An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

Abstract

Meshfree discretizations of state-based peridynamic models are attractive due to their ability to naturally describe fracture of general materials. However, two factors conspire to prevent meshfree discretizations of state-based peridynamics from converging to corresponding local solutions as resolution is increased: quadrature error prevents an accurate prediction of bulk mechanics, and the lack of an explicit boundary representation presents challenges when applying traction loads. In this paper, we develop a reformulation of the linear peridynamic solid (LPS) model to address these shortcomings, using improved meshfree quadrature, a reformulation of the nonlocal dilitation, and a consistent handling of the nonlocal traction condition to construct a model with rigorous accuracy guarantees. In particular, these improvements are designed to enforce discrete consistency in the presence of evolving fractures, whose {\it a priori} unknown location render consistent treatment difficult. In the absence of fracture, when a corresponding classical continuum mechanics model exists, our improvements provide asymptotically compatible convergence to corresponding local solutions, eliminating surface effects and issues with traction loading which have historically plagued peridynamic discretizations. When fracture occurs, our formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. We provide rigorous error analysis and demonstrate convergence for a number of benchmarks, including manufactured solutions, free-surface, nonhomogeneous traction loading, and composite material problems. Finally, we validate simulations of brittle fracture against a recent experiment of dynamic crack branching in soda-lime glass, providing evidence that the scheme yields accurate predictions for practical engineering problems.

References in corpus (6)

• Numerical methods for nonlocal and fractional models
• Minimal positive stencils in meshfree finite difference methods for the Poisson equation
• An Asymptotically Compatible Formulation for Local-to-Nonlocal Coupling Problems without Overlapping Regions
• An Asymptotically Compatible Approach For Neumann-Type Boundary Condition On Nonlocal Problems
• Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
• Classic dynamic fracture recovered as the limit of a nonlocal peridynamic model: The single edge notch in tension

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