An asymptotically compatible bond-based peridynamics with Gaussian kernel

TL;DR

The paper addresses the mismatch between nonlocal bond-based peridynamics and local elasticity and the difficulty of volume corrections in direct discretizations. It proposes a bond-based peridynamic model with an unbounded Gaussian kernel that yields asymptotic compatibility under meshfree discretization without requiring compatibility or volume-correction techniques, and derives material parameters via energy equivalence. A nonlinear damage model and a linearized Gaussian-kernel formulation are developed for 2D and 3D, with validations demonstrating convergence and accuracy on smooth problems and through crack propagation scenarios. Numerical experiments, including a plate with a pre-existing crack and the Kalthoff-Winkler impact, show that the approach reproduces known solutions and experimental crack patterns, highlighting its practical effectiveness as an alternative to volume-corrected PD schemes.

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.

Figures (14)

• Figure 1: Plots of the Gaussian function $G(\bm{\xi})$ (in two dimensions) with $\sigma= 1/10$ (left) and $\sigma= 1/20$ (right), respectively.
• Figure 2: Evaluation of fracture energy $G_0$. Here red line is the fracture surface. $\mathbf{A}$ and $\mathbf{B}$ are material points and $0\leq z \leq \delta$.
• Figure 3: Two models discretized by the meshfree method: (a) Truncation of the influence region with the $\chi_a^2(2)=36$ and $\sigma=\Delta x$; (b) The volume correction in PD with horizon size $\delta=3h$.
• Figure 4: Geometry of a plate with displacement boundary condition.
• Figure 5: Displacement variation in y-direction when $x=0$: (a)By BB-PD with FA method; (b)By BB-PD with LAMMA method; (c) By BB-PD with QWJ method; (d) By BBPD-GK with no volume correction.

Theorems & Definitions (1)

• Remark 1