Analysis of Cone Wave Reflection in Finite-Size Elastic Membranes and Extension of the Ballistic Impact Problem From Elastic to Viscoelastic Membranes

Abstract

The transverse ballistic impact on a two-dimensional (2D) membrane causes a truncated deformation cone to develop in the wake of tensile implosion waves. Here, the cone wave reflected from the finite boundaries of the elastic membrane has been studied analytically. A first-order linear nonhomogeneous differential equation for the ratio of the reflected cone wave front velocity to the speed of tensile waves is derived, which is further used to calculate the traveling time taken by the reflected cone wave to reach to the projectile surface. Since the reflected wave starts when the membrane is already in a deformed configuration, the speed of the reflected cone wave is a function of radius r in the cylindrical coordinates as opposed to almost constant speed of the incoming cone wave studied in the literature. The analytical results are validated with molecular dynamics (MD) simulations of the ballistic impact of projectiles onto a single layer of coarse-grained (CG) graphene. In the second part of the paper, we analyze the membrane impact problem for linear isotropic viscoelastic materials and find that the tensile wave speed for stresses and displacements is the same as that obtained in the case of a linear isotropic elastic material. We also show that only under special conditions, self-similar solutions for the cone wave are possible in viscoelastic materials modeled by Maxwell, Kelvin–Voigt, or a combination of similar models. Our findings lay some grounds on which further studies on the ballistic response of viscoelastic materials can be performed.