Longitudinal impact into viscoelastic media

We consider several one-dimensional impact problems involving finite or semi-infinite, linear elastic flyers that collide with and adhere to a finite stationary linear viscoelastic target backed by a semi-infinite linear elastic half-space. The impact generates a shock wave in the target which undergoes multiple reflections from the target boundaries. Laplace transforms with respect to time, together with impact boundary conditions derived in our previous work, are used to derive explicit closed-form solutions for the stress and particle velocity in the Laplace transform domain at any point in the target. For several stress relaxation functions of the Wiechert (Prony series) type, a modified Dubner–Abate–Crump algorithm is used to numerically invert those solutions to the time domain. These solutions are compared with numerical solutions obtained using both a finite-difference method and the commercial finite element code, COMSOL Multiphysics. The final value theorem for Laplace transforms is used to derive new explicit analytical expressions for the long-time asymptotes of the stress and velocity in viscoelastic targets; these results are useful for the verification of viscoelastic impact simulations taken to long observation times.