Exact analytical solutions for elastodynamic impact

We consider the one-dimensional impact problem in which a semi-infinite flyer collides with (and adheres to) the front face of a stationary target plate of finite thickness, with the back face of the target bonded to another semi-infinite medium. All three bodies are assumed to be linear elastic and homogeneous. Our interest is in explicit expressions for the stress and velocity in the target at all times after impact. The analysis of this problem is simplified by reducing it to an initial-boundary value problem for the target only, which is solved by combining the d'Alembert solution of the wave equation with the Laplace transform method. An appropriate impact boundary condition is required on the front face of the target. In the literature this is usually taken as a prescribed step in stress or velocity, but the correct boundary condition involves a linear combination of the unknown stress and velocity at the impact face. Our solutions are expressed in an apparently new, compact form involving the floor (or greatest-integer) function. The results are amenable to asymptotic analysis; in particular, solutions for stress-free or rigid back faces follow easily as limiting cases of the backing impedance, and the long-time asymptotes of stress and velocity in the target are seen to be independent of the target's elastic properties. All of our results are corroborated by derivation of exact discrete solutions from recursive equations for the impact problems.