Asymptotically compatible meshfree discretization of state-based peridynamics for linearly elastic composite materials

Abstract

State-based peridynamic models provide an important extension of bond-based models that allow the description of general linearly elastic materials. Meshfree discretizations of these nonlocal models are attractive due to their ability to naturally handle fracture. However, singularities in the integral operators have historically proven problematic when seeking convergent discretizations. We utilize a recently introduced optimization-based quadrature framework to obtain an asymptotically compatible scheme able to discretely recover local linear elasticity as the nonlocal interaction is reduced at the same rate as the grid spacing. By introducing a correction to the definition of nonlocal dilitation, surface effects for problems involving bond-breaking and free surfaces are avoided without the need to modify the material model. We use a series of analytic benchmarks to validate the consistency of this approach, illustrating second-order convergence for the Dirichlet problem, and first-order convergence for problems involving curvilinear free surfaces and composite materials. We additionally illustrate that these results hold for material parameters chosen in the nearincompressible limit.

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

“…For classical (local) linear elasticity, FEM solution obtained from the pure displacement form often deteriorates and becomes unstable when ν is close to 0.5. For the peridynamic Navier equation, however, numerical results in [37] show that the meshfree discretization converges to the local limit with a second-order convergence rate even for ν = 0.495. It is a challenging question to answer, but nevertheless worthwhile, to ask why the peridynamic Navier equation does not have an instability?…”

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

“…In comparison, nonlocal models characterized by fractional Laplacians have infinite nonlocal interactions [50]. Mathematical analysis and numerical methods have been developed for nonlocal diffusion models [1,25,29,39,56], linear and nonlinear nonlocal advection [27,30,37], nonlocal convectiondiffusion models [15,28,54,55], nonlocal Stokes equations [32], and peridynamics models [3,40,45,49,51,53,59,60]. We refer the readers to the monograph [24] and the survey work [14] for a more detailed discussion on nonlocal models.…”