A constructive peridynamic kernel for elasticity

Different kernel formulations have been proposed and used for bond-based peridynamic models of elastic materials. While all such kernels produce results that, in the limit of the nonlocal region (horizon) going to zero (while grid spacing relative to the horizon size is constant), converge, not all converge to the classical solution. Here, we introduce a constructive path to arriving at a peridynamic kernel for 1D, 2D and 3D elasticity. We focus on the particular one-point Gauss quadrature method of spatial discretization and study the convergence properties of different kernels with respect to the classical, local, model for dynamic elasticity in 1D, where exact representation of geometry is possible. We analyze the influence of two ways of imposing boundary conditions and of the “skin effect” on the solution. We show that, similar to the diffusion case, the peridynamic kernel derived based on physical principles for dynamic elasticity is the only one whose convergence to the classical solution does not depend on the fineness of the discretization grid, relative to the nonlocal region size. The results presented demonstrate that special care must be taken when claiming validation of peridynamic results by specific comparisons with classical, local model solutions.