Closed-form series solutions to peridynamic rod equations: Influence of kernel function

Peridynamics is a generalized continuum theory that takes into account long range internal force/moment interactions. The aim of this paper is to derive closed form analytical solutions for peridynamic rod equations with fixed-fixed and fixed-free boundary conditions. A family of kernel functions is introduced to analyze the influence on the results for rods under distributed static load and free vibrations for different initial conditions. To validate the derived solutions, nonlocal results are compared against classical results for different boundary conditions and horizon sizes. One can observe that when the horizon size approaches zero, the results according the non-local and local theories converge to each other. Introduced kernels indeed play different roles in both statics and dynamics. In particular, the Gauss type kernel function gives the results which are closest to the classical solutions.