Weak form of peridynamics

This study presents the weak form of peridynamic (PD) equations of motion to impose nonlocal essential and natural boundary conditions. The PD equilibrium equation is derived by using the principle of virtual work along with the divergence theorem while invoking the PD form of the first- and second-order derivatives of the displacement components. Therefore, the PD equations are valid at a point in the bulk or near the surface with an arbitrary interaction domain (variable horizon) arising from a nonuniform discretization. This capability enables the nonlocal PD representation of the internal force vector and the stress components without requiring surface and volumetric corrections. The implicit solution to the discrete form of the algebraic equations is achieved by employing BiConjugate Gradient Stabilized (BICGSTAB) method. The implementation of the implicit solver reduces the computational cost especially in solving large-scale PD models. The capability of the weak form of PD governing equations is demonstrated by considering an isotropic and a bimaterial plates with/without a pre-existing crack under a combination of different types of boundary conditions.