Shock Wave Structure and Sub-Shock Formation in a Hyperbolic Nonlinear Viscoelastic Model

Abstract

Shock wave propagation in nonlinear viscoelastic solids is fundamental to understanding their response to rapid dynamic loading. We analyze shock wave structures within a recent symmetric hyperbolic nonlinear viscoelastic model, proposed by Ruggeri (in Int. J. Non-Linear Mech. 160, 2024) and derived within the framework of Rational Extended Thermodynamics. An explicit expression is obtained for the critical Mach number at which a smooth traveling wave loses its \(C^{1}\) regularity and a sub-shock forms. For shocks propagating into an undeformed equilibrium state, the critical Mach number depends solely on the linearized material response, in particular on the ratio of the elastic moduli in the Zener constitutive model. The model admits both compressive and tensile (expansive) shock waves. Numerical simulations based on a Mooney–Rivlin elastic potential coupled with a quadratic viscous energy confirm the theoretical predictions for shock propagation in vulcanized rubber.