Compression of elastomer elements are nonlinear due to the cross-linked molecular structure owing to a property known as hyperelasticity. Hyperelasticity is defined as the nonlinear stress-strain behavior shown by rubber like materials which can be strained up to a range of 700% in tension and up to 40% in compression. The stress-strain behavior is modeled by using different material models which predict the behavior very precisely.  Linear actuators from Cascade Drives AB uses a patented load sharing mechanism using elastically deformable elements to distribute the torque evenly between all the gears interacting with a common gear rack. An accurate model predicting the response of elastomer under compression has been developed in this thesis project. The elastomers were loaded in compression to provide flexibility for the system. First a static model was developed, where both a rectangular and a cylindrical roller model were analyzed.  The two models were derived using a continuum mechanics approach and the stiffness of the elastomers along with the torque output of the gearbox was calculated. A MATLAB model and an FEA model using ANSYS was created, and the results were compared. An error estimate between the MATLAB and FEA results for the rectangular and roller model was plotted for a certain β° of rotation of the gear. The models were also checked for different materials and the output torque for the different materials was plotted and analyzed. Finally, the experimental results were compared with the MATLAB results for the rectangular and roller models. The rectangle and roller model can be both used to predict the behaviour of using elastomers as the load sharing elements in applications. / Kompression av elastomerelement är olinjär till följd av den tvärbundna molekylära strukturen, en egenskap som kallas hyperelasticitet. Hyperelasticitet definieras som det icke-linjära spännings-töjningsbeteendet som uppvisas av gummiliknande material vilka kan töjas upp till av 700% och upp till komprimeras upp till 40%. Spänningsbelastningsbeteendet modelleras med hjälp av olika materialmodeller som förutsäger beteendet.  Cascade Drives linjäraktuatorer använder elastiskt komprimerbara element i sin lastfördelningsmekanism för att använda multipla plinjonger på ett kuggrack utan att få ett överbestämt system. Lastfördelningsmekanismen ger även en viss flexibilitet för systemet.  En modell som förutsäger responsen hos elastomerer under kompression har utvecklats i denna avhandling. Två geometriska former undersöktes modeller togs fram för både en rektangulär och cylindrisk rulle. De två modellerna härleddes med en kontinuummekanisk metod och elastomerernas styvhet beräknades. En MATLAB-modell och en FEM-modell i ANSYS skapades och resultaten jämfördes och en feluppskattning modellerna gjordes. Modellerna undersöktes också för olika material och utmattningsegenskaperna för de olika materialen analyserades.  Rektangel- eller rullmodellen kan båda användas för att förutsäga hur en elastomer skulle bete sig i en växellådsapplikation.

Die vorliegende Arbeit befasst sich mit der Entwicklung physikalisch fundierter, variations-basierter Modelle zur Beschreibung von magneto-aktiven Polymeren bei finiten Deformationen. Die Erarbeitung einer theoretisch und numerisch instruktiven Abhandlung zur Modellierung von nicht-dissipativen sowie dissipativen magneto-mechanischen Systemen stellt dabei einen zentralen Aspekt dieser Arbeit dar. Die konstitutive Modellierung erfolgt in kanonischer Form im Rahmen einer Theorie erster Ordnung auf Basis zweier skalarwertiger Tensorfunktionen -- der totalen Energiedichtefunktion und dem Dissipationspotential -- welche das komplexe konstitutive Verhalten des Systems vollständig charakterisieren. Die Entwicklung fundierter, konstitutiver Makroskalenmodelle für nicht-dissipative sowie dissipative magneto-aktive Polymere erfolgt dabei mikrostrukturgeleitet und im Falle nicht-dissipativer magneto-aktiver Polymere explizit auf Basis eines umfangreichen -- mittels numerischer Homogenisierung erzeugten -- Datensatzes. Das im Rahmen der konstitutiven Modellierung genutzte energie-basierte Setting stellt die natürliche Umgebung für Stabilitätsbetrachtungen dar und erlaubt die Diskussion der materiellen Stabilität der entwickelten konstitutiven Modelle auf Grundlage verallgemeinerter Konvexitätskonzepte. Im Mittelpunkt der numerischen Behandlung der vorliegenden Problemstellungen steht die Konstruktion primaler Variationsprinzipien auf Basis der entwickelten konstitutiven Funktionen in Verbindung mit deren konformen Finite-Elemente-Approximationen. Die zugehörigen diskreten Variationsformulierungen zeichnen sich durch ihre Minimaleigenschaften aus, wodurch diese nicht durch die diskrete inf-sup Stabilitätsbedingung eingeschränkt sind. Die Leistungsfähigkeit und Validität der entwickelten Modelle wird anhand von Benchmark- und Konvergenzstudien im Detail untersucht. Der Fokus weiterer numerischer Studien liegt auf der makrostrukturellen Analyse des magnetostriktiven und magnetorheologischen Effektes. / The present work covers the development of physically motivated, variational-based models for the description of magneto-active polymers at finite deformations. A key aspect of this thesis is to develop a unified theoretical and computational framework for the modeling of non-dissipative and dissipative magneto-active polymers. Within a classical first order theory, a strong emphasis is put on a unifying constitutive modeling framework based on two scalar valued tensor functions, namely the total energy density function and the dissipation potential. This two potential ansatz allows the canonical derivation of thermodynamically consistent constitutive models for a broad spectrum of complex systems in a mathematically elegant manner. The development of a family of constitutive macroscale models for non-dissipative and dissipative magneto-active polymers is based on a microstructure-guided approach, whereby the parametrization of the developed non-dissipative constitutive model is explicitly based on a comprehensive material data set generated via computational homogenization. The energy-based setting provides the natural environment for stability analysis and allows the discussion of the material stability of the developed constitutive models based on generalized notions of convexity. The focus of the variational and computational framework lies on the construction of primal variational principles and their conforming finite element approximations. The associated discrete variational formulations are characterized by their minimization structure and are therefore not restricted by the discrete inf-sup stability condition. The performance and validity of the developed models are investigated in detail by benchmark and convergence studies. The main emphasis of further numerical studies lies on the macrostructural analysis of the magnetostrictive and magnetorheological effect.

A constitutive model has been developed to model the shock response of single crystal aluminum from peak pressures ranging from 2-110 GPa. This model couples a description of higher-order thermoelasticity with a dislocation-based viscoplastic formulation, both of which are formulated for single crystals. The constitutive model has been implemented using two numerical methods: a plane wave method that tracks the propagating wave front; and an extended one-dimensional, finite-difference method that can be used to model spatio-temporal evolution of wave propagation in anisotropic materials. The constitutive model, as well as these numerical methods, are used to simulate shock wave propagation in single crystals, polycrystals, and pre-textured polycrystals. Model predictions are compared with extensive existing experimental data and are then used to quantify the influence of the initial material state on the subsequent shock response. A coarse-grained model is then proposed to capture orientation-dependent deformation heterogeneity, and is shown to replicate salient features predicted by direct finite-difference simulation of polycrystals in the weak shock regime. The work in this thesis establishes a general framework that can be used to quantify the influence of initial material state on subsequent shock behavior not only for aluminum single crystals, but for other face-centered cubic and lower symmetry crystalline metals as well.

A Shock wave as represented by the Riemann problem and a Point-blast explosion are two key phenomena involved in a supernova explosion. Any hydrocode used to simulate supernovae should be subjected to tests consisting of the Riemann problem and the Point-blast explosion. L. I. Sedov's solution of Point-blast explosion and Gary A. Sod's solution of a Riemann problem have been re-derived here from one dimensional fluid dynamics equations . Both these problems have been solved by using the idea of Self-similarity and Dimensional analysis. The main focus of my research was to subject the CHIMERA supernova code to these two hydrodynamic tests. Results of CHIMERA code for both the blast wave and Riemann problem have then been tested by comparing with the results of the analytic solution. / by Abu Salah M. Ahsan. / Thesis (M.S.)--Florida Atlantic University, 2008. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2008. Mode of access: World Wide Web.

As formulações baseadas na mecânica do contínuo, embora precisas até certo ponto, por vezes não podem ser utilizadas, ou não são conceitualmente corretas para o entendimento de fenômenos em escalas reduzidas. Estas limitações podem aparecer no estudo dos fenômenos tribológicos em escala nanométrica, que passam a necessitar de novos métodos experimentais, teóricos e computacionais que permitam explorar estes fenômenos com a resolução necessária. Simulações atomísticas são capazes de descrever fenômenos em pequena escala, porém, o número necessário de átomos modelados e, portanto, o custo computacional - geralmente torna-se bastante elevado. Por outro lado, os métodos de simulação associados à mecânica do contínuo são mais interessantes em relação ao custo computacional, mas não são precisos na escala atômica. A combinação entre essas duas abordagens pode, então, permitir uma compreensão mais realista dos fenômenos da tribologia. Neste trabalho, discutem-se os conceitos básicos e modelos de atrito em escala atômica e apresentam-se estudos, por meio de simulação numérica, para a análise e compreensão dos mecanismos de atrito e desgaste no contato entre materiais. O problema é abordado em diferentes escalas, e propõe-se uma abordagem conjunta entre a Mecânica do Contínuo e a Dinâmica Molecular. Para tanto, foram executadas simulações numéricas, com complexidade crescente, do contato entre superfícies, partindo-se de um primeiro modelo que simula o efeito de defeitos cristalinos no fenômeno de escorregamento puro, considerando a Dinâmica Molecular. Posteriormente, inseriu-se, nos modelos da mecânica do contínuo, considerações sobre o fenômeno de adesão. A validação dos resultados é feita pela comparação entre as duas abordagens e com a literatura. / Formulations based on continuum mechanics are generally accurate in a macroscale level, but sometimes cannot be used, or it is not conceptually correct to use for the understanding of small scale phenomena. These limitations may be observed in the study of tribological phenomena at the nanoscale, which may consequently require new experimental, theoretical and computational methods in order to analyze these phenomena with the required resolution. Atomistic simulations may describe phenomena at small scale, but the required number of atoms to be modeled, i.e. the computational cost, usually becomes very high. Moreover, simulation methods associated with continuum mechanics (such as the Finite Element Method - FEM) are more interesting in relation to computational cost, but they are not accurate for atomic scale studies. The combination of these two approaches can then enable a more realistic understanding of tribological phenomena. This work discusses the basic concepts of friction and models for friction at atomic scale. This work also presents studies, by means of numerical simulation, for the analysis of friction and wear mechanisms in the contact of materials. The problem is approached considering different scales, and basing the analysis both on Continuum Mechanics and Molecular Dynamics (MD). For both methods, numerical simulations with increasing complexity were conducted to reproduce the contact between surfaces, starting from an initial model that simulates the effect of crystalline defects during the MD analysis of pure slip. In a second stage, adhesion phenomenon were implemented through continuum mechanics models. The validation of the models and the coupling between the two approaches were conducted by comparison with literature.

Im Rahmen der geometrisch und physikalisch nichtlinearen Kontinuumsmechanik werden Beobachterkonzepte und Darstellungsformen tensorieller Größen diskutiert und zum Teil neu eingeführt, die insbesondere die Modellierung und FEM-Implementierung komplizierter Modelle für inelastisches Materialverhalten bei großen Deformationen wirkungsvoll unterstützen. / Observer concepts and several kinds of representations of tensorial quantities are discussed and partly introduced within the framework of geometrically und physically nonlinear continuum mechanics. They are intended to support the modelling of complicated inelastic materials undergoing large deformations.

Im Rahmen der geometrisch und physikalisch nichtlinearen Kontinuumsmechanik werden Beobachterkonzepte und Darstellungsformen tensorieller Größen diskutiert und zum Teil neu eingeführt, die insbesondere die Modellierung und FEM-Implementierung komplizierter Modelle für inelastisches Materialverhalten bei großen Deformationen wirkungsvoll unterstützen. / Observer concepts and several kinds of representations of tensorial quantities are discussed and partly introduced within the framework of geometrically und physically nonlinear continuum mechanics. They are intended to support the modelling of complicated inelastic materials undergoing large deformations.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The present work shows Cauchy's Theorem on its classical form, and it has the objective of weakening their statements, and provides aplications in continuum mechanics. The metodology used here is the axiomatic, that is, a presentation of basic concepts with fundamental results and theirs proofs. The main result here is Theorem 5, since it shows that for a Cauchy ux weakly balanced with density f there is a field such that f is linear in almost every points of R. The conclusion obtained is we can substitute the statement that f is a continuum function of position and have similar conclusions. / O presente trabalho apresenta o Teorema de Cauchy em sua forma clássica e tem por finalidade enfraquecer as suas hipóteses, proporcionando sua aplicação na mecânica do contínuo. A metodologia empregada é a axiomática, ou seja, é apresentada uma listagem de definições básicas com vistas ao desencadeamento lógico das demonstrações que foram realizadas para atingir os objetivos dessa dissertação. O resultado principal é teorema 15, pois mostra que para um Fluxo de Cauchy Fracamente Balanceado com densidade f existe um campo T:R−→L(R3) tal que f é linear em quase todos os pontos de R. A conclusão obtida é que podemos substituir a hipótese de que a referida função e contínua na variável espacial e obter conclusão semelhante.

Dans les régions de montagne, la prévision des évènements gravitaires reste un défi pour la gestion des risques. Des méthodes de calcul telles que la méthode des éléments discrets (DEM), où les particules interagissent les unes avec les autres pour restituer un comportement global d’une masse granulaire, ont été utilisées pour aborder ce type de problématique. L’application de la DEM reste normalement limitée aux évènements de petits volumes impliquant un nombre de blocs plutôt faible, puisque les temps de calcul peuvent devenir rapidement prohibitifs avec l’augmentation du nombre de particules. Les méthodes de calcul continues sont donc une alternative intéressante car elles permettent de réduire les temps de calcul. Elles nécessitent cependant la définition d’une loi de comportement macroscopique capable de représenter correctement les principaux traits de comportement mécanique du matériau au sein de la masse. L'objectif principal du travail de thèse réside dans le développement d’un outil numérique permettant de modéliser certains aléas gravitaires tels que les écoulements en masse. Notre choix s’est porté sur une méthode Lagrangienne-Eulérienne (méthode des points matériels – MPM) capable de gérer de grandes déformations tout en bénéficiant des principaux avantages de la méthode des éléments finis (FEM). La méthode utilise une grille Eulérienne fixe sur laquelle se déplacent des points matériels pendant les simulations. Un outil numérique, nommé MPMbox (2D et 3D), a été développé entièrement durant la thèse en C++. Le code a été validé à l'aide d'une série de solutions analytiques en quasi-statique (tests géotechniques standards) ainsi que par des applications de la littérature incluant des déformations importantes et rapides (tests d'affaissement). Après validation, le code a été confronté aux prédictions d’un outil de calcul DEM (DEMbox) dans le cadre de simulations numériques impliquant l'écoulement (initiation, régimes transitoires, propagation et arrêt) d'un matériau granulaire (particules sphero-polyhédriques) sur un plan incliné. Les résultats ont été comparés en termes de distance de propagation, de forme du dépôt et d'énergies dissipées à l'interface et dans la masse pendant l'écoulement. Pour les applications qui ont suivies, des éléments discrets ont été couplés à la MPM afin qu'un bloc rigide (DEM) puisse interagir avec un sol déformable (MPM). Cette application a consisté en l'analyse (2D) de la collision entre un bloc rocheux rigide (rond ou carré) et un sol bicouche élastoplastique. Les investigations ont été largement basées sur la mesure de coefficients de restitution (rapport des énergies cinétiques avant et après impact) qui reste difficile à déterminer expérimentalement. / In mountainous regions, the prediction of gravitational phenomena remains a challenge for the management of risk. Computational methods such as the Discrete Element Method (DEM) have been used for the modeling of these types of phenomena, where particles interact with each other to give an overall behavior of the mass. Its application can be somewhat restricted to small and medium number of blocks, since the computational time can easily become too large. Continuum analyses are therefore an attractive approach, which can reduce the computational times, but that rely on a constitutive law to represent the behavior within the mass. The main objective of this PhD was to develop a numerical tool that allowed the modeling of some specific gravitational hazards, such as the flowing of mass. A Lagrangian-Eulerian method such as the Material Point Method (MPM) is able to handle large deformations, while preserving most of the capabilities of the Finite Element Method (FEM). The method uses an Eulerian grid which is only used as a numerical scratch-pad, and remains fixed during simulations. A numerical tool named MPMbox (2D and 3D) was then developed from the ground up using C++. The code was validated using a series of analytical solutions for quasi-static analysis (some standard geotechnical tests), as well as simulations including large and rather rapid deformations (slump tests). After validation, the code was first used to make a numerical comparison with the DEM. In the comparison, a parametric survey was carried out during which the flow of a granular material on a sloped surface was simulated. Results were compared in terms of run-out distance, spread of the deposit and energy dissipated at the interface and within the mass during the flow. For a second study, discrete elements were coupled with MPM so that a rigid block could interact with a deformable soil. This application consisted in the (2D) analysis of the collision between a discrete block (round and squared) and a bounded elasto-plastic double-layered soil (soft over hard layers). The investigations were largely based on the measurement of the restitution coefficient (ratio of kinetic energies before to after the impact), which cannot be easily determined experimentally.

As formulações baseadas na mecânica do contínuo, embora precisas até certo ponto, por vezes não podem ser utilizadas, ou não são conceitualmente corretas para o entendimento de fenômenos em escalas reduzidas. Estas limitações podem aparecer no estudo dos fenômenos tribológicos em escala nanométrica, que passam a necessitar de novos métodos experimentais, teóricos e computacionais que permitam explorar estes fenômenos com a resolução necessária. Simulações atomísticas são capazes de descrever fenômenos em pequena escala, porém, o número necessário de átomos modelados e, portanto, o custo computacional - geralmente torna-se bastante elevado. Por outro lado, os métodos de simulação associados à mecânica do contínuo são mais interessantes em relação ao custo computacional, mas não são precisos na escala atômica. A combinação entre essas duas abordagens pode, então, permitir uma compreensão mais realista dos fenômenos da tribologia. Neste trabalho, discutem-se os conceitos básicos e modelos de atrito em escala atômica e apresentam-se estudos, por meio de simulação numérica, para a análise e compreensão dos mecanismos de atrito e desgaste no contato entre materiais. O problema é abordado em diferentes escalas, e propõe-se uma abordagem conjunta entre a Mecânica do Contínuo e a Dinâmica Molecular. Para tanto, foram executadas simulações numéricas, com complexidade crescente, do contato entre superfícies, partindo-se de um primeiro modelo que simula o efeito de defeitos cristalinos no fenômeno de escorregamento puro, considerando a Dinâmica Molecular. Posteriormente, inseriu-se, nos modelos da mecânica do contínuo, considerações sobre o fenômeno de adesão. A validação dos resultados é feita pela comparação entre as duas abordagens e com a literatura. / Formulations based on continuum mechanics are generally accurate in a macroscale level, but sometimes cannot be used, or it is not conceptually correct to use for the understanding of small scale phenomena. These limitations may be observed in the study of tribological phenomena at the nanoscale, which may consequently require new experimental, theoretical and computational methods in order to analyze these phenomena with the required resolution. Atomistic simulations may describe phenomena at small scale, but the required number of atoms to be modeled, i.e. the computational cost, usually becomes very high. Moreover, simulation methods associated with continuum mechanics (such as the Finite Element Method - FEM) are more interesting in relation to computational cost, but they are not accurate for atomic scale studies. The combination of these two approaches can then enable a more realistic understanding of tribological phenomena. This work discusses the basic concepts of friction and models for friction at atomic scale. This work also presents studies, by means of numerical simulation, for the analysis of friction and wear mechanisms in the contact of materials. The problem is approached considering different scales, and basing the analysis both on Continuum Mechanics and Molecular Dynamics (MD). For both methods, numerical simulations with increasing complexity were conducted to reproduce the contact between surfaces, starting from an initial model that simulates the effect of crystalline defects during the MD analysis of pure slip. In a second stage, adhesion phenomenon were implemented through continuum mechanics models. The validation of the models and the coupling between the two approaches were conducted by comparison with literature.