A theory of inelastic behavior of materials Part II. Inelastic materials

A Theory of Inelastic Behavior of Materials Part I. Ideal Inelastic Materials

MIROSLAV SILHAV'f & JAN KRATOCHVfL

Communicated by W. NOLL

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2. Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3. Body Elements, Material Elements . . . . . . . . . . . . . . . . . . . . . . . . . 100 4. Generalized Elastic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5. Instantaneous Symmetry Group . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6. Plastic Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7. Symmetry Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8. Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9. Flow Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

10. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

1. Introduction

In this two-par t paper we at tempt to describe within the framework of NOLL'S new theory of simple materials [1] some aspects of mater ial behavior which are usually called inelastic. In the proposed theory our a t ten t ion is focused on three basic features of inelastic materials 1 (see e.g. [2]):

(a) Such materials behave elastically under certain sufficiently fast deforma- tions.

(b) Part of the deformat ion of an inelastic material , called here "plast ic d is tor t ion" , brings the material to a physically equivalent state.

(c) Deformat ions of inelastic materials are generally accompanied by changes in material structure.

Here in Part I we int roduce the concept of an ideal inelastic material and describe its behavior. Such a material exhibits features (a) and (b) bu t not (c). Inelastic materials which in addi t ion behave as in (c) will be studied in Part II.

The term "inelastic material" denotes here a class of materials which exhibit elastic, viscous and plastic behavior. A model of inelastic materials based on the properties (a)-(c) has been described in [3, 4].

98 M. SILHAVY & J. KRATOCHV~L

An ideal inelastic material is a simple material [1] endowed with additional mathematical structure that is intended to describe what may be called "elastic- ideal viscous-plastic" behavior. The simple material employed in Part I is specified by the first three axioms of NOEL [1] and a weak form of the accessibility axiom (Axiom VI in El]). (These axioms are denoted here as Axioms I-IV.) The re- maining two axioms which enter NOEL'S definition of simple materials in [1] are not necessary for the present considerations. The additional mathematical structure associated with elastic-ideal viscous-plastic behavior is expressed in Axioms V-XI. Among them Axioms VII, X, and XI concern some technical aspects of the theory and do not seem to be very restrictive. Axioms V, VI, VIII, and IX express the significant physical properties of ideal inelastic materials.

Axiom V formulated in Section 4 postulates the existence of a generalized elastic range, originally suggested in [5], which is analogous to the concept of elastic range proposed by OWEN E6] and considered further by KOSINSKI [7]. Axiom V provides that ideal inelastic materials behave elastically when subjected to very fast deformation processes which remain within the generalized elastic range.

In Axiom VI, stated in Section 6, we assume that the physical state reached by an ideal inelastic material at the end of a deformation process may be considered as resulting from two transformations. The material initially is subjected to a suitably selected symmetry transformation and then undergoes a fast elastic jump into the final configuration of the deformation process. The change of state associated with a deformation process is thereby resolved into what may be called "plastic" and "elastic" parts.

Using appropriate symmetry concepts introduced in Section 7, we study the non-uniqueness of plastic deformation and specify a condition under which the ideal inelastic material degenerates into an elastic material.

Axioms VIII and IX are important for the derivation of an evolution equation for the distortion corresponding to plastic deformation. Axiom VIII provides an ideal inelastic material, is isotropic, and its symmetry group is contained in the unimodular group. As a consequence we show in Section 8 that the symmetry group of a material which satisfies Axioms I-VIII must be the whole unimodular group and its elastic response is isotropic.

Axiom IX stated in Section 9 asserts that the initial state of an ideal inelastic material is completely determined by its final state and the previous deformation

\ process. This axiom is satisfied in the theories of rate-sensitive plastic materials which employ "internal variables" in formfilating constitutive relations (e.g. [8-12]). It is possible to show, however, that Axiom IX is not valid for materials with elastic range [6]. Thus Axiom IX excludes from our consideration the rate- independent theories of plastic behavior which rest on the assumption of the existence of an elastic range. However, the results obtained in Sections 1-8, which are derived without Axiom IX, are valid in both the rate-dependent and rate- independent cases.

Using Axioms I-XI, we prove two main results. First we show that each state of an ideal inelastic material is completely determined by its current configuration and a certain element of the symmetry group of the material. The element of the symmetry group connects two physically equivalent states and is called, in

Ideal Inelastic Materials 99

agreement with KRONER [2], plastic distortion. Second we prove that the evolution of plastic distortion during a deformation process is governed by a differential equation called a flow rule. It turns out that the analytical form of the flow rule is determined by a functional which depends on deformation process and values of plastic distortion only through their values in the immediate past. The flow rule derived here is similar to constitutive equations for measures of plastic deformation assumed in the internal variable theories of plasticity e.g. [8-12].

2. Basic Notation

Through this paper we use NOLL'S notation [1] concerning finite-dimensional real vector spaces. In particular, the set of real numbers is denoted by IR and the set of non-negative reals by IR+; IR ++ denotes the set of all positive reals. If

and ~22 are finite-dimensional real vector spaces, then Lin(J~, ~2) denotes the set of all linear transformations from ~ into ~ and Invl in(~ , Y22) is the set of all invertible linear transformations from Yll onto ~ (of course, Invlin ( ~ , ~-~2)4= if and only if dimension of ~ = dimension of J22). If Y is a finite-dimensional real vector space, we write Lin(3-) and Invlin(3-) instead of Lin(J,,3-) and Invlin (J,, 3-). The dual of 3- is the finite-dimensional real vector space 3-*=Lin(Y,,IR). The following subsets of Lin(J,, 3-*) and Lin(3-* 3-) are significant in continuum mechanics: (a) The set Sym+(Y,, 3-*)oLin(3- , 3-*) of all positive definite symmetric mappings from 3- onto 3-*; the elements of Sym+(3-,3- *) will be identified with the intrinsic configurations of the body element; (b) The set Sym(3-*3-) of all symmetric linear transformations from 3-* onto 3-; the elements of Sym(3-* 3-) are the possible values of an intrinsic stress of the material element.

The unimodular group of 3- will be denoted by Unim (3-); we have Unim(3-) c Invlin (3-).

If G0eSym+(3-,,3-*), then the orthogonal group of G o will be denoted by Orth(Go). We have Orth(Go) cUnim(3- ) .

If AeLin(3-) , then A* will denote the adjoint of A; we have A*eLin(3-*). If A~Invlin(3-), then also A*~Invlin(3-*) and (A*)-I=(A-1)*; we use the abbreviation A -* = (A*) - 1 = (A - 1)..

In addition to the above notation introduced in [1], we mention also the set of all transformations symmetric with respect to Go eSym+(3-,3-*). If G o ~Sym+(3-,,3-*), we define the Go-transpose of an element AeLin(3-) as follows:

A r = G o l A * Go e Lin (Y).

If A~Invlin(3-), then also AT~Invlin(3-) and (AT)-I=(A-1)T; w e use the abbreviation (At) -1 = ( A - I ) T = A -r . We say that A~Lin(3-) is symmetric with respect to G o if A r = A, i.e.

A = Go 1 A* G O .

We define

Sym(Go) = {A ~ Lin(3-)lA = G o l A * Go}.

100 M. ~ILHAV~( 8,~ J. KRATOCHViL

We say that A~Sym(Go) is Go-positive definite if (GoA v, v) is positive for all v 4=0, ve~--. The set of all elements of Lin(Y) which are positive definite and symmetric with respect to G O will be denoted Sym+(Go). We have

Sym + (Go) c Sym (Go) ~ Lin (~'~).

3. Material Elements

We now recall the concepts introduced by NOLL [-1] which are essential for consideration in the present paper.

In continuum mechanics a continuous body ~ is usually defined as a set endowed with a unique structure of a differentiable manifold. Therefore one can associate with each material point X e ~ a tangent space ~x, which is a finite- dimensional real vector space. Roughly speaking, the members of ~x describe the material points in an infinitesimal neighbourhood of X; therefore Yx is called the (infinitesimal) body element at X of the (global) body N.

In the new theory of simple materials [-1] NOLL uses a suitable class A of mappings 6 : ~ x N - - * I R § called distance function, to describe the structure of a continuous body on N. The distance functions in A are the possible (intrinsic) configurations of the body N. A configuration 6cA assigns to any two material points X, Y e ~ a number ~(X, Y)elR +, which is called the distance between X and Yin the configuration 6. As the elements of A do not involve extrinsic frames of reference, the description of the body structure in terms of the class A is completely intrinsic.

A configuration 6 : . ~ x . ~ l R + induces, by localization, on each of the tangent spaces Jx a positive definite symmetric bilinear form 6ix: ~x x Yx~IR. Such bilinear forms are called (intrinsic) configurations of the body element ~ . We see that each configuration of ~x may be identified with a unique member of Sym+(Jx, ~x*). As we shall deal exclusively with body elements rather than with global bodies, the reference to X will not be indicated further.

By a deformation process we mean a one-parameter family of configurations of J . More precisely, the deformation process for a body element J is a function P:[O, dp] ~ S y m + ( ~ , ~-*), where the non-negative number de is called the duration of the deformation process and the values p i = p ( o ) and p S = p ( d e ) are referred to as initial and final values of the process.

It is useful to introduce the following special types of deformation processes. If GeSym+(~--,, J-*) and telR +, a deformation process G(t) defined by G(t)(r)= G for re[0 , t] is called the freeze of duration t at G.

If P is a deformation process and tl, t 2 E [ - 0 , de] , with t 1 < t2, we define a new deformation process P~t,, t21 of duration t 2 - - tl, by P~t,. t2~ (t) = P(t + t 1) for tE [0, t 2 - q]. PEt,,t2J is called a segment of the given deformation process P.

If P~ and P2 are deformation processes such that P11 = P2 i, then a new process P1 * Pz, of duration de, + de2, defined by

~P~ (t) if re[0, de, I, (el * P2)(t) = (P2 (t - de,) if t 6 [-de, , de, + dp 2] (3.1)

is called a continuation of P1 with P2.


If a, doelR +§ we call an increasing function q~ from [0, do] onto [0, a] a rescaling function of the time interval [0, a] and we denote the set of all such functions N,. Let P be a deformation process and ~oe~d~; then the new deformation process P o (p, of duration do, is referred to as a resealing of the given deformation process P (see [5]).

In the precise formal definition of a body element NOLL [1] specifies its detailed structure. The structure of a body element contains all information about local kinematics at X and consists of:

(a) The tangent space Y of the body manifold at X.

(b) The set f# of possible configurations of the body element.

(c) A class H of deformation processes of the body element.

The triplet (5, f#,/7) is subjected to a number of axioms which will be not repeated here.

In this paper we shall only deal with a special case of a body element, for which the set f# is identified with Sym+(~,, 3-*) and the class 17 contains all continuous and piecewise continuously differentiable deformation processes and deformation processes of duration zero. Similarly we shall only consider rescaling functions which are continuous and have a piecewise continuous derivative. Since the body element considered is determined fully by the real vector space ~,, we shall denote this body element by J . It is not difficult to prove that 5- satisfies all axioms defining a body element according to NOLL [1].

A material element, the main concept of the new theory of simple materials [1], is defined as a body element endowed with additional structure that is designed to describe mechanical material properties. The definition associates with the material element a space of possible physical states. The state determines everything about the material element, most importantly, the response of the material element in every possible test. In the present framework a " tes t" means subjecting the material element to a deformation process and measuring the stress at the end. We use a stress measure called the "intrinsic stress" S, which was introduced by NOLL [1]. S is an element of the stress spaeeSf = Sym(3-* 3-). S describes the contact forces acting on the material element intrinsically, without reference to any frame space.

Thus when speaking about a material element 3-- we consider it understood that 3- is endowed with a structure implied by prescription of the following objects:

(a) The set Z called the state space of the material element ~. The elements a of 2; are the possible physical states of 3-.

(b) The mapping G: Z ~ (r from the state space 2; into the configuration space (r The value G(a) is the configuration determined by the state a.

(c) The mapping S: Z ~ 5 f from the state space X into the stress space 5 f. The value S(a) is the (intrinsic) stress determined by the state a.

(d) The mapping ~ : ( Z x / 7 ) f i c - , , Z , called the evolution function of Y,, whose domain

(Z • {(0", P) l a ~ Z, P ~ /7, pi= ~(o')} (3.2)

102 M. ~ILHAVY 8~ J. KRATOCHViL

is the set of all state-process pairs such that the state "fits" the initial configuration of the process. The value/)(a, P) is the state reached by the material element if, starting from the state a, it is subjected to the deformation process P.

In addition to (3.2) the following notation will be useful:

(i) The set of all continuous and piecewise continuously differentiable deformation processes beginning at the configuration G e ~-- Sym + (~,, ~-*) is denoted by

H o = {P~HIP i= G}. (3.3)

(ii) If E c fq and G E E, then the set of all continuous and piecewise continuously differentiable deformation processes that begin at the configuration G and whose values are in the set E is denoted by

FI ( E)o = { p ~ / / [ p i= G, P([0, dp])c E}. (3.4)

(iii) The set of all states that fit the configuration G is denoted by

22o = {ae2;I O(a) = G} (3.5)

and called the G-section of 22.

(iv) The response functional S of the element is defined by

= S o/~: (22 X/-/)flt---~. (3.6)

In [1] the quantities ~,, X, (~, S, and ~3 are assumed to satisfy six axioms. Only the first three axioms and a weak form of the accessibility axiom (Axiom VI in [1]) enter the considerations of this article. These axioms are denoted here as Axioms I-IV.

Axiom I. For all (a, P)~(Z x//)fi t ,

~(o,P)sZv: , i.e., ~(~3(a,P))=P:. (3.7)

Axiom II. I f P1,P2~Z, ff~Zpi and Plf=P2 i, then

8 * 8), (3.8)

Axiom III. For any Ge~, if al, a2~X G and

S(r 1, P)= S(a 2 , P) (3.9)

for all P ~ H 6, then a 1 =or 2.

Before formulating the weak accessibility axiom we introduce the concept of natural uniformity on the sections of the state space and the concept of natural topology on the state space (see [1], Section 11).

Let ~ be a material element with response functional ~. For every G~Sym+(J,, ~--*), we call the natural uniformity of the G-section X o the coarsest uniformity on S o which renders the mappings

S(., P): Xo~ .Y (3.10)


uniformly continuous for all PeH~ . We call the natural topology of the state space S=U{SGI GeSym+(W, J*)} the sum of the topologies on the sections S G induced by the natural uniformities of these sections.

Axiom IV. There is a 2o~S such that for each a~S, we can find a mapping i~"*Pi from some directed set into the set /Td~o) which satisfies

a = lim ~ (20, P~). (3.11) i

NOLL'S original formulation of the accessibility axiom postulates that each state a~S, is accessible from some relaxed state. This assumption is unnecessarily strong for our purpose, and therefore we require only that all states of the material element be accessible from a certain state which need not to be relaxed.

According to NOLL [1], the symmetry group of a material element 3- is defined as the set g of all A e l n v l i n ( J ) for which there exists a bijection za: S ~ Z such that

~(cr) = A* (~ (ZA (or)) A (3.12)

= A * , (3.13)

(ii) for all (a, P)e(S x/7)fit

~3 (,A (a), A - * PA =1) = tA([~(a, P)). (3.14)

The following propositions are stated and proved in [1]:

Proposition 3.1. The bijection t A of (3.12)-(3.14) is uniquely determined by the mapping A eg.

Proposition 3.2. I f A 1, A 2 e g, then

IAxAt=IA2O lA1 and IAI-t=(IAt) -1 . (3.15)

Let a e S be any state. The set

g~ = {A e g I ZA (a) = or} (3.16)

of all symmetries that leave the state cr invariant will be called the symmetry group of the state a. Of course g~ is a subgroup of g.

The following propositions are relevant for considerations in this paper.

Proposition 3.3 (see Proposition 10.2 in [1]). The symmetry group g, of a state a is a subgroup of the orthogonal group of G(a) and hence

g~ c g ~ Orth ((~ (a)). (3.17)

Proposition 3.4 (see Proposition 11.3 in [1]). A net (i~--~a 3 in S, converges to a 6 S if and only if (i) ~r i belongs eventually to some fixed section SG, (ii) a6SG, and (iii)

lim S(a,, P)= S(a, P) (3.18)

for all P ~/7~.

all a c S (i) for

and

104 M. SILHAVY & J. KRATOCHViL

Proposition 3.5 (see Proposition 11.4 in [1]). The state space Z with its natural topology is a Hausdorff space, i.e., nets in Z cannot have more than one limit.

The direct consequence of the definition of the symmetry group g and the natural topology of Z is the following proposition:

Proposition 3.6. For each A ~g the mapping /a: -Y---~S is continuous.

4. Generalized Elastic Range

An ideal inelastic element is defined in the present article as a material element which satisfies additional conditions expressed in Axioms V-XI. The conditions are designed to describe characteristic properties of ideal elastic-viscoplastic materials. Axiom V, stated in this Section, ensures that an ideal inelastic element behaves in certain limiting cases elastically.

In Axiom V we shall assume the existence of a generalized elastic range. The notions of the generalized elastic region and the generalized elastic range have certain similarities with the concepts of the elastic region and elastic range defined by OWEN [6]. Roughly speaking, OWEN defines an elastic region corresponding to a given deformation history as a set E of deformation gradients such that the stress functional is path independent on the set of all continuations of the given history which remain in E. The elastic range corresponding to a given deformation history is then defined as a maximal elastic region corresponding to this history.

The main difference between the concept of generalized elastic region and that of elastic region is that we require in the definition of generalized elastic region corresponding to a state a path independence of the functional ~3(a, .) only in the limit of fast processes. To specify this limit we shall use the concept of rescaling given in Section 3. If P is a deformation process and (p a rescaling function of the time interval [0, de], such that d r < dp, then the rescaling P o ~0 assumes the same sequence of configurations as the deformation process P but at higher average speed. One may then consider limits as d r ~ 0 of quantities depending upon P o q~ (see Condition E4 of the following definition).

Definition 4.1. Let a ~ Z and let E be a non-empty subset off#. We say that E is a generalized elastic region corresponding to 6, if the following four conditions are satisfied: 2

El. E = c l i n t E;

E2. For any two elements G1, GE~E there exists at least one deformation process

P ~ I I such that p i = G1 ' p f = G2 and P([0, de] )~E;

E3. G(a)~E;

E4. There is a mapping p(tT, E; .): E ~ Z satisfying a=p(a , E; G(o')) such that for every P 6 l l (E)d~o ) with dp>0

lim0t3 (a , p o q3)= p(a, E; Pf), (4.1)

2 If B is any subset of ~, then the symbols cl B and int B denote, respectively, the closure and the interior of the set B in the topological space ft. In particular, clC~ = in t a J= f f .


where q~6~dp, i.e., for every neighborhood Y of p(a, E ;P r there is a 3 > 0 such that ~(a, P o q~)eYwhen cp~tdp and d~<6. (The limit (4.1) is unique by Proposition 3.5.)

Conditions E l - E 3 delimit the basic topological nature of the generalized elastic region. Condition E4 contains the main idea of the definition of generalized elastic range. It requires that the material element which has generalized elastic range behaves elastically in the limit of increasingly fast processes from II(E)~(~).

We say that a generalized elastic region E corresponding to a is a generalized elastic range corresponding to a, if E contains all other generalized elastic regions corresponding to a. The generalized elastic range corresponding to a will be denoted by E(a).

Axiom V. For every a e Z there is a generalized elastic range corresponding to ~.

The following notation will be used:

(i) The mapping p(a, E(a);.): E ( a ) ~ Z from Definition 4.1 is denoted by

p+(a, .): E ( a ) ~ Z . (4.2)

(ii) The range of p + (a, .) is denoted by

g(a) = p + (a, E(a)). (4.3)

(iii) We shall use the abbreviation H~ for the symbol Fl(E(a))e(~ ), i.e.

II~=17(E(a))~(~). (4.4)

(iv) The instantaneous response function S+(a, .) corresponding to a is defined by

s+(~, . )=~ o p+(~, .): E(~)-. ~ . (4.5)

For every GeE(o), the value S+(a, G) is the stress produced in the limit of fast deformation processes PeH~ applied to the state ~ and with P: equal to G.

Using arguments similar to those given in [5] (see Proposition 4.1 and 4.2 in [5]), we are able to prove certain natural properties of the generalized elastic range E(a).

Proposition 4.1. Let o, 2~Z, with 2Eg(a); then

E(~)=E(2), g (a )=g(2) , (4.6)

p+(~, . )=p+(2, . ) , S+(~, .)= S+(2, .). (4.7)

In the next proposition we establish the transformation laws for the mappings a~-~E(cr), G~--~8(cr), ~--~p+(a,.), a~-~S+(~,.) under a symmetry transformation of the state space tT~--~ lA(t~ ).

Proposition 4.2. Let A ~g and a~Z. Then

E ( I A ( ~ ) ) = A - * E ( ~ r ) A -1 , O~(IA (0")) = 'A (g(cr)) (4.8)

106 M. ~ILHAV~" & J. KRATOCHVIL

and for every GeE(a) p + (za(tr), A-* GA =1)= Za(p+(cr, G)), (4.9)

S+ (zA(tr), A-* GA =1) = AS+(tr, G)A*. (4.10)

Proof. First let us show that A-*E(~)A -1 is a generalized elastic region corresponding to la(a), i.e. Conditions E l - E 4 from Definition 4.1 must be verified.

E 1. Since E(a)= cl int E(a), and since the mapping G~-*A-* GA-1 is a homeo- morphism for each invertible linear mapping A from Y- onto ~--, we conclude that also A-* E(a)A-I=clintA-* E(a)A -1

E2. Let G1, GzeA-*E(a)A-1; then A*GaA, A*G2AeE(a ). As E(tr) satisfied E2, there exists a process PelI such that A* GA =pi, A* G2A =P: and P([0, dp]) cE(tr). The process A-*PA -1 then satisfies (A-*pA-I)~=G1, (A-*PA-~): = G 2 and A-*PA-I([O, dp])cA-*E(a)A -1.

E3. From G(~r)eE(a) it follows that GOa(a))=A-*G(a)A-IeA-*E(a)A -1. E4. Let us define a mapping p(ZA(a),A-*E(a)A-X;.): A-*E(a)A-a~X by

p(Za(a),A-* E(a) A-1; a-* GA =a)= lA (p + (tr, G)) (4.11)

for every GEE(cr). From (4.11) we have

p(tA(tr),A-* E(tr)A-1; (~ (L4(tr)))= p (za (tr), A-* E(a)A-~; A-* G(a)A =1) = ( p +

= ,~ (G) .

Further, each process /5 from the set II(A-*E(a)A=I)~oA(~) ) is of the form /5=A-*PA -1, where PdI(E(a))o(~). Hence, if q ~ n ~ , , d ~ 0 , then

lim r (ta(tr),/50 ~p)=limt~0a(g), A-*(p o ~o)A =1)

= lim z a (~ (~r, P o q~))

= ta(lim r P o ~0)) = tA(p+(~, p:)) (4.12)

= p(ta(~r),A-* E(tr) A-1; A-* Pf A ~1)

= p(ta(tr), A-*E(~r) A-1;/sf).

In the above calculation the continuity of the mapping ta: I ~ I , proved in Pro- position 3.6, has been employed.

Thus, by Definition 4.1, the set A-*E(a)A -1 is a generalized elastic region corresponding to z a (cr). Axiom V then implies that

A-* E(cr) A -1 =E(ta(cr)). (4.13)

If we apply the inequality (4.13) to the state a '=ta(cr ) and the symmetry a ' = A - ~ g , we have (A')-*E(~')(a')-l=E(ta,(~r')); that is, A*E(ta(ir))A = E(ta-, (ta(~r))), or equivalently

E(tA(G)) = A - * E(a) A-1. (4.14)

From the last relation and (4.13) the relation (4.8)1 follows immediately.


The equations (4.8)1 and (4.12) indicate that the mapping

p(ta(a),A-* E(a)A-1;.): E(IA((7))--> Z

given by (4.11) is identical with the mapping p+(lA(a),. ). This fact yields the proof of (4.9) and the relation (4.8)2 is then an easy consequence of (4.9) and the definition of g(a) and g(zA(a)). The equation (4.10) follows directly from (4.5), (4.9), and (3.13). Q.E.D.

5. Instantaneous Symmetry Group

To each state a sX we now associate an instantaneous symmetry group which can be considered as a symmetry group of instantaneous (or elastic) response of the element.

Let A ~g,; then, by the definition of the symmetry group g, of a state a, zA(a ) = a. Using tA(a)=~r on the left hand sides of(4.8)-(4.10), we get for every A~g,,

E(a)=A-*E(a)A- ' , ~(a)= ,a(o~(a)) (5.1)

and for every GeE(a)

p + ( a , A - * C A - ' ) = , A ( p + ( a , G)), (5.2)

S+(a,A-*GA-1)=AS+(a, G)A*. (5.3)

One cannot generally expect that g~ is the maximal set for which (5.1)-(5.3) hold. Therefore we introduce the following definition.

The instantaneous symmetry group of the state a~X is the set ~,~ of all A~g for which (5.1)-(5.3) hold. It follows easily from this definition that Ae~, if and only if Aeg and

E(a) = A-* E(a) A - ' (5.4)

and for all GeE(a)

p+(~, A-* a A - ' ) = zA (p +(o, a)).

In fact, if (5.4) and (5.5) hold, then

e (o) = p + (o, E (a))

=p+(a,A-*E(a)'A-') .

= ,a(p +(a, E(~))) = z~ (e(a)).

Similarly, if GeE(a), then

S + (a, A-* GA -') = ~S(p + (a, A -* GA -'))

= S (t a (p + (a, G)))

= A ~ (p + (a, G)) A*

=AS+(a, G)A*.

(5.5t

(5.6)

(5.7/

108 M. ~ILHAV~" & J. KRATOCHViL

Somewhat more suprising are the following characterizations of ~,~.

Proposition 5.1. Let aeS . Then the following three conditions are equivalent:

(i) A e ~ . (ii) A e g and for all 2eg(a) we have zA(2)Eg(a ).

(iii) A e g and for some 2oeg(a ) we have la()~o)eg(O).

Proof. By the previous considerations (see (5.6) and (5.1)2) (i)~(ii). The implication (ii)~(iii) is trivial. We now prove (iii)~(i). If there is a 2oeg(a) such that L4()~0)eg(G), then, by Proposition4.1, E(la()~o))=E()~o)=E(a), and for each GeE(a) we have p+(la( )~O) , G)=p+(2o, G)=p+(a, G). On the other hand, Pro- position4.2 yields EOA(XO))=A-*E(2o)A -1 and for every GeE(2o)=E(a ) we have p+0a(2o), A - * G A - ' ) = ta(p+(20, G)). Hence

E(a)= E(ta(2o))= A - * E()oo) A -1 = A - * E(G) A -1, (5.8)

and for every GeE(a)

p + (a, A - * GA -1) = p + (tA (2o), A - * GA -1)

= t A (p + (20 , G)) (5.9)

= ,A (p + c ) ) .

From the relations (5.8), (5.9) and the characterizing relations (5.4), (5.5) of ~,~ we have Aeg~. Q.E.D.

Proposition 5.2. I f a e Z and )~eS(a), then

gx =~,~ c~ Orth (G (2)). (5.10)

In particular, for each a e S we have g, = go c~ Orth (()(a)).

Proof. Let Aegx. Then the relations (5.1)-(5.3) yield Ae~z. Now, by Proposi- tion4.1, p+(2 , . )=p+(a , . ) ; hence ~,z=~,~ and consequently Aeg, . Moreover, Proposition 3.3 yields AeOrth(G(2)); that is,

A e~,,~ c~ Orth (G (2)). (5.11)

Conversely, if Ae~,,~ and AeOrth(G()o)), then

= ,A (p + (G, A* ()0 A)) = tA(p + (a, G(2))) (5.12)

=za(2 ). Q.E.D.

The relation (5.10) of Proposition 5.2 is analogous to the relation (14.4) of Proposition 14.1 in [1], proved by NOEL for semi-elastic (and hence also for elastic) elements. We now attempt to explain the sense of the analogy.

With an inelastic element in a fixed state a e S we associate an elastic element ~ (a ) . The set of all configurations of ~ ( a ) is identified with E(a), the class of all


deformation processes for the element ~ ( a ) will be identified with the class H ~ of all processes with duration zero and all continuous and piecewise continuously differentiable deformation processes with values in E(cr), the state space of ~ ( a ) is taken to be the set g(a), the configuration and stress mapping of ~ ( a ) are identified with the restriction GIg(~) of G to the set ~(~) and the restriction Slg(~r) of S to the set #(or), respectively, and the evolution mapping r of ~(cr) is defined as ~(2 , P)=p+(cr, PY), where 2eg(a) and P e r t ~ with U=(~(2). It is not difficult to show that ~ ( a ) is an elastic element in NOLL'S sense [1]. Now according to general definition in [1] we are able to introduce the symmetry group g~(~r) of the element ~ ( a ) as the set of all invertible linear mappings A: 3--~ 3- such that there is a bijection kA: g(O')---*g(O') which satisfies the conditions:

(i) for all 2ed~(cr)

(ii) for all GeE(a)

~(;0 = A* ~(k~(;~)) A, (5.13)

(5.14)

k a (p + (a, A* GA)) = p + (a, G). (5.15)

The symmetry group g~z(a) of the state # e # ( a ) of ~ ( a ) is the set

(5.16)

Then Proposition 14.1 in [1] yields for all #eg(~r) a relation analogous to (5.10):

g~(~r) = gE(a) c~ Orth (G(#)). (5.17)

However, Proposition 5.2 is not a consequence of (5.17), since neither g~ can be identified with the symmetry group of the state 2 of ~ ( a ) (this follows directly from (5.16)) nor is ~ the symmetry group of ~(cr). If Ae~,~ and if the mapping k A of (5.13)--(5.15) is identified with the restriction iAIg(a ) of the mapping ta to d~(a), then (5.13)-(5.15) hold and we have A~gE(a), i.e. gE(a)~,~. If gE(e)=~,~, then (5.17) and (5.10) would imply gu =g~(~) for each #~g(~); that is, symmetries of a state #~#(~) as a state of 3- and as a state of ~ ( a ) would be the same. This cannot generally be expected.

Remark 1. For elements which satisfy Axioms I-V the dependence of the instantaneous symmetry group ~,~ on cr can be interpreted as a change of symmetry of elastic response caused by the deformation process. Within this category of effects falls a change of texture of inelastic materials during plastic deformation. Proposition 7.3 will show, however, that as a consequence of Axiom VI in ideal inelastic materials such effect is excluded.

Remark 2. Using the symmetry group ~ , we can speak about solid-like or fluid-like instantaneous behavior of the element which satisfies Axioms I-V. Modifying NOLL'S concept of a solid element (Eli Section 14) we say that an element which satisfies Axioms I-V has at (r solid-like instantaneous behavior, if

110 M. ~ILHAVr & J. KRATOCHViL

its instantaneous symmetry group g,~ at a is contained in the orthogonal group of some configuration from E (a). Similarly, an element, which satisfies Axioms I-V is said to have at a fluid-like instantaneous behavior if ~, contains the unimodular group of 3-'.

6. Plastic Distortion

In this Section we state and discuss Axioms VI and VII. Axiom VI postulates that the state reached by the element at the end of the deformation process P, if starting from the initial state aES, may be obtained as a superposition of two transformations of the initial state a. First, the state is subjected to a suitably selected symmetry transformation a ~ l B l(a), Beg, and then the state t~_,(~) undergoes a jump t B_I (a)~-~p § (l B 1 (a), Pf). Thus the change of the state is naturally resolved into plastic and elastic parts.

Axiom VI. Let (a, P)s(Z x H)fit. Then there is a B in g such that PYsE(~B-~(a)) and

= p + (o), PY). (6.1)

The following axiom is technically important. It simply says that the limit of states of the form tacl (p+(a, Ai-* GAi-~)), where i~--,A i is a mapping from some directed set into g, is again of the form IA-~(p +(tr, A - * GA-1)), where A eg.

Axiom VII. Let a sS , Gsf~. Let i~--~A~ be a mapping from some directed set into g such that for all i

Let there exist a limit

A i * GAi -a sE(a). (6.2)

lim tA-l(p +(a, A t * GAi-1)) = r ~S G . (6.3) i

Then there is an A~g such that

tA-,(p +(a, A -* GA -1))= r. (6.4)

The assumption of Axiom VI that the plastic part of the change of the state represents just a symmetry transformation is very restrictive. As a consequence we shall show in the following proposition that the instantaneous response of the element may alter during deformation only in a very limited way. The generalized elastic range E and the instantaneous response functions S § are expressible (see (6.6), (6.8)) in terms of a subset E o of ~ and a functions So, respectively. Both E o and S O are independent of a. This property remains the characteristic feature of classical ideal plastic materials. For this reason we use for the element considered here the term "ideal".

Proposition 6.1. There is a closed subset E o of f# and a mapping Po: Eo ~ S such that the following conditions hold:

(i) if a s S , then there is a D in g satisfying

a = to 1 (Po (D-* d(cr) O- 1)); (6.5)


(ii) /f tr satisfies (6.5), where DEg, then

E(a) = D* E o D (6.6)

and for all G~E(a)

p+(a, G)= t o ,(po(D-* GD 1)). (6.7)

Moreover, if we denote by S o: E o --+ 5 P the mapping given by S o = S o Po, then for all GeE(a)

S+(~r, G)=D 1 So (D- , 6 0 - 1 ) O-* . (6.8)

Proof. By Axiom IV there is a 2 o E_r such that for each a ES we can find a mapping i~--~Pi from some directed set to//C(Xo) which satisfies for all i, PJ =~(~r) and (3.11). Define Eo=E(2o), po=p+(2o,.) . According to Axiom VI, for each i there is a B i E g such that (in the following calculation we use that P/Y= t~(a) and (4.9))

(&, ~)= p +(t,,., (;~o), ~ ) = p + (lB., (2o), (~ (a)) (6.9)

= l B . , ( p + ( ) ~ o , Bi-* (~ (o) Bi-l)). By Axiom IV and (6.9) we have

cr = lim t Bc, (p + (),o, B(* G (a) B71)), (6.10) i

and, hence, by Axiom VII, there is a symmetry transformation Deg such that

a=to_,(p+(),o,D *G((z)D ~)) (6.11)

~--lo-l(po (D-* G(a) D-t)).

Now let a satisfy (6.11); then by (4.8)1 and Proposition 4.1 we have

E (~) = E ( t o , (p + (;,o, D -* C (~) D-')))

= D * E (p + (2 0 , n - * (~ (~) n - 1)) D (6.12)

= D* E ()'o) D

=D*EoD.

Similarly, if GzE(a), then

p+(t7, G)=p+(tD ,(P+(},o, D-* 0(o)D-l ) ) , G)

= ,o_,(p+(p+(}.o, D-* 0((~) D-l) , D-* GD 1)) (6.13)

= to_,(p+(2o, D-* GD- ' ) )

-= I D - I ( p o ( D - * GD-a)),

s+(G, G)=~(p+(~, ~)) =S(tD-I(P~ GD-i))) (6.14) = D - I ~ ( p o ( D - * O D 1))D-*

= D - 1 S o ( D - * G D - 1 ) D -*. Q.E.D.

112 M. ~ILHAV~' & J. KRATOCHViL

Each mapping Po: Eo ~ Z satisfying conditions (i) and (ii) of Proposition 6.1 will be called a structural mapping of the given element. The function So: Eo ~5~ defined by S O = S o p will be referred to as the stress-mapping corresponding to the structural mapping Po.

If Po is a structural mapping of the element and if aeZ, then each symmetry DEg satisfying the relation (6.5) of Proposition 6.1 will be called a plastic distortion corresponding to o and Po (or, briefly, a distortion corresponding to a). The set of all plastic distortions corresponding to 0 and Po will be denoted by @po(a) or @(0).

There are many structural mappings of the given element satisfying our axioms. In fact, if Po: Eo-OZ is a structural mapping of the element, and if A~g, then the mapping p~): E~--,Z given by Eo=A*EoA and zA(P'o(A*GA))=po(G ) is again a structural mapping of this element, and we have the relation

A -1 @0(0)= @~(0). (6.15)

7. Symmetry Considerations

In this Section we introduce two additional useful symmetry concepts: a symmetry group g(Po) of the structural mapping Po and a symmetry group g(So) of the stress mapping So. Symmetry of the structural mapping is closely related (see Proposition 7.1) to the non-uniqueness of plastic distortion. Further we shall show in Proposition 7.3 that the instantaneous symmetry group ~ and the symmetry group g(Po) of the structural mapping are conjugate in g. Using the symmetry group g(So) we shall be able, in Proposition 7.4, to specify a condition under which the ideal inelastic element degenerates into an elastic element.

The symmetry group g(Po) of the structural mapping Po is defined as the set of all A ~g for which A* E o A = E o, and for all G~Eo, Po (G)= z.4(Po (A* GA)).

Similarly, the symmetry group g(So) of the stress mapping S o is the set of all A e Invlin (J-) for which A* E o A = E o and for all G ~ E o , S o (G) = A S o (A* GA) A*.

The definition of g(Po) and g(So) easily yields

g(So) ~ g(po). (7.1)

In the following proposition we show that the larger is the group g(Po), the larger is the number of plastic distortions in ~(a).

Proposition 7.1. Let a ~ X, A elnvlin (J ) . Then the following three conditions are equivalent:

(i) A~g(po ). (ii) For all 0 ~ ( o ) we have ADen(a) .

(iii) For some D ~ ( a ) we have A D ~ ( a ) .

Proof. Implication (i) ~ (ii): Let A ~ g (Po) and D ~ (a). Then

= lv-,(Po (D-* (~(o) D-'))

= to-, (tA-' (Po (A * D-* G (a) D-1 A-'))) (7.2)

= Z(AV)-~ (Po ((AD)-* d (a)(AO)-~)),


that is, ADen(a). Implication (ii)~(iii) is trivial. Implication (iii)~(i): For some De~(a ) we have ADen(a). Then, by definition of the structural mapping, we have

E(a) = D* E o D = D* A* E o AD (7.3)

and for all GeE(a) p+(~, G)= tD-,(po(D-* GD-1))

(7.4) = l(AO)_ l(po((AD)-* G(AD) 1)).

From (7.3) and (7.4) it follows easily that A*EoA = E o and for all GeEo, po(G) = iA(Po(A* GA)); consequently, A6g(po). Q.E.D.

Further we establish for a fixed state a a "transformation law" for plastic distortion under symmetry transformation A ~g.

Proposition 7.2. If a~Z and A~g, then

(l A (a ) ) ~--- 9 (0" ) A -1 (7.5)

Proof. In view of the definition of ~(a) we have De ~ ( a ) if and only if

tA(a) = ZAOD-~ (po(D-* 0 (a)D-l))) (7.6)

=t(DA_,)_~(po((DA 1)-. O(ZA(a))(DA-1)-I))

which is valid if and only if DA-le~(ZA(a)). Q.E.D.

In the next proposition we establish a relation between the instantaneous symmetry group ~ , the symmetry group g(Po), and plastic distortion corresponding to a and Po.

Proposition 7.3. Let aEZ, De~(a). Then

~,,, = D-1 g(Po) D. (7.7)

Proof. In view of the definitions of ~,, g(Po) and ~(a) the following propositions (i)-(v) are equivalent:

(i) A ~,~.

(ii) For all GeD*EoD

p+(a, G)= tA(p+(a, A* GA)).

(iii) For all G~D*EoD

to I(Po( D-* GD-'))= tA ID-'(po(D-* A* GAD-')).

= ,Ao ~ (po ( D - * A D D - * a D - 1 D A D-1)).

(iv) For all HeE o

PO (O) = 1DA D , (Po ((DAD -1), H(DA O - 1))).

(v) DAD-16g(po). Q.E.D.

(7.8)

(7.9)

(7.10)

114 M. SILHAV'r 8r J. KRATOCHViL

The last proposition of this section shows that for ideal inelastic elements which are not elastic the symmetry group g cannot be a subgroup of the symmetry group g(So) of the stress mapping S o.

Proposition 7.4. Let S o be a stress mapping corresponding to the structural mapping Po of the element J . I f g(So)~ 8, then 3- is an elastic material element. In particular, /f g(po)=g, then 3"- is elastic.

Proof. Let (a,P)~(X • H)fiv By Proposition 6.1 there exists an element D~(~(0.,P)). Then the definition of plastic distortion yields

(0., P) = 'D-, (Po (D -* Pf D - a)), (7.11)

and since D ~ ( ~ ( a , P))cgcg(So) we have from the definitions of S o and g(So) further that

(0., P) = S (zD-, (Po (D-* P: D- 1))) =D-i S(po(D-* Pf n-i))n-*

(7.12) =D -1So(D-*PgD -a) D-*

= S o (Pf).

Hence if 0.1, 02 G Z~G, then for all P e//G

(0.1, P) = So (pI) = ~ (0.2, P). (7.13)

Consequently, by Axiom III, o" 1 = 0" 2. Therefore for each G~f# the G-section S o is a singleton. It follows from NOEL'S definition of elastic material element that this property is characteristic for these elements. (See Definition 14.1 in [-11 and also the paragraph which follows Remark 2 of Section 14 in [-1].) In particular, if g(po)=g, then according to (7.1) g(So)~g(po)=g, and 3-- is elastic. Q.E.D.

Remark 3. If the ideal inelastic element is not elastic, then, by Proposition 7.4, g(Po)~g. Using (?.7) of Proposition 7.3, we then have for each 0.~Z, D ~ ( 0 . ) , ~,r The last relation is nicely illustrated by plastic behavior of crystalline materials. The symmetry group ~, may be identified with symmetry of elastic response at 0., so ~, is determined in these materials by the symmetry of their crystal lattice. On the other hand, both basic modes of plastic deformation in crystals, namely, slip and twinning, are symmetry transformations (slip and twinning transformations for some classes of crystal lattices are listed e.g. in [-13, 14]) which need not belong to the symmetry group of the lattice. Hence there are plastic distortions D which represent these slipping and twinning modes and for which D ~ but D~g, i.e. ~,,~g.

Remark 4. If we restrict an ideal inelastic material by the additional condition g ~ Unim (3-), we conclude from the relation ~, ~ g and Remark 2 in Section 5 that the element cannot have fluid-like instantaneous behavior at any 0.. More- over, if g(Po) is a subgroup of Orth(G0) , where G O is a configuration Go~Eo, then by Proposition 7.3 we have for 0.~Z, D ~ ( a )

~,~ = D-1 g(Po) D ~ D-10r th (Go) D = Orth (D* GOD), (7.14)


where by (6.6), D*GoD~E(a ). Hence the element has solid-like instantaneous behavior at all a~2;.

Remark 5. Ideal inelastic materials represent an example of material elements which can have a symmetry group as large as the orthogonal group or even the unimodular group and yet need not to be isotropic (see Remark 1, Section 14 in [1]). By Proposition 5.2, gac~,~, and if for all a, ~,~ is a proper subgroup of Orth (G), with GeE(a), then there are no isotropic states. (For an isotropic state g~=Orth G(a), [1].)

8. Isotropy We now restrict our attention to isotropic elements satisfying g c Unim(J-)

which are not elastic elements.

Axiom VIII. The element ~-- is isotropic, not elastic, and g c Unim (Y).

Proposition 8.1. (i) The symmetry group g of the element must be the whole unimodular group:

g = Unim (Y). (8.1)

(ii) There is a GoEE o such that

g(Po) = g(So) = Orth (Go). (8.2)

First we shall prove a preliminary result.

Lemma 1. There is a Go6E o such that

g(Po) ~ Orth (Go). (8.3)

Proof. Let a be an isotropic state of ~,, i.e.

g~ = Orth(G(a)). (8.4)

Since, by Proposition 5.2, g~ = ~ c~ Orth(~(a)), it follows from (8.4) that

~,~ = Orth (G(a)). (8.5)

Let DeN(a) . Proposition 7.3 then yields

D-1 g(Po) D ~ Orth (G (a)), (8.6)

or equivalently

g(Po) ~ D Orth (G (or)) D-I = Orth (D -* (~(a) D- l ) . (8.7)

If we define G o = D - * G(a)D -~, we have then that GoeEo, and, hence from (8.7), g(po) ~ Orth (Go). Q.E.D.

Proof of Proposition 8.1. According to Axiom VIII the element J - is isotropic. By Proposition 10.3 in [-1] we have then two possibilities for g:

(a) g = Orth ((~(0-)), where a is some state of ~-,

(b) g ~ Unim (~-).

1 16 M. ~ILHAV~( & J. KRATOCHV[L

Suppose that (a) is valid. By Lemma 1, there is a GoeE o such that g(Po) =Orth(Go). On the other hand, g(po)cg=Orth(G(a)); therefore Orth(Go) c Orth(G(a)), which is possible only if Orth(Go)=Orth(G(a)). Consequently

g (Po)= g = Orth (~ (a)). (8.8)

However, by Proposition 7.4, the element satisfying (8.8) is elastic, which con- tradicts Axiom VIII. Hence only possibility (b) remains. It follows from the assumption g c Unim (J-) in Axiom VIII that g = Unim (J-), which proves item (i) of Proposition 8.1.

The assumed inelasticity of the element ~-- and Proposition 7.4 imply the g(Po) #: Unim (J-) and g(So)~ Unim (J-). By (8.3) and (7.1) we have then Orth(Go) c g (Po) ~ g(So), and by the maximality of the orthogonal group in the unimodular group we obtain (8.2). Q.E.D.

Axiom VIII has another consequence important for the derivation of a flow rule in Section 9. (Through the rest of the article the configuration satisfying condition (ii) of Proposition 8.1 will be denoted by Go; for each BeLin(Y-) the symbol B T will represent the transpose of B with respect to G o, i.e. BT= Go 1 B* G o e Lin (Y); see Section 2.)

Proposition 8.2. There is a unique mapping (f: X~Unim(Y)c~Sym+(Go) such that 0 ( a ) ~ ( a ) for all ae,Y,. The mapping (J has the following properties: I f a eX and AeUnim(J-), then

0(1A (a)) = [A- T • (a)2 A -']�89 (8.9)

In particular, if Q e Orth (Go) , then

0 (~e(~)) = Q 0 ( ~ ) Q - ' (8.10)

Proof. For each aeZ the set ~(a)c~Sym+(Go) is a singleton, i.e., it contains just one element. To prove this statement, let us point out that according to Proposi- tions 7.1 and 8.1, D, and D2e~(a ) ,~DzD~ 'eOr th (Go) . Hence if De~(a), where D=QU, QeOrth(Go), UeSym+(Go), then also

U = Q r Q U e ~ ( a ) . (8.11)

Consequently, ~ (a) c~ Sym + (Go) ~ ~[. If U1, U2e~(a)c~Sym+(Go), then U 2 ui-leOrth(Go), i.e.

(U 2 U,-')r=(U2 U, ,)-1; (8.12) therefore

U22= UT U2= UT U1= U? (8.13)

and finally Ul= U 2. Hence the set ~(a)c~Sym+(Go) is of the form

~(a) c~ Sym +(Go)= { 0(a)}, (8.14)

where 0(a)eSym+(Go)c~ Unim(~--). The uniqueness of the mapping 0 follows from (8.14).


Let us further prove the transformation law (8.9). If aeZ and AeUnim(Y), then by definition of 0 and by Proposition 7.2, 0(a)A-~e~(tA(a) ). On the other hand 0(a) A - 1 = Q 0 0 a (a)), where Q e Orth (Go). Consequently

=

= (Y Q

= E(~f (O') A -1)T ~] (o') A -1]�89 (8.15)

= [A- T 0(a)r O(a) A-l]�89

=[A-r(j(a)2A 1]�89

The relation (8.10) is then a consequence of (8.9); as for QeOrth(Go) we have [Q- T 0(a)2 Q-l]�89 = [QO(a)2 Qr]�89 = QO(a) QT. Q.E.D.

According to Proposition 8.2 for each state a there is a unique plastic distortion U, UsUnim(f )c~Sym+(Go) corresponding to a. We now introduce an evolution function U of a symmetric plastic distortion U. The evolution function 0 is a mapping 0: agi-,Unim(3-)c~Sym+(Go), whose domain

agi={(U,P)~[Unim(J-)c~Sym+(Go)]xlIlpieU*Eo U} (8.16)

is the set of all pairs (U,P), where UEUnim(Y--)c~Sym+(Go), PelI, such that the initial configuration U of the process P fits the set U* E 0 U. The value O(U, P) for (U, P)ear i is

O(U, P)= (J(P('v-,(Po (U-* P' U -1)), p)). (8.17)

Proposition 8.3. The mapping 0 has the following properties: (i) If aEZ, U6~(a)c~Sym+(Go) and P6H~(~), then(U,P)6d i and

O(U, P)E~(~(a, P)) c~ Sym+(Go). (8.18)

(ii) If (U, P1)~N i, P2~II and Pf =P2 ~, then (U, P1 *P2), (0(U, P~), P 2 ) ~ " and

O(U, P~ �9 Pz)= O(O(U, P~), P2)- (8.19)

(iii) If A~Unim(J-) , (U, p)~r then ([A-TU2A-I] �89 A-*pA-1)Eaff i and

~]([A-TU2A-I]�89 A-*PA-1)=[A-r(J(U,p)2A-I] �89 (8.20)

In particular, if QeOrth (Go), (U, P)es~#, then also (QUQ -1, Q-* PQ-1)e~r and

(_](QUQ-1, Q - , p Q 1)=Q~f(U, p)Q-1 (8.21)

If U is a symmetric plastic distortion corresponding to a, then, by property (i), (J(U, P) is a symmetric plastic distortion corresponding to r P). Property (ii) shows that the mapping 0 has a similar semi-group property as the mapping/3 in Axiom II. Property (iii) represents a transformation law for 0 under a transformation (U,P)~-~,([A-TU2A-X] �89 A-*pA-1). In the next section we shall show that for a class of materials restricted additionally by Axioms IX-XI we

118 M. ~ILHAV'~ ~r J. KRATOCHViL

shall construct another mapping /3 with similar properties as the mapping 0 which, however, satisfies a more natural transformation law (Proposition 9.5).

Proof of Proposition 8.3. (i) If U ~ ~ (a) c~ Sym + (Go), then U - * G (a) U - 1 ~ Eo and tr = 'v-, (Po (U- * G (a) U - 1)). Since we have P' = G (a), therefore U - * pi U-1 ~ Eo and a = zv_,(po(U-* pi U- ~)). The definition of d i and 0 then yield (U, P) e d ~ and

U ( U, P)= U(P(tv- I(Po(U-* UU- ~)), P) ) (8.22)

= O(~(a, P))e~(fi(a, n)).

(ii) The relation (U, P1)e~/i means that UeUnim(Y)c~Sym+(Go) and PlieU*Eo U. If P: is a process such that Plf= P2 i, then (Px * P2)i=P1 i and therefore also (P1.P2)eU*EoU; that is, (U,P~*P2)edk Further, using (8.18), with r 1 u _ , ( p o ( U - * P1 i u-l ) ) , we have

t?(u, P,)-* d(~(a, P1)) t)(u, P1)= t)(u, PO-* PS O(u, P1) (8.23)

= O(u, PO-*P~' O(u, Pi)~eo. Thus (U(U, P0, P 2 ) ~ " .

Let us prove the relation (8.19). By the definition of 0 and Axiom II we have

gr(U,/]1 �9 P2)= 0(~(~(a, P1), P2)). (8.24)

If we use the abbreviation W= U(U, P1), then ~(o, P1)= ,w-~(po(W-* PI ~ w-l))

(8.25) = 1W_l(Po(W--*P2 i W - l ) ) ,

and therefore by the definition of 0

o(~ (~(o, P0, P~))= o(~ (,W-l(po (w-,P; w-')), p~)) = U(W, P2) (8.26)

= 0(C7, Px), P~).

The relations (8.24) and (8.26) then imply (8.19).

(iii) Let A~Unim(Y-) and (U, P ) ~ / k Let G=U and a = zv_,(po(U-* GU-1)). Then the relation (8.9) in Proposition 8.2 implies

U([A - - T U2A-I]~, a -* PA -1)= 0(~ (lA(a), A-* PA -1))

: O(ta(~(a, P))) =[A_VO(f)(a,p))2A_I] ~ (8.27)

:[A-TO(U,p)2A-1]}.

The relation (8.21) may be deduced from (8.20) simply by evaluating the square roots appearing in (8.20). Q.E.D.

9. Flow Rules

In this Section we show that the evolution of the plastic distortion D during a deformation process P is governed by an ordinary differential equation of the


fo rm

/)(t) D(t) -i = L(D(t)-* Ploa] D (t)-i), (9.1)

where only the behavior of the mapping D(t)-*P[o,,lD(t)-':[O,t]~(~ in an arbitrarily small past neighborhood of time t can influence the values of the functional L. In agreement with classical terminology, the Equation (9.1) will be called the flow rule.

Among the assumptions sufficient for derivation of the flow rule Axiom IX is basic.

Axiom IX. For each P~H the mapping

~(., P): Xp~-~Xps (9.2)

is bijective, i.e., for a fixed PeII , if af eXp1,then there is a unique (r~eXp~ such that (a t, P) = uJ'.

It is possible to show that Axiom IX is not valid for materials with elastic range [-6]. Thus we cannot expect results analogous to those derived in this Section to be valid for rate-independent theories of plasticity which rest on the assumption that an elastic range exists.

Let ~ ' f denote a set

d f={ (U,P)~[Unim(g)c~Sym+(Go)] • (9.3)

Proposition 9.1. Let (U I, P ) e d I. Then there is a unique U i such that (U i, P)es4 i and

U(U i, P )= U I. (9.4)

Proof. If we denote af=lvs_,(po(Uf-*PIUI-1)) , then by Axiom IX there is a u' such that ~(a', P)=G I. We define U~= U(a~). Then crY= lv,-,(po(U~-*P ' U~-')) and

U := U (u:)= O (~)(a i, P))

= O(~Ov,-,(po(U'-*P' u-t)), P)) (9.5)

= f~(U i, P).

Suppose that U~, UjeUnim(3-)c~ Sym+(Go) are such that

U(U, ~, P )= U(Uj, P )= U y. (9.6)

If we denote for a = 1, 2, ~, = ,v;-,(po(U~-*U U~-')) and U J = O(U~, P), then by item (i) of Proposition 8.3

U~e ~ (~ (a':,, P)). Consequently,

by (9.6), U[ = Uf, so

~ ((r~, P)= ,vl-,(po(Uf-*PIUf-i));

t~ (u~, P) = ~ (cry, P) = al,

(9.7)

(9.8)

(9.9)

120 M. SILHAV~ &: J. KRATOCHViL

and consequently, by Axiom IX, ~r] =~r~. But the last relation is valid only if Ua ~ = Ug. Q.E.D

The next axiom specifies the analytical properties of the mapping t~O(u, qo, O. Axiom X. Let ( U, P)~ d i and d e > O. Then the mapping tv-~ O(U, Pw, tl) is continuous and pieeewise continuously differentiable on [0, de].

For (U, P ) E d ~, de>O, we denote by r P) the left-hand derivative of the mapping t~O(U, PEo, O at t=de.

In the following proposition we shall show that the time derivative of symmetric plastic distortion at the end of the deformation process P depends only on the final value of plastic distortion and on the behavior of P in an arbitrarily small past neighborhood of de.

P r o p o s i t i o n 9.2. Let (Ui, P1), (U2, P2) ~ ~r with P1, P2 E II of duration d a > O, d 2 > O, respectively. Suppose that 6 > 0 such that

Then

U(Ui, Pll)= tQ(U2, Pz). Suppose further that there is a

/]1 td, - ~, d~l = P2 [d: - 6, d23" ( 9 . 1 0 )

(J (U 1 , P1)= r ( U 2 , 1'2). (9.11)

Proof. Let us denote the common segment of P1 and P2 in (9.10) by P. By Pro- position 9.1 there is a unique U such that

O(U, P ) = / ) ( U 1 ,/]1) = r 2 , P2). (9.12)

For z r [0, 3] and e = 1, 2 we have, by Property (ii) in Proposition 8.3,

O (E, V~Lo.~_~+ 0 = O ( O (E, e, Eo.~-~j), et,o-~.~-~+O (9.13)

= O(OIU~, e, ro. ~_~j), Pro. 0.

For z = 6 the identity (9.12) becomes

O(U1,/]1) = tQ(U2, Pz)= O(tQ(U~, P~w, d~-61), P). (9.14)

Consequently, by (9.13) and Proposition 9.1 we must have

O(u,,e~[o.~, ~])=0(u2,e~to.~2 ~j)=v (9.15)

and hence, by (9.13) for z > 0 and e = l , 2

O(E, e~Eo, ~-~+0 = 0(v, rio, 0" (9.16)

Differentiating the identity (9.1 6) with respect to z at z = 6, we obtain for a = 1, 2, (J(U,, P~)= r P), and hence also (9.11). Q.E.D.

In view of Proposition 9.2 we now introduce an important equivalence relation on the set of all processes of non-zero duration. Let 11' be the set of all processes f r o m / / w i t h dp > 0. We say that/]1 and P2 ~ 11' of duration d 1 and d2, respectively,


are equivalent if there is a ~ > 0 such that

Pl [d~ -6 , dj ] = P2 [d2- & a2]' (9.17)

For each PeFI' the equivalence class of the element P with respect to this equivalence relation will be denoted by {P}. The corresponding quotient space of /7 ' will be denoted by S = {{P} IP~/7'}.

If n e e and AeInvlin(g-), then A*~AGff will denote the equivalence class whose members are of the form A*PA, Pe=.

Let us further define for each neff the final value of the class 7: as the common final value of the members of this equivalence class, i.e. let us write =*= pI, where PG~.

Finally, let us define the set A,

A = {~e~] hie E0} (9.18)

of all equivalence classes ~z whose final values n I belong to E o. Now we are ready to introduce the mapping L: A ~ L i n (Y) which enters the

final form of the flow rule. If ~eA, then the value L(~)eLin(Y) is defined as follows. We choose an arbitrary element P e ~ and find a U e Unim (~--) c~ Sym + (Go) such that (U, P)E~r i and C'(U, P ) = I (the existence of such U is guaranteed by Proposition 9.1). Having such U, we write

L(n)= U(U, P). (9.19)

This definition is unambiguous, for if P' is another member of ~ and U' is such that U(U', P ' )= 1, then by Proposition 9.2, U(U, P)= @(U', P').

Proposition 9.3. Let ~GA and QGOrth(Go). Then also Q-* gQ-1GA and

QL(~) Q-1 =L(Q * ~Q-1). (9.20)

Proof. The relation riGA means that gIGEo, but then using the definition of g(Po) and (8.2) we have also

(Q-*nQ-1)I=Q-*~IQ-1GQ *EoQ-I=Eo; (9.21)

that is, Q-* nQ -1 cA. By the definition of L, if Pe~, then L(~)= U(U, P), where U is a symmetric plastic distortion which satisfies U(U, P)= 1. Then Property (iii) in Proposition 8.3 yields

(J(QUQ-1, Q-*PQ-1)=QO(U,P)Q-I=Q1Q-I=I (9.22)

and Q- * PQ- 1G Q-* 7: Q- 1. Hence we have

L(Q-*zQ-1)=O(QUQ I,Q-*pQ-') . (9.23)

Now using again Property (iii) in Proposition 8.3, we obtain for te[0, de]

(J (Q UQ- 1, (Q-. pQ- 1)to, tl) = Q u ( u , PEo, ,7) Q -1 (9.24)

122 M. ~ILHAV~f • J. KRATOCHViL

By differentiating the identity (9.24) with respect to t at t =d e we obtain

( j(QUQ-I, O - . pQ-1)=Q(j (U, p) Q-1 (9.25)

and by (9.19) and (9.23) we get (9.20). Q.E.D.

Proposition 9.4. Let (U i, P)Ed i, dp>O. I f we denote for tel0, de]

V(t) = U(U i, P~o, 0), (9.26)

then the mapping V: [0, de] --* Sym + (Go) is a solution of the initial-value problem

�89 V(t) -1 + V(t) -1 l?(t)] =L(V(t)-*{Ptom} V(t)-~), (9.27)

V(0) = U i. (9.28)

Proof. From the definition of L we have for t~(0, dp]

L(V(t)-* {Pto, tl} V(t)-~) = (J(U(t), V(t)-* Pro, o V(t)-x), (9.29)

where the value U(t) is that symmetric plastic distortion for which

(J(U(t), V(t)- * Pro, ,1 V(t) -1) -- 1. (9.30)

By Property (iii) in Proposition 8.3

O([V(t)- W V(O)2 V(t)-1]~, V(t)-* Pro, tl V(t)-1)

= [V(t)- r (J(V(O), P~o, o) z V(t)-1]~- (9.31)

= IV(t) -1 V(t) 2 V(t)-l] ~ = 1

and hence

U(t) = [V(t)- r V(0): V(t)-l] ~. (9.32)

Using (9.32) and again Property (iii) in Proposition 8.3, if t~(O, de] and re(O, t] we see that

0 (U (t), V(t)- * Pro, ~1 V(t)-l)2 = V(t)- r 0 (V(O), Pro, ~1) 2 V(t)- ' (9.33)

= v ( t ) - T v(~)2 v ( t ) - l .

By differentiating the identity (9.33) with respect to z at z=t , and keeping in mind (9.30), we obtain

20(U(t), V(t)-* Pro, o V(t)- ' )= (/(t) V(t) -1 + V(t) -1 l)'(t). (9.34)

The relations (9.34) and (9.29) then imply (9.27). Q.E.D.

Conversely, we shall assume that each solution of the initial value problem (9.27), (9.28) is of the form (9.26).

AxiomXI. For each (U i, P ) ~ d i, de>O there is a unique solution of the initial- value problem (9.27), (9.28).


We are now able to state and prove in the following proposition the main result of this Section. Let c~ denote the set

cg~= {(D, P)~ Unim (9--) x H IP'e D* E o D} (9.35)

of all pairs (D, P)~Unim(~--) x H such that the initial value of the process is in the set D* E o D.

Proposition 9.5. There is a mapping D: cgi ~ Unim (Y) such that the following five conditions hold:

(i) I f (or, P)e(Z'xH)fit and D ~ ( a ) , then

b (D, P) e ~ (~3 (o, P)). (9.36)

(ii) I f (D i, p)~cgi and d e > O, then the mapping tF--~D (D i, P~o. t]) is the unique solution of the initial-value problem

b (t) D (t)-i = L (D (t)- * {Pro, ,7} D (t)- 1) (9.37)

D (0) = D ~ . (9.38)

(iii) If(D, P1)~cg i and P2~ H are such that PlY=P2 i, then (D, 111. Pz)~f~, (/3(D, P0, Pz) ~cgi and

/)(D, P1 * P2)=/3 (/)( D, P0, Pz)- (9.39)

(iv) I f A 6 Unim ( J-) and ( D, p)~cgi, then also ( D A - a, A -* p A-1)~cg i and

1)(D A -1, A - * PA -1) =/) (D, P) a -1 (9.40)

(v) I f Q~Orth(Go) and (D, p)~cgi, then also (Q O, P)~Cg i and

/) (Q D, P) = Q/) (D, P). (9.41)

Proof. Let us define the mapping/3 in the following way. Suppose that (D, p)~c#i. There are two possibilities:

(a) dv = 0. We then define

/) (D, P) = D. (9.42)

(b) dp>O. If Q, U is that unique pair for which Q~Orth(Go) , U~Sym+(Go) and D=QU, then ( U , P ) ~ d i and it is then possible to define a mapping V: [0, d~,] ~ Sym+(Go) by

V(t)=(J(U, Pto.,~), for t~[O, dp]. (9.43)

By Axiom X, this mapping is piecewise continuously differentiable. By use of theorems on ordinary differential equations, it is possible to prove that there is a unique continuous and piecewise continuously differentiable function R: [0, dr]--. Orth (Go) such that Ri= Q and for t e (0, dr]

R (t) T R (t) = �89 [ V(t)- 1 (/(t) - V(t) V(t)- 1 ]. (9.44)

124 M. SILHAV~/ t~r J. KRATOCHVIL

We then define

/3 (D, P) = R y V z = R I U(U, P). (9.45)

Let us show that the mapping/) just defined has the properties (i)-(v). (i) If ( a ,P )e (Zx H)rit and De~(a) , then the value D(D,P) is of the form

R U(U, P), where ReOrth(Go), U e ~ ( a ) c~ Sym +(Go). By Proposition 8.3 then O(U, P)e~(~(a , P)) and by Propositions 8.1 and 7.1 then also

b (D, P) = R (f(U, P) ~ ~ (~ (a, P)). (9.46)

(ii) Let (D i, P)e cgi and d e > 0. Denote for t 6 [0, de]

D (t) = b (D i, Pw, tl). (9.47)

If (Q, U) denotes the polar decomposition of D i, i.e., if Di= Q U, Q~Orth(Go), U~ Sym + (Go), then

D (t) = R (t) V(t), (9.48) where

V(t) = (J (U, P(o. ,1) (9.49) and

R(t)T[~(t)=�89 -1 ( l ( t ) - (/(t) V(t) -1] (9.50)

Ri=R(O )=Q . (9.51) Now

D (t) D (t)- 1 = [[~ (t) V(t) + R (t) l?(t)] V(t)- 1 R (t)- 1

= R (t) R (t) ~ + R (t) ~'(t) V(t)- 1 n (t)- 1 (9.52)

= R (t) [R(t) r R(t) + l/(t) V(t) -1] R(t)- 1.

Using (9.50) Propositions 9.4 and 9.3, we have

D(t) D(t) -1 =�89 [V(t)- ' l?(t)+ l)(t) V(t) -1] R(t) -1

= R (t) L(V(t) -* {P[o, t]} V(t)-l) R (t)-i

= L (R (t)- * V(t)- * {P[o, q} V(t)- ' R (t)- ') (9.53)

= L ([R (t) V(t)] - * {P(o, o} [R (t) V(t)]- 1) = L(D (t)- * {P[o, t]} D (t)- 1).

Hence the mapping D(.) is a solution of the initial-value problem (9.37), (9.38). Suppose that D1, D 2 are two solutions of the initial-value problem (9.37), (9.38).

We write for ~ = 1, 2

D~(t) = R~(t) V~(t) (9.54) where

R,( t)eOrth (Go), V,(t) sSym + (Go). (9.55) Then

R~(t) [R,(t) r/~,(t) + ~ (t) V~(t)-'] R~(t) -1

=D~(t)D~(t)-' =L(D~(t)-* {P[o,,]) D~(t)-') (9.56)

=R,( t ) L(v~(t)-* {P[o, o} V,(t) -1) R , ( t ) - '

Ideal Ine las t ic Mate r ia l s 125

and consequently

R,( t ) r [~( t )+ (/~(t) V,(t) -1 =L(v,(t)-*{Pto,,]} V~(t)-l). (9.57)

If we take the symmetric part of the last relation, we obtain

�89 [12~(t) V,(t) -~ + V,(t) -1 12,(t)] = L(V,(t)-* {Pto.,]} V~(t) -~) (9.58)

since by the definition the values of L are symmetric and R~(t) r/~(t) is antisymmetric. Moreover V~i= Vg= U. Then Axiom XI yields V 1 = V z . Further, taking the antisymmetric part of the relation (9.57), we obtain

R,(t)r [~,(t)=�89 -1 (/ ,( t)- (/,(t) V ~ ( t ) - l ] . (9.59)

By the above considerations V~ = V 2 and hence

R~ (t) r/~1 (t) = R z (t) r/~2 (t), (9.60)

R ~ - - i - R 2 - Q . (9.61)

As we have already pointed out the initial-value problem (9.44) has a unique solution and hence R 1 = R2, and consequently, by (9.54), D 1 = D 2 .

(iii) Suppose that (D, P1) e cg ~, P2 eH and PlS= Pzk Further, suppose for simplicity that both processes P1 and P2 are of non-zero durations d~ and d 2 . By Condition (ii) of the present proposition, if

D3(.): [0, d I +d2] ~ U n i m ( J ) (9.62)

is the solution of the differential equation

/)3 (t) D 3 (t)-I = L (D 3 (t)- * {(P1 * P2)[o, ,l} D3 (t)-l), (9.63)

with initial value D~=D, then /)(D, P~, Pz)=DI. Similarly, if

Da : [0, dl] ~ Unim (~-) (9.64) and

D2: 1-0, d2] ~ Unim (Y) (9.65)

are solutions of the initial-value problems

b~ (t) D 1 (t)-i = L (D 1 (t)- * {Pl[o, ,l} D~ (t)- 1), (9.66)

b 2 (t) D z (t)- 1 = L(D2 (t)- * { P2Eo, ,]} Dz (t)- 1) (9.67)

with the initial values D] =D and D~ =/)(D, P~), respectively, then

/) (D, P~ ) = D1 I, (9.68)

b(D(D1 P1), P2)= D2 I- (9.69)

126 M. ~ILHAVY • J, KRATOCHViL

But the uniqueness of the solution of these initial-value problems implies that

D1 = D3to, dlj, (9.70)

Dz = D3[al, al +ad (9.71)

for these mappings do solve these differential equations, and

D i =D, (9.72) 3 [0.dl]

D' = b (D, P~). (9.73) 3 [d2,dl + d2]

Relations (9.70), (9.71), (9.72) and (9.73) then imply (9.39).

(iv) Let (D,P)eCd, AeUnim(Y) . Then P~eD*EoD and hence

(A-* PA-1)i=A-* U A - l e A - * D* EoDA -1 (9.74)

=(D A-1)* EoD A-1; that is,

(D A-1, A- * p A-1)ecd. (9.75)

By Condition (ii) of the present proposition, if

D(.): [0, de] --' Unim ( J ) (9.76)

is a solution of the differential equation

D (t) D (t)-I = L(D (t)- * {Pro, tl} D (t)-a) (9.77)

with initial value D ~= D, then

D(D, P) = D I. (9.78)

But then D': [0, dp] ~ Unim (Y-) given by D' = D (.) A- 1 is a solution of

D'(t) D'(t) -~ = L(D'(t)-* {A- * Pro, ,1 A-l} D'(t)-i) (9.79)

with initial value D'i= D A-a, since

D' (t) O' ( t)- ' = b (t) O (t)-i (9.80) and

D'(t)-* {A-* P~o,q A x} O,(t)-I (9.81)

= O ( t ) -* { ~ o , . } o(t) -1. Hence

[~(DA -1, A-* PA 1)=D'S=DIA-1 =D(D, P)A -~ . (9.82)

(v) If (D, P)eCd and Q~Orth(Go), then, by Proposition 8.1 also

(Q D, P)e cd. (9.83)

Now, if D1 is a solution of the differential equation

b I (t) D i (t) -1 = L(a I (t)-- * {V[o, t]} 01 ( t ) - 1) (9.84)


with the initial value D] =D, then D'I: [0, de-] ~ Unim (5-) given by D' 1 =QD 1 is a solution of the differential equation

"t v -1 D 1 (t) D1 (t) = L(D'I (t)-* {Pro, q} D~ (t)- 1) (9.85)

with initial value D'li= Q D since

b~ (t) D' 1 (t) -1 = Q 1)1 (t) D 1 (t)-I Q -1 (9.86)

=Q L(DI (t)-* {Pro, q} Dl (t) -1) Q-1

and hence by Proposition 9.3,

bi (t) Di (t)-' = L (Q - * D , ( t ) - * { e to ' ,1} D1 ( t ) - ' g - 1) (9.87)

= L(O' 1 (t)-* {Pro, q} D'I (t)- 1).

By (ii), then,

and D(D, P~)= D{

D(Q D, P~)= D~=Q D:~ =Q D(D, PO. Q.E.D.

(9.88)

(9.89)

10. Summary

The purpose of this section is to show, somewhat informally, that the axiom- atic structure set up in previous sections provides a rational basis for what is called "the internal variable approach" to plasticity [3, 4]. This approach may be briefly phrased as follows: In the internal variable theory of plasticity each state of a material is determined by the corresponding configuration G, plastic distortion D (which is an invertible linear transformation from Y onto 5") and by a vector ~, the so called structural-parameter vector (ct is an element of some finite-dimensional real vector space). The corresponding intrinsic stress is then given by

S = D -1 So(D-* GD -1, ~) D-* , (10.1)

where S O is a function determined by the material. The evolution of plastic distortion and structural-parameter vector during the deformation process P is determined by the differential equations

D (t) D (t)-1 = Lo (D (t)- * P(t) D (t)- 1 0~ (t)) (10.2)

~t (t) =fo (D (t)- * P(t) D (t)- 1, ~ (t)) (10.3)

where Lo and fo are fixed functions. The structural-parameter vector is introduced in the theory to describe changes in the structure of material. Accordingly, if in the internal-variable theory one neglects changes in the structure of material, the state is then determined only by the configuration G and the plastic distortion D. The intrinsic stress S is then given by

S = D -1 So(D-* GD -1) D-* , (10.4)

128 M. SILHAV'( 8r J. KRATOCHViL

and the evolu t ion of plast ic d i s to r t ion is governed by the equa t ion

D (t) D ( t ) - ' = L o (D (t)- * P(t) D ( t ) - l ) . (lO.5)

Now, it is a consequence of our ax ioms tha t each state of ideal inelast ic mate r ia l e lement is de te rmined , in a well defined sense, by the co r r e spond ing con- f igurat ion and plast ic d i s to r t ion (see P ropos i t i on 6.1). Moreover , we have shown that the evolu t ion of plast ic d i s to r t ion dur ing a de fo rma t ion process P is governed by the o rd ina ry differential equa t ion

b (t) D ( t ) - i m_ L (D (l)- * P[o, t] D ( t ) - 1), (10.6)

where L i s a funct ional defined on a cer ta in class of de fo rma t ion processes. The values of L are affected only by values of its a rgument in an a rb i t ra r i ly small pas t n e i g h b o r h o o d of t ime t. In terms more suggestive than precise we can say that L has shor t - range memory . The flow rule (10.6) is a genera l iza t ion of (10.5), for within the ca tegory of funct ionals with shor t - range m e m o r y fall for ins tance funct ionals L for which

L (D ( t)- * PW, tl D ( t ) - ' ) = L 1 (D (t)- * P(t) D (t)- 1, D (t)- * P (t) D (t)- 1)

with a fixed funct ion L 1. Our Ax ioms I - X I impose cer ta in res t r ic t ions on the mapp ings S O and L.

A m o n g them we ment ion the " i s o t r o p y " of S o and L der ived in P ropos i t ions 8.1 and 9.3. I t wou ld be poss ible to der ive further res t r ic t ions which our ax ioms impose on S o and L and finally to find a comple te list of these res t r ic t ions (in the sense tha t mapp ings So and L satisfying this comple te col lec t ion of res t r ic t ions can serve as a basis for the cons t ruc t ion of general ideal inelast ic mate r ia l element). We have no t p roceeded in this way since our goal in the present ar t ic le was to demons t r a t e the existence of the flow rule in the theory descr ibed by Axioms I - X I .

Acknowledgement. The authors wish to express their deep thanks to Professor D. R. OWEN, Carnegie-Mellon University, for stimulating and helpful discussion during the course of this work and for many valuable remarks concerning the paper. The care taken by Mrs. M. KtJKRALOV~. in preparing the manuscript has been also most helpful.

References

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2. KR6NER, E., Kontinuumstheorie der Versetzungen und Eigenspannungen. Berlin-G6ttingen- Heidelberg: Springer 1958.

3. KRATOeHViL, J., Finite-Strain Theory of Inelastic Behavior of Crystalline Solids. p. 401-415 in Symposium on Foundation of Plasticity (A. SAWCZUK, Ed.). Leyden: Noordhoff 1973.

4. KRATOCHViL, J., Thermodynamics of Elastic-Inelastic Materials at Finite Strain. p. 5-62 in Dynamika o~rodk6w niesprezystych (P. PERZYNA, Ed.). Wroclaw-Warsawa-Krak6w-Gdafisk: Ossolineum 1974.

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Institute of Solid State Physics Czechoslovak Academy of Sciences 16253 Prague 6, Cukrovarnickfi 10

(Received March 4, 1977)

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A theory of inelastic behavior of materials Part II. Inelastic materials