mechanics of the inelastic behavior of materials— part...

MECHANICS OF THE INELASTIC BEHAVIOR OF MATERIALSÐPART 1, THEORETICAL UNDERPINNINGS

K. R. Rajagopal* and A. R. Srinivasa

Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, U.S.A.

(Received in ®nal revised form 20 March 1998)

AbstractÐThis is the ®rst of a two-part paper that is concerned with the modeling of the behaviorof inelastic materials from a continuum viewpoint, taking into account changes in the elasticresponse and material symmetry that occur due to changes in the microstructure of the material.The ®rst part discusses some of the fundamental issues that must be addressed when modeling theelastic response of these materials. In particular, we discuss in detail the far reaching e�ects of thenotion of materials with families of elastic response functions with corresponding natural con®gura-tions that was introduced by Wineman and Rajagopal (1990, Archives of Mechanics, 42, 53±75) andRajagopal and Wineman (1992, Int. J. Plasticity 8, 385±395) for the study of the inelastic behaviorof polymeric materials and later generalized and extended to the study of deformation twinning ofpolycrystals by Rajagopal and Srinivasa (1995 Int. J. Plasticity 11(6), 653±678, 1997, 13(1/2) 1±35).For these materials, a de®nition of material symmetry is introduced, that makes it possible to dis-cuss the concept of ``evolving material symmetry''.# 1998 Elsevier Science Ltd. All rights reserved

I. INTRODUCTION

It has long been known that a fundamental feature of liquids is that they can be made to¯ow by the application of the smallest shear stress and hence can take the shape of anycontainer, whereas solids retain a de®nite shape and size irrespective of the nature of thecontainer that holds them1. Many solids, ranging from crystalline ones like steel or alu-minum all the way to polymers like rubber, can be induced to permanently alter theirshape by the application of su�ciently large forces. The entire metal and polymer formingindustry exploits this fact.

This is the ®rst of a two-part paper that is concerned with the modeling of themechanical behavior of such materials from a continuum viewpoint taking into accountthe changes in the microstructure of the material as it deforms. This provides a uni®edframework for modeling a wide range of material behavior. No speci®c constitutivefunctions are advocated here, though some are invoked in our illustrative examples for the

International Journal of Plasticity, Vol. 14, Nos 10±11, pp. 945±967, 1998# 1998 Elsevier Science LtdPergamon

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*Corresponding author. Fax: +1-409-845-3081; e-mail: [email protected] Rajagopal (1995) for a fundamental re-examination of the notion of solids, ¯uids and gases from a con-tinuum viewpoint especially in the light of the manufacture of a variety of man-made materials that seeminglydefy conventional de®nitions.

sake of clarity; instead only those features that are common to a wide variety of micro-structural changes are considered.

I.1. Previous work

The current work has much in common with metal plasticity (although it is meant to beapplied to a larger class of materials) in the sense that its aim is to model hystereticbehavior that occurs even in the limit of quasistatic processes. Thus, in order to presentthe background in which the current theory is set, we shall brie¯y recount the salientdevelopments in the theory of plasticity. Tresca (1867) was the ®rst to observe that, undersu�ciently high pressures, solids are capable of ``¯owing''. Saint-Venant (1870), LeÂ vy(1870) and later von Mises (1913) developed the governing equations for the motion of``rigid-plastic'' materials2 that is the basis for much of the studies of metal forming.

A key assumption that is made in the above theories that are valid for large defor-mations) is that the material does not exhibit elastic behavior under any circumstances.This assumption may be satisfactory when one is interested in large deformations that aretypical of metal forming but is quite unsatisfactory when one is interested in modelingrecovery processes such as ``spring-back'' that occur in many sheet metal operations. Theneglect of the elasticity of the material also has unexpected consequences in certain situa-tions where the stress in the rigid regime cannot be uniquely deduced from the boundaryconditions. In such cases one may be unable to determine uniquely the conditions underwhich yielding occurs.

Prandtl (1924) and Reuss (1939) accounted explicitly for the elasticity of the materialprior to yield by modifying the Levy±Mises model, within in®nitesimal deformations, anddistinguishing between temporary recoverable deformations of the material (termed the`èlastic strain'') and the permanent deformations (termed the ``plastic strain''). The (line-arized) total strain was then assumed to be the sum of the elastic and plastic strains. Theresulting theory, termed the ``Prandtl±Reuss theory'' has been widely utilized with minormodi®cations to cater to speci®c needs. Thus, the mechanical response of such elastic±plastic materials is governed by

1. their elastic response;2. the conditions under which inelasticity is actuated (the yield criteria);3. the nature of the shape change (plastic strain) and its evolution.

Within the context of large deformations, Eckart (1948) appears to have been the ®rstone to theorize that, unlike elastic bodies, plastic materials such as copper, steel etc. pos-sess multiple stress-free (or natural) shapes and that these shapes play a fundamental rolein the mechanical behavior of the material, evolving gradually with the deformation. Heconsidered the geometry of the `ùnloaded'' (stress-free) shape of the material and de®nedthe `èlastic strain'' through the metric tensor associated with the unloaded state withrespect to the current state. He did not propose any de®nite equation to describe theyielding phenomena. Following Eckart (1948) several authors (see e.g. KroÈ ner (1960, p.286, eqn 4) (within the context of a linearized theory), Backman (1964), Lee and Liu(1967) etc.) have utilized the notion of unloaded or stress-free con®gurations to de®ne the

946 K. R. Rajagopal and A. R. Srinivasa

2See e.g. Hill (1950) and more recently Casey (1986) and Naghdi and Srinivasa (1994a) in this regard.

plastic strain of the material. An alternative approach has been advocated by Green andNaghdi (1965) who assumed that the ``plastic strain'' was a primitive tensorial variable,with certain of its properties stipulated. Fox (1968), like Green and Naghdi (1965), intro-duced a tensorial variable as a fundamental primitive in the theory and later de®ned a``local moving reference con®guration'' through it. He then introduced the notion of``elastic strain'' by means of a multiplicative decomposition.

There has been extensive debate on the notion of ``plastic strain'' and the way by whichit is to be identi®ed. For example, there has been considerable debate regarding theapplicability of the notion of the stress-free strain, as de®ned by Lee (1969), to materialsthat cannot be unloaded to a stress-free state.3 On the other hand, the ``plastic strain'' is aprimitive in the theory of Green and Naghdi (1965) as originally proposed by them.Recently, Casey and Naghdi (1992) have used a notion of ``maximal unloading'' to pre-scribe the ``plastic strain'' of the material. This notion depends upon the norm used for thestress so that di�erent prescriptions could be conceived, leading to di�erent de®nitions of``plastic strain''.

The formulations mentioned above are concerned with rate-independent hystereticbehavior of crystalline metals and as such are not directly extendable to other kinds ofinelastic behavior.

I.2. The current approach: a family of elastic response functions and multiple naturalcon®gurations

The main aim of the present paper is to identify and exploit features and principles thatare common to a wide range of inelastic phenomena, including metal plasticity, polymerinelasticity, twinning, phase transitions etc. (some of which require microstructural con-siderations). Although these phenomena have di�erent physical origins, their manifesta-tions have some striking similarities. We wish to exploit these similarities.

We shall deal only with issues concerning the elastic response of such materials in Part Iof the paper, deferring the study of the dissipative behavior to Part II.

As summarized in Section I.1., traditional theories of plasticity ®rst introduced thenotion of ``plastic strain''. This then served as the foundation for subsequent develop-ments. Thus, the stress is given by an expression of the form

T � T̂�E; ��; �1�

where E is a measure of strain from some con®guration and a is a scalar, vector or ten-sorial variable whose change is caused by the inelasticity of the material. Irrespective ofthe nature of the variable a we make the following observation:

The eqn (1) can be viewed as a family of elastic response functions, parametrized by thevariable a

This simple observation has far reaching consequences, many of which are entirelyindependent of the nature of the parametrization. Thus, the central tenet of the currentapproach is that the material possesses not one, but a family of elastic response functionsÐ

Mechanics of the inelastic behavior of materialsÐpart 1, theoretical underpinnings 947

3See also the discussion by Casey and Naghdi (1980) as well as that by Lee (1996) and Lee and Liu (1967) in thisregard.

each with its own natural con®guration (see Section III). While this idea is not newÐit hasbeen used implictly or stated explicitly by many authors e.g. Eckart (1948), Green andNaghdi (1973), Rajagopal and Wineman (1980), Casey and Naghdi (1992)Ðwe system-atically exploit this idea. In the process, we clarify issues of symmetry, ``plastic strain''etc., and provide a uni®ed framework for the discussion of a wide range of inelasticbehavior, including twinning, phase transition, crystallization and anisotropic ¯uids.

In this approach, it is the family of elastic responses with their attendant natural con-®gurations (which are taken as primitives just as in the case of elastic materials) that takecenter stage when discussing the elastic response, with various measures of deformation(such as ``plastic strain'') being only of secondary importance.

We show that one (and by nomeans the only) way to prescribe natural con®gurations for aclass of materials possessing instantaneous elasticity (see Section IV.1.1) is by an instan-taneous removal of all applied forces by means of a ``rapid path''.4 By de®ning the notion of``equivalent families of response functions'' (see de®nition (2) in Section IV.1.2), we show thatother prescriptions are possible and that they give rise to the same response for the material.

In order to di�erentiate between inelastic processes such as slip and twinning that leavethe lattice structure unaltered, from those such as martensitic transformations that alterthe microstructure, we de®ne the notion of ``similar response functions'' (see de®nition (3)in Section IV.1.3) using which we develop a special representation called the ``canonicalrepresentation'' (see Theorem (1) in Section IV.1.3) in which all the elastic responses withthe same microstructure have the same functional form for the stress, with their respectivenatural con®gurations being unique to them.

We ®rst develop the basic ideas within the setting of homogeneous motions so as tohighlight the many new aspects of the theory in as direct and intuitive a way as possible.Once the basic concepts have been developed, we then extend the results to inhomoge-neous motions as described in Section IV.2.

Finally, classical notions of material symmetry for simple materials (Noll, 1957)5 areextended to materials possessing multiple natural con®gurations. We then demonstrate(see Section V) that if we use the ``canonical representation'', then the material symmetrygroup of the elastic responses of the material will be the same if all the elastic responsesre¯ect the same microstructure, although they may have di�erent natural con®gurations.Such is the case for crystallographic slip and the result derived here can be specialized tore¯ect the observations of Taylor and Elam (1925) regarding this issue.

Here, it must be pointed out that Noll (1972), recognizing the shortcomings of the ori-ginal de®nition of simple materials (Noll, 1957), has extended the de®nition to explicitlyincorporate families of simple materials (according to his original de®nition (Noll, 1957))into a general framework. In this framework, each member of the family is identi®ed by acertain number of ``state'' variables. He then goes on to de®ne material symmetry atconstant ``state'', so that the material can have changing symmetry as the ``state changes''.No speci®c phenomenon is considered and hence no constitutive equation is provided forthe evolution of the ``state'' variables of the material. Our approach is more speci®callyoriented towards inelasticity of solids, and while our approach can be recast into thegeneral format proposed by Noll (1972), the restriction to inelasticity of solids highlights


4The concept of a rapid path is quite well known and widely used in the theory and application of viscoelasticity.A formal de®nition of this notion has been provided by Naghdi (1984) in the case of plasticity.5The class of simple materials includes elastic materials, Newtonian ¯uids and most visco-elastic ¯uids and solids.In this paper, unless otherwise stated, by a simple material we mean that according to Noll (1957)

the central assumptions and results of the theory and provides some insights that mightotherwise be obscured in a more general framework.

Section VI of the paper compares this theory with traditional plasticity and demon-strates how the current approach overcomes several shortcomings inherent in the otherapproaches. Section VII provides a summary of the advantages of the current approachand sets the stage for the discussion of the inelastic response.

The principal results of the current approach are the following:

1. Development of a general framework (utilizing the notion of ``families of elasticresponse functions'') that uni®es a wide range of disparate inelastic phenomena. Thisframework provides a sound basis for the discussion of permanent shape changesthat accompany such inelastic processes.

2. Rationalization of the notion of ``plastic strain'' and its extension to other processes(other than crystallographic slip) and to other materials (such as polymers) by usingthe notion of ``multiple natural con®gurations'' (see Section IV.1.1).

3. A demonstration that di�erent de®nitions of ``plastic strain'' could be used, all ofwhich represent the same inelastic response. We do this by the de®nition of``equivalent representations'' (see Section IV.1.2).

4. Classi®cation of changes in natural shape into those with and without accompanyingchanges of microstructure by constructing ``similar sets''. Members of each similarset share the same elastic response function (and hence the same microstructure) butpossess di�erent natural states. This naturally leads to the development of special (orcanonical) representations of elastic response (see Section IV.1.3). For example, thecurrent approach allows us to model crystallographic slip of single crystals withoutthe use of additional microstructural considerations such as those suggested by e.g.Mandel (1973), Casey and Naghdi (1992) and Naghdi and Srinivasa (1994b).

5. Extension of the classical notion of material symmetry (see Section V) to materialsthat possess multiple natural con®gurations and families of elastic response func-tions. This feature allows for the modeling of materials that change their symmetrygroup as a result of microstructural change (e.g. cubic to tetragonal transformationsin some materials). Using the notion of ``similar sets'', we also prove that the sym-metry group associated with the elastic response remains unaltered as long as themicrostructure remains unaltered, completely in line with established results, e.g.crystallographic slip (see Taylor and Elam, 1925).

II. PRELIMINARIES

Let X denote a typical material particle in a body B. A con®guration k(B) is the positionin Euclidean space occupied by the body. One of the fundamental concepts that we needto analyze the properties of material bodies, is the intuitive notion of the ``nearness'' oftwo con®gurations of a given body.

We consider a con®guration k1(B) to be in the E neighborhood of another con-®guration k(B) whenever the distance between the positions X0:=k0(X) and X1:=k1(X)of every material particle X in B is less than E, i.e. if

jX0 ÿ X1j < �: �2�


The above de®nition of ``nearness'' of two con®gurations will allow us to talk about var-ious aspects of the topology of the con®gurations such as open and closed sets of con®g-urations, etc.6

Let us assume that initially the material body, with a speci®c microstructural patternoccupies a con®guration k0(B).7 For simplicity, we shall assume that the body is initiallyhomogeneous in the sense that the microstructure is the same around every particle in thebody.8 When it is deformed, suppose that its initial response is that of an elastic solid; butonce its deformed (or current) con®guration is su�ciently ``far'' from its initial con®g-uration, suppose that certain microstructural changes take place in the body. For exam-ple, crystalline metallic materials may undergo crystallographic slip wherein the latticeplanes within the material slip or slide over one another like a deck of cards, or they mayundergo twinning or martensitic transformations wherein there is a rearrangement of thelattice structure within each grain (see Nishiyama, 1978; Holt et al., 1994, etc.). Polymericmaterials, on the other hand undergo network scission and re-formation wherein thearrangement of the nodes of the long chain polymer network are altered (see e.g. Fongand Zapas, 1976; Peterlin, 1976).

In spite of the disparate ways in which these structural changes occur, a striking mac-roscopic manifestation of these changes is that the body does not retrace its path in con-®guration space, and return to its original con®guration upon the removal of the loads. Aschematic representation of the one-dimensional response of such materials upon the slowapplication and removal of su�ciently large load is shown in Fig. 1.

In Fig. 1, point A corresponds to the material being in its initial con®guration. As longas the displacement gradient is less than the value corresponding to B, the response iselastic. From B to C, the response is inelastic and dissipative and the material goes to thecon®guration corresponding to point D when the load is removedÐthe material has suf-fered a ``permanent shape change''. It should be noted that the con®guration that thebody goes to upon removal of the load may depend upon the rapidity with which the loadis removed.

III. ELASTIC MATERIALS

At the outset, it is instructive to consider the response of a purely elastic material andrecall some fundamental facts regarding their behavior (for a detailed account of this, seeTruesdell and Noll (1992)). This will help us to understand the nature of the changesdescribed at the end of the previous section and represented schematically in Fig. 1, andhelp identify the fundamental variables which characterize the behavior of materials suchas the ones discussed above.

The purely mechanical behavior of elastic and elastic±plastic materials is intimately asso-ciated with the response to cycles of deformation. Thus, for our purposes, it is convenient to


6Of course, other de®nitions of ``nearness'' of con®gurations can be used, but the de®nition used here is quiteintuitive and su�cient for our purposes. With the above description of nearness of con®gurations, two con®g-urations that di�er by a translation and rotation will be considered far from each other if the translation orrotation is su�ciently large. Indeed, we can eliminate this by ®rst removing the translation and rotation of anarbitrary particle and then carrying out the comparison. Of course, this creates needless complications and thecurrent simple de®nition will su�ce for our purpose.7Henceforth we shall suppress B in the notation for a con®guration and simply write k(B) as k.8The notion of a homogeneous body is sometimes introduced through the requirement that the stress response bethe same at every material point in the body. We shall touch upon this in Section IV.2.

start with the notion of con®gurational paths and cycles. We de®ne a con®gurational path ofa body as a one parameter family of con®gurations k(l) which depend continuously on l.9

For elastic bodies, corresponding to a con®gurational path of the body, for each pointbelonging to it, there exists a corresponding path in the space of stresses. This stress pathis uniquely determined by the con®gurational path of the body and is independent of therate at which the con®gurational path is traversed. Moreover, if the con®gurational pathbegins and ends at the same con®guration (a con®gurational ``cycle'') then so does thestress path. Said di�erently, a con®gurational cycle corresponds to a stress cycle. It is theresponse to con®gurational cycles that is used to de®ne the class of elastic materials as asubset of the class of hypoelastic materials. Indeed, a hypoelastic material for which everycon®gurational cycle corresponds to a stress cycle is an elastic material. We shall onlyconsider hyperelastic materials, that is, materials for which the work done vanishes in anyclosed con®gurational cycle in which the velocity ®eld is the same at the start and the endof the cycle.

In order to mathematically represent the stress response of elastic materials we intro-duce the notion of a reference con®guration kr, which is usually a known con®guration,i.e. a con®guration for which the stress at each material particle is known. If we refer tothe position in kr of a typical material particle X as X, then its position x in any othercon®guration can be represented by

x � ��r�X; t�; �3�


Fig. 1. Schematic diagram of the stress±strain curves corresponding to di�erent kinds of inelastic behavior.

ABCD represents the response of a typical elastic±plastic material under ``slow'' loading. ABEFG represents the

response of some materials that may not elastically unload to zero stress under slow loading conditions. ABIJK is

a representation of the response of a typical shape memory alloy. AB0, DC0 and EF0 represent the instantaneouselastic response of materials that possess instantaneous elasticity. For such materials the responses AB0, DC0 andEF0 can be achieved in the limit of very ``rapid'' processes.

9The notion of ``nearness'' discussed in eqn (2) allows us to talk of continuous dependence on the parameter.

where the subscript kr indicates that the form of the function ��r depends upon the choiceof the reference con®guration. In view of the properties of elastic materials described inthe previous paragraph, it is easy to show that once a con®guration kr is chosen, the stressdepends only upon the deformation ��r from kr. In particular, for the class of ``simpleelastic materials'' (see Noll, 1957), the Cauchy stress tensor T at X depends upon ��r onlythrough the deformation gradient F�r at X and is of the form

T � T̂�r�F�r�; �4�

where

F�r :� @��r�@X

: �5�

In the constitutive eqn (4) the subscript kr indicates that the form of the function T̂�rdepends upon the choice of the reference con®guration.

III.1. Equivalent representations

The elastic response of a given body can have more than one representation, eachrepresentation depending upon the reference con®guration chosen. For instance, if a dif-ferent reference con®guration �� is used to represent the constitutive equation for thesame body, then the constitutive equation T̂�� is related to T̂�r , by the relationship

T � T̂�r �F� � T̂�� FPÿ1�; �6�

for all F in the domain of de®nition of T̂�r and where P is the gradient of the mappingfrom kr to �� and the function T̂�� is di�erent from T̂�r . The relationship (6) is a result ofthe fact that the value of the stress is determined solely by the current con®gurationoccupied by the body and is independent of the choice of the reference con®guration. Weshall refer to the ordered pair formed by a given reference con®guration k and its asso-ciated stress response function T̂� as a ``response pair'' and denote it by ��; T̂��.

The above considerations leads us to the followingDe®nition 1. Equivalent representations of the stress response: Two stress response pairs

��1; T̂�1� and ��2; ~T�2� for a given body are said to be equivalent if they satisfy eqn (6) with Pbeing the gradient of the mapping from k1 to k2.

Indeed, for elastic bodies, all the representations are equivalent. As we shall see laterthis de®nition will also play an important role in the discussions of the stress response ofinelastic materials in the next section.

A restriction on the form of the constitutive equations comes from frame-indi�erence.We shall just summarize the results here and refer the interested reader to Truesdell andNoll (1992) for a full discussion of the issue. The central result of the concept of frame-indi�erence is that the constitutive equation for the stress in an elastic material is given bythe form

T � R�rT�RT

�r� R�r

~T�r �E�r��RT�r�; �7�


where E�r , is the Green±St. Venant strain tensor de®ned by E�r � 1=2�FT�rF�r ÿ I�, R�r is

the rotation tensor obtained in the polar decomposition of F�r , T* is a tensorial quantitythat is referred to as the rotated stress tensor and the notation (.)T denotes the operationof transposing the tensor.

It is very important to realize that the form of the function ~T�r is in general dependenton both the shape and orientation of the reference con®guration kr.10 We shall get back tothis issue when we discuss aspects of material symmetry in Section V.

IV. MATERIALS THAT POSSESS MULTIPLE NATURAL SHAPES

In the light of our brief summary of the response of elastic materials, we are now in aposition to interpret the response of materials that permanently alter their shape. In orderto make the ideas that follow as clear as possible, we shall develop them within the contextof homogeneous motions of homogeneous bodies, extending them in Section IV.2 toinhomogeneous motions.

IV.1. Response to homogeneous deformations

We consider a material which is initially in a con®guration k0 (in which it is homo-geneous) corresponding to the point A in Fig. 1. By the initial elastic domain of thismaterial we mean a path connected set of con®gurations over which the response is that ofan elastic material. In other words, corresponding to any con®gurational path within thisdomain, there exists a unique stress path (independent of the rate at which the path istraveled) such that every con®gurational cycle corresponds to a stress cycle. Clearly k0 isan element of this elastic domain. It should be observed that we have introduced nonotion of a reference con®guration and no kinematical quantities for inelastic materials asyet, choosing to work directly with the set of con®gurations as a whole instead. The stressresponse of the material corresponding to the initial elastic domain will be referred to asthe ``initial elastic range'' of the material.

Now, in order to mathematically represent the response of the material within the initialelastic domain, we need to follow the procedure outlined for elastic materials in the pre-vious section and choose a convenient reference con®guration and write the response ofthe material in the form given in eqn (7). After this has been done we now consider theconsequences of deforming it beyond the elastic domain.

Once the material is homogeneously deformed so that its con®guration lies outside itsinitial elastic domain, certain microstructural changes take place. The body now displaysa di�erent elastic responseÐthe material has a new elastic domain and range. In order torepresent this new response mathematically, we repeat the procedure used for the initialelastic response function and choose a new (and possibly di�erent) reference con®guration�1 that is appropriate to this new elastic response and stipulate the response function to beof the form

T � T�1 �F�1 �: �8�


10The notion of shape and orientation relative to a ®xed reference frame are taken to be primitives. However,changes in shape and/or orientation can be quanti®ed by means of the stretch and rotation tensors.

We note that since the deformation is homogeneous, the con®guration k1 can be chosensuch that the material is homogeneous in that con®guration. Thus, for the elastic response(to homogeneous deformations) of the material after each permanent change of shape, weneed to (i) identify the new elastic domain and range (ii) choose an appropriate homo-geneous reference con®guration for the mathematical representation of the responsefunction and (iii) obtain a function of the form given in eqn (8).

In order to clearly distinguish the various con®gurations that arise in the study of thesematerials, we ®rst introduce the concept of Natural con®gurations.

These are the reference con®gurations chosen to represent the elastic response functionsof the materials and are the primary con®gurations of interest even in our study ofinelastic behavior.

The use of multiple reference con®gurations for the representation of the elasticresponse functions of the material is crucial for the de®nition of evolving material sym-metry, as will become clear in the subsequent sections.

IV.1.1. Instantaneous elasticity and the natural con®gurations The question naturallyarises as to what might be the nature of the natural con®gurations of the body and howone might identify them. Philosophically, all that is required is that it be a con®guration inwhich the stress is known.

In view of the wide range of inelastic phenomena that we wish to model, rather thanproviding a single universal de®nition for the natural con®gurations, we shall contentourselves with providing several examples. Speci®c prescriptions will depend upon theparticular inelastic phenomenon in question. Incidentally, a similar sentiment is expressedby Naghdi (1990) regarding the identi®cation of the ``plastic strain''. We shall considerthree possible cases:

1. Within the class of homogeneous elastic bodies, it is easy to see that any one of thestress-free con®gurations (if available) of the material may serve as a suitable candidate.The situation is quite a bit more complicated for inelastic materials since, at ®rst sight itmay seem impossible to unload some materials elastically to a state of zero stress, sincethe stress-free con®gurations may not lie within the current elastic domain of the body.Such is the case for materials (such as polymers and polycrystals) whose response issimilar to the curve ABEFG in Fig. 1. The ability to elastically unload a body to a stress-free state has been a hotly debated issue within rate-independent plasticity (see e.g. Lee,1969, 1996; Casey and Naghdi, 1980; Naghdi, 1990). The experimental evidence for qua-sistatic response of many metals seem to indicate that, under slow loading conditions, thestress-free state lies outside the elastic domain.

The above considerations do not include the possibility that the response of suchmaterials may depend upon the speed at which the experiment is performed so that, whenthe experiment is performed su�ciently rapidly, the response may proceed along EF0

instead of EFG in Fig. 1. This leads us to the next possibility.2. Every process inside the elastic domain of the material is non-dissipative in the sense

that such processes engender an elastic response for the material and the mechanical workis not transferred as thermal energy. Outside the elastic domain, most processes are dis-sipative in that a part of the mechanical work supplied is transferred as thermal energy.However, it is not unreasonable to suppose that, even outside the elastic domain, there arespecial classes of processesÐextremely rapid onesÐthat are non-dissipative. To be moreprecise, given a ®xed con®gurational path that is outside the current elastic domain, we


consider a sequence of processes, each traversing the path at a faster rate than the other. Thelimit of such a sequence corresponds to traversing the path instantaneously. Such limitingprocesses are called ``rapid processes'' and materials for which such processes are non-dissipative are said to possess instantaneous elasticity.

Thus, we consider bodies that can be instantaneously brought to a state of zero stress ina non-dissipative manner, i.e. whose instantaneous elastic domain includes states of zerostress. For such materials these stress-free con®gurations become candidates for the nat-ural con®gurations. Of course the resulting stress-free state may be ¯eeting and themicrostructure may change if given su�cient time, but all we need here is the ability toelastically bring the body to a stress-free state, however ¯eeting it might be.

The above considerations do not mean that the classical elastic domain of the materialexpands or contracts with di�erent rates of deformation. Indeed, as we shall see in SectionIV.1.3, the extent of the classical elastic domain is independent of the rate of deformation,but that a certain amount of time is needed for the microstructural change to manifest itself.

3. This leaves only the case when the body cannot be instantaneously unloaded to astress-free con®guration. In this case, any other con®guration with a known stress, such asa maximally unloaded con®guration (see Casey and Naghdi, 1992) can be used.

In some cases, the natural con®gurations may be selected based on other (micro-structural) considerations. Such is the case for example in martensitic transformationswhere the states of the material at the Austenite start temperature and the Martensite starttemperatures may be chosen.

We must also point out that many other prescriptions, similar to the maximally unloa-ded con®guration, are possible, all of which give rise to equivalent representations. Thiswill become abundantly clear in Section III of Part II where we show that a di�erent (andpossibly more convenient or ``natural'' set of natural con®gurations) can be de®ned aposteriori for a certain class of elastic±plastic materials. To drive home this point, Raja-gopal and Srinivasa (1995b) have provided an illustrative example wherein they haverecast the classical Prandtl±Reuss constitutive equations into an equivalent form in termsof natural con®gurations that are not stress-free. There is thus some ``¯exibility of choice''with regard to natural con®gurations (see Rajagopad (1995)).

IV.1.2. Equivalent representations of the stress response functionsOnce the natural con®gurations are chosen, the stress response of these materials may

be represented as a set of ordered pairs

��p; T̂�p �jp 2 Pn o

�9�

of natural con®gurations kp and the elastic response functions T̂�p associated with them.The set P represents the index set of all the possible natural con®gurations for the mate-rial. Care must be taken in discussing issues of frame indi�erence for each of the orderedpairs introduced in eqn (9). If we consider a motion which di�ers from the original motiongiven by eqn (3) by a time dependent translation and rotation, frame indi�erence assertsthat the value of the stress tensor for the second motion di�ers from that for the ®rst onlyin orientation. It should be noted that while the current con®guration changes its orien-tation, the set of natural con®gurations are the same for both the motions. This thenimplies that the form of the response functions can be reduced to eqn (7) with kr replacedby kp. A detailed analysis of this issue has been presented by Rajagopal and Srinivasa


(1995b) and the reader is referred to Section III of that paper. This is one of the funda-mental concepts associated with materials with multiple con®gurations and forms thebasis of the theory proposed by Rajagopal (1995).

The assumption of eqn (9) indicates that the stress response of inelastic materials con-sidered here is akin to that of not one but an entire family of elastic materials. The aboveidea of a family of elastic response functions is in line with the notion that, for metals atleast, the elastic response function, its elastic domain and range, as well as its naturalcon®guration are indicators of the nature of the interatomic forces. Indeed, it is theinteratomic forces that govern the crystal structure and lattice arrangement and determinemany of the properties of the material (such as material symmetry). On the other hand,the microstructural changes are strongly in¯uenced by the presence of lattice defects etc.,that do not a�ect the elastic response of the material. Of course a special case of the aboveis that of elastic solids wherein there is only one ordered pairÐa single natural con®g-uration and the stress response function associated with it.

For the case of inelastic materials, to make things more ``concrete'', consider deforma-tion-induced twinning, where the material changes its crystal structure to a ``twin'' when itis su�ciently deformed. This twinned structure possesses a di�erent elastic response thanthe original material in the sense that its stress-free con®guration di�ers from that of theoriginal by a shear deformation (the so-called twinning shear).11 Moreover, its newmaterial symmetry di�ers from that of the untwinned structure by a rotoinversion.Finally, there are several intermediate structures which are a result of a juxtaposition (ona very ®ne scale) of the twinned and untwinned structures. Such a material has beenmodeled by Rajagopal and Srinivasa (1995a), (1995b) by associating two elastic responsesand two natural con®gurations for the material, that is, two ordered pairs, one for theuntwinned material and another for the twinned material as well as an additional variablerepresenting the volume fraction of the twinned material.

In the case of metal plasticity, there are an uncountably in®nite number of naturalcon®gurations that are usually assumed to have the same volume. On the other hand, forthe case of the multinetwork theories of polymers, Wineman and Rajagopal (1990) andRajagopal and Wineman (1992) utilized a material with multiple elastic responses, withthe current con®guration of the material being chosen as the natural con®guration for thenew networks as they form. Thus, while the concept of a natural con®guration is commonfor these three cases, their manifestation is di�erent in each of them.

Of course, no two ordered pairs in a given family of elastic response functions areequivalent in the sense of elastic materials as de®ned by de®nition 1. However, we canextend the notion of equivalent representations to two families of response functions for agiven inelastic material by means of the following.

De®nition 2. Equivalent representations of the elastic response of inelastic materials: Twofamilies of ordered pairs ��p; T̂�p�; p 2 P and ��q; ~T�q�; q 2 Q are said to form equivalentrepresentations if they can be put in one to one correspondence in such a way that corre-sponding pairs are equivalent in the sense de®ned for elastic materials.

Once the constitutive equations have been developed using a given set of natural con-®gurations, using the above de®nition of equivalent representations, it can be recast aposteriori by using a di�erent set of natural con®gurations.


11Recall that we are discussing only homogeneous deformations here.

IV.1.3. Similar response pairs and canonical representations of the response functionsFor inelastic materials, a related and useful de®nition to that of ``equivalent repre-

sentations'' is the followingDe®nition 3. Similar stress response functions: Two stress response pairs ��0; T̂�0� and

��; ~T�� , from among a family of response functions for a given inelastic body (see eqn (9))are said to be similar, if

T̂�0�F� � ~T�� FP�; �10�

for all invertible F in the domain of T̂�0 and for some invertible tensor P. The tensor P will bereferred to as the ``similarity transformation'' between the two pairs.

At ®rst glance, the above eqn (10) seems identical to the eqn (6) that was used to de®neequivalent representations; however, there is a fundamental di�erence, i.e. unlike eqn (6),the tensor P used in the right hand side of eqn (10) is completely unrelated to the gradientof the mapping between the con®gurations �� and k0. Indeed, since T̂�p and ~T�� belong tothe same family, they are non-equivalent and hence do not satisfy eqn (6). However, somemembers of a given family may be similar to one another. For example, in the case ofcrystallographic slip of single crystals, all the members of a given family of stress responsefunctions are similar to one another. It should be observed that if the similarity transfor-mation P is the identity tensor, then the functional form of the two response functions T̂�pand ~T�� are identical. A schematic representation of the notion of similar and dissimilarresponse pairs is shown in Fig. 2.

It can easily be shown that the notion of ``similarity'' is an equivalence relation amongthe members of a family of response functions. Thus, given a family of stress responsefunctions, we can group their members into sets using similarity. We shall refer to suchsets as ``similar sets''. Within each set, we can further simplify the representation by meansof a special representation which we shall refer to as a ``canonical representation''.

Theorem 1. Canonical representation of the stress response. Given a family of responsepairs for an inelastic solid, an equivalent family (in the sense de®ned in de®nition (2)) can beconstructed such that for each similar set, the form of the response functions is the same forevery member of that set. Such a representation is called a canonical representation. In otherwords, when the canonical representation is used, each similar set has a unique stress-response function associated with it.

Proof. We prove the above theorem by constructing the required canonical representa-tion as follows:

1. Constructing equivalent representations of the members:Given a family of elastic response functions, we ®rst use the notion of similarity (see

de®nition (3)) to divide it up into similar sets. Now in each similar set we pick a membersay, ��p; T̂�p� which we shall refer to as the ``pivot'' member. Any other member in thatsimilar set, say ��; ~T�� is related to ��p; T̂�p� as described in eqn (10).

We now map the con®guration �� to a new con®guration k1 by means of a homo-geneous deformation with gradient12 P. If we now de®ne a new function T̂�1 by

T̂�1 �F� :� ~T�� FPÿ1�; �11�


12It should be observed that k1 cannot be identical to kp since no two members of a given family are equivalent.

then a routine calculation reveals that ��; ~T�� and ��1; T̂�1� are equivalent in the sense ofde®nition (1). Moreover, in virtue of de®nition (3), it is easy to see that this equivalentrepresentation is ``similar'' to ��p; T̂�p�, with the ``similarity transformation'' being theidentity tensor. Thus, the functional forms of T̂�p and T̂�1 are identical.

2. Construction of an equivalent representation for each similar set:


Fig. 2. Schematic stress±strain curves to illustrate the notion of ``similar'' and ``dissimilar'' elastic response

functions. (A) represents a family of similar response functionsÐshown with equal slopes to emphasize that the

response functions are the same although the natural con®gurations are di�erent. (B) represents a family of dis-

similar response functions (each shown with a di�erent slope).

Using the above procedure, we can construct equivalent representations for eachmember of a given similar set such that the similarity transformation between any two ofthe newly constructed representations is the identity tensor, so that, in the newly con-structed representation, all the members have the same functional form for the responsefunction. These new representations, together with the ``pivot'' member form an equiva-lent representation (in the sense of de®nition 2) of the similar set. By its very construction,all the members of this representation have the same form for the stress response functionbut di�erent natural con®gurations. Thus, repeating the above two steps for each similarset in the family, we can create a new equivalent family with the desired properties.&

The concept of canonical representations de®ned above, embodies the assumption thatthe elastic response is ``unchanged'' (i.e. that the elastic constants are unaltered) in metalplasticity (see Fig. 2).

The form of the response functions in the canonical representation then depend onlyupon the ``pivot'' member chosen, and di�erent (equivalent) canonical representations can beconstructed by using di�erent pivot members. For elastic materials, since there is only onemember in the family, any con®guration can be chosen as a pivot and, as is well known,the form of the response depends upon the choice of this single pivotal con®guration.

It cannot be overstated that not only the shapes but also the orientations of the naturalcon®gurations kp are important in determining the form of the elastic response functions.For example, in the case of crystallographic slip of single crystals (see Fig. 3), the membersof the family of stress response functions are similar to one-another and hence a singlepivot member su�ces to construct a canonical representation of the stress response. If wechoose initial con®guration A in Fig. 3 as our pivot member, a quick look at con®gura-tions B and C reveals that the associated response functions are similar but non-equiva-lent and that, the ``similarity transformation'' between B and C is an orthogonaltransformation. Consequently, if one constructs a canonical representation for the stressresponse of this material, the natural con®gurations corresponding to B and C will notdi�er in their shape but will di�er in their relative orientation.

IV.1.4. Physical interpretation of the ``similar sets'' As mentioned earlier, the elasticresponse functions represent the e�ect of the microstructural arrangement of the material,not accounting for the point, line and surface defects in the material that do not mucha�ect the elastic response but play a signi®cant role in determining the inelastic responseof the material. Thus, one may consider the members of a similar set to represent all thepossible macroscopic shapes that the material is capable of assuming without changing itsmicrostructural arrangement. Hence, for crystallographic slip of single crystals, all the mem-bers are similar since there is no change in the microstructural arrangement of the material,the change being only in the natural con®guration.13 The same is true in the multi-networktheories of polymers as well as deformation induced twinning. Thus one could considereach similar set as representing a single microstructural arrangement in the material.

On the other hand, when the microstructural state is fundamentally altered (as in thecase of martensitic transformations, where the lattice may change from a cubic to tetra-gonal arrangement, and partially twinned states during deformation induced twinningwhere there is a juxtaposition of the twinned and untwinned states), the resulting responsefunctions are no longer similar to each other and hence do not belong to the same similar set.


13This is not strictly true because the dislocation structures do change even in a single crystal.

IV.1.5. The de®nition of the ``natural deformation gradient'' tensor G and its role indetermining the natural con®gurations Once a canonical representation of the stressresponse has been constructed, it is clear that, within a given similar set, each member isuniquely identi®ed by its associated natural con®guration, while the stress response isuniquely identi®ed by the response of the pivot member. Indeed, in most cases, the pivotmembers may be chosen in such a way that the natural con®guration of a given memberuniquely identi®es it in the entire family. Such is the situation for both martensitic trans-formations, where the so-called ``Bain strain'' together with the appropriate rotation,identi®es the deformation that takes the cubic lattice to the tetragonal one. The di�erentvariants of martensite belong to a ``similar set'' and hence can be di�erentiated by theirrespective natural con®gurations. As seen in eqn (8), the role of the natural con®gurationis implicit in the sense that the constitutive equation for the stress response depends uponthe current natural con®guration.

In order to make this dependence explicit, we introduce some ®xed con®guration kr(say, the initial con®guration or one of the pivots in a canonical representation). Let G bethe gradient of the mapping from kr to kp. A knowledge of the ®eld G then su�ces toidentify the current natural con®guration once the con®guration kr is known. Then, wecan rewrite eqn (9) in the form

T � T̂G�F�p� :� T̂�F�p ;G�; �12�

where the functional form of T̂ depends upon the choice of all the natural con®gurationsassociated with the pivot members of a canonical representation. The above form of the


Fig. 3. Representation of three con®gurations of a single crystal demonstrating the evolution of the orientation

of the lattice. The cross hatching is meant to represent the lattice structure of the crystal. A is the initial con®g-

uration. B and C correspond to two con®gurations with the same strain but with di�erent lattice orientations.

constitutive equation describes the entire set of ordered pairs introduced in eqn (9) in asingle concise equation. Indeed, the ordered pairs can be recovered from eqn (12) bysubstituting particular values for G and obtaining the corresponding functions T̂�p . Ofcourse, the con®guration kp itself can be obtained from the knowledge of kr and G.

It is easy to see that, if the above framework is applied to classical plasticity, then thetensor G plays the role of the ``plastic deformation''. Several measures of strain immedi-ately become available to us by way of conventional de®nitions any of which may beinterpreted as the ``plastic strain''.

IV.2. Inhomogeneous motions

Recall that the previous subsection has been entirely devoted to the response of homo-geneous bodies to homogeneous motions. We now consider an alternative de®nition ofhomogeneous bodies to the intuitive one given in Section 3. We say that the body is elas-tically homogeneous, if the stress response function family (de®ned in eqn (8)) is the samefor every point in the body for homogeneous motions.

A complication immediately arises when one considers inhomogeneous motions sincedi�erent material points may undergo di�erent microstructural changes depending uponthe nature of the loading at each point. In particular, once a part of the material under-goes microstructural changes, it is usually not possible (except under very special circum-stances) to unload it elastically to a stress-free state even under rapid unloading. This isbecause, the stress-free con®guration would be incompatible and would not ®t together inan Euclidean space into a global con®guration.

The question thus arises as to how may one keep track of local changes in micro-structure of a material undergoing inhomogeneous deformations. We deal with suchsituations by considering each material point independently and constructing ®ctitioushomogeneous motions of the body, such that the deformation gradient associated with the®ctitious motion coincides with the deformation gradient of the actual motion at thematerial point in question at each instant of time. For this ®ctitious motion, we can con-struct canonical representations of the stress response and de®ne the tensor G that iden-ti®es the appropriate natural con®guration as described in the last subsection. We thenassign this tensor G to the material point in question. By repeating this procedure for theentire body we can construct a ®eld G(X,t) representing the local microstructure of everypoint in the body.

Before leaving this subsection, we would like to mention two alternative possibilities fordealing with inhomogeneous motions. The ®rst, developed by Eckart (1948), is to con-struct a globally unloaded con®guration which is a subset of a non-Euclidean space. Whilesuch an approach has the advantage of not requiring a separate investigation of inhomo-geneous motions, it is quite non-intuitive and requires an extensive excursion into themathematics of non-Euclidean spaces. Another alternative that has been extensively usedin the study of continuous distributions of dislocations (see e.g. Kondo, 1952; Eshelby,1956; Bilby, 1960; Nabarro, 1987; Naghdi and Srinivasa, 1994a, 1994b) is to considermappings of certain tangent vectors to the body (corresponding to the lattice vectors ofthe crystalline lattice) and thus construct G ``directly''. Not only can the current approachbe shown to be equivalent to this when we consider the case of single crystals, but it appliesto the case of polycrystals and polymers which do not have well-de®ned lattice structures.This is because, the current approach does not rest on the existence of lattice vectors.


Having completed the construction of the form in eqn (12) for the canonical repre-sentation of the response of inelastic materials, the only remaining issue to be discussed,insofar as the elastic response is concerned, is the notion of material symmetry and it is tothis aspect of the response that we turn to next.

V. MULTIPLE CONFIGURATIONS AND MATERIALS SYMMETRY

One of the main results in the theory of simple materials (see Truesdell and Noll, 1992)is that the material symmetry of the body is determined once and for all if its symmetrygroup with respect to some ®xed reference con®guration is known.

To elaborate, the functional form of the constitutive equations for a simple materialdepends upon the choice of a single reference con®guration kr. Once such a con®gurationis given, then the response of the material is a functional of the history of the deformationgradient F�r , with respect to that con®guration and is of the form (see Truesdell and Noll(1992, p. 60)

T�X; t� � F1�s�0��r �F�r �X; tÿ s�;X�: �13�

The symmetry group of the material with respect to this con®guration is the group G�r ofall transformations of the reference con®guration under which the functional form of theconstitutive eqn (13) remains ®xed. A fundamental result in the theory of simple materials(see Noll, 1957) asserts that if P is the gradient of the mapping between two con®gurationsk0 and k1, then the symmetry group of the material with respect to the new con®g-uration k1 is given in terms of that associated with k0 by ``Noll's Rule'' (see Truesdell andNoll (1992, p. 77, eqn. (31.6))) and is of the form

G�1 � PAPÿ1 for all A 2 G�0 : �14�

Thus once the symmetry group with respect to the reference con®guration is given, Noll'srule enables one to calculate the symmetry group with respect to any other con®gurationfor any simple material.

For the case of the inelastic materials discussed in the previous section, there is no singleresponse function of the form given in eqn (13). Rather, there is a whole set of orderedpairs of response functions as described by eqn (9) each of which represents a simplematerial whose constitutive equation is of the form of eqn (13).

It is well accepted that the material symmetry group of a crystalline solid re¯ects (insome sense) the lattice structure of the material. Since the lattice structure may be alteredby the application of force, it stands to reason that so can the material symmetry group.With this in mind, we shall de®ne the symmetry group G(kp) for the elastic response of thematerial whose natural con®guration is kp, i.e. the response pair ��p; T̂�p�, to be the set ofall tensors H such that

T̂�p �F�p � � T̂�p �F�pH�: �15�


It should be observed that owing to the assumption of instantaneous elasticity of thematerial, all values of F�p with non-zero determinants are allowed.14 Thus for an inelasticmaterial, possessing a family of response pairs, there will be as many symmetry groups asthere are members in the familyÐeach unrelated to the other.

In order to see the relationship of the above de®nition with that for simple materials, wenote that as long as the natural con®guration is ®xed and no microstructural changes takeplace, the response of the material is akin to that of a simple material. Thus Noll's rule(eqn (14)) is applicable between any two response pairs that are ``equivalent'' in the senseof de®nition 1. This is because the equivalent representations denote the same elasticresponse.

A further simpli®cation of the symmetry groups of the inelastic solid may be e�ected ifone considers a canonical representation of the stress response as de®ned in theorem (1) ofthe previous section. Since the response functions for each response pair in any givensimilar set is determined purely by its pivot member, the symmetry group of the individualresponse pair is also determined by that of the pivot member. Therefore if G�p is thesymmetry group of the pivot, all members of the similar set also possess the same sym-metry group. Hence in a canonical representation, there are only as many di�erentsymmetry groups as there are similar sets.

With this background, we once more consider the stress±strain response of a metallicsingle crystal whose crystalline lattice is cubic (for example) in its initial state as shown inFig. 3(A). Thus, the symmetry group for the elastic response with respect to its initialnatural con®guration is the cubic group. When the crystal is subject to a su�ciently largedeformation it undergoes an inelastic process. Experimental investigations have beencarried out since the early 1920's on a variety of single crystals (see e.g. Taylor and Elam,1925) and it is now well accepted that the lattice structure is unchanged during theinelastic process. In other words, the elastic response of the crystal after permanentdeformation is similar to that before the deformation. For such materials there is only onesimilar set in the canonical representation and all the response pairs are similar to oneanother; hence, there is only one symmetry group that is unaltered by the permanentdeformation. For other microstructural changes such as martensitic transformations, thesymmetry group changes since the lattice structure changes.

Notice that the de®nition of the symmetry group of the elastic response function allowsone to specify the symmetry group of each response function independently. If one were tomodel this inelastic single crystal by means of a theory of simple materials with the usualnotion of symmetry, Noll's rule would make it impossible for a material which has a cubicsymmetry in its reference con®guration to retain its cubic symmetry group with respect to anew con®guration that is obtained (say) as a consequence of a simple shear. A clear exampleof the symmetry group which results as a result of an uniaxial extension of any simplematerial which is isotropic in its reference con®guration is given by Wineman et al. (1988)and demonstrates that the new symmetry group contains certain non-orthogonal elements.

Thus, in the context of the present approach, wherein we consider a set of ordered pairsof elastic response functions, a clear de®nition of the symmetry group associated with theelastic response emerges naturally.


14On the other hand, if we restrict the domain of de®nition of T�p , then the de®nition of eqn (15) still holdsprovided that we stipulate that the equality in eqn (15) holds for all F�p in the domain of de®nition of T�p forwhich F�pH also belongs to the domain. Such is the situation, for example, when we restrict attention to rateindependent plasticity wherein we cannot reach states outside the elastic domain.

VI. COMPARISON WITH TRADITIONAL APPROACHES TO PLASTICITY

Traditional theories of plasticity seek to model the permanent deformation of thematerial by directly introducing the notion of ``plastic strain''. For example, Eckart(1948), KroÈ ner (1960, p. 286, eqn 4), Backman (1964), Lee (1969) and others introduce astress-free intermediate con®guration that is stipulated to di�er from the current con®g-uration by a pure stretch Ve, (see e.g. Lee (1969, p. 3), lines following eqn (20)) called the``elastic stretch'' of the material. The constitutive equation for the stress is then of theform

T � T�Ve�: �16�

Apart from questions regarding the existence of such con®gurations, it is immediatelyobvious that the form of eqn (16) is highly restrictive in the sense that the functional formof the material is the same for all such stress-free con®gurations, (thus being somewhatsimilar to the canonical representation introduced in the current work). Furthermore, dueto the tacit assumption that the function T in eqn (16) depends only upon the con®gura-tion kr and not upon the current stress-free con®guration, it can be shown that T has to bean isotropic function of Ve.

15 The above approach can be shown to be a special case of ourapproach if one restricts attention to materials whose elastic response is that of an iso-tropic solid and whose response pairs (as de®ned in eqn (9)) are similar to one-another (inthe sense of de®nition (3)). A canonical representation for such materials then results in aconstitutive equation of the form of eqn (16).

In contrast to this procedure, Casey and Naghdi (1992) postulate a response function ofthe form

T � T�F�r ;Ep�; �17�

where Ep is ®rst introduced as a primitive variable. Con®gurations corresponding to kp arededuced in terms of Ep by using the notion of ``maximal unloading'' (see Casey andNaghdi, 1992). It can easily be shown that the approach adopted by Casey and Naghdi(1992) reduces to the one discussed in the previous paragraph, whenever the material canbe unloaded to a stress-free con®guration and hence extends the usual notion of stress-freecon®gurations to a broader class.

The proposal of Casey and Naghdi (1992) does not provide a universal and uniquede®nition for the ``plastic strain''. This is because, as noted by Casey and Naghdi (1992),the notion of maximal unloading is not unique in the sense that there could be variousmeasures of ``maximality'' used to de®ne the maximally unloaded state, each giving rise toa di�erent con®guration kp and a di�erent de®nition of the ``plastic strain'' dependingupon the choice of the norm. It is thus possible to prescribe di�erent ``maximal'' criteriafor di�erent materials, so that the issue now becomes one of deciding which norm to use.

Other entirely di�erent de®nitions are possible. For example, we could use a strain thatcorresponds to the minimum strain energy amongst all the allowable strains in the elasticdomain (i.e. the strain corresponding to ``the most stable state'' within the elastic domain)or an appropriately de®ned ``centroidal strain'' corresponding to the centroid of the elastic


15See Casey and Naghdi (1980) for a discussion of these issues.

domain etc. In fact a de®nition can be chosen that corresponds to any point in the elasticdomain. It is to this ``¯exiblity'' that we referred when we addressed the issue in SectionIV.1.1. Furthermore, the maximally unloaded state may not be the physically most ``nat-ural state'' for these materials (e.g. the case of multinetwork theories where the currentstate is the most natural one for the new networks, since the new networks are unloaded inthe current con®guration).

In the eqn (16) and (17), it is unclear whether the function T depends upon one con-®guration or many con®gurations. Because of this, there has been considerable debateregarding the invariance properties of Ep (see the extensive discussion of this issue byNaghdi (1990, section 4C, p. 128)). Moreover, it has been recognized (see Casey andNaghdi, 1992) that the constitutive eqn (17) cannot be used to model certain materialswhose material symmetry changes independently of the strain in the material. We shallillustrate this with the example of a single crystal undergoing crystallographic slip. InFig. 3, the con®gurations (B) and (C) corresponding to two unloaded states of a crystal,have the same strain but di�erent lattice structures. The ``plastic strains'' associated withboth the con®gurations are the same and, in view of eqn (17), so is their subsequent elasticresponse as calculated by eqn (6). But the lattice structures corresponding to (B) and (C)are clearly di�erent so that their actual response functions must be di�erent. Casey andNaghdi (1992) postulate an additional variable to explicitly capture this change in thelattice orientation, somewhat akin to a proposal made by Mandel (1973). They thenchoose a con®guration that is crystallographically meaningful. But, in order to bring thisin line with the maximally unloaded states, they require that this con®guration be (a)stress free and (b) within the elastic domain at all times. The resulting theory is subject tomany of the same limitations that apply to the approach of Lee (1969).

The above comments are particularly relevant in view of our intention to develop atheory for inelastic behavior of solids that can also model phase transitions in solids.Experiments suggest that in many technologically signi®cant situations (such as thosefound in shape memory alloys), the new phases disappear upon unloading and the mate-rial returns to its original con®guration, although during the process, the internal struc-ture of the material is altered. The resulting load±deformation curve is similar to the curveABIJK in Fig. 2 (see e.g. Nishiyama, 1978). In such cases, the notion of an intermediatestress-free con®guration may be inappropriate. Moreover, symmetry changes play animportant role in the process of phase transition. It thus becomes essential in these situa-tions to be able to quantify the notion of changing material symmetry.

A limitation of a di�erent sort occurs when we consider permanent shape change inpolymeric solids. Here, during the loading process, a part of the network of polymersbreaks and reforms into a new undistorted network (see, e.g. Tobolsky and Andrews,1945; Peterlin, 1976; Fong and Zapas, 1976). For these cases, the physically relevantcon®guration is the current con®guration, since it is this con®guration at which the newnetwork is formed. Thus, although a stress-free intermediate state may exist, it is not thestate of interest when dealing with the inelastic response of such polymeric materials. Inthis case, no explicit parametrization of the families of response functions is needed.Instead, the appropriate member of the family is directly invoked when the conditions areappropriate.

The approach presented here allows for the use of di�erent prescriptions for the naturalcon®gurations under di�erent conditions since they are not derived quantities. Thus, thecurrent approach allows for the use of stress-free con®gurations when possible and


convenient, and other crystallographically signi®cant ones if necessary. Moreover, thenotion of evolving material symmetry is rendered meaningful within the context of thede®nition of material symmetry that is provided in Section 6. Finally, the currentapproach is capable of modeling discrete as well as continuous changes in the micro-structure by allowing for the possibility of ®nite, countable as well as uncountable numberof elastic response functions.

VII. CONCLUSION

The de®nition of the canonical representation of the family of stress responses of thematerial and considerations of changing material symmetry are the central results of PartI of this paper and set the stage for the development of the constitutive equations for theinelastic response of the material to be discussed in Part II that follows.

Having discussed the capabilities of the current approach in modeling the elasticresponse functions of a wide range of materials, we now turn our attention to a discussionof the evolution of the natural con®gurations of the material during a particular process.In what follows, we shall consider the case of a material that possesses an uncountablein®nity of ordered pairs of elastic response functions, a typical example of such a materialbeing a metallic single crystal, or a polycrystalline material.

AcknowledgementsÐThe authors thank National Science Foundation for its support of this work.

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