### Dissertations

in progress

in progress

Numerical simulations represent a powerful tool for the economic and sustainable design as well as the analysis of structures. Numerous steps are necessary to perform successful numerical simulations. These include the experimental investigation of the material behavior, the development of suitable material models, their implementation and validation, and finally the application in structural simulations. The present work deals all of these aspects in the context of quasi-brittle materials, which are encountered in many engineering disciplines.

First, a numerical study is performed for investigating the effect of nonlinear creep of shotcrete on the structural response of a sprayed concrete lining in the

context of the New Austrian Tunneling Method (NATM). The findings from this study motivate the development of an extended time-dependent damage plasticity model for concrete, which is formulated in the framework of the gradient-enhanced continuum theory. The main innovations of the novel model concern nonlinear creep as well as the development of damage due to creep under high sustained loads. On the basis of a comparison between experimental results on nonlinear creep of concrete from the literature and the response of the extended model the predictive capability is assessed at both material point and structural level.

Furthermore, the nonlinear creep behavior of normal strength concrete is studied by means of uniaxial compressive creep tests. Due to the lack of evidence of the effects of the moisture exchange of concrete with the ambiance on nonlinear creep, this influence is systematically investigated for the first time. The results show that the nonlinearity of creep is significantly different between sealed and drying conditions. In addition to the experimental study, it is shown that the extended model is able to predict the creep behavior observed in the experiments up to failure.

Moreover, the present thesis deals with the robust and efficient implementation of generalized continuum models, e.g., gradient-enhanced continuum models, into Finite Element programs. This is of particular important, since for the respective implementations first and higher order partial derivatives of the governing potential functions are required, which are often hard to determine analytically. Thus, a semi-analytical split approach using automatic differentiation is proposed. The superior performance compared to numerical differentiation techniques is shown in 2D and 3D Finite Element studies.

The mechanical behavior of anisotropic quasi-brittle materials, such as layered rock or 3D printed concrete, is characterized by irreversible deformations, pressuredependent hardening plasticity in the pre-peak regime, and strain softening in the post-peak regime accompanied by degradation of the material stiffness due to damaging processes. In addition, the mechanical response of layered quasi-brittle materials depends on the loading direction with respect to the orientation of the layering. The aim of the present thesis is the development of a mathematical constitutive framework suitable for modeling such anisotropic quasi-brittle materials. To achieve this objective, the mechanical behavior of two distinct material types is analyzed. Hence, the thesis is divided into two major parts.

The first part deals with the development of a constitutive model for layered intact rock and rock mass and its application to finite element simulations of deep tunneling. Based on a critical review of popular modeling approaches for considering transversely isotropic behavior in elasto-plastic models, the isotropic rock damage plasticity (RDP) model by Unteregger et al. (2015) is extended to inherent transversely isotropic material behavior. For obtaining mesh-insensitive results in finite element simulations, the softening behavior is regularized by an over-nonlocal implicit gradient-enhancement following Schreter et al. (2018b). The novel model, denoted as TI-RDP model, is implemented on the basis of an implicit time integration algorithm within the framework of the Material Modelling Toolbox Marmot (Neuner et al. (2021)). The TI-RDP model is calibrated and validated by integration point simulations and 3D finite element simulations of triaxial compression tests on Tournemire Shale by Niandou et al. (1997), performed for different inclination angles of the stratification planes with respect to the direction of axial loading and different confining pressures. For investigating the influence of the direction-dependent behavior of layered rock mass on deep tunneling, the TI-RDP model is applied to 2D finite element simulations of tunnel advance.

The second part of this thesis deals with the mathematical description of the mechanical behavior of hardened 3D printed concrete (3DPC), characterized by in- herently direction-dependent material behavior from the printing process. However, in contrast to layered rock structures, the resulting material behavior is orthotropic, regardless of the employed printing technique. To this end, the well-established concrete damage plasticity (CDP) model for conventional concrete by Grassl and Jirásek (2006) is used as a basis and extended to orthotropic material behavior. Thereby, special attention is paid to a straightforward calibration of the additionally introduced material parameters for considering orthotropic plastic behavior by standard laboratory tests. The novel model, denoted as 3DPCDP model, is calibrated by uniaxial compression tests on specimens of cast concrete used for 3D printing and on extrusion-based 3DPC specimens loaded in the principal directions of the material. For validation, three-point bending tests with different loading directions with respect to the printing layers, performed by Shkundalova et al. (2022), are used.

In this cumulative dissertation, experimental investigations on a wet shotcrete mix, which is currently also used in the construction of the Brenner Base Tunnel, are presented. The required test specimens for the test programme were obtained directly on a construction site of the Brenner Base Tunnel in order to determine values that are as close to practice as possible, since the spraying process has a decisive influence on the quality of the shotcrete. The developments of the unconfined compressive strength, the splitting tensile strength and the modulus of elasticity were determined for shotcrete ages from 8 hours to 365 days. In addition, the specific mode I fracture energy and material parameters required for the calibration and validation of multi-physics models were determined, such as the development of the temperature due to hydration, the desorption isotherm and the developments of the porosity and the water content. Furthermore, the shrinkage and creep behaviour of shotcrete was investigated. For this purpose, tests were carried out on sealed and drying test specimens with different sprayed concrete ages at the beginning of the test. The creep tests were carried out for different shotcrete ages at load application and for different stress levels. This extensive data set allows the calibration, validation and further development of material models for shotcrete for use in numerical simulations of tunnel driving according to the new Austrian tunnelling method within the framework of the finite element method.

During specimen extraction and determination of the fracture energy on shotcrete, it became apparent that the currently used wedge splitting method has several weaknesses. Therefore, an improved test set-up for wedge splitting tests to determine the fracture energy was developed. A modified test set-up with claw-shaped load introduction enables simple loading of the specimen without the need for a dedicated groove for load introduction. In addition, a special mechanism allows a slight preloading of the specimen and leads to improved crack propagation.

The design and dimensioning of concrete structures require adequate consideration of the material behaviour of concrete in terms of efficiency and sustainability. Especially for complex engineering structures, shrinkage and creep are of great importance both in the young concrete age and with regard to long-term behaviour.

A promising method for physically based modelling of the material behaviour of concrete is the description of concrete based on the mechanics of porous materials. Here, a multiphase system is considered, which consists of a solid microstructure with a branched pore system. Within the framework of multiphase modelling, the multiphysical properties of concrete can be described realistically according to hygric, thermal, chemical and mechanical phenomena. The development of the material properties due to the hardening of concrete is described as a function of the chemical process of hydration. Shrinkage is modelled according to the underlying hygric mechanism based on capillary stresses. Creep is described according to the viscoelasticity theory.

In the present work, the model formulation corresponding to the state of the art is first evaluated. For this purpose, the data set of a comprehensive test programme is considered, which includes experimental observations for different physical properties of a selected concrete mix. Various weaknesses of the modelling are identified, which are related to the hygro-mechanical behaviour. Subsequently, improved methods for consistent modelling and associated model calibration are developed based on the comprehensive experimental data set. These model extensions allow (i) an improvement of the description for autogenous shrinkage in young concrete age by an extended consideration of the hygro-mechanical coupling, (ii) an improvement of the description of the long-term behaviour of shrinkage by a decoupling of the model formulation of shrinkage and viscous creep, as well as (iii) a consideration of the change of the pore moisture under loading and the load-induced shrinkage related to it. Based on numerical simulations, it is shown that these improvements allow, in particular, a consistent representation of autogenous shrinkage and drying shrinkage as well as of basic creep and drying creep under compressive and tensile loading. Overall, multiphase modelling thus enables a realistic description of the multiphysical material behaviour of concrete.

This work is mainly concerned with predicting the behaviour of concrete at the material level and subsequently of reinforced concrete structures or components under different cyclic loading scenarios. For this purpose, the behaviour of an existing damage plasticity model for monotonic loading of concrete is first further investigated and then extended to a material model for cyclic loading. Empirical models are then proposed to predict the maximum lateral ductility and strengths of reinforced concrete columns because these play an important role in the transfer and distribution of loads respectively. These empirical models are validated against available experimental results in the literature.

In the first part of this paper, the application of a damage plasticity model for concrete proposed by Grassl and Jirásek to predict the nonlinear behaviour of reinforced concrete frames is investigated. Subsequently, in the second part of this work, the above material model is extended to describe the nonlinear cyclic behaviour of concrete under multiaxial loading. The new model is called the Enhanced Concrete Damage Plasticity Model (ECDP Model). Like some of its predecessors, it is based on the coupling of damage and plasticity theory and is characterised by isotropic hardening and softening behaviour, the capture of the development of inelastic strains and stiffness degradation, and the consideration of damage under compressive and tensile stresses. The performance of this material model for predicting the cyclic behaviour of concrete is investigated on the basis of various monotonic and cyclic test results (uniaxial tension and compression tests, biaxial and triaxial compression tests with low values of lateral confinement) and compared with the predictions of the cyclic material models of Lee and Fenves and Grassl et al.

The third and fourth parts of the present work are devoted to the development of empirical models for the prediction of the maximum drift ratio and the shear strengths of reinforced concrete columns under earthquake action, respectively. For this purpose, extensive reliable databases are used and divided into two groups, namely training and test sets, to develop and validate the empirical models. Furthermore, a new definition is presented to consider the influence of the required displacement ductility of columns with rectangular and circular cross-sections on their shear strength. The corresponding lateral ductility models (LDMe) and shear strength models (SSMe) are established for rectangular and circular column cross-sections through regression analyses on the training sets for the different failure modes (axial, bending, shear-bending, shear) of the columns. The accuracy of the proposed LDMs and SSMe is investigated using the test sets and also compared with the predictions of other existing LDMs and SSMe (FEMA 273, 1997; Elwood, 2004; Sezen and Moehle, 2004; Biskinis et al, 2004; Zhu et al, 2007; Pan and Li, 2012; ACI 318, 2014). In addition, the Monte Carlo method is used to further evaluate the SSMe. This can provide significant advantages over existing models.

The age structure of the bridge stock and the steadily increasing traffic volume over the last decades pose a great challenge for the maintenance of the existing bridge structures and make retrofitting measures, such as the expansion of conventional bridges with expansion joints and bearings to integral bridges with an additional reinforcement of the existing deck by means of a concrete layer, particularly important. Due to the time-dependent shrinkage and creep of concrete, the reinforcement method by means of top concrete is characterised by the build-up and release of constraining stresses.

Against this background, the shrinkage and creep behaviour of a concrete representative of top-layer concrete under compressive and tensile stress and taking into account the development of the moisture content is investigated in this dissertation. For this purpose, a suitable test programme is developed that allows a comparison of the results obtained from the individual compression and tensile creep tests with each other. Based on this comparison, a statement can be made, among other things, as to whether the creep of the concrete under compressive stress differs from that under tensile stress, as partly contradictory statements can be found in this respect in the literature.

The effects of different creep under compressive and tensile stress should also be evident in components subjected to bending stress, which is why the creep behaviour of a specimen subjected to bending stress is investigated in a further test programme.

The moisture content in the concrete and its influence on the creep behaviour of the concrete under different stresses are investigated by means of a measuring system based on the measurement of the electrolytic resistance.

The extensive set of measurement data allows, on the one hand, a comparison with the predictions based on the shrinkage and creep models contained in Eurocode 2, Model Code 2010 and in model B4 and, on the other hand, serves as a valuable basis for the calibration and validation of numerical models for the simulation of structural reinforcements using top concrete.

For this dissertation, Dr. Schreter was awarded the FCP Innovation Prize 2018 "for sustainable developments in civil engineering" by FCP - Fritsch, Chiari & Partner ZT GmbH and the Hypo Tirol Bank Dissertation Prize at TU Wien.

In addition, Dr Schreter was awarded the "Award of Excellence 2019" by the Federal Ministry of Education, Science and Research for this dissertation.

Finite element simulations of the excavation of deep tunnels make it possible to predict the deformations occurring during the excavation process, to dimension the necessary support means and to identify critical situations close to the failure state. The constitutive laws of the materials involved, such as the rock mass, are important influencing factors for the predictive capability.

The focus of the investigations is the isotropic damage plasticity model for intact rock and rock mass (RDP model) developed by Unteregger et al. (2015), which is extended by an advanced regularisation method. The RDP model considers linear elasticity and plasticity with the Hoek-Brown failure criterion as well as hardening and softening material behaviour, the latter based on continuum damage theory. The required material parameters for intact rock can be determined from laboratory tests. The extension to the material behaviour of rock is done with an isotropic equivalent continuum model, which considers the influences of discontinuities on the mechanical behaviour by empirical relations depending on geological parameters.

In the present work, a finite element model is developed which represents the excavation of a deep tunnel. A section of the Brenner Base Tunnel located in Innsbruck quartz phyllite is investigated, which was constructed by conventional blasting. During driving, triaxial compression tests on rock samples and displacement measurements were carried out.

A comprehensive numerical study of the tunnel section is presented for the evaluation of the RDP model. Furthermore, the comparison with elasto-plastic rock models with the failure criterion according to Hoek-Brown is carried out. Based on the RDP model, larger deformations and the localisation of deformations in shear bands in the immediate vicinity of the cavity are predicted as the rock begins to deteriorate. This makes it possible to describe the transition from a quasi-continuum to a quasi-discontinuum, which is an indicator of possible failure. Together with the application of an advanced material model for shotcrete, a good agreement of the calculated displacements with the measured data is achieved.

The tunnel simulations carried out show a complex distribution of damaged zones in the rock mass, which is why the softening behaviour of the RDP model is investigated in more detail. In order to obtain mesh-independent results in this context, a suitable regularisation procedure of the RDP model, which is based on the implicit gradient expansion of the damage formulation, is presented. By means of the gradient-extended RDP model, it is possible to represent the mechanical behaviour both objectively with respect to the finite element discretisation and realistically with respect to measured data. These decisive criteria are investigated by means of simulations of wedge splitting tests, triaxial compression and extension tests as well as the excavation of the tunnel section.

For this dissertation, Dr. Neuner was awarded the FCP Innovation Prize 2018 "for sustainable developments in civil engineering" by FCP - Fritsch, Chiari & Partner ZT GmbH at the Vienna University of Technology.

In addition, Dr Neuner was awarded the "Award of Excellence 2018" by the Federal Ministry of Education, Science and Research for this dissertation.

Numerical description and experimental investigation of the material behaviour of shotcrete and application in finite element analyses of deep tunnel excavations.

The present dissertation covers the description of the material behaviour of shotcrete, with a focus on applications in numerical simulations of tunnel driving. For this purpose, a new material model is presented, which is based on three continuum models for normal concrete. These three models include the damage plasticity model according to Grassl and Jirásek, the solidification theory according to Bažant and Prasannan, and the semi-empirical model according to Bažant and Panula to describe the shrinkage behaviour.

In this new model for shotcrete, hereinafter referred to as the SCDP model, the temporal development of the material properties due to hydration, hardening and softening material behaviour due to mechanical loading, plastic deformations, the decrease in material stiffness due to damage, shrinkage and non-linear creep are described. To evaluate the new model, a detailed study is presented, which includes a comparison with two other shotcrete models from the literature, namely the models according to Meschke and Schädlich and Schweiger. This comparison is based both on a validation using data from laboratory tests available in the literature and on a benchmark finite element model of the excavation of a deep-seated tunnel to evaluate the material models at the structural level.

The presented finite element model for a benchmark study is based on a section of the Brenner Base Tunnel for which a measurement programme was carried out to determine the deformations of the surrounding rock during excavation. The comparison of the measured data with the numerical results shows the realistic prediction based on the new SCDP model. Furthermore, a new test programme for shotcrete is presented, which includes the determination of the temporal development of the elastic modulus and the unconfined compressive strength as well as the shrinkage and creep behaviour of young shotcrete. It is shown that the material properties of the investigated shotcrete formulation differ significantly from the properties of the older formulations discussed in the literature. Based on the new test data, the calibration of the investigated shotcrete models is carried out.

This thesis deals with the development and application of a three-phase-model for concrete on the basis of the theory of porous materials. Starting from the microscopic balance equations the macroscopic balance equations of a multi-phase-system are derived. Due to the existing pore structure, caused by hydration, concrete is a porous material and moisture within the pores can be transported and stored. The content and the distribution of moisture inﬂuences the mechanical behaviour of concrete, for example the progress of drying shrinkage strains. By means of a three-phase-model for concrete the coupled hygric, thermal and mechanical behaviour of concrete can be numerical simulated. From the moment of concrete setting the progress of the distribution of moisture and temperature within a concrete member can be calculated with respect to diﬀerent initial and boundary conditions. The dependence of the reaction rate of hydration from the temperature and the moisture is considered. The latter result in the evolution and the spatial distribution of stiﬀness and strength. One important item of this thesis is the realistic calculation of drying shrinkage strains. They are determined with the calculated moisture distribution by means of the concept of eﬀective stress. For the long-time behaviour of concrete not only drying shrinkage strains are of note but also concrete creep, which is determined with the microprestress-solidiﬁcation theory of Bažant [14].

For the developed three-phase-model of concrete a lot of material parameters are necessary. To determine some of them and to validate the three-phase-model tests, which has been carried out at the university of Innsbruck at the unit of Strength of Materials and Structural Analysis in line with a research project, are computed numerical. To validate the creep model publicized test results are used. Bridge structures strengthened by concrete overlays are a ﬁeld of interest for an application of the three-phase-model. Particular the mechanical behaviour of the concrete overlay is investigated, because strains of the concrete overlay are restrained due to the existing concrete structure, which results in restraint stress. By means of the developed three-phase-model of concrete two numerical computations of concrete structures strengthened by concrete overlays are presented. One numerical computation deals with the retroﬁt of a real bridge structure with adding a concrete overlay. Thereby the environmental conditions as temperature, humidity and wind speed have been recorded and are considered as boundary conditions for the numerical computation.

This work deals with the presentation of a constitutive model for isotropic rock and its extension to anisotropic material behaviour. Beside the model definition, the focus is put on the stress update procedures, the identification strategies for the model parameters and the validation of the model by laboratory results.

In the first part a constitutive model for isotropic rock is presented. It is based on the works of Grassl and Jirasek (2006), Valentini (2011) and Unteregger (2015). In this model strain hardening is formulated within the framework of plasticity theory whereas strain softening is formulated within the framework of damage theory. A new, more flexible formulation of the damage evolution law is proposed, leading to a redefinition of the regularization scheme to assure mesh independent results in finite element analyses. Subsequently, an implicit update scheme for the elasto-plastic evolution equations is presented. Based on the results from triaxial tests on Innsbruck quartz phyllite an identification strategy for the model parameters is proposed and the model is validated.

The second part deals with approaches for considering plastic anisotropic material behaviour by means of so-called anisotropic variables. Those scalar variables render the e˙ect of the loading direction with respect to the material orientation and are based on the formulation of fabric tensors. Two approaches, one proposed by Pietruszczak and Mroz (2000, 2001) and the other one proposed by Gao et al. (2010) and Gao and Zhao (2012), are presented, critically evaluated and compared with each other. An identification strategy for the parameters of the anisotropic variables is proposed and parameters are identified for different rock types.

Finally both formulations of the anisotropic variable are used to extend an isotropic Drucker-Prager yield function and the aforementioned constitutive model for isotropic rock to anisotropic material behaviour. For each model formulation a parameter identi-fication strategy is devised and the models are calibrated for Innsbruck quartz phyllite. For both calibrated models meridional and deviatoric sections of the yield function are presented. It is shown that the anisotropic variable according to Pietruszczak and Mroz (2000, 2001) leads to loss of convexity of the yield surface, whereas the anisotropic variable according to Gao et al. (2010) and Gao and Zhao (2012) results in a discontinuous yield surface. This, in turn, leads to loss of convergence of the implicit stress update algorithm and the violation of Drucker’s stability postulate as well as the postulate of maximum plastic dissipation.

The mechanical behaviour of intact rock and fractured rock is of great interest for the production of cavity structures. Numerical methods such as the finite element method allow the complex, three-dimensional system consisting of rock body and support means to be captured, but it is difficult to obtain reliable predictions of the expected mechanical behaviour. One of the main reasons for this is the complex, non-linear constitutive behaviour of the rock and rock mass.

In this dissertation, a powerful three-dimensional constitutive model for rocks and its extension for mountains are proposed in the framework of continuum mechanics. The rock model is based on the combination of plasticity theory with damage mechanics. The developed damage plasticity model for rock allows the modelling of essential features of the mechanical behaviour. The strength is predicted analogously to the Hoek-Brown failure criterion. In addition, the model takes into account non-associated plastic flow and non-linear hardening, non-linear behaviour under predominantly hydrostatic compressive loading, as well as softening and the associated reduction in stiffness. Two different approaches are presented for extending the rock model to model rock masses. Both approaches are based on a rock mass classification in terms of the geological strength index and the disturbance factor introduced by Hoek and Brown.

A parameter identification procedure is presented to determine the material and model parameters for the rock damage plasticity model. The procedure is based on the manual determination of some parameters and the inverse determination of the remaining parameters from uniaxial and triaxial compression tests. In terms of inverse parameter identification, a hybrid optimisation algorithm based on the combination of evolutionary optimisation and gradient-based optimisation is developed. Two different formulations are investigated, starting from the least squares method and a smooth least squares formulation to minimise the deviations between experimental and numerical results. Furthermore, two approaches in terms of simultaneous and sequential determination of the required parameters are investigated.

The parameter identification procedure is verified by re-identifying material and model parameters from synthetic test results generated using the rock model. The validation of the parameter identification procedure is done by determining the material and model parameters (see figure) for different rock types. The validation of the damage plasticity model for rock is carried out using test results that were not used for parameter identification. Finally, the predicted mechanical behaviour according to the two presented approaches for rock modelling is presented and discussed.

In the present work, a multi-phase model for partially saturated soils is implemented in the open-source FE program OpenSees [76]. The code is deployed on a high performance computer system, the efficiency of the program is optimised and tested through a numerical study focusing on numerical simulations of geotechnical problems.

Within the multi-phase formulation, a modified constitutive model for the description of partially saturated soils, originally presented in [50], is applied. It is formulated as a function of two independent stress variables, namely the generalised effective stresses and the capillary stress. It has been successfully used to numerically simulate the behaviour of unsaturated (silty) sandy soils. This constitutive model has been improved in several ways, with the aim of improving stability during the transition from unsaturated to saturated states and extending the range of application from sandy soils to clays of low and medium plasticity.

The non-linear system of equations resulting from the finite element formulation of the multi-phase model is solved by Newton's method using the consistent tangent moduli. Newton's method is extended by a relaxation method and by a line-search algorithm along the directions given by the consistent tangent moduli. The massively parallel direct multi-front solver MUMPS [61] is used to solve the linear system of equations within the Newton iteration at the structural level.

The implemented multi-phase model is used for the numerical simulation of two geotechnical problems. In addition to a 3D simulation of the transient flow of water through a homogeneous earth dam, a pumping test for a well in the urban area of Tianjin is recalculated using both a two-dimensional and a three-dimensional discretisation, and the applicability of this numerical model to predict the subsidence in the vicinity of the well and the water flow in the soil is demonstrated.

Finally, large 3D simulations are run on two different HPC (High Performance Computing) cluster architectures, namely the shared memory system MACH [41] and the distributed memory systems Leo II [39] and Leo III [40] (see figure). The latter system is currently the newest supercomputer at the University of Innsbruck. It consists of 162 nodes with a total of 1944 cores. Its performance is measured by a number of indicators, such as speed-up, efficiency and scaling.

In this dissertation, a constitutive material model for boundary layers between cement-bound material layers is developed and its mechanical and fracture mechanical parameters are determined. The boundary layer model is implemented in a finite element program and validated by numerical calculations of small-scale tests.

The properties of the boundary layer result from the adhesion between the two contacting boundary surfaces and the material properties of the adjacent porous interface transition zone (accurate overlay interface zone) of the new cementitious material layer.

The mechanical and fracture mechanical properties are determined with the three-point bending tensile test, a torsion shear test modified for this purpose (see figure) and the splitting tensile test and compared with the identically determined properties of the monolithic test series. The tensile and shear strengths and the specific fracture energies under mode I and mode II are determined experimentally for this purpose and the parameters of the softening functions under mode I and mode II are calculated inversely. For the characterisation of the interface, the influence of three surface post-treatments of the interface: high pressure water blasted, brushed and smooth cleaned are investigated. These surface post-treatments are characterised with surface topographic parameters and the dependencies between these parameters and the boundary layer properties are evaluated.

The cohesive zone model for the macroscopic description of the boundary layers with the finite element method, which was further developed for the simulation of the boundary layer failure, is based on the work of Caballero et al [2008] and Carol et al [1997]. The boundary layer model is modified with a fracture mechanics failure approach that is consistent with the experimental methodology. This combines the failure modes I, II and IIa and allows a good representation of boundary layers that are predominantly loaded in shear. The elasto-plastic boundary layer model is implemented as a traction-separation law in the context of a zero-thickness interface element formulation in Abaqus [Simulia Corp., 2012]. The robustness, accuracy and efficiency of the implemented algorithm are important properties for the use and expressiveness of the interface model. They are assessed by contour plots for a wide range of given relative displacement increments and improved by an adaptive sub-stepping method, building on the work of Pérez-Foguet and others [2001], and an alternative starting point of Newton iteration.

Finally, the validation of the implemented interface model shows very good agreements of the recalculations with the three-point bending tensile tests and the torsional shear tests for mode I, II and mixed-mode failure I-II.

For this dissertation, Dr. Valentini was awarded the FCP Innovation Prize 2011 "for sustainable developments in civil engineering" by FCP - Fritsch, Chiari & Partner ZT GmbH at the Vienna University of Technology.

In addition, Dr Valentini was awarded the Scientific Computing Thesis Award 2011 of the University of Innsbruck for this dissertation.

In this dissertation, three-dimensional constitutive laws for concrete are evaluated, compared and further developed. These are a single-surface plasticity model developed by Etse [1992] and called the Extended Leon model, an extended version of this model by Pivonka [2001] and a combined damage-plasticity model presented in Grassl [2006]. In the context of this work, the latter model is referred to as the Modified Leon model. Each of these three material models was developed to reproduce the non-linear hardening and softening behaviour of concrete under uniaxial, biaxial and triaxial loading. In order to better represent the material behaviour of concrete, especially under uniaxial compressive loading and combined compressive-tensile loading, the author of this paper extended the softening law of the constitutive model by Grassl [2006]. Furthermore, a scheme for the regularisation of the material softening is presented, which guarantees the objectivity of the results, independent of the size of the finite elements.

In order to implement the advanced material models for concrete in a finite element program system, the solution algorithm for updating the stress tensor at integration point level must be robust, accurate and efficient. Three different solution algorithms based on the backward Euler method are implemented and evaluated in this thesis. These are the Newton method, a damped Newton method proposed by Deuflhard [2004], and a multi-stage method applied in Perez-Foguet [2001]. In addition, a strategy for adaptive control of the multilevel method and a strategy for controlling the errors of the voltage integration are presented. The robustness, accuracy and efficiency of the different solution strategies are compared by evaluating the convergence, the relative errors and the time required to compute the stresses for different sets of predictor stresses.

The individual constitutive models are validated by the recalculation of small-scale tests. For this purpose, uniaxial tensile tests and uniaxial, biaxial and triaxial compression tests of unreinforced concrete specimens are numerically simulated. The results predicted by the extended version of the Extended Leon model, the Modified Leon model and the extended version of the Modified Leon model are compared with experimental results. The validation of the extended version of the Modified Leon model and the verification of the implemented solution algorithms on the structural level are carried out by three-dimensional numerical calculations of four small-scale benchmark tests of unreinforced and reinforced concrete (see figure). Finally, the applicability of the extended version of the Modified Leon model for large three-dimensional finite element calculations is demonstrated by the ultimate load analysis of a 1:200 scale model test of the Zillergründl arch dam. By comparing the predicted and experimentally determined structural response, the performance of the developed numerical model is demonstrated.

Within the scope of the present work, the theoretical principles for a multi-phase model for the description of partially saturated soils are presented, the corresponding equations are prepared for a finite element formulation and implemented in an FE programme. Furthermore, the numerical model was tested in numerical studies and applied to solve geotechnical problems.

In the presentation of the theory underlying the multi-phase model, the balance equations applicable to an equivalent continuum are derived according to both the mixing theory and the averaging theory. After the introduction of the kinematic equations and the treatment of the corresponding material laws, the equations valid for a multi-phase model are simplified and summarised, taking into account corresponding restrictions for partially saturated soils. The Barcelona Basic Model (Alonso et al., 1990), formulated in terms of net stresses and capillary stress, is used to describe the mechanical behaviour of the grain structure. The variational formulation obtained from the basic differential equations is linearised for the use of the Newton method and spatially and temporally discretised for the geometric linear theory based on the finite element method. The corresponding equations are implemented in the FE program Abaqus for a monolithic solution approach. The modular structure of the code, which is divided into two levels, is described and supplemented by explanations of the practical implementation.

The developed multi-phase model is verified by comparison with simulation results of other authors with two numerical studies. The drainage of a sand column according to the experiment of Liakopolous (1965) is numerically simulated with two-dimensional and three-dimensional elements considering both a passive and an active air phase. The benchmark problem of the irrigation of a soil column (MUSE) defined within the Muse project was calculated with two- and three-dimensional elements under consideration of a passive air phase. Thereby, plastic settlements are predicted with the Barcelona Basic Model according to the irrigation.

The developed multi-phase model is applied to two geotechnical problems. The simulation of the transient flow through a homogeneous earth dam is carried out under consideration of a passive air phase. With the Barcelona Basic Model, settlements are calculated for the dam crest as well as for the air-side dam shoulder, as they can be observed in reality. The Essen air permeability test is numerically simulated with a rotationally symmetrical and a three-dimensional discretisation. The numerical model is very good at describing the drainage of the soil by the applied compressed air and the associated deformations (see figure). The results of the numerical simulation show quite good agreement with the experimental data documented by Kramer and Semprich (1989).

Essential features of the numerical model are that it can be used to treat two-dimensional, rotationally symmetrical and three-dimensional problems in geotechnical engineering, taking into account a material law for partially saturated soils. In the programming implementation, emphasis is placed on an appropriately structured and modular design. This is motivated on the one hand by the objective of also enabling the processing of practical problems through a robust code and on the other hand by the basic idea that later extensions of the model can be realised on the basis of this code without considerable difficulties in the programming implementation. For example, further material laws can be implemented both on the basis of generalised effective stresses and on the basis of net stresses.

The integration of elasto-plastic material laws to determine the stresses is an important part of non-linear calculations with the finite element method. This work deals with the stress calculation of elasto-plastic material models for partially saturated materials.

The consideration of partially saturated states gives rise to complex material laws. In the last two decades, some material models have been proposed to describe partially saturated soils (see figure). This dissertation examines the Barcelona Basic Model (BBM) (Alonso et al., 1990), a cap model extended for partially saturated states (Kohler, 2006) and the possibility described by Sanchez et al. (2005) to extend these models to model swellable soils.

Integration algorithms described in the literature are adapted for stress calculation of partially saturated soils and compared with each other. Besides the (most widely used) general projection method, explicit and semi-explicit methods are used. Furthermore, an implicit integration scheme formulated in terms of stress invariants is developed. These methods are combined with Richardson extrapolation to obtain error control. The Radau5 algorithm is used to compare and compute very accurate solutions. Besides the robustness and accuracy of the integration methods, their efficiency plays an essential role. With the integration methods developed in this dissertation, the computation time of the general projection method can be significantly reduced with the same accuracy.

Most elasto-plastic material models for partially saturated soils require a large number of material parameters, e.g. 11 parameters in the case of the BBM. Not all of them have a physical meaning and therefore cannot be determined directly from tests. For this reason, parameter identification using an optimisation procedure is important. In this work, the parameters are determined by minimising an objective function. Since this optimisation problem has several local minima, this task is solved with a global optimisation method, the particle swarm method.

This paper presents a numerical model for the simulation of cracking in concrete. It is a further development of a crack model based on the strong discontinuities method and formulated within the concept of elements with embedded discontinuities.

With this crack model it is possible to represent discontinuities in the displacement course caused by the crack formation, which can cut through a chosen discretisation in almost any form. The two-dimensional formulation of the crack model for unreinforced concrete structures presented in [Feist, 2004] is supplemented, further developed and also used for the simulation of the load-bearing behaviour of reinforced structures.

The crack model is extended within the scope of the present work by hardening and softening laws to take into account the shear force transfer along rough crack surfaces on the basis of the multi-surface plasticity theory. This makes it possible to realistically capture the formation of a continuous crack and to numerically describe the behaviour under combined tensile-shear loading.

To take into account local unloading conditions as well as unloading and reloading due to cyclic loading, the crack model with embedded discontinuities is coupled with an isotropic damage model. Furthermore, the crack model is extended in the present work by alternative formulations for the determination of the crack path.

Considering the crack evolution in the course of a numerical simulation using a smeared crack model with rotating crack directions, it can be observed that the initially predicted crack direction usually changes as the localisation progresses. Therefore, it can be expected that the simulation of a macroscopic crack is improved by describing the localisation process at the beginning with a smeared crack model and introducing a discontinuity only after a certain limit value of the crack opening is reached.

This alternative formulation, presented in [Jirasek and Zimmermann, 2001] in the context of a damage model, is applied to the concept of embedded discontinuities in the present work by combining a crack model based on the concept of smeared cracks with the latter.

As a second alternative formulation of the crack path, the procedure described in [Sancho et al., 2005] is implemented and investigated in the crack model with embedded discontinuities in the present work. Within the framework of this formulation, the crack direction is recalculated in each load increment up to a certain limit value of the crack opening and the crack direction is kept constant only after reaching the limit value. Furthermore, a continuous crack progression across the element boundaries is dispensed with. In the original crack model, the requirement of a continuous crack progression is fulfilled by the application of a crack tracking algorithm, which, however, can cause a kind of crack locking. It is shown that especially the combination of the crack model with embedded discontinuities with a smeared crack model developed within the scope of this work has significant advantages with regard to the prediction of crack formation.

With this model, called the delayed embedded crack model, a more realistic modelling of the cracking in unreinforced concrete structures is possible, starting with the damage in the form of distributed microcracks and the later transition to one or more macroscopic cracks. This crack model is also well applicable for the numerical simulation of reinforced concrete structures.

The extensions mentioned are verified by the recalculation of both laboratory tests known from the literature, such as the three-point bending test, the combined tension-shear test according to Hassanzadeh and the anchor pull-out test (see figure), and tests carried out at the Leopold-Franzens University of Innsbruck.

The application of the crack model with delayed embedded discontinuities for the numerical simulation of cantilever slab reinforcements using top concrete proves the suitability of the crack model extended within the scope of this work for the simulation of the load-bearing behaviour of more complex structures.

The present work deals with the development and verification of a material law to describe the mechanical behaviour of partially saturated soils and its implementation in a three-phase FE formulation. Such numerical models are suitable for the simulation of a variety of geotechnical problems such as consolidation, irrigation and drainage of soils under atmospheric conditions such as the flow through earth dams and drainage by means of compressed air e.g. in the context of tunnel excavations below the water table.

Starting from a multi-surface plasticity model for drained conditions, different approaches for the description of the stress state of partially saturated soils are investigated and compared by reformulating the underlying model depending on an effective stress tensor for partially saturated soils on the one hand and extending it on the basis of two stress state variables on the other hand. The latter is done by considering the development of the shear strength as well as the stiffness with increasing capillary stress. Based on the recalculation of tests, it is shown that only the extended material model allows an adequate modelling of the behaviour of partially saturated soils, such as the irreversible volume decrease under irrigation (see figure).

In this paper, two versions of the model formulated in two stress variables are proposed and investigated. The first one is characterised by the use of the net stresses and the capillary stress as independent stress state variables while for the second one the Bishop stresses in combination with the capillary stress are used as plastic internal variables. The validation of the material models is carried out using a series of experiments reported in the literature with different values of capillary stress. While both versions of the extended material model are suitable for modelling under partially saturated conditions, only the latter allows an automatic transition to the concept of effective stresses for water-saturated soils within the FE formulation, as shown by a consolidation example.

Based on the application of the implicit Euler backward method, a projection procedure for the proposed material models is derived and the elasto-plastic material tensors are determined. In contrast to classical material models, a second material tensor is obtained with respect to the capillary stress.

The verification of the developed FE model is done by numerical simulation of examples and comparisons of the numerical results with analytical solutions or experimental data. The numerical study includes the simulation of

(i) a consolidation problem,

(ii) a laboratory experiment on the drainage and irrigation of a soil column, and

(iii) the behaviour of an earth dam during and after impoundment.

The present work is devoted to the development of a family of finite elements for the simulation of the failure of unreinforced concrete structures due to the cracking of the material. The formation of a discrete crack can be considered from a mathematical point of view as the formation of a discontinuity surface characterised by a discontinuity in the displacement field. The description of the kinematics of a solid through which such a discontinuity surface runs is carried out with the help of the concept of strong discontinuity kinematics. The numerical model presented is based on the strong discontinuity approach (SDA) and is formulated within the concept of elements with embedded discontinuities. These elements allow the mapping of discontinuity surfaces, which can intersect a chosen spatial discretisation in almost any form. Thus, the application of complex adaptive methods can be dispensed with. The numerical model is formulated for quasi-static tasks under the assumption of small displacements and small distortions for monotonic loading scenarios. RANKINE's criterion is used as the breaking hypothesis. The presented elements are implemented for two- and three-dimensional problems in a commercial FE package.

The numerical model is also based on the fixed crack concept. It is shown that the SDA model in its basic form provides mesh-dependent solutions insofar as the numerical resolution of a discrete crack is influenced by the mesh topology. Tracking strategies offer the possibility to describe continuous discontinuity surfaces via neighbouring elements in order to eliminate this mesh dependency. A new crack tracking algorithm, which allows the treatment of problems with multiple cracks as well as problems with spatial crack surfaces, is presented. Furthermore, the determination of the propagation direction of crack surfaces is improved by a nonlocal averaging method.

The numerical model is verified by means of laboratory tests known from the literature. The subsequent analysis of a gravity dam proves the applicability of the model to simulations of large structures. Load investigations of dams also provide the impetus for considerations regarding the consideration of water pressure in opening cracks. Based on this, an extension of the SDA model with respect to this effect is proposed.

In addition to the tests known from the literature, an own test programme is also carried out and documented. The included tests on notched concrete beams with three-dimensional loading conditions are of particular value for the verification of the three-dimensional SDA formulation.

Robert STELZER (2003)

Adaptive Finite Element Analysis of Multi-Phase Problems

Gerhard ÖTTL (2003)

A Three-Phase FE-Model for dewatering of soils by means of compressed air

Bernhard Josef WINKLER (2001)

Ultimate Load Investigations of Unreinforced and Reinforced Concrete Structures on the Basis of an Objective Material Law for Concrete

Roland HINTERHÖLZL (2000)

Modelling the time dependent behaviour of fibre reinforced plastics and particulate composites by the theory of viscoelasticity and damage mechanics

Günther SCHULLERER (2000)

A numerical investigation for the analysis of fiber reinforced composites with respect to macromechanical and micromechanical modeling

Johannes JÄGER (1996)

Numerical treatment of anisotropic plastic equations for soils

Siegfried EBENBICHLER (1996)

Plane calculation of the continuous casting process in mixed Euler-Lagrangian coordinates

Herbert HALLER (1995)

Model for elasto-plastic damage