Phase-field peridynamics

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.