Multi-method global sensitivity analysis (MMGSA) for modelling floodplain hydrological processes

When studying hydrological processes with a numerical model, global sensitivity analysis (GSA) is essential if one is to understand the impact of model parameters and model formulation on results. However, different definitions of sensitivity can lead to a difference in the ranking of importance of the different model factors. Here we combine a fuzzy performance function with different methods of calculating global sensitivity to perform a multi-method global sensitivity analysis (MMGSA). We use an application of a finite element subsurface flow model (ESTEL-2D) on a flood inundation event on a floodplain of the River Severn to illustrate this new methodology. We demonstrate the utility of the method for model understanding and show how the prediction of state variables, such as Darcian velocity vectors, can be affected by such a MMGSA. This paper is a first attempt to use GSA with a numerically intensive hydrological model

Reduced-order modelling, circuit-level design and SOI fabrication of microelectromechanical resonators

This thesis deals with two important research aspects concerning radio frequency (RF) microresonators and switches. First, a new approach for compact modeling and simulation of these devices is presented. Then, a combined process flow for their simultaneous fabrication on a SOI substrate is proposed. Compact models for microresonators and switches are extracted by applying mathematical model order reduction (MOR) to the devices finite element (FE) description in ANSYS c° . The behaviour of these devices includes forms of nonlinearities. However, an approximation in the creation of the FE model is introduced, which enables the use of linear model order reduction. Microresonators are modeled with the introduction of transducer elements, which allow for direct coupling of the electrical and mechanical domain. The coupled system element matrices are linearized around an operating point and reduced. The resulting macromodel is valid for small signal analysis around the bias point, such as harmonic pre-stressed analysis. This is extremely useful for characterizing the frequency response of resonators. Compact modelling of switches preserves the nonlinearity of the device behaviour. Nonlinear reduced order models are obtained by reducing the number of nonlinearities in the system and handling them as input to the system. In this way, the system can be reduced using linear MOR techniques and nonlinearities are introduced directly in the reduced order model. The reduction of the number of system nonlinearities implies the approximation of all distributed forces in the model with lumped forces. Both for microresonators and switches, a procedure for matrices extraction has been developed so that reduced order models include the effects of electrical and mechanical pre-stress. The extraction process is fast and can be done automatically from ANSYS binary files. The method has been applied for the simulation of several devices both at devices and circuit level. Simulation results have been compared with full model simulations, and, when available, experimental data. Reduced order models have proven to conserve the accuracy of finite element method and to give a good description of the overall device behaviour, despite the introduced approximations. In addition, simulation is very fast, both at device and circuit level. A combined process-flow for the integrated fabrication of microresonators and switches has been defined. For this purpose, two processes that are optimized for the independent fabrication of these devices are merged. The major advantage of this process is the possibility to create on-chip circuit blocks that include both microresonators and switches. An application is, for example, aswitched filter bank for wireless transceiver. The process for microresonators fabrication is characterized by the use of silicon on insulator (SOI) wafers and on a deep reactive ion etching (DRIE) step for the creation of the vibrating structures in single-crystal silicon and the use of a sacrificial oxide layer for the definition of resonator to electrode distance. The fabrication of switches is characterized by the use of two different conductive layers for the definition of the actuation electrodes and by the use of a photoresist as a sacrificial layer for the creation of the suspended structure. Both processes have a gold electroplating step, for the creation of the resonators electrodes, transmission lines and suspended structures. The combined process flow is designed such that it conserves the basic properties of the original processes. Neither the performance of the resonators nor the performance of the switches results affected by the simultaneous fabrication. Moreover, common fabrication steps are shared, which allows for cheaper and faster fabrication.

Modelling and analysis of internally BFRP reinforced beams at elevated temperatures

The aim of the work is to conduct a finite element model analysis on a small – size concrete beam and on a full size concrete beam internally reinforced with BFRP exposed at elevated temperatures. Experimental tests performed at Kingston University have been used to compare the results from the numerical analysis for the small – size concrete beam. Once the behavior of the small – size beam at room temperature is investigated and switching to the heating phase reinforced beams are tested at 100°C, 200°C and 300°C in loaded condition. The aim of the finite element analysis is to reflect the three – point bending test adopted into the oven during the exposure of the beam at room temperature and at elevated temperatures. Performance and deformability of reinforced beams are straightly correlated to the material properties and a wide analysis on elastic modulus and coefficient of thermal expansion is given in this work. Develop a good correlation between the numerical model and the experimental test is the main objective of the analysis on the small – size concrete beam, for both modelling the aim is also to estimate which is the deterioration of the material properties due to the heating process and the influence of different parameters on the final result. The focus of the full – size modelling which involved the last part of this work is to evaluate the effect of elevated temperatures, the material deterioration and the deflection trend on a reinforced beam characterized by a different size. A comparison between the results from different modelling has been developed.

Modelling and analysis of thin-walled beams in the context of the Generalized Beam Theory

In this work, the Generalized Beam Theory (GBT) is used as the main tool to analyze the mechanics of thin-walled beams. After an introduction to the subject and a quick review of some of the most well-known approaches to describe the behaviour of thin-walled beams, a novel formulation of the GBT is presented. This formulation contains the classic shear-deformable GBT available in the literature and contributes an additional description of cross-section warping that is variable along the wall thickness besides along the wall midline. Shear deformation is introduced in such a way that the classical shear strain components of the Timoshenko beam theory are recovered exactly. According to the new kinematics proposed, a reviewed form of the cross-section analysis procedure is devised, based on a unique modal decomposition. Later, a procedure for a posteriori reconstruction of all the three-dimensional stress components in the finite element analysis of thin-walled beams using the GBT is presented. The reconstruction is simple and based on the use of three-dimensional equilibrium equations and of the RCP procedure. Finally, once the stress reconstruction procedure is presented, a study of several existing issues on the constitutive relations in the GBT is carried out. Specifically, a constitutive law based on mirroring the kinematic constraints of the GBT model into a specific stress field assumption is proposed. It is shown that this method is equally valid for isotropic and orthotropic beams and coincides with the conventional GBT approach available in the literature. Later on, an analogous procedure is presented for the case of laminated beams. Lastly, as a way to improve an inherently poor description of shear deformability in the GBT, the introduction of shear correction factors is proposed. Throughout this work, numerous examples are provided to determine the validity of all the proposed contributions to the field.

Soil-structure interaction during tunnelling in urban area: observations and 3D numerical modelling

This work illustrates a soil-tunnel-structure interaction study performed by an integrated,geotechnical and structural,approach based on 3D finite element analyses and validated against experimental observations.The study aims at analysing the response of reinforced concrete framed buildings on discrete foundations in interaction with metro lines.It refers to the case of the twin tunnels of the Milan (Italy) metro line 5,recently built in coarse grained materials using EPB machines,for which subsidence measurements collected along ground and building sections during tunnelling were available.Settlements measured under freefield conditions are firstly back interpreted using Gaussian empirical predictions. Then,the in situ measurements’ analysis is extended to include the evolving response of a 9 storey reinforced concrete building while being undercrossed by the metro line.In the finite element study,the soil mechanical behaviour is described using an advanced constitutive model. This latter,when combined with a proper simulation of the excavation process, proves to realistically reproduce the subsidence profiles under free field conditions and to capture the interaction phenomena occurring between the twin tunnels during the excavation. Furthermore, when the numerical model is extended to include the building, schematised in a detailed manner, the results are in good agreement with the monitoring data for different stages of the twin tunnelling. Thus, they indirectly confirm the satisfactory performance of the adopted numerical approach which also allows a direct evaluation of the structural response as an outcome of the analysis. Further analyses are also carried out modelling the building with different levels of detail. The results highlight that, in this case, the simplified approach based on the equivalent plate schematisation is inadequate to capture the real tunnelling induced displacement field. The overall behaviour of the system proves to be mainly influenced by the buried portion of the building which plays an essential role in the interaction mechanism, due to its high stiffness.

A mathematical framework for finite strain elastoplastic consolidation. Part 1: Balance laws, variational formulation, and linearization

A mathematical formulation for finite strain elasto plastic consolidation of fully saturated soil media is presented. Strong and weak forms of the boundary-value problem are derived using both the material and spatial descriptions. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. Balance laws are written for the soil-water mixture following the motion of the soil matrix alone. It is shown that the motion of the fluid phase only affects the Jacobian of the solid phase motion, and therefore can be characterized completely by the motion of the soil matrix. Furthermore, it is shown from energy balance consideration that the effective, or intergranular, stress is the appropriate measure of stress for describing the constitutive response of the soil skeleton since it absorbs all the strain energy generated in the saturated soil-water mixture. Finally, it is shown that the mathematical model is amenable to consistent linearization, and that explicit expressions for the consistent tangent operators can be derived for use in numerical solutions such as those based on the finite element method.

Modelling of chloride transport in non-saturated concrete : from microscale to macroscale

Corrosion of steel bars embedded in concrete has a great influence on structural performance and durability of reinforced concrete. Chloride penetration is considered to be a primary cause of concrete deterioration in a vast majority of structures. Therefore, modelling of chloride penetration into concrete has become an area of great interest. The present work focuses on modelling of chloride transport in concrete. The differential macroscopic equations which govern the problem were derived from the equations at the microscopic scale by comparing the porous network with a single equivalent pore whose properties are the same as the average properties of the real porous network. The resulting transport model, which accounts for diffusion, migration, advection, chloride binding and chloride precipitation, consists of three coupled differential equations. The first equation models the transport of chloride ions, while the other two model the flow of the pore water and the heat transfer. In order to calibrate the model, the material parameters to determine experimentally were identified. The differential equations were solved by means of the finite element method. The classical Galerkin method was employed for the pore solution flow and the heat transfer equations, while the streamline upwind Petrov Galerkin method was adopted for the transport equation in order to avoid spatial instabilities for advection dominated problems. The finite element codes are implemented in Matlab® . To retrieve a good understanding of the influence of each variable and parameter, a detailed sensitivity analysis of the model was carried out. In order to determine the diffusive and hygroscopic properties of the studied concretes, as well as their chloride binding capacity, an experimental analysis was performed. The model was successfully compared with experimental data obtained from an offshore oil platform located in Brazil. Moreover, apart from the main objectives, numerous results were obtained throughout this work. For instance, several diffusion coefficients and the relation between them are discussed. It is shown how the electric field set up between the ionic species depends on the gradient of the species’ concentrations. Furthermore, the capillary hysteresis effects are illustrated by a proposed model, which leads to the determination of several microstructure properties, such as the pore size distribution and the tortuosity-connectivity of the porous network. El fenómeno de corrosión del acero de refuerzo embebido en el hormigón ha tenido gran influencia en estructuras de hormigón armado, tanto en su funcionalidad estructural como en aspectos de durabilidad. La penetración de cloruros en el interior del hormigón esta considerada como el factor principal en el deterioro de la gran mayoría de estructuras. Por lo tanto, la modelización numérica de dicho fenómeno ha generado gran interés. El presente trabajo de investigación se centra en la modelización del transporte de cloruros en el interior del hormigón. Las ecuaciones diferenciales que gobiernan los fenómenos a nivel macroscópico se deducen de ecuaciones planteadas a nivel microscópico. Esto se obtiene comparando la red porosa con un poro equivalente, el cual mantiene las mismas propiedades de la red porosa real. El modelo está constituido por tres ecuaciones diferenciales acopladas que consideran el transporte de cloruros, el flujo de la solución de poro y la transferencia de calor. Con estas ecuaciones se tienen en cuenta los fenómenos de difusión, migración, advección, combinación y precipitación de cloruros. El análisis llevado a cabo en este trabajo ha definido los parámetros necesarios para calibrar el modelo. De acuerdo con ellas, se seleccionaron los ensayos experimentales a realizar. Las ecuaciones diferenciales se resolvieron mediante el método de elementos finitos. El método clásico de Galerkin se empleó para solucionar las ecuaciones de flujo de la solución de poro y de la transferencia de calor, mientras que el método streamline upwind Petrov-Galerkin se utilizó para resolver la ecuación de transporte de cloruros con la finalidad de evitar inestabilidades espaciales en problemas con advección dominante. El código de elementos finitos está implementado en Matlab® . Con el objetivo de facilitar la comprensión del grado de influencia de cada variable y parámetro, se realizó un análisis de sensibilidad detallado del modelo. Se llevó a cabo una campaña experimental sobre los hormigones estudiados, con el objeto de obtener sus propiedades difusivas, químicas e higroscópicas. El modelo se contrastó con datos experimentales obtenidos en una plataforma petrolera localizada en Brasil. Las simulaciones numéricas corroboraron los datos experimentales. Además, durante el desarrollo de la investigación se obtuvieron resultados paralelos a los planteados inicialmente. Por ejemplo, el análisis de diferentes coeficientes de difusión y la relación entre ellos. Así como también se observó que el campo eléctrico establecido entre las especies iónicas disueltas en la solución de poro depende del gradiente de concentración de las mismas. Los efectos de histéresis capilar son expresados por el modelo propuesto, el cual conduce a la determinación de una serie de propiedades microscópicas, tales como la distribución del tamaño de poro, además de la tortuosidad y conectividad de la red porosa.

Modelling of chloride penetration into non-saturated concrete: Case study application for real marine off shore structures

The aim of this paper is to explain the chloride concentration profiles obtained experimentally from control samples of an offshore platform after 25 years of service life. The platform is located 12 km off the coast of the Brazilian province Rio Grande do Norte, in the north-east of Brazil. The samples were extracted at different orientations and heights above mean sea level. A simple model based on Fick’s second law is considered and compared with a finite element model which takes into account transport of chloride ions by diffusion and convection. Results show that convective flows significantly affect the studied chloride penetrations. The convection velocity is obtained by fitting the finite element solution to the experimental data and seems to be directly proportional to the height above mean sea level and also seems to depend on the orientation of the face of the platform. This work shows that considering solely diffusion as transport mechanism does not allow a good prediction of the chloride profiles. Accounting for capillary suction due to moisture gradients permits a better interpretation of the material’s behaviour.

Modelling of chloride transport in non-saturated concrete : from microscale to macroscale

A model for chloride transport in concrete is proposed. The model accounts for transport several transport mechanisms such as diffusion, advection, migration, etc. This work shows the chloride transport equations at the macroscopic scale in non-saturated concrete. The equations involve diffusion, migration, capillary suction, chloride combination and precipitation mechanisms. The material is assumed to be infinitely rigid, though the porosity can change under influence of chloride binding and precipitation. The involved microscopic and macroscopic properties of the materials are measured by standardized methods. The variables which must be imposed on the boundaries are temperature, relative humidity and chloride concentration. The output data of the model are the free, bound, precipitated and total chloride ion concentrations, as well as the pore solution content and the porosity. The proposed equations are solved by means of the finite element method (FEM) implemented in MATLAB (classical Galerkin formulation and the streamline upwind Petrov-Galerkin (SUPG) method to avoid spatial instabilities for advection dominated flows).

Inception of debris avalanches: remarks on geomechanical modelling

Debris avalanches are complex phenomena due to the variety of mechanisms that control the failure stage and the avalanche formation. Regarding these issues, in the literature, either field evidence or qualitative interpretations can be found while few experimental laboratory tests and rare examples of geomechanical modelling are available for technical and/or scientific purposes. As a contribution to the topic, the paper firstly highlights as the problem can be analysed referring to a unique mathematical framework from which different modelling approaches can be derived based on limit equilibrium method (LEM), finite element method (FEM), or smooth particle hydrodynamics (SPH). Potentialities and limitations of these approaches are then tested for a large study area where huge debris avalanches affected shallow deposits of pyroclastic soils (Sarno-Quindici, Southern Italy). The numerical results show that LEM as well as uncoupled and coupled stress–strain FEM analyses are able to individuate the major triggering mechanisms. On the other hand, coupled SPH analyses outline the relevance of erosion phenomena, which can modify the kinematic features of debris avalanches in their source areas, i.e. velocity, propagation patterns and later spreading of the unstable mass. As a whole, the obtained results encourage the application of the introduced approaches to further analyse real cases in order to enhance the current capability to forecast the inception of these dangerous phenomena.

Modelling of chloride penetration into non-saturated concrete: case study application for real marine offshore structures

The aim of this paper is to explain the chloride concentration profiles obtained experimentally from control samples of an offshore platform after 25 years of service life. The platform is located 12 km off the coast of the Brazilian province Rio Grande do Norte, in the north-east of Brazil. The samples were extracted at different orientations and heights above mean sea level. A simple model based on Fick’s second law is considered and compared with a finite element model which takes into account transport of chloride ions by diffusion and convection. Results show that convective flows significantly affect the studied chloride penetrations. The convection velocity is obtained by fitting the finite element solution to the experimental data and seems to be directly proportional to the height above mean sea level and also seems to depend on the orientation of the face of the platform. This work shows that considering solely diffusion as transport mechanism does not allow a good prediction of the chloride profiles. Accounting for capillary suction due to moisture gradients permits a better interpretation of the material’s behaviour

Computational and Experimental Modelling of Mooring Lines Dynamics for Offshore Floating Wind Turbines

El cálculo de cargas de aerogeneradores flotantes requiere herramientas de simulación en el dominio del tiempo que consideren todos los fenómenos que afectan al sistema, como la aerodinámica, la dinámica estructural, la hidrodinámica, las estrategias de control y la dinámica de las líneas de fondeo. Todos estos efectos están acoplados entre sí y se influyen mutuamente. Las herramientas integradas se utilizan para calcular las cargas extremas y de fatiga que son empleadas para dimensionar estructuralmente los diferentes componentes del aerogenerador. Por esta razón, un cálculo preciso de las cargas influye de manera importante en la optimización de los componentes y en el coste final del aerogenerador flotante. En particular, el sistema de fondeo tiene gran impacto en la dinámica global del sistema. Muchos códigos integrados para la simulación de aerogeneradores flotantes utilizan modelos simplificados que no consideran los efectos dinámicos de las líneas de fondeo. Una simulación precisa de las líneas de fondeo dentro de los modelos integrados puede resultar fundamental para obtener resultados fiables de la dinámica del sistema y de los niveles de cargas en los diferentes componentes. Sin embargo, el impacto que incluir la dinámica de los fondeos tiene en la simulación integrada y en las cargas todavía no ha sido cuantificada rigurosamente. El objetivo principal de esta investigación es el desarrollo de un modelo dinámico para la simulación de líneas de fondeo con precisión, validarlo con medidas en un tanque de ensayos e integrarlo en un código de simulación para aerogeneradores flotantes. Finalmente, esta herramienta, experimentalmente validada, es utilizada para cuantificar el impacto que un modelos dinámicos de líneas de fondeo tienen en la computación de las cargas de fatiga y extremas de aerogeneradores flotantes en comparación con un modelo cuasi-estático. Esta es una información muy útil para los futuros diseñadores a la hora de decidir qué modelo de líneas de fondeo es el adecuado, dependiendo del tipo de plataforma y de los resultados esperados. El código dinámico de líneas de fondeo desarrollado en esta investigación se basa en el método de los Elementos Finitos, utilizando en concreto un modelo ”Lumped Mass” para aumentar su eficiencia de computación. Los experimentos realizados para la validación del código se realizaron en el tanque del École Céntrale de Nantes (ECN), en Francia, y consistieron en sumergir una cadena con uno de sus extremos anclados en el fondo del tanque y excitar el extremo suspendido con movimientos armónicos de diferentes periodos. El código demostró su capacidad para predecir la tensión y los movimientos en diferentes posiciones a lo largo de la longitud de la línea con gran precisión. Los resultados indicaron la importancia de capturar la dinámica de las líneas de fondeo para la predicción de la tensión especialmente en movimientos de alta frecuencia. Finalmente, el código se utilizó en una exhaustiva evaluación del efecto que la dinámica de las líneas de fondeo tiene sobre las cargas extremas y de fatiga de diferentes conceptos de aerogeneradores flotantes. Las cargas se calcularon para tres tipologías de aerogenerador flotante (semisumergible, ”spar-buoy” y ”tension leg platform”) y se compararon con las cargas obtenidas utilizando un modelo cuasi-estático de líneas de fondeo. Se lanzaron y postprocesaron más de 20.000 casos de carga definidos por la norma IEC 61400-3 siguiendo todos los requerimientos que una entidad certificadora requeriría a un diseñador industrial de aerogeneradores flotantes. Los resultados mostraron que el impacto de la dinámica de las líneas de fondeo, tanto en las cargas de fatiga como en las extremas, se incrementa conforme se consideran elementos situados más cerca de la plataforma: las cargas en la pala y en el eje sólo son ligeramente modificadas por la dinámica de las líneas, las cargas en la base de la torre pueden cambiar significativamente dependiendo del tipo de plataforma y, finalmente, la tensión en las líneas de fondeo depende fuertemente de la dinámica de las líneas, tanto en fatiga como en extremas, en todos los conceptos de plataforma que se han evaluado. ABSTRACT The load calculation of floating offshore wind turbine requires time-domain simulation tools taking into account all the phenomena that affect the system such as aerodynamics, structural dynamics, hydrodynamics, control actions and the mooring lines dynamics. These effects present couplings and are mutually influenced. The results provided by integrated simulation tools are used to compute the fatigue and ultimate loads needed for the structural design of the different components of the wind turbine. For this reason, their accuracy has an important influence on the optimization of the components and the final cost of the floating wind turbine. In particular, the mooring system greatly affects the global dynamics of the floater. Many integrated codes for the simulation of floating wind turbines use simplified approaches that do not consider the mooring line dynamics. An accurate simulation of the mooring system within the integrated codes can be fundamental to obtain reliable results of the system dynamics and the loads. The impact of taking into account the mooring line dynamics in the integrated simulation still has not been thoroughly quantified. The main objective of this research consists on the development of an accurate dynamic model for the simulation of mooring lines, validate it against wave tank tests and then integrate it in a simulation code for floating wind turbines. This experimentally validated tool is finally used to quantify the impact that dynamic mooring models have on the computation of fatigue and ultimate loads of floating wind turbines in comparison with quasi-static tools. This information will be very useful for future designers to decide which mooring model is adequate depending on the platform type and the expected results. The dynamic mooring lines code developed in this research is based in the Finite Element Method and is oriented to the achievement of a computationally efficient code, selecting a Lumped Mass approach. The experimental tests performed for the validation of the code were carried out at the `Ecole Centrale de Nantes (ECN) wave tank in France, consisting of a chain submerged into a water basin, anchored at the bottom of the basin, where the suspension point of the chain was excited with harmonic motions of different periods. The code showed its ability to predict the tension and the motions at several positions along the length of the line with high accuracy. The results demonstrated the importance of capturing the evolution of the mooring dynamics for the prediction of the line tension, especially for the high frequency motions. Finally, the code was used for an extensive assessment of the effect of mooring dynamics on the computation of fatigue and ultimate loads for different floating wind turbines. The loads were computed for three platforms topologies (semisubmersible, spar-buoy and tension leg platform) and compared with the loads provided using a quasi-static mooring model. More than 20,000 load cases were launched and postprocessed following the IEC 61400-3 guideline and fulfilling the conditions that a certification entity would require to an offshore wind turbine designer. The results showed that the impact of mooring dynamics in both fatigue and ultimate loads increases as elements located closer to the platform are evaluated; the blade and the shaft loads are only slightly modified by the mooring dynamics in all the platform designs, the tower base loads can be significantly affected depending on the platform concept and the mooring lines tension strongly depends on the lines dynamics both in fatigue and extreme loads in all the platform concepts evaluated.

Advanced interpretation of subsidence in Murcia (SE Spain) using A-DInSAR data – modelling and validation

Subsidence is a natural hazard that affects wide areas in the world causing important economic costs annually. This phenomenon has occurred in the metropolitan area of Murcia City (SE Spain) as a result of groundwater overexploitation. In this work aquifer system subsidence is investigated using an advanced differential SAR interferometry remote sensing technique (A-DInSAR) called Stable Point Network (SPN). The SPN derived displacement results, mainly the velocity displacement maps and the time series of the displacement, reveal that in the period 2004–2008 the rate of subsidence in Murcia metropolitan area doubled with respect to the previous period from 1995 to 2005. The acceleration of the deformation phenomenon is explained by the drought period started in 2006. The comparison of the temporal evolution of the displacements measured with the extensometers and the SPN technique shows an average absolute error of 3.9±3.8 mm. Finally, results from a finite element model developed to simulate the recorded time history subsidence from known water table height changes compares well with the SPN displacement time series estimations. This result demonstrates the potential of A-DInSAR techniques to validate subsidence prediction models as an alternative to using instrumental ground based techniques for validation.

An anisotropic viscous representation of Mohr-Coulomb failure for use is modelling coupled mantle-continent dynamics

In mantle convection models it has become common to make use of a modified (pressure sensitive, Boussinesq) von Mises yield criterion to limit the maximum stress the lithosphere can support. This approach allows the viscous, cool thermal boundary layer to deform in a relatively plate-like mode even in a fully Eulerian representation. In large-scale models with embedded continental crust where the mobile boundary layer represents the oceanic lithosphere, the von Mises yield criterion for the oceans ensures that the continents experience a realistic broad-scale stress regime. In detailed models of crustal deformation it is, however, more appropriate to choose a Mohr-Coulomb yield criterion based upon the idea that frictional slip occurs on whichever one of many randomly oriented planes happens to be favorably oriented with respect to the stress field. As coupled crust/mantle models become more sophisticated it is important to be able to use whichever failure model is appropriate to a given part of the system. We have therefore developed a way to represent Mohr-Coulomb failure within a code which is suited to mantle convection problems coupled to large-scale crustal deformation. Our approach uses an orthotropic viscous rheology (a different viscosity for pure shear to that for simple shear) to define a prefered plane for slip to occur given the local stress field. The simple-shear viscosity and the deformation can then be iterated to ensure that the yield criterion is always satisfied. We again assume the Boussinesq approximation - neglecting any effect of dilatancy on the stress field. An additional criterion is required to ensure that deformation occurs along the plane aligned with maximum shear strain-rate rather than the perpendicular plane which is formally equivalent in any symmetric formulation. It is also important to allow strain-weakening of the material. The material should remember both the accumulated failure history and the direction of failure. We have included this capacity in a Lagrangian-Integration-point finite element code and will show a number of examples of extension and compression of a crustal block with a Mohr-Coulomb failure criterion, and comparisons between mantle convection models using the von Mises versus the Mohr-Coulomb yield criteria. The formulation itself is general and applies to 2D and 3D problems, although it is somewhat more complicated to identify the slip plane in 3D.

On the stress analysis of cracked bodies by means of finite elements

The work described in this thesis deals with the development and application of a finite element program for the analysis of several cracked structures. In order to simplify the organisation of the material presented herein, the thesis has been subdivided into two Sections : In the first Section the development of a finite element program for the analysis of two-dimensional problems of plane stress or plane strain is described. The element used in this program is the six-mode isoparametric triangular element which permits the accurate modelling of curved boundary surfaces. Various cases of material aniftropy are included in the derivation of the element stiffness properties. A digital computer program is described and examples of its application are presented. In the second Section, on fracture problems, several cracked configurations are analysed by embedding into the finite element mesh a sub-region, containing the singularities and over which an analytic solution is used. The modifications necessary to augment a standard finite element program, such as that developed in Section I, are discussed and complete programs for each cracked configuration are presented. Several examples are included to demonstrate the accuracy and flexibility of the technique.