An analytical model for the Kelvin–Helmholtz instability in solids

Although Rayleigh–Taylor (RT) and Richtmyer–Meshkov(RM) instabilities in solids have long been confirmed to play a significant role in high-energy-density physics and to serve as tools to diagnose material properties under high-strain and high-strain-rate conditions, the Kelvin–Helmholtz (KH) instability in solids under extreme conditions has yet to be fully investigated. In addition, according to experiments driven by high explosives or by high-intensity lasers, the interface behavior is frequently determined by the KH instability with RT and RM instabilities, as well as by gravity decay waves. Therefore, a comprehensive analysis is required to study the mechanical properties of solids under extreme conditions and the multiple stability mechanisms that affect the interface behavior. This paper proposes a simplified and analytical mathematical model for stability analysis by considering surface tension, viscosity, elasticity, viscoelasticity, and perfectly rigid plasticity as a function of the specific physical conditions. The model is used to examine how the KH instability, RT instability, and gravity decay wave affect the evolution of the interface. We first recover the well-known dispersion relations based on the viscous potential flow method for the KH instability, and in particular investigate the stability threshold with varying shear velocities and in the presence of gravity decay waves. For the KH instability in elastic solids, once the velocity difference exceeds an asymptotic value (i.e., when the effect of elasticity cancels that of shear velocity), the interface always remains unstable for all perturbed wavelengths—a result that departs from the traditional conclusion about the RT instability between elastic solids. Unexpectedly, the viscosity in Kelvin–Voigt materials can enhance the resistance to the hydrodynamic instability, while the elasticity has the opposite effect. The growth rates and the cutoff wave numbers increase gradually as the viscoelastic parameter decreases, and the inhibition of the high-wave number disturbance by the viscoelasticity is correspondingly reduced. In addition, the cutoff wave numbers depend only on densities and the shear modulus of the material and are independent of the viscous properties of the materials. Finally, the evolution of the perturbed amplitude is studied in rigid plastic materials, which offers physical insights of interest to further study the KH instability in viscoplastic and elastoplastic materials. Based on the models developed herein, experiments are proposed for further research into the mechanical properties of materials under extreme conditions.