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IV. An Explanation of the Experimental Quasistatic Curves

We seek situations where the differential forms (13) are exact differentials in one parameter, t say. The experience shows that this parameter may be time. However, other physical interpretations are by no means excluded and, in cases where we are specifically interested in details we have to carefully assess the meaning of t. For the moment we are not interested in such details and we will eliminate t directly with the result of an explicit dependence between force and deformation. This is what we call the experimental constitutive law.

If the differential forms from equations (13) can be written as exact differentials in one parameter, then one has

with a_1,2,3 – constants. The equation (5) for force is then an ordinary Riccati equation

which admits a straightforward integration, with three possible results:

Here D and t₀ are defined by the equations

for the first case (19) and a, b, c, d are constants – not all of them arbitrary. One of the equations (19) – depending on what the case may be – describes a deformation process at constant strain – a relaxation process, but only for the set of circumstances where the physical parameters a, b, l₀ vary according to equations (17). Those equations, again, turn out to be easy to integrate giving

with t₀ having the same meaning as before and A, l₀, a₀, b₀ – some constants. Here only the first case of equations (19) has been considered, the other cases being entirely similar.

The equations (19) and (21) show that we do not need, at least for a certain level of scrutiny, to get into details of physical meaning of the parameter t. This parameter can be algebraically eliminated, leading to natural conclusion that the physical parameters l₀, a and b can be expressed in terms of force – a parameter which we can control in experiments. The same is true for the more important conclusion we want to present in this work, which stays a little further away in the course of analysis. Namely, had we worked with the first equation from (3), i.e. at constant force we would obtain the very same solutions as before. This time, however, they would describe the creep process possibly with respect to some other time parameter. More specifically (19) and (21) would be the same, the only difference is that we would have to switch a₁ with a₃ anything else remaining in place with some other integration constants. At least formally the creep and relaxation processes are dual – not only in the sense of Laplace transform, but also by their very definition.

We are thus naturally led to the idea of eliminating the parameter t by declaring the time as the same for the two processes then extracting tan [(t-t₀)Ö(D)] from a creep process and inserting it in a relaxation process. Everything from equations (19) and (21) will then be expressed in terms of the experimentally measure of accessible strain l. Especially the experimental force will be given by the following rational function

where the six constants a, b, c, a₀, b₀, c₀, five of which are independent, can be left to experimental evaluation from uniaxial tests. The function (22) is exhibited, with a high level of confidence, for biological tissues – soft as well as hard. The Figures 2 to 6 show this type of curve fit for some soft tissues of our own experiments as well as for human bone data taken from [8]. Curve – fitting quality, which can be seen directly on the figures, as well as the quality of assessment of the coefficients of (22), not shown in this work, are indeed exquisite.

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