Variational modelling of fracture in high-contrast microstructured materials: mathematical analysis and computational mechanics

在Francfort和Marigo的开创性工作之后，Mumford-Shah型自由不连续泛函已经被建立为研究脆性材料断裂的简化但相关的数学模型。对于有限对比成分，脆性能量的均匀化是现在很好理解，并提供了一个严格的微观到宏观的放大脆性断裂。直到最近，明确的高对比度脆性显微结构已被提供，这表明，已经为简单的Mumford-Shah型的自由不连续能，成分的高对比度的性质可以诱导一个复杂的有效行为超越了单一的成分。特别是，宏观粘聚力区模型和损伤模型可以通过均匀化纯脆性微观能量与高对比度系数。在这个框架中，简单到复杂的过渡起源于微观体-表面能量耦合，这是可能的，由于简并的泛函。由于需要了解机械材料失效的数学基础并开发计算上易于处理的数值技术，该项目的主要目标是通过均匀化简单的高对比度脆性材料来模拟所有可能的材料。在数学上，这相当于确定一组高对比度自由不连续泛函的变分极限闭合。这个问题在弹性力学的背景下有着悠久的历史，而如果允许断裂的话，人们对这个问题的理解就少得多了。对于变分分析，这将是至关重要的，以确定新的均匀化公式，“量化”的微观体表面能量耦合。此外，高对比度成分对宏观各向异性的影响将通过提供明确的微观结构实现极限模型与首选的裂纹方向进行研究。相关的数学工具将来自变分法和几何测度理论。沿着的方式，新的特设扩展和SBV-函数的近似结果将被建立。后者将在自己的权利的数学兴趣，并出现在尺度相关的自由不连续性问题的分析中广泛适用。计算力学的结果将建立在数学理论的基础上，并在分析变得不切实际时用相关的见解来补充它。高性能的快速傅立叶变换和自适应树为基础的计算方法将被开发，以评估新的细胞公式。识别出的损伤和粘聚区模型将被转移到组件规模的模拟中。预计这些发现将大大提高对材料失效的来源和机制的理解，并为识别可用于估计工业部件强度的各向异性材料模型提供计算工具。

After the seminal work of Francfort and Marigo, free-discontinuity functionals of Mumford-Shah type have been established as simplified and yet relevant mathematical models to study fracture in brittle materials. For finite-contrast constituents, the homogenisation of brittle energies is by-now well-understood and provides a rigorous micro-to-macro upscaling for brittle fracture. Only recently, explicit high-contrast brittle microstructures have been provided, which show that, already for simple free-discontinuity energies of Mumford-Shah type, the high-contrast nature of the constituents can induce a complex effective behaviour going beyond that of the single constituents. In particular, macroscopic cohesive-zone models and damage models can be obtained by homogenising purely brittle microscopic energies with high-contrast coefficients. In this framework, the simple-to-complex transition originates from a microscopic bulk-surface energy-coupling which is possible due to the degeneracy of the functionals. Motivated by the need to understand the mathematical foundations of mechanical material-failure and to develop computationally tractable numerical techniques, the main goal of this project is to characterise all possible materials which can be obtained by homogenising simple high-contrast brittle materials. In mathematical terms, this amounts to determine the variational-limit closure of the set of high-contrast free-discontinuity functionals. This problem has a long history in the setting of elasticity, whereas is far less understood if fracture is allowed. For the variational analysis it will be crucial to determine novel homogenisation formulas which “quantify” the microscopic bulk-surface energy-coupling. Moreover, the effect of high-contrast constituents on macroscopic anisotropy will be investigated by providing explicit microstructures realising limit models with preferred crack-directions. The relevant mathematical tools will come from the Calculus of Variations and Geometric Measure Theory. Along the way, new ad hoc extension and approximation results for SBV-functions will be established. The latter will be of mathematical interest in their own right, and appear to be widely applicable in the analysis of scale-dependent free-discontinuity problems. The computational mechanics results will build upon the mathematical theory, and will complement it with relevant insights when the analysis becomes impracticable. High performance fast Fourier transform and adaptive tree-based computational methods will be developed to evaluate the novel cell formulas. The identified damage and cohesive-zone models will be transferred to simulations on component scale. The findings are expected to significantly enhance the understanding of the sources and mechanisms of material-failure and to provide computational tools for identifying anisotropic material-models useful for estimating the strength of industrial components.