Analysis of Properties of Ramp Stress Relaxation Curves Produced by the Rabotnov Nonlinear Hereditary Theory

Abstract

The possibilities and applicability limits of the Rabotnov physically nonlinear constitutive equation containing two arbitrary material functions for nonaging materials are investigated. Analytically examined are the general properties of the resulting stress relaxation curves with an initial stage of constant strain rate and their dependence on duration of the stage, strain rate, and characteristics of the two material functions. Investigated are the monotonicity and convexity intervals and asymptotics of the relaxation curves, the jump of stress derivative at the end of the initial stage, the character of convergence of the family of relaxation curves as duration of the initial stage tends to zero, and other properties. Found are estimates for the deviation of the relaxation curves with an initial stage from those obtained in an instantaneous loading, and it is proved that this deviation tends to zero if the time tends to infinity or duration of the initial stage approaches zero. Revealed are the necessary restrictions on both material functions that allow an adequate modeling of the typical properties of experimental stress relaxation curves. Indicated are the effects which principally cannot be described whatever the material functions used. The possibilities of the Rabotnov nonlinear constitutive equation are compared with those of the Boltzmann–Volterra linear viscoelasticity theory (which were generalized introducing a second material function), and the additional effects that can be described by the nonlinear theory owing to presence of the second material function in it.