Analytical solution to the plate impact problem of layered heterogeneous material systems

Abstract

An analytical solution to the problem of one-dimensional high amplitude wave propagation in layered heterogeneous material systems has been developed, based on Floquet's theory of ODEs with periodic coefficients. The problem is formulated based on a conventional plate impact experimental configuration. In a plate impact test, the boundary condition at the plane of the impact varies with time as a result of multiple wave interactions at the interfaces of the layered target material. The approach of the solution is to convert the initial velocity boundary value problem to a time-dependent stress boundary value problem and then obtain the stress time history by means of superposition. By taking this approach, we explicitly consider multiple wave interactions at the heterogeneous interfaces. A characteristic steady-state stress σ mean for heterogenous material has been identified which is quite different from σ 0 the stress at the initial time of impact. It is shown that σ mean can be obtained by summing up the stress increments at the interfaces or by using mixture theory. The late-time (steady-state) solution procedures for the plate impact problem are presented for impact velocities corresponding to elastic as well as shock wave loading conditions. Results from the analytical model compare well with both numerical results obtained from a shock wave based finite element code and experimental data.