Elastic gradient theories and dislocation theories and their application to defects and microstructures

具有位错结构和微结构的材料在自然界和科学技术中起着重要的作用。从技术的角度来看，由微观结构引起的物理性能是重要的。从数学的角度来看，描述位错结构的基本模型是令人感兴趣的（例如偏微分方程理论）。因此，该项目位于材料科学，力学，固态物理和应用数学之间。在原子水平上，位错微观结构的描述是非常复杂的。如果微观结构被模拟为具有额外自由度的连续体，则可以理解许多重要性质。因此，微观结构可以用广义力学连续理论，如梯度理论，非局部理论和位错场理论来描述。该项目致力于微观尺度上的位错方面，以及位错连续理论的推广。特别地，考虑了诸如延迟和各向异性效应的动态效应。像准晶和石墨烯这样的材料目前引起了极大的兴趣。它为更好的建模开辟了新的视角，并考虑了新的方面（例如位错的新数学解决方案）。尽可能多地采用解析理论方法，并扩大和加强数值实现的基础。

Materials with dislocation structures and microstructures are of fundamental interest and play an important role in nature and technology. From the technical point of view, physical properties caused by the microstructure are of importance. From the mathematical point of view, the underlying models which describe the dislocation structures are of interest (e.g. theory of partial differential equations). Consequently, the project is situated between areas of materials science, mechanics, solid state physics, and applied mathematics. At the atomic level, the description of the dislocation microstructure is very complex. Many important properties can be understood, if the microstructure is modeled as a continuum with additional degrees of freedom. Thus, the microstructure can be described by generalized mechanical continuum theories such as gradient theories, nonlocal theories, and dislocation field theories.The project is devoted to aspects of dislocations on the micro-scale, and the generalization of dislocation continuum theories. In particular, dynamic effects such as retardationand anisotropic effects are considered. Materials like quasicrystals and graphene are currently of great interest. It opens new perspectives for a better modeling, and considers new aspects (e.g. new mathematical solutions for dislocations). As much as possible analytical theoretical methods will be used.In addition, the foundations for numerical implementations shall be expanded and strengthened.