Advances in Applied Mechanics, Volume 34

Advances in Applied Mechanics Volume 34 Editorial Board Y. C. FUNG AMES DEPARTMENT OF CALIFORNIA, SANDIEGO UNIVERSITY...

Author: Erik van der Giessen | Theodore Y. Wu

49 downloads 789 Views 6MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Advances in Applied Mechanics Volume 34

Editorial Board Y. C. FUNG AMES DEPARTMENT OF CALIFORNIA, SANDIEGO UNIVERSITY LA JOLLA,CALIFORNIA PAULGERMAIN ACADEMIE DES SCIENCES PARIS,FRANCE C.3. YIH(Editor, 1971-1982) JOHN W.

HUTCHINSON (Editor 1983-1997)

Contributors to Volume 34 ELENAV. BOFDANOVA-RYZHOVA PEDROPONTECASTAGEDA PAULR. DAWSON ESTEBAN B. MARIN EDUARD RIKS OI.EGS. RYZHOV PIERRESUQUET WEI H. YANG

ADVANCES IN

APPLIED MECHANICS Edited by Erik van der Giessen

Theodore Y. Wu

DELFT UNIVERSITY OF TECHNOLOGY DELFT, THE NETHERLANDS

DIVISION OF ENGINEERING AND APPLIED SCIENCE CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA

VOLUME 34

ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto

This book is printed on acid-free paper. @ Copyright 0 1998 by ACADEMIC PRESS All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

All brand names and product names mentioned in this book are trademarks or registered trademarks of their respective companies. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use, or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 019231, for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-1998 chapters are as shown on the chapter title pages; if no fee code appears on the chapter title page, the copy fee is the same as for current chapters. 0065-2165/98 $25.00

ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 1300 Boylston Street, Chestnut Hill, MA 02167, USA http://www.apnet.com

United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX http://www.hbuk.co.uk/ap/

International Standard Serial Number: 0065-2165 International Standard Book Number: 0-12-002034-3 Printed in the United States of America 97 98 99 00 01 IP 9 8 7 6 5 4 3 2 1

Contents vii

CONTRIBUTORS

ix

PREFACE

Buckling Analysis of Elastic Structures: A Computational Approach Eduard Riks I. 11. 111. 1V. V.

2

Introduction Basics, the Geometrical Point of View Basics, the Stability Point of View Computations Examples and Conclusion References

6 22 40 61 72

Computational Mechanics for Metal Deformation Processes Using Polycrystal Plasticity Paul R. Dawson and Esteban B. Marin I. 11. 111. IV. V. VI . VII. VIII. IX. X.

Introduction Orientations and Orientation Distributions Evolution of Texture and Strength Field Equations for Deformation Computing the Deformation by Using the Finite-Element Method Application to Forming Processes Studies of Microstructure Summary Notation Appendix: Matrix Representations Acknowledgments References

78 81 88 99 113 121 133 152 157 161 162 163

Nonlinear Composites Pedro Ponte Castaiieda and Pierre Suquet I. Introduction 11. Effective Behavior and Potentials 111. Variational Methods Based on a Homogeneous Reference Medium IV. Variational Methods Based on a Linear Comparison Composite V

172 175 187 192

Contents

vi V. VI. VII. VIII. IX.

A Second-Order Theory A Selection of Results for Linear Composites Applications to Nonlinear Composites and Discussion Concluding Remarks Appendices Acknowledgments References

216 228 231 280 28 1 295 295

The Mathematical Foundation of Plasticity Theory Wei H. Yang I. 11. 111. IV. V. VI.

Abstract Introduction Minkowski Norms and Holder Inequality Generalized Holder Inequality Constructing the Dual Norm Application to Plasticity A Duality Theorem for Plane Stress Problems References

303 304 307 308 309 311 313 315

Forced Generation of Solitary-Like Waves Related to Unstable Boundary Layers Oleg S. Ryzhou and Elena I.: Bogdanoua-Ryzhoua 1. Introduction: Historical Perspective 11. The Triple Deck 111. The BDA System IV. The KdV System V. Solitons and the Onset of Random Disturbances Acknowledgments References

AUTHOR INDEX SUBJECT INDEX

318 326 335 316 403 413 413

419 421

List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

ELENA V. BOGDANOVA-RYZHOVA (3171, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180

PEDROPONTECASTA~~EDA (171), Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6316 PAULR. DAWSON (77), Sibley School of Mechanical and Aerospace Engineering, College of Engineering, Cornell University, Ithaca, New York 14853-3466

ESTEBAN B. MARIN(77), Beam Technologies, Inc., Ithaca, New York 14850 EDUARD RIKS(l),Delft University of Technology, Faculty of Aerospace Engineering, 2629 HS Delft, The Netherlands OLEG S. RYZHOV(317), Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180 PIERRESUQUET(171), Laboratoire de MCcanique et d’Acoustique/ C.N.R.S., 13402 Marseille Cedex 20, France WEI H. YANG(303), Mechanical Engineering Auto Laboratory, The University of Michigan, Ann Arbor, Michigan 48109-2121

vii

This Page Intentionally Left Blank

Advances in Applied Mechanics has a history of publishing comprehensive, state-of-the-art articles in numerous subfields of applied mechanics. But in no way does this imply that this particular area has been fully excavated. The articles in the present volume give convincing evidence that the developments often continue, requiring an update of previous advances. Eduard Riks' article gives an up-to-date overview of the advancements made in the area of (post)buckling analysis since the often-quoted article in this series by B. Budiansky in 1974 (Vol. 14, pp. 2-65). In addition to providing a thorough outline of modern, computational techniques for buckling and postbuckling analysis of structures, this article also discusses recent methods of carrying out transient analyses after loss of stability. The latter offers new interesting insights into the notorious phenomenon of mode jumping. The article by Paul R. Dawson and Esteban B. Marin is related to R. J. Asaro's contribution to this series (Vol. 23, 1983, pp. 1-11.9,which marked the beginning of a rapid expansion of numerical applications of crystal and polycrystal plasticity. The article in this volume gives an exposition of some important refinements in computational methods that make polycrystal plasticity a viable tool for actually solving engineering forming processes. Particular emphasis is placed on recent innovations in the description of crystal orientations. In addition, this paper presents numerous examples of metals with a hexagonal close packed (HCP) crystal structure. Owing to seminal work in the 1960s, the linear-elastic properties of composite materials can now be estimated analytically from the properties of their constituents, along with the microstructure, with remarkable accuracy. Estimates of composite properties in the case of nonlinear material behavior, such as creep or plasticity, of one or more of the components are much more difficult to obtain. Analytical approaches to this problem that incorporate microstructural information have only been attempted during the last decade or so. The article by Pedro Ponte Castefiada and Pierre Suquet gives an overview of the latest developments

X

Preface

in this field based on variational methods. This scholarly work summarizes the key theoretical tools and presents applications to numerous model materials, with an emphasis on the effect of microstructure. The Chapter by Oleg S. Ryzhov and Elena V. Bogdanova-Ryzhova is a pioneering study of the fully developed nonlinear instability of viscous boundary layer during the final stage of transition into turbulent flow. This study is based on the important discoveries of several remarkable phenomena, first found by experiment at the Novosibirsk branch of the Russian Academy and now by theory, that the underlying mechanism actually involves generation of solitary waves under resonant forcing from flowboundary roughness and vibration. It appears that these results will be of general interest to researchers in this important field. In the article by Wei H. Yang, the author presents a task of integrating the existing and new bases of the mathematical theory of plasticity. In addition to the previously established conditions of convexity and normality as two pillars of support, an additional pillar, called the duality, is introduced here in terms of an equality inclusive condition which is claimed to bring the foundation to completion for the constitutive modeling of the mathematical theory of plasticity. This series had the fortune of being cultivated during the period 1971-1982 under the editorship of Professor Chia-Shun Yih, who made outstanding contributions in realizing advances of this series with distinction and subsequently continued serving as a wise counsel until his passing on 25 April 1997. It is with our sincere appreciation of his dynamic leadership and guidance that we pay our warm tribute to his memory. Theodore Y. Wu and Erik van der Giessen

Advances in Applied Mechanics Volume 34


ADVANCES IN APPLIED MECHANICS. VOLUME 34

Buckling Analysis of Elastic Structures: A Computational Approach EDUARD RIKS Faculty of Aerospace Engineering Delft University of Technology Delft. The Netherlands

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

I1. Basics. the Geometrical Point of View . . . . . . . . . . . . . . . . . . . . . . . A . Notation. Assumptions. Governing Equations . . . . . . . . . . . . . . . . .

6 6 9 B . Adaptive Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Special Points: Bifurcation and Limit Points . . . . . . . . . . . . . . . . . . 12 D . Implicit Representation of the Bifurcation Branch . . . . . . . . . . . . . . . 17

22 I11. Basics. the Stability Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A . Stability and Loss of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . B . Critical Equilibrium States as Special Cases . . . . . . . . . . . . . . . . . . 24 25 C . The Notion of Attraction and Repulsion . . . . . . . . . . . . . . . . . . . . 29 D . Stability at the Critical States . . . . . . . . . . . . . . . . . . . . . . . . . . 36 E . Reactive Forces at the Critical States . . . . . . . . . . . . . . . . . . . . . . 38 F . Incipient Motion at Unstable Points . . . . . . . . . . . . . . . . . . . . . . . IV . Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 40 A . Local and Global Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Perturbation versus Path-Following Methods . . . . . . . . . . . . . . . . . . 43 49 C . Linearized Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . D . Buckling and Postbuckling Analysis . . . . . . . . . . . . . . . . . . . . . . 52 54 E . TransientBehavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 F . Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Transient Analysis After Loss of Stability . . . . . . . . . . . . . . . . . . . 58 V . Examples and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A . ShellModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 62 B . Mode Jumping of a Plate Strip . . . . . . . . . . . . . . . . . . . . . . . . . . C . The Esslinger Cylinder Buckling Experiment . . . . . . . . . . . . . . . . . 67 D . Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

I ISBN 0-12-0(12034-3

ADVANCES I N APPLIED MECHANICS. VOL. 34 Copyright 0 1998 by Academic Press. All rights of reproduction in any form reserved . 0065-2165/98 $25 00

2

Eduard Riks

I. Introduction Twenty-two years ago, Budiansky presented in this periodical a very comprehensive overview of the theory of elastic stability as it had been established up to that date (Budiansky, 1974). This exposition of the state of the art in stability analysis was, to a large extent, based on the general theory of buckling and postbuckling behavior founded by Koiter some 30 years earlier (Koiter, 1945), a theory that had suffered from a very slow start in becoming known and accepted in the engineering community after its appearance. In 1945, this new theory represented an important step forward in the understanding of the buckling behavior of structures as it was observed in engineering practice and experiments. It was this theory that was able to explain why the determination of the bifurcation point in the initial load deformation response could not provide enough information to predict the stability behavior of a structure with sufficient accuracy. As it was demonstrated by Koiter, the bifurcation point could not, by itself, be used to predict the failure load of the structure. To come to a better evaluation of this load, it was also necessary to establish the intrinsic properties of the bifurcation point itself. If the bifurcation point were stable, as in the case of a simply supported flat plate, the system could be loaded beyond the critical load that is associated with this point. On the other hand, if the bifurcation point turned out to be unstable, as in the case of a thin-walled cylindrical shell in compression, the structure could be expected to fail long before the critical load was reached, although it was difficult to predict with precision the load at which this would occur. This so-called sensitivity of the failure load for initial imperfections in the geometry, boundary conditions, etc., existed if the bifurcation point itself was unstable. In contrast, imperfection sensitivity did not play a detrimental role if the bifurcation point turned out to be stable. Apart from this important observation, the theory also provided the keys to the determination of the equilibrium states that bifurcate from the initial state and a means of assessing the strength of the imperfection sensivity in the case where this sensitivity could be established. The available methods of solution in 1974 were primarily analytical in nature. Computers were already in use, but their influence on the development of the solution methods was at first more in the area of the solution of complicated analytical formulations rather than systematic discretization of the governing equations from the start. At that time, there were three basic difficulties connected with the application of the theory. In the first place, the theory was an asymptotic theory, i.e., it relied on series expansions of the governing equations and therefore the solutions had a restricted range of validity. Second, if the primary solution path

Buckling Analysis of Elastic Structures: A Computational Approach

3

turned out to be nonlinear, the problem could turn out to be inaccessible for analysis, and this situation presented itself automatically if the structure was governed by a limit point rather than a bifurcation point. There was also a third difficulty. This had to do with the structural complexity of the structures that could be analyzed. The solution of the governing equations was only possible if the geometry and material build-up of the structure under investigation remained relatively simple. If this was not the case, the obstacles for analysis could soon become insurmountable. Thus it is not surprising that the practical applications of the theory remained restricted to structures with a simple geometry such as plates, cylindrical shells, and curved stiffened panels, and this in conjunction with an elementary type of loading: uniform compression, for example. Since then, many years have passed and the situation has gradually changed. This change was brought about by the advent and evolution of the digital computer, which made it possible to develop computational tools with a range and power that were unheard of before this evolution started. The emergence of the finite-element method is undoubtedly one of the most important advances in numerical analysis of this time, and since 1974 it has also had an impact on the modeling of the stability behavior of structures. Two schools of thought on modeling arose. The first is based on the discretization of Koiter’s asymptotic theory, at times amended with extras that the increased freedom of this numerical approach allows. Early accounts of this treatment can be found in Haftka et al. (1971), and more recent contributions are given in Damil (1992), Arbocz and Hol(1990), Casciaro et ul. (1992), Azrar et al. (1993), and Lanzo and Garcea (1996), which also contain further references. The other school is a more radical departure from the perturbation theory and is based on the continuation principle (see, for example, Riks, 1973, 1984a; Rheinboldt, 1977; Seydel, 1989; Crisfield, 1991; Kouhia, 1992), which in turn uses the principles of the numerical solution of nonlinear equations (Ortega and Rheinboldt, 1970). The continuation approach as a general and practical tool for the solution of elastic stability problems was probably first considered in 1970 (Riks, 1970, 1972), but the finite element modeling capabilities of that time were not yet fully developed (in the nonlinear range), so that the first applications appeared years after these capabilities became available. Since then, progress in the further development and implementation of these techniques has been steady. In what respect does the continuation method (also called the incremental method) differ from the classical perturbation method? The basic difference is that the solutions are no longer restricted to a small domain of the configuration space as with the perturbation method, but can be obtained everywhere in this space. The continuation method is thus a global method, whereas the perturbation

4

Edua rd Riks

method is a local method. With the continuation method, the initial load deformation path, as pictured in Figure 1, can be computed irrespective of degree of the nonlinearity of the problem. With this approach the stability of the solutions can be monitored during the computations, and by doing this the critical point at which loss of stability occurs (A or B) can be determined. The technique also offers a way to compute the branches of bifurcation points (under certain restrictions), so that it becomes possible to assess the postbuckling behavior of the structural model. Because of the versatile finite element modeling capabilities that are currently available, the applications of this type of analysis are no longer confined to standard problems with simple geometries, but can be extended to real world systems. Recent examples of such analyses can be found in Vandepitte and Riks (1992), Young and Rankin (1996), and Nemeth et al. (1996). Thus the continuation principle in combination with the finite-element discretization method enables one to determine the static equilibrium branches (for as far as is deemed necessary) of a very general class of nonlinear structural models. Once these solutions are obtained, the stability behavior of these structures can be explained by examining the particular geometrical properties of these branches.

load

A B

I I deformation

I

deformation F I G . 1. Initial stable response followed by collapse.

>


5

But a stability analysis as described above is still a product of the classical quasistatic approach whereby only the solutions of the equilibrium equations are reviewed. According to the philosophy behind this approach, loss of stability is a dynamical transformation of state (see Figure 1) that starts at an unstable critical equilibrium state (a bifurcation point or limit point) but will end in a state in which the structure is no longer usable. Consequently, the critical state at which the motion starts is of interest, but what happens after this state is reached is not. However, it can be asked whether this quasistatic point of view can always be maintained. It has been known for a long time that unstable buckling does not always lead to an unserviceable state of the structure. To the contrary, it can happen that after such an event further loading is still possible. This occurs, for example, with plates and stiffened panels, and in these particular cases the phenomenon is called mode jumping. Thus it is not possible, at least not in the general case, to predict beforehand what will happen when an unstable critical point is reached. The passage through the critical state may have the result that the structure will end up in a new state with irreparable damage, but it may also happen that the structure remains in operation with no damage at all. In general, the actual outcome is dependent on the problem at hand and can only be predicted by an extended analysis, i.e., by taking the transient motion into account. The methods for integrating the equations of motion in the field of solid mechanics are actually quite well developed (Belytschko and Hughes, 1983; Argyris and Mlejnek, 1991), and the computer resources that are available at the present time no longer hinder the use of these methods. Consequently, in this chapter we not only consider the use of continuation methods for the solution of the equations of equilibrium, but also the use of transient methods to provide an answer to the question of how an actual buckling process works and where to it will lead. We believe that there are many problems in engineering practice where the answer to this question is badly needed. The discussion will closely follow the ideas that were developed in two recent publications (Riks et al., 1996; Riks and Rankin, 1997), but we aim here at a more complete presentation. Just as in the two references mentioned, we will introduce the numerical procedures by first giving a review of the elementary bifurcation theory for ordinary nonlinear equations that depend on a single parameter. The first part of this review is purely a geometrical introduction of this subject that is meant to serve as a preparation for the development of the static methods to be discussed later. The second part is focused on the stability or loss of stability that occurs at bifurcation or limit points, which is necessary for the understanding of what happens when such a point is reached.

6

Eduard Riks

The introduction to the computational procedures begins with a short description of the path-following method to be used for the computation of the static equilibrium branches, together with additional techniques that are needed for the analysis of these solutions. The computation of the buckling motion that starts at the point of loss of stability is the next subject, and the synthesis of this technique with the previous methods ultimately leads to the appropriate computational strategy, which makes it possible to numerically simulate the quasi-static as well as the dynamic elements of a complete buckling event. A good way to demonstrate the feasibility of a new strategy is to verify its predictive power by means of test results of well-documented buckling experiments. Therefore, the chapter will end with the description of the numerical simulation of two well-known buckling tests: the mode-jumping experiment of a plate strip carried out by Stein (19594 and the classical tests on cylindrical shells in compression that were carried out by Esslinger and collaborators in 1970 (Esslinger, 1970; Esslinger et al., 1977).

11. Basics, the Geometrical Point of View A. NOTATION, ASSUMPTIONS, GOVERNING EQUATIONS The structural models that will be studied here are supposed to be purely elastic. It is further assumed that an appropriate discretization procedure is available that allows us to represent the state of the structure in terms of a vector of finite dimension. In the following two sections we will first concentrate on the solutions of the static equations of equilibrium. To understand what type of loadings will be considered here, it is convenient at first to make the distinction between the configuration space C.N+K, in which the state d" of the structure is described and the computional space D.Nin which the freedoms, d, that are to be determined are described. Here N is the dimension of DN and ( N K) is the dimension of C N + K .Thus the computational space DN,which is a subspace of C.N+K, refers to all of the freedoms that are determined by the governing equations, while the configuration space refers to the same freedoms plus the variables that are prescribed by the kinematical boundary conditions. In this manner, the configuration of the structure under load can, at all times, be represented by an ( N K)-dimensional vector,

+

+

N+K

diei,

d" = i=l


7

+

where ei (i = 1 , 2 , 3 , . . . , N K ) are the natural base vectors that span C N + K . The subset {ei} (i = 1 , 2 , 3 , . . . , N ) spans DN. We assign to the base vectors ei the following properties: e'ej

= 6ij

(where 6 i j is the Kronecker delta)

(2.2)

where aTb denotes the inner product of any two vectors a, b E DNor C N + K . In the general case, the structure is loaded by forces and by the prescription of certain displacements. The loading of the first type can be described as follows: N

L =CLiei.

(2.3)

i=l

The loading of the second type is given by N+K

b=

C

biei

i=N+1

and is thus in a subspace Di of CN+Kcomplementary to DN. In principle, the load system (2.3,2.4) can be varied in a completely independent manner, in which case it is defined by M 5 ( N K ) parameters. But in the practical situation it is customary to vary the load system proportionally for certain parts of the analysis. In this case we write (2.3,2.4) in the form

+

N

i=l

N+K

b=hbo = h

boiei r=N+l

where the (fixed) combination of parameters {Loi)and {boj}define a base load that describes the shape of the loading, and h is the intensity that belongs to this system. It is noted that the load intensity h can be seen as a function of the time, but only in the sense that when time passes, the variation of h is infinitely slow. Thus we will consider only loads that are applied in a quasi-static fashion. The structures and the load systems that will be acting on them are supposed to be quasi-conservative. This means that when a structure is loaded and unloaded in an infinitely slow fashion, the initial state can be recovered without the dissipation of energy. The structure then behaves as if it is completely elastic. Quasiconservative also means that the external forces (2.3) that are prescribed can be

8

Eduard Riks

derived from a potential energy function. Only when the structure is in motion can dissipation of energy in the form of Rayleigh damping take place. The assumption of a quasi-conservative class of structures implies that we can define a potential energy function

P = Pud; hbo; hLoD = W[d; hboD - hdTLo

(2.6)

from which the static equations of equilibrium can be derived. The function W in the right-hand side of this expression stands for the internal elastic energy of the structure, while the second term denotes the potential of the force system L. Please note that the notation P = Pud; hb; hD with the arguments d; hb; h between open-faced brackets [ D means that P is a function of these arguments. This notation will be used here and in the following to avoid confusion with the use of the brackets of the type [ [ ( ) } ] that are reserved for the purpose of grouping objects and algebraic terms. After these preliminaries it is now no longer necessary to carry along the explicit distinction between the displacement type of loading and the force type of loading. This means that the potential energy function will simply be written as' :

P = P[d; AD.

(2.7)

However, it should always be kept in mind that both types of loadings can be applied simultaneously at all times. The equations of equilibrium follow from the requirement that P must be stationary:

ap 6d = Pdud; hD6d = f T 6d = 0 6P = ad

for V6d E IIDN

(2.8)

and 6d stands for an arbitrary virtual variation of d in DN. This leads to the set of equations

f[d; AD = 0

(2.9)

where it is understood that

d = diej

'

(sum i = 1, . . ., N ) .

(2.10)

It should be noted that on the basis of the previous assumptions, the loading is generally dependent on the state d, which is due to the displacement-induced part of the load. This stands in contrast to many treatments of these type of problems, where only dead-weight loads are considered.


9

In formal terms, the vector function f is a mapping, f: DN x IW1 -+ FN,where FN is the space (dual to DN) in which the forces are defined. In the context of the computations described here, a distinction between these two N-dimensional spaces is not required, and it is convenient therefore to write RN for both of them. It is this convention that will be adopted hereafter.

B. ADAPTIVE PARAMETRIZATION One aspect of the formulation that needs consideration is the parametrization of the governing equations. In the case of a one-parameter loading, the load intensity h emerges as the natural parameter that steers the deformation process in the actual physical case. But for the analysis of (2.9) and for development of the computational process, the load parameter h is not always suitable, and therefore it is useful to discuss some other choices for the parametrization of the solutions of (2.9). As mentioned, it is natural to choose h thus to view the solutions of (2.9) as

d = d[hD

(2.1 1)

but this leads to difficulties when we reach a state d* for which the corresponding value of h = h* satisfies h- < h* and h+ < h*, where the - and + denote the solutions dub-), d@+D just before and after d*. This is of course a limit point for h, and the failure of the description (2.11) in this case has to do with the fact that in the neighborhood of [d*,h*]no solutions of (2.9) exist for h > h*. A possible way out of this predicament is to exchange one of the displacement variables d~ (1 5 K 5 N ) and write

(2.12)

+

Equations (2.9) admit such a change, because there are (N 1) unknowns for N equations. A more general and, for our purposes, preferable choice is a parametrization that is adapted to the properties of the solution path in the domain where the solutions are described. This is a so-called local parametrization, where the term local implies that its definition is only in effect in a restricted domain of the solution space (see, for example, Rheinboldt, 1981; Riks, 1970, 1972; Seydel, 1989).

Eduard Riks

10

Suppose we have to describe the solutions in the neighborhood of a configuration [dl, h l ] ,which is a solution of (2.9). A change of basis for the decomposition of d in this neighborhood can be introduced by

d=dl+an+c, d l , n , c E RN, 0 E R1, cTn = O , n Tn = l ,

(2.13)

where the base vector n (or director) is chosen in some advantageous way, and the vector c is arbitrary, except that it must be independent of n. To introduce this new basis in the equations of equilibrium, the potential energy function is written as

P* = P[dl + a n

+ C; h] + KCTn

(2.14)

where the orthogonality condition in (2.13) is enforced by means of the Lagrange multiplier K . The stationary value of this function with respect to a,c, and K determines the equilibrium states of the structure, and this leads to the equations

+ a n + c; h] = 0, f[dl + o n + c; h ) + = 0 ,

(2.15b)

cT n = 0 .

(2.15~)

nTf(dl

K n

(2.15a)

+

This is a set of ( N 2) equations (completely equivalent to (2.9)) in the unknowns {a,ci (i = 1 , 2 , . . . , N ) , K , and h},one of which can be chosen as the path parameter. Suppose now that this is a. The solution of (2.15) in the neighborhood of [dl; hl] can then be presented as

d = duo],

d = dl

h = h[o],

h = hi

+ an + cQaD, + Ah[a].

(2.16)

Inspection of eqs. (2.15a) and (2.15b) shows that K = 0; this means that we can eliminate (a) from (2.15) and write for the part that remains,

fad; AD = 0 ,

(2.17a)

nT[d - dl) - a = 0.

(2.17b)

In comparison with the original formulation, the governing equations are now extended by one equation, and the extension defines the parameter a by the choice of n. The solution path in the neighborhood of [dl , h l ] is thus defined through (2.17) for some range of values of a. The geometrical meaning of this device is pictured in Figure 2.


11

It can now be deduced from Figure 2 and (2.17) that the equilibrium points [d[uD, h[uD]for a certain range of a: a* < a < u**, are defined by the intersection of equation (b) with the solution of (a) as long as nTd’, # 0, where d; = {dd[aD/doJ,=,, is the tangent to the path duo) in IWN at d[olD. The parameter choice a is thus “optimal” (at least to some extent), if we choose n = d ; , because in that case the range in which we can use the definition of a by (2.17b) can be expected to be “the largest.” The use of a suitable path parameter by the means of an extended system can further be generalized by setting

f [ d ;hD = 0 , h [ d ;AD - D = 0.

(2.18)

where h is a function h: RN x IW1 + IW1 that is chosen appropriately in the sense discussed above. We will call a the path parameter of the solution (d[uD,h(aD), and its precise meaning will depend on what will be chosen for h. Note that in principle, h is now also included in the definition of a. After these preliminaries we are now ready to discuss the solutions of (2.18). A loading process initiated from the unloaded state (d, h ) = (0, 0) will generally result in deformation states that are initially stable in the sense of the stability theory (see, for example, Koiter, 1945; and Thompson and Hunt, 1973). In the context of the assumptions of this theory, continued loading may result in loss of stability, and this will then take place at a particular point (d, h) = (dc;A,) of the solutions of (2.18). This particular point can be identified as either a limit point or a bifurcation point. It is not always realized, however, that bifurcation points and limit points, which are solutions of (2.18) with very special geometrical properties, do not automatically coincide with the collection of solutions of (2.18) at which loss of stability occurs. The coincidence occurs only a for a small subset of the bifurca-

FIG.2. Definition of the path parameter u .

Eduard Riks

12

tion and limit points that are determined by (2.18). In other words, loss of stability takes place at either a bifurcation point or at a limit point, but it is not true that every bifurcation point or limit point is connected with stability loss. To make this clear, we will first derive the geometrical conditions that determine bifurcation and limit points as special solutions of equations (2.18). The concept of the stability and loss of stability that leads to a restricted class of these special solutions will be introduced later.

c. SPECIAL POINTS: BIFURCATION AND LIMITPOINTS The notation for the governing equations (2.18) can be shortened to

F[x, a] = 0,

F E RN+I

(2.19a)

where

and we will use this abbreviation (2.19a) instead of the extended form (2.18) whenever it is convenient to do so. It is now advantageous to make a distinction between the path parameter a that is introduced by (2.18b) and a special choice of o,denoted by s, which is the arc length of the path x[aD. The latter satisfies the condition

dx Tdx -=(x)

(ds)

ds

x =1

(2.20)

#1

(2.21)

while for our choice of a,

(g)Tg

= (x) i T xi

Suppose now that a branch of the solutions of (2.18) c.q. (2.19) is known and is given by

xi = xi[aD

(2.22)

for a certain range of a,a* < a < a**.The governing equations are then identically satisfied, so that

F[x~;

= 0.

Repeated differentiation of X I [o)with respect to o,and noting the meaning (2.18) of F, results in the sets of equations


13

where F, is the Jacobian of the extended system of equation, a matrix with ( N 1) x ( N 1) components,

+

+

(2.24a) defined by (2.24b) It is noted that F, is nonsymmetrical in general, as a result of the arbitrariness of the auxialiary equation (2.18b). It is further noticed that we use the notation

(2.25) etc., where y=yae,,

a = 1 , 2 ,..., N + 1 ,

z=zaea,

a = 1 , 2 ,..., N + 1 ,

and it is understood that summation convention is implied. Equations (2.23) determine the so-called path derivatives xi; xr; x;”; etc. along the solution branch xl[aD, from which the tangent xo, the curvature xoo, etc. of x1 [a)can be constructed.They thus represent the local geometrical characteristics of x1 [a)at a. As long as the Jacobian F, (2.24) is nonsingular, the solutions of these equations exist and are unique, but this situation changes when F,(xl (a)) becomes singular at some value of a = a,. It is this special case that we will consider below. 1. Bifurcation According to the notions of linear algebra, the solutions of (2.23, i = 1, 2, . . .) exist and are unique when Rank{F,} = Rank{F,; R(’)] = N

+1

(i = 1 , 2 , 3 , . . .)

(2.26)

Eduard Riks

14

where {F,; R(')}is the extended matrix of the system (i) of (2.23). We will call the points of the equilibrium path x(oD at which this condition holds, regular solution points. Of particular interest are the points of xl[aD where loss of uniqueness of eqs. (2.23) occurs. Suppose this happens at a = a,. If multiple solutions of (2.23) exist, we must have Rank(F,} = Rank(F,; R(')] < N

+1

(i = 1 , 2 , 3 , .. .).

(2.27)

This condition means that multiple solutions of (2.23) are possible if F, is singular and the nonhomogeneous terms, the vectors R(') of (2.23), stay in the range of F,. Please note that this condition must hold for all suitable but otherwise arbitrary choices of h[xD in (2.18, 2.19). In this chapter it suffices to consider only a particular case of (2.27), i.e., Rank(F,) = Rank{F,; R(')} = N .

(2.28)

As will be shown, this special case corresponds to a so-called simple bifurcation point where only two solution branches cross. When (2.28) holds, the solution for the derivatives x ( ~of ) eqs. (2.23) can be written as - K(i)b+ y(')

.(i)

where K

( ~ is )

(2.29a)

an as yet undetermined parameter and b and y(') are the solution of (2.29b)

Thus b is the right null vector of F, at a, and y(') are particular solutions of (2.23(i)). The singular F, also has a left null vector, which we denote by c:

cTFx= 0.

(2.29~)

Condition (2.28) implies that C T ~ ( l= ) cT e N + 1

c'R'') = 0,

=o,

(2.30a)

i = 2 , 3 , 4 , ..., N ,

(2.3Ob)

because R(') = e N + 1 and R(') should be in the range of the singular F,. From (2.30a) it follows that c can be written as

c=(K)

EIwN+~,

aER,v.

(2.3 1)


15

Noting the structure of F, (2.24b), eqs. (2.29) then lead to the observation that

aT f d = o , T

a

(2.32a) (2.3 2b)

fh=O,

and because fd = symmetric (since fd = (Pd):):

fda = 0.

(2.33)

In the following we will often denote fd by

K =fa

E

X

RN.

(2.34)

We can then summarize the conclusion as following. If the system of eqs. (2.23( 1)) has two solutions, it is not only follows that F, is singular, but also that K = f d is singular, K having a one-dimensional null space spanned by a (2.32b) and a‘f, = 0 (in general). It also means that the Taylor expansion

x = x[c] = x[c,-D

+ x’[a,D(a

+ 21 x”[a,-D(c - a,)’ + . . .

- c,-)

-

(2.35)

has two solutions implying that two solution branches of (2.18) or (2.19) are intersecting at x[c,-D. The path derivatives of these branches can be obtained from the sets of equation

where R(’) are defined in (2.23), ~ ( 1 )are parameters and y(’) are particular solutions of equations (1, 2, 3, . . . , N ) . Note that the singular condition of K at the bifurcation point x[ecD implies that K is semidefinite, i.e., it can take values V’KV 5 0 for any vector v # a in RN.It is also of interest to note that the construction of the solution of equation (i = 1) is particularly simple if the derivative of the primary path at x[acD is known beforehand. This situation presents itself automatically when arc-length continuation is used for the computation of these branches, because in that case, the direction of the path that is traversed is always known (to some approximation or accurately) (see, for example, Riks, 1984b; Riks et al., 1990).

2. Stationary Points, Limit Points Apart from bifurcations, the solutions x[cD can have special features, such as a stationary point with respect to the direction eN+1, where the path derivative x’[e*D satisfies ei+lx’[c*D = h’[e*D = 0 (see Figure 3). At these points, the

Eduard Riks

16

II d II

w FIG.3. Stationary points with respect to h.

load parameter may have reached a maximum or a minimum, and if this property can be established, such points are referred to as limit points. It will be seen later that among the equilibrium states at which the load h[aD reaches a maximum or minimum, there are “some” that represent loss or gain of stability, and these points are then the proper limit points in the sense of the stability theory. We consider now a regular solution (F, is nonsingular) along the path x[aD, where I’ = 0. Inspection of (2.23) in this particular case shows

Kd’ = 0

(2.37a)

hdd‘ = 1 Kd”

+ fddd’d’ 4-fhh.” = 0 hdd” + hhh” = 0

(2.37b)

etc. Thus it follows that at a solution where I’ = 0, the tangent x’ is given by (2.38) where a must be the null vector of K = fd. Consequently, at stationary points of h = h[aD for a = a,, K = fd is also singular and, just as before, semidefinite. It can be further seen that at such points, the question whether IQa,Dis a maximum or minimum is decided by the solution of (2.37b). Again by inspection, it follows from this equation that at a = ac, can be written as


17

(2.39) The stationary point is thus a proper maximum if (2.40) and a minimum if (2.41)

D. IMPLICIT REPRESENTATION OF THE BIFURCATION BRANCH There are two contrasting ways to represent the solutions of the branching diagrams at bifurcation points. One form is explicit, i.e., in terms of a (multiple valued) Taylor series, as given by (2.35). The other is implicit, in terms of the socalled fundamentalbifurcation equation. For our exhibition,the latter formulation is preferable because the important issues to be discussed here are determined by this equation. As follows from the previous considerations, at bifurcation points of the solutions of (2.18) or (2.19), we are confronted with the conditions (2.32), which are equivalent to fd(a,] is singular,

fh E Range{fd[a,)}

(2.42)

where as before, the attention is restricted to the case where the null space of fd is one-dimensional, so that only simple bifurcation points are considered. In the same terms, the conditions that are connected with the occurrence of a stationary point with respect to the load intensity h are (see (2.37a), (2.39(2)): fd[acD is singular,

fh $ Range{fd[a,D}.

(2.43)

As before, the null vector of fd[a,D is denoted by a E RN.It is clear that this vector plays an important role in the solutions of equations near the point XI(ac),because it is one of the base vectors that determines the tangents to the two paths that are crossing at XI(a,]. The other base vector is a particular solution of (2.23( I)), for which we will take the tangent to the primary path x;[ac,-. In the case of a stationary point h; = 0 (2.42), a is the direction of the primary path at xi (a,) (see also (2.38)).

Eduard Riks

18

To describe the solutions near XI [o,),it is useful to make use of the LyapunovSchmidt-Koiter2 reduction of the equilibrium equations (2.9). For an in-depth exposition of this method, we refer to Golubitsky and Schaeffer (1985). A convenient starting point for the construction of the reduction is the principle of the stationary value of the potential energy. The solutions in the neighborhood of the special point x@,D can be written as

d=d,+pa+v,

aT v = O , (2.44)

h=h,+Ah

where

Note that this transformation involves a change of basis and that p measures the distance along the mode a, while the additional term v is forced to be perpendicular to a. The equilibrium equations that correspond to this transformation can now be derived from the modified potential energy P*:

P* = P[d,

+ pa + v; h, + Ah) + KaTv

(2.45)

in exactly the same way that this was done earlier in Section 1I.B. The governing equations follow from the stationary value of P* with respect to arbitrary variations of the amplitude p, the vector v, and the Lagrange multiplier K . The result is the system

+ v; h, + A h ) = 0, f[d, + pa + v; h, + A h ) + Ka = 0, aTf[d, + p a

aT v = O ,

(2.46a) (2.46b) (2.46~)

which is again completely equivalent to the system in (2.9). In comparison with the appearance of the transformation (2.15) that we derived in Section II.B, there is little that has changed, except that the direction vector n in (2.13) is now specified as n = a. The state at which equations (2.15) are evaluated is different, however, because here we have fd[x,D is singular, whereas in (2.15), the condition of fd was not specified. The power of the reduction (2.46) in the singular case is that the ( N 1)dimensional set of equations in v, K given by (b) and (c) at (2.46) can still be

+

2Koiter devised this method independently in terms of a variational procedure, which is the reason we also attach his name to it (Koiter, 1945).


19

solved uniquely in terms of p and Ah, at and in a small neighborhood of x,. This follows because the Jacobian of this system

J" =

[

K

aT

a . 0] is nonsingular

(2.47)

near and at x,. The solution of (2.46b, c) can therefore be assumed to exist in a small neighborhood of x, and can be written as

v = V ( p ;Ah].

(2.48)

Once V is constructed, the completion of the solution can be obtained by solving the so-called reduced equation (a): (2.49) which, in the present case, has the properties (2.50) The reduced equation is thus singular at x,, and it satisfies either the bifurcation condition gh = 0 or the condition for a stationary point with respect to A,h; = 0. The similarity transformation that stands at the basis of the LyapunovSchmidt-Koiter reduction splits the original (singular) N-dimensional set of equilibrium equations (2.9) into a small subset of equations containing the singularity and a large complementary set of equations that are regular. In the case of only one zero eigenvalue of fd, the case we consider here, the dimension of the reduced set is one. The interesting (and essential) part of the behavior of the solutions is completely described by this reduced set, and because it is small, it is more easily accessible than the original set of equations. It is now useful to consider the asymptotic solution of (2.49). An approximation of this kind can be obtained if we first solve (2.46b, c) for small variations of Ah and p. The solution can be carried out using a standard pattern, starting with the assumption that v in (2.46b, c) can be represented by the power series

+ + pAhw11 + Ah2w02 + higher order terms = AhKO1 + p2K20 + pAhK11 + Ah2K02 + higher order terms

v = Ahwo~ p2w20

(2.51a)

K

(2.51b)

Introduction in (2.46b, c) then results in four sets of equations:

(2.52a)

20

Eduard Riks

where

The underscore ( J is used here to remind the reader that the derivative terms are evaluated at the singular point xc = x(acD. Because the Jacobian J” (2.47) is nonsingular at xc = x[acD,the individual terms in the approximation for v defined in (2.51) can be obtained from these equations. Substitution of the result in (2.46a) and expansion then leads to gap: Ah) = A’, Ah

+o

+ 2AiAhp + 3A3p2 + 4 A i p 3 ( ~a h 2~, ah2p) ,

(2.53a)

where:

1 A; = -[aTfdha 2

A’, = aTfA; ‘

T

A3 = - a fddaa; 3!

+ aTfddaw01];

A: = $ aTfdddaaa-

I

(2.53b)

wlofdw20.

The reduction process can of course be extended in a very general way to cover a whole wealth of phenomena. Here we deliberately confined it to the three basic forms of singular behavior that are the most elementary among the solutions of the equations of elasticity. A natural way to represent the solutions is to plot them in the (A,p) plane, as has been done in Figure 4 for very small values of Ah and p. The three cases to be distinguished can then be specified as follows. (i): Limit point: A’, # 0

A’,A3 > 0 -+ branch (1); AiA3 < 0 + branch (1’).

(2.54)

(ii): Skew-symmetric bifurcation: A’, = 0

A;A3 > 0 -+branch ( l ) , (2); AkA3 < 0 -+ branch (I), (2’).

(2.55)

(iii): Symmetric bifurcation (pitchfork): A’, = 0

A i A i > 0 + branch (l),(2); A i A i < 0 -+ branch ( l ) , (2’).

(2.56)


I I L------

21

1

Limit points

Skew - symmetric bifurcations

Pitchfork bifurcations

FIG.4. Basic forms: singular behavior.

Thus the solutions of (2.53) represent: (i) a maximum or minimum with respect to A, (ii) a simple bifurcation point where two solutions cross at x, with A{,* # 0, and a simple bifurcation point with A{ # 0; A; = 0. It is noted that the latter type is often referred to as pitchfork bifurcation. The analysis of the foregoing concerned the geometrical form of the solutions of the equilibrium equations in the neighborhood of points where fd becomes singular. These results are derived under the assumption that the external circumstances such as the loading or kinematical boundary conditions are absolutely static and thus are in no way dependent on the time. However, in the actual physical situation, the structure is always subject to finite time-dependent disturbances that are induced by the environment. No matter how small these disturbances are, they will induce motions around the equilibrium state, which may or may not stay in the neighborhood of this state when time evolves. There is thus the question whether the equilibrium states determined by (2.9) are physically attainable, i.e.,

Eduard Riks

22

whether the time-dependent configurations of the structure in the actual physical situation will stay close to the equilibrium state that is determined in the imaginary situation where no disturbances can affect this state. The question of the existence of attainable states of equilibrium is, of course, equivalent to the question of the stability of these states. The need to distinguish stable solutions from unstable solutions, and to understand the mechanical significance of the notion of loss of stability is not only important for the computation of the load-carrying capacity of a given structure; it turns out also to be important for computation of the behavior of the structure after loss of stability. This is why we will now investigate the stability of the solutions presented in (2.54)-(2.56).

111. Basics, the Stability Point of View A.

STABILITY AND

LOSS OF STABILITY

The dynamical concept of stability of elastic structures is attributed to Lyapunov. Lyapunov’s criterion considers the motion of a structure initially at rest that is disturbed by a small perturbation (see, for example, Koiter, 1965, 1967a). If the incipient motion grows without bounds, independent of how small the initial disturbances are taken, the equilibrium state is declared unstable. On the other hand, if the motion remains bounded for initial disturbances below a certain finite measure, no matter how small this measure is, the equilibrium is declared stable. However, Lyapunov’s criterion is hardly ever applied in structural analysis. Far more popular is the much older energy criterion (attributed to Lagrange and Dirichlet), which can be proved to be equivalent to Lyapunov’s criterion, at least when we deal with a conservative mechanical system of finite degrees of freedom. The energy criterion states that if the potential energy of the structure at the equilibium state under consideration is a proper “minimum” (a “well”), no matter how shallow or small it is, the equilibrium is stable. If this property does not exist, the equilibrium state under consideration is unstable. It is noted here that the energy criterion can be interpreted as a geometrical analog of Lyapunov’s dynamical criterion, because it can be seen as a rule that connects the geometrical properties of the potential energy function of the mechanical system at the equilibrium state to the dynamical properties of the system at this state. In more specific terms, the energy criterion states that a given equilibrium state d, which is a solution of eqs. (2.9) for a certain load value h, is stable if and only if the potential energy function Pud; hD satisfies

Pad

+ 6d; hD - Pud; hD > 0

(3.1)


23

for V6d E RN: 0 < 116dl) < ,z2, where ,z2 can be as small as we please. (The notation 11 11 denotes here a suitable vector norm, for instance: llall = (aTa)'/2.) If the inequality (3.1) is not satisfied, the equilibrium configuration d is unstable. For practical applications, and thus for the inspection of the solutions of (2.9), this form of the criterion does not turn out to be very convenient, and this is the reason that one usually resorts to a weaker form of (3.1). Assuming that the energy function can be differentiated as many times as is needed, one can write

Because d is an equilibrium state, the first term behind the equality sign disappears, so that the role of the leading term in this expansion is taken over by the second term, the second variation:

where as before, Kid; AD = fdud; AD.

(3.4)

It can now be shown that the minimum of P[d; AD and thus stability at d is ensured if and only if the quadratic form (3.3) is positive definite, or lT2[6dD > 0

(3.5)

for VSd E R,. If the quadratic form is indefinite, that is if n2[6dD

P0

(3.6)

where the sign < holds for some Ad, the potential energy is not a minimum at d and the equilibrium is unstable. The intermediate state occurs when the quadratic form is semipositive definite, i.e., when nz@dD ? 0

(3.7)

where the equality sign holds for some 6d # 0. In this case no decision on the existence of the minimum can be obtained from (3.7) (see Koiter, 1945, and Thompson and Hunt, 1973). In the context of our formulation in terms of discrete variables, conditions (3.5) and (3.6) are sufficient for the existence of a proper minimum, or the exclusion of such a minimum, respectively. They are thus sufficient conditions for the existence

24

Eduard Riks

of stability or the absence of stability of the solution d. No verdict is obtained when (3.7) holds. In that case, the potential energy may still be a minimum, but this property can then be verified only by a return to the original form of criterion

(3.1). Condition (3.7) thus determines solutions that are neither definitely stable or definitely unstable. A solution d = d, that satisfies (3.7) is therefore considered to be at the stability boundary, which separates stable from unstable behavior. Such a state is called a critical state and will henceforth be denoted by d = d, or by x = xc. An equilibrium path ].[x that goes through a critical state x, can thus be divided into two parts. If one part is stable, the other is presumably unstable, because x, signifies the stability boundary. Indeed, this separation is often encountered. Traversing the equilibrium curve from the stable part toward xc will then result in a so-called loss of stability when xc is passed. However, exceptions are possible. There are special cases where the equilibrium curve that passes through x, is not divided into stable and unstable parts. An example of this behavior is found in the symmetrical branch of the pitch fork bifurcation point, as will be seen at a later stage of the discussion in Section E.

B. CRITICAL EQUILIBRIUM STATES AS SPECIAL CASES Simple critical equilibrium states in the sense of the stability theory are thus marked (see condition (3.7)) by a semipositive definite quadratic form (the second variation) l72[6dD. This means (from linear algebra) that the Jacobian K has the following property:

fdux,] = K[xcD is singular (with one-dimensional null space), V ~ K [ X , ] V > 0, VV: a'v = 0,

(3.8)

where a is the null vector of K(lxcD.In contrast, the simple bifurcation points and limit points that we considered in Section I1 were connected with the condition that the Jacobian K has the property

fdux,] = K[x,] V'K[X,]V

# 0,

is singular (with one-dimensional null space), VV:

a'v = o

(3.9)

(see Section 1I.C). It is thus observed that the condition of loss of stability (3.7) or (3.8) is a special case of the condition for singular behavior of the solutions that are given by (3.9). This special case occurs in (3.9) when the quadratic form v'Kv, under the side condition a'v = 0, remains strictly positive. Only in this particular case is a


25

“singular point” in the geometrical sense also a “singular point” in terms of the concept of stability. Points at which loss of stability occurs are thus special members of the family of singular points that are part of the solution set of equations (2.9). Solution branches of equations (2.9) that are stable are thus also special, in view of the circumstancethat unstable branches are the rule rather than the exception (see also (3.6)). Of course, solutions that are unstable do not have much practical meaning in the assessment of the load-carrying capacity of a structure, and therefore they can usually be left out of consideration.

c. THENOTIONOF ATTRACTIONAND REPULSION For what follows it will be helpful to expand our perception of the energy criterion by introducing a mechanical interpretation of the notion of stability. For this purpose we will consider an equilibrium state d = dl at some (fixed) value of the load intensity hl . The potential energy change that occurs if we perturb the state d (by artificial means) is then defined as

hl =constant where y E IWN is any perturbation from dl. The equilibrium equations in this notation follow from

~ ~ U Y D=~ 0Y

(3.11)

f[yD = 0.

(3.12)

which with fly] = ny[yDTgives

These equations are thus a shorthand notation for eqs. (2.9) if we take into account the shift d = dl y and leave out the reference to A. Please note that

+

(3.13) The equilibrium state dl under consideration is thus defined by y = 0, so that f[OD = 0. With the energy criterion, it is the geometrical concept of the shape of the energy surface P[dl + yD in the neighborhood of dl that determines the existence or the absence of stability of this state. It is now useful to investigate

26

Eduard Riks

whether it is possible to connect this geometrical concept to a mechanical concept of stability that is explained in terms of forces. The energy criterion (3.1) in its general form is a necessary and sufficient condition for stability. To determine whether a minimum of P exists at dl ,it is possible to consider the following scenario. We introduce a closed and convex surface around the point dl in RN that can be expanded or shrunk at will and determine the minimum of n[yD on this surface. Once the minimum n* is determined, we let this surface shrink and make it as small as we please. If this minimum, for an arbitrarily small but finite distance from dl, is still positive, the potential energy function P must be a proper minimum at d ~However, . if n*is negative, the potential energy is no longer a minimum at dl . The formulation of this analysis can be based on the introduction of the convex surface

where the matrix T E RN x IWN is symmetric, positive definite, but otherwise arbitrary, and r determines the “volume” of fi. To find the minimum of P (or, equivalently, on this surface, we introduce the modified potential

n)

n* = n[yD - 21 ~ [ y ~ -T r2] y -

(3.15)

where K E Rl is the Lagrange multiplier that enforces the constraint (3.14). Here the minus sign before the additional term is chosen for convenience. The condition for the minimum of n[yD on fi[y; D r is then given by the variational equations

[n,[yD - ~ y ~ T 1 G-y[yTTy - r 2 1 8 = ~0

(3.16)

or by the set of ordinary equations

fUyD - K T =~ 0, YTv - r 2 = 0 .

(3.17)

If it is possible to find a solution y* of this problem that satisfies Vy # y*: n[y*D < n[yD, then the value of n[y*D determines the minimum of l 7 on Q[rD. Stability of the configuration dl is ensured if it is possible to find an open interval for r: 0 < r < E , where E may be arbitrarily small, in which n[y*[rDD is positive and monotonically increasing with r. On the other hand, if the minimum l lBy*(rDD is negative, no matter how small E is chosen to be, stability of the equilibrium state dl does not exist.


27

It is now useful to point out that problem (3.17) corresponds to the formulation of a virtual experiment that is applied to our structural model. The experiment consists of a perturbation of the original equilibrium state dl , by forcing the perturbations y to be on the surface S2[y; rD. The enforcement of this kinematical constraint is “frictionless,” so that the structure is able to seek new equilibrium states on the surface a.These equilibrium states correspond to stationary values of the potential n*. Consequently, eqs. (3.17) represent the equilibrium equations of the structure under an additional loading given by K T ~which , is the force exerted on the structure by the frictionless surface S2 on which the configuration y is forced to stay. At the equilibrium states determined by eqs. (3.17), this force is in equilibrium with -f[yD, which is the force that the structure exerts on the frictionless surface S2 in return. It is now noted that the vector n = ‘Q is normal to the surface if y is on a:yT‘Q - r 2 = 0. Thus, according to eqs. (3.17), at the equilibrium states of the perturbed model the reaction force R = -f(y*D is parallel to this normal. Because S2 is convex, it can also be concluded that R has a component that points = -y*Tf[y*D is negative, and in the opposite to the center y = 0 when Y * ~ R direction when Y * ~ isRpositive. This observation can, of course, be rephrased by saying that the restoring force R = -fly*) is pointing at the interior of S2 when y*Tf[y*D > 0 and pointing to the exterior when y*Tf(y*D < 0. In the following we will call the reaction force R a force of attraction if it points in the direction of the interior as defined above. If R points toward the exterior, it is called a force of repulsion. It can now be shown that when the equilbrium state of the unperturbed structure dl is stable, and not critical, the reaction forces induced by the constraint i2 are always attractive for 0 < r < E where E + 0, whereas this property is lost when dl becomes unstable. This observation also holds for perturbations that are constrained by i2 but do not necessarily satisfy eqs. (3.17). In other words, it also holds when the surface Q[rD has friction to the extent that no motion along it is tolerated. To show that this assertion is correct, it is sufficient to consider the measure

(3.18) If G > 0, the force R is attractive, and if G < 0, it is repulsive. It is useful to introduce y = ru, where u is arbitrary under the restriction that it lies on i2[r = 1): uTTv - 1 = 0, and r is always positive: r > 0.

Eduard Riks

28 Expansion of uTf[ruD yields

+

uTf[rvD = uT[f[OD rfd[o]u

+ S[ruD]

(3.19)

where S[rnD = r 2 p [ u ,rD denotes the tail (remainder) of the expansion. Note that the leading term in p [ u ; rD is of order 2 in u and independent of r . With (3.19) it follows that

+

(3.20)

r-'G = uTfd[ODu rA[u; rD where

This expression provides the estimate

where

Cur] = Max(lA[u; rDI) vv

(the existence of this bound is assumed)

The value ml is here the smallest eigenvalue of K = fd[OD, defined by

[K - mlT]x

0.

1

(3.2 1b)

If the configuration dl = 0 is stable, mi > 0. This means that if we choose r so small that r < E , where E: &CUED= ml, (3.22) regardless of the direction of the perturbation y = ru. Consequently, at a stable, but noncritical equilibrium state, the reaction forces R = -fly) are guaranteed to be forces of attraction as long as y remains in Q (E):y T ' Q - c2 = 0. As mentioned, most solutions of the equilibrium equations (2.9) are unstable. What can be said about the reaction forces to arbitrary perturbations when dl is unstable? It follows from previous considerations that in this case there must be at least one particular direction u of the perturbation y = ru for which the reaction forces R are repellent. This follows because the Jacobian K = fd[OD has at least one eigenvalue ml < 0. Let the corresponding eigenvector be denoted by a, again with the normalization aTTa = 1. The value of the measure H in the direction

Buckling Analysis of Elastic Structures: A ComputationalApproach

29

v = a is then

so that

GI [ml

+ rlA[a; plllr.

(3.23)

This result shows that if r is chosen small enough, we can make sure that G < 0, and this, in turn, implies that a small perturbation in the direction n = a induces a reaction force that pulls the configuration y away from the equilibrium state y = 0, irrespective of the sign of a. It is thus clearly a reaction that will lead to an escape motion, and therefore it is associated with unstable behavior of the system.

D.

STABILITY AT T H E CRITICAL STATES

The test for the stability of the solutions of the equilibrium equations, in numerical simulations for example, can thus be carried out by examining the condition of the Jacobian K = fd[x[sDD. If K is positive definite, stability is ensured. If K is indefinite, the equilibrium is unstable. Only in the case of an encounter with a critical point, where K is semipositive definite, can the decision on the stability not immediately be provided. In these special cases it is necessary to return to the general criterion (3.1). It turns out that the investigation into the stability of the critical points is of considerable interest for the numerical treatment of the problems that we discuss in this paper. It was Koiter (1945), in his dissertation (see also Koiter, 1963), who studied this problem for the first time. He showed how the stability or instability of the critical points was indicative of the character of the behavior of the structure in the neighborhood of these points. His results also make it possible to classify the critical points as stable or unstable by simply examining the shape of the solutions in the neighborhood of these points. The investigation of the stability of the branches of a critical state x, (away from x,) can be carried out on the basis of the quadratic form

+

+

A2P = [aTfd[xDa]Ap2 [aTfd[xD]AvAp AvTfd[xDAv where

(3.24)

represents the solutions of eqs. (2.46), i.e., they represent the branches connected with x,, and Ap, Av are arbitrary variations of p and V, whereby aTTAv = 0.

Eduard Riks

30

The sign of A2P for 0 < (pl < EO, where EO can be arbitrarily small, determines the stability of the solutions connected with xc. It turns out that if one is interested only in the solutions in the immediate neighborhood of xc,i.e., when EO + 0, the analysis of (3.24) can be replaced by inspection of the sign of (3.25) where gup, V[pD, k[pDD is the reduced form (2.53) that is derived in the reduction process in Section 1I.D. The proof can be constructed by first considering the minimum of

l7* = n*[ApD =

Min

VAv: A k = f x e d

{[aTfd[x)]AvAp

+ AvTfd[xDAv + KaTTAv}.

(3.26)

This problem leads to the set of equations (3.27) which is compatible for reasons discussed before. The solution can be written as Av = Apzo; K = A ~ K oIntroduction . into (3.26) yields

A 2 P = [aTfd[X]a - z~fd[x)zo]Ap2.

(3.28)

It is now not difficult to see that when EO -+ 0 and thus 1pl + 0; Ak -+ 0, the above expression comes arbitrarily close to (3.25). This result is classical and can already be found in Koiter (1945,1963). To complete the discussion, we will now turn to the stability of the critical points themselves. In agreement with our expectations and following from inspection, at such points the reduced form A2P disappears. In this case it is thus necessary to go back to the general criterion (3.1), i.e., at x, it is necessary to inspect the sign of

~ [ Y D = P[dc + Y; &D

- P[d;

LD,

(3.29)

A, = constant for arbitrary variations y = Ad. To do this, it is again advantageous to look at the minimum of l 7 on the ball: !&: yTTy - r 2 = 0. But this time we will also introduce the change of basis that proved to be so useful in the determination of the branches of xc:

y = pa

+ v,

a T Tv = 0.

(3.30)

Buckling Analysis of Elastic Structures: A ComputationalApproach

31

To determine the minimum of

n* = n[yD - -21 ~ [ y ~ T r2] y

(3.31)

in this setting we have to consider the minimum of the modified potential: 1

IT** = n[pa + vD - 5 K[(pa + v)TT(pa+ v) - r2] + uaTTv

(3.32)

which is determined by

+ vDTa - KaTT(pa+ v)]6p + [f[pa + v] KT(pa + v) + uTaIT6v 1 - - &[(pa + v)TT(pa+ v) - r2] + 6uaTTv = 0 2

[fupa

-

(3.33)

for arbitrary variations 6p, 6v, 6 v , and J K . Using aTTa = 1, the resulting equations are written as

+ vD - ~p = 0, fapa + VD KT(pa + v) + vTa = 0, a T f[pa

(3.34a) (3.34b)

-

a T Tv = (pa

0,

(3.34c)

+ v)TT(,ua+ v) - r 2 = 0.

(3.34d)

These equations determine the minimum of n on a closed surface S2 (r) around the state y = 0. It is natural to consider the solutions of this set to be parameterized by r, the radius of S2[r), but we have the freedom to change this at will and replace r by any other freedom. The parametrizationin terms of p has a distinct advantage because in that case, r follows simply from r 2 = (pa v)TT(pa v). Another simplification follows by observing that the Lagrange multiplier u = 0 (which can be seen by projecting (3.34b) on a and comparing the result with (3.34a)). The equations that interest us most are thus contained in

+

f[pa

+

+ vD - KTV- KpTa = 0,

(3.35a)

aTTv = 0.

(3.35b)

Solution of these equations is again sought in terms of v = Vup);K = K [ p ) ,and this is possible because these equations are compatible even at p = 0, i.e., the Jacobian J:

J=

[

fd

aTT

71

is nonsingular

32

Eduard Riks

Once the solution is determined, the value of ll attained by this solution satisfies n,[rD = n[pUrD, V[w[rDDD < n[p-,v-D, where [p-, v-I # [p[rD, VUp[rDI is any combination of p and v that lies on Q[rD. It further follows that if ll, [rD > 0, stability is ensured, but in contrast, if n, < 0, stability is lost. We will now determine the solution of (3.35) for values of Ipl + 0 by assuming that the solution can be written in a Taylor series:

Repeated differentiation of (3.35) with respect to p, and evaluation of the resulting expressions at p = 0, yields the sequence of sets:

fd[ODy’ - KTV’- ETa = 0, aTTy’ = 0,

(3.37a)

The underscore ( _ ) again indicates that the variables are evaluated at x,; thus: y’ = v’[OD, g = ~ ( 0 1etc. , It follows immediately that the first set of equations yields = 0; y’ = 0, so that (b) can be reduced to

fd[ODf’ - 2 ~ ’ T a= -fdd[ODaa, aTT$’ = 0.

(3.38)

In the following we write the solution y” of (3.38) as: x” = 2w20 and note that w20 becomes completely identical to the solution w20 of the system (2.52a) for ( i j ) = (20) in Section 1I.D if we take T = I (I is the identity matrix). The solution Y = p[rDa+ p[rD2w20 Our3)of the problem (3.34) determines the location of a point on the surface Q[rD at which Il is a minimum with respect to all other points y on Q[rD. It defines a curve for values of r: 0 < r < E , where E is some small positive number. Along this curve the potential energy can be evaluated as a function of r or p. If, for vanishingly small values of r or p, this special function stays positive, the potential energy is also positive for all perturbations in the neighborhood Q[rD. In that case, the potential energy is still a minimum at y = 0 and the equilibrium state xc is stable. If the potential energy is negative along this curve for vanishingly small r , no minimum exists and the equilibrium of the critical state must be unstable.

+


33

Introduction of the notation ( )" = d / d r , n[OD = lT"Q0) = no0[OD= 0 at x,, the expansion of n[Y[r)Dat r = 0 gives

1 1 n=nooo[ODr3+ noo[ODr4+ . . . 3! 4! 0"

-

(3.39)

or equivalently, in terms of p:

1 n = -1 n ( 3 ) [ 0 ~+~3 n ( 4 ) 4 0 ~+ ~4 ... 3!

4!

(3.40)

where ( )(i) = d' / d p u " . For the purpose of determining the sign of l 7 in the critical case, (3.39) and (3.40) are equally applicable. According to our solution (3.37a and b), po = f l ; p"" = 0, and with this we arrive at ~ O O O [ O D

= n(3)~0~(~0)3,

00

I-IOO~OI)

W3)[oD = aTfddaa;

= n(4)~0~(~0)4,

(3.41)

~ I ( ~ ) [=o [aTGddaaa D - 12w0T2~wo21.

Please note that the coefficients (1/3!)II(3)QOD,(1/4!)n(4)[ODare equal to the coefficients A3, A:, etc. in (2.53). It now follows that

~I'~'[OD = aTfddaa# 0 j the critical state xc is unstable

(3.42)

because in that case there is one direction that yields a negative value for the leading term of (3.39) or (3.40). In the case that II(~)[OD = 0, the stability of d, is determined by the sign of ll(4)[OD, so that

I I ( ~ ) [ O D= [arfdddaaa- 1 2 ~ ~ ~ 2 f d w>o oz l j the

~I(~)[oD

critical state x, is stable;

= [aTbddaaaj the

I~W&&,WOZI
0 and attractive for A p < 0. In the case in which we deal with a symmetrical bifurcation, i.e., points of the type A3 = 0; A: # 0, the forces are repellent when A t < 0 and attractive when A: > 0 (see Figure 6). In the case of a proper limit point, we have A3 < 0, and

+


+

37

T: skew symmetric b i b t i o n

Limit point

+ +:+ h

11

symmetric bifurcation unstable

symmetric bifurcationstable

FIG.6. Perturbation forces.

the result is similar to that of the skew-symmetrical bifurcation point. Note that (3.49) can be written as G = Sign(ApDg(Ap; OD

(3.50)

where g[Ap.; OD = g ( A p ; AhDlah=o is the reduced form of (2.53) evaluated at Ah = 0. Thus the sign of ApguApL; OD determines whether the perturbations represented by & A p are attractive. To complete the picture of the perturbation forces, we should also investigate perturbations from the equilibrium states connected with the critical states, i.e., along the branches represented by y = 0; Ah # 0 + Adup) = d - d, = pa p2w20 . . . at Ahgp) = h [ p ]- hc. In this case it is sufficient to look at

+

+

Eduard Riks

38 perturbations

and determine the sign of

G=

GTf[Ad

+ 6; A@))

(3.52)

11611

It turns out that for small IApI, this expression can be reduced to

G = Sign[ApDg[p

+ A p ; Ah[p]D.

(3.53)

The force diagram of the branches of solutions around and at the critical states x, can thus be investigated by looking at the sign of G defined in (3.53). The result of this investigation is pictured in the force diagrams of Figure 6.

F. INCIPIENT MOTIONAT UNSTABLE POINTS Because one of the principal objectives of this paper is to consider the computation of the transient behavior of a structure after loss of stability has occurred, it is of interest to study the initial motion of a structure when it is perturbed from an unstable critical state. To keep the discussion as simple as possible, we will focus on a system that is truly conservative, so that there is no damping present. In that case the equations of motion are given by

MZ

+ f[d; hD = 0

(3.54)

where M is the mass matrix of the system and ( ) denotes differentiation with respect to the time t . We will now suppose that the structure is first loaded up to the critical point = [d,; h,], and after that it is forcefully kept in a position that = [d, z; A,]. corresponds to the critical state x, plus a small perturbation: The perturbation z is given by

XT

+

z = Ea

+

& 2 w20

(3.55)

where E is an arbitrary small number and a and w20 are the ingredients of the solution of the minimum problem (3.31), see pages 21-32. In this disturbed state at t = 0, the structure is suddenly released. For the solution of the motion that now follows, we have the following initial conditions:

d[OD = Z; d[OD = 0.

(3.56)


39

Let the solution be denoted by Q[tD = dut, ED, so that Q[tD satisfies (3.54) identically. Thus Q represents a curve in the N-dimensional space RN parameterized by the time t. This parameter choice is natural, but for what follows it is advantageous to introduce also the geometrical path parameter o that we introduced earlier and then let o be a function of the time t , thus: a = out) and Q = Q[o[tDD. If (.)’ denotes differentiationwith respect to the path parameters and, as before, ( ) denotes differentiation with respect to the time t , it is then possible to write the acceleration $ in the form

.. =

Q

Q’f?

+ Q”2.

(3.57)

It is now noted that because of the initial condition introduced at (3.56), the velocity 6 along the curve Q[aD at the very beginning of the motion ( t > 0) will will even be smaller. Consequently, the centrifugal term be small, and thus (c?)~ in (3.57) can be neglected with respect to the term that represents the acceleration along Q[aD during the initial stage of the motion. Thus, for small t the curve Quo) is approximated by the solution of the system:

(aTMd’[tD - 1 ) d t = 0;

~ [ o=)f?

(3.58)

where the second equation defines the path parameter o.It can now be demonstrated, that the particular solution of (3.58) for small excursions o from the critical state in the direction (3.55), is identical in form to the solution of the equations that determine the curve of the steepest descent or slowest ascent of the potential energy surface, pages 30-31. (If we choose T = M, assuming M is symmetric and positive definite). In other words, the path that the structure will follow upon a disturbance (3.55) from the critical state is initially of the form: Q = o a + o 2 w20

(3.59)

where a is the buckling mode and 2w20 = 1’’is given by (3.38). Having established this correspondence, it is now possible to use these observations in a simple characterization of initial motion in terms of the gain in the kinetic energy just after t = 0. The law of the conservation of energy states that the sum of the kinetic and potential energy is invariable, thus: T2[c$

+ P[dD = C = Constant

where 1 *T T2 = - d Md = the kinetic energy of the structure.

2

(3.60)

40

Eduard Riks

As argued, the path of the motion for t > 0 is approximately given by (3.59), and substitution in (3.60) gives T2 = -Pupa

+ p2w20D + C .

According to the initial conditions (see (3.56)), at t = 0, T2 = 0, so that C = PUEa E ~ w ~and o ] this , gives for the variation of T2 when t > 0,

+

T2 = -Pupa

+ p2w20D + P[Ea + E ~ W ~ O D

or (3.61) for p: I E J 5 IpI. Please note that this particular solution is only meaningful in the cases (i) A3 < 0; p > E > 0 or A3 > 0; p 5 E < 0 and (ii) A3 = 0; A: < 0. Thus it is possible to conclude from this construction that an arbitrarily small perturbation E from an unstable critical state, applied in the proper direction (3.59), induces a motion with a lunetic energy gain T2 in the early stages of the motion, which is described by (3.61). This gain is proportional to A p 2 or A p 3 (depending on whether A3 # 0), where A p = p - E is the distance traveled from the point at which the motion started at t = 0 (see (3.61)).

IV. Computations A. LOCALAND GLOBAL ANALYSIS The need to understand how a structure behaves under the loads for which it is designed and, in particular, how and at what intensity of the load it will fail is the primary motivation for conducting a stability analysis. Usually the question of the load-carrying capacity is not focused on one particular configuration of the structure alone, and it is almost never the case that only one particular set of loads must be considered. In general, the analysis must take into account a whole array of possibilities where variations in the eqs. (2.9) are thus also dependent on external parameters other than the load, and there can be many. Suppose for simplicity that there is only one design parameter v in addition to the load intensity A. The equations of equilibrium are then written as

fud; A;

UD

= 0.

(4.1)


41

This could be a characteristic length or anything that influences the design and therefore the properties of the structure. In this particular case we can picture the type of analysis that is required in a three-dimensional diagram (see Figure 7). Once the solutions of the equations are obtained, their characteristics can be described by plotting one of the components di of d against the values of h and v, as is done in the figure. A general representation of solutions in the case of the two external parameters h and v is

(4.2)

where p and t are some suitably chosen surface coordinates of this twodimensional equilibrium surface in R N + ~The . stability analysis in the practical situation can now be seen as the calculation of certain patches of this surface. For example, in Figure 7, this equilibrium surface is thought to belong to a structure, which for the design parameter v = 0 has a linear load deflection response I that is intersected by a bifurcating branch of equilibrium states I1 in A . The intersection point in the plane v = 0 is the bifurcation point. At this point the structure will become unstable; i.e., in the case drawn, it will buckle violently, a phenomenon called snap buckling. For values of the design parameter v > 0, the snap-buckling behavior at a bifurcation point is changed to a snap-buckling behavior at a limit point ( B , C ) , but the load at which this occurs is different, usually lower than in the case v = 0. Thus the influence of changes in the design on the structural response behavior is often an important part of the analysis, and it is natural make this analysis by computation of the responses, for example, of several traces of (4.2), as is given by

d = d[cj,

~ ( y ] ,

i = l , 2 , 3 , 4,...,

v,D,

h = h[uj,

(4.3)

~ ! = 1 , 2 ,,3. . .

The analyses mentioned earlier (Young and Rankin, 1996; Vandepitte and Riks, 1992) are based on these types of calculations. Other situations do present themselves, however. For example, if v is a measure of damage, like the length of a crack in a fuselage shell, one may want to compute the response of the structure under a stationary load but increasing length of the crack. In this way the surface is traced according to

d = d[v,D,

h = constant,

Q!

= 1 , 2 , 3 , .. .

(4.4)

42

Eduard Riks

Applications of this type can be found in (Rankin et al. 1993; and h o p s and Riks, 1994). There are situations where the calculation of the static solutions is only part of the complete analysis. This occurs when we are interested in computing the dynamic response of the structure after the static equilibrium is lost when h is about to exceed A,. In Figure 7 this occurs at the ridge of the hill that emanates from the bifurcation point h, = h,[v = OD. In that case, the transient behavior of the structure is computed from the equations of motion (3.54),augmented with some damping term. This process will lead to a change of configuration from d, to an end-state dnew (the existence of which will be assumed), which will then be stable. The new equilibrium state d,,, is in general associated with a far-field solution of (4.1) in the sense that Jld,- d,,, 11 is finite and usually large.

“t

FIG. 7. Global

analysis.


43

An analysis that is conducted by computing finite stretches of the equilibrium surface in Rn+2, or by computing a transient trajectory in this space with the time as path parameter, concerns global equilibrium states or global transient states that are computed point after point. This can be referred to as global analysis. Global analysis is typical for most of the finite-element analyses that are done in the nonlinear domain. But in addition to the global approach, it is also possible to use a local approach in which the solutions are sought in terms of a power series expansion that involves only local information belonging to a single solution point of eqs. (4.1). This type of solution presents an approximation of the equilibrium surface in the neighborhood of the point of which the series expansion is developed. It is a much older method, known as the perturbation method, asymptotic method, or the method of successive linearizations. In this paper we will refer to this type of method as a local approximation method. The theory described by Budiansky (1974) typically concerned the state of the art in local analysis methods, because it was exclusively concerned with the perturbation solution of bifurcation problems. Global methods often make use of local approximations, as predictors, for example. On the other hand, local methods of stability analysis sometimes rely on the use of global techniques. The perturbation method applied to the case of a compound bifurcation point generates multidimensional (reduced) equations of the type (2.59) (see, for example, Carnoy, 1980; Arbocz, 1987; Lanzo and Garcea, 1996), which are then easily solved by a global path-following technique. In this paper we will not dwell extensively on the further development and finite-element adaptations of the local approach that took place after 1974, because this would carry us to far from our main objective. However, we will use some of these techniques whenever this turns out to be an advantage. A local approximation technique that will be helpful here is the so-called linearized buckling analysis, by which estimates of the location of the critical state can be obtained in relation to the global state in which this method is used. Our main interest is, however, the global approach, and to introduce it properly we must start with Newton’s method. B. PERTURBATION VERSUS PATH-FOLLOWING METHODS The difference between local methods and global methods can best be explained by considering the solution of the response of a structure under the influence of a load system that is varied through a single parameter A. As before, we consider the equilibrium equations augmented with the equations that define the path parameter

F[x;

OD

= 0,

E RN+1

(4.5)

Eduard Riks

44 where

Suppose that a solution of (4.3, x = X I , is known beforehand at a = 01, and that Fx[xlD = nonsingular. In that case we can attempt to compute the Taylor series that we encountered earlier:

where the local quantities, the configurationxi and the path derivativesx', , x:, defined by

. ..,

are determined by a sequence of derivatives defined by (2.23). The construction of this solution x = E[oD is straightforward once the derivative terms F,,x',x;, Fxxxx',x~x',, etc., developed at xi, are obtained, because in that case, the solution of the sequence (4.7a, b, c, . . .) entails only one factoring of the matrix F, followed by a number of back-substitutions equal to the number of terms of that one wants to take into account in (4.6). However, as far as the range of validity of the solution (4.6) is concerned, nothing can be said in advance, and the outcome will depend on the type of problem at hand. The global way to construct the same solution is to develop this solution in terms of a number of points xi along the path defined by (4.1) for fixed values of the parameter oi: a1 < a2 < 0 3 < . . . . The way this is done is by using a predictor-corrector scheme that is referred to as the path-following method. Again, if x1 is known beforehand, we could use eqs. (4.7) to compute the derivative x', . With this information, it is possible to construct the approximation ai : K[x[a,DD becomes semipositive definite, or when x[acD satisfies

aTa = 1 , K[x[a,DDa = 0, v ~ K [ ~ [ >~ 0,~ D DVV:~ vTa = 0, see also Section 1II.B.

(4.23)

Eduard Riks

50 A necessary condition is

Det{K[x[ccDD} = 0.

(4.24)

We note at this point that a distinction between a bifurcation point and a limit point can be made by observing that at a bifurcation point F,[cT,)must also be singular (see Section 11),whereas at a limit point Fx[acDis still regular (in general). Ahead of xi, the solution x[aD is not yet known, but we have the approximation ~ [ C T ] , and thus it is possible to define an approximation to the critical state x, by setting

This is a nonlinear eigenvalue problem that can be solved by using Newton’s method, for example. It is easier, though, to use straightforward linearization, i.e.,

(4.26a) where (4.26b) The solution of this problem, the smallest eigenvalue ACT*= CT* - 01 and corresponding eigenvector a*,then presents an approximation to the values that belong to the still unknown critical state x,. The estimate of the latter is thus simply given by X*

= xi

+ AD;X~

(4.27)

+

ACT:^^. which includes, of course, the estimate of the critical load h* = hi Applied at the undeformed state xT = (dT,h ) = (0, O), the method coincides with the classical linearization, which assumes that the unbuckled state is given by dI = Ado, where do is the solution of K[ODdo

+ f h = 0.

(4.28)

Many practical structural designs that are optimized with respect to their weight (stiffened panels, for example) behave in this way, exactly or approximately. In these cases, the linearization process provides accurate solutions to the buckling load of these structures. Thus the (local) approximation method defined by (4.26) and (4.27) makes it possible to find estimates of the location of the critical state x, (which is part of

Buckling Analysis of Elastic Structures: A Computational Approuch

51

the curve x[aD) ahead of the known solutions xi ( i = 1 , 2 , 3 , . . . , k ) . From here it is now only a small step to design a procedure that automatically zooms in to the critical point x, (when such a point is anticipated). Several such approaches to this effect are possible (see, for example, Riks, 1973, 1978, 1984a; Seydel, 1989). However, even without an automatic search procedure, the simple feature described at (4.26) is already very useful to have at hand. Methods that are designed to compute the critical states of the solutions of the equilibrium equations are often divided in two categories, called “indirect” and “direct.” Direct methods are formulated in such a way that the solution of the critical state is obtained directly without use of the equilibrium path that is connected with this point (see, for example, Wriggers et al., 1988). An indirect method, on the other hand, is a method that makes use of the equilibrium path that is leading to this point x,. In practice, the distinction is not very significant (although this is not always realized), in particular because there are almost no differences in the type of operations that are required. In this paper we will restrict the discussion to a sketch of the indirect approach, because the steps of this method blend more naturally with the context of the analysis described here. An indirect method of calculating a critical point, starting from some point xi, can be constructed by observing that the distance An: obtained in (4.26) and (4.27) can immediately be used to compute the new step (point) xi+l = x[ai +Aa:) along x(a), using the standard continuation method discussed earlier.

I1 FIG.9. Computation of the critical state.

52

Eduard Riks

Because this new point xi+l is likely to be closer to x, than xi is, the new estimate xi*+1for x, computed from xi+l is also expected to be better than the previous estimate xi* (see Figure 9 for an illustration). The process can be repeated over and over again, with the result that one can find a sequence Xi+k, k = 1 , 2 , 3 , . . . , xi*+K,so that llxr,, - x,ll < E for K large enough. (In many practical situations, K = 2 or 3 is already sufficient to yield an acceptable result, depending the location of the starting point xi. See, for example, Rheinboldt (198 1)and Riks (1978)).

D. BUCKLING AND POSTBUCKLING ANALYSIS We are thus in the position to compute the quasi-static response of the structural model by using the path-following method up to and beyond the point at which loss of stability occurs. As indicated in the previous section, it is also possible to determine the critical state x, to some degree of accuracy. The critical state is a limit point if the component kQaDof the branch through x, is descending after A, is reached. Otherwise, x, is a bifurcation point. In either case, it makes little sense to compute the path x[aD far beyond the state x,, because for a > a,, x[aD is unstable. In the case of the limit point, the static response of the structure can be considered to be completed because there is no more information to be gained by further computation. Only when x, is a bifurcation point might it be of interest to compute a stretch of the intersecting branch. In this way, it is possible to identify the bifurcation point without computing the derivative terms (3.47) that determine its classification. The computation of the branch of a simple bifurcation point by continuation is just as difficult or easy (depending on one’s point of view) as the computation of the primary branch, provided the proper initialization can be provided. As explained, accurate estimates of the critical state x, can be obtained with the methods discussed earlier. The initialization of the path-following procedure for the computation of the branch 11, with x, as the starting point, can easily be carried out if the tangent to XII[CTD at a = a, is also known at this point. The procedure that is necessary for this type of computation is called brunch switching, because it switches the computations from one branch to the other. Thus it seems completely natural to use a predictor for branch I1 given by (4.29) noting that at the same time, direction associated with this step must be introduced into the auxiliary equation of (4. l), so that

Buckling Analysis of Elastic Structures: A Computational Approach h(d; h ) - o = x : . ~ ~ ~ (x,x) - c = 0.

53 (4.30)

This idea, which is illustrated in Figure 10, works well in the case that we have studied so far, i.e., the simple bifurcation point. During the calculation of x,, it is always possible to obtain a good estimate of the buckling mode a and the tangent xi to the primary path XI (see, for example, Riks, 1984a; Riks et al., 1990). The desired direction vector xLIIis then a linear combination of these two vectors:

xiII= PX:.~

+ pa.

(4.3 1)

The two scalars p and fi can be determined from the theory and depend on the factors aTfddaa,aTfdha.The problem here is the computation of these factors, because the higher order derivative terms fdd, etc., are not automatically available in a finite-element program. A possible way to circumvent this difficulty is to use instead of (4.31) the modified predictor: ["= X,

+ Aob,

g(d, h ) = bT(x- x,) - ACT= 0

(4.32a)

where b is given by

b = .xiI

+ pa,

bTXiI = 0,

bTb = 1.

See Figure 11 for an illustration.

bmlch X n ( 0 ) FIG. 10. Branch switch, the natural approach

(4.32b)

54

Eduard Riks

XdO)

FIG.1 1, Branch switch, a simplification.

The aforementioned description of the way switching of direction at a bifurcation point can be accomplished during the computations of the branching diagram concerns only mo examples of a large number of variants. In the computer program that is used here, a rather different method is implemented which is due to Thurston (Thurston et al., 1985),and which is essentially a Newton-Raphson iteration scheme applied to the transformed set of eqs. (2.53) (see also Riks, 1984b). This so-called equivalence transformation method is more difficult to implement than the simpler methods discussed above, but it has turned out to be a very robust method that can also be used for the analysis of compound bifurcation points (a phenomenon not studied in this paper) if the multiplicity of these points is not too large.

E. TRANSIENT BEHAVIOR The branching diagrams that can be computed by the methods described above enable us to deduce the extent to which we can load the structure before loss of stability takes place. This evaluation follows from the location of the critical points encountered and the shape of the solutions in their immediate surroundings. Although it cannot always be known beforehand, in many cases, loading beyond an unstable critical point will cause damage to the structure. Damage is inevitable when the gain of kinetic energy during the buckling process is so large that the structural system is no longer able to convert it properly. If this occurs, the motion will end in a state in which the system is no longer serviceable. A well-known example of a structure that will be destroyed after becoming unstable is the thin-walled cylindrical shell in compression, whereby the load system is of the dead-weight type.


55

On the other hand, the possibility also exists that loss of stability does not end in destruction. This takes place when the kinetic energy build-up during the motion remains relatively small (in comparison to the accumulated elastic energy of the structure, for example), and the motion ends in a new stable equilibrium state without accompanying damage (see Figure 12). The structure is then still serviceable in this state and in fact can be loaded even further. Examples of mechanical systems that continue to function after the first unstable critical point is passed can be found among certain stiffened panel configurations (see, for example, Stein, 1959a, b). In these cases, a loss of stability manifests itself as a sudden change in the buckling pattern a phenomenon like the load is further increased that can recur. The phenomenon of repeated mode changes that take place at certain instances while the load on the structure is steadily increased is called mode jumping. As mentioned in the Introduction, to be able to deal with these problems, we must take into account the transient nature of the jump or jumps. It should again be emphasized that the main theme of this chapter is the analysis of the behavior of a conservative structure under a load that is slowly varying. In

destruction

case a

case b

FIG. 12. Destruction dependencing on the amount of energy freed

56

Eduard Riks

the first stage of the loading program, the deformation of the structure will follow the single and stable equilibrium branch that passes through the undeformed state. Because the equilibrium along this path is stable, this slowly varying load factor implies that the response of the structure d = d[hD is also slowly varying, so that the velocity i and acceleration d can be neglected. The time scale during which this process takes place is thus very large. In contrast, when the equilibrium becomes critical and unstable, snap buckling will occur. This is an uncontrollable motion that starts from the neighborhood of the critical state and ends at some new stable equilibrium state. During this motion the load is still slowly varying in accord with our assumptions. However, the duration of the buckling motion now turns out to be very small for the cases to be considered here. For example, for the problems still to be discussed in Chapter 5, the time interval of the complete motion is seldom in excess of 40 milliseconds. The change in the load is thus negligible during the time the transient buckling process takes place, and it is this simplificationthat we will adopt in the remainder of this paper.

F. TRANSIENT ANALYSIS The transient change of deformation at buckling is governed by the equation of motion that we now introduce in its complete form, i.e., including a damping term:

Md + C i + f [ d ;AD = 0

(4.33a)

with the initial conditions t = 0:

d [ t = OD = Z,

hut = OD = VO.

(4.33b)

As before, M is here the mass matrix that represents the inertia of the structural model under consideration. For our purpose, it is sufficient to use viscous damping contained in the term C i . The damping term is necessary to simulate the dissipation of the kinetic energy that becomes available during the transient change of state. In this way it is possible to let the motion end at a new stable state of eq~ilibrium.~ We are thus faced with the problem of integrating this large system of equations of motion. This is a subject that received considerable attention in the field of 3H0w the amount of damping is determined in a particular analysis is discussed by Riks et al. (1996).


57

computational mechanics (see, for example, Belytschko and Hughes, 1983, and the text of Argyris and Mlejnek, 1991). Because the literature on these methods is extensive, we will only briefly comment on the type of method that we will be used here. Suppose the structure is in motion and that at the time t = tl the configuration dl = d[tlD and the velocity i~ = d(tlD are known. In that case, the acceleration ;[ti) = i l is also known through the equations of motion (4.33). To compute the state of the motion at the time t2 = tl At2, it can be assumed that the acceleration 8[tD between the time steps tl and t2 is reasonably well represented by the interpolation

+

(4.34) where a is the still unknown acceleration at (t2):

..

a = d(tD

(4.35)

and \yl and q 2 are interpolation functions, for example,

which must satisfy (4.36) The assumption (4.34)-(4.36) allows us to make use of the integrals

(4.37)

where it is noted that the integrals rn and w are explicit expressions in t and the still unknown vector a. The acceleration vector a can now be computed by requiring that a should satisfy (4.33a), thus:

This is a system of nonlinear equations in the unknown vector a , which can be solved by Newton's method, for example. Once a is computed, the process of integration can be repeated for the new step t2 -+ t3, and so on.

58

Eduard Riks

The example of the transient method given above belongs to the so-called Newmark family of methods. The method is called “implicit” because it requires the solution of the system (4.38), involving the nonlinear function f[0@2; a); h[t2DD, where a appears implicitly. Thus this is an iterative solution process comparable to the corrector operations of the path-following method. On the other hand, explicit methods, which we will not consider here, do not require such an elaborate process. These methods can be derived from the above scheme by replacing a in the argument o f f by an extrapolation based on the values of d, I and 8 in the previous step or steps. In this way an iterative solution of eq. (4.38) can be avoided. The method sketched in the foregoing makes use of the configuration d[tl D and its first and second time derivatives at t l . It is also possible to use other steps d[tojj, d[t-lD, etc., in the interpolation (4.36) that were computed before d[tlD. In that case, the method is called a multistep method. Multistep methods, such as the method of Park (1975) (see also Argyris and Mlejnek, 1991), which will be used here, thus use information of the past for the future, and in this way have a built-in tendency to dampen the motion, particularly the high frequencies of the motion. Implicit methods are numerically more stable than explicit methods, by which it is meant that implicit methods allow for larger time steps Ati to be taken before divergence of the method takes place. For a precise discussion of the concepts of numerical stability and accuracy of these methods we refer to Belytschko and Hughes (1983).

G . TRANSIENT ANALYSIS AFTERLOSS

OF STABILITY

To supplement a static analysis by computing the actual buckling process, we must be able to apply these transient integrators mentioned above, starting from the equilibrium points at which loss of stability occurs. In this way, the equilibrium solutions that are obtained by static analysis are used as an initialization for the transient analysis of the buckling process. It is possible, at least in principle, to use the scenario of a slowly increasing load, h = hut), starting from h = hut = OD = 0, and then study the passage (in time) through the points of loss of stability (see Marke, 1995). In view of the static methods of analysis that are available, it seems more practical, however, to use a perturbed equilibrium state in the neighborhood of the critical points in the initial conditions of the buckling motion. It is at this point that the question arises of how one should proceed, i.e., in what way should the static solution process be matched to the static part, or equivalently, how should the initial conditions (4.33b) for the buckling motion be formulated?


59

As can be concluded from the analysis in the foregoing sections, the unstable equilibrium states in the neighborhood of the critical points are actually degenerated attractors from which a perturbed configuration will be repelled if the perturbation is chosen in the proper direction, i.e., the direction for which the loss of stability takes place. We can deduce this from the force diagrams in Figure 6. For example, the (perturbed) configurations [d,.+ Ad; A, Ah] presented in Figure 13 by (A), (B), and (C) are appropriate for this purpose. The recipe for this type of initialization can thus be described as follows. Take the solution point nearest the critical state on the unstable secondary branch of the critical state that represents the point of loss of stability (B), (C). If no secondary branch exists, take the nearest point beyond the limit point (A). Use this configuration as the starting value of the transient procedure and apply a slight increase to the load. Thus if the nearest point is given by d = d,+, the initial conditions for the computation of the transient buckling motion are

+

(4.39) Please notice that in Figure 13, A-C, the configurations [d,+; A,+] are indicated by a shaded bullet and the jump configurations [d,+; A,+ Ah] by a blot. By studying the sign and modulus of the asymptotic form of the test function G in (3.53), it is not difficult to see why these choices will lead to the desired result. The configurations x,+ = [d,+; A,+] are unstable solutions of (4.5) close to the critical state. If the load is raised to h, at d,+, this configuration represents the particular perturbation that we encountered in Section III.E, a perturbation that induces a reactive force on x* = [d,+; A,] that is repellent. Raising the load just above the buckling load h, will only amplify this effect. The foregoing discussion is restricted to the cases of unstable critical points. If the point of loss of stability itself is stable, the secondary branches of this point are also stable and no jump will occur when the structure is loaded up to this point. Instead, the structure will follow one of the two rising paths 2 (case D). The computation of the rising paths 2 is possible with the path-following method discussed earlier, using a switch of direction at x, = [d,; A,]. But it is of interest to note that the transient method can also be used to switch branches if one wishes to do so. The steps that must be carried out in that case are pictured in Figure 13(D). The trajectory the time-stepping procedure is following is finite in length, because during this motion the damping that is present in the model will eventually destroy the kinetic energy build-up that occurred after loss of stability took place. The motion thus subsides and the dynamic configuration of the structure at low

+

Eduard Riks

60

-

stable unstable

I-I * Limit point unstable

Skew-symmetric bifurcation unstable

AS 1:

Symmetric bifurcation stable

Symmetric bifurcation unstable

FIG.13. The initialization of the buckling motion.

‘i

velocity can then be captured by a point of attraction that stands in the way of the motion. At least this is the way that we can interpret the outcome of this dynamic event. It is thus expected that when the kinetic energy T2 becomes smaller and smaller after some time, the configuration of the structure will move closer and closer to a new stable equilibrium state, so that eventually it will be possible to use the dynamic configuration ‘d@D at the step M : t = fM (where T 2 U f $ ~ ] ] < E * ) , as a starting value for the path-following procedure, and compute the new equilibrium point at A, Ah. The necessary condition for convergence to take place in this case is again that the configuration d[tMD should be close enough to the equilibrium solution, and this can be enforced by requiring E* to be small enough. It follows from experience that this initialization does lead to the expected effect, although it is not always possible to know how precise the test T2[‘h[t~)D < E* should be made.

+


61

V. Examples and Conclusion A. SHELLMODELING The theory of thin-walled shells plays an important role in structural stability analysis because structural components made of shells are the first to encounter instability phenomena. At the time of Budiansky’s paper, the shell formulations were already well established within the framework of the assumptions of small strains and moderate rotations. The main thrust in the development of the theory before that time had been toward the simplest possible formulation of the theory, to facilitate the solution of the resulting equations. For example, for the analysis of bifurcation problems in shell structures, the Von Karman-Donne11 equations were very popular (and still are; see the exposition in Budiansky, 1974). These equations are the outcome of one of the simplest possible formulations that still take the essential part of the geometrical nonlinearity of the shell behavior into account. They are capable of producing accurate results in many practical situations. With reference to the restrictions of the classical perturbation theory in elastic stability, Koiter (19674 developed shell equations specially suitable for the bifurcation analysis of shell structures. Like the Von Karman-Donne11 equations, these equations are also relatively simple in structure, but do not suffer from the typical deficiencies of the former in a number of situations. With the introduction of the computer, the development of the shell models by finite elements took a rather different turn. At first, attention was still directed at the implementation of existing theories. This implied, for example, the implementation of the Kirchhoff-Love assumptions and a strict adherence to the simplifications that had been so helpful in the development of the equations in the past. It was soon discovered, however, that the development of shell finite elements on the basis of the Kirchhoff-Love assumptions is very difficult and can be replaced by an approach that is based on an assumption that (superficially) looks like the introduction of shear strain effects through the thickness of the shell. This departure from the classical formulation, often called the Reissner-Mindlin assumptions (Reissner, 1945; Mindlin, 195l), facilitates the search for trial functions that satisfy interelement compatibility. A second departure from the classical approach in the formulation concerned the form in which the equations are derived. In the classical theory, the governing equations are almost always presented in terms of differential equations in their simplest possible form, using as many approximations as can be tolerated. In the current shell finite-element theory, these approximations are no longer necessary, so that the “exact” expressions can be retained. Moreover, the formulation is now no longer directed toward the development of

62

Eduard Riks

differential equations, but directly toward the development of ordinary nonlinear equations that can be seen as the discrete analog of the continuum equations. In this way, but also through fewer simplifications, the shell finite-element models that are currently available have a range of validity that surpasses that of the older nonlinear shell theories. Actually, the range of applicability of these models is only restricted by the condition that the strains in the shell should remain small and that the state of stress is approximately a state of plane stress. For example, the shell models developed by Bathe and Bolourchi (1988), Simo and Fox (1989a, b), Simo et al. (1989a, b), Sansour and Bufler (1992), IbrahimbegoviC and Frey (1994), Ramm and Matzenmiller (1988), and Biichter (1992) belong to the Reissner-Mindlin type of formulations. Recent examples of elaborations in terms of the classical Kirchhoff assumptions are given by Bout and van Keulen (1990) and van Keulen (1993). The shell finite-element model that is used in this paper can also be reckoned to fit into the classification of “small strains but arbitrarily large displacements and rotations.” It can be found in the STAGS code (Rankin et al., 1988). This code (structural analysis general shells), originally founded by Almroth and Brogan (Almroth, 1972; Almroth et al.; 1977), is in continuous development at Lockheed Martin and is sponsored by NASA. STAGS is a general-purpose finite-element program that is designed specifically for the stability analysis of shells, although it now has also other features. The shell model that is used here was originally based on shallow shell assumptions, the same assumptions that are the basis of the Von Karman-Donne11 equations mentioned earlier. However, this initial formulation was later extended to the large displacement and large rotation domain by a corotational framework developed by Rankin (see Rankin and Brogan, 1984; Rankin and Nour Omid, 1988). With this extension, the STAGS code is now capable of producing shell models that can be compared directly with the nonclassical developments mentioned above, and consequently they represent the state of art in this field. The development of the discrete modeling of shells has thus come a long way since its beginnings at the start of this century, and we will gratefully make use of its remarkable advances.

B. MODEJUMPINGOF A PLATE STRIP Stiffened panels under compressive loads can exhibit mode-jumping phenomena in their load deflection response diagram. Even the simplest example of this type of structure, the plate strip in axial compression, can show this behavior. Mode jumping is a phenomenon that can be explained on the basis of a con-


63

densation of bifurcation points (see Figure 14) in the neighborhood of the (first) buckling load, and the way the branches of these points interact. The analytical treatment of this type of problem poses considerable difficulties. In 1943, a constructive (approximate) solution to the plate strip problem was given by Koiter (1943).4 In this work, Koiter considered the case of an infinitely long plate strip in compression. In this idealized case, the mode changes are gradual so that the shape of the buckling mode can be taken as a continuous function of the applied load. The solution for the infinite plate that he obtained can serve as an approximate solution for a plate whose length-to-width ratio is considerably larger than 1. A number of years later in 1959, M. Stein also discussed these phenomena, which he observed in the course of a panel buckling test program. Since then, several authors have contributed to the analysis of the mechanism of this phenomenon. We can mention Supple (1970), Nakamura and Uetani (1979), Schaeffer and Golubitsky (1979), Golubitsky and Schaeffer (1985), and Suchy et al. (1983, who explained the jump behavior by analyzing the static branching diagram in the neighborhood of the first and second bifurcation points by using perturbation theory5 (see also Damil and Potier-Ferry, 1986).

F I G . 14. Perfect plate, clustered bifurcations

'There is no explicit reference to mode jumping in this report, but it is obvious from the way the problem is treated that the solution given, i.e., the load end-shortening diagram, is the approximate solution for a plate that jumps from one mode to another during loading. 'Golubitsky and Sheaffer took Stein's test article as an example.

64

Eduard Riks

Perturbation theory can explain why a structure under load will suddenly change its mode shape, but it does not offer the means to determine the load deformation diagram when several jumps occur in succession. As far as is known, Koiter’s solution6 of the plate strip remains the only example of an (approximate) analytical solution that provides this relation. In the general case, the discretization method in space and time, as it has been developed in the last 20 years, is at present the only technique available that is capable of computing this relation. The mode-jump problem of Stein (Figure 15) is a striking example of the capabilities of the tools of analysis discussed so far. A particular solution for an imperfect plate is presented by Riks et al. (1996). Here we will present another, similar solution of this problem. The plate strip (width 2b, length I ) is modeled by using symmetry with respect to a plane through the center perpendicular to the transverse direction y . The loaded edges at x = f1/21 are clamped but can move in the x direction. The longitudinal edge that is supported in this model is subject to simple support conditions. For the dimensions b, 1 and the thickness t that are chosen, we refer to the cited reference.

simple support

A \

FIG.15. Stein’s mode-jumping problem

6This solution is restricted to infinitely long plate strips in pure axial compression, with all kinds of longitudinal boundary conditions. The theory was later extended to shear loading as well (Koiter, 1946).


65

The solution of this problem is dependent on the initial state of the panel. If no impe$ections are taken into account, loading will result in bifurcation at the load intensity h, = 1 (the load factor is normalized with respect to the first buckling load). The critical state at h, = 1 turns out to be stable (as is common for plates). At a slightly higher value of the load, another bifurcation point exists along the basic state XI, and when the load is further increased other bifurcation points are encountered. We will now first describe what happens if the plate is perfect. The actual physical states that the perfect plate will follow during loading are the stable states. This means that at at a load value h slightly higher than h,, the structure will buckle in a stable manner along a branch XII with an initial mode shape that resembles the buckling mode computed at h,. This phenomenon can be compared to the equilibrium point moving from the basic state XI to the branch XII.Further loading, however, soon passes through an unstable critical point along the branch XII,and at that instant a mode jump occurs. The jump ends in a new stable state with different mode shape. For the plate configuration studied here, the branch XII exhibits five ‘‘half‘’-waves in the axial direction. The new state XIII,which is reached after the first jump has occurred, is marked by seven half-waves. Further loading will result in a second jump to XIV with nine half-waves, and so on. Thus when the panel is perfect (i.e., no imperfections are involved), the jumps only occur in uneven wave numbers. This behavior, which we do not present here in the form of illustrations, can be explained on the basis of the symmetry properties of the numerical model of the panel (see the discussion by Riks et al., 1996). For an imperfect plate, i.e., a plate that has initial imperfections, the behavior changes somewhat in that it is now no longer necessary that the deformations remain symmetrical (in the sense discussed above) as in the perfect case. The particular results that we present here in Figures 16, 18, Plate 1, correspond to a solution for a panel with geometrical imperfections. In this case, an arbitrary ensemble of buckling modes (the first four modes) that belong to the successive bifurcation points along the basic state XI of the perfect case are added to the structure to change its geometry slightly. Thus, imperfections are introduced by adding a small initial displacement wo to the original configuration. They were taken in the form wo = 0.01[al a2 a3 a4], where ai are the first four buckling modes computed along the basic state of the perfect plate. The result for this imperfect panel, the load deflection behavior, is given in Figure 16. There are in this case four jumps that occur at h = 1.38,4.50, 8.30, 11.83. A phase-plane portrait of one of the jumps is given in Figure 17. The changes in the deformation pattern during the transient phase can be judged from the frames (h = 0.01) + (h = 1.38) + (h = 4.5) + (h = 8.3) + ( h = 11.83) in Plate 1. The

+ + +

66

Eduard Riks

6

10

end-shortening F I G . 16. Load end-shortening plate.

first picture (1) shows the plate strip in its initial loading stage, which gradually goes over to the mode ( 2 ) . But the changes that then follow occur suddenly at the values of the load indicated above. The frames (2-5) show the states of the panel just before these jumps takes place. During the loading process the changes in wave form start with a change from 5 half-waves in the basic state Xf to 6 in the state XrI, from 6 to 8 from Xr, to the state X&, followed by the jump from XfII to Xfv, changing 8 into 10 half-waves and from Xfv to X$ with a change from 10 to 12. It is noted here that this particular outcome is different from that reported by Riks et al. (1996), because the combination of buckling modes that is used to represent the imperfections differs here with that of the ones taken in this reference. The initial geometry of the structure thus has a noted influence on the load at which the jumps take place and on the type of the jumps that occur.


67

FIG.17. A jump in the phase plane.

However, this is not surprising, because with this type of problem, the branching diagram, i.e., the stable static as well as the transient trajectory the structure will follow, is strongly influenced by the initial imperfections of the structure.

C. THEESSLINCER CYLINDER BUCKLING EXPERIMENT In Riks et al. (1996) we presented an example concerning a buckling experiment on carbon composite cylindrical shells conducted at NASA (Rankin et al., 1994). The agreement between the visually observed changes of deformation during the actual test and those produced by the numerical simulation were very

68

Eduard Riks

satisfactory in this case. In particular, the agreement between the actual and the computed shapes of the postcollapse state turned out to be very good. The NASA experiments were not monitored by a high-speed movie camera to register the transient part of the collapse process. This is not uncommon, because visual recording of the transient part of the buckling process is a very difficult undertaking and thus is seldom considered. However, there are a few examples of compression tests on cylindrical shells, where the transient part of the experiment was filmed. Such film experiments were carried out by Almroth (1963) and Esslinger (1970). On both occasions, the experiments were carried out in such a way that the tests on each specimen could be repeated over and over again. This made it possible to predict the moment at which the collapse motion would start, so that the operation of the high-speed camera (which is needed for this experiment) could be triggered at the appropriate time. It is not known whether the films Almroth made are still available, but Esslinger’s films are. In fact, they are available on video (Esslinger et al., 1977). The Esslinger experiments therefore offer a very suitable reference for numerical experiments of the kind considered here, and it is for this reason that we selected one case from the series recorded by Esslinger et al. (1977) and took this as a test case for the mode-jump procedure. The cylinder specimens were made of Mylar with dimensions radius R = 100 mm, thickness t = 0.254 mm, length L = 225 mm. The material was homogeneous and isotropic, with the following specifications: Young’s modulus E = 5500 N/m2, Poisson’s ratio u = 0.3, specific mass p = 1.3 x lop6kg/mm3. The shell edges were potted in some kind of epoxy material and were subject to a combination of axial compression, external pressure, and shear. In the numerical experiment that we present here, only one loading case is considered. This is the case of pure axial compression, which is the most common type of loading in tests conducted on cylindrical shells. To be able to reproduce the results, axial compression is induced by forcing one of the end plates to move inward (see Figure 18). The loading thus corresponds to displacement loading. In the testing machine, displacement control is never completely attainable, because the test fixtures absorb elastic energy, which is released when collapse sets in, thus disturbing the control mechanism that controls the position of the plate. It is an effect that can be simulated in the numerical model if we wish, but it is ignored here. The results we will show are thus obtained for the idealized case of a perfect end-shortening application. It is of interest to note that in the actual experiments the end shortening was enforced with a velocity of 1 m d m i n , which is in agreement with the notion of a quasi-static application of the load. For the finite-element model that emerged on the basis of these assumptions, a uniform mesh is used: 63 x 121 in axial and circumferential directions. For the


4

69

Controled displacement

R = 100 mm R/t = 393.7 t = ,254 mm

L = 225 mm

FIG. 18. The Esslinger experiment.

boundary conditions at the loaded ends, clamping is assumed, so that the edges do not deform in the axial direction and cannot rotate around the director’s tangent to the edge. The analysis begins with the determination of the critical load h,, which corresponds to the critical state of the basic solution XI = [dr (AD;h ] .This solution is most conveniently computed with the standard path-following method available in the STAGS code, together with the bifurcation prediction method (4.26). After this initial computation, the base load is adjusted so that the load factor h becomes h, = 1 at the critical state.7 As is well known (see, for example, Koiter, 1945), the cylindrical shell, with an R l t ratio as large as this one, is governed by clustered bifurcations with many nearly simultaneous modes for which the tools of analysis discussed here are not entirely suitable. This means that we did not attempt to compute the branches of the bifurcation first point. It also explains why the code had difficulties in ob7This load corresponds to the well-known formula of the buckling load of a long cylindrical shell:

70

Eduard Riks

taining accurate solutions for the eigen modes that belong to these bifurcations, although it produced estimates that are still useful in the construction of suitable initial imperfections in the second part of the analysis. In the actual Esslinger experiments, collapse of the specimens occurred at loads varying from A, = 0.8 to A, = 0.6, depending on the (unknown) imperfections of the shell and the precise nature of the boundary conditions that were applied. To provide a means of distinguishing between the two extreme cases, it was decided to conduct one simulation on the perfect geometry of the shell model and one on the imperfect model using the computed eigen modes a; mentioned above as geometrical initial imperfections. We will thus present two different cases that represent the extremes of the experimental outcome to some degree.

1. The Perject Case The static part of the analysis comprised 10 continuation steps along the basic equilibrium state of the shell in this case. At that point the load was hlo = 1.08, that is, 8% above the critical value h, = 1. This point, as a starting point for transient analysis, is rather far past the critical state, but in this way a rapid departure from the unstable state can be forced. The transient motion is triggered by an inevitable error in the solution [d[hloD; hlo],because the basic state is computed with the predictor corrector procedure, and thus fudlo; hlo) = E # 0. The computation of the transient motion can be speeded up somewhat if the buckling modes ai are used as initial perturbations (see Section IV.G), but we did not make use of this possibility here. The snap-buckling motion that started from this unstable state Plate 1 is pictured in Plate (2-6). The scale factor of the displacements displayed is 1, and the colors here indicate the magnitude of the displacement components u and w, i.e., the displacement component tangential to the circumference and the component in the direction of the normal to the shell middle surface, respectively. The time steps that the transient procedure takes varies between at the beginning of the motion and lop4 at the end.' The time that elapsed between the beginning of the motion and the state that was accepted as the postcollapse was At = 0.03 s. 2. The Imperject Case For the imperfect cylinder model, the first eight buckling modes were selected to serve as imperfections; the distribution of these modes was taken as

w = [ u , u , wIT = x a i a ; [ x . yD, 8The time integration procedure in STAGS has a step length control algorithm (Rankin el ul., 1988).


71

with q (i = 1, . . . , 8) arbitrarily chosen as cq: [0.04,0.02,0.04,0.02,0.04, 0.02,0.04,0.02]. The analysis begins with the computation of the imperfect basic state XI: [dr[sD; Ar(sD]. It took 15 steps to reach the collapse point (limit point) for this case at A, = 0.534. This is the deformation state shown in Plate 3.1. The jump was initiated by taking the deformation state one step beyond this point at the load value of A = 0.54 as the initial condition for the transient procedure, in accordance with the recipe for such initialization discussed in Section 1V.G. Snapshots of the ensuing motion are given in Plate 3.2-6.

D. DISCUSSION AND CONCLUSION The successive solutions that are obtained during computations of this kind can be displayed in sequence, as in a slow motion picture. These motion pictures of the analyses, which, unfortunately, cannot be shown here, display a striking similarity with the sequence of events recorded in the Esslinger experiments. In particular, the postcollapse state that is reached after the motion has died out compares with the states recorded in the experiment. For the slightly imperfect specimens, Esslinger found a postcollapse state characterized by a skew-symmetrical wave in the axial direction and nine waves in circumferential direction (see Plate 20, whereas for the case of larger imperfections (and a much lower collapse load), this postcollapse state is symmetrical in the axial direction, with eight waves in the circumferential direction (Plate 3f). It is, of course, tempting to speculate on the reason for the difference, but we skip this question here. In the case of the larger imperfections in the actual experiment, the shell could be loaded further, upon which further changes in the state of deformation of the shell occur (Esslinger et al., 1977). Unloading in the postcollapse state leads to snap-back to the initial basic state XI. It is true that the computations are very laborious. In particular, the transient parts of the analyses that we presented here take much computer time and are bulky in storage. At the present time routine calculations are not yet feasible on large models, but this situation is continually changing for the better. The model of the cylinder here had about 40,000 degrees of freedom, but we were able to obtain the results on a present-day workstation. The computer time for a complete simulation as presented here is on the order of 24 h on such a machine. Mode jumping plays a role in the design of stiffened panels in aerospace engineering, and therefore the combination of static and transient procedures will prove useful, now that they have become available. For example, there is an im-

72

Eduard Riks

portant problem for which these techniques will be welcomed as a tool of analysis. This is the problem of the residual strength of delaminated composite compression panels. Preliminary investigations have shown that the behavior of such panels is governed by mode-jump mechanisms caused by the delaminations, i.e., the delaminated areas can pop open or inversely implode during the loading sequence. In the computerized analysis of this phenomenon, the techniques described here will then become indispensible. We feel, therefore, that the techniques that we have presented in this chapter will find their practical applications in the near future.

References Almroth, B. 0. (1972). Collapse analysis for shells of general shape, Val. I. Tech. Rep. AFFDL TR-71-8. Almroth, B. O., Holmes, A. M. C., and Brush, D. 0. (1963). An experimental study of the buckling of cylinders under axial compression. Rep. 6-90-63-104, Lockheed Missiles and Space Company, Palo Alto, CA. Almroth, B. O., Stern, P., and Brogan, F. A. (1977). Future trends in nonlinear structural analysis. Comput. & Structures 10, 369-374. Arbocz, J. (1987). Post-buckling of structures, numerical techniques for more complicated structures. In: Buckling and post buckling: Four lectures in experimental, numerical and theoretical solid mechanics, Lecture Notes in Physics 288 (H. Araki, J. Ehlers, K. Hepp, R. Kippenhahn, H. A. Heidemuller, J. Wess, and J. Zittard, eds.). Springer-Verlag, Berlin. Arbocz, J., and Hal, J. M. A. M. (1990). Koiter’s stability theory in a computer-aided engineering environment. Internat. J. Solids Structures 26(9/10), 945-975. Argyris, J., and Mlejnek, H. P. (1991). In: Dynamics of structures: Texts on computational nieclzanics (J. Argyris, ed.), Vol. 5. North-Holland, Amsterdam. Azrar, L., Cochelin, B., Damil, N., and Potier-Ferry, M. (1993). An asymptotic-numerical method to compute the post-buckling behaviour of elastic plates and shells. Internat. J. Numei: Methods Engrg. 36, 1251-1277. Bathe, K. J., and Bolourchi, S. (1988). A geometric and material nonlinear plate and shell element. Comput. & Structures 1 1 , 2 3 4 3 . Belytschko, T., and Hughes, T. J. R., eds. (1983). Computational methods for transient analysis, Vol. I . Series of handbooks in Mechanics and Mathematical Methods. North-Holland, Amsterdam. Bidanid, N., and Johnson, K. H. (1979). Who was Raphson?, Short communication. Internat. J. Numer. Methods Engrg. 1, 148-152. Bout, A,, and van Keulen, F. (1990). A mixed geometrically nonlinear shell element, In: Theory and applications in applied mechanics (J. F. Dijkstra and F. T. M. Nieuwstadt, eds.). Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 269-276. Biichter, N., and Ramm, E. (1992). Shell theory versus degeneration-A comparison on large roattion finite element analysis. Int. J. Num. Methods Engrg. 34, 39-59. Budiansky, B. (1974). Theory of buckling and postbuckling of elastic structures. In: Advances in applied mechanics (C. S. Yih, ed.), Vol. 14. Academic Press, New York. Carnoy, E. (1980). Post-buckling analysis of elastic structures by the finite element method. Comput. Methods Appl. Mech. Engrg. 23, 143-174.


73

Casciaro, R., Salemo, G., and Lanzo, A. D. (1992). Finite element asymptotic analysis of slender elastic structures: A simple approach. Internat. J. Numer: Methods. Engrg. 35, 1397-1426. Crisfield, M. A. (1991). Non-linear finite element analysis of solids and structures, Vol. 1. Wiley, Chichester, England. Damil, N. (1992). De la thkorie de la bifurcation au calcul des structures. Doctoral thesis, University of Hassan 11, Faculty of Science 11, Casablanca, Morocco. Damil, N., and Potier-Ferry, M. (1986). Wavelength selection in the postbuckling of a long rectangular plate. Internat. Solids Structures 22(5), 5 11-526. Dvorkin, E. N., and Bathe, K. J. (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Engrg. Compur. 1, 77-88. Esslinger, M. (1970). Hochgeschwindigkeitaufnamen von Beulvorgang Dunwandiger, Axial Belasteter Zylinder, der Stahlbau, 31,73-76. Esslinger, M., Hall, R., and Klein, H. (1977). Beulen und Nachbeulen dunwandiger Schalen-Axial Belastung isotroper Kreiszylinder. Filme E2191 des IWF, Gottingen. M. Esslinger, Publ. Wiss. Film., Sekt. Techn. Wiss./Naturw., Ser. 3, Nr. 4E2191. (English spoken version available.) Golubitsky, M., and Schaeffer, D. (1985). Singularities and groups in bifurcation theory, Vol. 1. Applied Mathematical Sciences. Springer-Verlag. New York, Heidelberg, Tokyo. Haftka R. T., Mallet, R. H., and Nachbar, W. (1971). Adaptation of Koiter’s method to finite element analysis of snap through buckling behavior. Int. J. Solids and Structures 7, 1427-1447. IbrahimbegoviC, A. (1994). Stress resultant geometrically nonlinear shell theory with drilling rotations. Part I. A consistent formulation. Comput. Methods Appl. Mech. Engrg. 118(34),265-284. IbrahimbegoviC, A,, and Frey, F. (1994). Stress resultant geometrically nonlinear shell theory with drilling rotations. Part I. Computational aspects. Comput. Methods Appl. Mech. Engrg. llS(3-4). 285-309. Knops, H. A. J., and Riks, E. (1994). Mode-injection for the analysis of through cracks in pressurized fuselages. In: Advances in nonlinearjinite element methods (B. H. V. Topping and M. Papadrakakis, eds.). CIVIL-COMP Press. Koiter, W. T. (1943). De meedragende breedte bij grote overschrijdingen van de knikspanning voor verschillende inklemmingen van de plaatranden. (The equivalent width at loads far above the buckling load for various boundary conditions along the plate edges.) NLL Rep. 287. National Aerospace Laboratory, Amsterdam, The Netherlands. Koiter, W. T. (1945). On the stability of elastic equilibrium. H. J. Paris Publishers, Amsterdam. English translation: Rep. AFFDL-TR-70-25. Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, OH, 1970. Koiter, W. T. (1946). Het schuifplooiveld by grote ovcrschrijdingen van de knikspanning (The shear buckling mode at loads far above the critical value), Report X295. National Aerospace Laboratory, Amsterdam, The Netherlands. Koiter, W. T. (1963). Elastic stability and post-buckling behavior. In: Proceedings of the Symposium on Nonlinear Problems (R. E. Langer, ed.). University of Wisconsin Press, p. 257. Koiter, W. T. (1965). On the instability of elastic equilibrium in the absence of a minimum of the potential energy. Proc. Kon. Ned. Ak. v. Wetensch. (Proceedings of the Royal Netherlands Society of Arts and Sciences), Series B68. Koiter, W. T. (1967a). General equations of elastic stability for thin shells. Proceedings of the Symposium on the Theory of Shells, in Honour of Loyd Hamilton Donnell. University of Houston, Houston, TX. Koiter, W. T. (1967b). A sufficient condition for the stability of shallow shells. Proc. Kon. Ned. Ak.v. Wetensch. (Proceedings of the Royal Netherlands Society of Arts and Sciences), Series B, 70, No. 4. Kouhia, R. (1992). On the solution of nonlinear finite element equations. Comput. & Structures 44, 243-254.

74

Eduard Riks

Lanzo, A. D., and Garcea, G. (1994). Koiter’s analysis of thin walled structures by the finite element approach. Int. J. Num. Methods Engrg. 39, 3007-303 1. Mackens, W. (1987). Numerical differentiation of implicitly defined space curves, Bericht no. 45. Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, Germany, March. Marie, J. (199.5). Sudden change in second order nonlinear systems, slow passage through b$urcation. Ph.D. thesis, University of Wageningen. Mindlin, R. D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18, 31-38. Nakamura, T., and Uetani, K. (1979). The secondary buckling and post-buckling behaviours of rectangular plates. Internat. J. Mech. Sci., 265-286. Nemeth, M. P., Britt, V., Collin, T. J., and Stames, J. H. (1996). Nonlinear analysis of the space shuttle super light weight external fuel tank. A I M Pap. 96-1552-CP, 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April 15-17, Salt Lake City, UT. Collection of Technical Papers. Ortega, J. M., and Rheinboldt, W. C. (1970). Iterative solution of nonlinear equations in several variables. Academic Press, New York. Park, K. C. (1975). Evaluating time integration methods for nonlinear dynamic analysis. In: Finite element analysis for transient nonlinear behavior (T. Belytschko, J. R. Osias, and P. V. Marcal, eds.). Applied Mechanics Symposia Series. ASME, New York. Ramm, E., and Matzenmiller, A. (1988). Consistent linearization in elasto-plastic shell analysis. Eng. Comput. 5,289-298. Rankin, C. C. et al. (1988). The STAGS User’s Manual, NASA Contractor Report. Rankin, C. C., and Brogan, F. A. (1984). An element independent corotational procedure for the treatment of large rotations. In: Collapse analysis of structures (L. H Sobel and K. Thomas, eds.). ASME, New York. Rankin, C. C., and Nour Omid, B. (1988). The use of projectors to improve finite element performance. Comput. & Structures 10( 1-2), 257-267. Rankin, C. C., Brogan, F. A,, and Riks, E. (1993). Some computational toolsfor the analysis of through cracks in stifened fuselage shells. Computational Mechanics, Springer Inetmational, 13, Nr. 3, December 1993, pp. 143-156. Rankin, C. C., Riks, E., Stames, J. H., and Waters, W. A,, Jr. (1994). An experimental and numerical verification of the postbuckling behavior of a composite cylinder in compression. Euromed 3 17, LIVERPOOL, ENGLAND. (publication pending) Reissner, E. (1945). The effects of transverse shcar deformation on the bending of elastic plates. J. Appl. Mech. 12, 69-77. Rheinboldt, W. C. (1977). Numerical continuation methods for finite element applications. In: Formulations and computational algorithms injnite element analysis (K. J. Bathe, J. T. Oden, and W. Wunderlich, eds.). MIT Press, Cambridge, MA, pp. 599-631. Rheinboldt, W. C. (1981). Numerical analysis of continuation methods for nonlinear structural problems. Comput. & Strucrures 13, 103-1 13. Rheinboldt, W. C. (1982). The computation of critical boundaries on equilibrium manifolds. SIAM J. Numer. Anal. 19(3). Riks, E. (1970). On the numerical solution of snapping problems in the theory of elastic stability, SUDAAR No. 401. Stanford University, Stanford, CA. Riks, E. (1972). The application of Newton’s method to the problem of elastic stability. J. Appl. Mech. 39. Riks, E. (1973). The incremental solution of some basic problems in elastic stability. NLR Rep. TR 74005 U. National Aerospace Laboratory.


75

Riks, E. (1978). A unified method for the computation of critical equilibrium states of nonlinear elastic systems. Acta Tech. Actid. Sci. Hung. 87(1-2), 121-141 (also as NLR MP 77041 U, National Aerospace Laboratory, The Netherlands, 1978. Riks, E. ( I 984a). Some computational aspects of the stability analysis of nonlinear structures, Comput. Methods Appl. Mech. Engrg. 47, 2 19-259. Riks, E. (1984b). Bifurcation and stability, a numerical approach. NLR MP 84078 U, National Aerospace Laboratory, The Netherlands, 1984. (Also In: Innovative methods f o r nonlinear problems (W. K. Liu, T. Belytschko, K. C. Park, eds.). Pineridge Press, Swansea). Riks, E., and Rankin, C. C. (1997). Computer simulation of the buckling behavior of thin shells under quasi static loads. Arch. Coniput. Methods Engrg. In press. Riks, E., Rankin, C. C., and Brogan, F. A. (1990). Aspects of the stability analysis of shells, International Conference of Computational Engineering Science, April 10-14, 1988, Atlanta, GA. In: Static and dynamic stabiliv of shells (W. B. KrXtzig and E. Ofiate, eds.). Springer Series in Computational Mechanics, Springer-Verlag, Heidelberg. Riks, E., Rankin, C. C., and Brogan, F. A. (1992). The numerical simulation of the collapse process of axially compressed cylindrical shells with measured imperfections. Report LR-705. Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands. Riks, E., Rankin, C. C., and Brogan, F. A. (1996). On the solution of modejump phenomena in thin walled shell structures. Internat. J. Comput. Methods Appl. Mech. Engrg. 136. Sansour, C., and Bufler, H. (1992). An exact finite rotation shell theory, its mized variational formulation and its finite element implementation. Int. J. Num. Methods Engrg. 34, 73-1 15. Schaeffer, D., and Golubitsky, M. (1979). Boundary conditions and mode jumping in the buckling of a rectangular plate. Comm. Math. Phys. 69, 209-236. Seydel, R. ( 1 989). From equilibrium to chaos: Practical stabiliq analysis. Elsevier, New York, Amsterdam, London. Simo, J. C., and Fox, D. D. (1989a). On a stress resultant geometrically exact shell model. Part I. Formulation and optimal parametrization. Cornput. Methods Appl. Mech. Engrg. 72, 267-304. Simo, J . C., and Fox, D. D. (1989b). On a stress resultant geometrically exact shell model. Part 11. The linear theory; computational aspects. Comput. Methods Appl. Mech. Engrg. 73, 53-92. Simo, J. C., Fox, D. D., and Rifai, M. S. (1989a). On a stress resultant geometrically exact shell model. Part 111. Computational aspects of the nonlinear theory. Comput. Methods. Appl. Mech. Engrg. 79,21-70. Simo, J. C., Fox, D. D., and Rifai, M. S. (1989b). On a stress resultant geometrically exact shell model. Part VI. Variable thickness shells with through-the-thickness stretching. Comput. Meth0d.s. Appl. Mech. Engrg. 79. 21-70. Stein, M. (1959a). Loads and deformations of buckled rectangular plates. NASA Tech. Rep. R-40. National Aeronautics and Space Administration. Stein, M. (1959b). The phenomenon of change of buckling patterns in elastic structures. NASA Tech. Rep. R-39. National Aeronautics and Space Administration. Suchy. H., Troger, H., and Weiss, R. (1985). A numerical study of mode jumping of rectangular plates. Z. Angew Math. Mech. 65(2), 71-78. Supple, W. J. (1970). Changes of waveform of plates in the post-buckling range. Internat. J. Solids Structures 1243-1258. Thompson, J. M. T., and Hunt, G . W. (1973). A general theory ofelustic stability. Wiley, New York. Thurston, G. A,, Brogan, F. A,, and Stehlin, P. (1985). Postbuckling analysis using a general purpose code. AIAA Pap. 85-0719-CP. Presented at the AIAA/ASME/AHS 26th Structures, Structural Dynamics, and Materials Conference, Orlando, FL, April 15-17. Vandepitte, D., and Riks, E. (1992). Nonlinear analysis of Ariane 5 engine frame. Report LR 678. Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands.

76

Eduard Riks

van Keulen, F. (1993). A geometrical nonlinear curved shell element with constant stress resultants. Comput. Methods Appl. Mech. Engrg. 106, 3 15-352. Wriggers, P., Wagner, W., and Miehe, C. (1988). A quadratically convergent procedure for the calculation of stability points in finite analysis, Comput. Methods A&. Mech. Engrg. 70, 329-347. Young, R. D., and Rankin, C. C. (1996). Modeling and nonlinear analysis of large scale launch vehicle under combined thermal and mechanical loads. AIAA Pup. 96-1554-CP. 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April 15-17, Salt Lake City, UT. Collection of Technical Papers.


Computational Mechanics for Metal Deformation Processes Using Polycrystal Plasticity PAUL R . DAWSON and ESTEBAN B . MARIN* Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca. New York

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

I1. Orientations and Orientation Distributions . . . . . . . . . . . . . . . . . . . . A . Crystal Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Crystal Symmetries and Fundamental Regions . . . . . . . . . . . . . . . C . Crystal Orientation Distributions . . . . . . . . . . . . . . . . . . . . . . .

81 81 84 86

.

I11. Evolution of Texture and Strength . . . . . . . . . . . . . . . . . . . . . . . . . A . Single-Crystal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Orientation Distribution Conservation Equations . . . . . . . . . . . . . . . C. Evolving the Orientation Distribution by Using the Finite-Element Method D. Evolving Discrete Orientation Representations . . . . . . . . . . . . . . . . E . Texture Evolution during Homogeneous Deformations . . . . . . . . . . .

88 89 93 94 96 96

1V. Field Equations for Deformation . . . . . . . . . . . . . . . . . . . . . . . . . A . Macroscopic Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Single-Crystal Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . C . Viscoplastic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . D . Microscopic and Macroscopic Scales . . . . . . . . . . . . . . . . . . . . . E . Texture Comparisons for Several Partitioning Rules . . . . . . . . . . . . .

99 100 102 105

106 112

V . Computing the Deformation by Using the Finite-Element Method . . . . . A . Elastoviscoplastic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . B . Viscoplastic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 113

VI . Application to Forming Processes . . . . . . . . . . . . . . . . . . . . . . . . . A . Sheet Formability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Rolling of Polycrystalline Metals . . . . . . . . . . . . . . . . . . . . . . .

121 121 126

114 118

*Present address: Beam Technologies. Inc., Ithaca. NY

I1 ISBN 0-12-(X)2034-3

ADVANCES I N APPLIED MECHANICS. VOL. 34 Copyright 0 199R by Acadernrc Press. All rights of reproduction in any form reserved . 0065-2165198$25.00

Paul R. Dawson and Esteban B. Murin

78

VII. Studies of Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Aggregate Size Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Hybrid Element Polycrystal Simulations . . . . . . . . . . . . . . . . .

133 134 . . 140

VIII. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

IX. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. GreekSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 160 161

X. Appendix: Matrix Representations . . . . . . . . . . . . . . . . . . . . . . . . .

161

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

I. Introduction The mechanical properties of a polycrystalline metal depend on many attributes of its microstructure. A metal’s strength and formability are intimately connected to its crystalline structure, especially in terms of the slip systems that operate during plastic flow and the orientations of crystals that comprise the metal. Polycrystal plasticity theory embodies both of these features of the microstructure in a constitutive framework that can be incorporated in numerical formulations for analyzing deformation processes. In polycrystal plasticity, the actual movement of dislocations that results in the relative shift of crystallographic planes is idealized by the activity of a discrete number of slip systems. Each slip system contributes to one shearing mode of a crystal’s deformation. Various types of crystals exhibit different combinations of slip system geometry and resistance to slip, allowing for discrimination among metals and regimes in their behaviors. Polycrystal plasticity also provides a representation of the distribution of the crystal lattice orientations, referred to as the crystallographic texture. The slip systems and the crystallographic texture impart directionality to the microstructure. Mechanical properties at the macroscopic level, obtained from averaged microscopic responses, reflect this directionality in their anisotropy. A principal motivation for incorporating polycrystal plasticity in simulations of metal deformations is to benefit from the explicit link between anisotropic flow and its origins in the metal’s crystallographic structure. Asaro [ 3 ] reviewed experimental observations dating back to the turn of the century that document the nature of crystallographic slip in single crystals, as well as the theoretical developments that serve as the basis for modem polycrystal plasticity theory. The present paper summarizes developments in the effort to utilize this theory as the constitutive representation for

Mechanics of Metal Deformation Processes

79

large strain deformation of polycrystal solids, in essence addressing one of the outstanding challenges identified by Asaro. The use of polycrystal plasticity theory in deformation process simulation may be viewed as an example of applying state variable models for the constitutive description of the material. The texture, together with the strengths of the slip systems, acts as a characterization of the material state. The full constitutive theory also includes equations of the kinetics at fixed state, from which the relation between stress and deformation is obtained. Evolution of the state is accomplished with equations that prescribe the reorientation of crystal lattices and the changes in slip system strength induced by deformation. A link must be established between the macroscopic properties and the crystals that define the microstructure. It is the set of assumptions used in making this link that introduces many of the complexities of this approach, but provides much of the richness of behaviors as well. Applications of polycrystal plasticity with finite element formulations fall into two broad categories. One category includes those simulations that have a single orientation within each finite element. Elements represent crystals or portions of crystals. The second category includes simulations in which every element contains an approximation to the crystal orientation distribution. The orientations in each element may number as few as only tens of crystals, or may be in the thousands. The distinction between these categories principally consists of whether it is necessary to invoke an assumption for partitioning the deformation, given by the finite-element trial functions, among a population of orientations. For the first category, no such assumption is required because the elements are discretizing the volume within a single crystal. For the second category the elements each discretize volumes that contain large numbers of crystals. In this article we focus on the second category of formulations, with the intended application being macroscopic forming processes. However, we also discuss analyses that utilize a single orientation per element to illuminate how to better partition the macroscopic deformation among the crystals associated with that deformation. The earliest work combining finite elements and polycrystal plasticity fits within the first category. Gotoh [44] reported on the simulation of 125 facecentered cubic (FCC) crystals in a planar deformation as early as 1978. Needleman et d . [96] implemented rate-dependent single-crystal behavior in an elastoplastic formulation and studied the localization of thin strips under planar slip. Other examples include the studies reported by Harren and Asaro [46], Becker [ 161, Kaladindi et al. [64], Teodosiu et al. [127], Beaudoin et al. [14], Young et al. [ 1421, and Beaudoin et al. [ 151, all of whom have examined the issue of deformation heterogeneity over polycrystals with various implementations of crystal equa-

80

Paul R. Dawson and Esteban B. Marin

tions in finite-elementformulations. Havelicek et al. [52],Yao and Wagoner [ 1361, and Miller and Dawson [90] investigated issues of slip system activity. An important consideration in these implementations is the integration of the elastoplastic constitutive equations over an increment of deformation. Several methods have been reported, including approaches suggested by Peirce et al. [ 1131, Cuitiiio and Ortiz [32], Maniatty et al. [81], Rashid and Nemat-Nasser [115], Kalidindi et al. [65], and Steinmann and Stein [124]. These differ in the configurations in which the equations are written and the algorithms for performing the integrations. Simulations in the second category, ones that incorporate a full orientation distribution within every element, are more limited. Mathur and co-workers [85-871 reported on the use of Taylor and relaxed constraints assumptions in Eulerian simulations of rolling and drawing of FCC metals. Van Bael et al. [133] also examined rolling, but of body-centered cubic (BCC) metals, using a series expansion of the Taylor factors to compute the anisotropic stiffness. Chaste1 et al. [28] used a lower bound model to predict mineral behavior in simulations of the Earth’s mantle; Kalidindi et al. [65] simulated forging processes with a Taylor assumption; Beaudoin et al. [ 13, 371 examined three-dimensional sheet-forming applications. This brief list is not comprehensive, but rather indicates that applications to date have focused on two main areas: processes involving large strain deformations and deformations of materials that inherit texture from a prior process. Both of these applications require the capability to address heterogeneous deformations, provided by the finite-element method, and the description of anisotropic behavior, given by polycrystal plasticity theory. In this article we present a methodology for utilizing polycrystal plasticity as the constitutive description in simulations of large-strain metal-forming processes. We begin with a treatment of the description of crystal orientations and orientation distributions. We take this as a starting point because of the central role that texture plays in the quantification of state. It is most often either the feature we wish to predict or a feature that strongly influences the outcome of a process. We include a discussion of alternative formulations for the computation of texture evolution that stem directly from the choice of the representation of texture. This is followed by a summary of the field equations for the motion of a polycrystalline metal body. We present first an elastoviscoplastic formulation based on a polycrystal plasticity theory, followed by the assumptions used in arriving at a viscoplastic approximation. At this point it is instructive to detail some of the macro-to-micro linking assumptions that are possible. We next summarize the finite-element formulations for both the elastoviscoplastic and viscoplastic formulations. Here we deal with the computation of material stiffness for each of the linking assumptions.


81

In the second half of the article, several examples of these numerical formulations for forming applications are presented. Both sheet forming (stretching) and bulk forming (rolling) are discussed. The intent is to illustrate the preceding theory and numerical implementation with problems that bring to light some of the important issues that arise. We concentrate on simulations of metals of hexagonal close-packed (HCP) crystal structure because the high degree of single crystal anisotropy present in these metals forces us to confront those issues. Simulations of similar applications for FCC or BCC metals are cited. Finally, we present the results of two studies on the behavior of polycrystals and discuss the impact these have on the choices we must make in applying polycrystal plasticity as the constitutive description.

11. Orientations and Orientation Distributions A. CRYSTAL ORIENTATIONS Central to the mathematical quantification of crystallographic texture are the orientations of individual crystals. An unstressed crystal lattice always retains the same internal arrangement of crystal planes and directions regardless of the level of plastic strain. A crystal's orientation is simply that rotation needed to align a set of axes fixed to its lattice with a reference frame (or, inversely, to align the reference frame with the axes fixed to the lattice). The orientations of crystal lattices are discussed in terms of these rotations and the different parameterizations of such a rotation. The rotation tensor is a second-order quantity with nine components, but only three independent parameters. Most frequently in crystallographic computations, Euler angles are chosen to specify these parameters. By using Euler angles, the rotation is decomposed into a sequence of rotations, each about a prescribed axis'

There is not a unique set of axes el, e2, and e3 that permits this decomposition, and over the years many have been employed. The values of parameters @',$*, and @3 depend on the choice of rotation axes [21, 137, 139,721. While Euler angle parameterizations have been widely used, they possess a number of notable shortcomings. These are of two types: the difficulty in visualizing a crystal's orientation given the angles, and the poor quality of mathematical properties of the orientation space these parameters define [ 17,41,97,98,53,48]. ' A complete list of variables appears at the end of the article.

82


For most purposes in the crystallographic analyses, any of the Euler angle parameterizations serve adequately. As discussed in later sections of the paper, the fact that the metric of the orientation space may be singular makes this representation of crystal orientations unsuitable for certain techniques used in the simulation of texture evolution. For this reason we utilize an alternative method for representing orientations based on the invariants of rotation. The natural invariants of a rotation are its axis, n, and the angle of rotation about that axis, 4. The rotation necessary to define an orientation then can be written as

R($m) = n @n

+ cos@(I

-

n @ n)

+ sin4(1 x n).

(2)

This representation has been examined for describing orientations of individual crystals, as well as for the misorientation between two crystals [53]. The angle/ axis parameterization of crystal orientations may be viewed as the projection of a four-dimensional quaternion parameterization onto a three-dimensional volume given by the angle/axis representation of the general form r = r(4, n), where r is a vector in orientation space. This connection has been discussed elsewhere [77,96,75]; here we proceed directly to properties of the angle/axis spaces. In general, orientation spaces exhibit measures of length, area, and volume that are distorted owing to the non-Euclidean character of the space. Of particular concern is the distortion of the space as reflected through variations in the metric tensor determinant, as it scales the differential volume in all integrations performed over regions of orientation space. However, there are additional consequences of a space possessing a non-Euclidean metric that bear on the evolution of texture [94, 38, 751. In particular, calculation of the gradients of the reorientation velocity appears in the orientation distribution evolution equations, and these gradients depend not only on the physical attributes of plasticity, but also on the properties of the space in which the velocity is computed. A number of useful neo-Eulerian parameterizations have been proposed [41] in which the axis vector is scaled by a function of the angle of rotation (Figure 1). These may generally be written as

The function f(4)that scales n influences the mathematical properties of the orientation space, and is specified with certain properties in mind. To avoid a singularity at the origin of the space, two restrictions are imposed: lim

@+O

f(4)= 0

and

lim

@+O

f’(4)> 0,

(4)

83


X

Z

Z

Z

Z

Z

Z

Y

X

Y

FIG.1 . Anglehxis representation of orientations. Rotated cubes show orientations associated with corresponding r (qualitatively). Left column shows increasing rotation about [ n ]= [ 1 0 01. Right column shows rotation about [ n ]= [ I 2 21.

where f ' denotes the derivative of f (4) with respect to 4. Another criterion considered in the selection of f ( 4 )is the metric properties it endows the space. The metric, g, for the orientation space is associated with the measure of the distance between two rotations [94],

(ds')* = gij dr' d r j .

(5)

From this, the volumetric scaling can be written as

dv' = J d e t g d r ' d r 2 d r 3 . We employ a function f ( 4 )that defines a space having variable metric, but with the attractive feature that crystal symmetries will provide bounding surfaces that


84

are planes. This particular parameterization is the Rodrigues space [41] defined by

and having

and

where I". are the connexion coefficients, as needed for differentiation of quanti!k ties in orientation space.

B. CRYSTAL SYMMETRIES A N D FUNDAMENTAL REGIONS Crystals exhibit sets of orientations that are indistinguishable, and which define the symmetries that the crystals possess. With respect to the plastic deformation of crystals, equivalent orientations must exhibit both the identical geometric arrangement of atoms and identical strengths in the associated deformation modes (slip systems). This latter criterion places restrictions on the slip system strain hardening if orientational equivalency is to be preserved throughout a deformation. Granting that these restrictions are observed later when the mechanical response is considered, here we focus only on the geometric aspects of the symmetry. If we take H(@) to be a symmetry rotation, then an orientation equivalent to R can be written as

The existence of symmetrically equivalent orientations implies that there are subsets of orientation space that are adequate to define arbitrary crystal orientations. The smallest subset required to uniquely specify any possible orientation is referred to as a fundamental region. A fundamental region is determined by selecting one and only one orientation among the symmetrically equivalent orientations. For the angle/axis representation [53], a logical choice is to pick the orientation that lies closest to the origin of the parameter space (minimum I$). Considering


85

values of 4 and n associated with the symmetry operations delivers a set of surfaces given by

-

f r n 5 tan(4/4)

(1 1)

that are bounding planes for the fundamental region. This applies to every (4, n) for which the crystal symmetry exists, giving bounding planes that number in correspondence with the number of symmetries. The fundamental region R* is defined by the inner envelope of all bounding planes. In some instances, this means that planes generated by certain symmetries are inert, as they lie entirely outside the volume defined by planes generated by other symmetries. Such is the case for FCC crystals, where the fundamental region is depicted in Figure 2. Here only planes originating from the symmetries about the families of (100) and (111) crystal axes appear as facets of the fundamental region [41, 39, 511. The (110) axis symmetries lie at too great a distance from the origin to contribute to the definition of the fundamental region. For HCP crystal geometries, fewer symmetries exist than for FCC crystals. There are symmetries about the (0001) crystal axis (c-axis) as well as about the (1000) family of axes [77]. This gives a prismatic volume with 12 lateral facets, as shown in Figure 3. The bounding planes that form the surface of the fundamental region may be grouped into pairs, with the planes of each pair being equivalent under a symmetry operation. One plane of each pair is assigned to the portion and the other plane to the portion aR:, with U 872; = aR*.

an;,

A

F I G .2. Fundamental region for FCC crystals using Rodrigues parameterization.

86


X

FIG.3 , Fundamental region for HCP crystals using Rodrigues parameterization.

C. CRYSTAL ORIENTATION DISTRIBUTIONS A specimen of a polycrystalline metal that is large enough to be representative of the metal typically would contain thousands of crystals. The crystals in such a specimen would have a variety of orientations, reflecting the crystallographic texture of the metal sampled. An independent specimen taken from the same metal (same in the sense that it exhibits the same texture) would contain different crystals, each having its own lattice orientation. Although the two specimens would have different populations of crystals, they would be equivalent in a statistical sense if they were sufficiently large. In an ideal sense, the population of crystal orientations in any particular specimen might be thought of as having been selected from an orientation distribution. In simulations of textured materials, it is possible to work directly with the orientation distribution or to construct a sample of crystals from it that is faithful to its statistical features. We frequently refer to orientations as crystals, using the two terms interchangeably in much of the discussion of the mechanical properties derived from polycrystal plasticity theory. In this paper we discuss examples of both approaches, starting first with the orientation distribution.


87

Formally, the orientation distribution is a probability density, A ( r ) , describing the distribution of crystal orientations. Because A ( r ) is known, the volume fraction of crystals in a portion of orientation space, 6R*, can be evaluated [59, 1381 as

Here the texture specified by A ( r ) applies to one instant (fixed state); over a deformation, A ( r ) evolves in time. For textured metals, A ( r ) is often quite complex, and various techniques are used to approximate it. Series expansions based on spherical harmonics [67, 138,117] have been used extensively in quantitative texture analysis for the construction of A ( r ) from experimental data. More recently, expansions formed with tensorial basis functions [IOO, 110, 144, 1191 have been proposed. Here we approximate A ( r )with piecewise interpolation used in finiteelement formulations [76,75], defined over R*,

A h ( r )= N f ( r ) A j ,

(13)

where Ah (r) is the discretized approximation to A ( r ) ,Nj"( r )represents the interpolation functions, and A j represents the nodal point parameters (summation over j implied). Local interpolation offers the same advantages in representing A ( r ) as in the representation of field variables throughout finite-element applications: employing finite-element trial functions for the approximation of A ( r )makes accurate characterization of A ( r ) possible, even when it exhibits sharp gradients. We point out only that in contrast to using a discrete sample (discussed next), representing the orientation distribution itself retains information concerning the density of crystals at all points in orientation space. Representing the texture with a discrete number of orientations has been widely used in texture analyses, especially in modeling texture evolution (see, e.g., [66,22,23]). In some respects, it is most similar to a physical specimen that has a finite number of crystals. There are a number of ways in which the use of discrete sets of orientations expedites the simulation of the deformation of polycrystalline solids. Averaging to obtain the mechanical properties may be performed by simple summations. Evolution of texture may be computed by independently rotating each orientation in the discrete sample. The many independent computations that result from having a collection of individual orientations can be exploited with parallel computing architectures to allow for the use of large numbers of orientations at many points in a body [ 12, 131. Using a set of orientations as a means of approximating the orientation distribution is equivalent to partitioning orientation space into subregions equal in number

88


to the number of orientations. According to the magnitude of the probability density, we require that IJ f in eq. (12) be the reciprocal of the total number of crystals. This forces the volume of SR*to be small in regions where A(r) is high, and to be large in regions where A(r) is low. Thus many crystals are clustered together in an orientation space where the value of A(r) is high, as their respective volumes, SR*, are small. In regions characterized by low A(r), crystals are sparse because the corresponding volumes are large. Alternatively,it is possible to assign weights to orientations while keeping their corresponding volumes constant [73]. The weights themselves approximate the value of I J in ~ a region about the orientation of the crystal. This may reduce the number of crystals needed to accurately evaluate an orientational average, but can at the same time reduce the ability of the discrete orientation set to represent textures with many weak features (in contrast to a texture with a smaller number of strong components).

111. Evolution of Texture and Strength The inelastic response of a metal to an applied loading may involve one or more deformation mechanisms [43,5], such as crystallographic slip, diffusional creep, and twinning. The dominant mechanism at any instant depends on the regime of temperature and strain rate (or stress), as well as the microstructural condition of the material. Here we restrict our attention to conditions of moderate strain rates to lo3 s-'), cold to warm forming conditions (0.3 5 T / T m 5 0.8, where T is the absolute temperature and T, is the temperature at melt), and moderate grain sizes (lo@ 5 i 3 5 lop3 m, where L 3 is a characteristic grain dimension). In this regime, the inelastic strains are dominated by crystallographic slip [43, 711. Important for slip within a crystal are the orientations of its active slip systems relative to the applied stress, as well as the strength of these slip systems. The slip process itself alters the slip system strength through the accumulation of dislocations that act as an impediment to continued slip. The reorientation of the lattice of a deforming crystal, and with it the slip systems, is an integral part of the slip deformation mechanism. Lattice reorientation is a direct consequence of there being only a discrete number of deformation modes in a crystal having a limited number of slip systems. This immediately implies that the orientation distribution for a material is altered by the deformation, usually driving the material toward a highly textured and mechanically anisotropic condition. In this section we focus on the evolution of the crystallographic texture and the slip system strength. These are taken to be the relevant descriptors of the material state, which must be updated with deformation according to prescribed constitu-


89

tive equations. We begin by summarizing those equations and follow with alternative approaches for integrating them in correspondence with particular texture representations (as discussed in Section 1I.II.C).

A. SINGLE-CRYSTAL EQUATIONS Even though the full mechanical response includes both elastic and inelastic contributions, the evolution of crystallographic texture is a consequence of the inelastic deformations. Treatment of the combined elastoplastic response is delayed until Section 1V.B. For now, we note that assuming the elastic strains are small implies that the current and unloaded configurations are nearly identical. Consequently, for the sake of simplicity in discussing texture evolution, we do not discriminate between these configurations in this section. Rather, we proceed by considering only the viscoplastic response, whether or not any elastic deformations ultimately are considered. The slip-system orientations are fully described by the normal to the slip plane, ma, and the direction of slip on the plane, b", where a indicates a specific slip system [54, 137,3]. The Schmid tensor is constructed from these as Ta = b" @ ma. For the inelastic portion of the deformation alone, the crystal velocity gradient, L", is written as

L" = LP + Q*,

(14)

where

The slip-system shearing rates are denoted by Pa,and Q* is the lattice spin that brings Lc and LP into coincidence. L" is assumed for now to be prescribed externally for all crystal orientations, but does not have to be identical for all orientations. Discussion of the various relationships between Lc and the macroscopic average of Lc is deferred until Section 1V.D. As given by eq. (14), the lattice spin is the difference between the crystal and plastic velocity gradients, and is required to determine the rate at which the crystal lattice reorients. However, i2* cannot be computed directly by using eqs. (14) and (15) unless the slip-system shear rates, Ya,are known. To determine the slip-system shear rates, first the deviatoric crystal stress, t'",is determined by using a combination of several relations. The geometry of the slip systems is used to define the resolved shear stress, t",

90


A constitutive relationship relating slip [58, 102, 113,241

tffand

FLYis introduced from the hnetics of

where 1;" and m are material parameters and 2" is the strength of the slip system. Finally, the crystal deviatoric deformation rate, D", is a linear combination of the strain rates on the slip systems

where P" is the symmetric portion of T a .These equations, when combined, give a relationship between the (assumed) known crystal deformation rate and the crystal stress:

where SP is the crystal viscoplastic stiffness [ 5 8 , 8 5 ] :

and MP is the crystal viscoplastic compliance. Equation (19) is a nonlinear equation for the crystal stress in terms of a prescribed deformation rate and crystal state (orientation and strength), and is solved by using a Newton-Raphson procedure that includes a line search method [84,85]. Because the components of the crystal stress are known, the values for 1;" are determined from eqs. (16) and (17). The lattice spin is then available for computing the evolution of texture. The second aspect of the state that is essential for modeling plastic flow by slip is the strength of the slip systems. In eq. (17) the strength of each slip system was introduced as 2". Physical arguments regarding the mechanisms of strengthening (increases in 2") from the interactions of dislocations support the possibility that each slip system has a distinct 2* once plastic deformations have elevated the dislocation density [69, 31. From practical limitations of measuring distinct strength values in a polycrystal (for the purpose of initializing the state or verifying the model predictions), we restrict the slip-system strengths to being equal on all families of slip systems within a crystal. In fact, for crystals that have only one family of slip systems assuming a single 2 for all slip systems of all crystals within a complete aggregate makes little difference to the computed stress-strain

91


response [90]. The evolution equation is assumed to be of a generalized Voce form [135,68]:

where

ts

= 4, Ys o (")"I

and O,, to,t s 0Ysc1, , and rn' are material parameters. Equation (21) is integrated by a simple Euler method, which is adequate, given that the strain increment remains sufficiently small. Throughout this article we concentrate on HCP titanium to illustrate the application of the methods presented to polycrystalline metals. The predominant slipsystems, shown in Figure 4, are the basal systems, prismatic systems, and one family of pyramidal systems [101, 56, 131, 1381. Slip-system normals and slip directions are indicated in the conventional indices for HCP systems. Typically, the pyramidal slip systems are stronger than either the basal or prismatic systems, and in this paper we consider cases of initial strength ratios with factors of 5 and 10 [ 1371. In some cases, the pyramidal strength is assumed to be sufficiently high to disable the systems entirely. The slip-system hardening parameters in eq. (21) have been estimated from experimental data [7], for the w x m compression of commercial purity titanium at three strain rates. The data and model curves are shown in Figure 5. Only the basal and prismatic slip systems were activated in determining the model parameters given in Tables 1 and 2. C

C

a2

Pyramidal m = (1071) b =

C

a2

Basal rn = {OOOl) b =

a2

Prismatic m = {ioio) b =

F I G . 4. Pyramidal, basal, and prismatic slip systems used in titanium simulations.

92


CP Titanium Grade 3 Temperature = 675 "C A

} Experimentaldata

Effective strain (E) FIG.5 . Compression data and model computation for commercial purity titanium at 675OC. Model parameters determined using constrained hybrid assumption.

TABLE1 ELASTICITY PARAMETERS (HCP)

44.0 GPa

114.5 GPa

TABLE 2 VISCOPLASTICITY PARAMETERS (HCP)

0.06

1.0 s-'

370 MPa

31.3 MPa

75.8 MPa

0.005

5 x 10" s-'


93

B. ORIENTATION DISTRIBUTION CONSERVATION EQUATIONS The rotation of individual crystals during deformation modifies the orientation distribution of crystals that comprise the material. The probability density, A (r), introduced in Section II.II.C, is written for some particular fixed-state condition, while recognizing that deformation alters the state. Now we allow A to depend on time, t , in addition to the orientation, r, and express the texturing of the metal in terms of the evolution of A . The integral of the probability density over a fundamental region must be unity to guarantee that all crystal orientations do reside within the fundamental region [3I, 761,

1,.

A(r, t ) dv' = 1.

This constraint must be true at all times, and in rate form [31, 116, 110, 33, 1341 becomes at

')

+ V(A(r, t)v(r, t ) ) = 0,

where v(r, t ) is the reorientation velocity of a crystal located at position r at time t . The relationship of the reorientation velocity to the crystal spin depends on the particular parameterization of orientation. For an angle/axis representation [75] as given by eq. (3), we have for v,

v = fn

+ f'in,

(26)

where $=n.w, n=2

(27)

1 sin $ nxwxn+-wxn 2 (1 - cos $)

and

w = vect(Q*).

(29)

A prescribed crystal velocity gradient enters the evolution of the orientation distribution through the dependence of the reorientation velocity v(r, t ) on the lattice spin Q*. The method employed to integrate the evolution equation for the orientation distribution depends on the choice made for its representation. Here we present two approaches stemming from the finite-element discretization of the probability density and the discrete set of orientations.


94

c. EVOLVINGTHE ORIENTATION DISTRIBUTION BY USING THE FINITE-ELEMENT METHOD

A direct approach to formulating the finite element discretization of the probability density conservation equation (eq. (25)) [29] is first to form a residual over the fundamental region as

-+v*VA+AV.V

+

where is a weighting function. Solving this system is made difficult by the presence of the convective term and by the tendency for extremely strong peaks to develop. The difficulty introduced by the convective term is common to many physical systems that are governed by hyperbolic conservation laws. In eq. (30), the divergence of the reorientation velocity (V v) may approach a jump discontinuity in the limit of vanishing rate sensitivity in the slip system kinetic expression (eq. (17)). Although the rate sensitivity is always nonzero, when it is small there are sharp gradients in the divergence of the reorientation velocity, and numerical oscillations in the computed values of A can easily arise. Artificial diffusion and shock-capturing techniques [57,61,62] have been developed that enable accurate solutions for systems with these characteristics, and have been employed for integrating eq. (30) [76]. Instead of attempting to integrate the evolution equation in its Eulerian form, it may be cast first in a Lagrangian form. This has the advantage of eliminating the problematic convective term, but introduces the complexity that the original fundamental region itself convects, and becomes highly distorted if the texturing is pronounced (which is usually the case). Compromises between the Eulerian and Lagrangian approaches exist; here we employ an updated Lagrangian system in which the current reference frame convects with the crystal trajectories. This may be formulated with an incremental approach [63, 47, 771 that permits the direct computation of the Eulerian probability density as

-

/

R* ( A ( A - A ) + A V . v ) + d v r

-/,*V.(tVA)11.dvr

=0,

(31)

where 2 is the value of A projected upstream along the characteristic over the step At , as given by

A = A(r - v a t , t - Ar).

(32)

The second integral of eq. (3 1) is an artificial diffusion modification in which the diffusivity E is based on the element size, h, and local residual, R A , E

=c,h"R~;

R A = /(A - A)/At

+ AV * v I .

(33)


95

This updated Lagrangian formulation avoids the problem of a highly distorted reference region (and finite-element meshes) by employing the incremental convection scheme built into the computation of ( A - i ) / A t . We proceed to develop the finite-element discretization of eq. (3 1) by introducing piecewise interpolants for the trial functions and weight functions. As given in Section ILC, the approximation to A by finite-element discretization is

A h(r,t ) = N f ( r ) A j( t ) ,

(34)

and, employing a Bubnov-Galerkin approach,

W-,f)

= Nf(r)llrj(t).

(35)

In the computations presented in this paper, four node tetrahedral elements are prescribed. Requiring that the residual, J A , vanish gives a matrix equation for nodal values of A ,

[ K A ] { A l= { F A ) , where elemental contributions to [ K A ]and ( F A ) respectively, , are

and

where

gives the covariant derivative of the reorientation velocity field. The symmetry relations for the boundaries of the fundamental region must be preserved [41,94,75] as the texture evolves. This may be stated as

A @ , t ) = A ( R ( r ) ,t )

(40)

where R(r)specifies the symmetry operation for the crystal on aR:. This condition may be explicitly enforced by equating the nodal values of A to the values at the corresponding symmetrical points of the boundary, or by enforcing eq. (40) in a weak sense over the boundary [ A ( r ,t ) - A ( R ( r ) ,t)]~,b~da' = 0.

(41)

96


The explicit approach given by eq. (40) necessitates constraints on the construction and numbering of the finite-element mesh. These constraints are not necessary with eq. (41), but are removed at the expense of introducing a constraint condition in the matrix equation for A via either a penalty or Lagrange multiplier approach.

D. EVOLVING DISCRETE ORIENTATION REPRESENTATIONS The task of updating a discrete set of crystal orientations is straightforward. The crystal reorientation velocities are simply integrated over time, as given by

In general, all orientations in the set experience lattice rotations during deformation. Throughout the reorientation process, however, each orientation continues to constitute the same volume fraction of the crystal population. It is common for orientations to exit the fundamental region, in which case they may be repositioned at the symmetrically equivalent location within the fundamental region. Actually, it is not necessary to relocate a crystal that is outside the fundamental region unless some operations are performed explicitly over the fundamental region (such as plotting of the texture). Quantities used in determining the averaged responses of all crystals in the set typically are not influenced by whether the crystal is within the fundamental region or at a symmetrically equivalent location outside the fundamental region. Instead of updating the parameterization of the crystal orientation, such as the n and $J values in the angle/axis representation, it is possible to update the rotation R (eq. (1) or (2)) directly, if the lattice spin is known [ 8 5 ] :

R=Q*.R.

(43)

This has the advantage of working directly with the rotation throughout a deformation history, which eliminates the need to compute the rotation from the orientation parameters except at the initial time. However, it increases the number of parameters that must be updated and introduces the need to use an integration scheme that ensures that the lattice axes remain orthogonal [123].

E. TEXTURE EVOLUTION D U R I N G HOMOGENEOUS DEFORMATIONS Two commonly studied deformation modes are compression under uniaxial stress (called uniaxial compression) and plane strain compression (the ideal defor-


97

mation path of a flat rolling process). These two modes of deformation allow large strains, since stability limits are not encountered, and failure mechanisms, such as void growth and coalescence, are suppressed. Strains of approximately unity are often obtained experimentally under deformation conditions that are very close to homogeneous. Even larger deformations can be reached experimentally, as is well known for rolling, but confidence that the deformations always remain close to ideal is not high. Nevertheless, these deformation modes are an extremely important link between theory and experiment. Considering first plane strain compression, the velocity gradient is given by

LP = L o

(

1.0 0.0 0.0 sym 0.0 0.0 -1.0

)

s-1,

(44)

where, for the example shown, Lo is unity. Here we consider only the Taylor assumption (discussed in Section 1V.D) using an initial pyramidal slip system strength five times the basal and prismatic strengths. This gives a strength difference of 125.2 MPa, which is maintained throughout the deformation. The initial orientation distribution corresponds to uniform A (no preferred texture). The deformation continues until an effective strain of unity is reached ( i , f f = 4(2/3)Tr(D D) ). Texture develops monotonically throughout the deformation. When the Taylor assumption is used together with a single slip system hardness, the reorientation velocity field does not depend on the texture itself, but only on the deformation mode and the single crystal properties. Thus the reorientation velocity field does not vary in time for constant velocity gradient. The magnitude of the velocity, shown in Figure 6, has a number of equilibrium points where the velocity vanishes. However, not all of these are stable equilibria, which is required for a texture component to persist. The texture evolution is computed both by using the finite-element formulation for the probability density and by tracking the histories of a discrete set of 1000 orientations. The textures, presented in Figures 7 and 8, respectively, are qualitatively similar and exhibit strong peaks at the stable equilibrium points as expected. The most noticeable difference is in the greater smoothness of the finiteelement distribution, which stems in large measure from the quality of the initial distributions. Each is intended to represent an initially uniform distribution. The finite-element discretization does this exactly; the discrete aggregate does not. It should be noted that the orientations of the discrete aggregate are probably not optimal, so the conclusion is not that the discrete aggregate is incapable of representing a uniform distribution, but rather that it is not simple to choose the

98


0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

-

__

-

FIG.6 . Magnitude of the reorientation velocity field for plane strain compression shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value indicated to the right of the plane. Taylor assumption; pyramidal slip system strength 5 times basal strength.

orientations that represent a uniform distribution well. New methods are currently being tested that sample randomly from a space defined by a homochoric scaling [391 of an angle/axis representation (rather than from Euler angle representation used to generate this sample) and offer greater potential for rendering high-quality samples [89]. Next we consider uniaxial compression. The crystal velocity gradient is imposed as

LC= L"

(

0.5 0.0 0.0 sym 0.5 !)loo)

s-l,

(45)

and again the texture evolution is computed for an L , of unity. The texture components develop at stable equilibrium point locations evident in the reorientation velocity field shown in Figure 9. The finite-element and discrete aggregate methods predict similar textures in terms of the locations and strengths of texture components Figures (10 and 11). These are contrasted later with the predictions made by using other modeling assumptions.

99


A 16.0 14.2 12.4 10.7 8.9 7.1 5.3 3.6 1.8 0.0

FIG. 7. Finite-element orientation distribution A for plane strain compression at an effective strain of unity shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value indicated to the right of the plane. Taylor assumption; pyramidal strength 5 times basal strength. Peak value = 15.5.

IV. Field Equations for Deformation The intended application for the simulation tools described in this article is to predict the motion (deformations) of a polycrystalline solid and the concurrent evolution of its microstructural state under loadings that induce large strains. The previous two sections of the article concentrate on the characterization and evolution of the state in terms of the orientations of constituent crystals and the strengths of slip systems in the crystals. For the evolution of state, the motion experienced by the crystals is assumed to be known. In this section we focus on the systems of equations needed to compute the motion. We make use of equations of equilibrium at the macroscopic level, rules for computing macroscopic averages over the orientation distribution of crystals, and relations for single crystal behaviors, especially assumed kinematics. The methodology for solving the set of equations for the motion with a finite-element discretization is presented in Section V. We proceed initially by allowing for combined elastic and viscoplastic responses; the simplifications arising from neglecting elasticity are presented following the

100


FI G .8. Discrete aggregate orientation distribution A for plane strain compression at an effective strain of unity shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Taylor assumption; pyramidal strength 5 times basal strength. Peak value = 17.7.

more general case. Some applications make use of the purely viscoplastic approximation, whereas others utilize the full elastoviscoplastic development.

A. MACROSCOPIC RELATIONSHIPS We limit the analyses to quasi-static motions and to isothermal conditions, although the latter restriction is easily removed by solving for a changing temperature distribution concurrently with the motion. The field equations of interest for the isothermal, quasi-static case are those of equilibrium and the associated boundary conditions [45],

V.a=O -

in B ~ - a = t on aB, U=U on aB,. Here the body forces are neglected and the Cauchy stress is taken to be symmetric.

101


0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

L 4 FIG.9. Magnitude of the reorientation velocity field for uniaxial compression shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Taylor assumption; pyramidal slip system strength 5 times basal strength.

In addition to the equilibrium requirement placed on the stress distribution at the macroscopic scale, we also require that the stress at any macroscopic position be consistent with the average of the associated crystal stresses, with the distribution of crystals coincident with that macroscopic position

Similarly, the macroscopic motion at any point in the body is the averaged response of the crystals in the orientation distribution. For the deformation gradient, this condition is written as

F = (F') =

IR

or in rate form for the velocity gradient as

*

F'Adv',

(48)

102


A

FIG.10. Finite-element orientation distribution A for uniaxial compression at an effective strain of unity shown on cutting planes through the HCP fundamental region of Figure 3 . Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Taylor assumption; pyramidal strength 5 times basal strength. Peak value = 13.1.

The dependencies of c C , F Cand , Lc on the orientation are discussed in Section 1V.D as part of the task of defining rules that partition the macroscopic deformations to individual crystals.

B.

sINGLE-CRY STAL RELATIONSHIPS

The complete response of a single crystal to an imposed loading includes both elastic and inelastic strains. We assume that the elastic strains are associated with stretching of the crystal lattice and the inelastic strains are associated with crystallographic slip (see Figure 12). The deformation gradient is the combination of these responses, plus a rotation, and can be written in multiplicative form [80,4, 128,82, 181 as FC

= V* .R*.FP = v*.@p.

(50)

103


A

FIG. 1 1 . Discrete aggregate orientation distribution A for uniaxial compression at an effective strain of unity shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Taylor assumption; pyramidal strength 5 times basal strength. Peak value = 32.7.

R' FIG. 12. Configurations of a body undergoing plastic deformation by crystallographic slip, a rotation, and elastic stretching.


104

Fp is all of FC,except the elastic stretching, and defines the relaxed configuration B obtained by elastically unloading a differential volume within a crystal from the current configuration B to a stress-free condition without rotation. Using eq. (.SO), we can write the crystal velocity gradient as =V*.v*-l + v * . L p . v * - ' ,

[email protected]'

(51)

where LP

=FP

.F P - '

R .R*T + R* .F P .FP-'

=

.R*7

= R .R*T +cFb@ lY ma.

(52)

lY

The dyadic Pa @ ma defines the orientation of the crystal slip system at B. Equation (52) can also be written as

8P --( F P \;vp

= (Fp

-

.$-I)

.$-'),

S -

=*

C,,(bff

@ma)s

(53)

ff

.R*T +cl;lY((i;lY @ma)* (54) (Y

where the subscripts S and A denote the symmetric and antisymmetric parts of a second-order tensor. We assume that the elastic strains are always small in comparison to unity, which is reasonable for most metals, since the yield stress is several orders of magnitude smaller than the elastic shear modulus. Using the small strain assumption, we write V* as [lo, 391

v*= I + € * ,

l€*l - unidirectional rolling - 59% reduction ND

ND

RD

m a . int. = 62.66 s = 0.02

0.40

ND

RD

RD

m a . int. = 62.37 s = 0.50

max. int. = 65.92 s = 0.98

1.00 2.51 6.32 15.91 40.00

FIG.24. Near-surface (s = 0.98), midheight (s = 0.50). and near-centerplane (s = 0.02) textures computed using constrained hybrid assumption. Total reduction of 59%in one pass. (0001)equal area projection; scale is multiple of random. s is the normalized distance from the centerplane.

tilt is not large in any of the sets of simulations, but the angle does increase in magnitude in going from several lighter reductions to one heavy reduction. Furthermore, the strength of the surface texture relative to the centerplane texture depends on the rolling schedule. When the full reduction is achieved in a single pass, the surface texture has a maximum intensity that exceeds the centerplane texture; for other cases, the intensity of texture is greater at the centerplane than at the surface. The influence of repeated reversals in the shears near the surface with multiple passes has the effect of producing a less intense texture. Overall, the influence of through-thickness gradients in the deformation rate is to produce a tilt in the c-axes that corresponds to a shift toward a pure shear or simple shear texture. With fewer passes to obtain the same reduction, there is both larger tilt of the c-axes toward the pure shear orientation and an increase in intensity. The textures predicted using the constrained hybrid assumption are extremely strong and, as discussed in Section VII, tend to have the crystal c-axes aligned with the compression (ND) direction to a far greater extent than do other models. For this reason, it is useful to compare the trends concerning texture gradients obtained with the constrained hybrid assumption to those generated using a different partitioning assumption. From such a comparison we can determine whether the predicted trends are peculiar to the specific modeling assumptions. The uniform stress assumption is also capable of handling HCP crystals and is used for comparison with the constrained hybrid assumption.


131

- unidirectional rolling - 36% reduction/pass pass 1 -total reduction 36% ND ND ND

RD

RD

max. int. = 8.02

max. int. = 7.56

max. int. = 8.77

ND

Dass 2 -total reduction 59% ND

ND

RD

max. int. = 62.29 s = 0.02 0.40

max. int. = 53.13 s = 0.50 1.00

2.51

RD

max. int. = 56.35 s = 0.98

6.32 15.91 40.00

FIG. 25. Near-surface (s = 0.98), midheight (s = 0.50),and near-centerplane ( s = 0.02) textures computed using constrained hybrid assumption. Total reduction of 59% in two passes. (0001) equal area projection; scale is multiple of random. s is the normalized distance from the centerplane.

We evolve texture along the streamlines at s = 0.02, 0.5, and 0.98, using the velocity gradient histories from the simulations performed using the constrained hybrid assumption. By using the same velocity gradient histories, we can evaluate the impact of the partitioning rule on the texture gradients for identical sets of velocity fields. For the uniform stress calculations, the pyramidal slip system strength is five times the basal strength, and the other model parameters are given in Table 1. The textures predicted using the uniform stress assumption are considerably weaker than the constrained hybrid textures. However, the trends with respect to the through-thickness texture gradients are quite similar. As is evident from Figure 27, the near-centerplane textures are nearly identical for all of the reduction


132

- unidirectional rolling - 20% reduction/pass pass 1 - total reduction 20%

FIG.26. Near-surface (A= 0.98), midheight (s = 0.50). and near-centerplane (.I. = 0.02) textures computed using constrained hybrid assumption. Total reduction of 59% in four passes. (0001) equal area projection: scale i n multiple of random. s is the normalized distance from the centerplane.

schedules. However, the near-surface textures do show differences. For the case involving the full reduction in a single pass, there is a definite tilting of the c-axes in the near-surface texture, and the greatest texture intensity appears at the surface. For the case involving four reductions of 20% per pass, the tilt of the c-axes is less pronounced, and centerplane and surface texture intensities are comparable. In fact, for this case the variations in texture through the thickness of the workpiece are small, in terms of both the positions and magnitudes of the principal components. Thus although the details of the textures are different for the two partitioning assumptions, the trends they predict regarding the influence of rolling schedules on texture variations are comparable.


133

coo01 > - unidirectional rolling - 20% reductionlpass pass 3 - total reduction 48.8% ND ND ND

RD

RD

pass 4 - total reduction 59%

RD

m a . int. = 61.48 s = 0.02 0.40

m a . int. = 58.67 s = 0.50 1.00 2.51

max. int. = 53.41 s = 0.98

6.32 15.91 40.00

FIG 2 6 . -continued

VII. Studies of Microstructure At several points in the preceding sections of this article, we have noted that assumptions must be made to link the macroscale and microscale motions. That is, we must choose a rationale to partition the macroscopic straining among the crystals of an aggregate. The results presented in Section I11 show the sensitivity of the computed textures to the linking hypothesis. In the example of the rolling of HCP titanium, the strong basal textures are predicted in part because of the characteristics of the assumption employed (constrained hybrid). We must also decide on the level of detail used to represent the orientation distribution, whether this is in terms of the number of elements in the finite-element discretization of orientation space or the number of crystals in a discrete aggregate. In this section of the article we present two studies in which we examine these issues via numerical simulation.

Paul R. Dawson and Esteban B. Murin

134

(a)

cOOOl> - unidirectional rolling - 59% reduction in 1 pass ND

ND

ND

RD

max. int. = 2.31 s=0.02 0.50

max. int. = 4.74 ~=0.98

max. int. = 2.65 ~=0.50 1.08

1.66 2.24

2.82 3.40

(b) - unidirectional rolling - 59% reduction in 2 passes ND

ND

ND

RD

max. int. = 2.36 s=0.02 0.50

max. int. = 2.51 ~=0.50

RD

max. int. = 3.40 ~=0.98

1.08 1.66 2.24 2.82 3.40

FIG.27. Near-surface (s = 0.98), midheight ( s = 0.5). and near-centerplane ( s = 0.02) textures using the uniform stress assumption. a) Total reduction in one pass; b) total reduction in two passes; c) total reduction in [our passes. (0001) equal area projection; scale is multiple of random. s is the normalized distance from the centerplane.

A. AGGREGATE SIZESTUDY In the simulation of metal-forming processes, large numbers of elements are commonly needed to resolve the workpiece deformation accurately. In such cases, it is advantageous to quantify the texture with as few orientations as possible to

Mechanics of Metal Deformation Processes (c)

135

cOOOl> - unidirectional rolling - 59% reduction in 4 passes ND

max. int. = 2.35 s=0.02

ND

ND

max. int. = 2.74 ~=0.50

max. int. = 3.10

~=0.98

0.50 1.08 1.66 2.24 2.82 3.40 FIG.

27. -continued

lessen the computer execution times. Clearly, the smaller the number of crystals in an aggregate, the more difficult it becomes to represent a texture well, so the minimum acceptable size usually is dictated by the complexity of the texture. A measure of the robustness of a representation is the sensitivity of derived mechanical properties to modifications in the representation, such as deleting a few crystals or altering their orientations. For example, two sets of crystals may be selected randomly from the same orientation distribution. If the sample size is sufficiently large, the mechanical properties derived from both are indistinguishable. However, if the sample size is too small, the derived properties differ. Should small samples be assigned to different elements in a body, the properties in the body vary spatially. Consequently, the deformation is inhomogeneous, even if the boundary conditions normally give rise to a homogeneous state of stress. Such variations in deformation ultimately have an impact on the texture evolution. Beaudoin et al. [ 121 simulated the compression of cylindrical aluminum specimens taken from a rolled plate. The specimens were discretized with finite elements and assigned different aggregates of crystals sampled from the same initial orientation distribution for the rolled aluminum. A number of simulations were performed, using aggregates of different sizes, but for any one simulation the same number of crystals was assigned to every element. The influence on texture evolution was substantial. Simulations using large aggregates (- 1000 crystaldaggregate) showed nearly uniform deformation over the entire sample, and texture evolution equivalent to a homogeneous deformation. Simulations having small aggregates in every element showed substantial inhomogeneity in the

136

Paul R. Duwson and Estebun B. Murin

straining over the sample and developed more diffuse textures-ones actually in better agreement with experiment [ 131. More diffuse texture with smaller numbers of crystals in every aggregate is not a surprising result, once it is evident that the deformation differs locally from the macroscopic average. A less obvious issue is whether the number of orientations can affect the manner in which the individual crystals reorient for a particular local deformation. For simulations based on strict application of the Taylor assumption (and identical slip system strengths), this issue does not arise. Every crystal deforms with the macroscopic deformation rate, and the lattice reorientation is not influenced by the behavior of other crystals in the aggregate. The aggregate size does become important, however, for partitioning rules in which the crystal deformation rate depends on the texture, and thus on the texture representation. In such instances, the number of crystals in an aggregate may affect the texture evolution, both through the variability in the properties from aggregate to aggregate and through its influence on the behavior of the aggregate itself. For HCP crystals that are inextensible along the c-axis, the constrained hybrid assumption offers a method for determining the crystal deformation rates from the macroscopic values. With this assumption, both effects just discussed are possible. The latter behavior is examined first by comparing the texture evolution of isolated aggregates under identical imposed deformation histories. To concentrate on the influence of geometric hardening alone, hardening of the slip system is suppressed (slip system hardening is not substantial for this material, and although we do consider hardening elsewhere in the paper, it is not unreasonable to treat the slip systems strength as constant). A moderately high rate sensitivity is specified ( m = 1/3) [84]. Samples of 1024 crystals are constructed from aggregates of different sizes. For the largest aggregate size, only crystals from a single aggregate appear. For smaller aggregate sizes, the number of aggregates is simply 1024 divided by the number of crystals in the aggregate. The initial orientations for crystals in all aggregates of one size are chosen independently from the same isotropic distribution. Figure 28 shows the (0001) pole figures for a plane strain compression deformation to an effective strain of 0.75. The textures for the different cases are qualitatively similar: the c-axes rotate toward the compression (ND) axis, with some spread toward the transverse (TD) direction. However, the intensity of the texture is higher for a sample composed of smaller aggregates than a sample composed of larger ones. The maximum intensity increases from about 12.8 (times random) for a sample having a single aggregate of 1024 crystals to approximately 15.5 (times random) for samples composed of aggregates having only 16 crystals. As is evident in Figure 29, the stress rises during compression because the crystals

137

Mechanics of Metal Deformation Processes TD

TD

RD

1024 crystals I agg. max. int. = 12.75


TD

TD



0.53 1.00 1.90 3.61 6.85 13.00 FIG.28. Texture prediction using the constrained hybrid assumption for homogeneous plane strain compression textures computed with aggregates of various sizes. (0001) equal area projection; scale is multiple of random.

are collectively rotating toward an orientation with higher Taylor factors (principally toward having the c-axis aligned with the compression axis). Since the slip system strengths are constant, the increase in stress is due entirely to geometric hardening. The small aggregate size sample shows appreciably higher stress at strains above 0.75, consistent with its stronger texture. The cause of the more rapid texturing originates with the projection operator’s dependence on aggregate size. The projection operator’s magnitude is larger for smaller aggregates, rising sharply as the number of crystals in the aggregate drops below 100 (Figure 30). Consequently, there is a systematic influence of the aggregate size on the magnitude of the crystals’ deformation rates from the projection operator. The number of crystals chosen to represent a texture therefore affects the predicted rate of texture evolution.


138

---+--- 16crysIagg

--+-

0.0

0.2

64cryslagg 256cryslagg 1024 crys I agg

0.4

0.6

0.8

Macroscopic effective strain (E) FIG.29. Geometric hardening predicted using the constrained hybrid assumption for homogeneous plane strain compression computed with aggregates of various sizes.

We now return to the influence of aggregate size on the spatial variations in properties. Finite-element simulations of plane strain compression are performed with the viscoplastic finite-element formulation outlined in Section V.B, using a mesh composed of 1000 elements (Figure 31) in the same fashion as Beaudoin et al. [12]. In each element resides an aggregate with an identical number of orientations, but independently chosen from an initially uniform distribution. Simulations are performed using aggregates of 16, 64, 254, and 1024 crystals. The deformed meshes after a strain of unity for the 1024 and 16 crystaldaggregate cases, presented in Figure 3 1, illustrate the spatial heterogeneity introduced by the variations in mechanical properties with smaller aggregates. The corresponding texture evolution appears in Figure 32. Although there is a tendency for the texture to be more diffuse when the deformation varies spatially (as allowed in the finite-element simulation), the dominant effect on texture is from the projection operator being a function of the aggregate size. For simulation results to be insensitive to the aggregate size, the number of crystals in an aggregate should number at least 250 [84].


139

2.0

'A

a V ' 1.0

Q)

cn

s Q)

2 0.5

0.0 1

Number crystals / aggregate FIG.30. Average projection tensor predicted using the constrained hybrid assumption for various aggregate sizes and levels of strain under plane strain compression.

&=o.o

L

x=RD

y=TD z=ND

Y

FIG.3 1. Finite-element meshes for two aggregate sizes. Constrained hybrid assumption; plane strain compression. (A) Initial mesh: (B) Deformed meshes.

140

Paul R. Dawson and Esteban B. Marin A. FINITE ELEMENT SIMULATIONS

D

&=0.75

T

A

1

18.00 1024 crystals 1 agg. Max. Int. = 17.96

16 crystals / agg. Max. Int. = 24.56

B. MATERIAL POINT SIMULATIONS

I

8.74 4.24

2.06 1 .oo 0.49

1024 crystals 1 agg. Max. Int. = 18.39

16 crystals / agg. Max. Int. = 23.22

FIG.3 2 . Comparisons of textures from homogeneous and heterogeneous straining resulting from two aggregate sizes. Constrained hybrid assumption; plane strain compression. (0001) equal area projection; scale is multiple of random.

B. HYBRIDELEMENT POLYCRYSTAL SIMULATIONS Numerical modeling of polycrystals provides an approach for studying the behavior of polycrystalline metals that complements experimental data. Trends in the behaviors of polycrystals can be extracted from the simulation results, provided that there is confidence in the accuracy of the simulation tool. Such trends often are difficult or expensive to obtain experimentally,making simulation attractive as a useful alternative for experiment. Of course, simulation does not replace experiment; rather, simulation can reinforce experiment by assisting in the interpretation of the experimental findings. For polycrystalline solids, simulations of polycrystals are especially valuable in developing an understanding of the role of grain interactions in the deformation of constituent crystals and in the subsequent texture evolution. It has been appreciated since the first uses of Taylor and Sachs


141

assumptions that the assumed grain interactions are greatly simplified, and that the simplicity gained in the models comes at the expense of the accuracy of the predictions. This is not to say that such predictions have not been valuablequite the contrary. However, the predictions from these models often give texture components that are displaced in orientation space from the experimental observations and evolve too rapidly. To improve on the models is a difficult task, because the exact influence of the grain interactions on the texture evolution in particular is not easily abstracted. In this example we discuss simulations of collections of crystals that represent samples of the metal. Every crystal in a sample coincides with a distinct finite element, so that the properties within an element are those of a single crystal, the orientation of which is determined from sampling of an orientation distribution. Balance laws are enforced over the collection of crystals (elements) via the finite-element method. The macroscopic motion is imposed through boundary conditions on the body defined by the crystals, rather than a priori through a partitioning assumption such as the Taylor hypothesis. Such simulations are of the first category outlined in the Introduction. As summarized in the Introduction, finiteelement simulations of polycrystals of the first category have been conducted for many years. Recent advances in computer speeds, especially those enabled by parallel architectures, have made the simulation of large three-dimensional polycrystals a much more viable tool for studying the behaviors of polycrystals. The advantage of modeling a polycrystal with a finite-element discretization is that the crystal interactions emerge from the simulations as a consequence of enforcing the balance laws at the crystal interfaces. Using parallel computing strategies, sufficiently large numbers of crystals can be considered to provide physically meaningful environments for the crystals. That is, each crystal is surrounded in three dimensions by other crystals and is distant from the boundary of the domain. One example of studying three-dimensional polycrystals is the simulation of FCC metal behavior, using a viscoplastic approximation together with a hybrid finite-element formulation [14, 1201. The simulations give textures that are more diffuse than corresponding computations performed with a Taylor assumption. Furthermore, important components of the texture differ in strength and location in orientation space. In particular, the finite-element simulations develop the Brass component under plane strain compression, whereas predictions using the Taylor assumption do not. This is attributed to the heterogeneity in the straining from crystal to crystal, especially certain shear components of the deformation rate that are zero in the macroscopic average. Grain shape alters the strength of the heterogeneity for certain components of the deformation rate, and consequently further modifies the texture. The importance of the relative strength of a crystal’s neigh-

142


bors dominates over the orientation of the crystal in determining the deviation in its deformation rate from the macroscopic average [ 1201.

1. Hybrid Finite-Element Formulation The simulations presented in this section employ the hybrid element formulation of Beaudoin et al. [ 141. The more conventional velocity/pressure formulation described in Section V.B and used in the rolling application (Section V1.B) fails to converge smoothly if the number of crystals in each aggregate becomes too small. For a single crystayaggregate (that is, each element is an individual crystal), the solution usually fails to converge all together. Although the inclusion of elasticity can alleviate this difficulty [82], it complicates the formulation. The R-value analyses of Section V1.A use this latter approach, largely because the elastic response and the transition from elastic to elastoplastic behavior is a central issue in the evaluation of the R-value. Hybrid element formulations prove very effective in retaining a viscoplastic approximation, while achieving convergence robustly [132,6, 191. The hybrid formulation [ 141is constructed to obtain solutions that enforce equilibrium of traction across intercrystal boundaries. The crystals are the volumes associated with the domain decomposition at the heart of hybrid methods. The grain boundaries are domain boundaries. Within single crystals, equilibrium is ensured by the use of self-equilibrated stress trial functions. The elements, or domain subdivisions, are constrained to be in mutual equilibrium by requiring that, in a weighted residual sense, the integral of surface tractions over all surfaces vanishes. This constraint is written over all domains of the polycrystal

where 4 are weights and the superscript e denotes an elemental surface. The traction, t, at any point on the boundary is related to the stress via the Cauchy formula

t =a

*

g = (a’- PI) * g.

(119)

Substitution of the Cauchy formula into the traction equilibrium residual gives, after integration by parts, Tr[(a’ - PI) V 4 ] d v -

I ,4

*

1

t d a = 0.

(120)

Assumption of isochoric plastic deformations requires that the divergence of the velocity vanish (neglecting elastic deformations), from which a second weighted


143

residual is formed, using weights @,

ke

@(TrD)dv = 0.

(121)

The crystal constitutive behavior is obtained from the relation between the rate of deformation and the deviatoric stress (eq. (19)). A residual is constructed using this equation, together with appropriate weights, TC,

To estimate the crystal stiffness, M c ,eqs. (19) and (20) are solved, using the deformation rate corresponding to the previous iterate of the velocity field. Trial functions are introduced in the residuals for the interpolated field variables. Using the hybrid formulation, trial functions for the deviatoric stress appear, in addition to those for the velocity and pressure (eq. (89)).

Here [ N u] are piecewise discontinuous functions that satisfy equilibrium within an element a priori [ 191. For the crystal constitutive response relation, the residual from eq. (122) relates nodal stresses to the nodal velocities,

where

and

Using the above matrices, the traction equilibrium residual (eq. (120)) is given by

c

"Rl{sC} - IGel{pl - {f)]= 0.

(127)

r

The nodal stresses can be eliminated from this equation by inverting eq. (124), and then substituting the result into eq. (127). This yields

C [[Rl[HCl-'IRIT~Ul- [Gelif') - If)] = 0. e

(128)

144


This equation is solved simultaneously with the discretized residual for the incompressibility constraint

to give the velocity field for the polycrystal corresponding to its current geometry, state, and loading. Special attention is needed in solving this system by using the conjugate gradient method due to the poor conditioning resulting from the incompressibility constraint [ 121. With the velocity field known, the geometry and microstructural state are advanced over a time increment by Euler integration of the coordinate velocities and of the evolution equations for the lattice orientations and strengths. 2. HCP Polycrystal HCP metals show greater single crystal anisotropy than do most FCC metals. The difficulty of slip in the direction of the crystal c-axis accentuates the yield surface anisotropy. As mentioned previously, the difference in strength between prismatic and pyramidal slip systems can be sufficiently large to motivate the idealization that the HCP crystals are inextensible along their c-axes. Although the inextensible assumption is perhaps too severe, imposing the Taylor assumption creates difficulties as well. If every crystal must accommodate the macroscopic average, extension along the c-axis is, in general, required of every crystal. This results in both unrealistically high stress (needed to activate the pyramidal systems) and inaccurate texture predictions (since the distribution of slip over the slip systems is not accurate). Simulations of the compression of HCP titanium samples have been conducted to quantify the degree of the strain heterogeneity in a polycrystal, allowing but not requiring crystals to activate stronger pyramidal slip systems. A mesh of 4096 brick elements represents a sample of the metal having an equal number of crystals. The inner set of 1000 elements defines an aggregate of crystals referred to as a hybrid element polycrystal. The remaining outer 3096 elements serve as a medium through which a chosen deformation history is imposed. In particular, plane strain compression and uniaxial compression are examined. In both cases, crystals chosen randomly from an isotropic orientation distribution are assigned one to an element to initialize the texture. The slip system constitutive parameters correspond to those given in Tables 1 and 2. The deformation history is imposed, using macroscopic axial strain increments of approximately 0.006. Considerable heterogeneity arises during the deformation. As the sample compresses, elements distort from regular brick shapes, showing the influence of local shears. This is evident in the initial and deformed meshes for plane strain compression shown in


145

ND

FIG.33. Initial and deformed meshes for the hybrid element polycrystal under plane strain compression.

Figure 33.* The heterogeneity in deformation is strong, and its impact on the texture evolution is substantial in comparison to textures predicted using either the constrained hybrid or Taylor assumptions. Details of the plane strain and uniaxial compression simulations are discussed in the following paragraphs, along with the implications they suggest regarding various partitioning assumptions. The textures for the hybrid element polycrystal are presented over the HCP fundamental region in Figure 34 for plane strain compression using an initial ratio of the pyramidal to basal strengths of 5. The corresponding orientation distributions for the Taylor and uniform stress assumptions appear in Figures 8 and 15. Several points can be made concerning comparisons of these predictions. Texture components for the hybrid element polycrystal share some, but not all, of the texture components predicted with the Taylor assumption. In particular, the component associated with the c-axes oriented in the rolling (RD) direction predicted by the Taylor assumption is absent (or quite weak) in the hybrid element polycrystal orientation distribution. The component of the texture associated with a c-axis orientation toward the compression (ND) axis, but tilted from it, is similar in the location for the two assumptions, although it is weaker for the hybrid element polycrystal. This feature passes through the fundamental region nearly parallel to the ND-axis, but offset from it in either the plus or minus TD directions (indicating a c-axis tilt toward the rolling direction). Furthermore, the textures predicted with the hybrid element polycrystal are more diffuse than the corresponding Taylor assumption computations. Similar points can be made concerning the comparison 'Actually, the textures shown later correspond to simulations in which the mesh is repaired at intervals of approximately 10% strain so as to avoid such severe element distortions.

146


A 16.0 14.2 12.4 10.7 8.9 7.1 5.3 3.6 1.8 0.0

ITD

,RD

F IG .34. Hybrid element polycrystal orientation distribution A at an effective strain of unity shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Plane strain compression; pyramidal strength 5 times basal strength. Peak value = 7.0.

of uniform stress and hybrid element polycrystal computations. Although neither texture displays c-axis orientations oriented in the rolling direction, features of the fiber near the ND axis are different. For the uniform stress assumption, there are peaks at f0.66TDmaXthat are considerably stronger than the value at the origin. Texture variations predicted by the hybrid element polycrystal are more muted. The orientation distribution for the uniform stress assumption also generally displays greater texture intensities. For the case of pyramidal strength 10 times the basal strength, the hybrid element polycrystal texture (Figure 35) is more diffuse than the texture corresponding to a ratio of 5. Furthermore, there is a tendency for the c-axes to exhibit less tilt away from the TD axis. The texture for the Taylor assumption (Figure 36) with pyramidal strength 10 times basal does differ from the Taylor assumption texture with lower pyramidal strength, but to a lesser extent than differences in the hybrid element polycrystal results. In contrast to the hybrid element polycrystal, the Taylor assumption results show increased intensity in the texture with greater pyramidal strength. The uniform stress assumption gives virtually no difference

147


A

F I G . 3 5 . Hybrid element polycrystal orientation distribution A at an effective strain of unity shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Plane strain compression: pyramidal strength 10 times basal strength. Peak value = 5.4.

for the cases of pyramidal strength 5 and 10 times the prismatic strength (the latter texture is not shown). These stronger slip systems are not activated in either case. The trend toward less tilt of the c-axes in the hybrid element polycrystal textures is interesting in comparison to the constrained hybrid assumption predictions (shown in Figure 14). This assumption gives virtually no tilt, but rather shows a single component of texture positioned along the ND axis at ND = fNDma, (same point due to symmetry). It appears that the hybrid element polycrystal may indeed be approaching this limiting case for infinitely strong pyramidal slip systems (inextensible crystals), although this point has not been proved. A major distinction, however, continues to be the strengths of the orientation distributions. The constrained hybrid orientation distribution is far more intense than the hybrid element polycrystal distribution. The heterogeneity of straining among crystals of an aggregate plays a critical role in modifying the texture evolution. The Taylor assumption fixes the crystal deformation rates so that they are identical; the uniform stress and constrained hy-

148


A 16.0 14.2 12.4 10.7

8.9 7.1

5.3 3.6 1.8

0.0

ITD

,RD

FIG. 36. Discrete aggregatenaylor assumption orientation distribution A at an effective strain of unity shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Plane strain compression; pyramidal strength 10 times basal strength. Peak value = 22.4.

brid assumptions both allow for different deformation rates in every crystal. For both of these latter models, the variation in deformation rate depends only on a crystal’s own orientation plus the orientation distribution for the aggregate. In the hybrid element polycrystal, the additional dependence on crystal arrangement is introduced. Thus a crystal’s deformation rate depends on the orientation of crystals that neighbor it. The effect of this dependence has been explored for FCC polycrystals [ 1201 as well as for HCP polycrystals presented here. The methodology involves repeating the simulation of the response of the hybrid element polycrystal under plane strain compression using different assignments of orientations to elements, but keeping the same set of orientations. In this way, every crystal has a different set of neighbors in each simulation. Reassignments are repeated to give a total of 25 simulations. Each orientation then has associated with it 25 deformation rates corresponding to 25 sets of neighbors for the same plane strain compression loading. A summary of these data is shown in Figure 37, in which the largest, smallest, and average values of one deformation rate component are


149

o Maximum of 25 0

0.5 1.0 Theta [rad]

Minimum of 25 Average of 25

1.5

F I G . 37. Low, high, and average of the crystal’s extensional deformation rate components for 25 hybrid element simulations using inner 1000 crystals to define a polycrystal. Plane strain compression; pyramidal strength 5 times basal strength.

plotted as a function of the angle between the crystal c-axis and the compression (ND) axis. It is evident that the effect of neighborhood is very strong. Differences between high and low values of deformation rate are considerably larger than the variations in the average values of deformation rate with the c-axis orientation. A similar plot for the constrained hybrid and Taylor assumptions (Figure 38) shows a single value for the Taylor assumption and a distribution for the constrained hybrid assumption that is similar to the hybrid element polycrystal average values. The constrained hybrid assumption has only a single value for any orientation,

2.0 n v)

Taylor Constrained Hybrid

1.5

Y

Theta [rad] FIG.3 8. Crystals’ extensional deformation rate components for 1000 orientations computed using the constrained hybrid assumption. Plane strain compression.

150


owing to the assumption’s dependence on only the crystal’s orientation and the full orientation distribution. This is significant, as it shows that although variations in straining give a plausible distribution of deformation rate with crystal orientation, they are not adequate to model the full effect of strong anisotropy in the single crystal-behavior. The comparison between experiment and theory is made difficult for HCP systems by the broad range of behaviors the systems display. The crystal’s aspect ratio (ratio of c-axis to a-axis lengths) is important, so trends for zinc or zirconium may be different from those for titanium. Furthermore, the temperature of the deformation and alloying additives influences the texture evolution through their effects on the relative strengths of active slip systems [9]. At lower temperatures, especially in pure metals, twinning often is an important deformation mode. We restrict our attention here primarily to textures developing in the warm-to-hot regimen for single-phase (a)materials. Under these restrictions, twinning is not significant, although large differences in the relative strengths of basaUprismatic and pyramidal systems can exist. The textures reported for such conditions are fewer than those reported for lower temperatures where twinning occurs. Tanabe et al. [125] have reported data for warm rolling that show a strong tendency for c-axes to align toward the compression (ND) axis, but to be tilted toward the rolling (RD)direction. This is in agreement with the textures computed using the hybrid element polycrystal. Larson et al. [78], Thornburg and Piehler [129], and Peters and Luetjering [lo61 all reported similar c-axis tilt toward the rolling direction for titanium systems with sufficient alloying to suppress twinning. This is in contrast to pure titanium deformed at low temperatures in which twinning is significant and the c-axis tilt is toward the transverse (TD) direction [107, 1081. Uniaxial compression textures computed using the hybrid element polycrystal for pyramidal strengths of 5 and 10 times basal strength are shown in Figures 39 and 40. In these cases, the deformation is imposed only to an effective strain of 0.75 because of degradation in the performance of the elements. Again, greater pyramidal strength tends to orient the c-axes more in line with the compression ( z ) axis and to give a more diffuse texture. The texture components associated with the c-axes being perpendicular to compression axis tend to weaken as the pyramidal strength increases from 5 to 10. The texture obtained using the constrained hybrid assumption has a single fiber, located along the compression axis (Figure 41). This appears to be a limiting case for very large pyramidal strength for the hybrid element polycrystal. As with plane strain compression, the hybrid element textures are much weaker than those for the constrained hybrid assumption. Textures are shown in Figures 42 and 43 for the Taylor assumption (with


151

A 20.0 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0

FIG.39. Hybrid element polycrystal orientation distribution A at an effective strain of 0.75 shown on cutting planes through the HCP fundamental region of' Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Uniaxial compression; pyramidal strength 5 times basal strength. Peak value = 7.1.

strengths of 5 and 10 times pyramidal) and in Figure 44 for the uniform stress assumption (only for 5 times, as 5 and 10 times textures are virtually identical). Interestingly, the Taylor and uniform stress textures are quite similar, although the offset from the compression axis is larger for the Taylor assumption. Increasing the pyramidal strength tends to strengthen the Taylor assumption texture, but not to a significant extent. Neither of these partitioning rules leads to the elimination of the c-axis texture component perpendicular to the compression axes. A number of modeling efforts have examined the impact of the relative strengths of slip systems, including the effect of twinning [79, 107, 109, 138, 91 on texture evolution for various HCP metals. Whereas the self-consistent theories do allow for some variations in straining, Taylor analyses do not. This variation is important in that it leads to more diffuse textures as well as to changes in the relative strengths of texture components. Tools such as the hybrid finite-element formulation are a means of quantifying, the impact of this feature of the behavior.

152


FIG. 40. Hybrid element polycrystal orientation distribution A at an effective strain of 0.75 shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Uniaxial compression: pyramidal strength 10 times basal strength. Peak value = 4.1.

VIII. Summary Finite-element techniques offer the capability to simulate deformations that are spatially heterogeneous, that are governed by nonlinear equations, and that are controlled by properties that evolve with the deformation. Polycrystal plasticity can be used effectively in such finite-element formulations as the constitutive description of a textured metal to quantify its anisotropic properties from its crystallographic texture and the behavior of individual crystals. The advantages in combining finite-element methodology with polycrystal plasticity stem from being able to analyze complex problems, such as those that arise in metal deformation processes, with a theory that directly represents real features of the state, and to evolve the state as deformation proceeds. Critical to the use of such a state variable constitutive theory is the ability to initialize the state variables directly from tests and to compare simulation predictions to experimental observation. Here polycrystal plasticity has an advantage resulting from the intensive efforts


153

A 50.0 45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0

FIG.4 1. Discrete aggregatekonstrained hybrid assumption orientation distribution A at an effective strain of 0.75 shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Uniaxial compression. Peak value = 49.0.

in quantitative texture analyses over the past several decades. Textures can be determined readily from experiment and rendered as orientation distributions. Slip system strengths are less accessible, but when the theory is restricted to the use of average strengths, values can be estimated from routine mechanical tests (tension or compression) of macroscopic samples. In this article we discuss the implementation of polycrystal plasticity theory in finite-element formulations. The intent is to document a methodology for accomplishing this task, discuss some of the issues that arise, and demonstrate its application to metal deformation processes. Several capabilities are essential for the implementation to be successful. One is that the method for representing the texture is indeed adequate for describing the complex textures that are present in metals. To capture a texture with accuracy requires either large numbers (more than 1000) of individual orientations in discrete aggregates or a representation of the orientation distribution probability density with a comparable number of degrees of freedom. For this task we see a considerable advantage in the use of piecewise approximations from finite element discretizations.

154


A 20.0 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0

FIG. 42. Discrete aggregaternaylor assumption orientation distribution A at an effective strain of 0.75 shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Uniaxial compression; pyramidal strength 5 times basal strength. Peak value = 22.8.

Another capability that is important is the methodology for linking the continuum fields with their counterparts at the crystal level. Macroscopic properties are evaluated by averaging single-crystal properties over an orientation distribution. The properties depend on the behavior of the individual crystals, as well as the manner in which the macroscopic deformation rate is partitioned among the constituent crystals. Furthermore, the evolution of state, as defined by the texture and slip-system strength, is sensitive to the partitioning rule, as was demonstrated for HCP crystals by using a number of rules. There is not a single rule that provides optimal partitioning; rather, the single-crystal properties often are influential in determining which rules are successful and which are not. The magnitude of the rate dependence, the degree of anisotropy, and the hardening behavior all affect the partitioning of deformation. A third component of a successful implementation is a finite-element formulation for the motion of the body that is consistent with the microstructural characterization. There is no unique finite-element formulation that is universally supe-


155

A 20.0 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0

FIG.43. Discrete aggregate/Taylor assumption orientation distribution A at an effective strain of 0.75 shown on cutting planes through the HCP fundamental region of Figure 3. Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Uniaxial compression; pyramidal strength 10 times basal strength. Peak value = 21.2.

rior. The hybrid approach is particularly useful for the viscoplastic approximation when the number of orientations within an element is small and the discontinuities in properties at interfaces are appreciable. In the hybrid formulation, the residual of equilibrium focuses on interelement tractions, tending to eliminate imbalances at element interfaces that can lead to poor convergence. For instances when there is a large sample of crystals within every element and the average properties tend to be smooth, a more traditional velocity-based viscoplastic formulation performs well. The inclusion of elasticity, while complicating the implementation, improves the robustness if elements contain only a single orientation. The applications presented are intended to demonstrate instances where the approach is a particularly useful tool. Deformation of the sheet is one case in which the texture affects the stability of the deformation field and thus the formability of the sheet. Here we focus on the computation of R-values, but in other publications we show the dependence of sheet performance in hydroforming and limiting dome height tests on the metal texture. In rolling, texture variations arising from

156


A 20.0 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0

2.0 0.0

FIG.44. Discrete aggregate/uniform stress assumption orientation distribution A at an effective strain of 0.75 shown on cutting planes through the HCP fundamental region of Figure 3 . Cutting plane position relative to the maximum coordinate value is indicated to the right of the plane. Uniaxial compression; pyramidal strength 5 times basal strength. Peak value = 10.5.

the inhomogeneity of the deformation are of technical importance for the mechanical properties of the product. Together, the finite-element method and polycrystal plasticity provide a method for quantifying the strength of texture gradients. As discussed here for the rolling of HCP titanium, the simulations can predict the effect of the rolling schedule on the variations in texture within the rolled metal. Finite-element simulations performed at small scale can help expose the behavior of polycrystals and assist in the assessment of partitioning rules. In the examples presented, the heterogeneity of straining over an aggregate can be seen to diffuse the texture, as well as to modify the components predicted in comparison to Taylor, uniform stress, or constrained hybrid assumptions. Furthermore, the strong dependence on local neighborhood is demonstrated. Additional refinement is expected to reveal further insights into the behaviors of polycrystals. A major factor in the successful combination of polycrystal plasticity and finiteelement formulations for metal deformations is the availability of modem computing resources. Strategies that utilize parallel computer architectures are essential to being able to embed in every element aggregates of crystals, whether they


157

are represented by a discrete set of orientations or by a piecewise discretization (finite element) of orientation space, that have sufficient detail to capture realistic textures. With such strategies, simulating the deformation of a complex threedimensional part can include a complete characterization of the texture in every element and the evaluation of the mechanical properties from an appropriate averaging of single-crystal responses. The definition of “appropriate” depends in part on the intended use of the simulation results. For many applications the simple Taylor assumption suffices. For others it does not, and different assumptions must be used. To obtain increased accuracy or to model more complex materials, the variability of straining over the crystals in an aggregate is quite important. Although some models do exist for this purpose, as is discussed in this article for HCP metals, they are not entirely satisfactory. The expanding resource offered with parallel computing offers the potential to address this issue, in the short run by providing better understanding of polycrystal behaviors, and in the long run by eliminating the need to invoke partitioning assumptions.

IX. Notation Surface area of body Surface area in orientation space Orientation distribution Finite-element approximation of A Nodal parameters in finite-element approximation of A Slip direction on the (Y slip system Domain of the body Domain of the body in the natural (relaxed) configuration Crystal c-axis vector Diffusion parameter cc

@

cc

-

11 3

Macroscopic deformation rate Matrix form of D Crystal deformation rate Elastic portion of the crystal deformation rate Plastic portion of the crystal deformation rate Matrix form of DP Plastic portion of the crystal deformation rate in the relaxed configuration Matrix form of D p Base vectors

158

Paul R. Dawson and Esteban B. Marin Green strain tensor components Scaling function in angle/axis representation Elemental matrix in orientation distribution evolution formulation Elemental matrix in velocity formulation Macroscopic deformation gradient Crystal deformation gradient Plastic portion of the crystal deformation gradient Plastic portion of the crystal deformation gradient in the relaxed configuration Metric tensor for orientation space Elastic strain matrix in velocity formulation Elemental volumetric mode in velocity formulation Element size Crystal spdstrain matrix for velocity formulation Symmetry rotations Assembled elemental ([ S"]{h'])matrices Second-order identity tensor Fourth-order identity tensor Residual for probability density conservation equation Assembled elemental [ K A ] matrices in orientation distribution evolution formulation Elemental stiffness matrix in velocity formulation Assembled elemental [ K " ] matrices Macroscopic velocity gradient Crystal velocity gradient Elastic part of the crystal velocity gradient Scaling factor for the velocity gradient Plastic part of the velocity gradient Mean linear intercept measure of grain size Normal vector to the a! slip system Slip-system constitutive parameters Crystal compliance Crystal elastic compliance Crystal plastic compliance Crystal compliance matrix Mass coefficient matrix Crystal plastic compliance matrix Axis normal vector in angle/axis representation


159

Interpolation functions for Interpolation functions for uh Interpolation functions for p h Spatial derivative of [ N u ] Pressure Crystal projection tensor Matrix form of Pc Nodal point parameters in finite-element approximation of p Symmetric portion of the Schmid tensor, T, for a slip system Matrix form of p(Y Skew portion of the Schmid tensor, T, for a slip system Crystal orientation Set of crystal orientations comprising an aggregate Residual in orientation distribution approximation Rotation tensor Rotation in decomposition of F Elemental residual on deviatoric motion in velocity formulation Elemental residual on volumetric motion in velocity formulation Fundamental region Segments of the surface of R* Differential length in orientation space Average stiffness Crystal plastic stiffness Crystal elastic stiffness Crystal stiffness matrix Crystal plastic stiffness matrix Crystal elastic stiffness matrix Time Time step Traction vector Traction vector imposed on a B Schmid tensor Temperature Melt temperature Velocity vector Value of u imposed on the surface of B Finite-element approximation of u Nodal parameters in finite-element approximation of u Volume of body

160

Paul R. Dawson and Esteban B. Marin Volume of orientation space Volume fraction of grains Crystal reorientation velocity Elastic stretch Vector representation of Q* Macroscopic spin Crystal spin Plastic portion of crystal spin Plastic portion of the crystal spin in the relaxed configuration Matrix representation of spin terms in eq. (58) Body coordinates Transformation matrix for vector representation of D Partitioned stiffnesskompliance for relaxed constraints

A. GREEKSYMBOLS Slip-system identification Volumetric factor (det (I c)) Shearing rate on a slip system Sum of the slip-system shear rate magnitudes Slip-system constitutive parameters Christoffel connexion tensor Diffusivity used in numerical solution of orientation distribution Effective strain Elastic strain tensor at the current time Elastic strain tensor at the end of the previous time step Vector normal to the surface of B Vector normal to flat grain boundaries Trace operator in matrix form Weighting functions used in finite-element solution of orientation distribution Nodal parameters used in finite element approximation of $ Weights used on the surface of R* in the finite-element formulation for the orientation distribution evolution Weighting function used in equilibrium residual in velocity solution Weighting functions used in volumetric deformation constraint of velocity formulation Elastic bulk modulus

+

K


161

Elastic shear modulus Base vectors for grain axes Lattice spin Macroscopic Euler spin Crystal Euler spin Macroscopic Cauchy stress Crystal Cauchy stress Resolved shear stress on the (11slip system Macroscopic Kirchhoff stress Crystal Kirchhoff stress Slip-system strength Average slip-system strength Slip-system hardening parameters Angle between rolling direction and tensile axis of R-value specimen Slip-system hardening parameter

B. CONVENTIONS Vectors and second-order tensors are indicated with boldface lower- and uppercase letters, respectively, e.g., v and T; fourth-order tensors are denoted by calligraphic letters, e.g., P.The dyadic product of two vectors v1 and v2 is indicated as v1 8 v2. Tensor operations between two second-order tensors T1 and T2 are indicated as T1 Tz for the inner product (a second-order tensor), T1 8 T2 for the dyadic product (a fourth-order tensor), and TI : T2 for the scalar product (a scalar). Contraction operations over two indices between a fourth-order (P)and a second-order (T) tensor and between two fourth-order tensors PI and 732 are represented as P : T (a second-order tensor) and PI : P2 (a fourth-order tensor), respectively. The vector cross product between two vectors v1 and v2 is given by v1 x v2. Where index notation is used, we refer the tensor components to a general coordinate system, with the summation convention over repeated indices implied. Deviatoric quantities are denoted with primes. V indicates the nabla operator, and the trace operator for a second-order tensor A is indicated by Tr(A).

.

X. Appendix: Matrix Representations The various second order-deviatoric tensors used in the crystal constitutive equations, such as the crystal deformation rate tensor D'", are expressed as 5 x 1


162

vectors, using the convention

-

-

In this context, the term (d* WP - W P d * )in eq. (58) is represented in matrix form as [~?P](E’*},where [WP] is the following 5 x 5 skew-symmetric matrix: -

[ W P ]=

I

0

0

0

0

0

W13 aw’p a ,wl3

2Wf2

0

0

-W& -WF3

WIp3

-W&

-2W’p2 -W’p3

W& -aw;,

-awg

Wf3

0 W‘p2

-Wf2 0

.

Furthermore, the vector {a} and the matrix [XI in the finite-element equations (eqs. (90-105)) are used in the following transformations: T O ) = {6IT{D},

(D’) = [Xl(D),

where { D ) = (Dl1 0 2 2 0 3 3 D2l D31 D32}T and ( D ’ ) is the 5 x 1 vector representation of the deviatoric part of ( D } . They are given by

(6) = (1 1 1 0 0

O}T

Acknowledgments Support for this work has been provided by the Office of Naval Research under contract N O 0 0 14-95-1-0314, and by the National Science Foundation under grant MSS-9114861. Computing resources on the Connection Machine CM-5 were provided by the National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign. The authors are grateful to Ashish Kumar for performing the computations of texture evolution using the finite-element


163

orientation space formulation; to Vincent Prantil for original efforts in implementation of the constrained hybrid model; and Carlos T6me for use of codes for computing HCP yield surfaces. The authors wish to thank Armand Beaudoin, Nathan Barton, Donald Boyce, Fred Kocks, David Mika, Gorti Sarma, and Rudy Wenk for valuable discussions, comments, and suggestions. Many thanks to Karen Biesecker for her careful efforts in preparation of the manuscript.

References 1. S. Ahzi, R. J. Asaro, and D. M. Parks, Application of crystal plasticity thcory mechanically

processed BSCCO superconductors. Meclz. Muter: 15,201-222 (1993). 2. S. L. Altmann, Rotations, quaternions and double groups. Oxford Univ. Press, Oxford, 1986. 3. R. J. Asaro, Micromechanics of crystals and polycrystals. Ad\. Appl. Mech. 23, 1-1 15 (1983). 4. R. J. Asaro, and A. Needleman, Texture development and strain hardening in rate dependent polycrystals. Actrz Metall. 33, 923-953 (1985). 5 . M. F. Ashby, Mechanisms of deformation and fracture. Adv. Appl. Mech. 23, 117-177 (1983). 6. S. N. Atluri, On “hybrid” finite element methods in solid mechanics. In: Advances in cornpurer metliods for partial differential equations (R. Vichinevetsky, ed.). Publ. Association Intemationale Pour Lc Calcul Analogique (AICA), New Brunswick, NJ, 1975, pp. 346-355. 7. Atlris offormabiliiy. CP titanium grade 3. National Center for Excellence in Metalworking Technology, 1994. 8. W. A. Backofen, Deformation processing. Addison-Wesley, Reading, MA, 1972. 9. B. Bacroix, G. R. Canova, and H. Mecking, The prediction of deformation textures in a - ,8 titanium. In: Large plustic deformations: Fundanzental aspects and applications to metalforming (C. Teodosiu, J. R. Raphanel, and F. SidoroK, eds.). Balkema, Rotterdam, The Netherlands, 1993, pp. 101-108. 10. D. J. Bammann, and G. C. Johnson, On the kinematics of finite-deformation plasticity. Acta Mech. 70, 1-13 (1987). 1 1. F. Barlat, Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals. Muter: Sci. Engrg. 91.55-72 (1987). 12. A. J. Beaudoin, K. K. Mathur, P. R. Dawson, and G. C. Johnson, Three-dimensional deformation process simulation with explicit use of polycrystalline plasticity models. Internat. J. Plast. 9, 833-860 (1993). 13. A. J. Beaudoin, P. R. Dawson, K. K. Mathur, U. F. Kocks, and D. A. Korzekwd, Application of polycrystalline plasticity to sheet forming. Comput. Methods Appl. MecA. Engrg. 117, 49-70 (1994). 14. A. J. Beaudoin, P. R. Dawson, K. K. Mathur, and U. F. Kocks, A hybrid finite element formulation for polycrystal plasticity with considcration of macrostructural and microstructural linking. hteriiat. .I. Plust. 11, 501-521 (1995). 15. A. J. Beaudoin, H. Mecking, and U. F. Kocks, Development of shear bands and orientation gradients in FCC polycrystals. In: Simulation o f materiuls processing: Theory, methods and a~plicutions-NUMIFORM ’95(S. Shen and P. R. Dawson, eds.). Balkema, Rotterdam, The Netherlands, 1995, pp. 225-230. 16. R. Becker, Analysis of texture evolution in channel die compression. I. Effects of grain interaction. Acta Mefall. Muter: 39, 121 1-1230 (1991). 17. R. Becker, and S. Panchanadeeswardn, Crystal rotations represented as Rodriguez vectors. Textures and Microstructures 10, 167-1 94 ( I 989).

164


18. M. C. Boyce, G. G. Weber, and D. M. Parks, On the kinematics of finite strain plasticity. J. Mech. Phys. Solids 37,647-665 (1989). 19. C. Bratianu, and S . N. Atluri, A hybrid finite element method for Stokes flow. Part I. Formulation and numerical studies. Comput. Methods Aj’pl. Mech. Engrg. 36, 23-37 (1983). 20. J. D. Bryant, A. J. Beaudoin, and R. T. Van Dyke, The effect of crystallographic texture on the formability of AA 2036 autobody sheet. SAE Tech. Pap. 940161. 21. H. J. Bunge, Texture analy.sis in materials science. Butterworth, London, 1982. 22. H. J. Bunge, ed., Textures of materials (ICOTOM-9). In: Proceedirigs of the ninth internarional conference on textures of materials. Special Issues Textures Microstructures 14-18, 932 (1 991). 23. H. J. Bunge, ed. Textures of materials (ICOTOM-10). In: Proceedings of the tenth international conference on textures of materials. TransTech Publications, Aedermannsdorf, Switzcrland, 1994 (also in Mnter. Sci. Forum 157-162.

24. G. R. Canova, A. Molinari, C. Fressengeas, and U. F. Kocks, The effects of rate sensitivity on slip system activity and lattice rotation. Acta Metall. 36, 1961-1970 (1987). 25. G. R. Canova, H.-R. Wenk, and A. Molinari, Deformation modeling of multiphase polycrystals: Case of a quartz-mica aggregate. Actu Metall. Muter. 40, 1519 (1992). 26. K. S . Chan, and D. A. Koss, Deformation and fracture of strongly texture Ti alloy sheets in uniaxial tension. Meral. Trans. A 14A, 1333-1342 (1983). 27. Y. B. Chastel, and P. R. Dawson, An equilibrium-based model for anisotropic deformations of polycrystalline materials (ICOTOM- 10). In: Proceedings of the tenth international conference on textures ofmaterials, Germany, 2, 1747-1752 (1993). 28. Y. B. Chastel, P. R. Dawson, H.-R. Wenk, and K. Bennett, Anisotropic convection with implications for the upper mantle. J. Geophys. Res. 98 17, 17757-17771 (1993). 29. G. Y. Chin, and W. L. Mammel, Competition among basal prism and pyramidal slip modes in HCP metals. Metal. Trans. 1,357-361 (1970). 30. A. CICment, and P. Coulomb, Eulerian simulation of deformation textures. Scripta Metall. 13, 899-901 (1979). 31. A. CICment, Prediction of deformation texture using a physical principle of conservation. Muter. Sci. Engrg. 55, 203-210 (1982). 32. A. M. Cuitifio, and M. Ortiz, Computational modelling of single crystals. Model. SimuI. Muter. Sci. Engrg. 1,225-263 (1992). 33. Y. F. Dafalias, Planar double-slip inicromechanical model for polycrystal plasticity. J. Engrg. Mech. 199, 1260-1283 (1993). 34. P. R. Dawson, On modeling the mechanical property changes during flat rolling of aluminum. Internat. J. Solids Structures 23, 947-968 (1 987). 35. P. R. Dawson, A. .I.Beaudoin, and K. K. Mathur, Simulating deformation induced texture in metal forming. In: Numerical methods in industrial forming processes (J. L. Chenot, R. D. Wood, and 0. C. Zienkewicz, eds.). Balkema, Rotterdam, The Netherlands, 1992, pp. 25-33. 36. P. R. Dawson, Y. B. Chastel, G. B. Sarma, and A. J. Beaudoin, Simulating texture evolution in Fe-3%Si with polycrystal plasticity theory. In: Proceedings of the international conference on modeling of metal idling processes. The Institute of Materials, London, 1993. 37. P. R. Dawson, A. J. Beaudoin, K. K. Mathur, and G. B. Sarma, Finite element modeling of polycrystalline solids. European J. Finite Elements 3, 543-571 (1994). 38. P. H. Dluzewski, Crystal orientation spaces and remarks on the modelling of polycrystal anisotropy. J. Mech. Phys. Solids 39, 651-661 (1991). 39. G. M. Eggert, and P. R. Dawson, A viscoplastic formulation with elasticity for transient metal forming. Comput. Methods Appl. Mech. Engrg. 70, 165-190 (1988). 40. M. S . Engelman, R. L. Sani, P. M. Gresho, and M. Bercovier, Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements. Internat. J. Numer. Methods Fluids 2, 25 (1982).


165

41. F. C. Frank, Orientation mapping. In: Eighth international conference on textures of materials (J. S. Kallend and G. Gottstein, eds.). The Metallurgical Society, Warrendale, PA, 1988, pp. 3-13. 42. S. F. Frederick, and G. A. Lenning, Producing basal textured Ti-6A1-4V sheet. Metal Trans. B 6B, 601405 (1975). 43. H. J. Frost, and M. E Ashby, Defurmation-mechanism maps. Theplasticity and creep of metals and ceramics. Pergamon Press, Oxford, 1982. 44. M. Gotoh, A finite element formulation for large elastic-plastic deformation analysis of polycrystals and some numerical considerations on polycrystalline plasticity. Internat. J. Numer: Methods Engrg. 12, 101-1 14 (1978). 45. M. E. Gurtin, An introduction to continuum mechanics. Academic Press, New York, 1981, 46. S. V. Harren, and R. J. Asaro, Nonuniform deformations in polycrystals and aspects of the validity of the Taylor model. J. Mech. Phys. Solids 37, 191-232 (1989). 47. P. Hansbo, The characteristic streamline diffusion method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 96,239-253 (1992). 48. J. Hansen, J. Pospiech, and K. Liicke, Tables for texture analysis of cubic crystals. SpringerVerlag, New York, 1978. 49. E. W. Hart, Constitutive relations for the nonelastic deformation of metals. J. Engrg. Muter: Tech. 98, 193-202 (1976). SO. A. Hasegawa, T. Nishimura, and S. Ohtani, The texture hardening of titanium and its alloy sheets. In: Titanium science and technology (R. I. Jaffee and H. M. Burte, eds.). Plenum, New York, 1973, pp. 1349-1363. 51. A. J. Hatch, Texture strengthening of titanium alloys. Trans. Metall. SOC.AIME 233, 44-50 (1965). 52. F. Havlicek, M. Tokuda, S. Hino, and J. Kratochvil, Finite element analysis of micro-macro transition in polycrystalline plasticity. Internat. J. Plast. 8, 4 7 7 4 9 9 (1992). 53. A. Heinz, and P. Neumann, Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorhombic crystals. Acta Cryst. A47, 780-789 (199 1). 54. R. W. K. Honeycombe, The plastic deformation of metals, 2nd edn. Edward Arnold, London, 1984. 55. H. Honneff, and H. Mecking, A method for the determination of the active slip systems and orientation changes during single crystal deformation. In: Fifth international conference on textures of materials (G. Gottstein and K. Lucke, eds.). Springer-Verlag, Berlin, 1978, pp. 265-275. 56. W. F. Hosford, The mechanics of crystals andpolycrystals. Oxford Univ. Press, Oxford, 1993. 57. T. J. R. Hughes, M. Mallet, and A. Mizukami, A new finite element formulation for computational fluid dynamics. 11. Beyond SUPG. Comput. Methods Appl. Mech. Engrg. 54, 341-355 ( 1986). 58. J. W. Hutchinson, Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. Roy. SOC.London Ser: A 238, 101-127 (1976). 59. G. Ibe, and K. Liicke, Description of orientation distributions of cubic crystals by means of 3-D rotation coordinates. Texture 1, 87-89 (1972). 60. C. Johnson, Numerical solution of partial differential equations by the jinite element method. Cambridge Univ. Press, Cambridge, 1987. 61. C. Johnson, A. Szepessy, and P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Marh. Comp. 54, 107-129 (1990). 62. C. Johnson, A new approach to algorithms for convection problems which are based on exact transport + projection. Comput. Methods Appl. Mech. Engrg. 100,45-62 (1992).

166


63. C. Johnson, Streamline diffusion finite element method for compressible and incompressible fluid flow. In: Finite elements injuids (T. J. Chung, ed.), Vol. 8. Hemisphere, Washington, DC, 1992, pp. 75-96. 64. S. R. Kalidindi, C. A. Bronkhorst, and L. Anand, On the accuracy of the Taylor assumption in polycrystalline plasticity. In: Anisotropy and localization of plastic deformation. Proceedings of Plasticity '91 : The Third International Symposium on Plasticity and Its Current Applications (J. P. Boehler and A. S. Khan, eds.). Elsevier, London, 1991, pp. 139-142. 65. S. R. Kalidindi, C. A. Bronkhorst, and L. Anand, Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solids 40, 537-569 (1992). 66. J. S. Kallend, and G. Gottstein, eds., In: Eighth international conference on textures qf'materials (J. S. Kallend and G. Gottstein, eds.). The Metallurgical Society, Warrendale, PA, 1988, p. 1 127. 67. J. S. Kallend, U. F. Kocks, A. D. Rollett, and H.-R. Wenk, Operational texture analysis. Mater: Sci. Engrg. A132, 1-1 1 (1990). 68. U. F. Kocks, Laws for work-hardening and low-temperature creep. J. Engrg. Muter: Tech. 98, 76-85 (1976). 69. U. F, Kocks, and T. J. Brown, Latent hardening in aluminum. Acra Metal. 14, 87-98 (1966). 70. U. F. Kocks, and G. R. Canova, How many slip systems, and which? In: Deformation of polycrystals: Mechanisms and microstructures (N. Hansen et a/., eds.). Risg National Laboratory, Roskilde, Denmark, 1981, pp. 3 5 4 4 . 71. U. F. Kocks, Constitutive behavior based on crystal plasticity. In: Unified constitutive equations forplastic deformation and creep of engineering d o y s (A. K. Miller, ed.). Elsevier, Amsterdam, 1987, pp. 1-88. 72. U. F. Kocks, A symmetric set of Euler angles and oblique orientation space sections. In: Eighth international conference on textures of materials (J. S . Kallend and G. Gottstein, eds.). The Metallurgical Society, Warrendale, PA, 1988, pp. 3 1-36. 73. U. F. Kocks, J. S. Kallend, and A. C. Biondo, Accurate representations of general textures by a set of weighted gains. LA-UR-90-2828. Los Alamos National Laboratories, Los Alamos, NM, 1990. 74. U. F. Kocks, P. R. Dawson, and C. Fressengeas, Kinematics of plasticity related to the state and evolution of the material microstructure. J. Mech. Behavior Mater: 5(2), 107-128 (1994). 75. A. Kumar, and P. R. Dawson, Polycrystal plasticity modeling of bulk forming with finite elements over orientation space. Conzput. Mech. 17, 10-25 (1995). 76. A. Kumar, and P. R. Dawson, The simulation of texture evolution with finite elements over orientation space. I. Development. 11. Application to planar crystals. Cornput. Merhods Appl. Mech. Engrg. 130, 227-261 (1996). 77. A. Kumar, and P. R. Dawson, Modeling crystallographic texture evolution with finite elements over non-Eulerian orientation spaces. Compur. Methods Appl. Mech. Engrg. In Press. 78. F. R. Larson, A. Zarkades, and D. H. Avery, Twinning and texture transitions in titanium solid solution alloys. In: Titanium science and technology (R. I. Jaffee and H. M. Burte, eds.). Plenum, New York, 1973, pp. 1169-1973. 79. R. A. Lebensohn, and C. N. Tome, A self-consistent viscoplastic model: Prediction of rolling textures of anisotropic polycrystals. Muter: Sci. Engrg. A175,71-82 (1994). 80. E. H. Lee, Elastic-plastic deformation at finite strains. .I Appl. . Mech. 36, 1 4 (1969). 81. A. M. Maniatty, P. R. Dawson, and Y.-S. Lee, A time integration algorithm for elastoviscoplastic cubic crystals applied to modelling polycrystalline deformation. Internat. J. Numer: Methods Engrg. 35, 1565-1588 (1992). 82. E. B. Marin, and P. R. Dawson, On modeling the elasto-viscoplastic response of polycrytalline materials. Submitted for publication. 83. E. B. Marin, and P. R. Dawson, Elastoplastic finite element analyses of metal deformations using polycrystal constitutive models. Submitted for publication


167

84. E. B. Marin, P. R. Dawson, and J. T. Jenkins, Aggregate size effect on the predicted plastic response of hexagonal close-packed polycrystals. Modell. Simul. Muter. Sci. Engrg. 3,845-864 (1995). 85. K. K. Mathur, and P. R. Dawson, On modeling the development of crystallographic texture in bulk forming processes. Internat. J. Plast. 5,67-94 (1989). 86. K. K. Mathur, P. R. Dawson, and U. F. Kocks, On modeling anisotropy in deformation processes involving textured polycrystals with distorted grain shape. Mech. Muter. 10, 183-202 (1990). 87. K. K. Mathur, and P. R. Dawson, Texture development during wire drawing. J. Engrg. Muter. Technol. 112,292-297 (1990). 88. P. E. McHugh, R. J. Asaro, and C. F. Shih, Computational modeling of metal matrix composite materials. 1. Isothermal deformation patterns in ideal microstructures. Acta Metall. Muter. 41(5), 1461-1476 (1993). 89. D. Mika, Personal communication. 90. M. P. Miller, and P. R. Dawson, Influence of slip system hardening assumptions on modeling stress-dependence of work hardening. J. Phys. Mech. Solids. In press 91. A. Molinari, G. R. Canova, and S. Ahzi, A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall. 35, 2983-2994 (1987). 92. A. Molinari, and L. S. Toth, Tuning a self-consistent viscoplastic model by finite element results. I. Modeling, 11. Application to torsion textures. Acta Metall. Matei: 42, 2453-2458 (1994). 93. A. Morawiec, and J. Pospiech, Some information on quaternions useful in texture calculations. Textures Microstructures 10, 21 1-216 (1989). 94. A. Morawiec, The rotation rate field and geometry of orientation space. J. Appl. Cryst. 23,374377 (1990). 95. S. J. Murtha, J. M. Story, G. W. Jarvis, and H. R. Zonker, Anisotropy effects in the forming of aluminum sheet. In: Automotive sfamping technology, SAE SP-1067. Society of Automotive Engineers, Warrendale, PA, 1995, pp. 103-1 1 1. 96. A. Needleman, R. J. Asaro, J. Lemond, and D. Peirce, Finite element analysis of crystalline solids. Comput. Methods Appl. Mech. Engrg. 52,689-708 (198.5). 97. P. Neumann, Representation of orientations of symmetrical objects by Rodrigues vectors. Textures Microstructures 14-18, 53-58 (1991). 98. P. Neumann, The role of the geodesic and stereographic projections for the visualization of directions, rotations, and textures. Physica Status Solidi A: 131, 555-567 (1992). 99. R. W. Ogden, Nonlinear elastic deformations. Ellis Horwood, Chichester, 1984. 100. E. T. Onat, and F. A. Leckie, Representation of mechanical behavior in the presence of changing internal structure. J. Appl. Mech. 55, 1-10 (1988). 101. H. M. Otte, and A. G. Crocker, Crystallographic formulae for hexagonal lattices. Physica Status Solidi 9, 441450 (1965). 102. J. Pan, and J. R. Rice, Rate sensitivity of plastic flow and implications for yield surface vertices. Internat. J. Solids Structure 19, 973-987 (1983). 103. D. M. Parks, and S. Ahzi, Polycrystalline plastic deformation and texture evolution for crystals lacking five independent slip systems. J. Mech. Phys. Solids 38,701-724 (1990). 104. N. E. Paton, and W. A. Backofen, Plastic deformation of titanium at elevated temperatures. Metal. Trans. 1 , 2839-2847 (1970). 105. N. E. Paton, J. C. Williams, and G. P. Rauscher, The deformation of a-phase titanium. In: Titanium science and technology (R. I. Jaffee and H. M. Burte, eds.). Plenum, New York, 1973, pp. 1049-1069. 106. M. Peters, and G. Luetjering, Control of microstructure and texture in Ti-6A1-4V. In: Tiranium’80 science and technology (H. Kimura and 0. Izuma, eds.). Metallurgical Society of AIME, Warrendale, PA, 1980, pp. 925-935.

168


107. M. J. Phillipe, F. Wagner, and C. Esling, Textures deformation mechanisms and mechanical properties of hexagonal materials. In: Eighth international conference on textures of materials (J. S. Kallend and G. Gottstein, eds.). The Metallurgical Society, Warrendale, PA, 1988, pp. 837-843. 108. M. J. Phillipe, Texture formation in hexagonal materials. Muter. Sci. Forum. 157-162, 13371350 (1994). 109. M. J. Phillipe, M. Serghat, P. Van Houtte, and C. Esling, Modeling of texture evolution for materials of hexagonal symmetry. 11. Application to zirconium and titanium (Y or near (Y alloys. Acta Metall. Muter: 43(4), 1619-1630 (1995). 110. V. C. Prantil, J. T. Jenkins, and P. R. Dawson, An analysis of texture and plastic spin for planar polycrystals. J. Mech. Phys. Solids 41, 1357-1382 (1993). 11 1. V. C. Prantil, P. R. Dawson, and Y. B. Chastel, Comparison of equilibrium-based plasticity models and a Taylor-like hybrid formulation for deformations of constrained crystal systems. Model. Simul. Mater. Sci. Engrg. 3,215-234 (1995). 1 12. V. C. Prantil, J. T. Jenkins, and P. R. Dawson, Modeling deformation induced textures in titanium using analytical solutions for constrained single crystal response. J. Mech. Phys. Solids 43(8), 1283-1302 (1995). I 13. D. Peirce, R. J. Asaro, and A. Needleman, Material rate sensitivity and localized deformation in crystalline solids. Acta Metall. 31, 1951-1976 (1983). 114. J. L. Raphanel, and P. Van Houtte, Simulation of the rolling textures of BCC metals by means of the relaxed Taylor theory. Acta Metall. 33, 1481-1488 (1985). 1 15. M. M. Rashid, and S. Nemat-Nasser, A constitutive algorithm for rate dependent crystal plasticity. Comput. Methods Appl. Mech. Engrg. 94, 201-228 (1990). 116. M. M. Rashid, Texture evolution and plastic response of two-dimensional polycrystals. J. Mech. Phys. Solids 5, 1009-1029 (1992). 1 17. R. J. Roe, Description of crystallite orientation in polycrystalline materials. 11. General solution to pole figure inversion. J. Appl. Phys. 36, 2024203 I (1965). 118. G. Sachs, Zur Ableitung einer Fliessbedingung. Z. Ver. dt. Ing. 12, 134-136 (1928). 119. D. D. Sam, E. T. Onat, P. I. Etingof, and B. L. Adams, Coordinate free tensorial representation of the orientation distribution function with harmonic polynomials. Textures Microstructures 21, 233-250. 120. G. B. Sarma, and P. R. Dawson, Effects of interactions among crystals on the inhomogeneous deformation of polycrystals. Acta Mater. 44, 1937-1953 (1996). 121. G. B. Sarma, and P. R. Dawson, Texture predictions using a polycrystal plasticity model incorporating neighbor interactions. Infernat. J. Plast. 12(8), 1023-1054 (1996). 122. S. E. Schoenfeld, S. Ahzi, and R. J. Asaro, Elastic-plastic crystal mechanics for low symmetry crystals. J. Mech. Phys. Solids 43, 415446 (1995). 123. J. C. Simo, and L. Vu Quoc, Three dimensional finite strain rod model. Part 11. Computational aspects. Electronics Research Laboratory Memo UCBERL MS5/31. Univ. California, Berkeley, CA, 1985. 124. P. Steinmann, and E. Stein, On the numerical treatment and analysis of finite deformation ductile single crystal plasticity. Comput. Methods Appl. Mech. Engrg. 129, 235-254 (1996). 125. A. Tanabe, T. Nishimnra, and M. Fukada, The formation of hot rolled texture in commercially pure titanium and Ti-6A1-4V alloy sheets. In: Titanium’80 science and technology (H. Kimura and 0.Izuma, eds.). Metallurgical Society of AIME, Warrendale, PA, 1980, pp. 937-945. 126. G. I. Taylor, Plastic strains in metals. J. Inst. Metals 62, 307-324 (1938). 127. C. Teodosiu, J. L. Raphanel, and L. Tabourot, Finite element simulation of the large elastoplastic deformation of polycrystals. In: Large plastic deformations: Fundumenfal aspects and applications to metal forming (C. Teodosiu, J. R. Raphanel, and F. Sidoroff, eds.). Balkema, Rotterdam, The Netherlands, 1993, pp. 153-168.


169

128. C. Teodosiu, and F. Sidoroff, A theory of finite elastoviscoplasticity of single crystals. Internat. J. Engrg. Sci. 14, 165-176 (1976). 129. D. R. Thornburg, and H. R. Piehler, Cold rolled texture development in titantium and titanium alloys. In: Titanium science and technology (R. I. Jaffee and H. M. Butte, eds.). Plenum, New York, 1973, pp. 1187-1197. 130. D. R. Thomburg, and H. R. Piehler, An analysis of constrained deformation by slip and twinning in hexagonal close packed metals and alloys. Metal. Trans. A 6A, 1151-1523 (1975). 131. C. N. Tome, and U. F. Kocks, The yield surface of H.C.P. crystals. Acta Metall. 33, 603-621 (1985). 132. P. Tong, and T. H. H. Pian, A variational principle and convergence of a finite-element method based on assumed stress distribution. Internat. J. Solids Structures 5,463 (1969). 133. A. Van Bad, P. Van Houtte, E. Aemoudt, F. R. Hall, I. Pillinger, P. Hartley, and C. E. N. Sturgess, Anisotropic finite-element analysis of plastic metal forming processes. Textures and Microstructures 14-18, 1007-1012 (1991). 134. E. Van der Giessen, and P. Van Houtte, A 2D analytical multiple slip model for continuum texture development and plastic spin. Mech. Mater: 13, 93-1 15 (1992). 135. E. Voce, A practical strain-hardening function. Acta Metall. 51,219-226 (1948). 136. Z. Yao, and R. H. Wagoner, Active slip in aluminum multicrystals. Acta Metall. Mater: 41(2) 45 I 4 6 8 ( 1993). 137. H.-R. Wenk, ed., Preferred orientation in deformed metals: An introduction to modern texture analysis. Academic Press, London, 1985. 138. H.-R. Wenk, T. Takeshita, R. Jeanloz, and G. C. Johnson, Development of texture and elastic anisotropy during deformation of HCP metals. Geophys. Res. Lett. 15(1), 7 6 7 9 (1988). 139. H. R. Wenk, and U. F. Kocks, The representation of orientation distributions. Metal. Trans. 18A, 1083-1092 (1987). 140. H.-R. Wenk, K. Bennett, G. R. Canova, and A. Molinari, Modeling plastic deformation of peridotite with the self-consistent theory. J. Geophys. Res. 96, 8337 (1991). 141. P. D. Wu, K. W. Neale, and E. Van der Giessen, On crystal plasticity FLD analysis. Proc. Roy. Soc. London, to appear (1997). 142. M. J. Young, S. K. Choi, and P. F. Thompson, A finite element approach to modeling the development of deformation textures in FCC metals. In: Numerical predications of deformation processes and the behavior of real materials, 15th Rise, Symposium of Materials Science (S. Anderson, J. Bilde-Sclrensen, T. Lorentzen, 0. Pedersen, and N.Serrensen, eds.). Rise, National Laboratory, Roskilde, Denmark, 1994, pp. 621-626. 143. A. Zaoui, and J. L. Raphanel, On the nature of the intergranular accommodation in the modeling of elastoplastic behavior of polycrystalline aggregates. In: Lrrrge plastic deformations: Fundamental aspects and applications to metal forming (C. Teodosiu, J. L. Raphanel, and F. Sidoroff, eds.). Balkema, Rotterdam, The Netherlands, 1993, pp. 185-192. 144. Y. Zhang, and J. T. Jenkins, The evolution of the anisotropy of a polycrystalline aggregates. J. Mech. Phys. Solids 41, 1213-1243 (1993). 145. Y. Zhou, J. J. Jonas, M. Jain, and S. R. MacEwen, Calculation of polycrystal yield surfaces and forming limit diagrams of aluminum alloys sheets. In: Simulation of materials processing: Theory, methods and applications (S. F. Shen and P. R. Dawson, eds.). Balkema, Rotterdam, The Netherlands. 1995, pp. 369--374.



Nonlinear Composites PEDRO PONTE CASTANEDA Department of Mechanical Engineering and Applied Mechanics

University i$ Pennsylvania Philadelphia. Pennsylvaniu

and

PIERRE SUQUET Luborutoire de Micanique et d’Acoustique / C.N.R.S. Mar.seille, France

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172

175 I1. Effective Behavior and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . A . Individual Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 B . Local Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 C. Classical Variational Principles and Effective Potentials . . . . . . . . . . . 183 D . Classical Bounds of the Voigt and Reuss Type . . . . . . . . . . . . . . . . 185 I11. Variational Methods Based on a Homogeneous Reference Medium . . . . . . . 187 A . The Variational Principles of Talbot and Willis . . . . . . . . . . . . . . . . 187 B . Bounds via Piecewise Constant Polarizations . . . . . . . . . . . . . . . . . 189 191 C . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . VariationalMethods BasedonaLinearComparisonComposite . . . . . . . A . Nonlinear Composites with Isotropic Phases . . . . . . . . . . . . . . . B . Nonlinear Composites with Anisotropic Phases . . . . . . . . . . . . . . C . Bounds via Piecewise Constant Moduli . . . . . . . . . . . . . . . . . . . . D . Interpretation of the Variational Procedure as a Secant Method . . . . . E . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 192 . . 193 . . 198 205

. . 211

V . A Second-Order Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Weakly Inhomogeneous Nonlinear Composites . . . . . . . . . . . . . . . B . Nonlinear Composites with Arbitrary Phase Contrast . . . . . . . . . . . C . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213 216

. 216 . 220 227

VI . A Selection of Results for Linear Composites . . . . . . . . . . . . . . . . . . . 228 A . Hashin-Shtrikman Bounds for Random Composites 228 with Ellipsoidal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 ISBN 0-12-002034-1

ADVANCES IN APPLIED MECHANICS. VOL. 34 Copyright 0 1998 by Academic Press . All rights of reproduction in any form reierved . W65-2165/98$25 00

172

Pedro Ponte Castaiieda and Pierre Suquet B. Hashin-Shtrikman Bounds and Estimates for Random Composites with Particulate Microstructures . . . . . . . . . . . . . . . . . . . . . . . . 23 1 C. Self-consistent Estimates for Random Composites with Granular 233 Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Third- and Higher-Order Bounds for Random Composites . . . . . . . . . . 235

VII. Applications to Nonlinear Composites and Discussion . . . . . . . . . . . . . A. PorousMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Rigidly Reinforced Materials . . . . . . . . . . . . . . . . . . . . . . . . . . C. Two-Phase, Rigid Perfectly Plastic Composites . . . . . . . . . . . . . . . D. Additional Results and Comparison with Other Schemes . . . . . . . . . E. An Incremental Elastoplastic Model Based on the Variational Procedure . VIII. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 237 237 246 . 253 . 262 . 274

280

IX. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,281 A. Potentials, Convexity, Concavity and Duality . . . . . . . . . . . . . . . . . 28 1 B. Derivation of the Potential with Respect to a Parameter . . . . . . . . . . . 285 C. PTensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,288 D. A Brief Review of Other Schemes . . . . . . . . . . . . . . . . . . . . . . . 292 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.295

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.295

I. Introduction Heterogeneous materials are ubiquitous in nature, and increasingly so in manmade systems. Often the distribution of hetereogeneity is such that the material appears to be homogeneous at a large enough length scale. It is then of practical as well as theoretical interest to ask the question: What are the “average” properties of such heterogeneous materials? This review article is concerned with the theoretical prediction of the “effective properties” of such “composite materials,” directly from the properties of their constituent phases and their distribution, or “microstructure.” Of course, considerable progress has been made on this problem for the case where the constitutive behavior of the phases that make up the composite is linear elastic. The focus in this article will be on composites with nonlinear constitutive behavior, including plasticity and creep. Thus, for example, this article will be concerned with the determination of the effective stress-strain relations for metal-matrix composites. However, the emphasis will be on the development of “homogenization” methods that will apply to broad classes of nonlinear material behavior, as opposed to any specific material system. Although the effective behavior of the composite will be the main focus, it is also important to provide insight into the statistical distribution of local quantities, such as averages

Nonlinear Composites

173

or higher moments of the strains and stresses in each phase. These local quantities are extremely useful for understanding the evolution of nonlinear phenomena such as plasticity or damage. Research on composite materials with nonlinear constitutive behavior can be traced back to the classical work of Taylor (1938) on the plasticity of polycrystals, followed by the work of Bishop and Hill (1951a, b) and Drucker (1959) on ideally plastic polycrystals and composites, respectively. As will be discussed in the body of the paper, these estimates are the nonlinear equivalents of the VoigtReuss bounds for linear-elastic systems, and as such depend on the microstructure only through the proportions of the phases (or grain orientations). Recognizing the need to incorporate additional microstructural information to obtain improved estimates, several attempts have been made in the past to provide extensions of the self-consistent model for nonlinear composites, notably by Hill (1965a, b), who proposed an “incremental” version of the self-consistent method. This work spawned a large body of work, especially in the context of polycrystalline materials (see Hutchinson, 1970, 1976), including a simplified approach that has come to be known as the “secant” method (Berveiller and Zaoui, 1979). No attempt will be made here to review this work, although a brief discussion of the incremental and secant methods will be provided. In recent years, direct numerical simulations have been developed for composites with periodic microstructures (Christman et al., 1989; Tvergaard, 1990; Bao et al., 1991), as well as for composites with more general types of microstructures (Brockenborough et al., 1991; Moulinec and Suquet, 1995). Again, no attempt will be made to review this work here, although some comparisons with numerical simulations will be given in the text. The numerical approach has the advantage of high accuracy and has provided very useful insight into the relevant physical mechanisms at work in the deformation and failure of nonlinear composites, albeit at the expense of intensive computations, particularly for composites with complex random microstructures. Furthermore, the numerical approach has the disadvantage that, in general, it does not result in explicit forms for the overall constitutive relations, making them difficult to implement in stress analyses at the macroscopic level. The present article will be concerned almost exclusively with some closely related methods that have been developed recently to estimate the effective behavior of nonlinear composites with random microstructures. The methods are based on suitable approximations in the context of rigorous variational principles for the effective behavior of the nonlinear composites, and as such have the advantage of mathematical rigor, delivering bounds that improve, often significantly, on the classical bounds of the Voigt-Reuss type. In addition, they make effective use of the various estimates available for linear composites, allowing the

174

Pedro Ponte Castaiieda and Pierre Suquet

determination of corresponding estimates for nonlinear composites that are simple and which, in some cases, are even explicit. This explicit (or nearly so) character makes them extremely useful in the context of macroscopic simulations, allowing their straightforward implementation in standard finite-element codes (see Aravas et al., 1995). Next a brief outline of the paper is given. Section I1 deals with some preliminaries, including the definition of effective potentials, by means of the classical minimum energy principles, and their application to the determination of bounds of the Voigt-Reuss type for nonlinear composites. Section I11 gives a brief presentation of the variational principles of Talbot and Willis (1985). This pioneering work, which provided an extension of the Hashin-Shtrikman (1962a) variational principles for nonlinear composites, utilizing a linear homogeneous reference rnaterial, allowed the computation of the first improved bounds for nonlinear composites with random microstructures, incorporating statistical information of order two. Section IV presents new variational principles, originally developed by Ponte Castafieda (1991a) and Suquet (1993a), for isotropic and power-law composites, respectively, which allow the restatement of the effective energy function of a nonlinear composite in terms of that of an appropriately chosen linear heterogeneous reference material, where the distribution of moduli in the reference material is determined by the variational principle itself. Application of suitable approximations in the context of these variational principles then permits the determination of bounds for nonlinear composites, directly from corresponding bounds for linear comparison composites with the same microstructures as the nonlinear composites. In particular, these variational procedures allow the straightforward computation of the Talbot-Willis bounds, directly from the corresponding linear Hashin-Shtrikman bounds. But, more generally, they allow the determination of other types of bounds and estimates, which are not available from the Talbot-Willis procedure, again directly from their linear counterparts. These include, for example, higher-order bounds of the Beran (1965) type (Ponte Castafieda, 1992a). Section V starts with an asymptotic expansion, due to Suquet and Ponte Castaiieda (1993), which is exact to second order in the contrast of the properties of the phases and shows that the bounds and estimates obtained by the variational procedures of Section IV (and therefore those of Section 111) are only exact to first order in the contrast. This is unlike the Hashin-Shtrikman bounds and self-consistent estimates for linear composites, which are known to be exact to second order in the contrast. Motivated by this observation, a “second-order’’ procedure has been developed by Ponte Castafieda (1996a), which delivers estimates for nonlinear composites at finite contrast that are exact to second order. A selection of results for linear composites is presented in Section VI. Finally, Sec-


175

tion VII provides some applications to sample nonlinear composites, including porous materials, rigidly reinforced composites, and two-phase power-law and ideally plastic composites. This section also includes comparisons with the classical incremental and secant models, showing that the new procedures can yield significant improvements over the classical methods, which are sometimes found to give predictions that violate rigorous bounds (Gilormini, 1995). Throughout the text, vectors and second-order tensors will be denoted by boldface letters, whereas fourth-order tensors will be denoted by barred letters. In this connection, the various types of products will be denoted by dots (e.g., u * v = u i ~ l(L , :~ ) i = j I L i j k h ~ k h 11, , :: Q = L i j k h Q i j k h ) .

11. Effective Behavior and Potentials A. INDIVIDUAL CONSTITUENTS 1. Local Potentials

The constitutive behavior of the individual constituents of the composite is assumed to be governed by a potential, or strain-energy function, w ( E ) , in such a way that the (infinitesimal) strain and stress fields, E and a,are related by

aw a = -( E ) .

a&

This constitutive relation corresponds to nonlinear elastic behavior within the context of small strains. However, by interpreting E and a as the Eulerian strain rate and Cauchy stress, respectively, the above constitutive relation can be used equally well to model$nite viscous deformations. Assuming that w is a convex function of E , the relation (2.1) can be inverted with the help of the Legendre transformation,

u ( a ) = sup(a : E - ?a(&)}, E

defining a convex stressenergy function u , such that &

=

au -

am

(a).

(2.3)

The functions w and u are dual potentials; they are related by the classical reciprocity relations depicted schematically in Figure 1 and discussed in more detail in Appendix A. (Note that u = w* in the notation of the Appendix.)

176

Pedro Ponte Custafiedu and Pierre Suquet 3 00

”

0.0

0.04

0.02 E

FIG. 1. Schematic representation of the local potentials w ,u and of the secant and tangent moduli Es and Ec for the nonlinear constituents.

2. Isotropic Materials For isotropic materials, sufficiently general forms of w and u are given by W(E)

9

2

= - ks,

2

+

p(Eeq>,

(2.4)

and

where p and @ are dual convex potentials of a scalar variable. Also, a, and E, are the hydrostatic stress and strain, a, =

1

tr(u),

E,

=

1

tr(E),

a,, and E ~ ,are the Von Mises equivalent stress and equivalent strain,

177


ad and Ed being the stress and strain deviators, ad = a - ffmI, ~d = E - E m I . Note that the influence of the third invariant on the local potentials w and u has been neglected. The resulting stress-strain relations can be written,

f f m = 3k&m,

ad

= 2Ks(Eeq)Ed9

where aeq

p’(Eeq)

p r, (Eeq )=--=-------3~eq

3~eq

-

aeq 3+’(0eq)’

The response of the phases is therefore assumed to be linear for purely hydrostatic loadings (characterized by a constant bulk modulus k ) and nonlinear in shear (characterized by a strain-dependent secant shear modulus p S ) .

EXAMPLES.Two examples of constitutive potentials are worth mentioning explicitly. The first example is concerned with the J2 deformation theory of plasticity. In particular, the Ramberg-Osgood model corresponds to the choice

where po is the initial elastic shear modulus, a0 is a reference stress, EO is a reference strain, and n 2 1 is the hardening exponent. The second example is given by high-temperature creep of metals, which is commonly characterized in terms of a power law. Neglecting elastic effects and assuming incompressibility, the dissipation and stress potentials, p and $, of the material take the forms

(2.7) and +(aeq)= -ffoso ( y + l , n + l ao

where EO and a0 denote a reference strain rate and stress, respectively, and where n and rn are power exponents related by rn = l / n . Two special cases of powerlaw materials are of special interest. The Newtonian viscous material corresponds to n = m = 1, where p = a 0 / 3 denotes the viscosity. The Von Mises rigid ideally plastic material corresponds to the limit rn + 0 (or n + a), where a0 denotes the flow stress in tension. In this last case, the stress potential becomes

178

Pedro Ponte Castafieda and Pierre Suquet

unbounded for equivalent stresses exceeding 00, and it is useful to introduce the strength domain P , defined by the set

3. Crystalline Materials The general framework of potentials is also suitable to describe the creep of crystalline materials, which are, by definition, anisotropic materials. Consider a single crystal that undergoes creep on a set of K preferred crystallographic slip systems. These systems are characterized by the second-order tensors & k ) , k = 1 , . . . , K , defined by 1 P ( k ) = i ( n ( k )c3 m(k) m(k)c3 n(k)),

+

where n(k) and m(k) are the unit vectors normal to the slip plane and along the slip direction in the kth system, respectively, and c3 denotes the tensorial product of two vectors. Thus, when the crystal is subjected to an applied stress a, the resolved shear stress acting on the kth slip system is given by

The strain rate E in the crystal is the superposition of the strain rates on each slip system,

(2. I 1) where the shear strain rate y ( k ) on the kth system is assumed to depend on the applied stress only through the resolved shear stress r ( k ) ,in such a way that

(2.12) where the potentials + ( k ) are convex. A commonly used form for the slip potentials $ ( k ) is the pure power-law form

(2.13) where n(k) p 1 and (to)(k)are the creep exponent and reference stress of the kth slip system, respectively, and yo is a reference strain rate.


179

It follows that the constitutive relations (2.1 1) and (2.12) may be expressed in terms of the convex potential for the crystal (2.14) which is such that (2.15) The limit as all the n ( k ) + 00 corresponds to a rigid ideally plastic crystal, with strength domain given by (2.16) 4. General Form of the Potentials Motivated by these two classes of materials, it is assumed more generally that the potential w can be put in the form

for some appropriately chosen function F , where fined as 1 2

E

is a fourth-order tensor de-

(2.18)

E = - E @ E ,

possessing the usual diagonal symmetry and positive-definite property of an elasticity tensor

The function F is therefore defined on the space of fourth-order tensors Q with diagonal symmetry. Then the constitutive relation (2.1) can alternatively be written as d

=L,(E)

E,

(2.20)

where

aF ae

L,(E)= -(6)

(2.21)


180

is the secant modulus tensor of the material,' which also has diagonal symmetry. Under the assumption (2.17), the dual potential u can be put in the form

(2.22) where G is a function of fourth-order tensors s, which is related to F by explicit relations given in Appendix A. In terms of the secant compliance tensor of the material, the constitutive relation (2.3) may be expressed in the form E

= h'&(@)

aG

:6,

Ms(Q)= (Q). as

(2.23)

EXAMPLES.The potentials of the two classes of materials described above (isotropic or crystalline materials) can be written in the forms (2.17) and (2.22), respectively. For isotropic materials note that

2 =4 R :: 3

Eeq

2 = -J 2 :: Q, am

cTq:

3

6,

= 3K :: Q,

where J and IK are two orthogonal projection tensors:

Here that

I is the identity in the space of symmetric fourth-order tensors. It follows

W(E)

=

9 ke; 2

-

+ 9(eeq)= 3kJ :: + f(-43 R :: E ) , 6

with

f ( x ) = q(,E),

(2.25)

and u(u)=

1 2 2k

- Dm

+

+((aeq)

=

1 3k J :: Q + g(3K :: Q), with g(x) = $I&).

-

'Note that the extension F of w is well defined by (2.17) on tensors in the form o = (1/2)e @ E , but it is not uniquely defined for more general fourth-order tensors. The secant tensor L, is therefore not uniquely defined, as already noted by Gilormini (1995) and Suquet (1997a), directly from (2.21).


181

The corresponding secant tensors IL, in (2.21) and M, in (2.23) are obtained by straightforward derivations:

These secant tensors are isotropic. Similarly, for crystalline materials, note that

(2.28)

B. LOCALPROBLEM 1. Representative Volume Element Consider a representative volume element (r.v.e.) V of a composite where the size of the inhomogeneities is small compared to that of V . The composite is made up of N homogeneous phases V ( r ) ,r = 1, . . . , N , whose distribution is defined by characteristic functions x@). Let (.) and ( . ) ( r )denote spatial averages over V and V ( r ) respectively. , Then, for example,

where the notation

=

(x(')) are the volume fractions of the phases. The compact a(r)

= (u)(r),

g(r)

=p)(r)

(2.29)


182

will also be used to denote the average stress and strain in each phase. All phases are assumed to be homogeneous, with potentials w(I) and u @ ) ,and to be perfectly bonded at the interfaces. The potentials w and u depend on the position x inside V, in such a way that N

W(X,E ) =

N

C x ( ~ ) ( xw) ( ~ ) ( E ) ,

U(X, a) =

C

X ( ~ ) ( X )u " ' ( a ) .

(2.30)

r=l

r=l

2. Polycrystals A polycrystal will be regarded (in this study) as an aggregate of a large number of identical single crystals with different orientations. It can therefore be treated as a composite, where phase r is defined as the region occupied by all grains of a given orientation, relative to a reference crystal with potential given by (2.14). Letting Q(') denote the rotation tensor that defines the orientation of phase r , the corresponding potential u ( ~is) given by

where

3. Boundary Conditions and Averages The local stress and strain fields within V solve the so-called local problem consisting of the constitutive equations (2. l), the compatibility conditions satisfied by E , and the equilibrium equations satisfied by u:

aw u = -(e),

as

E

1 = - (Vu 2

+ Vu'),

div(u) = 0 in

V.

(2.31)

The problem is ill-posed in the absence of boundary conditions. Classically, two classes of boundary conditions can be considered on a V:

.

AfJine displacements

u(x) = E x on a V

Uniform traction

u(x) n(x) = Z n(x) on

.

-

(2.32)

a V.

(2.33)

The difference between the two types of boundary conditions has been discussed in detail by Suquet (1987). The mean strain and stress are the averages of the


183

local strain and stress fields (with usual extensions in the case of voided or rigidly reinforced materials): (2.34)

The relation (2.34) is either a mere consequence of the boundary condition (2.32) (when (2.32) is adopted) or a definition. Similarly, (2.35) is either a consequence of the boundary condition (2.33) (when (2.33) is adopted) or a definition. In either case, the following result (Hill, 1963) holds:

LEMMA(Hill). Let a be a divergence-free stressjield and u a displacement jield, and assume that one of the boundary conditions (2.32) or (2.33) is satisjied. Then, C : E = (a : E ( u ) ) .

(2.36)

It is emphasized that E and c are not necessarily related by the constitutive relation. Hill’s lemma is useful in determining the effective or homogenized behavior of the composite, which is defined here as the relation beween E and X.

c. CLASSICAL VARIATIONAL PRINCIPLES AND EFFECTIVE POTENTIALS The solutions u and a of the local problem (2.31), (2.32) can be given the following two equivalent variational representations:

Minimum potential energy. u is the solution of the problem (2.37) where

.

K(E) = {v = E x on a V).

(2.38)

Pedro Ponte Castatieda and Pierre Suquet

184

Minimum complementary energy. u is the solution of the problem

where

S(Z) = {t,div(t) = 0 in V,

(t)= Z}.

(2.40)

Effective strain-energy potential. Noting that the infimum problem in (2.37) defines the average strain energy in the composite, the effective strain-energy potential Weff is defined as

w e f f ( ~=)

inf ( w ( E ( v ) ) ) ) .

(2.41)

VEK(E)

Then,

a weff

-(E) aE

=

(-a w ( E ( u ) ) :

E(;))

= (u :

aE

~(2)).

But au/aE satisfies

au aE

- =II.x

on

av.

It follows from Hill’s lemma that

a weff(E) = ( u ) : II = x, aE

(2.42)

which is the effective stress-strain relation for the composite.

Effective stress-energy potential. Similarly, the effective stress-energy potential Ueff is defined as

in terms of which,

The potentials Weff and Ueff are convex (as a consequence of the convexity of w and u). It can can be further shown (see Suquet, 1987; Willis, 1989a), starting from (A2) of Appendix A and using Hill’s lemma, that they are in fact (Legendre) dual functions such that

Weff(E)+ Ueff(x) = ( w ( E ) )+ ( u ( u ) ) = (U : E ) = x : E.


185

The potentials (2.41) and (2.43) correspond to the boundary conditions (2.32). Adopting the boundary condition (2.33) would have led to different pairs of dual potentials, again given by the variational representations (2.41) and (2.43), but with different definitions of the set of admissible fields K(E) and S(X),namely,

K(E) = (v,

(E(v))

= E},

S(X) = {t,div(t) = Oin V,

t

. n = X VnonaV}.

However, under the additional assumption that the potentials w and u are strictly convex, the two types of boundary conditions are equivalent for the r.v.e., and are equivalent to the periodic boundary conditions used in the mathematical theory of homogenization (Bensoussan et al., 1978; Sanchez-Palencia, 1980). The limiting case of rigid ideally plastic materials for which the potentials are convex but not strictly convex requires special treatment, as in Bouchitte and Suquet (1991). Then, Weff is a positively homogeneous function of degree one in E, usually referred to as the plastic dissipation function, and it is useful to introduce the effective strength domain of the composite, defined as (Suquet, 1983,1987; de Buhan and Taliercio, 1991) Peff= { X such that there exists a(x) with ((r) = X, div((r(x)) = 0, and a(x) E P @ ) ,for x in phase r } .

(2.44)

Note that

Weff(E) =

{X : E},

SUP

x E peff and that

U " f f ( X )=

o +cc

i f X E peff, otherwise.

The boundary of Peffdefines the extremal surjfiuce of the composite (Hill, 1967).It can be characterized by a yield function F e f fsuch , that F e f f ( X 5 ) 0 if X E P e f f . The extremal surface depends on the flow stresses, the volume fractions and the arrangement of the individual phases. For a discussion of the general properties of P e f f ,refer to Suquet (1987).

D. CLASSICAL BOUNDS OF THE VOIGT AND REUSSTYPE The minimum energy principles (2.37) and (2.39) may be used to obtain rigorous bounds for the effective potentials Weff and Ueff. This was recognized long ago by Bishop and Hill (1951a,b) for rigid ideally plastic polycrystals and by

186

Pedro Ponte Custuiiedu and Pierre Suquet

Drucker (1966) for composites with elastic ideally plastic phases. Use of uniform trial fields in the variational principles (2.37) and (2.39) leads to the rigorous bounds of the Voigt (1889) and Reuss (1929) type,

and

c N

U"ff(Z;>5 ( u ) ( Z ) =

c(r)u(r)(Z),

(2.46)

r=l

or, equivalently,

where an upper * denotes the convex dual function. In the context of polycrystals, these bounds are often referred to as the Taylor (1938) and Sachs (1928) bounds, respectively.

EXAMPLE.The Reuss and Voigt bounds for incompressible, isotropic, power-law constituents read

where

When m = 0 (corresponding to rigid-plastic materials), the Reuss bound simplifies to

a ; =

inf

r = l , .... N

air).

As already observed, the Voigt and Reuss bounds incorporate only limited microstructural information, in the form of the volume fractions of the constituent phases. Because of this, they are not very useful in general, particularly when the contrast between the phases is large. In fact, they are known to be exact only to first order in the contrast (see Section V.A.l). The following sections are concerned with variational methods that have been developed to obtain sharper bounds incorporating additional microstructural information-beyond the phase


187

volume fractions-in the form of two- and higher-point statistics for the distribution of the phases in the composite.

111. Variational Methods Based on a Homogeneous Reference Medium Hashin and Shtrikman (1962a, b, 1963) introduced a variational procedure to estimate the effective modulus tensor of elastic composites with statistically isotropic microstructures. The Hashin-Shtrikman (HS) variational principles consist in an alternative representation of the effective potentials for the composite in terms of suitably chosen polarization fields relative to a homogeneous reference material with modulus tensor I,(”).Making use of the Green’s function for the elasticity problem associated with the linear homogeneous reference material, and using the hypothesis of statistical isotropy, Hashin and Shtrikman were able to obtain rigorous upper and lower bounds for the effective modulus tensor Leff of the composite, by respectively choosing I,(o)to be equal to the “maximum” and “minimum” modulus tensors of the phases I,(“).A generalization of the HS variational principles, which is suitable for nonlinear composites, was given by Talbot and Willis (1985), following Willis (1983). This extension of the HS variational principles, which can be used to obtain improved bounds (depending on up to two-point statistical information) for nonlinear composites, is discussed next. A. THEVARIATIONAL PRINCIPLES OF TALBOTAND W I L L I S Define w ( ” ) to be the potential function of a linear homogeneous reference material with uniform modulus tensor I,(’), such that

and assume that the “difference” potential (w- w(O)) is a concave function, so that the concave polar of this difference potential is defined as (see Appendix A)

*

It then follows from the fact that the difference potential is concave that

w(x, e ) - w(O)(e) = i n f { t : e - ( w - w(’))*(x, t)}, *A subscript * identifies a concave polar, whereas a superscript * denotes a convex polar.

(3.2)

188


which is nothing more than the statement that w - w(O) = (w - w(’))**. Therefore, substituting (3.2) for w in expression (2.41) for the effective potential Weff, and interchanging the order of the infima over E and t,one arrives at

Taking the “polarization field” t in the “inner” minimum problem as given, it follows that the minimizing displacement field u solves the boundary value problem:

div(IL(’) : E ( u ) )= -div(t),

u E IC(E).

(3.4)

Then, making use of the Green’s function G(’) associated with the system (3.4) in the domain V , it is possible to obtain the following expressions for the strain tensor:

where E is the average strain over V , and where r ( O ) is the linear integral operator defined by r(0)

*t=

1”

r(O)(x,x’) : [t(X’)

-

(t)] dV’,

(3.6)

whose kernel is related to G(O)via

In this last relation, the parentheses enclosing the subscripts denote symmetric parts, and in (3.6), the term (t)has been added to “regularize” the integral (see Willis, 1981). was essential in Note that the concavity of the difference potential (w attaining the equality in (3.3). If (w - w ( ~ is ) )convex, it is possible to obtain a corresponding result where the infimum over the polarization fields is replaced by a supremum, and where the concave polar is replaced by a convex polar (see Talbot and Willis, 1985). Typically, however, (w - w(O)) is neither concave nor convex-for example, for a power-law material with potential (2.7). In this case, the equality in (3.3) must be replaced by an inequality (either 5 or 2 , depending on whether (w - w(’)) has weaker-than-afine or stronger-than-afine growth at infinity, respectively). A dual variational principle may be obtained (see Talbot and Willis, 1985, for details), starting from the definition (2.43) of the stress potential Ueff, and


189

assuming that the function ( u - u(O)) is convex, where u(O) is the stress potential of the linear reference material, dual to (3.1). Then, using the fact that u - u(O) = ( u - uto))**, it can be shown that

where 9 now plays the role of a strain polarization. For a given polarization distribution, the inner problem can once again be solved formally by means of the relevant Green's functions in terms of 6. For brevity, these details will be omitted here (see Willis, 1981).

B. BOUNDSVIA PIECEWISE CONSTANT POLARIZATIONS Of course, the polarization field in (3.3) is difficult to determine exactly. Because of this, it is helpful to introduce the approximation of piecewise constant polarization: N r=l

Recalling that denote the volume fractions of the phases, and making = (x use of the fact that the average of a tensor field T over phase r is given by (T)@)= ((x(~)/&))T), it follows from (3.5) and (3.6) that the average E @ ) = ( E ) ( ' ) over phase r of the strain tensor E is given by .

N

where

( r , s = 1, . . . , N ) are tensors that depend only on the microstructure of the composite and on L(O).Furthermore, they are known (Kohn and Milton, 1986) to be symmetric in the superscripts r and s, and are not all independent, satisfying the relations N

N

r=l

s=l

190


Similarly, using integrations by parts, together with the system (3.4) and relation (3.3, it can readily be shown that ( W @ ) ( ~ ( U ) ) )=

1 2

w(")(E)- -(t : ( E ( u )- E)),

(3.11)

but since .

N

N

it follows, from (3.3), that N

_ -1

2

N

N 7; t(r) :r ( r s ) :

t(s)

(3.13)

r=I \ = 1

where (t)= C Nr Z~ l ( ' ) tThen, ( ~ )optimizing . over the polarizations 1, . . . , N ) , one obtains the conditions (for the t('))

t(')

(r =

( r = 1 , . ..,A'), (3.14)

so that, from (2.42), (3.13), and (3.14), one finally gets, replacing the inequality in (3.13) by an equality, the approximate stress-strain relation

C = L'O' : E + (t), where the t(') are obtained from relations (3.14). The upper bound (3.13) for Weff(E) was given by Ponte Castaiieda and Willis (1988), following an analogous development by Willis (1986) in the context of nonlinear dielectric composites. An alternative form of the result (Willis, 1991) may be obtained by noting, through the use of (3.9), that the optimality conditions (3.14) can be rewritten in the form

which can be inverted (see Appendix A) to give (3.15)


191

so that the (Legendre) dual variables E(') satisfy the conditions

( r = 1, . . . , N ) .

(3.16)

In terms of these variables, the bound (3.13) may be rewritten in the form

(3.17) where the t(') are given in terms of the E ( r ) by expression (3.15). It is possible to obtain nontrivial lower bounds for the effective stress potential U e f fby an analogous development, starting from (3.7), and taking the polarization field 7 to be piecewise constant. The details are omitted here (see, for example, Willis, 1991) C. DISCUSSION The expression (3.17) provides an upper bound for Weff, for any choice of w(O);the best bound is obtained by minimizing over IL(O). Note that the bound is finite only if (w- w(O)) has weaker-than-afJine growth at infinity, which would be the case, for example, for the class of power-law materials defined by (2.7). The minimization with respect to IL(O) is complicated by the fact that the computation of (w(') - w(O))** can be difficult. Ponte Castafieda and Willis (1988) and Willis (1989a, b) obtained nonoptimal bounds by restricting their attention to values of IL(O) for which (w(')- w(O))** = ( ~ ( ~ w(O)). 1 Later, Willis (1991, 1992) showed that improved bounds, agreeing with those of the variational procedure of Ponte Castafieda (1991a), could be obtained by eliminating this unnecessary restriction. For the special case of composites with linear constitutive behavior, it is possible to choose L(O),in two different ways, such that ( ~ ( ' 1 - w(O)) is either concave or convex, thus allowing, from (3.13), the construction of both upper and lower bounds for the effective modulus tensor of the composite (i.e., the classical Hashin-Shtrikman bounds). By contrast, for nonlinear composites, only a onesided bound is available for Weff by the method outlined above, the direction of which depends on the growth conditions on w.In this connection, it is emphasized that the upper bound (3.17) obtained from (3.3) for Weff and the corresponding lower bound obtained from (3.7) for Ueff can be shown to be exactly equivalent

192

Pedro Ponte CastaAeda and Pierre Suquet

(see Talbot and Willis, 1985). However, Talbot and Willis (1994,1995) have succeeded in obtaining improved bounds in the “other” direction, by the introduction of suitably chosen nonlinear reference materials. The classical upper bounds can be recovered formally from the above TalbotWillis variational principles in the limits as the reference modulus tensor L(O) tends to 0 or 00. Thus, for example, it is easy to see that letting tend to 00 in (3.7) and taking the polarization to be uniform leads to a trivial problem for c,which when solved leads to the Voigt upper bound (2.45). Similarly, the Reuss bound can be obtained directly from (3.7), with L(O)tending to 0. The estimate for Weff provided by (3.13) and (3.14), or (3.16) and (3.17), after optimizing over the choice of I,(’), is explicit except for the microstructural parameters r(rs), which must be determined for specific classes of microstructures. Explicit expressions for these microstructural parameters have been determined by Willis (1977, 1978) and Ponte Castaiieda and Willis (1995) for various classes of random microstructures with prescribed two-point correlation functions for the distribution of the phases, including microstructures of both the “particulate” and “granular” types (see Section VI). By using these results, bounds have been determined for special types of nonlinear composites and polycrystals, for example, by Ponte Castaiieda and Willis (1988), Willis (1989a, b, 1991), Talbot and Willis (1991), and Dendievel et al. (1991).

IV. Variational Methods Based on a Linear Comparison Composite Variational methods for obtaining improved bounds and estimates for the effective behavior of nonlinear composites, making use of the effective modulus tensor of suitably chosen linear-elastic comparison composites, were introduced by Ponte Castaiieda (1991a) for composites with isotropic phases and by Suquet (1993a) for composites with power-law constituents. In addition, a hybrid of the Talbot-Willis and Ponte Castaiieda procedures, using a linear thermoelastic comparison composite, was proposed by Talbot and Willis (1992). Finally, a procedure, also using a linear-elastic comparison composite, specifically designed for rigid ideally plastic composites, was developed by Olson (1994). Although all of these procedures are formally different (and were given different derivations by the various authors), they can in fact be shown to be equivalent under appropriate hypotheses on the local potentials. A distinct advantage of these variational procedures involving linear comparison composites, however, is that they can not only easily recover the nonlinear Hashin-Shtrikman bounds of the Talbot-Willis procedure, directly from the corresponding linear Hashin-Shtrikman bounds, but,


193

in addition, they are able to deliver higher-order nonlinear bounds, such as Berantype bounds, as well as other types of estimates that may not be available from the Talbot-Willis procedure. In this section, brief accounts will be given of the variational principles of Ponte Castaiieda (1992a) and Suquet (1993a) for composites with isotropic and powerlaw phases, respectively. This will be followed by a generalization, due to Suquet (1997b), of the variational procedure of Ponte Castaiieda, which is developed for nonlinear composites with anisotropic constituents. It will be shown that this generalization is capable of delivering as special cases, not only the variational principles of Ponte Castaiieda (1992a) and Suquet (1993a) for composites with isotropic constituents, but also those of deBotton and Ponte Castaiieda (1995) for polycrystalline aggregates of anisotropic single-crystal grains. The variational principles will then be used to generate a general class of bounds for nonlinear composites, by making suitable approximations in the set of admissible linear comparison composites. Finally, various interpretations of the variational procedures will be developed, including a recent interpretation of the resulting bounds in terms of secant moduli, due to Suquet (1995).

A. NONLINEAR COMPOSITES W I T H ISOTROPIC PHASES 1. Variational Principles for Materials with a Certain Concavity Hypothesis Consistent with relations (2.4), (2.25), and (2.30), the potential w of a nonlinear composite with isotropic phases may be written in the form

where N

N

k(X) =

X(r)(X)k(r), r=l

f ( X , E$)

=

X(r)(X)f(r)(E:q),

(4.1)

r=l

and where the functions f " ) , characterizing the deviatoric behavior of the material, are defined by the relations f ( ' ) ( p ) = v ( ' ) ( E ~ ~for ) p = E : ~ . It will be assumed that the functions f ( ' ) satisfy the following.

Concavity hypothesis. The f ( ' ) are assumed to be concave functions of p , such that f ( ' ) ( p ) = -co for p < 0, f ( ' ) ( O ) = 0, and f ( ' ) -+ 00 as p -+ 00.

194


By definition (see Appendix A), the concave dual function of f

( r ) is

given by

where the last equality follows from the fact that - f ( ' ) ( p ) = 00 for p < 0. Note that f:' is a concave, nonpositive function, such that f *( r )(4) = -00 for q 5 0. It follows from the concavity hypothesis that f::' = f ' " ) , and therefore that

Note that the above hypotheses on f ( r ) are consistent with weaker-than-quadratic growth for w@)at infinity, in agreement with the physical requirements for plasticity and creep. For example, for the power-law material (2.7), ~ p ( ~ ) (0 5 m 5 l), so that f ( ' ) p ( ' + m ) / 2is a concave function in the interval [0, 001, even if ~ p (is~itself ) convex. Next, introduce a linear comparison composite with potential wo,such that

-

- EL:^

Then, using the identity (4.2) with q = ip.0, it follows that the potential of the nonlinear composite w may be given the exact representation (4.4) where (4.5) Note that (4.6)

so that (4.7)


195

Now, substituting expression (4.4) for the local potential w into expression (2.41) for the effective potential Weff, it follows that

from which it is concluded, by interchanging the order of the infima over PO,that

E

and

where V ( p 0 ) = (v(x, po(x))) and Wit'" denotes the effective potential of the linear comparison composite: (4.9) This variational representation for the effective potential of a nonlinear composite in terms of the potential of an appropriately chosen "linear comparison composite" is due to Ponte Castaiieda (1992a). It is emphasized that, under the concavity hypothesis on the f'') functions, the variational representation (4.8) and the classical representation (2.41) are exactly equivalent. Thus, under these circumstances, the variational statement (4.8) can be given the following interpretation: the effective potential Weff of the nonlinear composite solid, as defined by (2.41), can be alternatively determined from the effective potential Wtff of a linear heterogeneous comparison solid, with bulk modulus k(x) and with shear modulus po(x), the precise variation of which is determined as the solution of the variational problem (4.8). Note that although the bulk modulus of the linear comparison composite is constant within each phase (the same as for the nonlinear material), the corresponding shear modulus will be nonuniformly distributed within each phase. Because of this, the variational principle (4.8) is, in general, at least as difficult to implement as the original classical variational principle (2.41). However, the variational principle (4.8) has the advantage that it allows useful approximations that are not accessible directly from the classical variational principles. One possible approximation in (4.8) is to replace the minimum over the set of arbitrarily variable, nonnegative shear moduli by the smaller set of piecewise constant, nonnegative moduli. The application of this approximation for the determination of bounds and other estimates for the effective behavior of nonlinear composites will be considered in a later subsection. This type of approximation was introduced, directly, by Ponte Castaiieda (1991a). It is also possible to start from the complementary energy representation (2.43) for Ueff to obtain a corresponding dual version of the variational statement (4.8).

196

Pedro Ponte Custufiedu and Pierre Suquet

To accomplish this, on account of (4.4), note that

where

is the stress potential of the linear comparison composite. From (4.7), also note that E

[

: u - u(x, u)}- wo(x, E ) )

= sup sup { E : a U

-

wo(x, E ) } - u(x, a)]

E

= sup {uo(x, u) - u(x, (7)).

(4.1 1)

U

Then, substituting expression (4.10) for u into expression (2.43) for U e f f ,and interchanging the infimum over u and the supremum over po, which is allowed by an appropriate version of the saddle point theorem, it is concluded that

Ueff(z) = sup

{u;"ff(z) - V(po)},

(4.12)

/I.O(X)>O

where it is recalled that V ( p o )= (v(x, po(x))) and where

is the effective stress potential of the linear comparison composite. Finally, it is emphasized that the two dual versions (4.8) and (4.12) are exactly equivalent (i.e., there is no duality gap). To see this note that

Ueff(C) =

SUP

{E : C - Weff(E)}

E

=

sup wn(x)>o

[ sup {E : C - Wiff(E)} - V ( p 0 )) E

(4.14)


197

2. Variational Principles for Power-Law Materials Special variational principles for pure power-law materials, also using linear comparison composites, have been proposed by Suquet (1993a). The composites under consideration here are made up of power-law materials with strain-energy functions (2.7) with the same exponent n and the same reference strain EO, but with different flow stresses q.For such composites, the variational characterization of the relevant effective strain potentials specializes to

Next, introducing a linear composite, which is incompressible and isotropic with an arbitrary nonnegative shear modulus po(x), at point x, and which occupies the same volume element V as the given nonlinear composite, it is possible to write that

Then, using Holder's inequality, one can show (see Suquet, 1993a, 1997a) that

for all positive fields po(x). The above inequality is in fact an equality when po(x) = ~ a o ( x ) ~ & - ~ ( v (and x ) ) therefore

The infimum of (aos&+'(v)) over all admissible fields v can be computed from (4.15) by interchanging the order of the infima, and one obtains, finally,

Note that when the constituents are power-law with the same exponent, the en) the effective energy Weff are positively homogeneous of ergy functions w ( ~ and degree rn 1 with respect to E and E, respectively.Therefore (4.16)compares two

+


198

functions, Weff and (W~"f)(mt1)/2, which have the same degree of homogeneity. The variational principle (4.16) can be given a dual and equivalent form,

B. NONLINEAR COMPOSITES WITH ANISOTROPIC PHASES 1. Generalized Variational Principles Assume that the functions F ( r j , defining the strain potentials w ( ~through ) relations of the type (2.17), are concave on the space of positive, symmetric fourthorder tensors e, i.e., they satisfy the following inequality:

F(')(te1

+ (1 - t ) Q ) 2 tF(')(e1) + (1 - t)F(')(e*)

ve1, e2,

0 5 t 5 1.

(4.18)

Note that this hypothesis implies weaker-than-quadratic growth for the potentials w ( ~on) the strain E , when e is set equal to E , as defined by (2.18). The concave dual function of F(') is defined (see Appendix A) as

F.$)(L) = inf(IL :: Q - F ( ' ) ( Q ) ) e

Note that F,'"

is concave, and that F(')(Q) = inf(L :: e - F : ~ ) ( I L ) ) , IL

from which it follows that

L=

~

aFi.1

a F(') (Q)

is the inverse of

ae

8

=

~

aL

(IL),

(4.19)

these relations being understood in the sense of subdifferentials when the functions F and F* are not differentiable. Next, define F ( x , e) as N

r=l

Recalling the definition (2.18) of F ( X , E < v ( x > ) )= inf

6 , at

every point x in V , one has that

{ I L ~ ( X ) ::

E ( v ( x ) )- F,(x,ILO(X))}.

M X )

The infimum is in fact restricted to stiffness tensors Lo(x) for which the function F* is finite. This restriction implies that ILo(x) is positive definite (see Ap-


199

pendix A). It follows, from the definition (2.41) of the effective potential Weff, that

Then, introducing a linear comparison composite with local potential,

1 wo(x, e(v)) = ILO :: E ( V ) = - E ( V ) : ILo(x) : E ( V ) , 2

(4.21)

and interchanging the infima in (4.20), one arrives at the following exact variational representation for the effective potential, namely,

where Woff is the effective potential (4.9) of the linear comparison composite defined by the local potential (4.21), and V(IL0) = (u(x,ILo(x))),with

u(x, ILo(x)) = -F*(x,ILo(x)) = sup [F(x, Q) - ILo(x) :: Q].

(4.23)

Q

Note that the optimality condition in (4.23) is precisely the definition of the secant modulus IL, (cf. relation (2.21)). The relation (4.22) expresses the nonlinear effective properties of the composite (through its potential Weff) in terms of two functions:

Wgff is the elastic energy of ajctitious linear heterogeneous solid (called the linear comparison composite) made up of phases with stiffness ILo(x) at point x; the linear comparison solid is chosen from among all possible comparison composites by solving the optimization problem (4.22).

u(x, .), the role of which is to measure the difference between the nonquadratic potential w (x,.) and the quadratic energy of the linear comparison solid. The relation (4.22) is exact and strictly equivalent to the variational characterization of Weff given in (2.41). It also provides an interpretation of the strain field in the actual nonlinear composite, which is the strain field in the optimal linear comparison solid. However, solving the optimization problem (4.22) exactly is a formidable task, and is often as difficult to achieve as the initial variational problem involved in the definition of Weff.The function V can be explicitly expressed in terms of the dual functions F:), which at least for simple potentials w ( ~are ) easy to evaluate. The difficulty lies in the precise determination of the energy Wtff

Pedro Ponte Custufieda and Pierre Suquet

200

for a linear comparison solid consisting of infinitely many different phases. Very little is known about this problem, and progress in the understanding of N-phase linear composites (with N 1 3) would certainly result in a better understanding of nonlinear composites, as clearly reflected by (4.22). For this reason, except for very specific situations, e.g., laminates (deBotton and Ponte Castaiieda, 1992; Suquet, 1993a), the optimal solution of (4.22) is not known, and only suboptimal solutions are determined. By “suboptimal” it is meant that the class of linear comparison composites considered must be restricted for explicit results to be obtained. One possible reduction of the problem consists in, instead of minimizing over the whole set of arbitrary variable L(x), considering only the smaller set of piecewise constant positive stiffness tensors. This is carried out in Section 1V.C. However, the general principle given can also be used in cases where the distribution of stiffness tensors does not coincide with the distribution of the phases. This was illustrated, for instance, in Suquet (1993a), where a well-known result by Drucker (1966) was recovered by choosing appropriately the linear comparison composite. This result is that the shear strength of a composite where a shear plane can be passed through the weakest phase is the strength in shear of the weakest phase. The appropriate choice of the linear comparison composite was a compliant band around the shear plane surrounded by a stiff matrix. Another example was given in Ponte Castaiieda (1992b), where exact results were derived for sequentially laminated composites, making use of nonuniform choices for the comparison moduli in the matrix phase of the laminates. A variational representation for the stress potential Ueff that is equivalent to (4.22) for Weff may be obtained by following an argument exactly similar to that followed in (4.14), to obtain the result that

U“ff(X) = sup {U;”f(X)- V(ILO)}.

(4.24)

Lo>O

Finally, note that the functions G(‘) in the alternative writing (2.22) of the stress potential are convex functions of the fourth-order tensor s (see Appendix A), such that U ( X , t)=

G(R) = SUP (Mo(x) :: R - G*(Mo(x))}, Mo

1

R = - t @ t. 2

It is then possible, directly from the variational representation (2.41) for Ueff and using the minimax theorem, to obtain the result that

Ueff(X) = SUP {Uiff(X)- (G*(Mo))}, Mo

(4.25)

20 1

Nonlinear Composites where

(4.26) Again the effective potential Ue' governing the nonlinear properties of the composite is expressed in terms of the quadratic stress potential of a suitably optimized linear comparison composite. 2. Special Variational Principles

i. Isotropic constituents.

When the individual constituents are isotropic with strain potentials of the type (2.4), the function F , as has already been shown, has the form

F ( Q )= 3kJ :: Q

F ( e ) = -co

+ f ( 43 JK :: e ) -

when JK :: Q ? 0,

otherwise,

where k and f are phase dependent. Then, the dual function F, is given by (the details of the calculation are given in Appendix A) F* (Lo) = f*

3

(5 PO),

when Lo admits the decomposition

Lo = 3kJ

+ 2p&,

with

PO > 0,

(4.27)

or -cc otherwise. The infimum over LOin (4.22) can therefore be restricted to isotropic tensors LOin the form (4.27), without loss of generality, and the variational statement (4.24) for Weff reduces to the simplified form (4.8) of Ponte Castaiieda (1992a). Similarly, both forms of the the dual variational principle (4.24) and (4.25) can be shown to reduce to (4.12) for composites with isotropic constituents.

ii. Polycrystals. When the individual constituents are single crystals (refer to Sections II.A.3, II.A.4 and II.B.2), the functions G(') have the form K k= 1

where it is recalled that

202


It can then be shown (see Appendix A) that

when the compliance tensor M(r)has the form (4.28) and +cc otherwise. The supremum in (4.25) can therefore be restricted to compliance tensors Mcr) having the decomposition (4.28). Thus the corresponding local stress potential for the linear comparison polycrystal is given by

It follows that the final form of (4.25) for polycrystals is

U"ff(X) =

{ U,"ff(X) - V ( a { ; i ) } .

sup

(4.30)

cy;~))(x)>o

r = l , ..., N , k = l , ..., K

where Uiff is the effective potential associated with the linear comparison polycrystal with grain potentials (4.29), and where

r=l k=l

This is the variational principle of deBotton and Ponte Castaiieda (1995) for nonlinear polycrystals, which was originally derived by assuming convexity of the functions g,(Li.Note that the functions a,(;;(x),k = 1, . . . , K , are defined over the region in space occupied by the crystals with fixed orientation r. iii. Power-law constituents. When the individual constituents are (incompressible) power-law materials with the same exponent m (recall that 0 5 m 5 l), the composite itself is a power-law materiaL3 In other words, the local potentials and the effective potential are positively homogeneous of degree rn 1:

+

W ( r ) ( h E= ) h ~ + l ~( (E ') ,)

Weff(hE) = Am+' Weff(E)

+

Vh >_ 0.

3T0 prove that Weff is positively homogeneous of degree m 1, it suffices to note that the solution of the local problem (2.31), (2.32) corresponding to an overall strain hE is Xu, where u is the displacement field for an overall strain E.


203

The function F defining the strain potential w is itself a power-law function of degree ( m 1)/2, and its dual function is a power-law function of degree (m l)/(m - 1). Next, letting Lo(x) = tf,o(x), for arbitrary positive t , and noting that W;" and V = - ( F * ) are homogeneous of degrees 1 and ( m l ) / ( m - 1) in Lo, respectively, it follows from the variational statement (4.22) that

+

+

+

+

Weff(E) = jnf inf tW;ff(E) t ( m + l ) / ( m - ' ) V ( & ) ) / . Lo>Ot>O

where W;" is the same as Wtff in (4.22), with Lo replaced by t o . Then the minimum over t can be easily evaluated to obtain another exact representation of Weff (note that V = - ( F * ) is positive; see Appendix A):

where the hats have been dropped for simplicity. A similar exact representation for Ueff can be given (the details are omitted for brevity):

iii.1 Isotropic power-law constituents. As already mentioned, when the individual constituents are isotropic, it is sufficient to consider isotropic linear comparison composites. For materials governed by the potential (2.7), the functions f , f * , g and g* take the forms

and

The variational principles (4.3 1) and (4.32) then reduce to the variational principles (4.16) and (4.17) of Suquet (1993a), which were originally derived by means of Holder's inequality, as seen previously.


204

iii.2 Power-law polycrystals. The corresponding formula for power-law polycrystals with slip-system potentials (2.13) is obtained similarly, directly from (4.32). The result is

r=l,,.,,N, k = 1 , ..., K

iv. Rigid ideally plastic constituents. In the ideally plastic limit ( m + 0), the variational representations (4.3 1) and (4.32) respectively reduce to

and

U"ff(Z) =

o

if U;"'(X) i v ( L ~ n) o +co otherwise.

MY^,'

> 0,

(4.35)

The right-hand side of (4.34) is, for every LO, a positively homogeneous function of degree one with respect to E. It is the dissipation function associated with the convex set P(Lo) = { X such that Utff(X) 5 V(Lo)}. Taking the infimum over Lo in (4.34) is equivalent to taking the intersection between the sets P(L0): peff =

n

P(L~).

LpO

This last expression can be useful when one is looking for P e f f rather , than Weff, and any choice of LOgives an estimate of Peff from the outside. Note that all of the sets P(L0) are defined by a quadratic equation in stress space, but that their intersection can have an arbitrary shape. In particular, it can have vertices, as illustrated by the case of composites reinforced by aligned fibers considered in Section VII. Similarly, simplified forms of the above results may be obtained for rigid ideally plastic composites with isotropic constituents, as well as for rigid ideally plastic polycrystals.


205

c. BOUNDSVIA PIECEWISE CONSTANT MODULI 1. Nonlinear Composites with Anisotropic Phases

i. Strain potential Weff. As mentioned in Section IV.B, the exact representation for the effective potential Weff requires the determination of the effective potential of a linear composite with infinitely many different phases, which is an extremely difficult problem. One possible simplification of the problem is to restrict the optimization over Lo(x)to the set of piecewise constant moduli,

c N

Lo(@ =

(r) (r)

x

(x),

(4.36)

r=l

ILt)

where the tensors are taken to be constant. The resulting optimization problem no longer provides an exact representation of the effective potential We' but, instead, gives an upper bound for this potential

where Woff is now the effective potential (4.9) of a linear composite with the same microstructure as the nonlinear composite. In this linear comparison composite, the domains V(') are occupied by linear phases with stiffness Lt).This comparison composite has an overall stiffness Liff,so that 1 Wiff(E) = - E : Lgff : E, 2

(4.38)

and the functions d r )are defined by (4.39) The bound (4.37) is a generalization for composites with anisotropic phases of a correspondingbound for composites with isotropic phases that was introduced by Ponte Castaiieda (199 la). An alternative generalization for anisotropic composites was given by Talbot and Willis (1992), which amounted essentially to (4.37), where the functions were defined by relations of the type (4.6) with generally anisotropic wt).It should be noted that these authors obtained their bounds as a special case of a more general result that does not require the hypothesis of concavity of the F ( r ) . Three important observations can be drawn from the definition (4.37) for the potential eff. First, the form (4.37) of the bound eff (E) involves an optimiza-

w

w

206


tion problem for the positive definite tensors LIT', which is (usually) solved by the set of stationarity conditions (4.40) Using these stationarity conditions, it follows by derivation of (4.37) with respect to E that the stress-strain relation for the linear comparison composite may be written in the form

aw eff aE

(E) = Lgff( L : ) )

: E,

(4.41)

where the )!L are now the modulus tensors satisfying the optimality relations (4.40). Thus the relation (4.41) is an approximation to the exact stress-strain relations of the nonlinear composite. Note that, in spite of its appearance, this effective stress-strain relation is nonlinear, because of the nonlinear dependence of the optimal Lr)on the average strain E. This observation is due to deBotton and Ponte Castaiieda (1992, 1993) in the context of nonlinear composites with isotropic phases. Second, note that the expression (4.38) for Woff can be rewritten in the form N

where

z(r)is the the second moment of the strain in phase r , defined as -

E

1 '"(v) = ( E ( v ) ) ( r ) = - ( E ( V ) 2

@ E (IV ) ) V

(4.42)

It then results, from (4.37) and (4.39), that N

.

r = l , ...,N

N

-

N

In the last equality, 2 = 2 ("(u), where u is the displacement field in the linear comparison composite. The form (4.44) of the upper bound (4.37) can be derived directly from the concavity of F ( r ) ,as originally done in Suquet (1995, 1997a) for composites with isotropic phases.


207

Third, the optimality conditions for Lt)in (4.43) read

Then, recalling the equivalence relation (4.19) and the definition (4.39) of the u ( ~ ) functions, the optimality conditions (4.40) for the Lf)can be rewritten in the form

The last equality in (4.45) expressing the second-order moment of the strain in each phase of the linear comparison composite in terms of its effective stiffness LEff has been obtained here as a consequence of the optimality conditions (4.40). It is interesting that one can directly obtain (see Appendix B) the expression of the second-ordermoment of the strain (or stress) in each phase in a linear composite by derivation of the effective strain energy of the composite (Suquet, 1995, 1997a):

These relations apply in particular to the linear comparison composite.

ii. Stress potential Ueff. A bound that is equivalent to (4.37) can be obtained by considering the stress potential Ueff and its variational representation (4.25). Thus, restricting the optimization in (4.25) to piecewise constant compliances ~ , f ) ,one obtains

where U;ff is now the effective stress potential associated with the same linear comparison composite as for Wlff above-one with the same microstructure as the nonlinear composite, but where the domains V(') are occupied by linear phases with compliances M". From this result, it follows that

a ueff(X)= M;ff(Mp) : x, ax

Ex-

(4.48)

208

Pedro Ponte Castaieda and Pierre Suquet

where the optimal M t ) are determinedby conditions analogous to those of (4.40). It also follows from (4.47) that N

N

z(r)

where = ( m ) @ ) = i(cr @ cr)@)is the second moment of the stress field in phase r of the linear comparison composite.The compliancesM (l ' of the comparison material are determined as the solution of the optimization problem (4.47), which can alternatively be written in terms of the solution of the following nonlinear problem for the variables ( r ) :

2. Nonlinear Composites with Isotropic Phases When the nonlinear constituents are isotropic, it has already been shown that the constituents of the linear comparison composite can be chosen to be isotropic. The bulk modulus is equal to the bulk modulus k(') of the nonlinear constituent, and the only modulus that has to be determined is the shear modulus &) in each phase. Thus the bound (4.47) reduces to

(4.51) where the functions are defined by the relations (4.6). The upper bound (4.51) for Weff,as well as the analogous lower bound,

r = l , ...,N

(4.52) was originally proposed by Ponte Castaiieda (1991a). From the associated optimality conditions, it follows that the effective stress-strain relations for the composite may be approximated by (deBotton and Ponte Castaiieda, 1992, 1993) C = LEff(pt)): E,

E = MEff(pt)): C.

(4.53)


209

Similarly, the nonlinear tensorial optimality conditions (4.45) can be shown to reduce to N scalar nonlinear equations:

(4.54)

a&)

= ( ( E : ~ ) ( ‘ ) ) ’ / ~ . These nonlinear equations can be given, alternaNote that tively, in terms of the stressxnergy potential,

(4.55)

Again, note that F&)= ( ( D : ~ ) ( ~ ) ) ’The / ~ . bounds for Weff and Ueff can then be rewritten:

(4.56)

where

w

u

These simplified forms of the bounds eff and eff, using the optimality conditions (4.54) and (4.55), were first given by Suquet (1995, 1997a). For power-law composites, one obtains the results

210


and

also due to Suquet (1993a). 3. Nonlinear Polycrystals Restricting the optimization in (4.47) to compliance tensors giving finite values for the functions u(") leads to the result (deBotton and Ponte Castafieda, 1995) that

ueff(c)2 u eff (X)=

sup LYj;;20

\..,

r=l

{fc

: Mi""(Q) :

e - y~c"'g:;;*(a; N

K

r = l k=l

,....N

k=1, ...,K

where Q denotes the whole set of positive slip compliances a:;;, and MIeff is the effective compliance tensor of the linear comparison polycrystal with grain compliances M(r),as given by (4.28) in terms of the slip compliances a. Here also the effective stress-strain relation of the polycrystal may be approximated by the stress-strain relation (4.48) of the linear comparison composite, with the optimal M): replaced by the optimal a:;;.The nonlinear optimality relations (4.50) read

(4.59) These nonlinear equations can be expressed more explicitly in terms of the slip compliances a:;; and the corresponding second moment of the resolved shears,

21 1

Nonlinear Composites The result is

(4.60) In addition, the bound can be rewritten in the form N

K

(4.61) r = l k=l

For power-law polycrystals, one also has the result

D.

INTERPRETATION OF T H E VARIATIONAL

PROCEDURE AS A SECANT METHOD It has already been noted that the stiffness tensors in the linear comparison composite are piecewise constant (by assumption) and are determined by the optimality conditions (4.40). This situation is reminiscent of the secant method (described in more detail in Appendix D). A rigorous connection between the variational bound (4.37) and the secant method can be made following Suquet (1995). The characterization (4.44) of involves a variational problem:

w

N

Under appropriate hypotheses on the growth of the function F @ ) ,the variational problem (4.63) admits a unique solution u, which satisfies

u E IC(E),

[

aJ

(u),v*] = 0,

vv* E IC(0).

(4.64)


212

The derivative a J/av is the Frtchet derivative of the functional J,and [ , ] is to be understood in the sense of a duality product. A straightforward calculation shows that

[

av

(v),

.*I

= (e(v) €4 E ( V * ) ) @ ) .

(4.65)

The variation of J with respect to v is

[g (u), where I$)

c N

=

c(‘)(e(u) :

ILp : e(v*))?

(4.66)

r=l

is the secant tensor in phase r , (4.67)

By virtue of (4.66), the solution u of (4.64) solves the following system of equations: (4.68) where

cx N

IL, (x) =

( T “1).

( r )(x)ILj.’)

r=l

Given the tensors , ):LI the above problem is a classical one for a linear elastic material. Note, however, that these tensors themselves depend on the strain field E through the second moments of the strain 7 which themselves depend on the tensors s = 1, . . . , N . Recall that the second moments of the strain in the individual phases are given by (4.46). The problem is therefore nonlinear, consisting of the set of equations (4.67) and (4.46), which are equivalent to (4.45). In conclusion, the variational procedure, when restricted to piecewise constant moduli, can be interpreted as a secant procedure. The nonlinear composite is replaced by a linear composite with constant (but unknown) stiffness tensors in each phase. These tensors are given by the constitutive law as the secant tensor for the some “effective” second moment of the strain 2 or of the stress dl- ( r ) , which performs (in a certain sense) an averaging of the strain or stress field in each phase. This is not the usual average (or first moment) classically used in most secant methods (Berveiller and Zaoui, 1979; Tandon and Weng, 1988), since in

ILt),

PLATE1. Changes of mode shapes during loading.

step 10

step 50

step 200

step 250

step 350

step 679

(3)

PLATE2. The buckling process, perfect cylinder.

step 37

step 50

step 70

step 100

step 120

step 262

PLATE3. The buckling process, imperfect cylinder.

PLATE4. Unidirectional composites with fibers aligned in the x3 direction. Cross-sectional

(x,-x*) maps of accumulated plastic strain corresponding to different macroscopic stresses. a) Pure shear. b) and c) Combined in-plane shear and axial tension. [b) corresponds to the state of stress at the vertex of the extremal flow surface.] d) Pure uniaxial tension along x3 (from Moulinec and Suquet, 1995).


213

r

general % ( r ) and ( r ) are different from the fourth-order tensor formed with the averaged strain or stress,

-

z(r)

p r ( u p = 21 c@) dr).

1 @ u ) ( r )# 1 - 2(u

@

-

@

The idea of considering an effective stress based on the second moment of the stress in each phase, instead of the usual first moment of the stress, is not entirely new. Something similar has been proposed by Qiu and Weng (1992) in the special context of porous materials. More specifically, these authors made use of the distortional energy in a linear comparison porous material to estimate the strain in the matrix phase, which allowed them to obtain improved estimates, relative to those obtained by the classical secant procedures, for the effective behavior of nonlinear porous materials. However, these authors took advantage of an approximation that is only exact when the matrix is incompressible, but which introduces errors relative to the “exact” (in the context of the piecewise constant moduli approximation) results of this section when the matrix phase is compressible (see Suquet, 1997a). Second moments have also been used earlier by Buryachenko and Lipanov (1989a, b) in the specific context of a generalization of the “multiparticle effective field scheme” for nonlinear composites. However, these authors apparently did not use (see Buryachenko, 1996) the exact expression for $(‘I, using instead an approximate formula that is also only exact for incompressible materials. The exact result for F ( r )was known and used by Kreher (1990), among others, to compute the stored energy in thermoelastic composites. Finally, after the present review was completed, the work of Hu (1996) was brought to the attention of the authors. This work, which is independent of the work of Suquet (1995, 1997a), concludes, similarly, that the variational method can be interpreted as as a secant method based on the second moment of the stress.

E. DISCUSSION

A few points are worth emphasizing at this stage. First, any estimate for the effective modulus tensor of a linear elastic composite (or class of composites) can be used to generate, by the variational procedures developed in this section, a corresponding estimate for a nonlinear composite with the same microstructure (or class of microstructures). This is a major advantage of these variational procedures using a linear comparison composite, because they are not tied in to any

214

Pedro Ponte Custuiieda and Pierre Suquel

specific type of estimate. Other competing schemes, on the other hand, are closely connected with specific types of estimates. For example, the Talbot-Willis procedure, discussed in Section 111, serves to provide only estimates of the HashinShtrikman type. Second, if the relevant estimate for the effective modulus tensor of the linear elastic composite happens to be an upper bound for Le", then an upper bound is generated for Weff. If, on the other hand, the linear estimate is a lower bound, then the variational procedure cannot be used, in general, to obtain a lower bound for the nonlinear composite. In this sense, the variational procedure has the same limitation as the Talbot-Willis procedure, in that only one-sided bounds can be obtained. However, if a fairly accurate estimate (not necessarily a bound) is available for a specific type of linear composite, the variational procedures of this section can be used to generate a corresponding estimate for a nonlinear composite with the same microstructure, or type of microstructures, where these estimates (for Weff) would tend to err on the high side, because of the approximations intrinsic to the variational procedure. In particular, the variational procedures can be used to reproduce exactly the classical nonlinear Voigt bounds (see Ponte Castaiieda, 1992a), as well as the Talbot-Willis bounds (see Ponte Castaiieda, 1991a, 1992a; Willis, 1991,1992), when the Voigt and Hashin-Shtrikman bounds are used, respectively, for the linear comparison composite. In addition, however, the variational procedures of this section can be used to generate higher-order (of grade higher than 2) bounds, such as Beran-type bounds (Ponte Castaiieda, 1992a), as well as other types of estimates, such as generalized self-consistent estimates (Suquet, 1993b), which would not be available from the Talbot-Willis extension of the Hashin-Shtrikman variational principles. Finally, it is mentioned that Smyshlyaev and Fleck (1995) have recently provided extensions of the variational procedures in the context of strain gradient plasticity (see also Fleck and Hutchinson, 1997). Third, the variational procedures of this section have made use of the concuvity hypothesis (4.18) on the function F associated with the local strain potential w through relation (2.17). Although this mild hypothesis is satisfied by the standard models of plasticity and creep, pathological examples, not satisfying this hypothesis, can be constructed for which the variational procedures fail to deliver Hashin-Shtrikman bounds that are as good as those that may be obtained via the Talbot-Willis variational procedure (see Willis, 1992; Ponte Castaiieda and Willis, 1993). On the other hand, when the concavity hypothesis is satisfied, Ponte Castaiieda ( 1 9 9 2 ~has ) shown, in the context of composites with isotropic phases, that the Talbot-Willis variational principles may be derived via the variational principles of Ponte Castaiieda, directly from the Hashin-Shtrikman varia-


215

tional principles for linear composites. As a way of preserving the advantages of both the Talbot-Willis and Ponte Castaiieda procedures, Talbot and Willis (1992) proposed a hybrid of these procedures, one that works for materials for which the concavity hypothesis is not satisfied, but at the expense of introducing a more complicated linear thermoelustic composite. Because of space restrictions, the variational principles of Talbot and Willis (1992) will not be discussed further in this work. It is noted, however, that Talbot and Willis (1997) have recently made use of these variational principles, together with a nonlinear comparison composite, to generate third-order bounds in the “other” direction. Fourth, an alternative, simpler way of accounting for materials for which the concavity hypothesis is violated has been proposed recently by Ponte Castaiieda (1996b, 1997b) (see also Kohn and Little, 1997, and Bhattacharya and Kohn, 1997, in the context of polycrystals). The idea of the method is similar to the “translation method” (Lurie and Cherkaev, 1986; Milton, 1990), except that the homogeneous quasi-convex reference material is replaced by a linear comparison composite, exactly as in the variational procedures developed in this section. Bounds for nonlinear composites can then be obtained by using a Reuss bound on the translated problem, which is now nonlinear, in combination with uny estimate for the linear comparison composite. Once again, space restrictions will not allow further discussion of this work. Suffice it to say that, at least for isotropic composites, the bounds resulting from this approach can be shown to be equivalent (when the concavity hypothesis is satisfied) to those obtained from the variational procedures reviewed in this section. In general, however, the newer method is more difficult to use than the earlier ones discussed in this section because they require an additional step involving the computation of Reuss-type bounds for nonlinear, nonconvex materials. This step can become computationally intensive, especially for polycrystalline aggregates. Fifth, the functions F(‘) in (2.17) are well defined for special “rank-one” fourth-order tensors Q of the form Q = 6 = ( 1 / 2 ) ~@ E . But their extension to arbitrary fourth-order tensors Q is not uniquely defined, even under the concavity hypothesis. The variational representation (4.22) is, of course, independent of the chosen extension, but the upper bound (4.44), valid for each possible F(‘), could depend on the chosen extension, since ( r ) is not necessarily rank one. In principle, it would be possible to optimize the upper bound by taking the infimum over all possible extensions F(‘) (the infimum of concave functions still being a concave function). This possibility and related issues about the nonuniqueness of the extension have not been yet explored in detail and will not be discussed further here.

r

weff

216

Pedro Ponte Castanleda and Pierre Suquet

V. A Second-Order Theory This section is devoted to exact results pertaining to the effective behavior of nonlinear composites with small contrast (or weakly inhomogeneous nonlinear composites) and to related estimates for arbitrary contrast of the phases. In the context of linear elasticity, it is well known that the effective modulus tensor of a weakly inhomogeneous composite with random microstructure can be determined exactly to second order in the contrast. In the first part of this section, an analogous result, due to Suquet and Ponte Castaiieda (1993), is derived for nonlinear composites. By comparing this exact asymptotic result with the corresponding expansion for the variational estimates of Section IV (and therefore of Section 111), it is found that the variational estimates are only exact to first order in the contrast, when estimates that are exact to second order are used to estimate the behavior of the linear comparison composite. In the second part of this section, a recently developed theory, due to Ponte Castaiieda (1996a), which does deliver estimates that are exact to second order in the contrast for nonlinear composites at arbitrary contrast, is developed.

A. WEAKLYINHOMOGENEOUS NONLINEAR COMPOSITES 1. Small-Contrast Expansions for the Exact Effective Potential

In this section, the contrast between the phases is assumed to be small. More specifically, the potential w is assumed to depend on a small parameter t , serving to characterize the contrast between the properties of the composite and those of a homogeneous nonlinear reference material with energy function w(O)( E ) , such that W(X, E ,

t ) = W(O)(E)

+ tsW(X, E).

(5.1)

The effective potential also depends on the parameter t :

where ut and E ( u ~ )are the local displacement and associated strain fields induced by appropriate boundary conditions generating an average strain E in V. It will be assumed that We""(., t ) and ut are continuously differentiable functions of t . Since t is small, it is appropriate to look for a perturbation series expansion of Weff about t = 0, given formally by

Weff(E, t ) = Weff(E, 0) + t

a weff ~

at

(E, 0)

a2weff + yt 2 7 (E, 0) + 0 ( t 3 ) .

(5.3)


217

The first few derivatives in this expansion may be computed by successive derivation of (5.2); this procedure will naturally involve derivatives of u,.The problem to be solved for ut is given by = 0,

uf E IC(E).

(5.4)

By differentiation of system (5.4), it is seen that uf (the dot denoting the derivative with respect to t) is, for all t , the solution of the system of equations

+

div(IL, : e(ur)) div(r,) = 0,

u, E K(O),

(5.5)

where

Note that IL, is the tensor of instantaneous or tangent moduli. The first two derivatives of Weff may be expressed in terms of only ui and u,.Indeed, it follows from (5.2) that

The first term vanishes by Hill's lemma, and therefore,

By means of the system of equations (5.5) satisfied by ut, one obtains the result that

a2weff

(-a

-(E, t ) = (6w)(x, E ( u , ) ) : e(u,)) = -(e(uZ) : ILt : e(u,) at2 ae

The relations (5.6) and (5.7), which hold for arbitrary values o f t , allow in particular the computation of the first two derivatives of Weff at t = 0. Using the fact that the material is homogeneous at t = 0, and therefore uo = E .x, it follows that

Weff(E, 0) =

w'O)

(E), (5.8)

where

21 8

Pedro Ponte CustaZeda and Pierre Suquet

In these relations, u o is the solution of the linear elasticity problem,

+

div(IL.") : E(UO)) div(t) = 0,

uo

E

K(O),

(5.10)

where

a

t ( ~ = )- ( ~ w ) ( x E). , aE

Given that the modulus tensor L(")is constant, problem (5.10)is a standard linearelasticity problem for a homogeneous material with a distribution of body forces determined by the polarization field t.Note that this is a special case of the polarization problem (3.4), encountered in the Hashin-Shtrikman variational principles of Section 111, with E = 0 and with Leo) specified by (5.9). When the composite is made up of N homogeneous phases (r = 1, . . . , N), the polarization field is piecewise constant (a constant in each phase), with

The determination of (a2Weff/at2)(E,0) then requires the computation of the elastic energy associated with uo (see (5.8)). This energy is obtained from (3.1 1) and (3.12), so that (5.1 1) "I

r=l s=l

are defined by (3.10). In conclusion, the where the microstructural tensors rcrs) following result, due to Suquet and Ponte Castaiieda (1993), is obtained: t2

Weff(E,f) = (w)(E) - 2

N

N

+ 0(t3).

t ( r :)F"") : t(")

(5.12)

r=1 s=l

Isotropic constituents. In this section, we restrict our attention to N-phase composites with isotropic constituents, with strain potentials defined by (5.1) and

219


Then the polarizations may be written in the form t(r)

= 36k(')E,I

+ 26p(")Ecl,

2 3

6p"') = - (6.f @I)'( E:J>

where Ed = E - E,I is the average strain deviator, and, using the spectral decomposition introduced by Ponte Castaiieda (1996a), the tangent modulus tensor L(")may be written in the form

L(O) = 3k(O)J

+ 2p(O)F+ 2h(O)lE,

(5.13)

where

and where Jl is the standard hydrostatic projection tensor, defined by (2.24). The tensors (5.14) and

F=K-E,

(5.15)

are also projections, such that

It follows from (5.3) (Ponte Castaiieda and Suquet, 1995) that, to second order in t , one has

2. Small-Contrust Expansion for the Estimate

weffof Section IV

Having obtained the small-contrast expansion to second order for the exact effective potential Weff, it is relevant to consider the corresponding expansion for

220


w

the approximate estimate eff arising from the variational procedures of Section IV. Thus, starting, for example, from (4.44), it is possible to obtain the result that

where the tensors H(") have a rather complicated form in general. Regardless of the explicit form of the the key observation is that these tensors are different from the tensors r(rs) in the corresponding expansion (5.12) for the exact effective potential Weff. This is because, whereas the tensors F(") depend on the tangent modulus tensor (5.9),the tensors can be shown to depend on the secant modulus tensor (4.67) and its derivative. In particular, for isotropic constituents, it has been remarked that the tangent modulus tensor is anisotropic, whereas the secant modulus tensor is isotropic, highlighting the difference between the secant and tangent modulus tensors. The important conclusion from this observation is that the estimate eff, obtained by the variational procedures of Section IV, for the effective potential Weff is exact only tofirst order in the contrast.

w

B. NONLINEAR COMPOSITES WITH ARBITRARY PHASE CONTRAST 1. Second-Order Estimates f o r the Effective Potential of N-Phase Composites Motivated by the findings of the previous subsection, a new method for estimating the effective behavior of nonlinear composites was proposed recently by Ponte Castaiieda (1996a). Like the variational methods of Section IV, 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 are in agreement with the smallcontrast asymptotic results of Suquet and Ponte Castaiieda (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 w ( ~ Thus, ). introducing reference strains E(r),the Taylor formula for w ( ~about ) E(') reads

&

(5.17)


22 1

where p(') and L@)denote an internal stress and a tangent modulus tensor, with components

+

respectively. Note that L (') depends on the strain E(') = A(') E(') (1 - A ( ' ) ) E , where A(') depends on E and is such that 0 < A ( r ) < 1. In terms of the average (E) and oscillating ( E ' ) components of E , the above expression for w(I) may be rewritten in the form

(5.19) where

(5.20) and where t(r)

= p ( r ) + L(r) . (E - E(r)).

(5.21)

It then follows from (2.41), by making the approximation that the strains E(r)are constant in each phase, that the effective potential W e f fof the nonlinear composite may be estimated as

where (5.23) and where the following definitions have been used: N

N

r=l

r=l

The advantage of approximation (5.22), relative to the exact expression (2.41), for the effective potential Weff of the N-phase nonlinear composite is that (5.22)

222


requires only the solution of a linear problem for an N-phase themoelastic composite, as defined by the Euler-Lagrange equations of the variational problem P in (5.23):

div(L : E ( u ' ) ) = -div(t),

u'

E

IC(0).

(5.25)

In general, estimates for N-phase linear-thermoelastic composites can be obtained by appropriate extension of the corresponding methods for N-phase linear-elastic composites (see, for example, Budiansky, 1970; Laws, 1973; Willis, 1981). Note that similar, but not equivalent, representations for the effective behavior of nonlinear composites, also using hetereogeneous thermoelastic reference materials, have been proposed by Molinari, Canova and Ahzi (1987) and Talbot and Willis (1992). Given an estimate for P , the expression (5.22) therefore provides a corresponding estimate for Weff,for all choices of the Ecr)and E@).A plausible prescription for the E(') is to set them equal to the averages of the strain field over the phases r . However, since the exact strain field is not known, the approximate field E , as determined by (5.25), is used instead, so that the proposed prescription for the E(') becomes

+

Physically, this is a where E ( r ) has been used to denote ( E ) ( ' ) = E (.d)(rj. good choice in the context of the estimate (5.22), because the strain E in phase r is expected to oscillate about its average in phase r 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 (5.23) (see Appendix B.3 for details), namely, (5.27) which can be used to obtain the phase-average strains .G('j, directly from P , via the relation (5.28) Therefore, the reference strains E"' may also be computed from P , by means of relations (5.26) and (5.28).


223

It is further remarked that the prescription (5.26) for the E(') is optimal in the sense that the estimate (5.22) for Weff is stationary with respect to the E('). To see this, note, from (5.20), that

and, from (5.27), that

so that, from (5.22),

which, because of the prescription (5.26), is identical to zero. One important consequence of the stationarity of the prescription (5.26) is that the overall stress-strain relation (2.42) for the composite may be approximated as

where the Z") are determined from relations (5.28), and where

This result is easily obtained by taking the derivative of relation (5.22) with respect to E, with E(') held fixed (because of stationarity), and enforcing (5.26). Notice that the "correction" term in the average stress-strain relation depends on the second moment of the difference between the strain field and its average over the phases. The prescription (5.26) also allows simplification of the estimate (5.22) for We". Note that the Euler-Lagrange equations (5.25) of the problem (5.23) for P imply, using integration by parts, that (t : E ( U ' ) )

= -(E(U') : L : E(U')),

(5.31)

224

Pedro Ponte Custaiiedu and Pierre Suquet

which, with the help of prescription (5.26) and definition (5.21) of the polarizations T ( ~ ) can , be used to rewrite estimate (5.22) in the simpler form

Weff(E)= Weff(E)

where the E(r) are determined by relations (5.28). Finally, the choice of E@)in definition (5.18) of the L(') is not as straightforward, and, in particular, stationarity of Weff with respect to E(') cannot be implemented. For this reason, Ponte Castaiieda (1996a) proposed the following physically motivated prescription for the E@),namely, E ( r ) = g(r) = ~ ( r ) .

(5.33)

There are other possibilities, but they have yet to be explored in full detail. One interesting consequence of prescription (5.33) is that it implies that

(5.34) 2. Exactness to Second Order in the Contrust It follows, from result (5.31), that it is also possible to rewrite estimate (5.22) for weffas N

(5.35) where E' is obtained from the solution of the boundary value problem (5.25) for u'. Now, for weakly inhomogeneous composites, with the parameter t in (5.1) set small, and assuming that

the following two observations can be made. First, note that the functions v ( ~ ) are in fact the second-order Taylor expansions of the functions w ( ~about ) Ecr), evaluated at E. Therefore.

Nonlinear Composites Second, the the form

t('),defined by

225

the relations (5.21), can be similarly shown to be of

(5.38) from which it follows that the solution of the boundary value problem (5.25) for u' is given by

u' = tuo

+ O( t2) ,

where uo is the solution of the boundary value problem (5.10) associated with the small-contrastexpansions of Section V.A.1. Combining these two observations, it is concluded that

r=l

which is in agreement with the small-contrast expansion (5.3) with (5.8). It can also be verified that prescriptions (5.28) and (5.33) for E(') and E(r),respectively, are consistent with hypotheses (5.36). Therefore, the second-ordertheory of Ponte Castaiieda (1996a) is exact to second order in the contrast. 3. 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 Weff. The result for P , which depends only on the effective modulus tensor Leffof a two-phase, linear-elastic composite with phase modulus tensors IL(') and L(2),is given by

1 P = - ( A t ) : (AIL)-' : (ILeff - (IL)) : (AIL.)-' : ( A t ) , 2

(5.39)

where AIL = L(')- IL.(2) and A t = t(')- ~ ( ~ It1 then . follows, from (5.28) and (5.33), that the prescriptions for E(') and E(') reduce to

E(') = E(') = E ( r ) = E + (A(') - 1) : (AIL)-' : ( A T ) ,

(5.40)

where the A(') denote the strain-concentration tensors (Hill, 1965a)for the linearelastic composite problem. They are such that c ( l ) ~ ( l )+ c ( 2 ) ~ ( 2 = ) 1,

Leff = c ( l ) ~ ( l ). ~

( 1+ ) , ( 2 ) ~ ( 2 ). ~ ( 2 )

which can be used to solve for the tensors A(r)in terms of the L(') and Leff. It is emphasized that estimates of any type for Leffcan now be used to gener-

226


ate corresponding estimates for Weff. For example, as pursued in Section VII, Hashin-Shtrikman and self-consistent estimates for Leffmay be used in (5.40) to generate, via (5.32), corresponding Hashin-Shtrikman and self-consistent estimates for We". Note, however, that prescription (5.40) for E(') is implicit in the sense that the expressions for I,(') and t('),appearing on the right-hand-side of (5.40), also depend on E('). 4. Isotropic Composites The application of the above procedure is briefly described here for composite materials with isotropic phase potentials of the form w ( r ) ( e )=

9 k(')&;L, 2

-

+ f"'(&:q),

(5.41)

where k(') denotes the bulk modulus and f ( ' ) characterizes the shear response of phase r . We note that the functions f ( ' ) are generally nonlinear, except that they are required to comply with the previously stated hypotheses on w. Given the above expression for w(') and recalling, from (5.33), the prescriptions that E(') = E(') = E ( " ) , one computes, from relations (5.18), ,(I'

= 3k(r)~,(;)1+ 2 p ( r ) 2 i ) ,

(5.42)

where

h ( r ) = FL(r)

and where J,

and IF(')

(5.45)

are projection tensors defined by (2.24), (5.46)

and

respectively. No further simplification is available in general for the second-order estimates (5.32), but fairly explicit results can be obtained for two-phase random composites


227

with statistically isotropic microstructures, in three and two dimensions, the latter physically corresponding to transverse shear of a fiber-reinforced composite with transverse isotropy. These simplifications for composites with isotropic phases and statistically isotropic microstructures arise because, in this case, the phase average strains E(') can be assumed to be aligned with the applied macroscopic strain E. Some more explicit results will be given in Section VII. C. DISCUSSION In this section, a perturbation expansion (Suquet and Ponte Castaiieda, 1993), exact to second order in the contrast, has been carried out for the effective potentials Weff of nonlinear composites. It is emphasized that the second-order term in the expansion depends only on the two-point statistics of the composite and completely specifies its effective behavior (to second order in the contrast). By comparison with this exact result, it was also found that the variational estimates of Section IV (and therefore those of Section 111)are exact only toJirst order in the contrast. In this connection, it is relevant to emphasize that the nonlinear HashinShtrikman bounds of Section IV are not "second-order'' bounds in the same sense as their linear counterparts. Like the linear bounds, the nonlinear bounds do incorporate second-order statistics, but unlike them, they are not exact to second order in the contrast. In addition, in this section, a second-order theory has been developed (Ponte Castaiieda, 1996a), that does deliver estimates that are exact to second order in the contrast. However, the approximations involved in this second-order theory are such that it is not possible to control the sign of the error in the approximations involved, so that the resulting estimates, unlike the variational estimates of Section IV, cannot be guaranteed to be bounds. Another important limitation of the second-order theory is the existence of a duality gap (i.e., fieffand do not satisfy the duality relation Ueff = (w"')*). More specifically, the corresponding estimates for the stress potential Ueff, obtained by a completely analogous procedure (to the one followed in Section V.B. 1 for Weff),are found to be nonequivalent to those for Weff. This is unlike the case for the variational estimates and U of Section IV, which were found to satisfy the duality relation. As a practical matter, in plasticity and creep, the second-order estimates based on the developments of Section V.B.l for Wefr are found to be more accurate than the analogous estimates for Ueff (see Ponte Castaiieda and Kailasam, 1997, for an analogous situation in the context of nonlinear conductivity). Roughly speaking, this is because Taylor expansions for w (which is subquadratic) are more accurate than the corresponding Taylor expansions for u (which is superquadratic).

+ff

weff

228


In theory, it would be desirable to develop methods that deliver bounds that are exact to second order in the contrast, although this may pose a formidable task, in general. In practice, however, as will be discussed in Section VII, the bounds of Section IV and the second-order estimates of this section provide complementary information, which can be used to estimate rather efficiently and accurately the overall behavior of nonlinear composites.

VI. A Selection of Results for Linear Composites The methods that have been developed in Sections IV and V.B. 1 make use of estimates for the effective modulus tensor of comparison linear-elastic and thermoelastic composites to generate corresponding estimates for nonlinear composites. Before presenting applications of these methods, a brief selection of various bounds and estimates for linear-elastic composites is given in the present section. For more comprehensive information, refer, for example, to the review articles by Hashin (1983), Walpole (1981), and Willis (1981), as well as the monograph by Nemat-Nasser and Hori (1993).

A. HASHIN-SHTRIKMAN BOUNDS FOR RANDOM COMPOSITES SYMMETRY WITH ELLIPSOIDAL Define the joint probability function of finding simultaneously phase r at x and phase s at x’ as

where statistical homogeneity has been assumed, so that P ( r s ) ( xx’) , = P ( ‘ ” ) ( xx’). Under the no-long-range order hypothesis, the finite-body (over V ) kernel in definition (3.6) of the operator k(O)may be replaced by the corresponding kernel constructed from the infinite-body Green function, which becomes a function of x - x’ only. Then, using the ergodic assumption allowing the replacement of ensemble averages by volume averages over the representative volume element of the composite V , it follows (Willis, 1977, 1981), by interchanging the order of integration in expressions (3.10) for the tensors K‘(rs), that


229

where r(O)(x) is now the infinite-body kernel associated with a homogeneous material with modulus tensor L(O). It is further assumed that the two-point probability function has "ellipsoidal symmetry," so that P('")(x) = P(")((IZ -XI) for some constant, symmetric tensor Z. Note that the special case where Z = I, such that the P ( r s )depend only on 1x1, corresponds to statistical isotropy. Thus ellipsoidal symmetry defines the class of microstructures that would be obtained by uniformly stretching statistically isotropic microstructures. Under this statistical hypothesis of ellipsoidal symmetry, Willis (1977, 1981) has shown that the microstructural tensors r(rs) reduce to

where

with

Note that P(O) has the diagonal symmetry of an elasticity tensor. It is therefore easier to use than the Eshelby tensor S(O) (Eshelby, 1957), to which it is related by P(O) = ScO): M(O), with M(O) = (Leo))-' denoting the compliance of the reference medium. The tensor P(") is known explicitly for a few cases of practical relevance. For example, when the reference medium is isotropic and the phases are assumed to be isotropically distributed, so that Z = I, the tensor P(O)is also isotropic, and

with

Other cases of special interest are given in Appendix C and are recalled in the following sections as needed. Given the above explicit expressions for the microstructural tensors F(") in terms of P(O),still within the context of the Hashin-Shtrikman variational procedure, it is possible to specialize to linear-elastic composites the expressions (3.9)

Pedro Ponte Custufiedu and Pierre Suquet

230

for the average strain in phase r , k c r )= A(’) : E, and (3.17) for the effective potential Weff. The result can be written in terms of the relevant effective modulus tensor, given by

(6.4)

As mentioned earlier, the choices LC0)= rnax,{IL(‘)} and LC0)= rnin,.{lL(‘)) lead, respectively, to rigorous upper and lower bounds for ILeff. It is emphasized that such bounds, although appearing to depend only on the volume fractions, also incorporate two-point statistics (through the microstructural tensors P(’)), and therefore have the potential of being sharper than the Voigt-Reuss bounds. The bounds (6.4) are due to Willis (1977) and are known to be exact to second order in the contrast SIL = IL - lLC0), so that Leff

- (IL)

-

(SIL : P(O) : (SlL - (SIL)))

+ O(SlL”),

which is in agreement with the general result (5.12) (simplified with the help of (6.1)). When the microstructure is statistically isotropic (with Z = I), the celebrated bounds of Hashin and Shtrikman (1962a, b, 1963) are recovered from (6.4). For example, for two-phase composites with isotropic phases,

L ( r ) = 3k‘r)j + 2pL(r)~,

Leff = 3keff~+ 2 p e f f ~ ,

where

with k p ) and pf’ given by (6.3). It is known (Francfort and Murat, 1986) that these bounds are attainable by composites with sequentially laminated microstructures, and are therefore optimal, provided that the phases are well ordered, > 0. One interesting feature of the sequeni.e., if ( p ( l )- p L ( 2 ) ) ( k (-’ ) tially laminated microstructures is that they are “particulate” in the sense that the constituents occupy clearly defined “matrix” and “inclusion” phases. The upper


23 1

bound requires the stiffer phase to occupy the matrix phase, and vice versa for the lower bound. This is suggestive of the fact that the Hashin-Shtrikman bounds may be appropriate for composites with particulate microstructures, at least for twophase systems. The next section develops this notion further. Another important case is that of two-phase composites with isotropic phases and “spheroidal symmetry” corresponding to transverse isotropy. In this case, the IF’() tensor is also known, and is given by relations (Cl) in Appendix C. The corresponding HashinShtrikman estimates (6.4) may be used to model laminated and fiber-reinforced composites (both continuous and chopped fibers).

B. HASHIN-SHTRIKMAN BOUNDSAND ESTIMATES FOR RANDOM COMPOSITES WITH PARTICULATE MICROSTRUCTURES Willis (1978, 1981) made an application of the Hashin-Shtrikman variational principles to N-phase composites with particulate microstructures consisting of a random distribution (in space) of families of aligned, self-similar inclusions of phase r (r = 2, . . . , N) in a matrix of phase 1, where the two-point probability function for the distribution of the centers of the inclusions was assumed to be statistically homogeneous and to satisfy the “ellipsoidal symmetry” hypothesis. This statistical hypothesis is defined by the requirement that the P ( r s )depend only on x” = x’ - x through the combinations IZ!”’ x”1, for some tensors Z!’), symmetric in their superscripts. In addition, no overlap of inclusions was assumed, so that P ( r s ) = 0 for overlap conditions. For two-phase systems, when the inclusions are ellipsoidal with identical shape, characterized by a tensor Z, and the distributional tensors Z:’) are all assumed to be equal to Z, Willis (1978) obtained Hashin-Shtrikman bounds for the pertinent effective modulus tensor Le“, by setting ILco) = L(’). The result was in precise agreement with the bounds (6.4), specialized for two-phase systems, which were derived originally for general classes of ellipsoidal microstructures (i.e., for microstructures that are not necessarily particulate). A generalization of this result for N-phase composites consisting of N - 1 types of aligned ellipsoidal inclusions, characterized by shape tensors Z j r ) , distributed with ellipsoidal symmetry in matrix of phase 1, with distribution shapes Z:”’ (different from the Z“), was given by Ponte Castafieda and Willis (1995). The result for the relevant microstructural tensors (3.10) is

-

232

Pedro Ponte Castatieda and Pierre Suquet

where IP':r) and)';'PI denote the P-tensors (6.2) associated with the inclusion and distribution shape tensors Z r ) and Z!", respectively. The corresponding estimates for Leff are obtained from the piecewise-constant polarization HashinShtrikman estimates (3.17), with L(O)set equal to the modulus tensor of the matrix L('),so that the polarization field vanishes exactly in the matrix phase. When L(') happens to be equal to the maximum or minimum of the L@), the resulting estimates are, respectively, rigorous upper and lower bounds for Leff;otherwise, they are plain variational estimates. When all of the distribution shape tensors Z:') are identical, say, equal to Z d , the result for Lefftakes the simple form N

Leff =

A(r) = (1+

IP'y:

C

c ( r ) L ( r ) : A('),

r=l

where IP'd is the IP' tensor (6.2), associated with the distribution shape tensor Zd, and 6L(') = L(r)- I,(').These estimates provide a generalization of Eshelby's (1957) estimates for composites with dilute concentrations of ellipsoidal inclusions, with which they agree to first order in the volume fraction of the inclusions. Note that the distribution effects appear only to second order in the volume fraction of the inclusions, whereas the particle shape effects are of first order in the volume fraction of the inclusions. These estimates also reduce to the estimates (6.4) of Willis (1977) when = P,! = IP' and when either the stiffest or the most compliant material occupies the matrix phase. (In this case, they also agree with the so-called Mori-Tanaka (1973) estimates, but not more generally, when the Mori-Tanaka scheme can give nonsymmetric, and therefore physically unacceptable, predictions for Leff.)Finally, note that the no-overlap hypothesis places restrictions on the maximum volume fractions of the various types of inclusions, depending on the aspect ratios, as well as on the orientations of the ellipsoidal inclusions and their distributions relative to each other (see Ponte Castaiieda and Willis, 1995). In a closely related development, Hervt et al. (1991) made use of the HashinShtrikman variational principles to obtain upper and lower bounds for the effective modulus tensor of elastic composites with statistically isotropic distributions of composite sphere assemblages (Hashin, 1962). (Note that the bounds of Hashin for this class of microstructures did not make use of the statistical isotropy hypothesis, and are therefore weaker.) A generalization of the work of Hervt et al. (1991) for composites with ellipsoidal distributions of more general "morphologi-


233

cally representative patterns” has been given by Bornert et al. (1996). For N-phase composites, when the morphologically representative patterns are “composite ellipsoids” with a common matrix phase, one of the bounds can be determined explicitly and is found to be in precise agreement with the corresponding one-sided bound (6.6) of Ponte Castaiieda and Willis (1995) (provided that the stiffest or most compliant material occupies the matrix phase). On the other hand, the other bound requires numerical computation in general (Bornert, 1996). For the special cases of composite spheres and cylinders distributed isotropically, however, the other bound may be computed exactly when the phases are isotropic (Hervt et al., 1991; Hervt and Zaoui, 1993). When the phases are additionally assumed to be incompressible, the result specializes to with

where the superscripts 1 and 2 have been used to denote the matrix phase and the spherical inclusions, respectively. One important practical implication of the work of Bornert et al. (1996) (see also Bornert, 1996) is that the one-sided bounds (6.6) of Ponte Castaiieda and Willis (1995) may be used as appropriate estimates for composites with particulate microstructures, at least for two-phase systems, provided that the contrast and the volume fraction of the inclusions are not too large. This observation is consistent with the previous comment on the optimality of the Hashin-Shtrikman bounds for statistically isotropic composites with isotropic constituents, and their attainability by microstructures that are particulate in character (Francfort and Murat, 1986). In particular, it means that expressions (6.6) may be used as valid estimates for particle- and fiber-reinforced composites. Finally, it is mentioned for completeness that Hashin-Shtrikman bounds have also been developed for other, more general types of microstructures, including the works of Lurie and Cherkaev (1986) and Milton and Kohn (1988) for composites with generally anisotropic microstructures.

c. SELF-CONSISTENT ESTIMATES FOR RANDOM COMPOSITES WITH

GRANULAR MICROSTRUCTURES

In the context of the Hashin-Shtrikman estimates (6.4), the observation was made by Willis (1977) that if the reference modulus tensor Leo) is chosen equal to the effective modulus tensor Leff,as given by relations (6.4), an implicit equation

234


is obtained for I,(’), which when solved for L(O) leads to an estimate for Leff. This prescription reproduces exactly the self-consistent estimates of Budiansky (1965) and Hill (1965b) for two-phase composites, as well as the self-consistent estimates of Hershey (1954) and Kroner (1958) for polycrystalline aggregates, without the need for making the explicit assumption that the grains are ellipsoidal in shape. Thus the variational estimates (6.4) for Leff,with the prescription L(O) = Leff,provide a generalization of the classical self-consistent method for random composites with prescribed two-point statistics. Similarly, it is possible to consider the limit as the volume fraction of the matrix phase vanishes in the particulate Hashin-Shtrikman estimates (6.6), provided that the shape and orientation of the inclusions are set equal to those of the distribution ellipsoids. The resulting microstructure is “granular” in character, being composed of aligned ellipsoidal grains of identical shape and orientation, but of varying size so as to fill space. In this case, there is no advantage in choosing the modulus tensor L(O) of the homogeneous reference material to be equal to that of the matrix phase, since there is no matrix phase and all of the phases are now inclusions. Then perhaps the only sensible prescription is once again to choose L(’)equal to Leff,as given by expression (6.6), so that the average polarization in the composite vanishes identically. The resulting estimate for Leffis, of course, not a bound, but has the advantage that it treats all of the phases symmetrically, which may be appropriate for certain classes of composites with granular microstructures, such as polycrystals. Furthermore, perhaps not surprisingly, the resulting estimate is found to agree (once the phase indices are relabeled) with the above-mentioned self-consistent estimates for general microstructures with prescribed two-point statistics. This suggests that the self-consistent estimates may be appropriate for composites with granular types of microstructures, in much the same way as the Hashin-Shtrikman estimates were found to be appropriate for composites with particulate microstructures. An alternative interpretation and generalization of the self-consistent estimates in terms of “perfect disorder” has been given by Kroner (1977). The fact that the self-consistent estimates may be appropriate for granular-type microstructures is also suggested by the numerical results of Suquet and Moulinec (1997), where the classical self-consistent estimates are shown to be in good agreement with the results of numerical simulations for a special class of granular microstructures. A related result (Milton, 1985; Avellaneda, 1987) that will be useful to keep in mind, in the context of nonlinear composites, is the fact that the classical self-consistent estimate is also attainable-that is, there is at least one microstructure for which the self-consistent estimate is an exact result. Again, not surprisingly, such a microstructure is granular in character.


235

Other types of self-consistent prescriptions have been proposed in the literature, including the generalized (or three-phase) self-consistent estimates of Christensen and Lo (1979), as well as the differential self-consistent scheme of McLaughlin ( 1977). The generalized self-consistent scheme is believed to provide adequate predictions for the overall properties for two-phase composites consisting of disordered assemblages of composite spheres and, in spirit, consists in considering a single composite sphere embedded in an infinite reference medium whose properties are precisely the unknown elastic properties of the composite. For two-phase incompressible phases with statistically isotropic distribution, the effective shear modulus predicted by this generalized self-consistent scheme is determined as the solution of the nonlinear equation

where the function F is defined by (6.7). Generally speaking, however, it is more difficult to give these types of estimates statistical interpretations, although Bornert (1996) has recently made progress along these lines in the context of the three-phase model. For brevity, these various estimates will not discussed be here, but it will be emphasized once again that the nonlinear homogenization procedures discussed in earlier sections have the capability of generating bounds or estimates for the overall properties of nonlinear composites from the corresponding bounds or estimates of any linear homogenization scheme.

D. THIRD-A N D HIGHER-ORDER BOUNDSFOR RANDOM COMPOSITES Beran ( 1965) introduced a procedure to obtain variational estimates depending on up to three-point statistics, directly from the minimum energy representations (2.41) and (2.43) for the effective potentials. These estimates were simplified and improved in various ways by Kroner (1977), Milton (1982), and Willis (198 1). However, for N-phase composites, these estimates are fairly complicated, involving microstructural tensors analogous to the k('")but , which depend on the threepoint probability functions P ( r s t )Note . that such bounds are known to be exact to third order in the contrast for weakly inhomogeneous composites. Here, for simplicity, only the simplification of Milton and Phan-Thien (1982) for two-phase composites with isotropic constituents and statistically isotropic microstructures will be mentioned. Thus, denoting the bulk and shear moduli of the phases by k(')

236


and p('), respectively, the third-order estimates for the effective shear and bulk moduli of the composite keff and peffmay be written in the forms

where f and jl are functions of the volume fractions c('), the phase moduli k(') and p @ )and , some third-order microstructural parameters q(') and {('I, lying in the interval [0, I]. For example, the third-order upper and lower bounds on keff are obtained by setting

respectively. Similarly, upper and lower bounds on peffare obtained by setting jl = 0 /6 and E/6, where 0 and E are given by rather complicated expressions (see eqs. (48) and (49) of Milton and Phan-Thien, 1982). When the phases are incompressible, however, the expression for j l for the upper bound reduces to

and q ( 2 )= 1 - q ( ' ) . ) (In these expressions, < ( 2 ) = 1 - {(I) Assuming for definiteness that k ( ' ) < k(2)and p ( ' ) < pLL(2), it is interesting to remark that the above third-order upper and lower bounds for keff and peffagree exactly with each other in the limits as q ( l ) and { ( I ) approach 0 and 1, reducing to the corresponding Hashin-Shtrikman (HS) upper and lower bounds in these limits, respectively. In particular, this means that the HS estimates are exact for statistically isotropic microstructures with q ( ' ) and (('1 equal to either 0 or 1. Examples of such microstructures are the sequentially laminated microstructures of Francfort and Murat (1986). Note that such microstructures are particulate in character, providing further evidence for the appropriateness of the Hashin-Shtrikman estimates for at least certain classes of particulate microstructures. Higher-order bounds incorporating statistical information of order higher than three have been proposed by a number of authors (see, for example, Kroner, 1977; Milton and Phan-Thien, 1982). Even-order bounds are obtained from the HashinShtrikman variational principles and odd-order bounds from the classical variational principles. As efficient computational methods are developed to measure


237

three- and higher-point statistics, the practical value of these higher-order bounds is expected to become more significant. In any event, higher-order bounds for linear composites can be as easily incorporated into the nonlinear homogenization procedures as the two-point bounds. In the next section on examples, however, only higher-order bounds of the Beran type will be developed, with the limited goal of illustrating the general ideas.

VII. Applications to Nonlinear Composites and Discussion This section presents some examples of the application of the procedures of Sections IV and V.B for a few specific cases, including porous and rigidly reinforced materials, as well as more general two-phase power-law and perfectly plastic composites. For simplicity, almost all of the nonlinear examples will be restricted to two-phase systems with isotropic phases, although a few results will be given for power-law polycrystals. A. POROUS MATERIALS Although porous materials with anisotropic matrix phases could be considered more generally, in this section for simplicity only porous materials with isotropic matrix phases will be considered. Thus the matrix phase strain and stress potentials will be taken to be of the form (2.4) and (2.5). Some selected bounds and estimates for the corresponding effective potentials Weff and Ueff are given below. 1. VariationalBounds and Estimates

Because of the assumed isotropy, the results of Section IV.C.2 for composites with isotropic constituents can be used to obtain bounds for the effective potential Weff of porous materials. Thus, specializing equations (4.56) and denoting the matrix by phase 1, one has that

(7.1) where

238


and where the first equation must be solved for ?

with

The corresponding lower bounds for U e f can be shown to have an analogous form. Use of any upper bound or estimate for the effective modulus tensor of a linear porous material, or class of porous materials, with an isotropic matrix then leads to corresponding upper bounds or estimates for the effective strain potential of nonlinear porous materials. No special simplificationof the resulting nonlinear estimates is available in general, but when the matrix phase is additionally assumed to be incompressible ( k ( ' )+ co),so that the effective modulus and compliance tensors of the linear porous comparison material can be written in the form

L

independent of for some microstructural tensors and 6, for Weff and the corresponding estimate for Ueff reduce to

w e f f ( ~I ) c(1)40(l)(E::)),

ueff(X)

the estimate (7.1)

c ( l ) ~ ( l ) ( ~eq (l)),

(7.3)

where

Some results of special interest for 2):

and/or

are listed below.

Hushin-Shtrikman bounds for statistically isotropic microstructures. When the distribution of the void phase is statistically isotropic, the linear HashinShtrikman (HS) bound (6.4) leads to a corresponding upper (lower) bound for Weff (Ueff).It is determined by the expressions

This HS bound for all statistically isotropic porous materials was first obtained by Ponte Castafieda (1991a) and Suquet (1992), directly from (4.51) and (4.57), respectively. Willis (199 1) showed that this result could be obtained, alternatively, by means of the Talbot-Willis procedure, thus improving an earlier result by Ponte Castafieda and Willis (1988), which made use of a nonoptimal choice of the ho-


239

mogeneous reference material in the context of the Talbot-Willis procedure. Talbot and Willis (1992) made use of the estimate (6.4), together with a version of (4.51), to obtain the more general result (7.4). The bound (7.5) was also derived as an ad hoe estimate (not as a bound) by Qiu and Weng (1992) by estimating the stress in the matrix from the energy in the porous material. In this connection, it is relevant to note that although the “energy method’ of Qiu and Weng gives results that are in agreement with (7.3) for porous materials with incompressible matrix phases, the method delivers less accurate results (different from (7.1)) for porous materials with compressible phases. Related ad hoc estimates, in the context of a “multiparticle effective field method,” have also been obtained by Buryachenko and Lipanov (1989a,b), by estimating the stress in the matrix from its second moment. Once again, these early estimates agree with the bound (7.5) for dilute porosity levels when the matrix is incompressible, but not more generally (see Buryachenko, 1996, for details). Incidentally, an alternative, simpler way of obtaining multiparticle effective field estimates for nonlinear composites would be to make use of the corresponding estimates for linear composites in the context of the general nonlinear estimate (7. I), interpreted as an approximation. Hushin-Shtrikman bounds for ellipsoidal voids distributed with ellipsoidal symmetry. Using the HS bounds (6.6) of Ponte Castaiieda and Willis (1995), specialized for porous materials with incompressible phases, it follows that Weff is given by (7.3) and (7.4), with

where P = (l/c(’))(Pi - c(’)Pd). When both the shapes of the pores and the distribution are spherical, this result specializes to (7.5). In an attempt to incorporate the effect of pore shape evolution during deformation processing, an application of this result for perfectly plastic composites with identical shapes for the voids and the distribution was given by Ponte Castaiieda and Zaidman (1994). More general results when the shape of the pores and distribution are different have been obtained by Kailasam et al. (1997a, b). Hushin-Shtrikman bounds for cylindrical voids with circular cross section. Taking the limit as the shape of the voids and distribution becomes cylindrical with circular cross section, the following result (Suquet, 1992) is obtained

where the axis of symmetry has been taken to be aligned with the x3 direction.

240

Pedro Ponte CastaZeda and Pierre Suquet

Hashin-Shtrikman boundsfor cracked solids. Another important case of the general result (7.6) is when one of the aspect ratios of the voids tends to zero, leading to cracks. (Note that d2)+ 0 in this case.) When the cracks are penny shaped, aligned, and distributed isotropically (in space), the following upper bound is obtained:

(7.8) = $nn(2)a3is a crack density parameter corresponding to n ( 2 )cracks where d2) of mean radius a per unit volume. The corresponding results for "flat" distributions of cracks (when the crack interactions are weak), which are obtained by linearizing (with respect to a ( 2 )the ) above result, were first given by Suquet (1992) and Talbot and Willis (1992). When the cracks are randomly oriented and distributed isotropically, the following upper bound is obtained:

which is due to Ponte Castaiieda and Willis (1999, in the linear context.

Self-consistent estimates. Making use of the self-consistent (SC) estimates of Hill (1965b) and Budiansky (1965), it follows (Ponte Castaiieda, 1991a) that expression (7.3) provides an estimate for Weff with (7.9) Note that the result is a rigorous upper bound for the special microstructure that attains the linear self-consistent estimate.

Beran third-order bounds. Finally, making use of the Milton (1982) simplified form of the McCoy (1970) third-order bounds for linear elastic composites, it is straightforward to derive the following third-order upper bound for porous materials with statistically isotropic microstructures

Bounds of this type for nonlinear composites were first proposed by Ponte Castaiieda (1992a) (see also Ponte Castaiieda, 1997b). Note that when the third-order


24 1

parameters (('1 and ~ ( ' 1are equal to 1, this result reduces to the HS upper bound ( 7 3 , but is generally tighter than (7.5) for other values of {('Iand q ( ' ) .

2. Second-Order Estimates As the second-order theory of Ponte Castaiieda (1996a) is very recent, not many results are currently available. However, counterparts of the HS estimate (7.5) and the SC estimate (7.9) are available for incompressible-matrix porous materials with statistically isotropic microstructures (or isotropic distributions of spherical pores), provided that the pores are additionally assumed to be incompressible, so that the composite is incompressible ( E m = 0). Thus these results physically correspond to the case where the pores are saturated with an incompressible fluid. Assuming isotropic behavior for the matrix phase, as characterized by the function f ( l ) in relation (5.41) (with k(') + oo),the second-order estimate (5.32) for the fluid-saturatedporous materials may be written in the form

(7.10)

+

where E ( ' ) = (1 c(')w)E is obtained from the consistent estimate (5.40) for the average strain in the matrix phase.

Hashin-Shtrikman estimatesfor statistically isotropic microstructures. Using the HS estimate (6.4) for the linear comparison porous material with (anisotropic) matrix modulus tensor given by (5.43), one obtains an estimate of the form (7.10) with

where r ( ' ) = p ( ' ) / A ( ' )and , where the function a , arising from the relevant P(O) tensor, is given by expression (C7) in Appendix C. Note that a depends on the material properties, through r ( ' ) ,as well as on the third invariant of the strain, through the variable 8 (recall from Appendix C that cos(38) = 4 det(Ed)). It is important to emphasize that the second-order estimates for statistically isotropic microstructures depend on the third invariant of the strain, whereas the corresponding variational bounds of Section IV do not.

242


Selj-consistent estimates for statistically isotropic microstructures. Proceedtensor in (6.4) now ing similarly, but taking into account the fact that the depends on the anisotropy of L(O) = Leff= 2heffIE 2peffF,one arrives at

+

w =-

[

-

1 c(2)a(8,

- seff

~

r ( ' ) a(6,sef f ) - seff

I' -1

where serf= peff/heffis obtained as a function of r ( * )and c(l) from the solution of the equation [seff

-

a(8,Seff)c(')] = r ( ' ) [ l- / 3 ( ~ , eff>c( 2 )

1 9

where the function /3, also defined by the corresponding tensor, is given by expression (C8) in Appendix C. Note that depends on r ( ' ) and 8. The above second-order estimates of the HS and SC types are due to Ponte Castaiieda (1996a) and Ponte Castaiieda and Nebozhyn (1997). Note that the above HS estimates also can be used to model the effective behavior of materials with dry pores under purely deviatoric loading conditions (with zero hydrostatic strain). However, the above SC estimates cannot be used to model nonhydrostatic loading of dry pores-in particular, the percolation limits of the saturated and dry pore problems are known to be different. (This is a well-known fact for linear-elastic composites.) Some preliminary results for hydrostatic loading of porous materials (with dry pores) have been given very recently by Nebozhyn and Ponte Castaiieda (1997). Making use of the (2-D) expressions (C10) for a and /3, instead of (C7) and (CS), the above formulae can also be used to obtain estimates for the plane strain transverse shear of a (fluid-saturated) porous material with aligned cylindrical pores.

3. Sample Results for Power-Law Porous Materials Purely deviatoric loadings. Figure 2 shows a comparison of the HashinShtrikman-type variational bounds (7.5) and second-order estimates (7.1 1) for statistically isotropic porous materials. The behavior of the (incompressible) matrix is characterized by the power-law relation (2.7), so that for purely deviatoric loading conditions (Em = 0), the effective potential Weff can be written in the form

Weff(E) =

m+l

(7.12)

where it is recalled that 6 depends on the determinant of the strain (6 = 0 corresponds to axisymmetric deformation, whereas 8 = 3r/6 corresponds to simple

243

Nonlinear Composites 1, . . . , . . . . . . . ,

. . .

I

. . .

I

0.8 -

0.6 -

0.4 -

0.2 -

n = 1000 U.0

F I G .2. Power-law porous materials with statistically isotropic microstructures. Various predictions for the effective flow stress for purely deviatoric loading as functions of the porosity c(*). Variational upper bounds (VB+) and second-order estimates (SOE) are shown. The Voigt upper bound VB+(V+) is microstructure independent. The upper hound VB+(HS+) was obtained by bounding the stiffness of the linear comparison composite by means of the Hashin-Shtrikman upper bound, and is valid for all isotropic microstructures. For the second-order estimate, the effective stiffness of the linear comparison composite was also estimated from the Hashin-Shtrikman upper bound and results are given for uniaxial tension (0 = 0) and simple shear (0 = n / 6 ) .a) n = 10. b) n = 1000.

~0"~'.

244


shear). Results are shown, in Figure 2, for two values of the power-law exponent n = l/rn equal to 10 and 1000. It can be seen that the second-order estimates (SOE) generally lie below the variational bounds (VB). It can also be seen that although the variational bounds are independent of the type of loading, the corresponding second-order estimates are different for the extreme cases of uniaxial tension (0 = 0) and simple shear (0 = n / 6 ) , with the shear results always lying below the tensile results. Note that the difference between the shear and tensile results becomes progressively larger, as the level of nonlinearity increases, with the second-order estimates remaining close to the variational estimates for tension, but with the second-order estimates predicting sharper drops in the load-carrying capacity of the porous material in shear. As first pointed out by Drucker (1959), this sharper drop for large values of n (tending to perfectly plastic behavior) is possible under shear loading because of the availability of localized deformation modes (i.e., slip bands) passing through the pores. There is also experimental evidence for this type of behavior (see Spitzig et al., 1988). On the other hand, for the axisymmetric deformation mode, the plastic deformation is diffuse through the matrix (see Duva and Hutchinson, 1984), and there are relatively small differences between the variational and second-order estimates (see Ponte Castaiieda, 1996a, for more details). It is emphasized that the second-order procedure has the capability of capturing more accurately the anisotropy of the localized deformation fields by means of the use of the anisotropic tangent modulus tensors (see Ponte Castaiieda, 1992a, for more details). Hydrostatic predictions. For hydrostatic loading conditions (E,, = 0), the Hashin-Shtrikman (HS) variational bounds (7.5) for the power-law porous materials can be written in the form

(7.13) where miff = (1 - c(~))(c(~))-('+,')/'. This result is a rigorous upper bound for all isotropic microstructures, including isotropic distributions of spherical pores. (Note that 0:" + 00 as c ( ~-+) 0, which makes sense because the matrix material has been assumed to be incompressible.) From the exact solution of the problem of a spherical shell subjected to hydrostatic loading (see, for example, Budiansky et al., 1982), an alternative bound can be constructed for composite-sphere (CS) types of microstructures. The result is

Note that this result coincides exactly with the HS bound when rn = 1, in which case the CS microstructure is known to attain the HS bound. However, for any


245

other value of in, the CS bound is always sharper than the HS bound. For example, in the limit as m + 0, the HS bound approaches (1 - c ( * ) ) / G ,whereas the CS bound approaches - In d2).It is noted that these two estimates can give very different predictions for dilute porosity levels (as d2)+ 0). Although it is reasonable that the CS bound should be sharper than the HS bound (because it applies for a very special type of statistically isotropic microstructure), it is important to note that the variational procedure can also be used to obtain a bound for the CS microstructure; unfortunately, the resulting bound is exactly the same as the HS bound above. The reason for the relative weakness of the HS bound is related to the fact that, for hydrostatic loading, the exact strain field in the CS microstructure varies as r P 3 (with Y denoting the radial coordinate), so that the exact secant modulus for this type of loading varies as r3(1-n')/2.Therefore, for an isolated void in an infinite matrix, the exact secant modulus can vary all the way from small values, in the vicinity of the void surface, to large values, far away from the void surface. It is thus seen that the approximation of a constant secant modulus in the matrix cannot be very good in this case. Fortunately, this is a very special type of situation; more typically, as, for example, for shear loading of a porous material, the secant modulus exhibits a narrower range of values-except in the vicinity of corners and other singularities, where the strains can get large locally (i.e., in regions of small volume)-and therefore the approximation of piecewise constant (by phase) secant modulus is more sensible in general. Of course, the advantage of the variational (and second-order) estimates is that they can be applied equally well for other, more general situations for which exact solutions-like the one for hydrostatic loading of a spherical shell-are not available. In the specific context of rigid perfectly plastic porous materials, a model that makes use of the exact solution for hydrostatic loading of a spherical shell was advanced by Gurson (1977). Of course, this model is very good for hydrostatic loading conditions, but it has been found to violate the HS bound (7.5) for purely deviatoric loading conditions (see Ponte Castaiieda, 1 9 9 1 ~Leblond ; et al., 1994). It is also restricted to pores of spherical shape, whereas the variational bounds can also be used for general nonspherical pore shapes (see Ponte Castaiieda and Zaidman, 1994). Analytical extensions of the Gurson model for aligned spheroidal pores have been given recently by Golaganu et al. (1993, 1994) and by Girirjeu (1996) following the work of Lee and Mear (1992a). An example of a porous material that is not statistically isotropic is provided by the periodic hexagonal patterns of voids considered in unit cell studies for creeping solids (Van der Giessen et al., 1995). Nonlinear self-consistent schemes for porous materials have been proposed by Michel and Suquet (1992) and Michel(l994).

246

Pedro Ponte Castaiiedu and Pierre Suquet B. RIGIDLYREINFORCED MATERIALS

In this section, composite materials with isotropic nonlinear matrix phases reinforced by a rigid phase are considered. As in the previous example, the phase strain and stress potentials will be taken to be of the form (2.4) and (2.5). Some selected bounds and estimates for the corresponding effective potentials Weff and Ueff are given below.

1. Variational Bounds and Estimates Because of the assumed isotropy, the results of Section IV.C.2 for composites with isotropic constituents can be used to obtain bounds for the effective potential Weff of rigidly reinforced composite materials. Labeling the matrix as phase 1 and specializing eq. (4.56), one obtains the following bound for Ueff:

(7.14) where

(7.15) and where the first equation must be solved for

a

with

Use of any lower bound or any estimate for the effective compliance tensor of a rigidly reinforced composite with an isotropic matrix then leads to corresponding lower bounds and estimates for the effective stress potential of nonlinear rigidly reinforced composites. When the matrix phase is additionally assumed to be incompressible ( k c ' )+ co),the resulting composite is also incompressible and the corresponding estimates for We"' and Ueff can be written in the form (7.3), (7.4), with E,, = 0, in terms of appropriate microstructural tensors i and = i-' . Some bounds and estimates for Weff and/or Ueff are listed below in terms of the variables g,!:' and/or respectively (the matrix being assumed to be incompressible).

a$,),


247

Hashin-Shtrikman estimates for ellipsoidal inclusions distributed with ellipsoidal symmetv. In this case, the HS estimates (6.6) of Ponte Castaiieda and Willis ( 1995) for ellipsoidal rigid inclusions distributed with ellipsoidal symmetry are lower and upper bounds for Liff and M8ff,respectively. Because of this fact, the expressions (7.3) do not yield rigorous bounds for Weff andlor Ueff, respectively. However, as mentioned earlier, the HS estimates (6.6) can be interpreted as appropriate variational estimates for particulate microstructures, and for this reason the corresponding nonlinear results can be interpreted as appropriate variational estimates for particulate microstructures. Thus expression (7.3), with the inequality replaced by an equality (in the sense of an approximation), gives estimates for Weff andor Ue“. The result is given by expressions of the form (7.4) with (7.16) where c(’)P= Pj - c(*)Pd.For spherical particles, distributed with statistically isotropic symmetry, the following estimate is obtained: (7.17) This “lower” estimate was proposed by Ponte Castaiieda (1991b, 1992a) for isotropic rigidly reinforced composites and generalized by Talbot and Willis (1992) for anisotropic composites. Talbot and Willis (1992) and Li et al. (1993) gave results for aligned spheroidal inclusions. GWjeu and Suquet (1997) have recently given an application to rigidly reinforced materials, where the matrix is taken to be of the Gurson type. Hashin-Shtrikman bounds for statistically isotropic microstructures. In this case, the HS upper bounds for linear elastic composites with arbitrary statistically isotropic microstructures are unbounded, and therefore the corresponding upper bounds for Weff are also unbounded. Physically, this corresponds to the fact that statistical isotropy does not exclude the possibility of rigid material lines or surfaces extending from one end of the specimen to the other, i.e., percolation. However, when the microstructures are additionally assumed to be particulate in character (which excludes the possibility of percolation, at least for small enough volume fractions of inclusions), it is possible to obtain finite upper bounds for the effective modulus tensor of rigidly reinforced composites. Thus linear HashinShtrikman bounds of this type were obtained by HervC, Stolz, and Zaoui (HSZ) (199 1) for composite-sphere assemblages and more generally for morphological patterns by Bornert et al. (1996). More recently, Bornert (1996) pointed out that in

248


fact the (lower) bounds for composite-sphere assemblages can also be interpreted as rigorous bounds for composites with the larger class of particulate microstructures considered by Ponte Castafieda and Willis (1995). When both the shapes of the rigid inclusions and the distribution are spherical, the upper bound can be computed explicitly from the corresponding linear bound (6.7) of Hashin (1962) and H e r d et al. (1991) (they are identical in this case) and is given by

This nonlinear HSZ bound for statistically isotropic rigidly reinforced materials with composite-sphere assemblage microstructures was first obtained by Suquet (1993a), directly from (4.57). Hashin-Shtrikman estimates for jiber-reinforced composites. Taking the shape of the inclusions and distribution to be cylindrical with circular cross section in the general result (7.16), the following result (Li et al., 1993) is obtained:

where the axis of symmetry has been taken to be along the x3 direction. Note that this composite is inextensible along the fiber direction and can only undergo shear in the transverse and longitudinal directions. Hashin-Shtrikman estimates for rigid disks. Another important case of the general result (7.16) is that in which one of the aspect ratios of the rigid inclusions tends to zero, leading to disks. (Note that c ( ~ + ) 0 in this case.) When the disks have circular cross section, and are aligned and distributed isotropically, the following result is obtained:

where = is the disk density parameter corresponding to n(2)disks per unit volume of mean radius a. The corresponding results for “flat” distributions of disks (when the disk interactions are weak) are obtained by linearizing the above results and were first given by Talbot and Willis (1992) and Li et al. (1993).


249

When the disks are randomly oriented and distributed isotropically, the following result (Ponte Castaiieda and Willis, 1995) is obtained:

Self-consistent estimates for statistically isotropic microstructures. Making use of the self-consistent estimates of Hill (1965b) and Budiansky (1965), it follows that (Ponte Castaiieda, 1991a) (7.20)

Beran third-order bounds for statistically isotropic microstructures. Finally, making use of the Milton (1982) simplified form of the McCoy (1970) thirdorder bounds (6.9) for linear-elastic composites, it is straightforward to derive the following third-order estimates for rigidly reinforced materials with statistically isotropic microstructures:

Note that when the third-order parameters = g**W = SUPb * Y - g*(y)). Y

If, in addition, g is differentiable, the following assertions (i), (ii), and (iii) are equivalent: ag (i) y = - (x), ax

ag*

+

-

(iii) g(x) g*(y) = x y. (y), aY When g is not differentiable, the above relations are to be understood in the sense of subdifferentials. The dual of a sum of two convex functions is not the sum of the two dual functions, but their inf-convolution, (Sl

(ii) x =

~

+ g2)*(y) = glVgz(Y) = z1+z2=y inf g;(zl) + gZ(z2).

643)

The concave dual function of an arbitrary function f defined on X is f*(Y) = i n f b . y - f ( x ) I .

(A4)

Note that when f is concave, -f is convex, so that all definitions or results valid for convex functions can be translated into similar results for concave functions. 2. Strain-Energy and Stress-Energy Functions Assume that the strain-energy potential w is a positive convex function, such that

w(0) = 0,

lim

Iel++m

W(E)

= +cc.

(A5)

Assume, in addition, that w may be written in the form W ( E ) = F ( E ) ,where P = ; E @ E and where F is a concavefunction defined on the vector space of symmetric fourth-order tensors Q . The dual function of F (in the sense of concave functions) is defined as

F,(lL,) = inf(IL :: Q - F ( e ) } . e

(A6)

Note that F* is negative (take Q = 0in (A6)) and that F* (L) = -cc when L is not positive definite. Indeed, assume that IL has a negative eigenvalue, i.e., that there exists EO such that EO : IL : EO 5 0. Then, taking Q = t2eo @ EO in the definition of

283


F* and letting t go to 00,we obtain (by virtue of the growth condition (A5)) that F,(L) = -CO. Since F is concave, F is the dual of F,, or

F ( e ) = inf (L:: Q - F*(L)}.

('47)

L>O

We claim that the dual potential w* of w can be written as w*(a) = G(ar), with ar = iu €4 u,where G is a convex function defined as G(J) = SUP

{IL-'

:: J + F*(L)}.

L>O

To prove this result, note that

It follows from (A7) that (with

E

= ; E €4

E)

Then

1

w * ( a ) = supsup 1( T : E - 2- - - E : L : E + F * ( L ) ) L>O

E

[-1

= sup L>O 2

(T

: L-': (T

+ F * ( L ) ) = lL>O sup {L-':: a~ + F*(L)} = G(ar).

(A9) equation Note that G is defined as the supremum of linear (therefore convex) functions of s. It is consequently a convex function of J. 3. Isotropic Constituents Let F be given in the form

F ( Q ) = 3kJ :: Q + f

(i R :: IE) when K ::

F ( I E )= -00 By definition, the concave dual of F is

F,(L) = itf (IL :: Q - F ( Q ) } K::e>O

otherwise.

Q

>_ 0,

Pedro Ponte Castaiiedu and Pierre Suquet

284

The optimality conditions for the problem (A10) can be expressed with a positive Lagrange multiplier h

4 3

4 3 h20,

IL = 3kJ + - f ' ( - R :: e)IK+hIK, R::ezO,

hR::e=O.

('411)

It follows from the optimality conditions (A1 1) that IL is necessarily an isotropic tensor,

4

IL = 3kJ + 2 p R , 2 p = ?f'(JK

:: e) + h .

Note that p is positive, (recall that f' is positive, since f is an increasing function). Therefore,

Finally,

F*@) =

{ f*(ip)

when

IL = 3kJ + 2pK,

p > 0,

otherwise.

--oo

(A13)

4. Crystalline Materials Consider first the stress potential for one slip system: G(k)(3) =

{

g(k)(2M(k):: 3)

when M(k) :: 3 2 0 ,

+Go

otherwise.

The convex dual of G(k)is, by definition,

GTk,(M) =

SUP

{M :: s - G ( ~ ) ( s ) )

3

M(k)::310

=

sup

{M :: s - g(k)(2M(k):: 3 ) ) .

S

M(k,::SzO

The optimality conditions for the problem (A14) can be expressed with a positive Lagrange multiplier h, = 2g;,)(2qk) :: 3 ) M ( k )

MI(,) :: s 3 0 ,

h 2 0,

+ hM(k), hM(Q :: s = 0.

('415)


285

Since M is necessarily proportional to M ( k ) ,

Finally: GTkj(M)

=

{ +oo

gTk,(a) when M = 2aMTk),

a > 0,

otherwise.

Next consider the case of a single crystal, where K

k= 1

Since G is the sum of convex functions of s, its dual is the inf-convolution of the corresponding dual functions:

where the infimum is over the a:;; such that

B. DERIVATION OF THE POTENTIAL WITH RESPECTT O A PARAMETER The derivation of the overall energy of a composite with respect to a parameter is used in various circumstances to obtain expressions for the second moment of the strain in each phase of a linear composite or for the small-contrast expansion of the effective potential of a nonlinear composite. We give here a general result applying to linear composites and to nonlinear composites as well.


286

Consider a composite governed by a pair of dual convex potentials w ( x , E , t ) and u (x,a,t ) depending on a parameter t. Then the localjelds Et and afand the overall potentials depend on t :

It also follows that

To prove this result, let i t denote the derivative of et with respect to t . Note that i , = E ( u j ) is a compatible strain field and its average is zero (since E does not depend on t ) . Differentiating (Bl) with respect to t yields

By the Euler-Lagrange equations associated with the variational principle (B 1)(1, the stress field a, = ( t , e t ) satisfies the equilibrium equations div(at) = 0 in V. According to Hill’s lemma,

2

(at : i t ) = (at)(it)= 0. This completes the proof of (B2),. The proof of the other equality is similar.

1. Application to Second Moments of the Strain in Linear Composites The result just derived can be used to obtain expressions for the second moments of the strain fields in the individual constituents of linear composites. Consider:

Then

and, according to (B2),,

287

Nonlinear Composites Thus the second moment of the strain field in phase ( r ) is given by

After rearrangement,

Similarly,

2. Second Moment of the Shear Stress Consider now:

au ( t ,

-

at

fJt)

= c(')u : M(" (k) . u3 '

@(t,

1 2

E) = - c : laeff :c.

Then, according to (B2)6,

3. Average Strain Due to Piecewise Constant Polarization Fields Finally, consider:

288


Then

This completes the proof of (5.27).

C. PTENSORS 1. Spheroidal Inclusion in an Isotropic Material Assuming that the matrix material is isotropic with bulk and shear moduli k(O) and p(O),respectively, and that the inclusion is spheroidal (i.e., ellipsoidal with circular cross section) with length-to-diameter aspect ratio w,the P tensor exhibits transversely isotropic symmetry, so that it may be written, using the notation of Walpole (1969, 1981), in the form

where (Eshelby, 1957)

+

+

2[4 - 3h(w) - 2w2]p(0) 3[2 - 3h(w) 2w2 - 3 ~ ~ h ( w ) ] k ( ~ ) 8( 1 - w 2 ) p ( 0 )(4p(O) 3k(O)) (C1) In these relations, (0)

Pp =

h(w)= h(w)=

+

w[arccos(w) - w % / i - T Z ] (1 - w2)3/2

w [ w d m- arccosh(w)] (w2 -

1)3/2

for w 5 1, for

w 2 1.

Asymptotic expressions for the above P tensor in the limits as w -+ 0 and 00 have been given by Willis (1981). These are useful in the computation of the effective modulus tensors for the cases of rigid disks and cracks. More general expressions,


289

assuming transversely isotropic behavior for the matrix material, have been given by Laws and McLaughlin (1979). When the matrix material is also incompressible, the above tensor simplifies and can be given the spectral decomposition

In this relation, IE [PI, IE In], and IE [dl are projection tensors, physically corresponding to transverse, longitudinal and axisymmetric shear (see deBotton and Ponte Castaiieda, 1992,1993),which are defined by

where aij = ninj and Bij = & j - ninj, with n denoting the axis of transverse isotropy. The coefficients p p ) , and correspond to the three possible modes of shear transverse to the fibers, shear along the fibers, and axisymmetric deformation, respectively, and they are given by the expressions

FA'',

2. Spherical Inclusion in an Anisotropic Material with Special Symmetiy Assuming that the modulus tensor of the matrix phase exhibits the material symmetry

+

+

= 3k(O)J 2p(O)IF 2h(O)IE,

where the tensors J, IE, and F are orthogonal projections defined by relations (2.24), (5.14),and (5.15), respectively, and assuming that the inclusion shape is spherical, so that the shape tensor Z = I, the associated tensor )'()'€I can also be written in the form

290


and

-=-I 1

1

2k(,O)

6n

/Ed I+I

p(0)

+ $(A(O)

.t12- a(O)(P - E d - e l 2

-

p(O))(lEd .,$I2 - a(O)(( . E d - 0 2 )

d s( P ) ,

(C5) with =

3k(0)+ p(o) 3k(0) + 4p(o)

'

These relations are due to Ponte Castaiieda (1996a, 1997a) and Ponte Castafieda and Nebozhyn (1997). When the matrix phase is incompressible (k(O) + OO), the above results for p(,O' and A!) can be simplified to give

and where 8 is the angle defined by (Kachanov, 1971) where r ( 0 ) = p(O)/AC0)

which therefore depends on the determinant of the strain. The functions a and B are, in turn, defined by the relations

29 1

Nonlinear Composites 2.0

71

A

C 1.8

n=+m

1.6 1.4

1.2 1.o

0.8 0.6 0.4

0.2

1

0.0 0.0

0.1

0.2

e

0.3

0.4

0.5

= n and of the loadF I G . 14. Constant C (C9) as a function of the exponent of nonlinearity ing parameter 8 . Reprinted from Suquet and Ponte Castaiieda (1993) with permission of the French Academy of Sciences.

and

where

is the integral introduced by Suquet and Ponte Castaiieda (1993). For high nonlinearities (large values of r(O)),C depends strongly on the third invariant of E d (ore), as shown in Figure 14. Finally, note that the corresponding two-dimensional version of the above result, which is useful in the analysis of fiber-reinforced microstructures, may be obtained formally by making the replacements

Note that these results are independent of the determinant in this case.

292


D. A BRIEFREVIEWOF OTHERSCHEMES 1. Secant Methods The starting point of all secant methods is the “local” problem to be solved for the local stress and displacement fields u and u:

u = L,(x): E(x),

div(u) = 0,

where the secant tensor L,depends on x through the strain field E(X) or through ~ ( xif) the form (2.17) is assumed for the strain-energy. Clearly L,ycan vary from one point to another, and this nonlinear problem is as difficult to solve as a linear problem with infinitely many different phases. An approximation is required to make further progress. The approximation for secant methods consists of replacing the secant tensors L,(x) with tensors that are constant over each phase:

(x) = L(‘),

in phase r.

In addition, it is reasonable to assume that the tensors L(‘)are given by the constitutive relations of phase r evaluated at some “effective strain” in charge of picking up the main features of the strain field in phase r. Assuming for a while that this effective strain is known, the approximate version of the local problem reads

a(x)= L(‘): E(X) 1 E = - (VU VU‘), 2

+

in phase r,

u E K(E).

div(u) = 0,

(D2)

For simplicity, the same notations u, u,E are used to denote the local field solutions of (Dl) and (D2), although it should be clear that these fields are different in general. In the rest of this appendix we consider only the solutions of (D2). Given the tensors L(’),the problem (D2) is now a classical one for a linear-elastic material. Note, however, that these tensors themselves depend on the strain field E (through the effective strain), which itself depends on the tensors L(“), s = 1, . . . , N . The problem is therefore nonlinear, but instead of having infinitely many nonlinear problems to solve (at each point x), there are “only” N nonlinear problems to solve in each phase r. In the most commonly used secant method (called classical hereafter), the effective strain of phase ( r ) is the average strain of this phase, E(‘) = (@)(’I, which can be expressed in terms of the overall strain E by means of the strain concentration tensor A(‘),

293

Nonlinear Composites Therefore the classical secant method may be summarized as follows:

1. A linear theory providing an expression for Leffand the N strain concentration tensors A(r)in each phase as functions of the secant tensors L(')Is=~,...,~ and of the microstructure.

2. The resolution of N nonlinear problems for the N unknown secant tensors L(r)(~('1): g ( r ) = A(r) . L'" = J$)(gW), . E, A@) = A(r)(L(\), " s = 1, . . . , N ) .

(D3)

3. Once the N nonlinear problems (D3) are solved, the overall stress-strain relation, given by

2. Incremental Method General procedure. The constitutive law (2.1) of the individual constituents may be alternatively written in the incremental form, u = Lt(x,E) : i,

a2w

L,(x,e) = -(x,E). ae2

Then the local problem, in incremental form, reads U

= L,(x): i(x),

1 i= - (vu 2

+ vuf),

div(u) = 0,

u

E

Ic(E).

(D6)

The tangent tensors Lr(x) can be different from one point to another, since they depend on the strain field E(x). Again, an approximation must be introduced to simplify the problem, and the tangent tensors Lt are replaced by tensors L:r), which are constant in each phase. These tensors are computed from the exact expressions of the tangent tensor in each phase ( r ) evaluated at some effective strain for this phase. The most obvious choice (and the one that has been uniformly adopted) for the effective strain of phase r is the average strain over this phase,

294


The local problem now reduces to u = L(') . i ( x ) ,

1 i = -(Vu 2

div(u) = 0,

+ Vu'),

u

E

K(E). (D8)

It is a linear problem for the increments iand u , which can be determined with any linear theory appropriate to the morphology of the composite and capable of predicting the effective properties of anisotropic linear composites. At each time t , the average strains E.(r) and the tangent tensors L(') are known. The overall elastic tangent tensor Leffis computed, and the overall constitutive law is found in incremental form as

2 = Leff(E): E.

(D9)

The increments of average strain in each phase are computed by =A (') : E,

where the tensors A(') are the elastic strain concentration tensors associated with the stiffness tensors L(').The average strains E ( " ) are then updated, and the procedure continues step by step. Under certain conditions, the above relations (D9) are integrable in time, and the overall constitutive law can be expressed as a relation (nonlinear) between stresses and strains. But nothing in the procedure guarantees that this integration is possible, and, when it is so, it is not clear that the resulting constitutive equations derive from a potential.

Isotropic composites. When the individual constituents are isotropic with a strain-energy given by (2.4), the tensor of the tangent moduli IL.' has the form (5.13) at each point x in the r.v.e.:

Lt(x) = 3kJ

+ 2P'F + 2hE,

Note that not only the coefficients and h, but also the tensor E depends on x. In addition, Lfis generally anisotropic (except when the strain is purely spherical). The linear theory used to derive the effective tangent stiffness must be able to predict the overall behavior of anisotropic phases, even when the geometry of the constituents does not favor any particular direction. The Hashin-Shtrikman bounds (6.4) can be used to estimate Leff.A first difficulty then is to compute the tensor P (') from the general relations (6.2). In general this determination must be done numerically. For random composites, the formulae (C3), (C4), (C5) can


295

be used. Another difficulty in the evaluation of (6.4) arises from the fact that the phases generally have different types of anisotropy. Indeed, the tangent tensors I,(') read = 3 k b - I ~+ 2 p ( r ) ~ ( r+ ) 2k(r)~(r),

. these computations simplify for composwhere IE(') = i E i )@ E i )Interestingly, ites with statistically isotropic microstructures. Then it can be assumed, following Hutchinson (1976), that the average strain deviators in each phase are parallel:

This last statement can be justified for spherical inclusions in a matrix, provided the loading applied to the composite is purely radial. It implies that the tensors IE(') are identical in all phases, i.e., that the tensors I,@) have exactly the same type of anisotropy and can be split on the same basis: J, IF, and IE.

Acknowledgments The authors are grateful to Professor J. R. Willis and A. Zaoui for many insightful discussions and valuable comments. They are also indebted to J. C. Michel and H. Moulinec for providing some of the numerical results presented here. The work of PPC was supported by the National Science Foundation (grants MSM88-09177, DMR-91-20668, and CMS-96-22714), the Air Force Office of Scientific Research (grant 91-0161), the Office of Naval Research (grant N00014-96-10681), and the Aluminum Company of America. The work of PS was supported by the Centre National de la Recherche Scientifique.

References Aravaa, N., Cheng, C., and Ponte Castaiieda, P. (1995). Steady-state creep of fiber-reinforced composites: Constitutive equations and computational issues. Internat. J. Solids Structures 32, 22 192244. Avellaneda, M. (1 987). Iterated homogenization, differential effective medium theory and applications. Comm. Pure Appl. Math. 40, 527-554. Bao, G., Hutchinson, J. W., and McMeeking, R. M. (1991). Particle reinforcement of ductile matrices against plastic flow and creep. Actu Metall. Muter: 39, 1871-1882. Bensoussan, A., Lions, J. L., and Papanicolaou, G. (1978). Asymptotic analysis forperiodic structures. North-Holland. Amsterdam.

296


Beran, M. (1965). Use of the variational approach to determine bounds for the effective permittivity of random media. Nuovo Cimento 38,771-782. Berveiller, M., and Zaoui, A. (1979). An extension of the self-consistent scheme to plastically-flowing polycrystals. J. Mech. Phys. Solids 26, 325-344. Bhattacharya, K., and Kohn, R. V. (1997). Elastic energy minimization and recoverable strains of polycrystalline shape-memory materials. To appear in Arch. Rational Mech. Anal. Bishop, J. F. W., and Hill, R. (1951a). A theory of the plastic distortion of a polycrystalline aggregate under combined stresses. Philos. Mug. 42,414427. Bishop, J. F. W., and Hill, R. (195 I b). A theoretical derivation of the plastic properties of a polycrystalline face-center metal. Philos. Mag. 42, 1298-1 307. Bornert, M. (1996). Morphologie microstructurale et comportement micanique: cuructirisations expkrimentales, approches par bornes et estimations autocohe'rentes gkniralise'es. Ph. D. thesis, Ecole Nutionale des Ponts et Chaussies. Bornert, M., Stolz, C., and Zaoui, A. (1996). Morphologically representative pattern-based bounding in elasticity. J. Mech. Phys. Solids 44, 307-33 I . Bouchitte, G., and Suquet, P. (1991). Homogenization, plasticity and yield design. In: Composite media and homogenization theory (G. Dal Maso and G. Dell' Antonio, eds.). Birkhauser, Basel, pp. 107-133. Brockenborough, J., Suresh, S., and Wienecke, H. (1991). Deformation of metal-matrix composites with continuous fibers: Geometrical effects of fiber distribution and shape. Acta Metall. Muter: 39,735-752. Budiansky, B. (1959). A reassessment of deformation theories of plasticity. J. Appl. Mech. 26, 259264. Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13,223-227. Budiansky, B. (1970). Thermal and thermoelastic properties of' composites. J. Comp. Muter: 4, 286295. Budiansky, B., Hutchinson, J. W., and Slutsky, S. (1982). Void growth and collapse in viscous solids. In: Mechanics of solids, The Rodney Hill 60th anniversary volume (H. G. Hopkins and M. J. Sewell, eds.). Pergamon Press, Oxford, pp 1 3 4 5 . Buryachenko, V. (1996). The overall elastoplastic behavior of multiphase materials with isotropic components. Actu Mech. 119,93-117. Buryachenko, V. A,, and Lipanov, A. M. (1989a). Effective field method in ideal plasticity theory for composite materials. Prikl. Mekh. Tekhn. Fiz. 3, 149-155. Buryachenko, V. A,, and Lipanov, A. M. (1989b). Prediction of nonlinear Row parameters for multicomponent mixtures. Prikl. Mekh. Tekhn. Fiz. 4, 6 3 4 8 . 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. Christman, T., Needleman, A,, and Suresh, S. (1989). An experimental and numerical study of deformation in metal-ceramic composites. Actu Metall. Muter. 37, 3029-3050. Chu, T., and Hashin, Z. (1971). Plastic behavior of composites and porous media under isotropic stress. Internat. J. Engrg. Sci. 9,971-994. deBotton, G. (1995). The effective yield strength of fiber-reinforced composites. Internat. J. Solids Structures 32, 1743-1157. deBotton, G., and Ponte Castafieda, P. (1992). On the ductility of laminated materials. Internut. J. Solids Structures 29, 2329-2353. deBotton, G., and Ponte Castafieda, P. (1993). Elastoplastic constitutive relations for fiber-reinforced solids. Infernat. J. Solids Srructures 30, 1865-1 890. deBotton, G., and Ponte Castaiieda, P. (1995). Variational estimates for the creep behavior of polycrystals. Proc. Roy. Soc. London Ser. A 448, 121-142.


297

de Buhan, P. (1983). HomogenCisation en calcul B la rupture: le cas du matCnau composite multicouche. C. R. Acad. Sci. Paris Ser II 296,933-936. de Buhan, P., and Taliercio, A. (1991). A homogenization approach to the yield strength of composites. European J. Mech. A Solids 10, 129-154. Dendievel, R., Bonnet, G., and Willis, J. R. (1991). Bounds for the creep behavior of polycrystalline materials. In: Inelastic deformation of composite materials (G. Dvorak, ed.). Springer-Verlag, New York, pp. 175-192. Drucker, D. C. (1959). On minimum weight design and strength of non-homogeneous plastic bodies. In: Non-homogeneity in elasticity and plasticity (W. Olszag, ed.). Pergamon Press, New York, pp. 139-146. Drucker, D. C. (1966). The continuum theory of plasticity on the macroscale and the microscale. J. Mate,: 1,873-910. Duva, J. M. (1984). A self-consistent analysis of the stiffening effect of rigid inclusions on a power-law material. J. Engrg. Muter. Tech. 106, 3 17-321. Duva, J. M., and Hutchinson, J. W. (1984). Constitutive potentials for dilutely voided non-linear materials. Mech. Mater. 3,41-54. Dvorak, G. J. (1992). Transformation field analysis of inelastic composite materials. Proc. Roy. SOC. London Ser. A 437.31 1-327. Dvorak, G. J., and Bahei-El-Din, Y. A. (1987). A bimodal plasticity theory of fibrous composite material, Acra Mech. 69,219-241. Dvorak, G. J., Bahei-El-Din, Y. A,, Macheret, Y., and Liu, C. H. (1988). An experimental study of elastic-plastic behavior of a fibrous boron-aluminum composite. J. Mech. Phys. Solids 36, 655687. Einstein, A. (1906). Eine Neue Bestimmung der Molekiildimensionen. Ann. Physik 19,289-306. Ekeland, I., and Temam, R. (1974). Analyse convexe et probl2mes variationnels. Gauthier-Villars, Paris. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. London Ser. A 241,376-396. Fleck, N. A., and Hutchinson, J. W. (1997). Strain gradient plasticity. Adv. Appl. Mech. 33,295-362. Francfort, G., and Murat, F. (1986). Homogenization and optimal bounds in linear elasticity. Arch. Rational. Mech. Anal. 94,307-334. GSrSjeu, M. (1996). Effective behaviour of porous viscoplastic materials containing axisymmetric prolate ellipsoidal cavities. C. R. Acad. Sci. Paris Ser. II 323, 307-314. Giirgjeu, M., and Suquet, P. (1997). Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles. J. Mech. Phys. Solids 45, 873-902. Germain, P., Nguyen, Q. S., and Suquet, P. (1983). Continuum thermodynamics. J. Appl. Mech. 50, 1010-1 020. Gilormini, P. ( 1995). Insuffisance de l’extension classique du mod& auto-coherent au comportement non-linkaire. C. R. Acad. Sci. Paris Ser. IIb 320, 115-122. Golaganu, M., Leblond, J.-B., and Devaux, J. (1993). Approximate models for ductile metals containing non-spherical voids+ase of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723-1754. Golagdnu, M., Leblond, J.-B., and Devaux, J. (1994). Approximate models for ductile metals containing non-spherical voids+ase of axisymmetric oblate ellipsoidal cavities. J. Engrg. Muter. Tech. 116,290-297. Gurson, A. L. (1977). Continuum theory of ductile rupture by void nucleation and growth. J. Engrg. Muter. Tech. 99,2-15. Hashin, Z. (1962). The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143-150. Hashin, Z. (1980). Failure criteria for unidirectional fiber composites. J. Appl. Mech. 47, 329-334. Hashin, Z. (1983). Analysis of composite materials: A survey. J. Appl. Mech. 50,481-505.

298

Pedro Ponte Castanleda and Pierre Suquet

Hashin, Z., and Shtrikman, S. (1962a). On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335-342. Hashin, Z., and Shtrikman, S. (1962b). A variational approach to the theory of the elastic behavior of polycrystals. J. Mech. Phys. Solids 10, 343-352. Hashin, Z., and Shtrikman, S. (1963). A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11, 127-140. He, M. Y. (1990). On the flow and creep strength of power-law materials containing rigid reinforcements. Muter: Res. Soc. Symp. Proc. 194, 15-22. Hershey, A. V. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. ASME J. Appl. Mech. 21, 236-240. HervC, E., Stolz, C., and Zaoui, A. (1991). A propos de l’assemhlage des sphkres composites de Hashin. C. R. Acad. Sci. Paris Ser: II 313, 857-862. HervC, E., and Zaoui, A. (1993). N-layered inclusion-based micromechanical modelling. Int. J. Engrg. Sci. 31, 1-10, Hill, R. (1963). Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11, 357-372. Hill, R. (1965a). Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13, 89-101. Hill, R. (1965h). A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213222. Hill, R. (1967). The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15, 79-95. Hom, C. L. (1992). Three-dimensional finite element analysis of plastic deformation in a whiskerreinforced metal matrix composite. J. Mech. Phys. Solids 40, 99 1-1008. Hu, G. (1996). A method of plasticity for general aligned spheroidal void of fiber-reinforced composites. Internat. J. Plasticity 12, 439449. Hutchinson, J. W. (1970). Elastic-plastic behavior of polycrystalline metals and composites. Proc. Roy. SOC.London Ser: A 319,247-272. Hutchinson, J. W. (1976). Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. Roy. SOC.London Ser. A 348, 101-127. Germain, P., Nguyen, Q.S., and Suquet, P. (1983). Continuum thermodynamics. J. Appl. Mech. 50, 1010-1020. Kachanov, L. M. (1971). Foundations ofthe theory ofplusticity. North-Holland, Amsterdam, p. 20. Kailasam, M., and Ponte Castdeda, P. (1997). A general constitutive theory for linear and nonlinear particulate media with microstructure evolution. Submitted for publication. Kailasam, M., Ponte Castaiieda, P., and Willis, J. R. (1997a). The effect of particle size, shape, distribution and their evolution on the constitutive response of nonlinearly viscous composites. Part I. Theory. Philos. Trans. Roy. Soc. London Ser: A 355, 1883-1900. Kailasam, M., Ponte Castaiieda, P., and Willis, J. R. (1997b). The effect of particle size, shape, distribution and their evolution on the constitutive response of nonlinearly viscous composites. Part 11. Examples. Philos. Trans. Roy. SOC.London Ser: A 355, 1901-1920. Kohn, R. V., and Little, T. D. (1997). Some model problems of polycrystal plasticity with deficient basic crystals. Submitted for publication. Kohn, R. V., and Milton, G. W. (1986). On bounding the effective conductivity of anisotropic composites. In: Homogenization and effective moduli of materials and media (J. L. Ericksen, D. Kinderlehrer, R. V. Kohn, and J.-L. Lions, eds.). Springer-Verlag, New York, pp. 97-125. Kreher, W. (1990). Residual stresses and stored elastic energy of composites and polycrystals. J. Mech. Phys. Solids 38, 115-1 28. Kroner, E. (19%). Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Z. Physik 151,504-518.


299

Kroner, E. (1977). Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids 25, 137-155. Laws, N. (1973). On the thermostatics of composite materials. J. Mech. Phys. Solids 21, 9-17. Laws, N., and McLaughlin, R. (1979). The effect of fibre length on the overall moduli of' composite materials. J. Mech. Phys. Solids 27, 1-13. Leblond, J.-B., Perrin, G., and Suquet, P. (1994). Exact results and approximate models for porous viscoplastic solids, Internat. J. Plasticiry 10, 2 13-235. Lee, B. J., and Mear, M. E. (1991). Effect of inclusion shape on the stiffness of nonlinear two-phase composites. J. Mech. Phys. Solids 39, 627-649. Lee, B. J., and Mear, M. E. (1992a). Axisymmetric deformation of power-law solids containing dilute concentration of spheroidal voids. J. Mech. Phys. Solids 40, 1805-1 836. Lee, B. J., and Mear, M. E. (1992b). Effective properties of power-law solids containing elliptical inhomogeneities. Part I. Rigid inclusions. Mech. Muter. 13, 3 13-335. Levin, V. M. (1967). Thermal expansion coefficients of heterogeneous materials. Mekh. Tverd. Tela 2, 83-94. Li, G., and Ponte Castaiieda, P. (1993). The effect of particle shape and stiffness on the constitutive behavior of metal-matrix composites. Infernat. J. Solids Structures 30, 3 189-3209. Li, G., and Ponte Castaiieda, P. (1994). Variational estimates for the elastoplastic response of particlereinforced metal-matrix composites. Appl. Mech. Rev. 47, S77-S94. Li, G., Ponte Castaiieda, P., and Douglas, A. S. (1993). Constitutive models for ductile solids reinforced by rigid spheroidal inclusions. Mech. Matea 15, 279-300. Lurie, K., and Cherkaev, A. (1986). Exact estimates of conductivity of a binary mixture of isotropic components. Proc. Roy. SOC.Edinburgh Sect. A 104, 21-38. Mandel, J. (1972). Plusticite' clussique et viscoplasticite'. CISM Udine Courses and Lectures, no. 97. Springer-Verlag, Berlin. McCoy, J. J . (1970). On the displacement field of an elastic medium: Random variations in material properties. In: Recent advances in engineering science (A. Eringen, ed.), Vol. 5. Gordon and Breach, New York, pp. 235-254. McLaughlin, R. (1977). A study of the differential scheme for composite materials. Infernat. J. Engrg. Sci. 15, 237-244. Michel, J. C. (1994). A self-consistent estimate of the overall behavior of viscous and porous materials. C. R. Acad. Sci. Paris Ser. I I 319, 1439-1446. Michel, J. C. (1996). A self-consistent estimate of the potential of a composite made of two power-law phases with the same exponent. C. R. Acnd. Sci. Paris Ser. II 322,447454. Michel, J.C., and Suquet, P. (1992). The constitutive law of nonlinear viscous and porous materials. J. Mech. Phys. Solids 40,783-8 12. Milton, G. W. (1982). Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids 30, 177-191. Milton, G. W. (1985). The coherent potential approximation is a realizable effective medium scheme. Comm. Math. Phys. 99,463-500. Milton, G. W. (1990). On characterizing the set of possible effective tensors: The variational method and the translation method. Comm. Pure Appl. Math. 43, 63-125. Milton, G. W., and Kohn, R. V. (1988). Variational bounds of the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36,597-629. Milton, G. W., and Phan-Thien, N. (1982). New bounds on the effective elastic moduli of twocomponent materials. Proc. Roy. Soc. London Ser. A 380, 305-33 I . Molinari, A,, Canova, G. R., and Ahzi, S. (1987). A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall. Muter. 35, 2983-2994. Mori, T., and Tanka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting dislocations. Acta Metall. Mater. 21, 57 1-574.

300


Moulinec, H., and Suquet, P. (1995). A FTT-based numerical method for computing the mechanical properties of composites from images of their microstructure. In: Microstructure- proper^ interactions in composite materials (R. Pyrz, ed.). Kluwer, Dordrecht, The Netherlands, pp. 235-246. Nebozhyn, M. V., and Ponte Castafieda, P. (1997). Second-order estimates for the effective behavior of nonlinear porous materials. In: Transformation problems in composite and active materials (Y. Bahei-El-Din and G. J. Dvorak, eds.). Kluwer, Dordrecht, The Netherlands. To appear. Nemat-Nasser, S., and Hori, M. (1993). Micromechanics: Overall properties of heterogeneous solids. Elsevier, Amsterdam. Olson, T. (1994). Improvements on Taylor’s upper bound for rigid-plastic bodies. Muter. Sci. Engrg. A 175, 15-20. Ponte Castaiieda, P. (1991a). The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39,45-7 I . Ponte Castaiieda, P. (199 I b). The effective properties of brittle/ductile incompressible composites. In: Inelastic deformafion ofcomposire materials (G. J. Dvorak, ed.). Springer-Verlag, New York, pp. 215-23 1. Ponte Castafieda, P. (1991~).Effective properties in power-law creep. In: Mechanics of creep brittle materials 2 (A. C. F. Cocks and A. R. S. Ponter, eds.). Elsevier, London, pp. 218-229. Ponte Castaiieda, P. (1992a). New variational principles in plasticity and their application to composite materials. J. Mech. Phys. Solids 40, 1757-1788. Ponte Castafieda, P. (1992b). Bounds and estimates for the properties of nonlinear heterogeneous systems. Philos. Trans. Roy. Soc. London Ser. A 340, 531-567. Ponte Castafieda, P. (1992~).A new variational principle and its application to nonlinear heterogeneous systems. SIAM J. Appl. Math. 52, 1321-1341. Ponte Castafieda, P. (1996a). Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 44, 827-862. Ponte Castaiieda, P. (1996b). Variational methods for estimating the effective behavior of nonlinear composite materials. In: Continuum models and discrete systems (CMDS 8) (K. Z. Markov, ed.). World Scientific, Singapore, pp. 268-279. Ponte Castaiieda, P. (1997a). Corrigendum: Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 45, 3 17. Ponte Castafieda, P. (1997b). Nonlinear composite materials: Effective constitutive behavior and microstructure evolution. In: Continuum micromechanics (P. Suquet, ed.). CISM Courses and Lecture Notes No. 377. Springer-Verlag, Wien, pp. 131-1 95. Ponte Castaiieda, P., and deBotton, G. (1992). On the homogenized yield strength of two-phase composites. Proc. Roy. Soc. London Ser. A 438, 4 1 9 4 3 1. Ponte Castafieda, P., and Kailasam, M. (1997). Nonlinear electrical conductivity in heterogeneous media. Proc. Roy. Soc. London Ser. A 453,793-816. Ponte Castafieda, P., and Nebozhyn, M. V. (1997). Exact second-order estimates of the self-consistent type for nonlinear composite materials. Mech. Muter. To appear. Ponte Castafieda, P., and Nebozhyn, M. V. (1998). Variational estimates of the self-consistent type for the effective behavior of some model nonlinear polycrystals. Submitted Ponte Castaieda, P., and Suquet, P. (1995). On the effective mechanical behavior of weakly inhomogeneous nonlinear materials. European J. Mech. A Solids 14,205-236. Ponte Castafieda, P., and Willis, J. R. (1988). On the overall properties of nonlinearly viscous composites. Proc. Roy. Soc. London Ser. A 416,217-244. Ponte Castafieda, P. and Willis, J. R. (1993). The effective behavior of nonlinear composites: A comparison between two methods. In: Confinirum models and discrete systems (CMDS 7) (K. H. Anthony and H.-H. Wagner, eds.). Trans Tech, Aedermannsdorf, Switzerland, pp. 35 1-360. Ponte Castafieda, P. and Willis, J. R. (1995). The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Solids 43, 1919-1951.


301

Ponte CastGeda, P., and Zaidman, M. (1994). Constitutive models for porous materials with evolving microstructure. J. Mech. Phys. Solids 42, 1459-1497. Ponte Castaiieda, P., and Zaidmdn, M. (1996). The finite deformation of nonlinear composite materials. I. Instantaneous constitutive relations. Internat. J. Solids Structures 33, 1271-1286. Qiu, Y. P., and Weng, C. J. (1992). A theory of plasticity for porous materials and particle-reinforced composites. J. Appl. Mech. 59, 261-268. Reuss, A. (1929). Calculation of the flow limits of mixed crystals on the basis of the plasticity of the monocrystals. 2. Angew. Math. Mech. 9,49-58. Rice, J. R. (1970). On the structure of of stress-strain relations for time-dependent plastic deformation in metals. J. Appl. Mech. 37, 728-737. Sachs, G. (1928). Zur Ableitung einer Fleissbedingun. Z. Ver: Dtsch. Ing. 72, 734-736. Sanchez-Palencia, E. (1 980). Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, 127. Springer-Verlag, Heidelberg. Smyshlyaev, V. P., and Fleck, N. A. (1995). Bounds and estimates for the overall plastic behavior of composites with strain gradients effects. Proc. Roy. SOC.London Ser: A 451, 795-810. Spitzig, W. A,, Smelser, R. E., and Richmond, 0. (1988). The evolution of damage and fracture in iron compacts with various initial porosities. Acta Metall. Muter: 36, 1201-121 1. Suquet, P. (1983). Analyse limite et homogCnCisation. C. R. Acad. Sci. Paris Ser: II 296, 1355-1 358. Suquet, P. (1987). Elements of homogenization for inelastic solid mechanics. In: Homogenization techniques for composite media (E. Sanchez-Palencia and A. Zaoui, eds.). Lecture Notes in Physics 272. Springer-Verlag, Berlin, pp. 193-278. Suquet, P. (1992). On bounds for the overall potential of power-law materials containing voids with arbitrary shape. Mech. Res. Comm. 19, 51-58. Suquet, P. (1993a). Overall potentials and extremal surfaces of power law or ideally plastic materials. J. Mech. Phys. Solids 41,981-1002. Suquet, P. (1993b). Bounds and estimates for the overall properties of nonlinear composites. In: Micromechanics of materials (J. J. Marigo and G. Rousselier, eds.). Eyrolles, Paris, pp. 361-382. Suquet, P. (1995). Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Castaiieda’s nonlinear variational procedure. C. R. Acad. Sci. Paris Ser: II 320, 563-571. Suquet, P. (1 996a). Overall properties of nonlinear composites: Secant moduli theories and variational bounds. In: Continuum models ofdiscrete systems (CMDS 8) (K. Z. Markov, ed.). World Scientific, Singapore, pp. 290-299. Suquet, P. (1996bj. Overall properties of nonlinear composites: Remarks on secant and incremental formulations. In Micromechanics of plasticity and damage of multiphase materials (A. Pineau and A. Zaoui, eds.). Kluwer Academic, Dordrecht, The Netherlands, pp. 149-156. Suquet, P. (1997a). Effective properties of nonlinear composites. In: Continuum micromechanics (P. Suquet, ed.). CISM Courses and Lecture Notes No. 377. Springer-Verlag, Wien, pp. 197264. Suquet, P. (1997bj. Work in progress. Suquet, P., and Moulinec, H. (1997). A numerical method for computing the overall response of nonlinear composites with complex microstructure. In: Disordered materials (C. Milton et al., eds.). IMA Lecture Notes. Springer-Verlag, New York, pp. 277-286. Suquet, P., and Ponte C iieda, P. (1993). Small-contrast perturbation expansions for the effective properties of nonlinear composites. C. R. Acad. Sci. Paris Ser. II 317, 1515-1522. Talbot, D. R. S., and Willis, J. R. (1985). Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math. 35, 39-54. Talbot, D. R. S., and Willis, J . R. (1991). The overall behavior of a nonlinear fiber reinforced composite. In: Inelastic deformation of composite materials (G. J. Dvorak, ed.j. Springer-Verlag. New York, pp. 527-545.

302

Pedro Ponte Custaiieda and Pierre Suquet

Talbot, D. R. S., and Willis, J. R. (1992). Some explicit bounds for the overall behavior of nonlinear composites. Internat. J. Solids Structures 29, 1981-1987. Talbot, D. R. S., and Willis, J. R. (1994). Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. Roy. Soc. London Ser. A 447, 365-384. Talbot, D. R. S., and Willis, J. R. (1995). Upper and lower bounds for the overall properties of a nonlinear elastic composite. In: Anisotropy, inhomogeneity and nonlinearity in solid mechanics (D. E Parker and A. H. England, eds.). Kluwer Academic, Dordrecht, The Netherlands, pp. 409414. Talbot, D. R. S., and Willis, J. R. (1997). Bounds of third order for the overall response of nonlinear composites. J. Mech. Phys. Solids 45, 87-1 I I . Tandon, G. P., and Weng, G. J. (1988). A theory of particle-reinforced plasticity. J. Appl. Meclz. 55, 126-135. Taylor, G. I. (1938). Plastic strains in metals. J. Inst. Metals 62, 307-324. Tvergaard, V. (1990). Analysis of tensile properties for a whisker-reinforced metal-matrix composite. Acta Metall. Muter. 38, 185-194. Van der Giessen, E., Van der Burg, M. W. D., Needleman, A,, and Tvergaard, V. (1995). Void growth due to creep and grain boundary diffusion at high triaxialities. J. Mech. Phys. Solids 43, 123-1 65. Van Tiel, J. (1984). Convex analysis. Wiley, New York. Voigt, W. (1889). Ueber die Bezienhung zwischen den beiden Elasticitlts-constanten isotroper. Ann. Physik 38,573-587. Walpole, L. J. (1969). On the overall elastic moduli of composite materials. J. Mech. Phys. Solids 17, 235-25 1. Walpole, L. J. (1981). Elastic behavior of composite materials: Theoretical foundations. Adv. Appl. Mech. 21, 169-242. 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. (1978). Variational principles and bounds for the overall properties of composites. In Continuum models and discrete systems (CMDS 2J (J. Provan, ed.). Univ. Waterloo Press, Waterloo, ON, Canada, pp. 185-2 15. Willis, J. R. (1981). Variational and related methods for the overall properties of composites. A d v Appl. Mech. 21, 1-78. Willis, J. R. (1983). The overall response of composite materials. ASME J. Appl. Mech. 50, 12021209. Willis, J. R. (1986). Variational estimates for the overall response of an inhomogeneous nonlinear dielectric. In: Homogenization and Efective Moduli of Materials and Media (J. L. Ericksen, D. Kinderlehrer, R. V. Kohn, and J.-L. Lions, eds.). Springer-Verlag, New York, pp. 247-263. Willis, J. R. (1989a). The structure of overall constitutive relations for a class of nonlinear composites. IMA J. Appl. Math. 43, 23 1-242. Willis, J. R. (1989b). Variational estimates for the overall behavior of a nonlinear matrix-inclusion composite. In: Micromechanics und inhomogeneity: The Toshio Mura anniversary volume (G. J. Weng, M. Taya, and H. Abe, eds.). Springer-Verlag, New York, pp. 581-597. Willis, J. R. (1991). On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids 39, 73-86. Willis, J. R. (1992). On method for bounding the overall properties of nonlinear composites: Correction and addition. J. Mech. Phy". Solids 40,441445. Willis, J. R. (1994). Upper and lower bounds for nonlinear composite behavior. Muter. Sci. Engrg. A 175,7-14. Zaidman, M., and Ponte Castaiieda, P. (1996). The finite deformation of nonlinear composite materials. 11. Evolution of the microstructure. Internat. J. Solids Structures 33, 1287-1 303.

ADVANCES IN APPLIED MECHANICS, VOLUME 34

The Mathematical Foundation of Plasticity Theory WE1 H. YANG The University of Michigan Ann Arbor; Michigan

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.303

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.304

11. Minkowski Norms and Holder Inequality . . . . . . . . . . . . . . . . . . . . . .

307

............................

308

IV. Constructing the Dual Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309

V. Application to Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 11

111. Generalized Holder Inequality

VI. A Duality Theorem for Plane Stress Problems . . . . . . . . . . . . . . . . . . . 313

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 315

Abstract Supporting the mathematical theory of plasticity, two pillars constituting the modeling of yield behavior are the convexity and normality conditions on a yield function. The models so established have been time tested to hold for a wide class of ductile materials in many practical applications of the theory. Their logical basis was the belief that a material does not release energy during plastic deformation (nonnegative dissipation). Even though the theory is sound, the lack of a mathematical proof on normality still causes some researchers to call it a postulate. In this paper, we shall dismiss this notion of assumption by introducing a generalized Holder inequality. The normality relation becomes a requirement for the inequality to be sharp (equality inclusive). This inequality also reveals the primal-dual relation of stress and strain rate. Even before this connection of the mathematical and physical implications, different duality relations were used in many known minimax theorems in plasticity. We shall claim that convexity, normality, and duality, all properties of the generalized Holder inequality, complete the foundation for the constitutive modeling of the mathematical theory of plas303 ISBN 0-12-002034-3

ADVANCES IN APPLIED MECHANICS. VOL. 34 Copyright Q 1998 by Academic Press. All rights of reproduction in any form reserved. 0065-2165/98$25.00

304

Wei H.Yang

ticity. By using these models, the minimax formulations of plasticity problems can be derived in a systematic and unified approach.

I. Introduction Convexity and normality are the two pillars of a constitutive model that is the basis of the mathematical theory of plasticity (Hill, 1950). The yielding behavior of metals has been modeled satisfactorily by convex yield functions more than a century ago (Tresca, 1864,1868). Such models remain valid, even in light of new materials and experimental results. Although it was conjectured earlier (Prandtl, 1924; Reuss, 1930), a logical basis of normality was not in place until 1959 (Drucker, 1959). Based on the existing foundation, we shall add another pillar to complete this useful constitutive model. First, we shall present a yield function f as a norm (or a seminorm) on the 3 x 3 symmetric stress matrix B. An initial yield criterion may be modeled by

f (c)= llall - 0 0 5 0 ,

or

l b l l 5 009

(1)

where 00 is a material constant. Both the form of the norm and the material constant are allowed to vary with plasticity deformation history to model subsequent yield behavior. This representation not only gives a sense of bounded stress, but will also lead to a foundamental inequality that encompasses the Cauchy-Schwarz and Holder inequalities as special case. These inequalities pertain to the inner product of vectors, matrices, and functions. We shall restrict our discussion to finite dimensional spaces that involve only vectors and matrices. The upper bound of an inner product is expressed in terms of dual norms of vectors and matrices. Although the Cauchy-Schwarz inequality applies only to Euclidean norms, the Holder inequality extends the applications to a family of Minkowski norms. Norms that model yield behavior are generally non-Euclidean. The inner product in plasticity involves the stress and strain rate matrices. The generalized Holder inequality for such an inner product may take the form

where the colon denotes the inner product operator between two matrices and the subscripts of the norms, ( p ) and ( d ) , suggest the primal-dual relation. A primal norm is determined to fit experimental data. Its dual norm is determined mathematically to establish the theorem of generalized Holder inequality. We shall first

Mathematical Foundation of Plasticity Theory

305

introduce the background leading to the theorem, then present its application to plasticity. The well-known normality relation in plasticity between the strain rate and stress becomes the natural sharpness condition of this generalized Holder inequality. This new inequality is fundamental to all minimax (duality) theorems in plasticity. A special case for plane stress problems will be presented as an example. The familiar Cauchy-Schwarz inequality for the Euclidean vector space is commonly stated as

where x, y are vectors in Rn; T transposes a vector, and clidean norm. The equality holds if y=ax,

aER,

11 . 112 denotes the Eu(4)

or the two vectors are colinear. The inequality (3) is often used in the upper bounding process of a mathematical analysis. A “sharp” upper bound that includes the equality case is vitally important in the field of functional analysis and its applications. Obviously, a function that is bounded above has a finite supremum. The supremum of a function is contained in the range of a sharp upper-bound function. Therefore, a search for the least upper bound will recover the maximum of the original function. This indirect method of finding the maximum of a function will fail if its upper-bound function is not sharp. The Cauchy-Schwarz inequality does not hold in a general non-Euclidean space. Holder introduced an extension so that a modified inequality is valid in a vector space in which a family of Minkowski norms (Birkhoff and MacLane, 1953) is defined. The colinearity condition (4) for sharpness must also be modified. We shall use a simple example to illustrate these points. Consider first the norm i n R 2 , Ilxllm = max(lx11, Ix21}.TwospecificvectorsxT = (1, 1) andy’ = (1, 1) produce the inner product xTy = 2. This inner product cannot be less than or equal to llxllmllyllm = 1. Now let y be measured by another norm, llylll = Iy1( 1 ~ 2 1 It . is well known that lx‘yl I IlxmIIIIyII1 for all x, y E R2,and that equality holds under a certain condition that, in general, is not the colinearity of x and y. This can be illustrated in geometric terms. Let

+

Wei H. Yang

306

Clearly, all such vectors lie on two “unit circles,” one in the shape of a square for x and the other in the shape of a diamond for y, as shown in Figure 1. The well-known inequality T

Ix YI I llXllo0llYlll = 1

(6)

holds for any x and y defined in (5). Here, the vectors x and y are said to be a pair of dual variables. Consider a specific vector, x=(l

a)?

o QT. According to computed results, bifurcation occurs at that instant that corresponds to the incipient transition time Ti, M 150. Thus the transition time shortens by a factor of 2 when passing from Q = 3.92 through the critical value Q = QT to Q = 3.93. This happens because an earlier (and weaker) phase of disturbance generation still belongs to the subcritical regime I; however, as soon as the mass of disturbances upstream of the hump amounts to about half of the mass of the BDA soliton, the process turns to the next (and more intensive) phase related to the supercritical regime 11. Noteworthy is the specific steplike behavior of a curve in Figure 15 drawn for Q = 3.93. Each step is associated with the birth of a soliton whose mass M rgrows, by virtue of (3.19), from 0 to 4n as a steplike function. So the number of steps determines the number of upstream emitted solitary waves. Hence it follows that the period T re-

0, even though there are features in common to both cases (Bogdanova-Ryzhova and Ryzhov, 1996). First of all, the amplitude of waves in the two systems ra-

Oleg S. Ryzhov and Elena K Bogdanova-Ryzhova

372

diated in opposite directions, downstream and upstream, continuously grows as Q decreases from zero, without being subject to radical changes through bifurcations at some critical values of the similarity parameter. The generation process goes smoothly and the disturbance amplitudes never become significantly larger compared to the sizes of pulsations developing inside a cavity. This property explains the absence of the first bifurcation. The mass-production rates in Figure 27 are convincing testimony to the general statements advanced without proof. Next, from the steplike run of all the curves associated with upstream soliton radiation, we may conclude that the incipient transition time Ti, as well as the period T of an individual disturbance birth shorten in relation to the corresponding quantities typical of humps with the same absolute value of Q . Further evidence can be picked up by comparing two plots in Figure 28 that exhibit the instantaneous displacement-thickness distributions for a hump with Q = 25 and a dent fixed by Q = -25. We see a much greater number of solitary waves emitted by the dent, although the specific comb-like arrangement of their peaks downstream and upstream of the roughness is inherent in both disturbance

0

10

15

20

t

25

30

35

27. Mass M - b transported by the wave system upstream of the dent in a broad range of 0. No bifurcation occurs in the gradual evolution of the disturbance+excitation process.

FIG.

Q

5

i

373

Forced Generation of Solitaly-Like Waves 0=25

-100

-50

0

50

100

50

100

X

(a)

Q=-25 10

-

X

c

z

5

o -5 I

-100

-50

0 X

(b) F I G . 28. Comparison of the instantaneous disturbance patterns generated by a hump and a dent of equal mass (1 Ql = 25) at the same time ( t = 30). A strong negative peak centered around x = 0 separates the depression area from the front part of the dent.

patterns under comparison. Regardless of the sign of Q , any one of these waves is, in effect, a nonlinear eigenmode of the homogeneous BDA model defined by its intrinsic amplitude and phase velocity, but carrying the same mass 4n. A distinctive feature of disturbances created by the dent derives from a strong negative quasi-steady peak in front of the short depression area. The boundary layer becomes significantly thinner within the peak zone at the bottom of the dent. Unlike the process associated with humps, the disturbance radiation by dents is sustained via strong pulsations taking place at the leading edge x = -b of the roughness rather than occurring at its crest x = 0. The strong negative quasisteady peak in the displacement-thickness distribution serves to prevent the tail

Oleg S. Ryzhov and Elena V Bogdanova-Ryzhova

374

end of the cavity from being involved into pulsation development. Therefore a function & b ( t ) = 8 ( t , -b), instead of & ( t ) = 6 ( t , O), proves to be more relevant as a dent characteristic. The behavior of this function, shown in Figure 29, reveals no kink forming on the leeward slope of any peak. So there is no reason for an event analogous to the second bifurcation to happen. From what has been said, it might be assumed that a typical phase portrait of the pulsation process in a plane (S-b, &, = dJ-b/dt) should be of the limit-cycle type. The phase planes shown in Figure 30 confirm this conclusion. What is more, all of the limit-cycle-type portraits computed for an extremely broad interval of Q < 0 consist of a single thin band and do not involve lesser loops, as was the case with regard to humps within the range QT < Q c Q;. The simplification of the phase portraits in Figure 30 stems from the absence of low-amplitude spikes alternating with neighboring main peaks in the distribution of 8-b = 8 - b ( t ) .

Q=-15.

&-3.

I

0

50

100

150

-2

200

'

20

0

t

40

t

Q=-25.

Q=-45. 15

10 10 c

v

0

5

5

Ld

0

0 0

10

20

t

30

40

-5

'

0

5

10

15

20

t

(a 29. Displacement thickness 6-h at the leading edge of the dent as a function of time the distributions, computed for a broad range of Q i0, consist of peaks only. FIG.

2.

Ail of

375

Forced Generation of Solitary-Like Waves Ck-3.

Q=-15. 20

1 0.5

10

0

0

-0.5

-10

5 2 'D

-1

-20

0

1 s-b(t)

2

-

3

2

0

2

4

6

8

s-b(t)

(a)

(b)

Q=-25.

0-45.

50

100 50

F; 2

0

0

'D

-50

-100

-50 0

(c)

5 6-b(t)

10

-5

0

5

10

15

s-b(t)

(d)

F I G . 30. Phase portraits in the plane (6-b. i - b ) featuring the smooth evolution of intense pulsations at the leading edge of the dent in a broad range of Q < 0. A single limit-cycle-type loop is intrinsic to the disturbance-excitation process.

As Figure 31 demonstrates, the pulsation crest Gmax(t) defined in (3.40) starts developing, being mild in size, upstream of the cavity bottom (x = 0) and sweeps a short distance downstream. Then it makes a sharp turn toward the leading edge, never passing through the central point of the dent. The amplitude of grows reaches the top height, not far from the leading as it moves upstream. As,,,a, edge, its intensity drops off slightly, and it leaves the cavity in the form of the peak of a solitary-like wave discharged against the oncoming stream. Then the cycle is repeated, but not in exactly the same way, because of deviations in successive traces. Solitary-like waves are radiated by a dent when amax is close to its top height, contrary to a hump, in which case the emission of a solitary wave occurs when , , ,a is about the minimum. For this reason trajectories in Figure 31 turn out to be disconnected (cf. Figure 21, associated with a hump).

376

Oleg S. Ryzhov and Elena V Bogdunova-Ryzhova Q=-3.

-3

Q=-15.

-2

-1

-2'

0

'

-3

-2

-1

0

-1

0

Xmax

Xrnax

Q=-25.

Q=-45. 15

J1~~'~~~111,1~11~~~1 ,

lo 5

~

0

0 I

-3

-2

-1

-5

0

-3

%ax

-2 %ax

(4

(4 FIG. 3 1. Maps on the plane (xmax, ),,,S

of intense pulsations at the leading edge of the dent, illustrating the salient properties of the solitary-wave radiation in a broad range of Q < 0. These pulsations are confined to the front of the dent, being isolated from its rear by a strong negative peak in the displacement-thickness distribution.

N.The KdV System A. CROSSFLOW INSTABILITY

A completely different scenario in which truly nonlinear disturbances are built up takes place in a 3D boundary layer in connection with crossflow instability (Ryzhov and Terent'ev, 1995, 1996). In this case the crossflow acts like a spanwise near-wall jet and excites initially linear, unstable eigenmodes that may be isolated from streamwise oscillations. The basic mechanism at the heart of the crossflow jetlike behavior stems from the pressure gradient, which strikes a balance with centrifugal forces in the main body of the boundary layer, because of the large curvature of stream surfaces. As for the integral term in the interaction

Forced Generation of Solitary-Like Waves

377

law caused by stream-surface slopes, it becomes negligibly small and drops out of (2.4) if a reference length in the spanwise direction falls to 0( ~ ~ ~ 1It’ is ) . remarkable that exactly the same interaction law and resulting scaling are pertinent to the viscous/inviscid interaction of a thin jet with a flat plate (Smith and Duck, 1977; Ryzhov, 1982).The near-wall jet bears a close resemblance to the crossflow of a 3D boundary layer in that their velocities decay to zero when approaching the upper reaches (however, the oncoming stream gives rise to the streamwise velocity of the boundary layer in question). Another example of an interacting boundary layer derives from a viscous flow past a flat plate mounted vertically in the gravity field and heated (Messiter and LiiiBn, 1976). Of these three types of disparate environmentalconditions, the 3D boundary layer involving crossflow as an intrinsic element is, certainly, of prime importance. Upon entering the truly nonlinear stage, disturbances in each of the fluid motions mentioned obey the celebrated KdV equation which was indicated by Korteweg and de Vries as far back as 1895 in the context of shallow water waves. The disturbance generation process provoked by an external forcing source consists predominantly of the excitation of nonlinear eigenmodes in the form of periodic cnoidal waves and KdV solitons. Let us proceed from a preliminary simplified version (2.9) of the Prandtl equations. Upon supplementing the limiting condition at the upper edge ( y + 00) of the near-wall sublayer (labeled I11 in Figure 2) by a constraint for the vertical component of velocity, we can recast (2.1 1) as W’

-yf

+ A’(t’,x’),

(4.la) (4.1b)

Again, if d,w’, p’, and A’ satisfy both (2.9) and (4.la), then (4.lb) is met automatically. The interaction law and the no-slip conditions at a solid surface y’ = yL(t’, z’) are given by (2.10) and (2.12), respectively. A further procedure in exploring a truly nonlinear stage of crossflow disturbances parallels that set forth above in connection with nonlinear waves propagating in the streamwise direction. The triple deck shown in Figure 2 transforms into a four-deck structure, since a new intermediate (adjustment) sublayer emerges to separate the main tier I1 from the viscous sublayer I11 within the narrow highamplitude-spike zone. As a result, an effectively 2D oscillation pattern takes on the form sketched in Figure 6, with z substituted for x (actually, the upper sublayer is missing from the disturbance field). The size of the spike is scaled in terms of a ratio A/&, where the second small parameter A >> E . Inside the adjustment

378

Oleg S. Ryzhov and Elena V. Bogdanova-Ryzhova

sublayer (2.1) and (2.2) fail; an appropriate rescaling of independent variables becomes (4.2a, b, c) The corresponding rescaling of desired functions leading to 0 (A4) pressure variations reads

(4.3a, b, c, d) It should be emphasized that the affine transformation (4.2), (4.3) differs from the one given by (3.2), (3.3) for truly nonlinear waves propagating in the streamwise direction. Insofar as the intermediate sublayer is much shorter and thicker than the initial near-wall deck, the term with viscous tangential stresses in (2.9b) becomes negligible and truncated equations (Ryzhov and Terent’ev, 1995, 1996)

-av + - aazw = o , ay aw

aw

-+v-+wat ay

aw aZ

(4.4a)

aP =--

az ’

(4.4b)

where p = p ( t , z ) on the strength of a p / a y = 0, control the inviscidhnviscid interaction process. Thus the adjustment sublayer in both extremes featuring 3D boundary-layer flows is akin to a nonlinear critical layer in the proximity of a solid wall. The excess pressure p on the right-hand side of (4.4b) is related to the instantaneous displacement thickness 6 = -A through (2.10), with z and A standing in place of z’ and A’, respectively. As (4.2) shows, in the limit A + 1, the adjustment sublayer becomes as thick as the initial boundary layer, whereas its dimension in the spanwise direction is comparable with the thickness. According to (4.3), the normal-to-wall and spanwise components of the velocity vector as well as the pressure tend to grow to 0 (1) quantities in this limit. Thus a nonlinear stage governed by the full system of the Euler equations, with the normal pressure gradient included, comes into operation. At the outer reaches ( y + co) of the adjustment sublayer, the limiting conditions (4.1) still hold true (with t , y , z and v , w ,p , A substituted for t’, y’, z’ and v ’ , w ’ , p’, A’, respectively) and define an explicit solution to the system of truncated equations (4.4), irrespective of the dependence of p on A . To meet the


379

boundary conditions at the bottom of the adjustment domain, we must introduce a thin near-wall viscous sublayer with the normal coordinate

within the spike zone. This sublayer is a continuation of the corresponding sublayer involved in the triple deck (Figure 6, with the upper sublayer missing). The oscillation parameters are given by

A 4

PI=

(1)P e d ,

A

(4.6a, b, c, d)

They satisfy the classical 2D boundary-layer equations in t , y e d , z , where the excess pressure p e d = p ( t , z ) is expressed in terms of the predetermined displacement thickness - A t d = -A(t, z ) by the second derivative (2.10). The no-slip conditions must be imposed at the wall. With the pressure distribution known in advance, the near-wall viscous sublayer plays a passive role. However, its solution is used in a matching procedure to derive a missing boundary condition,

for the adjustment sublayer containing predominantly inviscid motion. At a certain stage a strong singularity can develop within the near-wall sublayer and put an end to the asymptotic scheme adopted. As mentioned above, the disturbance pattern in the lower viscous deck is beyond the scope of this review. Substitution of (4.1) into (4.7) yields the forced KdV evolution system (4.8a) where A,,, = yw

+ A and (4%)

The corresponding homogeneous equation

was first derived by Korteweg and de Vries as far back as 1895 in connection with shallow water waves, and in the context of 3D boundary-layer stability (of

380

Oleg S. Ryzhov and Elena V Bogdanova-Ryzhova

main concern here), this equation appeared in Ryzhov and Terent’ev (1995,1996). General concepts as applied to a wide variety of viscous shear flows are exposed in Zhuk and Ryzhov (1982) and Smith and Burggraf (1985). No attempt has been made so far to establish a link between this recent discovery and experimental data that are associated with truly nonlinear phenomena at the threshold to transition in a 3D boundary layer on a swept wing, rotating disk, or cone at incidence. All that is available at present relates to completely different environments. Thus, solitary-like disturbances typical of theoretical findings are definitely present in observations by Baines (1984) on two-layer flow over topography and recorded by Farmer and Denton (1985) in a tidal stream over a sill. Careful measurements of Lee et al. (1989) of water-wave parameters show a broad agreement with physical models including, in particular, the forced KdV equation. Drawing on a mathematical analogy that applies under various environmental circumstances, we suggest that both the homogeneous and inhomogeneous KdV systems can be employed as the basis for the theoretical description of the excitation and the subsequent development of nonlinear crossflow eigenmodes in the form of periodic oscillations and solitons. Three qualitative conclusions parallel those stated for the BDA model, they follow from the asymptotic analysis without solving the governing equations. First, different normalizations of the normal-to-wall coordinate, fixed by (2. lc), (4.2b), and (4.5), show that the subdivision of the near-wall viscous sublayer, which controls the size of the crossflow eigenmodes, does not occur if A / & = 1. Therefore this sublayer is the site of the spiky signal birth at the triple-deck stage. In line with (2.2), (4.3) and (4.6), the spanwise and normal velocities as well as the self-induced pressure are of the same order of magnitude throughout the lower deck, remaining unsplit in the case A / E = I. Second, the adjustment sublayer resulting from the later subdivision at the truly nonlinear stage is thicker, in view of (4.2b), than the initial near-wall zone of the crossflow disturbance. The adjustment-sublayer thickening implies, in point of fact, that the spike elevates from the wall toward the upper edge of the 3D boundary layer as A / & grows. The other part of the subdivided disturbance continues to propagate close to the wall. Third, a rise in the crossflow-disturbance amplitude with A / & increasing is accompanied by acceleration of the spike relative to the waves that move in the spanwise direction within the entire near-wall viscous sublayer. Together, these conclusions could form the groundwork for the first direct check in wind-tunnel tests upon the validity of the theory as applied to 3D boundary layers.


381

B. DESCRIPTION OF FREEOSCILLATIONS I N TERMS OF ELLIPTIC FUNCTIONS There is an extensive literature on the homogeneous KdV equation covering conservation laws, direct and inverse scattering, Lax pairs, Hirota function, pole expansions in a complex plane, Backlund transformations, and other general topics. All of these elegant and powerful mathematical techniques are far beyond the scope of the present paper; a recent book by Ablowitz and Clarkson (1991) and references therein are recommended for a comprehensive survey of their basic properties. However, the disturbance emission process controlled by an external forcing is hardly amenable to analysis by any of the aforementioned methods. Moreover, in the context of boundary-layer oscillations, one-periodic and soliton solutions seem to be of primary significance in representing nonlinear crossflow eigenmodes excited by steady obstacles. For this reason we confine ourselves only to a brief discussion of the simplest results related to cnoidal waves and KdV solitons. Let us consider a traveling-wave-type solution,

A = F(6),

6 = k z - wt,

(4.10)

to (4.9), with the first derivative d F / d < determined by a cubic polynomial,

3k2 (”,)?

= P ( F ) e F3 -3cF2+clF+c2.

Here c = w / k is the phase velocity, and c1 and c2 stand for two arbitrary constants of integration. Suppose their values to be chosen in such a way that

P = ( F - F l ) ( F - F2)(F - F3), where F1 > F2 > F3 are three distinct real zeroes of P . Then we have an expression

to implicitly define a desired solution (4.10) through an elliptic integral of the first kind. Making use of the standard notations,

in Abramowitz and Stegun (1964) allows us to reduce (4.1 1) to the form 0=

h --(c &k

- TO),the motion must become effectively independent of the second parameter To, because g = 1, by virtue of our assumption. Thus the only similarity parameter S turns out to control distinctive features of the oscillation pattern at later stages after switching on the forcing source. This parameter has a simple physical meaning, being proportional to the moment of the area (mass) of a disturbing obstacle positioned on an otherwise flat plate. Because of the specific form of a function f in (4.8), the similarity law is stated in terms intrinsic to the crossflow-disturbance generation. We see that in the same 3D boundary layer, nonlinear eigenmodes radiated in the streamwise and spanwise directions depend on the roughness mass and the moment of the mass, respectively. Insofar as disturbing agencies can be disparate in nature, a variety of similarity laws feature the disturbance emission in different environments (shallow water, atmospheric and oceanic phenomena), with the forced KdV equation at the bottom. In what follows we leave aside an in-depth discussion of similarities arising from outer sources indicated by Akylas (1984), Cole (1983, Grimshaw and Smyth (1986), and Wu (1987) for such environmental conditions.

D. STABILITY ANALYSIS The shape of a stationary surface that corresponds to (4.28), (4.29) with a2 = 0 is given by at

yW = -[lnch(kx) k2

+ kx + In 21.

(4.32)

This approaches a flat plate aligned with the oncoming crossflow, as x -+ -00, and tends to another flat plate y w = (2al /k)x at infinity x -+ 00. In consequence, we may treat the resulting motion as a special type of ramp flow. It is worthy of note that surface geometry defined by (4.32) seems to be artificial as applied to


387

spanwise disturbances in a 3D boundary layer. On the contrary, the same geometry is quite natural for a problem on a thin jet developing in the vicinity of a flat plate and impinging against a ramp (Smith and Duck, 1977). One more natural formulation stems from considering a viscous flow past a flat plate mounted vertically in the gravity field and heated (Messiter and LiiiBn, 1976). On the strength of a requirement 6 = -A + 0, as x + -00, a constant entering the expression for A, becomes zero, therefore (4.29) yields al = 48k4. Let us slightly perturb the base solution and write down

A, = A,,y(x)

+ ~ ( tx),,

A,,

= -12k2 sech2(kx),

(4.33)

assuming q in (4.33) to be small compared to the leading term. Then the correction function comes from a linear equation,

1

?! - ?[fi + 12sech2(x)~ = 0, at

ax an2

upon taking advantage of an affine transformation t -+ t/k3, x + x/k. The independent variables can be separated by putting

D = e"'{(x), where { satisfies the following ordinary differential equation:

+ 12sech2(x){

1

= w{,

(4.34)

subject to regular conditions at infinity, i.e.,

3+ o

dx'l

exponentially fast for n = 0 , 1 , 2 , as 1x1-+ oo.

(4.35)

In (4.34), (4.35) we have exactly the same eigenvalue problem in w as one studied by Camassa and Wu (1991a) in connection with the wave systems generated and sustained at near-resonance in a shallow channel of uniform water depth. The difficulty of the problem derives from the fact that the governing operator is of the third order and non-self-adjoint. The stability of the base solution, and the corresponding ramp-type flow, depend on the sign of the real part of w . We note that if (4.34), (4.35) have an eigenvalue w and an eigenfunction {(x), they must admit of the other three eigenvalues, -w and f w * , along with their eigenfunctions {(-x) and {*(&x), respectively. As usual, the asterisk implies the complex conjugation. Therefore, the ramp flow under examination is absolutely unstable whenever an eigenvalue with a nonvanishing real part exists. Thus we are led to address, first of all, the question

388

Oleg S. Ryzhov and Elena V. Bogdanova-Ryzhova

of whether w = 0 can represent a simplest real eigenvalue. In this case (4.34) reduces to the second-order equation d2

Advances in Applied Mechanics, Volume 34

Advances in Applied Mechanics, Volume 34

Advances in Applied Microbiology, Volume 34

Advances in Applied Microbiology, Volume 34