Phase-field modeling of continuous fatigue via toughness degradation

We present a phase-field formulation to model fatigue crack growth over large numbers of cycles. Building upon a recently introduced phase-field formulation by the authors, fatigue is modeled phenomenologically by degradation of the fracture toughness, treated as a spatiotem-porally evolving material property, inside a region around the crack tip with size R-fatigue. The present formulation, however, treats cycle number N as a continuous variable, allowing for crack growth prediction over large numbers of cycles of experimental relevance in arbitrary geometries containing one or several cracks, as well as under various loading conditions. The phenomenological form of the degradation law is analytically motivated by first deriving a relationship between the crack growth rate per cycle da/dN and the stress intensity factor (SIF) variation amplitude delta K in the sharp-interface limit where R-fatigue is much larger than the phase-field regularization length xi. This relationship reproduces salient features of experimentally measured fatigue growth curves, including the existence of a minimum delta K for growth, a power law over an intermediate range of delta K, and a sharp increase of growth rate when the peak SIF value approaches the Griffith threshold K-c. Phase-field simulations are shown to reproduce similar growth curves with quantitative differences depending on the ratio R-fatigue/xi. The ability of the model to predict realistic crack paths is demonstrated by various examples in two and three dimensions including crack kinking under mode I+II loading, "en-passant "interacting cracks, and crack twisting in a three-point bending geometry with mode I+II+III loading.