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.