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<p>In this work we propose and validate a computational method for reconstructing constitutive relations (material properties) in complex multiphysics phenomena based on incomplete and noisy measurements which is applicable to different problems arising in nonequilibrium thermodynamics and continuum mechanics. The parameter estimation problem is solved as PDE–constrained optimization using a gradient–based technique in the optimize–then–discretize framework. The reconstructed material properties taken as an example here are the transport coefficients characterizing diffusion processes such as the viscosity and the thermal conductivity, and we focus on problems in which these coefficients depend on the state variables in the system. The proposed method allows one to reconstruct a smooth constitutive relation defined over a broad range of the dependent variable. This research is motivated by questions arising in the computational analysis and optimization of advanced welding processes which involves modelling complex alloys in the liquid phase at high temperatures.</p> / Doctor of Philosophy (PhD)

The apoptosis and dysfunction of vascular smooth muscle cells in the human descending thoracic aorta is often associated with cardiovascular diseases like aortic dissection and aneurysms. Knowledge of the mechanical effects of contractile smooth muscle cells plays a crucial role in the understanding these potentially lethal conditions. Located in the medial layer, vascular smooth muscle cells are arranged in the so-called herringbone pattern. In regards to the mechanics of the aorta, the consequences of this type of anisotropy have not been fully discussed in literature so far. In this end, a novel hyperelastic constitutive law is proposed which accounts for the dispersive, transmural alignment of vascular smooth muscle cells and their characteristic length-tension behaviour. The model is calibrated with experimental data and is applied to the simulation of an aortic ring under in vivo conditions. By approximating the geometry of the aorta as a layered, thick-walled cylinder, the corresponding quasistatic, mechanical boundary value problem is solved semi-analytically. It is shown that the herringbone pattern induces shear deformation and equalises the normal stress gradients in the aortic wall. Since arterial vessels are able to actively adapt and alter the alignment and activity of smooth muscle cells, the existence of the herringbone pattern is in accordance with Fung's principle of optimal operation.

In this study, two major internal erosion initiation processes, suffusion and concentrated leak mechanisms, which lead to both defect formation in a dam’s body and its foundation and high throughflow in dams subjected to internal erosion were studied. This understanding has the potential to facilitate numerical modelling and expedite dam safety assessment studies.  The throughflow properties of coarse rockfill material were studied by; analysing filed pump test data, performing extensive laboratory experiments with a large-scale apparatus and numerically simulating the three-dimensional flow through coarse rock materials, replicating the material used in the laboratory experiments. Results from the tests demonstrate that the parameters of the nonlinear momentum equation of the flow depend on the Reynolds number for pore Reynolds numbers lower than 60000.  Numerical studies were also carried out to conduct numerical experiments. By applying a Lagrangian particle tracking method, a model for estimating the lengths of the flow channels in the porous media was developed.  The shear forces exerted on the coarse particles in the porous media were found to be significantly dependent on the inertial forces of the flow. Suffusion and concentrated leak mechanisms were also studied by means of laboratory experiments to develop a theoretical framework for continuum-based numerical modelling. An erosion apparatus was designed and constructed with the capability of applying hydraulic and mechanical loading. Results were then used to develop constitutive laws of the soil erosion as a function of the applied hydromechanical load for both suffusion and concentrated leak mechanisms. Both the initiation and mass removal rate of were found to be dependent on the soil in-situ stresses. A three-dimensional electrical-resistivity-based tomography method was also adopted for the internal erosion apparatus and was found to be successful in visualising the porosity evolution due to suffusion. / <p>QC 20161006</p>

In der vorliegenden Arbeit werden kontinuumsmechanische Probleme mit nichtmateriellen Randbedingungen untersucht. Randbedingungen gelten dabei als nichtmateriell, wenn sie im Zeitverlauf nicht ein und demselben materiellen Punkt zugeordnet werden können. Die Erweiterung der klassischen kontinuumsmechanischen Feldgleichungen um solche Randbedingungen erfolgt unter Anwendung einer Arbitrary-LAGRANGE-EULER-Kinematik. Hierbei wird eine Notation entwickelt, bei der Feldgrößen und Operatoren ihre jeweilige Platzierung eindeutig zugeordnet wird. Insbesondere in Hinblick auf eine konsistente Darstellung von Ableitungsoperatoren werden die Vorteile dieser Schreibweise dargelegt. Zur Ermittlung und Untersuchung (semi-)analytischer Lösungen dienen Beispiele eindimensionaler Kontinua, die sich zwei unterschiedlichen Problemklassen zuordnen lassen. In der ersten Problemklasse gelingen analytische Lösungen mit Hilfe eines Integrations- und eines Separationsansatzes für das Modell einer axial unbewegten, schwingenden Saite. Als nichtmaterielle Randbedingungen werden dabei die transversalen Verschiebungen an zwei zeitabhängigen Positionen zu null vorgeschrieben. In der zweiten Problemklasse sind eine Saite sowie ein Seil, die einer vorgegebenen axialen Führungsbewegung unterliegen, Gegenstand der Untersuchung. In diesem Fall sind die zwei vorgegebenen, räumlich festen Verschiebungsrandbedingungen nichtmateriell. Es finden (semi-)analytische Verfahren Anwendung. Die Relativgeschwindigkeit zwischen den Randbedingungen und dem jeweils betrachteten Kontinuum wird dabei als beliebig zeitabhängig angenommen. Eine experimentelle Studie zum Schwingungsverhalten eines Monochords mit nichtmateriellen Randbedingungen vervollständigt die Analyse eindimensionaler Kontinua. Aus den ermittelten (semi-)analytischen Lösungen werden Rückschlüsse auf das Transformationsverhalten der Bewegungsgleichungen dreidimensionaler Kontinua gezogen. Damit sind die entwickelten Methoden in vielen technischen Anwendungen einsetzbar. Als ein wirtschaftlich bedeutendes Beispiel ist die Schwingungsanalyse axial bewegter Papierbahnen in Papierproduktionsmaschinen zu nennen. / Within this work, problems of continuum mechanics with non-material boundary conditions are investigated. Boundary conditions are classified as non-material if they can not be assigned to one and only one material particle over time. The extension of the classical field-equations of continuum mechanics by such boundary conditions is realized by application of Arbitrary-LAGRANGE -E ULER -Kinematics. Therefore a notation, which assigns the particular placement to field quantities and operators, is developed. The advantages of this notation can be identified particularly with regard to a consistent representation of derivative operators. Examples of one-dimensional continua, which can be assigned to different problem categories, are used to determine and investigate (semi-)analytical solutions. In the first category, analytical solutions can be found using an integral and a separation formulation for the model of an axially non-moving, vibrating string. As non-material boundary conditions the transverse displacements at two time-dependent positions are prescribed to zero. A string and a wire, which are moved axially, are investigated within the second problem category. In this case, the prescribed, spatially fixed displacement conditions are non-material. The applied methods are (semi-)analytical. The relative velocity between the boundary conditions and the considered continuum is assumed to be arbitrary time-dependent. An experimental study on the vibration behaviour of a monochord with non-material boundary conditions completes the analysis of one-dimensional continua. Conclusions on the transformation of the equations of motion of three-dimensional continua are derived from the determined (semi-)analytical solutions. For this reason the developed methods are usable in many technical applications. The vibration analysis of axially moving paper sheets in papermaking machines can be stated as an economical important example.

Gegenstand dieser Arbeit ist die Bereitstellung eines geeigneten Simulationswerkzeuges für die numerische Untersuchung der beim Rollkontakt zwischen Rad und Schiene auftretenden Phänomene. Hierbei liegt das Hauptaugenmerk auf der kontinuumsmechanischen Formulierung des mechanischen Feldproblems kontaktierender Körper sowie dessen numerischer Lösung mittels der Finite-Element-Methode. Zur Reduzierung des bei der Simulation von Rollkontakt aus der notwendigen sehr feinen Diskretisierung der Kontaktgebiete resultierenden numerischen Aufwandes wird eine relativkinematische Beschreibung herangezogen. Diese gemischte LAGRANGE-EULER-Betrachtungsweise beruht auf der Zerlegung der Bewegung in einen Starrkörperanteil und eine dazu relative Deformation. Die Herleitung der Bewegungsgleichung für das Kontaktproblem erfordert die relativkinematische Formulierung der kontinuumsmechanischen Grundgleichungen, d.h. der Bilanzgleichungen sowie der konstitutiven Beziehungen. Eine geeignete Kontaktmechanik einschließlich der Berücksichtigung des Kontakts rauer Oberflächen und veränderlicher Kontaktrandbedingungen ist ebenfalls notwendig. Die physikalische Einbindung der Körper in die Umgebung erfolgt über NEUMANNsche und DIRICHLETsche Randbedingungen. Auf dieser Basis können die Bewegungsgleichungen der Elastomechanik hergeleitet werden, welche sich jedoch einer analytischen Lösung verschließen. Somit werden sie in ihrer schwachen Form im integralen Mittel formuliert, was der Anwendung des Prinzips der virtuellen Verschiebungen als Ausgangspunkt für die numerische Lösung entspricht. Die rechentechnische Umsetzung erfordert die inkrementelle und diskrete Formulierung der Bewegungsgleichungen unter besonderer Beachtung der Trägheits-und Kontaktterme, wobei auf die Unterscheidung zwischen Haften und Gleiten beim Tangentialkontakt besonderes Augenmerk gelegt wird. Die numerische Lösung des Finite-Element-Gleichungssystems liefert den aktuellen Beanspruchungszustand zweier Körper im Rollkontakt. Die Funktionsfähigkeit der entwickelten Algorithmen wird abschließend anhand aussagekräftiger Beispielrechnungen zum statischen Kontakt und zum stationären Rollkontakt demonstriert, deren Ergebnisse gute Übereinstimmung mit analytischen Vergleichslösungen, soweit verfügbar, aufweisen. / Scope of this work is the preparation of a suitable simulation tool for the numerical investigation of rolling contact phenomena. The main focus lies on the continuum–mechanical formulation of the mechanical field problem of contacting bodies and its numerical solution within the framework of the Finite Element Method. For reducing the numerical effort in rolling contact simulation, induced by the necessity of a very fine discretization within the expected contact area, a relative–kinematical description is utilized. This arbitrary LAGRANGian–EULERian approach is based upon the decomposition of the total motion into a rigid body motion and a superimposed deformation. The derivation of the equation of motion for the contact problem requires the relative–kinematical formulation of the continuum–mechanical fundamental equations, i. e. the balance equations and the constitutive relations. A suitable contact model including the contact of rough surfaces and varying contact boundary conditions is also necessary. The physical embedding into the environment is accomplished by NEUMANN and DIRICHLET boundary conditions. Based upon that foundation the elastomechanics’ equations of motion are derived, which however can not be solved analytically in general. Hence, the equations of motion are transferred into their weak form by the application of the principle of virtual displacements serving for the numerical solution. The implementation of the problem demands for an incremental and discrete formulation of the equations, especially regarding the terms of inertia and the contact terms. Thereby, special attention has to be paid to the distinction between sticking and sliding within the framework of the tangential contact analysis. The numerical solution of the finite elements’ system of equations provides the state of stress, displacement and contact of two bodies in rolling contact. The reliability of the developed algorithms is finally verified by means of meaningful numerical examples for both static contact and for stationary rolling contact, whereby the numerical results coincide well with available analytical reference solutions.

In der vorliegenden Dissertation wird auf der Grundlage einer kontinuumsmechanischen Herangehensweise die Formulierung eines mechanischen Modells in Verbindung mit der Umsetzung in ein Schalenelement im Rahmen der Finite-Element-Methode zur Simulation des Tragverhaltens geschichteter Flächentragwerksstrukturen unter Berücksichtigung der Schädigungsart Delamination vorgestellt. Grundlage des Mehrschichten-Modells ist die Entwicklung einer geometrisch nichtlinearen oberflächenorientierten Schalentheorie mit schub- und dickenelastischem Verhalten ausgehend von der vollständigen Kinematik einer Multidirektor-Theorie. Der Oberflächenbezug gewährleistet eine auf Kontaktprobleme angepasste mechanische Modellbildung. Innerhalb der Schichten wurde ein Materialgesetz für linear elastisches, orthotropes Material verwendet, dessen Dreidimensionalität durch die Schalenformulierung nicht eingeschränkt wird. Das Hauptaugenmerk der Arbeit liegt auf der Entwicklung eines auf verschiedene Materialien anpassbaren Schichten-Verbundmodells. Das Versagen des Schichtenverbundes – Delamination genannt – wurde durch ein einfaches Spannungskriterium beschrieben. Die Delamination wird durch Modifikation der kinematischen Bedingungen diskret berücksichtigt. Zur Beschreibung des Tragverhaltens nach Ausbildung der Delamination wurde ein als „innerer Kontakt“ bezeichnetes Kontakt-Modell entwickelt, durch das Adhäsion zwischen den Schichten berücksichtigt werden kann. Das vorliegende Schalenmodell kann bei Berücksichtigung von Delamination auf Probleme, in denen kleine Relativverschiebungen zu erwarten sind, für beliebige elastische Materialien angewendet werden. Der Rahmen, in dem diese Arbeit entstand, gab den hauptsächlichen Einsatzbereich, die Simulation von Flächentragwerksstrukturen mit einer Verstärkungsschicht aus textilbewehrtem Feinbeton, vor. / This publication introduces, in a continuum-mechanical approach, the formulation of a mechanical model in connection with the transformation into a shell element using the finite element method for the simulation of the load-bearing behaviour of laminated shell structures thereby considering delamination as a type of damage. This multi-layer model is based upon the development of a geometrically nonlinear surface-related shell theory with shear-elastic behaviour and variable thickness, beginning with the complete kinematics of a multi-director theory. The surface relationship ensures a mechanical modelling which is adaptable for contact problems. A linear-elastic orthotropic material law, whose three-dimensionality is not restricted by the shell formulation, applies within the layers. The main focus of the thesis is on the development of a layer-bond model that can be adjusted for different materials. The debonding of layers – called delamination – is described by a simple stress criterion. Delamination is discretely taken into account by modifying the kinematic conditions. A contact model, called „inner contact“, that can be used to account for adhesion between layers, has been developed to describe the load-bearing behaviour after delamination has occurred. The present shell model is restricted to elastic material behaviour and can preferably be applied to such problems where small relative displacements are expected. The environment, in which this research has been conducted, established the primary of application area, which is the simulation of shell structures within a strengthening layer comprised of textile-reinforced concrete.

The research presented in this work addresses open questions regarding (i) the fundamental understanding of powder compaction, and (ii) the complex mechanical response of particle-binder composites under large deformations. This work thus benefits a broad range of industries, from the pharmaceutical industry and its recent efforts on continuous manufacturing of solid tablets, to the defense and energy industries and the recurrent need to predict the performance of energetic materials. Powder compacts and particle-binder composites are essentially confined particulate systems with significant heterogeneity at the meso (particle) scale. While particle mechanics strategies for modeling evolution of mesoscale microstructure during powder compaction depend on the employed contact formulation to accurately predict macroscopic quantities like punch and die wall pressures, modeling of highly nonlinear, strain-path dependent macroscopic response without a distinctive yield surface, typical of particle-binder composites, requires proper constitutive modeling of these complex deformation mechanisms. Moreover, continued loading of particle-binder composites over their operational life may introduce significant undesirable changes to their microstructure and mechanical properties. These challenges are addressed with a combined effort on theoretical, modeling and experimental fronts, namely, (a) novel contact formulations for elasto-plastic particles under high levels of confinement, (b) a multi-scale experimental procedure for assessing changes in microstructure and mechanical behavior of particle-binder composites due to cyclic loading and time-recovery, and (c) a finite strain nonlinear elastic, endochronic plastic constitutive formulation for particle-binder composites.

Die vorliegende Arbeit stellt eine mehrskalige Modellierungs-Strategie für die Beschreibung magnetorheologischer Elastomere (MRE) vor. Diese ermöglicht die Betrachtung von MRE sowohl auf der Mikroskala, wo die heterogene Mikrostruktur bestehend aus Partikeln und Matrix explizit aufgelöst ist, als auch auf der Makroskala, in welcher das MRE als homogener magnetisch aktiver Körper aufzufassen ist. Auf beiden Skalen kommt dabei eine Kontinuumsformulierung des gekoppelten magneto-mechanischen Feldproblems mit Gültigkeit für finite Deformationen zum Einsatz, wobei die Lösung des Systems partieller Differentialgleichungen mittels der Finite-Elemente-Methode erfolgt. Ausgehend von einer experimentellen Charakterisierung der Konstituenten werden Materialmodelle für die elastomere Matrix sowie Carbonyleisen- und Neodym-Eisen-Bor-Partikel formuliert und mittels dieser Daten kalibriert. Im nächsten Schritt erfolgt die Analyse des effektiven Verhaltens hart- und weichmagnetischer MRE auf Basis von numerischen Homogenisierungen verschiedener mikroskopischer Partikelverteilungen und den Materialmodellen für die Konstituenten. Um weiterhin die effiziente Simulation makroskopischer MRE-Proben und -Bauteile zu ermöglichen, ist daran anschließend die Entwicklung und Parametrisierung eines Makromodells ausgehend von mikroskopisch generierten Datensätzen beschrieben. Mit diesem für isotrope, weichmagnetische und elastische MRE gültigen Modell werden abschließend Simulationen des magnetostriktiven sowie des magnetorheologischen Effektes verschiedener Proben durchgeführt. / In this contribution, a strategy for the multiscale modeling of magnetorheological elastomers (MREs) is presented. It allows to consider these materials on the microscopic scale, where the heterogeneous microstructure consisting of an elastomer matrix and embedded magnetizable particles is explicitly resolved, as well as the macroscopic scale, where the MRE is considered to be a homogeneous magneto-active body. On both scales, a continuum formulation of the coupled magneto-mechanical boundary value problem valid for finite strains is applied. The solution of the system of partial differential equations is calculated by using the finite element method. Starting with an experimental characterization of the individual constituents, constitutive models for the elastomer matrix as well as carbonyl iron and neodymium-iron-boron particles are formulated and adjusted to experimental data. In a next step, basic effective properties of magnetically soft and hard MREs are analyzed by using a computational homogenization scheme, where different geometrical arrangements of the particles on the microscale are considered. In order to enable the efficient simulation of macroscopic MRE samples and components, the developement and parametrization of a macroscopic model based on a microscopically generated data basis is described. With this model which is applicable for isotropic, magnetically soft and elastic MREs, simulations of the magnetostrictive and magnetorheological effects of several sample geometries are performed.

The aim of this thesis is the fast and exact simulation of modern materials like fibre reinforced thermoplastics and fibre reinforced elastomers. These simulations are in the scope of large strain deformations and contain anisotropic and viscoelastic behaviour. The chapter Differential geometry outlines the necessary tensor analysis and differential geometry. We present the weak formulation in the undeformed domain and use Newton’s method to approximate the solution of this formulation, cf. Section 3.1 and Chapter 4, respectively. For the viscoelasticity we use a special ansatz for the internal variable. Next, we compute all necessary derivations for the Newton system, cf. Sections 4.2 and 4.3. We also investigate the symmetry of the material tensors in Section 4.4. Further, we present three methods to improve the convergence of Newton’s method, cf. Section 4.5. With these three methods we are able to consider more problems, compute them faster and in a more robust way. In Chapter 5 we concisely discuss the FEM and show the appearing matrices in detail. The aim of Chapter 6 is the application of the a posteriori error estimator to this complex material behaviour. We present some numerical examples in Chapter 7. In Chapter 8 the problems that arise in the simulation of fibre-reinforced elastomers are analysed and tackled with help of mixed formulations. We derive a symmetric mixed formulation from a reduced form of the energy density. Also, we reformulate the mixed variable for inextensibility to avoid the numerical cancellation in Section 8.3. The Section 8.4 is about a joined mixed formulation to solve problems with inextensible fibres in an incompressible matrix, like fibre-reinforced rubber. The succeeding section Section 8.5 deals with the arising indefinite block matrix system.:Contents Glossary 5 1 Introduction – motivation 13 2 Differential geometry 15 2.1 From parametrisations to the Lagrangian strain 15 2.2 Derivatives of tensors 20 3 Physical foundations 25 3.1 Large Deformation 25 3.1.1 Balance of forces 25 3.1.2 Energy minimisation 28 3.2 Anisotropic energy density 29 3.3 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Newton’s method 37 4.1 Newton system 37 4.2 Anisotropic material tensor 40 4.3 Viscoelastic material tensor 41 4.4 Symmetry of the material tensor 44 4.5 Load steps and line-search 47 4.5.1 Load steps – time steps 47 4.5.2 Backtracking for det ℱ > 0 48 4.5.3 Line search for energy minimisation 49 5 Implementation 53 5.1 Numerical Integration 53 5.2 Finite element discretisation 54 5.3 Voigt notation 56 6 Mesh control 65 7 Numerical results 69 7.1 Semi-analytical example 69 7.2 Cook’s membrane 71 7.2.1 Viscoelastic example 72 7.3 Chemnitz hook – Chemnitzer Haken 72 8 Mixed formulation 75 8.1 Motivation 75 8.2 General considerations 78 8.3 Smooth square root 81 8.4 Joined mixed formulation 84 8.5 Matrix representation 86 9 Conclusion 91 10 Theses 93 11 Appendix 95 11.1 Derivatives of the distortion-invariants with respect to the pseudo invariants 95 Bibliography 101