Classical and nonclassical discontinuities and their structures in nonlinear elastic media with dispersion and dissipation

Abstract

PREFACE This paper studies problems related to the propagation of one-dimensional nonlinear waves in elastic media by analytic and numerical methods. The equations of nonlinear elasticity theory are classified as hyperbolic system expressing conservation laws. Their solutions can be uniquely constructed only when the equations are supplemented with terms making it possible to adequately describe real smallscale phenomena, in particular, the structure of arising discontinuities. We consider the behavior of nonlinear waves in two cases, where the small-scale processes are caused by viscosity alone and where an important role is also played by dispersion. Solutions in these cases differ drastically. In cases with dispersion the complicated behavior of solutions containing discontinuities is described, which qualitatively differs from the behavior of nonlinear waves in cases without dispersion. The revealed special features of the behavior of solutions are not related to any specifics of the equations of nonlinear elasticity theory and are generic for systems of partial differential equations with complex hyperbolic part. This work was supported by the program “Mathematical Methods of Nonlinear Dynamics” of the Russian Academy of Sciences, by the Russian Foundation for Basic Research, and by the program “Leading Scientific Schools.”