A Variational–Based Formulation of Regularized Brittle Fracture
A Variational–Based Formulation of Regularized Brittle Fracture
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基于变分的规则化脆性断裂公式
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
C. Miehé
中科院分区：
文献类型：
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作者：
M. Hofacker;F. Welschinger;C. Miehé
本文对基于变分的脆性断裂及其梯度型正则化进行了比较研究，并概述了两种方法的数值实现方面。后者平滑了裂缝的尖锐位移不连续。在离散裂纹建模方面，我们提出了一种构型力驱动裂纹扩展的变分框架。后者为材料节点力的计算提供了依据，并驱动了本文提出的自适应节点加倍有限元框架的裂纹扩展。这种公式对于无缺陷均匀体和裂纹分支复杂情况下的裂纹起裂建模适用性有限。这可以通过正则化裂纹建模来克服。这里，裂纹表面项的椭圆近似产生一个正则化的双场泛函，其中一个附加的标量场近似不连续集。我们对这种从尖锐裂纹到正则化环境的转变提供了一个简单的解释。根据代表裂纹周围区域宽度的长度尺度，得出具有特定梯度损伤和硬化条件的光滑连续损伤型理论。这种变分框架是由一个耦合的两场有限元框架以一种直接的方式实现的。我们比较了两种方法对所选裂纹模式的代表性数值结果，并突出了两种网格的优缺点。（©2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim）
The paper performs a comparative study of variational‐based brittle fracture with its gradient‐type regularization, and outlines aspects of the numerical implementations of both approaches. The latter smoothes out sharp displacement discontinuities of cracks. On the side of discrete crack modeling, we propose a variational framework of configurational‐force‐driven crack propagation. The latter provides the basis for the computation of material nodal forces and drives the crack propagation in our proposed finite element framework with adaptive nodal doubling. Such a formulation is of limited applicability for the modeling of crack inititation in homogeneous bodies without defects and in situations with complex crack branching. This can be overcome by a regularized crack modeling. Here, an elliptic approximation of the crack surface term yields a regularized two field functional, where an additional scalar field approximates the set of discontinuities. We provide a simple interpretation of such a transition from the sharp crack to the regularized setting. It results in a smooth continuum‐damage‐type theory with a specific gradient‐damage and hardening terms, depending on a length scale that represents the width of a zone that surrounds the crack. Such a variational framework is implemented by a coupled two‐field finite element framework in a staightforward manner. We compare representative numerical results obtained by both methods for selected crack patterns and highlight the pro and contra of both meshes. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)