IUTAM Symposium on Transformation Problems in Composite and Active Materials (Solid Mechanics and Its Applications)

IUTAM SYMPOSIUM ON TRANSFORMATION PROBLEMS IN COMPOSITE AND ACTIVE MATERIALS

SOLID MECHANICS AND ITS APPLICATIONS Volume 60 Series Editor:

G.M.L. GLADWELL Solid Mechanics Division, Faculty of Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3G1

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

IUTAM Symposium on

Transformation Problems in Composite and Active Materials Proceedings of the IUTAM Symposium held in Cairo, Egypt, 9–12 March 1997

Edited by

YEHIA A. BAHEI-EL-DIN Structural Engineering Department, Cairo University, Giza, Egypt

and

GEORGE J. DVORAK Centre for Composite Materials and Structures, Rensselaer Polytechnic Institute, Troy, NY, U.S.A.

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Contents Preface Sponsors List of Contributors List of Participants

ix xi xiii xvii Inelastic Behavior of Composite Materials

Modelling Stress-Dependent Transformation Strains of Heterogeneous Materials A. Zaoui andR. Masson

3

A Micromechanical Model for Predicting Fatigue Response of Metal Matrix Composites Subjected to Environmental Degradation J. W. Foulk, D.H. Alien and D.C. Lagoudas

17

On The Micromechanical Modelling of the Viscoplastic Behavior of Single Crystal Superalloys J.L. Chaboche, S. Lhuillier, D. Nouailhas

33

Micromechanical Analysis of Inelastic Fibrous Laminates Y.A. Bahei-EI-Din, I.A. Ibrahim andA.G. Botrous

45

Fourier Transforms and Their Application to the Formation of Textures and Changes of Morphology in Solids W.H.Müller

61

Second-Order Estimates for The Effective Behavior of Nonlinear Porous Materials

M.V. Nebozhyn and P. Ponte Castañeda

73

Shape Memory Effects

Energetics in Martensites O.P. Bruno

Modeling of Cyclic Thermomechanical Response of Polycrystalline Shape Memory Alloys

Z. Bo and D.C. Lagoudas

91

109

vi

The Taylor Estimate of Recoverable Strains in Shape-Memory Polycrystals K. Bhattacharya, R. V. Kohn and Y.C. Shu

123

Thermomechanical Modeling of Shape Memory Alloys and Applications C. Lexcellent and S. Leclercq

135

On the Design of Hydrophones Made as 1-3 Piezoelectrics O. Sigmund, S. Torquato, L.V. Gibiansky and I.A. Aksay

147

Functionally Graded Materials A Stress Function Formulation for a Class of Exact Solutions for Functionally Graded Elastic Plates

A.J.M. Spencer

161

Micromechanical Modeling of Functionally Graded Materials T. Reiter and G.J. Dvorak

173

Thermal Fracture and Thermal Shock Resistance of Functionally Graded Materials

Z. -H. Jin and R. C. Batra

185

Micromechanics and Nonlocal Effects in Graded Random Structure Matrix Composites V. Buryachenko and F. Rammerstorfer

197

Transformation Problems in Composite Structures

Design of Composite Cylinder Fabrication Process

M. V. Srinivas and G.J. Dvorak

209

Dynamic Response of Elastic-Viscoplastic Sandwich Beams with Asymmetrically Arranged Thick Layers C. A dam and F. Ziegler

221

Design of Materials with Extreme Elastic or Thermoelastic Properties Using Topology Optimization O. Sigmund and S. Torquato

233

vii

Adaptive Structures

An Exact Solution for Static Shape Control Using Piezoelectric Actuation

H. Irschik, C. Adam, R. Heuer and F. Ziegler

247

On the Theory of Smart Composite Structures A.L. Kalamkarov and A.D. Drozdov

259

Smart Hinge Beam for Shape Control D. Perreux and C. Lexcellent

271

Elasticity Issues

Optimality of Dilute Composites Under Shear Load

S. Serkov, A. Movchan, A. Cherkaev, and Y. Grabovsky

285

A Hashin-Shtrikman Approach to the Elastic Energy Minimization of Random Martensitic Polycrystals V.P. Smyshlyaev and J.R. Willis

301

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Preface The field of composite materials has seen substantial development in the past decade, New composite systems are being continually developed for various applications. Among such systems are metal, intermetallic, and superalloy matrix composites, carbon-carbon composites as well as polymer matrix composites. At the same time, a new discipline has emerged of active or smart materials, which are often constructed as composite or heterogeneous media and structures. One unifying theme in these diverse systems is the influence that uncoupled and coupled eigenfields or transformation fields exert on the various types of overall response, as well as on the respective phase responses. Problems of this kind are currently considered by different groups which may not always appreciate the similarities of the problems involved. The purpose of the IUTAM Symposium on Transformation Problems in Composite and Active Materials held in Cairo, Egypt from March 10 to 12, 1997 was to bring together representatives of the different groups so that they may interact and explore common aspects of these seemingly different problem areas. New directions in micromechanics research in both

composite and active materials were also explored in the symposium. Specifically, invited lectures in the areas of inelastic behavior of composite materials, shape memory effects, functionally graded materials, transformation problems in composite structures, and adaptive structures were delivered and discussed during the three-day meeting. This book contains the printed contributions to the IUTAM Symposium. The time and effort spent by the authors in participating in the meeting and preparing the manuscripts for this book is greatly appreciated. Thanks are due to the IUTAM Bureau and Cairo University for sponsoring the meeting and providing partial funds for local and international participants. Special thanks are due to the local organizing committee for the fine local arrangements during the meeting.

Yehia A. Bahei-El-Din Cairo, Egypt

George J. Dvorak Troy, New York, USA

March 1997

ix

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Sponsors of the IUTAM Symposium on

Transformation Problems in Composite and Active Materials International Union of Theoretical and Applied Mechanics (IUTAM) Cairo University Faculty of Engineering Civil Engineering Center Center of Advancement of Graduate Studies and Research in Engineering Science

International Scientific Committee of the Symposium Professor Yehia A. Bahei-El-Din (Egypt), Co-Chairman Professor George J. Dvorak, (USA), Co-Chariman Professor J. L. Chaboche (France) Professor R. M. Christensen (USA) Professor F. D. Fischer (Austria) Professor K. Herrmann (Germany) Professor V. Kovarik (Czech Republic) Professor G. A. Maugin (France) Professor G. W. Milton (USA) Professor A. J. M. Spencer (UK) Professor F. Ziegler (Austria )

Local Organizing Committee Prof. Y.A. Bahei-El-Din Dr. A.M. Saleh Dr. I. A. Ibrahim Monte Carlo Tours

xi

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List of Contributors Adam, Christoph, Department of Civil Engineering, Technical University Vienna, A-1040 Vienna, AUSTRIA Allen, David, Center for Mechanics of Composites, Texas A&M University, College Station, Texas 77843-3141, USA

Bahei-El-Din, Yehia A., Structural Engineering Department, Cairo University, Giza, EGYPT Batra, Romesh C., Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0219, USA Bhattacharya, Kaushik, California Institute of Technology, Mail Code 104-44,

Pasadena, California 91125, USA

Bruno, Oscar P., Department of Applied Mathematics, California Institute of Technology, Pasadena, California 91125, USA

Buryachenko, V.A., Institut fur Leicht- und Flugzeugbau, Technische Universitat Wien, 1040 Wien, AUSTRIA Castaneda, Pedro Ponte, Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6315, USA

Chaboche, J.L., Office National d'Etudes et de Recherdes Aerospatiales (ONERA),

F-92322 Chatillon Cedex, FRANCE

Dvorak, George J., Center for Composite Materials and Structures, Rensselaer

Polytechnic Institute, Troy, New York 12180-3590, USA

Heuer, Rudolf, Department of Civil Engineering, Technical University Vienna, A-1040 Vienna, AUSTRIA

Kalamkarov, Alexander L., Department of Mechanical Engineering, Technical University of Nova Scotia, Halifax, Nova Scotia B3J 2X4, CANADA Lagoudas, Dimitris C., Aerospace Engineering Department, Texas A&M University,

College Station, Texas 77843-3141, USA

Lexcellent, C., L.M.A., UFR Sciences et Techniques, 24 rue de l’Epitaphe, 25030

Besancon Cedex, FRANCE

xiii

xiv

Mueller, Wolfgang H., Laboratorium fur Technische Mechanik, Universitat-GHPaderborn, Pohlweg 47-49, 33098 Paderborn, GERMANY Serkov, Sergei, School of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK Sigmund, Ole, Department of Solid Mechanics, Technical University of Denmark, DK 2800 Lyngby, DENMARK Smyshlyaev, Valery P., School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

Spencer, A.J.M., Department of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, UK Zaoui, Andre, Laboratoire de Mecanique des Solides, Ecole Polytechnique F 91128 Palaiseau Cedex, FRANCE

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List of Participants Abd-El-Hady, Faissal, Automotive Engineering Department, Ain Shams University, Cairo, EGYPT

Abo-Hamda, Mustafa, Department of Mechanical Design and Production Engineering, Cairo University, Giza, EGYPT Abuelfoutouh, Muhamed N., Aeronautical Engineering Department, Cairo University, Giza, EGYPT

Adam, Christoph, Department of Civil Engineering, Technical University Vienna, A-1040 Vienna, AUSTRIA Alien, David, Center for Mechanics of Composites, Texas A&M University, College Station, Texas 77843-3141, USA Anis, Ahmed R., Structural Engineering Department, Cairo University, Giza, EGYPT Bahei-El-Din, Yehia A., Structural Engineering Department, Cairo University, Giza, EGYPT

Batra, Romesh C., Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0219, USA

Bhattacharya, Kaushik, California Institute of Technology, Mail Code 104-44, Pasadena, California 91125, USA Botrous, Amany G., Civil Engineering Department, Tanta University, Tanta, EGYPT Bruno, Oscar P., Department of Applied Mathematics, California Institute of Technology, Pasadena, California 91125, USA

Buryachenko, V.A., Institut fur Leicht- und Flugzeugbau, Technische Universitat Wien, 1040 Wien, AUSTRIA Chaboche, J.L., Office National d'Etudes et de Recherdes Aerospatiales (ONERA), F-92322 Chatillon Cedex, FRANCE Dvorak, George J., Center for Composite Materials and Structures, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA Dwedar, Mohammed S., Automotive Engineering Department, Ain Shams University, Cairo, EGYPT xvii

xviii

Dzenis, Yuris A., Department of Engineering Mechanics, University of NebraskaLincoln, Lincoln, Nebraska 68588-0347, USA El-Hakeem, Farouk Aly H., Structural Engineering Department, Cairo University, Giza, EGYPT

El-Sheikhy, Refat A., Housing & Building Research Center, 56 Tahrir Street, Dokky, Giza, EGYPT Francfort, Gilles, L.P.M.T.M., Institut Villetaneuse, FRANCE

Galilee, Universite Paris 13, 93430

Gendy, Atef S., Structural Engineering Department, Cairo University, Giza, EGYPT Ghabrial, Nabil, Structural Engineering Department, Cairo University, Giza, EGYPT

Heuer, Rudolf, Department of Civil Engineering, Technical University Vienna, A-1040 Vienna, AUSTRIA Ibrahim, Iman A., Structural Engineering Department, Cairo University, Giza, EGYPT

Ibrahim, Samir S., Housing & Building Research Center, 56 Tahrir Street, Dokky, Giza, EGYPT Kalamkarov, Alexander L., Department of Mechanical Engineering, Technical University of Nova Scotia, Halifax, Nova Scotia B3J 2X4, CANADA Lagoudas, Dimitris C., Aerospace Engineering Department, Texas A&M University, College Station, Texas 77843-3141, USA

Lexcellent, C., L.M.A., UFR Sciences et Techniques, 24 rue de l’Epitaphe, 25030 Besancon Cedex, FRANCE

Luciano, Raimondo, Department of Industrial Engineering, University of Cassino, 03043 Cassino, ITALY Mahmoud, Mahmoud K., Housing & Building Research Center, 56 Tahrir Street, Dokky, Giza, EGYPT Megahed, Mohammed M., Department of Mechanical Engineering, Cairo University, Giza, EGYPT

Design and Production

xix

Milton, Graeme W., Department of Mathematics, The University of Utah, Salt Lake City, Utah 84112, USA

Mueller, Wolfgang H., Laboratorium fur Technische Mechanik, Universitat-GHPaderborn, Pohlweg 47-49, 33098 Paderborn, GERMANY Nagy, Badr S., Department of Mechanical Design and Production Engineering, Cairo

University, Giza, EGYPT

Nassef, Muhamed El-Adawy, Structural Engineering Department, Cairo University, Giza, EGYPT

Sadek, Edward A., Aeronautical Engineering Department, Cairo University, Giza,

EGYPT

Saleh, Ahmed M., Structural Engineering Department, Cairo University, Giza, EGYPT

Sayed, Hesham S., Structural Engineering Department, Cairo University, Giza, EGYPT Serkov, Sergei, School of Mathematical Sciences, University of Bath, Bath BA2 7AY,

UK

Shoukry, Khaled M., Housing & Building Research Center, 56 Tahrir Street, Dokky, Giza, EGYPT

Sigmund, Ole, Department of Solid Mechanics, Technical University of Denmark, DK

2800 Lyngby, DENMARK

Smyshlyaev, Valery P., School of Mathematical Sciences, University of Bath,

Claverton Down, Bath BA2 7AY, UK

Spencer, A.J.M., Department of Theoretical Mechanics, University of Nottingham,

Nottingham NG7 2RD, UK

Swellam, Mohammed H., Structural Engineering Department, Cairo University, Giza,

EGYPT

Youssef, Mahmoud A. Reda, Structural Engineering Department, Cairo University, Giza, EGYPT

Zaoui, Andre, Laboratoire de Mecanique des Solides, Ecole Polytechnique, F 91128

Palaiseau Cedex, FRANCE

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Inelastic Behavior of Composite Materials

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MODELLING STRESS-DEPENDENT TRANSFORMATION STRAINS OF HETEROGENEOUS MATERIALS

A. ZAOUI and R. MASSON

Laboratoire de Mécanique des Solides, Ecole Polytechnique, CNRS F 91128 Palaiseau Cedex

Abstract

This paper aims at drawing attention to some difficulties which may arise when using the concept of "stress-free strain" or "transformation strain" for the prediction of the macroscopic behaviour of rate-independent or rate-dependent elastoplastic polycrystals and

composite materials, especially in the context of the self-consistent homogenization procedure. These difficulties are due to the stress-dependence of plastic and viscoplastic strains, which can hardly be considered as true "stress-free strains", contrary to other

physically well-identified (thermal, magnetic...) eigenstrains: when omitting this stressdependence within an elastic framework, a direct self-consistent-type derivation of the overall behaviour leads to an overestimation of the mechanical interphase interactions and of the resulting overall stiffness. An alternative treatment, based on a tangent-type linearization process, is proposed which leads to the introduction of fictitious "stress-free strains" that do not reduce to the plastic or the viscoplastic ones and consequently yield

softer predictions. Nevertheless, the limits of Hill's approximation of the nonlinear selfconsistent modelling are acknowledged and discussed.

1. Introduction

Since the solution given by Eshelby (Eshelby, 1957) to the basic problem of an ellipsoidal eigenstrained inclusion embedded in an infinite elastic matrix, the concept of "stress-free strain" or "transformation strain" has been used intensively in the field of continuum micromechanics to analyse the elastic coupling of physically well-identified

(thermal, magnetic, electric...) eigenstrains with the associated elastic strains and of deriving their contribution to the overall behaviour of heterogeneous materials. The corresponding well-known homogenization treatments are quite unambiguous, due to the preservation of the linearity of the overall behaviour; the basic equations are recalled first

(section 2), with specialization to the self-consistent approach. New difficulties arise when nonlinear constitutive behaviour is considered, especially when plasticity is present. The first attempt to predict the overall response of rate-independent elastoplastic polycrystals through a self-consistent-type procedure (section 3) is due to Kröner (Kröner, 1961): it makes use of the concept of stress-free strain applied to the plastic strain. This procedure is debatable: it leads to reduction of the 3 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials , 3–15.

© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

4 mechanical interactions between the constituent phases to elastic ones, which does not

take into account their plastic relaxation due to the overall plastic flow. This treatment results in an overestimation of the overall stiffness which is consequently close to the

one predicted by the (uniform strain-based) Taylor model. An alternative treatment was

proposed more than thirty years ago by Hill (Hill, 1965) which did not rely anymore on

the concept of "transformation strain" but used Eshelby's solution of the inhomogeneity (instead of inclusion) problem within a linearized incremental formulation: as expected, this treatment yields softer predictions. A similar formulation has been developed too for the case of rate-dependent plasticity (without elasticity). The conclusion that the assimilation of the plastic strain to an eigenstrain when deriving self-consistent (stress or strain) concentration factors was not an adequate procedure seemed to be widely accepted up to the early eighties; at that time, it was argued by Weng (Weng, 1981) that this conclusion was not to be applied to ratedependent elastoplasticity, due to the fact that the viscoplastic strain would not obey the

same stress-dependence as the plastic one. Consequently, several authors have developed

rate-dependent elastoplastic self-consistent-type models relying, in an explicit or implicit manner, on the simulation of the viscoplastic strain by an eigenstrain: it will be shown

(section 4) that, like Kröner's, this approach still reduces the mechanical interactions

between the phases to elastic ones and results in too stiff predictions of the Taylor type. This conclusion is illustrated in the simpler case of linear viscoelasticity where the exact solution can be derived thanks to the use of the Laplace transform technique. An alternative treatment is proposed (section 5) for rate-dependent elastoplasticity: it combines Hill's linearization process and the use of the Laplace transform technique at each step, following a given macroscopic load path. This procedure leads to the introduction of a fictitious "stress-free strain" which does not reduce to the viscoplastic one in the general nonlinear case. Illustrations are given which compare the consequences of these two ways of considering the viscoplastic strain. The same method may be defined for viscoplasticity, but also for elastoplasticity, which offers an alternative treatment to Hill's one, associated with softer predictions. Nevertheless, the limits of Hill's framework itself must be acknowledged, as discussed in conclusion. 2. Heterogeneous Elasticity With Eigenstrains

To begin with, let us consider the simple case of an elastic heterogeneous material with moduli and compliances undergoing the known and fixed eigenstrain field The local constitutive equations read:

The basic homogenization problem, consisting in the derivation of the overall moduli

or compliances

is supposed to have been solved in the absence of eigenstrains

through the strain or stress concentration tensors A or B respectively, according to the

classical relations:

5

where stands for the transposed tensor of A. When an eigenstrain field is present, the overall moduli and compliances are unchanged and the homogenized constitutive equations read:

So, the problem is completely solved from the knowledge of A or B, which depend on the model chosen. If a self-consistent scheme is prefered, the eigenstrain field has to be stepwise uniform per phase (r) and the (averaged per phase) strain concentration tensor is given by

where

is the modified strain Green operator of the effective medium, I is the fourth

order unit tensor and

is the ellipsoidal inclusion which represents the phase (r). The local stress or strain state is derived from the associated concentration (or interaction) law, namely

where

is an auxiliary uniform strain determined by the condition

When the phases have the same elastic moduli

, and the interaction law reduces to

For ellipsoids with the same shape and orientation, and (6) may be written in the more classical following form which makes use of Hill's (Hill, 1965) overall constraint tensor and Eshelby's tensor

For spheres and isotropy (with and k the shear and bulk moduli respectively) and for isochoric eigenstrains, (7) reduces to

6

where

is Eshelby's well-known coefficient, ranging from .4 to .6 according to k. Note that this treatment is a purely elastic one, which implies that the eigenstrain field is considered as known and fixed and does not depend on the mechanical loading. This is true for thermal, electric, magnetic, ... strains which are clearly stressindependent eigenstrains. Conversely, this eigenstrain field does not modify the mechanical behaviour of the medium: the mechanical interactions between the phases are elastic and, within the self-consistent scheme, the underlying inhomogeneity problem refers to an elastic ellipsoidal inhomogeneity with a uniform eigenstrain embedded in an infinite elastic matrix with a uniform overall eigenstrain too. This conforms with the elastic nature of the considered constitutive behaviour and of the associated boundaryvalue problem, as well as with the basic assumptions which lead to Eshelby's solution of the inclusion and inhomogeneity problems. 3. Is Plastic Strain A True Eigenstrain?

No later than four years after the publication of Eshelby's solution of the inclusion problem, Kröner (Kröner, 1961) proposed to use it directly for the derivation of a rateindependent plasticity version of the self-consistent scheme for metallic polycrystals, by considering the plastic strain as a "stress-free strain". The basic idea looked undeniable: since a plastic strain imposed on a volume element can subsist after unloading, it can be dealt with as an eigenstrain and Eshelby's solution can be used for an ellipsoidal inclusion suffering a uniform plastic strain embedded in an infinite elastic matrix with the same moduli C. The resulting stress in the inclusion reads, according to (7):

or, for a spherical inclusion, isotropy and isochoric plasticity:

If this internal stress in the inclusion is such that the plastic yield criterion is violated in

it, has to be revalued so as to obey this criterion, but the matrix is supposed to remain elastic according to Eshelby's assumptions. The difficulty arises when the matrix behaviour has to be assimilated to the overall behaviour of the polycrystal, according to the self-consistent procedure. Since a plastically flowing polycrystal was considered, with the overall plastic strain Kröner proposed to consider this plastic strain as an eigenstrain too and to assign it uniformly to the matrix. Assuming spherical grains, uniform isotropic elasticity and isochoric plasticity, this leads, from (10), to Kröner's well-known concentration equation:

7

which is nothing but (8) with instead of The general case of ellipsoidal shapes, non uniform elasticity and non isochoric plasticity could be considered as well according to the same approach through equations (3) to (5). This treatment is not beyond reproach: the debatable point lies in the description of the overall behaviour, which is supposed to be exhibited by the matrix in the

inclusion/matrix problem. The foregoing procedure clearly reduces this overall behaviour to an elastic one, whether it is associated with an eigenstrain or not: the overall constitutive equations are nothing but the following: with

where A and B are the elastic concentration tensors. Whereas such equations are right for a given distribution of plastic strains they cannot be considered as constitutive equations for a plastically flowing body : the overall plastic strain is actually stressdependent in a way which must be explicitly stated in (12) in order to transform these elastic equations into elastoplastic ones. Otherwise, when deriving the concentration equation, they lead to an elastic analysis of the mechanical interactions between the phases, i.e. to a strong overestimation of the internal stresses. This effect can be seen more easily on the forms (8) or (11): since the internal stresses cannot exceed the yield stress, which is usually

lower than the shear modulus

this means

that the (plastic as well as total) strain deviations cannot exceed themselves and the resulting homogenization scheme reduces practically to Voigt's or Taylor's one, as has been checked in several applications. An important comment must be made on that point: the drawbacks of dealing with the plastic strain as with an eigenstrain are magnified by the classical selfconsistent procedure whose basic operation refers to a two-body problem (a single inclusion and an infinite matrix) whose solution is strongly dependent on the assumption of uniform plastic strains. Of course, the overestimation of the associated interactions decreases when more and more interacting bodies are considered such as in the "transformation field analysis" method (Dvorak, 1992), or in the context of generalized self-consistent schemes (Bornert, Stolz and Zaoui, 1996) (or of the finite element method...). Going back to the classical self-consistent scheme, we know that Kröner's treatment was first criticized by Hill (Hill, 1965) who proposed another approach relying on Eshelby's solution of the inhomogeneity (instead of inclusion) problem, without any reference to the concept of stress-free strain. It has recourse to a linearization procedure along the prescribed macroscopic loading path and to the associated local and overall elastoplastic instantaneous moduli The corresponding concentration equation reads:

with

8 where the overall elastoplastic constraint tensor depends on and on the shape and orientation of the ellipsoid. A similar treatment can be used (Hutchinson, 1976) for the case of viscoplasticity (without elasticity) by simply replacing in (13) and by and . A number of applications and developments (Hutchinson, 1970; Berveiller and Zaoui, 1979; Iwakuma and Nemat-Nasser, 1984) have shown that such a procedure is more relevant than Kröner's one and the corresponding model conforms better with the classical self-consistent point of view. Nevertheless, as will be discussed in conclusion, Hill's treatment is not beyond any reproach itself because of the assumption of uniform instantaneous moduli in the matrix. 4. Is Viscoplastic Strain A True Eigenstrain? The application of the foregoing considerations to elastoviscoplasticity is not straightforward. The main difficulty stems from the coupling between elasticity and viscosity which is responsible for the simultaneous occurrence of derivatives of different orders of and in the corresponding constitutive equations. This circumstance makes it inefficient to use Hill's linearization procedure since no instantaneous linearized moduli or compliances can be defined. That is the reason why the problem of the self-consistent modelling of rate-dependent elastoplastic inhomogeneous materials has seemed to be unapproachable for more than fifteen years, while Hill's criticism of Kröner's approach looked undeniable (when Brown (Brown, 1970) used Kröner's model to model the creep behaviour of polycrystals in 1970, he argued from its computational simplicity and admitted that his work was open to Hill's criticism). On the other hand, the case of linear viscoelasticity was solved elegantly by Laws and McLaughlin (Laws and McLaughlin, 1978) by use of the Laplace transform technique and the correspondence principle, thus circumventing the difficulty of a definition of instantaneous moduli, but obviously the use of the Laplace transform method was not allowed due to nonlinearity. Nevertheless, the seductiveness of the concept of eigenstrain still exerts its influence: first Weng (Weng, 1981) noticed that, according to a time-incremental procedure, the viscoplastic strain rate to be derived for a given time step depended on stress and not on stress rate so that it could be determined by the (known) stress state at the begining of the considered time step. He concluded that, unlike the plastic strain, the viscoplastic one could be dealt with as eigenstrain within a self-consistent model, and that Kröner's approach, while unadequate for rate-independent plasticity, could be applied without change to rate-dependent elastoplasticity. This conclusion was corroborated a few years later (Nemat-Nasser and Obata, 1986) in the context of finite strain from a Greentype analysis of the underlying inclusion problem by implicitly assigning a uniform viscoplastic strain rate to the matrix. Finally Harren (Harren, 1991) used a similar formulation and performed powerful numerical simulations of metal forming responses and texture developments for 3-D polycrystals: comparison (see Fig 1) of these predictions with those derived from the Taylor model proved that they were almost indistinguishable. The reason for that is quite clear. Weng is wrong when he states that "the crux of the matter is that, creep, unlike plastic deformation, is a truly "stress-free" process in the sense of Eshelby, because the creep strain rate, at any generic state, depends only on the current stress and deformation history and is independent of the stress rate... This

9

subtle point appears to have not been realized." Actually, the "crux of the matter" is that creep, much like plastic deformation, is a stress-dependent process and it cannot be considered that the overall viscoplastic strain and strain rate in the matrix of the elementary problem of the self-consistent scheme are insensitive to the presence of the inclusion. Similarly, Nemat-Nasser and Obata as well as Harren are not right when they

consider the definition of the elastic strain, namely the equations

as rate-dependent elastoplastic constitutive equations. This confusion leads them, through the classical Green procedure, to solve an inclusion problem where the matrix obeys an elastic behaviour and has suffered the uniform eigenstrain rate Consequently, the mechanical interactions between the phases, as expressed in Kröner's concentration law (written for time stress and strain rates and extended to finite strains), are of an elastic nature and lead to a strong overestimation of the overall stiffness, quite similar to what is obtained from the Taylor model, as discussed earlier. A simple way to check this inadequacy is to consider the special case of linear viscoelasticity which should be treated in the same way. For the sake of simplicity and to use closed form results, let us consider a two-phase isotropic material whose

constituents obey an isotropic incompressible Maxwellian law, according to the equations :

10 where e and s are the local strain and stress deviators, and are the elastic shear modulus and the viscosity coefficient of the constituent (i), which corresponds to one relaxation time for each phase. First let us use a Kröner-type concentration equation, which would read in the special case where

The overall shear modulus can be derived easily through the Laplace transform technique; from (15), (16), and the average relations we find for its Laplace-transformed with p the complex variable:

where c is the volume fraction of phase (2). This corresponds to the superposition of two exponential functions, and so to two relaxation times, which means that the overall shear relaxation spectrum consists of two discrete spectrum lines at these relaxation times. On the other hand, the same problem can be solved exactly according to a rigorous self-consistent scheme (Rougier, Stolz and Zaoui, 1993; Zaoui, Rougier and Stolz, 1995). With arbitrary, the overall shear relaxation spectrum is found to be composed of a continuous spectrum lying between with the intensity:

with

and of additional lines at (with the intensity for (with the intensity Figure 2 gives an illustrative comparison of both predictions of the normalized overall relaxation spectra there is no longer any doubt that Kröner-type approaches have the same limitations in the rate-independent and rate-dependent cases. 5. A New Formulation Based On The Use Of Fictitious Eigenstrains

It cannot be concluded from what precedes that the concept of eigenstrain is of no use when modelling the overall behaviour of nonlinear heterogeneous materials. If the original Eshelby's problem is examined in detail, it turns out that it could be used in an even broader sense: the crucial point is that, at any step of an incremental procedure, the eigenstrain field must be known a priori, independently of the mechanical loading. So, it can even depend on time, provided that this time-dependence is known in advance. Let us consider for example (Zaoui, Rougier and Stolz, 1995; Rougier, Stolz and Zaoui, 1994) the following simplified local nonlinear viscoelastic (or elastic-viscoplastic) equations:

11

with the elastic compliances s. We are supposed to have determined the local and overall responses to some given loading path from time and we are looking for

the response on the next infinitesimal time interval

equations can be linearized as follows:

The constitutive

where H(t) is the unit step function. Note that, for these linearized equations describe a Maxwellian behaviour with the fixed and known eigenstrain rate the additional term is no longer constant, so as to allow us to recover the actual constitutive equations (19): this is essential in order to keep, from the initial stage up to the memory of the actual nonlinear behaviour for the prediction of the response after since it influences the subsequent response at any time. The important point is that such a function is completely known in advance and depends in no way on the external loading applied beyond So, it can be

12 dealt with as an eigenstrain rate associated with the linearized Maxwellian constitutive behaviour. The subsequent analysis is quite straightforward since, by use of the LaplaceCarson transform technique defined by

the homogenization problem can be converted into a (symbolically) elastic one with eigenstrains. This problem has a classical solution which has been recalled above (eqns (1) to (5)): now all the quantities must be followed by an asterisk and considered as the Laplace-Carson transforms of time-dependent functions. An inverse transformation is needed at each step: it can be performed numerically, e.g. through a collocation method. Note that this formulation conforms with Hill's general approach insofar as uniform elastic-viscoplastic moduli or compliances are assigned to the matrix in the underlying matrix/inclusion problem. But the mechanical interactions are no longer elastic, and obey a viscous relaxation associated with a Maxwellian behaviour. The treatment can be extended to more general constitutive equations, such as these:

where the internal variables may be adapted to the description of crystalline materials (Navidi, Rougier and Zaoui, 1996). Figure 3 gives an illustration of the fact that, as expected, the overall creep compliance of an isotropic FCC polycrystal as predicted from Weng's treatment is lower than that derived from this approach.

13

Note that the combination of rate-independent and instantaneous plastic properties could be considered as well. This general formulation can be restricted to more specific situations, such as viscoplasticity (without elasticity) and rate-independent elastoplasticity: - in the first case, the linearized local constitutive equations at any step (n) read:

and the auxiliary strain

can still be dealt with as a fictitious eigenstrain. Of course,

the Laplace transform technique is no longer necessary. This treatment is akin to that of

Molinari et al (Molinari, Canova and Ahzi, 1987) for small strains, and it differs from it only in the final homogenization procedure. It yields softer predictions than those of Hutchinson (Hutchinson, 1976) and does not tend towards a Taylor-type model for power law creeping isotropic FCC polycrystals when the exponent tends to infinity, as Hutchinson's model is likely to do (Masson and Zaoui, 1997). - in the second case, we can define an alternative treatment to Hill's by considering a quasi-secant formulation instead of an incremental one. At any stage (n),

each phase is considered as a fictitious (multibranched) linear medium with the instantaneous compliances

and the eigenstrain

such that

The overall behaviour is still ruled by the same kind of equations. Figure 4 illustrates the fact that such a formulation yields softer responses than Hill's for the stress-strain tensile curve of an untextured FCC polycrystal whose grains obey a Schmid criterion with no hardening.

6. Conclusion It can be concluded that the concept of eigenstrain has to be used carefully when

modelling the overall nonlinear behaviour of heterogeneous materials, especially in the

context of the self-consistent scheme. On the one hand, the direct simulation of stressdependent strains such as the plastic or the viscoplastic ones by eigenstrains can lead to a considerable overestimation of the overall stiffness, except if the considered medium is divided into a sufficient number of subdomains (Dvorak, 1992). On the other hand, fictitious eigenstrains can be useful in order to approximate better the actual nonlinear

behaviour by a linear one. It has been found that such treatments could yield softer

predictions that incremental ones. This property is to be emphasized with regard to the limitations of Hill's treatment itself, whether rate-independent or rate-dependent nonlinear constitutive behaviours are considered. In any case, the concentration equation, which is the clue to

14

the homogenization modelling, is derived from the solution of a boundary-value problem which is somewhat inconsistent; as a matter of fact, in this inclusion/matrix problem the matrix is considered as a uniform linear medium though it is not uniformly stressed and strained. A rigourous treatment would have needed to consider the matrix as a homogeneous equivalent material with linearized moduli or compliances variable from point to point and depending on the local resulting stress and strain state... Some of the drawbacks resulting from this inconsistency concerning the connection beetween the overal moduli and compliances have been stressed by Dvorak (Dvorak, 1992); some others, related to the possible violation of rigourous upper nonlinear bounds for the overall moduli, have been emphasized by Gilormini (Gilormini, 1996). In this respect, it is reassuring that the new formulation proposed here leads systematically to softer responses compared to former ones. Nevertheless, a basic difficulty remains and new developments are needed in this still open field.

Acknowledgements Part of this work (section 5) was funded by EdF for nuclear applications. We are grateful to Dr Y. Rougier for his decisive initial contribution to the matter of this paper.

15 References Berveiller M. and Zaoui, A. (1979) An extension of the self-consistent scheme to plastically flowing polycrystals, J. Mech. Phys. Solids, 26, 325-344. Bornert, M., Stolz C. and Zaoui A. (1996) Morphologically representative pattern-based bounding in elasticity, J. Mech. Phys. Solids, 44, 307-331. Brown, G.M. (1970) A self-consistent polycrystalline model for creep under combined stress states, J. Mech. Phys. Solids, 18, 367-381. Dvorak, G.J. (1992) Transformation field analysis of inelastic composite materials, Proc. Roy. Soc. London, A 437, 311-327. Eshelby, J.D. (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. Roy. Soc. London, A 241, 376-396. Gilormini, P. (1996) A critical evaluation for various nonlinear extensions of the self-consistent model, in A. Pineau and A. Zaoui (eds), Micromechanics of Plasticity and Damage of Multiphase Materials, Kluwer Academic Publishers, Dordrecht , pp. 67-74. Harren, S.V. (1991) The finite deformation of rate-dependent polycrystals I, A self-consistent framework, J. Mech. Phys. Solids, 39, 345-360. Hill, R. (1965) Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 13, 89-101. Hutchinson, J.W. (1970) Elastic-plastic behaviour of polycrystalline metals and composites, Proc. Roy. Soc. London, A 319, 247-272. Hutchinson, J.W. (1976) Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. Roy. Soc. London, A 348, 101-127. Iwakuma, T. and Nemat-Nasser, S. (1984) Finite elastic-plastic deformation of polycrystalline metals, Proc. Roy. Soc. London, A 394, 87-119. Kröner, E. (1961) Zur plastischen Verformung des Vielkristalls, Ada Metall, 9, 155-161. Laws, N. and McLaughlin, R (1978) Self-consistent estimates for the viscoelastic creep compliances of composite materials, Proc. R. Soc. Lond., A 359, 251-273. Masson, R. and Zaoui, A. (1997) From rate-dependent to rate-independent self-consistent modelling of elastoplastic multiphase materials, in A.S. Khan (ed), Physics and Mechanics of Finite plastic and Viscoplastic Deformation, Neat Press, Maryland (USA), pp. 209-210. Molinari, A., Canova, G.R. and Ahzi, S. (1987) A self-consistent approach of the large deformation polycrystal viscoplasticity, Acta Metall, 35, 2983-2994. Navidi, P., Rougier, Y. and Zaoui, A (1996) Self-consistent modelling of elastic-viscoplastic multiphase materials, in A. Pineau and A. Zaoui (eds), Micromechanics of Plasticity and Damage of Multiphase Materials, Kluwer Academic Publishers, Dordrecht , pp. 123-130. Nemat-Nasser, S. and Obata, M. (1986) Rate-dependent finite elastoplastic deformation of polycrystals, Proc. Roy. Soc. London, A 407, 343-375. Rougier, Y., Stolz, C. and Zaoui, A. (1993) Représentation spectrale en viscoélasticité linéaire des matériaux hétérogènes, C. R. Acad. Sci. Paris, II 316, 1517-1522. Rougier, Y., Stolz C. and Zaoui, A. (1994) Self-consistent modelling of elastic-viscoplastic polycrystals, C. R. Acad. Sci. Paris, II 318, 145-151. Weng, G.J. (1981) Self-consistent determination of time-dependent behaviour of metals, J. Appl. Mech., 48,41-46. Zaoui, A., Rougier, Y. and Stolz C., (1995) Micromechanical modelling based on morphological analysis; Application to viscoelasticity, in R. Pyrz (ed), Microstructure-Property Interactions in Composite Materials, Kluwer Academic Publishers, Dordrecht, pp. 419-430.

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A MICROMECHANICAL MODEL FOR PREDICTING FATIGUE RESPONSE OF METAL MATRIX COMPOSITES SUBJECTED TO ENVIRONMENTAL DEGRADATION

J.W. FOULK D.H. ALLEN D.C. LAGOUDAS Center for Mechanics of Composites Texas A&M University College Station, Texas 77843 U.S.A.

Abstract

It has been observed in titanium matrix continuous fiber composites that structural life at elevated temperatures is significantly shorter in air than in inert gas. The physical

reasons for this degradation in life are dependent on the titanium alloy considered. In

this paper the metastable matrix alloy is considered, and is embedded with SCS-6 fibers to produce a four ply unidirectional laminate. Experimental results are briefly reviewed for laminates subjected to cyclic loading to failure at It is demonstrated that life is reduced by a factor of about five when a specimen is first degraded by placing it in an oven for 24 hours at and that this reduction is not due to oxidation. Rather, it is due to the development of a graded structure of brittle alpha grains that form due to oxygen diffusion into the matrix along the boundaries of the beta grains. These alpha grains induce microcracks near the surface of the composite that propagate into the interior of the composite and ultimately lead to premature failure of the structure. A computational model is employed herein based on the above experimental observations. This model utilizes the finite element method to predict the thermomechanical response of a unit cell of the laminate subjected to cyclic loading. The algorithm accounts for phase heterogeneity, matrix thermoviscoplasticity, and damage evolution due to surface cracking and fiber-matrix debonding. Crack growth is modelled via the inclusion of cohesive zones wherever experimental observations indicate that cracks may grow. Environmental degradation is accounted for by degrading the properties of the cohesive zone near the composite free surface, in accordance with experimental determinations of fracture toughness of the degraded material. Predictions obtained with the model indicate several useful observations. First, the effect of matrix viscoplasticity is important and should not be neglected in modelling. Second, there are significant differences between model predictions for the degraded and as-received material. Thus, with more advanced computers it may be possible to utilize the approach described herein to predict the effect of environmental degradation 17 YA. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 17–31. Kluwer Academic Publishers. Printed in the Netherlands.

18

on structural life in continuous fiber metal matrix composites. 1. Introduction

Over the last three decades, metal matrix composites (MMC’s) have been considered as candidate materials for numerous advanced technology initiatives. Although the properties of titanium matrix composites (TMC’s) are promising, they are generally life limited at elevated temperatures due to either the development of oxide layers on the surface, such as in matrix, or the development of an embrittled layer near the surface, as in the case of matrix. Experimental research indicates that both of these mechanisms contribute to the early development of multiple modes of microcracking that ultimately cause premature failure (Majumdar and Newaz, 1991; Foulk, 1994). Since these experiments are extremely costly and time consuming, it is submitted that model development is warranted in order to provide more cost effective means of evaluating life and the effect of environment on it. Furthermore, model development will help to understand these mechanisms so that improved performance can be attained through surface coating and/or alloy improvement. Towards this end, a methodology is developed herein for predicting failure of continuous fiber metal matrix composites undergoing environmental degradation. The predictions will utilize the finite element method coupled with models for material inelasticity, surface embrittlement, and crack propagation. The methodology will be described further in the body of the paper. 1.1 RECENT RESEARCH

Considerable research has focused on predicting the life of MMC systems. These predictions are complicated by substantial mechanical loads and harsh operating environments. Nicholas (1995) reviewed the mechanisms of failure and discussed many empirically based life prediction models. He noted the primary mechanisms of failure in [0] composites were fiber breaking and matrix cracking. In addition, monitoring the fiber stress was crucial to the prediction of life. Nicholas also concluded that environmental degradation played a significant role at elevated temperatures and that time must enter the model explicitly. Several researchers have studied the failure mechanisms associated with load and environment for continuous fiber metal matrix composites. Neu (1995) reviewed various experimental efforts and developed damage mechanism maps. Surface initiated damage due to oxidation was discussed and general conclusions were drawn with respect to crack propagation, fiber bridging, and eventual failure of the composite. Probably the most popular method of analysis involves the concentric cylinder model (CCM). Coker et al. (1993) coupled the concentric cylinder model with the Bodner-Partom viscoplasticity model (Chan et al.; 1988) to characterize [0] and [0/90] composites subjected to in-phase and out-of-phase thermomechanical fatigue. In addition to thermal stresses, the three dimensional stress-state of the [0] ply was calculated via FIDAP (Finite-Difference Code for Elastic-Plastic Analysis). Tamin et al. (1994) proposed a four-phase CCM for The phases included the fiber, fiber coating, interphase region (reaction zone), and matrix. The study focused on thermal stresses upon cool down and concluded that the carbon coating was favorable to the coating because it induced compressive stresses in the brittle interphase

19 region. Neu et al. (1994) included damage within the realm of the CCM through modulus degradation. Due to the geometric limitations of the model, fiber/matrix debonding was the only damage mechanism considered. Kroupa et al. (1996) compared various micromechanical models used to characterize MMC’s. Although damage was not considered in the analysis, the study sheds light on the validity of physical assumptions. The following models were reviewed for a composite laminate: 1) VISCOPLY (Thermoviscoplastic Response of Composites) based on the vanishing fiber diameter model (VFD); 2) FIDEP (Finite-Difference Code for Elastic-Plastic Analysis) multiple concentric cylinder model; 3) METCAN (Metal Matrix Composite Analyzer) laminated plate theory with nonlinear constitutive models; 4) LISOL (Laminate Inelastic Solver) similar to Aboudi’s method of cells; 5) ELAM (Elementary Analysis Method) concentric cylinder model; and 6) FEM (Finite Element Method). The two methods which best match the experimental stress-strain behavior were FEM and LISOL. Although the finite element method was more accurate, the authors recommended LISOL on the basis of numerical efficiency. In conjunction with the concentric cylinder model, Neu and Nicholas (1993) proposed a damage model incorporating the effects of matrix fatigue, surface-initiated environmental damage, and fiber-dominated damage for Environmental damage was assumed to correlate well with the diffusion of oxygen in both the matrix and the fiber. Consequently, Arhennius-type laws governed the evolution of damage. Given the initial stress state, S-N curves were generated for inphase, out-of-phase, and isothermal thermomechanical fatigue. The law governing cyclic failure is based on constituent life fractions. Nicholas (1995) used a linear life fraction model to predict cyclic life in composite. In addition, Nicholas presented a nonlinear life fraction model which varies with the square root of time-dependent damage. Although more computationally intensive, Kroupa et al. (1996) concluded that FEM analysis was the most accurate method. In addition, the finite element method provides an explicit framework for thermal gradients, material inelasticity, crack propagation, and environmental degradation. Few studies have focused on modeling the effects of environmental degradation on metal matrix composites. More specifically, with regard to titanium matrix composites, fewer studies have concentrated on the effects of oxidation. Wittig and Alien (1994) coupled a diffusion model with the finite element method to simulate the formation of a brittle oxide layer during cyclic fatigue of metal matrix composite. Hurtado and Alien (1994) used a similar model to predict oxide layer thickness and its effect on damage evolution in and Alien et al. (1996) modeled oxidation layer cracking in Comparing an oxidized and unoxidized specimen, fiber/matrix debonding occurred sooner in the oxidized specimen. Other researchers concentrated on the formulation for predicting the diffusion of oxygen and the formation of an oxide layer. Lagoudas et al. (1995) modelled the oxidation by assuming Fickian diffusion and a jump discontinuity in the oxygen concentration at the oxide interface. Xu et al. (1995) predicted the diffusion of oxygen through Fick’s second law and employed a fixed grid direct finite element method (FGDFEM) to located the oxide/metal interface. It is apparent from the recent literature that while considerable research has been undertaken to model life in titanium matrix composites, there is as yet no general consensus regarding the cause of premature failure due to environmental degradation.

20 2. Solution Approach

The research proposed herein focuses on the effects of oxidation on the life of [0]4 metal matrix composite. Thermal residual stresses, matrix inelasticity, crack propagation, and environmental degradation are all taken into account in the analysis. However, instead of choosing a micromechanical model and a corresponding law governing cyclic failure, these phenomena are included explicitly through the finite element method in an attempt to predict the life of the composite based on critical fiber stress. In order to employ this model some experimental guidance is necessary. Therefore, a limited experimental research program has been undertaken by the authors, and this research is first summarized insofar as it impacts on the model development. 2.1 EXPERIMENTAL RESEARCH

The following issues have been studied experimentally: 1) the effects of oxidation on the microstructure of 2) the initiation of surface cracks and typical crack spacings in oxidized specimens; and 3) stress-strain ratchetting and energy. These experimental studies were carried out by thermomechanically testing four ply

unidirectional specimens and then performing destructive examinations of the specimens by various techniques including optical and scanning electron microscopy, as well as an electron microprobe. As shown in Figure 1, when is heated for 24 hours at 700°C very little surface oxidation is observed. This led the research community to adoption of this material over less oxidation resistant alloys such as which oxidizes very rapidly (Allen et al., 1995). However, closer inspection of the figure indicates that a very fine grained alpha phase has formed along the beta grain boundaries, and the volume fraction of alpha phase is a function of the distance from the free surface, thus indicating that oxygen, which serves as an alpha stabilizer, has diffused into the interior of the composite. Wallace et al. (1992; 1993) have shown that this development of alpha phase inherently embrittles the matrix material. In order to examine the effects of environmental degradation, specimens were fatigued in air at One specimen was soaked in air at for 24 hours, while the other was not. Figure 2 illustrates the difference in life. In addition, scanning electron microscopy illustrates the extremely brittle nature of the fracture surface for the material tested in air, as shown in Figure 3. The observed fracture patterns such as mist and velocity hackles are indicative of a brittle rather than a ductile material. These observations provide justification for modeling the embrittled region as elastic rather than viscoplastic. By contrast, the specimen tested ni inert gas displays no brittle surface fracture, as shown in Figure 4. As can be seen from the figure, the crack surface is everywhere ductile. Much of the model development proposed below is based on the above observations.

21

22

23 2.2 MODEL DEVELOPMENT

This section details the development of a life prediction model. The authors have chosen to model the metal matrix composite using the finite element method coupled with models for matrix inelasticity and damage. Figure 5 illustrates the metal matrix composite. To simplify the analysis and save computational time, a repetitive unit cell is extracted. The selected unit cell is shown in Figure 6. Because the finite element method is well understood, this section will focus on the material inelasticity and crack growth models. The following field equations are solved within the finite element algorithm. If there are no body forces and inertial effects, the conservation of linear momentum is satisfied if:

Additionally, the stress tensor, is assumed to be symmetric. Assuming the materials of interest undergo small strains, higher order terms can be neglected. The linearized form of the strain tensor, is:

where is the displacement vector. Due to the nature of the definition, the strain tensor is also symmetric.

In addition to conservation laws and kinematic constraints, constitutive equations are needed to characterize the constituents. Both thermoelastic and thermoviscoplastic material models are needed in the analysis. In addition, constitutive relations must be developed for the cohesive zone. The mechanical constitution for a linear thermoelastic material is:

where

is the elastic modulus and

is the thermal strain tensor given by:

where is the tensor governing the coefficient of thermal expansion and T is the temperature. The above thermomechanical constitutive model is utilized in the fiber and the embrittled surface layer of the oxidized metal matrix. A more complicated formulation is required to model plasticity and rate dependence, with the stress-strain relation given by:

24

25

where is the inelastic strain tensor. The inelastic strain tensor and additional internal variables are assumed to be governed by internal variable evolution laws of the form:

where

represents a generic list of internal variables, and n is the number of internal

state variables. Bodner’s unified anisotropic viscoplasticity model (Chan et al., 1988) has been chosen to model the evolution of the inelastic strain. In addition to developing continuum constitutive models, the material behavior for the cohesive zones must also be developed. The normal and tangential tractions and

are defined by (Tvergaard, 1990):

where and are the normal and tangential crack opening displacements, respectively, and is a measure of bond strength. In addition, 1 is the normalized quantity coupling normal and tangential behavior:

and are length scales associated with debonding and and are material properties relating shear to normal strength. Furthermore, it is assumed that the cohesive zone is fully debonded when Both tractions and displacements are specified on the boundary of the unit cell.

26

Figure 6 illustrates the mixed boundary conditions imposed on the unit cell. The above equations have been implemented in the three dimensional finite element code SADISTIC (Structural Analysis of Damage Induced Stresses in Thermolnelastic Composites) (Allen et al., 1994). The interested reader is referred to this research for further details regarding the finite element algorithm. 3. Model Results

Results are presented herein for monotonic and cyclic loading of at with and without the effects of damage and surface embrittlement. The evolutions of inelastic strain and surface cracking are predicted, as well as their effects on the fiber stress. The fiber failure stress is well documented at 3840 MPa (Gambone and Wawner, 1994), and as will be shown herein, fiber fracture is a good indication of composite life. Figure 7 contrasts the elastic and viscoplastic behavior at The surface crack initiates in both analyses around 800 MPa. In the elastic case, the crack propagates sooner and farther. In fact, the crack partially bridges the fiber. In the viscoplastic case, the crack is arrested and begins to bridge the fiber upon failure. In both cases, fiber failure is preceded by fiber matrix/debonding. Results were also predicted for an unoxidized and oxidized specimen. Notice the global effect of the increased modulus in the embrittled region. The unoxidized case is simulated by characterizing the entire matrix as viscoplastic and giving the cohesive zones a common strength. This strength corresponds to the ductile interior, Layer 5, in Figure 8. Cyclic fatigue was predicted by assuming that the surface was ductile and viscoplastic in the undegraded case, and that the surface was elastic and fracture toughness was decreased in accordance with the cohesive zone parameters shown in Figure 8. The homogenized stress vs. strain curves for the undegraded and degraded predictions are given in Figures 9 and 10. Only fourteen cycles were predicted due to limits on computational resources. Fourteen cycles corresponds to 50 CPU hours on a Silicon Graphics Power Challenge XL Supercomputer (running in serial). As can be seen from the results, no surface crack was predicted in the undegraded composite, whereas significant surface cracking was predicted in the degraded composite on the first cycle. These predictions are in general agreement with the above experimental observations. Through viscoplasticity, both the oxidized and unoxidized predictions shed load to the fiber. However, the initiation and propagation of a surface crack due to environmental degradation significantly increases the fiber stress in the oxidized case, as shown in Figure 11. Although surface cracking does not occur after the first cycle, the fiber stress for the oxidized case is consistently higher than for the unoxidized case. 4. Conclusion

Model results herein suggest that experimentally observed matrix inelasticity and surface cracking caused by formation of embrittled alpha grains along the boundaries of the beta grains sheds load to the SCS-6 fiber which causes premature failure of the degraded composite. Although only a limited number of cycles have been modelled, the trends in the analysis suggest that the effects of environmental degradation on life may be modelled by the methodology proposed herein. The veracity of this

27

supposition awaits further investigation. 5. Acknowledgement

The authors gratefully acknowledge the support provided for this research by The United States Air Force Office of Scientific Research under grant no. F49620-94-10341. We also wish to thank Lockheed-Martin Aerospace Corporation - Fort Worth Division for providing the specimens used in this research, and to Dr. Golam Newaz for performing the fatigue test in Argon. 6. References Allen, D. H., Jones, R. H., and Boyd, J. G., 1994, “Micromechanical Analysis of a Continuous Fiber Metal Matrix Composite Including the Effects of Matrix Viscoplasticity and Evolving Damage,” Journal of Mechanics and Physics of Solids, Vol. 42, No. 3, pp. 502-529. Allen, D. H., Eggleston, M. R., and Hurtado, L. D., 1995, “Recent Research on Damage Development in SiC/Ti Continuous Fiber Metal Matrix Composites,” to appear in Fracture of Composites, E. A. Armanios, E., in Key Engineering Materials, Trans Tech Publications, 1995.

Allen, D.H., Foulk, J.W., Helms, K.L.E., 1996, “A Model for Predicting the Effect of Environmental Degradation on Damage Evolution in Metal Matrix Composites,” to appear in the proceedings on Applications of Continuum Damage Mechanics to Fatigue and Fracture, Orlando, FL. Chan, K. S., Bodner, S. R., and Lindholm, U. S., 1988, “Phenomenological Modeling of Hardening and Thermal Recovery in Metals,” Journal of Engineering Materials and Technology, pp. 1-8.

Coker, D., Ashbaugh, N. E., and Nicholas, T., 1993, “Analysis of the Thermomechanical Behavior of [0] and [0/90] SCS-6/Timetal®21S Composites,” in the proceedings of the ASME Winter Annual Meeting. Foulk, J.W., 1994, “Isothermal Behavior of Metal Matrix Composite at Undergraduate Summer Research Programs, Texas A&M University, pp. 256-264. Gambone, M.L. and Wawner, F.E., 1994, “The Effect of Elevated Temperature Exposure of Composites

on the Strength Distribution of Reinforcing Fibers,” Intermetallic Matrix Composites III, J.A. Graves, R.R.

Bowman, and J.J. Lewandowski, eds., MRS, Pittsburgh, PA, pp. 111-118.

Hurtado, L. D., and Allen, D. H., 1994, “Effect of Oxidation on Damage Evolution in Titanium Matrix MMC’s”, in the proceedings of the Symposium on Inelasticity and Micromechanics in Metal Matrix Composites, Twelfth U.S. National Congress of Applied Mechanics, Seattle, WA. Kroupa, J. L., Neu, R. W., Nicholas, T., Coker, D., Robertson, D. D., Mall, S., 1996, “A Comparison of Analysis Tools for Predicting the Inelastic Cyclic Response of Cross-Ply Titanium Matrix Composites,” Life Prediction Methodology for Titanium Matrix Composites, ASTM STP 1253, W.S. Johnson, J.M. Larsen, and B.N. Cox, Eds., ASTM, pp. 297-327.

Lagoudas, D. C., Ma, X., Miller, D. A., and Allen, D. H., 1995, “Modelling of Oxidation in Metal Matrix Composites,” International Journal of Engineering Science, Vol. 33, pp. 252-263. Majumdar, B.S., and Newaz, G.M., 1991, “Thermomechanical Fatigue of a Quasi-Isotropic Metal Matrix Composite,” Composite Materials: Fatigue and Fracture (Third Volume), ASTM STP I I , T.K. O'Brien, Ed., pp. 732-752, American Society for Testing and Materials, Philadelphia, PA.

Neu, R. W., and Nicholas, T., 1993, “Thermomechanical Fatigue of SCS-6/Timetal 21S Under Out-ofPhase Loading,” in the proceedings of the 1993 ASME Winter Annual Meeting, New Orleans, LA.

28 Neu, R.W., Coker, D., Nicholas, T., 1994, “Cyclic Behavior of Unidirectional And Cross-Ply Titanium Matrix Composites,” International Journal of Plasticity, Vol. 12, No. 3, pp. 361-385. Neu, R. W., 1995, “Thermomechanical Fatigue Damage Mechanism Maps for Metal Matrix Composites,” Thermo-Mechanical Fatigue Behavior of Materials: 2nd Volume, ASTM STP 1263, M. J. Verrilli and M. G.

Castelli, Eds., American Society for Testing and Materials, Philadelphia, PA.

Nicholas, T., 1995. “Fatigue Life Prediction in Titanium Matrix Composites,” Journal of Engineering Materials and Technology, 117:440-447.

Tamin, M. N., Zheng, D., and Ghonem, H., 1994, “Time-Dependent Behavior of Continuous-FiberReinforced Metal Matrix Composites: Composites Technology and Research.

Modelling and Applications,” submitted to the Journal of

Tvergaard, V., 1990, “Effect of Fiber Debonding in A Whisker-Reinforce Metal,” Materials Science & Engineering A: Structural Materials: Properties, Microstructure, and Processing, Vol. A125, No. 2, p. 203-213. Wallace, T. A., Wiedemann, K. E., and Clark, R. K., 1992, “Oxidation Characteristics of Beta-21S in Air in the Temperature Range 600 to 800°C," National Aeronautics and Space Administration, Langley

Research Center, Langley, VA.

Wallace, T.A., Bird, R.K., and Wiedemann, K.E., 1993, “The Effect of Oxidation on the Mechanical Properties of Beta-21S,” Beta Titanium Alloys in the 1990's, Eylon, D., Boyer, R.R., and Koss, D.A., eds., The Minerals, Metals, & Materials Society, pp. 115-126. Wittig, L. A., and Allen, D. H., 1994, “Modeling the Effect of Oxidation on Damage in SiC/Ti-15-3 Metal Matrix Composites,” Journal of Engineering Materials and Technology, Vol. 116, pp. 421-427. Xu, S., Lagoudas, D. C., and Allen, D. H., 1995, “Effects of Oxidation and Damage on the Mechanical Response of Metal Matrix Composites,” in the proceedings of the ASME Winter Annual Meeting, New

Orleans, LA.

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ON THE MICROMECHANICAL MODELLING OF THE VISCOPLASTIC BEHAVIOR OF SINGLE CRYSTAL SUPERALLOYS

J.L. CHABOCHE, S. LHUILLIER, D. NOUAILHAS O.N.E.R.A. 29 av. de la Division Leclerc, 92320 Châtillon, France

Abstract

A micro-macro modeling of the single crystal viscoplasticity behavior is discussed, based on a composite approach and continuum mechanics based constitutive equations at the level of the precipitate and matrix It is shown how such an approach cannot simulate correctly both the tensile and shear loading conditions. An attempt is done that reconciliate the two loading conditions, by introducing a volumetric term in the viscoplastic strain in order to redistribute the hydrostatic pressure in the matrix

channels. 1. Introduction

The turbine blade inelastic analysis is generally performed by using macroscopic

constitution equations that have been developed to describe the cyclic viscoplastic behaviour of single crystal superalloys. These models, based on a phenomenological formulation or on a crystallographic one, have demonstrated their ability to predict quite well the effects of the initial and plastic strain induced anisotropies (cubic symmetry). However, assuming the material as homogeneous, they cannot take into influence of the evolution of microstructure that can be observed at high with the appearance of the so-called rafting process (Biermann et Experiments performed on rafted microstructures (Pessah-Simonetti et al.,

account the temperature al., 1996). 1993) have

demonstrated a significant influence on the macroscopic behaviour, compared to the initial cuboidal microstructure. This is the reason why we are currently developing a micro-macro analysis based on a "composite" approach.

In the micro-macro analysis of the mechanical behaviour, especially for composite materials, we generally use one of the three following approaches : (i) The periodic homogenization procedure, that allows an accurate description of the local events provided the localization step can be performed by the finite element method. This method is clearly well suited for a quasi-periodic microstructure. 33 Y.A. Bahei-El-Din and G.J. Dvorak (eds.),

IUTAM Symposium on Transformation Problems in Composite and Active Materials, 33–44. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

34 (ii) The self consistent methods, and their numerous variants, that describe the presence of neighbouring reinforcements by the (unknown) macroscopic behaviour. They generally use constant fields (stress, strains) in each material phase. Such methods are indicated for random microstructures. (iii) The transformation field analysis, fairly developed by Dvorak and Benveniste

(1992), that proceeds as a bridge between the two previous methods : subphases can be considered with uniform inelastic or thermal strains (eigenstrains).

In the past, we have exploited the method (iii) with simplifying assumptions, in order to extract quasi-analytically the composite behaviour from the phase behaviours. Acceptable results were obtained for a SiC/Al composite with a low volume fraction of

whiskers (16%) (Chaboche et al., 1994). However, we know the limitations of the approach for larger volume fractions due to the over simplifications (uniform plastic strain in the matrix) that lead to a much too stiff inelastic behaviour. In the self consistent methods, it is well recognized to use a tangent stiffness approach (or a secant stiffness), in order to describe a plastic accomodation possibility (see Zaoui and Raphanel (1993) and Suquet (1996) for example). In the present paper, we w i l l demonstrate that the "composite approach" itself can be deficient (for large volume fractions), even for the more sophisticated way (i), the periodic homogenization, clearly an acceptable assumption for the considered microstructures.

The calculations presented here are performed on the i n i t i a l cuboidal microstructure, in order to set up the material constants for and phases. The first section describes the representative unit cell that has been selected, together with the assumptions made for the mechanical behavior of and phases. In the following section, results of the finite element calculations are discussed, for tensile and shear loadings. Inconsistencies between the two types of loadings are pointed out with these calculations. The last section presents the attempts made to improve the modelling, by modifying the geometry of the unit cell and by modifying the viscoplastic constitutive equations of the matrix.

35 2. Representative unit cell The study is made based on experimental results (Nouailhas, 1994) obtained on the MC2 single crystal at 1050°C. After an optimal thermal treatment, cuboidal precipitates are homogeneously distributed in the matrix (Caron and Khan, 1989), with a very high volume fraction of about 70%. Due to this regular microstructure, we have placed our approach in the framework of the homogenization of periodic media. The finite element analysis is performed in 3D. The unit cell is composed of a precipitate, surrounded by a thin layer of matrix, such as the volume fraction is 68%. At 1050°C, it is well known for this class of superalloy single crystals that the dislocations are not shearing the precipitates (Brien et al., 1996). The behavior of this phase is then modelled by a cubic elasticity law. Concerning the matrix, its behavior is supposed to be anisotropic viscoplastic, and we have introduced at this level the constitutive equations developed to predict the macroscopic cyclic behavior of single crystal superalloys. The analysis is mainly performed with a phenomenological model based on invariant formulation (Nouailhas and Culié, 1991), but we have systematically compared the predictions obtained with those delivered by a crystallographic model (Nouailhas et al., 1995). Figure 1 shows the unit cell that has been selected for the present analysis. It is composed of 20 nodes quadratic 3D isoparametric elements, using 27 integration points. Due to the high CPU time required for a calculation on a complete cell, the number of element must be limited. A good convergence of the calculation is obtained with only two rows of element in the matrix channels. The precipitate is schematically represented by a cube in the first calculations. The shape of the precipitate will be discussed later again.

36 3. Finite element calculations Two types of calculations have been systematically performed. The first one corresponds to a tensile loading in a crystallographic orientation. The second one is a shear loading. 3.1 TENSILE LOADING

In that situation, due to the symmetries of the problem, only of the unit cell is computed. Boundary conditions are such that opposite faces remain planars and parallels. An uniform displacement is imposed on one face. No misfit stresses are introduced in these calculations, preliminary studies having shown that the misfit stresses have no influence on the subsequent macroscopic curve (Nouailhas and Cailletaud, 1996). The material constants are determined by solving an inverse problem, in order to fit the best the experimental curve. Figures 2 and 3 show the result of this calculation. The agreement could be improved, but we consider this result as good enough for the purpose of the present study. At the end of the tensile loading, the stress state is mostly triaxial in the matrix channels, and the maximum stresses are obtained in the middle of the matrix channel perpendicular to the loading direction (Fig. 2-a). Low shear stress components are localized in the first row of elements that surround the precipitate. A very large plastic flow is produced in the matrix. The axial component is in tension in all the matrix, except at the intersection between horizontal and vertical channels (horizontal channels are perpendiculars to (he loading axis ) where it as a very high negative value. An important strain gradient is developed from the edges of the precipitate toward the center of cross-channels (Fig. 2-b).

37 Material constants fitted from this calculation, lead to a very soft viscoplastic law for the matrix. When using a crystallographic model for the matrix, similar stress and strain distributions are obtained, and only a small difference is observed on the overall behavior (Fig. 3). 3.2 SHEAR LOADING

For that loading condition, the complete cell must be computed, with the boundary conditions such as opposite faces deform but remain parallels. The imposed displacements correspond to a pure macroscopic shear loading. Since it is not possible to obtain such a curve experimentally (torsion test produces heterogeneous deformation (Nouailhas and Cailletaud, 1995)), we have used a numerical experiment delivered by a macroscopical model identified from cyclic tests on MC2 alloy (again, phenomenological and crystallographic models deliver the same curves). The validity of these macroscopic models have been checked by finite element analysis of tensiontorsion tests on tubular specimens (Nouailhas and Culié, 1991 ; Nouailhas and Cailletaud, 1995). The F.E. calculation, performed with the material constants fitted from the tensile loading totally disagrees with the "experiment" (Fig. 4). One can see, with the stress and strain contours (Fig. 5-a) that there is no important strain gradient. The overall behavior more or less corresponds to the behavior of the matrix alone. The use for the matrix law of the crystallographic model leads to the same result (Fig. 4).

For the selected configuration, it is clear that the material constants can not be set from the axial loading. The presence of a high volume fraction of elastic precipitate, together with the boundary conditions lead in that case to concentrate the deformation

38 at the intersection of the matrix channels, by Poisson's effect. Schematically, the system works as an assemblage of elementary models "in parallel". In order to reduce this overstiff behaviour, we need to select a very soft matrix behaviour which, at the same time, is inconsistent with the shear loading condition, the Poisson's effect being not present in that case ( i t works like an assemblage "in series").

4. Improved solutions

The above inconsistency between the tensile and shear conditions appears to be mainly

due to the over constraining situation of the tensile loading in the cubic axes : in that case, the high stress triaxiality in the matrix channel cannot be redistributed and the plastic strain takes place only in the very narrow region near the corner of the

precipitate. In order to improve the stress redistribution, two types of modifications have been studied. 4.1 GEOMETRICAL MODIFICATIONS There are three possible variations : ( i ) The volume fraction of precipitates itself. It influences the strength, with the same difference in the two considered conditions, but is

not really a free parameter, as being measured quite precisely, at least for the initial state, ( i i ) The assumption of regular matrix channels may not be representative enough of the reality, but additional F.E. analysis, introducing disalignements of precipitates have not produced modifications in the sence of reducing the overall strength (Nouailhas and Lhuillier, 1997). ( i i i ) The last possibility was to consider a rounded shape for the precipitate corner. This is the only one giving significant differences.

39

Additional calculations have been performed by using a unit cell made of one rounded precipitate, with a curvature radius of which is consistent with observed microstructures. The volume fraction is still 68%. The macroscopic tensile curve shows a less resistant behavior than the one obtained with sharp corners (Fig. 6). A new inverse problem has then been solved with the cell with rounded precipitate, leading to the macroscopic dashed curve of Fig. 6. This new identification nevertheless does not allow to solve the problem observed for shear loading. Again, the inconsistency between the two types of loadings is exhibited. If we observed the strain distribution in tension, we see that the strain localization at the intersection of the matrix channels still occurs, with lower magnitudes however. For the stress distribution, no real changes are observed with regard to Fig. 2.

4.2 A MODIFIED CONSTITUTIVE EQUATION FOR THE MATRIX In order to redistribute the important hydrostatic pressure, the solution we propose here is to take into account the effect of the hydrostatic pressure in the matrix yield criterion, which leads, via the classical normality assumption to a plastically compressible matrix. The set of constitutive equations expressed in the crystallographic axes is given in

Table I . This model has been detailed elsewhere, in its incompressible version (Nouailhas and Culié, 1991). In the present version, instead of writing the deviatoric stress tensor in the yield function equation, as classically done, we have introduced the stress tensor, without any other change. This model has been applied with a rounded precipitate. Since we have no guide to set the influence of the hydrostatic pressure, we have first solved an inverse

40 problem from the shear test, with the complete cell. In that case, there is almost no influence of the hydrostatic pressure. Then, the material constant (eqn. 2) is fixed from the tensile test. Results of the identification are given in Fig. 7, where we have

also plotted the tensile curve obtained with the identification from shear test without introducing the influence of the hydrostatic pressure. Material constants are given in Table 2. Fig. 7 clearly demonstrates that the present modification allows to predict both types of loadings, tensile and shear, with a unique set of material constants.

41

The volume change induced by the modification is highly non linear with the total strain. At 2% it is of the same order as the one due to elasticity. We consider this variation as being acceptable.

Figure 5-b shows the plastic strain distribution in the matrix channels, and we observe now an almost uniform deformation in the matrix channel perpendicular to the loading direction, that is much more consistent with the distribution of the dislocations experimentally observed (Caron and Khan, 1989). The computations a posteriori of a tensile loading on the matrix alone (Fig. 9) show that the last identification that has been performed is the most realistic compared with experimental results of Jouiad (1996) in the crystallographic orientation . These experiments indicate stresses between 120 and 160 MPa during plastic flow at moderate strain lower than 1%, which is very consistent with the behaviour shown by the matrix for the compressible law (fig. 8). In that case, due to the low level of stresses, the model predicts absolutely no influence of the hydrostatic pressure for the matrix alone.

42

Predictions performed for cyclic tests on MC2 single crystal superalloy at two different strain rates show also a very good agreement between the F.E. model and the experiments (Fig. 9). In particular, it was not possible to predict properly the strain rate influence with the two previous identifications, because of the too small viscous stresses.

43 5. Concluding remarks

The present micro-macro analysis of the

single crystal behaviour via a "composite

approach" has shown several new results, that can be summarized as follows : - There is actually a clear deficiency of the classical composite analysis when the matrix

inelastic behaviour is assumed incompressible. In the tensile loading in the cubic axes the boundary conditions, together with the 68% of clastic hard phase, lead to concentrate the plastic deformation in a narrow region. - The introduction of a more realistic rounded precipitate did not solve that problem completely. - The solution we propose is to introduce a plastic compressibility of the matrix via the influence of the hydrostatic pressure in the yield criterion. This modification of the matrix constitutive equations has only a small influence on the volume change but allows the hydrostatic stress redistribution and a wider plastified region. - This solution was able to reconciliatc the two predictions f o r the overall tensile and shear loadings. Moreover, it leads to an identified matrix behaviour that meets approximately the experimental results obtained on pure phase.

- Another way to redistribute the hydrostatic pressure could be to allow the plastic deformation of the phase as in Espié (1996). However, at high temperature, the shearing of the precipitates is never observed. - In this application, the d i f f i c u l t y is due to the high volume fraction of the precipitates, that leads to the confinement of the matrix on a very small distance. In that situation, the dislocation motion is mainly governed by the presence of the precipitate, and it docs not exist actually constitutive modelling taking into account the interaction between dislocations and precipitates.

- The modified model we propose in this work is a compromise that will allow to

compute different geometries of precipitates (rafts, rods...) and to quantify the influence of several parameters (dimension of the in the 3 directions, matrix channels thickness...). The final aim being to calibrate, from these calculations, macroscopic

models accounting for rafted microstructures. REFERENCES

Biermann. H., von Grossmann, B., Schneider. T., Feng, H. and Mughrabi, H. (1996) Investigation of the morphology and internal stresses in a monocryslalline turbine blade alter service : determination of the local thermal and mechanical loads, Proc. of Superalloys 96, eds. R.D. Kissinger et al., The Minerals. Metals &

Materials Society, pp. 201. Brien, V., Décamps, B., and Morton, J. (1996) Microstructural behaviour of a superalloy under repeated or alternate L.C.F. at high temperature, Proc. of S u p e r a l l o y s 96, eds. R.D. Kissinger el al., The Minerals. Metals & Materials Society, pp. 113.

44 Caron, P., and Khan, T. (1989) Development of a new nickel based single crystal turbine blade alloy for

very high temperatures. Proc. 1st European Conference on Advanced Materials and Processes EUROMAT'89, Aachen, FRG, eds. H.E. Exner and V. Schumacher, Vol. 1. Chaboche. J.L., Kruch, S., El Mayas. N. (1994) Thermo-elasto-viscoplastic constitutive equations for metal matrix composites, C.R. Acad. Sc. Paris, t. 319, Serie I I , pp. 971-977 Dvorak, G.J., Benveniste, Y. (1992) On transformation strains and uniform fields in multiphase elastic media, Proc. R. Soc. London, A, Vol. 437, pp. 291-310. Espié, L. (1996) Etude expérimentale et modélisation numérique du comportement mécanique de monocristaux de superalliages. Doctoral thesis, ENSMP, Paris. Jouiad, M. (1996) Caractéristiques mécaniques et étal d'ordre de la phase du superalliage base nickel. Doctoral thesis. CEMES/CNRS. Toulouse. Nouailhas, D. (Sept. 1994) Development of microstructure based viscoplastic models for an advanced design of single crystal hot section components, Brite Euram project BRE2-CT92-0176, second annual report. Nouailhas. D., and Cailletaud, G, (1995) Tension-torsion behaviour of single-crystal superalloys : Experiment and finite element analysis, Int. J. of Pluxticity, Vol. 1 1 , pp. 451 Nouailhas, D., and Caillelaud (1996) G., Finite element analysis of the mechanical behaviour of two-phase single crystal superalloys, Scripta Materialia, Vol. 34. pp. 565. Nouailhas. D., and Culié, J.P. (1991) Development and application of a model for single-crystal superalloy, in High Temperature Constitutive Modeling - Theory and Application. eds. A.D. Freed and K.P. Walker, ASME. MD-26, AMD- 1 2 1 , New-York, pp. 151. Nouailhas. D., Culié, J.P., Cailletaud, G., and Méric. L. (1995) Finite element analysis of the stress-strain behaviour of single-crystal tubes. Eur. J. Mech., A/Solids, Vol. 14, pp. 137. Nouailhas. D., and Lhuillier S. (1997) On the micro-macro modelling of single-crystal behaviour. submitted to Computational Materials Sciences. COMMAT 436, Vol. 10/3. Pessah-Simonetti. M., Caron, P., and Khan. T. (1993) Effect of a long term prior aging on the tensile behaviour of a high-performance single crystal superalloy, Journal de Physique IV, Vol. 3, pp. 347-350. Suquet, P (1996) Overall properties of nonlinear composites : Remarks on secant and incremental formulations, in A. Pineau, A Zaoui, eds, Plasticity and Damage of Multiphase Materials, Kluwer Acad. Pub, pp. 149-156. Zaoui, A., Raphanel, J.L. (1993) On the nature of the intergranular accomodation in the modelling of elastoviscoplastic behaviour of polycrislalline aggregates, Proc. MECAMAT'91, Teodosiu. Raphanel and Sidoroff eds, Balkema, Rotterdam.

MICROMECHANICAL ANALYSIS OF INELASTIC FIBROUS LAMINATES

Y.A. BAHEI-EL-DIN, I.A. IBRAHIM and A.G. BOTROUS Structural Engineering Department Cairo University Giza, Egypt

Abstract

Analysis of the inelastic behavior of fibrous composite laminates with detailed representation of the lamina microgeometry is described. Loading is limited to uniform in-plane stresses and out-of-plane normal stress, and to uniform changes in temperature. The objective is to predict the overall strains and the local fields in the constituents. This has been achieved by analysis of the laminates on two interacting structural scales, a microscopic scale which models the individual plies, and a macroscopic scale which provides the plies loading path. Macromechanical analysis of the laminate was conducted with the transformation field method, while micromechanical analysis of each ply was performed with the finite element method. Implementation of this methodology for laminates with a viscoplastic matrix is described. 1. Introduction

Analytical prediction of the overall behavior of composite materials involves modeling of complex geometrical, mechanical and physical properties. Significant progress has been made over the past two decades in modeling these properties for particulate as well as fibrous composites. In particular, two classes of micromechanical models can be found for unidirectionally reinforced fibrous composite materials with either elastic

or inelastic phase properties, averaging models and periodic array, or unit cell, models. Reviews of these models are offered by Bahei-El-Din and Dvorak (1989) and Dvorak (1991). While averaging models provide governing equations of the local fields and overall response in closed form, periodic array models utilize the finite element method for numerical evaluation of these quantities (Dvorak and Teply, 1985; Brockenbrough et al., 1991). The geometrical details captured in the latter approach, however, deliver more realistic predictions compared to the averaging models, particularly when local

stress-dependent inelastic flow or damage is present. Inelastic analysis of the unit cell with the finite element method can be performed using either the standard

displacement formulation which utilizes instantaneous properties of the phases (Bahei-El-Din, 1996; Bigelow, 1992), or the transformation field method which 45 Y.A. Bahei-El-Din and GJ. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 45–60. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

46

models the composite aggregate as an elastic medium subjected to local instantaneous inelastic strains (Dvorak et al., 1994). In actual applications, fibrous composites are utilized in a laminated construction in which several unidirectionally reinforced plies are bonded together to achieve the required product shape and properties. Analysis of complex shapes with either elastic or inelastic properties can be accomplished using a finite element approach in which constitutive behavior of the individual fibrous plies is derived from an averaging micromechanical model (Bahei-El-Din, 1996). Finite element modeling of laminated structures considering a more detailed representation of the microstructure, on the other hand, is difficult even for simple laminated structures such as symmetric plates, particularly when the layup deviates from that of a crossply. The difficulty lies mainly in generating the finite element mesh for the actual microstructure and layup in which the fibers of the individual plies intersect at various angles. To make progress, unit cell solutions developed for unidirectional plies are combined with the governing equations of the laminated plate found either from the classical lamination theory (Dvorak and Bahei-El-Din, 1995), or from a finite element model of the laminate (Bahei-El-Din, 1996). These solutions, however, involve evaluation of certain elastic transformation

factors which must be regenerated if either the phase properties change with temperature, or the distribution of local elastic fields changes due to damage. In this paper, a new approach to inelastic analysis of laminated plates with microstructural details is presented. The proposed method uses a standard nonlinear finite element procedure for the unit cell of unidirectional fibrous composites, while the loading path for the individual plies is found from a transformation field analysis of the laminated plate. In this way, local phenomena, whether physical or geometrical, related to the plies are incorporated in the finite element solution, while the corresponding overall ply deformation is accounted for in the laminate analysis. For completeness, the paper first summarizes the rate equations for the overall lamina stresses derived from macroscopic analysis of the laminate using the transformation field method. Next, micromechanical analysis of the individual plies is described, and the rate equations for the local fields and the overall transformation fields are provided. Finally, implementation of the two sets of equations considering viscoplastic phases is presented with an application. 2. Transformation Field Analysis of Laminated Plates

The behavior of a symmetric laminated plate consisting of 2N fully bonded thin elastic plies, Fig. 1, under thermomechanical loads is considered. Referred to a Cartesian coordinate system, in which the coincides with the midplane of the laminate, in-plane membrane forces and the corresponding uniform stresses, and are applied, together with a uniform normal stress, in the thickness direction Let lists the in-plane stresses applied to the laminate, and lists the corresponding laminate strains. The latter are caused by

47

the applied stresses,

and

in addition to the transformation strains generated in

the individual plies, such as thermal and inelastic strains, which are not recovered by

removal of the mechanical load. Assuming additive decomposition of the various effects (Dvorak, 1991), and adopting the notation of Bahei-El-Din (1992), the time

rates of the laminate in-plane stresses and strains are written as

where

and

are the in-plane transformation stress and strain, respectively, and L

and M are the elastic stiffness and compliance matrices for in-plane loading. The k vector lists the in-plane stresses caused in the laminate by a unit out-of-plane normal stress, in the absence of both the total in-plane strains and the transformation stress whereas n is the elastic compliance vector associated with the out-of-plane normal stress, On the other hand, the transformation strain represents the total

strain that remains in the laminate after complete unloading to zero stress, and the transformation stress

is seen to represent the total stress caused in a fully constrained

laminate by the transformation strain

Equations (1) and (2) provide the relations,

48 In analogy with (1) and (2), the uniform in-plane stress and strain rates of a ply in the local coordinate system Fig. 1, can be written as,

where

loading,

and

and

are the elastic stiffness and compliance matrices for in-plane

are the ply in-plane stress and strain caused by a unit out-

of-plane normal stress, and and are the ply in-plane transformation stress and strain. For a transversely isotropic ply with overall elastic longitudinal, and transverse moduli, and Poisson's ratios, and and longitudinal shear modulus the matrices found in eqs. (4) and (5) are given by (Bahei-El-Din, 1992; Dvorak and Bahei-El-Din, 1995),

where

n and pare Hill’s moduli (Hill, 1964), and

When expressed in the overall coordinate system

written as (Bahei-El-Din, 1992)

where

eqs. (4) and (5) are

49

and

is the angle between the local and the overall Fig. 1. The ply stresses in a laminate loaded by overall in-plane stresses out-of-plane normal stress and ply transformation stresses introduced by certain prescribed in-plane transformation strains can now be determined using the transformation field analysis method (Dvorak, 1992). The laminate is regarded as elastic, and the ply stresses are written as the sum of the overall stress and local transformation stress contributions (Dvorak and Bahei-El-Din, 1995),

We note that the lamina out-of-plane transformation stresses and do not necessarily vanish, but they are not introduced in eq. (14) since the in-plane equi-strain condition imposed on the perfectly bonded plies can be maintained under these transformation stresses without introducing additional ply stresses. The and matrices are stress distribution factors for in-plane overall stresses, and out-of-plane normal stress, respectively, and is transformation influence coefficient. The kth column of matrix provides the in-plane stresses and caused in lamina (i) by a unit transformation stress applied to lamina 0) while the overall stresses and are absent. The distribution factors and and the influence coefficients are evaluated by realizing the in-plane strain compatibility of the perfectly bonded and the force equilibrium condition, From these conditions, and using eq. (14), one can establish that (Bahei-El-Din, 1992; Dvorak and Bahei-El-Din, 1995)

50

where

is Kronecker's tensor, I is identity matrix, and

3. Analysis of a Unidirectional Lamina

We now consider a unidirectionally reinforced lamina subjected to the overall stress components which are found in the individual plies of the symmetric laminate considered above. Referred to the local coordinate system where the is parallel to the fiber longitudinal direction, and the -plane coincides with the transverse plane of the lamina, these are the axial normal stress the transverse normal stresses and the longitudinal shear stress Equations and (14) provide the time rates of the ply stresses under the laminate stress rates and and in the presence of transformation stress rates

in the individual plies. Our goal is to evaluate the local stresses in the fiber and matrix of a unidirectional lamina under the lamina overall stress rates, and to compute the rates of the lamina transformation stresses. The analysis is performed for a selected subdivision of the composite material into subvolumes. The local elastic and transformation fields are then sought either in terms of averages within the fiber and matrix phases of volumes or by piecewise uniform approximations in subvolumes or subelements of discretized phases. In particular, a representative volume V of the composite is subdivided into subelements Q of volume where so that each subelement resides in only one phase r. Conversely, each phase may contain one or more subelements. In the sequel, these two methods of subdivision of the phases are considered in evaluation of the required local fields. 3.1 TWO-PHASE AVERAGING MODELS

These models utilize Eshelby's solution (Eshelby, 1957) of an ellipsoidal inhomogeneity embedded in an infinite matrix under remotely applied uniform fields to estimate the average stresses and strains in the matrix and the fiber. Hence, the

51

number of subdivisions in these models is Denoting the volume of the matrix and reinforcement by and respectively, the volume fractions of the phases are given by where V is a representative volume of the composite material. The local stresses are found by superposition of two load systems applied to the elastic lamina, one consisting of the applied mechanical loads, in the form of uniform overall stress or strain, and one associated with internal transformation strains (Dvorak, 1992). Consequently, the elastic constitutive relations for the phases with transformation strain and stress are written in a rate form as

The and are phase elastic stiffness and compliance matrices. Superposition of the two load states, or on the boundary of V, and or in provides the local fields in the following form (Dvorak, 1992),

The coefficients and are mechanical strain and stress concentration factor matrices, and and are transformation influence factors. The kth column, of the strain transformation factor evaluates the contribution to the local strain in phase caused by a uniform transformation strain component of unit magnitude present in Similarly, the kth column of the stress transformation factor evaluates the contribution to the local stress in caused by a uniform transformation stress of unit magnitude present in Both the mechanical and transformation factors depend only on elastic moduli and the selected microgeometry of the lamina, and thus remain constant, except under temperature variations which affect the elastic moduli. In this case, the concentration factors need to be recalculated. The mechanical strain and stress concentration factors for two phase materials are given by Hill (1963);

52

and the transformation strain and stress influence coefficients are provided by Dvorak (1991);

where and are the overall elastic stiffness and compliance matrices. The overall elastic moduli can be either measured, bracketed by the Hashin-Rosen (1964) bounds, or estimated by approximate methods. In the present work, we considered and implemented the self-consistent (SC) (Hill, 1965a,b) and the Mori-Tanaka (MT) (Mori and Tanaka, 1973; Benveniste, 1987; Chen et al., 1992) models depicted in Figs. 2a,b, respectively. Both models center on Eshelby's solution of the inclusion problem and provide accurate predictions of the overall elastic moduli.

53

3.2. PERIODIC ARRAY MODELS

In periodic array models, the actual material geometry in the transverse plane is replaced with a certain periodic approximation. Under overall uniform fields and uniform temperature change, the local fields posses certain symmetric features such that a unit cell can be selected for evaluation of the local fields and the overall response. Figures 2c,d show two examples of periodic microgeometries in the transverse plane of a unidirectionally reinforced composite material with a periodic square array (Aboudi, 1986), and a periodic hexagonal array (PHA) (Dvorak and Teply, 1985; Teply and Dvorak, 1988). Evaluation of the local stresses and strains for this class of models under the lamina overall stresses and (c.f. Section 2) is performed for the selected representative volume V using the finite element method. In the present work, we utilized the unit cell derived from the PHA model and developed a mesh generator for subdivision of the fiber and matrix regions with various degrees of refinement. Figure 3 shows the PHA representative volume element and a sample of the finite element mesh in the transverse plane. Also indicated in Fig. 3a are the constrained degrees of freedom required for elimination of the rigid body motions. In addition,

54

invariance of the local fields under coordinate transformation derived from the periodic geometry in the transverse plane, as well as the condition of generalized plane strain provide displacement boundary conditions that are applied to the boundary nodes of the representative volume (Dvorak and Teply, 1985). Assuming a linear displacement field in an equivalent macroscopically homogeneous volume V, the method of virtual work can be used to compute the nodal forces, applied at the independent degrees of freedom indicated in Fig. 3a from the uniform overall stresses.

The result is

where

The volume V of the unit cell is computed using the dimensions given

in Fig. 3a, where the axial dimension H is selected such that the largest aspect ratio of the finite elements is in the order of 10. The finite element solution is obtained using the initial, or transformation, strain formulation given by Zienkiewicz and Cormeau (1974). In this method, the overall stiffness matrix of the representative volume V is assembled from the stiffness matrices of the subvolumes assuming elastic behavior of the phases, while nodal loads equivalent to the transformation strains in the subvolumes are computed with the method of virtual work and superimposed on the forces (28). In this case, the time rates of the nodal displacements, where W is the total number of nodes, are given by

where and

denotes the nodal forces derived from the overall lamina stresses, eq. (28), is the elastic stiffness matrix. Matrix defines the strain in element in

terms of the nodal displacements

where u is the displacement vector, S is a linear differential operator and

represents

the displacement shape function. The corresponding stress rate is then given by

55

The finite element procedure described above has been implemented by

Bahei-El-Din (1994,

1996) for a

given time record

of the ply stresses

and and the temperature In this case, the local stresses in the phases or subelements along the loading path are obtained by integration of eqs. (28)-(32). In the present study of inelastic laminates, on the other hand, the ply load history is not known a priori. Instead, only the current rates are known as defined by eqs. (10) and (14), and will depend on the development of inelastic strains in all plies.

Consequently, implementation of the governing equations and their integration is

performed simultaneously for all plies of the laminate. This includes the governing equations of the laminate, eqs. (1)-(18), and the governing equations of the individual

plies, eqs. (20)-(27) for a two-phase model, and eqs. (28)-(32) for a periodic array model. 3.3 OVERALL TRANSFORMATION FIELDS

Depending on the method of subdivision of the representative volume, time rates of the local fields can be either found in closed form, or computed with the finite element method as described above. In any case, the overall transformation stress and strain of the lamina are computed from the transformation fields generated in the subvolumes using the following connections (Levin, 1967; Dvorak, 1992);

where

and

are

vectors denoting the lamina overall transformation stress

and strain, respectively, and

and

are (6x1) vectors representing the

transformation stress and strain, respectively, in subvolume

There are several physical phenomena that give rise to the local transformation fields, and in a fibrous system. For example, transformation strains can be caused by thermal expansion of the phases, volume change of polymeric matrices due

to moisture, plastic deformation of metallic phases, creep of metallic as well as

polymeric matrices, and local damage and decohesion at the fiber interface. In the present work, we considered transformation strains caused by thermoviscoplastic

deformation of the phases which could be present in high temperature fibrous systems subjected to thermomechanical loads. In this case, the tensor components of the transformation strain rate can be written as the sum of thermal and inelastic strains;

56

where is the tensor of thermal expansion coefficients, and is the elastic compliance tensor. The parameters and are material constants, is an internal stress variable, and the tensor specifies direction of the inelastic strain rate in the stress space. Assuming the existence of an equilibrium yield function, and considering isotropic and kinematic hardening as well as thermal recovery of the yield surface, Bahei-El-Din et al. (1991) provided a specific form of the internal variable and described evolution of the yield surface under nonproportional loading. Their theory, which has been implemented in the present work is omitted here for brevity. A survey of other theories that can be used to describe the viscoplastic behavior of the phases is offered by Chaboche (1989). 4. Implementation and Application

When combined, the rate equations described above result in a system of first order differential equations (ODE) is the form

The unknown functions are identified with the laminate overall strain ply stresses and and transformation strain phase stress and transformation strain in all N plies as well as any internal variables required to define the rate-dependent deformation of the phases, e.g. the scalar function in eq. (34). The number of unknown functions and depends on the model used for representing the local fields. If averaging models are used, and represent the fiber and matrix average fields. In periodic array models with subdivision of the unit cell V into Q elements, and represent the average fields in the elements. In this case, the unknown functions in eq. (35) include the nodal displacements where W is the number of nodes selected in the finite element model of the unit cell. Assuming elastic response of the phases in the initial state, eq. (35) can be integrated for a specified time period using an ODE solver that is appropriate for stiff differential equations which are normally encountered in viscoplastic response modeled with the power law assumed in eq. (34). Our implementation of the procedure described above utilized the Gear method (Gear, 1971; Hindmarsh, 1974). To illustrate the main features of the method, the overall strains and fiber axial stress in a laminate caused by in-plane axial and shear stresses applied in a proportional path at 565°C are computed. The load-time record is shown in Fig. 4. Material properties of the elastic Sigma fiber and the viscoplastic Timetal-21S matrix provided by Bahei-El-Din and Dvorak (1997) are used, and a fiber volume fraction of 0.325 is assumed. The computed axial and shear strains are plotted in Figs. 5 and 6, and the axial fiber stress in the 0°-ply is shown in Fig. 7. The figures compare the

57

predictions obtained with the Mori-Tanaka averaging model and the periodic hexagonal array model. In the latter, a refined version of the mesh of Fig. 3b with 48 matrix elements and 24 fiber elements was used for each ply. It is seen that averaging the local fields over the fiber and matrix phases, as modeled by the Mori-Tanaka scheme, underestimates the overall strains and the fiber stress in comparison with the more refined representation of the local fields offered by the finite element solution of the PHA unit cell. Since axial deformation of the laminate is dominated by the elastic

58

0°-fiber, the Mori-Tanaka estimates of the laminate maximum axial strain and fiber axial stress in the 0°-ply are smaller than the finite element estimates by only 10%. In contrast, a much stiffer shear response is obtained with the Mori-Tanaka model leading to an estimated laminate maximum shear strain that is smaller than the finite element value by 60%.

59 5. Conclusions

A new approach for analytical prediction of the thermomechanical response of symmetric fibrous laminates with detailed representation of the lamina microgeometry has been presented. Under in-plane stresses, out-of-plane normal stress, and a uniform change of temperature, the method combines a transformation analysis of the laminate with a finite element solution of a unit cell of the individual plies to compute the overall strains and a piecewise uniform approximation of the local stresses and strains. Any constitutive law of the phases can be accommodated in the analysis. The present work included implementation for viscoplastic phases. Comparison of the overall strain and local stress predictions with a unit cell model and an averaging model for the unidirectional plies emphases the significance of the proposed approach. 6. References Aboudi, J. (1986) Elastoplasticity theory for composite materials, SolidMech. Archives 11, 141-183. Bahei-El-Din, Y.A. (1992) Uniform fields, yielding, and thermal hardening in fibrous composite laminates, Int. J. Plasticity 8, 867-892. Bahei-El-Din, Y.A. (1994) V1SCOPAC Finite Element Program for Viscoplastic Analysis of Composites,

User's Manual, Structural Engineering Department, Cairo University, Giza, Egypt. Bahei-El-Din, Y.A. (1996) Finite element analysis of viscoplastic composite materials and structures, Mechanics of Composite Materials and Structures 3, 1-28. Bahei-El-Din, Y.A. and Dvorak, O.J. (1989) A review of plasticity theory of fibrous composite materials, in

W.S. Johnson (eds.), Metal Matrix Composites: Testing, Analysis, and Failure Modes, ASTM STP 1032, ASTM, Philadelphia, pp. 103-129.

Bahei-El-Din, Y.A. and Dvorak, G.J. (1997) Isothermal fatigue behavior of Sigma/Timetal 21S laminates,

part ii: modeling and numerical analysis, Mechanics of Composite Materials and Structures 4, 131-158.

Bahei-El-Din, Y.A., Shah, R.S., and Dvorak, G.J. (1991) Numerical analysis of the rate-dependent behavior of high temperature fibrous composites, in S.N. Singhal, W.F. Jones, T. Cruse, and C.T. Herakovich (eds.),

Mechanics of Composites at Elevated and Cryogenic Temperatures, ASME, New York, AMD-vol. 118, pp. 67-78. Benveniste, Y. (1987) A new approach to the application of Mori-Tanaka's theory in composite materials,

Mech. of Mater. 6, 147-157.

Bigelow, C.A. (1992) The effect of uneven fiber spacing on thermal residual stresses in a unidirectional

SCS-6/Ti-15-3 laminate, J. Composites Technology & Research 14, 211-220. 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. 39, 735-752. Chaboche, J.L. (1989) Constitutive equations for cyclic plasticity and cyclic viscoplasticity, Int. J. Plasticity 5, 247-302. Chen, T., Dvorak, G.J. and Benveniste, Y. (1992) Mori-Tanaka estimates of the overall elastic moduli of certain

composite materials, J. Appl. Mech. 59, 539-546. Dvorak, G.J. (1991) Plasticity theories for fibrous composite materials, in R.K. Everett and R.J. Arsenault (eds.), Metal Matrix Composites, Mechanisms and Properties, vol. 2, Academic Press, Boston, pp. 1-77. Dvorak, G.J. (1992) Transformation field analysis of inelastic composite materials, Proc. R. Soc. Lond. A437, 311-327. Dvorak, G.J. and Bahei-El-Din, Y.A. (1995) Transformation analysis of inelastic laminates, in R. Pyrz (ed.), IUTAM Symposium on Microstructure-Property Interactions in Composite Materials, Kluwer Academic

Publishers, Netherlands, pp. 89-100.

Dvorak, G.J., Bahei-El-Din, Y.A. and Wafa, A.M. (1994) Implementation of the transformation field analysis for inelastic composite materials, Computational Mechanics 14, 201-228.

60 Dvorak, G.J. and Teply, J.L. (1985) Periodic hexagonal array models for plasticity analysis of composite materials, in A. Sawczuk and V. Bianchi (eds.), Plasticity Today: Modelling, Methods and Applications, W. Olsazak Memorial Volume, Elsevier Scientific Publishing Co., Amsterdam, pp. 623-642. Eshelby, J.D. (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. London A 241, 376-396.

Gear, C.W. (1971) Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. Hashin, Z. and Rosen, B. W. (1964) The elastic moduli of fiber-reinforced materials, J. Appl. Mech. 31, 223-232. Hill, R. (1963) Elastic properties of reinforced solids: some theoretical principles, J. Mech. Phys. Solids 11, 357-372.

Hill, R. (1964) Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour, J. Mech. Phys. Solids 12, 199-212.

Hill, R. (1965a) Theory of mechanical properties of fibre-strengthened materials: III. Self-consistent model, J. Mech. Phys. Solids 13, 189-198. Hill, R. (1965b) A self-consistent mechanics of composite materials, J. Mech. Phys. Solids 13, 213-222. Hindmarsh, A.C. (1974) GEAR: Ordinary Differential Equations System Solver, Lawrence Livermore Laboratory, Report UCID-30001, Revision 3. Levin, V.M. (1967) Thermal expansion coefficients of heterogeneous materials. Mekhanika Tverdogo Tela. 2,

88-94, English Translation: Mech. of Solids 11, 58-61. Mori, T. and Tanaka, K. (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions, Ada. Metall. 21, 571-574.

Teply, J.L. and Dvorak, G.J. (1988) Bounds on overall instantaneous properties of elastic-plastic composites, J. Mech. Phys. Solids 36, 29-58. Zienkiewicz, O.C. and Cormeau, I.C. (1974) Viscoplasticity-plasticity and creep in elastic solids - a unified

numerical solution approach, Int. J. Numer. Methods Engng. 8, 821-845.

FOURIER TRANSFORMS AND THEIR APPLICATION TO THE FORMATION OF TEXTURES AND CHANGES OF MORPHOLOGY IN SOLIDS

W.H. MÜLLER Laboratorium für Technische Mechanik Universität-GH-Paderborn Pohlweg 47-49 33098 Paderborn - Germany

1. Introduction The presence of inhomogeneities in solids very frequently leads to the generation of eigenstresses and eigenstrains. One particular example is provided by ceramic materials reinforced with Zirconia particles which undergo phase transformations accompanied by a change in shape and volume (e.g., Stevens, 1986). Another example arises in Ni-base

superalloys where a so-called forms in a matrix. The lattice parameters of both phases are different and, consequently, high internal stresses and strains occur near the coherent interface boundary (see, e.g., Hazotte et al., 1992, or Hazotte and Lacaze, 1994). Moreover, in both cases, externally superimposed mechanical stresses may locally trigger a phase transformation and, over time, lead to morphological changes of the microstructure. Fourier transforms, in particular in their discrete version, are an effective tool for the mathematical analysis of eigenstress problems in heterogeneous materials. In fact, Fourier transforms have been used before to determine microstresses and -strains around precipitates as well as to study their influence on the change of the local morphology in a solid (e.g., Khachaturyan, 1983, or Mura, 1987). In Section 2, the continuous and discrete versions of Fourier transforms (CFT and DFT) will be presented and applied to obtain a formal solution of linear-elastic eigenstress problems. Section 3 is devoted to various applications; in particular, an analytical, closed-form solution for the elastic fields in- and outside of a cylindrical inclusion in a cubic matrix will be derived. Moreover, the stress/strain fields of heterogeneities of complex shape subjected to complex loading conditions will be studied numerically. It will be demonstrated that DFT is capable of assessing the influence of an arbitrary degree of anisotropy, thermal mismatch, ordering, lattice mismatch, particle interaction, as well as elastic mismatch in a solid. Section 4 concentrates on the modeling of the formation of textures and of the change in morphology observed in solids that are subjected to internal and external loads. To this end the stresses and strains obtained through application of CFT or DFT are used to compute and minimize stored energies in heterogeneous materials. In par-

61 Y.A. Bahei-El-Din and G.J. Dvorak (eds.),

IUTAM Symposium on Transformation Problems in Composite and Active Materials, 61–72. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

62

ticular, a closed-form solution for the energetic interaction of two cylindrical inclusions in a cubic matrix will be presented. Moreover, by means of a numerical solution of coupled partial differential equations for the concentration and for a scalar orderparameter (Dreyer and Olschewski, 1995) the shape evolution of a coherent tetragonal precipitate in partially stabilized Zirconia will be simulated. 2. Essentials of Continuous and Discrete Fourier Transforms Consider an infinitely large region in space of dimension

tinuous Fourier transform,

of a field variable,

Then the con-

is defined as follows (see

Bracewell, 1986):

where

denotes the position vector in Fourier space. Provided

corresponding quantity rem as follows:

is known the

in physical space can be obtained through Fourier's theo-

Particularly useful for computations in Fourier space are the differentiation rule as well as the power theorem:

where bars denote complex conjugates. Consider now an array of N points, in a physical space of dimension d, arranged equidistantly over a square unit cell lattice of length L, (see Fig. 1):

Let discrete field variables, be defined in each of these points. Then the discrete Fourier transform for these variables can be obtained by summation:

63

If periodicity conditions hold across the representative volume element, RVE:

the corresponding quantities

theorem provided that

in physical space can be obtained through Fourier's

is known:

Note that all summations are finite and can be performed exactly, e.g., by fast Fourier transform. The sums do not represent an approximation of the Fourier integrals shown in eqns (1/2). A differentiation rule holds analogously to eqn

This rule is an approximation of spatial differentiation by a central difference quotient and it is based on the shift theorem which requires the periodicity conditions (6) to hold (see Bracewell, 1986). Second derivatives can be treated similarly:

64

Note that by linearization of these equations the acoustic limit is obtained, e.g.:

(for other discretization procedures used in literature see, e.g., Moulinec and Suquet, 1994, 1996). However, it will lead to numerical inaccuracies if the number of sampling points, N, is not high enough. The use of eqns (8, 9, 10) is clearly advantageous. In what follows static, linear-elastic problems with small deformations will be considered, i.e.:

where

denotes the Cauchy stress tensor,

are the total strains,

are the dis-

placements, are the eigenstrains, and is the stiffriess matrix which, for the time being, is assumed to be constant in space. In particular for cubic symmetry it reads:

where denote Kronecker symbols and and are Lamé’s constants. By mutual insertion of the equations shown in (11) and application of the continuous / discrete Fourier transforms the following formal solutions are obtained:

In particular, in two dimensions and for cubic materials the following relations

hold:

65 The constants of integration, or follow from constraints that need to be established in addition to the aforementioned periodicity conditions. A legitimate constraint is to prescribe the mean strains in infinite physical space or in the RVE, respectively (also see Moulinec and Suquet 1994, 1996). To this end the inverse Fourier transforms is applied to eqns (13-15):

and used to compute the average values of the strains as follows:

which clearly illustrates that the constants are nothing else but the mean averages of the strains. A possible choice is to put them equal to zero. If no external stresses are applied this is, in fact, a particularly reasonable choice in the continuous case since the average of the strains due to the misfit of a finite number of inclusions in an infinitely large volume of space must be equal to zero.

If the heterogeneities stiffnesses, in other words, if

and the surrounding matrix is a function of position:

have different elastic

the equivalent inclusion method (Mura, 1987) can be used to determine the stress/strain fields iteratively from an additional strain field

such that (for vanishing

if the matrix is stiffer than the heterogeneities or:

in the inverse case. The symbols tuted by either

or

can be computed from eqn (14) if

is substi-

66 3. Eigenstrain Problems Solved by Fourier Transforms For simplicity consider dilatoric eigenstrains within a cylindrical region of radius R :

Then eqn (13) can be rewritten (for vanishing

to become:

If the material is slightly anisotropic the denominator of eqn (24) can be expanded

resulting in a closed-form solution for the strains:

This solution is of the Eshelby-type, i.e., it is homogeneous-isotropic within the cylinder. The underlined terms correspond to Lamé’s solution for a misfit cylinder in an infinite matrix, both made of the same isotropic material. Details of the proof of these equations together with expressions of the missing strain components are given in Dreyer et al. (1997).

The usefulness of this approximation, even for comparatively

strong anisotropy, is demonstrated in Fig. 2. which shows cross-sectional cuts of the strain component in and outside of an inclusion made silver It is interesting to note that the maximum strains can be represented fairly well by eqn (26) even in this highly anisotropic case. An analogous plot for slightly

67 anisotropic material (e.g., aluminum,

between the analytical and numerical results.

would show hardly any differences

68

The applicability of DFT to eigenstrained inclusions of complicated shape is demonstrated in Fig. 4 which investigates the “Pentagonal Star” problem recently analyzed by Mura et al. (1995). Some, but not total (as claimed in that paper), uniformity of the elastic field, in the interior of a single star is clearly visible. Figure 5 shows the stresses resulting from DFT after 20 iterations when applied to the problem of a cylindrical in a Ni-base superalloy. The elastic data were taken from Socrate and Parks (1993): which

indicates

strongly

anisotropic

behavior:

4. Formation of Textures and Changes in Morphology

Fig. 6 shows two cylinders of different radii,

and

embedded in a matrix. The

cylinders as well as the matrix are made of the same cubic material with the same orientation of the main crystallographic axes. The dilatoric homogeneous eigenstrains, and are given by:

The elastic energy,

can be computed from (cp., eqns

(13)):

69 The integration can be performed analytically provided that the anisotropy of the material is not too strong:

Details of the integration can be found in Dreyer et al. (1997). Fig. 7 shows for the choice This function assumes a minimum at 45° and it has two maxima at 0° and 180°. If

is positive, an arrangement of the inclusions at

is preferable from an energetic point of view. On the other hand, if and alternating signs the energy is minimized when both inclusions are located at

have

According to Dreyer processes in which eigenstresses, ordering phenomena and diffusion are coupled on a micro-mechanics level can be described as follows, mechanical equilibrium is assumed to be reached faster than thermodynamical equilibrium. Consequently, the stresses resulting from misfit of crystal lattice will be computed as outlined in Section 2. The mathematical description of diffusion is based on the classical diffusion equation for the concentration of component A

and on a postulated rate equation for a scalar order parameter S:

†

provided

is negative as is the case for most metals.

70

Based on an evaluation of the entropy principle the diffusion flux, and the production of the order parameter, are related to the basic variables, i.e., strains,

concentrations and order parameter, as follows:

The underlined terms correspond, in part, to Fick’s law, and can also be found in the work by Wang el al. (1993). To describe the growth of tetragonal Zirconia precipitates in MgO-stabilized PSZ the configurational part of the free energy density, is expanded into a Landau polynomial:

with constants and reover, the eigenstrains are related to the order parameter according to:

Mo-

71

where are the eigenstrains of the stress-free configuration. Equations (31)-(36) were discretized in time and solved in combination with equations (11), for each time-step, to study the evolution of rhombus-like tetragonal Zirconia precipitates from originally circular shapes. A few results are presented in Fig. 8.

By application of the same techniques Dreyer (1995) has simulated the rafting phenomenon in Ni-base superalloys.

72 5, Conclusions and Outlook

It was demonstrated how continuous and discrete Fourier transforms can be used for an effective treatment of eigenstress problems in linear-elastic solids. Consequently, the next step should consist of the application of Fourier transforms to the description of elastic-plastic processes in heterogeneous solids. In a first attempt, the author computed the stresses/strains around a circular hole in a plate made of bilinear elastic-plastic material. The analysis was based on classical Mises flow theory in combination with the Fourier treatment outlined in eqn (21). The resulting plastic flow agreed with results obtained with commercial finite element codes. However, the time performance of such codes is still superior. 6. References Bracewell, R.N. (1986) The Fourier Transform and its Applications, 2nd edn., revised, McGraw-Hill, New

York. Dreyer, W. (1995) Development of microstructure based viscoplastic models for an advanced design of single crystal hot section components, in: J Olschewski (ed.) Periodic Progress Report, Development of Micro-

structural Based Viscoplaslic Models for an Advanced Design of Single Crystal Hot Section Components. Brite/Euram Programme, A. 1 - 17 - A. 1 -29.

Dreyer, W., Müller, W.H. and Olschewski, J. (1997) An approximate analytical 2D-solution for the stresses and strains in eigenstrained cubic materials, submitted to I.J.S.S., in review. Hazotte, A., Bellet, D , Ganghoffer, J.F., Denis, S., Bastie, P. und Simon, A. (1992) On the contribution of internal mismatch stresses to the high-temperature broadening of gamma-ray diffraction peaks in a Nibased single crystal, Philosophical Magazine Letters 66 (4), 189-196.

Hazotte, A. und Lacaze, J. (1994) Caractérisation quantitative de la microstructure des superalliages à base de nickel, La Revue de Metallurgie-CIT/Science et Génie des Malériaux, Février, 277-294. Khachaturyan, A G. (1983) Theory of Structural Transformations in Solids, John Wiley & Sons, New York. Moulinec, H. und Suquet, P. (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites, C. R. Acad. Sci. Paris, 318 (II), 1417-1423. Moulinec, H. and Suquet, P. (1996) A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comp. Meth. Appl. Mech. Engng., in print.

Mura, T. (1987) Micromechanics of Defects in Solids, second revised edition, Martinus Nijhoff Publishers,

Dordrecht, The Netherlands.

Mura, T., Lin, T.Y., Qin, S. (1995) Thermal stress in triangular and rectangular inclusion, in R.B. Hetnarski, N. Noda, T Tuji (eds.), Proceedings of the First International Symposium on Thermal Stresses and Related Topics, Thermal Stresses '95, Shizuoka University, 207-210. Stevens, R. (1986) An introduction to Zirconia - Zirconia and Zirconia Ceramics, Second Edition, Magne-

sium Elektron Ltd., Magnesium Elektron Publication No. 113, Litho 2000, Twickenham, UK.

Wang, Y., Wang, H., Chen, L.-Q. and Khachaturyan, A.G. (1993) Shape evolution of a coherent tetragonal precipitate in partially stabilized cubic A computer simulation, The Journal of the American Ceramic Society, 76 (12), 3029-3033.

SECOND-ORDER ESTIMATES FOR THE EFFECTIVE BEHAVIOR OF NONLINEAR POROUS MATERIALS

M. V. NEBOZHYN AND P. PONTE Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, PA 19104, U.S.A. Abstract. This work is concerned with the application of a new general procedure for estimating the overall constitutive behavior of nonlinear composites to porous materials with statistically isotropic microstructure. For two-phase systems, the procedure involves a linear-elastic comparison com-

posite with the tangent moduli of the constituent phases evaluated at appropriately chosen estimates for the average strains in the phases. The procedure can thus be used to generate estimates for the effective behavior of a nonlinear composite, directly from corresponding estimates for a linear comparison composite. One significant advantage of the procedure, over other procedures that are currently available in the literature, is that it leads to estimates that are exact to second order in the contrast. In addition, the predictions for the effective behavior of isotropic composites with isotropic nonlinear phases are found to depend on the third invariant of strain. The procedure will be used here to obtain estimates of the Hashin-Shtrikman and self-consistent types for isotropic, power-law, porous materials.

1. Introduction Porous materials, along with rigidly reinforced materials, are of special interest because they correspond to the limiting case of strongly heterogeneous two-phase composite materials. Unlike the case for weakly inhomogeneous composites, the range of possible behaviors for the effective properties of these composites can be quite broad, as suggested by the fact that one of the Voigt-Reuss bounds tends to either zero or infinity. For this 73 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 73-88. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

74 reason, porous materials constitute a good testcase for the accuracy of any homogenization procedure. In this paper, we consider the application of a recently proposed nonlinear homogenization method (Ponte Castañeda, 1996) to porous materials with power-law constituents and statistically isotropic microstructures. In addition, general results for two-phase statistically isotropic composites with fairly general nonlinear constitutive behaviors will be given, the porous case being considered as a limiting case of the more general two-phase result. Several methods have been developed for predicting the effective behavior of nonlinear porous materials. Among them may be cited, for example,

the works of Gurson (1977), Duva and Hutchinson (1984) and Duva (1986). These were based on the the use of approximate trial fields in the context of the classical minimum energy formulation of the problem of a spherical void in an infinite matrix. In addition, Ponte Castañeda and Willis (1988) made use of an extension of the Hashin-Shtrikman variational principles for nonlinear composites, due to Talbot and Willis (1985), to estimate the effective behavior of power-law porous materials. This work generated the first bounds of the Hashin-Shtrikman type, depending on two-point statistics, for nonlinear porous materials. Improved bounds of the HashinShtrikman type, as well as more general types of bounds and estimates, for nonlinear porous materials were developed by Ponte Castañeda (1991, 1992) by means of new variational principles involving “linear comparison composites”. This method has the advantage that it allows the computation of any bound or estimate for a nonlinear composite, directly from a corresponding bound or estimate for a linear composite with identical microstructure. Willis (1991) (see also Talbot and Willis, 1992) showed that the Hashin-Shtrikman bound of Ponte Castañeda (1991) for nonlinear porous materials could also be obtained from the Talbot and Willis (1985) variational principles. Finally, Suquet (1992) and Olson (1994) were able to derive the bound of Ponte Castañeda (1991) by means of closely related variational methods, specifically designed for power-law and ideally plastic materials, respectively. As will be seen here, the new homogenization method of Ponte Castañeda (1996) also makes use of a linear comparison composite, but unlike the earlier procedures, the choice of the linear comparison composite is different — involving the anisotropic “tangent” moduli of the phases, instead of the isotropic “secant” moduli, as in the procedure of Ponte Castañeda (1991) (see Suquet, 1995). 2. Effective constitutive relations

Consider a representative volume element of a composite, whose size is large compared to the size of the hetereogeneities in The composite is

75

made up of N homogeneous phases whose distribution is defined by characteristic functions The constitutive behavior of the the composite is characterized by the potential function

such that the strain and stress,

and

at point x in

are related by

In the above relation, the denote the potentials of the homogeneous phases. This constitutive relation corresponds to nonlinear elastic behavior within the context of infinitesimal strains. However, it may also be used to describe finite viscous deformations by interpreting and as the Eulerian strain rate and Cauchy stress, respectively. The effective behavior of the composite may then be described by the effective potential function

where

denotes the appropriate set of trial displacement (velocity) fields

so that the average stress and strain,

and

are related by

In the above expression for the effective potential denotes a spatial average over Similarly, will denote spatial averages over such that, for example,

where the denote the volume fractions of the phases. The local behavior of the phases will be assumed to be isotropic, with

76

where the are functions of a scalar variable consistent with the convexity condition on Also, is the hydrostatic strain, is the strain deviator, and is the Von Mises equivalent strain. A particular example of the above material model is that corresponding to incompressible pure power-law behavior, defined by

where the hydrostatic strain is set equal to zero as a consequence of the incompressibility The limits of the exponent and correspond to linear-elastic and rigid-perfectly plastic (of the Mises type) behaviors, respectively. In the first case, denotes the infinitesimal strain and is the shear modulus. In the second case, is taken to be the Eulerian strain-rate tensor, and corresponds to the yield strength of the material in tension. 3. The second-order theory 3.1. N-PHASE COMPOSITES

Like the variational procedure of Ponte Castañeda (1991), this procedure makes use of a linear heterogeneous comparison material, but the choice of the comparison material is different — involving the tangent modulus tensors of the constituent phases, instead of the corresponding secant modulus tensors. This choice of comparison material ensures that the resulting nonlinear estimates are exact to second order in the contrast, and thus in agreement with the small-contrast asymptotic results of Suquet and Ponte Castañeda (1993), provided that the corresponding estimates for the linear comparison composite are also exact to second order. The new method is based on the Taylor formula with remainder for the phase potentials Thus, introducing reference strains the Taylor formula for about reads

where the and denote an internal stress and a tangent modulus tensor, with components

Note that the depend on the strains where the depend on and are such that

77

Letting form

we may rewrite the above expression for

in the

where

and where

It then follows from (3) — by making the approximation that the reference strains

are constants

in each phase — that the effective potential

of the nonlinear composite may be estimated as

where

and where the following definitions have been used:

The advantage of approximation (15), relative to the exact expression (3), for the effective potential of the N-phase nonlinear composite is that (15) requires only the solution of a linear problem for an N-phase thermoelastic

composite, as defined by the Euler-Lagrange equations of the variational problem P in (16) :

In general, estimates for N-phase linear-thermoelastic composites can be obtained by appropriate extension of the corresponding methods for Nphase linear-elastic composites (see, for example, Budiansky, 1970, Laws,

1973 and Willis, 1981).

78 Given an estimate for P, the expression (15) therefore provides a corresponding estimate for for all choices of the and A plausible choice for the is to set them equal to the averages of the strain field over the phases However, since the exact strain field is not known, the approximate field from (18), is used instead, so that the proposed prescription for the becomes

where

denotes the average strain in the phase

Physi-

cally, this is a good choice in the context of the estimate (15), because the strain in phase is expected to oscillate about its average in phase in such a way that large deviations would only be expected in regions of relatively small measure. In this connection, it is useful to note the following identity, obtained from relation (16), namely

Therefore, the reference strains may be also computed from P, by means of relations (19) and (20). It is further remarked that the prescription (19) for the is optimal in the sense that the estimate (15) for is stationary with respect to the To see this, from (13) and (16), note that

and

respectively, so that from (15) it leads to

which, on account of the prescription (19), is identically zero. The prescription (19) also allows simplification of the estimate (15) for

Note that the Euler-Lagrange equations (18) of the problem (16) for P imply, using integration by parts, that

79

which, with the help of prescription (19) and the definition (14) of the polarizations can be used to rewrite the estimate (15) in the simpler form

where the are determined by relations (20). Finally, the choice of the in the definition (11) of the is not straightforward and, in particular, stationarity of with respect to the cannot be implemented. For this reason, Ponte Castañeda (1996) proposed the following physically motivated prescription for the

It is interesting to note, from (23), that the condition (26) implies that

Also, it follows from (25) together with (26), that the overall stress-strain relation (5) for the composite may be approximated by

3.2. TWO-PHASE COMPOSITES For two-phase composites, a well-known result by Levin (1967) permits further simplification of the thermoelastic problem P, and therefore of the corresponding estimate for The result for P, which depends only on the effective modulus tensor of a two-phase, linear-elastic composite with phase modulus tensors and is given by

where and and (26), that the prescriptions for

and

It then follows, from (20) reduce to

80 where the denote the strain-concentration tensors (Hill, 1965) for the linear-elastic composite problem. They are such that

which can be used to solve for the tensors

in terms of the

and

It is emphasized that estimates of any type for can now be used to generate corresponding estimates for For example, Hashin-Shtrikman and self-consistent estimates for will be used in (30) to generate via (25) corresponding Hashin-Shtrikman and self-consistent estimates for

For statistically isotropic microstructures, the Hashin-Shtrikman estimates for may be written in the form (Willis, 1977)

where is the modulus tensor of a homogeneous reference material. The choices equal to the maximum and minimum of the respectively, lead to the upper and lower Hashin-Shtrikman bounds (1963). On the other hand, the choice leads to the self-consistent estimates of Budiansky (1965) and Hill (1965), as shown by Willis (1977). In the above expression for the strain-concentration tensors, is a tensor with components

where

has components

and

is the in-

verse of the acoustic tensor with components The tensor is related to the well-known Eshelby (1957) tensor, arising in the solution of the elasticity problem for a spherical inclusion embedded in an infinite

matrix with modulus tensor 3.3. ISOTROPIC COMPOSITES

The application of the above-described procedure is briefly described here for composite materials with isotropic phase potentials of the form (8). Given such expressions for the and recalling, from (26), the prescriptions that one computes, from relations (11),

81

where

and where J, defined by

and

are projection tensors (Ponte Castañeda, 1996)

and

No further simplification is available, in general, for the second-order estimates (25). However, for the particular case of composites with isotropic phases and statistically isotropic microstructures, simplifications are possible because, in this case, the phase average strain deviators can be assumed to be “aligned” with the applied macroscopic strain deviator leading to the relations

It follows that all r, which, in turn, lead to:

for all r, so that

and

for

Given the definition (8) of the potentials the second-order estimate (25) for two-phase, isotropic composites can then be shown to reduce to

where

82 and where we have used the corresponding definitions:

Finally, setting with the effective coefficients and obtained from (31) and (32), and making use of (30), the variables and are found to be determined by the relations

and

In these relations, the moduli of the linear comparison composite are determined by the expressions (36) and (37), respectively.

and

4. Application to power-law porous materials

In this section, the general results obtained in the previous section will be specialized to porous materials. Thus, the porous materials will be considered to be two-phase composites with statistically isotropic particulate microstructures, where the inclusions (denoted by 1) will be occupied by the void phase and the matrix (denoted by 2) will be assumed to be made of a material with a potential of the form (9). Note that even though the matrix phase is incompressible, the porous material is compressible (because of the pores). In addition, because of statistical isotropy, the porous material will be isotropic and therefore its effective potential will be a function of the three principal invariants of the strain. Finally, the fact that the expression (9) is a homogeneous function of degree in implies a similar result for and, in conclusion, we have that

where the effective parameter is a function of the triaxiality and of the determinant of the normalized deviatoric strain ê, as determined by the variable such that ê by ( Kachanov, 1971). For the

83

particular case when the applied strain is purely deviatoric, therefore

and

where In this section, estimates of both the Hashin-Shtrikman and self-consistent type will be obtained for Proceeding first with the calculation of the Hashin-Shtrikman estimate for the porous material, note that it follows from relations (36) and (37), using expression (9) for that

Then, using (32) for with

with

and expression (61) for

as determined by (65) in the Appendix, we obtain the Hashin-Shtrikman estimate for which is defined by (31), namely

Substituting it into the general expression (47) for

we obtain

where C is given by expression (68) of the Appendix. Note that C depends

on the third invariant of the strain, as determined by Then, the overall potential obtained from (43), is of the form (49) and is given by:

where

and

84 with

The above second-order Hashin-Shtrikman estimates for are plotted, as functions of the volume fraction of the porous phase in parts (a) and (b) of the Figure, for two different values of n (10 and 1000). Note that the new simple shear estimates lie below the axisymmetric tension estimates, especially for the larger value of n. As discussed in Ponte (1996), this is because, for low-hardening materials, the flow fields in shear tend to become “localized” in a plane, whereas in tension the fields are “diffuse”. In addition, it is seen that the new secondorder Hashin-Shtrikman (HS) estimates not only satisfy the classical bound of the Voigt type, but also the HS bound of Ponte (1991). Of course, when the new second-order HS estimates and the HS upper bounds agree exactly, since both homogenization method reproduce the corresponding linear results from which they derive. On the other hand, the corresponding predictions of the second-order procedure for high-triaxiality loading conditions appear to violate the HS bound of Ponte (1991). To see this, note, from (49) and (55), that in this limit which is in disagreement with the expected prediction (see Duva and Hutchinson, 1984 and Ponte 1991). This weak result, which can be traced to the lack of convexity of the expression (49) with (55), can be improved by considering the convexification of namely, (see Ponte and Willis, 1988, for a similar situation in the context of the Talbot-Willis homogenization procedure). Thus, in the high-triaxiality limit, we obtain the result where

Unfortunately, even though now has the expected dependence on in the high-triaxiality limit, and the associated estimates for are better than those of Ponte and Willis (1988), these new predictions for still lie above corresponding Hashin-Shtrikman bound of Ponte (1991). In other to make the comparison with the bound of Ponte (1991) more explicit, it is useful to introduce an appropriately

normalized variable, namely, ods predict that varies almost linearly with

Then, both methfrom some finite limit,

85

at to a zero value at In particular, the variational method of Ponte (1991) yields the upper bound for all values of n, the second-order procedure gives 1.205, 1.159 and 1.152 for 10, 100 and 1000, respectively. Thus, in general, the new

86 second-order estimate of the Hashin-Shtrikman type are found to violate the upper bound of Ponte (1991). This deficiency of the secondorder procedure for high-triaxiality loading conditions in porous materials

suggests the possibility of obtaining improved predictions by means of better choices for the reference strains This possibility is currently under consideration. The computation of the corresponding self-consistent estimates for is similar, except that the choice must now be made in the determination of the effective modulus tensor of the linear comparison composite, from (31) and ( 3 2 ) . Because this choice leads to implicit relations for the components and of the effective modulus tensor the computation of the effective potential of the nonlinear porous composite is more complicated in this case and must be carried out numerically. In particular, because is now compressible, the computation of the pertinent P tensor must be done using the general relations (62) to (64). Illustrative results of the self-consistent type for are plotted in parts (c) and (d) of the Figure, for two different values of n (10 and 1000). Note, once again, that the new shear estimates lie below the axisymmetric tension estimates, especially for the larger value of However, it appear that the new second-order self-consistent estimates do not always lie below the corresponding self-consistent estimates of Ponte (1991). It is not clear why this is so presently, and it is hoped that further research will elucidate this point. 5. Acknowledgments

This work was supported by the National Science Foundation under grant number CMS-96-22714 and by the Aluminum Company of America. References Budiansky, B. (1965) On the elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids, 13, pp. 223–227. Budiansky, B. (1970) Thermal and thermoelastic properties of composites, J. Comp. Mater., 4, pp. 286–295. Duva, J. M. and Hutchinson, J. W. (1984) Constitutive potentials for dilutely voided non-linear materials, Mech. Mater., 3, pp. 41–54. Duva, J.M. (1986) A constitutive description of nonlinear materials containing voids, Mech. Mat., 5, pp. 137-144. Eshelby, J. D. (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. R. Soc. Land. A, 241, pp. 376–396. Gurson, A.L. (1977) Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criteria and flow rules for porous ductile materials Int. J. Eng. Mail. Tech., 99, pp. 2-15. Hashin, Z., and Shtrikman, S. (1963) A variational approach to the theory of the elastic

87 behavior of multiphase materials, J. Mech. Phys. Solids, 11, pp. 127–140. Hill, R. (1965) Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 13, pp. 89–101. Kachanov, L. M. (1971) Foundations of the Theory of Plasticity. North-Holland, Amsterdam, p 20. Laws, N. (1973) On the thermostatics of composite materials, J. Mech. Phys. Solids, 21, pp. 9–17. Levin, V. M. (1967) Thermal expansion coefficients of heterogeneous materials, Mekh. Tverd. Tela, 2, pp. 83–94. Olson, T. (1994) Improvements on Taylor’s upper bound for rigid-plastic bodies, Mater. Sc. Engng. A, 175, pp. 15–20. Ponte P. (1991) The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids, 39, pp. 45–71. Ponte P. (1992) New variational principles in plasticity and their application to composite materials, J. Mech. Phys. Solids, 40, pp. 1757–1788. Ponte P. (1996) Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. Mech. Phys. Solids, 44, pp. 827–862. Ponte P., and Nebozhyn, M. (1997) Exact second-order estimates of the selfconsistent type for nonlinear composite materials, Mech. Mater., submitted. Ponte P., and Willis, J.R. (1988) On the overall properties of nonlinearly viscous composites, Proc. R. Soc. Land. A, 416, pp. 217–244. Suquet, P. (1993) Overall potentials and extremal surfaces of power law or ideally plastic materials, J. Mech. Phys. Solids, 41, pp. 981–1002. Suquet, P. (1995) Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte nonlinear variational procedure, C.R. Acad. Sc. Paris II, 320, pp. 563–571. Suquet, P., and Ponte P. (1993) Small-contrast perturbation expansions for the effective properties of nonlinear composites, C.R. Acad. Sc. Paris II, 317, pp. 1515–1522. Talbot, D.R.S., and Willis, J.R., (1985) Variational principles for inhomogeneous nonlinear media, IMA J. Appl. Math., 35, pp. 39–54. Talbot, D.R.S., and Willis, J.R., (1992) Some explicit bounds for the overall behavior of nonlinear composites, Int. J. Solids Struct., 29, pp. 1981–1987. Willis, J. R. (1977) Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. Mech. Phys. Solids, 25, pp. 185–202. Willis, J. R. (1981) Variational and related methods for the overall properties of composites, In: Advances in Applied Mechanics, Vol. no. 21, (C. Yih, ed.), Academic Press, New-York, pp. 1–78.

A. P tensors Assuming that the modulus tensor of the reference material exhibits the material symmetry

where the tensors J, E and F are orthogonal projections defined by relations (38), (39) and (40), respectively, and assuming that the inclusion shape is spherical so that the shape tensor the associated tensor of the inclusion embedded in a matrix of reference material can also be written in the form

88 where

and

with

These relations are due to Ponte

Nebozhyn (1997). When the matrix phase is incompressible and

(1996) and Ponte

and

the above results for

can be simplified to give

and

with

and the angle

defined by (Kachanov 1971)

depending on the determinant of the strain, and where

is the integral introduced by Suquet and Ponte (1993). For high nonlinearities (large values of C depends strongly on the third invariant of ê

Shape Memory Effects

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ENERGETICS IN MARTENSITES

O. P. BRUNO Caltech Applied Mathematics, 217-50 Pasadena, CA 91125

1. Introduction Martensitic transformations are shape-deforming phase transitions which can be induced in certain alloys as a result of changes in the imposed

strains, stresses or temperatures. The interest in these alloys, which undergo a shape-deforming phase transition from a high temperature phase (austen-

ite) to a low temperature phase (martensite), stems in part from their appli-

cability as elements in active structures. In this paper we focus on the energy transfers that accompany the martensitic phase change. We discuss, in three

concrete examples, the ways in which temperature, together with the elastic and dissipated energies, determine the equilibria as well as the quasi-static dynamics in martensites. Thus, in we consider the pseudoelastic hysteresis in shape-memory wires; our treatment draws from (Leo et al., 1993; Bruno et al, 1995). In on the other hand, we follow (Bruno et al., 1996) and discuss equilibrium configurations in polycrystalline martensitic poly-

crystals. In finally, we present some new theoretical computations for certain typical microstructural lengthscales, the twin widths, observed in single-crystalline martensite twinning. The discussion of explains observed rate dependencies of hysteresis as resulting from generation, conduction and convection of latent heat. A main element in this discussion is an experimentally obtainable material constant, the dissipative force which determines the hysteresis size in very slow experiments. The treatment in completely neglects dissipation; it seeks to obtain the energy values necessary to achieve a given overall deformation in a martensitic polycrystal. The twin widths calculation of finally, assumes dissipative and elastic forces as essential elements. This computation, which incorporates simplifying approximations such as isotropic elasticity and special deformation modes, produces correctly the 91 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 91–108. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

92

order of magnitude of reported lengthscales without use of adjustable parameters.

2. Hysteretic Dynamics Both the shape-memory effect and the pseudoelastic behavior of shapememory alloys arise from the interplay of temperature and stress in the free energy of the alloy. In the shape-memory effect, an alloy deformed in

its martensitic phase recovers its initial shape when heated into a stable austenite regime. Pseudoelasticity occurs when an alloy that is austenite at zero stress is stressed in such a way that martensite becomes stable; the ensuing transformation results in a large straining of the material at an essentially constant stress. In this section we deal with the dynamics of shape-memory materials in one of its simplest settings, i.e., that of one dimensional wires under quasi-static deformations. The present discussion draws from our recent experimental and theoretical work on NiTi wires (Bruno et a/., 1995; Leo et al., 1993), indicating that the pseudoelastic hysteresis in their strainstress curves increases when the imposed strain rate is increased. Our model of these wires combines nonlinear thermoelasticity with a problem of heat flow determined by latent heat release at a two-phase interface. This model provides a law for the motion of a martensite—austenite interface which results, in particular, in excellent predictions of a variety of experimental results, as explained at the end of this section.

A purely mechanical explanation of this effect does not appear possible, and the cause of this phenomenon must therefore be closely related to the thermodynamics of the wire. To confirm this hypothesis, in addition to the tension tests for the wire in an air environment, additional tests were performed, at various strain rates, with the wire in a water bath. These experiments test the dependence of the observed phenomenon on temperature. Indeed, the heat of transformation released as austenite transforms into martensite diffuses away more rapidly through water than through air, and, therefore, the temperature of the wire is kept more nearly constant in this configuration. Comparison of the wire-in-air and wire-in-water experiments showed that the effect of the water bath for very slow experiments is negligible. For higher strain rates, on the other hand, the hysteresis widths are significantly reduced in the wire-in-water experiments. We thus concluded that thermodynamics is an important element in the pseudoelastic behavior under consideration, since the increase in hysteresis at the faster rates is greatly reduced by the increase in heat transfer between the wire and the water. A number of models have been proposed for the study of the phe-

93

nomenon of hysteresis in tension experiments involving SMA's. The experimental results mentioned above motivated us (Leo et al., 1993) to introduce yet another model of thermoelasticity. Our model, which is very simple indeed, can predict quantitatively experimental results such as those mentioned above. Points of contact as well as some essential differences between our approach and others are discussed in (Bruno, 1995). In this section a brief review of our approach is presented. Our model is based on a set of four assumptions which have been discussed extensively (Bruno, 1995; Bruno et ai, 1995; Leo et al., 1993). Fol-

lowing (Ericksen Ericksen, 1975), 1975), our model assumes that 1) the Helmholtz free energy contained in a mass element in the wire for a given strain e and at a given temperature T is given by a function which, for each fixed T, is non-convex in e. Further, this model assumes that the strain at any given point in the material constitutes a small departure from one of the local minimizers of (that is, a small departure from the natural strain of one of the phases), and that the corresponding stresses are linearly related to such departures. In what follows we consider cases which admit only two phases: the austenite (low strain, high temperature phase) and the martensite (high strain, low temperature phase). Experimental results indicate that the martensite phase grows by motion of martensite-austenite interfaces. By symmetry, two such interfaces,

one starting from each end, should occur -a fact that is indeed experimentally observed. (The action of the grips thus appears as an agent which favors nucleation.) We thus assume that 2) at any instant in time, a mixture of two phases occurs in the wire which takes the form of an austenite and a martensite region separated by two interfaces. In addition, our model assumes 3) the existence of a dissipative stress which is a constant associated with the material and independent of temperature, stress or other variables, and which acts on the wire at points coincident with the position of the interfaces. This is a very important element in our model. The necessity of existence of such dissipative forces and the mechanisms that produce them in our configurations were discussed in (Bruno, 1995). Let us note that, as the aforementioned interface moves within the wire, that is, as more austenite is transformed into martensite, latent heat is released. This heat release warms up the interfaces, and, therefore, changes the temperature dependent energy function Since latent heat is only released at the transformation fronts where the phase transition takes place, the elongation experiment sets up a non-constant temperature profile in the wire. Our model therefore assumes that 4) the stress in the wire is less than or equal to and bigger than or equal to where is the Maxwell stress corresponding to the interface temperature. Further, this

94 model assumes that the stress is equal to whenever the interface moves in a loading experiment, and that it is equal to whenever

the interface moves in an unloading experiment. It is not difficult to see that these assumptions imply the following set of equations for the temperature profile and the interface in a wire of length (see (Bruno et al., 1995) for details):

and

with initial and boundary conditions given by

In equations (1) and (2), is the mass density of the unstrained wire, its specific heat, and k, h and r its thermal conductivity, coefficient of convection of heat and radius, respectively. Equations (l)-(4) reflect the experimental observation, mentioned above, of two interfaces, one starting from each end. Indeed, symmetry considerations on a wire of length show that we must have at all times. The Maxwell stress may safely be assumed to be a known linear function of temperature (which can be determined easily from very slow experiments), and the stress finally, is determined by the overall elongation and the position s ( t ) of the interface. It is not difficult to utilize these equations to model both strain controlled and stress controlled configurations. For sufficiently long wires it is possible to derive, from equations (l)-(4), an analytic expression relating the hysteresis width as a function of the imposed strain or stress rates:

see (Bruno et al., 1995). Table 1 gives a comparison of theoretical and experimental results. The second and third columns in this table correspond to numerical simulations of wires of length L with and Inspection of these figures shows that the wire used in the

experiments is already in the asymptotic domain of validity of equation (5). The experimental results are those of (Leo et al., 1993); the agreement between theory and experiment falls within the experimental error.

95

In Figures 1 and 2 we show strain-stress curves and the temperature profiles predicted by our theory. These curves were obtained by numerical integration of equations (l)-(4) by the method of lines (Meyer, 1973); com-

parison with the experimental curves of (Leo et al., 1993) again shows good agreement. The temperature profiles shown on the right side of the Figures are snapshots taken at the time for which half of the wire had transformed into martensite. Further confirmation of our theory has been given by the experiments

of Shaw and Kyriakides (Shaw and Kyriakides, 1995) and Shield, Leo and Grebner (Shield et al., 1997). The first of these papers shows a variety of experiments for wires in water and in air, with temperature measurements in close agreement with those given above. The experiments of (Shield et al., 1997), finally, test the validity of the theory in a stress controlled configuration. For this case our theory predicts the existence of a critical stress above which the speed of transformation is infinite. This prediction

was confirmed and, further, very good agreement of predicted and observed stresses arid strain rates was found.

96

3. Polycrystals and martensitic transformations The patterns of transformation in martensitic polycrystals result as a compromise between two factors. On one hand they recognize a tendency which, in order to avoid conflicts with the imposed boundary conditions, would have the grains transform with an average transformation strain as close as possible to the applied strain. A second tendency, on the other hand, would have the grains not transform at all in order to avoid increases in energy resulting from mismatch between the transformation strains of neighboring grains. In we discuss numerical calculations and rigorous bounds for overall elastic energies in martensitic polycrystals. Such overall energies should prove useful in a finite element studies of macroscopic phenomena, such as the occurrence of two (macroscopic) interfaces, mentioned in the previous section, which occur during the elongation of polycrystalline NiTi wires. 3.1. ELASTIC ENERGY

We consider a polycrystal consisting of disjoint crystallites covering all space, each one of which can undergo a shape-deforming phase transition. The smallness of the grains when compared to macroscopic dimensions will be modeled by means of a homogenization limit, in which the crystallites are infinitely smaller than the polycrystal itself. The geometry of the grains and their crystallographic orientations (which is denned by an angle in two dimensions, or by a pair of angles, in three dimensions) are given random objects. In what follows we assume the

97 distribution of angles is statistically independent from that of the grain shapes. Thus, a realization of a polycrystal is an arrangement of grains covering space together with a crystallographic orientation assigned to each grain; the given probability measure on the set of all such realizations will be denoted by P. For any given realization and for certain spatially dependent quantities related to we will consider rescaled versions of /, where is a small parameter. The homogenized energy will result as the value of the energy in the limit as the dependence on will not be indicated explicitly when clear from the context. Each grain in this martensitic polycrystal can undergo a phase deforming phase transition and it transforms, under given solicitations, in order to reduce the total elastic energy in the structure. To characterize the possible modes of transformation of the various grains it suffices to prescribe the set of all strains that may arise as a result of transformation of a fixed single crystal whose axes are, say, parallel to the coordinate axes. The set of admissible transformation strains in a given grain is given, then, by conjugation of the set by the rotation matrix associated with its crystallographic orientation and restriction to In a two dimensional case, for example, calling

the planar rotation of angle in the grain is given by

the set of admissible transformation strains

see (Bruno et al., 1996) for a more detailed mathematical description of random polycrystals. In this paper we make the simplest assumption that all the phases are isotropic with identical elastic constants; these will be denoted by

and Assume now a homogeneous displacement

is applied on the boundary of our body B. Further, let be a given admissible assignment of transformation strains, and call the displacement which results from both the boundary displacement and the set of transformation strains on the rescaled grains In other words, calling

98 the strain in B and

the associated stress

satisfies the equations and In the homogenization limit

the elastic energy produced by a

given distribution of transformation strains under the boundary conditions (7) is given by

Here strain

is the free energy of the phase associated with the transformation and the bracket notation indicates volume averages:

For definiteness, our present discussion is restricted to estimation of overall energies at the critical temperature at which the free energies of Austenite and Martensite coincide. We thus take

for all phases the general case can be handled similarly. The overall (or homogenized) energy E is defined by the minimum value of W over all admissible stationary distributions of transformation strains. That is

where, calling

the value of

is stationary and

on

for all

and all

see (6). It has been shown (Bruno et al., 1996) that W is given by

99

Here 0 superindices denote quantities associated with the imposed boundary conditions, denotes the transformation strain of the k-th grain,

denotes the overall average of the transformation strains, and denotes the stress produced by transformation of the grain. Except for the stresses all quantities in (8) can be computed explicitly for a given

distribution of transformation strains. The stresses on the other hand, can be obtained, from Eshelby's formula, by integration, without need of solving a partial differential equation. It is thus seen that W is a quadratic form in the array of transformation strains The overall energy E associated with given boundary conditions results from W by minimization in the allowable set of transformation strains. 3.2. NUMERICS AND BOUNDS

The expression (8) can be used to derive rigorous upper bounds on the overall energy E. Alternatively, statistical optimization methods such as simulated annealing can be used to minimize this expression numerically. A number of interesting conclusions can be drawn by comparison of such bounds and numerical results. We refer to (Bruno et al., 1996) for such considerations and complete details on our approach. In what follows we comment briefly on the nature of our bounds, and we give examples demonstrating the quality of our numerics.

The derivation of our bounds on the overall energy E can be roughly described as follows. As we said, the patterns of transformation in martensitic polycrystals result as a compromise between two factors. On one hand

they recognize a tendency which, in order to avoid conflicts with the imposed boundary conditions, would have the grains transform with an average transformation strain as close as possible to the applied strain. A second tendency, on the other hand, would have the grains not transform at all in order to avoid increases in energy resulting from mismatch between the transformation strains of neighboring grains. In particular, the transformation strains of different grains in a minimum energy configuration are likely to exhibit statistical correlations. Thus, a calculation of the overall energy based on a hypothesis of non-correlation in the transformation strains of different grains must lead to an upper bound for this quantity. As it happens, the hypothesis of uncorrelated grains (UG) leads to explicitly computable quantities. Indeed, the only portion of (8) that is nonlinear on the transformation strains is its last (quadratic) term. Using ensemble averages, the UG hypothesis allows us to transform the average of this quadratic term into a quadratic expression for the average of the transformation strains. This leads to an upper bound which has been evaluated

100

explicitly for a number of configurations. To illustrate our comments we present, in Figure 3, a comparison of numerical results and bounds. In this simplified example the polycrystal is assumed to be a square array of circular grains in two dimensions. (See (Bruno et al., 1996) for details, where a number of additional examples including deviatoric and hydrostatic transformation strains may be found as well.) The “Taylor bound” mentioned in the caption, is the upper bound that results by assumption of a hypothesis of constant strain, such as that one used by G. I. Taylor in the context of polycrystalline plasticity (Taylor, 1938). We note from the figure that, here, the Taylor's hypothesis may lead to errors of the order of 50% . The quantity AE, finally, is the “Austenite Energy”, that is, the elastic energy that results in the polycrystal under the same boundary conditions if none of the grains transforms to martensite.

4. Lengthscales A fundamental element in the development of the crystallographic theory of martensite was the consideration of a planar surface, the habit plane, which separates twinned martensite from untransformed austenite in certain single crystals, cf. (Wayman, 1964, p. 170) and (Wayman, 1953, p. 1505). Such configurations certainly do not constitute the only fashion in

101 which austenite and martensite may coexist: a variety of regular as well as irregular patterns generally occur. But habit-plane morphologies are indeed observed in carefully monitored experiments, as demonstrated by a number of compelling micrographs; see (Wayman, 1964, p. 82), (Basinski and Christian, 1954). Here we follow (Wayman, 1953, p. 1505) in their consideration of such especially simple configurations. Our focus is on elasticity and dissipation of elastic energy into other mechanical observables. We present a theory which, based on the existence of the simple patterns mentioned above, attributes the finiteness of the observed microstructures (which would be deemed infinitesimal from an unqualified application of the crystallographic theory) not to competition between elastic and surface energies but, instead, to an interplay between elasticity and energy dissipation. The resulting computations, presented in show quantitative agreement with the observed twin sizes. 4.1. ELASTICITY

Martensitic transformations give rise to a number of martensite variants, whose associated deformations are characterized by a finite set of constant distortions The tensors are defined by

where I is the identity tensor and where is the homogeneous deformation associated with transformation into the variant. Note that, with this convention, would indicate null deformation. Also, the tensors need not be symmetric; their symmetric parts equal the corresponding Bain strains

As pointed out in (Wayman, 1953, p.1511), the set must be invariant under conjugation by rotations in the symmetry group of the parent phase. In a general configuration, the transformation distortions vary in space, and the spatial distribution of such quantities within the body is given by a tensor valued function The actual transformation distortion at a given point in the material is not necessarily given by an element of since the phase transition may give rise to relative rotations between the various parts of the body (Wayman, 1953). Further, the true deformation at a point in the material is generally not equal to since additional (small) elastic deformations occur as a result of

102 transformation of the various parts of the body. Thus, the true displacement vector at a point in the material results as a small elastic f t . i . 11 11 i that further changes the shape of the material element around deformation point r after it has undergone a distortion of the form

for some k and some rotation matrix R. The overall deformation vector D is given by . As we discuss configurations which consist of a planar interface separating austenite on one hand, and fine alternating layers of martensite on the other, we will find it convenient to use coordinates (x, y, z) with the ( x , y ) coordinate plane orthogonal to both the inter-twin planes and the twin-austenite interface. In such coordinate system the transformation distortions are independent of the variable

Assuming, further, that the x axis coincides with the twin-austenite interface, our three dimensional x-periodic configuration is described by Figure 4: the plane separates the austenite and the twinned martensite and the axis is perpendicular to the plane of the figure. The distortion vanishes for and it is constant and equal to a rotation of an element of in each one of the twin bands in the region Clearly, the stress at point in the material vanishes if Thus, for the small elastic deformations that the austenite and the martensite can undergo without additional phase change, it is reasonable to assume a linearly elastic law

where relation

is the stiffness tensor of the phase at point and calling

In view of the

we also have Our subsequent analysis assumes isotropic elasticity and identical elastic constants for the austenite and martensite phases, with Lame constants

103

where is the shear modulus and where, denoting by the bulk modulus, is given by Let us now introduce a transformation displacement which satisfies

and it is continuous outside the interface We take for To define for we note that equations (10) admit a unique solution up to an arbitrary constant. Indeed, on each martensite band we must have

and, in view of the compatibility conditions implied by the crystallographic theory of martensite, it is clear that constants may be chosen to make up a function which is continuous in the region and periodic, of period d, in the x direction; generically, however, is discontinuous at By periodicity, the vector admits a Fourier series expansion

with vector coefficients, and with

104

In an example below we will assume a simplified configuration in which both types of twins have the same width. In this case the Fourier coefficients are given by

for a certain vector

Note that in this case we have

Calling our problem then becomes

with the boundary conditions

u is everywhere continuous and

on

Equivalently,

It can be shown that, by consideration of stresses corresponding to a problem of plane strain and a stress corresponding to an anti-plane shear problem, the system (9) can be reduced to a decoupled pair of scalar equations in the plane.

For simplicity, we compute the elastic energy in the antiplane shear case only. Assuming transformation strains equal in magnitude to those observed in experiment, we then use our antiplane-shear energy expression in a calculation of the twin widths. Naturally, only order of magnitude predictions should be expected from this analysis. Complete results for the general case will be presented elsewhere. It is easy to check that, calling in the antiplane-shear case under consideration we have Fourier expansions

The total elastic energy in the configuration of Figure 4 (per unit length in the x- and z-directions) is given by

105

For our specific example this reduces to

where t is the projection of the strain of transformation on the habit plane. The infinite sum in this expression equals approximately 8.4, so that the energy is Estimating t by the magnitude of the transformation strain given in (Burkart and Read , 1953) we find With regards to the elastic modulus, we follow these authors and use 4.2. TWIN WIDTHS The calculation of the twin widths now proceeds from three main assumptions: 1) Energy dissipation is associated with the formation of the intertwin interfaces, and this is the only dissipative mechanism in the experiment; 2) The first occurrence of an austenite-martensite interface is as indicated in Figure 5. That is, calling b the side of the square base in the tetragonal bar of Figure 5, a fully formed austenite-martensite interface of length and corresponding twins filling a triangular region T, arise not as a propagation of a continuously moving interface, but, instead, as a single nucleation event. After this nucleation event the austenite-martensite interface moves continuously (as observed experimentally). And, 3) The forementioned triangular configuration is formed exactly when the excess energy obtained by cooling in the triangular region reaches a value consistent with the formation of a austenite-martensite interface (which stores a certain amount of energy in elasticity) as well as the formation of the inter-twin interfaces (which, we have postulated, involves dissipation). We provide some justification for assumptions 1) and 2); assumption 3) appears quite natural but will not otherwise be discussed. With regards to 1), clearly no dissipation occurs in the austenite region. If the main energy dissipation took place at the twin-autenite interface, on the other hand, then such dissipation would diminish with diminishing twin widths, and would be very small if the twin widths are small. Of course, this does not rule out the possibility that dissipation takes place both at the austenitemartensite interface, and at the inter-twin interfaces. We think this possibility unlikely in the Indium-Thallium configurations of (Burkart and Read , 1953), in view of their small transformation strains. A better understanding in this connection, however, will probably require further experimental investigations.

106

Assumption 2), on the other hand, is based on the following simple symmetry argument. If a small interface develops on the left hand corner of the bar of Figure 5, then a corresponding interface should develop on the right hand corner as well. As these interfaces grow they would intersect each

other, and eventually produce an X -interface -which is crystallographically admissible, and experimentally observed in some circumstances (Burkart and Read , 1953; Basinski and Christian, 1954). In the bars considered in these experiments, however, such X interfaces tend to occur in specimens which have been mishandled, or which have not been annealed properly. As reported in (Burkart and Read , 1953; Basinski and Christian, 1954), in well annealed bars of dimensions of those used in these papers, X interfaces tend not to occur. Single nucleation events involving regions substantially larger than the one of Figure 5 would probably not have gone unnoticed, on the other, hand, an therefore our assumption 2) seems fairly well substantiated. A calculation of the twin widths on the basis of these hypothesis follows easily, now, from the results of the previous section. The energy W of the twin-austenite interface per unit length of interface and per unit length in the direction equals

with The total length of twin-twin interfaces in Figure 5, on the other hand, is given by where s is the length of the twin-austenite interface. It follows that the energy dissipated in the formation of this

107

configuration can be expressed in the form

where Q is the total dissipated energy per unit length of twin-twin interface. Thus, using assumption 3), the configuration of Figure 5 may not have been

formed unless the triangular region T was cooled in such a way that an amount

of excess energy was made available within T. The configuration will form as soon as the available energy in the triangular region reaches a value equal to the minimum of equation (14) for all positive values of d. Differentiation shows that this minimum is achieved when

or

It follows that the minimum energy value is

(Notice, parenthetically, that the energy needed to form a small triangular region scales with which is much larger than the area of the corresponding triangle, (and thus the energy contained in the triangle) for sufficiently small s. This remark provides further support for our assumption 2), in that it establishes that some triangular configuration must be nucleated as a single event.) These are our main results. We now evaluate the expression (15) for parameter values corresponding to the indium-thallium system of (Burkart and Read , 1953). From Figure 8 of (Burkart and Read , 1953, p. 1521) the hysteresis width equals so that the dissipative force equals or equivalently, Half of the hysteresis force times the strain should give a reasonable estimate for the dissipation per unit length of bar on the cooling transformation. Since from (Burkart and Read , 1953, p. 1520) the total elongation of the bar is 0.37%, the total dissipation per unit length of bar equals

108

Because the area of the triangular region equals the dissipation involved in its formation equals one half the dissipation involved in propagating the austenite-twin interface by an amount b. So, the dissipation involved in the formation of triangular region equals

Since

and

(cf. (Burkart and Read , 1953)), and using equation (15) gives

or and the twin width is therefore equal to

Roughly, this value is in agreement with the experimental observation of (Burkart and Read , 1953), which gives a twin width of about References Basinski, Z. S. and Christian, J. (1954) Experiments on the martensitic transformation in single crystals of indium-thallium alloys, Acta Met. 2, 148–166. Bruno, O. P. (1995) Quasi-static dynamics and pseudoelasticity in polycrystalline shape–

memory wires, Smart Materials and Structures 4, 7–13; see also Mathematics and Control in Smart Structures, Proc. SPIE, 2192, 370–379, (1994). Bruno, O. P., Leo, P. and Reitich, F. (1995) Free boundary conditions at austenitemartensite interfaces, Phys. Rev. Lett. 74, 746–749. Bruno, O. P., Reitich, F. and Leo, P. (1996) The overall elastic energy of polycrystalline martensitic solids, J. Mech. Phys. Solids 44, 1051–1101, 1996. Burkart, M. W. and Read, T. A. (1953) Diffusionless phase change in the IndiumThallium system, Transactions AIME 197 pp. 1516–1524. Ericksen, J. L. (1975) Equilibrium of bars, Journal of Elasticity 5, 191–201,. Leo, P. H., Shield, T. W. and Bruno, O. P. (1993) Transient heat transfer effects on the pseudoelastic behavior of shape-memory wires, Acta metall. mater. 41, 2477–2485. Meyer, G. (1973) SIAM J. Numer. Anal. 10, 522. Shaw, J. A. and Kyriakides, S. (1995) Thermomechanical aspects of NiTi, J. Mech. Phys. Solids, 43, 1243–1281. Shield, T. W., Leo, P. H. and Grebner, W. C. (1997) Quasi-static extension of shapememory wires under constant load, Acta Materialia, 45, 67–74. Taylor, G. I. (1938) Plastic strain in metals, J. Inst. Metals, 62, 307–324. Wayman, C. M. (1964) Introduction to the crystallography of martensitic transformations, Macmillan. Wechsler, M. S., Lieberman, D. S. and Read, T. A. (1953) On the theory of the formation of martensite, Trans. AIME, 197, 1503–1529.

MODELING OF CYCLIC THERMOMECHANICAL RESPONSE OF POLYCRYSTALLINE SHAPE MEMORY ALLOYS

ZHONGHE BO AND DIMITRIS C. LAGOUDAS Center for Mechanics of Composites Aerospace Engineering Department Texas A&M University College Station, TX 77843-3141

Abstract In this paper, the evolution of plastic strains and Two-Way Shape Memory (TWSM) effect with respect to thermally induced cyclic phase transformation is

investigated for NiTi shape memory alloys. It is observed by Bo and Lagoudas (1997b) that the accumulation of plastic strains continues beyond a large number of cycles (2000), while the TWSM is saturated and remains stable after a few hundred transformation cycles, depending on the magnitude of the applied load. Motivated by the experimental observations, evolution equations for the accumulation of plastic strains and plastically related back and drag stresses, which govern the

evolution of TWSM, are proposed. Model predictions are successfully compared with experimental data. 1. Introduction

Unlike conventional metals, which are working in an elastic range under normal

operating conditions, Shape Memory Alloy (SMA) actuators are operating in an inelastic range, in which repeated martensitic phase transformations occur. Experiments (Miyazaki et al., 1981, 1986, Perkins and Sponholz, 1984, Perkins and

Bobowiec, 1986, Liu and McCormick, 1990, 1994, Stalmans et al., 1992a,b, Lim and McDowell, 1994, Hebda and White, 1995, Bo and Lagoudas, 1997b) show that substantial plastic strains will be induced during either stress induced or thermally induced cyclic phase transformations. A typical strain-temperature history of a SMA wire under a constant applied load, obtained in the Active Materials Lab at Texas A&M University, is shown in Fig. 1. Distinguished from regular ductile metals, where the plastic strains are created by extensive loading beyond the yield limit of the material, plasticity created in SMAs under regular operating conditions is caused by repeated phase transformations, and can occur at a relatively low stress level, for example, as low as 10% of the real yield limit of the martensitic phase. The accumulation of plastic strains in SMA actuators is, therefore, unavoidable. Besides the creation of plastic strains, material properties, such as

transformation start and finish temperatures, transformation hardening behavior, and even total hysteresis of SMAs are also undergoing significant changes during cyclic phase transformation. Thus, the modeling of cyclic thermomechanical re-

109 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 109–122. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

110 sponse of SMAs is of importance for successful application of SMA as actuators in active structures.

Most of the experimental results reported in the literature for NiTi SMAs

(Miyazaki et al., 1981, 1986, Perkins and Sponholz, 1984, Perkins and Bobowiec, 1986, Liu and McCormick, 1990, 1994, Rogueda et al., 1991, Stalmans et al., 1992a,b, Lim and McDowell, 1994, Hebda and White, 1995) are limited to small number of cycles, aiming to investigate the creation of the TWSM effect of SMAs. Hebda and White (1995) have performed tests up to 3000 cycles. However, their tests are focused on the stability of the two way shape memory effect created by low cycle training procedures, and not on the evolution of plastic strains. The objective of the present research is to investigate the evolution of plastic strain under different levels of applied load, and the relation between the accumulation

of the plastic strain and TWSM. Tests of thermally induced phase transformation at constant applied load are selected for our investigation since they better sim-

ulate real operating conditions of SMA actuators than the stress induced phase transformation. Relatively few results have been reported in the literature on the modeling of cyclic loading of SMAs as well as the creation of TWSM (Zhang et al., 1991, Rogueda et al., 1991, Patoor et al., 1991, Tanaka et al, 1992, 1995, Bo and Lagoudas, 1995, Lexcellent and Bourbon, 1996). The evolution of plastic strains has been modeled by Tanaka et al. (1995). In their approach, however, the accu-

mulation of plastic strains is assumed to be saturated after being subjected to a certain number of transformation cycles, which is not the case as observed in the

recent experiments of thermally induced phase transformation at constant applied

111 load by Bo and Lagoudas (1997b) (refer to the experimental data presented in Section 3.1.2 of this paper). Motivated by the new experimental observations gained in the tests performed in the Active Material Lab at Texas A&M University, the evolution equations for plastic strain and plasticity related back and drag stresses, which govern the evolving of the material properties, will be developed in this

work. The structure of the present paper is as follows: The constitutive equations describing the thermomechanical response of SMAs during a single transformation cycle (Bo and Lagoudas, 1997a, Lagoudas and Bo, 1997) is summarized in Section 2. The evolution equations of plastic strains and the plastically related internal state variables with respect to the number of loading cycles is presented in Section 3. where the theoretical results are compared with the experimental data, and a numerical scheme for implementing the model is also discussed. Conclusions are given in Section 4. 2. Summary of the Constitutive Equations of the Previous Model

In this section the constitutive equations in a 1-D form are summarized. The Gibbs free energy obtained by Bo and Lagoudas (1997a) can be reduced to the following 1-D form:

In the above, is the uniaxial elastic compliance, and . and are Young's moduli of austenite and martensite, respectively; is the uniaxial thermal strain; and are the corresponding 1-D back stresses, while and are the drag stresses; finally, and correspond to mass density, heat capacity, reference entropy and free energy, respectively. The constitutive equations are given by (Bo and Lagoudas, 1997a, Lagoudas and Bo, 1997)

In the above, E is the total strain in the above equation; is the uniaxial transformation strain at the reversal of the phase transformation, and is the

112 corresponding total martensitic volume fraction; is the maximum transformation strain for the specific transformation cycle and given applied stress; Y is a material constant describing the total dissipation of a SMA; the sign function sign is defined by sign where is the magnitude of is the 1-D reduction of the transformation tensor finally, is the transformation function, and is the thermodynamic driving force for the phase transformation. The uniaxial effective stress is defined by

In the above, and are the corresponding uniaxial back stresses due to phase transformation and the interaction of the phase transformation with plastic strains. The explicit expressions for the back and drag stresses are given by

where sign effective stress,

Substituting equation (4) into equation (3), the uniaxial is given by

where can be considered as the detwinned martensitic volume fraction, and and are material parameters. The thermodynamic force, conjugate to is defined by

In the above, where and are the average thermal expansion coefficients of austenite and martensite in loading direction, respectively; T and are current temperature and a reference temperature, respectively; and are the difference of the specific heat per unit volume and the difference of the specific entropy per unit volume, respectively. The drag stresses and are induced by the phase transformation and the interaction between the phase transformation with plastic strain. The evaluation of the drag stresses is given by

113 Substituting equation (7) into equation (6),

where the material parameter material parameters, i.e.,

Traditionally,

can be explicitly written as follows:

and are material parameters. For convenience, is introduced in equation (8) as a combination of other

is referred to as martensitic start temperature (see Section 3

for determination of material constants). The material constant in equation (8), representing the value of Y for an untrained SMA, is also introduced here for convenience. Equations (2) combined with equations (8) and (5) form a set of constitutive equations governing the thermomechanical response of SMAs. Equation (2a) is

a generalized Hooke’s law, equation (2b) is the flow rule for the transformation strain, and equation (2c) is the transformation function. 3. Modeling of Cyclic Loading of SMAs In section two, we have summarized the constitutive equations describing the thermomechanical response of SMAs during a single thermomechanical loading cycles. To model the cyclic loading of SMAs, there are two major questions that need to be answered: (1) the specific evolution law for the plastic strain during loading cycles; (2) the effect of the evolving microstructure due to plastic deformations on the phase transformation. The first problem is of practical importance because the accumulation of plastic strains will affect substantially the performance of SMA

actuators. The second issue is also crucial in predicting TWSM and the change of the strain-temperature or stress-strain hysteresis curves. In the present model, the characteristics of the microstructure and their effect on the phase transformation is described by the back and drag stresses. The specific evolution equation of the plasticity induced back and drag stresses controls the change of the transformation hardening behavior and the shifting of the hysteresis loops caused by cyclic loading. The methodology used in constructing the constitutive model is similar to that used in the theory of viscoplasticity without yield surface (Bodner, 1975, 1987,

Miller, 1987). An internal time, which is defined to be the accumulation of the absolute increment of the martensitic volume fraction, is introduced to replace the real time used in viscoplasticity models, since the phase transformation is nondiffusive (Wayman, 1983). In Section 3.1., an evolution equation for the plastic

114 strain is given. In Section 3.2., the evolution of the plastically related back and drag stresses will be discussed. The evolution of the total dissipation of a SMA will also be discussed in Section 3.2. The theoretical prediction will be compared to the experimental data. Finally, a procedure for the numerical implementation of the model is given in Section 3.3.

3.1. EVOLUTION OF PLASTIC STRAINS The plastic strain rate is assumed to relate stress through a flow rule in one dimensional form as follows:

where

is the flow factor. In the above, the plastic strain is defined to be where is the extension ratio of the wire at the austenitic state, and the stress is computed by where P is the magnitude of the applied load and is the initial cross-section area of the wire. Equation (10) implies that

Distinguished from conventional plasticity, the plastic strain produced in the present circumstances is due to phase transformation. It is, therefore, assumed after compared with experimental results that depends on the stress applied, the current total plastic strain, the martensitic volume fraction, and its time derivative as given by

where

is an internal time defined by

where t is the real time, and can be considered to be a detwinned martensitic volume fraction, where has the same meaning as in equation (2). In the theory of dislocation kinetics, the dislocation velocity is represented as a power function or an exponential function of the stress (Gilman, 1969). In the present formulation, a form for equation (12) is assumed to be given by

In the above equation, is a material constant, which is related to the saturation rate of the plastic strains, i.e., with increasing the plastic strain rate decreases faster; the function accounts for the possibility that plastic strains may not be created uniformly during phase transformation, and equally during the forward and reverse phase transformations; is a hardening function depending on developed plastic strain, A functional form for z is assumed to be given by a Taylor series expansion truncated after second order terms, i.e.,

115 where are material constants. The material constant describes the material initial resistance to the creation of plastic strains, and and represent the effect of hardening due to the plastic strain

For a given

and its increment, the increment of

can be computed using

equations (13). Then equations (14) and (15) can be utilized to solve and z, involved in these two equations. Specifically, equation (15) can be substituted into equations (14), and then the nonlinear ordinary differential equation (14) can be solved by utilizing any available numerical algorithms.

3.2. EVOLUTION OF PLASTICITY RELATED BACK AND DRAG STRESSES During a specific cycle of the phase transformation, the material parameters involved in the evolution equations for the back and drag stresses, characterizing the average microstructural changes induced during cyclic loading, are assumed to be given. In this section, their evolution laws with respect to loading histories will be investigated. We rewrite equation (4b) as follows

where

and

is given by

As an approximation, we assume that a specific distribution of defects induced during cyclic phase transformation is self-similar, but the strength of the defects increases with loading cycles. This assumption implies that the parameter is constant during cyclic loading history. The increasing strength of the defects is described by the evolution of the parameter Based on the experimental observation that the maximum transformation strain

is saturated after being subjected to a certain number of cycles, a form of an evolution equation for the parameter is assumed to be given by

where and are material constants; governs the increasing rate of with respect to the accumulation of plastic strains, and is its maximum magnitude. Both and are positive. Note that the sign of eventually represents the sign (orientation) of the back stress which is assumed to be equal to the direction of the plastic strain. For pure tensile or pure compressive loading conditions, using the flow rule given by equation (10), equation (18) can be integrated explicitly as follows

where

and

are absolute magnitudes of

and

respectively.

116 Similar arguments can be applied to the drag stress tion (7b) as follows

First we rewrite equa-

where

An evolution equation for

where

and

is proposed similar to

are material parameters, and

as follows:

the accumulation of the total

martensitic volume fraction, is given by

Different from

depends on the specific type of loading. Experiments

show that during the stress induced cyclic phase transformation, the transformation start stress for the forward phase transformation always decreases (or

equivalently the start temperature always increases) with respect to loading cycles (Miyazaki et al., 1986, Lim and McDowell, 1994). For NiTi SMAs undergoing thermally-induced self-accommodating martensitic phase transformation, the martensitic start temperature monotonically decreases (McCormick and Liu, 1994). For the detwinned thermally-induced phase transformation imposed by an applied load, the martensitic start temperature decreases, but the amount of decrease in transformation temperature is usually less than that of the case of selfaccommodating martensitic phase transformation (McCormick and Liu, 1994). This different trend in the shifting of transformation temperatures exhibited in stress induced and thermally induced phase transformations may result from the interaction between the following two training effects (i.e., positive training effect and negative training effect). The positive training effect, phenomenologically corresponding to the decrease of the martensitic start stress or increase of the martensitic start temperature, is attributed to the development of well-aligned groups of dislocations, which retain martensitic variants even after a SMA is heated above the original austenitic finish temperature. During subsequent cooling, martensite can easily nucleate and

grow from these retained martensitic variants. The average orientation of these retained martensitic variants will ultimately determine the orientation of the TWSM strain. The configuration of these groups of dislocations is relatively stable, resulting in repeatable transformation sequences in successive transformation cycles (Miyazaki et al., 1989, Hebda and White, 1995). The elastic energy stored in the neighborhood of these defects can be released as a driving force, promoting the forward phase transformation. Macroscopically, the martensitic start temperature is increased. The negative training effect may be caused by the creation of many randomly

orientated dislocations, which are introduced by the randomness in the orientation

117 of the martensitic variants formed during phase transformation. These randomly orientated dislocations pile up and build local energy barriers, making accommodation of the same configuration of martensitic variants more difficult in the next transformation cycle. Consequently, the formation of martensite in successive cycles tends to take different local paths, resulting in the increase of the local elastic energy induced by the phase transformation. Microscopically, the martensitic start temperature decreases (McCormick and Liu, 1994). Motivated by the experimental observations discussed above, is assumed to have three distinct values for stress induced phase transformation, thermally induced non-self-accommodating phase transformation, and self-accommodating phase transformation. For each case, equation (22) can be integrated as follows:

The total internal dissipation of SMAs can also be affected during cyclic phase transformations. However, this change is not significant during cyclic thermally induced phase transformations (Liu and McCormick, 1994). During stress induced phase transformations, the observation (Miyazaki et al., 1986, Lim and McDowell, 1994) that the total hysteresis area enclosed by a stress-strain curve decreases substantially with respect to the number of loading cycles may be due to the fact that partial transformation cycles are performed in their experiments. Thus, the shrinking of the hysteresis loop observed in their tests may indicate that the change

of the transformation hardening behavior will significantly change the shape of inner loops for trained SMAs. In the present formulation the total hysteresis, Y, is assumed to be constant.

3.3. RESULTS AND DISCUSSIONS A procedure to determine the material parameters used in the model can be found in the papers by Lagoudas and Bo (1997) and Bo and Lagoudas (1997b,c). Using the procedure described in these papers, and utilizing required experimental data, the material constants used in the present model are determined and given in Table 1. Note that the material constants given in Table 1 represent a full set of material constants used in the model to describe the cyclic behavior of SMAs.

To model the thermomechanical response of SMAs during a single transformation cycle, the constants related to the evolution of plastic strains and the evolution of back and drag stresses with respect to loading cycles are not needed. Thus the total number of constants will be reduced from 30 to 18. To model a trained SMA during a single transformation cycle, one more constant, at minimum, is needed in additional to the constants used for an annealed SMA. If one is only interested in modeling qualitatively the characteristics of the hysteresis response of SMAs, simpler transforamtion hardening function can be used, and the number of constants can be further reduced. For a detail discussion, please refer to the paper by Lagoudas and Bo (1997). A comparison of the evolution of plastic strains between experimental results and model prediction is shown in Fig. 2. The data used in calculating these curves

118

is given in Table 1. The experimental curves used in determining these parameters are the ones corresponding to the applied stresses of 35 MPa and 70 MPa. The latter one is used only up to 1000 cycles. It is shown in Fig. 2 that the prediction of the present model captures the main feature of the evolution of the plastic strain at different levels of the applied stresses. The overall prediction agrees quite well with the experimental measurements. The model simulations of the evolution of the plastic strain, the maximum transformation strain, and the two way recoverable strain under 35 MPa applied stress are compared with experimental results in Fig. 3, while the comparison of the model predictions with the experimental data for the case of 50 MPa applied stress are shown in Fig. 4. It can be seen that the theoretical simulations and predictions agree well with the experimental measurements.

Note that in the current formulation, the hardening function z is assumed to depend on only. A possible improvement could be achieved by introducing the dependence of z on the other state variables, such as the total martensitic volume fraction. Another modification could be introduced by proposing more complicated functional forms for which is now assumed to be 1. The function

119

120

is important when partial (inner) transformation loops are concerned, because the accumulation rate of the plastic strain may vary substantially at different stages of the phase transformation, i.e., may depend strongly on To motivate this modification, further experiments need to be performed.

4. Conclusions To model cyclic themomechanical response of SMAs, evolution equations for plastic strains and plasticity related back and drag stresses are proposed by adopting the methodology similar to that used in viscoplasticity models. The model simulations and predictions are compared with the experimental data, and good agreement is observed. The 1-D cyclic behavior of SMAs can be fully characterized using the present model. 5. Acknowledgments

The authors acknowledge the financial support of the Army Research Office, contract DAALo3-92-G-123, monitored by Dr. G.L. Anderson. 6. References Bo, Z. and Lagoudas, D.C., 1995, A thermodynamic constitutive model for cyclic

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International Journal of Engineering Science. Bo, Z. and Lagoudas, D.C., 1997b, Thermomechanical modeling of crystalline SMAs under cyclic loading, part III: Evolution of plastic strains and two-way shape memory effect, Submitted for publication in International Journal of Engineering Science. Bo, Z. and Lagoudas, D.C., 1997b, Thermomechanical modeling of crystalline SMAs under cyclic loading, part IV: Modeling of minor hysteresis loops, in preparation. Bodner, S.R. and Partom, Y., 1975, Constitutive equations for elastic-viscoplastic strain-hardening materials, ASME Journal of Applied Mechanics, 42, 385-389. Bodner, S.R., 1987, Review of unified elastic-viscoplastic theory, in Unified Constitutive Equations for Creep and Plasticity, A.K. Miller, ed., Elsevier, New York. Gilman, J.J., 1969, Micromechanics of Flow in Solids, McGraw-Hill, New York. Hebda, D. and White, S.R., 1995, Effect of training conditions and extended thermal cycling on nitinol two-way shape memory behavior, Smart Materials and Structures,

4, 298-304. Lagoudas, D.C. and Bo, Z., 1997, Thermomechanical modeling of crystalline SMAs under cyclic loading, part II: Material characterization and experimental results for a specific transformation cycle, Submitted for publication in International Journal of Engineering Science Lexcellent, C. and Bourbon, G., 1996, Thermodynamical model of cyclic behavior of Ti-Ni and Cu-Zn-Al shape memory alloys under undulated tensile tests, Mechanics of Materials, 24, 59-73. Lim, T.J., and McDowell, D.L., 1994, Degradation of an Ni-Ti alloy during cyclic loading. Proceedings of the 1994 North American Conference on Smart Structures and Materials, SPIE, Orlando, Florida, pp. 153-165. Liu, Y., and McCormick, P.G., 1990, Factors influencing the development of two-way

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122 Miller, A.K., 1987, The MATMOD equations, in Unified Constitutive Equations for Creep and Plasticity, A.K. Miller, ed., Elsevier, New York. Miyazaki, S., Otsuka, K., and Suzuki, Y., 1981, Transformation pseudoelasticity and deformation behavior in Ti-50.6at%Ni alloy, Scripta Metallurgica, 15, 287-292.

Miyazaki, S., and Otsuka, K., 1986, Deformation and transition behavior associated

with the R-phase in Ti-Ni alloys, Metallurgical Transactions, 17 A, 53-63. Patoor, B., Barbe, P., Eberhardt, A., and Berveiller, M., 1991, Internal stress effect in the shape memory behavior, Journal de Physique IV, 1, 95-100, European Symposium on Martensitic Transformation and Shape Memory Properties. Perkins, J and Sponholz, R.O., 1984, Stress-induced martensitic transformation cycling

and two-way shape memory training in Cu-Zn-Al alloys, Metallurgical Transactions

A, 15A, 313-321. Perkins, J. and Bobowiec, P., 1986, Microstructural effects of martensitic transformation cycling of a Cu-Zn-Al alloy: Vestigial structures in the parent phase, Metallurgical

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Tanaka, K., Hayashi, T., and Itoh, Y., 1992, Analysis of thermomechanical behavior of shape memory alloys, Mechanics of Materials, 13, 207-215. Tanaka, K., Nishimura, F., Hayashi, T., Tobushi, H., and Lexcellent, C., 1995, Phenomenological analysis on subloops and cyclic behavior in shape memory alloys under mechanical and/or thermal loads, Mechanica of Materials, 19, 281-292.

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effect, AD-24/AMD-123, Smart Structures and Materials, ASME, 79-90.

THE TAYLOR ESTIMATE OF RECOVERABLE STRAINS IN SHAPE-MEMORY POLYCRYSTALS K. BHATTACHARYA*, R.V. KOHN** and Y.C. SHU* * California Institute of Technology, Pasadena, CA 91125, USA. ** Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA.

1. Introduction Shape-memory behavior is the ability of certain materials to recover, on heating,

apparently plastic deformation sustained below a critical temperature. Some materials have good shape-memory behavior as single crystals but little or none as polycrystals, while others have good shape-memory behavior even as poly crystals.

Bhattacharya and Kohn (1996, 1997) have proposed a framework to understand this difference. They use energy minimization and the Taylor estimate to argue that the recoverable strains in a polycrystal depend not only on the texture of the polycrystal and the transformation, but critically on the change in symmetry during the underlying martensitic phase transformation. Their results agree with the experimental observations. Shu and Bhattacharya (1997) have also used the

Taylor estimate to study the effect of texture in polycrystals of Nickel-Titanium and Copper based shape-memory alloys. The use of the Taylor estimate was evaluated in some detail in Bhattacharya and Kohn (1997) and more recently in Shu and Bhattacharya (1997) and Shu (1997). In this short report, we summarize the model of recoverable strain and discuss some results that allow us to evaluate the

Taylor estimate. Saburi and Nenno (1981) have given a very insightful discussion about recoverable strains. Our work can be seen as an attempt to make their ideas more quantitative. The source of the shape-memory effect is a martensitic phase transformation. A material which undergoes a martensitic phase transformation has two distinct crystalline structures: the parent or austenite phase, preferred at high temperatures; and the product or martensite phase, preferred at low temperatures. Typically, the austenite has greater symmetry than the martensite; consequently, the martensite phase occurs in k different symmetry-related variants. The variants have the same crystalline structure, but have a different orientation relative to the

austenite lattice. The number of variants k depends on the change of symmetry. There are 3 variants for a cubic-tetragonal transformation, 4 for cubic-trigonal, 6 for cubic-orthorhombic, and 12 for cubic-monoclinic.

123 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 123–134. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

124 Consider a single crystal of the high temperature austenite phase. As it is cooled below the transformation temperature, the transformation takes place, creating a mixture of variants of martensite. While the transformation strain of each

variant is different, the macroscopic effect of the transformation is negligible due to self-accommodation between the different variants. Now deform the sample. The variants rearrange themselves, if they can, so as to remain essentially stress-free by changing the microstructure. The resulting deformation appears macroscopically plastic: there is no restoring force since the variants in their new configuration are not stressed. But in fact, it is recoverable: heating the crystal above its transformation temperature turns each variant of martensite back to austenite and the crystal springs back to its original shape. Notice, however, that only certain strains can be recovered: those that can be achieved by the rearrangement of martensite variants. Larger strains will introduce stress, leading to lattice defects and nonrecoverability. Therefore, the recoverable strains for a single crystal can be identified with the strains achievable by stress-free mixtures of martensite variants. Now consider a polycrystal, consisting (in the austenite phase) of a large number of grains with different orientations. On cooling, each grain transforms to

a self-accommodated mixture of martensite variants. As the polycrystal is deformed, each grain tries to accommodate the strain by adjusting its microstructure of stress-free variants. The deformation is recoverable if they succeed. Therefore, the recoverable strains for a polycrystal are the macroscopic averages of locally varying strain fields which can be accommodated within each grain by rearrangement of the martensite variants. We wish to calculate the set of recoverable strains in a polycrystal. Energy minimization provides a very convenient framework to formulate the ideas stated above.

2. Energy Minimization and Recoverable Strain

2.1. THE RECOVERABLE STRAIN OF A SINGLE CRYSTAL Consider a single crystal of the austenite. Choose this as the reference configuration. Therefore, the austenite has stress-free strain The stress-free strain of the ith variant of martensite, can be calculated from the lattice parameters. We assume that given some temperature and boundary condition the single crystal will occupy a state that minimizes the total energy

Here is the region occupied by the crystal; is the linear strain associated with displacement u. We shall refer to the integrand as the microscopic elastic energy. It depends on temperature. We are interested in temperatures below the transformation temperature. Here, the microscopic energy is

125

nonconvex with multiwell structure - one well for each variant. For our purposes, it suffices to assume that the microscopic elastic energy is given by:

Notice that we have assumed that each variant is elastic. As a matter of convenience, we have assumed that the elastic modulus is identical (and equal to identity) in each of the variants. We have also ignored the austenite well (since it is higher than the martensite wells at this temperature and we are going to minimize the energy). Energy minimization with this multiple well energy leads to minimizing sequences which we interpret as microstructure or fine-scale mixtures of the different variants of martensite (Ball and James 1987). Therefore, the behavior of a single crystal is governed not by the microscopic energy, but the mesoscopic energy Physically, is the average stored energy when the average strain is e after taking into account the microstructure. Mathematically, it is obtained by the relaxation of and may be defined as

We note that the definition (3) does not really depend on the choice of domain

There are equivalent definitions using periodic rather than affine boundary conditions, or Young measures rather than spatial averages. Also note the elementary but important fact that the minimum value of is the same as that of We can now give a mathematically precise formulation of recoverable strains. The recoverable strains for a single crystal are the minimizers of its mesoscopic energy. Therefore, we can define the set of recoverable strains for a single crystal,

In general, it is very difficult to evaluate the set (see for example, Bhattacharya et al 1994). However, the following result from Bhattacharya (1993) allows us to evaluate it in physically relevant cases including cubic-tetragonal, cubic-trigonal and cubic-orthorhombic transformations. Proposition 1 Let us call two linear strains f and g “ elastically compatible” if

they satisfy for some vectors a and n. Suppose the stressfree strains are pairwise elastically compatible. Then the associated set , defined by (4), is simply their convex hull. The cubic-monoclinic case is different: its stress-free strains are not pairwise compatible, and the associated set is not the convex hull. This complicates the analysis, but we can study this case by embedding it into the cubic-orthorhombic case (Bhattacharya and Kohn 1996). We can already begin to see that the number of variants is crucially important. Note that the convexification of is always a lower bound for Therefore is

126 always contained in the convex hull of Since the martensite variants are symmetry related, they all have the same trace, i.e. Thus lies in a -dimensional subset of the 5-dimensional “deviatoric hyperplane” When e.g. when the phase transformation is cubic-tetragonal

or cubic-trigonal,

is dimensionally deficient: its dimension is strictly smaller

than that of the deviatoric hyperplane. When on the other hand, counting suggests that should be a set with interior in the deviatoric hyperplane. Another important fact is that when the austenite is cubic and the martensite is tetragonal, trigonal, orthorhombic, or monoclinic, contains the isotropic strain

This follows from the characterizations of given in Bhattacharya and Kohn (1996). It is linked to the phenomenon of self-accommodation (Bhattacharya 1992).

2.2. THE RECOVERABLE STRAIN OF A POLYCRYSTAL A polycrystal is an assemblage of grains, each composed of the same shape-memory material in a different orientation. We describe the texture of a polycrystal by a rotation-valued function R (x ). R (x) gives the orientation of the crystal at the point x relative to some fixed reference crystal. In a typical polycrystal, R is piecewise

constant, though we shall not assume any such restriction in what follows. The total elastic energy stored in a polycrystal is given by

in place of (1). However, we know that energy minimization leads to minimizing sequences and microstructure. Therefore, the minimization problem does not change if we replace the microscopic energy in (6) with the mesoscopic energy (Acerbi and Fusco 1984). Physically, we are assuming that the length scale of the martensitic microstructure is smaller than that of the grains. In order to discuss the behavior of the polycrystal, we introduce the macroscopic energy of the polycrystal is obtained from by homogenization. A convenient definition is based on affine boundary conditions:

This has the advantage of being easy to use, however the resulting depends on the details of the domain and the texture R (x ). One might say that it describes a “specific sample” of polycrystal. One can provide alternate definitions based on periodic polycrystals, random polycrystals or Gamma convergence. We do not fuss about which distinctions between these; instead we use the definition which is most convenient for the purposes at hand. A more careful reader is referred to

Bhattacharya and Kohn (1997).

127

We are now in a position to give a mathematically precise meaning for recoverable strains in a polycrystal. The recoverable strains for a polycrystal are the minimizers of its macroscopic energy. Therefore, we define the set of recoverable strains for a polycrystal Notice that we have implicitly assumed that has minimum value 0. This is a consequence of (5). The isotropic tensor is in so for every x. It follows that 3. The Taylor Estimate and its Implications

Our task is now clear. To assess the recoverable strain of a shape-memory polycrystal we must consider the associated macroscopic energy and estimate the set where This task might at first appear hopeless. However, there is an elementary but fundamental bound, based on the use of a constant-strain test field:

Proposition 2 The set

of recoverable strains contains at least the set

We call this the “Taylor bound” to highlight the analogy with Taylor’s uniformstrain hypothesis from polycrystal plasticity (Taylor 1938 and Bishop and Hill 1951). The physical meaning of the Taylor bound is clear. It describes the strains which can be accommodated without making use of cooperative effects between grains.

Proposition 2 says only that It is silent concerning how much bigger might be. We believe, however, that is usually a good indicator for More precisely, we conjecture that for polycrystals with sufficient symmetry (and under appropriate hypothesis on the set has the same dimension as We therefore like to call the Taylor estimate. For materials of interest, the set is never empty: it always contains at least the strain by (5). We say the Taylor bound is trivial if consists of just this one point. For such polycrystals Proposition 2 says only that but our conjecture says that In other words, such polycrystals should have no recoverable strain. Let us explain heuristically why should be trivial if is. We start with the observation that for to be trivial, must be dimensionally deficient, i.e. it must not span the entire deviatoric strain space. So the constraint restricts the mesoscopic strain to lie in a lower dimensional set, varying from grain to grain. This is a severe algebraic restriction. We expect it to be inconsistent with the linear differential constraints which must be satisfied if e(x) is to come from a deformation.

128 Our conjecture provides the essential link between the symmetry of the phase transition and the recoverable strains of polycrystals. When there are fewer than 6 variants (tetragonal or trigonal martensite) the set is dimensionally deficient, and is trivial. We conjecture that is also trivial. Therefore such materials lose their shape-memory behavior when formed into unstructured polycrystals. When there are 6 or more variants (orthorhombic or monoclinic martensite) the set contains a neighborhood of in the deviatoric hyperplane tr constant, so the same is true of and Polycrystals made from such materials can recover some strain in any deviatoric direction. The maximum recoverable strain is unclear, but we expect it to exceed the conservative estimate 4. Model Problems in Two Dimensions

In this section, we consider two model problems, the 2D diagonal, trace-free elastic

material and the 2D diagonal elastic material. For polycrystals made from first material we prove that has the same dimension as For polycrystals made from the second we show that the dimension of can be larger than that of This demonstrates that the accuracy of the Taylor estimate is a problem-specific matter rather than a universal one.

4.1. 2D DIAGONAL, TRACE-FREE ELASTIC MATERIAL In this case the microscopic energy is given by (2) with

and

It is possible to show that the mesoscopic energy is

It follows that

is the line segment from

to

and the Taylor estimate is trivial:

We can prove the following theorem for this material (Bhattacharya and Kohn 1997).

Theorem 3 For any polycrystal made from the 2D diagonal, trace-free material,

the macroscopic energy satisfies

129 Here when

and

are texture-independent constants, and the estimate is only asserted is sufficiently small.

This theorem tells us that for any polycrystal sufficient symmetry, And thus this result is consistent with our conjecture that the dimension of agrees with the dimension of This theorem can be proved using the translation method with the translation that has been used by Lurie and Cherkaev (1984) as well as

Avellaneda et al. (1996) to bound the effective shear modulus of a 2D linear elastic polycrystal. However the argument has a small twist. There, the analog of the mesoscopic energy is quadratic at the origin and one simply needs to use the non-negativity of the translated integrand. Here, the mesoscopic energy is not quadratic at the origin and hence one needs to use the boundedness of the Fenchel transform of the translated integrand. This is what gives the rather unexpected exponent of 4 in the lower bound (14). We wondered whether the exponent is an artifact of the method. We have not completely resolved this; however we will now argue that the behavior of a

polycrystal with sufficient symmetry is subquadratic near the origin. The essence of the matter is that linearization and homogenization should commute (Geymonat et al 1993). To explain what this means, we note that the quadratic approximation of the mesoscopic energy near 0 is

Hence this linear material is degenerate in the sense that the shear moduli are 0 and 1. “Linearization then homogenization” means starting with the linear polycrystal whose local energy is then passing in the usual way to its effective energy “Homogenization then linearization” means considering our nonlinear macroscopic energy then taking its quadratic approximation near If the two operations commute then we should have

We can use the results by Lurie and Cherkaev (1984) for any polycrystal with sufficient symmetry, with So we conclude that is flatter than quadratic. The general statement that linearization and homogenization commute requires strict convexity of the energy however, we can adapt the proof to this problem (Bhattacharya and Kohn 1997).

4.2. 2D DIAGONAL ELASTIC MATERIAL In this case the microscopic energy is given by (2) with

and

130

The mesoscopic energy is

The set of recoverable strains for the single crystal is the convex hull of

a

two-dimensional square in the three-dimensional space of symmetric matrices:

This is a consequence of Proposition 1, applied separately to and to The Taylor estimate contains only multiples of the identity:

in particular, it is one-dimensional. This example is interesting because the Taylor estimate is not reliable in this case, not even for polycrystals which are macroscopically isotropic. Consider the

polycrystal obtained by layering the basic crystal with its rotation by using layers normal to (1,1) and equal volume fraction. It is very easy to show that

in a neighborhood of 0 for this laminated polycrystal. Though this rankone laminate is highly anisotropic we can use it to make isotropic polycrystals by mixing it with itself in different orientations. Thus, one can get isotropic polycrystals of the material with in a neighborhood of 0. Thus can be three-dimensional even though is one-dimensional.

5. Cubic-Tetragonal Material

Consider a material that undergoes cubic to tetragonal transformation. It has variants and

It is then a matter of calculation to show (Bhattacharya and Kohn 1996) that

There is no loss of generality if we restrict ourself to the volume-preserving case We may also suppose Then is an equilateral triangle centered at the origin in the two-dimensional subspace of diagonal, trace-free strains,

131 The vertices of

are

and

5.1. UNIAXIAL POLYCRYSTAL We now consider polycrystals with texture, in other words those in which all the rotations leave the unchanged. No restriction is placed on the geometry of the grains. It is not convenient to work with the convexification of this time. Instead we shall use the more symmetric function

where

and

are the orthogonal projections of e onto V and

and is the radius of the smallest ball in V containing This is a convex lower bound for the relaxation of (2). The sets and associated with the real cubic-tetragonal energy are different from those associated with (24). To avoid confusion we shall write for the sets defined using the more symmetric energy (24). Since this is a lower bound for the true cubic-tetragonal energy we have

We have evaluated the Taylor estimate in Bhattacharya and Kohn (1996) for a polycrystal with this texture. When specialized to the incompressible case that

calculation gives

and a similar calculation gives

In this case, we can prove the following theorem (Bhattacharya and Kohn 1997) using the translation method.

Theorem 4 For the symmetrized cubic-tetragonal energy (24), any uniaxial polycrystal satisfies

Here

and

are absolute constants, independent of texture and the value of

The estimate is asserted only when

is sufficiently small.

132

This theorem tells us that for any uniaxial polycrystal with sufficient transverse symmetry, the dimension of is at most one. Therefore, according to (26), the dimension of is also at most one. This agrees with the dimension of and hence this result is consistent with our conjecture.

5.2. GENERAL POLYCRYSTAL We now consider polycrystals which do not have a Taylor estimate

texture. In this case, the

is trivial (Bhattacharya and Kohn 1996). We conjecture in a

polycrystal with sufficient symmetry, the set will also be trivial. Unfortunately, we have not been able to prove (or contradict) such a result. However, we present the following example in support of our conjecture. Consider the rank-two laminated polycrystal shown in Figure 1. It is possible to show that there exists such that

Therefore, for this polycrystal, the set is trivial. Also note that we have a quadratic lower bound in contrast to the situations in the 2D diagonal, tracefree elastic material and the uniaxial polycrystal. At this time we are unable to comment on the reasons or the significance of this difference. Laminated polycrystals have been studied in some detail in (Shu and Bhattacharya 1997 and Shu 1997). We also show that for any finite-rank laminated polycrystal made of this material,

In other words, the dimension of

can be no greater than the dimension of

133 Further, for any such polycrystal with sufficient symmetry, the dimension of indeed zero and hence equal to the dimension of

is

6. Discussion

We have conjectured that for polycrystals with sufficient symmetry, same dimension as Here is a summary of the results in its favor.

has the

• Bhattacharya and Kohn (1997) give a very detailed analysis of a scalar model problem where the Taylor estimate is trivial and for any polycrystal with sufficient symmetry is also trivial. • We have discussed a similar example in Section 4.1 in two-dimensional elasticity.

• The uniaxial polycrystal made from the cubic-tetragonal material has a onedimensional Taylor estimate (Section 5.1). We have shown that for polycrystals with sufficient (transverse) symmetry is also one dimensional. • We construct an example of a laminated polycrystal of a material undergoing cubic-tetragonal transformation where the Taylor bound is trivial. We show that the set is also trivial (Section 5.2). The behavior of laminated polycrystals of a material undergoing cubic-trigonal transformation is also very similar. There is also one result running contrary. For our 2D diagonal elastic material, the Taylor estimate is one-dimensional, however there are polycrystals, even isotropic ones, for which is three-dimensional (Section 4.2). Thus our conjecture is false in this case. In particular, it cannot be taken as a universally valid assertion about nonlinear homogenization. Rather, it must reflect some (as yet undetermined) feature(s) of the set We wonder whether the following “lamination test” is one such feature. Let us denote by the set of recoverable strains for any rank-one laminated polycrystal. The lamination test would say that if for every rank-one laminated polycrystal, then the in a polycrystal with sufficient symmetry. In other words, our original conjecture will fail exactly when our set is such that we can increase the set of recoverable sets by simple lamination. This is consistent with the 2D diagonal elastic material in Section 4.2. And this is in some sense the spirit behind the use of lamination in the cubic-tetragonal material in Section 5.2. However, we are far from proving such a result. Indeed, the fact that the class of rank-one convex functions is different from the class of quasiconvex functions (Šveràk 1992) says that one should be cautious about such a lamination test. Acknolwedgment We gratefully acknowledge the partial financial support of the

AFOSR, ARO and NSF.

134

References [1] Acerbi, E. and Fusco, N. (1984) Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal. 86, 125-145.

[2] Avellaneda, M., Cherkaev, A.V., Gibiansky, L.V., Milton, G.W., and Rudel-

son, M. (1996) A complete characterization of the possible bulk and shear

moduli of planar polycrystals, J. Mech. Phys. Solids 44, 1179-1218.

[3] Ball, J.M. and James, R.D. (1987) Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal. 100, 13-52.

[4] Bhattacharya, K. (1992) Self-accommodation in martensite, Arch. Rat. Mech. Anal. 120, 201-244. [5] Bhattacharya, K. (1993) Comparison of the geometrically nonlinear and linear theories of martensitic transformation, Cont. Mech. Thermodyn. 5, 205-242. [6] Bhattacharya, K., Firoozye, N.B., James, R.D. and Kohn, R.V. (1994) Restrictions on microstructure, Proc. Roy. Soc. Edinburgh 124A, 843-878. [7] Bhattacharya, K. and Kohn, R.V. (1996) Symmetry, texture, and the recoverable strain of shape-memory polycrystals, Acta Mater. 44, 529-542. [8] Bhattacharya, K. and Kohn, R.V. (1997) Elastic energy minimization and recoverable strains of polycrystalline shape-memory materials, Arch. Rat. Mech. Anal. 139, 99-180. [9] Bishop, J.F.W. and Hill, R. (1951) A theory of the plastic distortion of a polycrystalline aggregate under combined stresses, Phil. Mag. 42, 414-427. [10] Geymonat, G., Müller, S., and Triantafyllidis, N. (1993) Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity, Arch. Rat. Mech. Anal. 122, 231-290.

[11] Lurie, K.A. and Cherkaev, A.V. (1984) G- closure of some particular sets of admissible material characteristics for the problem of bending of thin plates, J. Optim. Th. Appl. 42, 305-316.

[12] Saburi, T. and Nenno, S. (1981) The shape memory effect and related phenomena, in H.I. Aaronson et al. (eds.), Proc. Int. Conf. on Solid-Solid Phase Transformations, The Metall. Soc. AIME, New York, pp. 1455-1479. [13] Shu, Y. C., PhD. Thesis, California Institute of Technology, in preparation. [14] Shu, Y. C., and Bhattacharya, K. (1997) The effect of texture on the shapememory effect in polycrystals, in preparation. [15] Šveràk, V. (1992) Rank-one convexity does not imply quasiconvexity, Proc. Royal Soc. Edin. 120A , 185-189.

[16] Taylor, G.I. (1938) Plastic strain in metals, J. Inst. Metals 62, 307-324.

THERMOMECHANICAL MODELING OF SHAPE MEMORY ALLOYS AND APPLICATIONS C. LEXCELLENT AND S. LECLERCQ* Laboratoire de Mécanique Appliquée R. C., UMR 6604 CNRS-UFC UFR Sciences et Techniques - 24 rue de I'Epitaphe 25030 BESANCON CEDEX - FRANCE * Electricité de France, Direction des Etudes et Recherches Les Renardières - Ecuelles 77250 MORET SUR LOING - FRANCE

Abstract

The aim of the present paper is a general macroscopic description of the thermomechanical behavior of shape memory alloys (SMA). We use for framework the thermodynamics of irreversible processes. This model is efficient for describing the behavior of "smart" structures as a bronchial, a tentacle element and an prosthesis hybrid structure made of Ti Ni SMA wires embedded in a resin epoxy matrix. 1. Introduction

In order to account for the true behavior of shape memory alloys (SMA) (including stress-strain-temperature coupling), a thermodynamically consistent formulation has been devised. Leclercq and Lexcellent (1996) presented a macroscopic description that allows the simulation of the global thermomechanical behavior of SMA. We use the framework of the thermodynamics of irreversible processes. Two internal variables are taken into account : the volume fraction of self-accommodating (pure thermal effect) and the oriented (stress-induced) product phase. A specific free energy, valid in the total range of phase transition, is defined with particular attention to the interaction term. A study of the thermodynamic absolute equilibrium during phase transition proves its instability, and hence explains the hysteretic behavior of SMA. It is shown that the kinetic equations for the internal state variables, proposed in this paper satisfy the second law of thermodynamics. The postulate of five yield functions (each of them being related to one process) permits the phase transition criteria to be defined and the kinetic equations related to each process through consistency equations to be derived. In the stress-strain temperature space, this efficient model is capable of predicting the behavior of a SMA element used as actuator in a smart structure. The first application is devoted to a bronchial prosthesis structure made of a silicon sheet with embedded Ti Ni thin plates (Leclercq et al. ; 1996). In this case, the pseudoelastic behavior is considered. In the second application, we present an experimental and theoretical study of a tentacle element made of a flexure pivot structure and a predeformed shape memory Ni Ti actuator wire controlled by Joule's effect (Lexcellent

135 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 135–145. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

136 et al. ; 1996). The third application concerns a Ni Ti fibre embedded in a resin epoxy matrix (Thiebaud et al. ; 1996). First, we present the foundations of the SMA model, and then give the results obtained for the three smart structures.

2. Thermodynamic Principles and Constitutive Equations of the General Model The model presented here rests upon the classical local state postulate ; this assumes that there exists a representative volume where one can define and measure some internal quantities called internal variables. In the following, the elastic strain the temperature T, the volume fraction of self-accommodating martensite and the volume fraction of oriented martensite will be taken as variables of the Helmholtz free energy expression. Following an idea of Brinson (1993), we split the total martensite fraction z into two parts : (i) obtained under pure thermal action (austenite martensite) ; this is selfaccommodating (each variant having its own "complement") and hence without any macroscopic phase transition strain.

(ii) obtained by an external mechanical action (namely a stress tensor and associated with a macroscopic phase strain. The second basic assumption that is laid down concerns the partition of strains. The terms of the second order in the calculation of strains are neglected, so that the total strain is assumed to satisfy the following equation :

where

and

are, respectively, the elastic and the transformation strain.

2.1. SPECIFIC FREE ENERGY The Helmholtz free energy of the three-phase system considered in this study is chosen in the following form : where

The term

is the total volume fraction of the product phase :

from (2) is the free energy of the phase : corresponds to parent phase (austenite) corresponds to self-accommodating martensite corresponds to oriented martensite

Equation (4) gives the expression for this energy :

137

Here

and

are the specific energy and entropy of the -phase,

order tensor of the intrinsic elastic strain of the -phase.

is the second

corresponds to the fourth-

order tensor of isotropic elasticity.

Moreover, the terms and are, respectively the mass density, the actual temperature, the equilibrium temperature and the specific heat at constant volume. As an extension of the ideas of Raniecki et al. (1992), Huo and Müller (1993), the "configurational energy" which represents the interaction that appears between the phases, typically the incompatibilities between deformations, can be expressed by :

where

and

are constants.

2.2 SOME CONSTITUTIVE ASSUMPTIONS The intrinsic strain

is split in elastic and transformation parts :

Since we assume here that both parent and product phases have the same elastic constants represented by we can show that the elastic parts of the intrinsic

deformations are the same for each phase, so that:

Since only, the oriented product phase is responsible for the phase transition strain, the following holds :

Several questions arise regarding

rises. With resistance electrical measurement in situ

(giving the martensite fraction evolution) during pseudoelastic tensile tests

Vacher and Lexcellent (1991) established the proportionality between

and z.

Raniecki et al. (1992) proposed an extension of Vacher-Lexcellent's statement to

comply with the three dimensional situation

For this case of isothermal pseudoelasticity

and

and

where is the total pseudoelastic uniaxial strain obtained with the tensile test, classical von Mises equivalent strain and dev the stress deviator tensor.

the

138 One can easily note that the expression of complies with the normality rule to an ellipsoid in the case of proportional loadings. By writing it in rates, and assuming the rate of to be zero, one obtains

This choice of the expression may change in the future to take the following behavior into account : (i) asymmetry observed in tension-compression (Vacher and Lexcellent (1991), Orgeas and Favier (1995)), (ii) non proportional loadings.

2.3 SECOND LAW OF THERMODYNAMIC As usually done, we assume that the thermal dissipation term is non-negative, so that, the mechanical dissipation part coming from the Clausius-Duhem inequality may be written in a classical form.

with Introducing (10) and dividing by

where

we may write (1 la) as

is defined as a chemical potential of phase transition

and will be referred to as the thermodynamical forces associated with respectively.

and

Note that does not explicitly depend on the applied stress, this agrees with the fact that the oriented product phase appears only under a mechanical action. In case of orientation of the self-accommodating martensite, Thus

(12a) reads

139

where

Here, is referred to as the thermodynamical force associated with reorientation of the self-accommodating product phase. 2.4 EQUILIBRIUM CONDITIONS AND STABILITY

In their first paper about the model (devoted to pseudoelasticity), Raniecki et al. (1992) developed a systematic study of the equilibrium related to such solid-solid phase transition. In this paper, we will not repeat this study, but following their reasoning, we note that the two equations and give the absolute equilibrium states of the system. The stability of this equilibrium is related to the sign of the second derivative of the free energy with respect to the internal variables and which corresponds to the first

derivative of matrix

and

whose determinant

with respect to

and

Hence one has to deal with the

is

The matrix S is symmetric and thus has real eigenvalues. The product of these eigenvalues is equal to For classical transformation, so that the

eigenvalues have opposite sign and the equilibrium is unstable ; the material will not follow a stable equilibrium state during phase transition. For the intermediate phase transition in Ti Ni alloys, is positive and the states of equilibrium are stable and the behavior can be considered as reversible (Lexcellent et al. ; 1994). In the particular case of the reorientation of self-accommodating martensite, the equilibrium is represented by The instability of this equilibrium follows from the differentiation of

with respect to

140 2.5 SYSTEM EVOLUTION AND KINETICS

The instability of the equilibrium implies that there is as yet no thermodynamic relations that could give the equation of the branches of the hysteresis loop. Nevertheless, one needs such relations in order to write explicitely the evolution kinetics of and These equations have to be combined with the behavior equations (10) and (11b). Let us assume that there are five functions and that are respectively linked to the forward phase transition (F : austenite martensite) the reverse phase transition (R : martensite austenite) and the reorientation process of self-accommodating martensite. Taking a similar framework as in classical plasticity, we assume that the functions defined below are assumed to be constant during the phase transition they are linked with.

The constants and are non negative and and are functions taking zero value at the beginning of forward or reverse phase transition. They are chosen to obtain classical kinetic forms and also play a role in the phase transition criteria. By differentiation of the consistency equations one obtains for forward phase transition

and for reverse phase transition

Reorientation (self-accommodating martensite into reoriented martensite)

141

Some appropriate isothermal and anisothermal mechanical tests permit the determination of the fourteen material parameters :

3. Application to a Bronchial Prosthesis (Leclercq et al. ; 1996)

A bronchial prosthesis is a small device used in medecine when the diameter of a patient's bronchial tube is diminished due to cancer, for example. Figure 1 shows a photograph of this prosthesis. It is made of curved thin plates of metals embedded in two sheets of silicon. The prosthesis is used as follows : it is rolled up on itself to a very small diameter, in order to be inserted in the catheter. Then, the catheter is put in the human body and the prosthesis is pushed out at the right place. The tube returns to its initial shape (large diameter) and applies a stress on the walls of the bronchial tube in order to prevent it

from shrinking.

To avoid applying an excessive stress to the walls and hence a possible damage to the human body, thin plates made of SMA Ti Ni are used in pseudoelastic effect conditions. The isothermal pseudoelastic version of the present model is implemented in a finite element code POLYFORM (Daniel et al. ; 1993) and hence the stress applied by the prosthesis can be calculated. Thus, the dimensions of the curved plate can be established in accordance with the medical specification.

142 4. Experimental and Theoretical Study of a Tentacle Element Made of a Flexure Pivot Structure and a Shape Memory Alloys Actuator Wire (Lexcellent et al. ; 1996) The device is made of a flexure pivot structure (hinge) and a predeformed shape memory alloy as an actuator controlled by Joule's effect (Fig. 2). A simple heating of a Ti Ni wire allows a recovery stress to be created in the wire and hence a flexion of the hinge which is considered as elastic. In a first stage, appropriate thermomechanical tests are performed on the wire itself in order to obtain the material parameters of the unified model in non-isothermal conditions. In a second stage, modeling of the tentacle element is performed to allow for the control of the flexion of the hinge for an applied current intensity. Finally, one measures the forward (and reverse) path of the wire acting on the tentacle in the space with respect to the heating (and cooling) of the SMA (fig. 3, 4).

143

5. Smart Material Characterization

This section is concerned with thermal characterization of a Ti Ni SMA embedded in a epoxy resin matrix (Thiebaud et al ; 1996). When an electrical current is applied to the Ti Ni wire, an infrared camera measures of the surface temperature at the lateral faces. The integration of the heat equation in a finite element code gives the local temperature of the resin and consequently predict its viscoelastic behavior (fig. 5). Furthermore, the local stress-strain-temperature field can be obtained when a

predeformed Ni Ti wire embedded in resin is heated and hence produces recovery stress. Hence, the stiffness (and consequently the resonance frequencies) of this hybrid system change with the applied electrical current. The modeling is performed using viscoelastic laws for the epoxy resin (Le Moal and Perreux ; 1994) and the present model for the SMA fibre. In these structures, the main technological problem concerns the quality of the interface between the SMA wire and the epoxy resin matrix. Ni Ti fibres are sanded, anodized... and the surface state observed by profilometry. The quality of the interface is evaluated with "pull out" tests and the stress repartition is evaluated with photoelasticity measurements.

144

145 6. Conclusion

The aim of the present paper was to present a global approach including a thermomechanical modeling of SMA, and to use this model to describe the behavior of "smart" structures. Some improvements must be performed to accommodate non proportional loadings. To this end, Boyd and Lagoudas (1996) give a key which could be used in the writing of in eq (9). The description of the non-symmetric behaviour in tension compression will be soon included in the formulation of the transformation strain (Raniecki and Lexcellent; 1998). One has now to try to develop applications using SMA as actuators. A possible way could be to use SMA elements acting outside the structure. This solution seems more efficient than the classical one using SMA wires embedded into an epoxy resin matrix.

Acknowledgement The authors wish to thank the participants to the three applications and particularly : H. Benzaoui, F. Thiebaud, B. Zeghmati, J.F. Charmoillaux and B. Gabry. References Boyd, J.G., Lagoudas, D.C. (1996) A thermodynamical constitutive model for shape memory materials, Part I : the monolithic shape memory alloy, Int. J. of Plasticity 12, 6 805-842. Brinson, L.C. (1993) One dimensional constitutive behavior of shape memory alloys : thermomechanical derivation with non-constant functions and redefined martensite internal variable, J. Intelligent Mater. Syst. Structures 4 229-242. Daniel, J.L., Gelin, J.C., Paquier, P. (1993) Polyform-logiciel 3D pour la simulation du forgeage et de 1'embou-tissage des matériaux, Proc. Coll. Nat. Calcul Struct. Huo, Y., Müller, I. (1993) Non equilibrium thermodynamics of pseudoelasticity. Continuum Mech. Thermodyn. 5 163-204. Le Moal, P., Perreux, D. (1994) Evaluation of creep compliances of unidirectional fibre-reinforced composites, Composites Sci. and Tech. 51 469-477. Leclercq, S., Lexcellent, C. (1996) A general macroscopic description of the thermomechanical behavior of shape memory alloys, J. Mech. Phys. Solids, 44, 6 953-980.

Leclercq, S., Lexcellent, C., Gelin, J.C. (1996) A finite element calculation for the design of devices made of shape memory alloys, J. de Phys. IV 6 225-234.

Lexcellent, C., Benzaoui, H., Leclercq, S., Bourjault, A. (1996) Experimental and theoretical study of a tentacle element made of a flexure pivot structure and a shape memory alloy actuator wire, Proc. of 3rd France-Japan Congress and 1st Europe-Asia Congress on Mechatronics 2 566-570.

Lexcellent, C., Tobushi, H., Ziolkowski, A., Tanaka, K. (1994) Thermodynamical model of reversible Rphase transformation in Ti Ni shape memory alloys, Int. J. Press. Ves. Piping 58 51-57. Orgeas, L., Favier, D. (1995) Non-symmetric tension-compression behaviour of Ni Ti alloy, J. de Phys. IV 605-610. Raniecki, B. and Lexcellent, C. (1998) Thermodynamics of isotropic pseudoelasticity in shape memory alloys, to appear in Eur. J. Mech. Solids/A.

Raniecki, B., Lexcellent, C., Tanaka, K. (1992) Thermodynamic models of pseudoelastic behaviour of shape memory alloys, Arch. Mech., 44 261-284. Thiebaud, F., Zeghmati, B., Charmoillaux, J.F., Lexcellent, C. (1996) Smart material thermal characterization : Ni Ti shape memory alloy embedded in a resin epoxy matrix, Proc. of 3rd Int. Conf. on Int. Mat. 535-540. Vacher, P., Lexcellent, C. (1991) Study of pseudoelastic behaviour of polycristalline shape memory alloys by resistivity measurements and acoustic emission, Proc. of ICM VI Kyoto 3 231-236.

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ON THE DESIGN OF HYDROPHONES MADE AS 1-3 PIEZOELECTRICS O. SIGMUND 1 , S. TORQUATO2, L. V. GIBIANSKY 2 and I. A. AKSAY2 1

Department of Solid Mechanics, Technical University of Denmark,

DK-2800 Lyngby, Denmark 2

Department of Civil Engineering and Operations Research and Princeton Materials Institute, Princeton University Princeton, N.J. 08544, USA

Abstract We compare two different approaches to the design of 1 -3 piezocomposites with optimal performance characteristic for hydrophone applications. The performance characteristics we focus is the

hydrostatic charge coefficient The piezocomposite consists of piezoelectric rods embedded in an optimal polymer matrix. In the first approach, we allow the material parameters of the polymer matrix to be free design variables and we find the polymer parameters which give the optimal hydrophone performance. In the second approach, we use the topology optimization method to design the optimal (porous) matrix microstructure. Using the former approach, we find theoretical bounds for hydrophone behavior and using the latter, we design practically realizable but less efficient hydrophones. For both approaches, the optimal transversally isotropic matrix material has negative Poisson’s ratio in certain directions.

1. Introduction

Piezoelectric transducers have been employed as sensors and transmitters of acoustic signals in ultrasound medical imaging, non-destructive testing and underwater acoustics (Newnham and Ruchau 1991, Smith 1991). Besides these traditional applications, miniaturizing transducers opens up for a variety of new applications. Low-frequency transducers for underwater acoustics are known as hydrophones. This paper studies a class of composite piezoelectric transducers for hydrophone applications: composites consisting of an array of parallel piezoceramic rods embedded in a polymer matrix with electrode layers on the top and bottom surfaces (see Fig. 1). Following Newnham’s connectivity classification (Newnham 1986), we refer to this structure as a 1-3 composite. The composite is poled in the longitudinal (vertical or direction. Under an incident underwater acoustic field, the composite transmits strain to the ceramic rods which is transformed into a longitudinal voltage difference and thus acts as a sensor. Conversely, application of an alternating current will give rise to an acoustic field by the converse piezoelectric effect.

147 Y.A. Bahei-El-Din and G.J. Dvorak (eds.),

IUTAM Symposium on Transformation Problems in Composite and Active Materials, 147–158. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

148

One may ask why one would want to make a composite to begin with or, in other words, why is pure piezoceramic not used since it is the only material with piezoelectric properties? The basic problem is that under hydrostatic load, the anisotropic piezoelectric response of pure PZT is such that it has poor hydrophone performance characteristics. Specifically, consider a PZT rod poled in the axial direction subjected to hydrostatic load. The induced polarization field in the axial direction is found to be proportional to the applied pressure, i.e.,

where is the dielectric displacement in the T is the amplitude of the applied pressure, is the hydrostatic coupling coefficient, and are the longitudinal and transverse piezoelectric coefficients characterizing the dielectric response for axial and lateral compression, respectively. Unfortunately, and have opposite signs, thus resulting in a relatively small hydrostatic coupling factor For instance, PZT5A has

and

Therefore,

which is

small compared to As we will see in this paper, a polymer/piezoceramic composite can have a sen-

sitivity that is orders of magnitude greater than a pure piezoceramic device. Using a piezo/polymer composite, the factor of 2 on the transverse piezoelectric coefficient in Eq. (1) can be lowered, or even change sign, if we use a soft matrix material or a matrix

material with negative Poisson’s ratio (e.g. Smith 1991), thereby ensuring a much higher hydrostatic charge coefficient. The use of piezocomposites in hydrophone design has been studied in several pa-

pers. Hydrophones composed of piezoelectric rods in solid polymer matrices have been tested experimentally (Klicker et al. 1991, Newnham and Ruchau 1991). Using simple models in which the elastic and electric fields were taken to be uniform in the different phases, Newnham (1986), Chan and Unsworth (1989), and Smith (1991) qualitatively explained the enhancement due to the Poisson’s ratio effect. A more sophisticated analysis has recently been given by Avellaneda and Swart (1995) using the so-called differentialeffective-medium approximation. They found the effective performance factors: the hy-

drostatic charge coefficient the hydrostatic voltage coefficient and the electromechanical coupling factor as functions of the effective moduli of the composite and simple structural parameters. In Avellaneda and Swart (1995), it is assumed that the matrix

149

material is isotropic. This paper compares the results of two different approaches to the design of anisotropic matrix materials for 1-3 piezoelectrics. The first approach is the one by Gibiansky and Torquato (1997), who found theoretical bounds for hydrophone design using the elastic properties of the matrix material as design variables. In contrast to Avellaneda and Swart (1995), they allowed the matrix material to be transversely isotropic. However, they did

not consider finding the actual matrix microstructure corresponding to the optimal elastic properties. The second approach (Sigmund et al. 1997) takes the first steps towards closing this gap by designing the optimal microstructural matrix topology simultaneously with the optimization of the hydrophone performance. In this paper we will briefly discuss the two different approaches and make a comparison of the results obtained from the two methods. In section 2, we review how the effective hydrophone behavior is found using the homogenization theory (based on Avellaneda and Swart (1995)). Section 3 reviews the free material approach by Gibiansky and Torquato (1997) and section 4 reviews the topology

optimization based approach by Sigmund et al. (1997). In section 5 we compare the results of the two methods and section 6 discusses how the matrix microstructure can be build in practice. 2. Effective properties of the piezocomposite

The constitutive relations for elastic, piezoelectric media can be written as

where

sors and

are the stress, strain, dielectric displacement and electric field ten-

and

are the effective stiffness and clamped-body permitivity tensors,

respectively, and

is the effective piezoelectric stress matrix related to the effective

piezoelectric strain matrix through the relation A commonly used measure of hydrophone performance is called the hydrostatic charge coefficient which can be written as

where

For later use, the dilatational compliance is defined as

where

is the effective transverse bulk modulus.

150 2.1 Effective hydrophone properties The effective properties of the piezocomposite are calculated under the assumptions that: the length-scale associated with the microscopic variation of the matrix material is well below the diameter of the piezoelectric rods; the wavelengths of the incident acoustic

field is much longer than the size of the rods; we have perfect bonding between the matrix phase and the rods and that the stiffness of the matrix material is much lower than the stiffness of the piezoelectric rods. Using these assumptions, Avellaneda and Swart (1995) found the effective properties of 1-3 piezocomposites

where f is the volume fraction of piezoelectric rods, indices, (i) and (m) refer to piezoelectric and matrix properties, respectively, is the transverse bulk

modulus and p is a structural parameter defined as

Following Gibiansky and Torquato (1997), the hydrophone performance is optimized when the effective transverse bulk modulus is taken as the lower Hashin-Shtrikman bound

implying, that the piezoelectric rods should be ordered in a hexagonal array to ensure optimality of the composite. 3. Free material design approach

To optimize the hydrophone performance, (Gibiansky and Torquato 1997) allows the matrix properties 1 as well as the volume fraction of the piezoelectric rods to be free design variables. We assume that the matrix is comprised of an isotropic polymer with stiffness tensor and a void phase. The only restriction is given by the inequality

1

A free material design method similar to this was introduced for optimal compliance design by BendsØe et

al. (1994)

151 where

By a tensor inequality of the form

we mean that the difference

is a positive semi-definite matrix. The lower bound is introduced to prevent very low stiffnesses of the matrix material. The matrix properties and volume fraction of piezo-electric rods that maximize the hydrostatic charge coefficient with constraints given by Eq. (9) are found semianalytically using symbolic manipulation (Maple V). The procedure is described in detail in Gibiansky and Torquato (1997). 4. Topology optimization approach

For this approach, the microstructural topology of the matrix material and the volume fraction of piezoelectric rods are the design variables. To find the optimal matrix topology we use the so-called numerical topology optimization method originally suggested by BendsØe and Kikuchi (1988). The general topology optimization procedure determines for every point in space whether there is material at that point or not. Alternatively, discretizing the design domain by finite elements, every element is either solid or void. The topology optimization method has previously been applied to the design of materials with extreme elastic properties in Sigmund (1994a), (1994b) and (1995). A topology optimization method for practical three-dimensional material design has been described in Sigmund (1996) and is here used for the design of piezo-composites. The method, for design of optimal piezocomposites (Sigmund et al. 1997), essentially follows the steps of conventional topology optimization procedures. It is an iterative pro-

cedure, each iteration implying a homogenization procedure (finite-element analysis) to determine the effective properties of the porous matrix material, an evaluation of the effective piezoelectric properties (using equations from Avellaneda and Swart (1995)), a sensitivity analysis determining the change in objective function subject to matrix microstructural change and finally determining the optimal change in the porous matrix topology using linear programming. 4.1 Design discretization and matrix properties The design problem consists in finding the optimal microstructural topology of the matrix material which is a porous polymer. We start by discretizing the periodic base cell by a number, N, of 8-node cubic linear-displacement finite elements (using from several hundred to several thousand elements). The design procedure will determine whether

each of the elements should be either solid or void, allowing us to define a microstructure composed of small boxes (finite elements) as sketched in Fig. 2. To allow for the design of a detailed microstructure, N should be at least several thousand. Following the idea of standard topology optimization procedures, the problem is relaxed by allowing the material in a given element to have intermediate densities. The elastic tensor of each element e is written as a function of the design variable (element density), i.e.,

where is a penalty factor which ensures that the solution consists of entirely solid and ”void" elements. The vector of design variables x is defined as an N-vector containing the

152

design variables. The value of each design variable is bounded to the domain where is a small number is greater than zero to ensure non-singularity of the finite element stiffness matrix). The effective elastic properties of the matrix material as a function of the vector of element densities x, can be computed using numerical homogenization methods, based on finite-element calculations, as described in Bourgat (1977). Having found the effective matrix properties in the preceding subsections, the hydrostatic charge coefficient can be found as a function of the design variables and the volume fraction of the rods using Eqs. 6, 7 and 8. 4.2 The optimization problem

We consider the following optimization problem

Minimize Subject to

where is the global objective function and ε is a small number. The differential-effective-medium approach used to calculate the hydrophone properties assumes that the matrix material is transversally isotropic. Specifying horizontal, and two vertical symmetry planes in the base cell ensures orthotropy of the matrix material. To ensure transversal isotropy, two additional criteria must be fulfilled, namely that and that For computational reasons the equality constraints are implemented as inequality constraints in the actual optimization problem, implying that the squared error in obtaining isotropy should be less than a given small value. The square normalized error can be written as

153 For some design examples the in-plane bulk modulus approaches zero when we try to maximize the hydrophone performance. This would result in an impractical (very soft) design, hence, we introduce the lower bound constraint on the in-plane bulk modulus. The optimization problem Eqs. (11) is non-linear, and must be solved iteratively. To solve it, we use a sequential linear programming (SLP), which consists in the sequential solving of an approximate linear subproblem, obtained by writing linear Taylor series expansions for the objective and constraint functions.

The optimal volume fraction of piezoelectric rods f is found by a golden sectioning method in each iteration step. Having found the optimal f, the sensitivities of the hydrostatic performance coefficients and of the inplane matrix bulk modulus with respect to change in design variable are derived analytically as functions of the matrix constitutive tensor and the sensitivities thereof The sensitivities can be found directly from the strain fields already computed by the homogenization procedure and are calculated locally for each element (e.g. Sigmund and Torquato 1997b). This means that no additional finite-element problems have to be solved to find the sensitivities needed. To avoid numerical problems in the topology optimization method, such as "checkerboards" and mesh-dependencies, we use the filtering or mesh-independency technique suggested in Sigmund (1994a) and (1997). The mesh-independency algorithm is also used to control that the microstructure only has one (manufacturable) length-scale. 5. Design examples In this section, we apply the two approaches to the design of a piezo composite with maximum For other design examples and discussions of other hydrophone performance criteria, the reader is referred to Gibiansky and Torquato (1997) and Sigmund et al. (1997). 5.1 Properties of the piezoceramic and polymer The actual properties of the PZT-ceramic rods are taken as (upper 3 by 3 part of the 6 by 6 matrix)

The Young’s modulus of a typical amorphous polymer material is 2.5· and the Poisson’s ratio is 0.37 which give the following values of the polymer stiffness tensor

The minimum value of the in-plane bulk modulus of the matrix material is chosen as 3% of solid polymer i.e. for the free material approach and

154 Table 1: Effective values for pure piezoceramic, optimal piezocomposite with solid matrix and optimal piezocomposites obtained using the free material approach and the topology optimization approach.

for the topology optimization approach. The minimum volume fraction of the piezorods is

5.2 Example: free material approach The analytically obtained properties are shown in Tab. 1. The resulting effective properties

of the matrix material are

or

and the horizontal and vertical Young’s moduli

and respectively. We note that the vertical Poisson’s ratio is negative, which means that horizontal forces are inverted and act like compressive

forces and result in the enhancement of the hydrostatic charge coefficient. This means that the negative Poisson’s ratio of the matrix material makes the effective positive, thus enhancing the overall hydrostatic behavior. 5.3 Example: Topology optimization approach The base cell is discretized with 16 by 16 by cubic finite elements. By variable linking due to symmetry, the number of design variables (element densities) can be

decreased to The resulting optimal microstructure for maximization of the hydrostatic charge coefficient is seen in Fig. 3. The resulting effective properties of the matrix material are

or

ratio is negative.

and the horizontal and vertical Young’s moduli respectively. Again we note that the vertical Poisson’s

155 The negative Poisson’s ratio behavior of the microstructure in Fig. 3 can be difficult to imagine. To visualize the mechanism behind the negative Poisson’s ratio behavior, we show a two-dimensional interpretation in Fig. 3. Seen from the front (1-3 plane), the negative Poisson’s ratio behavior is seen to resemble the mechanism behind the inverted honeycomb structure (Almgren 1985, Kolpakov 1985). Seen from the side (2-3 plane), the mechanism is seen to be slightly different.

5.4 Comparison of the approaches By comparing the effective properties of the two alternatives from the free material approach and the topology optimization approach, we note that the mechanisms behind the enhancement are the same. Both approaches result in matrix properties with negative Poisson’s ratio’s which eliminate the factor 2 in Eq. 1. Comparing the actual value of the hydrostatic charge coefficients we see that free material design gives a three times higher value. On the other hand, the free material designed devise is half as stiff (the dilatational

compliance is doubled) as the topology optimized devise. Although the two materials are not directly comparable, we can conclude that the free material based parameters are more extreme than the topology based parameters. This can be explained by the fact that the free material based approach allows the full set of material properties only constrained by Eq. 9, whereas the matrix properties for the topology optimization approach come from a physically realizable microstructure which is restricted to have variation on one length scale. How the obtainable properties are limited when restricting the design domain to one length-scale composites was discussed in Sigmund and Torquato (1997a) (in these

proceedings). As a conclusion, we can say that the approach by Gibiansky and Torquato (1997) gives theoretical limits on hydrophone design whereas the approach by Sigmund et al. (1997) gives physically realizable devises bul less extreme hydrophone properties. 6. Manufacturing

Various options exist for the fabrication of the optimal three-dimensional microstructures. Our approach is based on a stereolithography method developed by 3-D Systems, Inc. (Jacobs 1992). In this method, a laser beam is focussed onto a photocurable liquid polymer solution. The microstructure obtained from the topology optimization approach is built layer by layer by spreading a thin film, with layer thicknesses between 50 to 200 microns, and then laser curing the film to define a pattern. The layering is repeated multiple times until the desired three dimensional body is completed. A prototype consisting of one base cell in larger scale (8 mm cubed) is shown in Fig. 4. Currently, we are working

on miniaturizing and repeating the periodic microstructure. Two dimensional microstructures with negative Poisson’s ratios have been manufactured in microscale (base cell size 50 microns) at Mikroelcktronik Centret, Denmark Technical University (Larsen et al. 1996).

156

157 7. Conclusions

This paper has shown how hydrophone performance can be increased by orders of magnitude by use of systematic methods for the design of polymer matrix material with piezoelectric rod inclusions. Our goal was to design a hydrophone for maximum hydrostatic charge coefficient

We compared two different approaches. In the first we allowed the elements of the constitutive tensor of the matrix material to be free design variables. The resulting devise enhances the value of the hydrostatic charge coefficient by a factor of 47 compared with the performance of solid piezoelectrics and a factor of 22 compared with an optimal composite of piezo rods and solid polymer. In the second approach we used a topology optimization method to design practically realizable microstructures and obtained enhancement factors of 16 and 7, respectively. Except for a prototype of a single unit cell, the suggested hydrophone devices have, so far, not been built in practice. We expect that imperfect interface bonding, packaging and other practical problems will degrade the overall performance of the hydrophone designs suggested here. Nevertheless, the suggested microstructures can provide guidance for further developments of hydrophone design. We considered fixed topology of the rods (vertical rods). The next step will be to let the shape of the rods be free to vary as well. This can be done using the three-phase topology method developed in Sigmund and Torquato (1996, 1997b) and (1997a) (these proceedings).

Acknowledgements

Most of this work was done while the first author was visiting Princeton University. The authors express their gratitude to M.P. Bendsøe and P. Pedersen for helpful discussions. R. Garg’s help in producing prototype cells is also gratefully acknowledged. This work received support from the ARO/MURI Grant DAAH04-95-1-0102 (OS, ST and IA) and from Denmark’s Technical Research Council (Programme of Research on Computer-

Aided Design) (OS). References Almgren, R. F.: 1985, An isotropic three-dimensional structure with Poisson's

Journal

of Elasticity 12, 839–878. Avellaneda, M. and Swart, P. J.: 1995, Calculating the performance of 1-3 piezocomposites for hydrophone applications: an effective medium approach, Working paper, Courant Inst. of Math. .

Bendsøe, M. P., Guedes, J. M., Haber, R. B., Pedersen, P. and Taylor, J. E.: 1994, An analytical model to predict optimal material properties in the context of optimal structural design, Transactions of the ASME, Journal of Applied Mechanics 61(4), 930–937. Bendsøe, M. P. and Kikuchi, N.: 1988, Generating optimal topologies in optimal design using a homogenization method, Computational Methods in Applied Mechanics and Engineering 71, 197–224.

158 Bourgat, J. F.: 1977, Numerical experiments of the homogenization method for operators with periodic coefficients, Lecture Notes in Mathematics, Springer Verlag, Berlin, pp. 330–356.

Chan, H. L. W. and Unsworth, J.: 1989, Simple model for piezoelectric ceramic/polymer in ultrasonic applications, IEEE, Transaction of Ultrasonic Ferroelectrics in Frequency Control 36,434–441. Gibiansky, L. V. and Torquato, S.: 1997, Optimal design of 1-3 composite piezoelectrics, Structural Optimization 13, 23–28. Jacobs, P.:

1992, Rapid Prototyping and Manufacturing – Fundamentals of Stereolithography,

SME, Dearborn, MI, USA. Klicker, K. A., Biggers. J. V. and Newnham, R. E.: 1991, Composites of PZT and Epoxy for hydrostatic transducer applications, Journal of the American Ceramics Society 64, 5. Kolpakov, A. G.: 1985, Determination of the average characteristics of elastic frameworks, PMM Journal of applied Mathematics and Mechanics, U.S.S.R. 49, 739–745. Larsen, U. D., Sigmund, O. and Bouwstra, S.: 1996, Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio, IEEE, International Workshop on Micro Electro Mechanical Systems, MEMS-96.

Newnham, R. E.: 1986, Composite electroceramics, Ferroelectrics 68, 1–32. Newnham, R. E. and Ruchau, G. R.: 1991, Smart electro ceramics, Journal of the American Ceramics Society 74(3), 463–480. Sigmund, O.: 1994a, Design of material structures using topology optimization, PhD thesis, Department of Solid Mechanics, Technical University of Denmark. Sigmund, O.: 1994b, Materials with prescribed constitutive parameters: an inverse homogenization problem, International Journal of Solids and Structures 31 (17), 2313–2329. Sigmund, O.: 1995, Tailoring materials with prescribed elastic properties, Mechanics of Materials

20,351–368. Sigmund, O.: 1996, Design and manufacturing of material microstructures and micromechanisms, in P. Gobin (ed.), Proceedings of the third international conference on intelligent material, ICIM96, Lyon, June, SPIE vol. 2779, pp. 856–866. Invited paper. Sigmund, O.: 1997, On the design of compliant mechanisms using topology optimization, Mechanics of Structures and Machines 25(4).

Sigmund, O. and Torquato, S.: 1996, Composites with extremal thermal expansion coefficients, Applied Physics Letters 69(21), 3203–3205. Sigmund, O. and Torquato, S.: 1997a, Design of materials with extreme elastic and thermoelastic

propeties using topology optimization, in these proceedings. Sigmund, O. and Torquato, S.: 1997b, Design of materials with extreme thermal expansion using a three-phase topology optimization method, Journal of the Mechanics and Physics of Solids

45(6), 1037–1067. Sigmund, O., Torquato, S. and Aksay, I. A.: 1997, On the design of 1–3 piezocomposites using topology optimization, Journal of Materials Research (to appear).

Smith, W. A.: 1991, Optimizing electromechanical coupling in piezo composites using polymers with negative Poisson’s ratio, proceedings of IEEE Ultrasonics symposium, IEEE.

Functionally Graded Materials

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A STRESS FUNCTION FORMULATION FOR A CLASS OF EXACT SOLUTIONS FOR FUNCTIONALLY GRADED ELASTIC PLATES

A J M SPENCER Department of Theoretical Mechanics University of Nottingham

Nottingham NG7 2RD, UK

Abstract We consider isotropic linearly elastic plates in which Young’s modulus and Poisson’s ratio vary in an a r b i t r a r y specified manner through the thickness. If this dependence is continuous the material may be regarded as a functionally graded elastic material; the case in which it is discontinuous

represents a. laminate. It is shown that a large class of exact solutions of the three-dimensional elasticity equations for this material is generated by stress functions which satisfy the two-dimensional biharmonic equation. The theory is illustrated by a solution for a semi-infinite crack in an in homogeneous elastic plate.

1.

Introduction

There is considerable current interest in materials that are deliberately constructed so as to he in homogeneous on the macroscopic scale. Such materials may have discontinuous mechanical properties, as with a laminated material, or have continuously varying properties, in which case they are termed graded materials. In the case of laminates, the main interest

is in anistropic laminae but functionally graded materials are frequently isotropic on the macroscopic scale. In this paper we consider elastic plates of isotropic, in homogeneous material with elastic moduli that vary continuously in the through-thickness direction (although with slight modifications the theory is also applicable to laminates in which the elastic moduli are piecewise continuous in the through-thickness direction) but are uniform in the in-plane directions. In previous papers (Rogers 1990, Spencer 1990 a n d Rogers and Spencer 1989), a large class of solutions of the three-dimensional elasticity equations 161 Y.A. Bahei-El-Din and GJ. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials. 161–172.

© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

162 was derived for in homogeneous isotropic linearly thermoelastic plates whose Lamé elastic constants and µ are specified (not necessarily continuous) functions of the through-thickness coordinate z. The case considered was

that of a symmetric plate in which and are even functions of z, so that the stretching and bending deformation modes uncouple. Spencer and Mian (1996) extended this work to include non-symmetric plates, in which case the stretching and bending solutions are coupled. The ideas underlying the solutions originate in a paper by Michell (1900) (see also Love, 1927), which gave exact plane stress solutions for homogeneous moderately thick elastic plates. In the above papers it was shown that Michell’s solutions can be extended to include through-thickness inhomogeneity. The result is that any solution of the classical two-dimensional equations for stretching and bending of thin plates generates, in a simple manner, an exact solution of the three-dimensional elasticity equations for an isotropic in homogeneous plate. There is no restriction on the plate thickness.

The formulation in the papers cited above was in terms of average inplane displacements and the mid-surface d e f l e c t i o n . In t h i s paper we present an alternative f o r m u l a t i o n in terms of a stress f u n c t i o n , w h i c h is described in Sections 2 - 5. In Section 6 the theory is applied to the problem of a

semi-infinite crack in a plate. 2.

Notation and General Theory

In this section, all vector and tensor q u a n t i t i e s are referred to a system of rectangular cartesian coordinates 0xyz. Displacement components are denoted by where

The stress tensor

and the i n f i n i t e s i m a l strain components by

is given by

etc.

163

The equations of equilibrium are, with negligible body forces

The stress-strain relations for an isotropic elastic material can be expressed as

where E is Young’s Modulus and is Poisson’s ratio. Alternatively the stress-strain relations ran be expressed in terms of the Lamé elastic moduli and The strain components must satisfy the strain compatibility relations

and four similar relations obtained by cyclic permutation of x, y, z in (5). For present purposes it is convenient to express ( 5 ) in terms of the stress components, using (4), as

164 with given by cyclic permutations of x, y, z. These six compatibility conditions are not completely independent, but collectively are sufficient for compatibility of the strain components. We consider inhomogeneous materials in which E and are specified functions of z (subject to the conditions of positive-definite strain energy). These functions need not be continuous. If E and are piecewise continuous we have the important special case of a laminated or layered material. When E and vary continuously with z the material may be regarded as a functionally graded material. The theory may be applied to, for example the half-space or to a plate bounded by the surfaces z = ±h. In the case of a plate, if E and are even f u n c t i o n s of z

then the plate is symmetric: otherwise it is non-symmetric. Laminated plates are often constructed so as to be s y m m e t r i c , but graded materials are frequently non-symmetric. The case of a homogeneous material is recovered by choosing E and to be constants. 3.

Classes of Exact Solutions

Spencer and M i a n (1996) gave t h e following classes of solutions of the threedimensional elasticity e q u a t i o n s (a) Quasi-stretching solutions:

where and commas denote differentiation with respect to the indicated variable. (b) Quasi-bending solutions:

and

is the two-dimensional Laplacian operator.

165 These are exact solutions of the three-dimensional elasticity equations provided that and satisfy the two-dimensional thin plate theory equations

(and hence

where

and the coefficients

and

are given by

Here primes denote differentiation with respect to z, and and are constants formed from weighted through-thickness averages of functions of and µ(z) (for details see Spencer and Mian (1996)). Equations (12) can be solved by simple quadratures. The integration constants that arise can be chosen to satisfy boundary conditions on the plate lateral surfaces, for example zero traction on the surfaces It is important to note that any solution of the thin-plate two-dimensional equations (10) generates an exact three-dimensional solution of the elasticity equations for any (not necessarily continuous) variation of and through the thickness of the plate.

4.

Stress Function Formulation

The solutions (7) and (9) are expressed in terms of and as primary variables; it is straightforward to determine the stress components once u, v and w are known. However in some problems it is convenient to work directly in terms of the stress components. In plane elasticity this is normally achieved by introducing the Airy stress function

such that

166 The equations of plane elastic stress or plane elastic strain are then satisfied provided that obeys the biharmonic equation

We here seek solutions of the three-dimensional elasticity equations, w i t h E and depending on z, in terms of a stress function This is essentially a reformulation of the solutions ( 7 ) a.nd (9). Consideration of (7), (9) and the corresponding stresses suggests looking for solutions for the stress of the form

By substituting in (3) it follows that t h e equilibrium equations are satisfied provided t h a t and

It follows immediately that

where α and β are constants t h a t can be chosen to satisfy certain boundary conditions on the lateral surface. In particular, if on then and so It is also necessary to satisfy the strain compatibility conditions. By substituting (15) into (6) and expressions derived from (6) by cyclic permutation of x, y, z it follows that the compatibility conditions (5) and their cyclic permutations are all satisfied if (16) holds and P(z), Q(z), R(z), M(z), K(z) and J(z) satisfy

167 With (17), these form a set of six equations for the six coefficients P(z), Q(z), R(z), M(z), K(z) and J(z). They can be integrated consecutively by quadratures. For the remainder of this paper we consider a plate bounded by with traction-free lateral surfaces, so that

Then it follows from (15) and (17) that

and hence, from (17),

Then integrating

and

gives

where c, d, e are integration constants and

From (17),

and (23)

where

and which determines one of the constants c, d, e in terms of the other two. It then follows from that

168 where

and m are further integration constants and

In the case of a laminated material, E(z) and are discontinuous at interlaminar interfaces, and their derivatives are not defined at the corresponding values of z. The integration of ( 1 9 ) then requires f u r t h e r consideration. For simplicity we assume for the remainder of this paper that E(z) and are diflerentiable functions of z. When the material is homogeneous, so that E and are constants, the integration of (17) and ( 1 9 ) is elementary. For a symmetric plate, the constants c, e and relate to solutions in which and are even functions of z and are odd functions of z (ie stretching solutions), whereas d and m relate to solutions in which are odd in z, and are even in z (ie bending solutions). For non-symmetric plates the stretching a n d bending solutions do not uncouple in this way. After taking note of (27), four of the five constants remain disposable. Without loss of generality, one of these may be absorbed into the stress function χ. The others may be chosen to satisfy various additional conditions. For example, if we choose them so that

then the bending moments

are all zero, and it also follows from shear force resultants

( 2 2 ) and

that the

are zero. Thus in this case the plate is subject only to in-plane (stretching) stress resultants, although in the case of a non-symmetric plate these will, in general, give rise to out-of-plane displacements.

169

Alternatively the choice

(together with (22)) ensures that the in-plane stress resultants are zero, and the plate undergoes only bending moments and shear force resultants, although for a non-symmetric plate these, in general, cause in-plane stretches. Of course, these quasi-stretching and quasi-bending solutions can be superposed if so required by the edge boundary conditions. Major simplification arises if the plate is symmetric, and the results become even simpler if the plate is homogeneous, in which case they become equivalent to exact solutions given by Love(1927) for plane stress of moderately thick plates. The significance of these results is t h a t any solution of the biharmonic equation in two dimensions generates an exact three-dimensional solution of the elasticity equations with general inhomogeneity in the zdirection. Since many solutions of the biharmonic equation are known, the process generates a large class of exact, three-dimensional, inhomogeneous elasticity solutions.

5.

Cylindrical Polar Coordinates

The relations (15) are readily transformed to give the stress components referred to cylindrical polar coordinates r, θ, z in terms of χ. The expressions are

170 where

and as before, χ satisfies the biharmonic equation

and P(z), Q(z), R(z), M(z), K(z) and J(z) remain as before. For future reference, it is noted that in the two-dimensional theories of elastic plane stress and plane strain, the in-plane stress components are given in terms of a biharmonic stress function as

6.

Semi-infinite Crack in a Plate

In plane elasticity, the stress f u n c t i o n

satisfies the biharmonic equation and gives rise through (33) to stress

components

In particular, on

and on The stress function therefore describes the solution to the problem of an opening crack lying along the x-axis from to the origin. More generally it gives the asymptotic solution for the stress near a crack tip at the origin, for an opening crack lying in the negative x axis. The square root singularity in (35) ensures that the crack has finite strain

energy density. It is natural to apply a stress function χ of the form (34) to find an analogous solution, through (32), for a crack in an inhomogeneous material. However inspection of (32) and (34) shows that this procedure will give rise to crack-tip stress components of order as well as of order through the terms in R (z) in (32). This implies unacceptable singular strain energy

171

density at the crack-tip. At least a partial solution of this difficulty is to seek solutions of the form

The additional term with coefficient is harmonic and so generates stress of order but no higher order. Then by substituting (36) in (32) we obtain

where

Ideally, the stress component terms of order

should be eliminated. This

is not possible pointwise, but their through-thickness averages can be made

zero by choosing

such that

Then, on the crack faces

It is also required that the resultant shear force on the crack surface be

zero, so that

which, from ( 1 7 ) , and

The stress in the plane

is equivalent to

ahead of the crack is then

172

and on this plane

Thus

may be regarded as a mean stress intensity factor, but this mean may conceal large through-thickness variations in the crack-tip stress. Four further conditions are required to determine the ratios of the constants c, d, e, l, m. Two of these are given by (27) and (41). To specify

l and m it is necessary to impose two further conditions on R(z), or equivalently on S(z). Acknowledgement The author thanks the Leverhulme Trust for the award of an Emeritus

Fellowship. References Love, A.E.H. (1927) The Mathematical Theory of Elasticity, 4th

edn. Cambridge

University Press. Mitchell, J . H . (1900) On the direct determination of stress in an elastic solid, with applications to the theory of plates, Proc. London Math. Soc. 31, 100-124.

Rogers, T.G. (1990)Exact three-dimensional bending solutions for inhomogeneous and laminated elastic plates, in G. Eason and R.W. Ogden (eds), Elasticity, mathematical methods and applications, Ellis Horwood, Chichester, pp. 301-313.

Rogers, T.G. and Spencer, A . J . M . (1989) Thermoelastic stress analysis of moderately thick inhomogeneous and laminated plates, Int. J. Solids Structures 25 1467-1482. Spencer, A . J . M . (1990) Three-dimensional elasticity solutions for stretching of inhomogeneous and laminated plates, in G. Eason and R.W. Ogden (eds), Elasticity, mathematical methods and applications, Ellis Horwood, Chichester, pp. 347 - 356. Spencer, A.J.M. and Mian M.A. (1996) Exact solutions for functionally graded and laminated elastic materials. X I X t h International Congress of Theoretical and Applied Mechanics, Abstracts p445, Kyoto, Japan.

MICROMECHANICAL MODELING OF

FUNCTIONALLY GRADED MATERIALS THOMAS REITER * and GEORGE J. DVORAK Center for Composite Materials and Structures Rensselaer Polytechnic Institute

Troy. NY 12180-3590, U.S.A.

Abstract Thermoelastic response of graded composite material is examined for both uniform changes in temperature and steady-state heat conduction in the gradient direction. Detailed finite element studies of the overall response and local fields in the discrete

models were conducted, using large plane-array domains with simulated skeletal and particulate microstructures. Homogenized layered models with the same composition gradient and effective properties, derived from the Mori-Tanaka and/or self-consistent

methods, were analyzed under identical boundary conditions. Comparisons of temperature distributions and the overall and local fields predicted by the discrete and homogenized models were made using a C/SiC composite system with very different Young’s moduli of the phases, and relatively steep composition gradients. Close agreement with the discrete model predictions is observed for homogenized models which derive effective properties estimates from several averaging methods: In those

parts of the graded microstructure which have a well-defined continuous matrix and

discontinuous reinforcement, the effective moduli, expansion coefficients and heat

conductivities are approximated by the Mori-Tanaka estimates. In skeletal microstructures that often form transition zones between clearly defined matrix and reinforcement phases, the effective properties are approximated by the self-consistent estimates. Subject to these selection rules, the averaging methods originally developed for statistically homogeneous aggregates under uniform overall fields may be applied to graded material subjected to nonuniform overall loads. A complete description of this investigation was presented by T. Reiter, G. J. Dvorak and V. Tvergaard, J. Mech. Phys. Solids, 45, 1281-1302, and in a forthcoming paper in the same volume. The results do not suggest that nonlocal or new micromechanical theories are needed for modeling functionally graded materials. Such theories appear appropriate only in those limited volumes of the material where the field averages are very small and their gradients very large.

* On leave from the Institute of Lightweight Structures, Technical University of Vienna, Austria.

173 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 173–184. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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1. Introduction

This paper examines thermomechanical behavior of graded composite materials, consisting of one or more dispersed phases of directionally variable volume fractions embedded in a matrix of another phase. The solution domains under consideration are

subdivided by internal percolation thresholds or wider transition zones of skeletal microstructures between the different matrix phases. A detailed description of the actual geometry of graded microstructures is usually not available, except perhaps for information on volume fraction distribution and approximate shape of the dispersed phase or phases. Therefore, evaluation of overall response and local stresses and strains in graded materials must rely on analysis idealized models. Several micromechanical models have been developed over the years for analysis of macroscopically homogeneos representative volumes composite materials under uniform overall stress. Of course, such representative volumes are not easily identified in systems with variable phase volume fractions. Also, admissible applications of the standard micromechanical models to loading by nonuniform overall fields have not been explored. However, application of these method appears to be justified by the slow density changes, and by the relatively small ratios of field gradients to field averages found in most graded systems.

The purpose of the present study is to is to examine if the available models can be applied with reasonable degree of confidence to graded microstructures subjected to nonuniform loading. To this end, comparisons of the overall thermoelastic response, local mechanical fields, and temperature fields under steady-state heat conduction, are made for selected discrete models and homogenized layer models of the graded microstructures, where the homogeneos layer moduli are estimated by available micromechanical methods, in terms of the local phase properties and volume fractions. In order to make good agreement more difficult to achieve, a C/SiC composite system with large differences in phase properties and steep composition gradients was used in the comparisons.

2. Discrete and homogenized models

The graded material models selected for the comparative studies are based on a planar hexagonal array of inclusions in continuous matrices. Figure 1 shows three discrete microstructures, designated as Model 1_2 with a distinct percolation threshold, Model 2 with a wide skeletal transition zone, and Model 3 which has both a wide transition region and a threshold. Five computer-generated Model 2 microstructures were evaluated and the results averaged for presentation. Figure 2 shows details of the two overlapping arrays used in generating the graded material models, with a relative displacement in the equal to one half width of one hexagonal cell. Also shown is a single-layer array which was not used because of its propensity to early clustering. Both arrangements provide for rows of hexagonal cells parallel to the the number of inclusions in each row defines the row volume fraction. Volume fraction gradients in the are simulated by changing the number of inclusions in subsequent rows; many different gradient magnitudes can be generated in this manner. Each of the discrete models consist of 50 rows of hexagonal cells, with 40 cells per row. The five end rows are homogeneous, then one inclusion is added in each next row. The resulting gradient is uniform, equal to

175

0.025/row, of the same magnitude in all three models. The composition gradients observed in actual microstructures are usually much smaller, equivalent to about 0.005 to 0.0025/row. As implied by the use of planar arrays, the typically particulate microstructure of a graded material was replaced by a graded fibrous system which was more easily implemented in a discrete model.

Realization of the graded microstructures in numerical calculations was made using the C/SiC system (Sasaki, et al.,1991). Both phases were regarded as isotropic with the

176

following phase elastic moduli conductivity

and

coefficients of thermal expansion

and heat

Phase properties of the carbon/silicon carbide system

These properties were also used in the homogenized models, which consisted of many layers. One such layer is identified in Fig. 2. The layer thermoelastic properties and heat diffusivities were estimated by either the self-consistent or Mori-Tanaka averaging methods. The relevant formulae are listed in the Appendix. The C/SiC system appears to be particularly suitable for comparisons of different micromechanical models because of the large differences in the Young's modulus, thermal conductivities and expansion coefficients. Together with the relatively steep composition gradients used in the models, the choice of dissimilar phase properties serves to enhance the heterogeneity of the model materials. Two replacement schemes were employed in development of the homogenized models. First, the properties of all layers in the model domain were estimated by each of the methods. The Mori-Tanaka method was used in two versions, once with the matrix properties equal to those of phase 1, and once with those of phase 2; these estimates are labeled as MTM1 and MTM2 in the figures below. SCS denotes homogenization by the self-consistent method. The second replacement scheme employed combination of methods, as shown in Fig. Ic. One of the appropriate version of the Mori-Tanaka methods was employed in each of the subdomains with a well-defined matrix, and the self-consistent estimate in the transition domain with a skeletal microstructure; certain transition functions were utilized in this case to make the property variation continuous. Figure 3 illustrates the variation of the transverse Young's modulus of Model 3, shown in Fig. 1c, predicted by these replacement schemes. The second replacement scheme provides a step-wise change of the modulus, with COMBS. 1 corresponding to the domain subdivision indicated in Fig. 1. The COMB3.2 model was similar to COMB3.1, but without the self-consistent estimate in the skeletal zone. Both traction, displacement and mixed boundary conditions involving uniform and linearly varying distributions of tractions or displacements were applied to the material models. Also, constant or variable temperature changes were applied. As an example, Fig. 4 shows the conditions that allow cylindrical bending of a cantilever layer of the graded material under a temperature gradient generated by steady-state heat flow. Similar conditions were applied to simulate a uniform temperature change of Other boundary conditions used in related parts of the study corresponded to a linear gradient in the in-plane shear stress on planes const., also to a single surface shear component applied to a fully supported gradient coating, and to both constant

177

and linearly varying distribution of the transverse normal strain with zero tractions on the lateral surfaces.

on plane

178 In each comparison, the same mesh geometries and boundary conditions were applied to both homogenized and discrete models. The thermal and mechanical fields were obtained from two-dimensional finite element solution, using ABAQUS

generalized plane strain elements. The domain was bounded in the thickness direction by

two planes parallel before deformation, that could move relative to each other. The

deformation in the thickness direction was assumed limited to a uniform normal strain;

resultants of the external forces and moments on the bounding planes were equal to

zero. The fine subdivision of the mesh and the small thickness of the homogenized layers relative to particle size caused oscillations in the computed average stresses; these were reduced in the figures by plotting a three-layer moving average of the computed results. 3. Selected comparisons of discrete and homogenized models

Figure 5 shows the distribution of the transverse normal stress, computed in each of the three discrete models of a graded layer in Fig.l, under loading by surface displacements on a plane const. that cause uniform normal strain through the thickness of the layer. The lateral surfaces are free of tractions. Also shown are estimates of the said stress by the self-consistent and both Mori-Tanaka methods. This comparison uses the first replacement scheme described in the previous section, with each estimate of layer elastic constants evaluated through the thickness the layer. As

179 expected under the prescribed boundary conditions, in those parts of the layer where there is a distinct matrix phase, the computed transverses stress is closely approximated by the appropriate Mori-Tanaka estimate. In the presence of a sharp percolation boundary, as in Model 1_2, there is a steep transition of the stress magnitude from the MTM1 estimate, valid in the phase 1 (carbon) matrix part, to the MTM2 estimate valid in the phase 2 (silicon carbide) matrix part. In contrast, the stress distribution in the discrete Model 2, with a extensive skeletal zone, is well approximated by the self-consistent prediction through the entire thickness. On the other hand, the presence of both skeletal zone and percolation threshold in Model 3 is responsible for the transition from MTM1 to SCM and later to MTM2, as seen in the figure. The indicated stress distributions through the thickness of the layer were also followed by the average stresses in the phases. Similar agreements between discrete and homogenized model predictions were found under linearly varying transverse overall normal strains, and also under linearly varying overall shear stresses in the transverse plane. Of course, such loading conditions define the longitudinal stress or strain in each layer and thus limit mutual interactions between layers. However, when the loading conditions do not provide such well-defined distribution of a transverse field, the first replacement scheme that estimates properties of all layers by a single averaging method becomes less reliable. This was observed, for example, when a supported graded layer was subjected to a uniform longitudinal shear stress on its surface. Figure 6 shows both overall and local average transverse strains in a graded layer of Model 3in Fig. 1c, under a uniform change of temperature. The mechanical boundary conditions are those shown in Fig.4. The second replacement scheme was used, with

180 layer properties estimated by the different methods indicated in Fig. 1c. Recall that

the estimates of transverse Young's moduli of the layers appear in Fig. 3, and that other moduli, as well as the expansion coefficients and heat conductivities follow similar step wise distributions. Here, Phase 0 denotes the effective medium, deformed by overall strains. Note the close agreement between the overall transverse strain computed for the discrete Model 3 and homogenized model COMB3.1. Very good agreement is also observed between the phase strain averages. In contrast, when the first replacement scheme was used in this comparison, both MTM1 and MTM2 homogenizations of Model 3 showed significant deviations from the discrete model response. However, a satisfactory agreement was seen between the Model 2 response and the self-consistent model.

Next, the graded layer of Model 3 was subjected to the thermomechanical boundary

conditions shown in Fig. 4. Assuming that the thermal and mechanical responses are

not coupled, the steady state temperature distribution caused by the prescribed heat flow is evaluated first, and then applied together with the mechanical constraints to

the graded layer. Figure7 presents the temperature distributions obtained in the discrete Model 3, together with several homogenized model predictions. Note again the good agreement between Models 3 and COMB3.1. The COMB3.2 model, which is similar to COMB3.1., but lacks the self-consistent approximation in skeletal zone, shows less satisfactory agreement. The SCS prediction deviates also, but had shown good agreement with the response of the discrete Model 2. Both MTM1 and MTM2

181 homogenizations through the entire thickness show considerable differences from discrete model predictions.

182

Figure 8 compares the overall and local transverse strain predictions by discrete Model 3 and model COMB3.1. Note the constant overall (Phase 0) strain value through the thickness of layer; this is the desirable response of a graded material, however, at the small temperature difference of 100 C. To appreciate the complexity of the local stress distributions obtained with the various models, we present in Figure 9 the stresses in Phase 2 (SiC) caused by the loads described in Fig. 4. Note again the good agreement between the predictions of Model 2 and SCS, and Model 3 and COMB3.1.

4. Conclusions The results indicate that overall response and average overall and local fields in materials with a single composition gradient can be reliably evaluated by judiciously selected homogenized layer models. The layer properties can be estimated by standard averaging methods. In those parts of the microstructure with a well-defined matrix, by the Mori-Tanaka method with the appropriate matrix phase. In parts with skeletal microstructure, the layer property estimates should be obtained from the self-consistent method. In any event, the results do not indicate any need for development of new micromechanical methods for analysis of graded materials. However, as noted by Dvorak and Zuiker (1995), overall response of even statistically homogeneous composites is sensitive to large field gradients in those material volumes where the field averages are of negligible magnitude. Of course, such volumes must be, by definition, very small.

Acknowledgement: The work of TR was supported by a grant from the Max Kade Foundation, and that of GJD by the DARPA/ONR University Research Initiative project on Mechanism-based design of composite structures at Rensselaer. Partial funding came from the Direktor Ib Henriksen's Fulbright Grant to GJD at the Department of Solid Mechanics, Technical University of Denmark. Appendix Here we summarize expressions needed to estimate elastic moduli, thermal expansion coefficients and thermal conductivity of the homogenized layer material by the two methods used in the text. As in Table 1, both phases are assumed to be isotropic and the particulate reinforcement distribution random; the effective medium that represents; the composite is also isotropic. Each layer is regarded as a representative volume of the composite material, with phase volume fractions derived from the ratio of the phase volumes contained in the finite elements that subdivide each layer of the discrete models of Fig. 1. The denote the bulk and shear moduli and the volume fraction of the matrix phase. The denote the elastic constants and volume fraction of the particle phase; The Mori-Tanaka estimate of the overall bulk and shear moduli K and G of such particle-reinforced composite was derived by Benveniste (1987) as,

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with the coefficients derived by Berryman (1980),

The self-consistent estimate of the bulk and shear moduli of the above composite system was obtained by Hill (1965) as,

where Note that these are implicit expressions for the unknown K and G, and that they are invariant with respect to phase exchange. It turns out that for has the range After substituting for the first equation can be solved for K in terms of G:

while G can be obtained by solving the quartic equation,

Analogous expressions for composites reinforced by aligned or randomly oriented isotropic short fibers or platelets were derived with the self-consistent method by Walpole (1969). The Mori-Tanaka estimates of the moduli for such systems are given by (A1), with appropriate and taken from Berryman (1980); for anisotropic reinforcement see Chen, et al., (1992). The overall moduli of fiber composites can be estimated in a similar manner. In the present work, the replacement of the two-phase model by the homogenized model utilized the Mori-Tanaka estimates by Chen, et al., (1992), and the self-consistent estimates by Hill (1965). The coefficient of thermal expansion for isotropic composites with isotropic phases follow exactly from the overall and phase moduli and phase expansion coefficients (Levin 1967),

where K is the overall bulk modulus, determined either by the self-consistent or one of the Mori-Tanaka methods. The effective thermal conductivity of the layers was found with the same method used in approximating the elastic moduli, under the assumption of zero thermal resistance at interface boundaries. The Mori-Tanaka estimates for particulate and fibrous composites with isotropic phases were found by

184

Hatta and Taya (1986), in terms of phase conductivities an isotropic two-phase composite with reinforcement matrix the results is,

and volume fractions For by spherical particles in a

The self-consistent counterpart to (A5) was derived by Hashin (1968), in the implicit form,

In addition to the overall thermomechanical properties, averages of local stress and strain fields caused by uniform overall stresses or strains and uniform changes in temperature can be evaluated by the self-consistent or Mori-Tanaka methods. The relevant procedures can be found in the original references; their summary appears in Reiter, Dvorak and Tvergaard (1997). References Benveniste, Y., 1987, A new approach to the application of Mori-Tanaka's theory in composite materials, Mechanics Mater., Vol.6, pp. 147-157. Berryman, J. G., 1980, Long wavelength propagation in composite elastic media, II.Ellipsoidal inclusions, J. Acoust. Soc. Am. 68, 1820-1831. Chen, T., G. J. Dvorak and Y. Benveniste, 1992, Mori-Tanaka estimates of the overall moduli of certain composite materials, J. Appl. Mech., 59,539-546. Dvorak, G. J. and J. R. Zuiker, 1995, Effective local properties for modeling of functionally graded composite materials, IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, ed. By D. F. Parker and A. H. England, Kluver Acad Publishers, Dordrecht, pp.103108. Hashin, Z.,1968, Assessment of the self consistent scheme approximation: conductivity of particulate composites, J. Compos. Mat. 2, 284-300. Hatta, H., and Taya, M, 1986, Equivalent inclusion method for steady state heat conduction in composites. Intl. J. Engng. Sci., 24, 1159-1172. Hill, R.,1965, A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13,213-222. Levin, V. M., 1967, Thermal expansion coefficients of heterogeneous materials, Mekhanika Tverdogo Tela 2,88-94. Mori, T.and Tanaka, K., 1973, Average stress in matrix and average energy of materials with misfitting inclusions, Acta Metall., 21,571-574. Reiter, T, Dvorak, G. J. and Tvergaard, V., 1997, Micromechanical models for graded composite materials, J. Mech. Phys. Solids, 45,1281-1302. Reiter, T. and Dvorak, G. J., 1997, Micromechanical models for graded composite materials, II Thermomechanical loading, J. Mech. Phys. Solids, to appear. Sasaki M. and Hirai.T., 1991, Fabrication and properties of functionally gradient materials, J. Ceram. Soc.Jap., 99,1002-1013. Walpole, L. J., 1969, On the overall elastic moduli of composite materials, J. Mech. Phys. Solids, 17,235-251.

THERMAL FRACTURE AND THERMAL SHOCK RESISTANCE OF FUNCTIONALLY GRADED MATERIALS Z.-H. JIN and R.C. BATRA Department of Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0219, USA Abstract

We first analyze thermal stresses and thermal cracking in a strip of a functionally graded material (FGM) subjected to sudden cooling. It is assumed that the shear modulus of the material decreases hyperbolically with the higher value occurring at the surface exposed to the thermal shock and that thermal conductivity varies exponentially. It is shown that the maximum tensile thermal stress induced in the strip is substantially reduced by the presumed thermal conductivity gradient.

Thermal stress intensity factors (TSIFs) are also calculated for an edge crack at the surface exposed to the thermal shock and results show that while the TSIF is relatively insensitive to the shear modulus gradient, it is significantly reduced by the thermal conductivity gradient. The crack growth resistance curve of a ceramic-metal FGM is also studied and it is found that the FGM exhibits strong R-curve behavior when a crack grows from the ceramic-rich region into the metalrich region. Finally, the thermal shock resistance of FGMs is discussed. 1. Introduction

Functionally graded materials (FGMs) for high temperature applications are special composites usually made from ceramics and metals. The ceramic in an FGM offers thermal barrier effects and protects the metal from corrosion and oxidation. The FGM is toughened and strengthened by the metallic composition. The compositions and the volume fractions of the constituents in an FGM are varied gradually, giving a nonuniform micro-structure with continuously graded macroproperties. The macro-nonhomogeneous properties of an FGM reduce thermal stresses in it when it is subjected to thermal loading (Hasselman and Youngblood, 1978). Thermal residual stresses can be relaxed in a metal-ceramic layered material by inserting an FGM interface layer between the metal and the ceramic (Drake et al., 1993). When subjected to thermal shocks, FGM coatings suffer significantly less damage than conventional ceramic coatings (Kuroda et al., 1993). The knowledge of crack growth in FGMs is important in order to evaluate their integrity. Assuming an exponential spatial variation of the elastic modulus, Delale and Er-

185 Y.A. Bahei-El-Din and G.J. Dvorak (eds.),

IUTAM Symposium on Transformation Problems in Composite and Active Materials, 185–195. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

186

dogan (1983) analysed a crack problem for a nonhomogeneous solid subjected to mechanical loading. By further assuming exponential variations of thermal properties, Jin and Noda (1994a) calculated thermal stress intensity factors (TSIFs). For general nonhomogeneous materials, Jin and Noda (1994b) showed that the crack tip fields are identical to those in homogeneous materials if the material properties are continuous and piece-wise continuously differentiable. Hence, the stress intensity factor concept can still be used to study the fracture behavior of FGMs. Here we first investigate thermal stresses and the TSIF in an edge-cracked strip of an FGM and explore the effects of shear modulus and thermal conductivity gradients on thermal stress and the TSIF. Then, the R-curve of an FGM is studied based on the crack-bridging concept. Finally, the thermal shock resistance of FGMs is discussed. 2. Thermal Stresses Consider a long FGM strip of width b and assume the following variation in its

material properties:

where E is Young’s modulus, Poisson’s ratio, the shear modulus, α the coefficient of thermal expansion (CTE), the thermal conductivity, and the thermal diffusivity. Constants

and

are given by

and

and are, respectively, the values of and at and and their values at The assumed Poisson’s ratio (1b) is subjected to the constraint and the thermal diffusivity is assumed to be constant for mathematical convenience. With assumptions (1)-(3), the basic nonhomogeneous thermoelasticity equations governing the Airy stress function F and the temperature T under plane strain conditions are

Here we have assumed that the mechanical deformations occur slowly and inertia effects are negligible. Now assume that the FGM strip is initially at a uniform

187 temperature the surface is suddenly cooled to a temperature and the surface remains at temperature The temperature distribution in the strip is given by

where Furthermore

and

For the one-dimensional temperature field

is the nondimensional time.

given in Eq. (6), the

thermal stress is (Jin and Batra, 1996a)

where E,

and

are given in Eqs. (l)-(3) and

Figure 1 shows the thermal stress normalized by in the strip for various inhomogeneity parameters and at a nondimensional time The constant is taken as zero and is calculated from Eq. with and It is seen that the maximum tensile stress occurring at is significantly reduced by increasing that describes the thermal conductivity gradient. The thermal stress is relatively insensitive to The evolution of the normalized tensile stress at (which is maximum in the strip at a given time) is depicted in Fig. 2. The tensile stress decreases with increasing and its all-time maximum value occurs at 3. Thermal Cracking

We consider a stationary edge crack of length in an FGM strip as shown in Fig. 3. It has been pointed out that the surface cracking is a common failure mode in FGMs (Kawasaki and Watanabe, 1993). The cracked FGM strip is subjected

to the thermal shock at as discussed in Section 2. The integral equation for the thermal crack problem can be written as

188

in which

is the dislocation density along the crack face, is the displacement in the -direction at the crack surface, is a known Fredholm type kernel, and is given by Eq. (7). According to the singular integral equation method (Gupta and Erdogan, 1974), Eq. (9) has a solution of the form

where

by

is continuous and bounded on the interval

the normalized TSIF,

Normalizing

at the crack tip is obtained to be

The TSIF can be calculated from Eqs. (11) and (12) after the integral equation (9) has been solved. The general variation of the normalized TSIF, with time and crack length is similar to that in a homogeneous material (Nied,

1983). For a given crack length , the TSIF increases from zero, passes through a peak value at a particular time which increases with an increase in the crack length, and decreases to zero subsequently. There is a critical normalized crack length at which the peak value of the TSIF reaches a maximum. However, the TSIF may be negative at the initial stage of the thermal shock for some values of and reaches its peak value when steady state is reached. Figure 4 shows the

189

normalized peak TSIF as a function of

for various values of with and Also shown in Fig. 4 is the peak TSIF for a homogeneous material. The peak TSIF increases with reaches its maximum at about and then decreases with further increase in It is seen that the peak TSIFs for the FGM are significantly reduced as compared with those for a homogeneous material. 4. Crack Growth Resistance Curve

Crack growth in FGMs is a complex phenomenon due to their complex microstructures. It has been experimentally observed that surface cracking in the material gradient direction at the ceramic side is the most common failure mode in a ceramic-metal FGM subjected to a thermal shock (Kawasaki and Watanabe, 1993). In the present study, we only consider Mode I crack growth in the direction of the material gradation. Also, it is assumed that the crack will grow from the ceramic-rich region into the metal-rich region as cracks are more likely generated first in the ceramic-rich region. We determine the fracture toughness of FGMs based on the crack-bridging concept. Crack-bridging is believed to be a major toughening mechanism in ductile particulate reinforced brittle matrix composites (Krstic, 1983). When a ceramic-metal FGM is fabricated with the metallic

grains dispersed in a continuous ceramic phase (Rabin and Heaps, 1993), the FGM

may be regarded as a metal particulate reinforced ceramic composite with graded compositions. In this case, the crack-bridging concept may be used to study the

toughness behavior or R-curve of the FGM. However, the crack-bridging concept may not be always appropriate to FGMs as the microstructure in an FGM may be very different from that in a particulate composite.

190

Consider a ceramic-metal FGM strip of width b with an edge crack of length a subjected to remote pure bending M as shown in Fig. 3, and with material properties given by (1)-(3). After the crack has initiated, it will grow in the ceramic with plastically stretched metal grains behind the crack tip bridging the crack faces. It is assumed that the metal elsewhere does not undergo plastic deformations. Previous studies (Mataga, 1989) have shown that for ductile particulate reinforced brittle matrix composites, the bridging exhibited softening behavior, i.e., the bridging stress decreased with an increase in crack opening The bridging law may then be modeled by the linear softening relation

where is the maximum bridging stress, and the maximum crack opening displacement of the bridging zone at which the bridging stress drops to zero. The values of and are taken from Bao and Zok (1993) who approximated the complex bridging law of Mataga (1989) by a linear softening relation. For the crack face bridging law (13), the integral equation for the dislocation density along the crack face, is

where

is the volume fraction of the metal in the FGM, H( ) is the Heaviside step function, is the initial crack length, is the current crack length, is the bridging length, and

Note that the volume fraction in Eq. (14) is a function of For the FGM with material properties given by (1)-(3), we use the three-phase model for macrohomogeneous composites (Christensen and Lo, 1979) to determine with end conditions at and at we note that the model is only approximate for FGMs. The solution of Eq. (14) still has the form (11) and the stress intensity factor, at the crack tip

is given by

191 The R-curve,

can then be evaluated from (Jin and Batra, 1996b)

where is the solution of Eq. (14) normalized by and without considering bridging, and is the bending stress corresponding to with

where is the fracture toughness of the ceramic in the FGM. Figure 5 shows the R-curve for an FGM with and Three initial crack lengths are considered, i.e., and 0.2 with specimen width mm. It is seen from Fig. 5 that the FGM exhibits strong R-curve behavior as the crack grows from the ceramic-rich region into the metal-rich region. Since the initial crack size influences the R-curve, therefore, the R-curve is not a material property. 5. Thermal Shock Resistance Knowledge of thermal shock resistance of FGMs is essential for their successful high temperature applications. For a macro-defect-free ceramic specimen, a wellestablished thermal shock parameter is

where and are Young’s modulus, Poisson’s ratio and the CTE of the ceramic and is its strength. represents the maximum sudden temperature drop that the specimen can withstand without macro-crack initiation and propagation. For a ceramic-metal FGM strip of width b subjected to a sudden temperature drop at the edge the maximum tensile thermal stress is given by

where the constant depends on the gradients of material properties. We assume that when equals macro-cracks will initiate at the ceramic side of the FGM. Hence, the thermal shock parameter for the FGM is

The ratio for various inhomogeneity parameters and is shown in Table 1. It is clear that R can be enhanced by increasing Hence, the FGM has a better thermal shock resistance in terms of R.

192

We now consider an edge-cracked FGM strip, shown in Fig. 3, and subjected to the thermal shock The TSIF has been obtained in Section 3. Here we study the minimum temperature drop to grow the crack. From Eq. (12) the peak TSIF for a given crack length is given by

where is the time at which attains its peak value. When reaches given by Eq. (18), the crack will start to grow, and the temperature drop, corresponding to the crack initiation can be computed from

The corresponding critical temperature drop for the cracked ceramic is

where corresponds to the peak TSIF for the ceramic specimen. It follows from Eqs. (18), (23) and (24) that

Figure 6 shows the normalized critical temperature drop,

for the

FGM ve/rsus the nondimensional crack length It is apparent that is significantly higher for the FGM than that for the ceramic. Hence, a cracked FGM can withstand higher temperature drop than the cracked ceramic without propagating the crack.

Finally, we qualitatively investigate the thermal shock damage in the cracked FGM specimen. The peak TSIFs for both the cracked FGM and the cracked

ceramic strips are shown in Fig. 7. At a given initial crack length

thermal shock

reaches the threshold

when the

for the cracked ceramic, the crack

in the ceramic will grow unstably until it reaches a length

(Fig. 7a). Hence,

the ceramic is severely damaged. However, the cracked FGM does not suffer any

193

194 damage now since its threshold Inversely, if both the cracked FGM and ceramic strips are subjected to a thermal shock equal to the threshold of the cracked FGM, the cracks in both the FGM and the ceramic will grow (Fig. 7b). The crack will grow to in the ceramic and to in the FGM, where is the final crack length in the FGM without considering its crack growth curve. It can be seen that is significantly smaller than Since the FGM exhibits strong crack growth curve behavior, may be significantly smaller than 6. Concluding Remarks

We have analysed the thermal shock resistance properties of an FGM that has a hyperbolically decreasing shear modulus with the higher value at the surface exposed to the thermal shock, and the exponentially increasing thermal conductivity. Such an FGM may be obtained by dispersing metal particulates in a ceramic matrix with an appropriate particulate volume fraction gradation. It is shown that the maximum tensile thermal stress in a strip of the FGM is substantially reduced by the assumed thermal conductivity gradient. The TSIF for an edge crack in the FGM strip is relatively insensitive to the shear modulus gradient but is significantly reduced by the thermal conductivity gradient. The FGM exhibits strong R-curve behavior when a crack grows from the ceramic-rich region into the metal-rich region. The FGM has superior thermal shock resistance to the ceramic. Acknowledgement

This work was partially supported by the ONR grant N00014-94-1-1211 with Dr. Y. D. S. Rajapakse as the program manager. 7. References Bao, G. and Zok, F. (1993) On the strength of ductile particle reinforced brittle matrix composites, Acta Metal. Mater, 42, 3515-3524. Christensen, R.M. and Lo, K.H. (1979) Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids, 27, 315-330. Delale, F. and Erdogan, F. (1983) The crack problem for a nonhomogeneous plane, ASME J.

Appl. Mech., 50, 609-614.

Drake J.T., Williamson, R.L. and Rabin, B.H. (1993) Finite element analysis of thermal residual

stresses at graded ceramic-metal interfaces, Part II: interface optimization for residual

stress reduction, J. Appl. Phys., 74, 1321-1326.

Gupta, G.D. and Erdogan, F. (1974) The problem of edge cracks in an infinite strip, ASME J. Appl. Mech., 41, 1001-1006.

Hasselman, D.P.H. and Youngblood, G.E. (1978) Enhanced thermal stress resistance of structural ceramics with thermal conductivity gradient, J. Am. Ceram. Soc., 61, 49-52.

Jin, Z.-H. and Batra, R.C. (1996a) Stress intensity relaxation at the tip of an edge crack in a functionally graded material subjected to a thermal shock, J. Thermal Stresses, 19,

317-339. Jin, Z.-H. and Batra, R.C. (1996b) Some basic fracture mechanics concepts in functionally graded materials, J. Mechs. Phys. Solids, 44, 1221-1235. Jin, Z.-H. and Noda, N. (1994a) Transient thermal stress intensity factors for a crack in a semiinfinite plane of a functionally gradient material, Int. J. Solids Struct., 31, 203-218.

195 Jin, Z.-H. and Noda, N. (1994b) Crack-tip singular fields in nonhomogeneous materials, ASME J. Appl. Mech., 61, 738-740. Kawasaki, A. and Watanabe, R. (1993) Fabrication of disk-shaped functionally gradient materials by hot pressing and their thermomechanical performance. Ceramic Transactions, Vol. 34: Functionally Gradient Materials (ed. J.B. Holt, M. Koizumi, T. Hirai and Z.A. Munir), pp. 157-164. American Ceramic Society, Westerville, Ohio. Krstic, V.D. (1983) On the fracture of brittle-matrix/ductile-particle composites, Phil. Mag. A, 48, 695-708. Kuroda, Y., Kusaka, K., Moro, A, and Togawa, M. (1993) Evaluation tests of functionally gradient materials for regeneratively cooled thrust engine applications. Ceramic Transactions, Vol. 34: Functionally Gradient Materials (ed. J.B. Holt, M. Koizumi, T. Hirai and Z.A. Munir), pp. 289-296. American Ceramic Society, Westerville, Ohio. Mataga, P.A. (1989) Deformation of crack-bridging ductile reinforcements in toughened brittle materials, Acta Metall., 37, 3349-3359. Nied, H.F. (1983) Thermal shock fracture in an edge-cracked plate, J. Thermal Stresses, 6, 217-229. Rabin, B.H. and Heaps, R.J. (1993) Powder processing of FGM. Ceramic Transactions, Vol. 34: Functionally Gradient Materials (ed. J.B. Holt, M. Koizumi, T. Hirai and Z.A. Munir), pp. 173-180. American Ceramic Society, Westerville, Ohio.

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MICROMECHANICS AND NONLOCAL EFFECTS IN GRADED RANDOM STRUCTURE MATRIX COMPOSITES

V. BURYACHENKO 1,2 AND F. RAMMERSTORFER1 1

Institute for Light Weight Structures and Aerospace Engineering, Vienna Technical University 27–29, A–1040 Vienna, Austria 2 Permanent address: Department of Mathematics, Moscow Institute of Chemical Engineering, 107884 Moscow, Russia. Abstract. We consider Functionally Graded Materials (FGMs) as a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of ellipsoidal inclusions, when the concentration of the inclusions is a function of the coordinates and the meso–stress boundary condition are nonuniform. The micromechanical approach is based on the generalization of the “multiparticle effective field” method (MEFM), previously proposed for statistically homogeneous random structure composites by one of the present authors. The hypothesis of effective field homogeneity near the inclusions is used. The nonlocal dependences of the effective elastic moduli as well as of conditional averages of the strains in the components on the concentration of the inclusions are demonstrated. An explicit representation for the operator of effective properties in the form of a differential operator of second order acting on a sufficiently slowly varying ensemble average strain field is obtained.

1. Introduction

In micromechanics of random structure composites the assumption is usually made that all characteristic lengths associated with the spatial variations of the statistical mean field quantities are significantly large compared to all characteristic lengths associated with the spatial variations in the materials properties (such as the size of the inclusions, the mean distance 197 Y.A. Bahei-El-Din and G.J. Dvorak (eds.),

IUTAM Symposium on Transformation Problems in Composite and Active Materials, 197–206.

© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

198

between them and the characteristic length of the variation of the inclusion number density). If this assumption is valid, the governing equations for the ensemble–averaged fields are identical in form with the usual local thermoelastic equations for homogeneous media. At the same time it is known, that the eventual abandonment of this so–called hypothesis of statistically homogeneous fields leads to a nonlocal coupling between statistical averages

of the stress

and strain

tensors when the statistical average

stress is given by an integral of the field quantity weighted by some tensorial

function, i.e. the nonlocal effective elastic properties. Therefore the value of the statistical average field will locally depend on it value at the other points in its vicinity. If the field is varying sufficiently slowly in the

neighborhood of an arbitrary point y, then its can be expanded about x in a Taylor series and, therefore, the integral operator of nonlocal effective

elastic properties can be considered as a differential one. The construction of an effective operator was carried out by Beran and McCoy (1970) by perturbation method with small fluctuations of elastic mismatches by the

use of correlation functions of all orders involving the material properties of members of the assemblage. At the same time for statistically homogeneous

composites with considerable elastic mismatches, one particle methods are used, which are reduced to Mori–Tanaka method and Hashin–Shtrikman estimations both for particular cases of microstructures and for a slowlyvarying mean fields. So Willis (1985) analysed the nonlocal influence of

density variations in an otherwise homogeneous medium, assuming an ex-

ponential form for the two–point correlation function. For pure mechanical loading, Zuiker and Dvorak (1994) derived nonlocal relations for analysing

local stress states in an elastic composites under macrostress gradients. Drugan and Willis (1996) derived a nonlocal constitutive equation and estimated the minimum representative element size for the use of the “effective

modulus” concept. A more general case of a nonlocal constitutive equation at the micro level was analysed by Smyshlyaev and Fleck (1994).

It is interesting to note in this respect that recently a new method has been introduced, namely the multiparticle effective field method (MEFM) [references may be found in the survey of Buryachenko and Kreher (1995), and Buryachenko (1996)]. MEFM, leads not only to quantitatively reasonable results (in comparison with experimental dates and exact analytical solutions for some regular structures) but MEFM also has quality benefits

following immediately from the consideration of multiparticle interactions. From such considerations it transpires that final relations for effective properties depend explicitly not only on the local concentration of the inclusions but also on at least binary correlation functions of the inclusions. Therefore, effective properties as well as local statistical average stresses in the

199

components are nonlocal functions of the inclusion concentration as well as of the applied inhomogeneous mean field. That was shown by Buryachenko and Parton (1990) under some additional assumptions for pure

mechanical loading of FGMs at slowly-varying mean fields. By the use of

pseudo-differential operator techniques Buryachenko and Lipanov (1992) obtained an explicit representation for the stress concentration operator in the form of a differential operator of second order acting on sufficiently slowly varying ensemble average stress fields. In the present work the results of the two last mentioned papers are generalized in the framework of the same method. The joint actions of both nonlocal effects, produced by an inhomogeneous inclusion number density and inhomogeneous average applied stress and temperature fields varying over distances that are comparable to the particle size, are analysed. Binary interaction effects of the inclusions are considered in detail. 2. Preliminaries

Let stresses be strains are related to each other via the constitutive equation is the second order tensor of local eigenstrains and L is the fourth-order anisotropic

elasticity tensor, which for isotropic materials is given by

and are the bulk and shear modulus, respectively; and are the unit second-order and fourth-order tensors. The local strain and stress tensors satisfy the linearized straindisplacement relations and the equilibrium equation, respectively. We consider a mesodomain subjected to the nonuniform traction boundary

conditions where T(x) is the traction vector at the external boundary is its unit outward normal, and is a given symmetric tensor, representing the macroscopic stress state. Common notations for tensor products have been employed: In the mesodomain

ellipsoids

containing a set

with characteristic functions

and an aggregate of Euler angles

of

centers

semi-axes

a characteristic function W is

defined. It is assumed that all inclusions have identical mechanical and geometrical properties and are grouped into the component In the matrix and in the inclusions

assumed to be constant:

for

the tensor

is

and

The upper index of the material properties tensor put in parentheses shows the number of the respective component. The

200

subscript 1 denotes a jump of the corresponding quantity (e.g. of the material tensor). The phases are perfectly bonded. We introduce a conditional probability density which describes the probability density of finding the inclusion in the domain with the center the inclusions in the domains with the centers being treated as fixed. We will consider statistically inhomogeneous media, when the conditional probability density is noninvariant with respect to translations, this means is a number density of inclusions in the point and is the concentration, i.e. volume fraction, of the component in the point x: Here the notation will be used for the average taken for the ensemble of a statistically inhomogeneous field in the point The notation denotes

the average over the component As in Buryachenko and Parton (1990) a general integral equation for statistically inhomogeneous composites can be obtained

The integral operator kernel is defined by the Green tensor G of the Lame’ equation of a homogeneous medium: is the Dirac delta function.

3. Multiparticle effective field method After conditional statistical averaging Eq. (2) turns into an infinite system of integral equations. In order to close and approximately solve this system we now apply the MEFM hypotheses (see details in Buryachenko and Kreher, 1995): H1) Each inclusion has an ellipsoidal form and is located in the effective field which is homogeneous over the inclusion H2) Each pair of the inclusions and is located in an effective field and

According to hypothesis H1 and to Eshelby’s theorem we get

where and the tensor Q is associated with the well–known Eshelby

201

tensor

Taking hypothesis H2) into account, we get

In Eqs. (5)–(7) for two inclusions and a successive approximation method within the limits of a point approximation of binary interactions of the inclusions is used. For statistically inhomogeneous media as well as for inhomogeneous boundary conditions const (1) the system (5) is a system of integral equations. The solution of this system is extremely difficult for the general case of and In order to obtain explicit relations for effective properties we will consider cases with different simplified assumptions.

4. Particular cases Perturbation method. In the case of a dilute concentration of the inclusions as well as for a weakly inhomogeneous medium

the perturbation method is appropriate. Then instead of hypothesis H2) (3) we can use assumption and from (5), taking Eq. (4) into account, we get the overall constitutive equation

where the local thermoelastic parameters are described by the tensors

and the nonlocal part of effective compliance is defined by

For the representation of the integral operator (12) in differential form we expand about in a Taylor series and integrate term by term over the whole space:

202

where

and for the multi-index

the following notations where used

Strongly inhomogeneous media. For the general case of inhomogeneity of the composite material, when the condition (8) is not valid, the hypotheses H1), H2) in conjunction with Fourier transform and Taylor expansion techniques can be used for an approximative solution of Eq.

(5) and subsequent estimation of effective parameters. Then, in analogy to Buryachenko and Parton (1990) we obtain

where the physical meaning of the tensor

for statistically homogeneous media, when const., is explained by Buryachenko and Kreher (1995). For statistically inhomogeneous media the tensor

and

(18) and, therefore, the effective parameters

depend on the parameters of the inclusion distribution

not only at the considered point x, but also in a certain neighborhood

of that point (leading to a so–called nonlocal effect); the diameter of this region is estimated as three times the characteristic dimensions of the inclusions. As a result, a statistically inhomogeneous composite medium be-

haves like a macroscopically inhomogeneous medium with local effective parameters

determined for a nonlocal distribution of the

inclusions. Relations (16) and (17) can be used for finding the stress distribution in the mesodomain for prescribed boundary conditions (1), and , with (4) and (5), a conditional statistical average stress field inside the inclusions can be estimated. Therefore, the present approach explicitly couples the microstructural details with the global analysis.

The nonlocal part of effective compliance operator

for sufficiently weakly varying field

can be approximated by the

203

differential operator

Statistically homogeneous media in varying stress fields. For statistically homogeneous composites the odd terms in the Taylor series (13) and (19) are zero by virtue of the fact that the generalized function will be an even homogeneous function in the considered case. Moreover, for statistically homogeneous media the calculation of the second order differential approximation of the nonlocal operator is even practical. Thus, similarly to (14), we present the first term of the expansion. No details of the tedious but straightforward calculation, analogous

to Buryachenko and Lipanov (1992), are given

Buryachenko and Lipanov (1992) used a step radial distribution function

where is the “included volume” (since inclusions cannot overlap). More general cases of two-point correlation functions (21) and any comparison moduli where considered by Drugan and Willis (1996) at the “quasi-crystalline” approximation, which is equivalent to the identity

[see also Zuiker and Dvorak (1994), Smyshlyaev and Fleck (1994)]. Remarks. The obtained relations depend on the values, associated with the mean distance between inclusions, and do not depend on the other characteristic size, e.i. the mean inclusion diameter. This fact may be explained by the initial acceptance of the hypothesis H1 dealing with the homogeneity of the field inside each inclusion. In the case of a variable representation of for instance in polynomial form, the mean size of the inclusions will be contained in the nonlocal dependence of microstresses on the average stress Furthermore, for simplifications of the derivations we consider only the first two terms of the series (6), (7) in the problem

solution of binary interactions of the inclusions. As a consequence, the final representation of (7) does not contain another characteristic size of the problem, namely one having the order of long-range action between two inclusions. Hovewer this values appears in the representations for the tensor D (18) and, therefore, for the nonlocal operator (19), (20).

204

5. Numerical results As an example we consider a composite material consisting of isotropic incompressible components and having identical rigid spherical inclusions. At first we analyze the statistically homogeneous case const. (21). We estimate the effective shear modulus by different methods:

Equations (23) and (24) are obtained for a dilute concentration of the inclusions by the perturbation method (6), (7), (10) and by Chen and Acrivos (1978), respectively. Relations (25) and (26) are calculated by MEFM in the framework of two–particle (6) and one–particle (or Mori–Tanaka) (22)

approximations, respectively. Recently Buryachenko (1996) has shown that the use of Eq. (25) leads to an underestimating evaluation of the effective shear modulus by 2.6 times for c = 0.43 compared to the experimental data as well as to the more exact point approximation of weakly interacting inclusions (26). At the same time the Mori–Tanaka solution (25) differs from the dilute approximation (23) and (24) by not more than five per cent for the concentration of the inclusions Therefore, even in the limiting case of an infinite elastic mismatch and the perturbation method (6), (7), (10) provides the same accuracy as the Mori–Tanaka approach (17), (18), (22) and makes it possible to estimate nonlocal dependencies of effective properties on the varying concentrations of the inclusions and

the average stress

as well as to analyze nonstep binary correlation

functions

As an example of statistically inhomogeneous media, let us consider a spherical cluster of inclusions, called cloud with center and radius containing homogeneously distributed centers of spherical inclusions (with the radius for and for In Fig. 1 the estimations of normalized effective moduli as functions of the dimensionless coordinate

are represented for

where

represents

the relative distance from the cloud boundary. In the interior of a large cloud sufficiently far away from its boundary, coincides with the isotropic effective moduli for the statistically homogeneous medium.

Near the boundary of the cloud, the tensors of the effective moduli are transversally isotropic and vary significantly within the boundary layer

205

(nonlocal boundary layer effect). The character of the dependence varies (increases or decreases monotonically or nonmonotonically with r) with the variation of the cloud size (scale

effect). For

the values

coincide with

the Mori–Tanaka approach (7), (17), (18), (22) (degeneration of binary interaction effect). Of course, in the case of using the Mori–Tanaka approach (22) we would obtain , that is to say the boundary layer and scale effects can not be identified fundamentally. Other popular one particle methods, such as self–consistent methods, differential methods and so on, considered by Zuiker (1995), have the same quality of prognostic potentials: Up to now we demonstrated the difference between the approaches (6) and (22) for the estimation of effective elastic moduli For statistically homogeneous media with in the interest of obtaining maximum difference between the nonlocal effective properties, estimated

by one-particle (22) and two-particle (6) approaches, respectively, we will evaluate the nonlocal effective elastic operator in the governing equation where and In Fig. 2 a comparison between the normalized parameters calculated by the use of far–field (6) (solid curve) and the “quasi-crystalline” (22) (dotted curve) approximations is presented. For sufficiently large concentrations of the inclusions, the values (solid and doted curves) differ considerably.

206

Thus, for statistically homogeneous media in the considered cases taking binary interaction of the inclusions into account leads to a significant improvement in estimating the effective elasticity tensor (factor 2.6) and to an increase of some components of nonlocal effective elastic operator by 6 times. For statistically inhomogeneous media, the principally new boundary layer and scale effects were estimated.

Acknowledgments– The first author acknowledges the financial support from the Christian Doppler Forschungisgesellschaft, and from the Fonds zur Förderung der wissenschaftlichen Forschung (under grant P12312-NAW). Parts of this work are related to the COST 512 action, financially supported by the

Österreichisches Bundesministerium für Wissenschaft und Verkehr under grant GZ 49.935/3–II/4/94.

References Beran, M. J. and McCoy, J. J. (1970) Mean field variations in a statistical sample of heterogeneous linearly elastic solids. Int. J. Solid Structures 6, 1035–1054.

Buryachenko V. A. (1996) The overall elastoplastic behavior of multiphase materials with isotropic components. Ada Mechanica 119, 93–117. Buryachenko V.A. and Kreher W.Z. (1995) Internal residual stresses in heterogeneous solids —A statistical theory for particulate composites. J. Mech. Phys. Solids 43, 1105–1125. Buryachenko V. A. and Lipanov A. M. (1992) Thermoelastic stress concentration at ellipsoidal inclusions in matrix composites in the region of strongly varying external stress and temperature fields. Deformation and Fracture of Structural-Inhomogeneous materials, (eds. O. B. Naimark and S. E. Evlampieva), pp. 12–19. AN SSSR, Sverdlovsk. (In Russian.) Buryachenko V. A. and Parton V. Z. (1990) Effective parameters of statistically inho-

mogeneous matrix composites. Izv. AN SSSR, Mekh. Tverd. Tela. (6), 24–29. (In Russian. Engl. Transl. Mech. Solids 25, 22–28.) Chen H. S. and Acrivos A. (1978) The effective moduli of composite materials containing spherical inclusions of non–dilute concentration. Int. J. Solid Structures 14, 349–364. Drugan W.J. and Willis J. R. (1996) A micromechanics–based nonlocal constitutive equation and estimates of representative volume elements for elastic composites. J. Mech.

Phys. Solids 44, 497–524. Smyshlyaev V. P. and Fleck, N. A. (1994) Bounds and estimates for linear composites with strain gradient effects. J. Mech. Phys. Solids 42, 1851–1882.

Willis J. R. (1985) The nonlocal influence of density variations in a composite. Int. J.

Solids Structures 21, 805–817.

Zuiker J. R. (1995) Functionally graded materials: Choice of micromechanics model and limitations in property variations. Composites Engineering 5, 807–819. Zuiker J. R. and Dvorak G. J. (1994) The effective properties of functionally graded composites-I. Extension of the Mori–Tanaka method to linearly varying fields. Composites Engineering 4, 19–35.

Transformation Problems in Composite Structures

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DESIGN OF COMPOSITE CYLINDER FABRICATION PROCESS M.V. SRINIVAS1 AND G.J. DVORAK Center for Composite Materials and Structures Rensselaer Polytechnic Institute Troy, NY 12180-3950, U.S.A.

Abstract Fiber prestraining is often employed to reduce fiber waviness while fabricating composite cylindrical structures by filament winding or fiber placement. The effect of fiber prestraining on the final residual stress state in a closed-end laminated cylinder fabricated on a steel mandrel is analyzed here. The elements influencing the residual stress state are identified and the corresponding influence functions relating the ply stresses to the fiber prestrain are obtained. The ply residual stresses are evaluated for a constant prestrain distribution in all the layers. An optimization procedure is implemented to find prestrain distributions that produces minimal residual stresses in layers. The role of mandrel stiffness on final residual stress state is also studied by selecting different mandrel thicknesses. 1. Introduction:

Multilayered composite laminates are utilized in cylindrical structures such as submersibles, pressure vessels, etc., to withstand high compressive stresses under applied external pressure loading. Such cylinders are generally manufactured by filament winding process on a steel mandrel or by fiber placement with in-situ curing, or layup and curing of either new layers on already cured layers, or all layers at once. During the winding process, the fibers can be subjected to tensile prestraining force for reducing fiber waviness. Such prestraining forces cause significant residual stress fields which are preserved in the structure after manufacturing and during its lifetime. The residual fields are superimposed with local fields caused by external pressure, and could thus contribute to premature failure of the structure. The present study investigates the effect of fiber prestraining on residual stress state in the cylindrical laminate by modeling the fabrication process and finding 1 Currently at G.E. Corporate Research and Development Center, Schenectady, NY 12301, U.S.A.

209 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 209–219. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

210 the desired fiber prestrain for optimal distribution of residual stresses in the cylinder wall. The effect of cooling from curing temperature on final residual stress state is also examined. The residual stress state in the cylinder is related to the applied prestraining forces through certain influence functions. The evaluation of influence function is described in Section 2; they are obtained using the multilayer composite cylinder analysis developed by Dvorak et al. (1997), Dvorak and Prochazka (1996). The influence functions are utilized in deriving an optimum prestrain distribution, discussed in Section 3, such that the total residual and mechanical stresses in all plies are bounded by certain strength criteria. Finally, numeral results are presented in Section 4. 2. Fabrication Model:

The composite cylinder fabrication process model considers successive ply layup

on a steel mandrel of certain axial and radial stiffness. Prestraining forces are applied in the fibers during the ply deposition sequence, which includes either in-

situ curing or placement and curing of all layers at once. After laying the required

number of layers, the mandrel is removed from the cylindrical structure. For analysis purposes, let the mandrel be denoted as layer and the composite laminate layers as The contribution of prestraining forces to the final residual stress state consists of three components, namely, the self-stress in the fiber, including the thermal stresses created during curing, the relaxation stress caused in the already deposited layers by the prestress applied in the currently made layer, and the stress due to mandrel removal. The superposition of the self-stress and the relaxation stress yields the stress-state in the completed structure before removing the mandrel. The superposition of all three components provides the final residual stress state in the structure. The self-stress is caused by applying the prestraining force in the ith layer before curing and is retained in the structure even after curing. Consider the layer (i) which has an inner radius outer radius and thickness It is composed of number of fibers, each of diameter oriented along the winding angle and distributed uniformly throughout the ply. Prestraining force is applied in each layer along its winding direction which can be resolved in the cylinder's hoop and axial directions as,

Stresses caused by the prestraining force components in the corresponding ply are

the hoop self-stress and the axial self-stress expressed as (Srinivas et al., 1998),

which can be

211 where

is the volume fraction of the ith ply.

Let the hoop and axial self-stress averages in each layer be written in the form,

Substituting for the stress components in (3) and (4) from (2),

where

are the pre-straining force vectors in the and z directions, respectively. The and are matrices that represent the self stress influence functions. Matrix form of these influence functions and

the other similar ones to follow later are described in Srinivas et al. (1998). Thermal stresses created during fabrication of the structure can also be included as part of the self-stress. For example, in an in-situ fiber placement fabrication process with temperature changes due to local curing, the thermal force components can be approximated as (Srinivas et al., 1998),

where is the axial Young’s modulus of the fiber and is the axial thermal expansion coefficient of the fiber. The thermal prestraining force is added to the prestraining force in (1) and the total self-stress is evaluated from (2). The relaxation stress is caused in the already cured layers by applying the pre-straining forces in the layer j that is being laid. The hoop component of the pre-straining force applies a radial traction where the radial pressure can be derived from the equilibrium in the layer as,

Similarly, the axial component axial stress therefore,

generates an axial traction

The

in the new layer (j) is equivalent to the self axial stress

Internal stresses due to unit radial pressure and unit axial stress are first obtained from the solution procedure described by Dvorak et al. (1997).

212 These unit stress solutions are then used to write the hoop and axial stresses caused in each layer by the actual fiber prestrain forces

and

Note that the radial traction and axial force both generate radial, hoop and axial stresses in the layers. The stresses caused by unit radial traction are denoted as

The stresses caused by the unit axial stress

are denoted as,

The relaxation stresses (denoted by the superscript ) caused in the layers of the completed structure by the actual pre-straining forces and are obtained by superposition of the contributions from pre-straining of each layer as,

Expressing the hoop and axial relaxation stresses in the vector form,

and substituting for the stress components

from (10 - 12) will yield,

where and are the relaxation stress influence functions of dimension Superposition of the self-stress and relaxation stress yields the average stresses in the mandrel and plies of a cylinder supported by the mandrel. These stresses are,

where

213 and

The mandrel supporting the cylindrical structure exerts radial and axial reaction forces. These can be evaluated from (10) and (12) as,

Mandrel removal from the structure is accounted for by applying the forces

at the inner surface and at both ends of the cylindrical structure. The corresponding components of average stresses in the layers are,

where the

and

are the layer average stresses due to unit radial

pressure applied at and and the layer average stresses caused by an unit axial stress respectively. Using the matrix form, Equations (22) and (23) are written as,

where the

are

stress vectors are,

and are stress influence influence matrices. The total average stresses in the layers due to the reaction forces and in (22) and (23) are,

214 where

Finally, adding the ply stress averages due to self-stress, relaxation stress and reaction stresses provides the total residual stress state in all plies of the composite structure. These stresses can then be written as,

where again, the influence functions have the dimensions are given by,

and

The stress influence functions are restricted by certain constraints imposed by the overall force and moment equilibrium conditions. These constraints indicate

that the influence function matrices are singular with one rank deficiency (Srinivas et al., 1998). 3. Optimal Solution:

The direct determination of the residual stresses for a known distribution of prestraining forces follows directly from the equations derived in the previous Section. However, from the design point of view, it is necessary to determine the distribution of the prestraining forces such that the residual stresses in the structure are well within the strength limits under both loading and no-load conditions. Hence, we need to determine the distribution of pre-strain in layers that maintains a desired level of residual stresses. The dominant residual stress components that are of interest here are the hoop and axial components. They have to be kept within required levels by adjusting the pre-strain applied in the fiber direction. The solution involves conditional equations representing more stress levels than the unknown variables, which are

the pre-straining forces. We seek an optimal solution for pre-straining forces that yield desired distribution of axial and hoop stresses in layers. The objective is to keep the residual stresses in all the layers at a minimum. The optimization problem is the minimization of an objective function involving the stress variance, defined by,

Certain constraint conditions are imposed on the solution. straining forces can be applied in the layers, therefore,

Only tensile pre-

215 Further, the residual stresses caused in the layers should not attain the strength limits. The maximum stress failure criterion chosen for predicting the strength limits imposes additional constraints on the residual stresses in (33). The prestraining force is applied in the direction of winding, which is inclined at to the axis of the cylinder. The hoop stress and the axial stress in the global system can be resolved along the principal material axes and equated to

the corresponding strengths as

where

and

are the tensile and compressive strengths in the axial direction, are the tensile and compressive strengths in the transverse direction, and is the shear strength. The solution to the problem was obtained through quadratic programming by active solution (Gill et. al, 1981,1984) method using NAG Fortran library routine (1993).

and

4. Results and Discussion: The results presented here are for a composite cylinder having an inner radius

an outer radius ness, arranged in a repeating

and composed of 100 layers of equal thicklayup. This layup guarantees that in

the composite system chosen here, all layers of the laminate experience the same

compressive stresses under external hydrostatic pressure. The layers are made

216 of AS4/3501 – 6 Carbon/Epoxy composite whose properties are listed in Table 1. In addition to the residual stresses due to pre-straining forces, the results are superimposed with the stresses caused by application of a hydrostatic pressure Such pressure can be realized under water at a depth of 2500m. The results shown in the figures that follow apply to the finished cylindrical structure. Three mandrel thicknesses were considered in modeling the fabrication process, all made of steel plate, 0.05m, 0.15m and 1.00m thick. The latter thickness represents an essentially rigid mandrel, but the former may approximate the

stiffness of an actual supporting structure. In each case, the mandrel was regarded as an additional layer in the analysis of stresses prior to mandrel removal.

Figure 1 shows the distribution of local transverse normal stresses that act in the direction perpendicular to the fiber axis in each ply, for the three mandrel thicknesses. The fiber prestrain was applied at a constant value in all plies through

217 the cylinder wall, such as to cause stress of 1,000 MPa in each fiber, this can be achieved by applying about 113 lbs in tension to a 10,000 filament tow. The results in the left column show a superposition of the residual and applied stresses due to external pressure of 25MPa. The right column present the residual stresses acting alone. Each figure also shows the allowable stress limits for the plies. The implication is that in structures supported by relatively compliant mandrels, the prestrain causes transverse stresses that exceed the allowable limits, especially in

the absence of external pressure. The large stresses are found in the 60 plies, while the stresses in other plies are essentially constant and within the limits. We also

found high shear stresses in the plies under similar support and loading conditions. To avoid the high stresses while still maintaining fiber prestress in the layers, optimal prestrain distributions were evaluated using the procedure described in Section 3. Again, the effect of three mandrel thicknesses on the optimal prestrain distribution was examined. It is evident from Figure 2 that a variable distribution of the prestrain is needed to keep the local stresses within prescribed limits. How-

218

ever, less than optimal but smooth prestrain distributions can be identified that do not cause excessive stress magnitudes. The final stresses generated by the external pressure and/or optimized prestrain distribution in the above cylindrical structure, for the most compliant mandrel, are shown in Figure 3. All three in-plane stress components in each ply are plotted. Note that the normal stresses in the axial or fiber direction are constant in all plies when the cylinder is subjected to the external pressure of 25 MPa. Moreover, all the local components are well within the allowable limits, and very small when the cylinder is in unloaded state. This is a desirable situation that should limit

long-term creep of the matrix. In conclusion, the results show that rather unfavorable stress distributions may be induced in the laminate by constant fiber prestrain. Quite possibly, analogous

219 undesirable effects may result in other composite structures because of either intended of inadvertent prestrain of the fibers during fabrication. Therefore, careful analysis of the prestrain effects and corresponding control of the prestrain magnitudes in manufacture are required for efficient and safe use of fibrous composites in structures. Acknowledgement: This work was supported by the Ship Structures and Systems S & T Division of the Office of Naval Research, Dr. Yapa Rajapakse served as program monitor.

References: Daniel, I.M. and Isahi, O. (1994). University Press, New York.

Engineering Mechanics of Composite Materials, Oxford

Dvorak G. J. and Prochazka P. (1996). Thick-walled Composite Cylinders with Optimal Fiber Prestress. Composites Part B, 27B, 643-649. Dvorak, G.J., Srinivas, M.V. and Prochazka, P. (1998). Design and Fabrication of Submerged Cylindrical Laminates. Int. J. Solids Structures, to appear.

Srinivas, M.V., Dvorak, G.J. and Prochazka, P.,Design and Fabrication of Submerged Cylindrical Laminates. II Effect of Fiber Prestress. Int. J. Solids Structures, to appear.

Gill, P.E., Murray, W. and Wright, M.H. 1981. Practical Optimization, Academic Press, New York.

Gill, P.E., Murray, W., Saunders, M.A. and Wright, M.H. (1984). Procedures for Optimization Problems with a mixture of Bounds and General Linear Constraints, ACM Transactions on Mathematical Software, 10, 282-298. NAG Fortran Library Manual. (1993). NAG Ltd., Oxford.

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DYNAMIC RESPONSE OF ELASTIC-VISCOPLASTIC SANDWICH BEAMS WITH ASYMMETRICALLY ARRANGED THICK LAYERS C. ADAM, F. ZIEGLER Department of Civil Engineering, Technical University of Vienna Wiedner Hauptstr. 8-10/E201, A-1040 Vienna, Austria

Abstract. Composite beam structures with asymmetrically arranged, perfectly bonded and thick layers are analyzed. Layerwise continuous and linear in-plane displacement

fields are implemented, and as such model both the global and local elastic-viscoplastic response of laminate beams. By definition of an effective cross-sectional rotation the complex problem reduces to the simpler case of a homogenized shear-deformable beam with effective stiffness and corresponding set of boundary conditions. Inelastic defects of the material are equivalent to eigenstrains in an identical but elastic background structure of the homogenized beam with proper effective virgin stiffness. Proper resultants of these eigenstrains are defined. Since the incremental response of the background is considered linear within a given time step, solution methods of the linear theory of flexural vibrations can be applied. The confirmed load consists of the given external forces and the eigenstrain resultants. The latter are updated in space in the course of time. 1.

Introduction

Composite materials are used as structural components in various engineering applications. These components can be beams, plates or shells. In the last decades numerous laminate theories have been developed to describe approximately the kinematics and stress states of these composite structures. Many of these theories are extensions of the conventional single-layer theories of homogeneous beams, plates and shells. The only difference between the so-called equivalent-single-layer theories (e.g. Whitney & Pagano, 1970, Gordaninejad & Bert, 1989, Reddy, 1984) and homogeneous structural theories is in accounting for the varying layer thicknesses and material properties by means of effective beam properties. There are also layerwise laminate theories. These are derived by admitting a separate displacement field within the individual layers of the composite. Besides the equations of motion an additional set of equations is obtained by prescribing the continuity of the transverse shear stresses across the interfaces. The latter can be formulated either by expressing the shear stress 221 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 221–232.

© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

222 distribution according to the law of conservation of momentum (e.g. Di Taranto, 1965, Swift & Heller, 1974, Yu, 1995), or by defining the interlaminar shear stresses by

means of Hooke's law (e.g. Yan & Dowell, 1972, Durocher & Solecki, 1976, Heuer, 1992). In particular, a comparative study of different theories of laminates is given by

Reddy(1993).

A survey of nonlinear vibrations of composite beams is given by Sathyamoorthy

(1982a, 1982b), a review of the analysis of localized impacted sandwich structures is performed by Abrate (1997). Flexural vibrations of redundant viscoplastic sandwich

beams with thin faces are discussed by Fotiu & Irschik (1989). In Brunner & Irschik (1994) the elastic-plastic response of multi-layered composite beams using the so-called second order theory is analyzed.

This paper relates to layerwise laminate beam theories of Heuer (1992) and Adam & Ziegler (1997), where the complete dynamic elastic and elastic-plastic flexural response of symmetrically designed beam structures under external loading is discussed. These theories are subsequently extended in order to analyze inelastic composite beams with asymmetrically arranged and perfectly bonded layers. According to the layerwise laminate theories, the governing equations are derived by application of

the Timoshenko theory of shear-deformable beams to each individual layer. The continuity of the transverse shear stress across the interfaces is considered by defining the interlaminar shear stress, and hence, by means of the generalized Hooke's law. An

effective cross-sectional rotation is introduced, which reduces the complex problem to the simpler case of an equivalent homogeneous beam with effective stiffness and corresponding set of boundary conditions. Furthermore, the dynamic response of the homogenized beam due to imposed fields of eigenstrains is analyzed. The practical importance of eigenstrains is well known from their application in thermoelasticity. Within a multiple field approach, thermal strains, piezoelectrically induced strains, as well as additional fields of strains that are due to plastic deformation (with all of them from their nature incompatible within the elastic strain field) can be interpreted conveniently as eigenstrains acting in the background beam (Irschik & Ziegler, 1995, 1996). Such an interpretation is especially important for applications, because efficient solution strategies exist in the related field of thermoelasticity. Since beam theories deal with stress and strain resultants, proper resultants of these eigenstrains, i.e. imposed curvatures and averaged imposed shear angles, are defined. They constitute the response of the elastic structure when loaded by the imposed curvatures and the averaged imposed shear angles. Under the above aspects and within such a multiple field approach, the nonlinear problem

turns incrementally into two linear problems, where the first one is simply the response

of the associated linear elastic background structure to the given external loads. The second part accounts incrementally for the effects of the physical nonlinearities in the structure. Since the response in both cases is linear within a given time step, solution methods of the linear theory of flexural vibrations can be applied. This problem oriented semi-analytic algorithm has been applied in a series of papers, to vibrations and to waves, in various types of elastic-plastic structures, e.g. Fotiu et al. (1992, 1994), Adam & Ziegler (1997).

223 Application studies verify the accuracy and demonstrate the advantages of the proposed laminate layerwise theory in analyzing the dynamic response of inelastic composite beams. An elastic-viscoplastic material is considered, since in dynamic problems the rate dependence of plastic deformation becomes dominant. The intensity and the distribution of the á priori unknown physical imposed fields of strain acting in the background are determined by the constitutive law in an iterative procedure.

2.

Governing Equations

Composite beams of three perfectly bonded layers with constant rectangular crosssections of area and of modulus of elasticity and of span l in principal bending about the y-axis are considered. The origin of the neutral axis is determined by

where is the distance of the center of gravity of the i-th layer measured in the direction of the lateral coordinate z from a common origin, and where the effective longitudinal stiffness is just the sum of the individual stiffness,

The governing equations are derived by applying the assumptions of Timoshenko's theory of shear deformable beams to each individual layer. Consequently, the displacement field in the i-th layer is assumed to be of the form, Yu (1995),

224 where represents the horizontal displacement at distance z from the central axis, is the portion of at denotes the cross-sectional rotation of the i-th layer and w is the lateral deflection common to all layer axes. The components of the faces can be expressed in terms of and of the cross-sectional rotations in order to satisfy the interface displacement continuity relations, see Figure 1,

In equation denotes the vertical coordinate from the central axis to the interface of the corresponding face and the core, see Figure 1. Within a geometrically linearized theory, total strain and shear angle are given by the gradients

and they are directly derived from equations (3) and (4). The comma in the subscripts indicates partial differentiation.

The normal stress component is neglected and the remaining stress components are related to the strain and shear angle by the generalized Hooke's law, in general the rate form applies,

where is the shear modulus of the homogeneous i-th layer and the imposed fields and denote inelastic strain and inelastic shear angle, respectively. Since beam theories deal with stress and strain resultants, it is important at this stage to determine layerwise proper resultants of these eigenstrains, i.e. plastic stretches plastic curvatures and average plastic shear angles The layerwise eigenstrain resultants are related to the axial force bending moment and shear force of each individual layer. Substituting equations (3) - (6) into the integrals of the stress resultants,

where respectively, and

are the vertical coordinates to the uppermost and lowest fiber, the following relations are obtained,

The imposed eigenstrain resultants in equations (8) are given by

225

They describe layerwise the gross influence of the inelastic strain and the inelastic shear angle on the deformation of the beam. The geometric cross-sectional properties of the layers entering equations (8) and (9) are defined by

where and are the layer's width and height, respectively, and denotes the vertical coordinate of the center of gravity of the i-th layer, Figure 1. The factor is a shear coefficient. The proper choice of its value is discussed by Yu (1995). The total amount of the cross-sectional resultants is determined by summation,

The equations of motion are derived by considering the free-body diagram of an infinitesimal beam element, loaded by a given transverse force per unit of length q(x,t) and an external moment per unit of length m(x,t). Conservation of the angular momentum about the y-axis and conservation of momentum in z- and x-direction renders, after some algebra together with the relations given in (8), the following coupled set of partial differential equations,

with the mass per unit of length

and the mass density

of the individual

layer. The boundary conditions can be prescribed by the following products,

An additional set of equations is obtained when the transverse shear stress is specified to be continuous across the interface. Two types of approximations are acknowledged in the literature. If the "correct" shear stress, that is its value expressed

226 by means of the law of conservation of momentum, is used, the in-plane equilibrium of the faces is automatically satisfied. The resulting beam theory is of the sixth order, see e.g. Di Taranto (1965), Mead (1982). Alternatively, a simplified boundary value problem can be derived by prescribing the shear stress continuity according to the generalized Hooke's law (Heuer, 1992, Adam & Ziegler, 1997),

In analogy to the Timoshenko theory for homogeneous beams also equation (16) exhibits the simplified assumption that the shear stress distribution is uniform throughout the layer.

Subsequently both, the longitudinal as well as the rotatory inertia are neglected,

thus, limiting the analysis to the lower frequency band of structural dynamics. Eliminating the cross-sectional rotations and the horizontal displacements from equations (12) and (13) together with equations (4) and (16) renders a fourth-order differential equation for the deflection w,

Equation (18) can be interpreted as the equation of a homogeneous shear-deformable background beam with effective flexural stiffness effective shear stiffness and mass per unit length forced by given loads q and m and by the two fields, the effective inelastic curvature and the inelastic shear angle The distribution and time evolution of the latter are to be determined by considering the constitutive relations. At this stage they are assumed to be known forcing functions. The effective properties and the effective resultant eigenstrains of the three-layered beam of Figure 1 are given by

with

227 In order to obtain a complete analogy to the homogeneous shear-deformable beam, an effective cross-sectional rotation is defined (Heuer, 1992, Adam & Ziegler, 1997), which is connected to the layer deformation by means of the following relation,

The condition of shear stress continuity, equation (16), together with equation (22) eventually leads to the following expression for

When inserting equations (19), (20) and (23) into equation (11b), it turns out that also the relation of the bending moment shows the form of that of a homogeneous beam,

Consequently, by means of equations (22) - (24) the equation of motion (18) can be separated to form a set of two second order equations,

Equations (25), (26) describe the higher order problem of an elastic-viscoplastic composite beam with piecewise continuous in-plane displacement fields in full analogy to the lower order engineering theory of a homogeneous elastic-plastic shear beam. Classical homogeneous boundary conditions are specified in analogy to the homogeneous shear-deformable beam. - Simply supported end:

- Free end:

- Partly clamped end:

228 Note, that the clamped-end boundary condition for which cannot be formulated in a consistent manner within this effective beam theory. The coupled set of differential equations (25), (26) is solved together with the actual boundary conditions for w and in an incremental procedure, which will be described later. Subsequently, the cross-sectional rotation of the core and of the faces are to be determined. Decomposition of equation (23) yields the cross-sectional rotation of the core,

The cross-sectional rotations of the faces are calculated from equation (16),

Contrary to a homogeneous shear beam theory or to an equivalent-single-layer theory, the elongation of the neutral axis is not only due to inelastic deformation but there is also an elastic elongation, also in the case of vanishing axial in-plane forces. To determine this horizontal displacement equation (14) is used. Introducing (4) and (16) into (14) and considering the assumptions of (17) yields an ordinary differential equation of second order for

with the effective inelastic stretch of the central axis,

and the abbreviations

Equation (32) must be solved with the proper boundary conditions taken into account.

For horizontally immovable supports there is

and, for vanishing axial forces the corresponding boundary condition can be expressed as

229

After evaluating the cross-sectional rotations and the normal stresses are expressed by means of Hooke's law, equation (6a). Finally, from the equilibrium equation in the the shear stress distribution is given by

3.

Constitutive Relations

The rate dependent plastic strain distribution is determined by a viscoplastic law of metals, which is similar to that of Perzyna (1963). For plane stress it reads,

with the yield surface

In (38) denotes Macauley's bracket, and m are constant viscosity parameters and k is the constant radius of the static yield surface. The equivalent eigenstrain distribution is obtained from the equivalence

4.

Numerical Solution

In the present analysis the solution of the coupled set of equations of motion (25) and (26) is found by superposition of two linear elastic contributions,

where are the deflection and effective cross-sectional rotation, respectively, of the elastic homogenized background beam due to the given loads q and m, while are the deflection and effective cross-sectional rotation, respectively, produced in the homogeneous background by imposed effective eigenstrain resultants and The response thus is evaluated in advance by the well-known procedure of linear elastodynamics. Since the distribution of imposed fields of eigenstrains in the background beam is not known in advance and depends on the current state of overall stress and strain,

230 have to determined incrementally by stepping the time and updating the strength of the eigenstrain resultants iteratively in each time step. In Adam & Ziegler (1997) an appropriate solution procedure has been established for this special case of internal loading. 5.

Application

The proposed layerwise beam theory is applied to a three-layered simply supported beam with rectangular cross-section. The left support is horizontally fixed, and the right support is free to move in axial direction. The eigenfrequencies and the normalized mode shapes of such a simply supported shear beam with effective stiffness can be found in e.g. Adam & Ziegler (1997). The ductility of the faces is considered in the form of an ideally elastic-perfectly viscoplastic material, equation (38), while for

simplicity's sake the core is assumed to remain unlimited elastic. The numerical results are compared with those predicted by an equivalent-single-layer (e-s-l) theory (Adam, 1997), to illustrate its merits, and to quantify the significant improvements towards the equivalent-single-layer approximation, where only a single linear cross-sectional rotation describes the displacement of the layered beam. In all subsequent calculations the mechanical properties of the laminate are characterized by the following parameters: longitudinal wave speed ductility parameter Poisson's ratio viscosity parameters and and the ratios with and with denoting the linear fundamental period. If it is assumed that the Young's modulus of the faces is the foregoing parameters describe a laminate beam, whose faces consist of aluminum and the core of polyvinyl chloride. The dimension of the beam is determined by the ratios and The shear coefficient is chosen to be so that the

231

fundamental frequency of the equivalent shear beam and that derived by the plane stress theory takes on the same value. At time instant the composite is subjected to a distributed load according to a sine-half wave, which is also harmonic in time, The ratio of excitation frequency versus fundamental eigenfrequency is

the amplitude of the applied force is determined by the non-dimensional ratio Figure 2 shows the normalized time evolution of the plastic drift and of the midspan deflection. The lower order engineering approximation (e-s-l) underestimates the plastic drift and overestimates the total deflection. Also the moment-(effective) curvature relations (Figure 3), as well as the distribution of at (Figure 4) illustrate the improvement achieved by the present method of analysis.

232 Acknowledgement

Support through the grant P09533-TEC, 1993/95, of the Austrian National Science Foundation FWF is gratefully acknowledged.

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Brunner, W., and Irschik, H. (1994) An efficient algorithm for elasto-viscoplastic vibrations of multi-layered composite beams using second-order theory, Nonlinear Dynamics 6, 37 - 48. Di Taranto, R.A. (1965) Theory of vibratory bending for elastic and viscoelastic layered finite-length beams, J. Appl. Mech. 32, 881 - 886.

Durocher, L.L., and Solecki, R. (1976) Harmonic vibrations of isotropic, elastic, and elastic/viscoelastic threelayered plates, J. Acoust. Soc. Am. 60, 105 - 112.

Fotiu, P., and Irschik, H. (1989) Modal analysis of vibrating viscoplastic composite beams on multiple supports, Earthq. Eng. Struct. Dynamics 18, 1053 - 1064.

Fotiu, P.A., Irschik, H., and Ziegler, F. (1992) Large dynamic deflections of elastic-plastic structures including material damage due to void growth, in D. Besdo and E. Stein E. (eds.), Finite Inelastic Deformations - Theory and Applications, Springer, Berlin, Heidelberg, pp. 67 - 80. Fotiu, P.A., Irschik H., and Ziegler F. (1994) Modal analysis of elastic-plastic plate vibrations by integral equations, Engineering Analysis with Boundary Elements 14, 81 - 97. Gordaninejad F., and Bert, C.W. (1989) A new theory for bending of thick sandwich beams, Int. J. Mech. Sci. 31, 925 - 934. Heuer, R. (1992) Static and dynamic analysis of transversely isotropic, moderately thick sandwich beams by analogy, Acta Mechanica 91, 1 - 9. Irschik, H., and Ziegler, F. (1995) Dynamic processes in structural thermo-viscoplasticity, Appl. Mech. Rev. AMR 48, 301 - 316. Irschik, H., and Ziegler, F. (1996) Maysel's formula generalized for piezoelectric vibrations: application to thin shells of revolution, AIAA J. 34, 2402 - 2405.

Mead, D.J. (1982) A comparison of some equations for the flexural vibration of damped sandwich beams, J. Sound Vibration 83, 363 - 377. Perzyna, P. (1963) The constitutive equations for rate sensitive plastic materials, Quart. Appl. Math. 20, 321 332. Reddy, J.N. (1984) A simple higher-order theory for laminated composite plates, J. Appl. Mech. 51, 745 752. Reddy, J.N. (1993) An evaluation of equivalent-single-layer and layerwise theories of composite laminates, Composite Structures 25, 21 - 35. Sathyamoorthy, M. (1982a) Nonlinear analysis of beams, Part I: A survey of recent advances, Shock and Vibration Digest 14, 7 - 18. Sathyamoorthy, M. (1982b) Nonlinear analysis of beams, Part II: Finite element methods, Shock and Vibration Digest 14, 19 - 35. Swift, G.W., and Heller, R.A. (1974) Layered beam analysis, J. Engng. Mech. Div. ASCE 100, 267 - 282. Whitney, J.M., and Pagano, N.J. (1970) Shear deformation in heterogeneous anisotropic plates, J. Appl. Mech. 37, 1031 - 1036. Yan, M.-J., and Dowell, E.H. (1972) Governing equations for vibrating constrained-layer damping sandwich plates and beams, J. Appl. Mech. 39, 1041 - 1046.

Yu, Y.-Y. 1995: Vibrations of Elastic Plates, Springer, New York.

DESIGN OF MATERIALS WITH EXTREME ELASTIC OR THERMOELASTIC PROPERTIES USING TOPOLOGY OPTIMIZATION

O. SIGMUND1 and S. TORQUATO2 1

Department of Solid Mechanics, Technical University of Denmark,

DK-2800 Lyngby, Denmark 2

Department of Civil Engineering and Operations Research and

Princeton Materials Institute, Princeton University Princeton, N.J. 08544, USA

Abstract

Isotropic composites with extremal elastic or thermoelastic properties are designed using a two or three-phase topology optimization method. The elastic composites are made of two different material phases and the thermoelastic composites are made of of two different material phases and a void phase. The composite microstructures are restricted to one length-scale. The topology optimization method is used to find the distribution of material phases that extremizes an objective function (e.g., shear modulus or thermal expansion coefficient) subject to constraints, such as isotropy and volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using a numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming. The design method is first used to design two-phase composites with extremal values of bulk and shear moduli. The properties of the optimal composites are quite far away from theoretical bounds which is explained by the fact that we only allow one length-scale of the microstructure. Then the design method is used to design three-phase materials with extremal thermal expansion coefficients. For this case, the obtained thermal expansion coefficients are very close to theoretical bounds. Furthermore, it is demonstrated how materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.

233 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 233–244. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

234 1. Introduction

In this paper, we use a topology optimization procedure developed in Sigmund (1994a), (1994b) and Sigmund and Torquato (1997) to determine the distribution of two or three phases (two different bulk material phases and a void phase) in order to design composites with extremal elastic or thermoelastic behavior. For the design of extreme thermal expansion composites, three phases are used (as opposed to the two phases for the pure elastic design ) since one can achieve effective properties of the composite beyond those of the individual components. Microstructural variation is limited to one length scale in a unit cell as this is most easily manufacturable. Finally, the obtained values are compared with theoretical bounds. Materials with specific or extreme thermoelastic properties are important in many engineering applications. An example is in the design of functionally graded materials, where we want to explore the possible range of material properties that can be made from mixtures of different base materials. Materials with high bulk modulus, high shear modulus and low weight are important especially in transportation and aerospace appli-

cations. Materials with zero thermal expansion are of importance in structures subject to temperature changes such as civil engineering and space structures as well as piping

systems. Examples are bridges, where temperature changes between day and night, and summer and winter, cause big structural changes, and space applications, where temperature differences between sunny and shady sides of a structure are extreme. There are a few known examples of isotropic materials with negative thermal expansion coefficients (e.g. glasses in the titania-silica family), there are more examples of materials that have

negative thermal expansion at very low temperatures (< 100K) such as silicon and germanium. Examples of materials with directional negative thermal expansion coefficients at room temperature are Kevlar, carbon fibers, plastically deformed (anisotropic) Invar (Fe-Ni alloys). For design of composites with extremal values of bulk and shear moduli the theoretical bounds on pairs of effective bulk and shear moduli are given in Cherkaev and Gibiansky (1993) who improved the bounds by Hashin and Shtrikman (1963). The problem of finding the structures that extremize the effective elastic properties of two-phase media has a long history beginning with the composite-sphere assemblages of Hashin and Shtrikman (1963) for the bulk modulus problem. Certain hierarchical finite rank laminates

were shown to realize the Hashin-Shtrikman bounds for maximum bulk and shear moduli of isotropic two-phase composites (Francfort and Murat 1986). More recently, Milton and Cherkaev (1995) have found multi-length scale materials possessing elastic proper-

ties ranging over the entire range compatible with thermodynamics assuming infinitely high stiffness of the stiff phase and infinitely low stiffness of the soft phase. Vigdergauz (1989) has studied single-inclusion microstructures of extreme rigidity. Sigmund (1994a)

and (1994b) has designed material structures with specific elastic properties (including isotropic negative Poisson's ratio material), where the microstructure is restricted to one length scale. For three-phase materials the theoretical bounds for effective thermal expansion co-

efficients were first given by Schapery (1968) and Rosen and Hashin (1970). Recently,

Gibiansky and Torquato (1997) have improved upon the Rosen-Hashin bounds using the

235 so-called translation method. This improvement was actually motivated by the topology optimization results of the present study. In this paper, we use the two and three-phase topology optimization methods proposed in afore mentioned works of Sigmund and Sigmund and Torquato to design isotropic composites with extremal elastic or thermoelastic properties. The results are compared with the available bounds on thermoelastic properties for two and three-phase materials. The basic goal is to maximize or minimize the effective shear modulus for the elastic case (two phases) and the thermal expansion coefficient for the thermoelastic case (three phases), subject to constraints on phase volume fractions, isotropy and bulk modulus. The topology optimization procedure used here is described in detail in aforementioned works by Sigmund and Sigmund and Torquato. The method essentially follows the steps of conventional topology optimization procedures (e.g. Bendsøe and Kikuchi 1988) (or Bendsøe (1995) for an overview of methods). The design problem is initialized by defining a design domain discretized by a number of finite elements. The optimization procedure then consists in solving a sequence of finite-element problems followed by

changes in density and material type of each of the finite elements, depending on the local strain energies. At each step of the topology optimization procedure, the effective thermoelastic properties of the microstructure are determined using a finite element based numerical homogenization procedure as developed in Bourgat (1977). 2. Procedures for two or three-phase topology optimization

This section briefly describes the numerical procedure for topology optimization of two or

three-phase material structures in two dimensions. For more details, the reader is referred to the references by Sigmund. Assuming two-dimensional linear elasticity (i.e. small strains), perfect bonding between the material phases, uniform temperature distribution and constant material properties, the thermoelastic behavior of materials can be described by the constitutive relations given as

where and are the elasticity, stress, strain and thermal strain, respectively, and is the temperature change. We refer to as the “thermal strain tensor” (the resulting strain of a material which is allowed to expand freely and which is subjected to increase in temperature of one unit). For the two or three-phase composites of interest, the constitutive equation Eq. (1) is valid on a local scale (with superscripts (0), (1), and (2) appended to the thermoelastic properties, i.e. and and the macroscopic scale (with superscript (*) appended to the properties). In the latter case, the stresses and strains are averages over local stresses and strains, respectively, i.e.

where overbar denotes the volume average. The effective thermoelastic properties,

and

of the composites are computed using a numerical homogenization method.

236 The goal of this work is to extremize the effective shear modulus (two phase materials) and the effective isotropic thermal expansion coefficient (three phase materials) by distributing, in a clever way, given amounts of the two material phases and possibly void within the design domain representing a base cell of a periodic material. As will be seen later, materials with extreme shear modulus or thermal expansion tend to have low bulk modulus. Thus, for practical applications, one must bound the effective bulk modulus from below. The optimization problem An optimization problem for solving above mentioned design problem can be written as Minimize : subject to :

where and are N-vectors containing the design variables and and are small numbers and and are lower and upper bounds on the volume fractions of material 1 and 2, respectively. The individual parts of the optimization problem Eq. (3) are discussed in the following. Design variables and mixture assumption The base cell is discretized by N finite elements. The basic idea of the topology optimization method is that the material type, i.e., material phase 1, phase 2 or void, can vary from finite element to finite element as seen in Fig. 1. With a fine finite-element discretization, this allows us to define complicated bimaterial topologies within the design domain. The design problem consists in assigning either phase 1, 2 or void to each element such that the objective function is minimized. Following the idea of standard topology optimization procedures, the design problem is relaxed by allowing the material at a given point to be a mixture of the three phases. This makes it possible to find sensitivities with respect to design changes, which in turn allows us to use mathematical programming methods to solve the optimization problem. Using a simple artificial mixture assumption, the local stiffness and thermal strain coefficient tensor in element e can be written as a function of the two design variables and

where is a penalization factor which penalized intermediate values of the design variables. The variable . can be seen as a local density variable with

237

meaning that the given element is “void” and

meaning that the given element is solid material. The variable is a “mixture coefficient” with meaning that the given element is pure phase 1 material and meaning that it is pure phase 2 material. By experience the penalty parameter is set to 3. For the design of two-phase elastic composites, the design variable vector is a zero vector and the only variables are the element densities

Constraints on volume fractions The volume fractions of the three phases can be calculated as the sums

where Y is the volume of the base cell and

is the volume of element e.

Isotropy constraint

The composites are constrained to be isotropic. Orthotropy of the materials can be obtained simply by specifying at least one geometrical symmetry axis in the base cell. Assuming that the material structure is orthotropic, the conditions for isotropy of the elasticity tensor under plane stress assumption are that The condition for thermal expansion isotropy is that For technical reasons, these conditions are implemented as in-equality

238 constraints imposed by constraining the squared error in obtaining elastic or thermal

isotropy. The normalized error in obtaining isotropy can be written as

The error in obtaining isotropic thermal expansion can be defined as

Lower bound constraints on bulk modulus

Extreme elastic and thermoelastic properties can be obtained if we allow the overall stiffness of the material to be small. Low stiffness is generally undesirable and therefore we will introduce a lower bound constraint on the bulk modulus of the material. The lower bound constraint on the bulk modulus as a function of the elastic tensor is

Lower bound constraints on design variables For computational reasons (singularity of the stiffness matrix in the finite element formulation), the lower bound on design variable is set to not zero Numerical experiments show that the “void” regions have practically no structural significance and can be regarded as real void regions. The bounds on the design variables can

thus be written as

and

3. Bounds on effective thermoelastic properties

Rigorous bounds for pairs of effective bulk and shear moduli and bounds on the effective coefficients of three-phase, isotropic composites will serve to benchmark the design algorithm. For simplicity, we assume that the constituent phases are isotropic which implies that they can be described by their Young's moduli and their Poisson’s ratios and and their thermal strain coefficients and The bulk and shear moduli of the phases are then and respectively. Elastic (two-phase) bounds

The best bounds on the isotropic effective pairs of bulk and shear moduli , given volume fraction information only, were derived by Cherkaev and Gibiansky (1993) who improved the bounds of Hashin and Shtrikman (1963). A bounded domain of possible effective shear and bulk moduli for a specific choice of constituent phases is shown in Fig. 2.

239 Thermoelastic (three-phase) bounds

Bounds on the isotropic effective pairs of bulk modulus and thermal strain coefficient and bulk modulus of three-phase, isotropic composites were first found by Schapery (1968) and Rosen and Hashin (1970). A bounded domain of possible effective bulk moduli and thermal strain coefficients for a specific choice of constituent phases is shown in Fig. 3. We found that the proposed design method did not yield pairs that were close to the Schapery-Rosen-Hashin bounds. There were two possible explanations for this discrepancy: either the design method could not find the optimal solutions, or the bounds themselves could be improved upon. Indeed, the latter explanation turned out to be true. Inspired by this discrepancy Gibiansky and Torquato (1997) recently found

improved bounds, which are also shown in Fig. 3. As will be seen in the subsequent section, the solutions obtained by the design procedure are very close to the new bounds. Examination of the elastic and thermoelastic bounds in Figs. 2 and 3 reveals that extreme values of shear modulus (i.e. negative Poisson’s ratios) or extreme values of the thermal strain coefficients only are possible for low bulk moduli. Therefore, there is a tradeoff between extremizing the shear modulus or thermal strain coefficients on the one

hand and ending up with a stiff material on the other. 4. Design examples 4.1 Comparison with elastic (two-phase) bounds In this example we consider the design of two-phase materials with extreme values of the shear modulus. The material data for the two phases are chosen as and the volume fractions are prescribed to be Six design examples are considered. Fig. 2 shows 3 by 3 arrays of the optimal microstructures and the effective pairs of bulk and shear moduli are plotted together with the theoretical bounds. The microstructures where discretized by 3600 finite elements and the design domains where rectangular for examples 1, 4-6 and quadratic for examples 2 and 3. Microstructures 1-3 are obtained for maximization of the shear modulus subject to varying the bulk modulus constraints and microstructures 4-6 are obtained for minimization of

the shear modulus. The obtained values are seen to be quite far away from the theoretical bounds. This is explained by the fact that we only allow one length scale in the microstructures. As an option in the optimization code, finer variation in the microstructures can be allowed. The influence of allowing several length-scales in the microstructures

is currently under investigation. As a curiosity, microstructure 3 in Fig. 2 can be said to be a one length-scale version of Milton’s negative Poisson’s ratio fishbone microstructure

(Milton 1992) and microstructures 1 and 4 can be said to be one length-scale versions of the rank-3 materials of (Francfort and Murat 1986).

4.2 Comparison with thermoelastic (three-phase) bounds The material data for the two phases are chosen as and the volume fractions are prescribed to be

The last phase is void.

240

241

We consider the following three three-phase design examples: (a) minimization of the isotropic thermal strain coefficient with a lower bound constraint on the effective bulk modulus given as 10% of the theoretically attainable bulk modulus and four-fold geometric symmetry; (b) maximization of bulk modulus for fixed zero thermal expansion and horizontal reflection symmetry; (c) maximization of isotropic thermal stress coefficient and four-fold symmetry. The resulting topologies are shown in Fig. 4 and their effective properties are plotted as small circles in Fig. 3. Studying the graph in Fig. 3, we see that the obtained effective values are far away from the original Schapery-Rosen-Hashin bounds. This discrepancy inspired Gibiansky and Torquato to try to improve the bounds and indeed improvement was possible as also seen in Fig. 3. The effective values of the examples (a)–(c) are still somewhat away from the improved bounds. This can be explained by the fact that the new bounds by Gibiansky and Torquato have not been proven to be optimal. Furthermore, it is our experience that a finer finite-element mesh makes it possible to get closer to the bounds. The actual mechanisms behind the extreme thermal expansion coefficients of the materialstructures can be difficult to understand. To visualize one of the mechanisms, the (exaggerated) displacements, due to an increase in temperature of the microstructure in Fig. 4 (top), is shown in Fig. 5. When allowing low bulk moduli [as in examples (a)], the main mechanism behind the extreme (negative) thermal expansion is the reentrant cell structure having bimaterial components which bend and cause large deformation when heated. The bimaterial interfaces of design example (a) bend and make the cell contract, similar to the behavior of negative Poisson’s ratio materials (Lakes 1993). If a higher effective bulk modulus is specified, as in example (b), the intricate bi-material mechanisms

242

243 are less pronounced resulting in a less extreme expansion Finally, maximizing the expansive stress, as in example (c), results in a structure without bimaterial mechanisms, where the high expansion phase (cross hatched phase) is arranged such that it maximizes the expansion.

5. Conclusions A material design method based on topology optimization techniques has been used to design material microstructures with effective properties getting close to theoretical bounds for isotropic elastic and thermoelastic composites. The variation in microstructural topologies was constrained to one length-scale which explains why it was impossible to obtain effective shear moduli very close to the theoretical bounds. For the design of thermoelastic materials, the optimization procedure has been shown to be very accurate in producing the optimal microstructures. Indeed, the results of this study motivated Gibiansky and Torquato to improve upon the 29-year old Schapery-Rosen-Hashin bounds on the thermal expansion of three-phase media. Our obtained values are close to the GibianskyTorquato bounds. We have shown that extreme shear and thermal expansion behavior can be obtained but at the cost of a low bulk modulus. Therefore, there is a tradeoff between extremizing elastic or thermal strain coefficients on the one hand and ending up with a stiff material on the other. In practice our optimally designed materials can be fabricated using different techniques. Negative Poisson’s ratio materials have been built using surface micromachining techniques with cell sizes down to 50 microns (Larsen et al. 1996). For larger cell sizes the microstructures can be built using stereolithography techniques (e.g. Jacobs 1992). The method can be modified to handle three-dimensional microstructures. The extension to three dimensions is straight forward, but computer time increases dramatically. Extensions to three dimensions for two material phases have been done in Sigmund(1995) and Sigmund et al. (1997) (in these proceedings).

Acknowledgements Part of this work was done while the first author was visiting Princeton University. The work was supported by Denmark's Technical Research Council (Programme of Research on Computer-Aided Design) (OS) and the ARO/MURI Grant DAAH04-95-1-0102 (OS and ST). References Bendsøe, M. P.: 1995, Optimization of Structural Topology, Shape and Material, Springer.

Bendsøe, M. P. and Kikuchi, N.: 1988, Generating optimal topologies in optimal design using a homogenization method, Computational Methods in Applied Mechanics and Engineering 71, 197–224. Bourgat, J. F: 1977, Numerical experiments of the homogenization method for operators with periodic coefficients, Lecture Notes in Mathematics, Springer Verlag, Berlin, pp. 330–356.

244 Cherkaev, A. V. and Gibiansky, L. V: 1993, Coupled esimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite, Journal of the mechanics and physics of solids

41(5), 937–980. Francfort, G. and Murat, F.: 1986, Homogenization and optimal bounds in linear elasticity, Archieves of Rational Mechanical Analysis 94, 307–334.

Gibiansky, L. V. and Torquato, S.: 1997, Thermal expansion of isotropic multi-phase composites and polycrystals, Journal of the Mechanics and Physics of Solids (to appear). Hashin, Z. and Shtrikman, S.: 1963, A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of the Mechanics and Physics of Solids pp. 127–140. Jacobs, P.:

1992, Rapid Prototyping and Manufacturing - Fundamentals of Stereolithography,

SME, Dearborn, MI, USA.

Lakes, R.: 1993, Materials with structural hierarchy, review article, Nature 361, 511–515. Larsen, U. D., Sigmund, O. and Bouwstra, S.: 1996, Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio, IEEE, International Workshop on Micro Electro Mechanical Systems, MEMS-96. Milton, G. W.: 1992, Composite materials with Poisson’s ratios close to -1, Journalofthe Mechan-

ics and Physics of Solids 40(5), 1105–1137. Milton, G. W. and Cherkaev, A. V: 1995, Which elasticity tensors are realizable?, Journal of Engineering Materials and Technology, Transactions of the ASME 117(4), 483–493.

Rosen, B. W. and Hashin, Z.: 1970, Effective thermal expansion and specific heat of composite materials, International Journal of Engineering Science 8, 157–173.

Schapery, R. A.: 1968, Thermal expansion coefficients of composite materials based on energy principles, Journal of Composite Materials 2(3), 380–404.

Sigmund, O.: 1994a, Design of material structures using topology optimization, PhD thesis, Department of Solid Mechanics, Technical University of Denmark. Sigmund, O.: 1994b, Materials with prescribed constitutive parameters: an inverse homogenization problem, International Journal of Solids and Structures 31( 17), 2313–2329.

Sigmund, O.: 1995, Tailoring materials with prescribed elastic properties, Mechanics of Materials 20,351–368. Sigmund, O. and Torquato, S.: 1997, Design of materials with extreme thermal expansion using a three-phase topology optimization method, Journal of the Mechanics and Physics of Solids 45(6), 1037–1067.

Sigmund, O., Torquato, S., Gibiansky, L. V. and Aksay, I. A.: 1997, On the design of hydrophones

made as 1–3 piezoelectrics, in these proceedings.

Vigdergauz, S. B.: 1989, Regular structures with extremal elastic properties, Mechanics of Solids 24(3), 57–63.

Adaptive Structures

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AN EXACT SOLUTION FOR STATIC SHAPE CONTROL USING PIEZOELECTRIC ACTUATION H. IRSCHIK*, C. ADAM**, R. HEUER**, F. ZIEGLER** * Institute of Mechanics and Machine Design, Johannes-Kepler-University Linz, A-4040 Linz-Auhof, Austria ** Department of Civil Engineering, Technical University of Vienna Wiedner Hauptstr. 8-10/E201, A-1040 Vienna, Austria

Abstract. Exact solutions for the static shape control of smart structures by piezoelectric actuation are derived. Skeletal structures that are assembled of beam- and truss-type members are considered. The desired deformation of the structural

assemblage is exactly matched by means of piezoelectric actuators that are subjected to spatially non-uniform distributions of effective input voltages. The modeling of piezoelectric structures is based on eigenstrain analysis. Additionally, it is demonstrated how to actuate a structural system in order to enforce a desired deformation that is affine to a force-loaded structure with additional boundary constraints. 1.

Introduction

The present paper is concerned with the problem of quasi-static shape control of a structural assemblage made of beam- and truss-type smart structures, where the piezoelectric effect is utilized for the sake of structural actuation. Especially in the fields of application to large space structures and to smart structures, static shape control forms a topic of current interest, cf. Haftka and Adelman (1985), Austin et al. (1994), Varadajan et al. (1996). The problem of static shape

control in general forms a complex inverse problem, and may be stated as follows: Which kind of spatial input voltage distribution should be imposed to a structure in order to match the resulting field of deformation of the structure to a desired one? So far, this question has been tackled by non-linear optimization techniques, where a finite number of actuator patches has been applied to the structure, see Varadajan et al. (1996) and Agrawal et al. (1994). Since a finite number of actuator patches is used in the cited papers, a desired deflection of a flexible distributed-parameter system cannot be achieved exactly. In the present formulation, the actuating piezoelectric strains are considered as eigenstrains acting upon the assemblage of structural members. 'Eigenstrain' represents a generic name devoted by Mura (1991) to non-elastic strains such as thermal 247 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 247–258. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

248 expansion strains. The identification of piezoelectrically induced strains as eigenstrains is not a formal question. It has been stated by Vinson (1992) in the literature that the analogy between the piezoelectric effect and the thermal and hygrothermal effect is extremely important because it enables the analyst and designer to utilize all available thermoelastic and hygrothermal solutions to solve problems involving piezoelectric materials. Eigenstrain analysis is known to represent a convenient tool for flexural

control of smart beams, cf. Irschik, Belyaev and Schlacher (1994, 1995), Irschik and Ziegler (1996). In the present paper, the eigenstrain analysis is extended to a piezoelectrically actuated multi-body assemblage of beam- and truss type structural elements. In order to develop exact benchmark solutions for the problem of structural shape control, distributed piezoelectric actuators with a non-uniform spatial distribution of input voltage are considered subsequently. A review of various technologies for generating piezoelectric actuators with a non-uniform spatial distribution of input voltage, e.g. by shaping the actuator electrodes, has been presented by Lee (1992). Of course, distributed piezoelectric actuators also can be imagined as the result of smearing out a large number of densely packed single actuator patches with continuously varying input voltage.

It is within the scope of the present contribution to derive an exact analytic solution for the static shape control problem of an assemblage of beam and truss-type structural elements. The desired deformation of the structure is considered to be produced by external forces acting upon the background structure which is free of eigenstrains. In developing the corresponding exact solution of the inverse problem, we follow results recently obtained by Irschik and Ziegler (submitted for publication), where the problem of static shape control has been treated in the three-dimensional setting of anisotropic bodies in the presence of imposed non-uniform distributions of eigenstrain. Briefly, it has been shown by Irschik and Ziegler (submitted for publication) that compatible strains generated by imposed external forces, when imposed upon the body as a spatially varying distribution of eigenstrain, produce deformations identical to the deformations generated by imposed forces, and that these eigenstrains form an impotent field. Such distributions of eigenstrains not producing any stress are called 'impotent eigenstrains'. In the present paper, the results of Irschik and Ziegler (submitted for publication) are utilized for the shape control of structures at the level of the classical theory of trusses and beams, i.e. the terms curvature, bending moment, mean strain and normal

force are used instead of three-dimensional strain and stress, and imposed piezoelectric curvature and piezoelectric mean strain play the role of eigenstrain. Especially, it is demonstrated in the present contribution how to actuate a structural system in order to enforce a desired deformation that corresponds to a force-loaded structure with additional boundary constraints, such as additional supports or clamped ends. As special examples, a piezoelectrically actuated cantilever beam with a desired deflection corresponding to a force-loaded clamped-fixed beam, and a piezoelectrically actuated statically determinate truss with a deformation of the joints that is equal to the deformations of a redundant truss are considered. Recently, the exact solution of a

249 piezoelectric cantilever beam with a space-wise sinusoidal desired deflection has been considered in some detail by Irschik, Hagenauer and Ziegler (1997). 2.

Constitutive modeling of a uni-axial structural member with actuating piezoelectric layers

In the following, a single structural member composed of attached or embedded piezoelectric layers and a substrate material is considered. Perfect bond between the layers and the substrate is assumed, and the piezoelectric layers are considered to operate as actuators. Beam-type structural members with rectangular cross-section of area A are studied, where the axial coordinate of the member is denoted by The deformation of the member is assumed to take place in the (x, z)-plane, z denoting the transverse coordinate of the laminate. The constitutive modeling of an actuating piezoelectric layer is based on the following assumptions: The piezoelectric layer acts as a capacitor, where the electric voltage V is constant along the metallic electrodes of the capacitor and is connected to the non-vanishing component of the quasi-static electric field density by

where h denotes the (small) thickness of the piezoelectric layer. The components of the field density tangential to the layer vanish, . An isothermal, uni-axial linear constitutive equation is set up between the respective component of the electric flux vector, and the axial strain in the layer:

where and represent effective constitutive parameters. These parameters are obtained from the full set of three-dimensional constitutive equations by setting together with and inserting the result in order to replace the effect of the non-axial strains in Eq. (2), Within a isothermal approximation, the mechanical constitutive equations are analogously written in the uni-axial form,

again denoting an effective elastic modulus. Inserting Eqs.(l) and (2) into Eq.(3) yields:

250 where the piezoelectric eigenstrain is given by

with effective modulus of elasticity,

Note the analogy between the piezoelectric eigenstrain in Eq. (4), and thermal eigenstrains, cf. Ziegler (1995). It has been stated in the literature that the analogy between the piezoelectric and the thermal effect makes it possible to utilize all available thermoelastic solutions to solve problems involving piezoelectric materials, Vinson (1992). Following the geometric assumptions of the Bernoulli–Euler beam theory, cf. again Ziegler (1995),

the total strain in a fiber located in the distance z from the axis is given by

where u and w denote the axial and transverse displacement of the member axis, respectively. The location of the axis is selected in the elastic centroid of the

cross-section with area A

The constitutive equations between normal force and mean strain, and between bending moment and curvature of the beam axis in the presence of piezoelectric eigenstrain

become, respectively,

251

where extensional and flexural stiffness of the structural member are given by crosssectional integration

respectively. Weighted mean eigenstrain and eigen-curvature occurring in Eq. (10) and (11) are defined by cross-sectional integration

Finally, the distribution of the axial stress within the cross-section becomes

3.

Maysel's formula applied to the piezoelectric structural assemblage

Noting the analogy between the piezoelectric and the thermal effect, piezoelectrically induced deformations of an assemblage composed of truss- and beam-type structural members are efficiently calculated by means of Maysel's formula, cf. Ziegler and Irschik(1987):

where the integration first is separately performed for the j-th structural member s, and the summation afterwards extends the n members of the structural

assemblage. In Eq. (16), the dummy variable of integration is denoted by The component of structural displacement due to an imposed electric voltage in the direction of an arbitrary unit vector and at the location x , is denoted by .

252 Furthermore, is the isothermal normal stress at applied at x, in the direction and and bending moment and normal force, respectively:

4.

due to a single unit force denote the corresponding

The dummy force method applied to force–induced structural deformations

In this section, the dummy force method is shortly reviewed for the case of structural deformations that are produced by external forces in the background (i.e. in the absence of piezoelectric strains). The principle of virtual work is applied to the dummy force

problem at first:

where a variation is denoted by the symbol

The virtual work of the unit dummy

action is given by

and the virtual work of the dummy internal forces is

A special field of virtual displacements is used in the following derivation, namely the virtual displacements produced by external forces that are imposed upon the background structure (in the absence of piezoelectric strains)

where in the present case

253

The virtual work of the dummy external and internal forces, done with respect to the force-induced displacements, now becomes, Eq. (20),

and the work of the internal forces is, Eq. (21),

In Eqs. (22) - (25), the index (f) refers to the force loading. Inserting Eqs. (22) and (25) into Eq. (19) leads to the following integral statement, which is known as the principle of virtual forces, cf. Ziegler (1995):

5.

Static shape control by applied electric voltage

The inverse problem: which distribution of imposed electric voltage should be applied in order to annihilate or at least to minimize deformations due to imposed forces, can be exactly solved, just by comparing the integral statements given in Eqs. (16) and (26). It is seen that any structural deformation due to imposed forces is exactly equal to the deformation produced by piezoelectric actuation

if the imposed piezoelectric curvature is chosen as

254 and the piezoelectric mean strain is taken in the form

Applying distributed piezoelectric actuation according to Eqs. (28) and (29) yields an exact solution of the underlying static shape control problem. Note that, since the

piezoelectrically induced and the force-induced deformations of the whole structure are equal, so are the axial and the transversal displacements of the structural members:

Inserting Eqs. (30) and (31) into Eqs. (28) and (29) and applying the result to Eqs. (10) and (11), it is seen that the piezoelectrically induced bending moment and normal force vanish, also in the case of a redundant structure:

In the particular interpretation of Eq. (32), the spatial distribution of piezoelectric actuation presented in Eqs. (28) and (29) represent an impotent field.

6.

Desired deflection of a structural system with additional kinematic constraints

This Section is concerned with the case of a desired deflection corresponding to a structural system with additional kinematic constraints, such as additional supports. Note that it is kinematically admissible to use the force induced deformation of such a system as a virtual deformation in Eq. (22), since the deformation of the system with constraints satisfies the kinematic conditions of the released original system:

where a hat refers to the system with additional constraints. Running again through the derivations of Section 5, one eventually ends with the result

255 It is thus possible to achieve a desired deformation of redundant structure by piezoelectric actuation of a statically determinate structure:

7.

Example

As an example, we consider a cantilevered beam, x = 0, and the free end at x = L, see Fig. 1.

with the clamped end at

The desired deflection of this cantilevered beam is considered as the deflection of a clamped-hinged (C-S) beam, Fig. 2, with the simple support at x = L, due to uniformly distributed lateral force load . Assuming B(x) = B = const., the desired deflection is

cf. Roark (1989). In order to achieve this deflection, we follow the suggestion given in Eq. (34), and apply in the form

256

where the bending moment of the clamped-hinged beam is

see again Roark (1989), and Fig. 2.

The deflection of the cantilever that is due to the piezoelectrically induced curvature of Eq. (38) is subsequently calculated using Mohr's conjugate beam method, cf. Ziegler

257 (1995). The conjugate beam is a cantilever, however with the clamped end at x = L and the free end at x = 0, Fig. 3. Since the original beam is statically determinate, its bending moment due to vanishes:

Following Mohr's conjugate beam method generalized to include the presence of piezoelectric curvature, the conjugate beam must be loaded by the fictitious forces

cf. Eqs. (38) and (39). The bending moment of Mohr's conjugate beam is equal to the deflection of the original one, and indeed, it gives the desired deflection

as it has been predicted in the previous section.

Thus, it becomes possible, to produce a desired deflection of the statically determinate beam by means of spatially distributed piezoelectric actuation, that is equal

to the deflection of the statically indeterminate clamped-hinged beam when loaded by the same forces.

258 References Agrawal, S., Tong, D., and Nagaraja, K. (1994) Modeling and shape control of piezoelectric actuator embedded elastic plates, Journal Intelligent Material Systems and Structures 5, 515-521. Austin, F., Rossi, M.J., Van Nostrad, W., Knowles, G., and Jameson, A. (1994) Static shape control of adaptive wings, AIAA-Journal 32, 1895-1901. Haftka, R.T., and Adelman, H.M. (1985) An analytical investigation of static shape control of large space structures by applied temperature, AlAA-Journal 23, 450-457. Irschik, H., Belyaev, A.K., and Schlacher, K. (1994) Eigenstrain analysis of smart beam-type structures, in M. Acar, J. Makra and E. Penney (eds.), Mechatronics: The Basis for New Industrial Developments, Comp. Mechanics Publ., Southampton, pp. 487-492. Irschik, H., Belyaev, A.K., and Schlacher, K. (1995) Distributed control of structures using eigenstrain analysis, in G.W. Housner, S.F. Masri and A.G. Chassiakos (eds.), Proc. 1st World Conference on Structural Control, Vol.1, WP3, Association for Structural Control, Los Angeles, pp. 73-82. Irschik, H., and Ziegler, F. (1996) Maysel's formula generalized for piezoelectric vibrations: application to thin shells of revolution, AIAA-Journal 34, 2402-2405.

Irschik, H., and Ziegler, F. (submitted for publication) Static shape control of structures by applied stressfree eigenstrains, AIAA-Journal. Irschik, H., Hagenauer, K., and Ziegler, F. (1997) An exact solution for static shape control by piezoelectric actuation, in U. Gabbert (ed,), Proc. Second Scientific Conference on Smart Mechanical Systems-

Adaptronics, VDI-Verlag, Dusseldorf.

Irschik, H., Heuer, R., and Ziegler, F. (1997) Static shape control of redundant beams and trusses by thermal strain, in R.B. Hetnarski (ed.), Proc. Thermal Stresses '97, Rochester, New York, pp. 469-472. Lee, C.-K. (1992) Piezoelectric laminates: theory and experiment for distributed sensors and actuators, in H.S. Tzou and G.L. Anderson (eds.), Intelligent Structural Systems , Kluwer, Dordrecht, pp. 75-168. Mura, T. (1991) Micromechanics of Defects in Solids, 2nd ed., Kluwer, Dordrecht. Roark, R.J. (1989) Formulas for Stress and Strain, 6th ed., McGraw-Hill, New York. Varadajan, S., Chandrashekara, K., and Agarwal, S. (1996) Adaptive shape control of laminated composite plates using piezoelectric materials, in D. Martinez and I. Chopra (eds.), Proc. AIAA/ASME/AHS Adaptive Structures Forum, Salt Lake City, UT, AIAA-Paper No. 96-1288, pp. 197-206. Vinson, J.R. (1992) The Behavior of Shells Composed of Isotropic and Composite Material, Kluwer, Dordrecht. Ziegler, F. (1995) Mechanics of Solids and Fluids, 2nd ed., Springer-Verlag, New York.

Ziegler, F., and Irschik, H. (1987) Thermal stress analysis based on Maysel's formula, in R.B. Hetnarski (ed.), Thermal Stresses II, Elsevier, 120-188.

ON THE THEORY OF SMART COMPOSITE STRUCTURES

A.L. KALAMKAROV and A.D. DROZDOV Department of Mechanical Engineering Technical University of Nova Scotia P.O. Box 1000, Halifax, Nova Scotia Canada, B3J 2X4

Abstract The present paper is concerned with the basic aspects of a newly suggested theory of smart composite structures based on the continuum mechanics approach. The governing

equations describing the behavior of a smart composite structures incorporating sensors

and actuators are derived, and the basic optimization problems in the design of these controllable structures are formulated. This theory deals mainly with the extremal features of the controllable smart structures. The objective of modeling is to determine limiting properties of the smart structure. This also allows to determine whether the properties of the presently existing materials, sensors and actuators are sufficient for the optimal design of smart structure, or the development of some new materials, sensors or actuators is required. The basic optimization problems for the smart composite structures are illustrated by three examples in which the three main sources of control are emphasized. These are the residual strains, material properties, and the geometry of a structure. In the first example, we derive the optimal residual stress in an actuator which provides the minimum deflection of a composite cantilevered beam under static loading. It is shown that the effect of actuator allows to reduce the maximum deflection by 28 times compared with the same beam without active control. The second example is concerned with the optimal design of the controllable Winkler's foundation in the problem of

vibration damping for a simply supported beam under the dynamic loading. The controllable property here is a rigidity of foundation. It is shown that by using the optimally designed controllable foundation, the maximum deflection of a beam can be reduced by about 8 times. The third example deals with the optimal design of an actuator for a smart composite beam. The objective is to reduce the maximum deflection by applying a constant residual strain to the actuator. It is shown, in particular, that for the strains which exceed the obtained critical value, the optimal length of the actuator is smaller than the length of the beam, and it diminishes up to zero with the growth of the applied strain. 259 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 259–270. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

260

1. Introduction The major objective of the present paper is to introduce a new theory of smart composite structures in the framework of the continuum mechanics. Under a smart

composite structure we mean a structure with sensors and actuators which is actively controlled, and which performs a required motion that is optimal in a class of admissible motions. The theory of smart structures may employ mathematical methods similar to the methods of the optimal control theory, but it is essentially focused on the dependence of the cost functional on the system parameters and on the optimal design of a system which provides the optimal (with respect to a cost functional) properties of an active control. For example, in the simplest problem of active damping of vibrations for a cantilevered beam, see e.g., Su and Tadjbakhsh

(1991), and Irschik and Ziegler (1996), the problem of optimal control is to find the signals applied to actuators which minimize the deflection, whereas the problem in the theory of smart structures is to choose the properties of actuators and their distribution that ensure the minimal deflection under assumption that all the actuators work in their optimal regimes. In order to estimate structures in progress, it is important from the engineering standpoint to predict their ultimate (limiting) features bearing in mind that any appropriate changes in properties of the main material of a structure, as well as of sensors and actuators, are admissible. This allows us to decide whether properties of the presently existing materials and devices are sufficient for the structure, or new materials and devices are required for the project. In the present paper we discuss the governing equations for a smart structure with sensors and actuators. We formulate the corresponding optimization problems, and we solve three applied examples of a practical interest.

2. Governing Equations and Optimization Problems Let us consider a smart composite structure consisting of an elastic solid and a set

of sensors and actuators. Generally, both the main solid and auxiliary devices are assumed to be made of anisotropic composite materials. The structure occupies a domain with a boundary At the instant external forces are applied to the body. The load consists of body forces B and surface tractions b. The surface forces 6 are applied to a part of the boundary The other part of the boundary, is assumed to be fixed.

Denote by u(t,x) the displacement vector at point with coordinates x = at moment We assume that the vector field u is sufficiently smooth, and the strain tensor can be defined at any point Confining

261 ourselves to the case of infinitesimal strains we can write

where is the gradient operator, and T denotes transpose. Denote by the stress tensor. This tensor satisfies the equation of motion

where is the mass density, and the dot denotes the inner product. Eqns. (1) and (2) should be fulfilled both for controllable (with the use of actuators), and for uncontrollable motions. The constitutive equations depend essentially on the presence (or absence) of sensors and actuators. In the absence of actuators, we treat the material as anisotropic elastic composite. Accordingly, the stress can be expressed in terms of the strains by generalized Hooke’s law Here C is a fourth-rank tensor of elastic coefficients. To take into account the presence of sensors and a heterogeneous structure of the composite material, we assume that the main material is inhomogeneous which is modeled by the dependence

C = C(x), see e.g., Kalamkarov (1992). In the presence of actuators, the material is modeled as a two-phase blend of the main material (3) and distributed actuators. The latter means that the number

of actuators can be rather large, and we can assume that in any small (in the sense of continuous mechanics) domain with volume V, the actuators occupy a subdomain with volume Dividing by V we define volume density of actuators As it is common in mechanics of blends, we assume that the volume density coincides with the surface density. We treat actuators also as anisotropic composite elastic solids with tensor of elastic coefficients Similarly to Eqn. (3) we write

where

is the tensor of residual strains in actuators produced under the action of control signals. The total stress in material is equal to the sum of stresses in the main material and in the actuators

Substitution of expressions (3) and (4) into Eqn. (5) yields

262 Eqn. (6) demonstrates three sources of control for a smart structure. First, we can choose the tensor as a function of time t and spatial coordinates whereas other parameters are prescribed. In this case the material characteristics do not change, and the motion is controlled only by residual strains in actuators. This kind of control occurs, for example, when electrical signals are applied to piezoelectric actuators, or when thermal loads axe applied to shape memory alloys. Second, we can choose the tensor as a function of time t and spatial coordinates whereas other parameters remain prescribed. From the physical standpoint, this means that we vary only elastic properties of material, without changing other characteristics. This kind of control occurs in electro-mechanical systems where some sub-systems are turned on and off under the action of electric signal, or in merely mechanical systems in physical fields (temperature, humidity, radiation, etc.), as well as in structures with electrorheological actuators. Third, we can choose parameter as a function of spatial coordinates whereas other parameters are prescribed. This corresponds to the problem of optimal design of a geometry of a structure with the distributed actuators. Unlike the standard problems of optimal design, we seek here not a shape of the structure, but

optimal (in some sense) spatial distribution of actuators.

It is evident, that the aforementioned sources of control can be combined. As a result, we obtain the basic problem in the theory of smart structures: to determine the optimal mechanical properties of the main composite material and the optimal spatial distribution of sensors and actuators which ensure (under the optimal control of the actuators with the use of information from sensors) the achievement of a required performance. The theory can be generalized by bearing in mind nonlinearity in the response for both the main material and actuators, and the inelastic properties of the main material (which lead to more complex constitutive equations). Nevertheless, the above three main sources of control: residual stresses, material properties, and geometry, remain unchanged. The Eqns. (1), (2), and (6) together with the boundary conditions

determine the behavior of a smart composite structure. The problem consists in establishing such control parameters which provide an optimal performance in a class of admissible motions.

3. Optimization of Residual Strains in a Smart Structure As a first example of application of the general theory, let us consider a cantilevered beam with length l and rectangular cross-section of a unit width. The beam consists of two perfectly bonded layers, see Fig. 1. The upper layer is made of a linear elastic

263 material. It has thickness h and Young’s modulus E. The lower layer is made of a piezoelectric elastic material with thickness and Young’s modulus The subscript a means that the lower layer is employed as an actuator. We assume that electric potentials applied to the actuator surfaces produce a residual strain

lower layer. This strain depends only on time, to provide an optimal conduct of the whole structure.

in the

and it can be controlled

Introduce coordinates x and y as it is shown in Fig. 1. Axis x coincides with the interface between the upper and lower layers. Denote by u(x) the displacement in x direction, and by w(x) the beam deflection on the interface. The non-zero components and of the displacement field are the following:

where prime denotes the differentiation with respect to x. According to Eqn. (8), the only non-zero component of the strain tensor equals

Neglecting the Poisson’s effect, we assume that the only non-zero component of the stress tensor is The stress is related to the strain by Hooke’s law, cf. Eqns. (3) and (4),

For the longitudinal force

264 after simple algebra we obtain

It follows from the equilibrium equations that N = 0 This equality together with expression (11) implies that

We now substitute expression (10) into the formula for the bending moment

and find

For

and

Eqn. (13) yields the well-known formula

Let us assume now that a time-independent transverse force P is applied to the free end of the cantilevered beam. It follows from the equilibrium equations that

Substitution of expression (13) into this equality yields

where

Integrating Eqn. (14) with boundary conditions

we obtain

265 Let us consider the following optimization problem: find such a value strain which minimizes the maximum deflection of the beam

of

To solve this problem we, first, fix and calculate for this strain, and, afterwards, minimize the obtained value with respect to As s result of solution (details are omitted here) we obtain that the optimal strain in the actuator, which minimizes the beam deflection can be calculated according to the formula

The maximum deflection of the beam is equal to

Expression (17) allows to predict the limiting properties of the actuator. It follows from Eqn. (16) that in the absence of control, the maximal deflection of the beam equals Comparison of expressions (17) and (18) shows that the optimal control of the actuator allows to reduce the maximum deflection by 28.3 times.

4. Optimization of Material Properties of a Smart Structure In the second example, let us consider a simply supported elastic beam lying on a smart Winkler elastic foundation. The beam has length l, cross-sectional area S, moment of inertia I, and mass density These parameters are assumed to be independent of the longitudinal coordinate x. At instant t = 0 a distributed transverse load q(x) is applied to the beam. Denote by w(t,x) the beam deflection at point x at moment t. Under the standard assumptions of the technical theory of bending, function w obeys the following equation

with the boundary conditions

266

and the initial conditions

Here g(t,x) is the reaction of foundation at point x at instant t. We assume that this reaction is controllable due to changes in the rigidity of foundation. Namely, we assume that

Here c(t) is a piece-wise continuous function taking its values from the interval

where c1 and c2 are given positive constants,

From the practical point of view, such a foundation can be designed as a

Winkler foundation including two kinds of springs. The springs of the first kind are non-controllable, their deposit to the total rigidity is characterized by term c1. The springs of the other kind are controllable. They can be ”turned on” and ”turned off” by a control electrical signal. When the signal is absent all these srpings are turned off, and no additional reaction arises in the foundation. When the signal is present, some springs are turned on, and their number is proportional to the signal intensity. The maximum reaction means that all these springs work, and their contribution into the total rigidity increases its value up to Introduce the dimensionless variable and parameters being a characteristic deflection of the beam)

In this notation Eqn. (19) can be written as follows (for simplicity asterisks are omitted):

Our objective is to find the optimal control of the foundation rigidity which minimizes the maximum deflection of the beam

and which satisfies the restrictions

267 We confine ourselves to a particular case when

For load (27), it is natural to seek the solution of Eqn. (24) in the form

where W (t) is a function to be found. It is easy to show that function (28) satisfies

the boundary conditions (20).

Figure 2. Ratio

versus the limitation on the controllable rigidity of the foundation.

Substitution of expressions (22) and (28) into Eqn. (24) yields I

Introduce the new control function Z(t) satisfies the inequality

According to Eqn. (26), the

As a result of solution (details are omitted here), we obtain that the dependence of the dimensionless parameter

268 on is significant. This dependence is plotted in Fig. 2. The results show that the parameter decreases in and reaches its minimal valuefor This leads to two important conclusions: (i) by using the optimal control of the rigidity of foundation we can reduce the maximum deflection of the beam by 8.3 times ( (ii) to achieve the maximum effect of damping, it is not necessary to increase the foundation rigidity ad infinity, but it is sufficient to increase the initial rigidity only by 8 times with the same efficiency.

5. Optimization of a Geometry of a Smart Structure

Let us consider a simply supported elastic beam with the length l and with an actuator. The actuator is modeled as an elastic beam with length 2a, which is perfectly bonded to the main beam and is located symmetrically with respect to the beam center, see Fig. 3. At instant t = 0 distributed transverse load with a constant intensity q is applied to the beam, and an electrical signal is transmitted to the piezoelectrical actuator. This signal produces a residual compressive strain in the actuator. The beam deforms under the action of external load and actuator's compression. The bending moment M relates to the beam deflection w through the expression

where parameters A and B are determined by Eqn. (15).

269

The equilibrium equation is written as follows:

Integration of Eqn. (32) with the boundary conditions M(0) = M(l) = 0 implies

that

where

is the Green function for an appropriate boundary problem. Substitution of expression (31) into Eqn. (33) yields

Integration of these equations with the boundary conditions w(0) = w(l) = 0 yields

Introduce the dimensionless variables

In the new notation, calculating the integrals in Eqn. (35) we obtain

where

The optimization problem is formulated as follows: for a given intensity of the

residual strain

find parameter

which minimizes the maximal deflection

and which satisfies the natural restriction 0 Problem (36) and (37) is not difficult from the mathematical viewpoint. The dependence of the optimal length of the actuator on the residual strain can

270 be obtained numerically. A non-evident conclusion follows from the plot of this dependence, see Fig. 4. According to the numerical analysis, a long actuator with the length equal to the length of the beam is optimal only for relatively small residual strains, For the optimal length of the actuator

diminishes with an increase in

and tends to zero when

Returning to

the dimensional variables, we find the following formula for the critical value of the strain:

For the optimal length of the actuator is smaller than the length of the main beam, and it can be determined from the plot of Fig. 4. 6. References Irschik, H. and Ziegler, F. (1996) Maysel’s formula generalized for piezoelectric vibrations: application to thin shells of revolution, AIAA Journal. 34,

2402-2405. Kalamkarov, A.L. (1992) Composite and Reinforced Elements of Construction, Wiley, Chichester, N.Y. Su, Y.-A. and I.G. Tadjbakhsh, I.G. (1991) Optimal control of beams with dynamic

loading and buckling, Trans. ASME. J. Appl. Mech. 58, 197-202.

SMART HINGE BEAM FOR SHAPE CONTROL

D. PERREUX and C. LEXCELLENT Laboratoire de Mécanique Appliquée R. C., UMR 6604 CNRS-UFC UFR Sciences et Techniques - 24 rue de l'Epitaphe 25030 BESANCON CEDEX-FRANCE

Abstract The deflexion of a (uniform) beam depends on four quantities : the load, the length, the Young's modulus and the second moment of area, I. This paper presents a smart beam with a second moment of area which can be varied by the use of a shape memory Alloy (S.M.A.) ; this smart beam performs better than classical smart beams in which the actuator operates directly on the beam. A thermomechanical modeling of the beam is proposed, and predictions from it agree well with experimental results. 1. Introduction A smart structure or material is one that is able to adapt to varying service conditions (Martin ; 1994). Usually, a smart structure is composed of an actuator, an inactive structure, a set of transducers which provide the strain in the structure and an electronic device for controlling. A promising candidate for the actuator in a smart structure is a shape memory alloy (SMA). To be effective as an actuator, the SMA must be able to produce significant work ; there must be displacement (or strain) and stress. It seems to be possible to produce one or the other, strain or stress, but not both in significant amounts. Thus, if we start with a prestrained sample in the martensitic state, then simple heating and no stress action will, by the one way shape memory effect (O.W.S.M.E.), produce large deformation. On the other hand, if we start with a prestrained sample, then with deformation held constant, simple heating will develop a significant stress, called recovery stress (Tobushi et al ; 1991). In both of these extreme situations the external work is zero. Knowing the law describing the interactions between stress, strain and temperature, we can find conditions under which stress and strain will both exist, but even then the external work produced will be insignificant compared to the electric energy which is supplied for heating the SMA elements by the Joule-Thomson effect.

271 Y.A. Bahei-El-Din and GJ. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 271–281. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

272

There are other situations to consider. In the isothermal pseudoelastic case which corresponds to the creation of "oriented martensite" under external stress action, the work done can be important (Tobushi et al. ; 1991), but in this case the SMA element is a passive one and could not be used as an actuator. There are smart materials consisting of a resin epoxy matrix with embedded prestrained Ni Ti fibres (making up a few percents of the total mass) (Lin and Rogers, 1991 ; Thiebaud et al. ; 1996). Heating of the SMA fibres can produce a shift in the resonance frequencies by the recovery stress effect (Tobushi et al ; 1994). However, if the matrix has an appreciable stiffness, the change in the geometrical shape of the body will be small. Similar comments apply to piezoelectric ceramics used for shape control . the deflection of a sandwich beam by a PZT ceramic is small because the actuactor produces negligible work (Hodar and Perreux ; 1996). We conclude that in spite of its attractive properties, OWSME etc, a SMA element, when used directly, is not an efficient actuator. In this paper, we consider an indirect use of a SMA element . We consider a hybrid element composed of an elastic structure (a beam) and a SMA element. The combined hybrid structure is smart because heating of the SMA element changes the physical parameters specifically the second moment of area of the elastic structure. From a very general point of view, we can see the role of a SMA element as producing a redistribution of stress in a structure ; this redistribution of stress will bring about a corresponding redistribution of displacement. This presents a design problem : how should SMA elements be deployed to produce amplified displacements of certain points of a structure ? In this paper, we develop the theory relative to a single hybrid beam element ; this element can be viewed as one module of a complex intelligent structure. This paper describes a beam with a second moment of area which can be varied by the use of a Shape Memory Alloy (SMA). 2. Principle of Change of second Moment of a Beam

Consider a simple beam with constant rectangular section loaded in flexure (Fig. 1). For elastic loading, the maximum deflection is given in the classical form:

273

where P is the load, L is the length of the beam, E is Young's modulus of the material and I is the second moment of area of the beam. For a rectangular beam

Here, h is the thickness and b the width of the beam. If we assume that I is a variable, the rate of change of the maximal deflection is obtained by:

If by an external device, the thickness of the beam can be modified, the change in

second moment is expressed by:

then:

The nonlinear equation (5) shows that a small variation of h provides a large variation of the maximum deflection.

The stress distribution at the cross section x of the beam

where

and

is the bending moment at the cross section x. We conclude that a change

in the thickness implies a change of second moment and modifies the stress distribution ; this is the basic idea which we will use. Fig. 2 depicts a "hinge beam" made of two beams hinged along their length.

274

The second moment of area is a function of the angle of rotation :

Figure 3 shows that the moment changes in a strong way with The material data and other parameters are given in Table 1, the material used is PMMA.

275

If the SMA wire is shrunk by a thermomechanical process, the angle will change and consequently the deflection of the beam. The use of SMA seems very interesting ; to obtain a significant change in the rotating angle, the strain of the actuator between the edges of the beam need only be a few percents. The system consisting of the « hinge beam » and the SMA wire can be considered as a module. A number of modules can be assembled to obtain a complex structure. The hinge between the two beams can be made of some rubbery material ; alternatively the hinge could be produced by reducing the cross section of the beam along the mid-line, as shown in Fig. 5a. Fig. 5b shows another possibility for the use

of an SMA wire to produce a change in beam cross section.

3. Modeling of the rotating angle in smart hinge beam

Consider the cross section of the hinge beam presented in figure 4. Projecting the beams and arms in the horizontal we find Thus the rate of change of is:

276

The strain of the SMA wires is

where is the unstretched length of the wires (without prestrain).The strain rate the SMA wire can be introduced into (10) to give

of

By writing the force balance of the total system, it can be easily shown that the stress in the wires is:

where S is the total cross section surface of the wires. Equation (13) shows that (under constant P)

Equations (1) and (8) show that to obtain the angle must be determined. Equation (12) relates the change of this angle to the strain in the SMA. The angle can be obtained by introducing the SMA thermomechanical behavior laws which link the strain rate with the temperature rate and the stress rate. 4. Modeling of Shape Memory Alloys

The general modeling of SMA anisothermal behavior has already been described by Leclercq and Lexcellent (1996). The present paper gives the main equations used in the modeling of the smart hinge beam.

With the classical hypothesis of one-dimensional small deformation, the total strain rate is assumed to satisfy the following equation:

where

is the elastic strain rate:

277

and is Young's modulus of the wires, and is the phase transformation strain rate. In fact in (15) the thermal dilatation strain is neglected. The equations governing the process are:

where a, c and d are material parameters, is the transformation strain for complete phase transition ; in our case, it corresponds to the prestrain of the initial wire. is the starting reverse transformation temperature (Martensite Austenite). Equation (17) is valid only for the reverse transformation. The equivalent equations for the direct transformation (Austenite Martensite) are established in the publication of Leclercq and Lexcellent (19%). This paper is focused on the reverse transformation because it is the phenomena which is observed when the wires are heated by the JouleThomson effect. The direct transformation

is used for cooling from the highest

The initial conditions of the wires are

this gives a length equal to

temperature. We follow the idea of Brinson (1993), in which the total martensite fraction z is divided into two parts: the self accommodating one called obtained under pure thermal loading and the oriented martensite one called obtained by the combination of stress and temperature effects. The self accommodating martensite gives a negligible macroscopic transformation strain, whereas the oriented martensite given by equation (17) produces an important associated strain.

and By solving the set of equations (12, 14-17), we can obtain the change of as a function of temperature (Fig. 6). For this calculation, the parameters are given in Table 1.

278

The normalized deflection of the beam defined by

can now be obtained as

a function of the temperature through equation (8) which gives the second moment in terms of the angle Figure (7) presents this result.

For a temperature change of 30° C, is divided by 4. The performance of this system seems very attractive, because the volume of SMA in this device is very low, and nevertheless the change of the normalized deflection of the smart hinge beam is large.

279 5. Experimental Validation

The beam was made of P.P.M.A., the wires were NiTi and were heated by JouleThomson effect. Figures 8 and 9 show photographs of the system for two temperatures and Figure 10 presents the change of the deflection of the smart hinge beam versus P for both temperatures. The theoretical curve is shown on the same figure and exhibits a good agreement.

280

The temperature of the wires as a function of the current intensity was measured by an

infrared camera. Figure 11 presents the results. The temperature control was made by controlling the current intensity during the test.

6. Conclusion

Although it is theoretically possible to control a structure by using a direct SMA system, it is impractical because of the insignificant work that such a SMA system can do. We have proposed an indirect system. The basis of our method is the use of a prestrained structure ; the strain energy stored in the material allows a significant

281

shape modification to be obtained. There are various ways to obtain the prestrain ; we used a simple load, but springs or internal temperature-produced stresses could also be used. The proposed system has many possible applications ; its principal feature is that a small actuator volume produces a large change in deflection. Our group is currently developing systems composed of several smart beams. Acknowledgement The authors thank Professor G.M.L. Gladwell for his valuable help in improving the English language text. References Brinson, L.C. (1993) One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non constant functions and redefined internal variables, J. Intelligent Mater. Syst. Structures 4, pp. 229-42. Hodar, F. and Perreux, D. (1996) Characterization and modelling of a smart sandwich structure, Proceedings of J.N.C. 10, 2, pp. 883, 892. Leclercq, S. and Lexcellent, C. (1996) A general macroscopic description of the thermomechanical behavior of shape memory alloy, J. Mech. Phys. Solids, 44, 6, pp. 953-80. Lin, M.W., Rogers, C.A. (1991) Analysis stress distribution in a shape memory alloy composite beam, 32nd SDM Conference, Baltimore MD, April 8,10.

Martin, W. (1994) Adaptive materials-2nd short course on smart structures and materials Glasgow. Thiebaut, F., Zeghmati, B., Charmoillaux, J.F. and Lexcellent, C. (1996) Smart material thermal characterization: NiTi shape memory alloy embedded in a resin epoxy matrix, Proceedings I. CIM 96, P.F. Gobin and J. Tatibouet Editors, pp. 535-40. Tobushi, H., Iwanaga, Tanaka, K., Hori, T. and Sawada, T. (1991) Deformation behavior of TiNi shape memory alloy subjected to variable stress and temperature cont. mech. therm. 3, pp. 79-93. Tobushi, H.,Tanaka, K., Sawada, T., Hattori, T. and Lexcellent, C. (1994) Representation of recovery stress associated with the R-phase transformation inTiNi shape memory alloy (property under constant residual strain), trans J.SME. 59, 557, pp. 171-177.

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Elasticity Issues

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OPTIMALITY OF DILUTE COMPOSITES UNDER SHEAR

S. SERKOV1, A. MOVCHAN 1 , A. CHERKAEV2 and Y. GRABOVSKY2 1 Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK 2 Departments of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

Abstract. The problem of optimal dilute composites subjected to uniform loading is analysed. This problem is equivalent to specification of the shape of

a single cavity in an infinite elastic plane subjected to a uniform load at infinity. A cavity of fixed area is said to be optimal if it provides the minimal energy change from the homogeneous plane to the plane with the cavity. We show that for the case of shear loading the contour of the optimal cavity is not smooth. The shape is specified in terms of conformal mapping coefficients and explicit analytical representations for components of the dipole tensor associated with the cavity are employed. The applications are to the problems of optimal control for dilute composites.

1. Introduction The problem of optimal shape of solids has been initiated in (Prager, 1968) where the sufficient conditions of optimality were established. The shape of optimal cavities and the optimal properties of composites have been studied for a uniform hydrostatic loading and a biaxial loading with the principal stresses of the same sign (Vigdergauz, 1994), (Grabovsky and Kohn, 1995). The optimal configuration of several cavities has been described by (Vigdergauz, 1988). Also we refer to the monograph (Sokolowski and Zolesio, 1992), which includes mathematical and numerical analysis of shape sensitivity associated with elliptic boundary value problems. 285 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 285–300. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

286

One would, perhaps, regard the case of an infinite plane containing a single cavity to be a classical one. However, when the stress tensor prescribed at infinity has eigenvalues of different signs, the shape of the cavity (of fixed area) minimizing the energy increment seems to be unknown; it is appropriate to refer to the numerical experiments performed by (Vigdergauz and Cherkaev, 1986), where some characteristic features of the optimal shape are discussed. Here it may be important to remark that we are looking for an optimal cavity with a simply connected boundary. The case of multiple cavities (”second rank” laminate structures) has been considered earlier (Gibiansky and Cherkaev, 1996), (Milton, 1986). It has been shown that these structures corresponds to ”optimal” lower energy. At the same time if the additional constraint, uniqueness of the cavity in periodic cell, is imposed, the ”optimal”s lower bound (Gibiansky and Cherkaev,1996) for energy is not reachable. However we can prescribe the lower bound for energy, which characterizes single cavities only. The shape of cavity having this energy will be found further. One of the practical applications of the single cavity problem is in the mathematical modelling of dilute composites. Optimization problems for composite elastic media have been intensively studied (see, for example, the papers (Kohn and Strang, 1986), (Gibiansky and Cherkaev, 1996), (Gibiansky and Cherkaev, 1996), (Milton, 1986), (Bendsoe, 1995), (Grabovsky and Kohn, 1995)). In the present work we specify the optimal shape of a cavity (of fixed area) which minimizes the absolute value of the energy increment for the case of the shear load (the eigenvalues of the stress tensor have different sign). The algorithm employed in this work can be effectively applied to the case of any uniform load. We show that for the case of pure shear the optimal cavity is a curved quadrilateral, and the angle near the corners is equal to the critical value 102.6° (also, see (Carothers, 1912)). The structure of the paper is as follows. We begin with the formulation of the optimization problem and discussion on the necessary conditions of optimality and applications to models of dilute composites. Then we present the minimization algorithm involving the analytical and numerical technique. It uses the concept of the dipole matrices corresponding to a remote field associated with finite cavities in an elastic plane (it enables one to produce explicit representation of the energy increment via the dipole coefficients). The complex variable technique is employed in this work, and the solution is written in terms of the Kolosov-Muskhelishvili potentials obtained in a series form. Direct minimization procedure was used to specify the coefficients of the expansion of the conformal mapping function and the optimal shape of the cavity (we also verify that this solution satisfies the

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necessary conditions of optimality). Finally, we verify the answer by solving the inverse elasticity problem and find the shape of the cavity that satisfies the optimality condition. 2. Problem formulation 2.1. ELASTICITY PROBLEM

First, we consider the boundary value problem in a domain with a single finite cavity G. The elastic material is characterized by the Lame constants and We impose free-traction boundary conditions on and the uniform shear stress field is specified at infinity. The displacement field satisfies the following boundary value problem

where and R is sufficiently large, are components of the unit outward normal. Following the variational approach (see (Sokolowski and Zolesio, 1992)) we define the energy space for the boundary value problem (1) and introduce the norm:

The elastic energy of the region bounded by

can be evaluated as

We remark that it can be represented as a difference between the potential energy and the work of external forces. The solution of the boundary value problem (1) minimizes the energy (3):

Define the increment of energy as a difference between two functionals associated with the full energy in the homogeneous disk R} and in the disk weakened by the cavity G:

288

Here are components of the stress and the strain tensors in the homogeneous disk, and are the stress and strain in the disk with the cavity 2.2. OPTIMIZATION PROBLEM

The main objective is to find the shape of the cavity which provides the minimal absolute value of the energy increment. The constraints on the fixed area and boundness of the domain are imposed. When the principal stresses associated with have the same sign it is known that the optimal shape is an ellipse. Here we concentrate our efforts on the case of shear, i.e. the principal stresses have different signs. Let us now discuss the set of admissible minimisers. An infinite plane with an arbitrary single cavity can be mapped to the exterior of the unit disc by the conformal mapping represented in the form

where R and are constant coefficients. The constraint of the fixed area can be written via the conformal mapping coefficients:

We determine the set of admissible cavities by their coefficients R and of conformal mapping to the unit disc. The optimization problem becomes:

find the set of coefficients R and to a cavity which minimizes (5).

(see (6)) restricted by (7) that lead

This formulation corresponds to our goal of finding an optimal simply connected cavity rather then an array of smaller cavities of the same area. 1

Note that the boundary value problem is solved in the bounded domain

\G.

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2.3. THE ENERGY CHANGE

The energy increment can be found explicitly in the form (Zorin et al, 1988), (Babich et al, 1989)

where is the 4-th order “polarisation” tensor (Pólya and Szegö, 1951), (Movchan and Movchan, 1995), (Movchan and Serkov, 1991). It has the same symmetry properties as the Hooke’s tensor Namely,

and the number of independent elements is six for plane strain formulations. The “polarisation” tensor characterises the remote displacement field associated with the presence of a cavity G. If we specify the constant-strain fields corresponding to the following displacements (biaxial tensile and shear loading at infinity) then the displacement fields in the region admit the following representation and at infinity the “polarisation” fields admit the following asymptotic expansions: 2

where is the Somigliana tensor, and tives of the component IJ with respect to

and denote the derivaor respectively.

The “polarisation” tensor characterises the morphology of the cavity and elastic constants of the material. It depends on coefficients and elastic constants only 2

Repeated indices are regarded to be the indices of summation

290

To obtain explicit approximate formulae for the tensor we consider the truncation of expansion (6) and keep the first N terms of series. It allows one to reduce the problem to the system of N linear algebraic equations. In particular, when N = 3 we obtain 3:

where

Thus, the mathematical formulation of the optimization problem reduces to the maximization problem for the function of N variables

3. Minimization technique

The optimization problem (10) reduces to the minimization of a function of several variables. The unknown variables are the coefficients of the conformal mapping subject to the restriction (it corresponds to the one-to-one mapping of the unit disk to the exterior of the cavity). Our purpose is to find the set of conformal mapping coefficients minimising the absolute value of the energy increment (8). Specifically, our problem reduces to multidimensional minimisation of a function of 2N variables ( the conformal mapping coefficients are complex). 3

Formulae for arbitrary N have been published in (Movchan and Serkov, 1997).

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To do this we use the downhill simplex method (Press et al, 1987). This method requires the values of the function only, and it does not require the derivatives. The downhill simplex method must start not just with a single point, but with 2N + 1 points, defining an initial simplex in 2N-dimensional space. We decide that the initial starting point corresponds to a unit circle and take the other points to be where are 2N unit basis vectors and are constants representing the problem characteristic length scale along the direction of modified simplex is less then small parameter The following convergence criterion is used with the simplex method

where is a small positive parameter and In this work we used the Numerical Recipes routine “amoeba” (Press et al, 1987) realising the downhill simplex method. 4. The cavity of the optimal shape 4.1. CONFORMAL MAPPING COEFFICIENTS

Now consider the cavity under the shear loading. The optimal bound for the energy increment is well known (see, for example (Gibiansky and Cherkaev, 1996)). It corresponds to the ”second rank” laminated composites:

292

For a circular cavity we find that the absolute value of the energy increment is in two times greater than the optimal one (11). Our aim is to find the geometry of the region, which is specified by a smaller energy change than the circular cavity and which is the best among simply-connected defects. Using the exact representation for the energy increment (8) and the minimisation procedure described in the previous section in the case of N = 3 we obtain the following optimal mapping function

Here the constant R is chosen in such a way that the cavity has a unit area. Increasing the number of terms in (6) and taking into account 7, 11, 15 and 19 terms correspondingly we calculate the non-zero conformal mapping coefficients for the optimal domain. The results are presented in Table 1 and corresponding plots in Figure 1.

The energy increment for such a domain can be specified as

where the coefficient is given in Table 2 as a function of the number N of conformal mapping coefficients. For a circular cavity under pure shear the coefficient is equal to 4.

in formula (13)

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We can see (Figure 1) that by increasing the number of coefficients the optimal domain shape approaches a shape which looks like a square. Note that the interesting case is the case of non-pure shear. If the principal stresses have the same sign, the optimal shape will be an ellipse. But in the case of opposite signs the optimal cavity will be close to a rectangular one with the sides ratio given in Figure 2. The ratio of the energy increment and the optimal energy (11) is presented in Figure 3. 4.2. THE DIRECT METHOD OF OPTIMIZATION

We describe the properties of the displacement field corresponding to the optimal cavity under shear loading. Using the standard Kolosov-Muskhelishvili technique (Muskhelishvili, 1953) one can find the complex potentials and

294

for the conformal mapping function with coefficients from Table 1 as a solution of the following boundary integral equations:

Omitting technical calculations we write the series representation for the complex potential

where the coefficients

solve the linear system:

where

Note, that non-zero conformal mapping coefficients have indices It corresponds to non-zero coefficients and and all remaining coefficients vanish. The complex potential admits the representation:

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Using (15) and (16) we calculate the tensile stress on the boundary for a different number of terms in the conformal mapping function (Table 1). The results are presented in Figure 4 for N = 7, 11, 15, 19. It is possible to see that the modulus of the tangential stress is constant along the contour except a small neighbourhood of corners, where it vanishes, and the diameter of this neighbourhood vanishes as we increase N. This observation suggests that for an optimal domain under pure shear the tangential component of the stress tensor is piecewise constant on the boundary:

The conformal mapping is sought in the form of modified ChristoffelSchwartz integral

where is different from It corresponds to a transformation of the unit disk to the symmetric domain with the angle near the corners. The opening angle near the vertex of the point can be calculated as

296

If we choose

then (18) can be rewritten in the form

which agrees with Table 1. It suggests that the quantity angle in the Carothers problem (Carothers, 1912).

is the critical

4.3. THE SOLUTION OF THE INVERSE PROBLEM The objective of this section is to find the shape of domain G satisfying the optimality conditions const on the boundary. Here we do not use the energy evaluation but begin with the Kolosov-Muskhelishvili boundary integral equation for the complex potential

where is the conformal mapping function, is the boundary of unit disk and is the shear stress at infinity. Now we can apply the condition of the piecewise constant tensile stresses on the boundary. The points in the map to the corners of the domain G in Conditions (17) can be rewritten as

297

where A is an unknown constant. The unit circle boundary , therefore

is the image of the

Using the Schwartz formula we obtain

Therefore, expression (22) is the explicit representation for the complex potential satisfying the condition (17). Taking the derivative of (19), and using the identity and the expression for the derivative of the Cauchy integral (Muskhelishvili, 1953) we reduce the integral equation (19) to

where dev

We also use the expression for the real part of the potential the symmetry of the conformal mapping

and the Cauchy formula for a function the holomorphic in the unit disk

After some lengthly but simple calculations the following integral equation has been derived for the unknown function

298 Now, we represent the conformal mapping function expansion:

by the Laurent

Integrands in (24) can be expanded in series of different powers of . The function can be represented as a Taylor series as well. By integrating the series term by term and collecting the coefficients near the same powers of we obtain the following linear system of equations for coefficients

In Table 3 we show the values of the conformal mapping coefficients for different N and note that these data are consistent with Table 1.

Analysing the behaviour of the inverse problem providing the piecewise constant tensile stresses on the boundary, we can make a conclusion. Proposition. The optimal domain G that provides the minimal absolute value of the energy increment in the state of pure shear satisfies the optimality condition on the boundary and can be described by the conformal mapping (25) with coefficients (26). The complex potential under such conditions has the explicit representation (22) and the conformal mapping (25) corresponds to the convex domain, which is close to a square. However, the corner angle is equal to the critical Carothers value (Carothers, 1912). It provides zero singularity at the corners for the case of a shear loading.

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Acknowledgements

The authors would like to thank Prof. G.W.Milton for valuable discussion. AVC gratefully acknowledges the support from National Science Foundation through the grant DMS-9625129. SKS acknowledges the financial support of Bath University. Both ABM and SKS are grateful to the University of Utah for financial support and provision of computer facilities. Finally our thanks are due to Prof. S.B. Vigdergauz for reading the manuscript and making useful comments. References Babich, V.M., Zorin, I.S., Ivanov, M.I., Movchan, A.B. and Nazarov, S.A. (1989) Integral characteristics in problems of elasticity, Steklov Mathematical Institute (LOMI). Preprint P-6-89, Leningrad (in Russian). Bendsoe, M.P. (1995) Optimization of Structural Topology, Shape and Material, SpringerVerlag, Berlin, New-York. Carothers, S.D. (1912) Plane strain in a wedge, Proceedings of the Royal Society, Edinburgh 23, 292.

Gibiansky, L.V. and Cherkaev, A.V. (1996) Design of composite plates of extremal rigidity, in A. Cherkaev and R. Kohn (eds.), Topics in the Mathematical Modelling of Composite Materials, Birkhausen, New-York, pp. 95-138. (Russian version in Report 914, Phys.-Tech. Inst.Acad.Sci.USSR, Leningrad, 1984)

Gibiansky, L.V. and Cherkaev, A.V. (1996) Microstructures of composites of extremal rigidity and exact bounds on the associated energy density, in A. Cherkaev and R. Kohn (eds.), Topics in the Mathematical Modelling of Composite Materials, Birkhausen, New-York, pp. 273-317. (Russian version in Report 1115, Phys.Tech.Inst.Acad.Sci.USSR, Leningrad, 1987). Grabovsky, Y. and Kohn, R.V. (1995) Microstructures minimising the energy of a two phase elastic composite in two space dimensions. II: The Vigdergauz microstructure.

Journal of the Mechanics and Physics of Solids 43(6), 949-972. Kohn, R.V. and Strang, G. (1986) Optimal design and relaxation of variational problems. Communications on Pure and Applied Mathematics 39, 113-182, 353-377. Milton, G.W. (1986) Modelling the properties of composites by laminates, in J.L. Ericksen, D. Kinderlehrer, R. Kohn and J.L. Lions (eds.), Homogenization and Effective Moduli of Materials and Media, Springer-Verlag, New-York, pp. 150-174. Movchan, A.B. and Movchan, N.V. (1995) Mathematical Modelling of Solids with Nonregular Boundaries, CRC Press. Movchan, A.M. and Serkov, S.K. (1997) The Pólya - Szegö matrices in asymptotic models of dilute composites, European Journal of Applied Mathematics (in press). Movchan, A.B. and Serkov, S.K. (1991) Elastic polarisation matrices for polygonal domains, Mechanics of Solids 26(3), 59-64. Muskhelishvili, N.I. (1953) Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen. Prager, W. (1968) Optimality criteria in structural design, Proceedings National Academy of Natural Science, USA 61(3), 794-796. Pólya, G. and Szegö, G. (1951) Isoperimetric Inequalities in Mathematical Physics, Princeton University Press. Press, W.H., Flannery, B.P., Tenkolsky, S.A., Vetterling, W.J. (1987) Numerical Recipes. The Art of Scientific Computing, Cambridge University Press. Sokolowski, J. and Zolesio, J.P. (1992) Introduction to Shape Optimization. Shape Sensi-

300 tivity Analysis, Springer-Verlag, Berlin, New York.

Vigdergauz, S.B. (1988) Stressed state of an elastic plane with constant-stress holes,

Mechanics of Solids , 23(3), 96-99.

Vigdergauz, S. (1994) Two-dimensional grained composites of extreme rigidity, Journal

of Applied Mechanics. Transactions of the ASME 61(6), 390-394. Vigdergauz, S.B. and Cherkaev, A.V. (1986) A hole in a plate, optimal for its biaxial

extension - compression, Applied Mathematics and Mechanics(PMM) 50(3), 401-404. Zorin, I.S., Movchan, A.B. and Nazarov S.A. (1988) Use of the elastic polarisation tensor in problems of crack mechanics, Mechanic of Solids 23(6), 120-126.

A HASHIN-SHTRIKMAN APPROACH TO THE ELASTIC ENERGY MINIMIZATION OF RANDOM MARTENSITIC POLYCRYSTALS

V.P. SMYSHLYAEV

School of Mathematical Sciences University of Bath Bath BA2 7AY, U.K. AND J.R. WILLIS

Department of Applied Mathematics and Theoretical Physics Cambridge CB3 9EW, U.K.

1. Introduction This paper proposes a Hashin-Shtrikman type variational approach for the elastic energy minimization of martensitic polycrystals and derivation of associated rigorous bounds for the overall elastic energy. Detailed analysis will be given in a forthcoming paper (Smyshlyaev & Willis, 1997a) identified

henceforth as (SW97a). The present study has been partly motivated by recent work of Bruno et al. (1996), who have developed a numerical scheme for calculation of the nonlinear overall energy of certain martensitic polycrystals, and derived upper bounds on the overall energy which considerably improved the Taylor bounds. The “uncorrelated grains” bounds of Bruno et al. (1996) are derived from the assumption that each grain of the polycrystal is of circular shape and is allowed either not to transform at all or to transform fully into a crystal's recoverable strain. The “uncorrelated grains” assumption implies that transformation modes of different grains are statistically uncorrelated and their spatial distribution is therefore statistically isotropic. In the present work we propose an approach which in a sense develops and generalizes that of Bruno et al (1996) and extends it to more general grain shapes with certain overall statistics of distribution of grains

301 Y.A. Bahei-El-Din and G.J. Dvorak (eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, 301–317. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

302

shapes and their orientations. To this end a variational principle of HashinShtrikman type is developed for the polycrystal’s energy. This involves a non-local functional with a Green’s function related kernel operating on trial “transformation fields” which are appropriately constrained to accommodate both the single crystal's constitutive law and the polycrystal's texture. For a statistically uniform polycrystal the variational principle is reformulated to require minimization with respect to all possible two point correlation functions of “sub-microstructure”, compatible with the texture. This new variational principle is applied to derive upper bounds for a statistically uniform polycrystal by using a particular set of trial fields, with the property that the scale of the trial sub-microstructure is much finer than the scale of the polycrystal's texture. Subsequent optimization with respect to this sub-microstructure for each particular orientation reveals a connection with relaxation of a single crystal “with fixed volume fractions” and associated H-measures as discussed by Kohn (1991). The new bounds also demonstrate improvement in the examples con-

sidered by Bruno et al. (1996) as a result of an improved optimization procedure, and reveal the effect of incompatibility of transformation strain within a single crystal for the polycrystal’s overall properties.

2. Energy Minimization Problem 2.1. SINGLE CRYSTAL The constitutive behaviour of a reference elastic crystal which can undergo

an austenite-martensite phase transformation is specified in terms of a finite set of possible modes of transformation:

Here is the strain that would arise in the stress free reference crystal if it were fully transformed into the "martensite" phase . The zero strain is also included in for convenience to interpret the untransformed “austenite“ state as the “zero” phase. If the full crystal has undergone a transformation with index , its further elastic deformation is in accordance with the Hooke’s law, with elastic tensor which may generally depend on

Here are components of the stress tensor e is related to displacement through

and the infinitesimal strain

303

(comma in subscript denotes differentiation and summation is applied to repeated suffixes). The elastic energy density in such a crystal is

In this paper the elastic tensors are taken to be isotropic and independent of (being denoted C henceforth). In the context of energy minimization approaches (see e.g. Kohn, 1991) a conventional formulation is based on introduction of the microscopic elastic energy with a “multiwell” structure

The relaxed (or “mesoscopic”) energy is defined as an infimum of the energy functional over a representative volume V with prescribed boundary conditions at the boundary of V:

(see e.g. Kohn, 1991, and further references therein). Here the set of displacement fields on V such that on

denotes :

One can introduce the set S of “recoverable” strains as the set of all e such that . The set S describes all possible overall strains which can, loosely speaking, be achieved by a stressfree configuration (see e.g. Bhattacharya &. Kohn, 1997, for more precise discussion). 2.2. POLYCRYSTAL

A polycrystal occupying a volume is regarded as a collection of perfectly bonded single crystals with large number N of distinct orientations. The orientations of the crystals are characterized by rotation tensors N. Therefore, the rotation-valued function R(x) in describing the polycrystal’s texture can be represented in the form

304

Here

are the characteristic functions of the orientations

orientation and The volume fraction

if the crystal’s grain containing the point otherwise. of orientation is therefore

has

Rotation of the reference crystal correspondingly rotates the basic set of the transformation modes:

i.e. any

from

transforms into

.The

set of the recoverable strains S transforms in similar fashion into

If

we will also denote

, where

with similar notation applied to

denoting the set of recoverable strains in the orientation The microscopic energy and the relaxed energy formed accordingly under the rotation

are also trans-

The polycrystal is subjected to boundary conditions

The energy minimization problem for the polycrystal is formally analogous to that for the single crystal: for each find the normalized minimum energy function defined as the infimum over all possible displacement fields satisfying the boundary conditions (9):

305

3. Hashin-Shtrikman Type Variational Principle In this section, which is central for our construction, the polycrystal's energy is expressed in terms of minimization of a nonlocal functional with respect to trial “transformation fields” or, by analogy with the classical Hashin-Shtrikman method developed for elastic composites, “polarization

fields”. The nonlocality is due to involvement of the elastic Green's function, and accounts for the effect of spatial correlation of the orientation distribution statistics on In effect, our construction develops and generalizes that of Bruno et al. (1996) with Green’s function kernels serving instead of the Eshelby solutions, which has the advantage of applicability to arbitrary grain shapes with certain overall (statistical) symmetry, and clarifies the connection with the relaxation problem for single crystals. We sketch a derivation of the Hashin-Shtrikman variational principle. A technically more detailed derivation is found in our recent paper (SW97a). By substituting the multiwell representation (8), (4) into the definition (10) of and specializing to (under the above assumption of the elastic moduli being isotropic and independent of the transformation mode), one arrives at

Here the set

consists of all strain fields

such that for any

belongs to for some from point to point. Interchanging the orders of infima in (11) results in

which may vary

Assuming now fixed, evaluate the internal infimum. This is done following the pattern of the Hashin-Shtrikman (1962) construction (see e.g. Willis, 1981) where plays the part of “strain polarization”. It is noticed (see e.g. SW97a) that (12) is minimized by displacement field

that satisfies

in . The above equation can be viewed as the equation of equilibrium for a homogeneous elastic medium with elastic modulus C, occupying volume and subjected to a body force

306

Standard considerations result in representation of the solution the Green's function in as follows:

A representation for

via

follows immediately by differentiation

with denoting symmetrization with respect to indices and Upon substituting (15) into (12), via routine manipulation (SW97a) one arrives at the representation central for our purposes:

Here

denotes an integral operator whose kernel is related to the

Green’s function via

The new variational statement (16) for the polycrystal's energy may be regarded as a variant of the Hashin-Shtrikman (1962) variational principle adapted to the present context. It states that is the infimum value of the functional on the right hand side of (16) with respect to all choices of the microscopic transformation strains compatible with the polycrystal’s texture. Importantly, this functional contains a nonlocal (integral) operator whose the kernel is related to the Green’s function. 4. Statistical Variational Principle

We wish to develop the basic variational principle (16) further under the assumption that the volume contains a very large number of grains whose spatial distribution and orientations are prescribed by certain statistical parameters. The trial “transformation” fields can be represented as

where In turn,

are the orientation characteristic functions introduced above.

307

where are the characteristic functions corresponding to r-th transformation mode in orientation , i.e. if belongs to a grain with orientation and the trial field takes value in , and otherwise. Obviously,

Substitution of (18), (19) into (16) yields

where the following notation has been introduced:

The trial fields therefore are all possible divisions of the domains with prescribed orientations into subdomains of the allowable transformation modes. The infimum in (21) is taken with respect to all characteristic functions , which enter the right hand side of (21) through and via (22)-(23), subject to constraint (20). The polycrystal is assumed to be statistically uniform and under the “ergodicity” hypothesis the volume fractions can be also regarded as “local” expectations to find a point x belonging to phase k. Also, twopoint correlation functions are introduced: is the probability to find the point x in phase and the point x´ in phase l, or, in other words, the probability that . As a consequence of statistical uniformity is actually a function of and if the polycrystal is statistically isotropic, will depend only on The hypothesis of no long range order is also usually imposed, which requires

as

308

In the statistical context it is reasonable to take the trial fields

to be also statistically uniform, to regard as corresponding local expectations, and to introduce the “submicrostructural” two-point correlation function

as the probability that

Under

the hypothesis of no long range order now implying the volume averaging can be replaced (see e.g. Willis, 1981) by statistical ensemble averaging at a point, and the Green's function entering through (17) may be replaced by the translation-invariant infinite body Green’s function with integration over the volume replaced by integration over the infinite Euclidean space As a result (23) transforms to

In t u r n , (21) can be transformed into

where the infimum is now taken with respect to all possible volume subfractions and two-point correlation functions statistically compatible with the prescribed statistics of the polycrystal (characterized partly by and . The statement (25), which may be regarded as a statistical version of (16), indicates that the energy minimization in polycrystals may be regarded as a constrained optimization where the twinning-type submicrostructure develops on the background of a given texture. 5. Upper bounds

The variational representation (25) requires characterization of a set of possible two-point correlation functions compatible with the orientation distribution statistics. Full characterization of such set does not seem possible. Instead, we seek to restrict the minimization to a a tractable subset of possible statistics and to minimize within this subset to derive upper bounds for The basic restriction we are making is considering only those submicrostructures which are much finer than the underlying microstructure

309

. The sought sub-microstructures are of the form

where for any is a set of “microscopically” statistically uniform characteristic functions in the whole space for any Due to the statistical uniformity one can introduce a set of the two-point correlation functions corresponding to , as the probability that . Obviously, , and are the volume fractions corresponding to the phases of the microstructure . The“no long range order” hypothesis requires

as

In the present language the above assumption that the sub-microstructures are “much finer” than the original orientation distribution microstructure

means that the“correlation length” associated with

is

much smaller than that corresponding to Relation (26) interprets the trial microstructures as an “intersection” of the given microstructure with N other sub-microstructures

We assume also that for any

tically independent from and

the sub-microstructure

and, moreover, that for different

is statis-

and

are also mutually independent.

The above independence conditions imply:

and (no sum on and Let also the polycrystal’s texture have an ellipsoidal symmetry (see e.g. Willis, 1977), i.e. the correlation functions actually depend only on where A is a linear positive definite operator in the Euclidean space. The above assumption means that are constant on ellipsoidal surfaces . In particular, if (with I denoting the identity operator), then are functions of only, which characterizes geometrically isotropic statistics.

310 The key tensors entering the variational principle (25) are transformed after some manipulation (SW97a) into

Here the tensor Q is related to the Eshelby tensor P associated with an via

ellipsoidal inclusion

(see Willis, 1977, 1981),

Substitution of (29) into (25) yields the following upper bound for

Here

are the total mean of the transformation field and its conditional mean in the orientation respectively. The upper bound (31) holds for any and for any realizable by a microstructure with volume fractions 6. Optimization and Related Issues

The upper bound (31) is to be optimized with respect to all possible trial microstatistics As a result, the optimal upper bound for (31) assumes the form:

311

where

and

are determined by (32). Here

are defined by (30), and

are two-point correlation functions of all possible microstruc-

tures

with fixed volume fractions

6.1. RELATION TO THE SINGLE CRYSTAL RELAXATION PROBLEM

An important feature of the upper bound (33) is the reduction to evaluation of the internal infimum (34) for each orientation k. We make an important observation that the evaluation of (34) is directly associated with the socalled “relaxation with fixed volume fractions” for single crystals (see e.g. Kohn, 1991). A way to see this is to specialize our construction to a degenerate polycrystal with which corresponds to the single crystal, as only one orientation is allowed (see also Smyshlyaev & Willis, 1997a,b). This observation establishes the connection between evaluation of the Hashin-Shtrikman type upper bound (33) for the polycrystal and the relaxation for the single crystal via H-measures (Kohn, 1991).

6.2. RELATION TO H-MEASURES It has been shown (SW97a) that the key infimum in (34) can be represented in the form

where

and is the so-called H-measure on the unit sphere (Gerard, 1991; Tartar, 1990) associated with the weakly convergent microstructures In this language, evaluation of translates directly to the necessity for each crystal's orientation to optimize (34) with respect to H-measures, exactly as discussed by Kohn (1991) in the context of the relaxation problem for a single crystal:

312 The minimization in (36) is with respect to all H-measures on the unit sphere realized by a weakly convergent sequence of microstructures. Realizable H-measures satisfy certain attractive properties, convenient for purposes of optimization (see e.g. Kohn, 1991, Section 8):

for any continuous “test functions”

As Kohn (1991) has discussed, properties (37)–(40) are necessary to characterize possible H-measures but may be insufficient. However, as Kohn (1991) has also shown, for (the double-well problem) restrictions (37)-(40) are sufficient as every related H-measure can be approximated by a sequential lamination. The minimization can then be executed explicitly which in our language results in

In the case of more than two wells there is no guarantee that (37)–(40) is sufficient for characterizing an H-measure. For the case of three wells (K = 2) we have performed (Smyshlyaev & Willis, 1997b) minimization directly with respect to H-measures subject to (37)–(40) and have given a geometric description of the extremal measures, i.e. those measures which deliver the minimum to (35), for arbitrary elastic tensor C. We have also described a realizable subclass of possible extremal measures. Minimization with respect to this subclass provides, in general, upper bounds for the relaxed energy which, in principle, can be used to derive upper bounds for

via (33), (35). The problem of evaluation of I trivializes of course in the case of full kinematic compatibility of the basic set i.e. if any convex combination definition,

forms a recoverable strain (Section 2.1). In this case, by if and from (33)

313

6.3. RELATION TO WORK OF BRUNO ET AL. (1996) Return to the polycrystal to establish connection of the bound (33) with the bounds obtained by Bruno et al. (1996). Note first that the statistical approach developed here demonstrates that the restriction of Bruno et al. to polycrystals with grains of circular shape appears to be redundant, as only “circular overall symmetry”, i.e. statistical isotropy of the geometry of orientations distribution (even not necessarily isotropy of the orientations themselves!), is required. Moreover, the present approach may also be adapted to more general statistical distributions not necessarily geometrically isotropic, e.g. to those with ellipsoidal statistics (Willis, 1977). To establish further connections with the work of Bruno et al. (1996) notice first, that the central upper bound (33) may itself be bounded from above by restricting the search only to those for which

is a recoverable strain,

ments, the term containing

(see (7)). Then, by the above argu-

vanishes and the “cruder” bound is

The above “cruder” bound (42) may be interpreted as a result of developing the Hashin-Shtrikman procedure to a polycrystal if any single orientation k is only allowed to transform fully into a transformation strain where i.e. by disregarding in-grain kinematically incompatible transformation fields. The above “cruder” bound (42) may be interpreted as a result of applying the Hashin-Shtrikman procedure to a polycrystal if any single orientation k is only allowed to transform fully into a transformation strain where By analogy, our bound (33) may be viewed as that obtained by extending the above notion of a single crystal transformation strain for not necessarily compatible combinations of the “basic” strains (O.P. Bruno, private communication). It will be shown via an example that the bound (33) does improve upon (42) which demonstrates the importance of the single crystal relaxation in the case of incompatibility of the crystal’s transformation strains.

314

Bruno et al. allowed for each isotropically distributed grain orientation two possibilities: each grain either fully transforms into a uniform transformation strain or it does not transform at all. The spatial distribution of the grains of these two types was assumed “uncorrelated”, i.e. still isotropic. Therefore, the “untransformed” part of each orientation may itself be regarded in our language as a separate “fictitious” orientation, with the right hand sides modified accordingly. Bruno et al. (1996) did not consider in their paper incompatible trial fields although (O.P. Bruno, private communication) they have been aware of the possibility of including them into their scheme. Their explicitly derived bounds, however, (see e.g. Bruno et al., 1996, formulae (82)–(84)) (which do cover incompatible strains for certain relations between the involved parameters) correspond, in fact, in

our language to the “cruder” bound (42). Further, what Bruno et al. do to implement their scheme, in our language means the following. First, fix a “generalized” transformation strain for a reference crystal, then find its

representation

in every orientation

and optimize with respect to the two possibilities: (orientations are supposed to be distributed continuously). Finally, optimize with respect to (see formula (54) of Bruno et al., 1996). Obviously, this algorithm may lead to loss of accuracy of the bounds as better bounds could be obtained by allowing also to vary with orientation. So, even for a fully compatible single crystal the bound (42) appears to improve that established by Bruno et al. (1996). It is perhaps worth remembering also that the bound (??) as discussed above, was actually derived from a more general variational principle (16) and its statistical version (25), by using the restrictions of “separation of scales”. This in principle leaves a possibility of further improvement by relaxing these restrictions in some way. 7. Examples

The above strategy has been implemented for two-dimensional purely deviatoric statistically isotropic polycrystal with an isotropic elastic modulus characterized by Lamé parameters as considered by Bruno et al (1996, Fig. 9b). Three compatible wells are assumed: The deviatoric applied strain is characterized by Therefore, even for a fully compatible single crystal the bound (42) appears to improve that established by Bruno et al. (1996). More importantly, the bound (42) becomes intrinsically cruder than our new bound (33) for a non-trivial single crystal relaxation where the term plays a non-trivial role, as is demonstrated on Fig. 2 for a

315 two-well incompatible single crystal. In this example the isotropic elastic moduli have been taken as in Fig. 1, but the martensitic transformation strain was chosen to be incompatible with as both eigenvalues

of

are positive. As a result, the term

does not vanish and

is calculated explicitly (Kohn, 1991). Figure 2 displays our bound (see SW97a for technical details of optimization) which is a non-convex two-well function. For comparison, we have also calculated from (6.42) the bound which would have been predicted by the algorithm disregarding the in-grain incompatibility. As expected, it is considerably cruder for intermediate loading values, where the in-grain relaxation effect plays significant role. References Eshelby, J.D. (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems Proc. Roy. Soc. London A 241, 376-396.

Gerard, P. (1991) Microlocal defect measures. Comm. Partial Diff. Equat. 16, 1761–1794. Hashin, Z. & Shtrikman, S. (1962) On some variational principles in anisotropic and inhomogeneous elasticity, J. Mech. Phys. Solids 10, 335-342. Kohn, R.V. (1991) The relaxation of a double-well energy. Continuum Mech. Thermodyn.

3, 193-236.

Smyshlyaev, V.P. & Willis, J.R. (1997a) A “nonlocal” variational approach to the elastic

energy minimization of martensitic polycrystals, to appear in Proc. Roy. Soc. London

A. Smyshlyaev, V.P. & Willis, J.R. (1997b) On relaxation of a three-well energy, submitted to Proc. Roy. Soc. London A.

Tartar, L. (1990) H-measures, a new approach for studying homogenization, oscillation and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh 115A, 193–230. Willis, J.R. (1977) Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25, 185–202. Willis, J.R. (1981) Variational and related methods for the overall properties of composites. In Advances in Applied Mechanics (ed. C.-S. Yih), pp. 1–78. New York: Academic

Press.

316

317

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