Early-time dynamics of the Richtmyer-Meshkov instability at Kelvin-Voigt viscoelastic interfaces

Abstract

The Richtmyer-Meshkov instability (RMI) is initiated when a shock wave interacts with a perturbed interface between two materials, imparting an initial velocity to the interfacial perturbation. When the materials are viscoelastic, the subsequent evolution is influenced by the coupled effects of inertia, viscous dissipation, and elastic resistance. The interplay of these effects remains poorly understood and is important for predicting the dynamic response of shock-loaded soft materials, e.g., in blast-induced injury. Using the incompressible Cauchy momentum equation with a Kelvin-Voigt constitutive model, we conduct a linear stability analysis for the early-time post-shock evolution of a single-mode interface perturbation of two viscoelastic materials. We derive an implicit dispersion relation for the temporal growth rates of the perturbation. This relation depends on density, viscosity, and elasticity contrasts between the two materials and an viscoelastic parameter which is a bounded ratio of the viscous dissipation and elastic resistance response rates. Four material cases, spanning symmetric to highly contrasted interfaces, are examined to show how the viscoelastic number affects the growth rate structure as complex-dominated (oscillatory) or real-dominated (monotonic relaxation). The critical viscoelastic number separates these two regimes and is primarily governed by the density and viscosity contrasts. These results provide a theoretical criterion for predicting the early-time response of shock-driven interfaces involving fluids and viscoelastic solids.