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Modeling of ductile fracture in polycrystalline structures is challenging, since it requires integrated modeling of cracks, crystal plasticity, and grains. Here we extend the typical phase-field framework to the situations with constraints on the order parameters, and formulate two types of phase-field models on ductile fracture. The Type-I model incorporates three sets of order parameters, which describe the distributions of cracks, plastic strain, and grains, respectively. Crystal plasticity is employed within grain interiors accommodated by  J  2  plasticity at grain boundaries. The applications of the Type-I model to single crystals and bicrystals demonstrate the influences of grain orientations and grain boundaries on crack growth. In the Type-II model,  J  2  plasticity is assumed for the whole system and grain structures are neglected. Taking advantage of the efficiency of the fast Fourier transform, our Type-II model is employed to study low cycle fatigue. Crack closure and striation-like patterning of plastic strain are observed in the simulations. Crack growth rate is analyzed as a function of the  J -integral, and the simulated fatigue life as a function of plastic strain agrees with the Coffin–Manson relation without a priori assumption.

Phase-field framework with constraints and its applications to ductile fracture in polycrystals and fatigue