Finite element implementation of coupled temperature-rate dependent fracture using the phase field model

In this paper, a coupled temperature-rate dependent fracture of ductile materials is investigated using a phase field model. The model is proposed based on the strain energy of materials, in contrast to a stress-based method, to follow paths of possible cracks through adding a scalar field as the phase field variable. In this case, two sets of governing equations account in the displacement and phase field are obtained, then solved using the nonlinear finite element method and an implicit integration algorithm. The model is implemented in ABAQUS commercial finite element code through a user element subroutine (UEL). The effects of temperature and loading rate on the initiation and propagation of cracks for ductile solids are examined based on the von Mises criterion. Besides, the threshold parameter as a function of temperature is presented to control the effect of plastic deformation on cracks initiation and propagation for different temperature levels. Moreover, new equations are suggested for the initiation and propagation of cracks as a function of temperature when temperature levels are lower and higher than the ductile-brittle transition temperature. Finally, the influence of loading rates and temperatures on the responses of different specimens is evaluated using the proposed model. The comparison of obtained results with other experimental and numerical data confirms the accuracy of the suggested model. The results also verify that the new model can simulate the influence of lower temperatures on cracks' paths in which the change of the fracture phase from ductile to brittle occurs.