On the treatment of boundary conditions for bond-based peridynamic models

TL;DR

This paper introduces two novel methods for applying classical boundary conditions in bond-based peridynamic models, addressing a key challenge in non-local simulation of structural mechanics.

Contribution

It proposes and compares two approaches for incorporating boundary conditions into peridynamics, enhancing the model's compatibility with classical boundary treatments.

Findings

Both methods effectively impose boundary conditions in numerical experiments.

The second method ensures compatibility with local models as the horizon approaches zero.

Numerical results demonstrate the viability of the proposed approaches.

Abstract

In this paper, we propose two approaches to apply boundary conditions for bond-based peridynamic models. There has been in recent years a renewed interest in the class of so-called non-local models, which include peridynamic models, for the simulation of structural mechanics problems as an alternative approach to classical local continuum models. However, a major issue, which is often disregarded when dealing with this class of models, is concerned with the manner by which boundary conditions should be prescribed. Our point of view here is that classical boundary conditions, since applied on surfaces of solid bodies, are naturally associated with local models. The paper describes two methods to incorporate classical Dirichlet and Neumann boundary conditions into bond-based peridynamics. The first method consists in artificially extending the domain with a thin boundary layer over which the displacement field is required to behave as an odd function with respect to the boundary points. The second method resorts to the idea that peridynamic models and local models should be compatible in the limit that the so-called horizon vanishes. The approach consists then in decreasing the horizon from a constant value in the interior of the domain to zero at the boundary so that one can directly apply the classical boundary conditions. We present the continuous and discrete formulations of the two methods and assess their performance on several numerical experiments dealing with the simulation of a one-dimensional bar.