Phase field modeling of general imperfect interfaces

材料界面在许多学科中起着重要的作用，例如在工程或材料科学中。说明性的例子包括准脆性材料中的裂纹或多晶体中的晶界。如果预先知道这些界面的几何形状，则可以方便地将它们建模为尖锐界面。然而，这些信息并不总是可用的。例如，裂纹的位置以及几何形状是事先不知道的，但却是要解决的边值问题的一部分。这类问题--也称为自由不连续问题--可以通过相场理论有效地建模。针对机械载荷，在文献中确实可以找到几种不同的尖锐界面相场近似。然而，他们中的大多数是基于严重的简化（格里菲斯型模型）。通过对比，其他模型基于相对简单的内聚区方法。此外，仅部分考虑了几何方面，并采用了几何线性化设置。因此，在本项目中将阐述一个概括。更准确地说，这个项目的目标是一个一致的相场近似的复杂尖锐的界面模型经历大变形。关于尖锐的界面，最近提出的一个框架的申请人代表的出发点。它是第一个允许通过同时满足物理学中的所有基本平衡定律来捕获任意材料对称性（各向异性）的方法-也适用于大变形的情况。这个框架是由相场理论近似。为了保证相场理论收敛于底层尖锐界面模型的平衡方程，将考虑并显著修改和扩展申请人最近提出的相场理论。它是基于一个秩-1分解的变形梯度，并表现出更高的收敛速度比以前的相场模型（对尖锐的界面问题）。虽然动力学效应，即，那些与接口传播有关的问题，在提交的建议中将被忽略，它们当然必须包括在第二步中。通过这样做，最终可以建立一个建模框架，允许有效地分析散装材料和界面之间的相互作用，并最终支持开发新材料。

Material interfaces play an important role in many disciplines such as in engineering or in materials science. Illustrative examples include cracks in quasi-brittle materials or grain boundaries in polycrystals. If the geometry of such interfaces is known in advance, they can be conveniently modeled as sharp interfaces. However, this information is not always available. For instance, the position as well as the geoemtry of cracks are not known beforehand but are part of the boundary value problem to be solved. This type of problems - also referred to as Free Discontinuity Problems - can be effectively modeled by means of phase field theory.Focusing on mechanical loading, several different phase field approximations for sharp interfaces can indeed be found in the literature. However, most of them are based upon severe simplification (Griffith-type models). By way of contrast, the other models are based on relatively simple cohesive zone approaches. Furthermore, thermodynamical aspects are only partly considered and a geometrically linearized setting is adopted. For this reason, a generalization is to be elaborated within this project. To be more precise, the goal of this project is a consistent phase field approximation of complex sharp interface models undergoing large deformations.Concerning sharp interfaces, a framework recently proposed by the applicant represents the starting point. It is the first one allowing to capture arbitrary material symmetries (anisotropy) by simultaneously fulfilling all fundamental balance laws in physics - also in the case of large deformations. This framework is to be approximated by phase field theory. In order to guarantee convergence of the phase field theory to the balance equations of the underlying sharp interface model, a phase field theory recently presented by the applicant will be considered and significantly modified and extended. It is based on a rank-1 decomposition of the deformation gradient and shows higher convergence rates than previous phase field models (towards the sharp interface problem).This project represents the first step towards deriving physically sound constitutive models for phase field approaches. While kinetic effects, i.e., those related to the propagation of interfaces, will be ignored in the submitted proposal, they certainly have to be included in a second step. By doing so, a modeling framework can eventually be established which allows to effectively analyze the interaction between bulk materials and interfaces and finally, to support developing new materials.