CONSTITUTIVE MODELING FOR THE UNSTABLE STATES OF MATERIALS BY WAVE DYNAMICS AND VARIATIONAL METHODS

When fracture happens, in several problems of engineering the selection of the constitutive relation has an essential role beyond the equations of motion and the kinematical equation. A wide range of studies published in this topic use various types of constitutive models being more or less appropriate approximations to the mechanical problem under consideration. In searching for a generalized approach our paper deals with the theory of constitutive modeling of solid materials. The studies presented int he following are based on the almos obvious concept that the set of basic equations descibing solid bodies should possess mathemetical consistency. In this work two basic reqiurements are considered. The one is the existence and regular propagation of waves, while the other is a so-called generic behavior at material instability. The first condition is of dynamic nature appearing at rapid, high rate loading. The second one is closely related to fracture: we assume that the onset of material instability is a starting point of the process, which leads to fracture. The methods we use are based on variational principles and the theory of dynamical systems. For this reason we should define an infinite dimensional dynamical system, which is determined by the set of fundamental equations of the solid body. Such equations are used to define differential opeartors acting on the basic field variables like displacement, velocity, stress and strain, which satisfy appropriate boundary conditions. Then we concentrate on a selected state S0 of the material, which satisfies all the basic equations and additionally all the bondary conditions. Now state S0 of the material is called stable, if the solution of the dynamical system satisfies the conditions of the Liapunov stability. By using the methods of the theory of dynamical systems we will study how material instability happens and we judge whether it is a generic type of loss of stability or not