MECHANICS OF THE INELASTIC BEHAVIOR OF MATERIALSÐ PART 1, THEORETICAL UNDERPINNINGS

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1 Pergamon International Journal of Plasticity, Vol. 14, Nos 10±11, pp. 945±967, 1998 # 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved /98/$Ðsee front matter PII: S (98) MECHANICS OF THE INELASTIC BEHAVIOR OF MATERIALSÐ PART 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 behavior of inelastic materials from a continuum viewpoint, taking into account changes in the elastic response 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 the elastic response of these materials. In particular, we discuss in detail the far reaching e ects of the notion of materials with families of elastic response functions with corresponding natural con gurations that was introduced by Wineman and Rajagopal (1990, Archives of Mechanics, 42, 53±75) and Rajagopal and Wineman (1992, Int. J. Plasticity 8, 385±395) for the study of the inelastic behavior of polymeric materials and later generalized and extended to the study of deformation twinning of polycrystals 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 discuss 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 any container, whereas solids retain a de nite shape and size irrespective of the nature of the container that holds them 1. Many solids, ranging from crystalline ones like steel or aluminum all the way to polymers like rubber, can be induced to permanently alter their shape by the application of su ciently large forces. The entire metal and polymer forming industry exploits this fact. This is the rst of a two-part paper that is concerned with the modeling of the mechanical behavior of such materials from a continuum viewpoint taking into account the changes in the microstructure of the material as it deforms. This provides a uni ed framework for modeling a wide range of material behavior. No speci c constitutive functions are advocated here, though some are invoked in our illustrative examples for the *Corresponding author. Fax: ; krajagopal@mengr.tamu.edu 1 See Rajagopal (1995) for a fundamental re-examination of the notion of solids, uids and gases from a continuum viewpoint especially in the light of the manufacture of a variety of man-made materials that seemingly defy conventional de nitions. 945

2 946 K. R. Rajagopal and A. R. Srinivasa sake of clarity; instead only those features that are common to a wide variety of microstructural changes are considered. I.1. Previous work The current work has much in common with metal plasticity (although it is meant to be applied to a larger class of materials) in the sense that its aim is to model hysteretic behavior that occurs even in the limit of quasistatic processes. Thus, in order to present the background in which the current theory is set, we shall brie y recount the salient developments in the theory of plasticity. Tresca (1867) was the rst to observe that, under su ciently high pressures, solids are capable of `` owing''. Saint-Venant (1870), Le vy (1870) and later von Mises (1913) developed the governing equations for the motion of ``rigid-plastic'' materials 2 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 deformations) 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 are typical of metal forming but is quite unsatisfactory when one is interested in modeling recovery processes such as ``spring-back'' that occur in many sheet metal operations. The neglect of the elasticity of the material also has unexpected consequences in certain situations where the stress in the rigid regime cannot be uniquely deduced from the boundary conditions. In such cases one may be unable to determine uniquely the conditions under which yielding occurs. Prandtl (1924) and Reuss (1939) accounted explicitly for the elasticity of the material prior to yield by modifying the Levy±Mises model, within in nitesimal deformations, and distinguishing between temporary recoverable deformations of the material (termed the ``elastic strain'') and the permanent deformations (termed the ``plastic strain''). The (linearized) total strain was then assumed to be the sum of the elastic and plastic strains. The resulting theory, termed the ``Prandtl±Reuss theory'' has been widely utilized with minor modi 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 rst one to theorize that, unlike elastic bodies, plastic materials such as copper, steel etc. possess multiple stress-free (or natural) shapes and that these shapes play a fundamental role in the mechanical behavior of the material, evolving gradually with the deformation. He considered the geometry of the ``unloaded'' (stress-free) shape of the material and de ned the ``elastic strain'' through the metric tensor associated with the unloaded state with respect to the current state. He did not propose any de nite equation to describe the yielding 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 2 See e.g. Hill (1950) and more recently Casey (1986) and Naghdi and Srinivasa (1994a) in this regard.

3 Mechanics of the inelastic behavior of materialsðpart 1, theoretical underpinnings 947 plastic strain of the material. An alternative approach has been advocated by Green and Naghdi (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), introduced 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 which it is to be identi ed. For example, there has been considerable debate regarding the applicability of the notion of the stress-free strain, as de ned by Lee (1969), to materials that cannot be unloaded to a stress-free state. 3 On the other hand, the ``plastic strain'' is a primitive 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 prescribe the ``plastic strain'' of the material. This notion depends upon the norm used for the stress 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 hysteretic behavior of crystalline metals and as such are not directly extendable to other kinds of inelastic behavior. I.2. The current approach: a family of elastic response functions and multiple natural con gurations The main aim of the present paper is to identify and exploit features and principles that are common to a wide range of inelastic phenomena, including metal plasticity, polymer inelasticity, twinning, phase transitions etc. (some of which require microstructural considerations). Although these phenomena have di erent physical origins, their manifestations 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 I of the paper, deferring the study of the dissipative behavior to Part II. As summarized in Section I.1., traditional theories of plasticity rst introduced the notion of ``plastic strain''. This then served as the foundation for subsequent developments. Thus, the stress is given by an expression of the form T ˆ ^T E; ; 1 where E is a measure of strain from some con guration and a is a scalar, vector or tensorial variable whose change is caused by the inelasticity of the material. Irrespective of the 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 the variable a This simple observation has far reaching consequences, many of which are entirely independent of the nature of the parametrization. Thus, the central tenet of the current approach is that the material possesses not one, but a family of elastic response functionsð 3 See also the discussion by Casey and Naghdi (1980) as well as that by Lee (1996) and Lee and Liu (1967) in this regard.

4 948 K. R. Rajagopal and A. R. Srinivasa each with its own natural con guration (see Section III). While this idea is not newðit has been used implictly or stated explicitly by many authors e.g. Eckart (1948), Green and Naghdi (1973), Rajagopal and Wineman (1980), Casey and Naghdi (1992)Ðwe systematically 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 inelastic behavior, 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 take center 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 no means the only) way to prescribe natural con gurations for a class of materials possessing instantaneous elasticity (see Section IV.1.1) is by an instantaneous 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 that other 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 leave the lattice structure unaltered, from those such as martensitic transformations that alter the 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 ``canonical representation'' (see Theorem (1) in Section IV.1.3) in which all the elastic responses with the same microstructure have the same functional form for the stress, with their respective natural con gurations being unique to them. We rst develop the basic ideas within the setting of homogeneous motions so as to highlight 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 inhomogeneous motions as described in Section IV.2. Finally, classical notions of material symmetry for simple materials (Noll, 1957) 5 are extended to materials possessing multiple natural con gurations. We then demonstrate (see Section V) that if we use the ``canonical representation'', then the material symmetry group of the elastic responses of the material will be the same if all the elastic responses re 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 to re 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 original de nition of simple materials (Noll, 1957), has extended the de nition to explicitly incorporate 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 a certain number of ``state'' variables. He then goes on to de ne material symmetry at constant ``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 for the evolution of the ``state'' variables of the material. Our approach is more speci cally oriented towards inelasticity of solids, and while our approach can be recast into the general format proposed by Noll (1972), the restriction to inelasticity of solids highlights 4 The 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. 5 The 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)

5 Mechanics of the inelastic behavior of materialsðpart 1, theoretical underpinnings 949 the central assumptions and results of the theory and provides some insights that might otherwise be obscured in a more general framework. Section VI of the paper compares this theory with traditional plasticity and demonstrates how the current approach overcomes several shortcomings inherent in the other approaches. Section VII provides a summary of the advantages of the current approach and 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 elastic response functions'') that uni es a wide range of disparate inelastic phenomena. This framework provides a sound basis for the discussion of permanent shape changes that 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 using the 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 of which 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 accompanying changes of microstructure by constructing ``similar sets''. Members of each similar set share the same elastic response function (and hence the same microstructure) but possess di erent natural states. This naturally leads to the development of special (or canonical) representations of elastic response (see Section IV.1.3). For example, the current approach allows us to model crystallographic slip of single crystals without the 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 materials that possess multiple natural con gurations and families of elastic response functions. This feature allows for the modeling of materials that change their symmetry group as a result of microstructural change (e.g. cubic to tetragonal transformations in some materials). Using the notion of ``similar sets'', we also prove that the symmetry group associated with the elastic response remains unaltered as long as the microstructure 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 position in Euclidean space occupied by the body. One of the fundamental concepts that we need to analyze the properties of material bodies, is the intuitive notion of the ``nearness'' of two con gurations of a given body. We consider a con guration k 1 (B) to be in the E neighborhood of another con- guration k(b) whenever the distance between the positions X 0 :=k 0 (X) and X 1 :=k 1 (X) of every material particle X in B is less than E, i.e. if jx 0 X 1 j <: 2

6 950 K. R. Rajagopal and A. R. Srinivasa The above de nition of ``nearness'' of two con gurations will allow us to talk about various aspects of the topology of the con gurations such as open and closed sets of con gurations, etc. 6 Let us assume that initially the material body, with a speci c microstructural pattern occupies a con guration k 0 (B). 7 For simplicity, we shall assume that the body is initially homogeneous in the sense that the microstructure is the same around every particle in the body. 8 When it is deformed, suppose that its initial response is that of an elastic solid; but once its deformed (or current) con guration is su ciently ``far'' from its initial con guration, suppose that certain microstructural changes take place in the body. For example, crystalline metallic materials may undergo crystallographic slip wherein the lattice planes within the material slip or slide over one another like a deck of cards, or they may undergo twinning or martensitic transformations wherein there is a rearrangement of the lattice structure within each grain (see Nishiyama, 1978; Holt et al., 1994, etc.). Polymeric materials, on the other hand undergo network scission and re-formation wherein the arrangement of the nodes of the long chain polymer network are altered (see e.g. Fong and Zapas, 1976; Peterlin, 1976). In spite of the disparate ways in which these structural changes occur, a striking macroscopic 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. A schematic representation of the one-dimensional response of such materials upon the slow application 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 long as the displacement gradient is less than the value corresponding to B, the response is elastic. From B to C, the response is inelastic and dissipative and the material goes to the con guration corresponding to point D when the load is removedðthe material has suffered a ``permanent shape change''. It should be noted that the con guration that the body goes to upon removal of the load may depend upon the rapidity with which the load is removed. III. ELASTIC MATERIALS At the outset, it is instructive to consider the response of a purely elastic material and recall some fundamental facts regarding their behavior (for a detailed account of this, see Truesdell and Noll (1992)). This will help us to understand the nature of the changes described at the end of the previous section and represented schematically in Fig. 1, and help identify the fundamental variables which characterize the behavior of materials such as the ones discussed above. The purely mechanical behavior of elastic and elastic±plastic materials is intimately associated with the response to cycles of deformation. Thus, for our purposes, it is convenient to 6 Of course, other de nitions of ``nearness'' of con gurations can be used, but the de nition used here is quite intuitive and su cient for our purposes. With the above description of nearness of con gurations, two con gurations that di er by a translation and rotation will be considered far from each other if the translation or rotation is su ciently large. Indeed, we can eliminate this by rst removing the translation and rotation of an arbitrary particle and then carrying out the comparison. Of course, this creates needless complications and the current simple de nition will su ce for our purpose. 7 Henceforth we shall suppress B in the notation for a con guration and simply write k(b) ask. 8 The notion of a homogeneous body is sometimes introduced through the requirement that the stress response be the same at every material point in the body. We shall touch upon this in Section IV.2.

7 Mechanics of the inelastic behavior of materialsðpart 1, theoretical underpinnings 951 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. AB 0, DC 0 and EF 0 represent the instantaneous elastic response of materials that possess instantaneous elasticity. For such materials the responses AB 0, DC 0 and EF 0 can be achieved in the limit of very ``rapid'' processes. start with the notion of con gurational paths and cycles. We de ne a con gurational path of a 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 point belonging to it, there exists a corresponding path in the space of stresses. This stress path is uniquely determined by the con gurational path of the body and is independent of the rate at which the con gurational path is traversed. Moreover, if the con gurational path begins and ends at the same con guration (a con gurational ``cycle'') then so does the stress path. Said di erently, a con gurational cycle corresponds to a stress cycle. It is the response to con gurational cycles that is used to de ne the class of elastic materials as a subset of the class of hypoelastic materials. Indeed, a hypoelastic material for which every con gurational cycle corresponds to a stress cycle is an elastic material. We shall only consider hyperelastic materials, that is, materials for which the work done vanishes in any closed con gurational cycle in which the velocity eld is the same at the start and the end of the cycle. In order to mathematically represent the stress response of elastic materials we introduce the notion of a reference con guration k r, 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 to the position in k r of a typical material particle X as X, then its position x in any other con guration can be represented by x ˆ r X; t ; 3 9 The notion of ``nearness'' discussed in eqn (2) allows us to talk of continuous dependence on the parameter.

8 952 K. R. Rajagopal and A. R. Srinivasa where the subscript k r indicates that the form of the function r depends upon the choice of the reference con guration. In view of the properties of elastic materials described in the previous paragraph, it is easy to show that once a con guration k r is chosen, the stress depends only upon the deformation r from k r. In particular, for the class of ``simple elastic materials'' (see Noll, 1957), the Cauchy stress tensor T at X depends upon r only through the deformation gradient F r at X and is of the form T ˆ ^T r F r ; 4 where F r : 5 In the constitutive eqn (4) the subscript k r indicates that the form of the function ^T r depends 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, each representation depending upon the reference con guration chosen. For instance, if a different reference con guration is used to represent the constitutive equation for the same body, then the constitutive equation ^T is related to ^T r, by the relationship T ˆ ^T r F ˆ^T FP 1 ; 6 for all F in the domain of de nition of ^T r and where P is the gradient of the mapping from k r to and the function ^T is di erent from ^T r. The relationship (6) is a result of the fact that the value of the stress is determined solely by the current con guration occupied by the body and is independent of the choice of the reference con guration. We shall refer to the ordered pair formed by a given reference con guration k and its associated stress response function ^T as a ``response pair'' and denote it by ; ^T. The above considerations leads us to the following De nition 1. Equivalent representations of the stress response: Two stress response pairs 1 ; ^T 1 and 2 ; ~T 2 for a given body are said to be equivalent if they satisfy eqn (6) with P being the gradient of the mapping from k 1 to k 2. Indeed, for elastic bodies, all the representations are equivalent. As we shall see later this de nition will also play an important role in the discussions of the stress response of inelastic 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 and Noll (1992) for a full discussion of the issue. The central result of the concept of frameindi erence is that the constitutive equation for the stress in an elastic material is given by the form T ˆ R r T R T r ˆ R r ~ Tr E r R T r ; 7

9 Mechanics of the inelastic behavior of materialsðpart 1, theoretical underpinnings 953 where E r, is the Green±St. Venant strain tensor de ned by E r ˆ 1=2 F T r F r I, R r is the rotation tensor obtained in the polar decomposition of F r, T* is a tensorial quantity that is referred to as the rotated stress tensor and the notation (.) T denotes the operation of transposing the tensor. It is very important to realize that the form of the function T ~ r is in general dependent on both the shape and orientation of the reference con guration k r. 10 We shall get back to this 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 a position to interpret the response of materials that permanently alter their shape. In order to make the ideas that follow as clear as possible, we shall develop them within the context of homogeneous motions of homogeneous bodies, extending them in Section IV.2 to inhomogeneous motions. IV.1. Response to homogeneous deformations We consider a material which is initially in a con guration k 0 (in which it is homogeneous) corresponding to the point A in Fig. 1. By the initial elastic domain of this material we mean a path connected set of con gurations over which the response is that of an elastic material. In other words, corresponding to any con gurational path within this domain, there exists a unique stress path (independent of the rate at which the path is traveled) such that every con gurational cycle corresponds to a stress cycle. Clearly k 0 is an element of this elastic domain. It should be observed that we have introduced no notion of a reference con guration and no kinematical quantities for inelastic materials as yet, choosing to work directly with the set of con gurations as a whole instead. The stress response of the material corresponding to the initial elastic domain will be referred to as the ``initial elastic range'' of the material. Now, in order to mathematically represent the response of the material within the initial elastic domain, we need to follow the procedure outlined for elastic materials in the previous section and choose a convenient reference con guration and write the response of the material in the form given in eqn (7). After this has been done we now consider the consequences of deforming it beyond the elastic domain. Once the material is homogeneously deformed so that its con guration lies outside its initial elastic domain, certain microstructural changes take place. The body now displays a di erent elastic responseðthe material has a new elastic domain and range. In order to represent this new response mathematically, we repeat the procedure used for the initial elastic 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 be of the form T ˆ T 1 F 1 : 8 10 The 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.

10 954 K. R. Rajagopal and A. R. Srinivasa We note that since the deformation is homogeneous, the con guration k 1 can be chosen such 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, we need to (i) identify the new elastic domain and range (ii) choose an appropriate homogeneous reference con guration for the mathematical representation of the response function 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 these materials, we rst introduce the concept of Natural con gurations. These are the reference con gurations chosen to represent the elastic response functions of the materials and are the primary con gurations of interest even in our study of inelastic behavior. The use of multiple reference con gurations for the representation of the elastic response functions of the material is crucial for the de nition of evolving material symmetry, as will become clear in the subsequent sections. IV.1.1. Instantaneous elasticity and the natural con gurations The question naturally arises as to what might be the nature of the natural con gurations of the body and how one might identify them. Philosophically, all that is required is that it be a con guration in which the stress is known. In view of the wide range of inelastic phenomena that we wish to model, rather than providing a single universal de nition for the natural con gurations, we shall content ourselves with providing several examples. Speci c prescriptions will depend upon the particular inelastic phenomenon in question. Incidentally, a similar sentiment is expressed by Naghdi (1990) regarding the identi cation of the ``plastic strain''. We shall consider three possible cases: 1. Within the class of homogeneous elastic bodies, it is easy to see that any one of the stress-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 it may seem impossible to unload some materials elastically to a state of zero stress, since the 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 is similar to the curve ABEFG in Fig. 1. The ability to elastically unload a body to a stressfree 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 quasistatic response of many metals seem to indicate that, under slow loading conditions, the stress-free state lies outside the elastic domain. The above considerations do not include the possibility that the response of such materials may depend upon the speed at which the experiment is performed so that, when the experiment is performed su ciently rapidly, the response may proceed along EF 0 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 work is not transferred as thermal energy. Outside the elastic domain, most processes are dissipative 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 are special classes of processesðextremely rapid onesðthat are non-dissipative. To be more precise, given a xed con gurational path that is outside the current elastic domain, we

11 Mechanics of the inelastic behavior of materialsðpart 1, theoretical underpinnings 955 consider a sequence of processes, each traversing the path at a faster rate than the other. The limit of such a sequence corresponds to traversing the path instantaneously. Such limiting processes are called ``rapid processes'' and materials for which such processes are nondissipative are said to possess instantaneous elasticity. Thus, we consider bodies that can be instantaneously brought to a state of zero stress in a non-dissipative manner, i.e. whose instantaneous elastic domain includes states of zero stress. For such materials these stress-free con gurations become candidates for the natural con gurations. Of course the resulting stress-free state may be eeting and the microstructure may change if given su cient time, but all we need here is the ability to elastically 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 material expands or contracts with di erent rates of deformation. Indeed, as we shall see in Section IV.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 a stress-free con guration. In this case, any other con guration with a known stress, such as a 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 (microstructural) considerations. Such is the case for example in martensitic transformations where the states of the material at the Austenite start temperature and the Martensite start temperatures may be chosen. We must also point out that many other prescriptions, similar to the maximally unloaded con guration, are possible, all of which give rise to equivalent representations. This will become abundantly clear in Section III of Part II where we show that a di erent (and possibly more convenient or ``natural'' set of natural con gurations) can be de ned a posteriori for a certain class of elastic±plastic materials. To drive home this point, Rajagopal and Srinivasa (1995b) have provided an illustrative example wherein they have recast the classical Prandtl±Reuss constitutive equations into an equivalent form in terms of 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 functions Once the natural con gurations are chosen, the stress response of these materials may be represented as a set of ordered pairs n p ; ^T o p jp 2P 9 of natural con gurations k p and the elastic response functions ^T p associated with them. The set P represents the index set of all the possible natural con gurations for the material. Care must be taken in discussing issues of frame indi erence for each of the ordered pairs introduced in eqn (9). If we consider a motion which di ers from the original motion given by eqn (3) by a time dependent translation and rotation, frame indi erence asserts that the value of the stress tensor for the second motion di ers from that for the rst only in orientation. It should be noted that while the current con guration changes its orientation, the set of natural con gurations are the same for both the motions. This then implies that the form of the response functions can be reduced to eqn (7) with k r replaced by k p. A detailed analysis of this issue has been presented by Rajagopal and Srinivasa

12 956 K. R. Rajagopal and A. R. Srinivasa (1995b) and the reader is referred to Section III of that paper. This is one of the fundamental concepts associated with materials with multiple con gurations and forms the basis of the theory proposed by Rajagopal (1995). The assumption of eqn (9) indicates that the stress response of inelastic materials considered here is akin to that of not one but an entire family of elastic materials. The above idea of a family of elastic response functions is in line with the notion that, for metals at least, the elastic response function, its elastic domain and range, as well as its natural con guration are indicators of the nature of the interatomic forces. Indeed, it is the interatomic forces that govern the crystal structure and lattice arrangement and determine many 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 above is that of elastic solids wherein there is only one ordered pairða single natural con guration and the stress response function associated with it. For the case of inelastic materials, to make things more ``concrete'', consider deformation-induced twinning, where the material changes its crystal structure to a ``twin'' when it is su ciently deformed. This twinned structure possesses a di erent elastic response than the original material in the sense that its stress-free con guration di ers from that of the original by a shear deformation (the so-called twinning shear). 11 Moreover, its new material 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 (on a very ne scale) of the twinned and untwinned structures. Such a material has been modeled by Rajagopal and Srinivasa (1995a), (1995b) by associating two elastic responses and two natural con gurations for the material, that is, two ordered pairs, one for the untwinned material and another for the twinned material as well as an additional variable representing the volume fraction of the twinned material. In the case of metal plasticity, there are an uncountably in nite number of natural con gurations that are usually assumed to have the same volume. On the other hand, for the case of the multinetwork theories of polymers, Wineman and Rajagopal (1990) and Rajagopal and Wineman (1992) utilized a material with multiple elastic responses, with the current con guration of the material being chosen as the natural con guration for the new networks as they form. Thus, while the concept of a natural con guration is common for 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 are equivalent in the sense of elastic materials as de ned by de nition 1. However, we can extend the notion of equivalent representations to two families of response functions for a given inelastic material by means of the following. De nition 2. Equivalent representations of the elastic response of inelastic materials: Two families of ordered pairs p ; ^T p ; p 2P and q ; ~T q ; q 2Q are said to form equivalent representations if they can be put in one to one correspondence in such a way that corresponding 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 a posteriori by using a di erent set of natural con gurations. 11 Recall that we are discussing only homogeneous deformations here.

13 Mechanics of the inelastic behavior of materialsðpart 1, theoretical underpinnings 957 IV.1.3. Similar response pairs and canonical representations of the response functions For inelastic materials, a related and useful de nition to that of ``equivalent representations'' is the following De nition 3. Similar stress response functions: Two stress response pairs 0 ; ^T 0 and ; ~ T, from among a family of response functions for a given inelastic body (see eqn (9)) are said to be similar, if ^T 0 F ˆ~T FP ; 10 for all invertible F in the domain of ^T 0 and for some invertible tensor P. The tensor P will be referred 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 ne equivalent 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 gradient of the mapping between the con gurations and k 0. Indeed, since ^T p and ~T belong to the same family, they are non-equivalent and hence do not satisfy eqn (6). However, some members of a given family may be similar to one another. For example, in the case of crystallographic slip of single crystals, all the members of a given family of stress response functions are similar to one another. It should be observed that if the similarity transformation P is the identity tensor, then the functional form of the two response functions ^T p and T ~ are identical. A schematic representation of the notion of similar and dissimilar response pairs is shown in Fig. 2. It can easily be shown that the notion of ``similarity'' is an equivalence relation among the members of a family of response functions. Thus, given a family of stress response functions, we can group their members into sets using similarity. We shall refer to such sets as ``similar sets''. Within each set, we can further simplify the representation by means of a special representation which we shall refer to as a ``canonical representation''. Theorem 1. Canonical representation of the stress response. Given a family of response pairs for an inelastic solid, an equivalent family (in the sense de ned in de nition (2)) can be constructed such that for each similar set, the form of the response functions is the same for every member of that set. Such a representation is called a canonical representation. In other words, when the canonical representation is used, each similar set has a unique stressresponse function associated with it. Proof. We prove the above theorem by constructing the required canonical representation 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 member say, p ; ^T p which we shall refer to as the ``pivot'' member. Any other member in that similar set, say ; ~T is related to p ; ^T p as described in eqn (10). We now map the con guration to a new con guration k 1 by means of a homogeneous deformation with gradient 12 P. If we now de ne a new function ^T 1 by ^T 1 F :ˆ ~T FP 1 ; It should be observed that k 1 cannot be identical to k p since no two members of a given family are equivalent.

14 958 K. R. Rajagopal and A. R. Srinivasa then a routine calculation reveals that ; ~T and 1 ; ^T 1 are equivalent in the sense of de nition (1). Moreover, in virtue of de nition (3), it is easy to see that this equivalent representation is ``similar'' to p ; ^T p, with the ``similarity transformation'' being the identity tensor. Thus, the functional forms of ^T p and ^T 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 dissimilar response functions (each shown with a di erent slope).

15 Mechanics of the inelastic behavior of materialsðpart 1, theoretical underpinnings 959 Using the above procedure, we can construct equivalent representations for each member of a given similar set such that the similarity transformation between any two of the newly constructed representations is the identity tensor, so that, in the newly constructed representation, all the members have the same functional form for the response function. These new representations, together with the ``pivot'' member form an equivalent 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 function but di erent natural con gurations. Thus, repeating the above two steps for each similar set 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 that the elastic response is ``unchanged'' (i.e. that the elastic constants are unaltered) in metal plasticity (see Fig. 2). The form of the response functions in the canonical representation then depend only upon the ``pivot'' member chosen, and di erent (equivalent) canonical representations can be constructed by using di erent pivot members. For elastic materials, since there is only one member 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 natural con gurations k p 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 members of the family of stress response functions are similar to one-another and hence a single pivot member su ces to construct a canonical representation of the stress response. If we choose initial con guration A in Fig. 3 as our pivot member, a quick look at con gurations B and C reveals that the associated response functions are similar but non-equivalent and that, the ``similarity transformation'' between B and C is an orthogonal transformation. Consequently, if one constructs a canonical representation for the stress response of this material, the natural con gurations corresponding to B and C will not di 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 elastic response 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 much a ect the elastic response but play a signi cant role in determining the inelastic response of the material. Thus, one may consider the members of a similar set to represent all the possible macroscopic shapes that the material is capable of assuming without changing its microstructural arrangement. Hence, for crystallographic slip of single crystals, all the members 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-network theories of polymers as well as deformation induced twinning. Thus one could consider each similar set as representing a single microstructural arrangement in the material. On the other hand, when the microstructural state is fundamentally altered (as in the case of martensitic transformations, where the lattice may change from a cubic to tetragonal arrangement, and partially twinned states during deformation induced twinning where there is a juxtaposition of the twinned and untwinned states), the resulting response functions are no longer similar to each other and hence do not belong to the same similar set. 13 This is not strictly true because the dislocation structures do change even in a single crystal.

16 960 K. R. Rajagopal and A. R. Srinivasa 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 guration. B and C correspond to two con gurations with the same strain but with di erent lattice orientations. IV.1.5. The de nition of the ``natural deformation gradient'' tensor G and its role in determining the natural con gurations Once a canonical representation of the stress response has been constructed, it is clear that, within a given similar set, each member is uniquely identi ed by its associated natural con guration, while the stress response is uniquely identi ed by the response of the pivot member. Indeed, in most cases, the pivot members may be chosen in such a way that the natural con guration of a given member uniquely identi es it in the entire family. Such is the situation for both martensitic transformations, 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 erent variants of martensite belong to a ``similar set'' and hence can be di erentiated by their respective natural con gurations. As seen in eqn (8), the role of the natural con guration is implicit in the sense that the constitutive equation for the stress response depends upon the current natural con guration. In order to make this dependence explicit, we introduce some xed con guration k r (say, the initial con guration or one of the pivots in a canonical representation). Let G be the gradient of the mapping from k r to k p. A knowledge of the eld G then su ces to identify the current natural con guration once the con guration k r is known. Then, we can rewrite eqn (9) in the form T ˆ ^T G F p :ˆ ^T F p ; G ; 12 where the functional form of ^T depends upon the choice of all the natural con gurations associated with the pivot members of a canonical representation. The above form of the