An Asymptotically Compatible Meshfree Approach for Neumann-Type Boundary Condition on Peridynamics

In this work we consider the Linear Peridynamic Solid (LPS) model with traction loads on a sharp boundary surface. Since peridynamics is a nonlocal model with a finite nonlocal horizon parameter (delta) characterizing the range of nonlocal interactions, its boundary conditions are naturally defined on a region with non-zero volume outside the surface, in contrast to more traditional engineering scenarios where the traction loads are only provided on a sharp co-dimension one surface. Therefore, the treatment of Neumann-like boundary conditions on a sharp boundary surface is practically useful on many engineering applications but has proven challenging for discretizations on peridynamics and the general nonlocal models.

We propose a new generalization of classical local Neumann conditions by converting the local traction load to correction terms in the Linear Peridynamic Solid (LPS) model for material points close to the sharp boundary surface, which provides an estimate for the nonlocal interactions of these material points with points outside the domain. When the domain boundary satisfies some geometric assumptions, it can be theoretically shown that the resultant formulation is independent of any rigid-body rotation between the deformed and reference configurations. Moreover, the formulation passes the linear patch test. On the numerical side, to obtain an asymptotically compatible scheme we utilize a recently introduced optimization-based quadrature framework to numerically discretize the nonlocal Neumann-type LPS model. In this meshfree framework, surface effects for problems involving bond-breaking are automatically captured without modifying the model and therefore requires no remeshing nor refinement when new fracture forms, which is especially attractive in handling the dynamic fracture of materials. We numerically verify the asymptotic compatibility of our approach with a series of static linear elastic problems with analytic solutions, including problems on domains involving curvilinear surfaces and corners. When the nonlocal interaction range (delta) is reduced at the same rate as the grid spacing, the numerical results demonstrate first order convergence (O(delta)) of nonlocal numerical solutions to the correct local limit in the L2 norm. To investigate the applicability of the proposed approach on real-world applications, we simulate the dynamic fracture/crack branching in glass induced by tensile loads, and quantitatively validate the proposed approach by comparing the simulation results with experimental observations and numerical results from state-of-the-art codes. We have also tested the proposed formulation on both Cartesian particle arrangements and unstructured quasi-uniform pointsets, and obtained consistent and symmetric crack branching patterns. Therefore, the proposed formulation is independent of pointset arrangements.