Phase-Field Peridynamics

2026-04-10

AI Summary

This work addresses the numerical instability in correspondence-based peridynamics caused by conventional bond-breaking fracture models, which disrupt the approximation of the nonlocal deformation gradient. To resolve this issue, the authors propose a phase-field peridynamic framework that continuously degrades bond energy through a bond-phase-field parameter and employs a separate kinematic degradation function to preserve the accuracy of the nonlocal deformation gradient. The method eliminates instabilities associated with abrupt bond deletion, rigorously satisfies thermodynamic consistency with Griffith’s fracture theory, and, for the first time, provides an analytical derivation of the normalization constant for general spherical influence functions. Numerical experiments demonstrate that the approach achieves high stability, accuracy, and theoretical fidelity across Mode I/II fracture scenarios, boundary traction problems, and the Kalthoff–Winkler dynamic fracture test.

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Abstract

Peridynamics formulates the balance of linear momentum as an integro-differential equation, making it naturally suited for fracture modeling without special treatment of discontinuities. The bond-associated correspondence formulation provides a highly accurate peridynamic framework by computing bond-wise deformation gradients that are free of zero-energy modes and yield accurate results even near boundaries. However, the traditional fracture approach based on irreversible bond deletion can compromise this formulation, as the progressive removal of bonds degrades the nonlocal approximation of the deformation gradient and can lead to numerical instabilities. In this work, a novel phase-field peridynamics approach is introduced that avoids these instabilities. Instead of deleting bonds, the energetic contribution of each bond is continuously degraded through a bond phase-field parameter, while a separate kinematic degradation function preserves the accuracy of the nonlocal deformation gradient approximation. The normalization constant ensuring thermodynamic consistency with Griffith's fracture theory is derived analytically for general spherical kernel functions as a ratio of two one-dimensional integrals. Numerical examples including mode I and mode II fracture, the boundary tension test with different kernel functions and horizon ratios, and the Kalthoff-Winkler experiment demonstrate the stability, accuracy, and consistency of the proposed approach.

Problem

• Peridynamics
• Fracture modeling
• Bond deletion
• Numerical instability
• Deformation gradient

Innovation

• Phase-Field Peridynamics
• bond-associated correspondence
• nonlocal deformation gradient
• thermodynamic consistency
• zero-energy modes