Convergence Analysis and Numerical Studies for Linearly Elastic Peridynamics with Dirichlet-Type Boundary Conditions

TL;DR

This paper analyzes how solutions of nonlocal linear elastic peridynamics models converge to local models as the interaction horizon shrinks, providing explicit convergence rates and verifying them through numerical benchmarks.

Contribution

It offers a detailed convergence analysis for nonlocal boundary conditions in peridynamics, including explicit rates based on boundary data extensions and numerical validation.

Findings

Constant extensions achieve 0.5 order convergence.

Linear extensions achieve 1.5 order convergence.

Numerical benchmarks confirm theoretical convergence rates.

Abstract

The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the body. Therefore, theoretical and numerical challenges arise in order to properly formulate Dirichlet-type nonlocal boundary conditions, while connecting them to the local counterparts. While a careless imposition of local boundary conditions leads to a smaller effective material stiffness close to the boundary and an artificial softening of the material, several strategies were proposed to avoid this unphysical surface effect. In this work, we study convergence of solutions to nonlocal state-based linear elastic model to their local counterparts as the interaction horizon vanishes, under different formulations and smoothness assumptions for nonlocal Dirichlet-type boundary conditions. Our results provide explicit rates of convergence that are sensitive to the compatibility of the nonlocal boundary data and the extension of the solution for the local model. In particular, under appropriate assumptions, constant extensions yield $\frac{1}{2}$ order convergence rates and linear extensions yield $\frac{3}{2}$ order convergence rates. With smooth extensions, these rates are improved to quadratic convergence. We illustrate the theory for any dimension $d\geq 2$ and numerically verify the convergence rates with a number of two dimensional benchmarks, including linear patch tests, manufactured solutions, and domains with curvilinear surfaces. Numerical results show a first order convergence for constant extensions and second order convergence for linear extensions, which suggests a possible room of improvement in the future convergence analysis.