A thermodynamically consistent phase-field framework for viscoelastic fatigue fracture with dissipated-energy-driven toughness degradation

Viscoelastic materials, widely used in mechanical systems, are prone to fatigue damage under cyclic loading, where the evolution of such damage has been shown to be closely linked to internal energy dissipation. In this paper, a thermodynamically consistent phase-field modeling framework is developed to simulate fatigue crack behavior in viscoelastic materials, in which a toughness degradation mechanism driven by the history-dependent dissipated energy is introduced to characterize the progressive reduction of fracture resistance under cyclic loading. The proposed model, implemented in COMSOL Multiphysics, accurately reproduces experimentally observed crack paths reported in the literature, and the predicted crack growth behavior follows the Paris law. Simulation analyses on dumbbell-shaped and single-edge notched specimens further reveal that the viscoelastic dissipation mechanism governs distinct crack evolution patterns under different loading frequencies: lower frequencies induce greater energy dissipation per cycle, leading to significant damage accumulation and crack growth within fewer cycles but over longer durations; whereas higher frequencies result in greater energy input per unit time, promoting earlier crack initiation and faster propagation. Additional simulations on a three-dimensional C-shaped structure further illustrate the model's numerical robustness and extendibility to more complex geometries.