Stability of shock wave structures in nonlinear elastic media

Abstract

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 studied. A procedure to analyze spectral (linear) stability of these solutions is described. The main focus is the stability analysis of special discontinuities, the stationary structure of which is represented by the integral curve connecting two saddle points corresponding to the states in front of and behind the discontinuity. This analysis is done using the properties of the Evans function, an analytic function on the right complex half-plane, which has zeros in this domain if and only if there exist unstable modes of linearization around a solution representing a special discontinuity with the structure.

Riemann problem for longitudinal-torsional waves in nonlinear elastic rods

“…The presence of special discontinuities may significantly affect the solution of the Riemann problem. Solutions of the Riemann problem that allow the formation of special discontinuities were constructed in [5,6,7] for various models of continuum mechanics. In these studies the authors showed that if the initial data allow the construction of a solution with a special discontinuity, then the numerical solution of the original equations yields an asymptotics containing a special discontinuity structure.…”