Mathematical theory of evolution of loaded solids and media

Abstract

The paper puts forward an evolutionary approach to describing the deformation response of loaded solids and media, which is based on the concepts of nonlinear dynamics. The deformation response is considered to mean destruction processes in solid media in external force fields, i.e., the processes of inelastic deformation and simultaneous failure. The mathematical theory of evolution of solids and media is shown to rest on the equations of solid mechanics as fundamental equations of mathematical physics describing the most general laws of conservation of mass, momentum, angular momentum and energy. The whole variety of physical mechanisms of inelastic (plastic) deformation and dilatation, i.e., the generation of discontinuities of different scales and physical nature (vacancies, pores, micro- and mesodamage, etc.) is integrally described through assigning nonlinear response functions of the loaded medium by evolutionary constitutive equations of the first and second group. These equations are thus derived on the basis of the leading physical mechanisms of the studied scale. It is shown that by varying only the correlation between positive and negative feedbacks (other things equal) the response of the loaded medium varies from the typical plastic flow to brittle failure. The procedure of introducing the real process time in the model is proposed. It allows solving problems on shock wave loading as well as problems of geodynamics and plate tectonics with characteristic times of millions of years.