Some closed form series solutions to peridynamic plate equations

Abstract

Peridynamics is a generalized continuum theory which is developed to account for long range internal force/moment interactions. Peridynamic equations of motion are of integro-differential type and only several closed-form analytical solutions are available for elementary structures. The aim of this paper is to derive analytical solutions to peridynamic Kirchhoff type (shear rigid) plates equations for both static and dynamic cases. Applying trigonometric series with respect to in-plane coordinates, solutions for the deflection function of a rectangular simply supported plate are derived. The coefficients in the series are presented in a closed analytical form such that the equations of motion are exactly satisfied. For the dynamic case the solution is derived applying the variable separation with respect to the time and the in-plane coordinates. Several numerical cases are presented to illustrate the derived solutions. Furthermore, results of peridynamic plate equations are compared against results obtained from the classical plate theory. A very good agreement between these two theories is obtained for the case of the small horizon sizes which shows the capability of the presented approach.