decomposition of strain, strain energy function and the yield criterion

Figure A1.

Shape optimization

1 ¼ 1e þ 1p 1 1 Cp ð1; 1 p ; j p Þ ¼ ð1 2 1 p ÞEð1 2 1 p Þ þ K p ðj p Þ2 2 2

ðA1Þ

Fp ðs; q p Þ ¼ jsj 2 ðsy 2 q p Þ # 0 The remaining equations of the model on constitutive relations for stress and evolution equations for internal variables can be obtained from standard thermodynamics developments and principle of maximum plastic dissipation with

s¼

›Cp ¼ Eð1 2 1 p Þ; ›1

qp ¼ 2

›Cp ¼ 2K p j p ›j p ðA2Þ

›Fp ¼ g_ p sgn ðs p Þ; 1_ ¼ g_ ›s p

p

›Fp j ¼ g_ ¼ g_ p ›q p _p

p

Similarly, the fundamental ingredients of damage model are the choice of compliance in order to describe the damage, the strain energy and the damage criterion: 1 x d ¼ sDs 2 1 cd ð1; D; j d Þ ¼ s1 2 x d þ K d ðj d Þ2 2

ðA3Þ

Fd ðs; q d Þ ¼ jsj 2 ðsf 2 q d Þ # 0 The remaining ingredients of constitutive and evolution equations are again obtained from thermodynamics and the principle of maximum damage dissipation:

s¼

›Cd ¼ D 21 1; ›1

qd ¼ 2

›Cd ¼ 2K d j d ›j d ðA4Þ

›Fd D_ s ¼ g_ d ¼ g_ d sgn ðs d Þ; ›s

›Fd j_ d ¼ g_ d ¼ g_ d ›q d

The equilibrium equations in the present case, with macroscale displacement field already known given as 0 and u (Figure A1), reduce to a set of equilibrium equations with unknown displacements at the micro-scale. For the mesh consisting of the elements with exact microstructure representation, where the first element covers the domain Vd and the second element covers the domain Vp, the only unknown displacement at the micro-scale is the displacement at the interface, u (Figure A1). The weak form of the equilibrium equations at the micro-scale can then be written as Gu ðu; V ; wÞ ¼

Z

dw sðu; 1 p ; j p ; u Þ dx þ Vp dx

Z Vd

dw sðu; D; j d Þ dx ¼ 0; dx

ðA5Þ

where _ j_ d ; g_ p ; g_ d Þ V ¼ ð1 p ; j p ; D; j d ; 1_ p ; j_ p ; D;

ðA6Þ

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There is a single design variable describing the interface position d between the plastic and the damage bars. The coupled non-linear mechanical-optimisation problem of this kind can then be rewritten as mind;u;V maxl Lðd; u; V ; lÞ;

Lðd; u; V ; lÞ ¼ J ðd; u; V Þ þ Gðd; u; V ; lÞ;

ðA7Þ

where

644

p

d

p

p

d

la ¼ {l eq ; l F ; l F ; l 1 ; l j ; l D ; l j }

ðA8Þ

are the Lagrange multipliers enforcing the constraints imposed by different mechanical equations. For clarity we can also write an explicit form of the last term in equation (A7) for the chosen model problem given as Z 8R 9 ›l eq ›l eq p p _p d d > > > > s ðd; u; 1 ; j ; l Þ dx þ s ðd; u; D; j ; g _ Þ dx p > V ›x > > > d ›x > > V > > > > > > p > > p p p F > > ½F ðs; q Þ·g_ l > > > > > > > > p > > p p p 1 > > ½ 1 2 g _ sgn ð s Þ l _ < = p Gðd; u; V ; lÞ ¼ : ðA9Þ p p j _ ½ j 2 g _ l > > > > > > > > d > > > > ½Fd ðs; q d Þ·g_ d l F > > > > > > > > > > d d D _ > > ½ D s 2 g _ sgn ð s Þ l > > > > > > > > d d jd : ; _ ½j 2 g_ l We note that only the first of these equations is global in the sense that it concerns the whole domain V ¼ Vp < Vd whereas the others are local equations which concern only a given quadrature point. The only ingredient which remains to specify is the cost function, corresponding to the first term in equation (A7). An explicit form of the cost function in equation (A7) above can be written in accordance with a given goal; one possible choice advocated in this paper is related to the dissipation, which can be written for damage D d and plastic component D p as 1 D d ðd; u; D; j d ; g_ d Þ ¼ sD_ s þ q d j_ d ; 2

D p ðd; u; 1 p ; j p ; g_ p Þ ¼ s1_ p þ q p j_ p

ðA10Þ

Such a choice would clearly involve all the variables and would allow for any preference in the solution procedure. An analytic solution of this problem can be provided in several cases (Melnyk, 2004), which is very useful for testing various phases of software development. In a more general case where the analytic solution is not available, one has to compute the solution numerically. One can start by first solving the equilibrium equation and computing the evolution of the internal variables, which can be written as 0¼

›L ¼ G ) ðu; 1 p ; j p ; g_ p ; D; j d ; g_ d Þ ›l

ðA11Þ

We indicated in equation (A11) that this first step allows us to compute all the values of the mechanical state variables. The next step is the computation of the Lagrange multipliers according to 0¼

›L ›J ›G ¼ þ lV ; ›V ›V ›V

0¼

›L ›J ›G ¼ þ l eq : ›u ›u ›u

ðA12Þ

The latter defines a linear problem as the consequence of the dual formulation we adopted herein. Moreover, the solution is already well prepared by the previous computational stage in equation (A11), where, for example, the last equation would simply call for the tangent matrix K to obtain K l eq ¼ 2

›J ; ›u

K¼2

›G ›u

ðA13Þ

Having computed the solution for the Lagrange multipliers we can then proceed to computing the values of the design variables according to 0¼

›L ›J ›G ¼ þl ›d ›d ›d

ðA14Þ

For example, for a given choice of the cost function ½J 2 J 0 2 ; we can write an explicit form (Melnyk, 2004) of this equation as (" # ninc X npg h i2 h i2 X ›leq;d ›leq;p a;n a;n d p J a;n 2 J 0 wa;n 2 J a;n 2 J 0 wa;n þ sa;n wa;n 2 sa;n wa;n ·d 2 › x › x n¼1 a¼1 2

" h

J da;n

i2

2 J 0 wa;n 2

h

J pa;n

2 J0

i2

# ›leq;d ›leq;p a;n a;n wa;n þ sa;n wa;n 2 sa;n wa;n ·l·d ›x ›x

! 2 X d 21 d b e _ ð21Þ ub wa;n d 2 2½J a;n 2 J 0 1_pa;n C þ ½J a;n 2 J 0 la;n sgn sa;n Da;n 2 b¼1

2 X

ð21Þb ueb wa;n d þ

b¼1

! 2 2 X X ›leq;d ›leq;p a;n a;n b e D21 C 2 ð21Þ u ð21Þb ueb wa;n d b wa;n d 2 a;n ›x › x b¼1 b¼1

2 X p d p p d d D21 ð21Þb ueb wa;n d þ lF 2 lF a;n g_ a;n sgn sa;n C a;n g_ a;n sgn sa;n a;n b¼1

2

2 X

! ð21Þb ueb

wa;n d þ

lDa;n D_ a;n

D21 a;n

2

b¼1

2 X

! ð21Þb ueb

ðA15Þ

wa;n d

b¼1

! 2 X 21 d d b e _ ð21Þ ub wa;n ·l 2 ½J a;n 2 J 0 ga;n sgn sa;n Da;n 2 b¼1

! 2 2 X X ›leq;d a;n 21 21 b e Fd d d b e þ Da;n ð21Þ ub wa;n ·l 2 la;n g_a;n sgn sa;n Da;n 2 ð21Þ ub wa;n ·l ›x b¼1 b¼1 2lDa;n D_ a;n

D21 a;n

2

2 X

! ð21Þb ueb

wa;n ·l

Shape optimization

) ¼0

b¼1

By computing the value of d we complete one computational cycle. Similarly, one can complete as many cycles as needed in order to obtain the solution for any particular choice of the cost function.

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Parameterization based shape optimization: theory and practical implementation aspects

646

Marko Kegl

Received September 2004 Accepted January 2005

Faculty of Mechanical Engineering, University of Maribor, Maribor, Slovenia Abstract Purpose – To present an approach to parameterization based shape optimization of statically loaded structures and to propose its practical implementation. Design/methodology/approach – In order to establish a convenient shape parameterization, the design element technique is employed. A rational Be´zier body is used to serve as the design element. The design element is used to retrieve the nodal geometrical data of finite elements (FEs). Their field geometrical data are obtained using the FE own internal functions. For practical implementation it is proposed to establish the optimization cycle by two separately running processes. The data exchange is established by using self-descriptive and platform-independent XML conforming data files. Findings – The proposed approach offers an unified approach to shape optimization of skeletal, as well as continuous structures. Structural shape may be varied smoothly with a relative small set of design variables. The employment of a gradient-based optimization algorithm assures computational efficiency. Research limitations/implications – The aspects of FE mesh deterioration are not considered in this work. This would be necessary if for the actual problem at hand major and excessively non-uniform shape changes of the FE mesh are expected. Practical implications – A useful source of information for someone who is planning to develop a general or special-purpose integrated structural analysis and shape optimization software. Originality/value – The paper offers a rather simple, but quite powerful approach to structural shape optimization together with practical hints for its computational implementation. Keywords Optimization techniques, Structures, Numerical analysis Paper type Research paper


1. Introduction The field of structural shape design, supported by optimization, began to develop more intensively about three decades ago. It soon turned out that shape optimization is accompanied by several new difficulties and pitfalls (Haftka and Grandhi, 1986) not known in conventional design procedures where only the so-called “sizing parameters” has been employed as design variables. Already in those early days, it became clear that most of the difficulties and pitfalls are mainly related to the parameterization of the shape and deterioration of the finite element (FE) mesh caused by the design changes. As a result different parameterization concepts began to emerge (Imam, 1982; Braibant and Fleury, 1984; Shyy et al., 1988; Chang and Choi, 1992; Reitinger et al., 1994) and the need for adequate design sensitivity analysis caused the development of new techniques (Haftka and Adelman, 1989; Haug et al., 1986). The synthesis of parameterization based shape design and FE codes turned out to be a rather complex undertaking and to mitigate the situation a lot of effort has been put into the integration of commercial FE codes into new structural design procedures (Chang et al., 1995).

Apart from the shape parameterization concept, another concept of evolutionary type began to develop (Bendsøe and Kikuchi, 1988; Rozvany et al., 1992; Hinton and Sienz, 1995; Bendsøe, 1995; Maute and Ramm, 1995; Reynolds et al., 1999). By evolutionary methods, shape and topology optimization is performed simultaneously. In a most simplistic view, these methods manipulate on material distribution and/or properties depending on a suitable criterion (stress level, compliance, etc.) in individual parts of the structure. Thus, these methods do not perform optimization in the strict mathematical sense (solving a problem of mathematical programming) but they effectively remove or add material in order to get the best possible topology and shape of the structure. These methods exhibit remarkable and attractive properties ranging from generality to the absence of the need for shape parameterization and they often yield very good results. By comparing both concepts one can say that both of them have their advantages and disadvantages. As a very raw estimate one can say that for problems where the final solution is completely unknown, an evolutionary approach might be a better choice. And for problems, where we already have an idea how the final result should approximately look like, a parameterization based approach seems to be the more attractive choice. Thus, the decision, which approach to use, depends mainly on the actual problem at hand. Recent developments in the last years leaded to a wide palette of new techniques where the ideas from evolutionary concept were mixed by the ideas involved in the parameterization concept. Thus, at the time it seems to be more reasonable to distinguish among those methods that handle shape and topology (Schwarz et al., 2001; Cho and Jung, 2003; Garcia and Gonzalez, 2004; Zhou et al., 2004; Wang et al., 2003) and those that handle only shape (Kim et al., 2002, 2003; Ohsaki et al., 2003; Shen and Yoon, 2003) of the structure. It is also interesting to note that for structural analysis, besides of the very well established FE method, mesh-free methods are becoming increasingly popular (Kim et al., 2002, 2003). In this paper, attention will be focused on parameterization based shape optimization, more precisely, on shape optimization by employing the design element technique (Imam, 1982; Kegl, 2000) and the discrete method of differentiation. In contrast to the continuum based differentiation, by this approach the design sensitivity analysis is performed by differentiating the discrete form of the structural governing equation. In practical applications the data needed for that computation might not be available easily if a general purpose FE code is used. However, if one employs a specialized FE code, the discrete differentiation approach is an attractive choice because the sensitivity analysis can be performed very efficiently with a relative small computational effort. The outline of the paper is as follows. Section 2 outlines briefly the addressed optimization problem. Section 3 discusses the shape parameterization concept. In Section 4 the response and sensitivity analysis is discussed. Section 5 focuses on the solution procedure and Section 6 presents some numerical examples. 2. Problem statement Let us consider an elastic FE modeled structure being properly supported and loaded by external static forces (Figure 1). Let the shape of the structure be dependent on parameters bi ; i ¼ 1; . . . ; N; being assembled in the vector b [ R N : We call these parameters, the design variables.

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In general, when the design variable values change, the structure changes its shape and possibly its support locations. External forces may also be influenced by the design change (e.g. in case of pressure loading). It is obvious that this change will also result in a change of the structural response u [ R M (Figure 1). The shape design problem can now be formulated as follows: find such values of the design variables that the shape of the structure will be the best possible with respect to some criteria. Mathematically, this can be formulated in a form of a non-linear mathematical programming problem as follows ð1Þ

min f 0 subject to constraints f i # 0;

i ¼ 1; . . . ; K

bLi # bi # bUi ;

i ¼ 1; . . . ; N

ð2Þ

Here f 0 ¼ f 0 ðb; uÞ denotes the objective function, which is often (but not necessarily) defined either as the volume or the compliance of the structure. The constrained quantities f i ¼ f i ðb; uÞ usually concern nodal displacements and rotations, element strains and stresses, geometrical constraints, technological limitations and so on. The symbols bLi and bUi denote the lower and upper limits of the design variables. It should be noted that in equations (1) and (2) the design variables have to be considered as the independent variables and the response variables as the dependent one. In other words, one actually has u ¼ uðbÞ: This dependency is established implicitly by the structural response equation F2R¼0

ð3Þ

where F ¼ Fðb; uÞ and R ¼ Rðb; uÞ are the vectors of internal and external forces, respectively. Internal forces obviously depend explicitly on b and u. On the other side, external forces are often constant. However, they might become dependent on b if, for example, the weight of the structure is taken into account. And if one additionally employs elastic supports, external forces will also depend on u. It is obvious that the solution of the above problem necessitates at least three things. First of all, we need a suitable shape parameterization concept. Then, we need some FE code that can implement the adopted shape parameterization concept and, finally, we need some optimization code that can handle the solution process of the problems (1)-(3). 3. Shape parameterization of the structure Let us consider a 3D structure and let the hull of the structure represent the boundary surface of a 3D body B. Let the structure be modeled by FEs of a continuum or discrete (skeletal) type (Figure 2). It should be noted that referring to the body B rather than to

Figure 1. A design change influences shape, support locations, loads and consequently the response of the structure

the structure itself has the benefit that continuum and discrete (skeletal) structures can be handled in a unified way. Let us now assume that the FE mesh is a convective mesh following automatically the changes of the shape of B. Then, by defining B as a conveniently parameterized body, one can expect that the shape of the FE mesh (and the structure) can be modified significantly in an elegant and efficient way. This can be achieved by regarding B as the range of some mapping VB : U £ R N ! B where U , R 3 is a convenient 3D domain, for example, a unit cube (Kegl, 2000). The mapping VB should be regarded as follows: VB maps a point from U into a point in B under the influence of the design variables (Figure 3). In other words, the design variables should be regarded as parameters influencing the image B of the domain U. It follows that a design-dependent and convective FE mesh can simply be derived by defining its geometrical parameters (e.g. location of nodes, orientation of direction vectors, etc.) in the domain U rather then directly in the real 3D space. Once the mapping VB is defined, the geometrical data of such a mesh can quickly be computed for any values of the design variables. The shape of the structure is parameterized in terms of b: A convenient parameterization should provide suitable flexibility of B by a minimal set of design variables. Since B has to represent the whole structure, the most efficient approach is usually to partition the body B into ND smaller parts Di, termed the design elements (Figure 4), so that


Figure 2. The body B is defined by the hull of the structure

Figure 3. The design variables act as the parameters of the mapping VB

Figure 4. Partitioning the whole body into geometrically simpler design elements

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ND

B ¼ < Di

ð4Þ

i¼1

The design element Di is supposed to exhibit a relative simple shape so that it can easily be parameterized by employing an appropriate standard mapping Vi : U £ R N ! Di : Similarly, one can expect to derive a convenient parameterization of the whole body B. The general form of the mapping Vi is determined by the type of the employed design element. A design element can be any suitable geometrical 3D object ranging from a simple parameterized cube to any parameterized free-form body. However, if one would need to select the one with very good all-round qualities, a rational Be´zier body would surely be one of the favorites. Therefore, in the following we will focus on that type of element whereat any other geometrical body might also represent a convenient choice in certain special situations – depending on the problem at hand. Let D , R 3 denote a rational Be´zier body representing a generic design element (the subscript i, denoting the number of the design element, will be skipped for brevity). Then, D is the range of the mapping V, defined by N3 N1 X N2 X X

p¼

BNi 1 BNj 2 BNk 3 vijk q ijk

i¼1 j¼1 k¼1 N3 N1 X N2 X X

ð5Þ BNi 1 BNj 2 BNk 3 vijk

i¼1 j¼1 k¼1

where ¼ and BNk 3 ¼ BNk 3 ðs3 Þ are the Bernstein polynomials 2 1 and N 3 2 1; respectively. The symbols s1, s2 (Farin, 1993) of the order and s3 denote the independent parameters representing the coordinates of a generic point s [ U ¼ ½0; 13 in the unit cube U so that s ¼ ½s1 s2 s3 T : The vector q ijk ¼ q ijk ðbÞ denotes a vertex (a control point) of the defining polygon net, of the body and vijk ¼ vijk ðbÞ is its corresponding weight (Figure 5). The defining polygon net is topologically rectangular, meaning that it is defined by N 1 £ N 2 £ N 3 vertices – the symbols N1, N2 and N3 denote the number of vertices in corresponding parametric directions. By adopting the above definitions, V maps a point s from U into the corresponding point p ¼ pðs; bÞ of the range D representing a rational Be´zier body (Figure 6). It should be noted that D inherits all the attractive geometrical properties of Be´zier curves in the same manner, as this is the case with Be´zier surfaces. So, the boundaries of D will generally follow the shape of its defining polygon net; the corner points of the defining polygon net and the body are coincident and, finally, the body will always be contained within the convex hull of its defining polygon net. The boundary surface of a BNi 1

Figure 5. A Be´zier body and its defining polygon net with 3 £ 2 £ 2 control points

BNi 1 ðs1 Þ;

BNj 2

¼ BNj 2 ðs2 Þ N 1 2 1; N 2

Be´zier body consists of six Be´zier surfaces and the boundary curves of these surfaces are Be´zier curves. These curves can easily be matched exactly to circular shapes because the Be´zier body is of a rational type. The control point positions and weights depend on the design variables b. Thus, a variation of the values of the design variables will result in a smooth variation of the shape of D. That means that if one defines the FE mesh in U rather than in D directly, one will get a smoothly varying FE mesh. Let us consider a generic FE element and let its nodes be defined in U by the positions s j ; j ¼ 1; . . . ; J where J is the total number of the element nodes (Figure 7). By knowing the nodal pre-images s j of the FE, the actual nodal positions in the real 3D space can easily be calculated as r j ¼ pðs j ; bÞ;

j ¼ 1; . . . ; J


ð6Þ

for any given values of design variables b. For many FEs (truss elements, volume elements, plane elements, etc.) this is the only geometrical data needed – with the exception of some geometry related parameters like the area of cross-section. For some FEs, however, a direction vector d j may also be needed (Figure 7). For example, for a beam element one has to define the orientation of the local element frame. And for a shell element one needs a vector being normal to the shell surface. Although these vectors may be obtained by other means, the most effective approach in a shape-changing situation is probably to derive these vectors directly from the geometry of the design element. At each point of the design element, there is a set {e i }i¼1;2;3 of three characteristic direction vectors

›pðs; bÞ ; i ¼ 1; 2; 3 ð7Þ ›s i being tangent to the parametric lines of the element. By taking the vector products of these vectors ei ¼

d1 ¼ e2 £ e3;

d2 ¼ e3 £ e1;

d3 ¼ e1 £ e2

ð8Þ

Figure 6. The point s is mapped into p

Figure 7. Mapping of the FE node positions

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one gets a set {d i }i¼1;2;3 of three non-orthogonal vectors being normal to the three characteristic design element surfaces defined by si ¼ const; i ¼ 1; 2; 3 (Figure 8). This set of direction vectors can be efficiently employed in defining FE-related directions. For example, for a shell element lying in the s3 ¼ const surface of the design element, its normal direction unit vector is defined by n ¼ d 3 =kd 3 k: It should be noted that such a definition of n is valid throughout the design process since d3 depends on design variables and it changes accordingly to the shape variation of the design element. This is because obviously e i ¼ e i ðs; bÞ and thus also d i ¼ d i ðs; bÞ: The direction vectors di thus represent a very convenient way to define FE-related directions. For the FE nodes mapped from s j ; j ¼ 1; . . . ; J the corresponding sets of nodal direction vectors are given by dji ¼ d i ðs j ; bÞ;

i ¼ 1; 2; 3;

j ¼ 1; . . . ; J

ð9Þ

So far we have all the needed nodal geometrical data of a FE. But to perform FE analysis one needs the corresponding geometrical field data (position and required direction at any internal point of the FE). For this purpose one can simply employ FE internal shape functions H j ; j ¼ 1; . . . ; J so that one gets r¼

J X j¼1

H jrj;

di ¼

J X

H j dji ;

i ¼ 1; 2; 3

ð10Þ

j¼1

Note that with these expressions at hand one can easily calculate all the geometrical quantities needed for the calculation of internal forces, stiffness matrix and loads of the FE. Thus, instead of reading fixed FE geometry data from a file, the required quantities are calculated according to the current values of the design variables (Figure 9).

Figure 8. Characteristic direction vectors of the design element at point p

Figure 9. The design element (DE) code calculates the nodal data and the FE code is employed to get the field data

The geometry retrieval scheme shown in Figure 9 can be most easily and efficiently implemented in a specialized FE code. This should not be very difficult since the differences to a conventional FE code are rather small. First of all, one can see that the FE input data is actually the same as in the conventional case, except that it has a different meaning (location of the node in U rather than in the real space). The second thing one can observe is that now the FE code has to hold the input data for the design elements and it must have build-in expressions to evaluate design element related quantities. And the third thing we may notice is that the FE code has to hold the current design data – this data is needed when the design element related geometrical quantities are evaluated. Next, it is worth to spend some words on establishing the design dependency of the design elements, more precisely, the dependency of qijk and vijk on b. In our applications we found it most practical to use a functional expression parser employing a FORTRAN-like syntax. In that way the components of qijk and vijk can be defined by arbitrary expressions in terms of the design variables b. The same parser is also responsible for evaluating the derivatives of these expressions with respect to the design variables. The last thing concerns the meshing procedure. Obviously the meshing has to be done in U rather that in the real space. If the mesh geometry changes due to design changes are rather small, this meshing can be done only once prior to the shape optimization procedure. However, if the mesh geometry may change significantly, mesh distortion may cause significant loss of accuracy and lead to doubtable results. The only way out in such a situation is a suitable mesh adapting procedure integrated into the design process. This topic is not addressed in this paper. 4. Response and sensitivity analysis In order to solve the problems (1) and (2) using a gradient-based method of mathematical programming, one needs a procedure for the calculation of the functions f i ; i ¼ 0; . . . ; M and their derivatives df i =db; i ¼ 0; . . . ; M at some given design. This calculation is called the response and sensitivity analysis. The functions f i ; i ¼ 0; . . . ; M are expressed in terms of b and u. Thus, in order to perform the response analysis, the main computational effort will surely be the calculation of u at some given b. This is done by solving the response equation (3). Assuming that the response equation is non-linear with respect to u, the Newton-Raphson procedure will typically be employed. By this procedure the response is obtained iteratively by improving it with the increments Du ¼ 2

›F ›R 2 ›u ›u

21 ðF 2 RÞ

ð11Þ

For that purpose in each iteration the stiffness matrix, the internal forces and the loads of the structure have to be computed. These quantities are obtained by summing the corresponding FE quantities. When computing the stiffness matrix, the internal forces and the loads of the FE, its geometry (r, di) has to be retrieved using equations (5)-(10). The derivatives df i =db; i ¼ 0; . . . ; M are expressed in terms of b, u and du/db. For the sensitivity analysis we therefore need u and du/db evaluated at current b. The response u is already known from the response analysis, while du/db has to be


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calculated from the sensitivity equation. This equation can be derived by differentiating the response equation with respect to b. By doing this and rearranging the terms, we obtain du ›F ›R 21 ›R ›F ¼ 2 2 : ð12Þ db ›u ›u ›b ›b

654

It should be noted that while the response equation may be non-linear with respect to u, the sensitivity equation is always linear with respect to du/db. Even more, the first term in parentheses on the right hand side is the tangential stiffness matrix of the structure. This matrix is known (and already decomposed) from the response analysis. Thus, the sensitivity equation can be quite easily solved with a relative small computational effort. The only thing we need is the partial design derivatives of internal and external forces – the terms in the second parentheses on the right side of equation (12). This necessitates the calculation of the partial design derivatives of the internal forces and loads of the FE. Again, for this calculation all geometry data of the FE has to be retrieved using the proper relations. The quantities r and di are obtained by using the already mentioned equations. And the design derivatives have to be obtained from dr X j dr j ¼ ; H db db j¼1 J

j J dd i X j ddi ¼ ; H db db j¼1

i ¼ 1; 2; 3

ð13Þ

where dr j dp ¼ ; db s¼s j db

deij d ›p ¼ db db ›si s¼s j

de j dd1j de2j ¼ £ e3j þ e2j £ 3 ; dbk dbk dbk

ð14Þ

k ¼ 1; . . . ; N

ð15Þ

and similar expressions can be obtained for ddj2 =dbk and ddj3 =dbk : The derivative dp/db can be expressed as N3 N1 X N2 X X dq ijk N 1 N 2 N 3 dvijk q ijk þ vijk Bi Bj Bk db db dp i¼1 j¼1 k¼1 ¼ 2 N3 N1 X N2 X db X N1 N2 N3 Bi Bj Bk vijk i¼1 j¼1 k¼1

ð16Þ N3 N1 X N2 X X

BNi 1 BNj 2 BNk 3 vijk q ijk

i¼1 j¼1 k¼1 N3 N1 X N2 X X

!2 BNi 1 BNj 2 BNk 3 vijk

N3 N1 X N2 X X i¼1 j¼1 k¼1

i¼1 j¼1 k¼1

In a similar way one can derive d½›p=›si =db:

BNi 1 BNj 2 BNk 3

dvijk db

!

The above relations can easily be implemented in a specialized FE code. This can be a stand-alone FE program with its own graphical user interface and pre- and post-processing capabilities. In addition to that, however, it should be possible to run the program non-interactively in a batch mode for the optimization process. In this case the program should accept two input files: the structural data file and the file containing current values of the design variables. It should output a single file containing the response of the structure: the functions f i ; i ¼ 0; . . . ; M and their derivatives df i =db; i ¼ 0; . . . ; M evaluated at current design (Figure 10). For practical purposes all the files should have a self-descriptive and platform-independent syntax. A very good choice is to use XML conforming files.


5. Solution procedure Since, the design variables are continuous in our case and assuming that the functions in equations (1) and (2) are differentiable with respect to the design variables, the problems (1) and (2) can probably most effectively be solved using a gradient-based optimization algorithm (like e.g. the recursive quadratic programming method or some approximation method). In that case the solution procedure is iterative and can be outlined as follows: 5.1 Solution procedure Set k ¼ 0; choose some initial b ð0Þ : Calculate f i ; i ¼ 0; . . . ; M at b (k) (response analysis). Calculate df i =db; i ¼ 0; . . . ; M at b (k) (sensitivity analysis). Submit the calculated values to the optimizer in order to get some improvement Db (k) and calculate the improved design b ðkþ1Þ ¼ b ðkÞ þ Db ðkÞ : (5) Set k ¼ k þ 1 and check some appropriate convergence criteria – if fulfilled exit, otherwise go to step 2.

(1) (2) (3) (4)

In practice the above solution procedure can be implemented in several different ways. In our work we found it most practical to use a separate (stand-alone) optimization program and a separate analysis program. The arrangement is as shown in Figure 11. This arrangement enables multi-case optimizations (the same simulator is run several times) as well as the solution of complex optimization problems (several different simulators are run as necessary). In any case the optimizer runs all the simulations, acquires all the necessary data, assembles the data into a single optimization problem and calls its own optimization algorithm to improve the design. All the steps of this procedure are repeated in each cycle of the optimization process.

Figure 10. Operation of the FE program in the non-interactive mode

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In our current implementation the whole solution procedure is driven by the optimizer, which controls the execution of the simulation processes and the data exchange. The data exchange among separately running programs is established by XML conforming data files. The build-in optimization algorithm of the optimizer is based on the approximation method of mathematical programming (described in Kegl and Oblak, 1997; Kegl et al., 2002).

656 6. Numerical examples Four numerical examples will be considered. The first one presents a bridge structure, which was build recently over the river Savinja in Mozirje, Slovenia. Its purpose is to illustrate in detail the preparation and solution of an optimal design problem involving shape and sizing design variables. The next three examples merely illustrate on what kind of problems the proposed approach may be employed. These examples consider a double layer truss structure, a shell structure and side-cutting pliers. 6.1 A bridge structure Let us consider a bridge structure consisting of a straight reinforced concrete span hanging on a reinforced concrete arch and steel bars (Figure 12). In order to simplify the situation, a 2D FE model is used as follows. The arch is modeled by plane stress eight node elements and the bars by truss elements. The span is modeled simply by beam elements because its purpose in this example is merely to model properly the loading of the arch and the bars. The data of the structure are as follows. The thickness of the arch is t ¼ 800 mm: The material properties are: E ¼ 210; 000 MPa; r ¼ 7:7 kg=m3 and n ¼ 0:28 for the

Figure 11. The optimizer runs the simulator and controls the whole solution procedure

Figure 12. The bridge structure

bars; E ¼ 41; 500 MPa; r ¼ 2:5 kg=m3 and n ¼ 0:2 for the arch and E ¼ 37; 000 MPa; r ¼ 2:5 kg=m3 and n ¼ 0:2 for the span. The structure is loaded by the weight of the span w ¼ 196; 200 N=m and by external forces F 1 ¼ 50; 000 N and F 2 ¼ 200; 000 N simulating two fully loaded trucks standing on the bridge. Let us now formulate the optimization problem. The objective is to determine the shape of the span and the cross-sectional areas of the bars so that the compliance of the arch and the thickness of the bars are minimal. The constraints are related to the volume of the arch, which is not allowed to increase and to the stresses in individual bars which must not become greater than the allowable value. First of all we have to parameterize the shape of the arch. For that purpose we only need one design element with 5 £ 2 £ 1 ¼ 10 control points. Since, we have N 3 ¼ 1; in this example the design element is degenerated to a Be´zier surface. The positions of some control points are defined in Table I, while the other positions may easily be found by taking into account the symmetry of the arch. The weights of all control points are equal to unity. So far we introduced five shape design variables being related to the control point positions. The next thing we have to do is to define the design dependent cross-sectional areas of the bars. These definitions are given in Table II. The missing data can be found by symmetry. It should be noted that the design dependencies given in Tables I and II are defined in such a way that the order of magnitude of individual design variables is around one. Moreover, by the introduced definitions the initial design of the bridge (reflecting the actual design of the structure) is recovered by setting all design variables equal to zero. The objective of the optimization is to minimize the compliance of the arch and the total cross-sectional area of the bars. It should be noted that the compliance of the arch depends solely on design variables one through five, while the cross-sectional area of the bars depends only on design variables six through nine. Thus, the objective function can simply be written as a sum f 0 ¼ c A þ aB Control point 111 211 311 121 221 321

Bar 1 2 3 4


ð17Þ

X

Y

0 11,000 22,000 0 11,000 22,000

0 8000 þ 400b2 9500 þ 500b4 1200 þ 400b1 8000þ 400b2 þ1000 þ 400b3 9500þ 500b4 þ1000 þ 500b5

Table I. Control point positions in (mm)

Area of cross-section 15168 þ 5000b6 15168 þ 5000b7 15168 þ 5000b8 15168 þ 5000b9

Table II. Cross-sectional areas of the bars in (mm2)

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Here cA ¼ cA ðb; uÞ is the compliance of the arch and aB ¼ aB ðbÞ is the total cross-sectional area of all bars. For numerical reasons both quantities are normalized so that their values are equal to one at the initial design b 0. The first imposed constraint is related to the volume of the arch, which must not increase. Thus, vA 2 v0A # 0

658

ð18Þ

where vA ¼ vA ðbÞ is the volume of the arch and v0A ¼ vA ðb 0 Þ is its volume at the initial design. The stresses in the bars are limited by

si 2 s # 0;

i ¼ 1; 2; 3; 4

ð19Þ

where si ¼ si ðb; uÞ is the stress in the ith bar and s ¼ 80 MPa is the maximal allowable stress. Finally, for aesthetic reasons the vertical co-ordinate of the arch vertex was limited by yV 2 ymax # 0

ð20Þ

where yV ¼ yV ðbÞ is the vertical co-ordinate of the vertex and ymax ¼ 10 m is its maximal allowable height. The optimization problem is now fully defined so that one can run the solution process. The initial design was selected as b0i ¼ 0; i ¼ 1; . . . ; 9 which approximately corresponds to the actually build structure. The optimization process was smooth and stable – the objective function was decreasing monotonically. After ten iterations the near-optimum design was obtained. And the numerical results obtained after 20 iterations are presented in Table III. One can see from Table III that optimization reduced both compliance of the arch as well as the cross-sectional area of the bars. At the same time the maximal constraint violation (at the initial design related to the stress in the bars) was reduced to zero. The initial and optimal structures are shown in Figure 13. One can see that by optimization the height of the arch increased by approximately 1 m and reached the maximal allowable value of 10 m. The arch became thicker in the middle and thinner at both ends. Quantity

Table III. Comparison of initial and optimal design

b1 b2 b3 b4 b5 b6 b7 b8 b9 cA aB Maximum violation (percent)

Initial

Optimal

0 0 0 0 0 0 0 0 0 1 1 14.8

0.0601 2.0535 2 0.0623 5.3899 0.0035 0.0815 2 0.3648 2 0.1212 2 0.1724 0.8048 0.9538 0.0


Figure 13. Initial and optimal shape of the arch

6.2 A double-layer roof truss The second example just illustrates the results of shape optimization of a double layer truss of a roof structure. The initial design of the truss is shown in Figure 14. The shape of the truss was parameterized using one design element with 3 £ 5 £ 2 ¼ 30 control points. The positions of these control points depend on 16 shape design variables. Additionally, there are two sizing design variables controlling the inner and outer radius of the hollow circular cross-section of the employed truss members. The structure is loaded by its own weight and by external vertical forces. The objective was to minimize the compliance of the structure while keeping the volume (weight) of the structure constant. Additionally, the constraints related to structural geometry, element stresses and local element buckling were imposed. The problem was solved successfully and the final design is shown in Figures 15 and 16. 6.3 A variable-thickness shell structure The third example illustrates the result of shell structure optimization. The initial design of the shell is a flat rectangle with constant thickness (Figure 17).

Figure 14. Initial design of the truss – side, front and top view

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The shape of the shell was parameterized using one design element with 5 £ 5 £ 1 ¼ 25 control points. The positions of these control points depend on eight shape design variables. Additionally, there are six design variables controlling the thickness distribution. The structure is loaded by a snow load. Shell elements with four nodes were used to model the structure. The objective was to minimize the compliance of the structure while keeping the volume (weight) of the structure constant. Additionally, the constraints related to structural geometry were imposed. The optimal design is shown in Figure 18. 6.4 Side-cutting pliers The last example illustrates the results of shape optimization of the cutter of side cutting pliers. The wire frame of the initial design of the cutter is shown in Figure 19. The shape of the cutter was parameterized using five design element with a total of 52 control points. Due to technological reasons the positions of some of these control points depend only on three shape design variables. The structure is loaded according to the prescribed standards. Volume elements with 20 nodes were used to model the cutter. The objective was to minimize the volume of the structure while taking into account the constraints related to structural geometry and to the stress levels. The optimal design is shown in Figure 20.

Figure 15. Optimal shape of the roof – side view

Figure 16. Optimal shape of the roof – front view

Figure 17. Initial design of the shell


Figure 18. Optimal design of the shell

Figure 19. Initial design of the cutter

Figure 20. Optimal design of the cutter

7. Conclusion Although in the field of shape optimization the use of general-purpose commercial FE codes has many advantages, the development of specialized FE code may also be an attractive alternative. A specialized modeler and a shape parameterization-aware FE code may lead to an efficient designer tool. Of course, such a tool cannot cover the whole range of engineering shape design problems. However, certain types of problems may be addressed and solved in a very efficient and elegant way. By adopting the strategy of a stand-alone optimizer which uses separate simulators, specialized simulators may be developed for certain types of problems. Thus, the library of simulators may be arbitrarily extended whereas the optimization procedure remains always the same.

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References Bendsøe, M.P. (1995), “Optimization of structural topology”, Shape and Material, Springer, Berlin. Bendsøe, M.P. and Kikuchi, N. (1988), “Generating optimal topologies in structural design using a homogenisation method”, Computer Methods in Applied Mechanics and Engineering, Vol. 71, pp. 197-224. Braibant, V. and Fleury, C. (1984), “Shape optimal design using B-splines”, Computer Methods in Applied Mechanics and Engineering, Vol. 44, pp. 247-67. Chang, K.H. and Choi, K.K. (1992), “A geometry based shape design parameterization method for elastic solids”, Methods of Structures and Machines, Vol. 20, pp. 215-52. Chang, K.H., Choi, K.K., Tsai, C.S., Chen, C.J., Choi, B.S. and Yu, X. (1995), “Design sensitivity analysis and optimization tool (DSO) for shape design applications”, Computing Systems in Engineering, Vol. 6, pp. 151-75. Cho, S. and Jung, H.S. (2003), “Design sensitivity analysis and topology optimization of displacement-loaded non-linear structures”, Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp. 2539-53. Farin, G. (1993), Curves and Surfaces for Computer Aided Geometric Design, 2nd ed., Academic Press, New York, NY. Garcia, M.J. and Gonzalez, C.A. (2004), “Shape optimisation of continuum structures via evolution strategies and fixed grid finite element analysis”, Structural and Multidisciplinary Optimization, Vol. 26, pp. 92-8. Haftka, R.T. and Adelman, H.M. (1989), “Recent developments in structural sensitivity analysis”, Structural Optimization, Vol. 1, pp. 137-51. Haftka, R.T. and Grandhi, R.V. (1986), “Structural shape optimization – a survey”, Computer Methods in Applied Mechanics and Engineering, Vol. 57, pp. 91-106. Haug, E.J., Choi, K.K. and Komkov, V. (1986), Design Sensitivity Analysis of Structural Systems, Academic Press, New York, NY. Hinton, E. and Sienz, J. (1995), “Fully stressed topological design of structures using an evolutionary procedure”, Engineering Computations, Vol. 12, pp. 229-44. Imam, M.H. (1982), “Three-dimensional shape optimization”, International Journal for Numerical Methods in Engineering, Vol. 18, pp. 661-73. Kegl, M. (2000), “Shape optimal design of structures: an efficient shape representation concept”, International Journal for Numerical Methods in Engineering, Vol. 49, pp. 1571-88. Kegl, M. and Oblak, M.M. (1997), “Optimization of mechanical systems: on non-linear first-order approximation with an additive convex term”, Communications in Numerical Methods in Engineering, Vol. 13, pp. 13-20. Kegl, M., Butinar, B.J. and Kegl, B. (2002), “An efficient gradient-based optimization algorithm for mechanical systems”, Communications in Numerical Methods in Engineering, Vol. 18, pp. 363-71. Kim, N.H., Choi, K.K., Chen, J.S. and Botkin, M.E. (2002), “Meshfree analysis and design sensitivity analysis for shell structures”, International Journal for Numerical Methods in Engineering, Vol. 53, pp. 2087-116. Kim, N.H., Choi, K.K. and Botkin, M.E. (2003), “Numerical method for shape optimization using meshfree method”, Structural and Multidisciplinary Optimization, Vol. 24, pp. 418-29. Maute, K. and Ramm, E. (1995), “Adaptive topology optimization”, Structural Optimization, Vol. 10, pp. 100-12.

Ohsaki, M., Ogawa, T. and Tateishi, R. (2003), “Shape optimization of curves and surfaces considering fairness metrics and elastic stiffness”, Structural and Multidisciplinary Optimization, Vol. 24, pp. 449-56. Reitinger, R., Bletzinger, K.U. and Ramm, E. (1994), “Shape optimization of buckling sensitive structures”, Computing Systems in Engineering, Vol. 5, pp. 65-75. Reynolds, D., McConnachie, J., Bettess, P., Christie, W.C. and Bull, J.W. (1999), “Reverse adaptivity – a new evolutionary tool for structural optimization”, International Journal for Numerical Methods in Engineering, Vol. 45, pp. 529-52. Rozvany, G.I.N., Zhou, M. and Birker, T. (1992), “Generalized shape optimization without homogenization”, Structural Optimization, Vol. 4, pp. 250-2. Schwarz, S., Maute, K. and Ramm, E. (2001), “Topology and shape optimization for elastoplastic structural response”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 2135-55. Shen, J. and Yoon, D. (2003), “A new scheme for efficient and direct shape optimization of complex structures represented by polygonal meshes”, International Journal for Numerical Methods in Engineering, Vol. 58, pp. 2201-23. Shyy, Y.K., Fleury, C. and Izadpanah, K. (1988), “Shape optimal design using high-order elements”, Computer Methods in Applied Mechanics and Engineering, Vol. 71, pp. 99-116. Wang, M.Y., Wang, X. and Guo, D. (2003), “A level set method for structural topology optimization”, Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp. 227-46. Zhou, M., Pagaldipti, N., Thomas, H.L. and Shyy, Y.K. (2004), “An integrated approach to topology, sizing, and shape optimization”, Structural and Multidisciplinary Optimization, Vol. 26, pp. 308-17.




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Multi-scale modeling of heterogeneous structures with inelastic constitutive behaviour

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Part I – physical and mathematical aspects

Received November 2004 Accepted January 2005

Damijan Markovic Ecole Normale Supe´rieure de Cachan, LMT Cachan, Wilson, Cachan, France

Rainer Niekamp Institute of Scientific Computing, Technische Universita¨t Braunschweig, Braunschweig, Germany

Adnan Ibrahimbegovic´ Ecole Normale Supe´rieure de Cachan, LMT Cachan, Wilson, Cachan, France

Hermann G. Matthies Institute of Scientific Computing, Technische Universita¨t Braunschweig, Braunschweig, Germany, and

Robert L. Taylor Department of Civil & Environmental Engineering, University of California at Berkeley, Berkeley, California, USA Abstract


Purpose – To provide a computational strategy for highly accurate analyses of non-linear inelastic behaviour for heterogeneous structures in civil and mechanical engineering applications Design/methodology/approach – Adapts recent developments on mathematical formulations of multi-scale problems to the recently developed component technology based on Cþ þ generic templates programming. Findings – Provides the understanding how theoretical hypotheses, concerning essentially the multi-scale interface conditions, affect the computational precision of the strategy. Practical implications – The present approach allows a very precise modelling of multi-scale aspects in structural mechanics problems and can play an essential tool in searching for an optimal structural design. Originality/value – Provides all the ingredients for constructing an efficient multi-scale computational framework, from the theoretical formulation to the implementation for parallel computing. It is addressed to researchers and engineers analysing composite structures under extreme loading. Keywords Elasticity, Numerical analysis, Structures Paper type Research paper

The work was supported by the French Ministry of Science, Ecole Normale Supe´rieure invited professors funding program and the Slovene Ministry of Science, Education and Sport, to which the authors acknowledge gratefully.

1. Introduction Composite materials have always played an important role in structural engineering. Either through normal manufacturing processes, since material processing cannot always avoid some final flaws such as porosity, or by engineered design in which one desires to improve material resistance, such as in hard inclusion composites (e.g. metal composites, concrete, etc.). Unfortunately, the heterogeneous nature of materials induces more significant heterogeneous field solutions in a structural loading framework, and consequently, in general, makes modeling more difficult. Often heterogeneous material behaviour can be obtained through experimental phenomena identification using fundamental thermodynamic formulations, such as in the case of damage and plasticity models (Lemaıˆtre and Chaboche, 1988). However, it sometimes happens that as soon as these phenomenological models attempt to describe real materials under extreme conditions, they become very complicated with many parameters that are difficult to identify experimentally. Another approach to deal with modeling of heterogeneous materials is analytical homogenization. Methods of this family have had considerable success in describing linear elastic behaviour for several types of microstructure (Bornert et al., 2001; Hori and Nemat-Nasser, 1999). A basic hypothesis of homogenization techniques is the complete separation of scales, i.e. the characteristic size of heterogeneities is assumed to be infinitely smaller than the characteristic structural dimension. The application of these methods to modeling of non-linear inelastic behaviour is not straightforward (Bornert et al., 2001 for overview) and remains a very active research field. Most of the studies of non-linear homogenisation methods are based on the transformation field analysis (TFA) introduced in (Dvorak, 1992). The original theory assumes that the inelastic strains (e.g. plastic strain or thermoelastic strain) are uniform in each phase and, using constitutive behaviour for each phase, one obtains a set of equations that completely define the macroscopic response. In general, this approach gives a good representation of the macroscopic stress-strain relation, but has two weak points: (1) the uniformity of inelastic strains is usually not confirmed by experiments; and (2) the elastic localization rule used in (Dvorak, 1992) is not an optimal one (Chaboche et al., 2001). Some improvements in regard to these two aspects exist in the literature (Chaboche et al., 2001; Buryachenko et al., 2002; Michel and Suquet, 2003). Due to limits in the analytical approximations, which are based on several hypotheses, and due to constantly increasing computer power in recent years, many numerical studies have been made using transformation field analysis. Most often the finite element method (FEM) is used for the calculations on the micro-scale (Brockenbrough et al., 1991; Ghosh and Moorthy, 1995; Jiang et al., 2001; 2002). Nevertheless, other numerical methods also have been developed for the same purpose (Moulinec and Suquet, 1994; Paley and Aboudi, 1992; Ladeve`ze et al., 2001; Ladeve`ze and Nouy, 2003). In many civil and mechanical engineering applications, the scales are either weakly or strongly coupled and the modeling algorithm has to be modified. If scales are separated far enough and the scales are only weakly coupled, a two-level FEM procedure called the FE2 method can be applied (Kruch et al., 1998; Miehe, 2002; Feyel,

Multi-scale modeling

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2003). In this approach we have one FEM model for the structural scale and a second one at each material point, i.e. the quadrature point in the FEM structural scale model. The material response is obtained from the second level FE analysis. In this paper we consider strongly coupled scales where we utilize equally a finite element model at both scales, however, the micro-scale is finitely smaller than the macro-scale rather than infinitely smaller (Ibrahimbegovic and Markovic, 2003; Ghosh et al., 2001). We apply a localized Lagrange multiplier method to couple the two scales (Park et al., 2002; Park and Felippa, 2000). In this context, the macro-scale is the element frame situated between different sub-domains which may be interpreted as the micro-scale because of their much finer FE representation. Thus, for each “macro element” we have a choice of using either a macroscopic model obtained in a conventional way or a very fine FE model that more accurately characterizes the material constitutive behaviour. An important characteristics of our approach is the local nature of the micro-scale calculations (Ibrahimbegovic and Markovic, 2003; Markovic and Ibrahimbegovic, 2004). The FE models representing the microstructure communicate between each other exclusively through the degrees of freedom of the macro-scale model. Hence, the sub-structuring scheme does not affect the macro-scale resolution procedure. In addition, the independence of micro scale calculations renders the micro-macro strategy easily parallelizable. The outline of the paper is as follows. In the next section we recall the theoretical and numerical framework of the proposed multi-scale strategy. A solution algorithm suitable for parallel computing is presented in Section 3. In Section 4 we consider technical issues of the parallel implementation. To illustrate the computational efficiency, some numerical examples are shown in Section 5. Some conclusions are drawn in Section 5. 2. Formulation of the strong coupling multi-scale interface 2.1 Concept of strongly coupled scales We consider a general class of heterogeneous structure problems subjected to an arbitrary loading and obeying non-linear physical laws resulting in non-linear phenomena such as damage and plasticity. It is assumed that the scales are strongly coupled and that their evolution has to be calculated simultaneously. The FEM for solid and structural mechanics is used to analyze the response at both scales. The structure is first modeled with a “macro” FEM mesh and, subsequently, the microstructure volume is represented by a “micro” FEM mesh (see Figure 1). Using an appropriate formulation the residual, stiffness, etc. for each “macro” element is obtained from a micro-scale finite element calculation. Using this approach we replace the macroscopic constitutive law at the finite element level rather than at the quadrature point level. The microstructure window (see Figure 1) is chosen such that its dimensions match those of the corresponding macro finite element. 2.2 Variational formulation The coupling of the scales in our multi-scale FE model is obtained through the framework of localized Lagrange multipliers (Park et al., 2002; Park and Felippa, 2000). The macro mesh plays the role of a “frame” which is connected to the micro mesh by the Lagrange multipliers (Figure 2).


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Figure 1. Micro- and macro-scale finite element model of a simple structure

As a model problem, we write an elastic potential, whose stationary value leads to the solution of the problem, as: P ¼ Pmicro þ Pinter f þ Pext 7 ! stat:;

ð1Þ

where the different terms are defined as:

Figure 2. Macro mesh FE connected to the micro mesh by Lagrange multipliers l

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Pmicro ¼ Pinter f ¼

Z Vm

Z

Cm ð1; jk Þ dV

Pext ¼ 2

Z

u M · t dS 2 ›s V

m

Z

u m · f dV V

ð2Þ

M

l · ðu 2 u Þ dS; G

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where u M is the macro displacement field (coarse discretization), u m is the micro displacement field (fine discretization), bart are specified boundary tractions and f are body forces. Accordingly, Cm is micro scale free energy, 1 is the strain field at the micro scale, j k are the internal variables at micro scale, and l is the Lagrange multiplier field that connects two different scales. We also note that V defines the total domain, (sV is the part of the boundary where tractions are applied, and G denotes the total interface surface between the two scales. Using the stationarity condition of the potential, dP ¼ 0; we obtain the weak formulation of the equations: . Micro-scale equilibrium: Z Z ðd1 m : s m 2 du m fÞ dV þ du m l dS ¼ 0; ð3Þ Vm .

G

Micro-macro displacement field compatibility: Z dl · ðu m 2 u M Þ dS ¼ 0;

ð4Þ

G

.

and Macro-scale equilibrium: Z Z du M · t dS þ du M · l dS ¼ 0: ›s V

ð5Þ

G

In equation (3) above we use the standard kinematic and thermodynamic relations: 1 ›ui ›uj þ 1ij ¼ ; 2 ›xj ›xi

ð6Þ

and

sij ¼

›Cð1kl ; jk Þ : ›1ij

ð7Þ

Remark 1. Strictly speaking, equations (3)-(7) are valid only for elastic material behaviour. In the case of plastic or/and damage behaviour, we would have to add to equation (1) some other stationarity conditions on inelastic dissipation

(Simo et al., 1989; Simo and Taylor, 1985; Ibrahimbegovic et al., 2003), in case the inelastic behaviour is considered as associative. If the constitutive laws are non-associative (Lemaıˆtre and Chaboche, 1988; Simo and Taylor, 1985), the corrections need to be done directly on the weak formulation (equations (3)-(7)). However, since the treatment of the inelasticconstitutive behaviour is carried out locally, i.e. on the integration point level of each micro-scale finite element, it does not affect the interface formulation. Therefore, with respect to the micro-macro interface, equations (3)-(7) are valid for any non-linear behaviour, s ¼ sð1; jk Þ: 2.3 Discretization of the fields The fields u M, l and u m are discretized as follows: uM < NM UM

l < NlL

um < NmUm;

ð8Þ

which further permits us to rewrite the system (3)-(5) as: Z

R m :¼

m

ðB mT s m ðU m Þ 2 N mT fÞ dV þ

V

R l :¼

Z

Z

N mT N l dSL ¼ 0

G T

N l ðN m U m 2 N M U M Þ dS ¼ 0

ð9Þ

G

R M :¼

Z

N ›Vs

MT

t dS þ

Z

T

N M N l dSL ¼ 0; G

where the micro strain projector Bm is defined as: 1 ›N m ›N m m m m dik þ djk ) 1^m Bijk ¼ ij ¼ Bijk U k ; 2 ›x j ›xi

ð10Þ

with 1^ m ðU m Þ ¼ 1 m ju m !N m U m as the approximation of the strain field. Note that the system of equations in (9) is non-linear. We choose to solve it by a Newton iterative method, which requires the linearization of each equation, according to: < RðkÞ Rððkþ1Þ ·Þ ð·Þ þ

›RðkÞ ð·Þ Dhi ; › hi

ð11Þ

where the superscript (k) denotes the value at the kth iteration, and hi the set of independent variables. Applying this operation to the system (9), we obtain the set of linear equations: 9 8 9 2 ðkÞ 38 2RðkÞ > DUðkÞ D E 0 > m > > > > > m > > > > > = < = 6 T 7< ðkÞ ðkÞ 6E 7 2R DL 0 2F ¼ ; ð12Þ l 4 5> > > > > > > ðkÞ > ðkÞ > > > > T 0 : DU ; : 2R ; 0 2F M

M


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where the matrices E and F define the projection between the Lagrange multiplier field and the micro and macro displacement field, respectively. D (k) denotes the value of the micro stiffness at the k-th iteration. These arrays are calculated according to: E¼

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Z N

mT

l

N dS

F¼

G

Z N

lT

M

N dS

D

ðkÞ

¼

Z Vm

G

m B mT C UðkÞ m B dV ;

ð13Þ

where the material stiffness tensor is given by: C ijkl ¼

›sij : ›1kl

ð14Þ

In (12) above we present the discretized micro-macro problem formulation for arbitrary interpolation functions of the three fields, u m, l and u M. If the displacement interpolations, N m and N M, are imposed by the choice of our macro and micro finite element models, the choice for the Lagrange multiplier interpolation functions, N l, is more open and can make a difference in the quality of the solution (Markovic and Ibrahimbegovic, 2004). For the macro and micro finite element models we use standard isoparametric bilinear quadrilateral element technology, with the shape functions being: 1 ðj; hÞ ¼ ð1 þ ja jÞð1 þ ha hÞ; N m;M a 4

a ¼ 1; 2; 3; 4

ð15Þ

where 21 # j; h # 1 and ja, ha are natural coordinates of the nodes. The Cartesian coordinates are given by: x¼

4 X a¼1

N a ðj; hÞxa and y ¼

4 X

N a ðj; hÞya :

ð16Þ

a¼1

For the Lagrange multiplier interpolation, we consider two types of interfaces: displacement based and force based. In the case of a displacement based interface, we assume that the micro displacement field, u m, will be constrained to follow to the macro displacement field, u M, on the interface surface G. This is obtained by taking the following interpolation function for the Lagrange multipliers, N la ¼ dðr 2 r a ÞLa ; ;r i [ ›V0 e0 ;

ð17Þ

where d(·) is the Dirac function, ra is the coordinate vector of the ath boundary node and V0 e0 denotes the macro element domain corresponding to a micro sub-domain (i.e. macro finite element). For this interpolation, the Lagrange multiplier parameters, La, represent nodal forces on the micro mesh boundary. From equation (17) we can easily deduce the following relation between the micro and macro fields on the interface surface:

u m j›V0 e0 ¼ T e u M ;

ð18Þ

where T e is the linear interpolation operator between the macro and micro fields. Instead of assuming a linearly distributed displacement field on the micro domain as the localization operator (equation (18)), we can alternatively try to linearly distribute the tractions. This will would lead us to a force based interface and occurs as a consequence of the following choice for the Lagrange multiplier interpolation: lðx; yÞ ¼ 2s M ðx; yÞ · nðx; yÞ;

ð19Þ

where n is a unit normal vector to the boundary surface and the macro stress field s M is defined as:

sxx ¼ L1 þ L2 y syy ¼ L3 þ L4 x sxy ¼ L5 :

ð20Þ

The choice of the interpolation for l, defined through (19) and (20), is inspired by the stress interpolation of the very accurate Pian-Sumihara hybrid stress finite element formulation, introduced in (Pian and Sumihara, 1984). The linear stress interpolation in (20) is self-equilibrated and as such is considered as optimal for the bilinear macro element. In fact, on account of the a priori self-equilibrated interface traction interpolation, the “floating body” micro problem becomes well posed (for a more detailed discussion refer Markovic and Ibrahimbegovic, 2004; Markovic, 2004). For linear elastic material behaviour, the force based interface always gives a better response than the displacement based interface (Ibrahimbegovic and Markovic, 2003). However, this is not necessarily true for non-linear or in-elastic material behaviour. The quality of the result with respect to the total external work and the total dissipation for the two types of micro-macro interface treatments turns out to be dependent on the micro sub-domain size, the behaviour of different phases and the macroscopic boundary conditions (for details refer Markovic and Ibrahimbegovic, 2004; Markovic, 2004). 3. Parallel solution of equations We solve the system (12) by Gaussian elimination, eliminating first the values of DUm, then DL, ending up with a non-linear system for DUM. By the elimination of variables DUm and DL we use an operator split approach (Ibrahimbegovic and Markovic, 2003), where for a given global-local problem, we first converge on the local equations with the global variables being fixed, and then correct the global variable approximation in order to solve the global equations. Here, we deal with three sets of variables (Um, L and UM) and hence with three operator split iterative processes. We note that by choosing l max ¼ mmax ¼ 1 below, the procedure degenerates to a simple iteration with standard Gaussian elimination. In our implementation we chose to converge each problem at each sub-loop level, which leads to a robust though not always the fastest algorithm (Ibrahimbegovic et al., 1999). Our algorithm was implemented according to:


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DO k ¼ 1. . .convergence ~ ðkÞ DUðkÞ ¼ 2R ~ ðkÞ D M l M ~ ðkÞ ¼ F T DðkÞ21 F; D l l

21

T ðkÞ DðkÞ E l ¼E D

~ ðkÞ ¼ RðkÞ þ F T DðkÞ21 RðkÞ R M l l M

672

21

21

2 F T DðkÞ E T D ðkÞ RðkÞ m l ðkþ1Þ ðkÞ ) UM ¼ UðkÞ M þ DUM

DO l ¼ 1. . .l max or convergence ~ ðk;lÞ ; D ~ ðk;lÞ ¼ ðE T D ðk;lÞ21 EÞ ~ ðk;lÞ DLðk;lÞ ¼ 2R D l ~ ðk;lÞ ¼ 2Rðk;lÞ þ E T D ðk;lÞ21 Rðk;lÞ þ FUðkÞ R l l m M

ð21Þ

) L ðk;lþ1Þ ¼ L ðk;lÞ þ DL ðk;lÞ DO m ¼ 1. . .mmax or convergence ðk;l;mÞ

~ ¼ 2R D ðk;l;mÞ DUðk;l;mÞ m m

~ ðk;l;mÞ ¼ Rðk;l;mÞ þ EL ðk;lÞ R m m ðk;l;mþ1Þ ) Um ¼ Uðk;l;mÞ þ DUðk;l;mÞ m m

ENDDO m ENDDO l ENDDO k

Remark 2. Gaussian elimination without pivoting is generally not well adapted to solve indefinite systems, (e.g. equation (12)), with a large number of zeros on the diagonal, since it could result in a solution of large full matrices and thus lead to inefficiency in the procedure. For such linear systems an iterative method is more appropriate, for example the Uzawa algorithm. However, in our case the number of micro equilibrium equations greatly exceeds those of the compatibility equations. Therefore, the linear system for DL associated with each macro element, obtained after eliminating the micro displacements, DUm, will be full, but is not a large system (5 £ 5 in the present formulation). On the other hand, elimination of the micro displacements can be done inexpensively since the matrix D, obtained from the micro finite element analysis, is sparse and already LU-decomposed.The highest level iterative process is analogous to an iterative process of a standard non-linear FEM approach (see Figure 3). The only difference is in the way the element residual and stiffness matrix are calculated. They are obtained through the micro finite element computational process instead of a numerical integration of the constitutive law over the macro element. For the global process the element residual vector is calculated as R e ¼ F T L and the 21 e T ðkÞ stiffness matrix as K ¼ F Dl F.


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Figure 3. Micro-macro computation scheme

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The two inner iteration loops in equation (21), for L and for U m, respectively, are carried out at the micro level. Unlike a standard FEM, where individual element stiffness and residual computation effort is negligible in comparison with the global solution of equations, in the proposed micro-macro strategy the individual element response computation, i.e. corresponding to the micro problem solutions, can be as computationally demanding as the macro equation solution. Nevertheless, the formulation is constructed in a manner such that all the micro problems are independent of each other for every macro iteration. Hence, they can be computed in parallel, one or a few as a separate process on one of the computer processors. In Section 4 we show briefly how the parallelized algorithm is implemented on a multi-processor computer system. 4. Software architecture The middle-ware used to implement the parallel micro-macro simulation is the Component Template Library (CTL), a recent development due mainly to the second author. The CTL is a component technology based on Cþ þ generic template programming (Alexandrescu, 2002). Similarly to CORBA (Henninga and Vinoski, 2002) it can be used to realize distributed component-based software systems. Here, a component is a piece of software which consists of a well defined interface and an implementation. Interface and implementation are connected through a communication channel, e.g. TCP/IP, MPI, PVM, or by dynamic linkage. In this way the use of a component is independent of its location. This library serves as an easy to use programming environment for distributed applications in an abstract manner. It allows one to define a new component, e.g. a simulation instance, using the functionality of an existing library without need of change nor recompilation of the library. Hence, if there is a library version of a simulation code, there is the possibility to use this code as a component with a standard subroutine call semantic. The Figure 4 demonstrates the master-worker architecture of this partitioned simulation. This architecture allows for an easy exchangeability of the simulation components. The complexity of message passing libraries like MPI, PVM, or of using sockets is completely hidden. Here the finite element program FEAP (Finite Element Analysis Program, refer Taylor, 2004) is used for the macro as well as for the micro scale. In order to use this simulation program as a component, it had to be converted into the form of a library offering functionalities such as: read input data, compute the residual, or the (decomposed) stiffness. Using this library a general component interface for simulations was implemented. To each finite element of the macro scale discretization, the location (path and machine) of the component computing its micro scale is assigned. In the initialization loop over the macro elements this location is used to start and initialize the micro components. In each computational step the macro elements are visited twice. First they call in a non-blocking manner the lower level components to compute K e and R e in parallel by invoking the corresponding subroutines defined in the interface. In the second loop these quantities are read and used to assemble the global macro scale system. The solution of the macro scale system is presently performed inside the master component in serial.


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Figure 4. Task distribution scheme on different processors

5. Numerical examples This section is divided into three parts, one describing the problem, the next one presenting an elasto-plastic porous matrix modeling, and the last replacing the voids by damageable inclusions. In this latter example, we also compare the exact finite element microstructure representation with a phenomenological macroscopic elasto-plastic damage model proposed in (Ibrahimbegovic et al., 2003) which has a significant practical interest. Namely, the exact microstructure representation, in spite of its efficient parallel implementation, is computationally very time and memory demanding, so we should use it only in specific parts of the structure where the stress gradients are high and the different scale effects are strongly coupled. 5.1 Problem description The proposed multi-scale strategy has been validated in (Markovic and Ibrahimbegovic, 2004) for small examples and the results have been compared to a complete micro-scale solution. We demonstrated there that the precision obtained by the approach is satisfactory under the condition that the micro subdomain, i.e. the macro element size, is properly chosen. The two interface possibilities considered, i.e. the displacement and the force based methods, sometimes gave very different response and we concluded that for non-linear material behaviour (e.g. an elasto-plastic model) neither of the two methods is superior in all situations with respect to the scale ratio, phase property and macro boundary conditions. However, the force based interface usually appeared to be more robust. In addition, in general it lead to a lower bound of

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Figure 5. Heterogeneous structure subjected to a three point bending test. Load driven by displacement control

Figure 6. Three different types of macro finite element models for the same structure: 1 £ 1; 2 £ 2; 4£4

the structural stiffness, whereas the displacement based interface lead to an upper bound with respect to an exact reference value of the stiffness. With a similar goal, here we show the convergence of the multi-scale approach and the influence of the micro-macro interface definition for an example with a more realistic size. The problem is one where no direct microscopic solution is available and also one where the parallel implementation is beneficial. The problem we consider is a three point bending test in which the corresponding boundary conditions are spread over a finite distance to avoid locally infinite energy in the system (see Figure 5). For simplicity, the microstructure is assumed to be regular, consisting of the elasto-plastic matrix with damageable circular inclusions. The scale ratio (see Figure 5) is taken to be < 100; which corresponds to a realistic concrete structure with a length of approximately 1 m and containing aggregates of < 1 cm: The micro-macro model of the structure shown in Figure 6 contains up to 1,000,000 degrees of freedom, and the problem was run on the local PC/Linux network at LMT-Cachan. Because of the multi-user network environment the computing time could not be exactly determined and is estimated as 3-5 h. If available, one should use a cluster of dedicated machines or a parallel supercomputer designed for these purposes.

5.2 Elasto-plastic matrix with voids We model the matrix (see Figure 6) by a von Mises elasto-plastic model, which uses the following yield function, rffiffiffi 2 p p ð22Þ ðsy 2 q p Þ; F ðs; q Þ ¼ kDev sk 2 3 Dev[·] is the deviatoric part of a tensor, the where sy is the uniaxial initial yieldpstress, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tensor norm is defined as, kð · Þk ¼ ð · Þij ð · Þij and the isotropic hardening behaviour is assumed to have the following form, q p ðj p Þ ¼ sy þ K p j p þ ðs1 2 sy Þð1 2 e 2b p

p p

j

Þ;

ð23Þ

p

where K , b and sy are the model parameters. In addition, the elastic behaviour is assumed as linear isotropic, i.e. s ¼ Cð1 2 1 p Þ; where 1 1 C ijkl ¼ 2m e ðdik djl þ dil djk Þ 2 dij dkl þ K e dij dkl : 2 3 The values of applied parameters are: K e ¼ 3:5 · 1010 ; m e ¼ 1:2 · 1010 ; sy ¼ 4:0 · 107 ; s1 ¼ 8:0 · 107 ; b p ¼ 1; 000 and K p ¼ 1:0 · 108 : Hence, using standard thermodynamic assumptions and the principle of maximum plastic dissipation, the behaviour of the matrix is completely defined (Simo and Taylor, 1985; Lubliner, 1990). Since the problem we consider is too large to have a tractable complete micro solution, we no longer have a reference result to compare with and cannot be positive, which of the micro-macro models is more reliable with respect to micro subdomain (macro element) size or interface definitions (displacement or force based). Therefore, we have to estimate the best values from the convergence characteristics. For different macro element and inclusion sizes, we observe the value of the force, F, defined as the stress integral over the loaded surface, as a function of the imposed displacement U. In Figure 7 we show how the upper bound of the displacement based interface formulation and the lower bound of the force based formulation converge with decreasing relative size of the voids. We note in passing that by decreasing the scale ratio (inclusion size vs dimension of the structure), we are approaching a standard homogenization theory assumption (Bornert et al., 2001), where the macroscopic mechanical response of the Representative Volume Element (RVE) is independent of the type of boundary conditions (displacements or tractions). The results in Figure 7 confirm this hypothesis, since as the inclusion size decreases the macro element corresponds exactly to the RVE. However, the observed convergence in Figure 7 has no practical interest, since the different macro meshes and micro-structures do not correspond to the same mechanical problem. Hence, in the following we present results for different micro-macro models corresponding to the same mechanical problem, i.e. a structure containing 16 £ 32 voids regularly distributed. It is evident from the forcedisplacement response shown in Figure 8 that the choice of the macro element size can be of significant importance. The major contribution to the error in the proposed micro-macro strategy comes from the interface field assumptions, which are the linearly distributed displacements for the displacement based formulation and the linearly distributed tractions for the force based formulation. Those assumptions can be justified only if the macro mesh is


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Figure 7. Three point bending test with porous elasto-plastic matrix

Figure 8. Three point bending test with porous elasto-plastic matrix

fine enough so that the stress or displacement field variations over the subdomain boundary are sufficiently close to a linear distribution and, at the same time, that the microstructure domain contains a representative number of heterogeneities so that the micro field characteristics near the boundary can be masked. In particular, at present the model implementation of the multi-scale method does not permit a refinement of the macro mesh and thus is not adapted for every multi-scale situation. Another improvement could involve introduction of higher order elements at the macro scale.

5.3 Combined exact and phenomenological microstructure representation Keeping the same geometrical properties of the problem containing the porous matrix (Figures 5 and 6), we replace the voids by inclusions with similar elastic properties to those of the matrix and with damage behaviour. The damage model used was first proposed in (Ibrahimbegovic et al., 2003), its formulation and numerical implementation being directly inspired by the classical plasticity models. Thus, as a major ingredient, it uses a fracture criteria with a maximum damage dissipation. Here we use the following damage threshold function for the inclusion behaviour, 1 Fd ðs; q d Þ ¼ Trs 2 ðsf 2 q d Þ; 3

ð24Þ

where sf is the fracture stress value and q d the damage hardening function. Ductile damage behaviour is assumed, so that q d is equivalent to the hardening function in plasticity, q d ðj d Þ ¼ sf þ K d j d þ ðsf 1 2 sf Þð1 2 e 2b

d d

j

Þ;

ð25Þ

where sf1, K d, b d are the damage hardening parameters and j d the damage hardening variable. The damage criteria in (24) leads to an isotropic damage model with k d as the single damage variable corresponding to the following damage tensor (for details see Ibrahimbegovic et al., 2003), Dijkl ¼ k d dij dkl ;

ð26Þ

where dij denotes the Kronecker delta. We assume the following inclusion damage material parameters: sf ¼ 0:5 · 107 ; sf 1 ¼ 1:0 · 107 ; b d ¼ 100 and K d ¼ 2:5 · 106 ; and linear elastic properties as: K ed ¼ 1:8 · 1010 ; m ed ¼ 1:0 · 1010 : We do a similar three point bending test as for the porous matrix with 16 £ 32 inclusions in total. In Figure 9 we observe a very good convergence of the 8 £ 4 elements and 16 £ 8 elements per macro mesh, but only for the force based interface. For the displacement based interface, both macro meshes considered lead to a response which is too stiff. We consider also a phenomenological approach to model this type of two-phase material. The approach consists of an elasto-plastic matrix with damageable inclusions using the elasto-plastic damage model proposed in (Ibrahimbegovic et al., 2003). We use the same type of plasticity/damage criteria as used above for the individual phases, but with different model parameters. The model parameters were identified from simple numerical tests in which homogeneous boundary conditions were applied (Markovic, 2004). Using an elasto-plastic damage model for the macroscopic behaviour of the two-phase material, we do not assume that the plastic part of the model describes the matrix and the damage part the inclusions; but only that the coupled model can represent the inelastic behaviour of the matrix and inclusion phases phenomenologically. After identifying the macroscopic model parameters using homogeneous boundary conditions tests, we validate the response using the three point bending test. We observe in Figure 10 a good agreement of the macroscopic model and the micro-macro approach result. If we use the micro representation for the microstructure only in


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Figure 9. Three point bending test with elasto-plastic matrix and damageable inclusions

Figure 10. Three point bending test with elasto-plastic matrix and damageable inclusions

the parts of the structure where we expect high stress gradients and the macroscopic model elsewhere, the force-displacement curve fits perfectly with the one where we use the fine micro-structural representation everywhere (see Figure 10). In Figure 11 we show the regions of the structure which were represented by an exact finite element model. In this example we know a priori to a good extent which part of the structure will be more severely stressed. In a general case one would need to use an error estimator and an adaptive algorithm to identify the critical region.


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6. Conclusions In this work we present a novel multi-scale strategy for strongly coupled scales which include significant nonlinear behaviour (including plasticity and damage). We also summarize its numerical implementation and show it is well suited for parallel and distributed computing. In the context of classical finite element analysis the approach presented makes it possible to replace a phenomenological model of a multi-phase material at each numerical quadrature point by a very fine finite element model that takes into account the exact micro-structure. It is further possible to choose selectively

Figure 11. Structural model combining microscopic and macroscopic representations

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the subregions which actually need a micro-scale based description from those where a phenomenological macroscopic model, in our example elasto-plastic damage model, offers a satisfying representation. The present selection criteria is ad hoc and the subject merits additional investigations to devise an appropriate automatic error estimator based selection procedure. Using the finite element code FEAP in combination with the distributed component interface library CTL we achieved an efficient parallel implementation, allowing us to run problems of realistic dimensions. The parallelization aspects, which are only sketched herein, are elaborated upon more in a subsequent paper (Niekamp et al., 2004). References Alexandrescu, A. (2002), Modern Cþþ Design, Addison-Wesley, Boston, MA. Bornert, M., Bretheau, T. and Gilormini, P. (2001), Homogeńeísation en mećanique des mate´riaux, I and II, Hermes-Science, Paris. Brockenbrough, J.R., Suresh, S. and Wienecke, H.A. (1991), “Deformation of metal-matrix composites with continuous fibers: geometrical effects of fiber distribution and shape”, Acta Metall. Mater., Vol. 39 No. 5, pp. 735-52. Buryachenko, V.A., Rammerstorfer, F.G. and Plankensteiner, A.F. (2002), “A local theory of elastoplastic deformation of two-phase metal matrix random structure composites”, J. Appl. Mech., Vol. 69, pp. 489-96. Chaboche, J.L., Kruch, S., Maire, J.F. and Potter, T. (2001), “Towards a micromechanics based inelastic and damage modeling of composites”, Int. J. of Plasticity, Vol. 17, pp. 411-39. Dvorak, G. (1992), “Transformation field analysis of inelastic composite materials”, Proc. R. Soc. Lond. A, Vol. 437, pp. 311-27. Feyel, F. (2003), “A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua”, Comp. Meth. Appl. Mech. Engrg, Vol. 192, pp. 3233-44. Ghosh, S., Lee, K. and Raghavan, P. (2001), “A multi-level computational model for multi-scale damage analysis in composite and porous materials”, Int. J. Solids Struct., Vol. 38, pp. 2335-85. Ghosh, S. and Moorthy, S. (1995), “Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite element method”, Comp. Meth. Appl. Mech. Engrg, Vol. 121, pp. 373-409. Henninga, M. and Vinoski, S. (2002), Advanced CORBA Programming with Cþ þ , Addison-Wesley, Boston, MA. Hori, M. and Nemat-Nasser, S. (1999), “On two micromechanics theories for determining micro-macro relations in heteregeneous solids”, Mech. Mat., Vol. 31, pp. 667-82. Ibrahimbegovic, A., Gharzeddine, F. and Chorfi, L. (1999), “Classical plasticity and viscoplasticity models reformulated: theoretical and numerical implementation”, Int. J. Numer. Methods Eng., Vol. 42, pp. 1499-535. Ibrahimbegovic, A. and Markovic, D. (2003), “Strong coupling methods in multiphase and multiscale modeling of inelastic behaviour of heterogeneous structures”, Comput. Meth. Appl. Mech. Engrg, Vol. 192, pp. 3089-107. Ibrahimbegovic, A., Markovic, D. and Gatuingt, F. (2003), “Constitutive model of coupled damage-plasticity and its finite element implementation”, European J. Finite Elem., Vol. 12-4, pp. 381-405.

Jiang, M., Jasiuk, I. and Ostoja-Starzewski, M. (2002), “Apparent elastic and elastoplastic behaviour of periodic composites”, Int. J. Solids Struct., Vol. 39, pp. 199-212. Jiang, M., Ostoja-Starzewski, M. and Jasiuk, I. (2001), “Scale-dependent bounds on effective elastoplastic response of random composites”, J. Mech. Phys. Solids, Vol. 49, pp. 655-73. Kruch, S., Chaboche, J.L., Feyel, F. and Roux, F.X. (1998), “Two-scale damage analysis of components reinforced with composite materials”, 4th World Conference on Computational Mechanics, Buenos Aires. Ladeve`ze, P., Loiseau, O. and Dureisseix, D. (2001), “A micro-macro and parallel computational strategy for highly heterogeneous structures”, Int. J. Numer. Meth. Engng, Vol. 52, pp. 121-38. Ladeve`ze, P. and Nouy, A. (2003), “On a multiscale computational strategy with time and space homogenization for structural mechanics”, Comput. Meth. Appl. Mech. Engrg, Vol. 192, pp. 3061-87. Lemaıˆtre, J. and Chaboche, J.L. (1988), Mećanique des Mate´riaux Solides, Dunod, Paris. Lubliner, J. (1990), Plasticity Theory, Macmillan, New York, NY. Markovic, D. (2004), “Multi-scale modeling of heterogeneous structures with nonlinear inelastic behaviours”, Phd thesis, ENS Cachan. Markovic, D. and Ibrahimbegovic, A. (2004), “On micro-macro interface conditions for micro-scale based fem for inelastic behaviour of heterogeneous materials”, Comput. Meth. Appl. Mech. Engrg, (in press). Michel, J.C. and Suquet, P. (2003), “Nonuniform transformation field analysis”, Int. J. Solids Struct., Vol. 40, pp. 6937-55. Miehe, C. (2002), “Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation”, Int. J. Numer. Meth. Engng, Vol. 55, pp. 1285-322. Moulinec, H. and Suquet, P. (1994), “A numerical method for computing the overall response of nonlinear composites with complex microstructure”, Comput. Meth. Appl. Mech. Engrg, Vol. 157, pp. 69-94. Niekamp, R., Markovic, D., Matthies, H., Ibrahimbegovic, A. and Taylor, R.L. (2004), “Multi-scale modeling of heterogeneous structures with inelastic constitutive behaviour: Part II – parallel and distribution algorithms” (in preparation). Paley, M. and Aboudi, J. (1992), “Micromechanical analysis of composites by the generalized cells model”, Mech. Mater, Vol. 14, p. 127. Park, K.C. and Felippa, C.A. (2000), “A variational principle for the formulation of partitioned structural systems”, Int. J. Numer. Meth. Engng, Vol. 47, pp. 395-418. Park, K.C., Felippa, C.A. and Rebel, G. (2002), “A simple algorithm for localized construction of non-matching structural interfaces”, Int. J. Numer. Meth. Engng, Vol. 53, pp. 2117-42. Pian, T.H.H. and Sumihara, K. (1984), “Rational approach for assumed stress finite elements”, Int. J. Numer. Meth. Engng., Vol. 20, pp. 1638-85. Simo, J.C., Kennedy, J.G. and Taylor, R.L. (1989), “Complementary mixed finite element formulations for elastoplasticity”, Comput. Meth. Appl. Mech. Engrg, Vol. 74, pp. 206-77. Simo, J.C. and Taylor, R.L. (1985), “Consistent tangent operators for rateindependent elastoplasticity”, Comput. Meth. Appl. Mech. Engrg, Vol. 48, pp. 101-18. Taylor, R.L. (2004), FEAP – User Manual, available at: www.ce.berkeley.edu/,rlt


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Some aspects of 2D and/or 3D numerical modelling of reinforced and prestressed concrete structures Pavao Marovic´, Zˇeljana Nikolic´ and Mirela Galic´

Received October 2004 Reviewed December 2004 Revised January 2005

Faculty of Civil Engineering and Architecture, University of Split, Split, Croatia Abstract Purpose – To provide an insight in one relatively simple and efficient numerical model for analysing reinforced and prestressed concrete structures, and to raise a discussion leading to the creation of one universal and robust 3D algorithm. Design/methodology/approach – A new numerical model for analysing reinforced and prestressed concrete structures is developed and main theoretical details are described to aid the understandings. The approach is clear, easily readable and the body of the text is divided into logical sections starting from theoretical explanations ending in the large number of different practical examples. Findings – Provides information about developing new and relatively simple numerical model for analysing reinforced and prestressed concrete structures, indicating what can be improved. Recognises the lack of knowing real behaviour of 3D concrete and starts a discussion on it. Research limitations/implications – The knowledge of the 2D and especially 3D concrete behaviour is still poor and the concrete model developers use many simplifications. So, many new experiments should be performed and better numerical models should be developed. There is large area for researchers but having in mind that experiments are very expensive. Practical implications – Obtained results of the 3D analysis of reinforced and prestressed concrete structures can stand as a benchmark for future researches in this field especially to the young researchers and concrete model developers. Originality/value – This paper presents new and very simple numerical model for analysing reinforced and prestressed concrete structures. Paper could be very valuable to the researchers in this field as a benchmark for their analyses. Keywords Concretes, Structures, Stress (materials), Numerical analysis, Finite element analysis Paper type Research paper

Introduction The finite element method offers a powerful and general analytical tool for studying the behaviour of reinforced and prestressed concrete structures. In civil engineering practice, numerical modelling of reinforced concrete and prestressed concrete Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 22 No. 5/6, 2005 pp. 684-710 q Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644400510603069

The partial financial support, provided by the Ministry of Science, Education and Sport of the Republic of Croatia under the projects Numerical Modelling of Engineering and Lightweight Concrete Structures, Grant No. 083133, Total Reinforced Lightweight Concrete Structures, Grant No. 083130, and Numerical and Experimental Models of Engineering Structures, Grant No. 0083061, is gratefully acknowledged.

structures is mainly used as a tool for the assessment of the structural safety of non-standard reinforced and prestressed concrete structures. During the last two decades it has been studied by many authors. The most important works for our investigations are the papers presented by Damjanic´ (1983), Gotovac and Jaramaz (1982), Philips (1992), Majorana et al. (1990), Mihanovic´ et al. (1993), Roca and Mari (1993), Hofstetter and Mang (1995), Nikolic´ and Mihanovic´ (1997) and Antoniak and Konderla (2000) while a lot of additional papers on this subject, among others, can be found in the Proceedings from the series of the International Conferences on Computer Aided Analysis and Design of Reinforced Concrete Structures (ICC, 1984; Hinton and Owen, 1986; IABSE CCMCS, 1987; SCI-C, 1990; EURO-C, 1994, 1998, 2003) and the Proceedings of the International Conferences on Fracture Mechanics of Concrete Structures (FramCos-1, 1992; FramCos-2, 1995; FramCos-3, 1998; FramCos-4, 2001). The modelling of concrete, reinforced concrete and prestressed concrete structures contains many difficulties due to the lack of the exact knowledge of its behaviour. The material characteristics itself vary enormously; concretes can be lightweight, of normal density or high density, can be of low, normal or very high strength and their material properties generally can exhibit considerable scatter. Reinforcing and prestressing steels can also have very different characteristics, ranging from hot rolled plain mild steel bars to cold worked high strength deformed bars and to very high strength prestressing wires, strands and cables. However, the behaviour of steel is much better defined. There are different loading conditions, such as various short or long term, static, dynamic or cyclic. Although a large number of research works have been performed, both experimentally and theoretically, a lot of answers are still missing. During the last two decades different models and modelling techniques, material laws and failure criteria including fracture mechanics and damage theories have been introduced and exploited but there is no general consensus which one is the most suitable for the numerical modelling of reinforced concrete and prestressed concrete structures. Nevertheless, certain modelling options have gained greater popularity than others (usually those based only on the model author’s image, marketing strength of the software developer or software vendor, engineer’s intuition, etc.). This paper presents, after discussing some general aspects of the numerical modelling of the reinforced and prestressed concrete structures, a numerical treatment of reinforcing bars and prestressing tendons in two-dimensional (2D) and three-dimensional (3D) numerical modelling of reinforced and prestressed concrete structures. The reinforcing bars and prestressing tendons are embedded into the concrete element and both are modelled by one-dimensional (1D) isoparametric 3-node elements. These elements make it possible to model arbitrarily curved reinforcing bars and prestressing tendons in space. A brief description is given in this paper, while a more detailed one can be found in Galic´ (2002) and Galic´ et al. (2002). Finally, three numerical examples: (1) prestressed prismatic girder with different boundary conditions, i.e. clamped at one end and freely supported at the other end; (2) prestressed non-prismatic girder clamped at one end and extended over the fixed support at the other end; and

2D and/or 3D numerical modelling 685

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(3) prestressed I-beam, are given to compare the results obtained by the described models with the published one, both numerical and experimental.

General considerations As mentioned before, during the last two decades different models and modelling techniques, material laws and failure criteria including fracture mechanics and damage theories have been introduced and exploited but there is no general consensus which one is the most suitable one for the numerical modelling of reinforced concrete and prestressed concrete structures (Philips, 1992; Damjanic´, 1979-1999). One can start with the analyses assuming linear-elastic isotropic laws for concrete and reinforcement, what is straightforward, and simply make use of standard linear-elastic programmes with material data corresponding to short-term or long-term working load levels. Such analyses, however, have a limited value because of the highly non-linear nature of concrete as a material. In particular, concrete cracks at low stress levels, causing early stress redistributions, and is often cracked before any load is applied due to such effects as shrinkage. On the other hand, one can start with the improved elastic analyses which can be obtained by assuming concrete has crack-induced anisotropic material properties in certain regions of the structure. However, this necessitates knowing beforehand which regions are cracked and in what direction the crack exists. Although this approach will give a first-order approximation to non-linear behaviour it is not unfamiliar. Many conventional methods of analysis and design of reinforced and prestressed concrete take into account some non-linear behaviour by assuming concrete cannot carry any tensile stress in certain directions. A fully rational analysis, therefore, needs to include the most important material non-linearities. Geometric non-linearity is less important in reinforced and prestressed concrete structures and is only associated with certain exceptional structural elements which are not often met in practice. Thus, the analysis of reinforced and prestressed concrete structures requires defining the behaviour of concrete, of the reinforcement (bars and tendons) and of the interaction (bond) between them. These can be modelled homogeneously by a smeared approach but usually are modelled separately. In this case, the discretisation then inherently combines them to produce the required overall response while at the same time information on the behaviour of the individual components is required. Thus, in modelling reinforced and prestressed concrete there are two distinct aspects to consider: (1) discretisation; and (2) material laws. Discretisation means how the individual components (i.e. concrete, reinforcement bars and tendons, bond) should be represented as finite elements. There are several basic approaches for discretising reinforced and prestressed concrete and these essentially depend on how the reinforcement is dealt with. Those approaches include: . discrete representation which uses separate 1D elements either as bar or beam elements;

.

.

.

.

embedded representation which incorporates an axial bar member into the concrete element so that its displacements are consistent with the parent element; distributed representation which assumes that the steel is distributed in a particular orientation over the concrete element and is used in conjunction with composite concrete-reinforcement material laws; layered techniques which assume layers to be either of concrete or steel what allows progressive non-linear responses to be followed through the thickness; and different bond elements allowing bond-slip behaviour to be modelled.

Material laws mean which laws should be used to represent the non-linear material behaviour of the individual components? In dealing with this, models usually distinguish between short-term and long-term effects. Short-term effects embrace: . cracking, crushing and non-linear stress-strain relations of concrete; . yielding and non-linear stress-strain relations of reinforcing bars and prestressing tendons; . interaction between concrete and steel (bond characteristics); and . interface behaviour at concrete/concrete joints, concrete/steel joints, etc. Long-term effects embrace: . creep, shrinkage and temperature changes; and . incremental microcracking and other material degradation such as breakdown of bond due to varying strain rates and repeated and cyclic loading. It can be concluded that the efficiency of the numerical analysis is closely linked to the availability of the reliable constitutive properties of concrete, of the reinforcement and of the interaction between them. Due to these complexities and including different loading conditions it is still extremely difficult, if not impossible, to establish a numerical model which can be adequate for every real situation. Hence, certain idealizations and simplifications have to be employed if the model is to be used in the numerical analysis of practical engineering problems. In fact, the final choice of a model to be used still depends on the application envisaged. Consequently, as mentioned earlier, numerous numerical models and approaches with considerable simplifying assumptions have been proposed, as well as a large number of very sophisticated one. However, in many cases the accuracy of these elaborated models far outweighs the accuracy of the available material properties and they often require extensive computations. In addition, the complexity of practical engineering problems makes numerical analysis more expensive. After these general considerations, it is difficult to expect to get answers to the following simple and final questions which model and approach are generally the best ones? Which model and approach are generally the most appropriate ones? Description of the proposed models This paper presents two numerical models for the computation of reinforced and prestressed concrete structures. The first one is a model for the analysis of plane structures (Nikolic´ and Mihanovic´, 1997; Nikolic´, 1993), while the second one is for the 3D analysis (Galic´, 2002; Galic´ et al., 2002).


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Generally, curved prestressing tendons and reinforcing bars are embedded into a 2D 8-node isoparametric element in the case of 2D analysis, while a 3D 20-node isoparametric element is used for 3D analysis (Zienkiewicz and Taylor, 2000). The developed models make it possible to compute friction losses and losses caused by short-term deformation of concrete. Within this modelling the basic assumptions are: (1) the reinforcing bars or prestressing tendons are modelled with 1D isoparametric element whose node coordinates are defined in the global coordinate system; (2) the bond between concrete and bars or tendons is perfect, i.e. the slip behaviour is neglected; and (3) the bar or tendon element can take only axial force. It follows the standard finite element procedure (Zienkiewicz and Taylor, 2000). In both models the computation for the post-tensioned prestressed structures is organized in three phases. The load can be applied incrementally in each phase. In the phase which precedes the prestressing of the tendons the structure is computed taking into account the dead load and one part of the permanent load. Concrete or reinforced concrete structures are analyzed herein. In the prestressing phase the tendons are tensioned individually. The prestressing force can be applied at once or incrementally. In the third phase which follows the tensioning of all tendons, the structure is computed taking into account the remaining part of the dead load and all kinds of the live load. The described models are implemented in the computer programs PRECON (Nikolic´, 1993) and PRECON3D (Galic´, 2002). In both cases, the advantage of the proposed modelling is complete freedom in prescribing the location and geometry of reinforcing bars and prestressing tendons. The full advantage of 3D modelling over 2D modelling is evident when the width of the cross-section over the height is not constant, e.g. when we have I, T, P or similar cross-sections, and when the prestressing tendon is placed out of the cross-section symmetry plane. Determination of the tendon geometry The proposed model for the numerical treatment of reinforced and prestressed concrete structure consists of 2D or 3D concrete elements with embedded reinforcing bars and/or prestressing tendons. The 2D 8-node elements and 3D 20-node elements are used for concrete modelling and one dimensional 3-node elements are used for reinforcing bars and prestressing tendons. Prestressed tendons can occupy a general position within the concrete element; they can be either straight or parabolic. They can also consist of straight and parabolic parts (Figure 1). All tendons can be simulated in this way, whether straight or parabolic, if the incontinuity of the first derivation is obtained during the transfer from the region of one curvature into another. The three nodes of the 1D elements are used here only for the tendon geometry description and the mapping of the integration points without introducing additional degrees of freedom of the global problem. For 2D analysis, geometry of the tendon is described by square parabola. The tendon position is determined by two boundary nodes whose coordinates are defined in global coordinate system. In the paper an iterative procedure for interpolating coordinates of the central node is used.

For 3D analysis, geometry of the tendon is described by the space function of second order (Figure 1). In this way any position of the tendon can be described, curved into one or more planes. This model offers different possibilities for cable description but it requires more input data necessary for defining its position. In this model the tendon position is defined by coordinates of two nodes and the location of the two tangents at each boundary node. The tendon position is determined by nodes whose coordinates are defined in a global coordinate system. In order to ensure the continuity between tendon elements, the boundary nodes have to be placed at the intersection points of the tendon and boundary plane of 3D concrete element. The 1D tendon element geometry described by a space function of the second order can be parametrically expressed as: xðtÞ ¼ ax t 2 þ bx t þ cx ;

yðtÞ ¼ ay t 2 þ by t þ cy ;

zðtÞ ¼ az t 2 þ bz t þ cz


ð1Þ

where t is the parameter defined in the interval t [ ½t1 ; t2 : Deformation matrix of the tendon As tendon can take only axial force the tendon element deforms only in the axial direction, i.e. differential increment ds changes into ðds þ 1s dsÞ: The displacements u, v or u, v and w are defined in the global coordinate system x-y or x-y-z, respectively, and tendon element deformation is defined in the line coordinate system x. For 2D analysis the specially derived deformation matrix BS of the tendon element embedded into 2D concrete element (Nikolic´, 1993) can be expressed as: h i ð2Þ B S ¼ B s11 B s12 B s21 B s22 · · · B si1 B si2 · · · B sn1 B sn2 where the members of the matrix are:

Figure 1. Different possibilities of the tendon geometry for 2D and 3D analysis

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›N i dx 2 ›N i dx dy ¼ þ ›x ds ›y ds ds

B si2

›N i dy 2 ›N i dx dy ¼ þ ›x ds ›y ds ds

ð3Þ

considering,

690

dx 1 ›x ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ›x ; ds ›x ›y þ ›x ›x

dy 1 ›y ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ›x : ds ›x ›y þ ›x ›x

In equations (2) and (3) n is the number of nodes of the 2D finite element (n ¼ 8 in this case) and Ni denote shape functions of that element. For 3D analysis the specially derived deformation matrix BS of the tendon element embedded into 3D concrete element (Galic´, 2002) can be expressed as: h i B S ¼ B s11 B s12 B s13 B s21 B s22 B s23 ··· B si1 B si2 B si3 ··· B sn1 B sn2 B sn3 ð4Þ where the members of the matrix are: ›N i ›x 2 ›N i ›x ›y ›N i ›x ›z B si1 ¼ þ þ ›x ›x ›x › x ›x ›x ›x ›x 2 ›N i ›y ›N i ›y ›x ›N i ›y ›z B si2 ¼ þ þ ›y ›x ›y › x ›x ›y ›x ›x 2 ›N i ›z ›N i ›z ›x ›N i ›z ›y B si3 ¼ þ þ ›z ›x ›z › x ›x ›z ›x ›x

ð5Þ

The expression for other members of the deformation matrix are evaluated analogously where n is the number of nodes of the 3D finite element (n ¼ 20 in this case) and Ni denote shape functions of that element. Stiffness matrix of the tendon The stiffness matrix of the tendon element is obtained by line numerical integration in the Gauss points of the 1D tendon element. The values of the Cartesian derivations ›Ni/›x, ›Ni/›y and ›Ni/›z in the deformation matrix of the 1D element are obtained in the Gauss points too. The positions of the Gauss points on the 1D element are defined in the line coordinate system by coordinate x. The Gauss point coordinates in the global coordinate system can be expressed by shape functions of the line element H k ¼ H k ðxÞ and its node coordinates: x g:p: ¼

n X k¼1

H k ðxÞxk ;

y g:p: ¼

n X k¼1

H k ðxÞyk ;

z g:p: ¼

n X

H k ðxÞzk

ð6Þ

k¼1

where n is the number of nodes ðn ¼ 3Þ; xk, yk and zk are the node coordinates of the 1D element in global coordinate system, and x g.p., y g.p. and z g.p. are the Gauss point coordinates in the global coordinate system.

The shape functions of the 3D concrete element depend on the local coordinate system j-h-z, while the position of Gauss points is defined in the global coordinate system x-y-z, equation (6). So, it is necessarily to map the positions of Gauss points from global coordinate system (x g.p., y g.p., z g.p.) into local coordinate system (j g.p., h g.p., z g.p.). Consequently, three non-linear equations are formed: n X N i ðj g:p: ; h g:p: ; z g:p: Þxi ¼ 0 x g:p: 2 y g:p: 2 z g:p: 2

i¼1 n X i¼1 n X

N i ðj g:p: ; h g:p: ; z g:p: Þyi ¼ 0

ð7Þ

N i ðj g:p: ; h g:p: ; z g:p: Þzi ¼ 0

i¼1

whose solution is: 8 g:p: 9jþ1 8 g:p: 9j 8 g:p: 9 j > j > > > > > Dj > > > > > = = > = < < < g:p: g:p: g:p: h h D h ¼ þ > > > > > > > > ; ; > ; : z g:p: > : z g:p: > : Dz g:p: >

ð8Þ

The increment of the Gauss point coordinates (Dj g.p., Dh g.p., Dz g.p.) in the jth iteration can be evaluated from the equation: 8 n 9 X > > g:p: g:p: g:p: g:p: > > >x 2 N i j j ; hj ; z j xi > > > > > > > 8 g:p: 9j i¼1 > > > > > > j D > > > n > > > > > X = < = < g:p: g:p: g:p: g:p: 21 y 2 N i jj ; hj ; zj yi Dh g:p: ¼ J j21 ð9Þ > > > > g:p: > i¼1 > > > > > ; : Dz > > > n > > > X > > g:p: g:p: g:p: > > g:p: > > z 2 N j ; h ; z z i i > > j j j > > ; : i¼1

The stiffness matrix of the tendon element is obtained in the known way as: Z KS ¼ BTS D S B S dV es

ð10Þ

V es

where DS is the elasticity matrix of the tendon, and dV eS is the differential volume of the tendon element which can be expressed as the function of the element length and cross-section area, dV s ¼ As dx: Modelling of the tendon forces After defining the tendon position and stiffness of the tendon it is necessary to determine the influence of prestress force upon the concrete element. The tendon force at any cross-section depends upon the applied force and prestress losses. Generally, it is necessary to determine the prestress influence in the internal points of the tendon if


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the external forces at the tendon ends are known. The influence of the prestressing tendons on the concrete is modelled by distributed normal and tangential forces along the tendons and two forces concentrated at the anchors. The developed models make it possible to compute friction losses and losses caused by the short-term deformation of concrete. The concentrated forces act in the points where the prestressing forces are applied. The tensile stresses occur in the tendon during the prestressing phase. After the anchorage the tendon tries to return to its original position what causes compression on the concrete element. This influence is modelled by a concentrated compressive force which acts in the point of the anchorage. Due to the prestressing, besides the concentrated compressive forces at the anchorages, there are forces along the tendon which are modelled as a distributed load with its normal and tangential components (Figure 2). Normal load pn(s) at any cross-section of the tendon depends upon of the curvature of the tendon and the intensity of the prestress force at that section, while tangential load pt(s) is the frictional force per unit length. From the equilibrium equation according to Figure 2, the normal and the tangential components of the continuously distributed load can be expressed as: pn ðsÞ ¼ kðsÞFðsÞ;

pt ðsÞ ¼ ^mpn ðsÞ

ð11Þ

As it can be seen it is important to determine tendon curvature k(s). For 2D model tendon curvature is always positive value and can be expressed as: 0 x y0 00 00 x y kðsÞ ¼ 0 ð12Þ ðx 2 þ y 0 2 Þ3=2 The values x0 ¼ dx=dx; y0 ¼ dy=dx are given according to:

Figure 2. The differentially small arc element of the tendon and acting forces for 2D analysis

3 dx X dH k ¼ xk ; dx k¼1 dx

3 dy X dH k ¼ yk dx k¼1 dx

ð13Þ

while, x00 ¼

d2 x d2 y and y00 ¼ 2 : 2 dx dx


From the equilibrium equation according to Figure 3, the normal and the tangential components of the continuously distributed load can be expressed by equation (11), but in this case k(s) is the tendon spatial curvature. So, it is necessary to perform double mapping of k(s) from line coordinate system x, firstly into the local coordinate system j-h-z, and secondly into the global coordinate system x-y-z of the 3D concrete element. So, the tendon spatial curvature k(s) can be expressed as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !2 !2ffi u 2 2 u d2 x dx2 dx d2 x 2 2 2 2 2 d y dx dy d x d z dx dz d x kðsÞ ¼ t þ þ þ þ þ dx 2 ds dx ds 2 dx 2 ds dx ds 2 dx 2 ds dx ds 2 ð14Þ where the components d2 x=ds 2 ; d2 y=ds 2 and d2 z=ds 2 for this operation are defined as: d2 i d di dx dx d2 i dx dx di d2 x ¼ ; i ¼ x;y;z ð15Þ ¼ þ 2 2 ds dx dx ds ds dx ds ds dx ds 2

Figure 3. Implementation of the tendon forces

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3 X di dH k ¼ ik ; dx dx k¼1

3 X d2 i d2 H k ¼ ik ; dx 2 dx 2 k¼1

i ¼ x;y;z

i ¼ x;y;z

ð16Þ

ð17Þ

Determination of the equivalent nodal forces In the general case, the tendon can occupy a general position within 2D or 3D concrete element. The point in which the tendon is anchored (point A or B) is located on the boundary plane of the concrete element not necessary at its nodes. As the problem is solved with the FEM approach these forces have to be transferred into the nodes of the parent concrete element, i.e. we have to determine adequate equivalent nodal forces, i.e. equivalent nodal forces due to anchorage forces and equivalent nodal forces due to distributed load along tendon. Determination of the equivalent nodal forces due to anchorage forces. The force point of application coordinates are defined in the global and line coordinate system by the geometry of the tendon. It is necessary to map this point into the local coordinate system of the parent concrete element to determine equivalent nodal forces of the concrete element. For 2D analysis the equivalent force at node i ði ¼ 1; . . . ; 8Þ; due to the concentrated force at the anchor, can be seen in the Figure 4. The force components in the nodal point F ix and F iy can be expressed as:

Figure 4. The equivalent nodal forces due to the anchorage forces for 2D analysis

0 @

F xi

1

0

A ¼ N i ð j k ; hk Þ @ i

Fy

F xk

1

A k

Fy

ð18Þ

where F kx and F ky are components of the force in the direction of the axes x and y in a global coordinate system, and N i ðjk ; hk Þ is the value of the shape function of a 2D 8-node element at the force point of application. For 3D analysis the number of equations is larger. So, it is necessary to compute three components Fx, Fy and Fz in the direction of the axis x, y and z in a global coordinate system (Figure 5) using shape functions Ni (jk, hk, zk) of 3D 20-nodes element. For 3D analysis the equivalent forces at node i ði ¼ 1; . . . ; 20Þ; due to the concentrated force at the anchor, what can be seen in the Figure 5, can be expressed as: F i ¼ N i ðjk ; hk ; zk ÞF k Afterwards, the force components in the nodal points can be expressed as: 0 i1 0 k1 Fx Fx B iC B kC BF C B C B y C ¼ N i ð j k ; hk ; z k Þ B F y C @ A @ A F zi F zk


ð19Þ

ð20Þ

Determination of the equivalent nodal forces due to distributed load along tendon. Due to the prestressing, beside the concentrated compressive forces at the anchorages, there are forces along the tendon which are modelled as distributed load with its normal and tangential components. These values are defined in the line coordinate system x. As the problem is solved with the FEM approach these forces have to be transferred into the nodes of the 2D or 3D concrete element, i.e. we have to determine adequate equivalent nodal forces. The force along the tendon changes during the prestressing, so, we have to determine the increments of the normal and the tangential components in the direction of the global coordinate axes x-y or x-y-z. It is necessary to map this load from local line coordinate system x into the global coordinate system x-y or x-y-z, and afterwards to define components of these

Figure 5. The equivalent nodal forces due to the anchorage forces for 3D analysis

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forces into the direction of x and y or x, y and z-axes (Figure 6 for 2D analysis and Figure 7 for 3D analysis). The total forces along the tendon in the direction of the global coordinate axes are obtained by the numerical integration. For 2D analysis, these components can be expressed as:

696 ¼ P g:p: x

Z K

pn

›y ›x 2 pt dx; ›x ›x

For 3D analysis the same components are:

Figure 6. Equivalent nodal forces due to the prestressing for 2D analysis

Figure 7. Equivalent nodal forces due to the prestressing for 3D analysis

P g:p: ¼ y

Z 2pn K

›x ›y 2 pt dx ›x ›x

ð21Þ

P g:p: x

Z

0

1 2

1 dx dx C B ¼ @ 2pn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 2 pt A dx 2y 2x 2z ds d x d d d K þ ds 2 þ ds 2 ds 2

P g:p: ¼ y

P g:p: ¼ z

Z

Z

0

ð22Þ

1 2

1 dy dy C B ffi r 2 2 pt A dx @ 2pn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dx d y d2 x d2 z ds K þ ds 2 þ ds 2 ds 2 0

ð23Þ

1 2

1 dz dz C B ffi r 2 2 pt A dx @ 2pn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ds d x d y d x d z K þ ds 2 þ ds 2 ds 2

ð24Þ

where r is radius of the tendon curvature, r ¼ 1=kðsÞ; shown in Figure 3, and k(s) is determined in the equation (14). Performing the Gauss numerical integration of equations (21)-(24) one can obtain the values of the distributed load components along the tendon in the Gauss points of the g:p: and pg:p: 1D tendon element ( pg:p: x ; py z ). To determine the influence of this distributed load along 1D tendon element on the concrete element it is necessary to map the coordinates of the Gauss points from the global coordinate system to the local coordinate system of the parent concrete element. Finally, the components of the equivalent nodal forces due to the distributed load along tendon defined in the global coordinate system can be expressed as: For 2D analysis: Pxi ¼

3 X

N i ðj g:p: ; h g:p: Þ P g:p: x ;

k¼1

Pyi ¼

3 X

N i ðj g:p: ; h g:p: Þ P g:p: y

ð25Þ

k¼1

For 3D analysis: Pxi ¼

3 X

N i ðj g:p: ; h g:p: ; z g:p: Þ P g:p: x ;

k¼1

Pyi ¼

3 X

N i ðj g:p: ; h g:p: ; z g:p: Þ P g:p: y ;

ð26Þ

k¼1

Pzi ¼

2D and/or 3D numerical modelling

3 X

N i ðj g:p: ; h g:p: ; z g:p: Þ P g:p: z

k¼1

Possibilities of the tendon prestressing Tendons can be prestressed at one end or at both ends. If the tendon is prestressed at one end applying force FA, then force F(s) at the any cross-section of the tendon can be computed according to expression (27) by integration along the appropriate length: Rs kðsÞ ds FðsÞ ¼ F A e A ð27Þ

697

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The force at any cross-section of the tendon and the distributed load can be obtained directly according to equations (1) and (27). If the tendon is prestressed from one end, and if the forces FA and FB at both tendon ends are known, then the friction coefficient m can be computed according to the expression:

m¼

1 FA ln ; gðsÞ F B

gðsÞ ¼

3 X dsðxi Þ rðxi Þ i¼1

ð28Þ

If the tendon is prestressed on both ends, the force decreases, due to friction between the tendon and concrete, if the distance from the end is increased. In symmetrical prestressing, the problem can be simply solved since the decrease in force is the greatest at the middle of the beam. For a beam with length l the force in the tendon at distance l/2 can be calculated according to equation (27), whereas the value g(s) is obtained by integrating from the beginning to the middle of the beam. If the tendon is asymmetric or if the prestressing forces at the tendon ends are not equal (FA– FB), the procedure is more complex. The minimum force will occur at the cross-section, which has not been previously known. Let us denote by x the distance of that cross-section from end A, and by l2 x the distance from end B. The force at that cross-section Fmin can be computed according to forces FA or FB by using one of the following expressions: 2m

F Lmin ¼ F A e

Rx A

kðsÞ ds

;

F Rmin ¼ F B e

2m

R l2x B

kðsÞ ds

ð29Þ

The forces should be equal regardless of the ends at which they were computed. By equating terms in equation (29), i.e. F Lmin ¼ F Rmin ; we shall obtain an equation where the integration limit is an unknown value. This equation is solved numerically. In this model, asymmetrical prestressing is performed by taking a cross-section with the greatest decrease in force. The prestressing force in a cross-section assumed according to equation (29) is calculated before computing the entire structure. According to the ratio between forces F Lmin and F Rmin ; the assumed cross-section is moved either to the left or to the right. This procedure is repeated for each tendon separately in the phase of input data preparation. Subsequently, the structure is computed and the possible difference between the two forces Fmin to the left or to the right from the selected cross-section can be neglected. Material model The assumptions adopted in the material modelling are based on the known phenomenological observations and include the simulation of concrete cracking, tension stiffening (or tensile strain-softening), non-linear multiaxial compressive behaviour of concrete and yielding of reinforcement, both bars and tendons (Damjanic´, 1983, 1986; Owen et al., 1983; Cervera, 1986). All these have been previously implemented into the computer programme PRECON for the 2D numerical analysis of reinforced and prestressed concrete structures (Nikolic´ and Mihanovic´, 1997; Nikolic´, 1993; Mihanovic´ and Nikolic´, 1993). Actually, we have enhanced this model into the 3D one, PRECON3D, and for the present moment we have solved only the geometrical part of the problem (Galic´, 2002). So, the new 3D model is tested only for the linear behaviour.

In 3D material modelling of reinforcing bars and prestressing tendons we do not expect any problems because they exhibit uniaxial behaviour in the longitudinal direction and they can both be modelled by a uniaxial elasto-visco-plastic model (Damjanic´, 1983; Nikolic´ and Mihanovic´, 1997; Nikolic´, 1993; Owen et al., 1983; Mihanovic´ and Nikolic´, 1993). Considering 3D concrete modelling the main problem will make numerical description of the before mentioned phenomenological observations due to the complexities of the concrete behaviour under different 3D conditions and lack of the experimental evidences. Although some experiences exist it is still a large unknown area.


Numerical examples The described modelling of the reinforcing bars and prestressing tendons in 2D and 3D are implemented in the computer programmes PRECON (Nikolic´ and Mihanovic´, 1997; Nikolic´, 1993) and PRECON3D (Galic´, 2002). The performance of the proposed models is illustrated by the solution procedure of three examples: (1) prestressed prismatic girder with different boundary conditions, i.e. clamped at one end and freely supported at the other end; (2) prestressed non-prismatic girder clamped at one end and extended over the fixed support at the other end; and (3) prestressed I-beam.

Example 1: prestressed prismatic girder with different boundary conditions This example shows a prestressed prismatic girder with different boundary conditions, i.e. clamped at one end and freely supported at the other end. It is originally taken from Antoniak and Konderla (2000). The geometry of the girder can be seen in Figure 8. Location of tendon embedded into concrete structure is defined by function z(x). Friction and own weight of the beam is neglected. Shown example was analyzed by Antoniak and Konderla (2000), both analytically and numerically. The analytical solution is obtained with the load equivalent method and numerical with FEM approach using beam elements and with FEM using volume (3D) elements. Reactions and displacement are expressed as a function of prestressing force F, where l is the length of the girder, h is the height of the cross-section of the girder, g is

Figure 8. Problem formulation

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constant defined as g ¼ Fl=Ebh 2 where E is Young’s modulus of elasticity and b is the width of the cross-section. In this paper the analysed girder is discretized with 10 3D 20-node elements with incorporated 10 1D 3-node tendon elements. This 3D discretisation follows original 3D one for the purpose of a better comparison of the performed analysis. The results of this analyses and comparisons are shown in Figure 9 and Table I. Table I shows the support reactions and mid-span displacement calculated by the mentioned four approaches. The shown results agree well for all approaches especially for 3D analyses given by numerical programme PRECON3D and numerical programme publish by Antoniak and Konderla (2000). The difference in obtained results is neglectable (0.24 per cent).

Figure 9. FEM volume solution

Solutions Numerically by PRECON3D Numerically, (Antoniak and Konderla, 2000) (FEM, beam elements) Numerically, (Antoniak and Konderla, 2000) (FEM, volume elements) Table I. Support reactions and mid-span displacements (compared results)

Analytically, (Antoniak and Konderla, 2000)

Reactions

Displacement (g ¼ Fl=Ebh 2 )

RA ¼ 0.1239 Fh/l MA ¼ 0.1239 Fh RB ¼ 0.1239 Fh/l

wðl=2Þ ¼ 0:19528gl

RA ¼ 0.1255 Fh/l MA ¼ 0.1255 Fh RB ¼ 0.1255 Fh/l

wðl=2Þ ¼ 0:18680gl

RA ¼ 0.1236 Fh/l MA ¼ 0.1236 Fh RB ¼ 0.1236 Fh/l RA ¼ 0.1250 Fh/l MA ¼ 0.1250 Fh RB ¼ 0.1250 Fh/l

wðl=2Þ ¼ 0:19530gl wðl=2Þ ¼ 0:18750gl

Example 2: prestressed non-prismatic girder The second example is prestressed non-prismatic girder clamped at one end and extended over the fixed support at the other end taken from El.-Mezaini and Citipitioglu (1991), refer Figure 10. The geometrical and material data are taken from El.-Mezaini and Citipitioglu (1991). The modulus of elasticity of the concrete is E c ¼ 28; 000 N=mm2 ; Poisson ratio is 0.25, the modulus of elasticity of the tendon is E s ¼ 22; 400 N=mm2 and the tendon cross-section area is As ¼ 2; 000 mm2 : The load is considered in three phases: Phase I. In the first phase the structure was computed taking into account the load it carried before the prestressing of tendons, the girder’s own weight g ¼ 25 kN=m3 and a uniformly distributed dead load q ¼ 20 kN=m: A concrete or reinforced concrete structure is analyzed herein. According to the known geometry and load we form the global stiffness matrix KI for Phase I according to the expression: KI ¼ KC þ KS


ð30Þ

where KC is the concrete stiffness matrix and KS is the reinforcement stiffness matrix, which is obtained by the numerical integration along the reinforcing bar according to the expression: Z ds BTS E S B S AS dx ð31Þ KS ¼ d x x In equation (31) BS is a strain matrix of the reinforcement element, ES is the tangential modulus of elasticity of the reinforcement, AS is the cross-section area of the bar, ds is a differential element of the length and x is the independent normalized coordinate. Global load vector (vector of residual forces) FI is determined according to the expression: FI ¼ FC þ FS

ð32Þ

where FC is a vector of external forces and residual forces on concrete element, while FS is vector of residual forces due to reinforcement strain: Z ds FS ¼ BTS sS AS dx ð33Þ dx x In equation (33) sS is the normal stress in reinforcement.

Figure 10. Geometry of the girder and loadings

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Phase II. Generally, in the second phase the tendons are tensioned individually. The prestress force can be applied at once or incrementally, and, thus, gradual prestressing procedures can be simulated. The previously applied force can be subsequently decreased, which is sometimes done in practice, in order to reduce high initial stress in one part of the tendon. During the prestressing phase the respective tendon is not treated as a structural element. Its geometry is used actually to compute the initial influence of prestressing which is modelled as a fictitious distributed load. In subsequent iteration, the tendon functions as a classical reinforcement with a given initial stress. During successive prestressing of tendons, the tendon which is currently being prestressed does not influence the stiffness of the structure, while the previously prestressed tendons take over the stresses as a classical reinforcement. The global stiffness matrix in this phase can be presented in the following form: KiII ¼ K I þ

i21 X

KjP

ð34Þ

j¼1

where i tendon index, i.e. index of a group of tendons which are being prestressed; KiII global stiffness matrix at the moment of prestressing the ith group of tendons; KI global stiffness matrix of Phase I, KjP stiffness matrix of one tendon or group of tendons which started functioning as a classical reinforcement. The loading vector can be presented in the following form: FiII ¼ F I þ

i X

DFjII

ð35Þ

j¼1

where FiII global vector of loading at the moment of prestressing the ith group of tendons; FI load vector after Phase I; DFjII vector of equivalent load which represents the influence of the prestressing force of a given tendon upon the concrete structure. When the prestressing force is introduced into the structure gradually, vector DFjII is applied incrementally and not at once. In this example, Phase II is the prestressing phase and the loading includes all loads from Phase I and a prestressing force F ¼ 2; 000 kN applied at one end of the tendon while the other end is anchored into concrete body. Phase III. The prestressing of all tendons is followed by the third phase in which the structure is computed taking into account the remaining part of the dead load and the live load. Concrete, reinforcement and all prestressed tendons which function as a classical reinforcement, contribute to the stiffness of the structure. The load is applied incrementally until failure. The stiffness matrix KIII in this phase is: K III ¼ K C þ K S þ

n X

KjP

ð36Þ

j¼1

where n is the number of prestressed tendons, i.e. the number of prestressed tendon groups. The loading vector can be presented as:

F III ¼ F I þ

n X

DFjII þ DF III

ð37Þ

j¼1

where DFIII is the part of the loading vector which resulted from the load taken over by the structure after completed prestressing. When the load is applied incrementally vector DFIII is applied in increments. In this numerical example Phase III is the phase considering service-load conditions and the loading includes all loads from Phase II, a uniformly distributed live load p ¼ 20 kN=m and a concentrated load P ¼ 200 kN: This example was previously analysed by El.-Mezaini and Citipitioglu (1991) in the linear domain with 2D discretisation, then it was analysed in the linear and non-linear domain with 2D discretisation by numerical programme PRECON (Nikolic´, 1993, 1995) and finally in the linear domain but with 3D discretisation with the developed computer programme PRECON3D (Galic´, 2002). The analysed girder is discretised with 36 3D 20-node elements with incorporated 18 1D 3-node elements describing reinforcement, refer Figure 11. Table II shows the support reactions in cross-section A calculated by the mentioned three approaches in the linear domain. The shown results (Table II) for Phase I agree well for all three approaches. This is probably because all three approaches use the same standard finite element procedure for calculating stresses and strains under own weight and dead load, i.e. when prestressing is not taken into account. However, the results for Phases II and III differ, what was expected, because the analysis with the developed approach is 3D (Galic´, 2002) while the analyses with two other approaches (El.-Mezaini and Citipitioglu (1991); Nikolic´, 1995) are 2D (plane stress conditions). We suppose that these differences are due to the influence of the real volume of the 3D finite element on the numerical integration, namely, the “volume” in 2D analysis is standardly depicted as the area of the plane finite element multiplied by the unit thickness. So, the influence of


Figure 11. Discretisation of the girder and boundary conditions

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3D analysis PRECON3D (Galic´, 2002) Phase I

704 Table II. Support reactions at cross-section A (Ri (kN), M (kNm))

Phase II Phase III

Rx ¼ 106 Rz ¼ 179 M ¼ 297 Rx ¼ 2430 Rz ¼ 275 M ¼ 677 Rx ¼ 2210 Rz ¼ 430 M ¼ 1,076

2D analysis El.-Mezaini and PRECON(2D) Citipitioglu (1991) (Nikolic , 1993) Rx ¼ 105 Rz ¼ 180 M ¼ 298 Rx ¼ 2479 Rz ¼ 273 M ¼ 647 Rx ¼ 2260 Rz ¼ 487 M ¼ 1,093

Rx ¼ 107 Rz ¼ 180 M ¼ 296 Rx ¼ 2 469 Rz ¼ 273 M ¼ 655 Rx ¼ 2251 Rz ¼ 485 M ¼ 1,093

the prestressing force on the results over the real volume and over the “plane” volume should differ. Additionally, non-linear analysis with 2D discretisation of the structure is performed with the programme PRECON(2D) and the deflection of the point C is observed up to the failure of the structure. With programme PRECON3D only the linear analysis is performed by increasing load intensity factor. Figure 12 shows load-deflection curves of the point C for both approaches and a very good agreement of the obtained results in the linear part can be seen. Deformed configurations of the girder under the loading at Phases I, II and III are shown in Figure 13, respectively, and the values of the displacement of point C are given in brackets. Example 3: Prestressed I-beam The prestressed beams and/or girders used in everyday engineering structures generally have I, T, P or similar cross-sections. The beams and/or girders with those cross-sections due to apparent 3D stress state cannot be analyzed exactly with the 2D model and code what was one of the reasons for developing 3D model and code (Galic´, 2002). In this example, prestressed I-beam taken from Nguyen (1998) is analyzed. The beam geometry and loading are shown in Figure 14.

Figure 12. Load versus deflection at point C


Figure 13. Deformed configurations of the prestressed non-prismatic girder

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706

Figure 14. Geometry of the analysed I-beam

The material characteristics of the I-beam are (Nguyen, 1998): the modulus of elasticity of the concrete E c ¼ 35; 000 N=mm2 ; Poisson’s ratio of the concrete n ¼ 0:25; the modulus of elasticity of the prestressed tendon E s ¼ 21; 0000 N=mm2 and the prestressed tendon cross-sectional area As ¼ 1962:5 mm2 : The I-beam concrete structure is discretised with 550 3D isoparametric 20-node finite elements and with 55 1D isoparametric 3-node elements for tendon discretisation. Figure 15(a) shows a deformed configuration of the I-beam under prestressing force only while Figure 15(b) shows a deformed configuration of the I-beam under concentrated force P ¼ 200 kN acting after prestressing. The load-deflection diagrams for three different analyses: (1) numerical analysis according to Nguyen (1998); (2) experimental investigations according to Nguyen (1998); and (3) numerical analysis according to the presented proposed model and the computer programme PRECON3D (Galic´, 2002), are shown in Figure 16. These lines present mid-span deflection under the second loading case. A very good agreement of the obtained results for all three analyses in the linear domain is evident. Conclusions This paper presents a numerical treatment of reinforcing bars and prestressing tendons in 3D numerical modelling of reinforced and prestressed concrete structures. The advantage of the proposed modelling is complete freedom in prescribing the location and geometry of reinforcing bars and prestressing tendons. The described modelling of the reinforcing bars and prestressed tendons in 3D is implemented in the computer programme PRECON3D (Galic´, 2002) based upon the previously developed programme PRECON for the same analyses in 2D (Nikolic´, 1993). Three numerical examples:


Figure 15. Deformed configuration of the I-beam: (a) under prestressing force; and (b) under concentrated force acting after prestressing occur

Figure 16. Load-deflection diagrams for different analyses

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(1) prestressed prismatic girder with different boundary conditions, i.e. clamped at one end and freely supported at the other end; (2) prestressed non-prismatic girder clamped at one end and extended over the fixed support at the other end; and (3) prestressed I-beam, are given to illustrate the possibilities of the developed models and computer programmes. The obtained results show good agreement with the published ones (El.-Mezaini and Citipitioglu, 1991; Nikolic´, 1995; Nguyen, 1998) for the same examples, both numerically and experimentally. The full advantage of the proposed 3D modelling is evident when the width of the cross-section over the height is not constant, e.g. when we have I, T, P or similar cross-sections (Marovic´ et al., 2003). References Antoniak, D. and Konderla, P. (2000), “General FEM model of prestressing tendons”, Computer Assisted Mechanics and Engineering Sciences, Vol. 7, pp. 435-48. Cervera, M. (1986), “Non-linear analysis of reinforced concrete structures using three dimensional and shell finite element models”, PhD Thesis, C/Ph/93/86, University of Wales, Department of Civil Engineering, Swansea. Damjanic´, F.B. (1979-1999), Private Communications, Split-Swansea-Ljubljana. Damjanic´, F.B. (1983), “Reinforced concrete failure prediction under both static and transient conditions”, PhD thesis, C/Ph/71/83, University of Wales, Department of Civil Engineering, Swansea. Damjanic´, F.B. (1986), “A finite element technique for analysis of reinforced and prestressed concrete structures”, in Bergan, P., Bathe, K.J. and Wunderlich, W. (Eds), Proceedings Europe-US Symposium on Finite Element Methods for Non-linear Problems, Trondheim, August 1985, Springer, Berlin, pp. 623-37. El.-Mezaini, N. and Citipitioglu, E. (1991), “Finite element analysis of prestressed and reinforced concrete structures”, Structural Engineering, Vol. 117, pp. 2851-64. EURO-C (1994) in Mang, H.A., Bicánic´, N. and de Borst, R. (Eds), Proceedings Int. Conf. on Computer Modelling of Concrete Structures – EURO-C 1994, Innsbruck, March 1994 1/2, Cromwell Press, Melksham. EURO-C (1998) in de Borst, R., Bicánic´, N., Mang, H.A. and Meschke, G. (Eds), Proceedings Int. Conf. on Computational Modelling of Concrete Structures – EURO-C 1998, Badgastein, March-April 1998 1/2, A.A. Balkema, Rotterdam. EURO-C (2003) in Bicánic´, N., de Borst, R., Mang, H.A. and Meschke, G. (Eds), Proceedings Int. Conf. on Computational Modelling of Concrete Structures – EURO-C 2003, St Johann im Pongau, March 2003 1/2, A.A. Balkema, Rotterdam. FramCos-1 (1992) in Bazant, Z.P. (Ed.), Proceedings 1st Int. Conf. on Fracture Mechanics of Concrete Structures – FramCoS 1, Beaver Run Resort, June 1992, Elsevier Applied Science, London. FramCos-2 (1995) in Wittmann, F.H. (Ed.), Proceedings 2nd Int. Conf. on Fracture Mechanics of Concrete Structures – FramCoS 2, Zurich, July 1995. FramCos-3 (1998) in Mihashi, H. (Ed.), Proceedings 3rd Int. Conf. on Fracture Mechanics of Concrete Structures – FramCoS 3, Gifu, October 1998.

FramCos-4 (2001) in van Mier, J.G.M., Mazars, J., Pijaudier-Cabot, G. and de Borst, R. (Eds), Proceedings 4th Int. Conference on Fracture Mechanics of Concrete and Concrete Structures – FramCoS 4, Cachan, May-June 2001, Elsevier Applied Science, London. Galic´, M. (2002), “Numerical 3D model of prestressed concrete structures”, MSc thesis (in Croatian), Faculty of Civil Engineering, University of Split, Split. Galic´, M., Marovic´, P. and Nikolic´, Zˇ. (2002), “Numerical model of prestressing tendons embedded into the 3D concrete element”, in Eberhardsteiner, J. and Mang, H.A. (Eds), Book of Abstracts of the 5th World Congress on Computational Mechanics, Vienna, July 2002, Volume I, Vienna University of Technology, Vienna, p. I-572, available at: http://wccm. tuwien.ac.at, 2002-2006. Gotovac, B. and Jaramaz, B. (1982), “The application of composite finite elements”, Gradevinar, Vol. 34 No. 7, pp. 259-66 (in Croatian). Hinton, E. and Owen, D.R.J. (Eds) (1986), Computational Modelling of Reinforced Concrete Structures, Pineridge Press, Swansea. Hofstetter, G. and Mang, H.A. (1995), Computational Mechanics of Reinforced Structures, Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden. IABSE CCMCS (1987), Colloquium on Computational Mechanics of Concrete Structures – Advances and Applications, IABSE Report, 54, Delft University Press, Delft. ICC (1984) in Damjanic´, F.B., Hinton, E., Owen, D.R.J., Bicánic´, N. and Simovic´, V. (Eds), Proceedings Int. Conference on Computer Aided Analysis and Design of Concrete Structures, Split, September 1984 1/2, Pineridge Press, Swansea. Majorana, C.,Natali, A. and Vitaliani, R. (1990), “Analysis of three-dimensional prestressed concrete structures using a non-linear material model”, Engineering Computations, Vol. 7 No. 2, pp. 157-66. Marovic´, P.,Nikolic´, Zˇ. and Galic´, M. (2003), “Comparison of two-dimensional and three-dimensional analysis of reinforced and prestressed concrete structures”, in Kompisˇ, V., Sladek, J. and Zˇmindak, M. (Eds), Proceedings of the Extended Abstracts of the 9th Int. Conference on Numerical Methods in Continuum Mechanics, Zˇilina, pp. 101-2. & CD Proc., 16 p. Mihanovic´, A. and Nikolic´, Zˇ. (1993), “Numerical model for posttensioning concrete structures”, Int. J. Engineering Modelling, Vol. 6 Nos 1-4, pp. 35-43. Mihanovic´, A., Marovic´, P. and Dvornik, J. (1993), Non-linear Calculations of Reinforced Concrete Structures (in Croatian), Society of Croatian Structural Engineers, Zagreb. Nguyen, K.T. (1998), “Non-linear analysis of concrete beams with unbonded tendons”, in de Borst, R., Bicánic´, N., Mang, H.A. and Meschke, G. (Eds), Proceedings Int. Conference on Computational Modelling of Concrete Structures – EURO-C 1998, 2, A.A. Balkema, Rotterdam, pp. 749-55. Nikolic´, Zˇ. (1993), “Development of the numerical model for post-tensioning of plane reinforced concrete structures”, MSc thesis (in Croatian), Faculty of Civil Engineering, University of Split, Split. Nikolic´, Zˇ. (1995), “Numerical modelling of prestressed structures”, Gradevinar, Vol. 47, pp. 121-9 (in Croatian). Nikolic´, Zˇ. and Mihanovic´, A. (1997), “Non-linear finite element analysis of post-tensioned concrete structures”, Engineering Computations, Vol. 14 No. 5, pp. 509-28. Owen, D.R.J., Figueiras, J.A. and Damjanic´, F.B. (1983), “Finite element analysis of reinforced and prestressed concrete structures including thermal loading”, Comp. Meths. Appl. Mech. and Engng, Vol. 41, pp. 323-66.


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Philips, D.V. (1992), “Numerical modelling of brittle materials: concrete and reinforced concrete”, Lecture notes on Non-linear Engineering Computations, 1st Short Course on Advanced Computational Engineering Mechanics, TEMPUS-ACEM, Ljubljana, pp. C/3-75. Roca, P. and Mari, A.R. (1993), “Numerical treatment of prestressing tendons in the non-linear analysis of prestressed concrete structures”, Computers and Structures, Vol. 46 No. 5, pp. 905-16. SCI-C (1990) in Bicánic´, N. and Mang, H.A. (Eds), Proceedings 2nd Int. Conference on Computer Aided Analysis and Design of Concrete Structures, Zell-am-See, April 1990 1/2, Pineridge Press, Swansea. Zienkiewicz, O.C. and Taylor, R.L. (2000), The Finite Element Method, Volume 1: The Basis, 5th ed., Butterworth-Heinemann, Oxford.



Coupling FEM and BEM for computationally efficient solutions of multi-physics and multi-domain problems

Coupling FEM and BEM

Boris Sˇtok and Nikolaj Mole

Received October 2004 Accepted January 2005

Laboratory for Numerical Modelling and Simulation, University of Ljubljana, Ljubljana, Slovenia

711

Abstract Purpose – To present numerical approaches to the solution of physically coupled non-linear problems, which frequently happen to be characterized by their multi-domain character. Design/methodology/approach – By adopting coupled solution strategies a considerable attention is devoted, in order to obtain a computationally efficient numerical algorithm, to the selection of appropriate space and time discretization, as well as to a proper discrete approximation method used. Findings – Coupling of two methods, the finite element method and the boundary element method, respectively, has proved to be computationally exceedingly advantageous, particularly in case of moving domains. Practical implications – As specific case studies computer simulation of an induction heating problem and a mushy-state forming problem are considered. A thorough discussion on the coupling effects, characterizing the evolutions of respective physical quantities’ fields, is given, and their impact on those evolutions is identified. Originality/value – This paper presents efficient numerical strategies for the solution of a certain class of multi-physics and multi-domain problems. Keywords Finite element analysis, Simulation, Numerical analysis, Linear programming, Mechanical systems Paper type Research paper

1. Introduction When considering continuum mechanics problems it is to emphasize that actions not necessarily of mechanical origin can contribute significantly to a development of the mechanical state in a continuum (Sˇtok, 1997). In fact, in mechanical engineering (Rojc and Sˇtok, 2003; Koc and Sˇtok, 2004), manufacturing technology (Mihelicˇ and Sˇtok, 1996; Mole, 2002; Poje et al., 2003) and materials processing (Sˇtok and Mole, 1990, 1993) there is a variety of technological multi-physics applications, that are based exclusively upon the established temperature field evolution, and having definite mechanical state as a final result or goal. Also, though the problem of temperature field determination in a body of interest is definitely a heat conduction problem, it is only by applying external heat agents, which may be themselves as well a result of the respective field variable evolution in a companion physical problem, that such a determination could be performed. The interdependence between individual physical phenomena which always exist in that kind of physically complex problems, usually exposing at a same time also a large degree of non-linearity, causes that the time evolution of a primary field variable in


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one physical system cannot be determined without prior knowledge of a respective primary field variable in the other physical system. Among a variety of multi-physics applications the induction heating seems to be a process which is at same time representative and physically clear enough to be used as a demonstration example to point out characteristic features of the solution to such problems. In this process, where the principal physical mechanism generating heat is electromagnetic induction, there are three physical phenomena – the mechanical, heat conduction and the electromagnetic one. Each of the three physical problems, i.e. the mechanical, heat conduction and the electro-magnetic one, are per se physically rather pretty clarified and mathematically well determined, irrespective of eventual complexities characterizing their description (Sˇtok and Mole, 2001). In principle, provided the respective boundary conditions and domain distributed actions are known in time each of the considered problems can be solved on its own. In reality, however, interdependence between individual physical phenomena always exists. From a computational point of view this characteristic, which is very often mutual with physical interdependence exhibited in both directions, demands definitely for a coupled solution strategy. Another type of coupling, not necessarily characterized by physical dissimilarity, is encountered when a problem domain consists of several interacting sub-domains. Within the same physical framework, again, physical response of a subsystem can be determined only upon simultaneous knowledge of respective physical behaviour of those subsystems that are adjacent to a considered one. Which obviously demands for a coupled solution strategy, as well (Sˇtok and Mole, 1990, 1993; Mole, 2002; Rus et al., 2003). As simplifications, that are rather frequently used in traditional engineering approach in order to reduce the complexity of multi-physics problems, and eventually multi-domain problems, are always disputable, the adoption of such simplifications being prone to loss of physical objectivity, we shall reconsider in this paper strategies that can be used for a solution of physically coupled non-linear problems. A considerable attention will be devoted to the issue of appropriate space and time discretization methods selection that would result in a computationally efficient solution. Two case studies, that will be considered numerically for the purpose of this presentation, contain all the characteristics discussed. In the first case, a workpiece will be heated by induction, and afterwards, when a required non-uniform temperature distribution will be attained, it will be quenched in order to obtain specific mechanical properties. In the second case, induction will be used to heat a billet to a required nearly uniform temperature distribution, bringing it thus in a mushy-state, which will be followed by a subsequent metal forming operation. The three physical problems associated with the considered case studies, i.e. the thermomechanical, heat conduction and the electromagnetic problem, respectively, will be considered in detail, and their coupling effects will be discussed upon results of the corresponding computer simulations. 2. Multi-physics and multi-domain field problems – formulation and characterization 2.1 Definition of a certain class of boundary value problems For the sake of simplicity let us consider first a one domain problem that is characterized by a single physical phenomenon. With x denoting Cartesian coordinates x, y, z of a point in space, let x [ V and x [ G denote, respectively, the domain of

interest and its boundary, Figure 1, while u ¼ uðxÞ denotes a primary variable of the problem which can be either scalar or vector quantity, and real or complex. In general, the field distribution u ¼ uðxÞ may change with time, therefore, u ¼ uðx; tÞ; where t stays for the time variable. Time evolution of the considered field distribution is governed by a corresponding problem equation, in accordance with given initial and boundary conditions, which can be written as

713

Auðx; tÞ ¼ f ðx; tÞ; x [ VðtÞ uðxÞ ¼ u0 ðxÞ; x [ V0 ; t ¼ 0


ð1Þ

Buðx; tÞ ¼ gðx; tÞ; x [ GðtÞ; t . 0: Here, u ¼ u0 ðxÞ is the initial field distribution at time t ¼ 0; while f ¼ f ðx; tÞ and g ¼ gðx; tÞ are the applied external actions, acting, respectively, within the domain V and on the boundary G, while A and B are respective differential operators associated with the governing equation and boundary conditions. For the class of physical problems we are going to address the differential operators A and B are of the following general form 3 X 3 X › › › ›2 aij þb þc 2 A¼ ›xi ›x j ›t ›t i¼1 j¼1 ð2Þ 3 X › B ¼ {Bu ; Bq }; Bu ¼ 1; Bq ¼ di ; ›xi i¼1 where, for convenience, we have used for the coordinates x, y, z the following representation form: xi, i ¼ 1; 2; 3 with x1 ¼ x; x2 ¼ y and x3 ¼ z: In the formulation of the boundary condition operator B nature of the possible boundary conditions has been taken into account, resulting thus in the splitting of the operator B into Bu and Bq, which give, respectively, the operational structure of corresponding essential and natural boundary conditions. Because of conjugacy of the primary and secondary problem variables entering the two types of boundary conditions the boundary G can be accordingly split in two parts, Gu and Gq, respectively, fulfilling Gu < Gq ¼ G and Gu > Gq ¼ B:

Figure 1. General problem description

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Coefficients aij, b, c and di are the problem constitutive parameters that depend to a great degree on material physical properties. The most simple case regarding the corresponding problem solution is undoubtedly encountered when those parameters are constant in time. However, very often the field distribution u ¼ uðxÞ does affect the physical properties, therefore C p ¼ C p ðuÞ;

ð3Þ

where symbolics Cp is introduced to represent the whole set of the above-mentioned constitutive parameters. This characteristic renders the problem solution definitely non-linear, but, in the context of the assumed single physical phenomenon one domain problem, with domain and boundary actions, f(x,t) and g(x,t), respectively, fully independent, it is still self-consistent. In other words, having in view physical coupleness in multi-physics problems we are going to address in the next subsection, self-consistency of a problem may be considered as equivalence to physical uncoupleness. Considering dependence (equation (3)) in the problem equations (1) the uncoupled problem may be represented symbolically in the form of PðuÞ ¼ 0 _ Pðu; u_ Þ ¼ 0 _ Pðu; u_ ; u€uÞ ¼ 0;

ð4Þ

where the three forms arise with regard to the established degree of the time derivatives, that are active in the governing equation operator A (equation (2)). 2.2 Characterization of multi-physics and multi-domain problems In a multi-physics problem each individual physical phenomenon could be considered roughly in the same way as presented above. However, we refer to a problem as a real multi-physics problem only when there is a rather strong interdependence established between the respective physical phenomena evolutions, which makes them coupled. Let us consider, for the sake of simplicity, a problem that is characterized by two physical phenomena within a common material domain V, and u1 ¼ u1 ðx; tÞ and u2 ¼ u2 ðx; tÞ being the respective primary variables, Figure 2(a). The respective field

Figure 2. Multi-physics and multi-domain problem description

evolutions are governed by the same type of equations as given by equation (1), provided all the appertaining quantities entering the definition equations (1) and (2) are supplemented by the respective indices, i.e. 1 or 2 for the considered cases. Which is certainly not enough for the multi-physics problem to be considered as a coupled one. Then, where the coupleness comes from? In the same way as in a single physical phenomenon problem the appertaining constitutive parameters can be affected by the respective field distribution (equation (3)), it is quite possible that in a physically coupled problem the constitutive parameters, determining individual physical problems, are affected, apart from the proper field distribution, also from the corresponding field distribution of the companion physical problem. Since this interaction can be exhibited, in general, in both directions, the following relations can be established ð1Þ ð2Þ ð2Þ C ð1Þ p ¼ C p ðu1 ; u2 Þ; C p ¼ C p ðu2 ; u1 Þ;

ð5Þ

which definitely brings coupleness between the considered physical phenomena into force. In some problems coupleness can be demonstrated also by the companion field distribution specifying directly the applied domain actions f(x,t), experiencing thus the following functional relations. f 1 ¼ f 1 ðu1 ; u2 Þ; f 2 ¼ f 2 ðu2 ; u1 Þ:

ð6Þ

Here, the highest degree of functional dependence, including also eventual dependence of the proper field distribution, has been taken into account. In contrast to the established physical interdependence, manifested through the whole domain V, the companion field distribution may affect also the applied boundary actions g(x,t) as well, which can be expressed for this particular case in the following functional form. g 1 ¼ g1 ðu1 ; u2 Þ; g2 ¼ g 2 ðu2 ; u1 Þ:

ð7Þ

A similar class of interdependence arises in a multi-domain problem, Figure 2(b), irrespective of the number of physical phenomena. The physical consistency of the respective primary and secondary field variables at a common interface boundary G12 between two adjacent sub-domains V1 and V2 is enforced by fulfilment of the corresponding interface continuity conditions, which can be written as G12 ¼ G12 ðu ð1Þ ; u ð2Þ Þ ¼ 0:

ð8Þ

Also, though without any physical background as discussed above, such a kind of problems can be generated purely for computational purposes (Rus et al., 2003). In analogy to the formulation equation (4) the coupled multi-physics problem can be formulated by defining the corresponding two equations sets, symbolically written as P 1 ðu1 ; u2 Þ ¼ 0; P 2 ðu2 ; u1 Þ ¼ 0

_

P 1 ðu1 ; u_ 1 ; u2 ; u_ 2 Þ ¼ 0; P 2 ðu2 ; u_ 2 ; u1 ; u_ 1 Þ ¼ 0

_

P 1 ðu1 ; u_ 1 ; u€ 1 ; u2 ; u_ 2 ; u€ 2 Þ ¼ 0; P 2 ðu2 ; u_ 2 ; u€ 2 ; u1 ; u_ 1 ; u€ 1 Þ ¼ 0:

ð9Þ


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Each of the particular functional relationships, i.e. P 1 ðu1 ; . . .Þ ¼ 0 and P 2 ðu2 ; . . .Þ ¼ 0; respectively, expresses a unique condition governing the respective single physical phenomenon under consideration. In particular, the relationship P 1 ðu1 ; . . .Þ ¼ 0 governs the field variable u1 ¼ u1 ðx; tÞ whilst P 2 ðu2 ; . . .Þ ¼ 0 is responsible for the field variable u2 ¼ u2 ðx; tÞ:

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2.3 Strategies for solution of coupled problems The above generally addressed multi-physics and multi-domain problems can be regarded as a coupled problem, the solution of which is applicable to both classes of problems. Solving of the complete coupled problem, as determined by functional relations (equation (9)), requires use of iteration techniques to couple the respective partial solutions. In the solution strategy different aspects, the problem is characterized with, are to be properly addressed. First, although the simplest case, not containing any time rates of the variables u1 ¼ u1 ðx; tÞ; u2 ¼ u2 ðx; tÞ; is considered, its solution may be required at several time/load steps in connection with an incremental continuation procedure if the history of the field variables is requested for timely variable boundary conditions and/or in order to facilitate the numerical analysis of strongly non-linear problems. In contrast to the referred time-independent problems, transient processes, on the other hand, are characterized by the appearance of the time rates of the primary variables in the governing equations. Both the variables u1 ¼ u1 ðx; tÞ; u2 ¼ u2 ðx; tÞ and their time rates are unknown quantities that are linked via the integration in time ua ¼

Z

u_ a dt; u_ a ¼

Z

u€ a dt; a ¼ 1; 2

ð10Þ

which is usually performed by approximation following an incremental scheme. When dynamic effects are not pronounced the single step formula _ t þ DtÞ Dt; _ tÞ þ zwð wðt þ DtÞ ¼ wðtÞ þ ½ð1 2 zÞwð

ð11Þ

may be considered as sufficient to allow stepping of a field variable w ¼ wðx; tÞ in time. Here, notation w ¼ wðx; tÞ is used to represent in turn either ua ¼ ua ðx; tÞ or its time rate u_ a ¼ u_ a ðx; tÞ. Provided the respective field values at time t ¼ t are known, the time integration formula allows to calculate the corresponding values at time t ¼ t þ Dt: Regarding the selected magnitude of the collocation parameter z, which can take the value within the range 0 # z # 1; either explicit ðz ¼ 0Þ or implicit ðz – 0Þ time integration scheme is obtainable. € In contrast to the time integration formula (11), where the time rate w€ ¼ wðx; tÞ does not enter in the scheme, there exists for the solution of dynamically characterized problems a series of integration schemes, either explicit or implicit. From the computational point of view explicit methods are particularly attractive since they require, in principle, less computational work. Here, we will make use of a modified € constant acceleration method with w€ ¼ wðx; tÞ assumed constant during time interval ½t; t þ Dt; and having the magnitude as determined at the start of the considered time ˙ (x,t) and w(x,t) interval. In consequence, the corresponding values of integrated fields w at the end of the considered time interval can be found in accordance with

_ t þ DtÞ ¼ wð _ tÞ þ wð € tÞDt; wð

_ tÞDt þ wð € tÞ wðt þ DtÞ ¼ wðtÞ þ wð

Dt 2 : 2

ð12Þ

The just discussed time/load integration aspect of the problem solution deals with the time dimension, covering thus the history of the respective field variables. The considered time integration algorithms are general, and can be used in the solution of uncoupled as well as coupled problems. The second aspect is, however, specific for coupled problems. It actually deals with the issue of achieving a physically consistent coupled solution that will provide each individual partial solution to be in agreement with the appertaining governing equations, and at the same time taking full account of the impact the companion field evolution exerts on it. In conjunction with the suitable approximate time integration scheme, equations (11) and (12), the governing equation (9) provide a system for the field computation of either the state variables or their time rates at each time instant, in the sequel denoted by v1 ¼ v1 ðx; tÞ; v2 ¼ v2 ðx; tÞ for convenience. In this context, the essential problem is considered to be the solution of a system of equations F 1 ðv1 ; v2 Þ ¼ 0; F 2 ðv2 ; v1 Þ ¼ 0;

ð13Þ

at distinct time instants during the course of an incremental continuation procedure. At this stage, analytical field solution of the considered system being not feasible, let us transform the problem (13) into a corresponding discretized matrix form. F 1 ðV 1 ; V 2 Þ ¼ 0; F 2 ðV 2 ; V 1 Þ ¼ 0:

ð14Þ

The introduced vectors V1 and V2 represent the finite sets that are assembled from the respective nodal values at discrete points of the domain V, used for the spatial approximation of v1 ¼ v1 ðx; tÞ; v2 ¼ v2 ðx; tÞ fields at considered time instant. Correspondingly, and provided each individual field problem is adequately linearized within the considered time step, the coupled problem formulation (14) represents actually two systems of linear equations for the unknowns V1 and V2, respectively. The coupleness between the two systems is manifested implicitly since impact of the companion field distribution, e.g. V2 in the equations set F 1 ðV 1 ; V 2 Þ ¼ 0 for the unknowns V1, is entering the system coefficients of the equations set F 1 ðV 1 ; V 2 Þ ¼ 0: This characteristic complicates the systems solution by rendering it non-linear and requiring an iterative procedure for achieving it. Considering the way how coupleness is manifested it proves advantageous to handle each of the equations set in equation (14) individually. A feasible technique involves then the consecutive solution of equation (14), applying an adequate iteration procedure with one of the sets of unknowns V1 and V2 assumed temporary fixed. Among different ways of how performing the required iteration loop in order to reach the coupled solution at considered time instant, two of them, respectively, Jacobi iteration and Gauss-Seidel iteration procedure, are rather familiar and easy to implement. If in the ith cycle of the iteration loop estimate V i is supposed to be known, then iþ 1 is a new estimate. Seeing that in Jacobi iteration, applied to the solution of V equation (14)


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i iþ1 i F 1 V iþ1 1 ; V 2 ¼ 0; F 2 V 2 ; V 1 ¼ 0;

ð15Þ

the considered systems of equations are actually uncoupled with respect to the iþ1 temporary unknowns V iþ1 1 ; V 2 ; the respective systems solutions within an iteration cycle may be treated also concurrently. On the contrary, in Gauss-Seidel iteration, when applied to the solution of equation (14). i iþ1 iþ1 ; V V ; V F 1 V iþ1 ¼ 0; F ¼ 0; ð16Þ 2 1 2 2 1 the iteration solution procedure must be performed consecutively. The iteration namely starts with an estimate to one of the sets of unknowns, say V i2 : The first equation in equation (16) is then solved for the other set of unknowns, V iþ1 1 ; and this updated information is used in the second equation to provide the new estimate V iþ1 2 : 3. Physical and mathematical description of considered physically coupled problems 3.1 Description of electromagnetic-thermal-mechanical coupling Beside cutting processes, heat treatment and metal forming are among the most frequently encountered processes in materials processing technologies. Both are characterized by the presence of different physical phenomena, which contribute decisively by their coupling to the resulting mechanical/metallurgical state of a manufactured product. In the sequel, we will consider such processes in which electromagnetic-thermal-mechanical coupling is established. In heat treatment both the chemical composition and a kind of heat treatment, that a workpiece is subjected to, allows for a great variety of properties to be obtained. Since kinetics of structural changes in a material of a specified chemical composition is exclusively conditioned with a time variation of the temperature field it is of fundamental importance in any heat treatment process to achieve controlled heating and cooling. The temperature field time variation, though generating structural changes in a direct and independent way, depends itself on the external factors above all. On one side it is the energy entering the system, and on the other side it is the heat leaving the system that determine the heat transfer. The major trouble of such a thermal analysis consists in the field determination of the energy sources and the corresponding time variation in the heating phase, while in the cooling phase it consists in the identification of real boundary conditions, taking the amount of heat leaving the system properly into account. When electromagnetic induction is used for heating, strong coupleness between the thermal and electromagnetic phenomena is proven. On the other hand, the imposed cooling rate gives rise to significant changes of the metallurgical/mechanical state, which is characterized by quite similar intensity of coupleness between the thermal and mechanical phenomena. In hot metal forming pretty the same physical phenomena can be evidenced, but with rather different degree of coupleness proven. In order to bring a workpiece to a desired temperature distribution again electromagnetic induction can be applied, while in the metal forming phase the heated workpiece is exposed to a considerable change of its initial geometry. In contrast to the established strong electromagnetic-thermal coupling, coupleness between the mechanical and thermal phenomena may not be so

intensive. Of course, in connection with the irreversible plastic work heat may be generated, the amount of it and its impact on the temperature field distribution being dependent on a great number of factors, that will not be considered here. In order to distinguish in the following mathematical descriptions the dimensional structure of a considered quantity let us introduce in the sequel the following convention: scalar, vector and tensor variables will be denoted, respectively, by italic, italic bold and simple bold notations, e.g. a, A for scalars, a, A for vectors and a, A for tensors. 3.2 Description of electromagnetic eddy current problem In an electromagnetic field (emf) problem with a relatively slow time variation, i.e. s ! v1; where s and 1 are, respectively, the electrical conductivity and permitivity, and v is the source frequency, the conducting currents dominate the problem, therefore, the displacement currents can be neglected. In consequence, the steady-state approximation may be used which simplifies the problem considerably (Sˇtok and Mole, 2001). Maxwell’s equations, which govern the emf distribution, read for the considered case as follows curl H ¼ J ; curl E ¼ 2

›B ; div B ¼ 0; ›t

ð17Þ

with H, E being vectors of magnetic and electric strength, B being vector of magnetic flux density, and J being vector of the conduction current density. These quantities are further related by the following constitutive relationships B ¼ mH ; J ¼ J source þ J eddy ¼ sE;

ð18Þ

with m being the magnetic permeability, and Jsource, Jeddy being, respectively, the source and eddy current densities. In order to give a complete mathematical description of the emf problem, in which a workpiece that is made of an electrically conducting material is passed in an induction heating process through coils carrying an alternating electric current, the whole space domain must be considered. Accordingly, the infinite domain V in which the emf exists can be decomposed in three non-overlapping sub-domains Va; a [ {I; W; A} with indices I, W, A referring to the corresponding domains, occupied, respectively, by the inductor, the workpiece and the air (Figure 3). The physical consistency of the emf, which exists in all three sub-domains Va; a [ {I; W; A}; is achieved, apart from obeying the governing domain equations (17) and (18), by fulfilment of the corresponding continuity relations at the boundaries Ga; a [ {I; W}: At these boundaries, which are actually common interfaces to the adjacent electrically nonconductive air domain, the continuity of the normal component of magnetic flux density and the tangential component of magnetic field intensity is implied ðB a 2 B A Þ·n a ¼ 0; ðH a 2 H A Þ £ n a ¼ 0; a [ {I; W};

ð19Þ

where vector na is exterior normal to the boundary Ga; a [ {I; W}: The behaviour of the emf at infinity is characterized by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð20Þ B A ! 0 as jxj ¼ x 2 þ y 2 þ z 2 ! 1: The magnetic flux being divergence free can be expressed in terms of a vector magnetic potential A which is itself, according to the Coulomb gauge, divergence free


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Figure 3. Disposition of sub-domains in a three-domain axisymmetirc electromagnetic induction problem

B ¼ curl A; div A ¼ 0:

ð21Þ

In consequence, a solution to the stated emf problem equations (17)-(20) can be found by the corresponding transformation of the governing equations. By assuming linear and isotropic materials with constant permeability the transformation yields ›A ›A ¼ 2mJ source ; J eddy ¼ 2s ; ð22Þ DA 2 ms ›t ›t with associated interface and boundary conditions in terms of the vector potential A. Here, potential A takes the role of the mathematically defined primary variable, instead of H or E, the magnetic and electric strength, respectively, which are actually the physical primary variables of the problem. If the applied source potential is assumed harmonic in time all the associated field quantities are time harmonic functions as well. Their distribution, therefore, depends only on the position and phase delay at each point in space, which can be mathematically managed in a simple way by introducing complex notation. The time variation of the potential A(x, t) at any point x can be obtained as the real part of a corresponding complex function ^ 0 ðxÞ expðjvtÞ; ð23Þ Aðx; tÞ ¼ A0 ðxÞ cosðvt þ wðxÞÞ ¼ Re½A ^ 0 ðxÞ includes the where the introduced complex magnetic vector potential A information about the field amplitude and its time delay with respect to the imposed source excitation. With respect to the introduced complex framework all physical quantities associated with the emf problem, which were considered above as real vector functions, can be understood from now on as complex functions. Their behaviour is still determined by the same system of equations, which however has now to be treated as a system of complex equations.

The emf, produced by the current-carrying coils, induces eddy currents in the workpiece, which is in consequence heated resistively by the Joule effect. The electric power qv, induced in a unit volume of a workpiece by eddy currents, is proportional to the intensity of the actual emf, and its effective value, when expressed in terms of the ^ 0 ðxÞ, is as follows vector potential A qV ¼

sv 2 ^ 2 jA0 ðxÞj : 2

ð24Þ

This electric power acts as an imposed heat source in the associated thermal problem within the sub-domain VW, and is actually the key driving force. 3.3 Description of heat conduction thermal problem Heat transfer in a continuum, that occupies an isotropic material domain V and is exposed to volume distributed heat source qV and eventual boundary heat flux qS, is governed by thermodynamic equilibrium. With rc and k being, respectively, the thermal capacity and thermal conductivity, the corresponding temperature field T(x, t) in the body is then subject to time variation in accordance with the heat conduction equation

rc

›T ¼ div ðk grad TÞ þ qV ; ›t

ð25Þ

and associated initial and boundary conditions Tðx; 0Þ ¼ T 0 ðxÞ; Tðx; tÞ ¼ T* ðx; tÞ;

x [ V; t ¼ 0 x [ GT ^ 2k grad Tðx; tÞ·n ¼ q* ðx; tÞ; x [ Gq ;

ð26Þ t . 0:

Here, it has been assumed that on the part GT of the boundary G essential boundary conditions with prescribed temperature variation T* are given, whilst on the remaining part Gq natural boundary conditions, i.e. convection and radiation heat fluxes q*, are known. The key driving force for the temperature field evolution, temperature T being the primary variable of the problem, is the application of external agents qV and qS. Considering the induction heating and subsequent rapid cooling, that will be addressed in the context of heat treatment process investigation, the intensities of those actions are provided by knowledge of respective companion physical processes, namely, electromagnetic induction on one hand and heat exchange between the workpiece and a quenchant on the other hand. In this regard heat source intensity, as determined by equation (24), will be taken into account. Also, it is worth mentioning that in contrast to the emf analysis, where the whole space domain composed of three material sub-domains had to be fully taken into consideration, the thermal analysis can be performed just by considering a sub-domain that is occupied by the workpiece. 3.4 Description of elastoplasticity and viscoplasticity contact mechanics problem Mechanical response of a deformable continuum, occupying an isotropic elasto-plastic material domain V and being exposed to volume and surface distributed forces, f and p *, respectively, and to eventual variation of the initially isothermal state T0, is governed by the set of equilibrium, deformation and constitutive equations


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div s þ f ¼ 0 1 1 ¼ ½grad u þ ðgrad uÞT 2 Ea s¼H:12 ðT 2 T 0 Þd; 1 2 2y

ð27Þ

where u is the displacement vector, and 1 and s are, respectively, the strain and stress tensor. Material properties E, y and a are in turn Young’s modulus, Poisson’s ratio and coefficient of thermal expansion. The fourth order tensor H is actually the stress-strain constitutive tensor which is composed in case of pure elastic response from the elastic coefficients E and y , whilst in case of inelastic response it is enlarged by incorporation of parameters characterizing such behaviour, e.g. hardening modulus H. Finally, d is the Kronecker tensor. Physical consistency of the mechanical response is achieved, in addition to equation (27), by fulfilment of corresponding initial and boundary conditions, which read for the assumed initially stress-strain free mechanical state as follows

sðx; 0Þ ¼ s 0 ðxÞ ¼ 0; 1ðx; 0Þ ¼ 1 0 ðxÞ ¼ 0; uðx; 0Þ ¼ 0; x [ V; t ¼ 0 ð28Þ u ¼ u* ðx; tÞ; x [ Gu ^ s·n ¼ p* ðx; tÞ;

x [ Gs ; t . 0:

Here, it has been assumed that on the part Gu of the boundary G essential boundary conditions with prescribed displacements are given, whilst on the remaining part Gs natural boundary conditions, expressed in terms of applied tractions, are known. Since displacement field u(x, t) acts as the primary variable of the mechanical problem, the set of governing equations (27) can be adequately reduced. Considering the structure of the strain tensor 1, 1 ¼ 1 s þ 1 T ; with 1 s and 1 T being, respectively, the stress and temperature induced strain, the stress induced strain tensor 1 s can be further decomposed into elastic and inelastic part, 1 s ¼ 1 e þ 1 p : Assuming also material properties constant this reduction yields vector equation governing time evolution of the displacement field.

m divðgrad uÞ þ ðl þ mÞ gradðdiv uÞ 2 ð2m þ 3lÞ a gradðT 2 T 0 Þ ð29Þ 2 2m div 1 p þ f ¼ 0: The above equation introduces, in regard to the Navier-Lame equilibrium equation, a modification due to the presence of inelastic response. The key driving force for the variation of mechanical quantities in the above stated heat treatment process investigation will definitely be thermal expansion, in fact, its unhomogeneous and possibly abrupt distribution through the body. Therefore, occurrence of plastic deformation at least in a part of the body is rather expected. In order to enable, at any stage of the process evolution, proper identification of the respective material response, i.e. either elastic or inelastic, in the constitutive equation of (equation (27)) the thermomechanical state function F, referred also as a yield function, is introduced in accordance with

Fðs 2 a; T; 1 p Þ ¼ f ðs 2 aÞ 2 K p ðT; 1 p Þ:

ð30Þ

The defined yield function depends directly on the actual state of stress s and accumulated plastic strain 1p, and indirectly on temperature T and eventual nonisotropic hardening, expressed by the so-called back-stress tensor a. As indicated, temperature affects the yield parameter Kp which is definitely of great importance. Yield function F is actually a uniaxial equivalent for the corresponding characterization of the otherwise spatial stress state. Accordingly, the loading function f ðs 2 aÞ is the corresponding mapping to uniaxial stress representation and yield parameter Kp becomes uniaxial yield stress. With respect to equation (30), where only non-positive values of the yield function F are admissible by definition, i.e. F # 0; characterization of the current material response that corresponds to eventual change of the applied loads can be uniquely determined. Accordingly, the mechanical states, proving at some level of loading F , 0; are characterized by elastic response, irrespective of either loading or unloading is applied. If hardening materials are considered, which means ›K p =›1 p . 0; then occurrence of plastic deformation is conditioned by the stress state corresponding to F ¼ 0 and stress change, associated with the given change of the applied loads, corresponding to dF ¼ 0: In order to describe properly conditions met in metal forming processes the above description of boundary conditions should be enlarged. Because in metal forming a change of initial workpiece geometry is imposed by direct action of tooling, i.e. punch and die, on the formed workpiece, the considered mechanical problem actually belongs to a class of contact mechanics problems. Accordingly, the usual division of the boundary G should be correspondingly enlarged by taking contact conditions at the common interface, say Gc, between the contacting bodies into account. This enlargement yields Gu < Gs < Gc ¼ G; by fulfilling simultaneously Gu > Gs ¼ Gu > Gc ¼ Gs > Gc ¼ B: In contrast to essential and natural boundary conditions that are defined explicitly, at the contact boundary Gc none of the problem variables are given explicitly. Since at that boundary stress-displacement compatibility between the variables of the respective bodies in contact must be respected, this relationship is established implicitly by considering coupling with the contacting body. This compatibility namely implies that kinematic and stress variables at material points on the adjacent surfaces follow the non-penetrability condition, kinematic stick-slip relations according to the actual frictional law, and contact stresses continuity (Sˇtok and Hudoklin, 1994). If the two bodies in contact are denoted, respectively, by BA ðx [ VA Þ and BB ðx [ VB Þ; and their common boundary by Gc ¼ GA > GB ; then the compatibility of respective contact stresses ða ¼ A; BÞ is imposed by p A þ p B ¼ 0; p a ¼ sa ·n a ; x [ Gc ;

ð31Þ

while consistency of the contact status poses additional constraints on the contact tractions

sa ¼ p a ·n a # 0 ^ ta ¼ jp a 2 sa n a j # Pðm; sa Þ; x [ Gc ;

ð32Þ

where sa and ta are, respectively, the normal and tangential component of the traction vector pa. The constraint posed on ta establishes actually its relation with respect to frictional resistance P(m, sa), with m being coefficient of friction.


723

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Kinematic compatibility of the contacting points x A ¼ x B ; which occupy at a given stage of loading the same position, is directly related to the actually established relationship of ta in equation (32). By an additional loading established inequality in the considered relationship imposes equality of displacements, while established equality imposes equality of displacements in the direction normal to the actual contact area, which can be formulated as follows: du A ¼ du B _ dunA ¼ dunB ; duna ¼ du a ·nA ; x [ Gc :

ð33Þ

In principle, a solution to the contact problem can be then achieved by coupling the two single domain solutions, provided compatibility (equations (31)-(33)) of the two mechanical states is respected. Another type of mechanical problem can be formulated with respect to hot metal forming, where elastic deformation is negligible and viscoplastic response dominates. In principle, governing equations (27) are fully applicable, provided that the stress and strain tensor, s and 1, are substituted by their deviatoric parts, s and e, respectively, and rate form of the respective equations is considered. The constitutive equation in equation (27) is adequately substituted by a corresponding viscoplastic one (Mole et al., 1996). Often, the Norton-Hoff law in the form of pffiffiffi m21 e_ ð34Þ s ¼ 2K 3evp is successfully applied. The physical parameters K and m are, respectively, the consistency of the material and the rate sensitivity index, while evp is the equivalent strain rate, similar to the previously mentioned equivalent plastic strain 1p. Their respective definitions are: Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z rffiffiffiffiffiffiffiffiffiffiffiffi 2 p p 2 1_ : 1_ dt; evp ¼ e_ : e_ dt: ð35Þ 1p ¼ 3 3 In the viscoplastic problem, as well as in elastoplastic one, material incompressibility is assumed with regard to inelastic deformation, yielding tr d1 p ¼ 0; div u_ ¼ 0;

ð36Þ p

where tr stays for the trace of differential of plastic strain tensor 1 . The time evolution of the velocity field v(x, t) or equivalently 1ðx; _ tÞ; the strain rate tensor 1_ ¼ e_ is namely primary variable of the viscoplastic problem, is then obtained by considering, along with the considered domain governing equations, the respective rate forms of initial, boundary and contact conditions. 3.5 Governing equations for axisymmetric case In cases with cylindrical symmetry the problem quantities are invariant with respect to the circumferential direction, the property that affects also the structure of the involved non-scalar physical quantities. Several components of the vector primary variables, such as displacement component uw in the mechanical problem, and magnetic and electric strength components Hw, Er and Ez, respectively, in the electromagnetic problem are null by definition. From the computational point of view the most significant consequence of cylindrical symmetry is evidenced in the electromagnetic

problem, because of the corresponding transformation of a rather complex vector problem to easy manageable scalar problem. On the other hand, the governing equations of the remaining two physical problems preserve their form. To enable further mathematical manipulations with respect to the new framework a cylindrical coordinate system (r, u, z) with er, eu, ez as unit basis vectors will be used. ^ z; tÞ Regarding the emf problem, Figure 3, it can be stated that the current density Jðr; of the form ^ z; tÞ ¼ J^0 ðr; zÞ exp ðjvtÞ; J^0 ðr; zÞ ¼ Jû ðr; zÞe u Jðr;

ð37Þ

gives rise to the emf the structure of which is on the contrary rotationally planar ^ z; tÞ ¼ H^ 0 ðr; zÞ exp ðjvtÞ; H^ 0 ðr; zÞ ¼ H^ r ðr; zÞe r þ H^ z ðr; zÞe z ; Hðr; ^ z; tÞ is circumferential while the corresponding magnetic potential Aðr; ^ 0 ðr; zÞ ¼ A ^ u ðr; zÞe u : ^ z; tÞ ¼ A ^ 0 ðr; zÞ exp ðjvtÞ; A Aðr;

ð38Þ

ð39Þ

The considered emf problem can be thus expressed in terms of a single complex scalar ^ u ðr; zÞ; which on the central axis always vanishes due to cylindrical function A ^ u ðr; zÞ ; Aðr; zÞ; Jû ðr; zÞ ; J ðr; zÞ for brevity, symmetry of the problem. By setting A and considering the above functional structure the governing differential equation (22) reduces to 1 þ jmvs A 2 mJ source ; ð40Þ DA 2 gðA; J source ; sÞ ¼ 0; gðA; J source ; sÞ ¼ r2 the solution of which yields, by fulfilment of the associated continuity conditions 1 ›Aa Aa 1 ›A A A A nr ¼ nr ; x [ Ga ; a [ {I; W} Aa ¼ AA ; þ þ ma ›n a r mA ›n a r ð41Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jxj ¼ r 2 þ z 2 ! 1 ) AA ! 0; ^ z; tÞ: the space distribution and time variation of the vector magnetic potential Aðr; Contrary to the vector quantities characterizing the emf problem, in the thermal problem the temperature field T(r, z, t) is a scalar field whose initial value is specified by the initial conditions. The time evolution of the temperature field is determined by the governing domain equation (25) and specified boundary conditions (equation (26)). In case that material properties are assumed constant the transformation of equation (25) yields: DT þ

qV r c › T ¼ : k ›t k

ð42Þ

This equation can be further transformed to the same form as obtained for the emf problem. In fact, from the numerical viewpoint any approximation of the considered field time derivative will be realized by an adequate combination of the corresponding field values T* ðr; zÞ ¼ Tðr; z; t* Þ computed at previous time instant, i.e. at time t*, and unknown field values T(r, z, t) at the actual time t considered. For the explicit time scheme the following equation is obtained


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DT 2 gðT; qV ; T*Þ ¼ 0; gðT; qV ; T*Þ ¼

1 rc qV 1 rc T* ; ð43Þ þ T2 t 2 t* k k t 2 t* k

the solution of which yields, by fulfilment of the associated boundary conditions the space distribution and time variation of the temperature field T(r, z, t). Similar conclusions can be made with respect to the governing equation of mechanical problem. For the two non-zero components of the displacement vector, ur and uz, equation (29) can be written in the componental scalar form Dua þ

1 fðl þ mÞ grad ðdiv uÞ 2 ð2m þ 3lÞa grad ðT 2 T 0 Þ m

ð44Þ

p

22m div1 þ f ga ¼ 0; a ¼ r; z where notation {· · ·}a is used to denote the corresponding vectorial component. Also those equations can be further transformed to the above obtained forms, equations (40) and (43). Symbolically, we obtain ð45Þ Dua 2 g ua ; f a ; u*a ¼ 0; a ¼ r; z; where in the term ua* the respective contributions of the displacement component adjacent to ua, temperature T and plastic strain 1 p are assembled. The above equations yield, by fulfilment of the associated boundary conditions the space distribution and time variation of the displacement field u(r, z, t). 4. Finite element and boundary element numerical implementation 4.1 Integral variational formulations In view of an approximative solution of the governing equations, respectively, equation (40) in the emf problem, equation (43) in the thermal problem and equations (45) in the mechanical problem, the problem can be mathematically reformulated. Taking advantage of formally identic structure, and introducing instead of A(r, z), T(r, z, t) and ua(r, z, t) a generalized variable u(r, z), omitting for convenience its time dependence, the following integral equation Z ½Du 2 gðu; f V ; w* Þ v dV ¼ 0; ð46Þ V

can be obtained, its equivalence with the above stated governing equations being established on the basis of arbitrary selection of admissible function v(r, z). The function gðu; f V ; w* Þ represents the corresponding g-terms in the respective governing equations. By applying Green’s theorems the above equation can be further reformulated to yield, respectively, a weak Z Z Z ›u dG 2 gðu; f V ; w* Þv dV ¼ 0; ð47Þ 2 7u7v dV þ v V G ›n V and inverse variational form of the basic integral formulation

Z

Z

Z ›u ›v 2u gðu; f V ; w* Þ v dV ¼ 0: dG 2 ›n ›n V

ð48Þ


These two integral equations are fundamental equations for the finite element (FEM) and boundary element (BEM) methodologies that will be applied in the approximative solution of the problem. To allow general treatment of the both approximation approaches we take again the generalized variable u(r, z) as our primary variable, which will be in accordance to the applied approach adequately discretized.

727

uDv dV þ V

v

G

4.2 Finite element approximation Field approximation of the generalized variable u(r, z) in the finite element domain VF ¼ SVe is based, Figure 4(a), on the respective nodal values U kF and corresponding local polynomial approximation within each finite element Ve. uFe ðr; zÞ ¼

Nfe X

fek ðr; zÞU Fk ; fek ðr i ; zi Þ ¼ dik ; ðr; zÞ [ Ve :

ð49Þ

k¼1

In order to obtain the corresponding FEM discrete formulation of the considered physical problems we subject, considering the specified approximation (equation (49)), the weak variational

formulation (47) to the application of a finite series of test functions vk ðr; zÞ [ vFk ; vFk ; fek that are associated to the given FE discretization nodes. The result of the test functions application is a corresponding system of linear equations with U Fk as unknowns. Its matrix structure, where internal domain nodal quantities are separated from the contour ones for convenience, regarding subsequent coupling with the corresponding BE set of equations when considering a multi-domain problem, is as follows: 9 8 9 2 F 38 V F = FI SII SFIC < U FI = < 4 5 ¼ : ð50Þ F F F F F SCI SCC : U C ; : V F C þ G F C ; In a single domain problem this system is solved by considering the corresponding essential and natural boundary conditions.

Figure 4. Finite element and boundary element discretization pattern with respective finite and boundary element approxiamtions of basic problem variations

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By having determined all the nodal values U Fk of the FE discretized problem the investigated direct problem can be considered as solved. In fact, with the corresponding local polynomial approximation (equation (49)) within each finite element Ve completely determined, the subsequent processing of the respective secondary field variables, which involves a space derivation of the primary field variable within a finite element domain Ve, is straight-forward, thus yielding the secondary field variables variation through the whole problem domain V. 4.3 Boundary element approximation Alternatively, the field approximation of the generalized variable u(r, z) can be performed exclusively upon respective approximation of its distribution on the domain’s boundary. Then, the approximation of the primary variable u(r, z) on the boundary GB ¼ SGe is based, Figure 4(b), on the respective nodal values U Bk and corresponding local polynomial approximation along each boundary element Ge. uBe ðr; zÞ ¼

Nbe X

cek ðr; zÞU Bk ; cek ðr i ; zi Þ ¼ dik ; ðr; zÞ [ Ge :

ð51Þ

k¼1

In addition, in the BE approach also normal derivative must be approximated on the boundary. This approximation is based, independently of equation (51), on the respective nodal values QBk and corresponding local polynomial approximation along each boundary element Ge

›u ›n

B ðr; zÞ ¼ e

Nbeq X

xek ðr; zÞQBk ; xek ðr i ; zi Þ ¼ dik ; ðr; zÞ [ Ge :

ð52Þ

k¼1

In order to obtain the corresponding BEM discrete formulation of the considered physical problems we subject, considering the specified approximations (51) and (52), the inverse variational formulation (48) to the application of a finite series of test functions vk(r, z), that are associated to the given BE discretization nodes. Those f unctions, known also as the fundamental solutions to a respective series of field problems, are determined in accordance with vk ðr; zÞ [

B related vk ; vBk [ DvBk 2 g vBk ; 0; 0 ¼ 2dðr 2 r k ; z 2 zk Þ : The result of the test functions application is a corresponding system of linear equations with U Bk and QBk as unknowns, which reads B

SBU U BC þ SBQ Q BC ¼ V F C :

ð53Þ

In a single domain problem this system is solved by considering the corresponding essential and natural boundary conditions. With respect to the structure of equation (53) this is certainly true, if the vector on the right side of this matrix equation is assumed to be known. By having determined all the boundary values U Bk and QBk of the BE discretized problem the investigated direct problem can be considered as solved. In fact, with the corresponding local polynomial approximations (51) and (52) within each boundary element Ge completely determined, the subsequent processing of the respective primary and secondary field variables in any interior point within the problem domain V can be readily achieved by considering once again the integral equation (48). In order

to obtain approximation uP to the primary field variable u(r, z) at an interior domain point P, the integral equation is subjected to the application of the corresponding fundamental solution vp ðr; zÞ [ {vp ; vp [ Dvp 2 gðvp ; 0; 0Þ ¼ 2dðr 2 r p ; z 2 zp Þ}; associated to the considered point P. This operation yields the approximation uP in the form of Z Z ›u ›v v 2u ð54Þ uP ¼ dG 2 ½gðu; f V ; w* Þv 2 ugðvP ; 0; 0Þ dV: ›n ›n G V


729

Unfortunately, the evaluation of the domain integral includes itself prior knowledge of the field variable u(r, z) through the domain V, which definitely complicates B the determination of the approximation uP. Actually, as vector V F C in equation (53) depends itself on the domain field distribution u(r, z), it tremendously complicates already the solution of this basic matrix system (equation (53)). In general, it is not to be expected that this trouble, which is certainly one of major defficiencies of the BEM, may be discarded. Fortunately, there are cases which result, due to specific behaviour of the field variables under certain conditions, in cancelation of the above domain integral contributions, thus causing a solution of the domain problem being exclusively expressed in terms of the boundary values of respective problem primary and secondary variables. Under such conditions the BEM may become most attractive and very competitive with regard to the FEM. Coming back to the determination of the secondary field variables in the problem domain V, a similar procedure involving a corresponding space derivation of the equation (48) is applied point by point. 4.4 Coupled finite element and boundary element approximation Coming back to our particular case of modelling the considered emf problem, Figure 5, there are two major reasons to use mixed FE-BE approach in a numerical solution to the considered multi-domain problem (Sˇtok and Mole, 2001). The domain of a workpiece could experience significant non-linear material behaviour, while the surrounding air domain, which is in principle infinite, is characterized by constant physical parameters. In addition, the relative position between the inductor and a workpiece is continuously changing during the induction process. In respective local reference frames the physical components, i.e. the coil and the heated piece (Figure 6), can be discretized through volume uniquely by finite elements, their varying relative position in the absolute reference frame not affecting the finite element structure within the local domain. On the contrary, as a linear domain problem can be always

Figure 5.

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730

transformed to a boundary problem, the respective air domain can be considered in an associated absolute reference frame just by a corresponding discretization of its boundary. Owing to the established behaviour of emf at infinity the only boundaries to be considered are those appertaining to the coil and the piece. Modelling of an infinite domain is so avoided in an elegant manner, but the greatest advantage to use boundary elements is certainly a facilitated management of the boundaries space variation, which can be done solely by considering it locally, i.e. within the air sub-domain. Therefore, no finite element remeshing is needed. In accordance to the most appropriate discretization methodology applied the investigated multi-domain problem can be solved by finding a solution of individual sub-domain problems in terms of nodal domain/boundary values of the respective physical variables, the physical consistency of the overall solution being ensured by imposing the continuity conditions (equation (41)) across the sub-domain interfaces. In principle, in a multi-domain problem two uncoupled systems of equations, regardless of whether FEM or BEM is used, can be built upon given discretization. Because of existing interdomain coupleness their physically consistent solution is, however, attained only by simultaneous consideration of the corresponding interface continuity conditions. In view of direct application to the particular induction-heating problem, which we consider, the coupling of a FEM discretized domain with a BEM discretized domain must be enforced. Accordingly, the respective FEM and BEM systems of equations, as determined by equations (50) and (53) 2 4

SFII SFCI

9 8 9 38 V F = SFIC < U FI = < FI 5 ¼ ; SFCC : U FC ; : V F FC þ G F FC ;

B

SBU U BC þ SBQ Q BC ¼ V F C ; 0

ð55Þ

are to be solved simultaneously. Here, reference is made directly to the considered B emf problem, therefore, V F C ¼ 0: The fulfilment of the continuity conditions (equation (41)) imposes coupling of the two systems, which is established in discretized form by corresponding connectivity matrices Ca U BC ¼ C U U F ; C UC U FC ;

G

F

F C ¼ C QU U BC þ C Q Q BC :

ð56Þ

The solution of the resulting coupled system of equations 2

SFII

6 F 6S 6 CI 4 0

SFIC SFCC 2 C QU C UC SBU C UC

38 F 9 8 V F 9 > > > > > > > FI > >UI > > > = = > < < 7> F F 7 2C Q 7 U C ¼ V F C > > > 5> > > > > > > QB > > ; > V B> SBQ : ; : F C C 0

ð57Þ

yields nodal values of the magnetic vector potential AðU ¼ AÞ upon which the heat generation over the volume of the workpiece can be computed.

5. Numerical simulation of physically coupled case studies 5.1 Case 1 – induction heating – quenching simulation In the induction heating-quenching set-up (Figure 6), where induction heating is followed by immediate quenching, the high-frequency electric source ðv ¼ 400 kHzÞ with the effective current variation, as shown in Figure 7(a), is applied for 70 s on a moving copper coil ðs ¼ 60:106 ðV mÞ21 ; mr ¼ 1Þ which encloses a workpiece of initial temperature 228C, the latter having the Curie point at temperature 7688C. The coil displacement in the axial direction is realized according to the given velocity profile (Figure 7(a)). Because of large variation of the temperature field due to the induced emf the physical properties are considered as temperature dependent, and when actual also magnetic field dependent (Figure 7(b)). In the thermal part of computation thermal convection and radiation is taken into account, with convection heat transfer coefficient being 10 W/m2K on the workpiece surface exposed to heating. Quenching of the workpiece, which is applied immediately after passing of the inductor, is performed corresponding to two different intensities of cooling, the respective heat transfer coefficients being 10 W/m2K (quenching on air) and 1,000 W/m2K (quenching by water). Emissivity coefficient of the workpiece is taken as 0.92. In the mechanical part thermo-elasto-plastic material response with linear temperature independent hardening ðH ¼ 23; 333_ MPaÞ; and constant Young’s modulus and Poisson’s ratio, their values being, respectively, 210,000 MPa and 0.3, is assumed. The yield stress and coefficient of thermal expansion are taken, however, as


731

Figure 6. Numerical model of induction heating-quenching set-up built upon mixed finite element – boundary element discretization

Figure 7. Time variation of induction heating process parameters and temperature dependence of material properties

EC 22,5/6

temperature dependent, obeying functional relations ~ ¼ 279 þ 6:616T~ 2 5:855T~ 2 þ 0:042T~ 3 þ 0:025T~ 4 ½MPa sp ðTÞ ~ ¼ ½12:0842 2 0:8601T~ þ 0:1811T~ 2 þ 0:0169T~ 3 2 0:0027T~ 4 £ 1026 ½K21 aðTÞ

732

Figure 8. Heat source time evolution

Figure 9. Temperature time evolution

˜ stays for T~ ¼ T £ 1022 , and where dimensionless temperature parameter T temperature T is to be taken in 8C. In the actual numerical model (Figure 6) a triangular FE discretization of the workpiece domain with a quadratic field approximation for the primary variable has been used in the electro-magnetic, thermal and mechanical analyses. In electro-magnetic analysis the inductor and air domain is taken into account, too. For discretization of the inductor domain we used the same FE as for the workpiece domain, while the air domain has been discretizated by BE with a quadratic function approximation, coinciding in nodes with boundary edges of the workpiece and inductor. For the approximation of the BE normal derivative on the boundary a sectionally constant approximation has been assumed.

From the displayed computed results in Figures 8 and 9, showing the time evolution of the generated heat source and the corresponding temperature variation across the workpiece’s cross-section beneath the coil during first 15 s, that is when the coil is fixed, the impact of several parameters affecting the magnetic induction can be clearly revealed, despite the fact that high source frequency causes sharp localization of the emf near the surface. Most evident is certainly the passing beyond the Curie point with abrupt change in the magnetic permeability, in consequence of which the emf spreads from a very thin surface layer (about 0.05 mm) with large heat source intensity into interior of the workpiece (to the depth of about 0.5 mm), which is characterized by a significant decrease of the generated heat (Figure 8). This behaviour is directly reflected in the evolution of the temperature field as it can be seen from the corresponding plots in Figure 9. Later on, with the coil in movement, the above behaviour is not manifested so abundantly. There are several reasons for a remarkable decrease of the intensity of generated heat source on the surface (Figure 10), all of them being a consequence of mutual interacting effects. First, there is a significant impact of the temperature field variation, established in the part of the workpiece in front of the inductor, on the electric conductivity, and in consequence on the induced eddy currents. Also, much broader region is affected by exceeding of the Curie temperature, diminuishing thus the intensity of the electromagnetic field at the surface. Finally, because of shorter exposure time of the surface points to the high emf less heat is generated by induced electric power, and consequently a smaller temperature variation is realized. The basis for the aforementioned discussion can be revealed by a careful inspection of the temperature time evolution (in case of air cooling) in a workpiece’s cross-section in Figure 10, taken in a position that corresponds to passing of the coil at time 34 s. Since influence of end effects on the emf response can be excluded, this position can be considered as a characteristic one. Therefore, this evolution is similar for all the cross-sections coming under the coil during its movement along the workpiece. From Figures 10 and 11 it is evident that cross-section points close to the surface temperature are subject to much faster change than those in the interior. Temperature time evolution is actually affected by the nature of the considered process which consists of two phases, the heating one with the coil moving toward and over the observed cross-section, and the cooling one, when the coil is moving away from the observed cross-section. Direct impact of the both is most evidenced just at the surface. Characteristic for the first phase is a rapid increase of the surface temperature,


733

Figure 10. Temperature time evolution

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734 Figure 11. Temperature radial distribution

while in the second phase the respective temperature decrease can be regulated by a corresponding cooling rate. Figure 12 shows a comparison of the temperature time evolution for two cases, when, the coil being removed from the observed cross-section, the workpiece is chilled by air or water, the considered cases clearly demonstrating strong dependence of temperature on the cooling intensity. In quenching processes the most important goal is to obtain a proper mechanical state in a workpiece. Influence of temperature time evolution on the stress-strain field in the workpiece after quenching can be seen from Figures 13 and 14. Although the

Figure 12. Temperature time evolution in dependence of cooling intensity

accumulated equivalent plastic strain in case of water quenching is higher than in air quenching, the resulted residual stresses are smaller. Smaller residual stresses are the result of more intensive cooling, where, in contrast to air quenching, additional plastic deformation occures with opposite sign of deformation than in the heating phase. 5.2 Case 2 – induction heating-forging simulation Induction heating is used also in connection with metal forming. In order to facilitate the forming operation a workpiece is often preheated to a desired temperature, thus


735

Figure 13. Equivalent plastic strain time evolution in dependence of cooling intensity

Figure 14. Residual stresses distributions due to different cooling rates

Figure 15. Numerical model of induction heating set-up

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736

Figure 16. Time variation of the effective electric current and temperature time evolution in the workpiece

Figure 17. Scheme of a forging process

diminishing considerably its resistance to forming. In the considered case (Figure 15) a workpiece of initial temperature 208C, its Curie point being at temperature 7688C, is heated by means of the low-frequency electric source ðv ¼ 60 HzÞ; applied on a fixed copper inductor that consists of 164 coil turns ðs ¼ 60 £ 106 ðV mÞ21 ; mr ¼ 1Þ: The respective effective current time variation is shown in Figure 16. Same material properties are assumed as in the previous case with temperature and magnetic field dependence taken fully into account, since the experienced variation of the temperature field is even greater. Owing to thermal insulation of the workpiece during heating no thermal convection and radiation at the workpiece’s surface is taken into account in thermal part of computation. Because uniform temperature field, obtained in the workpiece before forging is high enough to ensure a semi-solid material state, the visco-plastic material response according to Norton-Hoff’s law ðK ¼ 1 Pa sm ; m ¼ 0:1Þ is assumed. Since presence of mushy state essentially reduces the friction between the workpiece and the forging tool, a friction free contact is considered. Also, no temperature variation in the workpiece during forging is considered, due to short duration (Figure 17). The same combined FE-BE discretization, as used in case 1, has been applied in computer simulation of induction heating of the workpiece and subsequent forging

(Figure 15). Also, in order to reduce the computational time, the inductor domain has been divided in 32 sub-domains. Advantage of induction heating with low-frequency electric source is that emf spreads into interior of the workpiece, thus generating heat through the whole domain. This behaviour is reflected in more homogeneous temperature field evolution, with temperature difference between the workpiece’s surface and its interior being not so large, as when using a high-frequency electric source. As evident from Figure 16 an optimal time variation of the electric current resulting in efficient heating can be obtained, based exclusively on the corresponding computer simulation of induction heating. In the considered case we succe eded to heat the workpiece to the domain uniform temperature 1; 390 ^ 108C in 20 min. The subsequent forging is decisevely characterized by the strain rate distribution which, due to its direct relation with the stresses, determines actually the resistance to forming. Such a strain rate field distribution, taken at a position near the end of forging, is displayed in Figure 18.


737

6. Conclusions Issues regarding a solution of physically coupled non-linear problems, the latter being frequently characterized also by their multi-domain character, have been thoroughly addressed in the paper. Respective solution strategies have been reconsidered, along with the discussion on the selection of appropriate space and time discretization methods that result in a computationally efficient solution. In this regard, the coupling of the FEM and the BEM in a numerical solution of the considered physically complex problem has proved to give great advantage over the performance of a single numerical method. Namely, when solving a time varying multi-domain problem no remeshing was actually needed in the computational procedure with such an approach. In connection with the considered case studies three physical problems have been considered in detail. In view of the given physical background the coupling effects, characterizing the evolutions of respective physical quantities fields in thermomechanical, heat conduction and the electromagnetic problem, have been thoroughly discussed and their impact on those evolutions found out. The performed investigation, which has been abundantly supported by the simulation results, clearly

Figure 18. Equivalent strain rate distribution

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shows that in such physically complex cases a merely qualitative estimation, though based on the logical thinking, cannot reveal all the details. References Koc, P. and Sˇtok, B. (2004), “Computer-aided identification of the yield curve of a sheet metal after onset of necking”, Computational Materials Science, Vol. 31, pp. 155-68. Mihelicˇ, A. and Sˇtok, B. (1996), “Optimization of single and multistep wire drawing processes with respect to minimization of the forming energy”, Struct. Optim., Vol. 12 Nos 2/3, pp. 120-6. Mole, N. (2002), “Racˇunalnisˇka simulacija procesa preoblikovanja kovin v testastem stanju – computer simulation of metal forming in semi-solid state”, PhD thesis, University of Ljubljana. Mole, N., Chenot, J.L. and Fourment, L. (1996), “A velocity based approach including acceleration to the finite element computation of viscoplastic problems”, Int. J. for Num. Meth. in Eng., Vol. 39, pp. 3439-51. Poje, J., Sˇtok, B. and Mole, N. (2003), “By interconnecting and integration of knowledge toward a new technological quality”, Proceedings of IDDRG 2003 Conference, Bled, Slovenia, pp. 233-40. Rojc, T. and Sˇtok, B. (2003), “About finite element sensitivity analysis of elastoplastic systems at large strains”, Computer & Structures, Vol. 81, pp. 1795-809. Rus, P., Sˇtok, B. and Mole, N. (2003), “Parallel computing with load balancing on heterogeneous distributed systems”, Adv. Eng. Softw., Vol. 34 No. 4, pp. 185-201. ˇStok, B. (1997), “Racˇunalnisˇko simuliranje – podpora ucˇinkovitejsˇemu nacˇrtovanju tehnolosˇkih procesov¼ computer simulation – a means for improving the efficiency of technology processes design”, Strojnisˇki Vestnik, Vol. 43 Nos 11/12, pp. 463-82. Sˇtok, B. and Hudoklin, A. (1994), “How to tackle the compatibility constraints in a computational solution of frictional contact problem”, Finite Elem. Anal. Design, Vol. 18, pp. 111-9. ˇStok, B. and Mole, N. (1990), “Finite element modelling and simulation of electroslag remelting process”, Proceedings of the Third Int. Conf. on Tech. of Plasticity, Kyoto Vol. 2, pp. 1013-9. Sˇtok, B. and Mole, N. (1993), “Matematicˇno modeliranje rotacijskega litja – analiza termomehanskega stanja v orodju ¼ mathematical modelling of rotary casting – thermomechanical analysis of a mould”, Kovine Zlitine Tehnologije, Vol. 27 Nos 1/2, pp. 175-80. Sˇtok, B. and Mole, N. (2001), “Coupling FE and BE approach in axisymmetric eddy current problems solution”, Proceedings of the Int. Conf. on Comp. Eng. and Science, Puerto Vallarta, Mexico, p. 6.

6, Engineering structures: nonclinical analysis optimal design and identification

Sequential Analysis and Optimal Design

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Optimal Design: Theory and Applications to Materials and Structures

Plastic Analysis and Design of Steel Structures

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Analysis and design of flight vehicle structures

Tall Building Structures: Analysis and Design

Eurocode 6 - Design Of Masonry Structures

Digital Control Engineering: Analysis and Design

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Fire Safety Engineering - Design of Structures

Principles of optimal design

Optimal Design of Experiments

Optimal Device Design

Optimal shape design

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Organization Structures: Theory and Design, Analysis and Prescription

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Failure Analysis of Engineering Structures: Methodology and Case Histories

Analysis and Design of Plated Structures: Volume 2: Dynamics

Structures for Nuclear Facilities: Analysis, Design, and Construction

Optimal Experimental Design with R

Design and Analysis of Fatigue Resistant Welded Structures

Practical stress analysis in engineering design

6, Engineering structures: nonclinical analysis optimal design and identification

Sequential Analysis and Optimal Design

Sequential analysis and optimal design

Optimal Design: Theory and Applications to Materials and Structures

Plastic Analysis and Design of Steel Structures

Analysis and design of plated structures

Analysis and design of flight vehicle structures

Tall Building Structures: Analysis and Design

Eurocode 6 - Design Of Masonry Structures

Digital Control Engineering: Analysis and Design

Digital Control Engineering: Analysis and Design

Fire Safety Engineering - Design of Structures

Principles of optimal design

Optimal Design of Experiments

Optimal Device Design

Optimal shape design

Optimal Shape Design

Optimal Device Design

Dynamic system identification. Experiment design and data analysis

Organization Structures: Theory and Design, Analysis and Prescription

Quantum Engineering: Theory and Design of Quantum Coherent Structures

Environmental Wind Engineering and Design of Wind Energy Structures

Protein Engineering and Design

Protein Engineering and Design

Protein Engineering and Design

Failure Analysis of Engineering Structures: Methodology and Case Histories

Analysis and Design of Plated Structures: Volume 2: Dynamics

Structures for Nuclear Facilities: Analysis, Design, and Construction

Optimal Experimental Design with R

Design and Analysis of Fatigue Resistant Welded Structures

Practical stress analysis in engineering design

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