Approaches to Fracture Mechanics based on Local and Global Energy Minimization

Griffith提出的脆性断裂的能量框架不能预测无缺陷体的裂纹起裂。这个问题可以通过适当定义的具有体积和表面贡献的增量能量泛函的全局最小化来克服。由这些涉及体积和表面能的函数控制的问题出现在应用科学的各个领域，从图像和信号处理到断裂力学。为了在数值上处理这类问题，已经提出了许多近似泛函表示，使用高阶奇异摄动，有限差分或非局部能量。然而，文献中缺乏工程语言中裂缝增量最小化方法的综合理论处理及其适用于大规模工程应用的相关计算实现。该项目的目的是研究基于全局能量最小化的脆性和粘性断裂的理论和计算框架，该框架先验地规避了Griffith理论在裂纹起裂方面的主要缺点。在项目的第一期研究中，获得了以下成果：将基于构形力驱动的锐裂纹扩展的断裂局部能量最小化概念扩展到三维环境；本工作的结果为后续全球方法的发展提供了参考。采用梯度平衡方程驱动的相场来描述一组正则化的裂纹不连续面，提出了扩散裂纹扩展的全局能量最小化概念。该框架建立了具有特定本构函数的光滑连续损伤型断裂理论。建立了由位移场、断裂相场和双耗散力场组成的扩展三场模型，该模型的粘性过力结构为复杂三维裂纹拓扑的建模提供了一个非常可靠的扩散裂纹扩展计算环境，具有巨大的潜力。在项目的后续研究阶段，我们将对断裂相场模型进行改进，以模拟更复杂的裂纹拓扑结构，并将其扩展到多场环境。在第一个研究阶段，一个非常重要的方面是将其嵌入到自适应网格细化过程中，以解决扩散裂纹区域。为此，将针对梯度型断裂相场模型开发新的构形力驱动网格细化指标。此外，我们计划研究包括复杂裂纹分支在内的动态断裂问题。最后，我们将强调裂缝相场建模中固有的体本构建模的优势，因为它可以嵌入更复杂的多场问题，如耦合热-机械和电磁-机械问题。

The energetic framework of brittle fracture proposed by Griffith is unable to predict crack initiation in a body free of defects. This problem can be overcome by a global minimization of suitably defined incremental energy functionals having both volume and surface contributions. Problems governed by such functionals involving volume and surface energies appear in a variety of areas in applied sciences, ranging from image and signal processing to fracture mechanics. In order to deal numerically with these kind of problems, many approximate functional representations have been proposed, using high-order singular perturbations, finite differences or non-local energies. However, comprehensive theoretical treatments of incremental minimization methods for fracture in an engineering language and their associated computational implementations suitable for large-scale engineering applications are missing in the literature. The purpose of the project is to investigate theoretical and computational frameworks of brittle and cohesive fracture based on global energy minimization that a priori circumvent the main drawback of Griffith’s theory with regard to crack initiation. In the first period of the research project the following results have been obtained: Local energy minimization concepts of fracture based on configurational–force–driven sharp crack propagation were extended to the three–dimensional setting. Results obtained from this work provide reference solutions for the subsequent developments on global methods. Global energy minimization concepts were developed for a diffusive crack propagation, were a set of regularized crack discontinuities is described by a phase field which is driven by a gradient–type balance equation. The proposed framework results in a smooth continuum–damage–type theory of fracture with specific constitutive functions. An extended three–field model that consists of the displacement field, the fracture phase field and the dual dissipative force field was developed, whose viscous over–force structure provides a very robust computational setting of diffusive crack propagation with an enormous potential with regard to the modeling of complex three–dimensional crack topologies. In the subsequent period of the research project, we will improve the phase field model of fracture with regard to the modeling of more complex crack topologies as well as its extension to multi–field environments. As it turned out in the first research period, a very important aspect is its embedding into adaptive mesh refinement procedures which resolve diffusive crack zones. To this end, new configurational–force–driven mesh refinement indicators will be developed for the gradient–type phase field model of fracture. Furthermore, we plan to investigate problems of dynamic fracture including complex crack branching. Finally, we will underline the advantage of bulk constitutive modeling inherent in the phase field modeling of fracture with regard to its embedding into more complex multi–field problems such as coupled thermo–mechanical and electro–magneto–mechanical problems.