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Nicolae Mazilu PhD, J. Vossoughi PhD, T. A. Conway PhD

ABSTRACT: The problem of the interrelation between creep and relaxation for biological tissues is reanalyzed within the framework of the broadest definition of these two phenomena. The task we had in view is a closed form solution of the problem. It is found here that a reassessment of the meaning of the usual quasistatic loading curve is needed, in order to overcome the difference between the objective times of the two processes. The work concentrates in clarifying this basic problem.

Key words: biological materials, creep, relaxation, nonlinear behavior, quasistatic loading curve, homography

I. Introduction

The question has been recently reiterated as to the interconnection between creep and relaxation phenomena within the realm of soft tissue modeling [1]. Specifically, if the creep results can be recovered from relaxation data (or vice versa) a tremendous amount of work and trouble can be saved, with results worth considering for practical biomechanical applications – assessment of reliability of prosthetic parts for instance. Unfortunately the study reveals no real possibility of predicting creep from data on relaxation (or vice versa). The key point in approaching the interrelation of creep and relaxation is the usual Laplace transformation for linear viscoelastic materials. It takes the considerations of nonlinearity [2] to correlate the two phenomena the way we think they may be correlated. However, the discussion in [2] indicated us that, in approaching this correlation, there may be some problems transcending the Laplace transformation. For, it occurred to us that there is a missing link in the usual approach of the problem. Namely, such phenomena as creep and relaxation involve deformation processes during which the internal structure of the sample changes, and it seems to us that merely Laplace transform cannot take properly into account these changes of structure. The main reason is that these internal phenomena dictate the scale of time to be considered with a Laplace transform, and this scale of time is by no means accessible to the kind of experiments we are taking into consideration. Realizing this fact, we either have to supply some other data that give an estimation of the rate of processes of creep and relaxation, which again, is a new task that may make the approach economically unjustifiable, or find explicitly the experimental expression of eliminating the time from considerations even within the framework of structural changes. This fact will hopefully be able to give information in a closed form, thus avoiding the uncertainty of series expansion used in [2] to assess nonlinearities.

The subject matter of the present work is this hot point in the modern Science of Materials, especially biological tissues. We specifically want to explore the second avenue mentioned above, namely of finding the constitutive expression obtained by eliminating the time between the processes of creep and relaxation. This – we think – will give us the possibility to overcome the usual negative answer on the interconnection between them. However the price paid is a phenomenological definition of the creep and relaxation and, to the extent we want to maintain the Laplace transform in the picture, a possible extension of its definition too. The main idea is that the creep and relaxation are indeed dual phenomena but, and this is the most important thing, not apriori dual. The duality must be expressed with respect to the usual uniaxial loading curve, so familiar in any constitutive research, yet so hard to explain in its entirety. Here a phenomenological explanation is proposed, to the effect that the usual quasistatic-loading curve is simply a statistical intermingling between the two fundamental phenomena – creep and relaxation.

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