@phdthesis{,
    author = {Steinbrecher, Ivo Sebastian},
    title = {Mixed-dimensional finite element formulations for beam-to-solid interaction},
    editor = {},
    booktitle = {},
    series = {},
    journal = {},
    address = {},
    publisher = {},
    edition = {},
    year = {2022},
    isbn = {},
    volume = {},
    number =  {},
    pages = {},
    url = {},
    doi = {},
    keywords = {Mixed-dimensional interaction, Finite element method, Nonlinear beam theory, Mortar methods, Beam-to-solid volume coupling, Beam-to-solid surface coupling, Beam-to-solid surface contact, 1D-3D position and rotation coupling, 1D-2D position and rotation coupling, 1D-2D contact, Fiber-reinforced materials},
    abstract = {The interaction between slender fiber- or rod-like components, where one spatial dimension is much larger than the other two, with three-dimensional structures (solids) is an essential mechanism of mechanical systems in numerous fields of science, engineering and bio-mechanics. Examples include reinforced concrete, supported concrete slabs, fiber-reinforced composite materials and the impact of a tennis ball on the string bed of a tennis racket. Applications can also be found in medicine, where stent grafts are a commonly used device for endovascular aneurysm repair, and in many biological systems such as arterial wall tissue with collagen fibers. The different types of dimensionality of the interacting bodies, i.e., slender, almost one-dimensional fibers and general three-dimensional solids, pose a significant challenge for typical numerical simulation methods. Classical modeling techniques usually require a compromise between a detailed description of the one-dimensional structures and overall model complexity. The main focus of this work is the development of novel computational approaches to simulate the interaction between fiber-like structures and three-dimensional solids. The key idea therein is to explicitly model the slender components as one-dimensional Cosserat continua based on the geometrically exact beam theory, which allows for an accurate and efficient description of the slender fibers (beams). Since the dimensions of the coupled differential equations are not equal, the resulting combined interaction problem is a {mixed-dimensional} beam-to-solid interaction problem. Not only the governing equations of the beam but also the developed interaction schemes are exclusively formulated along the one-dimensional beam centerline. From a mechanical point of view, the resulting mixed-dimensional interaction of nonlinear geometrically exact beam finite elements with classical continuum finite elements introduces a singular solution, similar to the problem of a concentrated line load acting on a three-dimensional continuum. As one of the main contributions of this thesis, theoretical considerations and numerical examples verify that this singularity does not affect the usability of the proposed methods within the envisioned application range. Based on the considered applications, two different types of interacting geometry pairs can be identified: line-to-volume, e.g., beams embedded in solid volumes, and line-to-surface, e.g., beams tied or in contact with the surface of a solid volume. Within the present work, coupling (i.e., tying) of the beam centerline position to the underlying solid in line-to-volume problems is investigated first. As a next step, also the rotations of the Cosserat continua are coupled to the solid volume. This requires the construction of a suitable rotation (i.e., triad) field inside the solid (Boltzmann) continuum. For both, positional and rotational coupling, mortar-type methods, inspired by classical mortar methods from domain decomposition or surface-to-surface interface problems, are employed to discretize the coupling constraints. A penalty regularization is performed to eliminate the Lagrange multipliers from the global system of equations, which results in a robust coupling scheme. This is verified by several numerical examples, in which consistent spatial convergence behavior can be achieved and potential locking effects can be avoided. The second half of this thesis extends the previously developed algorithms for line-to-volume coupling to line-to-surface coupling. This introduces the additional complexity of having to account for the surface normal vector in the positional coupling constraints. It is demonstrated that only a consistent handling of the surface normal vector leads to physically accurate results and guarantees fundamental mechanical properties such as conservation of angular momentum. Finally, a Gauss point-to-segment beam-to-solid surface contact scheme that allows for the modeling of unilateral contact between one-dimensional beams and two-dimensional solid surfaces is presented. The previously mentioned building blocks constitute a novel mixed-dimensional beam-to-solid interaction framework, which is verified by theoretical discussions and numerical examples throughout this thesis. Possible extensions are outlined in this thesis and propose numerical and algorithmic improvements as well as the treatment of other physical effects such as delamination between embedded beams and the surrounding volume. However, already in the present state, the presented framework is an efficient, robust, and accurate tool for beam-to-solid interaction problems and can become a valuable tool in science and engineering.},
    note = {},
    
    school = {Universität der Bundeswehr München},
    
}