The main goal of this work is to model size effects, as they occur in materials with an intrinsic microstructure at the consideration of specimens that are not by orders larger than this microstructure. The micromorphic continuum theory as a generalized continuum theory is well suited to account for the occuring size effects. Thereby additional degrees of freedoms capture the independent deformations of these microstructures, while they provide additional balance equation. In this thesis, the deformational and configurational mechanics of the micromorphic continuum is exploited in a finite-deformation setting. A constitutive and numerical framework is developed, in which also the material-force method is advanced. Furthermore the multiscale modelling of thin material layers with a heterogeneous substructure is of interest. To this end, a computational homogenization framework is developed, which allows to obtain the constitutive relation between traction and separation based on the properties of the underlying micromorphic mesostructure numerically in a nested solution scheme. Within the context of micromorphic continuum mechanics, concepts of both gradient and micromorphic plasticity are developed by systematically varying key ingredients of the respective formulations.
The present thesis is concerned with the simulation of the loading behaviour of both hybrid lightweight structures and piezoelectric mesostructures, with a special focus on solid interfaces on the meso scale. Furthermore, an analytical review on bifurcation modes of continuum-interface problems is included. The inelastic interface behaviour is characterised by elastoplastic, viscous, damaging and fatigue-motivated models. For related numerical computations, the Finite Element Method is applied. In this context, so-called interface elements play an important role. The simulation results are reflected by numerous examples which are partially correlated to experimental data.
Tire-soil interaction is important for the performance of off-road vehicles and the soil compaction in the agricultural field. With an analytical model, which is integrated in multibody-simulation software, and a Finite Element model, the forces and moments generated on the tire-soil contact patch were studied to analyze the tire performance. Simulations with these two models for different tire operating conditions were performed to evaluate the mechanical behaviors of an excavator tire. For the FE model validation a single wheel tester connected to an excavator arm was designed. Field tests were carried out to examine the tire vertical stiffness, the contact pressure on the tire – hard ground interface, the longitudinal/vertical force and the compaction of the sandy clay from the test field under specified operating conditions. The simulation and experimental results were compared to evaluate the model quality. The Magic Formula was used to fit the curves of longitudinal and lateral forces. A simplified tire-soil interaction model based on the fitted Magic Formula could be established and further applied to the simulation of vehicle-soil interaction.
On the one hand, in the world of Product Data Technology (PDT), the ISO standard STEP (STandard for the Exchange of Product model data) gains more and more importance. STEP includes the information model specification language EXPRESS and its graphical notation EXPRESS-G. On the other hand, in the Software Engineering world in general, mainly other modelling languages are in use - particularly the Unified Modeling Language (UML), recently adopted to become a standard by the Object Management Group, will probably achieve broad acceptance. Despite a strong interconnection of PDT with the Software Engineering area, there is a lack of bridging elements concerning the modelling language level. This paper introduces a mapping between EXPRESS-G and UML in order to define a linking bridge and bring the best of both worlds together. Hereby the feasibility of a mapping is shown with representative examples; several problematic cases are discussed as well as possible solutions presented.
In contrast to the spatial motion setting, the material motion setting of continuum mechanics is concerned with the response to variations of material placements of particles with respect to the ambient material. The material motion point of view is thus extremely prominent when dealing with defect mechanics to which it has originally been introduced by Eshelby more than half a century ago. Its primary unknown, the material deformation map is governed by the material motion balance of momentum, i.e. the balance of material forces on the material manifold in the sense of Eshelby. Material (configurational) forces are concerned with the response to variations of material placements of 'physical particles' with respect to the ambient material. Opposed to that, the common spatial (mechanical) forces in the sense of Newton are considered as the response to variations of spatial placements of 'physical particles' with respect to the ambient space. Material forces as advocated by Maugin are especially suited for the assessment of general defects as inhomogeneities, interfaces, dislocations and cracks, where the material forces are directly related to the classical J-Integral in fracture mechanics, see also Gross & Seelig. Another classical example of a material - or rather configurational - force is emblematized by the celebrated Peach-Koehler force, see e.g. the discussion in Steinmann. The present work is mainly divided in four parts. In the first part we will introduce the basic notions of the mechanics and numerics of material forces for a quasi-static conservative mechanical system. In this case the internal potential energy density per unit volume characterizes a hyperelastic material behaviour. In the first numerical example we discuss the reliability of the material force method to calculate the vectorial J-integral of a crack in a Ramberg-Osgood type material under mode I loading and superimposed T-stresses. Secondly, we study the direction of the single material force acting as the driving force of a kinked crack in a geometrically nonlinear hyperelastic Neo-Hooke material. In the second part we focus on material forces in the case of geometrically nonlinear thermo-hyperelastic material behaviour. Therefore we adapt the theory and numerics to a transient coupled problem, and elaborate the format of the Eshelby stress tensor as well as the internal material volume forces induced by the gradient of the temperature field. We study numerically the material forces in a bimaterial bar under tension load and the time dependent evolution of material forces in a cracked specimen. The third part discusses the material force method in the case of geometrically nonlinear isotropic continuum damage. The basic equations are similar to those of the thermo-hyperelastic problem but we introduce an alternative numerical scheme, namely an active set search algorithm, to calculate the damage field as an additional degree of freedom. With this at hand, it is an easy task to obtain the gradient of the damage field which induces the internal material volume forces. Numeric examples in this part are a specimen with an elliptic hole with different semi-axis, a center cracked specimen and a cracked disc under pure mode I loading. In the fourth part of this work we elaborate the format of the Eshelby stress tensor and the internal material volume forces for geometrically nonlinear multiplicative elasto-plasticity. Concerning the numerical implementation we restrict ourselves to the case of geometrically linear single slip crystal plasticity and compare here two different numerical methods to calculate the gradient of the internal variable which enters the format of the internal material volume forces. The two numerical methods are firstly, a node point based approach, where the internal variable is addressed as an additional degree of freedom, and secondly, a standard approach where the internal variable is only available at the integration points level. Here a least square projection scheme is enforced to calculate the necessary gradients of this internal variable. As numerical examples we discuss a specimen with an elliptic inclusion and an elliptic hole respectively and, in addition, a crack under pure mode I loading in a material with different slip angles. Here we focus on the comparison of the two different methods to calculate the gradient of the internal variable. As a second class of numerical problems we elaborate and implement a geometrically linear von Mises plasticity with isotropic hardening. Here the necessary gradients of the internal variables are calculated by the already mentioned projection scheme. The results of a crack in a material with different hardening behaviour under various additional T-stresses are given.
Aim of this work was the extension and development of a coupled Computational Fluid Dynamics (CFD) and population balance model (PBM) solver to enable a simulation aided design of stirred liquid-liquid extraction columns. The principle idea is to develop a new design methodology based on a CFD-PBM approach and verify it with existing data and correlations. On this basis, the separation performance in any apparatus geometry should be possible to predict without any experimental input. Reliable “experiments in silico” (computer calculations) should give the engineer a valuable and user-friendly tool for early design studies at minimal costs.
The layout of extraction columns is currently based on experimental investigations from miniplant to pilot plant and a scale-up to the industrial scale. The hydrodynamic properties can be varied by geometrical adjustments of the stirrer diameter, the stirrer height, the free cross sectional area of the stator, the compartment height as well as the positioning and the size of additional baffles. The key parameter for the liquid–liquid extraction is the yield which is mainly determined at the in- and outlets of the column. Local phenomena as the swirl structure are influenced by geometry changes. However, these local phenomena are generally neglected in state-of-the are design methodologies due to the complex required measurement techniques. A geometrical optimization of the column therefore still results in costs for validation experiments as assembly and operation of the column, which can be reduced by numerical investigations. The still mainly in academics used simulation based layout of counter-current extraction columns is based at the beginning of this work on one dimensional simulations of extraction columns and first three dimensional simulations. The one dimensional simulations are based on experimental derived, geometrical dependent correlations for the axial backmixing (axial dispersion), the hold-up, the phase fraction, the droplet sedimentation and the energy dissipation. A combination of these models with droplet population balance modeling resulted in a description of the complex droplet-droplet interactions (droplet size) along the column height. The three dimensional CFD simulations give local information about the flow field (velocity, swirl structure) based on the used numerical mesh corresponding to the real geometry. A coupling of CFD with population balance modeling further provides information about the local droplet size. A backcoupling of the droplet size with the CFD (drag model) results in an enhancement of the local hydrodynamics (e.g. hold-up, dispersed phase velocity). CFD provided local information about the axial dispersion coefficient of simple geometrical design (e.g. Rotating Disc Contactor (RDC) column). First simulations of the RDC column using a two dimensional rotational geometry combined with population balance modeling were performed and gave local information about the droplet size for different boundary conditions (rotational speed, different column sizes).
In this work, two different column types were simulated using an extended OpenSource CFD code. The first was the RDC column, which were mainly used for code development due to its simple geometry. The Kühni DN32 column is equipped with a six-baffled stirring device and flat baffles for disturbing the flow and requires a full three dimensional description. This column type was mainly used for experimental validation of the simulations due to the low required volumetric flow rate. The Kühni DN60 column is similar to the Kühni DN32 column with slight changes to the stirring device (4-baffles) and was used for scale up investigations. For the experimental validation of the hydrodynamics, laser based measurement techniques as Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF) were used. A good agreement between the experimental derived values for velocity, hold-up and energy dissipation, experimentally derived correlations from literature and the simulations with a modified Euler-Euler based OpenSource CFD code could be found. The experimental derived axial dispersion coefficient was further compared to Euler-Lagrange simulations. The experimental derived correlations for the Kühni DN32 in literature fit to the simulated values. Also the axial dispersion coefficient for the dispersed phase satisfied a correlation from literature. However, due to the complexity of the dispersed phase axial dispersion coefficient measurement, the available correlations gave no distinct agreement to each other.
A coupling of the modified Euler-Euler OpenSource CFD code was done with a one group population balance model. The implementation was validated to the analytical solution of the population balance equation for constant breakage and coalescence kernels. A further validation of the population balance transport equation was done by comparing the results of a five compartment section to the results of the commercial CFD code FLUENT using the Quadrature Method of Moments (QMOM).
For the simulation of the droplet-droplet interactions in liquid-liquid extraction columns, several breakage and coalescence models are available in the literature. The models were compared to each other using the one-group population balance model in Matlab which allows the determination of the minimum stable droplet diameter at a certain energy dissipation. Based on this representation, it was possible to determine the parameters for a specific breakage and coalescence model combination which allowed the simulation of a Kühni miniplant column at different rotational speeds. The resulting simulated droplet size was in very good agreement to the experimental derived droplet size from literature. Several column designs of the DN32 were investigated by changing the compartment height and the axial stirrer position. It could be shown that a decrease of the stirrer position increases the phase fraction inside the compartment. At the same time, the droplet size decreases inside the compartment, which allows a higher mass transfer due to a higher available interfacial area. However, the shifting results in an expected earlier flooding of the column due to a compressed flow structure underneath the stirring device. In a next step, the code was further extended by mass transfer equations based on the two-film theory. Mass transfer coefficient models for the dispersed and continuous phase were investigated for the RDC column design.
A first mass transfer simulation of a full miniplant column was done. The change in concentration was accounted by the mixture density, viscosity and interfacial tension in dependence of the concentration, which affects the calculation of the droplet size. The results of the column simulation were compared to own experimental data of the column. It could be shown that the concentration profile along the column height can be predicted by the presented CFD/population balance/mass transfer code. The droplet size decreases corresponding to the interfacial tension along the column height. Compared to the experimental derived droplet size at the outlet, the simulation is in good agreement.
Besides the occurrence of a mono dispersed droplet size, high breakage may lead to the generation of small satellite droplets and coalescence underneath the stator leads to larger droplets inside the column and hence to a change of the hold-up and of the flooding point. A multi-phase code was extended by the Sectional Quadrature Method of Moment (SQMOM) allowing a modeling of the droplet interactions of bimodal droplet interactions or multimodal distributions. The implementations were in good agreement to the analytical solution. In addition, the simulation of an RDC column section showed the different distribution of the smaller droplets and larger droplets. The smaller droplets tend to follow the continuous phase flow structure and show a higher distribution of inside the column. The larger droplets tend to rise directly through the column and show only a low influence to the continuous phase flow.
The current results strengthen the use of CFD for the layout of liquid-liquid extraction columns in future. The coupling of CFD/PBM and mass transfer using an OpenSource CFD code allows the investigation of computational intensive column designs (e.g. pilot plant columns). Furthermore the coupled code enhances the accuracy of the hydrodynamics simulations and leads to a better understanding of counter-current liquid-liquid extraction columns. The gained correlation were finally used as an input for one dimensional mass transfer simulations, where a perfect fit of the concentration profiles at varied boundary conditions could be obtained. By using the multi-scale approach, the computational time for mass transfer simulations could be reduced to minutes. In future, with increasing computational power, a further extend of the multiphase CFD/SQMOM model including mass transfer equation will provide an efficient tool to model multimodal and multivariate systems as bubble column reactors.
The manuscript divides in 7 chapters. Chapter 2 briefly introduces the reader to the elementary measures of classical continuum mechanics and thus allows to familiarize with the employed notation. Furthermore, deeper insight of the proposed first-order computational homogenization strategy is presented. Based on the need for a discrete representative volume element (rve), Chapter 3 focuses on a proper rve generation algorithm. Therein, the algorithm itself is described in detail. Additionally, we introduce the concept of periodicity. This chapter finalizes by granting multiple representative examples. A potential based soft particle contact method, used for the computations on the microscale level, is defined in Chapter 4. Included are a description of the used discrete element method (dem) as well as the applied macroscopically driven Dirichlet boundary conditions. The chapter closes with the proposition of a proper solution algorithm as well as illustrative representative examples. Homogenization of the discrete microscopic quantities is discussed in Chapter 5. Therein, the focus is on the upscaling of the aggregate energy as well as on the derivation of related macroscopic stress measures. Necessary quantities for coupling between a standard finite element method and the proposed discrete microscale are presented in Chapter 6. Therein, we tend to the derivation of the macroscopic tangent, necessary for the inclusion into the standard finite element programs. Chapter 7 focuses on the incorporation of inter-particle friction. We select to derive a variational based formulation of inter-particle friction forces, founded on a proposed reduced incremental potential. This contribution is closed by providing a discussion as well as an outlook.
The main concern of this contribution is the computational modeling of biomechanically relevant phenomena. To minimize resource requirements, living biomaterials commonly adapt to changing demands. One way to do so is the optimization of mass. For the modeling of biomaterials with changing mass, we distinguish between two different approaches: the coupling of mass changes and deformations at the constitutive level and at the kinematic level. Mass change at the constitutive level is typically realized by weighting the free energy function with respect to the density field, as experimentally motivated by Carter and Hayes  and computationally realized by Harrigan and Hamilton . Such an ansatz enables the simulation of changes in density while the overall volume remains unaffected. In this contribution we call this effect remodeling. Although in principle applicable for small and large strains, this approach is typically adopted for hard tissues, e.g. bone, which usually undergo small strain deformations. Remodeling in anisotropic materials is realized by choosing an appropriate anisotropic free energy function. <br> Within the kinematic coupling, a changing mass is characterized through a multiplicative decomposition of the deformation gradient into a growth part and an elastic part, as first introduced in the context of plasticity by Lee . In this formulation, which we will refer to as growth in the following, mass changes are attributed to changes in volume while the material density remains constant. This approach has classically been applied to model soft tissues undergoing large strains, e.g. the arterial wall. The first contribution including this ansatz is the work by Rodriguez, Hoger and McCulloch . To model anisotropic growth, an appropriate anisotropic growth deformation tensor has to be formulated. In this contribution we restrict ourselves to transversely isotropic growth, i.e., growth characterized by one preferred direction. On that account, we define a transversely isotropic growth deformation tensor determined by two variables, namely the stretch ratios parallel and perpendicular to the characteristic direction. <br> Another method of material optimization is the adaption of the inner structure f a material to its loading conditions. In anisotropic materials this can be realized by a suitable orientation of the material directions. For example, the trabeculae in the human femur head are oriented such that they can carry the daily loads with an optimum mass. Such a behavior can also be observed in soft tissues. For instance, the fibers of muscles and the collagen fibers in the arterial wall are oriented along the loading directions to carry a maximum of mechanical load. If the overall loading conditions change, for instance during a balloon angioplasty or a stent implantation, the material orientation readapts, which we call reorientation. The anisotropy type in biomaterials is often characterized by fiber reinforcement. A particular subclass of tissues, which includes muscles, tendons and ligaments, is featured by one family of fibers. More complex microstructures, such as arterial walls, show two fiber families, which do not necessarily have to be perpendicular. Within this contribution we confine ourselves to the first case, i.e., transversely isotropic materials indicated by one characteristic direction. The reorientation of the fiber direction in biomaterials is commonly smooth and continuous. For transverse isotropy it can be described by a rotation of the characteristic direction. Analogous to the theory of shells, we additionally exclude drilling rotations, see also Menzel . However, the driving force for these reorientation processes is still under discussion. Mathematical considerations promote strain driven reorientations. As discussed, for instance, in Vianello , the free energy reaches a critical state for coaxial stresses and strains. For transverse isotropy, it can be shown that this can be achieved if the characteristic direction is aligned with a principal strain direction. From a biological point of view, depending on the kind of material (i.e. bone, muscle tissue, cartilage tissue, etc.), both strains and stresses can be suggested as stimuli for reorientation. Thus, whithin this contribution both approaches are investigated. <br> In contrast to previous works, in which remodeling, growth and reorientation are discussed separately, the present work provides a framework comprising all of the three mentioned effects at once. This admits a direct comparison how and on which level the individual phenomenon is introduced into the material model, and which influence it has on the material behavior. For a uniform description of the phenomenological quantities an internal variable approach is chosen. Moreover, we particularly focus on the algorithmic implementation of the three effects, each on its own, into a finite element framework. The nonlinear equations on the local and the global level are solved by means of the Newton-Raphson scheme. Accordingly, the local update of the internal variables and the global update of the deformation field are consistently linearized yielding the corresponding tangent moduli. For an efficient implementation into a finite element code, unitized update algorithms are given. The fundamental characteristics of the effects are illustrated by means of some representative numerical simulations. Due to the unified framework, combinations of the individual effects are straightforward.
Within the last decades, a remarkable development in materials science took place -- nowadays, materials are not only constructed for the use of inert structures but rather designed for certain predefined functions. This innovation was accompanied with the appearance of smart materials with reliable recognition, discrimination and capability of action as well as reaction. Even though ferroelectric materials serve smartly in real applications, they also possess several restrictions at high performance usage. The behavior of these materials is almost linear under the action of low electric fields or low mechanical stresses, but exhibits strong non-linear response under high electric fields or mechanical stresses. High electromechanical loading conditions result in a change of the spontaneous polarization direction with respect to individual domains, which is commonly referred to as domain switching. The aim of the present work is to develop a three-dimensional coupled finite element model, to study the rate-independent and rate-dependent behavior of piezoelectric materials including domain switching based on a micromechanical approach. The proposed model is first elaborated within a two-dimensional finite element setting for piezoelectric materials. Subsequently, the developed two-dimensional model is extended to the three-dimensional case. This work starts with developing a micromechanical model for ferroelectric materials. Ferroelectric materials exhibit ferroelectric domain switching, which refers to the reorientation of domains and occurs under purely electrical loading. For the simulation, a bulk piezoceramic material is considered and each grain is represented by one finite element. In reality, the grains in the bulk ceramics material are randomly oriented. This property is taken into account by applying random orientation as well as uniform distribution for individual elements. Poly-crystalline ferroelectric materials at un-poled virgin state can consequently be characterized by randomly oriented polarization vectors. Energy reduction of individual domains is adopted as a criterion for the initiation of domain switching processes. The macroscopic response of the bulk material is predicted by classical volume-averaging techniques. In general, domain switching does not only depend on external loads but also on neighboring grains, which is commonly denoted as the grain boundary effect. These effects are incorporated into the developed framework via a phenomenologically motivated probabilistic approach by relating the actual energy level to a critical energy level. Subsequently, the order of the chosen polynomial function is optimized so that simulations nicely match measured data. A rate-dependent polarization framework is proposed, which is applied to cyclic electrical loading at various frequencies. The reduction in free energy of a grain is used as a criterion for the onset of the domain switching processes. Nucleation in new grains and propagation of the domain walls during domain switching is modeled by a linear kinetics theory. The simulated results show that for increasing loading frequency the macroscopic coercive field is also increasing and the remanent polarization increases at lower loading amplitudes. The second part of this work is focused on ferroelastic domain switching, which refers to the reorientation of domains under purely mechanical loading. Under sufficiently high mechanical loading, however, the strain directions within single domains reorient with respect to the applied loading direction. The reduction in free energy of a grain is used as a criterion for the domain switching process. The macroscopic response of the bulk material is computed for the hysteresis curve (stress vs strain) whereby uni-axial and quasi-static loading conditions are applied on the bulk material specimen. Grain boundary effects are addressed by incorporating the developed probabilistic approach into this framework and the order of the polynomial function is optimized so that simulations match measured data. Rate dependent domain switching effects are captured for various frequencies and mechanical loading amplitudes by means of the developed volume fraction concept which relates the particular time interval to the switching portion. The final part of this work deals with ferroelectric and ferroelastic domain switching and refers to the reorientation of domains under coupled electromechanical loading. If this free energy for combined electromechanical loading exceeds the critical energy barrier elements are allowed to switch. Firstly, hysteresis and butterfly curves under purely electrical loading are discussed. Secondly, additional mechanical loads in axial and lateral directions are applied to the specimen. The simulated results show that an increasing compressive stress results in enlarged domain switching ranges and that the hysteresis and butterfly curves flatten at higher mechanical loading levels.
Discontinuities can appear in different fields of mechanics. Some examples where discontinuities arise are more obvious such as the formation of cracks. Other sources of discontinuities are less apparent such as interfaces between different materials. Furthermore continuous fields with steep gradients can also be considered as discontinuous fields. This work aims at the inclusion of arbitrary discontinuities within the finite element method. Although the finite element method is the most sophisticated numerical tool in modern engineering, the inclusion of discontinuities is still a challenging task. Traditionally within finite the framework of FE methods discontinuities are modeled explicitely by the construction of the mesh. Thus, when a fixed mesh is used, the position of the discontinuity is prescribed by the location of interelement boundaries and not by the physical situation. The simulation of crack growth requires a frequent adaption of the mesh and that can be a difficult and computationally expensive task. Thus a more flexible numerical approach is needed which leads to the mesh-independent representation of the discontinuity. A challenging field where the accurate description of discontinuities is of vital importance is the modeling of failure in engineering materials. The load capacity of a structure is limited by the material strength. If the load limit is exceeded failure zones arise and increase. Representative examples of failure mechanisms are are cracks in brittle materials or shear bands in metals or soils. Failure processes are often accompanied by a strain softening material behaviour (decreasing load carrying capacity with increasing strain at a material point). It is known that the inclusion of strain softening material behaviour within a continuum description requires regularization techniques to preserve the well- posedness of the governing equations. One possibility is the consideration of non-local or gradient terms in the constitutive equations but these approaches require a sufficiently fine discretization in the localization zone, which leads to a high numerical effort. If the extent of the failure zone and the failure process to the point of the development of discrete cracks is considered it seems reasonable to include strong discontinuities. In the framework of fracture mechanics the inclusion of displacement jumps is intuitively comprehensible. However, the modeling of localized failure processes demands the consideration of inelastic material behaviour. Cohesive zone models represent an approach which is especially suited for the incorporation within the finite element framework. It is supposed that cohesive tractions are transmitted between the discontinuity surfaces. These tractions are constitutively prescribed by a phenomenological traction separation law and thus allow for the modeling of different inelastic mechanisms, like micro-crack evolution, initiation of voids, plastic flow or crack bridging. The incorporation of a displacement discontinuity in combination with a cohesive traction separation law leads to a sound model to describe failure processes and crack propagation. Another area where the existence of discontinuities is not as obvious is the occurence of material interfaces, inclusions or holes. The accurate modeling of such internal interfaces is important to predict the mechanical behaviour of components. The present discontinuity is of different nature: the displacement field is continuous but there is a jump in the strains, which is denoted by the expression weak discontinuity. Usually in FE methods material interfaces are taken into account by the mesh construction. But if the structure exhibits multiple inclusions of complex geometry it can be advantageous if the interface does not have to be meshed. And when we look at at problems where the interface moves with time, e. g. phase transformation, the mesh-independent modeling of the weak discontinuities naturally holds major advantages. The greatest challenge in the modeling of discontinuities is their incorporation into numerical methods. The focus of the present work is the development, analysis and application of a finite element approach to model mesh-independent discontinuities. The method shall be robust and flexible to be applicable to both, strong and weak discontinuities.