Approaches to Fracture Mechanics based on Local and Global Energy Minimization

Griffith提出的脆性断裂的能量学框架不能预测无缺陷物体中的裂纹萌生。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。由这种涉及体积和表面能的泛函所支配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。为了数值处理这类问题，人们提出了许多近似的泛函表示法，使用高阶奇异摄动法、有限差分法或非局部能量法。然而，在工程语言中对断裂增量最小化方法的全面的理论处理及其适用于大规模工程应用的相关计算实现在文献中是缺失的。该项目的目的是研究基于全局能量最小化的脆性和粘结断裂的理论和计算框架，该全局能量最小化先验地绕过了Griffith理论关于裂纹萌生的主要缺陷。在研究项目的第一阶段，取得了以下成果：将基于位形力驱动的尖端裂纹扩展的断裂局部能量最小化概念推广到三维环境。研究结果为后续的全局方法研究提供了参考方案。提出了扩散裂纹扩展的全局能量最小化概念，即用一个由梯度型平衡方程驱动的相场来描述一组规则的裂纹不连续性。所提出的框架导致了具有特定本构函数的光滑连续损伤类型的断裂理论。建立了一个由位移场、断裂相场和双耗散力场组成的扩展三场模型，其粘性超力结构为扩展裂纹扩展提供了一个非常可靠的计算环境，在模拟复杂的三维裂纹拓扑结构方面具有巨大的潜力。在研究项目的后续阶段，我们将改进断裂相场模型，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解决扩散裂纹区的自适应网格加密过程中。为此，将为裂缝的梯度型相场模型开发新的构造力驱动的网格精化指标。此外，我们计划研究动态断裂问题，包括复杂的裂纹分支。最后，我们将强调断裂相场模型所固有的整体本构模型的优势，因为它嵌入到更复杂的多场问题中，例如热-机械和电磁-机械耦合问题。

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.