Two-Sided Estimates for the Relaxation Function of the Linear Theory of Heredity via the Relaxation Curves during the Ramp-Deformation and the Methodology of Identification

Abstract

The general properties of the relaxation curves with the initial stage of deformation and their dependence on the duration of the initial stage of deformation and the properties of the relaxation function are analytically studied. These curves are induced by the constitutive equation of viscoelasticity with an arbitrary relaxation. New accurate two-sided estimates for the relaxation curves and their absolute and relative deviations from the relaxation curves for instantaneous deformation are derived. The uniform convergence of the set of relaxation curves when the duration of the initial stage tends to zero is proved.

Effective universal two-sided estimates for the relaxation function (at any time instant) are obtained via the relaxation curves of the material during ramp-deformation. On their basis, simple and effective formulas for determining the relaxation function from the relaxation curves for ramp deformation obtained in the material tests have been proposed. The error estimations of these approximations are given. The uniform convergence of the set of approximations to the relaxation function when the duration of the initial stage tends to zero is proved. The higher accuracy of the estimates found and the proposed approximation in comparison with the known related approaches to the determination of the RF (relaxation function) is established.

The results of the analysis are useful for clarifying the many of the possibilities of the linear theory, the domain and indicators of its (non)applicability (and also of a number of its generalizing nonlinear constitutive equations of viscoelasticity, for example, proposed by Rabotnov,Ilyushin, Pobedrya, etc.) to improve the methods of selection, identification and adjustment of linear models. In particular, they are useful for obtaining reliable estimates of the lower bound for the observation window of the relaxation function from the experimental relaxation curve of the material in terms of the initial stage duration to clarify the “ten-times rule” and expand the observation window to the region of small time values.