Ce travail concerne l'analyse de la vulnérabilité des structures de protection et des ouvrages sensibles en béton soumis à des actions dynamiques sévères (impacts, explosions) dues à des risques anthropiques d'origine accidentelle ou non. L'objet est la mise au point d'outils prévisionnels de simulation capables de décrire de manière objective le comportement dynamique du béton. Pour cela, une approche numérique novatrice reposant sur la méthode des Eléments Discrets (MED) est développée. Une première partie de cette thèse concerne la simulation des essais quasi-statiques de compression et traction uniaxiales. Une loi de transfert de moment (LTM) a été introduite pour pallier au problème de fragilité en compression simple. Ensuite, la procédure d'identification des paramètres du modèle modifié a été optimisée pour bien reproduire le comportement macroscopique du béton. Enfin, le modèle a été validé en représentant correctement le comportement quasi-statique de plusieurs types de béton. La deuxième partie du travail traite la simulation des essais de traction dynamique du béton aux barres de Hopkinson. Les résultats ont montré la nécessité de prendre l'effet de vitesse de déformation dû au matériau pour bien reproduire le comportement expérimental. Ensuite, Les paramètres du modèle permettant de reproduire cet effet de vitesse ont été identifiés. Enfin, des essais avec des taux de déformation très élevés ont été simulés et les résultats numériques ont été en accord avec le comportement observé expérimentalement. / This work concerns the analysis of the vulnerability of sensitive concrete structures subjected to severe dynamic actions such as impacts due to natural hazards or human factors. The object is to develop a numerical tool that can describe objectively the dynamic behaviour of concrete. Then, a 3D discrete element method (DEM) was developed and used to perform the analysis. The first part of this thesis focuses on the simulation of quasi-static uniaxial compression and traction tests. A moment transfer law (MTL) was introduced to overcome the problem of brittle compressive behavior. Then, the identification procedure of the modified DEM model has been optimized in order to reproduce very well the macroscopic behaviour of concrete. Finally, the model has been validated by representing properly the real quasi-static behavior of different types of concrete. The second part of the study deals with the simulation of the dynamic Hopkinson traction bar tests of concrete. The results showed that a local rate effect has to be introduced to reproduce the strain rate dependency, which would then be a material-intrinsic effect. Then, the parameters of the model have been identified. Finally, simulations were run at high strain rates and showed consistent results with respect to experimental behaviour.

L'objectif principal de ce travail de thèse est d'améliorer la prise en compte des peuplements de taillis dans les logiciels d'analyse trajectographique.Dans un premier temps, un modèle numérique permettant de créer des peuplements virtuels de taillis à l'échelle du versant a été développé et validé sur la base d'inventaires forestiers réalisés dans des taillis.Deux modèles numériques permettant de simuler des impacts de blocs sur des franc pieds et des cépées ont été également développés en utilisant la Méthode des Éléments Discrets (MED).Ces modèles ont été calibrés par des essais d'impact sur des tiges de hêtre.Ils permettent d'intégrer l'influence du houppier et du système racinaire, de modéliser explicitement le contact entre le bloc et les tiges impactées, et d'intégrer les non-linéarités matérielles (rupture des tiges, délaminage) se développant dans le tronc au cours de l'impact.Ces travaux ont conduit à la construction d'un modèle trajectographique MED permettant de simuler la propagation d'un bloc dans une forêt de taillis à l'échelle du versant. Finalement, le rôle protecteur de différents peuplements de taillis contre l'aléa de chute de bloc a été caractérisé à l'aide de ce modèle. / This research work is dedicated to improve the integration of coppice stands in rockfall analyses.First, a model was built to create virtual coppice stands. This model was validated using field inventories in coppice stands.Two numerical models were developed to simulate impacts of blocks on single trees and coppice stools using the Discrete Elements Method (MED).These models were calibrated using laboratory impact tests on beech stems.They account for the influence of the root system and of the crown on the tree dynamic response, the explicit modeling of the contact between the block and the impacted stem and the non-linearity evolution into the trunk during impact.Finally, a DEM rockfall software was developed to model rockfall propagation in coppice stands.The protective role against rockfall hazard of different coppice stands was characterized with this model.

Le formage de matériaux multiphasés comprend des mécanismes complexes en lien avec la rhéologie,la morphologie et la topologie des phases.Du point de vue numérique,la modélisation de ces phénomènes en résolvant les équations aux dérivées partielles (EDP) décrivant le comportement continu des phases n'est pas trivial.En effet,de nombreuses discontinuités associées aux phases se déplacent et peuvent interagir.Ces phénomènes peuvent être conceptuellement déclicats à intégrer au modèlecontinu et coûteux en termes de calcul.Dans cette thèse,la méthode des éléments discrets (DEM) est utilisée pour modéliser phénoménologiquement les déformations finies inélastiques dans les multi-matériaux.Les lois d'interactions attractive-répulsive sont imposées à des particules fictives,dont les ré-arrangements collectifs modélisent les déformations irréversibles de milieux continus.Le comportement numérique des empilements de particules est choisi pourreproduire des traits caractéristiques de la viscoplasticité parfaite etisochore:contrainte d'écoulement,sensibilité à la vitesse de déformation,conservation du volume.Les résultats d'essais de compression de bi-matériaux simples,simulés avec la DEM,sont comparés à la méthode des éléments finis (FEM) et sont en bon accord.Le modèle est entendu pour pouvoir supporter des sollicitations de traction.Une méthode de détection de contacts et d'auto-contacts d'objets physiques estproposée,basée sur l'approximation locale des surfaces libres.Les capacités de la méthodologie globale sont testées sur des mésostructurescomplexes,obtenues par tomographie aux rayons X.La compression à chaud d'un composite métallique dense est modélisée.La co-déformation peut être observées à l'échelle spatiale des phases.Deux cas de matériaux ``poreux'' sont considérés.Premièrement la simulation de la compression puis traction d'alliagesd'aluminium présentant des pores.Ces pores proviennent du coulage du matériau,leur fermeture et ré-ouverture mécanique est modélisée,y compris la coalescence à grande déformation.Deuxièmement la simulation de la compression de mousse métallique de faibledensité.Typiquement utilisée dans le but d'absorber de l'énergie mécanique,la compression jusqu'à densification provoque de nombreuses interactions entreles bras de matière. / Forming of multiphase materials involves complex mechanisms linked with therheology,morphology and topology of the phases.From a numerical point of view,modeling such phenomena by solving the partial differential equation (PDE) system accounting for thecontinuous behavior of the phases can be challenging.The description of the motion and the interaction of numerous discontinuities,associated with the phases,can be conceptually delicate and computationally costly.In this PhD,the discrete element method (DEM) is used to phenomenologically model finite inelastic strain inmulti-materials.This framework,natively suited for discrete phenomena,allows a flexible handling of morphological and topological changes.Ad hoc attractive-repulsive interaction laws are designed betweenfictitious particles,collectively rearranging to model irreversible strain in continuous media.The numerical behavior of a packing of particles can be tuned to mimic keyfeatures of isochoric perfect viscoplasticity:flow stress, strain rate sensitivity, volume conservation.The results for compression tests of simple bi-material configurations,simulated with the DEM,are compared to the finite element method (FEM) and show good agreement.The model is extended to cope with tensile loads.A method for the detection of contact and self-contact events of physicalobjects is proposed,based on a local approximation of the free surfaces.The potential of the general methodology is tested on complex mesostructuresobtained by X-ray tomography.The high temperature compression of a dense metallic composite is modeled.The co-deformation can be observed at the length scale of the phases.Two cases of ``porous'' material are considered.Firstly,the simulation of the compression and the tension of aluminum alloys with poresis investigated.These pores stem from the casting of the material,their closure and re-opening is modeled,including the potential coalescence occurring at large strain.Secondly,the compression of a metallic foam,with low relative density,is modeled.Typically used in energy absorption applications,the compression up to densification involves numerous interactions between thearms.

Un modèle numérique décrivant le comportement de fibres souples en suspension dans un écoulement de fluide en régime inertiel a été développé au moyen d'un couplage entre la méthode des éléments discrets et la méthode des volumes finis. Chaque fibre est discrétisée en plusieurs éléments de type poutre permettant de prendre en compte une déformation (flexion, torsion, allongement) et un mouvement de corps rigide. Les équations du mouvement des fibres sont résolues au moyen d'un schéma explicite du second ordre (temps et espace). Le mouvement de la phase fluide est décrit par les équations de Navier-Stokes, qui sont discrétisées et résolues au moyen d'un schéma aux volumes finis non structurés, d'ordre 4 (temps et espace). Le couplage entre la phase solide (discrète) et la phase fluide (continue) est obtenue par une pseudo méthode IBM (Immersed Boundary Method) dans laquelle l'effort hydrodynamique est calculé analytiquement. Plusieurs modèles de force hydrodynamique issus de la littérature sont analysés et leur validité ainsi que leurs limites sont identifiées. Pour des nombres de Reynolds (Re) correspondant au régime inertiel (0.01 < Re < 100, Re défini à l'échelle de la fibre), des formulations non-linéaires de la force hydrodynamique exercée par un écoulement uniforme sur un cylindre infini sont utilisées. Le couplage a aussi été utilisé pour des fibres rigides en écoulement de Stokes, en utilisant l'expression de la force de traînée issue de la théorie des corps élancés (`slender body theory'). Une expression du moment hydrodynamique par unité de longueur est obtenu à partir de simulations numériques par volumes finis de l'écoulement autour d'un cylindre élancé.Le modèle développé a été validé par comparaison avec plusieurs résultats expérimentaux et analytiques, du régime de Stokes (pour des fibres rigides) jusqu'aux régimes inertiels. Dans le cas du régime de Stokes, des simulations numériques du cisaillement de suspensions de fibres semi-diluées ont été réalisées. Le modèle développé permet de capturer les interactions hydrodynamiques et non-hydrodynamiques entre les fibres. Les interactions élasto-hydrodynamiques pour $Re$ fini ont été validées dans deux cas. Dans le premier cas, la flèche d'une fibre encastrée-libre dans un écoulement uniforme a été obtenu par calcul numérique et le résultat validé par comparaison aux résultats expérimentaux de la littérature. Dans le second cas, la conformation de fibres élancées et très déformables dans un écoulement turbulent homogène et isotrope a été obtenu par calcul numérique et le résultat validé par comparaison aux résultats expérimentaux de la littérature. Deux études numériques ont été réalisées pour étudier l'effet de la présence de fibres en suspension sur la turbulence au sein du fluide suspensif. Le modèle numérique a permis de reproduire le phénomène de réduction/amplification de la turbulence dans un écoulement en canal ou en conduite, dû à l'évolution microstructurale de la phase fibreuse. / A numerical model describing the behavior of flexible fibers under inertial flows was developed by coupling a discrete element solver with a finite volume solver.Each fiber is discretized into several beam segments, such that the fiber can bend, twist and rotate. The equations of the fiber motion were solved usinga second order accurate explicit scheme (space and time). The three dimensional Navier-Stokes equations describing the motion of the fluid phase was discretizedusing a fourth th order accurate (space and time) unstructured finite volume scheme. The coupling between the discrete fiber phase and the continuous fluid phasewas obtained by a pseudo immersed boundary method as the hydrodynamic force on the fiber segments were calculated based on analytical expressions.Several hydrodynamic force models were analyzed and their validity and short-comings were identified. For Reynolds numbers (Re) at the inertial regime(0.01 < Re < 100, Re defined at the fiber scale), non linear drag force formulations based on the flow past an infinite cylinder was used. For rigid fibers in creeping flow, the drag force formulation from the slender body theory was used. A per unit length hydrodynamic torque model for the fibers was derived from explicit numerical simulations of shear flow past a high aspect ratio cylinder. The developed model was validated against several experimental studies and analytical theories ranging from the creeping flow regime (for rigid fibers) to inertial regimes. In the creeping flow regime, numerical simulations of semi dilute rigid fiber suspensions in shear were performed.The developed model wasable to capture the fiber-fiber hydrodynamic and non-hydrodynamic interactions. The elasto-hydrodynamic interactions at finite Reynolds was validated with against two test cases. In the first test case, the deflection of the free end of a fiber in an uniform flow field was obtained numerically and the results were validated. In the second test case the conformation of long flexible fibers in homogeneous isotropic turbulence was obtained numerically and the results were compared with previous experiments. Two numerical studies were performed to verify the effects of the suspended fibers on carrier phase turbulence and the numerical model was able to reproduce the damping/enhancement phenomena of turbulence in channel and pipe flows as a consequence of the micro-structural evolution of the fibers.

The current simulation method for granular dynamics used by the physics engine AGX Dynamics is a nonsmooth variant of the popular Discrete Element Method (DEM). While powerful, there is a need for close to real time simulations of a higher spatial resolution than currently possible. In this thesis a data-driven model reduction approach using machine learning was considered. A data-driven simulation pipeline was presented and partially implemented. The method consists of sampling the velocity and density field of the granular particles and teaching a machine learning algorithm to predict the particles' interaction with a bulldozer blade as well as predicting the time evolution of its velocity field. A procedure for producing training scenarios and training data for the machine learning algorithm was implemented as well as several machine learning algorithms; a linear regressor, a multilayer perceptron and a convolutional neural network. The results showed that the method is promising, however further work will need to show whether or not the pipeline is feasible to implement in a simulation.

No presente trabalho é apresentada uma versão do Método dos Elementos Discretos formado por barras (DEM) no estudo do colapso de estruturas. O Método dos Elementos Discretos foi introduzido, especialmente, para a simulação numérica de problemas de dano e fratura. Esse método tem habilidade natural para introduzir descontinuidades de uma maneira muito direta e intuitiva. Além disso, métodos discretos oferecem uma estrutura conveniente para dar conta da desordem da microestrutura do material por meio de modelos estatísticos. A versão do DEM utilizada neste trabalho consiste na discretização do contínuo em barras que formam uma treliça espacial regular, onde massas equivalentes são concentradas nos nós, e as rigidezes das barras são equivalentes ao contínuo que tentam representar. Leis uniaxiais de dano permitem modelar fratura e dano anisotrópico com relativa facilidade. Esta versão foi amplamente testada em diversos campos da Engenharia, entre eles, problemas dinâmicos, de impacto, geração e propagação de sismos, estudo de efeito de escala em rochas e concreto, análise da microestrutura de materiais. Este trabalho apresenta dois grandes temas, nos quais foram realizadas implementações no DEM que aumentam sua aplicabilidade. Também são implementadas modificações nas leis constitutivas antes utilizadas, e apresentadas, também, novas leis para dar flexibilidade na calibração dos modelos. São comparados os resultados utilizando as diversas leis na análise do efeito de escala de placas submetidas à tração. Também são analisados os resultados obtidos sob a óptica da teoria de escala multifractal. Neste campo, encontram-se respostas muito interessantes que explicam os mecanismos de fratura, assim como dão uma noção de que alterações deveriam ser realizadas no DEM para conseguir que o método fique completamente objetivo em relação à escala. Nesse processo, estudam-se diferentes formas de obter as dimensões fractais de placas de rocha submetidas à tração e é analisada a influência de alguns dos parâmetros do DEM, além da relação constitutiva utilizada. Finalmente, o DEM é introduzido dentro do sistema comercial Abaqus, com objetivo de resolver problemas com grande quantidade de graus de liberdade ou com as condições de contorno ou de carregamento muito complexos. Apresentam-se exemplos de validação e exemplos de aplicação que mostram as vantagens das inovações realizadas. / This paper presents a version of the Truss-like Discrete Element Method in the study of the collapse of structures. The discrete element method was introduced especially for numerical simulation of fracture and damage problems. This method has the natural ability to introduce discontinuities in a very direct and intuitive way. Moreover, discrete methods offer a convenient framework to account for the disorder of the material microstructure by means of statistical models. The truss-like Discrete Element Method (DEM) used in this work represents the continuum by means of a periodic spatial arrangement of bars with the masses lumped at their ends. The rigidity of the bars is equivalent to the continuum to trying to represent. Uniaxial damage Laws allow model fracture and anisotropic damage with relative ease. This version was widely tested in various engineering fields including: dynamics problems, impact, generation and propagation of earthquakes, study of scale effect in rock and concrete analysis of the microstructure of materials. This work presents two major issues in witch were performed DEM implementations that increase its applicability. To obtain a better description of the model modifications in the constitutive laws are implemented and new ones are presented. The scale effect results of plates of rock subjected to traction obtained to different laws are compared. These obtained results are examined under the Multifractal scaling law theory. In this field, very interesting answers that explain the mechanisms of fracture are found. They gives some notions of which changes should be made in DEM to obtain a fully scale objective method. In the process, different ways to obtain the fractal dimension of rock plates subjected to traction are studied. The influence of some DEM parameter and constitutive laws are also analyzed. Finally, the DEM has been implemented within the commercial system ABAQUS to solve problems with a large number of degrees of freedom or very complex contour or loading conditions. Presents examples of validation and application that show the benefits of innovations through.

No presente trabalho é apresentada uma versão do Método dos Elementos Discretos formado por barras (DEM) no estudo do colapso de estruturas. O Método dos Elementos Discretos foi introduzido, especialmente, para a simulação numérica de problemas de dano e fratura. Esse método tem habilidade natural para introduzir descontinuidades de uma maneira muito direta e intuitiva. Além disso, métodos discretos oferecem uma estrutura conveniente para dar conta da desordem da microestrutura do material por meio de modelos estatísticos. A versão do DEM utilizada neste trabalho consiste na discretização do contínuo em barras que formam uma treliça espacial regular, onde massas equivalentes são concentradas nos nós, e as rigidezes das barras são equivalentes ao contínuo que tentam representar. Leis uniaxiais de dano permitem modelar fratura e dano anisotrópico com relativa facilidade. Esta versão foi amplamente testada em diversos campos da Engenharia, entre eles, problemas dinâmicos, de impacto, geração e propagação de sismos, estudo de efeito de escala em rochas e concreto, análise da microestrutura de materiais. Este trabalho apresenta dois grandes temas, nos quais foram realizadas implementações no DEM que aumentam sua aplicabilidade. Também são implementadas modificações nas leis constitutivas antes utilizadas, e apresentadas, também, novas leis para dar flexibilidade na calibração dos modelos. São comparados os resultados utilizando as diversas leis na análise do efeito de escala de placas submetidas à tração. Também são analisados os resultados obtidos sob a óptica da teoria de escala multifractal. Neste campo, encontram-se respostas muito interessantes que explicam os mecanismos de fratura, assim como dão uma noção de que alterações deveriam ser realizadas no DEM para conseguir que o método fique completamente objetivo em relação à escala. Nesse processo, estudam-se diferentes formas de obter as dimensões fractais de placas de rocha submetidas à tração e é analisada a influência de alguns dos parâmetros do DEM, além da relação constitutiva utilizada. Finalmente, o DEM é introduzido dentro do sistema comercial Abaqus, com objetivo de resolver problemas com grande quantidade de graus de liberdade ou com as condições de contorno ou de carregamento muito complexos. Apresentam-se exemplos de validação e exemplos de aplicação que mostram as vantagens das inovações realizadas. / This paper presents a version of the Truss-like Discrete Element Method in the study of the collapse of structures. The discrete element method was introduced especially for numerical simulation of fracture and damage problems. This method has the natural ability to introduce discontinuities in a very direct and intuitive way. Moreover, discrete methods offer a convenient framework to account for the disorder of the material microstructure by means of statistical models. The truss-like Discrete Element Method (DEM) used in this work represents the continuum by means of a periodic spatial arrangement of bars with the masses lumped at their ends. The rigidity of the bars is equivalent to the continuum to trying to represent. Uniaxial damage Laws allow model fracture and anisotropic damage with relative ease. This version was widely tested in various engineering fields including: dynamics problems, impact, generation and propagation of earthquakes, study of scale effect in rock and concrete analysis of the microstructure of materials. This work presents two major issues in witch were performed DEM implementations that increase its applicability. To obtain a better description of the model modifications in the constitutive laws are implemented and new ones are presented. The scale effect results of plates of rock subjected to traction obtained to different laws are compared. These obtained results are examined under the Multifractal scaling law theory. In this field, very interesting answers that explain the mechanisms of fracture are found. They gives some notions of which changes should be made in DEM to obtain a fully scale objective method. In the process, different ways to obtain the fractal dimension of rock plates subjected to traction are studied. The influence of some DEM parameter and constitutive laws are also analyzed. Finally, the DEM has been implemented within the commercial system ABAQUS to solve problems with a large number of degrees of freedom or very complex contour or loading conditions. Presents examples of validation and application that show the benefits of innovations through.

While tribology involves the study of friction, wear, and lubrication of interacting surfaces, the tribosurfaces are the pair of surfaces in sliding contact with a fluid (or particulate) media between them. The ubiquitous nature of tribology is evident from the usage of its principles in all aspects of life, such as the friction promoting behavior of shoes on slippery water-lubricated walkways and tires on roadways to the wear of fingernails during filing or engine walls during operations. These tribosurface interfaces, due to the small length scales, are difficult to model for contact mechanics, fluid mechanics and particle dynamics, be it via theory, experiments or computations. Also, there is no simple constitutive law for a tribosurface with a particulate media. Thus, when trying to model such a tribosurface, there is a need to calibrate the particulate media against one or more property characterizing experiments. Such a calibrated media, which is the “virtual avatar” of the real particulate media, can then be used to provide predictions about its behavior in engineering applications. This thesis proposes and attempts to validate an approach that leverages experiments and modeling, which comprises of physics-based modeling and machine learning enabled surrogate modeling, to study particulate media in two key particle matrix industries: metal powder-bed additive manufacturing (in Part II), and energy resource rock drilling (in Part III). The physics-based modeling framework developed in this thesis is called the Particle-Surface Tribology Analysis Code (P-STAC) and has the physics of particle dynamics, fluid mechanics and particle-fluid-structure interaction. The Computational Particle Dynamics (CPD) is solved by using the industry standard Discrete Element Method (DEM) and the Computational Fluid Dynamics (CFD) is solved by using finite difference discretization scheme based on Chorin's projection method and staggered grids. Particle-structure interactions are accounted for by using a state-of-the art Particle Tessellated Surface Interaction Scheme and the fluid-structure interaction is accounted for by using the Immersed Boundary Method (IBM). Surrogate modeling is carried out using back propagation neural network. The tribosurface interactions encountered during the spreading step of the powder-bed additive manufacturing (AM) process which involve a sliding spreader (rolling and sliding for a roller) and particulate media consisting of metal AM powder, have been studied in Part II. To understand the constitutive behavior of metal AM powders, detailed rheometry experiments have been conducted in Chapter 5. CPD module of P-STAC is used to simulate the rheometry of an industry grade AM powder (100-250microns Ti-6Al-4V), to determine a calibrated virtual avatar of the real AM powder (Chapter 6). This monodispersed virtual avatar is used to perform virtual spreading on smooth and rough substrates in Chapter 7. The effect of polydispersity in DEM modeling is studied in Chapter 8. A polydispersed virtual avatar of the aforementioned AM powder has been observed to provide better validation against single layer spreading experiments than the monodispersed virtual avatar. This experimentally validated polydispersed virtual avatar has been used to perform a battery of spreading simulations covering the range of spreader speeds. Then a machine learning enabled surrogate model, using back propagation neural network, has been trained to study the spreading results generated by P-STAC and provide much more data by performing regression. This surrogate model is used to generate spreading process maps linking the 3D printer inputs of spreader speeds to spread layer properties of roughness and porosity. Such maps (Chapters 7 and 8) can be used by a 3D-printer technician to determine the spreader speed setting which corresponds to the desired spread layer properties and has the maximum spread throughout. The tribosurface interactions encountered during the drilling of energy resource rocks which involve a rotary and impacting contact of the drill bit with the rock formation in the presence of drilling fluids have been studied in Part III. This problem involves sliding surfaces with fluid (drilling mud) and particulate media (intact and drilled rock particles). Again, like the AM powder, the particulate media, viz. the rock formation being drilled into, does not have a simple and a well-defined constitutive law. An index test detailed in ASTM D 5731 can be used as a characterization test while trying to model a rock using bonded particle DEM. A model to generate weak concrete-like virtual rock which can be considered to be a mathematical representation of a sandstone has been introduced in Chapter 10. Benchtop drilling experiments have been carried out on two sandstones (Castlegate sandstone from the energy rich state of Texas and Crab Orchard sandstone from Tennessee) in Chapter 11. Virtual drilling has been carried out on the aforementioned weak concrete-like virtual rock. The rate of penetration (RoP) of the drill bit has been found to be directly proportional to the weight on bit (WoB). The drilling in dry conditions resulted in a higher RoP than the one which involved the use of water as the drilling fluid. P-SATC with the bonded DEM and CFD modules was able to predict both these findings but only qualitatively (Chapter 11)

Este trabalho apresenta uma formulação do Método dos Elementos Discretos (MED) utilizando uma abordagem vetorial para o tratamento das rotações. As rotações são calculadas com a parametrização de Rodrigues. As principais contribuições do trabalho são: o cálculo dos deslocamentos tangentes utilizando o vetor das rotações incrementais da parametrização de Rodrigues; e, a integração do movimento de rotação utilizando o método leapfrog com as expressões da parametrização das rotações de Rodrigues. A formulação é apresentada para partículas esféricas e superelipsóides. O cálculo do deslocamento tangente, que é utilizado para o cálculo das forças de atrito, é feito a partir da velocidade angular da partícula. Em geral, o deslocamento tangente é calculado a partir da velocidade linear instantânea do ponto de contato. Aqui, o deslocamento do ponto de contato é dado pelo movimento da partícula, tanto de translação quanto de rotação. Apesar da abordagem por meio de rotações, é mostrado este cálculo pode ser feito sem o uso de tensores de segunda ordem. O movimento da partícula é descrito por uma abordagem incremental. É apresentada uma formulação do método de integração leapfrog com a utilização da expressão das rotações sucessivas da parametrização de Rodrigues. A detecção do contato entre superelipsóides é feita por um método do tipo \"vetor normal comum\", resolvido como um problema de minimização. Os resultados mostram que a parametrização de Rodrigues pode ser utilizada com método dos elementos discretos tanto para a execução da rotação quanto para o cálculo de grandezas que envolvem este tipo de movimento como o deslocamento tangente. / This work presents a formulation for Discrete Element Method (DEM) adopting a vector ap-proach to solve rotations. Herein, rotations are solved using Rodrigues parameterization. The main contributions of this work are: tangential displacements using the incremental rotation vector from Rodrigues parameterization, and integration of the rotation movement using leap-frog method and Rodrigues rotation tensor. The formulations are presented to spheres and superelliptical particles. Tangential displacements, which are used to get friction forces, are calculated through angular velocity. In most of DEM implementations, tangential displacements are calculated through the instantaneous linear velocity of the contact point. Instead, here the displacement of the contact point is given through the rotation of the particle. It is showed that the vector of in-cremental rotations can be calculated through the angular velocity. Particle movement is described using an updated Lagrangian approach. Leapfrog method is formulated in such a way to use the Rodrigues expression for successive rotations. Contact detection between superellipsoids is solved using a technic called \"common normal approach\", and it is solved as a minimization problem. The results show that the Rodrigues parameterization can be applied to discrete element method to both execute rotations and to evaluate physical quantities that are related to this kind of movement as tangential displacement.

No presente trabalho é apresentada uma versão do Método dos Elementos Discretos formado por barras (DEM) no estudo do colapso de estruturas. O Método dos Elementos Discretos foi introduzido, especialmente, para a simulação numérica de problemas de dano e fratura. Esse método tem habilidade natural para introduzir descontinuidades de uma maneira muito direta e intuitiva. Além disso, métodos discretos oferecem uma estrutura conveniente para dar conta da desordem da microestrutura do material por meio de modelos estatísticos. A versão do DEM utilizada neste trabalho consiste na discretização do contínuo em barras que formam uma treliça espacial regular, onde massas equivalentes são concentradas nos nós, e as rigidezes das barras são equivalentes ao contínuo que tentam representar. Leis uniaxiais de dano permitem modelar fratura e dano anisotrópico com relativa facilidade. Esta versão foi amplamente testada em diversos campos da Engenharia, entre eles, problemas dinâmicos, de impacto, geração e propagação de sismos, estudo de efeito de escala em rochas e concreto, análise da microestrutura de materiais. Este trabalho apresenta dois grandes temas, nos quais foram realizadas implementações no DEM que aumentam sua aplicabilidade. Também são implementadas modificações nas leis constitutivas antes utilizadas, e apresentadas, também, novas leis para dar flexibilidade na calibração dos modelos. São comparados os resultados utilizando as diversas leis na análise do efeito de escala de placas submetidas à tração. Também são analisados os resultados obtidos sob a óptica da teoria de escala multifractal. Neste campo, encontram-se respostas muito interessantes que explicam os mecanismos de fratura, assim como dão uma noção de que alterações deveriam ser realizadas no DEM para conseguir que o método fique completamente objetivo em relação à escala. Nesse processo, estudam-se diferentes formas de obter as dimensões fractais de placas de rocha submetidas à tração e é analisada a influência de alguns dos parâmetros do DEM, além da relação constitutiva utilizada. Finalmente, o DEM é introduzido dentro do sistema comercial Abaqus, com objetivo de resolver problemas com grande quantidade de graus de liberdade ou com as condições de contorno ou de carregamento muito complexos. Apresentam-se exemplos de validação e exemplos de aplicação que mostram as vantagens das inovações realizadas. / This paper presents a version of the Truss-like Discrete Element Method in the study of the collapse of structures. The discrete element method was introduced especially for numerical simulation of fracture and damage problems. This method has the natural ability to introduce discontinuities in a very direct and intuitive way. Moreover, discrete methods offer a convenient framework to account for the disorder of the material microstructure by means of statistical models. The truss-like Discrete Element Method (DEM) used in this work represents the continuum by means of a periodic spatial arrangement of bars with the masses lumped at their ends. The rigidity of the bars is equivalent to the continuum to trying to represent. Uniaxial damage Laws allow model fracture and anisotropic damage with relative ease. This version was widely tested in various engineering fields including: dynamics problems, impact, generation and propagation of earthquakes, study of scale effect in rock and concrete analysis of the microstructure of materials. This work presents two major issues in witch were performed DEM implementations that increase its applicability. To obtain a better description of the model modifications in the constitutive laws are implemented and new ones are presented. The scale effect results of plates of rock subjected to traction obtained to different laws are compared. These obtained results are examined under the Multifractal scaling law theory. In this field, very interesting answers that explain the mechanisms of fracture are found. They gives some notions of which changes should be made in DEM to obtain a fully scale objective method. In the process, different ways to obtain the fractal dimension of rock plates subjected to traction are studied. The influence of some DEM parameter and constitutive laws are also analyzed. Finally, the DEM has been implemented within the commercial system ABAQUS to solve problems with a large number of degrees of freedom or very complex contour or loading conditions. Presents examples of validation and application that show the benefits of innovations through.