Phase field modeling of general imperfect interfaces

材料界面在工程学或材料科学等许多学科中都扮演着重要的角色。例如，准脆性材料中的裂纹或多晶体中的晶界。如果事先知道这些界面的几何形状，就可以方便地将它们建模为尖锐的界面。然而，这些信息并不总是可用的。例如，裂纹的位置以及几何距离事先是未知的，但这是有待解决的边值问题的一部分。这类问题--也称为自由间断问题--可以用相场理论来有效地模拟。聚焦于机械载荷，在文献中确实可以找到几种不同的尖锐界面的相场近似。然而，它们中的大多数都是基于严格的简化(Griffith类型的模型)。相比之下，其他模型基于相对简单的凝聚区方法。此外，只考虑了部分热力学方面，并采用了几何线性化的设置。出于这个原因，将在本项目中对其进行概括。更准确地说，这个项目的目标是对经历大变形的复杂尖锐界面模型进行一致的相场近似，申请人最近提出的一个框架代表了起点。这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捉任意材料对称性(各向异性)的装置--在大变形的情况下也是如此。这一框架可以用相场理论来近似。为了保证相场理论收敛到基本的尖锐界面模型的平衡方程，将考虑申请人最近提出的相场理论，并对其进行显著的修改和推广。它基于变形梯度的秩1分解，表现出比以前的相场模型更高的收敛速度(对于尖锐的界面问题)。这个项目代表了为相场方法推导物理合理的本构模型的第一步。虽然提交的提案将忽略动力效应，即与界面传播有关的效应，但它们肯定必须列入第二个步骤。通过这样做，最终可以建立一个建模框架，允许有效地分析块体材料和界面之间的相互作用，并最终支持新材料的开发。

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.