This paper is devoted to a study of problems involving the propagation of one-dimensional non-linear waves of small amplitude in elastic media, using analytic and numerical methods. The equations of non-linear elasticity theory belong to the class of hyperbolic systems expressing conservation laws. For the unique construction of solutions it is necessary to supplement these equations with terms that make it possible to adequately describe actual small-scale phenomena, including the structure of the discontinuities that arise. The behaviour of non-linear waves is considered in two cases: when the small-scale processes are conditioned by viscosity, and when dispersion plays an essential role in addition to viscosity.

https://iopscience.iop.org/article/10.1070/RM2008v063n02ABEH004516/pdf

Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory

Investigation of the shock adiabat of quasitransverse shock waves in a prestressed elastic medium

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.

https://link.springer.com/content/pdf/10.1134%2FS0965542517060069.pdf

Special discontinuities in nonlinearly elastic media

/pdf/klassicheskie-i-neklassicheskie-razryvy-v-resheniiakh-251vrozmha.pdf

Классические и неклассические разрывы в решениях уравнений нелинейной теории упругости@@@Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory

The stationary structure stability of discontinuous solutions to nonlinear hyperbolic equations describing the propagation of quasi-transverse waves with velocities close to characteristic ones are...

E.I. Sveshnikova

Papers

Simple waves in nonlinearly elastic media