Weak form of bond-associated non-ordinary state-based peridynamics free of zero energy modes with uniform or non-uniform discretization

The non-ordinary state-based peridynamics (NOSB PD) is attractive because of its ability to employ existing constitutive relations for material models. The deformation gradient tensor and the force density vector appearing in the equilibrium equations are expressed in terms of nonlocal integrals. The definitions of these nonlocal integrals affect the accuracy and stability of PD predictions. Therefore, this study introduces a more accurate representation of the deformation gradient and the bond associated (BA) force density vector by using the peridynamic differential operator (PDDO). Also, it presents the weak form of BA-NOSB PD governing equations in order to impose natural and essential boundary conditions without the use of Lagrange multipliers for implicit and explicit analysis. By considering a two-dimensional rectangular plate with and without a hole under tension, the numerical results demonstrate the accuracy of BA-NOSB PD with no oscillations and zero energy modes.