Continuously embedded manifold models in mechanics

Many materials feature embedded fibers or layers: Examples in nature are seen in wood, layered rocks, and biological tissues. In manufactured materials, such embedded structures are seen in textiles, carbon composites, and laminates. We propose a new concept for the modeling and simulation of these materials and thereby develop an understanding how they behave and how physical processes inside them take place. The project considers two distinct fields of applications. One is concerned with structure mechanics, centered around the question: How do materials with embedded sub-structures deform and which stresses and strains result as a consequence? The other field is fluid mechanics where the focus is on flow and transport in structured materials. The modeling of the mechanical behavior and the physical processes inside the materials with embedded sub-structures is a major challenge. One has to individually model the behavior of the bulk material on the one hand, and the embedded fibers and layers on the other. The latter requires sophisticated concepts from differential geometry, tensor calculus, and continuum mechanics. The resulting coupled models are formulated as mathematical boundary value problems which have to be solved using numerical simulation methods and high-performance computers. New variants of the Finite Element Method, the most popular simulation method, are required, tailored to the challenges of the coupled models formulated before. This project enables an improved description and understanding of complex materials with embedded sub-structures through modeling and simulation. Emphasis is placed on a generic framework that can be customized to all kinds of such materials in nature and engineering. Thereby, it shall also support the development of innovative, high-performance, and resource-efficient materials such as in composites and laminates.