On rate and gradient-dependence of solids as dynamical systems

Abstract

This paper aims at presenting a mathematical background of material instability problems such like strain localization or flutter, as an application of the theory of dynamical systems. The basic field equations of the solid continuum are the kinematic equations, the Cauchy equations of motion and the constitutive equations. This system of equations is completed with initial and boundary value conditions and can define a dynamical system. Then, a condition of material stability can be obtained using Lapuuov's indirect method. Also the basic material instability modes can he classified as static (divergence type) or dynamic (Hopf) bifurcations of dynamical systems. Such formulation gives a mathematical interpretation of rate and gradient-dependence in the constitutive equations and mesh dependence in numerical studies, pointing out their close relationship to the structure of the critical eigenspace of an operator defined by the dynamical system.