Energy integral in fracture mechanics (J‐integral) and Gauss‐Bonnet Theorem
Energy integral in fracture mechanics (J‐integral) and Gauss‐Bonnet Theorem
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断裂力学中的能量积分（J-积分）和高斯-邦尼定理
DOI：
10.1002/zamm.200700140
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
H. Nagahama
中科院分区：
文献类型：
--
作者：
K. Yamasaki;H. Nagahama
J积分（一种与路径无关的能量积分）形式是分析非线性断裂力学的标准方法。结果表明，变形场的能量密度在同伦算子中对应于位错向错场的J积分，并给出了位错向错场的力的物理解释。缺陷的连续统理论为理解J积分的拓扑方面提供了一个自然的框架。这种几何解释表明J积分是微分几何中著名定理的替代表达，即，高斯-博内定理（Gauss-Bonnet theorem）（亏格= 1）J积分的几何表达式表明Eshelby的能量动量（物质空间的物理量）与爱因斯坦3形式（物质空间的几何对象）密切相关。
The J‐integral (a path‐independent energy integral) formalism is the standard method of analyzing nonlinear fracture mechanics. It is shown that the energy density of deformation fields in terms of the homotopy operator corresponds to the J‐integral for dislocation‐disclination fields and gives the force on dislocation‐disclination fields as a physical interpretation. The continuum theory of defects gives a natural framework for understanding the topological aspects of the J‐integral. This geometric interpretation gives that the J‐integral is an alternative expression of the well‐known theorem in differential geometry, i.e., the Gauss‐Bonnet theorem (with genus = 1). The geometrical expression of the J‐integral shows that the Eshelby's energy‐momentum (the physical quantity of the material space) is closely related to the Einstein 3‐form (the geometric objects of the material space).