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II.    The Idea of Approach

It is commonly accepted, and indeed is hard to see otherwise, that the general states of stress and strain are described by 3×3 matrices (specifically tensors) – square tables of the form

Regarding the relationship of the two tables during a specific deformation process, it is a matter of hypothesis mainly about the material submitted to deformation. One has a very limited range of experimental possibilities to test such a hypothesis – constitutive law – and the most reliable one among these possibilities is the uniaxial loading test – tension as well as compression. Usually one considers that such a test offers a relationship between two eigenvalues of the tables from equation (1).

Here a subtle point occurs, which we need to discuss. The fact that a constitutive law relates two eigenvalues of 3×3 matrices is the fundamental limitation of the possible forms of a constitutive law; we might say almost the only one within the given hypotheses of representation of stresses and strains. The eigenvalues of the two matrices are always the roots of some cubic equations. Now, it is an algebraically known fact [3] that the relationship between the roots of two cubic equations – as long as it is rational – must be a homography. Specifically, if s_1,2,3 are the eigenvalues of s and e_1,2,3 are those of e, then we can have either one of the relations

at our convenience. Here A, B, C, D are some constants, only three of which are independent.

It is now clear that, if our representations of stress and strain by matrices has some degree of truth, then a relationship of the type (2) must be revealed in uniaxial experiments as a constitutive law, with A, B, C, D as ‘material constants’. The fact of the matter is that this is indeed the case for some soft tissues [4] or, in general, on limited portions of any experimental loading curve. In one word, any material exhibits in a specific way and to a specific extent, a behavior described by an equation of type (2). On the entire range of the experimental deformation, however, a constitutive relation of the type (2) is hardly exhibited and the reason for this is phenomenologically clear: every material changes its structure during the deformation process. This change is reflected in that of the material coefficients A, B, C, D, which makes the constitutive law vary with deformation. The mathematical description of the deformation process must then contain this variability of the constitutive law with deformation as an essential mathematical ingredient, and the constitutive laws of the type (2) turn out to be the most elegant way to account for such mathematics.

Indeed, let us suppose that we have experimentally

does not really matter what form we use, it is entirely up to our convenience. Here f is a measure of force and l is a measure of deformation. Either one of the experimental laws (3) allows us to relate the variations of the material coefficients to the experimental values of l and f , if we are able to perform two experiments: deformation at constant stress (creep) and deformation at constant strain (relaxation). By this theory the two deformation processes achieve a fundamental character that goes beyond that conferred to them within the usual Laplace transform framework. The first of these processes is described by equation df = 0, and the second is described by equation dl = 0. Using (3) we thus have either

for the creep process, or

for the relaxation process. Here w_1,2,3 are the following differential forms in the material coefficients

Our theoretical speculations here are mainly centered on the equation (5), due to the fact that the coefficients of the homography giving l in terms of f seem to have some attractive physical interpretations. These will be revealed in the next section. We shall use the equation (5) in cases where it is possible a continuum approach with respect to some parameter or parameters. For the moment we notice that, from a purely mathematical point of view, there are a few ways to use the equations (4) and (5). If the measured forces, as well as the material parameters, are stochastic processes then (4) and (5) can be interpreted as stochastic equations governing the two processes of creep and relaxation. It is well known the fact that the creep process in metals (at least some stages of the process) have a stochastic explanation [5] and that the stochastic constitutive model [6] for hard biological tissues has a noticeable success [7]. Another way to use the equations (4) and (5) is by a continuum approach. This approach, which we will be mainly our concern here, seems to offer a possibility to uncover the mechanism of appearance of constitutive laws as conservation laws.

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