Properties of the Set of Strain Diagrams Produced by Rabotnov Nonlinear Equation for Rheonomous Materials

Abstract

The Rabotnov nonlinear constitutive equation with two arbitrary material functions for rheonomous materials in the one-dimensional case is analytically investigated. An equation for a set of theoretical strain diagrams at constant loading rates is derived. Their general qualitative properties are analytically analyzed depending on the properties of material functions. The following points are investigated: the monotonic intervals and convexity of the strain diagrams, the nature of their dependence on the loading rate, the existence and type of limit curves when the loading rate tends to zero or infinity, the conditions of existence of the inflection point and the limiting stress (flow stress), the cutoff conditions (simulation of fraction), formulas for instantaneous and long-term moduli, the conditions of their finiteness and the difference from zero. Based on a comparison of the properties of theoretical diagrams with typical properties of experimental diagrams for rheonomous materials, the minimal restrictions on the material functions have been established to provide an adequate description of the main rheological effects. Indicators of the applicability of the constitutive equation and the effects that cannot be described using any material functions have been found. The characteristic features of the strain diagrams for three main classes of models, namely, with a regular, unbounded, and singular relaxation function are shown. The advantages of the Rabotnov constitutive equation compared with the linear constitutive equation of viscoelasticity (from which it was obtained by introducing the second material function) are discussed. The additional effects that can be described by using the mentioned nonlinear relation are pointed out.