Green's functions in non-local three-dimensional linear elasticity

In this paper, we compare small deformations in an infinite linear elastic body due to the presence of point loads within the classical, local formulation to the corresponding deformations in the peridynamic, non-local formulation. Owing to the linearity of the problem, the response to a point load can be used to obtain the response to general body force loading functions by superposition. Using Laplace and Fourier transforms, we thus obtain an integral representation for the three-dimensional peridynamic solution with the help of Green's functions. We illustrate this new theoretical result by dynamic and static examples in one and three dimensions. In addition to this main result, we also derive the non-local three-dimensional jump conditions, as well as the weak formulation of peridynamics together with the associated finite element discretization.