A phase field fatigue approach with equivalent stress accumulation for viscoelastic solids

Phase field fatigue fracture modeling through toughness degradation has become a promising method for predicting fatigue failure, but its application in viscoelastic solids is rare and a proper mechanical quantity is required for fatigue accumulation. In this study, a phase field approach for viscoelastic fatigue fracture is proposed by modifying the viscoelastic free energy functional. Specifically, a scalar fatigue degradation function is introduced to govern the toughness degradation with the fatigue history variable as the argument. Differing from conventional choices, the equivalent stress measure related to fatigue is taken to accumulate the history variable. In addition, to improve the numerical convergence, the phase field regularization employs the fourth-order extension of the AT1 model, and the crack tip singularity of the local stress measure is regularized by applying the nonlocal approach that couples an additional differential equation into the phase field formulations. The multi-field problem is solved in a staggered manner and the C¹ continuity multiresolution discretization is achieved in the isogeometric framework that integrates hierarchical B-splines. Numerical examples involving tension, shear, and diametral compression show the proposed approach can properly address the fatigue crack initiation and propagation, and reproduce the Paris behavior. The predicted fatigue life results agree well with the existing data, showing the validity of the approach.