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

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.

💡 Research Summary

This paper provides a rigorous mathematical and computational comparison of the two most widely used bond‑failure criteria in bond‑based peridynamics: the critical stretch criterion and the critical energy‑density criterion. The authors focus on brittle elastic materials modeled by a generalized prototype micro‑elastic brittle (PMB) model, where the material response is governed by an influence function ω(r) that depends only on the bond length r.

First, the critical energy‑density criterion is reformulated as a bond‑dependent critical stretch s_c(r) (Lemma 3.1 for three‑dimensional problems and Lemma 4.2 for two‑dimensional problems). This shows that the energy‑density criterion can be expressed in the same functional form as the stretch criterion, but with a stretch value that varies with r. By demanding that s_c(r) be independent of r, the authors derive the necessary and sufficient condition for the two criteria to be equivalent: ω(r)·r must be constant, i.e., ω(r)=β r⁻¹ for some β>0 (Theorem 1.1, formally Theorems 3.3 and 4.4). Consequently, the original PMB model (ω≡1) and virtually all practical influence functions do not satisfy this condition, proving that the two criteria are generally nonequivalent.

The paper then investigates a family of influence functions ω(r)=r⁻α with α<d+1 (d=2 or 3). Theorem 1.2 (formal Theorems 3.4 and 4.5) shows a striking dependence on α:

• For α<1, the critical‑stretch criterion tends to break shorter bonds first, while the critical‑energy‑density criterion may break longer bonds before the short ones.
• For 1<α<d+1, the opposite occurs: longer bonds are more likely to fail under the stretch criterion, whereas shorter bonds may fail first under the energy‑density criterion.

These analytical results are corroborated by extensive numerical experiments. In a simple isotropic extension test, different α values produce markedly different bond‑breaking patterns for the two criteria. In crack‑tip evolution simulations, the critical‑stretch criterion yields slower tip propagation and a smoother crack path, whereas the energy‑density criterion leads to faster propagation and, for certain α (e.g., α=2), early branching. A traction‑loaded pre‑notched plate demonstrates that, under the same loading, the stretch criterion produces a single horizontal crack while the energy‑density criterion generates a branched network; increasing the load causes branching in both cases but with significantly different branch angles and lengths.

The authors emphasize that the choice of failure criterion is not merely a numerical convenience; it fundamentally alters predicted fracture behavior, crack speed, branching patterns, and the relative importance of short versus long bonds. Since ω(r) encapsulates material microstructure (e.g., non‑local interaction range), selecting an appropriate influence function and failure criterion must be guided by experimental calibration or multiscale modeling.

In conclusion, the paper establishes that critical stretch and critical energy‑density criteria are not equivalent except for the special case ω(r)=β r⁻¹. This non‑equivalence has profound implications for peridynamic modeling of brittle fracture, especially when realistic, non‑uniform influence functions are employed. Future work should explore multi‑criterion approaches, damage‑softening extensions, and the integration of these findings into state‑based peridynamics and multiscale frameworks to achieve more accurate and physically consistent fracture predictions.