Diese Arbeit liefert ein dreidimensionales numerisches Homogenisierungskonzept, welches beim Übergang von der Mikro- zur Makroskala einen Wechsel in der Kontinuumsbeschreibung beinhaltet. Während für die Beschreibung der Makroskala das verallgemeinerte Cosserat-Kontinuum verwendet wird, basiert die Mikroskala auf der klassischen Cauchy-Theorie. Um das homogene Cosserat-Ersatzmaterial im Rahmen numerischer Simulationen nutzen zu können, erfolgt die Implementierung geeigneter Finiter Elemente in das Programmsystem Abaqus und deren Verifikation. Neben der Diskussion der bei der Homogenisierung beobachteten Effekte werden anhand eines idealisierten Modells eines biaxialverstärkten Mehrlagengestrickes die Vorteile gegenüber der klassischen Herangehensweise aufgezeigt. / This contribution provides a threedimensional homogenization approach which includes the switch of the continuum theory during the scale transition. Whereas the microscopic scale is described in the framework of the classical Cauchy theory, the macroscopic scale is based on the generalized Cosserat continuum. In order to use the obtained homogeneous Cosserat material, suitable finite elements are implemented in the commercial program system Abaqus followed by an appropriate verification. Beside the discussion of the arising effects the advantages of this approach compared to the classical procedure are shown by means of an idealized model of a biaxial woven fabric.

Almost all animal cells maintain a thin layer of actin filaments and associated proteins underneath the cell membrane. The actomyosin cortex is subject to internal stress patterns which result from the spatiotemporally regulated activity of non-muscle myosin II motors in the actin network. We study how these active stresses drive changes in cell shape and flows within the cortical layer, and how these cytoskeletal deformations and flows govern processes such as cell migration, cell division and organelle transport. Following a continuum mechanics approach, we develop theoretical descriptions for three different cellular processes, to obtain - in collaboration with experimental groups - a detailed and quantitative understanding of the underlying cytoskeletal mechanics. We investigate the forces and cortex flows involved in adhesion-independent cell migration in confinement. Many types of cell migration rely on the extension of protrusions at the leading edge, where the cells attach to the substrate with specific focal adhesions, and pull themselves forward, exerting stresses in the kPa range. In confined environments however, cells exhibit migration modes which are independent of specific adhesions. Combining hydrodynamic theory, microfluidics and quantitative imaging of motile, non-adherent carcinosarcoma cells, we analyze the mechanical behavior of cells during adhesion-independent migration. We find that the accumulation of active myosin motors in the rear part of these cells results in a retrograde cortical flow as well as the contraction of the cell body in the rear and expansion in the front, and we describe how both processes contribute to the translocation of the cells, depending on the geometric and mechanical parameters of the system. Importantly, we find that the involved propulsive forces are several orders of magnitude lower than during adhesive motility while the achieved migration velocities are similar. Moreover, the distribution of forces on the substrate during non-adhesive migration is fundamentally different, giving rise to a positive force dipole. In contrast to adhesive migration modes, non-adhesive cells move by exerting pushing forces at the rear, acting to expand rather than contract their substrate as they move. These differences may strongly affect hydrodynamic and/or deformational interactions between collectively migrating cells. In addition to the work outlined above, we study contractile ring formation in the actin cytoskeleton before and during cell division. While in disordered actin networks, myosin motor activity gives rise to isotropic stresses, the alignment of actin filaments in the cortex during cell division introduces a preferred direction for motor-filament interactions, resulting in anisotropies in the cortical stress. Actin filaments align in myosin-dependent shear flows, resulting in possible feedback between motor activity, cortical flows and actin organization. We investigate how the mechanical interplay of these different cortical properties gives rise to the formation of a cleavage furrow during cell division, describing the level of actin filament alignment at different points on the cortex with a nematic order parameter, in analogy to liquid crystal physics. We show that cortical anisotropies arising from shear-flow induced alignment patterns are sufficient to drive the ingression of cellular furrows, even in the absence of localized biochemical myosin up-regulation. This mechanism explains the characteristic appearance of pseudocleavage furrows in polarizing cells. Finally, we study the characteristic nuclear movements in pseudostratified epithelia during development. These tissues consist of highly proliferative, tightly packed and elongated cells, with nuclei actively travelling to the apical side of the epithelium before each cell division. We explore how cytoskeletal properties act together with the mechanics of the surrounding tissue to control the shape of single cells embedded in the epithelium, and investigate potential mechanisms underlying the observed nuclear movements. These findings form a theoretical basis for a more detailed characterization of processes in pseudostratified epithelia. Taken together, we present a continuum mechanics description of the actomyosin cell cortex, and successfully apply it to several different cell biological processes. Combining our theory with experimental work from collaborating groups, we provide new insights into different aspects of cell mechanics.

Industrial development processes as well as research in physics, materials and engineering science rely on computer modelling and simulation techniques today. With increasing computer power, computations are carried out on multiple scales and involve the analysis of coupled problems. In this work, continuum modelling is therefore applied at different scales in order to facilitate a prediction of the effective material or structural behaviour based on the local morphology and the properties of the individual constituents. This provides valueable insight into the structure-property relations which are of interest for any design process. In order to obtain reasonable predictions for the effective behaviour, numerical models which capture the essential fine scale features are required. In this context, the efficient representation of discontinuities as they arise at, e.g. material interfaces or cracks, becomes more important than in purely phenomenological macroscopic approaches. In this work, two different approaches to the modelling of discontinuities are discussed: (i) a sharp interface representation which requires the localisation of interfaces by the mesh topology. Since many interesting macroscopic phenomena are related to the temporal evolution of certain microscopic features, (ii) diffuse interface models which regularise the interface in terms of an additional field variable and therefore avoid topological mesh updates are considered as an alternative. With the two combinations (i) Extended Finite Elemente Method (XFEM) + sharp interface model, and (ii) Isogeometric Analysis (IGA) + diffuse interface model, two fundamentally different approaches to the modelling of discontinuities are investigated in this work. XFEM reduces the continuity of the approximation by introducing suitable enrichment functions according to the discontinuity to be modelled. Instead, diffuse models regularise the interface which in many cases requires even an increased continuity that is provided by the spline-based approximation. To further increase the efficiency of isogeometric discretisations of diffuse interfaces, adaptive mesh refinement and coarsening techniques based on hierarchical splines are presented. The adaptive meshes are found to reduce the number of degrees of freedom required for a certain accuracy of the approximation significantly. Selected discretisation techniques are applied to solve a coupled magneto-mechanical problem for particulate microstructures of Magnetorheological Elastomers (MRE). In combination with a computational homogenisation approach, these microscopic models allow for the prediction of the effective coupled magneto-mechanical response of MRE. Moreover, finite element models of generic MRE microstructures are coupled with a BEM domain that represents the surrounding free space in order to take into account finite sample geometries. The macroscopic behaviour is analysed in terms of actuation stresses, magnetostrictive deformations, and magnetorheological effects. The results obtained for different microstructures and various loadings have been found to be in qualitative agreement with experiments on MRE as well as analytical results. / Industrielle Entwicklungsprozesse und die Forschung in Physik, Material- und Ingenieurwissenschaft greifen in einem immer stärkeren Umfang auf rechnergestützte Modellierungs- und Simulationsverfahren zurück. Die ständig steigende Rechenleistung ermöglicht dabei auch die Analyse mehrskaliger und gekoppelter Probleme. In dieser Arbeit kommt daher ein kontinuumsmechanischer Modellierungsansatz auf verschiedenen Skalen zum Einsatz. Das Ziel der Berechnungen ist dabei die Vorhersage des effektiven Material- bzw. Strukturverhaltens auf der Grundlage der lokalen Werkstoffstruktur und der Eigenschafen der konstitutiven Bestandteile. Derartige Simulationen liefern interessante Aussagen zu den Struktur-Eigenschaftsbeziehungen, deren Verständnis entscheidend für das Material- und Strukturdesign ist. Um aussagekräftige Vorhersagen des effektiven Verhaltens zu erhalten, sind numerische Modelle erforderlich, die wesentliche Eigenschaften der lokalen Materialstruktur abbilden. Dabei kommt der effizienten Modellierung von Diskontinuitäten, beispielsweise Materialgrenzen oder Rissen, eine deutlich größere Bedeutung zu als bei einer makroskopischen Betrachtung. In der vorliegenden Arbeit werden zwei unterschiedliche Modellierungsansätze für Unstetigkeiten diskutiert: (i) eine scharfe Abbildung, die üblicherweise konforme Berechnungsnetze erfordert. Da eine Evolution der Mikrostruktur bei einer derartigen Modellierung eine Topologieänderung bzw. eine aufwendige Neuvernetzung nach sich zieht, werden alternativ (ii) diffuse Modelle, die eine zusätzliche Feldvariable zur Regularisierung der Grenzfläche verwenden, betrachtet. Mit der Kombination von (i) Erweiterter Finite-Elemente-Methode (XFEM) + scharfem Grenzflächenmodell sowie (ii) Isogeometrischer Analyse (IGA) + diffuser Grenzflächenmodellierung werden in der vorliegenden Arbeit zwei fundamental verschiedene Zugänge zur Modellierung von Unstetigkeiten betrachtet. Bei der Diskretisierung mit XFEM wird die Kontinuität der Approximation durch eine Anreicherung der Ansatzfunktionen gemäß der abzubildenden Unstetigkeit reduziert. Demgegenüber erfolgt bei einer diffusen Grenzflächenmodellierung eine Regularisierung. Die dazu erforderliche zusätzliche Feldvariable führt oft zu Feldgleichungen mit partiellen Ableitungen höherer Ordnung und weist in ihrem Verlauf starke Gradienten auf. Die daraus resultierenden Anforderungen an den Ansatz werden durch eine Spline-basierte Approximation erfüllt. Um die Effizienz dieser isogeometrischen Diskretisierung weiter zu erhöhen, werden auf der Grundlage hierarchischer Splines adaptive Verfeinerungs- und Vergröberungstechniken entwickelt. Ausgewählte Diskretisierungsverfahren werden zur mehrskaligen Modellierung des gekoppelten magnetomechanischen Verhaltens von Magnetorheologischen Elastomeren (MRE) angewendet. In Kombination mit numerischen Homogenisierungsverfahren, ermöglichen die Mikrostrukturmodelle eine Vorhersage des effektiven magnetomechanischen Verhaltens von MRE. Außerderm wurden Verfahren zur Kopplung von FE-Modellen der MRE-Mikrostruktur mit einem Randelement-Modell der Umgebung vorgestellt. Mit Hilfe der entwickelten Verfahren kann das Verhalten von MRE in Form von Aktuatorspannungen, magnetostriktiven Deformationen und magnetischen Steifigkeitsänderungen vorhergesagt werden. Im Gegensatz zu zahlreichen anderen Modellierungsansätzen, stimmen die mit den hier vorgestellten Methoden für unterschiedliche Mikrostrukturen erzielten Vorhersagen sowohl mit analytischen als auch experimentellen Ergebnissen überein.