Abstract A new form of M-integral associated with time dependence parameters, is presented herein for viscoelastic materials. Based on the equivalent domain integral method, this time-dependent M-integral is numerically implemented as an effective and accepted fracture mechanical parameter for damage induced by crack growth in viscoelastic materials. Based on the linear viscoelastic model defined through Prony series, the conservation of the time-dependent M-integral for viscoelasticity is verified by applying user defined Python scripts. The results show that the newly proposed time-dependent M-integral can be successfully calculated numerically. Furthermore, numerical examples are given to demonstrate the validity and relevance of the time-dependent M-integral in viscoelastic material. In particular, the variations of the time-dependent M-integral for different loading rates are considered, which show that the crack growth behavior over a period of time in viscoelastic material can be evaluated based on the value of the time-dependent M-integral. In addition, the time-dependent M-integral is calculated to assess the damage degree induced by two collinear cracks.

中文翻译：

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基于域积分法的裂纹粘弹性材料M积分的概念及数值评价

摘要 本文提出了一种新形式的与时间相关参数相关的 M 积分，用于粘弹性材料。基于等效域积分方法，该时间相关的 M 积分在数值上实现为有效且可接受的断裂力学参数，用于由粘弹性材料中的裂纹扩展引起的损伤。基于通过 Prony 系列定义的线性粘弹性模型，通过应用用户定义的 Python 脚本验证了粘弹性的瞬态 M 积分的守恒。结果表明，新提出的瞬态 M 积分可以成功地进行数值计算。此外，给出了数值例子来证明粘弹性材料中瞬态 M 积分的有效性和相关性。特别是，考虑了不同加载速率下随时间变化的 M 积分的变化，这表明可以根据随时间变化的 M 积分的值来评估粘弹性材料在一段时间内的裂纹扩展行为。此外，计算瞬态 M 积分以评估由两个共线裂纹引起的损坏程度。