Dynamical systems, rate and gradient effects in material instability

The main aim of the paper is to study the theoretical background of material instability in view of the theory of dynamical systems. In our work the system of the fundamental equations of solid continua defines a dynamical system for functions satisfying the boundary conditions. Material stability and instability of a state of the body are studied as the Lyapunov stability of a corresponding solution of such a dynamical system. We concentrate on the effects of rate- and gradient dependence on the onset of instability and (in some cases) on the post-localization behavior. The results show that the variables of the constitutive equation play an essential role in material instability properties. We performed calculations for various rate- and gradient-independent and- dependent materials and added higher gradient-dependent terms, too. We show how the inclusion of rate-dependent terms decouple the coexistent static and dynamic bifurcation of the classical (rate and gradient independent) case. Gradient dependence has an effect on the number of critical eigenvalues and eigenfunctions. For numerous types of constitutive equations the critical eigenspace is an infinite-dimensional one, but there are certain higher gradient-dependent cases, in which a simple critical eigenspace can be obtained and an analytic nonlinear bifurcation investigation is possible. That nonlinear case can give a mathematical interpretation of intrinsic material length as well.