A. G. Kulikovskii

The stability of a metastable shock wave in a viscoelastic medium under two-dimensional perturbations☆

The unsteady motions of a viscoelastic medium are considered, taking account of a small anisotropy and a small non-linearity. The behaviour of a metastable, quasi-transverse shock wave when it interacts with non-one-dimensional perturbations is investigated numerically. The stability of this wave under non-one-dimensional perturbations of large amplitude is demonstrated.

Uniqueness of self-similar solutions to the Riemann problem for the Hopf equation with complex nonlinearity

Solutions of the Riemann problem for a generalized Hopf equation are studied. The solutions are constructed using a sequence of non-overturning Riemann waves and shock waves with stable stationary and nonstationary structures.

The stability of quasi-transverse shock waves in anisotropic elastic media☆

Abstract The stability of weak quasi-transverrse shock waves in a weakly anistropic elastic medium with respect to arbitrarily oriented perturbations is investigated in the linear approximation. It is shown that fast quasi-transverse shock waves are stable.

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

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.”

Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation

Solutions of problems for the system of equations describing weakly nonlinear quasi-transverse waves in an elastic weakly anisotropic medium are studied analytically and numerically. It is assumed that dissipation and dispersion are important for small-scale processes. Dispersion is taken into account by terms involving the third derivatives of the shear strains with respect to the coordinate, in contrast to the previously considered case when dispersion was determined by terms with second derivatives. In large-scale processes, dispersion and dissipation can be neglected and the system of equations is hyperbolic. The indicated small-scale processes determine the structure of discontinuities and a set of admissible discontinuities (with a steady-state structure). This set is such that the solution of a self-similar Riemann problem constructed using solutions of hyperbolic equations and admissible discontinuities is not unique. Asymptotics of non-self-similar problems for equations with dissipation and dispersion were numerically found, and it appeared that they correspond to self-similar solutions of the Riemann problem. In the case of nonunique self-similar solutions, it is shown that the initial conditions specified as a smoothed step lead to a certain self-similar solution implemented as the asymptotics of the unsteady problem depending on the smoothing method.

Study of discontinuities in solutions of the Prandtl–Reuss elastoplasticity equations

Relations across shock waves propagating through Prandtl–Reuss elastoplastic materials with hardening are investigated in detail. It is assumed that the normal and tangent velocities to the front change across shock waves. In addition to conservation laws, shock waves must satisfy additional relations implied by their structure. The structure of shock waves is studied assuming that the principal dissipative mechanism is determined by stress relaxation, whose rate is bounded. The relations across shock waves are subject to a qualitative analysis, which is illustrated by numerical results obtained for quantities across shocks.

Shock waves in elastoplastic media with the structure defined by the stress relaxation process

We study nonlinear waves in a Maxwell medium in which residual strains and hardening occur. The properties of the medium are defined so that for slow processes with characteristic times much greater than the stress relaxation time, the medium behaves as an elastoplastic medium. We analyze continuous travelling waves in the form of smoothed steps regarded as discontinuity structures in an elastoplastic medium and demonstrate the dependence of relations at discontinuities on the definition of the stress relaxation process in the discontinuity structure.

A self-similar wave problem in a Prandtl-Reuss elastoplastic medium

We consider a self-similar piston problem in which stresses on the boundary of a half-space are changed instantaneously. The half-space is filled with a Prandtl–Reuss medium in a uniform stressed state. It is assumed that the formation of shock waves is possible in the medium. We prove the existence of a solution to the problem in the cases when two or all three stress components are changed at the initial moment.

Long nonlinear waves in anisotropic cylinders

Small-amplitude plane nonlinear waves in anisotropic cylinders are considered in the case of longitudinal and torsional waves having close velocities. Anisotropy corresponding to this condition can take place in specifically plaited ropes and in the case of anisotropy of other nature. The characteristic velocities are found, and simple waves are studied.