Bridging the gap between local and nonlocal numerical methods—A unified variational framework for non-ordinary state-based peridynamics

The paper aims to develop a unified variational framework to bridge the gap between the non-ordinary state-based peridynamics (NOSB-PD) and the classical continuum mechanics (CCM). First, a new force state vector is proposed by introducing the first Piola–Kirchhoff stress. This new force state vector enables the stress divergence of each material point to be expressed by averaging all the force state vectors in its support domain. The new force state vector also ensures the mathematical consistency between the strong form of PD and CCM when the horizon of a material point approaches to zero. Second, the displacement and traction boundaries in CCM are transformed into the non-local fictious boundary layers in PD, and a non-local Gauss's formulation is presented by transforming the displacement and traction boundaries in CCM into the non-local fictious boundary layers in PD, and this formulation unifies the variational framework and boundary conditions of PD and CCM. Third, a fully implicit algorithm is developed to obtain the general nonlinear problems such as fracture and large deformation of solid materials. Further, a penalty method is employed to eliminate the zero-energy mode oscillation inherently observed in NOSB-PD, and the penalty force and penalty stiffness matrix are derived for the proposed implicit algorithm and numerical implementation. Numerical results demonstrate that the proposed method is accurate and can well capture the fracture and large deformation of solid materials. Results also indicate that the method can effectively prevent the zero-mode oscillations inherently observed in the original NOSB-PD, and thus ensures the computational stability.