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III. Physical Interpretation of the Experimental Homography

We want the experimental homography to be as close as possible to the experiment situations, where the parameters are chosen in order to correspond to some specific arrangements. This is what we take here into consideration for an adequate physical interpretation of the parameters. As mentioned earlier, among the coefficients A, B, C, D only three are really independent (we can arbitrarily divide with a factor in both the numerator and the denominator of the homography). If we write l in terms of f in the form

(7)

then we have explicitly three parameters l₀, a, b which, moreover turn out to have attractive physical interpretations in terms of experimental situations. The equation (7) can be obtained from (5) by the following replacements:

(8)

One naturally takes the form (7) into account in case one wants to explicitly state the fundamental fact that there is no deformation without force. Indeed we have

(9)

giving us the first physical meaning: l₀ is the measure of deformation in absence of experimental force. Whether this measure is the initial length of the sample or a more involved state of strain – this is quite a specific experimental problem. Along the same line of ideas we have

(10)

giving the physical interpretation of the parameter b conditional, however, on the physical interpretation of a: it is an elastic stiffness. This can be deduced by a simple inspection of equation (10). The interpretation of parameter a is not quite so simple. It involves the other two parameters and, if they are taken to be what equations (9) and (10) show them to be, then this gives a physical meaning to a. Specifically we have

(11)

In words a^-1 is the measure of deformation for no initial length of the sample and no stiffness of its material. A well – known situation that corresponds to this is illustrated in Figure 1 at the end of this essay: a long molecular chain with close ends has initially no length and no stiffness, and can stretch under arbitrary force. It stiffens as the deformation proceeds. More general, this can be the situation of a defect, which has no resistance.

No doubt then that a can be related to the ‘health’ of the sample’s material, in the sense of its closeness to a continuum. It is this intuitive image that explains how the stochastic models work in the deformation processes: like in the primitive models of metal creep [5], where different slip systems are randomly activated by deformation as it proceeds, we may say that different ‘material defects’ in general enter or exit the process of deformation randomly. This very process of coming in and out of participation to deformation is a stochastic process. A quasistatic deformation curve is then a superposition of ‘sample paths’ of this stochastic process. As we will see later we can define this curve even more specifically.

We have now the interpretation of a = 0 in equation (10) and it seems to be in accordance with our intuition: in terms of a molecular chain a = 0 is the limit of a fully stretched chain. We will no longer deal with this aspect of the problem in the present work but only observe that, practically, a can be found from equation

(12)

In words: represent the deformation in terms of force and then take the limit of high forces. At least for polymers and soft tissues this method seems to be as good as it gets. The same practical evaluations can be found easily for b and l₀.

We can go now a little further to see what the differential forms w_1,2,3 are in terms of the new experimental parameters. Taking equation (8) into equations (6) gives

(13)

The last one of these differential forms has a clear physical meaning: it is the Hooke’s law. Indeed, in the limit of zero force the equation (5) becomes

(14)

and in view of the previous physical interpretation of b this equation is a Hooke law. Notice that here the Hooke’s law is referred to the initial ‘material’ length of the sample. This length however may not be independent of a and b  - material parameters of the current physical state. By the same token, in the limit of high forces we have from equation (5)

(15)

and for constant force we have approximately

(16)

This is a kind of incompressibility constraint, and needs further investigation. However we do not undertake this investigation here, but go along with a particular continuum development of the theory that seems to convey special meaning to experimental facts.

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