Material Systems - A Framework for the Description of Material Behavior

ALBRECHT BERTRAM

Communicated by D. OWEN

Contents

1. Introduction ...... 99 2. Material Systems ...... 102 3. The State Space of Material Systems ...... 104 4. Material Isomorphy ...... 108 5. Material Symmetry ...... 113 6. Inverse Systems ...... 119 7. Uniform Structures on the State Space ...... 122 8. NOLL'S Material Elements ...... 127 9. Example: Rigid-Plastic Materials ...... 128 10. References ...... 132

1. Introduction

In the last few decades, renewed interest in the phenomenological description of anelastic material behavior has resulted in the proposal of many different constitutive theories, and two basic approaches to the formulation of such theo- ries have emerged. The first requires that one introduce variables apart from the configuration and stress in order to describe the state of a material element. These additional variables are called "internal" or "hidden" (state) variables. Their values are determined by an "evolution function" which in most cases enters into an ordinary differential equation. A shortcoming of this approach is that these variables, unlike deformation and stress, do not have a physical meaning which is the same for all materials. Given the results of experiments on a single material, it may not be clear how to choose the internal variables, and it is not always the case that a finite number of internal variables suffice to determine the state of a material element. The second approach requires that one specify "consti- tutive functionals" defined on "histories", i.e., functions depending upon all past values of a deformation process, instead of the present deformation alone. Apart from the fact that we will never have complete knowledge of the entire history 100 A. BERTRAM of a material, there is good reason to assert that this approach describes only a restricted class of materials ~, namely, those with fading memory; classical de- scriptions of plastic materials are not in this class. ~ These two approaches to the description of anelastic behavior were related by NOEL in his "new theory of simple materials", published in 1972. This theory represents an important step towards establishing a satisfactory general frame- work for the description of anelastic behavior. I have chosen NOEL'S theory as a starting point for the present one*, although I have had to extend it in order to include non-revertible materials (such as aging ones) and non-mechanical behavior (thermodynamics, electrodynamics), and to describe non-classical constraints. In working out the present theory, I have attempted to begin with empirical ("natural") notions, such as time, stress, ahd deformation, and to avoid concepts introduced as purely mathematical formalisms. As a result, this framework for the description of material behavior is more general and simpler than NOEL'S and covers essentially all the known theories of materials such as "internal variables" (see Section 3) as well as NOEL'S new "simple materials" (Section 8). Because NOEL'S old "simple materials", i.e. the semi-elastic ones, are included in his "new theory", they also have a definite place within the present theory, and, hence, I can describe many well-known classes of materials: elastic, visco-elastic, hypo-elastic, plastic, and aging materials (e.g. concrete). By identifying variables in an appropriate way, one can apply the present concepts to thermo- and electrodynamic materials ~; even applications outside physics may be considered. In order to describe informally the concept of a "material system", I consider an arbitrary number of (homogeneous) samples of a material to be studied and imagine carrying out (configuration-) processes for these samples, i.e. trajectories in the space of the independent variables, or configurations. The set of all processes that may be performed with a certain material is called the class of processes of the material system. Each such process is assumed to determine the values of certain dependent variables, or effects (stresses, energy-flux, electric-current, for example). The relationship between processes and effects is expressed by means of a material function. A class of processes and a material function defined on it constitute a material system. The bulk of this paper deals with the problem of comparing and classifying material systems. Material systems can be transformed into new ones whose properties may be distinct from the original systems. The idea of transforming a material system leads to a derived concept of state. Following a suggestion of ONAT, we introduce states as equivalence classes of processes, a procedure which stems from systems theory.

NOEL now calls them "semi-elastic". ~ However, OWEN has shown that many features of classical models for plastic behavior arise naturally for materials with memory which possess an "elastic range". Another framework for describing materials is that of COLEMAN & OWEN WhO gave much emphasis to a general formulation of the laws of thermodynamics. This work gave me much inspiration, although I have not yet worked out a thermodynamical theory within the present context. ~'~ See BERTRAM (1) p. 146f. Material Systems 101

The concept of material isomorphy, introduced in Section 4, is a tool for com- paring two material systems and for deciding whether or not they describe the same material behavior; if they do they belong to the same material. Although formally quite similar, the concept of material symmetry (Section 5) plays a different role: it classifies a material system according to invariance properties under certain symmetry transformations. We obtain a natural distinc- tion between two classes of symmetry transformations: one forms a semi-group under composition; it contains the other collection of transformations, which forms a group. These two concepts are due to NOEL and have been reformulated for the pre- sent theory. They turn out to be simpler and more comprehensive than in NOEL'S work. The present theory differs from others in its treatment of internal constraints, i.e., restrictions on the class of processes (for example, on the admissible con- figurations). As a consequence, the effects cannot be considered as being deter- mined by the process. The following diagrams illustrate this possibility; in them, e is an appropriate independent variable (a process parameter) and a a dependent variable (an effect parameter). Figure 1 shows the behavior of a material due to

Fig. 1 Fig. 2 Fig. 3 a unilateral constraint. ~ This could be an elastic material reinforced by inextensible fibres that have no stiffness under compression, such as textile cords or thin steel wires. Figure 2 represents the characteristic curve of an ideal diode (e = voltage, a ---- current). It shows the somewhat unpleasant property of precluding a global functional dependance both of tr on e and of e on a. Figure 3 describes a classical constraint: one process parameter is fixed, one effect parameter is undetermined. An example of this type of constraint is incompressibility: e is a density-parameter, and a the pressure. In order to describe constraints of these types, we use here material relations instead of functions. Such a material relation maps each process into a set of effects. The values of material functions are assumed to be non-empty, closed subsets of the space of the dependent variables. Only if all values of a material function are singletons is the material free of constraints. This approach has two main advantages. First of all, we can maintain the principle of determinism in a slightly modified version: The process determines the set of all possible effects.

~" PRAGER and FICHERA have studied unilateral constraints. 102 A. BERTRAM

Secondly, a somewhat arbitrary distinction between reactions and constitutive effects is avoided. This saves us from having to enter discussions on whether or not the reactions produce entropy or energy) The three examples above lead to the question of whether or not there is a canonical scheme for identifying independent and dependent variables. The char- acteristic curve of Figure 1 can be described as a function e(a) but not as a function a(e). Figure 2 does not admit to a functional representation of either type. How- ever, if we employ set-valued functions, the three examples may be described as material functions in both directions. It is not always possible to exchange the dependent and independent variables, and in Section 6 we will establish conditions necessary and sufficient for this property of material systems. There we introduce a description of material behavior by constitutive relations which is equivalent to the one used in earlier sections of this paper. The reader who is only interested in unconstrained materials may omit Sec- tion 6 and regard the material functions as being single-valued for the entire paper. He who favors the classical distinction between reactions and constitutive effects may regard the material function as being single-valued and as determining only the constitutive part of the effects (whatever this may be). In Section 7 we extend NOEL'S method of constructing "natural" uniform and topological structures on the state space to the case of set-valued functions. These structures are necessary in order to define relaxation properties of material sys- tems. This theory rests, as was mentioned above, on NOEL'S new theory of simple materials. To illustrate this point, we define in Section 8 a subclass of our material systems which are essentially NOEL'S simple materials. Because NOEL and others ~~ have discussed many examples of special materials, I give here only one example of a class of materials, namely the rigid-plastic ones. Although quite well-known and rather simple, these materials do not fit into any of the usual theoretical frameworks. The reader who is interested in more examples, especially ones involving constraints, is referred to my doctoral thesis.g~

2. Material Systems

Let J- be a finite-dimensional real linear space and Y-* its dual. We call Y the space of dependent variables and ~--* the space of independent variables. We will later make use of the fact that these spaces are endowed with a standard topology and uniformity which renders addition and scalar multiplication uni- formly continuous operations. It is not postulated that these spaces are endowed with an inner product or a norm.

In BERTRAM (2) I showed that properties of this kind are material properties and do not obey a general principle. See GREEN, NAGHDI • TRAPP, ANDREUSSI & PODIO GUIDUGLI, GURTIN & PODIO GUIDUGLI, BERTRAM & HAUPT, ALTS. g~ See DEE PIERO, SILHAVY & KRATOCHV[L. ~$$ BERTRAM (1). Material Systems 103

Examples: If we assume that the body ~' is a differentiable manifold, we may consider at each point XE ~ the tangent space ~J-x~ and its dual space 3-*~', called the co- tangent space; both spaces lack a canonical inner product. Following NOLLe, we define an intrinsic configuration of XE ~' to be a linear symmetric positive-definite mapping G: J'x ~ ~ 3"*~'. We denote the set of all linear mappings from J'x ~ into 3-*~' by the tensor product J-*~ | ~-*~. According to NOLL, the stresses are described by linear symmetric mappings S: ~r~ ~ ~xM, i.e. by elements of the space ~-x ~' | J'x ~, which is defined analogously. Thus, in purely mechanical theories, we can make the identifications 3-* _: ~'*~ | g-*~ and 3- = ~x ~ | 3-x~. This intrinsic description in mechanical theories can be extended to other physical theories by making the following identifications: (i) the temperature and the internal energy are real numbers; (ii) the temperature gradient, the electromotive intensity, and the magnetic induction are covectors, i.e. elements of ~*~; (iii) the electric current, the polarization, the magnetization, and the energy flux are tangent vectors, i.e. elements of J'x M. Thus we can identify the space of dependent variables in continuum physics by

3- ~I~ X Jrx~X ~x~X J-x~X 3-x~X 3-x ~ | 3"x~ , and the space of all independent variables by ~r, --- a~ • ~-~ • ~-~ • ~-~.~ • ~-~ • ~-~ | ~-~.~.

Here • denotes the Cartesian product. (We can always replace given sets with "larger" spaces in order to obtain linear spaces and the duality between ~- and ~-*.)

A (configuration) process is a mapping of a closed (time-) interval into the space of independent variables.

Definition. Let d be a non-negative number. A process of duration d is a mapping P: [0, d]--> 3-*. The values of P are called configurations.

When we deal with more than one process, we use the same subscript for both a process and its duration. A process of zero duration is called a null process. If P1 and P2 are two processes, and there holds Pl(dl) = P2(0), we define the composition of P~ and P2 by

(Pl(t), if t --~ dl, P2o PI [P2(t -- d~), if dl -

Definition. A class of processes is a set ~ of processes which satisfies the following four conditions: (P1) ~ is not empty;

The intrinsic description is given in more detail in Section 9. 104 A. BERTRAM

(P2) all processes have the same initial value, i.e., P1, P2 C .~ ~ PI(0) : P2(0); (P3) all subprocesses of processes of ~ are again in ~; (P4) for each process P~ E ~ there exists a continuation P2 of non-zero duration such that P2 ~ P~ E ~'.

Examples for classes of processes are: (i) the set of all constant processes with a preassigned initial value and arbitrary dura- tion; (ii) the set of all processes with a preassigned initial value and arbitrary duration; (iii) the set of all processes with a preassigned initial value and with durations smaller than a given positive number; (iv) the set of all r-times continuously differentiable processes with a preassigned initial value.

We note that each class of processes contains a unique null process. The last primitive concept of this theory is that of a constitutive function.

Definition. Let ,~ be the set of all non-empty closed subsets of ~', called the set of effects. A constitutive function F is a mapping F: # -+ 6~. F(P) is called the effect of P.

As was explained in the Introduction, for unconstrained systems, F is always single-valued, i.e. the range of F is a subset of the set of singletons formed from elements of g-. For constrained systems the values of F may have many elements. This concept saves us from making arbitrary distinctions between constitutive effects and reactions.

Definition. Let g- be a finite-dimensional real linear space with dual space J-*, let ~ be a class of processes with values in ~--*, and let F be a constitutive function defined on ~ with values in 8, the set of all closed, non-empty subsets of ~'-. The triple MS := (J-, ~, F) is called a material system.

The rest of this paper is concerned with investigating the properties of material systems.

3. The State Space of Material Systems

Let X and Y be sets, let Xo be a subset of X, and let f be a function from X into Y. We denote the restriction off to Xo by f]xo.

Transformation Theorem 3.1. Let MS = (~'-, ~, F) be a material system and Pr be a process in ~ with duration dr. Then we can transform MS into a new material system MS' in a natural way by setting MS' : (3-, ~', F'), ~' := (P'] P' is process such that P' o PTC ~}, F'(P') := F(p'o Pr) for all P'C ~'. Material Systems 105

It is easy to verify that, for all Pr in ~ MS' really is a material system. It is called the system MS transformed by Pr. We introduce the transformation function HMs, which maps the transformation process Pr E ~ into the system transformed by Pr. Let J/be the range of HMs, i.e. the set of material systems that can be obtained from MS in this way. Then HMs : ~ ~ d/ is surjective; in general it fails to be injective. In order to remove this shortcoming, we introduce the notion of state. This is done by defining an equivalence relation on the class of processes.

Definition. Let MS = (~, ~, F) be a material system and P1, P2 be in ~. We call P1 equivalent to /'2, and write P1 "~ P2, if MS transformed by P1 equals MS transformed by P2, i.e., P1 "~ P2 <::>HMs(PI) = HMs(P2). We call the equivalence classes under --~ states; the collection of all states forms the state space .o~e of MS. The processes that are equivalent to the (unique) null process of ~ are called cyclic processes; their equivalence class is the initial state.

It is obvious that the relation ,.,o really is an equivalence relation. The physical interpretation of this definition is the following: two states of a material system are the same if and only if they cannot be distinguished by performing any process whatsoever and comparing the effects. In order to illustrate this concept, we give two examples.

Example 1. We define two processes PI and P2 to be similar if there exists a monotone bijection c,: [0, dl] ~ [0, d2], such that P1 =/'2 " c~. A material system is said to be rate-independent if similar processes are equivalent. Roughly speaking, such systems cannot distinguish between two processes that trace out a single trajectory of configura- tions at different rates. A subset of the class of rate-independent material systems is the class of elastic systems; for elastic systems, two processes are equivalent if they end at the same con- figuration.

Example 2 (aging systems). Two types of aging occur for these material systems: kine- matic aging and response aging. The first may be obtained by non-stationary constraints (see BERTRAM (1)) and is described by requiring that certain segments of processes can be performed at one time but not at another. However, in the present theory response aging is of more interest. Let ~ be a non-empty subset of the non-negative reals which is bounded above by the supremum of the durations of all processes of the class of processes (including oo). A material system is defined to be response aging at times in ~ if equiv- alent processes Pi with at least one duration d iE ~ have the same duration. For re- sponse aging systems with 0 E ~ there is obviously no cyclic process other than the null process. An example of response aging is the hardening of concrete, caused by time-dependent chemical reactions taking place in the material.

We define the effect at a state to be the effect of any transformation process from the equivalence class of that state; that effect is equal to the effect of the null process of the transformed material system. The assignment of effects to states is formalized by means of an output function for the effects of states E: ~e __> g.

The dot placed between the symbols for two mappings denotes composition. 106 A. BERTRAM

We define the configuration of a state to be the final value of any transforma- tion process from its equivalence class. By analogy, we introduce the output function for the configurations of states G: ~ -+ J'*.

A quite similar approach is the method of preparation suggested by BRIDGMAN and detailed by GILES, PERZYNA,and PERZYNA & KOSINSKI. Their basic idea can easily be described in this context. Let MS = (3-, ~, F) be a material system, Po a process and ~'(Po) := (PC ~' I Po o PE #}, i.e. the set of all processes in ~' that can be continued by Po. Of course, this set may be empty. In general, if PI and P2 are in ~(Po), we cannot expect that F(Po o P~) equal F(Po o P2) unless P1 and P2 are equivalent, and hence represent the same state. In this case we may say that P1 and P2 correspond to the same "method of preparation". "Two states or methods of preparation need not be distinguished if they are equivalent in respect of any prediction which might be made--that is, if they correspond to the same assertion concerning the result of any experiment which might be performed on the system." (GILES, p. 17.) This concept of state coincides essentially with mine here.

The following theorem is a consequence of a well-known theorem on the "natural function" of an equivalence relation, i.e., the function which maps each process into its equivalence class.

Theorem 3.2. Let MS = (~-', ~, F) be a material system with transformation function HMS, let ~l := HMS(~), let ~ be the state space of MS, and let o~ be the natural function of ~. Then there exists a unique bijection i such that i . ~o(P) = HMs(P) for all P E ~; the following diagram is then commutative:

HMs

Fig. 4

According to this theorem it is equivalent to talk either about material systems transformed by a certain process Pr or about the system being in the state ~o(Pr). The latter point of view is often simpler, because in many cases only a finite number of parameters (internal variables) determine the state completely. If MS = (~--, ~, F) is transformed by Pr into MS" = (J-, ~', F'), it is obvious that ~,/l :: HMS(~) ~ all' := HMS'(~'). It is quite reasonable by means of the bijection i between d/and 5( to identify the states in the following manner:

i~llS " HMS( P" ~ PT) ~ i~, " HMs'(P') for all P' ff ~'. For the state spaces there follows ~ ) ~'. Material Systems 107

Example. There are theories of granular media (dry sand, etc.) that permit only defor- mations which evolve towards a critical density, i.e., only dilatative or compressive chang- es can occur when the actual density is below or above the critical one, respectively. This is a nonstationary unilateral constraint that makes the system age kinematicaUy. The set of densities which are accessible via continuations of a process is a non-increasing function of the number of continuations.

It is interesting to ask under what conditions ~e, is identical to ~. This question leads to the following concept.

Definition. Let MS ---- (~--, ~, F) be a material system and P E ~. P is called revertible if there is a cyclic process in ~ that contains P as a sub-process.

In other words, one can completely undo the transformation by a revertible process by means of another transformation process which returns the system to its initial state. The following propositions are easy to verify.

Proposition 3.3. Let .~e be the state space of a material system, and let .~' be the state space of MS' = HMs(P) for any P E :~. P is revertible if and only if

Proposition 3.4. Equivalent processes are all revertible or all non-revertible.

Proposition 3.5. A process P3 = P2 o p~ is revertible with respect to a material system MS if and only if P1 is revertible with respect to MS and P2 is revertible with respect to HMs(P~).

Every class of processes contains at least one revertible process, the null process. More generally, every cyclic process is revertible. It may happen that all processes in a certain class of processes are revertible. We call such a material system revertible. In light of Proposition 3.5 we conclude that revertible systems can only be transformed into revertible systems. However, there are non-revertible systems which can be transformed into non-revertible or into revertible systems. The following proposition clarifies the relation between revertibility and aging of material systems.

Proposition 3.6. Response aging material systems are non-revertible.

Proof. Assume tllat a material system is response aging at a time t ~ 0 and revertible. Then there exists a process P1 with dl = t, and there is a continuation P2 of P1 such that P2 ~ P~ is a cyclic process. P2 ~ P~ can be continued by P1, and P~ o P2 o p1 is equivalent to P~. Because the duration of P1 o P2 ~ Pt is greater than the duration d~ of Pt, these processes cannot be equivalent, and the proposition is proved for the case t =# 0. The second possibility is that MS ages at t = 0. This means that there is no cyclic process other than the null process. Accordingly, no process of non-zero duration may be continued to a cyclic process. By Axiom P4 there exists at least 10 8 A. BERTRAM one continuation of the null process, and this continuation yields a non-revertible process for MS. Therefore, MS is non-revertible; q.e.d.

On the other hand, one can construct non-revertible systems that are not aging. Thus the aging systems are a proper subset of the non-revertible ones.

The following definitions are motivated by NOLL'S theory and are of importance in establishing the relation between that theory and the present one.

Definition. Let z C ~ be a state of a material system MS. We denote by ~z the class of processes of the system i(z), and define (:~, ~) := ((z, P) Iz ~ ~, P E ~z}. We call the function 0: (~, ~) --+ =@e defined by ~(z, P) ---- ~oi(2)(P) the evolution function for MS. Here, ~oi(z) denotes the natural function of the material system i(z) which maps each process in ~ into the corresponding state of the system i(z).

For a given material system it is often desirable to find a convenient representation for its states. In general, there are many such representations. It may happen that the states can be represented by a sequence of real numbers, or even by a finite set of reals, but of course this is not always assured. Let us now consider material systems that have the following properties: 1) the state space can be represented by an open subset of a finite-dimensional linear space, and 2) the evolution function can be formulated incrementially, i.e. by a first order differen- tial equation in time = ~(z, d) and, if necessary, by a mechanism that assures a unique solution as an integral along a configuration process starting at a certain initial state. By choosing a basis in the state space, the state can be represented by a certain number of components (al, ~2 .... ). It is always possible to do this in a way that c~1 to %, are the components of the configuration. We call the rest of the state parameters %,+1, ~+2 .-- internal or hidden variables. Although the two assumptions are rather restrictive, there are many applications of this theory in the literature on viscoelasticity, hypoelasticity, plasticity, and thermo- dynamics. In most cases the space of the internal variables is real and finite-dimensional.

4. Material Isomorphy

In this section we shall investigate isomorphisms between material systems. In doing so, we give a precise meaning to the notion that two systems exhibit the same physical behavior. Let MS, = (3--1, ~,, F,) and MS2 = (3-2, ~2, F2) be material systems. First, J-1 and ~'-2 are isomorphic if and only if they have the same dimension. Isomorphisms of vector spaces are, of course, linear bijections; we shall denote the collection of these isomorhisms by Iso (~'1, 3-2). If A E Iso (~J--1, ~--2), it follows Material Systems 109 that the adjoint A*, the inverse A -1, and the inverse of A*, A-*, satisfy the rela- tions A* E Iso (~-*, ~-*),

A -~ E Iso (3-2, Y-O,

a-* c Iso (:*, :~).

Therefore, we may take for the isomorphisms between classes of processes the mappings induced by elements A-* of Iso (~'-*, 3"*). The isomorphisms of the effects are the elements A of Iso (5"1, ~d'-2).

If we identify 3- as in Section 2, then A may physically be interpreted as being in- duced by an identification of the tangent vectors in one tangent space to those in a second tangent space.

The following results are immediate consequences of these definitions.

Proposition 4.1. Let P be a process with values in ~-'~', and let A E Iso (Y,, 3-2). Then A-*(P) is a process with values in ~--*. If #, is a class of processes with values in ~--~', then A-*(#I) is a class of processes with values in #-~'. If P2 ~ PI is a process with values in ~Y-*, then

A-*(P2 o P~) = A-*(P2) o A-*(P,).

Detinition. Two material systems MS, = (.Y-~, ~, F1) and MSz = (9"-2, "~2, 1;'2) are called materially isomorphie (relative to A), if there exists a mapping A such that (I1) A E Iso (3"~, :2),

(I2) ~2 = A-*(#,),

03) F2" A-*(P) = A. F~(P) for all P E #,.

In order to show that this definition is symmetric in the two material systems we verify that MS2 and MS, are materially isomorphic (relative to A -1) when- ever the conditions of the definition hold. First we have

(I1)' A-i E ISO (~-2, 6~1), as already mentioned. Applying A* from the left to equation (I2) yields

(I2)" #~ ---- A*(~2). By substituting P = A*(P') in (I3) and by applying A -1 from the left we get

(I3)' F, A*(P') = A -~ F2(P') for all P'E ~2-

The following theorem shows that isomorphisms really preserve the detailed structure of material systems. 1 10 A. BERTRAM

Theorem 4.2. A material isomorphism maps a) equivalent processes into equivalent ones, b) cyclic processes into cyclic ones, c) revertible processes into revertible ones, d) non-revertible processes into non-revertible ones.

Proof. a) Employing the definition of transformed systems and the isomorphy conditions I1-I3, one can easily verify that, for each PE ~, A is a material isomorphism between MS~ :-~ HMs~(P) and MS2 :---- HMs~(A-*(P)). Let P1, P2 in ~ be equivalent for MS~, so that HMsl(P~) equals HMsl(P2), and each of these transformed systems is isomorphic to the systems Hus~(A-*(P~))and H~ts~(A-*(P2)) relative to A. This can only be the case when these systems are equal, too. Hence A-*(P~) and A-*(P2) are equivalent. b) If A is a material isomorphism between MSI and MS2, then A-* maps the null process of ~1 into the null process of ~2. Because a process is cyclic if and only if it is equivalent to the null process, result (a) implies (b). c) If P~ is a revertible process of MSI, then there is a cyclic process P o P~ in ~. By b), A-*(P o Px) = A-*(/~) o A-*(P~) is a cyclic process in ~2 which contains A-*(PI) as a subprocess, and is therefore revertible. d) Assume that P is non-revertible and A-*(P) is revertible. By the symmetry property of the definition of material isomorphy mentioned above, the inverse material isomorphy maps the revertible process A-*(P) into the non-revertible one P. This contradicts c), and hence A-*(P) must be non-revertible; q.e.d.

We now make precise the statement that two material elements are composed of the same material:

Definition. Two material systems are called m-equivalent if each system can be transformed by a suitably chosen revertible process, so that the two resulting mate- rial systems are materially isomorphic. Each equivalence class under the relation of m-equivalence is called a material.

Proposition 4.3. m-equivalence is an equivalence relation on the set of all material systems.

Proof. a) The symmetry of the relation is obvious. b) To verify reflexivity, one can take both revertible processes to be the null process and the material isomorphism to be the identity on J-. c) In order to prove transitivity of m-equivalence, let MS1 and MS 2 be m-equiv- alent. There then exist revertible processes P1 E ~1 and P2 E ~i~2 and an iso- morphism A E Iso (5"1, J'2) such that HMs~(P~) and HMs,(P2) are materially iso- morphic relative to A. Similarly, let MS2 and MSa be m-equivalent, and choose P2 E ~2 and Pa E ~a, both revertible, and A' E Iso (J-2, 3"a), such that H~4s~(P2) and HMs3(P3) are materially isomorphic relative to A'. Recall that P2 is revertible if and only if P2 is a subprocess of a cyclic process Po ~ P2- This may be continued by any process in ~2, in particular, by P2. The process P~ o Po ~ P2 is also revertible Material Systems 1 11 and equivalent to P2 relative to MS 2. By Theorem 4.2, P2 ~ P0 is revertible if, and only if A*(P2 ~ Po) ~ P1 is revertible (for MS~). If HMs~(P1) is materially isomorphic to HMs2(P2) relative to A, then HMsI(A*(P2 o Po) ~ P1) is materially iso- morphic to HMs:(A-* A*(P2 o Po) ~ P2) = HMs~(P2) relative to A, because P2 ~ Po ~ P2 contains P2. On the other hand, HMs2(P2) is materially isomorphic to H~s~(P3) relative to A'. We can compose the two isomorphisms to obtain the isomorphism A'. A between HMs~(A*(P2 ~ Po) ~ P1) and HMs~(P3); q.e.d.

The following proposition simplifies the definition of material.

Proposition 4.4. Two material systems are m-equivalent if, and only if, at least one of the systems can be transformed by a revertible process into a system materially isomorphic to the other.

Proof. If MS1 and MS2 are two material systems belonging to the same material, then there exist two revertible processes P1 and P2 such that HMs~(P~) and HMs,(P2) are isomorphic relative to A. If Po ~ P1 is a cyclic process for MS1, then it is easy to verify that MS~ is m-equivalent to HMs2(A-*(Po) o P2) relative to A, and that A-*(Po) o P2 is revertible for MS2. Conversely, choose the null process Po as a revertible process for MSI. If HMs~(Po) = MSt is materially isomorphic to HMs,(P2), then the two systems are also m-equivalent; q.e.d.

In the foregoing proof we have used the following easily proven fact: If MSI is m-equivalent to MS2 relative to A, then H~csl(P) is m-equivalent to H~s,(A-*(P) ) for every P E 5~1, and the material isomorphisms are the same. If a material system is revertible, i.e. its process class contains only revertible processes, then by Theorem 4.2 c) we can easily see that every m-equivalent mate- rial system is again revertible. We call a material revertible, if a representing mate- rial system is revertible (and hence every one in the same equivalence class). Two material systems which are not m-equivalent, might still have the property that one can be obtained from the other by a transformation process. This is a generalization of the notion of "transformation" to materials.

Definition. Let MS1 and MS 2 be material systems. We say that we can transform the material of MSI into the material of MS2 if there is a process P E ~ such that HMsl(P) is m-equivalent to MS2. Of course, P does not have to be revertible. Otherwise this notion would coincide with that of m-equivalence. Next we show that the foregoing definition is independent of the choice of the material systems representing the two materials.

Proposition 4.5. If we can transform the material of a material system MS1 into the material of MS2, then the same is true for every other pair of material systems representing the same materials.

Proof. Let MSi be four material systems with classes of processes #i. Let MS1 be m-equivalent to MS3; i.e. there exists a revertible process P1 in #~ such that 112 A. BERTRAM

HMsI(PO is isomorphic to MS3 relative to A1, and P1 can be continued by P,~ such that /'4 ~ P~ is cyclic for MSt. Let MS2 be m-equivalent to MS4, i.e. there exists a revertible process P3 E ~4 such that MS2 is isomorphic to HMs,(Pa) relative to A a. Assume that we can transform the material of MS~ into the material of MS2, i.e. there exists a P2 E ~ such that HMsl(P2) is isomorphic to MSz relative to A2. It is left to the reader to show that the material system HMs3(AF*(P2 ~ P4) is materially isomorphic to H~ts,(Pa) relative to -4 3 A 2 All; q.e.d.

In light of this proposition, the following definition is meaningful.

Definition. We say that one material can be transformed into another, if this is the case with respect to at least one pair of material systems representing the materials (and, hence, for all such pairs).

It is easy to show that one can transform revertible materials only into re- vertible ones. But the converse is not true, i.e. transformation of non-revertible materials can lead to both revertible and non-revertible ones. If the transforming process is revertible, the transformed material surely is non-revertible.

Let us investigate the dependence of the concept of state on the concept of material isomorphism. Let MSi----(,~'i, ~i, Fi) be material systems with state spaces ~ei, let Pi be in ~i, and MS[ = (J-i, ~, F;) :-- HMsi(Pi). We recall from Theorem 3.2 the relations zi : o~i(Pi) : i[-l(MS~)~ ~i. If we now define a selection function q~i :Y'i-+ ~i of a material system MS i as a mapping that maps a state into an (arbitrary) process out of its equivalence class, then r ~0i is the identity on .~';. Of course, a material system can have many selection functions, but each one is a right inverse of the natural function o~i of the system.

Definition. Let MS1 be a material system isomorphic to MS2 relative to A. The bijection 7,4 (of the state spaces) induced by the material isomorphism A is the function ~'a : ,~e __~ ~2 defined by

7A(Z) := 09 2 " A-* 991(2) This definition is meaningful because A-* transforms equivalent processes in ~ into equivalent processes in ~2.

A-* l

Fig. 5 Material Systems 113

Theorem 4.6. The evolution functions satisfy

~A ~ el(Z, P) ~-- Q2(~a(Z), A-*(P)) for all (z, P)6 (~f, ~)1.

Proof. The definitions of 7A and 91 yield the relations: 7A " 9x(Z, P) = o2 " A-* q:l " 9~(z, P) = 0,2. A-* .q~. o~,(e o ~0~(z)). The following pairs of processes are then equivalent: ~l" ~ol(P o ~01(z))~ p o q~l(z), A-* . ~ . o~(eo ~(z)) ,.~ A-*(e) o A-*(~0,(z)), and it follows that YA" ex(Z, P) = r o A-*" ~l(z)]

= o~2[A-*(P) o ~2" ~A(Z)] = e2(TA(Z), A-*(P)); q.e.d.

5. Material Symmetry

In continuum physics it is convenient to classify material systems by means of properties which are invariant under certain transformations, called symmetry transformations.

Definition. Let MS = (~J', ~, F) be a material system. A function A is called a symmetry transformation for MS if there is a process PA C ~ such that, with

MSa := HMs(P A) =- (,~', ~aA, FA), the following conditions hold: (Si) A E Iso (•, J),

(s2) ~A = A-*(~),

($3) A" F(P) ---- FA(A-*(P)) for all PC ~.

We say that MS is A-symmetric to MSA.

The conditions S1-$3 are equivalent to the assertion that MS and MSa are materially isomorphic. However, we intentionally concealed this fact in the above definition in order to keep distinct two concepts having similar mathematical descriptions and yet quite different physical meanings. In the last section we com- pared two different material systems by investigating whether a material isomor- phism exists or not. Here we study a single material system by considering all the isomorphisms that do exist in the above sense. 114 A. BERTRAM

We denote the set of all symmetry transformations of a given material system by WMs, called the symmetry semigroup of MS. This terminology is justified in the following proposition.

Proposition 5.1. Z,vfMS forms a semigroup (with unity) under composition.

Proof. Let A1 and A 2 be in ~MS and let P1 and P2 be the transformation processes in the statement that AI and A2 are symmetry transformations, respectively; i.e. MS is At-symmetric to HMs(P1) = (~--, ~',F') and A2-symmetric to HMs(P2) z (J-, ~", F"). By $2 and the definition of the class of processes of transformed material systems we have the implications:

P2 ~ ~ ~ AI*(P2) ~ ~' ~ P3 : = AI*(P2) o Pl C ~.

Let HMs(Pa) = (~--, g~'", F'"). The classes of processes are related by the follow- ing conditions, which are equivalent: PE~ <=~ A~-*(P) E ~" ~ Ai*(P) o P2 E r AI*(Az*(P) ~ P2) = A~* As (p) o Ai-*(P2)9 ~' ~ AI* A~*(e) o A-*l(e2) o el G r A~-* A2*(P)G ~"'. By $3 and by the definition of the constitutive function of a transformed material system we conclude that F'"(A?*. A~*(P))

= F(A{*. Af*(P) o AI*(P2) o P,)

= F(A{* A2*(P) o A?*(e2)) = A1 F(A2*(P)o P2)

= Aa" F"(A2*(P))

= A1 A2 F(P) for all PE ~.

Thus we have shown that A1 m2 is in Yt~MS. The associativity of the composition results from the same property of the composition of linear functions and of the operator o on processes. The unity of 3r is the identity of ~-" with corresponding process equal to the null process; q.e.d.

Just as was the case for material isomorphisms, each symmetry transformation yields an induced function on states.

Proposition 5.2. For every A E ~MS there is a unique injection 7A: ~ ---> Kr such that ~'A " C~ = tOA " A-*(P) Material Systems 115 holds for every PC-~. In other words, the following diagram is commuta- tive:

A-*

Fig. 6

The mapping 7n is called the injection induced by A and is given by the formula

7A(z) = tOMs(A-* " q~Ms(z) o t"4).

Proof. Let P be in ~1 and let z = tOMs(P). According to the formula for YA in the statement of the proposition, we have

74 " tOMs(P) = tOMs(A-*" Cp,wS" toMs(P) ~ P4)

= toMs(A-*(e) o e~)

= toZa-*(e)).

The left-inverse of 74 is given by 7J-I = toMS" A* r defined on -~A :---- 74(~). In fact, the defining relations for 74 and y~ yield for all z

74' " 7A(z) = tOMS" A* " q~4 " [,OMs(A-* " ~Ms(Z) o P A)]

= tOMS" A* q~A " tOA " A-* ~0Ms(Z)

~Z.

Therefore 74 is injective. The uniqueness of 74 follows from the fact that WMs and to4" A-* are well-defined functions on 2; q.e.d.

We now consider the case where the transformation process P.4 is revertible.

Definition. Let A be in the symmetry semigroup JZ'Ms of a material system MS, such that MS is A-symmetric to HMs(P4) and P4 is revertible. We call the set of all functions A with this property the symmetry group cg, s of MS.

The symmetry group is never empty, because it always contains the identity on ~ relative to the null process. The symmetry group is contained in the symmetry semigroup, and, for revertible systems, both sets coincide. The following proposi- tion justifies the name "group".

Proposition 5.3. ~MS is a group under composition. 116 A. BERTRAM

Proof. We first consider the assertion that A1 A2 is in fgMS whenever there holds: A1, A 2 E ~MS. We have seen in Proposition 5.1 that A1 A2 is in ~ffMS whenever A~ and `42 both are in ~MS, which is certainly the case under the as- sumption from above. We still have to verify the statement that A[-*(PA2) ~ PA, ~ is revertible if PA~ and PA~ are revertible. The process A;-*(PA:) is revertible according to Theorem 4.2 c), and the composition of revertible processes is revertible according to Proposition 3.5. The associativity results from the fact that composition is an associative operation, and it is obvious that the identity on ~-- is in f~MS. Thus, it remains only to show that the inverse A -~ of a symmetry transfor- mation A C fqMs again is in ~Ms- Let PA be revertible, and choose a continuation /~ such that /~ ~, PA is a cyclic process. Moreover, P is in ~ if and only if P o/;o PA is in ~, and this is equivalent to the assertion that P o/3 is in ~a and, hence, to the condition that A*(P o fi) is in ~. Now let MSa-~ = (~-, ~A-,, FA-O := HMs(A*(P)). According to Theorem 4.2 c) A*(P) is revertible and a process P is in ~ whenever A*(P) is in ~A-~, i.e. ~A-~ = A*(~). On the other hand, the definitions of FA and FA-~ yield

A .F A ~. A*(P) = A. F(A*(P) o A*(/~))

= FA(`4-*" A*(P) o .4-*..4*(/')) = FA(eo ~)

~- F(P o t;o PA) ---- F(P) for all P C ~, and we conclude that A -I is in fqMS; q.e.d.

Proposition 5.4. Let A be in ~ffMS and 7A : ~e _+ ~ be the injection induced by A. 7A is bijective if and only if A is in the symmetry group ~#MS.

Proof. If A is in fqMS we have 7A(z) = ~OMs(A-* " q~Ms(Z) ~ PA) for all z C ~.

Because PA is revertible, we can choose a continuation/' such that /3 o PA is a cyclic process in ~. As shown in the proof of Proposition 5.3 a process P is in ~ if and only if A*(P o fi) is in ~. The surjectivity of tOMS and the relation 60Ms(P ) = ~oa(P o P) tell us that 7A is surjective and hence bijective. On the other hand, let us assume that A is in Jg~MS and that 7A is bijective. By definition, tO~s is surjective, so that 7A'O~MS = OJA" A-* is surjective as a function from into ~e. This is true only if~oA is surjective. According to Proposition 3.3 ~en := tnA(~ ) equals ~ if and only if PA is revertible; q.e.d.

The next proposition explains more precisely the relation between ~MS and (#Ms.

Proposition 5.5. The symmetry group flus is the maximal group in the symmetry semigroup ~MS, i.e. ~MS contains every other subgroup in ~MS. Material Systems 1 17

Proof. If (~ Q o~Ms is a group under composition, then there holds

A E ~ <=~ A -I C fr ~ A-I ~ ~MS.

We show that for each A in ~ the corresponding process P4 is revertible. This fact rests on the following relations: PG3 ~ <=> A* (P) C ~A-~

<::>A*(P) o PA-, C ~A-*(A*(e) o e41) = e o A-*(e4 ,)~ #4 <::>P ~ A-*(P4-,) ~ PA ~ and for every P E

F(P) = A . FA I(A*(P)) = A" F(A*(p) o P4-O -~ A " A -1" FA{A-*[A*(P) o P4-~])

= F[e o A-*(e4-,) o eA.

Therefore, whenever A is in ~, A-*(P4-1) o PA is a cyclic process. P4 is rcvcrtiblc, and A is in (gMS; q.e.d.

If ~V:MS itsclf is a group, thcn ~s = fg~s. This occurs, for cxamplc, if the material system is rcvcrtiblc. But this is not the only case; onc can also con- struct a material system for which :g'MS-~ ~MS--~ (Id:-}, and every process except the null proccss is non-revertible. The following theorem shows that the injections induced by symmetry trans- formations thcmsclvcs form a scmi-group (or group) under composition.

Theorem 5.6. a) At, A 2 C "~PMS ~ )'(A~.4~) = ~'4L " 7A=" b) Aa, A2, A3 C aff MS ~ 7(At'A2) " 7A3 = 7At " ~2(A2"Aa)" c) 7'~a:- = Ida. d) A C ~MS ~ 74-, = (TA)-*.

Proof. a) Let i = I, 2 and Ai be in ~MS, and let Pi be processes such that MS is Ai-symmetric to H~s(Pi). In the proof of Proposition 5.1 we have shown that P(A~4~) = A~*(P2) ~ P~, and, according to the construction of 74 in Propo- sition 5.2, we have

74i(Z) = r ~)MS(Z ) c: Pi).

Together these relations yield

7(A~.A,)(Z) = COMs[A~-* " A~* CPMs(Z) o A~*(P2)o Px]. 1 18 A. BERTRAM

On the other hand we have

~A, " )'A2(z) : tOMs[A 1 * " q;MS " O)Ms[A2-* " CflMs(Z) o P2] ~ PI} = OL~s{Ai*" A2*" 9Ms(Z) o Af-*(P2)~ P,}, where we have selected the identity on ~ for q~MS ~ tOMS. b) This result follows from a) and the associativity of composition of func- tions. c) For each state z

71dj-(2) = OJMs(Idj-* " qgMs(Z) ~ PId)" Because PId may be chosen to be the null process, we conclude that

71d~,-(z) = tOMs" q~Ms(Z) = Id.~. d) It was shown in the proof of Proposition 5.2 that

(~,4)-' (z) -- ~oMS " A* ~A(z). On the other hand 7A ,(Z) = OJms[A* " q;Ms(Z) c PA-~]. In proving the existence of inverses in the symmetry group, we have shown that PA .... A*(/3), where /3 ~ PA is a cyclic process, i.e.,

tOMs(P) = OJMs(P ' ])~ PA) = r176 ?) for all P 6 ~. Accordingly, if we set z : t~L~s(P), then the above relations yield

VA-,(2) : a)Ms[A* " ~Ms(Z) '~ A*(/~ = tOMS" A*(P ~ 1~)

= O~MS" A* ~A(z) = (TA)t(z); q.e.d.

Propositions 5.7 and 5.8 below describe the invariance of symmetry groups and symmetry semi-groups under revertible transformations and the relation between the groups of two material systems belonging to the same material. The proofs are straightforward and are omitted.

Proposition 5.7. Let ,ftcMS be the symmetry semigroup and ~MS the symmetry group of a material system MS. For each revertible P in ~, the material system MS" := HMs(P) has the same symmetry semigroup ~,MS and symmetry group fqms as does MS.

Proposition 5.8. Let MS1 and MS2 be materially isomorphic relative to A with symmetry semigroups ,)ffMS,, ~'ms2 and with symmetry groups ~MS~ and .~.ws~, respectively. Then ~MS2 : A ~MS, " A-l,

cBMS~ = A ~MSt " A-t. Material Systems 119

6. Inverse Systems

In order to use a constitutive function to describe material behavior, one must decide which variables are to be "independent" and which are to be "dependent". As we here consider constitutive functions which are set-valued, there can be numerous possibilities for this choice. In this section we introduce a description of material systems which avoids the distinction between dependent and indepen- dent variables, and we give conditions under which the roles of dependent and independent variables can be reversed. in this chapter, ~-- will denote a fixed vector space and 9"* its dual space. From now on we call processes with values in 3-* configuration processes in order to keep them distinct from what we shall call "effect processes".

Definition. A mapping /~: [0, d~] -~- ~-- with d~ ~ 0 is called an effect pro- cess of duration d~.

We will use the terms "effect null-process", "continuation of an effect process", and "effect subprocess" in a sense strictly analogous to that of Section 2. We denote the set of all configuration processes by C and the set of all effect processes by E.

Definition. The relation /~ (C• displayed below is called the constitutive relation # of a material system MS ---- (~-, v~, F):

/~ := {(P,/~)[ PE ,~, E-( E, de = d~:,/~(t)E F(Pito.,l) for all t in [0, dE.i}. (R1)

By condition (P1) (see Section 2) ~' is never empty. Moreover, F(P) is non- empty for every P in #, and hence/~ is non-empty. For an unconstrained material system, all effects are singletons, and the constitutive relation has the property that, for every P C ~, there is exactly one effect process/~ such that P/z/~. Never- theless, it is possible that two different configuration processes are related to the same effect process. By definition, the constitutive relation of a material system is uniquely deter- mined by that material system. This leads to the question: which relations on • are constitutive relations for some choice of material system?

Proposition 6.1. A relation u < C• is a constitutive relation of a material system if and only if the following conditions hold: (R2) The set ~' :=(PE C]':I/~E E with P/~/~} is a class of processes, i.e. (P1)-(P4) hold; (R3) related processes have the same duration, i.e.

# This notion shall be kept distinct from the one suggested by PERZYNA 8r KO- SII~SKI. 120 A. BERTRAM

(R4) if P#/~ and P#/~2, and /~3 E E with duration de is such that /~3(t)E (/~(t)} W (/~2(t)} for all t C [0, de], then P#/~a ; (R5) the sets (/~(d)[/~E E, P/~/~} are closed in 3- for every process P in (3.

Proof. In view of the definition of constitutive relation, we easily see that condi- tions (R2)-(R5) are necessary. In order to show their sufficiency, we take a rela- tion that obeys (R2)-(R5) and define ~ and F by := (P ~ C[3 E E E with P/z/?), (R6) F(P) := (E(d) I/~ E E and Ptt/~}. (R7) Then (R2) assures that # really is a class of processes and that, for each P E ~, there exists at least one effect process EE E such that Ptt/~. Thus, F is well defined on 2~, its values are non-empty (by (R2) and (R3)) and closed in J" (by (R5)). (R4) is a saturation condition on/z to assure that # does not contain fewer elements than it would have by (R1) if (R6) and (R7) hold; q.e.d.

Proposition 6. I tells us that the function (3J, ~, F) ~-> # (with # given by (R1)) maps the collection of material systems onto the collection of relations on C • E satisfying (R2)-(R5). We now show that this mapping establishes a one-to-one correspondence between the two collections.

Proposition 6.2. The mapping (J', ~, F) ~/t is injective.

Proof. Let (~, ~1, F~) and (~, ~2, F2) be material systems, and let/zx and/~2 be the corresponding constitutive relations. Suppose that/t~ equals/t2. For every PC #~, there exists an /~ in E such that PtzIE; therefore, P#2E and PE #2. Consequently, #1 Q ~2, and an analogous argument shows that ~2 Q #1. As a consequence of the definition (R1) Fa(P) equals the set (/~(d) ] J~E E, P/a~J~), as well as F2(P) = (&d) l S, em ). By fll = f12 El(P) and F2(P) coincide. Thus ~ -----#2 and F1 = F2, so that (~--, #~, F~) = (J-, ~2, F2); q.e.d.

Propositions 6.1 and 6.2 permit us to speak of a constitutive relation # and its corresponding material system. The description of material behavior by material systems or by constitutive relations is completely equivalent, and we could have formulated the entire theory of material systems in terms of constitutive relations. This fact is particularly useful in the discussion of inverse systems which follows. lf/~ is a relation on C • we denote its inverse by /~-1 C E • i.e. PtzE <=~ E~-Ip. If we interchange the roles of the independent and dependent variables, we can introduce material systems of the form (9"-*, ~, F) instead of (f .... ), such that the configuration processes have values in ~-" (instead of J-*), and the effects are subsets of ~--* (instead of ~). The constitutive relations of such a system are subsets of E• (instead of C• It may happen that a constitutive relation /z for a system (3-, ~, F) is such that/z -1 is a constitutive relation for a material system (5", #', F') (with interchanged variables). Material Systems 121

Definition. We call a constitutive relation/z invertible if #-1 also is a constitutive relation. We call a material system MS invertible if its constitutive relation is invertible. We call the material system corresponding to #-1 the inverse system of MS.

By Proposition 6.2, the inverse system of an invertible system is well defined, and the inverse of the inverse system is again the original one. Both systems describe the same material behavior in an obvious sense. For invertible systems the division of the variables into dependent and independent ones is arbitrary. This fact is expressed best by describing the system by means of a constitutive relation. Unfortunately, this is not always possible. The theory of plasticity without strain hardening provides an example of a non-invertible system. As we shall see in Section 9, the effect (stress) does not determine the configuration process (plastic flow), not even up to an arbitrary factor. If we want to know whether or not a given material system is invertible, we construct its constitutive relation by (R1), invert it, and find out whether or not the conditions (R2)-(R5) are fulfilled. If this is the case, the inverse system MS_ I can be constructed by (R6) and (R7). However, this procedure can be shortened by means of the following theorem.

Theorem 6.3. Let MS ---- (J-, ~, F) be a material system and # its constitutive relation. They are invertible if and only if the following conditions hold: (R8) the effect of the null process is single-valued; (R9) if Plt~E and P:tzE, and P3 C C with the same duration is such that P3(t)C (PI(t)}W {Pz(t)} for all tE [0, d~], the,, P3C ~ and E(d)~ F(P3); (R10) the sets {P(d) [ PE ~, PlzE) are closed for every EE E.

Proof. (R9) and (R10) for/~ are obviously equivalent to (R4) and (R5) for #-1, respectively. If (R3) holds for/~, it does for #-1. We still have to prove that the set A : ---- (E E E ] 3 P E C, P/~/~) is a class of processes with values in Y, if and only if (R8) holds. ~ is not empty, and by (R1)/~ is not empty, and hence A is not empty. All processes of A start with the same initial value if and only if (R8) holds. If PI~E, then by (R1)

P ]t0,tl #E ]t0,tl for every t G [0, de].

Thus A contains all subprocesses of its elements, as does ~, and for every /~E A there is surely a continuation, which also is the case for every P E ~ with P/z/~. Hence, A is a process class; q.e.d.

We note that if (R8), (R9), and (R10) hold for a material system, a transformed system does not inherit these properties automatically, especially not (R8). Thus the property of invertibility is not preserved under transformations. Conversely, a non-invertible material system can be transformed into an invertible one, even by a revertible transformation process. 122 A. BERTRAM

If a material system fails to obey only condition (R8), then we can remove this shortcoming by restricting the effect of the null process to a singleton, by declara- tion. The condition (R9) is not so easily circumvented; it is violated in the example of plasticity without strain hardening.

7. Uniform Structures on the State Space

Up to now the state space has not been endowed with a linear, metric, or topo- logical structure. Therefore we are not yet able to consider convergence of se- quences (nets, filters), or even Cauchy-sequences in ~, or continuous functions from ~e into a topological space. In order to define relaxation properties of material systems, it is very helpful to use topological tools, and in this section we will intro- duce topological structures on the set of effects and on the state space. Let us consider a material system in such a state that we can perform a constant process (freeze) of arbitrary duration. The accompanying trajectory in the state space will contain states that belong to the same configuration but do not have to be all identical. The principal goal of this section is to give a precise meaning to the statement that this trajectory converges to a "relaxed state". Therefore we have to make precise what it means for one state to lie in a neighborhood of an- other. The following suggestion seems to be quite natural: one state Zl is said to be neighboring another state z2 if two conditions hold:

(i) ~'=~ ---- N.,, i.e. we can perform the same processes starting from each state, and (ii) the effects of any process in ~zi starting from zl are neighboring those that we obtain if we start from z2 with the same process.

But what does it mean for effects to be neighboring one another? For single- valued effects this is clear: we have a standard uniform structure on the finite- dimensional linear space J-. We propose to lift the uniformity from f to the set g of all non-empty closed subsets of 9_. For the case in which the base space J- is endowed with a metric d, HAUSDORFF introduced the following metric on the set of all non-empty, closed and bounded subsets of J-:

D(X, Y) :-= max (sup inf d(x, y), sup inf d(x, y)). xEX yEY yEY xEX

This may be generalized to include unbounded sets if we put D' := D/(I + D), which is bounded above by 1 and equals 1 if and only if D takes the value + co. This metric generates a uniformity on ~. This procedure can be generalized to base spaces that have a uniformity U, but not a metric: If Vis in U, we define for X~d ~

V(X) := (c E 9-" I 3 a ( X with (a, c) (Is) C ~--. Material Systems 123

Then the following sets form a basis for a uniformity U on 8:

I7," : = {(X, Y) E g • g I X ~ V(Y) and Y Q V(X)} as V runs over U. The proof of this well known fact is straightforward and is given in BERTRAM (1) for example. Of course, U determines a topology on 6~, and, as we have restricted our considerations to closed subsets, o* is a Hausdorff space when given this topology. Because J- is metrisable and complete according to its standard uniformity as a linear space, 8 is also completefl From now on we consider 8 to be endowed with the uniformity U and asso- ciated topology, so that 6~ is a complete uniform Hausdorff space. If we restrict our attention to single-element subsets of ~d--, then the uniformity U on 3" and the appropriate restriction of U coincide: (a, b)E VE U ~=>((a}, (b})E I~E (7.

As an example we consider the traditional treatment of internal constraints by the assumption that the effects are determined only to within terms called "reaction effects" (see TRUESDELL• NOLL, Section 30). In mechanics the effects are described by symmetric Cauchy stress tensors, which form a six-dimensional vector space with inner product defined in terms of the trace. In this case, effects are sets of the form T = {Te + 2N l )l E 1R), where only T e and the direction N of the reaction stresses Tr = 2N are deter- mined by the material. (It is customary to normalize the extra stresses T e by employing the condition tr (Te N) = 0.) Let us consider two stress sets T i --- {Tei+ 2iN l 2i E R}, with i = 1, 2. We measure the distance between the extra stresses by d(T,,, Te) := ~/tr (Te, -- T~,) 2 (see Fig. 7). The Hausdorff metric introduced above measures the distance between 7"1 and 7"2

0 r~l u r~z =9ac~,rzl

Fig. 7

$ See MICHAEL. 124 A. BERTRAM by D(T1, T2) = max t sup inf d(tl, t2), sup inf d(h, t2) I [t~ET~ t2ET2 t2ET2 tlET1 J = max~sup inf 1/tr [(T~ + 21N)--(Te~- 22N)] 2.... } I).~E P~ 22E R = sup inf ]/tr [Tel -- Te2 + (al -- 22) N] z ),~ER 22(R = I/tr (Te, -- Te2)a ~-- d(Te~, Te). In this case the Hausdorff distance between the sets T1 and /'2 equals the standard distance between the extra stresses. The same holds if n constraints, 1 ~ n =< 6, are imposedandthestressesarerepresentedby T={Te 4- ~2jNjI~jER }. j=! We now introduce uniformities in the state space of a given material system MS by adapting to the present theory a procedure proposed by NOLL Let ~ be an arbitrary class of processes, not necessarily the one for the system MS. We call the set the ~-section of the state space. Of course, the configurations of two states belong- ing to the same section are the same. Every state z E ~ is in exactly one section; ~therefore we have obtained a partition of the state space. Let E be the output function of the effect (as defined in Section 3), let ~ be the evolution function, and let ~e be a non-empty section of the state space for MS. In the function E.~ 1~:r215 ~e~ •176 we can keep the second argu- ment fixed to obtain for each P E ~ the function E. r P) I~ : ~e ~ ~.

Definition. We call the natural uniformity of ~ the coarsest uniformity that renders the functions E" ~o(', P) 1~ : ~e _> # uniformly continuous for all P ~ ~. The induced topology is called the natural topology of ,.~.

The following propositions are similar to results of NOLL (Propositions 11.2, 11.4).

Proposition 7.1. Let .~'~ be a non-empty section of the state space and let ~ be the evolution function of a material system. It follows that for every P in ~o(., P) I~ : ~ -+ ~r~, is uniformly continuous. Here ~' is the set of all continuations of P in ~.

Proof. First we must show that the above function is well defined, i.e. that o(~, P) ( .~e, for all P in ~. By definition the class of processes for all states z in .~e is ~, and the subset ~' of all processes of ~ that contain a given P C as a subprocess depends only upon P and ~. Therefore ~(., P) I~r~ is a mapping of the uniform space ~ into the uniform space ~,. Then ~, has the uniformity induced by the mappings E. Q(., P)' I~,, P' E #', and the mapping E. ~(., P) I~v~ Material Systems 125 is uniformly continuous for all P ~ ~'. We now apply a theorem of topology (see BOUgBAKI,p. 190) : Let ~ be an index set, let P be in :~, and let (Ae} be a family of uniform spaces. Let X and Y be uniform spaces, let f: X-+ Y be a function, and for each P~, let ge: Y-+Ae be a function. If Yhas the uniformity induced by the functions ge, thenfis uniformly continuous if and only if the func- tions ge "f are uniformly continuous. If we take for X, Y, and Ap the sets ~, g~,, and ~, respectively, and for f and ge the functions 9(', P)Ig~ and E. r P') la'~,, then ge "f = E. ~(., P' o P) Ig~, is uniformly continuous, and, by the above theorem, so is f; q.e.d.

Proposition 7.2. Every section ~ with its natural topology is a Hausdorff space.

Proof. Let z~ and z2 be two different states in ~e; then there is at least one process P in ~ such that E. O(z~, P) ~ E. ~(z2, P). As 8 is a Hausdorff space, it contains two disjoint open neighborhoods V,. of E. O(zi, P), with i----1, 2. Their pre-images under E. ~(-, P) are open (because of continuity) and disjoint, so that the pre-images are neighborhoods that seperate zl and z2; q.e.d.

We now show that the material isomorphisms induce uniform isomorphisms for the corresponding sections of the state space.

Proposition 7.3. Let MSI and MS2 be materially isomorphic relative to A, and let 7A be the induced bijection of the state spaces, 7A : ~e _+ ~e2. The func- tion 7A Igl~: 0~1~-+-~2n-.(~) is a uniform isomorphism for all sections .~e1~ of the state space .~e1.

Proof. Each A in Iso (3"~, 3r2) is a uniform isomorphism between the two (uni- form) spaces 3"~ and 3-2. The same holds for A-* : 5~ -+ 3-~, and for the for- ward image function A : gl -+ 82. If -~1~ is a section of the state space .~f~, then we have 7A(~1~) ---- -~2A-*(~), for, if two states z~ and z2 have the same class of processes ~.~ = ~z~, then they have an identical image under A-*, and vice versa. Every one of the following functions is uniformly continuous for all P in ~: Et ~1(', P) [~, A, A -1, E 2 " e2(', A-*(P)) ]ya(~,~). As a composition of uniformly continuous functions, A" Et-Q~(', P)]zv,# = E2" Q2(7A('), A-*(P)) I~r~ also is uniformly continuous. Next we apply the same theorem as in the proof of Proposition 7.1. Here, the mentioned families of sets or functions have but one member each. Let us take for X, Y and A? the sets ~el~, .~eZA_,~) -----7a(.~l~ ), and o~2, respectively, and forfandg? the functions 7~ I~r~eand E2 e~(', A-*(P)) ]~r,a_,(#). We conclude that 7a [~ is uniformly continuous, and, as the proof is symmetric in the indices 1 and 2, the same holds for 751 = 7A-~; q.e.d.

We now have the tools to define relaxation properties of material systems.

Definition. Let z be a state in ~ from which we can perform constant processes Pa of arbitrary duration d ~ 0. z is called a relaxable state if {~(z, Pa) ] d > 0) 126 A. BERTRAM is a Cauchy net. If all states in ~e are relaxable, then the system is called relaxable. If the limit 2(z) :---- lim ~(z, Pal)G d-+e~ exists, it is called a relaxed state. In this definition we used: 1) the uniform structure in order to define Cauchy nets, 2) the topology in order to define the limit of a net, 3) the Hausdorff property to obtain a unique limit. Because we do not assume the sections of the state space to be complete, the existence of the limit is not assured. We do not wish simply to postulate the com- pleteness of the sections, because it is quite difficult, if not impossible, to describe in general the physical consequences of that postulate. We prefer to introduce the completion by the standard procedure of topology. In doing this, we make use of two results from topology (see SCHUBERT, pp. 126, 124): 1) For every Hausdorff space A there is a unique (up to isomorphisms) complete Hausdorff space A-, such that A is dense in A_ 2) Let A be a uniform space, B a complete Hausdorff space, and f: A-+/~ be a uniformly continuous function. Then f has a unique uniformly continuous extension 97 on A.

By using these facts, we can complete each of the sections ~e to ~e. We then define ~:= ~J ~--~ and extend E.~o(., P) to E. ~(-, P) on ~(--. The output function G is constant on each section ~e and therefore, extends to a function t7 on ~e--, which is constant on every completed section ~---,,j. The output function of the effects is extended by the formula ft,(z) "= E. ~(z, Po), where Po is the null process in Nz. We enlarge in an analogous manner the class of all transformable material systems ~ to a class of systems JC/and extend the associated bijection i to a uniform isomorphism i. It is not clear if the completions of the sections of state spaces have any phys- ical meaning. Let us consider a state z that is in ~e but not in ~e, i.e. this state is not (exactly) accessible from the initial state by a process. By mathematical for- malisms described above, we might let the system start in this state z and describe the behavior of the system theoretically, that is, we can describe which processes may be performed and what their effects are. However, this behavior does not have an exact counterpart in physical reality. Nevertheless, we consider this pro- cedure of completion in order to study the relation of the present theory to NOLL'S new theory of simple materials. Before we do this in the next section, let us see how the concept of material symmetry may be extended to the completed spaces.

Definition. Let MS be a material system and J/~ = i-Ms(~-). Then o~Ms is the set of all material isomorphisms between MS and a material system MS" = iMs(Z') with z' E ~e. If, furthermore, for the completed state space Y" of MS" there holds ~' = ~, then the isomorphism also is in ~Ms. Material Systems 127

For the case in which the sections of state space are themselves complete, i.e. ~' = if, the set ~Ms coincides with the symmetry semigroup and ~Ms coincides with the symmetry group. This follows from the fact that ~ -- ~ is uniformly isomorphic to ~', so that also ~e, = ~, holds. This means that we can reach every state from z'; in particular, the initial state can be reached and, thus, the transformation process of z' is revertible. For other systems we only have ~MS ~ ~MS, and fg~ s ~ ~MS"

8. Noll's Material Elements

The aim of this chapter is to indicate how NOLL'S new theory of simple ma- terials is included in the present theory. This is not very difficult, as most of the concepts here stem more or less from NOLL'S theory. This is not the appropriate place to summarize NOLL'S theory in detail. In- stead, I will point out the main differences and then define a special class of ma- terial systems within our theory that correspond to NOLL'S simple materials. NOEL'S theory describes only mechanical material behavior. Because his theory is easy to generalize to include other physical effects, this difference need not be considered further. A more important difference lies in the way internal constraints are treated. NOLL prefers the traditional way and distinguishes between reactions and extra effects (by normalization), but he recognizes this procedure as being somewhat arbitrary. We avoid this distinction by introducing at the outset set-valued functions. The other important difference between the two theories concerns the primitive concepts. In the present theory, a material system is comprised of the list of variables, the class of processes, and the material function. The concept of state is deduced by factoring the class of processes relative to an equivalence relation. Consequently, all of the subsequent concepts like material isomorphy, symmetry, and invertibility could be treated without this concept of state. In NOLL'S theory the concept of state is primitive. Each of his systems is a septuple with the following entries: 1) the space of the variables; 2) the set of possible configurations; 3) the class of processes; 4) the state space; 5) the output function of the configuration; 6) the output function of the effect; 7) the evolution function. NOEL's class of processes corresponds to the set ~J ~z of all segments of pro- cesses in the present text. It must obey four conditions that are stronger than those imposed on 9 ~. The output functions and the evolution function of NOLL'S material elements are required to obey three axioms that can be deduced for the corresponding functions in our theory. NOLL'S remaining axioms deal with topo- logical questions. If we restrict our attention to single-valued material functions, our topology and uniformity essentially coincide with NOLL'S. His Axiom IV requires the state space sections to be complete. This is also the case in our theory, when we make the completions and extensions described in the foregoing section. NOLL'S Axiom V states that every state is relaxable, which in the present theory 128 A. BERTRAM constitutes a restriction to a special class of materials, namely the relaxable ones. NOEL'S Axiom VI is important for understanding further the connection between the two theories. It states that there is a relaxed state from which every other state is approximately accessible. We can use such a state as our initial state. However, in our theory this state does not have to be relaxed, not even relaxable. This axiom excludes most types of aging materials where the starting state is neither relaxed nor can be reached approximately from later states. The simple materials in the sense of NOEL constitute a subclass of our material systems, and I refer to them as systems "of NOEL'S type":

Definition. A material system is of Noll's type if the following conditions are satisfied: (N1) every process P C ~ may be continued by a constant process of arbitrary duration with the value P(d); (N2) # is closed under continuation, i.e. if P~, P2 are in ~ and Pl(tl) : P2(t2) for a t~ in [0, d~] and a t2 in [0, d2], then

tPx(t) for t ~ tt P(t) := [P2(t2 -- tt + t) for tl < t ~ d2 -- t2 + t~ is in ~; (N3) For each P~, P2 in #, P2 can be continued by a process P3 such that Pa ~ P2 is in ~, P3(d3) = P~(dl); (N4) the material function F is single-valued for all P in ~; (N5) the material system is relaxable; (N6) the initial state is relaxed.

(N1) as well as (N2) exclude most non-trivial classes of processes where all of the processes are once or even several times continuously differentiable. Such processes are important for materials of the differential type. Moreover, (N1) excludes materials with finite life, (N3) restricts the variety of (non-classical) constraints that can be imposed, and (N6) excludes many aging materials.

9. Example: Rigid-Plastic Materials

NOEL viewed his new theory of simple materials as being capable of describing many types of material behavior, including plastic behavior. Although NOEL has not published articles specifically directed toward theories of plasticity, ~ILHAVY KRATOCHViL have studied plastic systems with viscoelastic range which are quite general and fit into the present theory. ~ However, their theory does not include a very simple form of plastic behavior known as "rigid-plastic" behavior. These important and rather simple materials are not included in any theory with a single-valued stress response function. Each rigid-plastic material is specified by means of three relations: a yield criterion, a flow rule, and a strain-hardening function.

See also DEE PIERO. Material Systems 129

If such a material is given in an initial state, no deformation (i.e., yielding) will occur as long as the stresses are below the yield limit, given by means of a yield criterion, represented by a real-valued differentiable function, which can be described in the intrinsic or in the reference description. In this chapter I shall use the reference description and the intrinsic description suggested by NOLL simultaneously. Although the latter seems to me more appro- priate and natural, it is yet not sufficiently well-known to warrant omitting here the more familiar reference description.

In the intrinsic description, a (continuous) body ~' is an n-dimensional differentiable manifold with boundary, and we can apply to it the concepts and results of the differen- tial geometry of manifolds. In particular, we have at each point XE g the tangent space 3-x:~ and the cotangent space J-*~', both n-dimensional vector spaces (without canonical inner product). A motion is a time-dependent imbedding k of the body mani- fold into a Euclidean space (with translation space ~e'). The local gradient of k is a linear function K(X, t) = grad k(X, t) : ~x~ -+ ~1/ called a (local) placement, and the (local) configuration is defined by G(X, t) = K* . K(X, t), a symmetric and positive-defmite linear mapping from J-x ~ to 3-}:~ which can serve to identify both spaces and induces the following inner product: u. v := (G(u), v), for all u, v E 3-x~. We denote the set of these mappings by Sym+(3-x&, 3"}:~), and the linear symmetric (indefinite) mappings in the other direction by Sym (J-.~..~, 3-x:~). The intrinsic stress S is an element of this latter set, so that S and G are dual variables. We identify J" with Sym (.~3-~:~, ~'x ~) and o~r* with Sym (J-x:~, J*~). In order to relate this description to the reference description, we choose an arbitrary reference placement ko(X) with gradient Ko(X) and define Go := Kff. K o as reference configuration. Then the classical deformation gradient is F(Xo, t) : = grad (k- k~- 1) (X0, t) =-K" K~-I(Xo, t). The intrinsic stresses are related to the Cauchy stress tensor T by the formula T = K . S . K * : ~F" -+ OV" , such that the stress power per unit volume is 1 tr(T. D) ---- tr (T./v. F -1) = -~-tr (S" G) with the rate of deformation tensor D := 1 (F. F_ 1 F_ t . i~t)" Hence the Cauchy

1 stresses are always defined in connection with an embedding in the Euclidean space, locally expressed by K. In what follows we will indicate the intrinsic description by the suffix i and the reference description by r. (This shall not mean that the variables indicated by r do always depend on the reference placement.)

Let the yield criterion be given by the real function ~r(T) = V',(S, a); the points where these functions equal zero form the yield surface, and for the points inside this surface these functions are negative. A well-known example 130 A. BERTRAM is the yon Mises yield criterion, which is given by a function of the stress deviator I T':-- T-- -~-(tr T)ld:

! ~,(T) = ]/tr T '2 --/~ = ]/@ (I2- 3lit)-/s

1/ 2 2 -I/~-(l~.c - 31Is.a) -- k [.3 = ~rs(s, G) where the principal invariants of Tand ofS. G are identical. (They are denoted by Roman numerals.) If hardening is included, the yield criterion depends on a hardening parameter c~ which may be real or tensor-valued and is a rate-independent function of the pro- cess: eq(P) = ~,(F(t)). The yield criterion then becomes 7'~(s, a, ~,.) -- ~Ur(T, o,,).

In the example, /~ is no longer a constant, but a function of the hardening para- meter a. No deformation can occur in a rigid-plastic material when T is inside the yield surface. Deformations with T on the yield surface are called "yielding" or "(plas- tic) flow" and are specified by means of a flow rule. The term "plastic potential" describes a class of flow rules for which plastic flow occurs in the direction of the outward normal of the yield surface. In connection with the yon Mises yield criterion we obtain the Levy-yon Mises flow rule: D--2T', 2>0. The intrinsic form of this rule is 1 (; = 2,,((G . S . G -- ~- Is.aG) .

If we take the stresses as the dependent variables, we interpret the flow rule as being an (implicit) material function S,(G, G, a,) or /~,(F, /~, ~r).

Both functions are defined for G -71- 0 and /) 4: O, respectively; their values are elements of the set of effects, i.e. closed sets of stresses. For the given example we have ~i(G,d, oci) = {S~ .y_ ~22 ( S ----~-Is.GG-1 a) =G-X.(S.G -~} if ).>0 I t and Material Systems 131

Note that only the deviatoric part of the stress occurs; the pressure remains arbitrary. (This material is assumed to be incompressible.) In general, we impose two conditions on the flow rule: t) the response is rate-independent,

S~(G, G, 0ct) = S~(G, 76, ~,) for all 7 > 0;

2) it is not possible to leave the yield surface during the flow, i.e. v,,(d,(a, 6, a, = (o}

throughout the domain of the function S~.

The second condition enables us to eliminate the factor 2 in the Levy-von Mises flow rule, so that we obtain the formulas

^ G-1.G.G -1 Si(G, G, oq) : [S = --pG -1 + K (tr (G-~" G)2) and fr(F, F, = {7" = --pld +/< ~tr~ p~ -1

with D := -2-I (I~. F -1 + F -t ./~t). We now come to the formal description of a rigid-plastic material system: Let the class of processes ~ be the set of all continuous and piecewise continuously differentiable processes with a preassigned initial value Go, with values in Sym+ (J'x ~, 3-*~), and with det (Go" P-~(d)) = 1, let ~i be a hardening function defined on ~, let ~(S, G, o~i) be a yield criterion defined on Sym (~d"*~, Yx~) • Sym+ (3-x.~ , ~-*~) • oci(# ), and let Si be a flow rule defined on Sym+ (J'x&, J-~)xSym(~--x~,~--*~)• The material function is given by

(S C Sym (~--*~, J'x~) [ ~Ji(S, P(d), oci(P)) ~ O) if P'(d) = 0, F(P) [~,(P(d),/'(d), ~(P)) if P(d) ~;~ 0;

//'(d) denotes the left-derivative of P, /6(0) :: 0 for all P in ~. This material is rate-independent and thus is not aging. A consequence of this property is that the hardening function cannot change its values during constant processes of arbitrary duration. Because constant processes are in and their effects are also constant, the material system is relaxable. It is not invertible, as the conditions (RS) and (Rg) are violated. Although we cannot say anything about revertibility in general, isotropic hardening surely does not de- fine a revertible system. We expect that deformations for systems with kinematic hardening can be reverted by means of reversals of deformations. In order to specify the state space we restrict ourselves to the case where no hardening occurs (perfect plasticity). The state of such a system can be represented 132 A. BERTRAM

by (i) the configuration P(d), and (ii) the direction of/~(d) or the information that it is zero. The sections of the state space are those that have the same configuration P(d). We cannot specify the natural uniformity and topology of the sections with- out more information about ~g and Si, but we do not expect them to be trivial.

Acknowledgment. I would like to thank RUDOLF TROSTEL, ARNOLD KRAWIETZ, my teachers, DAVID OWEN, and EKKEHARD TJADEN for their helpful suggestions and encouragement.

10. References

ALTS: Thermodynamik elastischer K6rper mit thermokinematischen Zwangsbedingun- gen -- fadenverst~irkte Materialien. Technische Universit/it Berlin. 1979. ANDREUSSI & PODIO GUIDUGLI: Thermomechanical constraints in simple materials. Bull. Acad. Polon. Sci., S6r. sci. techn. 21, 4. 1973. BERTRAM (1): Materielle Systeme mit inneren Zwangsbedingungen. Doctoral thesis. Technische UniversitRt Berlin. 1980. BERTRAM (2): An introduction of internal constraints in a natural way. ZAMM 60, p. 100, 1980. BERTRAM t~r HAUPT: A note on Andreussi/Guidugli's theory of thermomechanical constraints in simple materials. Bull. Acad. Polon. Sci., S6r. Sci. techn., 24, 1. 1976. BOURBAKI: Topologie g6n6rale. Chap. II, 2. Paris. 1965. BRIDGMAN: The thermodynamics of plastic deformation and generalized entropy. Rev. mod. phys. 22, 1. 1950. COLEMAN & OWEN: A mathematical foundation for thermodynamics. Arch. Rational Mech. Anal. 54, 1. 1974. FICHERA: Boundary value problems of elasticity with unilateral constraints. Handbuch der Physik VI a/2. Ed. TRUESDELL. Berlin, Heidelberg, New York. 1972. p. 391. GILES: Mathematical foundations of thermodynamics. Oxford, London, New York, Paris. 1964. GREEN, NAGHDI & TRAPP: Thermodynamics of a continuum with internal constraints. Int. J. Engng. Sci., 8. 1970. pp. 891-908. GURTIN & PODIO GUIDUGEI: The thermodynamics of constrained materials. Arch. Rational Mech. Anal. 51, 3. 1973. HAUSDORFF: Grundzi.ige der Mengenlehre. New York. 1949. MICHAEL: Topologies on spaces of subsets. Tran. Am. Math. Soc. 71. 1951. NOEL: A new mathematical theory of simple materials. Arch. Rational Mech. Anal. 48, 1. 1972. ONAT in IUTAM 1966. Symposium. Ed. PARKUS & SEDOV. Wien. 1968, p. 292-313. ONAT in IUTAM 1968. Symposium. Ed. BOLEY. Berlin, Heidelberg, New York. 1970. p. 213-225. OWEN: A mechanical theory of materials with elastic range. Arch. Rational Mech. Anal. 37, 2. 1970. p. 85. PERZYNA & KOSiNSKI: A mathematical theory of materials. Bull. Acad. Polon. Sci., S6r. sci. techn. 21, 12. 1973. PERZYNA: A gradient theory of rheological materials with internal structural changes. Arch. Mech. 23, 6. 1971. pp. 845-850. DEE PIERO: On the elastic-plastic material element. Arch. Rational Mech. Anal. 59, 2. 1975. p. 111. Material Systems 133

PRAGER: On elastic, perfectly locking materials, in: Applied mechanics. Ed. G6RTLER. Berlin. 1966. SCHUBERT: Topologie. Stuttgart. 1964. SILHAV~" & KRATOCHV~L: A theory of inelastic behavior of materials. Arch. Rational Mech. Anal. 65, 2. 1977. TRUESDELL & NOLL: The non-linear field theories of mechanics, kIandbuch der Physik. III/3. Ed. FLOGGE. Berlin, Heidelberg, New York. 1965.

Technische Universit/it Berlin (West) 2. Institut ftir Mechanik

(Received February 11, 1982)