Browsing by Author "Ehlers, Wolfgang (Prof. Dr.-Ing.)"

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Open Access
Application of a micropolar model to the localization phenomena in granular materials : general model, sensitivity analysis and parameter optimization
(2007) Scholz, Bernd; Ehlers, Wolfgang (Prof. Dr.-Ing.)
In the present work, the localization phenomena of granular material has been analyzed in order to provide a method for the computation of realistic boundary-value problems. The reason for this localization effects are material instabilities caused by the softening behavior of the material, which can be observed at homogeneous material tests. For this purpose, in the first step, the non-polar behavior of granular material is considered. Based on observations from material tests, a non-linear elastic law as well as hardening and softening laws in the context of the elasto-plasticity have been developed. In the next step, based on the described non-polar behavior, inhomogeneous biaxial tests are taken into account for the investigation of the micropolar behavior. As a consequence of the material softening, localization zones, so-called shear bands, appear. For the numerical computation of such localization phenomena, regularization techniques are required, since the standard continuum theory results in an ill-posed problem. Due to the fact that in the biaxial tests a finite thickness of the shear band can be observed, a regularization of the problem is naturally given by the effects due to the microstructure of the material. In case of granular material, the microstructure can be taken into account by application of the Cosserat or micropolar theory, which is applied in this thesis for the computation of localization problems. One of the main problems by use of the micropolar theory is, beside the numerically implementation, the identification of the associated material parameters. Considering inhomogeneous biaxial tests with shear banding, the effect of regularization can be observed by two facts. Firstly, by the gradual decrease of the stress after the appearance of the shear band and, secondly, by the finite thickness of the incoming shear band. Hence, both effects are used for the determination of the additional parameters of the Cosserat theory. Whereas the usage of stress-strain relations,in the context of the literature concerning the parameter identification procedure is a usual method, the consideration of the shear band thickness is a new challenge. Hence, a new method has therefore been developed. In addition to the Cosserat parameters, all the other material parameters, namely, the parameters of the elastic behavior as well as the parameters of the plastic hardening and softening must be identified. In consequence of the high number of material parameters given therewith, the overall identification process was performed by two major steps. Thus, in the first step, homogeneous material tests are applied for the determination of the parameters governing the non-polar behavior. In the second step, the material parameters of the micropolar behavior are determined basing on inhomogeneous biaxial tests, which enable the observation of the incoming shear band by use of the stereophotogrammetry. Generally, the identification procedure yields an inverse problem. For the solution of such problems, a lot of methods in theframework of the non-linear optimization exist. With a view to the high numerical costs for the computation of the boundary-value problem of the biaxial test, the gradient-based SQP method has been applied in this work. The computational cost can be further reduced by the application of the semi-analytical sensitivity analysis, which is also discussed in this work.The stepwise realization of the overall identification process as well as the semi-analytical computation of the sensitivities is demonstrated exemplarily by use of experimental data basing on Hostun sand. Finally, in the present work, with the particle model a completely different approach for the modeling of granular material is discussed which traces back to the Molecular Dynamics. Using this particle model, one is able to simulate the granular microstructure of the material, i.e., the single material grains, directly.Basing on a two-dimensional particle model, the biaxial test has been modeled, with a set of $30,000$ monodisperse, circular particles.With these simulations, the appearance of shear bands were shown and, furthermore, the occurrence of couple stress along the shear band was demonstrated in the same manner as done before with the continuum model. With these results, it has been illustrated that the Cosserat theory is a correct approach for the simulation of localization effects of granular media.
Open Access
Beschreibung und Anwendung eines elastisch-plastischen Materialmodells mit Schädigung für hochporöse Metallschäume
(2002) Droste, Alexander; Ehlers, Wolfgang (Prof. Dr.-Ing.)
Die Nachfrage nach intelligenten Leichtbaukonzepten für alle Sparten der modernen Transportindustrie steigt ständig. Dabei bestimmen Werkstoffe und Werkstofftechnologien in entscheidender Weise die Anwendbarkeit und Wirtschaftlichkeit der Produkte. Primäre Ziele sind die maximale Energie und Rohstoffeinsparung unter zusätzlichen ökologisch und ökonomischen Randbedingungen, beispielsweise niedrige Herstellungskosten, Gewichtsreduktion und Recyclingfähigkeit. Die Forderung nach geringeren Betriebs- und Fertigungskosten, höheren Nutzlasten, verbesserter Umweltverträglichkeit, erhöhter Unfallsicherheit und steigendem Komfort stellen dabei eine nicht immer leicht zu erfüllende Optimierungsaufgabe an die Produkte und an die verwendeten Materialien dar. Aus technischer Sicht werden Formstabilität und Biegesteifigkeit bei gleichzeitiger Dichtereduzierung als eine effektive Lösung dieser Problematik gesehen. Als Ergebnis solcher Forderungen erweitert die Herstellung und Anwendung metallischer Schäume das Spektrum bestehender Materialien wie Wabenstrukturen (Honeycombs) und Kunstoffschäumen, die bereits erfolgreich im großtechnischen Einsatz sind. Insbesondere die Kombination der hohen spezifischen Steifigkeit und der hohen Energiedissipation in 'Crash'-Situationen in Kombination mit einer geringen spezifischen Dichte führen zu einem deutlich erweiterten Eigenschaftsspektrum. Metallische Schäume bieten Vorteile bezüglich Festigkeit, Temperaturbeständigkeit, Umweltverträglichkeit und Wirtschaftlichkeit gegenüber den bestehenden Materialien. Dabei kann speziellen technischen Anforderungen durch neue Verfahren gezielt über den Herstellungsprozeß und das verwendete Matrixmaterial Rechnung getragen werden. In der vorliegenden Arbeit geht es um die numerische Darstellung des Materialverhaltens solcher hochporösen Metallschäume. Hierdurch lassen sich durch numerischen Verfahren Aussagen über das zu erwartende Verhalten eines solchen Materials bei Belastung treffen. Die Herleitung und Anwendung eines finiten Materialmodells mit Schädigung wird vorgestellt.
Open Access
Coupled deformation and flow processes of partially saturated soil : experiments, model validation and numerical investigations
(2013) Avci, Okan; Ehlers, Wolfgang (Prof. Dr.-Ing.)
The main focus of the presented thesis lies on realistic simulations of initial-boundary-value problems (IBVP) in the field of geomechanics using a partially saturated soil. To reach this goal, the deformation and flow behaviour of the partially saturated soil has been intensively analysed based on the topics of the experimental investigation, the constitutive modelling, the parameter identification and model validation. Due to the coupled deformation and flow process of partially saturated soils, accurate experimental investigations of their mechanical and hydraulic behaviour are very complex and sophisticated. For the modelling of the partially saturated soil in the framework of the Theory of Porous Media (TPM), the principle of phase separation is applied. Based on this principle, the mechanical and hydraulic properties of the soil can be simply experimentally investigated in a decoupled manner. That means the mechanical deformation-dependent properties of the test material GEBA sand are experimentally investigated on dry sand via drained triaxial experiments with homogeneous boundary conditions, whereas the hydraulic behaviour is determined with deformation-free experiments. In the context of the soil modelling, the mutual interactions of the individual phases of the soil are taken into account by additional production terms (physical coupling terms). On the basis of these experiments, all required constitutive equations for the triphasic soil model have been derived thermodynamically consistent within the TPM. A cruical point in the matter of material modelling is the experimental investigation of the test material, because false measurements or faulty experimental equipments produce faulty data sets. Based on faulty results, wrong conclusions and assumptions of the material behaviour would be drawn and, thus, would lead to incorrect constitutive modelling approaches. In this regard, in order to ensure a measurement of triaxial tests as error-free as possible, the employed triaxial test setup is optimised concerning measuring error sources. The yield as well as the failure behaviour of dense sand is investigated by use of drained triaxial experiments. Especially, it could be shown through triaxial stress-path-depending compression tests that the standard model approach to limit the hardening of the yield surface by a fixed failure surface is not correct. The experimental results show that the evolution of the yield surface is limited by a variable failure surface depending on the hydrostatic stress state. The good agreement of the simulations with the experiments shows that the presented model approach with a hydrostatic stress-dependent failure surface is promising for realistic simulations of quasi-static IBVP of cohesionless-frictional materials. Constitutive models for materials with an non-linear elastic and a plastic hardening and softening behaviour are complex and own many material parameters. For the identification of the large number of material parameters on the basis of experimental data, the FE tool PANDAS was coupled with the gradient-based SQP optimisation method. The required sensitivities of the fitted quantities of the non-linear restricted optimisation problem with respect to the optimised material parameters are computed semi-analytically. The validation of the triphasic soil model in regard to the coupled deformation and flow processes is carried out by numerical simulation of different slope failure scenarios at the technical scale. The numerical results showed that the presented TPM soil model is well suited to mimic the physical behaviour of multiphasic materials such as partially saturated sand and is also be able to reliably predict slope failure triggered by varying the hydraulic boundary conditions. Additionally, the triphasic soil model is applied for the simulation of natural slope movement and is tested for its capability to predict possible failure mechanisms. This investigation is carried out by numerical FE analysis of the Heumös hillslope in Ebnit (Austria). The triphasic model is further extended to model internal soil-erosion problems. Concerning this, an erosion phase is introduced, which represents the fluidised grains detached from the soil skeleton by the streaming pore water. The objective of the numerical investigation of erosion problems is focused on the analyses of embankment destabilisations induced by loosing solidity due to the internal erosion. In this regard, several numerical examples are presented and discussed.
Open Access
Coupled problems in the mechanics of multi-physics and multi-phase materials
(2015) Zinatbakhsh, Seyedmohammad; Ehlers, Wolfgang (Prof. Dr.-Ing.)
To select a suitable solution strategy is certainly a vital step towards successful simulation of physical phenomena. Considering this, the presented study was set out to explore the important aspects regarding numerical treatment of couple problems stemming from mathematical modelling of coupled multi-field systems. Such coupled systems are observed in a broad range of applications in distinct disciplines. The versatility of the applications has attracted many experts, who have examined coupled problems from different angles. Thus, the mission to fulfil the goals of the present project entailed scrutinising a complex field comprising a vast number of publications incorporating various nomenclatures, classifications, solution strategies, stability analysis procedures, etc. Reviewing the related works, it was noticed that in many cases either a proper explanation of the presented procedures is missing or authors interpret the methods in a rather mathematical way that makes it puzzling for researchers from more application-oriented disciplines. Recognising this flaw, the focus was especially directed towards avoiding abstraction and, instead, presenting an interpretation of the related concepts in an application-friendly style. To this end, a great amount of effort was invested in understanding and explaining the solution strategies proposed for the coupled problems and revealing the relations between them. In particular, several partitioning techniques, including the primal and the dual Schur complement methods, the global and the localised Lagrange-multiplier schemes, and also partitioning methods that follow a staggered integration were investigated and the relations between them were established. Apart from that, the notions related to the stability of the numerical schemes were introduced and their significance was explained. In particular, the terminologies, the stability criteria and the relation between them were reinterpreted in an application-related manner, where the explanations were supported by several examples. This study eventually led to the development of a stability analysis algorithm that can be employed to find the critical grid sizes in different scenarios with minimum difficulty and without any need to solve the whole problem. Furthermore, our attempt in employing the localised Lagrange-multiplier method for partitioning of the surface- coupled problem of fluid-porous-media interaction led to a parallel decoupled solution strategy, comprising a novel application of the concept of modified Eulerian description for describing the motion of a fluid body within moving boundaries, implementation of the penalty method for modelling of fluid as well as porous bodies using the FE-solver PANDAS, and generation of a workflow structure facilitating communication between existing modules of PANDAS.
Open Access
Damage in multi-phasic materials computed with the extended finite-element method
(2012) Rempler, Hans-Uwe; Ehlers, Wolfgang (Prof. Dr.-Ing.)
Material failure is in general a critical situation. It is accompanied with reduced load capacities. Thus, buildings, structures, configurations, etc. tend to collapse and by that loose their designated purpose. This can result in catastrophic consequences. Therefore, a reliable prediction of damage processes is necessary. Consequently, the structural information of crack configurations is of essential interest. The structural information of cracks is usually determined under the assumption of homogeneous, single-phase materials. Certainly, not all materials consist of just one single phase only. Actually, nearly all materials are - more or less - porous materials. Especially grown, biological tissue needs to be regarded as a multi-phasic material. Every biological tissue consists of structural cells, blood vessels, nerves and much more. Tissue rupture, or fracture, respectively, can become a direct hazard to life and living. So, these damage processes are of great interest. Interstitial fluid has to be taken into account when regarding living tissue. As a consequence, damage can result in fluid leaking, sucking, or exchange. This can become a serious danger for, e.g., internal organs. The Theory of Porous Media (TPM) is capable of a macroscopic description of multi-phasic continua. Therein, the information about the underlying micro-structure is obtained by the concept of volume fractions. Thus, the material microstructure can remain unknown. Furthermore, the TPM postulates fully coupled, thermodynamically consistent balance equations for multiple constituents. These characteristics make the TPM the ideal approach to describe biological tissue as immiscible multi-phasic aggregates. Nowadays, it has become common practise to compute material behaviour numerically. Therein, the Finite-Element Method (FEM) has proven to be well-suited for the numerical approximation of differential balance equations. But, the FEM is limited in the simulation of material failure. Thus, the extended FEM (XFEM) was lately developed to overcome this restriction. The XFEM bears the advantage that the finite-element mesh does not need to honour the geometric shape of discontinuities. On this basis, especially when targeting three-dimensional (3-d) problems, efficient finite-elements are crucial for a correct discretisation. Moreover, sophisticated tracking techniques are necessary to exploit the advantage of XFEM damage simulations. With focus on - but not limited to - grown, biological materials, the aim of this monograph is the development of a numerical methodology for the simulation of damage in multi-phasic materials. Therein, the goal is to present a consistent numerical method for the simulation of discontinuous boundary-value problems (BVP) within a 3-d non-linear setting. Representative examples from the field of bio-mechanical problems should reflect the numerical capabilities of the presented method. Due to the generality of the used methods, the presented methodology could also be used in - only at first sight - totally different application areas, e.g., in the context of CO2 sequestration. This thesis is structured into four main chapters. The fundamentals of continuum mechanics are briefly discussed in Chapter 2. Basic continuum-mechanical principles are presented within the framework of the Theory of Porous Media (TPM). A thermodynamically consistent biphasic material model is developed. Constitutive settings describe a fully fluid-saturated, porous solid skeleton. The discussion on boundary-value problems (BVP) closes the considerations on regular continuum mechanics. A brief introduction into the theoretical fundamentals of fracture mechanics is given in the first part of Chapter 3. The second part focuses on the correlation of these fundamentals to a continuum-fracture-mechanical framework. This framework is the basis of the subsequent discussion on the numerical methodology. Furthermore, the investigation of the localisation of discontinuities reveals a crack propagation criterion for the solid skeleton. Chapter 4 presents the numerical implementation of damage processes within the previously developed biphasic continuum-mechanical model. The numerical implementation focuses on the extension of the well-known Finite-Element Method (FEM). The basic principle of the extended Finite-Element Method (XFEM) is first introduced using an example from the field of elasto-inelastic material behaviour. This example yields to an augmented FEM (AugFEM). Sophisticated tracking techniques are presented for the successful numerical simulation of discontinuities. The last main chapter, Chapter 5, presents numerical examples that are computed on the basis of the theoretical aspects of the preceding chapters. A discontinuous 2-d numerical example simulates the fluid exchange within a tear opening of a hydrated tissue cross section. Finally, 3-d numerical examples address the problem of fracture of the human femur.
Open Access
An extended biphasic description of the inhomogeneous and anisotropic intervertebral disc
(2009) Karajan, Nils; Ehlers, Wolfgang (Prof. Dr.-Ing.)
It is the aim of this contribution to develop a finite element model, which is as simple as possible, but at the same time complex enough to capture many of the occurring tissue properties of the intervertebral disc (IVD). In order to better understand these properties from an engineering point of view, the needed basic anatomical knowledge is briefly reviewed in the beginning of this treatise, thereby addressing the lumbar spine with focus on the IVD and its material properties. In particular, the IVD appears as the largest avascular part of the body and its microstructure leads to an electro-chemically active material with anisotropic, inhomogeneous and strongly dissipative behaviour. In the following main part of this work, the complete continuum-mechanical modelling process is extensively discussed as well as the numerical treatment of the resulting governing equations. Starting from the thermodynamically consistent Theory of Porous Media (TPM), two phases and three components are introduced for the description of IVD tissue. In particular, this is the extracellular matrix (solid skeleton) carrying fixed negative charges which is saturated by a pore fluid consisting of a solvent (liquid) as well as anions and cations of a dissolved salt. Following the idea of superimposed continua, an individual motion function is introduced for each of the constituents, whereas the components of the pore fluid are always expressed relative to the deforming solid skeleton. In order to capture the finite kinematics of the inelastic solid skeleton, its deformation gradient is multiplicatively split into inelastic and elastic parts. Next, the materially independent balance equations are derived from the respective master balances and accustomed to the soft biological tissue under study. In order to keep the resulting set of equations as simple as possible, while still keeping the ability to reproduce osmotic effects, an assumption according to Lanir is made. In this context, the tissue is regarded to be always immediately in electro-chemical equilibrium, which allows to describe the electro-chemically active tissue using only an extended biphasic model. Applying van't Hoff's law finally allows to compute the occurring osmotic pressure as a function of the solid displacement. Moreover, in order to characterise the inhomogeneous anisotropic and viscoelastic solid skeleton as well as the viscous pore fluid, several constitutive equations need to be formulated, thereby depending on a thermodynamically admissible set of process variables. Herein, the endangerment of postulating nonphysical constitutive assumptions is avoided by strictly following the restrictions resulting from the evaluation of the entropy inequality. Finally, the chosen constitutive functions of the solid skeleton are based on Ogden-type strain energy functions, which automatically include several simpler material laws. The viscoelastic contribution is based on a generalised Maxwell model which is dominated by the concept of internal variables with linear evolution equations. Finally, the superimposed dissipative effect of the viscous pore fluid is captured using the famous Darcy filter law. As a last step, the applicability of the derived model is proven with realistic computations of the IVD. Herein, the resulting set of governing partial differential equations is discretised in time and space using the finite difference method and the mixed finite element method, respectively. The theoretically introduced material parameters are determined using experimental data as well as material parameters obtained from a vast collection of related literature sources. Since many parameters appear in a coupled manner, their identification is often only possible via inverse computations. Following this, a numerical sensitivity analysis is carried out yielding an indication for the relevant parameters in experiments concerning a motion segment in a short-duration compression-flexion experiment as well as in long-term loading situations. Subsequently, the efficiency of the implementation is demonstrated by a parallel simulation of a lumbar spine segment carried out on 84 processors simultaneously, thereby exhibiting almost one million degrees of freedom.
Open Access
Extended modelling of the multiphasic human brain tissue with application to drug-infusion processes
(2014) Wagner, Arndt; Ehlers, Wolfgang (Prof. Dr.-Ing.)
The brain is the most significant and complex organ of human beings and plays a key role as the control centre of the nervous system. At first glance, the brain seems to be adequately protected against external influences by the rigid skull. However, severe situations may arise if the functionality of the system is compromised within the intracranial cavity itself. For example, a life-threatening situation is caused by solid neoplasm, commonly known as brain tumours. It is obvious that an adequate theoretical modelling of the brain allows a simulation of the occurring biomechanical effects under certain circumstances. This contributes to a profound understanding of the complex processes within the tissue aggregate. Moreover, it provides the possibility to numerically study new medical treatment options and their clinical results in order to support and assist the practising surgeons. However, the biomechanical modelling of the brain is a challenging task. Certainly, this is caused by the patient-specific structural complexity of the three-dimensional anatomical shape of the brain. Moreover, the brain-tissue aggregate is a complex subject of multicomponent nature with electro-chemical properties. In this respect, the tissue characteristics of the brain-matter constituents show significant anisotropic and heterogeneous properties, which require an extended description within the framework of porous materials. In this monograph, the relevant anatomical and physiological aspects of the human brain are briefly summarised. Therein, the main focus is placed on the composition of the brain’s tissue-aggregate and the specific characteristics of its components, as far as needed for the modelling approach. The research rationale is considered by means of tumour diseases and their current treatment options. Related medical-imaging methods are introduced, which enable an insight into living tissues and, therefore, provide the possibility for a patient-specific determination of material parameters. Afterwards, the continuum-mechanical fundamentals, required for the description of the brain matter, are given. Therefore, the basic concept of the Theory of Porous Media (TPM) is applied to the multicomponent tissue-aggregate. In particular, a four-constituent model is investigated, which consists of three immiscible phases and one miscible component. The immiscible phases of the tissue-aggregate are represented by the solid skeleton (i. e. tissue cells and vascular walls), the blood and the overall interstitial fluid. Moreover, the interstitial fluid is constituted by a liquid solvent and a dissolved therapeutic solute (as a result of a medical administration). For this purpose, elements of the Theory of Mixtures are embedded in the standard TPM in order to enable the description of miscible components. Furthermore, the kinematical relations of superimposed constituents are provided, and the balance equations for the overall aggregate as well as for its particular constituents are presented. Based on that, the specific adaptation of the material-independent balance equations by an appropriate constitutive setting is discussed. Therefore, constitutive relations are derived, which describe the characteristic material behaviour of the brain’s tissue. In this regard, the constitutive assumptions for the constituents involved, is examined by means of a thermodynamically consistent framework in terms of an evaluation process of the entropy inequality. On this theoretical basis, the numerical realisation of the developed model is investigated. Therefore, the finite-element method is chosen as a suitable numerical methodology to approximate the solution of the arising set of coupled partial differential equations. For this purpose, the weak formulations of the governing balance relations are discretised in space and time. This numerical part is concluded by the description of the applied monolithic solution strategy. Finally, the application of the derived theoretical and numerical investigations to the human brain is carried out. Therein, capabilities for a patient-specific estimation of required simulation parameters, such as local anisotropic permeabilities and diffusivities, are studied in detail. Next, the possibilities for a customised creation of geometries for the simulation of realistic initial-boundary-value problems are discussed. This finally allows the study of selected numerical examples, demonstrating the feasibility of the presented modelling approach. These examples start with the basic material behaviour of brain tissue and then face the invasive delivery process of therapeutics. In this regard, the therapeutical distribution is shown for realistic geometries of the human brain and, afterwards, a survey on the influence (by a local numerical sensitivity analysis) of several involved simulation parameters is examined.
Open Access
Growth, modelling and remodelling of biological tissue
(2014) Krause, Robert Friedrich; Ehlers, Wolfgang (Prof. Dr.-Ing.)
It is the aim of this work to present an accurate way of modelling, growth and remodelling processes within the framework of continuum biomechanics. Therefore, a multiphase continuum approach is used for the material modelling, where the cells and the extracellular matrix are represented by the solid constituent, and the extracellular liquids are summarised as the fluid constituents. Furthermore, biological processes only occur if the involved cells are sufficiently supported by metabolites (oxygen, vitamins, nutrients, etc.), which are needed for cell metabolic processes. Therefore, a single, non-mechanical quantity is introduced to summarise the metabolites inside the extracellular liquids. This approach provides the necessary thermodynamic restrictions which are evaluated for avascular tumour growth and bone remodelling. To understand the mechanism of growth and remodelling and its consequences, the mechanics as well as the cell dynamics must be considered. In this regard, the Systems Biology aims at investigating the intra- and extracellular signalling pathways, which are involved in the mechanotransduction and trigger the metabolic processes. Using modern computational methods allows for the combination of systems-biological and biomechanical methods within an integrated approach. In this context, Scientific Workflow technology provides an excellent platform to allow for the integration of existing legacy applications from different vendors into an integrative simulation workflow. Therefore, the existing applications are extended or wrapped by a webservice interface, which is then invoked by the workflow instances. This radically new approach allows for a straight-forward merging of computational models from different scales and provides the possibility to further expand each model individually. To reveal the capability of the multi-field growth model, three-dimensional (3-d) simulation examples of both phenomena are presented. This allows the integrated use of separated numerical applications as webservices for each simulation part, which are controlled by a workflow management system. By replacing the phenomenological bone remodelling algorithm with a biologically motivated cell population model, the benefits of this method are demonstrated.
Open Access
Multi-field modelling and simulation of the human hip joint
(2014) Mabuma, Joffrey; Ehlers, Wolfgang (Prof. Dr.-Ing.)
A major biomedical problem is the poor self-healing and regeneration of cartilage. Therefore, cartilage tissues are easily prone to degenerate, leading to pain and working disabilities in middle-aged and older people. In particular, osteoarthritis (OA), a commonly occurring form of cartilage degeneration, is estimated to affect 630 million people worldwide, representing 15% of the global population. In Germany, the Robert-Koch Institute, responsible for national health data reporting, mentions only 1,6% of the under 30-year-old population displaying symptoms of OA. Until the age of 50, the OA prevalence reaches 14,9%, and after the age of 60, one-third of the female population and one-fourth of the male population suffer from OA, respectively. Moreover, between 2003 and 2010, the registered OA cases increased from 22,6% to 27,1% for women and from 15,5% to 17,9% for men. Obviously, the overall increase of OA cases is intimately connected to the rising cost of the healthcare. In 2004, diseases of the muscle-skeleton system occupied the third position in terms of generated costs at 24.46 billion euros, after cardiovascular and digestive diseases. From the costs related to muscle-skeleton diseases in 2012, 6.77 billion euros were incurred for OA, and 39% of the cases of missed work due to OA disease referred to hip-joint OA. In this general context, OA appears as a well-known clinical syndrome related to cartilage degeneration. Still, the mechanisms responsible for OA remain poorly understood. Up to now, the diagnoses are based on a combination of clinical, radiological and anamnestic criteria, mainly concentrating on the statistical occurrence of OA without a strong focus on the characteristics of the patient. One goal of this monograph is to extend the range of available possibilities for clinicians to strengthen their diagnoses. For this purpose, a numerical tool is provided to guarantee a valid representation of the OA occurrence for a real hip-joint anatomy. This process naturally requires a highly complex geometrical and constitutive modelling in order to represent the patient-specific, highly anisotropic and heterogeneous features of the hydrated cartilage tissue. Therefore, a thermodynamically consistent model of soft biological tissue based on the Theory of Porous Media (TPM) is presented and adapted to the specific case of articular cartilage. In this connection, a consistent calibration strategy for the complex computational model is elaborated, which is the second objective of this work. Next, the focus lies on the consideration of realistic boundary conditions applied at the cartilage surface of the femoral head. Based on the contact stresses at the articular surface, a novel visualisation tool is introduced to evaluate the influence of OA during normal and pathological walking processes.
Open Access
Multiphasic flow processes in deformable porous media under consideration of fluid phase transitions
(2008) Graf, Tobias; Ehlers, Wolfgang (Prof. Dr.-Ing.)
Within this contribution, a multiphasic, continuum mechanical model for the description of porous materials with several fluid constituents under consideration of non-isothermal conditions and phase transition processes between liquid and gaseous pore water was presented based on the well-founded framework of the Theory of Porous Media (TPM). The required thermodynamically consistent constitutive relations were derived via an evaluation of the entropy inequality. In the following, this general model was reduced to a triphasic one consisting of a solid, a liquid water and an overall gas phase, which was built by water vapor and air. Furthermore, a special attention was taken on the numerical treatment of multiphasic flow processes. Finally, the presented initial-boundary value problems showed the capability of the discussed model to simulate engineering problems of practical relevance. In particular, concerning the derived continuum mechanical model, each phase of the porous material is governed by its individual temperature. The solid skeleton is assumed to behave like an elasto-viscoplastic, the pore fluids like viscous materials. Furthermore, the solid skeleton as well as the pore liquids are described in a mechanical sense as materially incompressible constituents, whereby their effective densities are only a function of the respective temperatures. The gaseous pore fluid constituents are assumed to behave like ideal gases building together one overall pore gas phase. It could be shown that the ratio between the partial pressure of a gaseous component within the overall gas phase and the overall effective gas pressure is given by the respective molar fraction, whereas the overall effective gas pressure is given by the sum of the partial pressures of the gaseous constituents, which corresponds directly to Dalton's law. The numerical treatment of the presented triphasic model is based on the finite element method (FEM), whereas extended Taylor-Hood elements with quadratic ansatz functions for the solid displacement vector and linear ansatz functions for the pore fluid pressures and saturations as well as the temperatures are used. Furthermore, the special numerical treatment of multiphasic flow processes in porous materials was discussed, which led to the application of a certain stabilization technique to overcome the occurring numerical problems. Finally, the presented multiphasic porous media model was applied to several two- and three-dimensional initial boundary-value problems, where the FE tool PANDAS/M++ was used. Particularly, typical pollutant infiltration and slope failure problems, injection processes of heated pore gas into a water saturated porous material and the so-called heatpipe problem were discussed. It could be shown that the model is capable to describe the strong interaction between the pore fluids flow and the deformable soil matrix as well as the occurring thermal effects and phase transitions processes between liquid and gaseous pore water.
Open Access
Porous media viscoelasticity with application to polymeric foams
(2005) Markert, Bernd; Ehlers, Wolfgang (Prof. Dr.-Ing.)
The goal of this contribution is to merge the advances in porous media theories and the state of the art in single-phase finite viscoelasticity within a well-founded thermodynamical framework. In particular, a thermodynamically consistent constitutive setting is presented where, based on the internal variable concept, an extended Ogden-type viscoelasticity formulation is embedded into the macroscopic Theory of Porous Media (TPM). By focusing on immiscible binary solid-fluid aggregates, essential nonlinearities of the strongly coupled problem are included in the formulation. Thus, the developed biphasic continuum mechanical model accounts for the relevant physical properties stemming from the porous microstructure, the moving and interacting viscous pore fluid (compressible or incompressible), and the directly coupled intrinsic viscoelasticity of the skeleton material itself. In order to demonstrate its suitability, the presented model is especially adapted to the behavior of open-celled polymer foams, as these materials combine all nonlinearities under absolute finite viscoelastic deformations. Finally, after the numerical treatment of the governing model equations through the mixed finite element method (FEM), large strain 3-d simulations reveal the capabilities of the proposed macroscopic formulation and the efficiency of its numerical implementation.
Open Access
Saturated porous media dynamics with application to earthquake engineering
(2012) Heider, Yousef; Ehlers, Wolfgang (Prof. Dr.-Ing.)
The numerical modelling of fluid-saturated porous media dynamics within a continuum-mechanical framework is the ultimate aim of this dissertation. This purpose is achieved by exploiting the Theory of Porous Media (TPM) together with thermodynamically consistent constitutive laws for the material modelling. Additionally, the Finite Element Method (FEM) beside different monolithic or splitting time-stepping schemes are used for the numerical implementation. Within an isothermal and geometrically linear framework, the focus of this monograph is on fully saturated biphasic materials with immiscible phases. This covers the case of materially incompressible solid and fluid aggregates, and the case of a materially incompressible solid but compressible fluid constituent. Moreover, the treatment comprises two important incidents in porous media dynamics, namely, dynamic wave propagation in unbounded domains and liquefaction events.
Open Access
Simulation of charged hydrated porous materials
(2009) Acartürk, Ayhan Yusuf; Ehlers, Wolfgang (Prof. Dr.-Ing.)
It is the goal of this work, to understand the behaviour of charged, hydrated, multiphasic materials and to find a thermodynamically consistent model, in order to perform realistic numerical simulations. Hydrated, porous materials are build up of several constituents, i. e., they consist of a charged solid skeleton and a moving interstitial viscous pore-fluid. The pore-fluid itself is composed of the solvent and the ions of a dissolved salt. Due to this special structure such materials answer with swelling and shrinking processes to electrical fields and to changes of the chemical composition of their environment. These materials occur in both the geomechanics as well as biomechanics. As examples for geomechanical materials clay and shale are mentioned and for biomechanical ones the soft biological tissues, i. e., articular cartilage and the inner core of the intervertebral disk, the Nucleus Pulposus, are mentioned. As described above, these materials have a complicated microstructure. Such a microstructure can be described best by a continuum-mechanical model. Thus, the present thesis is based on the macroscopic Theory of Porous Media (TPM). After a short introduction to the topic of the charged, hydrated porous materials, the fundamental concepts of the TPM are briefly discussed. After this general part valid for all multiphasic continua, the axiomatically introduced balance equations are adapted to the given situation, i. e., the regarded continuum consists firstly of two unmiscible phases, the solid skeleton and the pore-fluid and, secondly, the pore-fluid itself is build up of miscible components, the solvent and the solutes. Moreover, the two strong restrictions on the overall aggregate, i. e. the saturation condition and the electroneutrality condition are incorporated into the entropy inequality by use of Lagrange multiplicators. Subsequently, the entropy inequality is evaluated. By this approach it is guaranteed that the constitutive assumptions do not contradict the entropy inequality. The result is a model, wherein the solid deformation is described on the basis of a Neo-Hookean law, the pore-fluid motion on the basis of an extended Darcy equation, the ion diffusion on the basis of an extended Nernst-Planck equation and the electrical potential by the Poisson equation. The governing set of partial differential equations (PDE) is solved in the frame of the finite element method (FEM). Therefore, the initial and the boundary conditions for the individual partial differential equations (PDG) are discussed. On closer inspection, it is noticeable that for the solution of the PDE different primary variable sets can be chosen. The corresponding sets of equations sets are discussed and the respective pros and cons are put out. Likewise, possible simplifications are discussed, where, by special assumptions, the number of primary variables and, thus, the number of equations to be solved may be reduced. Moreover, depending on the choice of the primary variable set, the boundary conditions depend on the current state of the domain. Such boundary conditions are also known in the area of free surface flows and, also, in fluid-structure interaction. The Dirichlet boundary conditions of these equations need to be fulfilled weakly. Subsequently, simulations are carried out on the basis of the model deduced in this work. In a first step, the different primary variable sets are examined numerically regarding accuracy and stability. It shows up that the primary variable set with most weakly fulfilled boundary conditions is to be preferred. In order to demonstrate the abilities of the model, three-dimensional simulations of a swelling hydrogel cylinder showing finite deformations and a gripper made of an electroactive polymer (EAP) bending under an electric field are shown. Finally, it may be concluded that the model based on the overall pressure and the molar concentration is to be preferred, although this formulation means a higher programming effort. This set of primary variables is numerically more stable and much faster than the other ones.
Open Access
Spannungs-Verformungsverhalten granularer Materialien am Beispiel von Berliner Sand
(2000) Müllerschön, Heiner; Ehlers, Wolfgang (Prof. Dr.-Ing.)
Die makroskopische Beschreibung des komplexen Materialverhaltens einer granularen Struktur erfordert die Berücksichtigung verschiedenster materialspezifischer Eigenschaften. Hierzu wird in der vorliegenden Arbeit ein elasto-plastisches Stoffmodell vorgestellt, basierend auf experimentellen, theoretischen und numerischen Untersuchungen. Im experimentellen Bereich ist die Durchführung von Triaxialversuchen mit geeigneten Randbedingungen zu nennen. Dabei ist die Einhaltung homogener Spannungs- und Verzerrungsfelder im Inneren der Probe zu gewährleisten. Desweiteren wird eine neue Methode zur exakten Messung von sehr kleinen Probenvolumenänderung vorgestellt. Bei der theoretischen Materialmodellierung spielt die Entwicklung eines geeigneten Elastizitätsgesetzes für Reibungsmaterialien eine zentrale Rolle. Dazu wird zuerst eine Literaturrecherche mit einer Beurteilung von vorhandenen Elastizitätsgesetzen durchgeführt. Im Anschluß daran wird ein Vorschlag für eine neue Verzerrungsenergiefunktion gemacht, deren Eigenschaften ausführlich diskutiert werden. Auf der Basis von Ergebnissen aus experimentellen Entlastungsschleifen bei Triaxialversuchen wird eine Parameteridentifikation für das vorgestellte Elastizitätsmodell durchgeführt. Aufbauend auf Vorarbeiten von Ehlers (1993) im Bereich der Plastizitätstheorie werden zur Modellierung des plastischen Deformationsverhaltens vorhandene Konstitutivgleichungen erweitert und spezialisiert. Dazu werden Evolutionsgleichungen zur Beschreibung der Materialverfestigung in Abhängigkeit der akkumulierten plastischen Arbeit eingeführt. Auf der Basis von triaxialen Kompressions- und Extensionsversuchen sowie von hydrostatischen Kompressionsversuchen erfolgt eine Parameteridentifikation der im Modell enthaltenen Materialparameter mit Hilfe der Formulierung von Least-Squares-Funktionalen.
Open Access
Thermomechanical modelling of non-isothermal porous materials with application to enhanced geothermal systems
(Stuttgart : Institut für Mechanik (Bauwesen), Lehrstuhl für Kontinuumsmechanik, Universität Stuttgart, 2016) Koch, David; Ehlers, Wolfgang (Prof. Dr.-Ing.)
Geothermal heat is the energy stored in the Earth that can be considered inexhaustibly over any reasonable time frame. Moreover, The higher temperatures of greater depths can be used to generate electricity and district heating. To use the geothermal heat in an economical way, temperatures of at least 80 to 100 degrees Celsius are necessary. Depending on the geological conditions, the geothermal gradient, that is the change of temperature relative to the depth, has a large range from one up to hundreds degrees Celsius per hundred metres. For an enhanced geothermal system (EGS), the permeability of the low-porosity rock usually has to be increased. This is commonly done by hydraulic stimulations. To operate an EGS, at least two wells must be drilled. After stimulation and, consequently, creation of the reservoir, the cold water is pumped down through the injection well (IW) into the subsurface. It flows through the porous rock and is, thereby, heated at the contact surfaces with the hot rock. Subsequently, the heated water is delivered back to the surface through the production well (PW), where the heat energy of the water can be used directly for heating applications, or can be converted into electricity. Finally, returning the cooled water into the IW closes the loop. Although the system is supplied continuously from the lower layers with heat, the temperature of the reservoir eventually decreases and, thus, the productivity of the reservoir diminishes in the course of years. In order to make predictions about the development of a reservoir in advance, not only the properties of the water and the rock must be accurately described, but also the propagation of heat, in particular the transition of heat from the rock to the water, must be taken into account. The subsurface consists of two components, namely the fractured rock and the water in the cracks. Driven by a pressure gradient, the water flows through the cracks. There are different mechanisms how the heat propagates within the reservoir. Depending on the temperature difference between the components and according to the second law of thermodynamics, the heat exchange takes place at the contact surfaces with the rock. Furthermore, a heterogeneous temperature distribution in each component results in a heat flow by conduction along the temperature gradient in the rock as well as in the water. Moreover, the heat is transported by the flow of water, thus, convectively. The system is further supplied with heat from deeper layers via conduction. The model is capable to describe the mechanical behaviour of the components, which is determined by the deformation of the rock due to a mechanical load, the flow velocity of the water within the cracks and the mechanical interaction between the components. Moreover, and this is the main focus here, the temperatures of the components are different in general and are, therefore, considered individually. Thus, the thermo-mechanical coupling, the heat transfer based on conduction within the components, the convection because of the flow of water, and the heat exchange between the constituents are considered. It is the aim of this contribution to derive a biphasic, thermodynamically consistent porous-media model, where a viscous fluid, namely water, flows through the pores of the elastic and incompressible solid skeleton, namely the rock, while both constituents are under non-isothermal conditions. The continuum-mechanical model is embedded in the framework of the TPM, where the rock and the water are respectively represented by a porous solid and a pore fluid. To achieve this goal, in addition to the kinematic relationships, which directly result from the spatial configurations and their changes over time, the balances of mass, linear momentum and energy are axiomatically introduced. These balance relations build a system of partial differential equations (PDE), which must be closed by means of constitutive assumptions. The constitutive assumptions have to satisfy the constraints arising from the evaluation of the entropy inequality. It ensures the thermodynamically consistency of the formulation. Furthermore, the primary variables for the solution of the PDE system are considered to be the solid displacement, the pore-fluid pressure, and the fluid and solid temperatures. To describe the materially incompressible solid, an elastic material behaviour is assumed. Due to the fact that the deformations of the solid are relatively small, it is sufficient to consider linearised stress and strain tensors. Furthermore, the thermal expansion of the solid constituent is neglected. The viscous fluid is also considered materially incompressible, but its density changes dependent on the temperature. Moreover, quasi-static conditions are considered, because of negligible accelerations of the constituents and of the overall aggregate. The numerical implementation of the model is achieved via the finite-element method (FEM), using the FE-Tool PANDAS. In the context of geothermal systems, the heat transfer via convection is usually dominant. Different techniques were implemented and their performance was investigated to overcome the problem of oscillations due to the convection-dominant system. Finally, an initial-boundary-value problem is solved according to a circulation test in a geothermal plant in Soultz-sous-Forêts (France).
Open Access
Weak or strong : on coupled problems in continuum mechanics
(2010) Markert, Bernd; Ehlers, Wolfgang (Prof. Dr.-Ing.)
The present work aims at giving a concise introduction to the vast field of coupled problems, particularly to those of importance in engineering and physics. Therefore, the common terminology and an appropriate classification of coupled equation systems is presented accompanied by some mathematical and computational issues. Attention is focused on volumetrically coupled multi-field formulations arising from the continuum mechanical treatment of multi-physics problems, but also geometrically coupled problems are addressed. Based on actual problems in the areas of poroelastodynamics, continuum biomechanics, and fluid-saturated porous media in general both the theoretical modeling by means of coupled continuum equations as well as the efficient numerical solution in the context of the finite element method (FEM) are presented and discussed in a problem-oriented fashion.