Constitutive relations for dynamic material instability at finite deformation

Abstract

This paper aims to present a mathematically consistent formulation of the second gradient dependence in the constitutive equations for material instability phenomena in case of finite deformations. Thus the set of fundamental equations of the solid continuum (the kinematic equations, the Cauchy equations of motion and the constitutive equations) should also be written for finite deformations. Two basic properties are required: the existence and regular propagation of waves and the generic behavior at the loss of stability. Firstly, the wave dynamics is studied. To encounter the second gradient effects, we should use the third order waves here. Secondly, the system of fundamental equations completed with initial and boundary value conditions forms a dynamical system. Then, identifying material stability with Lapunov stability of a state of the continuus body, the loss of stability should be one of the two basic types of instabilities of dynamical systems: a static or a dynamic bifurcation. These instability modes should be strictly different for a generic dynamical system.