Bond failure in peridynamics: Nonequivalence of critical stretch and critical energy density criteria

TL;DR

The paper addresses whether two widely used bond-failure criteria in bond-based peridynamics—critical stretch and critical energy density—are mathematically equivalent. It develops a generalized PMB (GPMB) bond-based framework with an influence function $\omega(r)$ and recasts the energy-density criterion as a bond-dependent stretch criterion, enabling a direct comparison. The authors prove that the criteria are not equivalent in general and identify the exact condition $\omega(r)=\beta r^{-1}$ under which they coincide; for other choices, the criteria yield different bond-breaking and fracture dynamics, with the bond-length dependence of breaking controlled by the exponent $\alpha$ in $\omega(r)=r^{-\alpha}$. Numerical 2D experiments on isotropic extension, crack-tip evolution, and crack propagation/branching corroborate the analytical findings, showing that the predicted fracture patterns and crack paths depend sensitively on both the failure criterion and the influence function. The work highlights the substantial modeling implications of criterion choice in peridynamic fracture simulations and provides a framework for translating energy-based criteria into stretch-based implementations, clarifying when such translations preserve the intended fracture behavior.

Abstract

This paper rigorously analyzes bond failure in the peridynamic theory of solid mechanics, which is a fundamental component of fracture modeling. We compare analytically and numerically two common bond-failure criteria:~{\em critical stretch} and~{\em critical energy density}. In the former, bonds fail when they stretch to a critical value, whereas in the latter, bonds fail when the bond energy density exceeds a threshold. By focusing the analysis on bond-based models, we prove mathematically that the critical stretch criterion and the critical energy density criterion are not equivalent in general and result in different bond-breaking and fracture phenomena. Numerical examples showcase the striking differences between the effect of the two criteria on crack dynamics, including the crack tip evolution, crack propagation, and crack branching.

Figures (15)

• Figure 1: Description of all bonds $\bm{\xi}$ connecting a point $\bm{x}$ located at a distance $z \in (0,\delta)$ below the fracture surface to points $\bm{x}'$ in the spherical cap $\Omega_z$ above the fracture surface.
• Figure 2: Uniform discretization with a grid spacing of $\Delta x = \Delta y = \delta/3$ in Example 1.
• Figure 3: Illustration of the bond-breaking patterns for the critical energy density bond-failure criterion for different values of the influence function parameter $\alpha$ in Example 1.
• Figure 4: Comparison of the crack tip evolution between the two bond-failure criteria for $\alpha = 0$ in Example 2.
• Figure 5: Comparison of the crack tip evolution between the two bond-failure criteria for $\alpha = 0$ in Example 2 (cont.).

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1: Isotropic extension
• Remark 2.2
• Remark 2.3
• Lemma 3.1: 3D critical energy density criterion as a critical stretch criterion
• proof
• Remark 3.2
• Theorem 3.3: Nonequivalence of bond-failure criteria for 3D problems
• proof