Some analytical solutions to peridynamic beam equations

Peridynamics (PD) has been introduced to account for long range internal force/moment interactions and to extend the classical continuum mechanics (CCM). PD equations of motion are derived in the form of integro-differential equations and only few analytical solutions to these equations are presented in the literature. The aim of this paper is to present analytical solutions to PD beam equations for both static and dynamic loading conditions. Applying trigonometric series, general solutions for the deflection function are derived. For several examples in the static case including simply supported beam and cantilever beam, the coefficients in the series are presented in a closed analytical form. For the dynamic case, the solution is derived for a simply supported beam applying the variable separation with respect to the time and the axial coordinate. Several numerical cases are presented to illustrate the derived solutions. Furthermore, PD results are compared against results obtained from the classical beam theory (CBT). A very good agreement between these two different approaches is observed for the case of the small horizon sizes (HSs), which shows the capability of the current approach.