Abstract In the present paper, we develope a 3-D Msrf-integral to evaluate the surface energy that is corresponding to formation of multiple surface cracks. The Msrf-integral is formulated by modifying the conventional 3-D M-integral with deliberate origination of the coordinate system and proper delimitation of the integral surface. The integral is demonstrated to be surface-independent in a modified manner and so can be accurately evaluated provided an integration surface remote from the crack fronts is selected. With this feature, the theory of integration can be straightforwardly incorporated into the finite element method. In fact, it is not necessary to simulate the near-front regions with very fine FE simulations to obtain accurate solutions of Msrf.

中文翻译：

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使用改进的 3-D M-Integral 对多个表面裂纹进行表面能评估

摘要 在本文中，我们开发了一个 3-D Msrf 积分来评估与多个表面裂纹的形成相对应的表面能。Msrf 积分是通过修改传统的 3-D M 积分来制定的，该积分系统是故意起源的坐标系和积分表面的适当定界。积分被证明是以一种修改的方式与表面无关，因此如果选择远离裂纹前沿的积分表面，则可以准确地评估积分。有了这个特点，积分理论就可以直接融入到有限元方法中。事实上，没有必要用非常精细的有限元模拟来模拟近前区域来获得 Msrf 的准确解。