Analytical solutions of peridynamic equations. Part II: Elastic wave propagation

Abstract

We use the separation of variables technique to construct analytical solutions for peridynamic models of dynamic elasticity. We show that, similar to the case of peridynamic models for transient diffusion, infinite series nonlocal solutions for peridynamic elasticity can be obtained directly from the solutions of the corresponding classical model by inserting “peridynamic/nonlocal factors” in the time-exponential part of the solution. The analytical solutions show that wave dispersion, caused by nonlocality, is contained in the horizon-dependent nonlocal factor. We obtain formulas for wave dispersion and group velocities for 1D and 2D peridynamic elastic wave propagation with three commonly-used peridynamic kernels. We observe interesting complexity in nonlocal solutions, generated by nonlocal wave dispersion, and “proportional” to the size of the nonlocal interaction region. Different from the diffusion case, as time goes to infinity, the nonlocal solution for elasticity does not converge to the classical one for a fixed horizon size, meaning that nonlocal effects persist in time. We solve several examples of wave propagation with Dirichlet boundary conditions and smooth or discontinuous initial conditions and compare these analytical solutions with those corresponding to the classical model, which is seen as a particular case of the PD model for horizon equal to zero. Interestingly, we find that in PD solutions, initial discontinuities in space persist at the same location, in time. While most of the analytical solutions we present here are formal, for some of the cases, we are able to prove uniform convergence of the series solutions. This work is the first presentation of a systematic analytical treatment of peridynamic problems in 2D finite domains.