Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity
Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity
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DOI：
10.1007/s00205-009-0271-4
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发表时间：
2010-08
影响因子：
2.5
通讯作者：
Duvan Henao;C. Mora-Corral
中科院分区：
文献类型：
--
作者：
Duvan Henao;C. Mora-Corral
在本文中，我们提出并分析了一个考虑空化和断裂的非线性弹性变分模型。把空化和断裂理论统一起来的主要思想是把空化和裂纹都看作是产生新表面的现象。因此，我们定义一个函数来测量创建的表面的面积。这个泛函与笛卡尔电流理论有关。我们证明了该泛函的有界性意味着变形梯度行列式的序贯弱连续性，并且证明了几乎处处一对一变形的弱极限也是几乎处处一对一的。然后，我们使用这些结果来获得包含弹性能和表面能的变分模型的最小值的存在性，同时考虑了取向保持和非穿透条件。
In this paper, we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one almost everywhere deformations is also one-to-one almost everywhere. We then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy, taking into account orientation-preserving and non-interpenetration conditions.