This research investigates fracture behaviour in the Cosserat materials. The Cosserat elasticity description of the materials incorporates a characteristic length scale (e.g. grains, particles, fibres, etc.) into the model. The characteristic length scale in such materials is known to significantly influence the macroscopic behaviour of the whole body. Simulation of the fracture processes, such as crack opening and propagation, in Cosserat materials still remains a challenge for the scientific community. The goal of this thesis is to propose and validate a two dimensional extended finite element method model of edge cracks within the Cosserat elasticity theory framework. The crack modelling was conducted using the Finite Element Method (FEM) and eXtended Finite Element Method (XFEM) implemented in the Matlab code. The strong and weak formulations of the problem and the discrete XFEM equations are presented in the thesis. Mode I and II edge crack models in a Cosserat medium are discussed and verified through a series of patch and convergence tests. In addition, the numerical evaluation of the J-integral for the Cosserat medium is presented, and the J-integral for the Cosserat medium is compared to the J-integral for the classical elasticity. The XFEM/Cosserat method is shown to be robust and able to effectively model the edge crack problems in a Cosserat medium. Moreover, the Cosserat elastic parameter is found to be a powerful coupling tool between the microrotations and the translations. The Cosserat J-integral differs from the classical J-integral by 2% to 40%, for a given crack depending on the micropolar elastic coupling constant.

This research investigates fracture behaviour in the Cosserat materials. The Cosserat elasticity description of the materials incorporates a characteristic length scale (e.g. grains, particles, fibres, etc.) into the model. The characteristic length scale in such materials is known to significantly influence the macroscopic behaviour of the whole body. Simulation of the fracture processes, such as crack opening and propagation, in Cosserat materials still remains a challenge for the scientific community. The goal of this thesis is to propose and validate a two dimensional extended finite element method model of edge cracks within the Cosserat elasticity theory framework. The crack modelling was conducted using the Finite Element Method (FEM) and eXtended Finite Element Method (XFEM) implemented in the Matlab code. The strong and weak formulations of the problem and the discrete XFEM equations are presented in the thesis. Mode I and II edge crack models in a Cosserat medium are discussed and verified through a series of patch and convergence tests. In addition, the numerical evaluation of the J-integral for the Cosserat medium is presented, and the J-integral for the Cosserat medium is compared to the J-integral for the classical elasticity. The XFEM/Cosserat method is shown to be robust and able to effectively model the edge crack problems in a Cosserat medium. Moreover, the Cosserat elastic parameter is found to be a powerful coupling tool between the microrotations and the translations. The Cosserat J-integral differs from the classical J-integral by 2% to 40%, for a given crack depending on the micropolar elastic coupling constant.

The drill-strings used in drilling operate under extreme condi-tions, therefore, an accurate understanding of their dynamics is necessary and has attracted much interest. Although a bottom hole assembly (BHA) is to a great ex- tent responsible for the dynamics of the system, the in uence of the drill-pipes has been increasingly neglected by current models. Their dynamics and geometrical behaviour should be better analysed for a deeper understanding of underlying phe- nomena. For example, under stick-slip oscillations, the torque on the drill-string may cause torsional buckling of the drill-pipes, incurring in helical con guration, in which the apparent length is reduced, a ecting the forces at the bit{rock interface. With such behaviour and interactions in mind, this work focuses on elaborating a comprehensive mathematical model to investigate the dynamics of drill-strings, with attention to the drill-pipes section. Firstly, lower dimensional models are used to analyse the stick-slip limit cycle and its limits of existence. Then, a model developed for MEMS is used as a base for a comprehensive model using the formu- lation of Cosserat rods. Relevant boundary conditions are applied and a numerical simulation procedure is established. Simulations are performed for a range of sce- narios under stick-slip occurrence, and the behaviour of the drill-pipes is analysed. Focus is then given to axial vibrations with bit-bounce and the in uence on stick- slip, later to lateral vibrations with whirling motion of the drill-pipes, and nally to helical con gurations, taken by the drill-string under combined torsional, axial and lateral loads, showing the consequent shortening of the drill-string.

In the effort to simulate the biologically inspired continuum robot’s dynamic capabilities, researchers have been faced with the daunting task of simulating—in real-time—the complete three dimensional dynamics of the the “beam-like” structure which includes the three “stiff” degrees-ofreedom transverse and dilational shear. Therefore, researchers have traditionally limited the difficulty of the problem with simplifying assumptions. This study, however, puts forward a solution which makes no simplifying assumptions and trades off only the real-time requirement of the desired solution. The solution is a Finite Difference Time Domain method employing an explicit single step method with cheap right hands sides. The cheap right hand sides are the result of a rather ingenious formulation of the classical beam called the Cosserat rod by, first, the Cosserat brothers and, later, Stuart S. Antman which results in five nonlinear but uncoupled equations that require only multiplication and addition. The method is therefore suitable for hardware implementation thus moving the real-time requirement from a software solution to a hardware solution.

Stuttgart, Univ., Diss., 2003.

One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation. The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation. The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.

Diese Arbeit liefert ein dreidimensionales numerisches Homogenisierungskonzept, welches beim Übergang von der Mikro- zur Makroskala einen Wechsel in der Kontinuumsbeschreibung beinhaltet. Während für die Beschreibung der Makroskala das verallgemeinerte Cosserat-Kontinuum verwendet wird, basiert die Mikroskala auf der klassischen Cauchy-Theorie. Um das homogene Cosserat-Ersatzmaterial im Rahmen numerischer Simulationen nutzen zu können, erfolgt die Implementierung geeigneter Finiter Elemente in das Programmsystem Abaqus und deren Verifikation. Neben der Diskussion der bei der Homogenisierung beobachteten Effekte werden anhand eines idealisierten Modells eines biaxialverstärkten Mehrlagengestrickes die Vorteile gegenüber der klassischen Herangehensweise aufgezeigt. / This contribution provides a threedimensional homogenization approach which includes the switch of the continuum theory during the scale transition. Whereas the microscopic scale is described in the framework of the classical Cauchy theory, the macroscopic scale is based on the generalized Cosserat continuum. In order to use the obtained homogeneous Cosserat material, suitable finite elements are implemented in the commercial program system Abaqus followed by an appropriate verification. Beside the discussion of the arising effects the advantages of this approach compared to the classical procedure are shown by means of an idealized model of a biaxial woven fabric.

[pt] O presente trabalho trata da aplicação da análise limite numérica (ALN) a problemas geotécnicos. Os meios (solo ou rocha) são considerados como contínuos convencionais e como contínuos de Cosserat. Da aplicação da formulação mista da análise limite e da discretização do meio por uma malha de elementos finitos é obtido um problema de programação matemática (PM). A aplicação desta metodologia nos contínuos de Cosserat (2D) fornece problemas de programação linear (PL) e nos contínuos convencionais (2D e 3D), problemas de programação não-linear (PNL). A solução do problema de PM foi através dos programas de otimização: LINDO (PL), LINGO (PNL), MINOS (PNL) e LANCELOT (PNL). Também foram implementados os algoritmos não lineares -Quase Newton com deflexão- e -Han-Powell-. A formulação é validada em problemas cuja solução analítica é conhecida ou em dados experimentais. Estes exemplos mostram a rapidez e a eficácia da ALN para a determinação da carga de colapso e do mecanismo de ruptura do problema. / [en] The present work treats of the application of the numerical limit analysis (NLA)to geomechanics problems. The soil or rock mass is considered as conventional continuous and Cosserat continuous. A mathematical programming (MP) problem is obtained through the application of the mixed formulation of limit analysis and the finite elements mesh. The application of this methodology in the Cosserat continuous (2D) supplies linear programming (LP) problems and in the conventional continuous (2D and 3D) nonlinear programming (NLP) problems. The solution of the problem of MP was through the LINDO (LP), LINGO (NLP), MINOS (NLP) and LANCELOT (NLP) programs. It was also implemented nonlinear algorithms -Quasi-Newton feasible point method- and -Han-Powell-.The formulation is validated in problems whose analytic solution is known or in experimental data. These examples show the speed and the effectiveness of NLA for the determination of the collapse load and of the mechanism of rupture of the problem. / [es] EL presente trabajo trata de la aplicación del análisis límite numérica (ALN) a problemas geotécnicos. Los medios (suelo o roca) son considerados como contínuos convencionales y como contínuos de Coserat. De la aplicación de la formulación mixta del análisis límite y de la discretización del medio por una malla de elementos finitos se obtiene un problema de programación matemática (PM). La aplicación de esta metodología en los contínuos de Coserat (2D) nos lleva a problemas de programación lineal (PL) y en los contínuos convencionales (2D y 3D), problemas de programación no lineal (PNL). La solución del problema de PM fue a través de los programas de optimización: LINDO (PL), LINGO (PNL), MINOS (PNL) y LANCELOT (PNL). También fueron implementados los algoritmos no lineares quase- Newton con deflexión y Han Powell . Se evalúa la formulación propuesta en problemas donde se conoce la solución analítica o en datos experimentales. Estos ejemplos muestran la rapidez y la eficacia de la ALN para la determinación de la carga de colapso y del mecanismo de ruptura del problema.

The term “granular material” describes any assembly of macroscopic particles. This broad definition encompasses a wide variety of everyday materials, for example sand, cereals, gravel and powders. However, despite their commonplace nature, to date no universally accepted set of constitutive equations exists to describe the behaviour of these materials. Thermomechanics and micromechanics are two modelling methodologies previously employed in separate efforts to represent granular behaviour. In this thesis, the two theories are integrated to develop new models of idealised granular materials. (For complete abstract open document)

Stuttgart, Univ., Diss., 2007.