A crystal plasticity-based and temperature-dependent multi-phase field model for the ductile fracture of single crystals at elevated temperatures

In this work, a temperature-dependent multi-phase field model coupled with a crystal plasticity framework is developed to investigate the ductile fracture of single crystals at elevated temperatures. The principle of virtual power at finite deformation is extended to derive multi-phase field formulations, yielding macroscopic and microscopic force balance equations. The accumulated plastic slip on each slip plane is introduced as the driving force for the ductile damage evolution of that plane, which provides clear physical significance for microscale damage analysis. Within the thermodynamic framework, constitutive equations for damaged crystals are derived, including the macroscopic stress constitutive equation and the microscopic phase field constitutive equation. For theoretically incorporating the temperature effect, a temperature-dependent fracture threshold energy governing damage initiation and a temperature-dependent degradation function controlling damage evolution are proposed to explicitly characterize thermal influences. The main contribution of the developed model lies in the explicit modelling of the temperature effect on damage through a thermally coupled free energy function, along with the rigorous derivation of damage constitutive equations within the thermodynamic framework. For numerical implementation, an efficient and robust explicit algorithm is developed to solve the phase field and deformation field. The comparisons between numerical simulations and experimental results demonstrate the good capability of the proposed model. Based on the proposed model, the influence of temperature on the coupling effect between microplasticity and microdamage on slip planes is revealed. This study provides a new insight for ductile fracture modelling of single crystals at elevated temperatures.