Stability of classical shock fronts for compressible hyperelastic materials of Hadamard type

TL;DR

This paper analyzes the stability conditions of classical shock fronts in compressible hyperelastic materials of Hadamard type, explicitly computing the stability function and establishing criteria for shock stability based on material parameters.

Contribution

It explicitly computes the stability function for shocks of arbitrary amplitude in hyperelastic materials and derives stability conditions in terms of shock parameters and elastic moduli.

Findings

Explicit computation of the stability function (Lopatinskieterminant) for arbitrary amplitude shocks.

Derivation of conditions for uniform and weak shock stability based on material parameters.

Extension of stability analysis to multidimensional, nonlinear shock fronts in hyperelastic materials.

Abstract

This paper studies the uniform and weak Lopatinski\u{\i} conditions associated to classical (Lax) shock fronts of arbitrary amplitude for compressible hyperelastic materials of Hadamard type in several space dimensions. Thanks to the seminal works of Majda (Mem. Amer. Math. Soc. 43 (1983), no. 281; Mem. Amer. Math. Soc. 41 (1983), no. 275) and M\'etivier (Trans. Amer. Math. Soc. 296 (1986), pp. 431-479; Comm. Partial Diff. Eqs. 15 (1990), no. 7, pp. 983-1028), the uniform Lopatinski\u{\i} condition ensures the local-in-time, multidimensional, nonlinear stability of such fronts. The stability function (also called Lopatinski\u{\i} determinant) for shocks of arbitrary amplitude in this large class of hyperelastic materials is computed explicitly. This information is used to establish the conditions for uniform and weak shock stability in terms of the parameters of the shock and of the elastic moduli of the material.