Stress-controlled hysteresis and long-time dynamics of implicit differential equations arising in hypoplasticity

Abstract

A long-time dynamic for granular materials arising in the hypoplastic theory of Kolymbas type is investigated. It is assumed that the granular hardness allows exponential degradation, which leads to the densification of material states. The governing system for a rate-independent strain under stress control is described by implicit differential equations. Its analytical solution for arbitrary inhomogeneous coefficients is constructed in closed form. Under cyclic loading by periodic pressure, finite ratcheting for the void ratio is derived in explicit form, which converges to a limiting periodic process (attractor) when the number of cycles tends to infinity.