A Free Discontinuity Approach to Brittle Fracture Mechanics: Analysis and Numerical Implementation

BourdinDMS-0605320 本文将G. Francfort和J-J. Marigo。这种表述方式从经典的Griffith理论出发，同时保留了它的本质，表面和体积项之间的竞争。 这样做，它允许解决裂纹路径确定，创建新的裂纹，和裂纹之间的相互作用问题。 它的数值实现是基于一个伽玛收敛近似，这给出了一个自然的方式来考虑全球最小的所有可能的裂纹集。 研究人员扩展了现有的模型，以考虑在热载荷和非均匀介质中裂纹的扩展，并在这些领域进行数值实验。 他建立了一个鲁棒的数值算法，避免了大多数局部极小，并研究了伽玛收敛和局部极小之间的联系，为这个模型。 他在超级计算机上实现了一种离散区域分解方法，并研究了离散化与正则化参数之间的关系。 断裂力学是一个非常活跃的研究领域，有着重要的应用. 近年来，法国戴高乐机场2F航站楼的意外倒塌，哥伦比亚号航天飞机在重返大气层时的解体，以及美国航空公司582号航班在纽约皇后区上空的坠毁，都与无法预测和无法解释的骨折有关。 在脆性断裂领域（包括陶瓷、玻璃和混凝土等多种材料），大多数公认的理论都是基于格里菲斯准则，并局限于一个孤立的、预先存在的裂纹沿沿着给定路径的扩展。 研究者扩展了Griffith理论的分析和数值实现。Francfort和J. J. Marigo，这消除了这些限制。 他扩展了目前的模型，以解决更一般的问题（热负荷，异质材料），并改进了目前在超级计算机上的数值实现。

BourdinDMS-0605320 The investigator extends the analysis and the numericalimplementation of a Free-Discontinuity approach to brittlefracture mechanics proposed by G. Francfort and J-J. Marigo. This formulation departs from the classical Griffith theory whilepreserving its essence, the competition between surface and bulkterms. Doing so, it allows to address the issues of crack pathdetermination, creation of new cracks, and interactions betweencracks. Its numerical implementation is based on aGamma-convergence approximation, which gives a natural way toconsider global minimizations over all possible crack sets. Theinvestigator extends the existing model to account for thepropagation of cracks under thermal loads and in heterogeneousmedia and conduct numerical experiments in these areas. Hebuilds a robust numerical algorithm avoiding most localminimizers, and studies the link between Gamma-convergence andlocal minimizers for this model. He implements an overlappingdomain decomposition method on supercomputers and studies therelation between discretization and regularization parameters. Fracture mechanics is a very active area of research, withvital applications. In recent years, the unexpected collapse ofterminal 2F at Charles de Gaulle airport in France, thedisintegration the Columbia space shuttle upon re-entry, and thecrash of American Airlines Flight 582 over Queens, NY were alllinked to unpredicted and unexplained fracture. In the area ofbrittle fracture (which encompasses materials as diverse asceramics, glass, and concrete), most commonly accepted theoriesare based on Griffith's criterion and are limited to thepropagation of an isolated, pre-existing crack along a givenpath. The investigator extends the analysis and the numericalimplementation of a generalization of Griffith's theory, proposedby G. Francfort and J.J. Marigo, that eliminates theserestrictions. He extends the current model to account for moregeneral problems (thermal loads, heterogeneous materials), andimproves the current numerical implementation on supercomputers.