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

MECHANICS OF THE INELASTIC BEHAVIOR OF MATERIALS.PART II: INELASTIC RESPONSE

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 28 April 1998)

AbstractÐThis is the second of a two part paper dealing with the inelastic response of materials.Part I (see Rajagopal and Srinivasa (1998) International Journal of Plasticity 14, 945±967) dealtwith the structure of the constitutive equations for the elastic response of a material with multiplenatural con®gurations. We now focus attention on the evolution of the natural con®gurations. Weintroduce two functions-the Helmholtz potential and the rate of dissipation functionÐrepresentingthe rate of conversion of mechanical work into heat. Motivated by, and generalizing the work ofZiegler (1963) in Progress in Solid Mechanics, Vol. 4, North-Holland, Amsterdam/New York;(1983) An Introduction to Thermodynamics, North-Holland, Amsterdam/New York) we thenassume that the evolution of the natural con®gurations occurs in such a way that the rate of dis-sipation is maximized. This maximization is subject to the constraint that the rate of dissipation isequal to the di�erence between the rate of mechanical working and the rate of increase of theHelmholtz potential per unit volume. This then allows us to derive the constitutive equations forthe stress response and the evolution of the natural con®gurations from these two scalar functions.Of course, the maximum rate of dissipation criterion that is stated here is only an assumption thatholds for a certain class of materials under consideration. Our quest is to see whether such anassumption gives reasonable results. In the process, we hope to gain insight into the nature ofsuch materials. We demonstrate that the resulting constitutive equations allow for responsewith and without yielding behavior and obtain a generalization of the normality and convexityconditions. We also show that, in the limit of quasistatic deformations, if one considers materialsthat possess yielding behavior, then the constitutive equations reduce to those corresponding tothe strain space formulation of the rate independent theory of plasticity (see e.g. Naghdi (1990)Journal of Applied Mathematics and Physics A345, 425±458.). Moreover, in this limit, the max-imum rate of dissipation criterion, as stated here, is equivalent to the work inequality of Naghdiand Trapp (1975) Quartely Journal of Mechanics and Applied Mathematics 28, 25±46). The mainresults together with an illustrative example are presented. # 1998 Elsevier Science Ltd. All rightsreserved.

I. INTRODUCTION

In Part I, we have articulated a framework that can account for the elastic response of awide range of materials that undergo microstructural changes. We introduced the notionof a family of response pairs (see Part I, Rajagopal and Srinivasa, 1998) each consisting ofan elastic response function T�p and a preferred con®guration �p with respect to which thisresponse function is de®ned. By considering certain equivalence relationships between

International Journal of Plasticity, Vol. 14, Nos 10±11, pp. 969±995, 1998# 1998 Elsevier Science LtdPergamon

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*Corresponding author. Fax: +1-409-862-3989; e-mail: [email protected]

these pairs, we were able to obtain a canonical representation in such a way that theelastic response can be represented as

T � T�p�F�p

;G�; �1�

where G is a tensor that, for homogeneous deformations, is the gradient of the mappingfrom �r to �p, with �r being a ®xed reference con®guration. For inhomogeneous defor-mations, G is determined as discussed in Part I, Section III.2.

Now, we shall concern ourselves with the inelastic behavior engendered by the evolu-tion of the current natural con®guration �p of the material. As mentioned in the intro-duction to Part I, the present approach allows for discrete as well as continuous changesin the natural con®gurations, deformation twinning being an example of the former whilemetal plasticity is an example of the latter. In this paper, we shall only be concerned withgradual (or ``continuous'') changes in the natural con®guration of the material. It is pos-sible to include parameters such as the strain hardening parameter, the back-stress tensor,etc., as additional variables in the constitutive equations such as (1) and the other con-stitutive equations that follow. However, for the sake of simplicity and to highlight thebasic structure of the theory we shall not include them here. Thus, the focus here is on theframework rather than any speci®c forms for the constitutive equations.

If the response of the material is purely elastic as discussed in Part I, it is well knownthat the stress response can be derived from a scalar stored energy function. This reducesthe speci®cation of the constitutive equations to a single scalar function. In the case ofinelastic behavior, however, constitutive equations must be speci®ed for the stressresponse as well as for the evolution of the natural con®gurations. It is natural to questionwhether one can make certain physically plausible assumptions regarding the nature ofthe response functions and hence obtain certain restrictions on these constitutive equa-tions. This approach has been particularly successful for rate independent plasticity,where the use of a work inequality (see e.g. Il'iushin, 1961; Naghdi and Trapp, 1975)reduces the speci®cation of the constitutive equations to two scalar functionsÐa storedenergy function and the yield function, the gradient of the former delivering the stresswhile that of the latter delivers the direction of the rate of change of the plastic strain (seeSrinivasa (1996) for further details regarding this issue).

In the current work, beginning with the notion of a rate of dissipation function that repre-sents the rate of conversion of the mechanical power supplied into heat, we (see Section IIIbelow) appeal to the use of a ``maximum rate of dissipation criterion'' (see condition (2)) toestablish the structure of the constitutive equation for the evolution ofG. In particular, the timerate of change of G will be shown to be along the gradient of the rate of dissipation function.

I.1. Dissipation and inelasticity

By far the most common constitutive assumption for the evolution of the natural con-®gurations is by means of a di�erential equation that relates the time derivatives of G tothe current state (speci®ed by the current deformation gradient F�r , the current value of Gand possibly a ®nite number of other variables). An overwhelming majority of con-stitutive equations for elastic±plastic materials are speci®ed in this manner. While ourapproach is similar in form, the philosophical underpinnings are di�erent and we arrive atthe di�erential equations in a slightly di�erent manner.

970 K. R. Rajagopal and A. R. Srinivasa

The fundamental distinguishing feature of inelastic materials from elastic ones is thatthe former are capable of converting a part of the energy supplied to them as heat whereasthe latter are not. Indeed both these material are capable of storing energy so that theconcept of dissipation or the rate of conversion of work into heat is a central notion in thestudy of inelastic materials. It is usual, within classical plasticity to specify some evolutionequation for the plastic strain and then to de®ne the rate of dissipation or ``rate of plasticwork per unit volume'' (see e.g. Hill, 1948, Lee, 1969) as the inner product of the stresswith the rate of plastic strain.

In our approach, however, the rate of dissipation takes center stage as a new indepen-dent notion (along with the stored energy). Thus, in the current theory we have threeenergetic quantitiesÐthe work done, the stored energy and the rate of conversion of workinto heat. The isothermal form of the energy balance equation then stipulates that the rateof dissipation is equal to the di�erence between the ratio of work and the rate of change ofthe Helmholtz potential. We make constitutive assumptions on the nature of the workdone, the Helmholtz potential and the rate of dissipation. By postulating that the rate ofdissipation per unit volume is the maximum possible, we arrive at the form of the evolu-tion equation for G in terms of quantities associated with the Helmholtz potential and therate of dissipation function. In this, we follow a line of thought originally suggested byZiegler (1963, 1983) and later by Ziegler and Wehrli (1987), extending and adapting it toour new uni®ed framework. As discussed in Section (6), Ziegler (1963, 1983) assumes acertain ``orthogonality condition'' and shows that, under certain circumstances, it impliesa criterion of maximum rate of dissipation. Our starting point is di�erent, in the sense thatwe start by assuming a physically plausible maximum rate of dissipation condition (SeeAssumption 2) and show that certain normality and convexity conditions are both neces-sary and su�cient for it (see Theorems 1 and 2).

II. DISSIPATIVE PROCESSES

We consider a material that initially occupies a con®guration �r, and is deformed su�-ciently so that it undergoes a continuous microstructural change.

To begin with, we can rewrite the constitutive equation (1) for the stress in the moremathematically suitable form

S � S�p�E�r;G�; �2�

where S is the symmetric Piola±Kirchho� stress de®ned by

S � det�F�r �Fÿ1�r TFÿT�r : �3�

In equation (2), the dependence on the deformation gradient F�p (measured from thecurrent natural con®guration) is implicit since F�p � F�r�G�ÿ1. In view of this, and frameindi�erence, any function which depends upon F�p and G can be rewritten in terms of E�rand G. While it is more intuitive to write the constitutive equations in terms of F�p , formathematical reasons, it is more convenient to write it in terms of the Green±St.Venantstrain E�r and G. We shall use this form in the remainder of this section.

The speci®cation of the constitutive equation is complete once the evolution of �p andhence of G is determined. Moreover, the size and shape of the elastic domains must also

Mechanics of the inelastic behavior of materials, Part II 971

be speci®ed by means of additional parameters. We shall concentrate only on the evolu-tion of the natural con®gurations here.

First, we shall assume that the rate of change of G depends only upon the currentnatural con®guration and the deformation gradient from it, so that we may write

Lp �> Lp�E�r;G�; �4�

where, for future convenience, we have de®ned

Lp :� _G�G�ÿ1; �5�

analogous to the ordinary velocity gradient and where the form of Lp depends uponall the natural con®gurations �p. For each ®xed G (corresponding to each elastic regime),the range of the function Lp represents all the possible values that Lp may take forvarious values of E�r . This set of values represents the allowable range of values of Lp

for this particular microstructure and natural con®guration. We shall refer to this setof allowable values of Lp as the admissible or accessible values of Lp for a particularmicrostructure.

II.1. The elastic domain

We shall assume that the material has a non-null elastic domain E for each G. For anyvalue of E�r that lies in the elastic domain for a given microstructure the only possible valueof Lp would be zero. Said di�erently, the inverse image of 0 (for ®xed G) of the mappingLp is the elastic domain corresponding to that microstructure, i.e., the elastic domain isgiven by the equation

Lp�E�r;G� � 0 �6�

Thus the elastic domain is implicitly de®ned by (6) and there is no needÐat this stageÐtointroduce a separate yield function as is usual in elastic±plastic materials. If Lp is stipu-lated to be continuous in E�r for ®xed G, then the above considerations indicate that theelastic domain (being the inverse image of a closed set) is closed, i.e. the boundary of theelastic domain also belongs to it. The sets

Lpj :�k Lp k�k Lp�E�r;G� k� const: > 0n o

�7�

will be referred to as constant rate sets and are a generalization of the traditional dynamicyield surfaces in plasticity. The fact that depends upon E�r and G but not on Lp or _E�rindicates that the response of the material is not homogeneous in the rate (as is the casefor traditional plasticity) and hence the response is rate dependent.

For future reference, we also de®ne the ``direction'' of Lp by � :� Lp= whose con-stitutive equation (from (4)) is given by

� � Lp�E�r;G�k Lp�E�r;G� k

:� ��E�r;G�; whenever 6� 0: �8�


II.2. Materials with yielding behavior

The development of the constitutive equations includes the possibility that the body inquestion has an elastic domain with a non-null interior. For these materials, as in tradi-tional plasticity, we shall assume that the classical elastic domain (de®ned in Part I) isenclosed by a closed and bounded surface called the yield surface and represent it bymeans of an equation of the form

g�E�rG� � 0: �9�

We hasten to add here, that in the current theory, the function g is not a new or additionalconstitutive function but only a representation of the boundary of the elastic domain thatis de®ned by eqn (6).

We shall further assume that for ®xed G, the elastic domain is characterized by theinequality g�E�r ;G� � 0. As in classical plasticity, we shall start with a mechanical statethat is within the elastic domain. We then consider an arbitrary smooth path E�r�s�parameterized by the arc length s along the path. As long as the path lies within the elasticdomain, i.e., as long as g�E�r�s�;G � 0, the response is elastic. As soon as the path crossesthe yield surface, inelastic behavior is initiated. The conditions under which this occurs arecalled the ``loading criteria'' and are exactly the same as that used in the strain-space for-mulations of plasticity. We shall not list the loading criteria here but refer the interestedreader to Naghdi (1990) for a detailed discussion.

II.3. The ``driving force function''

The constitutive equations (2) and (4) are the fundamental constitutive assumptionsregarding the behavior of inelastic materials whose overall response is rate-dependent. Wemay assume that Lp depends upon additional parameters, in which case additional con-stitutive equations for their evolution need to be speci®ed. However, these additionalparameters do not add any complexity to the problem and we shall suppress them so thatwe can highlight the most important features of the response.

The constitutive assumptions (2) and (4) together with the balance of mass, momentumand moment of momentum (together with appropriate boundary and initial conditions)form a complete set of equations for the determination of the response of the material. Nothermodynamical criteria have been used to restrict the constitutive assumptions as yetand it is to this aspect that we shall turn to next.

Typically, one assumes that the constitutive assumptions must be consistent with somerate of dissipation inequality (e.g. the isothermal form of the Clausius±Duhem inequality)for all processes that the material is capable of undergoing, leading to certain restrictionson the form of the constitutive functions. For example, the isothermal form of the Clau-sius±Duhem inequality (see Truesdell and Noll (1992)) stipulates the existence of a storedenergy function (the Helmholtz potential) that satis®es

S� _E�r ÿ %0 _ :� � � 0; �10�

where the rate of dissipation � is the di�erence between the mechanical power and the rateof increase of the Helmholtz potential per unit volume.


Here, we shall adopt a di�erent approach wherein we admit the existence of two ther-modynamical entitiesÐthe Helmholtz potential and the rate of dissipation function �.The Helmholtz potential depends upon the natural con®guration that is associated withthe current state of the material and the deformation gradient measured from it, i.e. uponthe variables E�r and G. It is the rate of dissipation function that would appear as a heatsource term in the equation for the determination of the temperature if we considered afully thermomechnical process. While both elastic and inelastic materials possess storedenergy functions, the feature that distinguishes inelastic materials from elastic ones is thedissipation of energy that is possible in the former. Moreover, since inelastic materials arealso capable of exhibiting elastic response, the dissipation occurs only during those partsof the process during which microstructural changes take place.

We thus assume the following functional forms for the Helmholtz potential and the rateof dissipation functions:

� �E�r ; G�; �11�

� � ��G;Lp� � 0: �12�

The form of the rate of dissipation function � is based on the fact that dissipation is inti-mately associated with changing con®gurations and is independent of the elastic response(and hence of the deformation gradient from the current natural con®guration). It shouldbe noted that for ®xed G, the rate of dissipation function � is de®ned only on the admis-sible values of Lp de®ned earlier in this section.

The Helmholtz potential is assumed to be de®ned for all values of E�r and at least twicecontinuously di�erentiable with respect to each of its arguments. For future convenience,we shall de®ne a continuously di�erentiable function A called the ``driving force func-tion''* by

A � A�E�r ; G� :� ÿ%0@

@GGT: �13�

The range of A for ®xed G is a subset A, of a nine dimensional Euclidean space,y the latterbeing referred to as the A-space. As we shall soon see, the A-space plays a signi®cant rolein the description of the inelastic response of the material. For future reference, the imageof the elastic domain E under the mapping A will be referred to as the elastic range in A-space. Since the elastic domain is non-empty, neither is the elastic range in A-space.

We shall further stipulate that the gradient @A=@E�r has full rank through-out the strainspace for ®xed G. This ensures that ``locally'' the set A is a six-dimensional submanifoldof the 9-dimensional Euclidean A-space. Furthermore, it ensures that A is a quotient map(see e.g. Munkres, 1975), i.e. any subset Y of A is open in A if and only if its inverse imageAÿ1(Y) is open. These topological facts play a signi®cant role in future developments andare crucial to the establishment of the physical signi®cance of the yield functions.


*The reason for the nomenclature will become clear presently when we shall show that A plays a signi®cant rolein determining the rate at which the natural con®gurations evolve. The ``driving force'' function, as de®ned herecan also be shown, for special constitutive assumptions, to reduce to the ``con®gurational force'' introduced byEshelby (1970) within the context of ®nite elasticity (see Naghdi and Srinivasa, 1993).yThe dimension of the codomain will be the same as the dimension of the space to which G belongs.

II.4. The rate of dissipation equation

We now require the constitutive theory to re¯ect the following restriction:Assumption 1. The constitutive functions S, Lp, and � must satisfy the rate of dis-

sipation equation

S� _E�r ÿ %0 _c � ��G; Lp�E�r ;G��; �14�

for ®xed Ekrand G and for all possible values of _Ekr

.In virtue of the non-negativity of the rate of dissipation function, any set of constitutive

functions that satisfy (14) automatically satis®es the Clausius±Duhem inequality for iso-thermal processes.

Upon substitution of the constitutive assumption (11) and the application of the chainrule for di�erentiation, it can be shown that the left hand side of (14) is a�ne in _E�r for®xed values of E�r and G while the right hand side is independent of _E�� . Thus, since therate of dissipation equation is assumed to be satis®ed for all possible values of _E�r , anecessary condition for the satisfaction of the rate of dissipation equation is that

S � S�E�r ;G� � %0@

@E�r: �15�

In other words, the stress is derivable from the Helmholtz potential, just as in the case ofelasticity. However, here the Helmholtz potential depends also upon the (changing) nat-ural con®gurations.

With the above restriction (15) on S, the rate of dissipation equation reduces to

A�E�r ;G��Lp � ��G;Lp�; �16�

where Lp � Lp�E�r ;G). Since the material is assumed to possess an elastic domain that isnon-dissipative, we can immediately conclude from (16) that ��G; 0� � 0. Of coursenothing can be said yet about the value of � when Lp is nonzero. In particular, at thepresent juncture, it is possible for changes in natural con®gurations to occur in an entirelynon-dissipative manner. We shall see that such behavior is eliminated by the assumptionsmade on the nature of the inelastic response in Section IV.

No further restriction can be obtained based on the non-negativity of the rate of dis-sipation alone. Indeed, the full potential of the introduction of the rate of dissipationfunction is not realized until a further assumption is made. Before embarking on addi-tional assumptions, it may be helpful to understand the various terms that appear in (16).

For ®xed E�r , the left hand side of (16) represents the rate of decrease of the storedenergy at constant total strain, i.e. with the current con®guration ®xed. It thus representsthe rate at which energy is released due to the changing natural con®gurations. Thus,given the functions S, and Lp, one can easily obtain the rate of energy releasedduring the inelastic process. Likewise, the function � can be used independently toobtain the rate of heat generated during any process. The rate of dissipation equation(14) then asserts that these two independent quantities happen to be equal for everyallowable process undergone by the material. The reduced rate of dissipation equation(16) thus represents a balance or accommodation between the energy release rate andthe rate of dissipation.


An interesting consequence of the assumed form for the function is that if we con-sider a special process during which the strain and G have the same values at the begin-ning and the end of a cycle, the Helmholtz potential also returns to its original value.Thus, in such cycles (if they exist) all the mechanical work done is converted into heat. Onthe other hand, if one were to consider a more elaborate form for , one that involves thepath history of G, for instance, then some of the external work supplied may be stored inthe material and only the remaining part converted into heat. Such situations are fairlytypical of metals where the number of dislocations increases dramatically during theirplastic response. These dislocations ``trap'' some of the energy in their strain ®elds. Theexperiments of Taylor and Quinney (1931) revealed that about 80% of the externalmechanical work was converted into heat, the other 20% presumably being ``trapped'' inthe dislocation networks. Current models accommodate this by multiplying the ``plasticwork'' by an ad-hoc factor whose value is approximately 0.8 (see e.g. Lee, 1969, eqns (44)±(46); Mason et al. 1994; Brown et al. 1989, eqn (9)). Instead, if we use a more elaborateform of the Helmholtz potential that accounts for the energy trapped in the strain ®eldsaround the dislocations, the resulting Helmholtz potential will then be physically moti-vated by the processes that occur at the microstructural level without modifying the formof the energy equation.

III. THE MAXIMUM RATE OF DISSIPATION CRITERION AND ITS CONSEQUENCES

The reduced rate of dissipation equation (16) places only a mild restriction on the formof Lp, ensuring only that the rate of dissipation of energy is non-negative in every process.

In order to further guide our choice of the form of Lp we need to look closely at themanner in which the energy is released and dissipated. A careful study of (16) reveals thatthe energy release rate A�Lp for ®xed values of E�r and G increases linearly with themagnitude of Lp whereas the rate of dissipation may be nonlinear in Lp in general. Thus,(16) represents one equation to be satis®ed by the nine components of Lp. Intuitivelyspeaking, the material ``chooses'' one of the values of Lp which satis®es the reduced rateof dissipation equation. The question is: What physical criterion determines the choicethat is made? We shall endeavour to answer this question next. Here we shall suppose thatthe following condition holds:

Assumption 2. The maximum rate of dissipation criterion: Given the function Lp, letLp0:�Lp�E�r ;G) be the actual value of Lp for a given value of E�r and G. Let�0 � ��G;Lp0� be the actual value of the rate of dissipation. If Lp1 is any other admissiblevalue of Lp with greater rate of dissipation than the actual value, then, for that value ofLp, the computed value of the energy release rate would be less than the computed valueof the rate of dissipation. In other words, if Lp 6� Lp0 is any other value such that

��G;Lp� � ��G;Lp0�; �17�then

A�E�r ;G��Lp < ��G;Lp�: �18�

In the above criterion, the admissible values of Lp are those in the range of Lp when E�r isvaried while G is held ®xed.


The above assumption is called the maximum rate of dissipation criterion because, itimplies that

Lemma 1. Given the function Lp, for ®xed values of E�r and G, of all admissible valuesof Lp that satisfy the reduced dissipation equation (16), the actual value of Lp, given byLp0 :� Lp�E�r ;G� is the one which corresponds to the maximum rate of dissipation.

Proof. Let Lp1 6� Lp0 be some other value of Lp that also satis®es the rate of dissipationequation (16) for the same G and E�r . Then in view of (17) and (18), the correspondingrate of dissipation must necessarily be less than the actual value of the rate of dissipationand the lemma is proved.&

Clearly, the above lemma isolates an unique value of Lp for given values of E�r and Gfrom among all those that satisfy (16), the criterion for the choice being that the rate ofdissipation be the maximum possible.

Before leaving this section, it is not di�cult to show that, for the class of materialsconsidered here, the assumption embodied in eqns (17) and (18) implies that for a ®xed �,the rate of dissipation increases faster with than the energy release rate. This will becrucial to the establishment of various results in the following sections. Of course, themaximum rate of dissipation criterion that is stated here is only an assumption that holdsfor a class of materials under consideration and we do not consider it to be a fundamentalprinciple that is valid for all materials. Our quest is to see whether such an assumptiongives reasonable results for certain classes of materials. In the process, we hope to gaininsight into the nature of such materials.

III.1. Preliminary consequences of the maximum rate of dissipation condition

From an elementary consideration of maximization through the use of calculus, we canconsider the maximization of the rate of dissipation function ��G;Lp) subject to the con-straint (16). Then, by a standard procedure (using Lagrange multipliers) we obtain thefollowing equation for the determination of Lp:

A�E�r ;G� � l@�

@Lp; �19�

where l is a Lagrange multiplier that is determined by the satisfaction of (14). In other words,the above result implies that the driving force A is directed along the normal to the surface ofconstant rate of dissipation.

Note that the above result only guarantees that the solution corresponds to a localextremum of � and does not really address the issue of the existence of a global maximum.Moreover, we obtain the value of Lp implicitly. In the remainder of this work, we use stan-dard techniques of convex analysis to establish global maxima as well as to obtain the valueof Lp explicitly as a function of the basic kinematical variables. In the process, we shallestablish the dual of (19), i.e. that Lp is directed along the normal to a surface in A-space.*

III.1.1. Nature of the elastic domain. One of the direct consequences of the assumption2, concerns the characterization of the elastic domain of the material in terms of the


*This is stated and proved in Theorem 3 of Section IV that follows. Readers are directly referred to that section ifthey would like get a quick look at the central results of this paper. They are also urged to look at the example inSubsection IV.1 in order to get a feel for the application of the results.

driving force A and the rate of dissipation �, and the conditions under which inelasticbehavior is initiated. We shall ®rst establish that

Lemma 2. Any process during which Lp 6� 0 is dissipative, i.e.,

��G;Lp� > 0 whenever Lp�E�r ;G� 6� 0: �20�

The above condition is complementary to the fact, derived in Section III immediately after(16), that any process in the elastic domain is non-dissipative. The result (20) seems soobvious that one is tempted to assume it as a basic requirement. However, recall that atany stage in the process, elastic response is always a possibility. The above lemma impliesthat ``given the possibility of deforming in either a dissipative or a non-dissipative man-ner, the material always chooses the dissipative route''.

Since the material is assumed to have a non-null elastic range, it is obvious that forevery G there is at least one value of E�r for which Lp�E�r ;G� � 0. We have already seen inSection III that ��G; 0� � 0. With this in mind we can now prove the above lemma bycontradiction as follows:

Proof. For some value of G and E�r , let Lp0 :� Lp�G;E�r� 6� 0 be such that��G;Lp0� � 0. Then, we have

��G; 0� � ��G; Lp�G;E�r�� 0; �21�and, in view of (16),

0 � A�E�r ;G��0 � ��G; 0�: �22�

Clearly, (21) and (22) contradict (17) and (18). &The above lemma immediately implies thatLemma 3. If E�r is such that A�E�r ;G� � 0, then Lp�E�r ;G� � 0. In other words, the

elastic domain contains all values of strain with vanishing driving force.Thus, we obtain a partial characterization of the elastic domain as a consequence of the

maximum rate of dissipation criterion. It is useful to compare this result with the usualassumption that points corresponding to vanishing stresses are in the elastic domain. Theabove remark clearly points to the fact that it is always possible to elastically unload toany zero driving force state corresponding to A � 0. In view of this we can a-posteriori usethe con®gurations corresponding to A � 0 as our natural con®gurations rather than thestress-free con®gurations. We can then rewrite all our constitutive equations using thenotion of equivalence de®ned in Part I.

In order to obtain a further characterization of the elastic domain in terms of the rate ofdissipation and the driving force, we now de®ne a scalar quantity called the ``driving forcemagnitude'' in a given direction � (which is a unit vector in a nine-dimensional Euclideanspace) as

d�E�r ;G; �� :� A�E�r ;G��: �23�

Physically, d is the potential for con®gurational change in the direction of �. This can becomputed irrespective of whether the natural con®guration changes or not and hence canbe evaluated even in the elastic domain. It is worth emphasizing that while the energyrelease rate is zero in the elastic domain, the same is not true for d.


Next, we observe from Lemma 2 that as long as > 0, the rate of dissipation and hencethe ratio of � to is bounded below by zero. Thus, for each ®xed G and �, we de®ne a``threshold function'' T (G,�) by

T �G;�� :� inf >0

��G; ��

� 0 �24�

The reason for the name ``threshold function` is that this function serves as the thresholdthat must be crossed by d for inelastic response to occur. We state this in the form of thefollowing lemma that characterizes the entire elastic domain:

Lemma 4. For ®xed G, for a point Ekrto belong to the elastic domain, it is necessary

and su�cient that d�E�rG; �� < T �G;�� for all admissible values of �.

Proof. We keep G ®xed and consider the set of all E�r such that d�E�r ;�� < T �G;�� forall admissible values of �. Since T is the in®mum of

��G; �� in each direction, it is then

clear that d�E�r ;G; �� < ��G; ��= . Multiplying both sides of this inequality by > 0and using (23), we arrive at

A�Lp < ��G;Lp� for all Lp 6� 0: �25�

Thus, we conclude that for these values of E�r the only value of Lp for which the rate ofdissipation equation is satis®ed is 0. Thus, these values of E�r belong to the elastic domain,proving the su�ciency part of the lemma.

To prove necessity, let E�r be some point in the elastic domain such that, for some �,the normalized energy release rate is greater than the threshold function, i.e.

d�E�r ;G; �� > T �G;��: Then, since T is the in®mum of ��G; �� , the latter being a con-

tinuous function of Lp for ®xed G, we obtain d > ��G; ��= > 0 for some > 0. For thisvalue of � and , we de®ne Lp0 :� �.

Thus, since E�r is in the elastic domain, we then have

��G;Lp0� > ��G; Lp�E�r ;G�� 0; �26�and

A�E�r ;G��Lp0 > ��G;Lp0�: �27�

Clearly, (26) and(27) contradict (17) and (18).&This completes the characterization of the elastic domain. Moreover, it is clear from the

above result that inelastic behavior is initiated as soon as the driving force magnitudeexceeds the threshold function in some admissible direction �.

III.1.2. Necessary and su�cient conditions for the satisfaction of the maximum rate ofdissipation criterion. Our next result deals with the construction of the rate of dissipationfunction f in A-space corresponding to �. This step is essential because, although the rate ofdissipation criterion as stated in (17) and (18) is physically well motivated, it is mathemati-cally inconvenient for the derivation of the central results of this paper. The following the-orem shows that there is an equivalent way of formulating the reduced rate of dissipationequation and the maximum rate of dissipation criterion, using only functions de®ned on A.


Theorem 1. Necessary and su�cient conditions for the satisfaction of the maximum rateof dissipation criterion.

Given the Helmholtz potential �E�r ;G� whose derivative A, with respect to G, (intro-duced in (13)), is an open map, there exist functions Lp�E�r ;G� � Lp and ��G;Lp� � 0,satisfying the reduced rate of dissipation equation (16) as well as the criteria (17) and (18)if and only if continuous functions f�A;G� � 0 and ~Lp�A;G� de®ned on the range A ofthe function A can be constructed such that, for ®xed G,

i: Lp � ~Lp�A�E�r ;G�;G� � Lp�E�r ;G�; �28�

� � f�A�E�r ;G�;G� � ��G; Lp�E�r ;G��; �29�

ii: A� ~Lp�A;G� � f�A;G�; �30�for all A in the range of A, and

iii. For ®xed values of G and � > 0, consider two points A and A0 in A, �A 6� A0�that satisfy

f�A;G� � � and f�A0;G� � �: �31�

Further, let Lp :� ~Lp�A;G� and Lp0 :� ~Lp�A0;G�: Then

�Aÿ A0��Lp> 0 if Lp 6� Lp0;� 0 if Lp � Lp0:

��32a; b�

Proof. Necessity:To prove necessity, we assume that there are constitutive functions ; Lp and � (the last

of which is de®ned on the range of Lp for ®xed G) that satisfy the reduced rate of dis-sipation equation (16) as well as the maximum rate of dissipation criterion embodied in(17) and (18).

To prove the necessity of part (i), we ®rst establish the following

1. Lemma 5. If two values of the strain E�r correspond to the same value of A, then thecorresponding values of Lp are also equal, i.e., if A��E�r�1;G� � A��E�r�2;G� � A,then Lp��E�r�1;G� � Lp��E�r�2;G� � Lp.

Proof. We shall prove this by contradiction. Let us assume thatLp1 :� Lp��E�r�1;G� 6� Lp��E�r�2;G� :� Lp2 and without loss of generality assume that

��G;Lp1� � ��G; Lp��E�r �2;G��: �33�

Then, the reduced rate of dissipation equation (16) together with the fact that value of A isthe same at both values of strain implies that

A��E�r �2;G��Lp1 � ��G;Lp1�: �34�

The conditions (33) and (34) contradict (17) and (18) thus completing the proof. &The above lemma indicates that for ®xed values of G, there is a function that relates

A and Lp. It follows from the above lemma that, for ®xed G, if two values of strain


correspond to the same value of A, then the rate of dissipation � at these two values of thestrain are also the same. In other words, the rate of dissipation can be found once the valueof A and G are given.

2. Construction of the functions Lp and f:LetG be ®xed. For eachA belonging toA, letE�r be any point in the inverse image of A underthe mapping A. We simply set ~Lp�A;G� � Lp�E�r ;G� and f�A;G� � ��E�r ;G�: That thesefunctions are well de®ned and independent of the choice of E�r follows immediately fromLemma 5. Moreover, by their very construction, the functions ~Lp and f satisfy (28) and (29),respectively.

The continuity of the two functions follows from the fact that Lp is continuous in E�r andthat A is an open map (see Munkeres, 1975, p. 139, Theorem 11.1). The function f in (29)delivers the rate of dissipation for each value ofA in the range of A and can thus be called therate of dissipation function inA space. This proves the necessity of the part (i) of the theorem.

3. Proof of necessity of part (ii):Next, substituting (28) and (29) into the reduced dissipation equation (16) immediatelydelivers (30) and part (ii) is proved.

4. Proof of necessity of part (iii):Finally, to prove part (iii), let us begin by assuming that Lp 6� Lp0. Then the condition (31)together with (28), (29) and (16) imply that

A�Lp � ��G;Lp� � A0 �Lp0 � ��G;Lp0�: �35�Finally, eqns (35) and (18), imply that

�Aÿ A0��Lp > 0: �36�Thus (32)a is proved.

On the other hand, if Lp � Lp0, the reduced rate of dissipation equation (16) appliedsuccessively using A and A0 delivers

A�Lp � ��G;Lp� � ��G;Lp0� � A0 �Lp0 � A0 �Lp; �37�which, in turn, delivers (32)b.

Su�ciency: Given the functions ; Lp and f satisfying the conditions (30)±(32), con-structing the functions Lp and � that satisfy (28) and (29) is simply a matter of substitutingA � A�E�r ;G� in ~Lp�A;G� and f�A;G�. This proves part (i). A similar substitution into(30) immediately assures the satisfaction of the rate of dissipation equation (16).

It remains only to prove that the maximum rate of dissipation criterion is satis®ed. Todo this, let us ®x a point A0 and let Lp0 � ~Lp�A0;G�. Now let Lp � ~Lp�A;G� be someother value such that (17) is satis®ed. Then, in virtue of the relationships (28) and (29), theconditions (31) for the application of the inequalities (32) are satis®ed. These inequalities,together with (28), (29) and (14), imply that the inequality (18) is satis®ed.&

IV. GEOMETRICAL INTERPRETATION OF THE MAXIMUM RATE OF DISSIPATION

CRITERION

The theorem established in Section III does not provide a clear insight into the natureof the constitutive equations. In particular, we do not yet have a characterization of theconstitutive function Lp. We shall focus attention on this in the following analysis.


In preparation for the main theorem, for ®xed G and for each � � 0, we shall de®ne alevel set L(a,G) by

L��;G� :� fA 2 Ajf�A;G� � �g: �38�

Clearly, these level sets are closed, being the inverse images of closed sets under the con-tinuous mapping f. Moreover, the level sets corresponding to � > 0 have non-emptyinteriors relative to A, since for each G the elastic domain, which is contained in each ofthe level sets, is non-empty.

We shall now show that the function f�A;G�, can be chosen in such a way that it deli-vers a constitutive equation for Lp that satis®es (16)±(18). In this endeavour, we shallmake use of Theorem 1 that was obtained in the previous section.

We ®rst present a result that provides us with a geometrical interpretation of the max-imum rate of dissipation condition:

Theorem 2. Convexity and normality conditions.The maximum rate of dissipation condition implies that (a) the level set L��;G� of thefunction f�A;G� for each value of � > 0 is contained in the intersection of a convex setC��;G� � E* with the range A of the function AÃ . In particular,

U��;G� :� fA 2 Ajf�A;G� < �g � C��;G�; �39�

B��;G� :� fA 2 Ajf�A;G� � �g � @C��;G�; �40�

where the notation @C��;G� represents the boundary of C��;G�. (b) At all points ofB��;G�Ðwhich, by (40) lie on the boundary of C��;G�ÐLp is along the direction of theoutward unit normal to @C��;G�:

Statement (a) of the above theorem is referred to as the convexity condition, whilestatement (b) is the normality condition. The proof of the above theorem is quite similarto that given by Srinivasa (1996) and is quite easy since many of the major results havealready been established in the previous subsection.

Proof. 1. Construction of the convex set C��;G� for � > 0:With each element A of B��;G�, we associate a half space H�A;G� in A-space given by

H�A;G� :� fY 2 Ej�Yÿ A��Lp�A;G� � 0g: �41�

This is a closed convex set. We now de®ne C��;G� to be the intersection of all the halfspaces H�A;G� as A ranges over the entire closed set B��;G�, i.e.

C��;G� :� \A2B��;G�H��;G�: �42�

This set is closed and convex (see e.g. Rockafellar, 1970, p. 10, Theorem (2.1) Corollary(2.11)). Of course at this stage we have no guarantee that C is non-empty.

2. Demonstration that C is non-empty:Recall that for each G, the elastic domain in strain space is non-null. Consequently, if E�ris some point in the elastic domain, consider its image Y under the mapping A. We knowthat f�Y;G� � 0. Now, for ®xed � > 0, consider any element A of B��;G�. We can now


*E stands for a nine-dimensional Eudidean space.

apply part (iii) of Theorem 1 with A0 set equal to Y and obtain from (32)a that�Yÿ A�� ~Lp�A;G� < 0. Thus every point in the image of the elastic domain belongs toeach of the sets H�A;G� and hence to their intersection. Therefore, we conclude that theset C��;G� is non-empty.

3. Demonstration that U��;G� and B��;G� satisfy (39) and (40), respectively:Next, consider any point A in L��;G�. In virtue of (32)a,b and the de®nition (41) we caneasily see that L��;G� belongs to each half space H�A;G� and hence is a subset of theconvex set C��;G�. Furthermore, each point A in B��;G� belongs to the convex setC��;G� as well as the boundary of the half space H�A;G�. Thus the boundaries of the halfspaces H�A;G�, de®ned by

@H�A;G� :� fY 2 Ej�Yÿ A�� ~Lp�A;G� � 0g �43�

are supporting hyperplanes (see Rockafellar, 1970) of the convex set C��;G�.Clearly, since the image of the elastic domain is in the interior of the half spaces H, no

single supporting plane contains the whole set L��;G� and hence C��;G� is a convex body.Indeed, in virtue of the result (32)a,b, the members of the set U��;G� are in the interior ofevery half-space H�A;G� and thus in the interior of C��;G�. This proves (39). The factthat L��;G� is contained in C��;G� and that every point of B��;G� is an element of thecorresponding supporting hyperplane @H�A;G� implies (see Rockafellar, 1970, p. 100,Theorem 11.6) that B��;G� is contained in the boundary of C��;G�, thus proving (40).

It remains only to prove that ~Lp�A;G� is normal to C��;G� at each point in the setB��;G�. We shall turn to this next.

4. Proving the normality condition:It should be observed that, by construction (cf. eqn (43)), at each point A in B��;G�, thevalue of ~L��;G� is normal to the corresponding supporting hyperplane @H�A;G�. Thus~Lp�A;G� is an element of the normal cone to C��;G� at the point A (see Rockafellar,1970, p. 15 and p. 100). We will now show that the normal cone to C��;G� at each point inB��;G� contains only one element, i.e., that the boundary of C��;G� does not containedges or corners at any point in B��;G� and the half-space H�A;G� through A is actuallya tangent half-space (see Rockafellar, 1970, p. 169).

We do this by contradiction as follows:Let A in B��;G� be some point such that the half space associated with it is not a tangenthalf space. Then, since A is on the boundary of the convex set C��;G�, and by the de®ni-tion of tangent half-space, it is clear that there exist at least two tangent hyperplanes thatpass through A, the direction of whose normals are not parallel. This means that by (43),there exist two unequal values of Lp, say Lp1 and Lp2, for which

A�Lp1 � A�Lp2 � f�A;G� � �; �44�

which is impossible by the fact that Lp is a function of A and G established in Lemma 5.In view of this, we can conclude that the half-space through every point of B��;G� is a

tangent half-space and hence possesses an unique normal.&

Thus, apart from the constitutive function representing the Helmholtz potential thatdelivers the stress response, we can obtain the form of the function ~Lp from the rate ofdissipation function f�A;G� in view of the properties described in the above theorem. These


properties are in the form of global convexity properties over the subsetA in a nine-dimensionalspace and are, as such, extremely di�cult, if not impossible to verify directly, for a candidatefunction. Moreover, the domain of the function f is determined by the derivative of the Helm-holtz potential and hence it appears that we will have to alter the rate of dissipation functionevery time we change the Helmholtz potential. Ideally we would like to specify the function f insuch way that the maximum rate of dissipation criterion is satis®ed for any choice of theHelmholtz potential that satis®es mild di�erentiability conditions. Said di�erently we would liketo be able to de®ne and f independently of each other and yet satisfy the rate of dissipation eqn(16) as well as the maximum rate of dissipation criterion in the form (17) and (18).

We achieve this by stipulating certain properties for the function f(A, G) that are su�-cient to ensure that the rate of dissipation equation and the maximum rate of dissipationcriterion are both met. These properties are in the form of conditions on derivatives andcan thus be easily veri®ed. Moreover, the function is de®ned over the whole nine-dimen-sional space and hence can be restricted to any subset of it. This ensures that the rate ofdissipation equation and the maximum rate of dissipation criterion are satis®ed for anychoice of the Helmholtz potential. Of course, the price we pay for this separation of theHelmholtz potential and the rate of dissipation function is that the properties are nolonger necessary for the satisfaction of the rate of dissipation equation and the maximumrate of dissipation condition, but merely su�cient.

We shall state these properties in the form of the following:Theorem 3. Consider a function f(A, G) de®ned on all of A-space that satis®es the fol-

lowing criteria:1.

f�A;G� � 0; f�0;G� � 0: �45�

2. The level sets L��;G� of the function f�A;G�, de®ned by fA 2 E j f�A;G� � �g areconvex for each � > 0

3. At all points where f�A;G� > 0, the function is di�erentiable with respect to A foreach G and its gradient @f=@A is non-zero.

4.lf�A;G� > f�lA;G�; �46�

for 0 < l < 1 and f�A;G� > 0.

If, for any choice of the Helmholtz potential that is twice di�erentiable with respectto E�r and G, the function f is restricted to the set A (which is the range of A de®nedthrough (13)) and if, at each point A of A, we de®ne

~Lp�A;G� :�0 whenever f�A;G� � 0

f�A;G�A�@f=@A

@f@A whenever f�A;G� > 0;

��47�

then, there are functions Lp � Lp�E�r ;G� and � � ��G;Lp�Ðthe latter being de®ned onthe range of the function Lp for ®xed GÐthat satisfy the reduced rate of dissipationequation (16) and the maximum rate of dissipation criteria (17) and (18).

Proof. We shall prove the above theorem by showing that the functions f and ~Lp meetall the criteria of Theorem 1 which in turn has been proved to be necessary and su�cientfor the satisfaction of the reduced rate of dissipation equation (16) and the maximum rateof dissipation criterion (17) and (18).


First, in virtue of the conditions (1)±(4) on the function f, it can easily be veri®ed thatthe denominator on the right hand side of (47) is non-zero. The satisfaction of part (i) ofTheorem 1 is then simply a matter of substituting (13) into the function f as well as the lefthand side of (47).

Furthermore, as can be veri®ed by taking the inner product of both sides of (47) with A,the eqn (47) also implies that the reduced rate of dissipation equation in the form (30) isautomatically satis®ed, thus proving part (ii) of Theorem 1. Also, in virtue of the convexityof the level sets of f and the fact that, by (47), Lp is directed along the outward unit normalto the level set, it is immediately obvious that the part (iii) of the conditions laid down inTheorem 1 is satis®ed as long as Lp�A;G� 6� Lp�A0;G�, i.e., we have proved (32)a.

It only remains to prove that condition (32)b of Theorem 1 is satis®ed whenLp�A;G� � Lp�A0;G� 6� 0, the case when Lp � 0 being trivially true.

We shall prove this by contradiction. Consider two points A and A0 in A such that~L�A;G� � ~Lp�A0;G� 6� 0. Now, if f�A;G� � f�A0;G� � �, then in view of the fact that@f=@A is non-vanishing, the points A and A0 both lie on the boundary of the level setL��;G�. By the convexity of the latter and the fact that Lp is directed along the normal,the condition (32)b is satis®ed.

Now consider the situation if � � f�A;G� > f�A0;G� � �. Let @H�A0;G� be the tangenthyperplane (de®ned in (43)) to the level set L��;G� at A0. In view of the convexity ofL��;G�, this tangent hyperplane does not have any point in common with the interior ofthe level set. Now, consider the point lA with 0 < l :� �

� < 1. In virtue of the fact that~Lp�A;G� � ~Lp�A0;G� it can easily be veri®ed that lA is an element of @H�A0;G� (see also(43)). However, the condition (46) implies that

f�lA;G� < lf�A;G� � �; �48�

and thus @H�A0;G� has a point in common with the interior of L��;G�, which is acontradiction.

A similar conclusion results if we interchange the roles of A and A0 in the preceding para-graph, so thatwe arrive at the fact thatwhenever ~Lp�A;G� � ~Lp�A0;G� then (32) is satis®ed.&

In this manner we have reduced the speci®cation of the constitutive equations for theinelastic material to two scalar functions; the ®rst being the Helmholtz potential whosederivative with respect to the strain delivers the symmetric Piola±Kirchho� stress tensor,and the second is the rate of dissipation f whose derivative with respect to the ``drivingforce'' A delivers the rate of change of the natural con®guration through (47). Moreoverthe functions and f can be chosen independently of each other although the maximumrate of dissipation criterion depends upon both and � in an intricate fashion.

IV.1. An illustrative example

Before concluding this section, we present here an example that illustrates the proce-dure for the speci®cation of constitutive functions that satisfy (16)±(18). We start byassuming that the Helmholtz potential is given* by


*Of course, frame indi�erence will impose restrictions on , but those restrictions are unimportant in the presentcontext. We also note that the form (49) for the constitutive equations may be easily recast in terms of E�r and G,but again, it is not necessary to do so.

� �F�p�: �49�

For this constitutive assumption, a routine calculation using (3), (13) and (15) gives

T � �=��det�F�r��@

@F�p�F�p �T; �50�

and

A � �det�F�r��F�p�TT�F�p�ÿT; �51�

where the symbol (.)ÿT stands for the inverse transpose of the tensor. Thus, for the parti-cular assumption (49), the driving force A is closely related to the Cauchy stress andvanishes if and only if the latter vanishes.

As far as the rate of dissipation function is concerned, a possible choice that satis®es thecriteria of Theorem 3 is a ``von-Mises like` form given by

f�A;G� � A�Aÿ K2; �52�

where K � K�G� > 0 corresponds to a simple strain-hardening parameter. It is not possi-ble to rewrite this constitutive assumption purely in terms of T and G unless some speci®cform for (49) is assumed. Of course, the non-negativity of f implies that the above equa-tion is valid only for values of A that satisfy A�A > K2. At all other values of A we simplyset f � 0. Thus, the boundary of the elastic range is given by the equation A�A � K2. Oncea speci®c choice for the constitutive function in (49) is made, then, by using (50) and(51) in the equation for the boundary of the elastic range, we can obtain the yield surface gin strain space.

Substitution of (52) into (47) delivers the constitutive equation for Lp in the form

~Lp�A;G� :� 0 whenever A�A � K2;A�AÿK2

A�A� �

A whenever A�A > K2:

(�53�

The above constitutive equations represent a generalization of the usual Prandtl±Reussequations to a rate-dependent model that is also valid for large deformations.

V. THE RESTRICTED (OR CONSTRAINED) MAXIMUM RATE OF DISSIPATION CRITERION

The maximum rate of dissipation criterion embodied in (17) and (18) are stringentconditions on the nature of the dissipative processes and are, as such, far stronger than theisothermal version of the Clausius±Duhem inequality which merely requires that the rateof dissipation be non-negative for all admissible processes. The question naturally ariseswhether the maximum rate of dissipation criterion can be ``relaxed'' or weakened in somesense, in such a way that the resulting constitutive equations correspond to physicallyrelevant behavior. One way to accomplish this would be to ``constrain'' the admissiblevalues of Lp over which the rate of dissipation is maximized.

In this regard, we ®rst observe that in many situations the allowable values of G andhence Lp are a-priori constrained in some manner. A classic example in plasticity would be


the requirement that the volumes of all the natural con®gurations be the sameÐthematerial is ``plastically incompressible''. In the present theory, this constraint would takethe form tr�Lp� � 0 where the symbol tr stands for the trace of the tensor in question.Constraints such as these are automatically enforced a priori on the set of all admissiblevalues of Lp and do not result in a weakening of the maximum rate of dissipation criter-ion. Such constraints correspond to ``holonomic'' or con®gurational constraints of clas-sical dynamical systems.

We wish to consider a di�erent alternative: The rate of dissipation is maximized onlyover some proper subset of the full admissible set of values of Lp. This weakens the max-imum rate of dissipation criterion, and, in the limit when the subset over which the max-imum rate of dissipation criterion holds consists of a single point other than the pointLp � 0, delivers no useful information apart from the requirement that the rate of dis-sipation be positive for inelastic processesÐi.e. we recover the Clausius±Duhem inequal-ity. We shall not consider the most general subset possible but instead restrict ourselves toconsidering subsets, the maximization over which still preserves convexityÐalbeit in amore relaxed form.

Before embarking on this line of enquiry, we ®rst note that, tacit in the assumption thatthe rate of dissipation � is a function of G and Lp alone (see eqn (12)) is the followingproperty: If, for a given G, the value of Lp is the same at two di�erent values of the strain,then so is the rate of dissipation corresponding to these two strains. We ®rst weaken thisassumption, by considering a broader class of materials whose rate of dissipation func-tions may depend explicitly on the strains. To elaborate, let N(G) be a given tensor valuedfunction of G whose values are all unit tensors and let

��E�r ;G� :� N�G��A�E�r ;G�: �54�

We shall now assume that the function � depends upon E�r through � so that we de®nethe rate of dissipation � by

� � ��;G;Lp�: �55�

For example, if N is the identity tensor, then the assumption (55) would imply that therate of dissipation function depends explicitly on the trace of A.

For future convenience, for any second order tensor B we de®ne

B? :� Bÿ �N�B�N: �56�and note that

B?�N � 0: �57�

We note that if we now consider the set of admissible values of Lp with both G and � held®xed, we get a subset of the full admissible set and hence we can consider a weakenedversion of the maximum rate of dissipation criterion. Of course, the above dependence israther specialized and does not address the entire gamut of possibilities. However, thisform is quite su�cient for illustrative purposes and is capable of modeling a wide range ofmaterials including dilatant, pressure sensitive materials such as those discussed by Spitziget al. 1973.


We now stipulate thatAssumption 3. Restricted maximum rate of dissipation criterion. The maximum rate of

dissipation criterion stipulated in condition 2 and embodied in eqns (17) and (18) are validonly over those admissible values of Lp obtained from Lp by varying E�r while holdingboth G and � ®xed.

Under the above condition, the results developed in Section III are valid with onlyslight modi®cations. Thus, instead of re-establishing all the results in Sections III and IV,we shall just point out the changes that need to be made to account for the restrictions.The reader can easily verify that the proofs established earlier carry over in a straightfor-ward manner.

In particular, Lemma 1 of that section is still valid but only over the subset of admis-sible values as de®ned in conditions (3). Moreover, Lemmas 2, 3 and 4 of Section III.1.1continue to hold provided we replace ��G;Lp� by ��;G;Lp� and ��G;�) (de®ned in (24))by ��;G;�� in eqns (20)±(27) and the in®mum on the right hand side of (24) is takenover the subset of admissible values of Lp obtained from Lp by keeping both � and G®xed.

The restricted maximum rate of dissipation criterion also implies that the parts (i) and(ii) of Theorem 2 continue to hold without modi®cation (other than the replacement of��G;Lp� and ��;G;Lp��. The main change occurs in part (iii) of Theorem 2 which ismodi®ed to:

iii. For ®xed values of G and � > 0, consider two points A and A0 in A, with A 6� A0

but with A�N � A0 �N, that satisfy

f�A;G� � � and f�A0;G� � �: �58�

Further, let Lp :� ~Lp�A;G� and Lp0 :� ~Lp�A0;G�: Then

�Aÿ A0��Lp � �A? ÿ �A0�?��L?p> 0 if L?p 6� �Lp0�?;� 0 if L?p � �Lp0�?;

(�59�

The proof of Theorem 2 with part (iii) modi®ed as shown above, is exactly the same asbefore except that we keep � ®xed. Speci®cally, Lemma 5 goes through unchanged so thatthe functions f and ~Lp can be constructed exactly as before.

The key observation that needs to be made here is that (59) is a restriction of (32) to thehyperplane N � A � const: Similarly, the statement of the convexity and normality condi-tions in Theorem 3, is modi®ed into the following form:

Theorem 4. Restricted convexity and normality conditions.The restricted maximum rate of dissipation condition 3 implies that

a. The restriction of the level set L��;G� of the function f�A;G� to the hyperplaneA de®ned by � � const: for each value of � � 0 is contained in the intersection of aconvex set C��;G� � A with the restriction of the range A of the function A to thehyperplane A. In particular, if we now de®ne A :� A \ A then

U��;G� :� fA 2 Ajf�A;G� < �g � C��;G�; �60�

B��;G� :� fA 2 Ajf�A;G� � �g � @C��;G�; �61�


where the notation @C��;G� represents the boundary of C��;G�.b. At all points of B��;G� (which, by (61) lie on the boundary of C��;G��;L?p is along

the direction of the outward unit normal to @C��;G�.The proof of the above theorem is along the same lines as that of Theorem 3, and so we

shall not repeat it here. We note in passing that the restricted maximum rate of dissipationcondition provides no information on Lp �N, which may be speci®ed independently.

Finally, in Theorem 4, we replace the function f�A;G� by f �A?;�;G�, stipulate that thelevel sets of f are de®ned with both G and � held ®xed, and assume, instead of item 4 inTheorem 4 that

lf��;A?;G� > f��; lA?;G�; �62�

for 0 < l < 1 and f ��;A?;G� > 0, and that the direction of �Lp�? is along the tensor@f=@A?. The value of Lp �N has to be speci®ed as a function of A and G independently.Once this is done, we calculate the magnitude of �Lp�? from the rate of dissipation equa-tion in the form (30). Then we can show that the restricted maximum rate of dissipationcriterion is satis®ed for any choice of the Helmholtz potential function that is twicedi�erentiable in E�r and G.

We can increase the number of restrictions on the set of admissible values by consider-ing many constraints of the form �=const. with di�erent values of N. Each such restric-tion implies that an additional component of Lp can be chosen independently, theremaining components being given by the normality condition.

VI. RELATIONSHIP TO CLASSICAL DEVELOPMENTS IN PLASTICITY AND RELATED

RESULTS

As we have noted in the introduction, the above developments have been inspired bydevelopments in plasticity such as the work inequality introduced by Il'iushin (1961) forsmall strains and later extended by Naghdi and Trapp (1975) to ®nite deformations aswell as the related work by Ziegler (1963, 1983) and Ziegler and Wehrli (1987).

In this section we point out the similarities and highlight the di�erences with otherdevelopments, beginning with classical rate-independent plasticity, followed by a com-parison of the present work with the work of Ziegler. We shall only consider materialswhose elastic domains have non empty interiors so that the discussion of Section II.2(particularly regarding the loading conditions) are relevant here.

VI.1. Quasistatic process and rate-independent plasticity

Within the context of the current developments, the constitutive equations that corre-spond to rate-independent response are not considered separately. Instead, these con-stitutive equations are obtained only as limiting cases when the strain trajectory alwayslies on or inside the classical elastic domain, i.e. as ``quasistatic'' limits. We shall rely onthe intuitive notion that such a response can be approached by considering very slowdeformations. During such processes, the trajectory in strain space approaches theboundary of the elastic domain from the outside.

Conditions at the boundary of the yield surface become very important for these pro-cesses and we shall explore this in detail in this section. We ®rst observe that the equation


f�A;G� � 0 represents the entire elastic range in A-space including the boundaries. Thecorresponding function in strain space is obtained by substituting (13) into the functionf�A;G�. We ®rst note that since the points in the elastic range satisfy f�A;G� � 0, thefollowing corollary of Theorem 1 part (iii) can be easily established:

Theorem 5. Corollary to Theorem 1iii. For ®xed values of G, let A0 be a point inside the elastic range in A-space. If A is any

other point outside the elastic range, and if Lp � ~Lp�A;G� 6� 0, then

�Aÿ A0�� > 0 �63�

where � is the unit tensor in the direction of Lp.Preliminary to discussing quasistatic processes, we ®rst consider the response of the

material to a step strain.

VI.1.1. Evolution of natural non®gurations at constant strain. In view of the constitutiveassumption (4) for Lp, it is clear that for ®xed E�r , the natural con®gurations evolve intime, and the resulting constitutive equations can be considered as a ``dynamical system''at each particle. Of course, nothing happens until the strain value is outside the currentelastic domain. The question arises as to the nature of the evolution of the natural con-®gurations if the material is deformed to a con®guration that is outside the current elasticdomain (for e.g. through a ``rapid path'') and then the shape is held ®xed. One wouldexpect on physical grounds (at least if E�r is ``close to the elastic domain'') that naturalcon®gurations initially evolve rapidly, gradually slowing down until the elastic domain``catches up'' and E�r is once more in the elastic domain. Clearly such behavior is deter-mined by the nature of the evolution equation (4). In particular, if the evolution equation isgiven by (47) so that it satis®es the maximum rate of dissipation condition, then the evolu-tion is governed by the nature of the rate of dissipation function f. The conditions underwhich the evolution is ``stable'' in the sense that G does not become unbounded dependsupon the ``plastic modulus'' (de®ned by @A=@G� and the hardening behavior. We shall notdiscuss this issue in detail here but instead simply assume that Lp approaches 0 with time ifE�r is held constant. When this happens, we see that the corresponding trajectory in A-spaceapproaches the elastic range in A-space from outside. The characteristic ``relaxation'' timefor this process is determined by the time constants associated with the dynamical system.

VI.1.2. Quasistatic processes. If we now consider processes that occur on a time scalethat is much slower than the ``relaxation time'' we arrive at the limiting case of ``quasi-static processes''. For such processes, the trajectory in A-space always lie on the boundaryof the yield surface while the magnitude of Lp tends to zero. We thus consider the limitingdirection of Lp in the following sections and demonstrate that this limiting direction isnormal to the yield surface in A-space.

VI.2. Limiting directions in A-space

We ®rst consider the situation in A-space for such processes. Now, by its very de®ni-tion, if any point Ab 2 A is on the boundary of the elastic range in A-space then, thereexists a sequence fAng of points, such that f�An;G� > 0 whose limit is Ab. Correspondingto this sequence of points, there exists a sequence of unit tensors, de®ned by


�n :�~Lp�An;G�k ~Lp�An;G� k

: �64�

Now, ~Lp�Ab;G� � 0 because ~Lp is a continuous function of A and Ab is in the elasticrange. Hence we cannot de®ne � at Ab through the function ~Lp evaluated at Ab. Indeed,the sequence �n de®ned in (64) may fail to converge. Moreover, if it does converge, theunit tensor to which it converges will in general but depend upon the particular sequenceconsidered. In the event that a sequence f�ng converges, we shall de®ne a limiting direc-tion of Lp at a point Ab on the boundary of the elastic range as

�b � limn!1;f�An;G�>0

~Lp�An;G�k ~Lp�Ap�An;G� k

�65�

where the An converge to Ab. By this de®nition, there may be many limiting directions fora given value of Ab.

Now since each of the �n satis®es (63) with A replaced by An, we can easily see that, inthe limit as n tends to 1 we arrive at

�Ab ÿ A0��b > 0; �66�

for all A0 in the elastic range and for all the limiting directions �b at Ab. Now a standardargument exactly along the lines used to establish the normality and convexity conditionsin Theorem 2 can be repeated to show that

Theorem 6. Convexity of the elastic range in A-space and normality of the limitingdirections

a. The elastic range in A-space is contained in the intersection of a convex set C�G� � Ewith the range A of the function A. In particular, the boundary with respect to A of theelastic range, is contained in the intersection of the boundary of C�G� and A.

b. At each point of the boundary of the elastic range, the limiting directions �b areelements of the normal cone to C.

Note that, the current approach allows for the presence of corners in the boundary ofthe elastic range although for points corresponding to f�A;G� > 0, the level sets have nocorners and Lp is uniquely de®ned (see Theorem 2). The above theorem is identical to thatestablished by Srinivasa (1996, Theorem 4.1) where it has been shown to be necessary andsu�cient for the satisfaction of a work inequality for rate-independent elastic-plasticmaterials (see Naghdi and Trapp, (1975)). Thus, the above theorem, together with thatestablished by Srinivasa (1996, Theorem 4.1) imply that the maximum rate of dissipationcriterion, when applied to processes wherein the strain trajectories are always along theboundary of the elastic range as described above, is identical to the work inequality.However, in a general process, the two conditions may not be the same.

VI.3. Comparison of the maximum rate of dissipation criterion with traditional approaches

The structure of our constitutive equations allows for a wide range of response includ-ing materials whose elastic domain in strain space have empty interiors. The currentapproach is valid for arbitrary deformations and, when the maximum rate of dissipationcriterion as stated in condition (2) is adopted, delivers normality and convexity conditions


in A-space. Moreover, in the current approach, the yield function in A-space has the clearphysical meaning of the rate of conversion of work into heat, associated with that value ofA. Finally, as pointed out in the previous subsection, it reduces to the strain-space for-mulation of the rate-independent theory of plasticity when we consider quasi-staticresponse. Furthermore, the classical normality and convexity conditions, which havehitherto been obtained by the use of the work inequality in the rate-independent theory,were also recovered as limiting cases.

This approach has certain advantages over the classical viscoplasticity models due toPerzyna (1963). In common with the current approach, the model proposed by Perzyna(1963) possesses instantaneous-elasticity, and, assumes an additive decomposition of thetotal in®nitesimal strain into an elastic and plastic part. A constitutive equation thatrelates the rate of plastic strain to the stress tensor is then assumed. This approach, whichis limited to small deformations, cannot be directly extended to large deformations with-out substantial changes. Moreover, it is a purely mechanical theory that has no thermo-dynamical framework. Speci®cally, the ``dynamical yield surfaces'' introduced there lackany physical signi®cance in terms of either the storage or rate of dissipation of energy.

Several other models based on the notion of `òverstress'' have been proposed (see e.g.Phillips and Wu, 1973), but we shall not discuss them here, instead referring the interestedreader to Naghdi (1990, Section 6B) for a detailed review. In some of these theories thematerials do not possess instantaneous elasticity and thus do not fall under the categoryof models that we are concerned with in this paper. Finally, since these theories are basedon the same fundamental assumptions that underlie classical plasticity theories, they haveno general framework for the modeling of changes in material symmetry.

The main thrust of Part II of this two-part paper has been a precise and careful state-ment of the maximum rate of dissipation criterion (see condition (1)) and the proof that itimplies and in turn is implied by the normality and convexity conditions for viscoplasticmaterials. The current procedure has some features in common with that of Ziegler (1963,1983) and indeed many of the developments here are motivated by these works. There arehowever many crucial and signi®cant di�erences in the approach both at a fundamentallevel regarding the meaning of the primitives in the theory as well as the mathematicallevel regarding the conclusions that can be drawn from certain assumptions.

We ®rst consider some minor errors in Ziegler's analysis that can be easily set right withonly minor modi®cations. At the outset we hasten to add that these errors notwithstand-ing, the work of Ziegler (1963, 1983) infuses new, insightful ideas that leads to a betterunderstanding of the inelastic behavior of materials. Our motivation in discussing theseissues is to set these ideas in a fairly rigorous mathematical framework.

Ziegler (1983) begins by considering in®nitesimal deformations6 and introduces two newprimitives into the theory other than the classical notions of forces and displacements:`ìnternal parameters'' denoted by �ij and corresponding `ìnternal forces'' denoted by �ij(see Ziegler, 1983, Chapter 4, pp. 65±69, as well as Ziegler and Wehrli, 1987, pp. 187±189)without any speci®c de®nition for these quantities. Moreover, after assuming that thestresses and the internal forces are both decomposed into a quasiconservative part that is


*As noted by Ziegler and Wehrli (1987, p. 186, para. 2) in a recent review,``...the review will be restricted, wherever the strain tensor becomes part of the development, to its ®rst

approximation, that is to in®nitesimal strains. The authors hope that the results may motivate future workextending the method to ®nite deformations''.

obtained by di�erentiating the free energy and a dissipative part that requires a separateconstitutive equation, the classical entropy inequality is modi®ed in an ad-hoc manner byadding to it the inner product of the rate of the internal variables with the corresponding``dissipative part'' of the `ìnternal forces'' (see Ziegler and Wehrli, 1987, eqns (2.14)±(2.16)). Finally it is stipulated that (cf. Ziegler and Wehrli, 1987, p. 189, lines followingeqn (16)):

``However, since the �ij are internal parameters, the �ij do not appear in the ®rst fun-damental law, and if we exclude gyroscopic components..., we have �ij � 0''.

Thus in their approach, the `ìnternal forces'' �ij are introduced only to be set to zero afew lines later.

The restriction to in®nitesimal deformations does not in any way restrict the funda-mental idea behind the approach of Ziegler (1983) and can be relaxed without signi®cantlymodifying the basic ideas. The introduction of the `ìnternal forces'' �ij, does not appear toconfer any additional generality to the theory (especially in view of the fact that they areset to zero). However, their presence raises fundamental questions as to the nature of theprimitive variables in the theory. Indeed, the entire theory can be developed without theintroduction of these `ìnternal forces'', thus reducing the number of primitives in thetheory without sacri®cing generality.

Focussing attention on the more fundamental issues, the internal parameters �ij,although similar in practice to G utilized here, are introduced directly as primitives withtheir identi®cation deferred until speci®c models are introduced. This approach is similarto that taken in classical plasticity theories and is subject to the same criticisms discussedin Part I of this paper.

In our approach, both the tensor G and its power conjugate in the form of the ``drivingforce function'' are derived quantities. Finally, instead of an ad-hoc addition of the powerdue to the change of the tensor G into the entropy production inequality, we derive itdirectly from the rate of dissipation equation (14).

Turning our attention now to the maximum rate of dissipation criterion, we state it inthe form of a physical assumption on the nature of the dissipative forces (see Assumptions(1) and (2) in Section III) and use that to show that normality and convexity in A-spaceare both necessary and su�cient for its satisfaction. In this approach, since neither G nor Aare primitives, the admissible values of Lp and A have to be delineated prior to theapplication of the maximum rate of dissipation criterion.

Ziegler (1963, 1983) and Ziegler and Wehrli (1987) do not use the maximum rate ofdissipation criterion as a restriction on the constitutive variables. Instead they assume thatthe ``dissipative forces'' (which are akin to A in the current approach) must be ``derivablefrom the dissipation function'' (cf. Ziegler (1983, p. 253, line 5) and hence claim (cf.Ziegler and Wehrli, 1987, p. 190, lines following eqn (2,20))

`Ìf this direction has to be given by �, it must as has been shown by (ZA4.3), bedetermined by the gradient @�=@qi:::.''

In the above quotation, � corresponds to the rate of dissipation function � in the cur-rent paper, qi are the `ìnternal parameters'' (corresponding to the components of G) and``(Z.14.3)'' stands for Chapter 14.3 in Ziegler (1983).

A close look at the relevant chapter in Ziegler (1983) reveals that the derivation of theresult is based on the use of a Taylor expansion that is not coordinate free. Speci®cally,Ziegler (1983) expands the function � in a Taylor series and then argues (cf. Ziegler, 1983,Chapter 14.3, pp. 253±255) that the `ònly vector'' in the Taylor series expansion is the


gradient @�=@qi and hence it follows that the dissipative forces must be directed along thenormal. He then goes on to dismiss inner products between higher gradients claiming thatthey are not invariant to coordinate changes. The last statement is false-since the Taylorseries expansion as well as the inner products can be done in an entirely coordinate freeway for any analytic function de®ned on a Hilbert space (cf. Cartan, 1983) so that theyremain valid for any curvilinear coordinate system. The root of Ziegler's (1983) erroneousconclusion lies in the fact that he did not take into account the metric tensor when car-rying out the inner products (see Ziegler, 1983, p. 254, eqn (14.34) and the lines immedi-ately following it). It is the metric tensor which ensures that the result of the Taylor seriesis coordinate independent. Ziegler's (1983) conclusion is all the more puzzling since thereis a discussion of curvilinear coordinates in the later chapters.

If one ignores Ziegler's (1983) argument regarding the direction of the ``dissipative for-ces'', his conclusion becomes nothing more than a purely mathematical assumption ofnormality. Then, using a number of assumptions regarding the smoothness and invert-ibility of the functions he arrives at the maximum rate of dissipation condition as anecessary consequence of the assumed ``orthogonality condition'' for certain special clas-ses of materials. Thus, Ziegler (1983) accomplishes one part of the proof, namely neces-sity. Conceptually, in Ziegler's (1983) approach, the mathematical notion of``orthogonality'' is assumed and the ``maximum rate of dissipation'' is merely a con-sequence of this notion.

It is clear that Ziegler (1963, 1983) and Ziegler and Wehrli (1987) have realized thefundamental importance of the rate of dissipation function � as well as the idea of themaximum rate of dissipation criterion in determining the constitutive equations thatdescribe the dissipative response, as evident from their numerous writings on the subject.The present approach clari®es the role played by the rate of dissipation function, and bystating the maximum rate of dissipation criterion in a precise form and combining it withthe concept of materials with multiple natural con®gurations (discussed in Part I), exploitsits full potential.

AcknowledgementsÐThe authors wish to acknowledge the support of the National Science Foundation incarrying out this research.

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