Abstract
The development of novel high-tech materials is a critical aspect of the engineering sciences and has significance in many branches of industry. Reliable prediction of material behavior commonly requires the description of material properties on multiple length scales (e.g., the microscopic, mesoscopic, or macroscopic scale). In the past, studying energy-based
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multiscale models has proven remarkably successful due to their amenability to tools in the calculus of variations. However, the standard variational theories are insufficient to gain a deeper insight into advanced mechanical features, often modeled via non-convex (inhomogeneous) differential constraints. It is therefore essential to combine novel techniques for constrained multiscale models with profound knowledge of the classical variational theories and materials science. In this spirit, the thesis presents new analytic results for three energy-based models whose common denominator is the constraint of incompressibility. First, we focus on the dimension reduction of models for locally volume-preserving thin strings and rods. With the help of Gamma-convergence, we derive lower-dimensional models that effectively describe the deformation behavior of the three-dimensional body for the entire hierarchy of scaling regimes of the external forces. We differentiate between regimes that lead to highly flexible string models and all remaining scalings, in which we derive constrained Kirchhoff- and von Kármán-type rod theories. In all physically relevant limit models, we find that the constraint of incompressibility only impacts the elastic energy itself (and thus its minimizers) and not the set of admissible deformations. Second, we characterize the effective deformation behavior of hyperelastic materials reinforced by parallel, long, thin, rigid fibers. Within a suitable homogenization framework, the set macroscopic responses to external forces correspond to weak Sobolev limits of functions that satisfy the challenging inhomogeneous, differential constraint of rigidity on the fibers. It turns out that the material behavior of fibered composites is highly anisotropic in the sense that the strain in the direction of the fibers has unit length and higher regularity and only depends on the cross-section variables. We illustrate several examples of admissible deformations and set this result apart from earlier work about elastic bodies reinforced by rigid layers. This work is the first fundamental step towards a complete homogenization result of hyperelastic bodies reinforced by fully rigid fibers. Third, we tackle a new variational model for single-slip polycrystalline finite plasticity based on earlier works on single-slip single-crystal finite plasticity. The goal here is to describe the deformation behavior of a collection of rotated copies of single crystals, called the grains of the polycrystal. The primary challenge is the solvability of a specific inhomogeneous, non-convex differential inclusion with affine boundary values representing the admissible macroscopic strains. We determine necessary conditions by combining well-known relaxation and convex integration results with a new characterization of globally affine solutions to a relaxed differential inclusion. Sufficient conditions are derived from a generalized Hadamard jump condition and the resulting compatibility conditions along the boundary grains. Under suitable assumptions on the polycrystal grains, these conditions coincide and yield a full description of the macroscopic deformation behavior.
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