Discretized peridynamics for linear elastic solids

Peridynamics is a theory of continuum mechanics employing a nonlocal model that can simulate fractures and discontinuities (Askari et al. J Phys 125:012-078, 2008; Silling J Mech Phys Solids 48(1):175-209, 2000). It reformulates continuum mechanics in forms of integral equations rather than partial differential equations to calculate the force on a material point. A connection between bond forces and the stress in the classical (local) theory is established for the calculation of peridynamic stress, which is calculated by summing up bond forces passing through or ending at the cross section of a node. The peridynamic stress and the constitutive law in elasticity are used for the derivation of one- and three-dimensional numerical micromoduli. For three-dimensional discretized peridynamics, the numerical micromodulus is larger than the analytical micromodulus, and converges to the analytical value as the horizon to grid spacing ratio increases. A comparison of material responses in a three-dimensional discretized peridynamic model using numerical and analytical micromoduli, respectively, is performed for different horizons. As the horizon increases, the boundary effect is more conspicuous, and the errors increase in the back-calculated Young's modulus and strains. For the simulation of materials of Poisson's ratios other than 1/4, a pairwise compensation scheme for discretized peridynamics is proposed. Compared with classical (local) elasticity solutions, the computational results by applying the proposed scheme show good agreement in the strain, the resultant Young's modulus and Poisson's ratio.