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

TL;DR

This paper introduces an improved meshfree discretization method for state-based peridynamics that guarantees asymptotic compatibility, accurately models fracture, and effectively handles traction loads, validated through rigorous analysis and practical experiments.

Contribution

The authors develop a reformulated peridynamic model with enhanced quadrature, boundary treatment, and fracture representation, ensuring convergence to classical solutions and accurate fracture simulation.

Findings

Achieves asymptotic compatibility with classical continuum mechanics.

Provides accurate fracture surface representation during crack propagation.

Demonstrates convergence and accuracy through benchmark tests and real-world experiments.

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.