A framework for the description of material behavior

Material Systems - A Framework for the Description of Material Behavior ALBRECHT BERTRAM

Communicated by D. OWEN Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Material Systems . . . . . . . . . . . . . . . . . . . . . . . The State Space of Material Systems . . . . . . . . . . . . . . Material Isomorphy . . . . . . . . . . . . . . . . . . . . . . Material Symmetry . . . . . . . . . . . . . . . . . . . . . . Inverse Systems . . . . . . . . . . . . . . . . . . . . . . . . Uniform Structures on the State Space . . . . . . . . . . . . . NOLL'SMaterial Elements . . . . . . . . . . . . . . . . . . . Example: Rigid-Plastic Materials . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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99 102 104 108 113 119 122 127 128 132

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 theories 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 "constitutive 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

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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 descriptions 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 framework 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 n o w 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.

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The concept of material isomorphy, introduced in Section 4, is a tool for comparing 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 distinction 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 present 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 configurations). As a consequence, the effects cannot be considered as being determined 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.

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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 characteristic 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. However, 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 Section 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 systems. 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 uniformly 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).

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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 cotangent 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 ~ ~ ~ x M , 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 m a p p i n g o f a closed (time-) interval into the space o f independent variables. Definition. Let d be a non-negative number. A process of duration d is a m a p p i n g P : [0, d]--> 3-*. The values o f P are called configurations. W h e n we deal with m o r e than one process, we use the same subscript for both a process and its duration. A process o f zero duration is called a null process. I f P1 and P2 are two processes, and there holds P l ( d l ) = P2(0), we define the composition o f P~ and P 2 by P2o PI

( P l ( t ) , if t --~ dl, [P2(t - - d~), if dl - < t :~ (d~ -k d2).

Clearly, composition is a n o n - c o m m u t a t i v e but associative operation: Pa ~ (P2 ~ P1) = (Pa ~ P2) ~ P~. I f P3 = P 2 o P1, we call P1 and P 2 segments o f Pa, P z a continuation o f P~, and P~ a subprocess o f Pa.

Definition. A class of processes is a set ~ o f processes which satisfies the following four conditions: (P1) ~ is n o t empty; The intrinsic description is given in more detail in Section 9.

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(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 duration; (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 M S : = (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 o f f to Xo by f]xo. Transformation Theorem 3.1. Let M S = (~'-, ~ , F) be a material system and P r be a process in ~ with duration dr. Then we can transform M S into a new material system M S ' in a natural way by setting M S ' : (3-, ~ ' , F'),

~ ' : = ( P ' ] P ' is process such that P ' o PTC ~}, F'(P') : = F ( p ' o Pr) for all P ' C ~ ' .

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It is easy to verify that, for all P r in ~ M S ' really is a material system. It is called the system M S transformed by Pr. We introduce the transformation function HMs, which maps the transformation process P r E ~ into the system transformed by Pr. Let J / b e the range of HMs, i.e. the set of material systems that can be obtained from M S 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 M S = ( ~ , ~ , F) be a material system and P1, P2 be in ~ . We call P1 equivalent to /'2, and write P1 "~ P2, if M S transformed by P1 equals M S 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 configurations 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 configuration.

Example 2 (aging systems). Two types of aging occur for these material systems: kinematic 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 equivalent processes Pi with at least one duration d iE ~ have the same duration. For response 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 from the equivalence class of that state; that process of the transformed material system. formalized by means of an output function

effect of any transformation process effect is equal to the effect of the null The assignment of effects to states is for the effects of states E: ~e __> g.

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

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We define the configuration of a state to be the final value o f any transformation process f r o m its equivalence class. By analogy, we introduce the output function for the configurations o f 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 M S = (3-, ~, F) be a material system, Po a process and ~'(Po) : = (PC ~' I Po o P E #}, 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 o f a well-known theorem on the " n a t u r a l function" o f an equivalence relation, i.e., the function which maps each process into its equivalence class.

Theorem 3.2. Let M S = (~-', ~ , F) be a material system with transformation function HMS, let ~ l : = HMS(~), let ~ be the state space o f M S , and let o~ be the natural function o f ~ . Then there exists a unique bijection i such that i . ~o(P) = HMs(P) f o r 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 P r or about the system being in the state ~o(Pr). The latter point o f view is often simpler, because in m a n y cases only a finite n u m b e r o f parameters (internal variables) determine the state completely. I f M S = (~--, ~ , F) is transformed by P r into MS" = (J-, ~ ' , F'), it is obvious that ~,/l : : HMS(~) ~ all' : = HMS'(~'). It is quite reasonable by means o f the bijection i between d / a n d the states in the following m a n n e r : i~llS " HMS( P" ~ PT) ~ i ~ , " HMs'(P') for all P ' ff ~ ' . F o r the state spaces there follows ~ ) ~ ' .

5( to identify

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Example. There are theories of granular media (dry sand, etc.) that permit only deformations which evolve towards a critical density, i.e., only dilatative or compressive changes 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 M S ---- (~--, ~ , 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 M S ' = 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 = P 2 o p~ is revertible with respect to a material system M S if and only if P1 is revertible with respect to M S 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 a n d 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 M S 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

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one c o n t i n u a t i o n o f the null process, a n d this c o n t i n u a t i o n yields a non-revertible process for MS. Therefore, M S is non-revertible; q.e.d. O n the o t h e r h a n d , one can c o n s t r u c t non-revertible systems t h a t are n o t aging. T h u s the aging systems are a p r o p e r subset o f the non-revertible ones. T h e following definitions are m o t i v a t e d by NOLL'S t h e o r y a n d are o f i m p o r t a n c e in establishing the relation between t h a t t h e o r y a n d the present one. Definition. L e t z C ~ be a state o f a m a t e r i a l system MS. W e d e n o t e by ~ z the class o f processes o f the system i(z), a n d 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). F o r 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 o f 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 differential 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 o f this theory in the literature on viscoelasticity, hypoelasticity, plasticity, and thermodynamics. In most cases the space of the internal variables is real and finite-dimensional.

4. Material Isomorphy

I n this section we shall investigate i s o m o r p h i s m s between m a t e r i a l systems. I n d o i n g so, we give a precise m e a n i n g to the n o t i o n t h a t two systems exhibit the same physical behavior. Let M S , = (3--1, ~ , , F , ) a n d MS2 = (3-2, ~ 2 , F2) be m a t e r i a l systems. First, J-1 a n d ~'-2 are i s o m o r p h i c if a n d only if they have the same dimension. I s o m o r p h i s m s o f vector spaces are, o f course, linear bijections; we shall d e n o t e the collection o f these i s o m o r h i s m s by Iso (~'1, 3-2). I f A E Iso (~J--1, ~--2), it follows

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that the adjoint A*, the inverse A -1, and the inverse of A*, A-*, satisfy the relations 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 induced 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 P 2 ~ 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 M S z = (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) whenever 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, 9 A*(P') = A -~ 9 F2(P') for all P ' E ~2-

The following theorem shows that isomorphisms really preserve the detailed structure of material systems.

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Theorem 4.2. A material isomorphism maps

a) b) c) d)

equivalent processes into equivalent ones, cyclic processes into cyclic ones, revertible processes into revertible ones, 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 P E ~ , 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 P x ) = 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 material 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 M S 2 be m-equivalent. There then exist revertible processes P1 E ~1 and P 2 E ~i~2 and an isomorphism A E Iso (5"1, J'2) such that HMs~(P~) and HMs,(P2) are materially isomorphic 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

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and equivalent to P2 relative to M S 2. By Theorem 4.2, P2 ~ P0 is revertible if, and only if A*(P2 ~ Po) ~ P1 is revertible (for MS~). I f HMs~(P1) is materially isomorphic to HMs2(P2) relative to A, then HMsI(A*(P2 o Po) ~ P1) is materially isomorphic to HMs:(A-* 9A*(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. I f 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. I f 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. I f HMs~(Po) = M S t is materially isomorphic to HMs,(P2), then the two systems are also m-equivalent; q.e.d. In the foregoing p r o o f we have used the following easily proven fact: I f 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. I f 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 material system is again revertible. We call a material revertible, if a representing material 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 M S 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. I f 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 9 A 2 9 A l l ; 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 revertible ones. But the converse is not true, i.e. transformation of non-revertible materials can lead to both revertible and non-revertible ones. I f 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. I f we now define a selection function q~i :Y'i-+ ~ i of a material system M S i as a mapping that maps a state into an (arbitrary) process out of its equivalence class, then r 9 ~0i is the identity on .~';. O f course, a material system can have m a n y 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 - *

9

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))

f o r all (z, P ) 6 (~f, ~)1.

Proof. The definitions of 7A and 91 yield the relations: 7A " 9x(Z, P) = o2 " A - * 9 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 M S = (~J', ~ , F) be a material system. A function A is called a symmetry transformation for M S if there is a process PA C ~ such that, with M S a : = 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 M S is A-symmetric to MSA. The conditions S1-$3 are equivalent to the assertion that M S and M S a 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 compared two different material systems by investigating whether a material isomorphism exists or not. Here we study a single material system by considering all the isomorphisms that do exist in the above sense.

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A. BERTRAM

We denote the set of all symmetry transformations of a given material system by WMs, called the symmetry semigroup of M S . 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. M S 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 following conditions, which are equivalent: PE~