No description available.

Large-eddy simulation (LES) and unsteady Reynolds-averaged Navier-Stokes (URANS) calculations have been performed to investigate the effects of different mathematical models for scalar variance and its dissipation rate as applied to both a non-reacting bluff-body turbulent flow and an extension to a reacting case. In the conserved scalar formalism, the mean value of a thermo-chemical variable is obtained through the PDF-weighted integration of the local description over the conserved scalar, the mixture fraction. The scalar variance, one of the key parameters for the determination of a presumed β-function PDF, is obtained by solving its own transport equation with the unclosed scalar dissipation rate modelled using either an algebraic expression or a transport equation. The proposed approach is first applied to URANS and then extended to LES. Velocity, length and time scales associated with the URANS modelling are determined using the standard two-equation k-ε transport model. In contrast, all three scales required by the LES modelling are based on the Smagorinsky subgrid scale (SGS) algebraic model. The present study proposes a new algebraic and a new transport LES model for the scalar dissipation rate required by the transport equation for scalar variance, with a time scale consistent with the Smagorinsky SGS model.

This thesis describes the application of the first-order Conditional Moment Closure (CMC) to the autoignition of high-pressure fuel jets, and to piloted and lifted turbulent jet flames using classical and advanced CMC submodels. A Doubly-Conditional Moment Closure (DCMC) formulation is further proposed. In the first study, CMC is applied to investigate the impact of C₂H₆, H₂ and N₂ additives on the autoignition of high-pressure CH₄ jets injected into lower pressure heated air. A wide range of pre-combustion air temperatures is considered and detailed chemical kinetics are employed. It is demonstrated that the addition of C₂H₆ and H₂ does not change the main CH₄ oxidisation pathways. The decomposition of these additives provides additional ignition-promoting radicals, and therefore leads to shorter ignition delays. N₂ additives do not alter the CH₄ oxidisation pathways, however, they reduce the amount of CH₄ available for reaction, causing delayed ignition. It is further shown that ignition always occurs in lean mixtures and at low scalar dissipation rates. The second study is concerned with the modelling of a piloted CH₄/air turbulent jet flame. A detailed assessment of several Probability Density Function (PDF), Conditional Scalar Dissipation Rate (CSDR) and Conditional Velocity (CV) submodels is first performed. The results of two β-PDF-based implementations are then presented. The two realisations differ by the modelling of the CSDR. Homogeneous (inconsistent) and inhomogeneous (consistent) closures are considered. It is shown that the levels of all reactive scalars, including minor intermediates and radicals, are better predicted when the effects of inhomogeneity are included in the modelling of the CSDR. The two following studies are focused on the consistent modelling of a lifted H₂/N₂ turbulent jet flame issuing into a vitiated coflow. Two approaches are followed to model the PDF. In the first, a presumed β-distribution is assumed, whereas in the second, the Presumed Mapping Function (PMF) approach is employed. Fully consistent CV and CSDR closures based on the β-PDF and the PMF-PDF are employed. The homogeneous versions of the CSDR closures are also considered in order to assess the effect of the spurious sources which stem from the inconsistent modelling of mixing. The flame response is analysed over a narrow range of coflow temperatures (Tc). The stabilisation mechanism is determined from the analysis of the transport budgets in mixture fraction and physical spaces, and the history of radical build-up ahead of the stabilisation height. The β-PDF realisations indicate that the flame is stabilised by autoignition irrespective of the value of Tc. On the other hand, the PMF realisations reveal that the stabilisation mechanism is susceptible to Tc. Autoignition remains the controlling stabilisation mechanism for sufficiently high Tc. However, as Tc is decreased, stabilisation is achieved by means of premixed flame propagation. The analysis of the spurious sources reveals that their effect is small but non-negligible, most notably within the flame zone. Further, the assessment of several H₂ oxidation mechanisms show that the flame is very sensitive to chemical kinetics. In the last study, a DCMC method is proposed for the treatment of fluctuations in non-premixed and partially premixed turbulent combustion. The classical CMC theory is extended by introducing a normalised Progress Variable (PV) as a second conditioning variable beside the mixture fraction. The unburnt and burnt states involved in the normalisation of the PV are specified such that they are mixture fraction-dependent. A transport equation for the normalised PV is first obtained. The doubly-conditional species, enthalpy and temperature transport equations are then derived using the decomposition approach and the primary closure hypothesis is applied. Submodels for the doubly-conditioned unclosed terms which arise from the derivation of DCMC are proposed. As a preliminary analysis, the governing equations are simplified for homogeneous turbulence and a parametric assessment is performed by varying the strain rate levels in mixture fraction and PV spaces.

This thesis describes the application of the first-order Conditional Moment Closure (CMC) to the autoignition of high-pressure fuel jets, and to piloted and lifted turbulent jet flames using classical and advanced CMC submodels. A Doubly-Conditional Moment Closure (DCMC) formulation is further proposed. In the first study, CMC is applied to investigate the impact of C₂H₆, H₂ and N₂ additives on the autoignition of high-pressure CH₄ jets injected into lower pressure heated air. A wide range of pre-combustion air temperatures is considered and detailed chemical kinetics are employed. It is demonstrated that the addition of C₂H₆ and H₂ does not change the main CH₄ oxidisation pathways. The decomposition of these additives provides additional ignition-promoting radicals, and therefore leads to shorter ignition delays. N₂ additives do not alter the CH₄ oxidisation pathways, however, they reduce the amount of CH₄ available for reaction, causing delayed ignition. It is further shown that ignition always occurs in lean mixtures and at low scalar dissipation rates. The second study is concerned with the modelling of a piloted CH₄/air turbulent jet flame. A detailed assessment of several Probability Density Function (PDF), Conditional Scalar Dissipation Rate (CSDR) and Conditional Velocity (CV) submodels is first performed. The results of two β-PDF-based implementations are then presented. The two realisations differ by the modelling of the CSDR. Homogeneous (inconsistent) and inhomogeneous (consistent) closures are considered. It is shown that the levels of all reactive scalars, including minor intermediates and radicals, are better predicted when the effects of inhomogeneity are included in the modelling of the CSDR. The two following studies are focused on the consistent modelling of a lifted H₂/N₂ turbulent jet flame issuing into a vitiated coflow. Two approaches are followed to model the PDF. In the first, a presumed β-distribution is assumed, whereas in the second, the Presumed Mapping Function (PMF) approach is employed. Fully consistent CV and CSDR closures based on the β-PDF and the PMF-PDF are employed. The homogeneous versions of the CSDR closures are also considered in order to assess the effect of the spurious sources which stem from the inconsistent modelling of mixing. The flame response is analysed over a narrow range of coflow temperatures (Tc). The stabilisation mechanism is determined from the analysis of the transport budgets in mixture fraction and physical spaces, and the history of radical build-up ahead of the stabilisation height. The β-PDF realisations indicate that the flame is stabilised by autoignition irrespective of the value of Tc. On the other hand, the PMF realisations reveal that the stabilisation mechanism is susceptible to Tc. Autoignition remains the controlling stabilisation mechanism for sufficiently high Tc. However, as Tc is decreased, stabilisation is achieved by means of premixed flame propagation. The analysis of the spurious sources reveals that their effect is small but non-negligible, most notably within the flame zone. Further, the assessment of several H₂ oxidation mechanisms show that the flame is very sensitive to chemical kinetics. In the last study, a DCMC method is proposed for the treatment of fluctuations in non-premixed and partially premixed turbulent combustion. The classical CMC theory is extended by introducing a normalised Progress Variable (PV) as a second conditioning variable beside the mixture fraction. The unburnt and burnt states involved in the normalisation of the PV are specified such that they are mixture fraction-dependent. A transport equation for the normalised PV is first obtained. The doubly-conditional species, enthalpy and temperature transport equations are then derived using the decomposition approach and the primary closure hypothesis is applied. Submodels for the doubly-conditioned unclosed terms which arise from the derivation of DCMC are proposed. As a preliminary analysis, the governing equations are simplified for homogeneous turbulence and a parametric assessment is performed by varying the strain rate levels in mixture fraction and PV spaces.

Large-eddy simulation (LES) and unsteady Reynolds-averaged Navier-Stokes (URANS) calculations have been performed to investigate the effects of different mathematical models for scalar variance and its dissipation rate as applied to both a non-reacting bluff-body turbulent flow and an extension to a reacting case. In the conserved scalar formalism, the mean value of a thermo-chemical variable is obtained through the PDF-weighted integration of the local description over the conserved scalar, the mixture fraction. The scalar variance, one of the key parameters for the determination of a presumed β-function PDF, is obtained by solving its own transport equation with the unclosed scalar dissipation rate modelled using either an algebraic expression or a transport equation. The proposed approach is first applied to URANS and then extended to LES. Velocity, length and time scales associated with the URANS modelling are determined using the standard two-equation k-ε transport model. In contrast, all three scales required by the LES modelling are based on the Smagorinsky subgrid scale (SGS) algebraic model. The present study proposes a new algebraic and a new transport LES model for the scalar dissipation rate required by the transport equation for scalar variance, with a time scale consistent with the Smagorinsky SGS model.

Depuis une dizaine d’années, l’évolution de la puissance des ordinateurs a permis de développer l’utilisation, dans les études d’ingénierie, des simulations 3D CFD (Computational Fluid Dynamics) pour l’étude de l’atmosphère à petite échelle, en particulier pour la dispersion de polluants sur des sites industriels et urbains complexes. Compte tenu de la complexité des domaines à étudier et des ressources de calcul généralement disponibles, ces études sont la plupart du temps réalisées à l’aide des modèles RANS (Reynolds Averaged Navier-Stokes), et particulièrement avec le modèle de fermeture k – e. Différents travaux de validation de l’approche RANS k – e ont mis en évidence quelques limitations à reproduire la dynamique de l’écoulement et de la dispersion dans des configurations géométriques complexes. Le travail de recherche réalisé dans le cadre de cette thèse a pour objectif une caractérisation expérimentale fine de l’écoulement et de la dispersion turbulente autour d’un obstacle bidimensionnel placé dans une couche limite de surface, afin d’évaluer la validité des modèles RANS en vue de leur application pour l’étude de la dispersion atmosphérique.Dans un premier temps, nous avons utilisé des techniques d’anémométrie à fil chaud, d’anémométrie laser Doppler et d’anémométrie par image de particules, pour déterminer le champ de vitesse dans une couche limite de surface rugueuse et autour d’un obstacle bidimensionnel de section carrée. Une attention particulière a été portée sur l’analyse des termes de l’équation évolutive de l’énergie cinétique turbulente (ECT) et sur la détermination de la viscosité turbulente vt. Différentes approches ont également été utilisées pour estimer le taux de dissipation e de l’énergie cinétique turbulente. Nous avons mis en évidence que ces différentes approches fournissent des résultats comparables dans le cas de la couche limite, tandis que seule la technique estimant e comme le résidu de l’ECT est applicable dans le sillage de l’obstacle. De plus, nos mesures ont permis d’évaluer les paramétrisations du modèle k – e et de montrer que la valeur du coefficient Cμ = 0.09 ne semble pas adaptée dans le cas de la couche limite, conduisant à une surestimation de vt, alors que cette valeur apparait plus adaptée dans le cas de l’obstacle. Une étude de sensibilité, portant la détermination de la constante σk du modèle k – e, indique une contribution non négligeable des termes de corrélation entre la vitesse et la pression dans le sillage de l’obstacle.Dans un deuxième temps, nous avons étudié la dispersion d’un scalaire passif, en mesurant les différents moments statistiques de la concentration, au moyen d’un détecteur à ionisation de flamme. Nous avons également déterminé les flux turbulents de masse, par un couplage entre les mesures de vitesse et de concentration, en prenant soin de contrôler les influences réciproques des deux techniques de mesure. Ces mesures nous ont permis de tester la validité de différents modèles de fermeture de l’équation d’advection-diffusion pour estimer les flux dans le sens vertical et dans le sens longitudinal. Nous avons également pu déterminer expérimentalement le coefficient de diffusivité turbulente Dt, nous permettant d’évaluer un nombre de Schmidt turbulent Sct, afin de mettre en évidence que la valeur Sct = 0.7 est adaptée à la majorité des zones étudiées, excepté dans la zone de recirculation induite par l’obstacle. Enfin, nous nous sommes intéressés aux différents termes de l’équation de la variance de la concentration et plus particulièrement à son taux de dissipation. À nouveau, les mesures nous ont permis de tester un modèle de fermeture disponible dans la littérature et de montrer la bonne cohérence entre le modèle et l’expérience. / In the last decades, there has been an increasing use of Computational Fluid Dynamics (CFD)simulations to evaluate the impact of atmospheric pollutants dispersion in within industrial and urban sites. Given the high geometrical complexity of these sites, these simulations are mainly performed adopting a Reynolds Averaged Navier-Stokes (RANS) approach and using k−e closure models. As is well known from previous studies, RANS k−e simulations are affected by some limitations that prevent them correctly reproducing the dynamics of the flow and the pollutant dispersion in complex geometrical configurations. The aim of the PhD is to provide a detailed experimental characterization of the flow and the turbulent dispersion around an idealized two-dimensional obstacle placed within a boundary layer flow. This is subsequently used to analyse the reliability of RANS closure models as predictive tools for the atmospheric dispersion of airborne pollutants. Initially we focus on the flow dynamics of a boundary layer flow developing over a rough wall and in the wake of a 2D obstacle. The velocity field is investigated experimentally by means of different measurement techniques, namely Hot Wire Anemometry (HWA), Laser Doppler Anemometry (LDA) and Stereo-Particle Imagery Velocimetry (PIV). A particular attention was devoted to the estimate of the turbulent viscosity nt as well as of the terms composing the turbulent kinetic energy budget (TKE), including its rate of dissipation e which was determined adopting different approaches. These measurements allowed us to analyse the accuracy of the parameterizations included in a standard k−e closure model. Our analysis show that a value of the coefficient Cμ = 0.09 leads to significant overestimation of nt in a boundary layer flow. Conversely, adopting Cμ = 0.09 provides a good agreement between modeled and direct estimates of nt in the wake of the obstacle. As a second step, we studied the dispersion of a passive scalar emitted by a ground level line source. To that purpose we measured the first four order moments of the concentration probability density function by mean of a flame ionization detector (FID). Furthermore, the coupling of the FID system with the LDA or HWA system allowed us to directly measure the turbulent mass transfer, i.e. the correlation between velocity and concentration fluctuations. The combination of these two techniques was carefully analyzed, in order to quantify eventual mutual disturbances of one measurement technique on the other. The measurements of the velocity/concentration correlations allowed us to determine experimentally the turbulent diffusivity Dt and the turbulent Schmidt number Sct , and therefore to test the accuracy of different closure models of the advection-distribution equation. Our results show that the value of the turbulent Schmidt number is approximately equal to 0.7 in most of the domain, except in the recirculation zone on the wake of the obstacle. Experimental data provide also a complete description of the spatial distribution of the concentration variance, and of the term composing its budget (with a focus on its dissipation). As for the velocity field, we test the reliability of different closure model proposed in the literature of the turbulent mass transfer terms, enlightening the shortcomings of simple gradient-law closer models.

Doppler radar wind profilers are already widely used to measure atmospheric winds throughout the free troposphere and stratosphere. Several methods have been developed to quantify atmospheric turbulence with such radars, but to date they have remained largely un-tested; this thesis presents the first comprehensive validation of one such method. Conventional in-situ measurements of turbulence have been concentrated in the surface layer, with some aircraft and balloon platforms measuring at higher altitudes on a case study basis. Radars offer the opportunity to measure turbulence near continuously, and at a range of altitudes, to provide the first long term observations of atmospheric turbulence above the surface layer. Two radars were used in this study, a Mesosphere-Stratosphere-Troposphere (MST) radar, at Capel Dewi, West Wales, and the Facility for Ground Based Atmospheric Measurements (FGAM) mobile boundary layer profiler. In-situ measurements were made using aircraft and tethered-balloon borne turbulence probes. The spectral width method was chosen for detailed testing, which uses the width of a radar's Doppler spectrum as a measure of atmospheric velocity variance. Broader Doppler spectra indicate stronger turbulence. To obtain Gaussian Doppler spectra (a requirement of the spectral width method), combination of between five and seven consecutive spectra was required. Individual MST spectra were particularly non-Gaussian, because of the sparse nature of turbulence at its observation altitudes. The width of Gaussian fits to the Doppler spectrum were compared to those from the `raw' spectrum, to ensure that non-atmospheric signals were not measured. Corrections for non-turbulent broadening, such as beam broadening, and signal processing, were investigated. Shear broadening was found to be small, and the errors in its calculation large, so no corrections for wind shear were applied. Beam broadening was found to be the dominant broadening contribution, and also contributed the largest uncertainty to spectral widths. Corrected spectral widths were found to correlate with aircraft measurements for both radars. Observing spectral widths over time periods of 40 and 60 minutes for the boundary layer profiler and MST radar respectively, gave the best measure of turbulence intensity and variability. Median spectral widths gave the best average over that period, with two-sigma limits (where sigma is the standard deviation of spectral widths) giving the best representation of the variability in turbulence. Turbulent kinetic energies were derived from spectral widths; typical boundary layer values were 0.13 m 2.s (-2) with a two-sigma range of 0.04-0.25 m 2.s (-2), and peaked at 0.21 m 2.s (-2) with a two-sigma range of 0.08-0.61 m 2.s (-2). Turbulent kinetic energy dissipation rates were also calculated from spectral widths, requiring radiosonde measurements of atmospheric stability. Dissipation rates compared well width aircraft measurements, reaching peaks of 1x10 (-3) m 2.s (-3) within 200 m of the ground, and decreasing to 1-2x10 (-5) m 2.s (-3) near the boundary layer capping inversion. Typical boundary layer values were between 1-3x10 (-4) m 2.s (-3). Those values are in close agreement with dissipation rates from previous studies.

Der Gegenstand der vorliegenden Arbeit ist die Simulation von Betonstrukturen beliebiger Geometrie unter überwiegender Zugbelastung. Die Modellierung erfolgt auf Makroebene als Kontinuum und zur Lösung des mechanischen Feldproblems wird die Finite-Elemente-Methode verwendet. Ein neues Materialmodell für Beton und eine Erweiterung der Bruchenergetischen Regularisierung werden vorgestellt. Die Arbeit ist in zwei Teile gegliedert. Im ersten Teil wird ein lokales, anisotropes Schädigungsmodell abgeleitet, wobei als Schädigungsvariable ein symmetrischer Tensor zweiter Stufe gewählt wird. Die Verwendung einer Normalenregel im Raum der dissipativen Kräfte zur Bestimmung der Schädigungsevolution und die Definition der Schädigungsgrenzflächen im Raum der dissipativen Kräfte gewährleisten die Gültigkeit der Hauptsätze der Thermodynamik und des Prinzips der maximalen Dissipationsrate. Vorteilhaft ist die Symmetrie der Materialtangente, die sich aus diesem Vorgehen ergibt. Eine Formulierung mit drei entkoppelten Schädigungsgrenzflächen wird vorgeschlagen. Eine wichtige Forderung bei der Ableitung des Materialmodells war die Verwendung einer möglichst geringen Anzahl von Materialparametern, welche darüber hinaus aus wenigen Standardversuchen bestimmbar sein sollten. Das Schädigungsmodell enthält als Materialparameter den Elastizitätsmodul, die Querdehnzahl, die Zugfestigkeit und die auf eine Einheitsfläche bezogene Bruchenergie. Im zweiten Teil der Arbeit stehen Lokalisierung und Regularisierung im Fokus der Betrachtungen. Aufgrund der lokalen Formulierung des Materialmodells tritt bei Finite-Elemente Simulationen eine Netzabhängigkeit der Simulationsergebnisse auf. Um dieser Problematik zu begegnen und netzunabhängige Simulationen zu erreichen, werden Regularisierungstechniken angewendet. In dieser Arbeit wird die Bruchenergetische Regularisierung eingesetzt, die durch die Einführung einer äquivalenten Breite in ein lokal formuliertes Stoffgesetz gekennzeichnet ist. Die spezielle Wahl eines Wertes für die äquivalente Breite beruht auf der Forderung, dass in der Simulation die korrekte Bruchenergie je Einheitsfläche für den Bruchprozess verbraucht wird, d.h. die Energiedissipation der Realität entspricht. In vorliegender Arbeit wird die neue These aufgestellt, dass die Energiedissipation nur für den Fall korrekt abgebildet wird, wenn die im Stoffgesetz enthaltene äquivalente Breite in jedem Belastungsinkrement der Breite des Bereiches entspricht, in dem in der Simulation Energie dissipiert wird. In einer Simulation wird in den Bereichen Energie dissipiert, in denen die Schädigung im aktuellen Belastungsinkrement zunimmt. In vorliegender Arbeit werden die energiedissipierenden Bereiche daher als Pfad der Schädigungsrate bezeichnet. Um Erkenntnisse über die Entwicklung des Pfades der Schädigungsrate über den Belastungsverlauf zu erhalten, wurden umfangreiche Untersuchungen anhand von Simulationen eines beidseitig gekerbten Betonprobekörpers unter kombinierter Zug-Schubbeanspruchung durchgeführt, wobei die gewählten Werte für die äquivalente Breite variiert wurden. Es wurde stets eine Diskretisierung mit linearen Verschiebungselementen verwendet, wobei die Bereiche mit zu erwartender Schädigung feiner und regelmäßig mit Elementen quadratischer Geometrie diskretisiert wurden. Die Ergebnisse der Untersuchungen zeigen, dass die Breite des Pfades der Schädigungsrate abhängig ist von der Schädigung am betrachteten Materialpunkt, dem von Schädigungsrichtung und Elementkante eingeschlossenen Winkel, der Elementgröße und den Materialparametern. Um die geforderte Übereinstimmung von äquivalenter Breite und der Breite des Pfades der Schädigungsrate zu erreichen, werden neue Ansätze für die äquivalente Breite vorgeschlagen, die die erwähnten Einflüsse berücksichtigen. Simulationen unter Verwendung der neuen Ansätze für die äquivalente Breite führen zu einer guten Übereinstimmung von äquivalenter Breite und der Breite des Pfades der Schädigungsrate in der Simulation. Die Ergebnisse der Simulationen, wie z.B. Last-Verformungsbeziehung und Rissverläufe, sind netzunabhängig und stimmen gut mit den experimentellen Beobachtungen überein. Basierend auf den gewonnenen Erkenntnissen wird eine Erweiterung der Bruchenergetischen Regularisierung vorgeschlagen: die Adaptive Bruchenergetische Regularisierung. Im abschließenden Kapitel der Arbeit werden mit der vorgeschlagenen Theorie, dem neuen Schädigungsmodell und der Adaptiven Bruchenergetischen Regularisierung, noch zwei in der Literatur gut dokumentierte Versuche simuliert. Die Simulationsergebnisse entsprechen den experimentellen Beobachtungen. / This doctoral thesis deals with the simulation of predominantly tensile loaded plain concrete structures. Concrete is modeled on the macro level and the Finite Element Method is applied to solve the resulting mechanical field problem. A new material model for concrete based on continuum damage mechanics and an extended regularization technique based on the fracture energy approach are presented. The thesis is subdivided into two parts. In the first part, a local, anisotropic damage model for concrete is derived. This model uses a symmetric second-order tensor as the damage variable, which enables the simulation of orthotropic degradation. The validity of the first and the second law of thermodynamics as well as the validity of the principle of maximum dissipation rate are required. Using a normal rule in the space of the dissipative forces, which are the thermodynamically conjugated variables to the damage variables, and the definition of the loading functions in the space of the dissipative forces guarantee their validity. The suggested formulation contains three decoupled loading functions. A further requirement in the derivation of the model was the minimization of the number of material parameters, which should be determined by a small number of standard experiments. The material parameters of the new damage model are the Young’s modulus, the Poisson’s ratio, the tensile strength and the fracture energy per unit area. The second part of the work focuses on localization and regularization. If a Finite Element simulation is performed using a local material model for concrete, the results of the Finite Element simulation are mesh-dependent. To attain mesh-independent simulations, a regularization technique must be applied. The fracture energy approach, which is characterized by introducing a characteristic length in a locally formulated material model, is used as regularization technique in this work. The choice of a value for the characteristic length is founded by the requirement, that the fracture energy per unit area, which is consumed for the fracture process in the simulation, must be the same as in experiment, i.e. the energy dissipation must be correct. In this dissertation, the new idea is suggested that the correct energy dissipation can be only attained if the characteristic length in the material model coincides in every loading increment with the width of the energy-dissipating zone in the simulation. The energy-dissipating zone in a simulation is formed by the integration points with increasing damage and obtains the name: damage rate path. Detailed investigations based on simulations of a double-edge notched specimen under mixed-mode loading are performed with varying characteristic lengths in order to obtain information concerning the evolution of the damage rate path during a simulation. All simulations were performed using displacement-based elements with four nodes. The range with expected damage was always finer and regularly discretized. The results of the simulations show that the width of the damage rate path depends on the damage at the specific material point, on the angle between damage direction and element edges, on the element size and on the material parameters. Based on these observations, new approaches for the characteristic length are suggested in order to attain the coincidence of the characteristic length with the width of the damage rate path. Simulations by using the new approaches yield a sufficient coincidence of the characteristic length with the width of the damage rate path. The simulations are mesh-independent and the results of the simulation, like load-displacement curves or crack paths, correspond to the experimental results. Based on all new information concerning the regularization technique, an extension of the fracture energy approach is suggested: the adaptive fracture energy approach. The validity and applicability of the suggested theory, the new anisotropic damage model and the adaptive fracture energy approach, are verified in the final chapter of the work with simulations of two additional experiments, which are well documented in the literature. The results of the simulations correspond to the observations in the experiments.

Der Gegenstand der vorliegenden Arbeit ist die Simulation von Betonstrukturen beliebiger Geometrie unter überwiegender Zugbelastung. Die Modellierung erfolgt auf Makroebene als Kontinuum und zur Lösung des mechanischen Feldproblems wird die Finite-Elemente-Methode verwendet. Ein neues Materialmodell für Beton und eine Erweiterung der Bruchenergetischen Regularisierung werden vorgestellt. Die Arbeit ist in zwei Teile gegliedert. Im ersten Teil wird ein lokales, anisotropes Schädigungsmodell abgeleitet, wobei als Schädigungsvariable ein symmetrischer Tensor zweiter Stufe gewählt wird. Die Verwendung einer Normalenregel im Raum der dissipativen Kräfte zur Bestimmung der Schädigungsevolution und die Definition der Schädigungsgrenzflächen im Raum der dissipativen Kräfte gewährleisten die Gültigkeit der Hauptsätze der Thermodynamik und des Prinzips der maximalen Dissipationsrate. Vorteilhaft ist die Symmetrie der Materialtangente, die sich aus diesem Vorgehen ergibt. Eine Formulierung mit drei entkoppelten Schädigungsgrenzflächen wird vorgeschlagen. Eine wichtige Forderung bei der Ableitung des Materialmodells war die Verwendung einer möglichst geringen Anzahl von Materialparametern, welche darüber hinaus aus wenigen Standardversuchen bestimmbar sein sollten. Das Schädigungsmodell enthält als Materialparameter den Elastizitätsmodul, die Querdehnzahl, die Zugfestigkeit und die auf eine Einheitsfläche bezogene Bruchenergie. Im zweiten Teil der Arbeit stehen Lokalisierung und Regularisierung im Fokus der Betrachtungen. Aufgrund der lokalen Formulierung des Materialmodells tritt bei Finite-Elemente Simulationen eine Netzabhängigkeit der Simulationsergebnisse auf. Um dieser Problematik zu begegnen und netzunabhängige Simulationen zu erreichen, werden Regularisierungstechniken angewendet. In dieser Arbeit wird die Bruchenergetische Regularisierung eingesetzt, die durch die Einführung einer äquivalenten Breite in ein lokal formuliertes Stoffgesetz gekennzeichnet ist. Die spezielle Wahl eines Wertes für die äquivalente Breite beruht auf der Forderung, dass in der Simulation die korrekte Bruchenergie je Einheitsfläche für den Bruchprozess verbraucht wird, d.h. die Energiedissipation der Realität entspricht. In vorliegender Arbeit wird die neue These aufgestellt, dass die Energiedissipation nur für den Fall korrekt abgebildet wird, wenn die im Stoffgesetz enthaltene äquivalente Breite in jedem Belastungsinkrement der Breite des Bereiches entspricht, in dem in der Simulation Energie dissipiert wird. In einer Simulation wird in den Bereichen Energie dissipiert, in denen die Schädigung im aktuellen Belastungsinkrement zunimmt. In vorliegender Arbeit werden die energiedissipierenden Bereiche daher als Pfad der Schädigungsrate bezeichnet. Um Erkenntnisse über die Entwicklung des Pfades der Schädigungsrate über den Belastungsverlauf zu erhalten, wurden umfangreiche Untersuchungen anhand von Simulationen eines beidseitig gekerbten Betonprobekörpers unter kombinierter Zug-Schubbeanspruchung durchgeführt, wobei die gewählten Werte für die äquivalente Breite variiert wurden. Es wurde stets eine Diskretisierung mit linearen Verschiebungselementen verwendet, wobei die Bereiche mit zu erwartender Schädigung feiner und regelmäßig mit Elementen quadratischer Geometrie diskretisiert wurden. Die Ergebnisse der Untersuchungen zeigen, dass die Breite des Pfades der Schädigungsrate abhängig ist von der Schädigung am betrachteten Materialpunkt, dem von Schädigungsrichtung und Elementkante eingeschlossenen Winkel, der Elementgröße und den Materialparametern. Um die geforderte Übereinstimmung von äquivalenter Breite und der Breite des Pfades der Schädigungsrate zu erreichen, werden neue Ansätze für die äquivalente Breite vorgeschlagen, die die erwähnten Einflüsse berücksichtigen. Simulationen unter Verwendung der neuen Ansätze für die äquivalente Breite führen zu einer guten Übereinstimmung von äquivalenter Breite und der Breite des Pfades der Schädigungsrate in der Simulation. Die Ergebnisse der Simulationen, wie z.B. Last-Verformungsbeziehung und Rissverläufe, sind netzunabhängig und stimmen gut mit den experimentellen Beobachtungen überein. Basierend auf den gewonnenen Erkenntnissen wird eine Erweiterung der Bruchenergetischen Regularisierung vorgeschlagen: die Adaptive Bruchenergetische Regularisierung. Im abschließenden Kapitel der Arbeit werden mit der vorgeschlagenen Theorie, dem neuen Schädigungsmodell und der Adaptiven Bruchenergetischen Regularisierung, noch zwei in der Literatur gut dokumentierte Versuche simuliert. Die Simulationsergebnisse entsprechen den experimentellen Beobachtungen. / This doctoral thesis deals with the simulation of predominantly tensile loaded plain concrete structures. Concrete is modeled on the macro level and the Finite Element Method is applied to solve the resulting mechanical field problem. A new material model for concrete based on continuum damage mechanics and an extended regularization technique based on the fracture energy approach are presented. The thesis is subdivided into two parts. In the first part, a local, anisotropic damage model for concrete is derived. This model uses a symmetric second-order tensor as the damage variable, which enables the simulation of orthotropic degradation. The validity of the first and the second law of thermodynamics as well as the validity of the principle of maximum dissipation rate are required. Using a normal rule in the space of the dissipative forces, which are the thermodynamically conjugated variables to the damage variables, and the definition of the loading functions in the space of the dissipative forces guarantee their validity. The suggested formulation contains three decoupled loading functions. A further requirement in the derivation of the model was the minimization of the number of material parameters, which should be determined by a small number of standard experiments. The material parameters of the new damage model are the Young’s modulus, the Poisson’s ratio, the tensile strength and the fracture energy per unit area. The second part of the work focuses on localization and regularization. If a Finite Element simulation is performed using a local material model for concrete, the results of the Finite Element simulation are mesh-dependent. To attain mesh-independent simulations, a regularization technique must be applied. The fracture energy approach, which is characterized by introducing a characteristic length in a locally formulated material model, is used as regularization technique in this work. The choice of a value for the characteristic length is founded by the requirement, that the fracture energy per unit area, which is consumed for the fracture process in the simulation, must be the same as in experiment, i.e. the energy dissipation must be correct. In this dissertation, the new idea is suggested that the correct energy dissipation can be only attained if the characteristic length in the material model coincides in every loading increment with the width of the energy-dissipating zone in the simulation. The energy-dissipating zone in a simulation is formed by the integration points with increasing damage and obtains the name: damage rate path. Detailed investigations based on simulations of a double-edge notched specimen under mixed-mode loading are performed with varying characteristic lengths in order to obtain information concerning the evolution of the damage rate path during a simulation. All simulations were performed using displacement-based elements with four nodes. The range with expected damage was always finer and regularly discretized. The results of the simulations show that the width of the damage rate path depends on the damage at the specific material point, on the angle between damage direction and element edges, on the element size and on the material parameters. Based on these observations, new approaches for the characteristic length are suggested in order to attain the coincidence of the characteristic length with the width of the damage rate path. Simulations by using the new approaches yield a sufficient coincidence of the characteristic length with the width of the damage rate path. The simulations are mesh-independent and the results of the simulation, like load-displacement curves or crack paths, correspond to the experimental results. Based on all new information concerning the regularization technique, an extension of the fracture energy approach is suggested: the adaptive fracture energy approach. The validity and applicability of the suggested theory, the new anisotropic damage model and the adaptive fracture energy approach, are verified in the final chapter of the work with simulations of two additional experiments, which are well documented in the literature. The results of the simulations correspond to the observations in the experiments.