Content

The focus of the lecture lies on geometrically exact beam theories as an example for nonlinear models in structural mechanics. Slender beam- or fiber-like components represent essential constituents of mechanical systems in countless fields of application and scientific disciplines. Examples are high-tensile industrial ropes and webbings, fiber-reinforced composite materials or synthetic polymer materials. On entirely different time and length scales, such slender components are relevant when analyzing the supercoiling process of DNA strands, the characteristics of carbon nanotubes or the cytoskeleton of biological cells, a biopolymer network of highly slender filaments that crucially influences the mechanics of cells. Often, these slender components can be described by means of geometrically fully nonlinear beam models. In many cases, the interaction between these beam- or fiber-like components—such as through mechanical contact or intermolecular forces—plays an important role.

In this lecture, the so-called geometrically exact beam theory will be considered, which is based on a fundamental kinematic assumption and accounts for large rotations and strains in a consistent manner, i.e., without additional approximations. In particular, the geometrically exact Simo-Reissner beam theory (kinematic assumption: beam cross-sections remain plain; nonlinear counterpart of the linear Timoshenko beam theory) and the geometrically exact Kirchhoff-Love beam theory (kinematic assumption: beam cross-sections remain plain and normal to the centerline; nonlinear counterpart of the linear Euler-Bernoulli beam theory) will be introduced as well as their interpretation as a 1D Cosserat continuum, i.e., a continuum theory with three translational and three rotational degrees of freedom.

Since the presence of large rotations is an inherent characteristic of fully nonlinear structural theories, the theory of larger rotations (in particular the theory of the special orthogonal group SO(3)) will be introduced in the beginning. Based on these fundamentals, the geometrically exact beam theory will be postulated as a nonlinear Cosserat continuum theory consisting of (translational and rotational) strain measures, stress resultants, constitutive relations and mechanical equilibrium conditions (in spatial and material form, respectively). In addition, it will be shown that the geometrically exact beam theory can be derived from the equations of nonlinear continuum mechanics supplemented by the fundamental kinematic assumption (e.g., plain beam cross-sections). For this purpose, a summary of the nonlinear continuum mechanics theory of solids (3D Boltzmann continua) will be presented, among others underlining the structural similarities with the geometrically nonlinear beam theory (1D Cosserat continuum).

Eventually, the lecture provides an overview of modeling approaches for mechanical contact interactions between beam- or fiber-like structures (e.g., in ropes or fabrics) and the embedding of such structures in a 3D Boltzmann continuum as required, e.g., for the fiber-scale modeling of fiber-reinforced composite materials. Finally, relevant numerical discretization and solution methods (in particular discretization schemes based on the Finite Element Method - FEM) will be presented. In a dedicated lab session, students will apply these methods using a FEM software developed at the institute to analyze selected problems of fully nonlinear beam mechanics.