Abstract 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 fields 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.

中文翻译：

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使用相场模型的耦合温度速率相关断裂的有限元实现

摘要 在本文中，使用相场模型研究了韧性材料的温度-速率耦合断裂。该模型是基于材料的应变能提出的，与基于应力的方法相反，通过添加标量场作为相场变量来跟踪可能的裂纹路径。在这种情况下，获得了位移场和相场中的两组控制方程，然后使用非线性有限元方法和隐式积分算法求解。该模型通过用户单元子程序 (UEL) 在 ABAQUS 商业有限元代码中实现。基于 von Mises 准则，研究了温度和加载速率对延性固体裂纹萌生和扩展的影响。除了，提出了作为温度函数的阈值参数，以控制塑性变形对不同温度水平下裂纹萌生和扩展的影响。此外，当温度水平低于和高于韧性 - 脆性转变温度时，建议使用新方程来解释裂纹的萌生和扩展与温度的关系。最后，使用所提出的模型评估加载速率和温度对不同试样响应的影响。获得的结果与其他实验和数值数据的比较证实了建议模型的准确性。结果还验证了新模型可以模拟较低温度对裂纹路径的影响，其中裂纹发生从韧性到脆性的变化。