The fractal scale-invariant structure of a temporal hierarchy in the relaxation and energy dissipation processes in a visco-elastic/capillary-porous medium

Abstract

The phenomena of elastic aftereffects during loading/unloading of viscoelastic and capillary-porous bodies, relaxation of their stresses is accompanied by the energy accumulation and dissipation to be taken into account in the theory of oscillations which also considers the behavior of materials when the force is applied to them. The elastic aftereffect and stress relaxation forms ostensibly opposite energy processes. In the first case, under constant load deformation, the work increases in course of time, and in the second case, under constant load deformation, the work (energy) decreases. While researching on the energy dissipation in the conditions of oscillations application, i.e. within the frame of internal friction theories, one can find that some theories are based on the dependence of friction on the oscillations’ velocity, other ones establish the dependence of friction on the amplitude. Research papers are based on the hypothesis of M.M. Davydenkov, according to which the energy when subjected to oscillations depends on the amplitude and does not depend on the velocity. According to E.S. Sorokin, the theory of internal friction is poorly consistent with the theories describing the inherited properties of materials (viscoelastic and capillary-porous ones). A tendency is observed: the better a theory reflects hereditary properties, the worse this theory is adapted to describe energy losses due to oscillations.In this paper, an attempt has been made to harmonize both these theories and numerous experiments on the destruction of materials described in the academic literature. It turns out that in order to remove contradictions, it is necessary to take into account the dependence of body deformation changing in the course of time.It is shown that the hierarchy of times determining shear and bulk relaxation in viscoelastic/capillary-porous medium has a fractal (scale-invariant) structure. It was observed that the presence of time fractality eases the modeling of viscoelastic/capillary-porous bodies resulting in the universal relaxation function of a rather simple kind. In particular, for the scale-invariant distribution of relaxation characteristics medium, the application of algebraic relaxation law for viscoelastic/capillary-porous materials is possible: this resulting in rheological models and state equations with the derivatives of fractional order.