Computational homogenization of brittle fracture

对于具有复杂微观结构的材料建模，多尺度方法尤为重要，即掌握各向异性材料难以确定的材料性能。对于材料的硬化行为，无论是在平均场背景下还是在计算均质化方面，多尺度方法已经达到了高度的成熟和复杂程度。该提案的目标是发展模拟技术，在计算上均匀化一类特殊的脆性断裂模型。基于近年来在力学中常用相场损伤模型近似的Francfort-Marigo模型的周期性和随机均匀化的数学结果，有效相场损伤模型的（各向异性）参数需要通过对复杂微观结构的模拟来确定。与迄今为止发表的方法相比，所提出的方法确保了独立于应用边界条件的适当尺度转换，因为（数学上）确保了代表性初等体积的存在。有效（各向异性）临界能量释放率在规定法向下，由最小面积的断口面（通过微观结构）计算。利用G. Strang的思想，可以将非凸极小曲面问题转化为具有唯一极小值的凸规划。与图像处理中使用的分割方法类似，需要确定执行算法，使工业复杂的异质材料的Francfort-Marigo模型的计算均匀化。为了证明所提出的技术的力量，该方法应在商业有限元代码Abaqus中实现，并增强以适用于复杂零件。该项目的目标是：计算断裂面和有效临界能量释放率-基于数字体图像的复杂微结构-复杂合成微结构-具有局部各向同性和各向异性材料行为的材料-各向异性相场损伤模型的快速和鲁棒宏观解。如果获得批准，该项目将为相场损伤模型的用户提供见解。假设各向异性相场模型的哪一种形式，以及如何在微观尺度上对相关参数进行鲁棒、高效的数值计算。

For modelling materials with complex microstructure multiscale methods are especially important, i.e. to get hands on material properties which are difficult to determine for anisotropic materials. For hardening material behavior, multiscale methods have reached a high level of maturity and sophistication, both in the mean field context and for computational homogenization.The goal of the proposal is to develop simulation technology to computationally homogenize a special class of brittle fracture models. Building upon recent mathematical results on the periodic and stochastic homogenization of the Francfort-Marigo model, which is often approximated by phase field damage models in mechanics, the (anisotropic) parameters of an effective phase field damage models shall be determined by simulations on complex microstructures. In contrast to methods published so far, the proposed methodology ensures a proper scale transition independent of the applied boundary conditions, because the existence of a representative elementary volume is (mathematically) ensured.The effective (anisotropic) critical energy release rate is computed, for prescribed normal, by the fracture surface (through the microstructure) of minimal area. Using an idea of G. Strang the non-convex minimal surface problem can be transformed into a convex program with unique minimal value. In analogy to segmentation methods used in image processing, performing algorithms are to be identified, enabling the computational homogenization of the Francfort-Marigo model for heterogeneous materials of industrial complexity.To demonstrate the power of the proposed technique, the method shall be implemented in the commercial finite element code Abaqus, and enhanced to work for complex parts.The objectives of the proposed project are:- computation of the fracture surface and the effective critical energy release rate for - complex microstructures based on digital volume images - complex synthetic microstructures - materials with locally isotropic and anisotropic material behavior- fast and robust macroscopic solution of an anisotropic phase field damage modelIn case of approval the project will provide insights for users of phase field damage models, which form of the anisotropic phase field model needs to be assumed and how the associated parameters can be robustly and efficiently computed numerically, provided the corresponding parameters on the microscopic scale are available.