An asymptotically compatible bond-based peridynamics with Gaussian kernel

TL;DR

This paper introduces a Gaussian kernel-based bond peridynamic model that inherently achieves asymptotic compatibility with local elastic solutions without complex corrections, validated through simulations and impact experiments.

Contribution

It proposes a novel unbounded Gaussian kernel peridynamic model that naturally aligns with local elastic solutions, simplifying numerical implementation and improving accuracy.

Findings

Model achieves asymptotic compatibility without correction

Demonstrates convergence and accuracy in 2D simulations

Successfully predicts displacements in impact scenarios

Abstract

In this paper, we introduce a novel bond-based peridynamic model that utilizes a Gaussian kernel function. Previous peridynamic models, when directly discretized, have exhibited a lack of asymptotically compatibility with their corresponding local elastic solutions. Additionally, these models have faced challenges in accurately computing the volume of intersecting regions between the horizon of the material point and its neighboring cells. These difficulties in numerical simulations within peridynamics have spurred numerous efforts to develop corrective numerical methods. While such corrective methods have addressed certain issues, they remain complex to formulate and computationally intensive. Instead of addressing these challenges through modified numerical discretization, this paper presents a novel approach: bond-based peridynamics with a Gaussian kernel . This model replaces the bounded kernel function commonly used in existing peridynamic models with an unbounded Gaussian kernel function. Through direct meshfree discretization, we demonstrate that the solution of the proposed model , without the need for compatibility correction or volume correction, aligns with the corresponding local elastic solution. Furthermore, we derive the relevant material parameters based on energy equivalence to an elastic continuum model. Building upon this foundation, we propose a damage model and a linearized version of the proposed peridynamic model. We validate our proposed model through simulations of 2D smooth problems, demonstrating its convergence and accuracy. Finally, we assess the applicability of our approach to realistic scenarios by predicting displacements caused by a 2D plate with pre-existing cracks and replicating high-velocity impact results from the Kalthoff-Winkler experiments.