Este trabalho propõe estudar o projeto de estruturas submetidas a grandes deslocamentos utilizando o Método de Otimização Topológica (MOT). O MOT é um método numérico capaz de fornecer de forma sistemática a distribuição ótima de material no domínio de uma estrutura de forma a atender a um dado requisito de projeto, por exemplo, o valor de flexibilidade máxima permitida em uma estrutura. Desde sua introdução, há quase três décadas, o MOT ganhou popularidade na área acadêmica e na indústria. Até o presente momento (2011), a maioria dos trabalhos relacionados com o método tem se preocupado com a otimização de estruturas com o comportamento linear, ou seja, pequenos deslocamentos. Um pequeno número de artigos e trabalhos tem sido relacionado com a modelagem e otimização topológica de estruturas submetidas a efeitos não-lineares. Este trabalho propõe compilar as formulações descritas na literatura e agregar novas técnicas na implementação da OT de forma a melhorar a robustez na obtenção de resultados sob não-linearidade geométrica. O MOT para o comportamento não-linear geométrico neste trabalho foi implementado utilizando o modelo de material SIMP. O comportamento não-linear geométrico é representado utilizando a formulação Lagrangiana para as leis de material de Kirchhoff-Saint Venant e neo-Hookiana. Ambas as leis de material foram implementadas utilizando o método de elementos finitos (MEF) e o equilíbrio estático da estrutura é obtido através de uma rotina incremental e iterativa de Newton incluindo todos os elementos (inclusive os de baixa densidade) dentro do domínio de projeto. A sensibilidade da função objetivo é deduzida utilizando o método adjunto e o problema de otimização é resolvido utilizando o Método das Assíntotas Móveis (MAM) em conjunto com uma função de Relaxação proposta para estabilizar a solução de OT não-linear. A função de projeção não-linear em conjunto com o Método da Continuação é utilizada para eliminar o problema de tabuleiro e independência de malha, melhorando a convergência dos resultados. A função objetivo para minimização da flexibilidade no ponto de aplicação do carregamento é testada, considerando um carregamento fixo. Neste trabalho, os exemplos mostram que as diferenças na rigidez das estruturas otimizadas utilizando modelagem linear e não-linear são geralmente pequenas para pequenos carregamentos, mas elas podem ser grandes em certos casos envolvendo grandes cargas, acarretando em instabilidades na estrutura, o que pode degenerar a solução obtida. / This work proposes studying the design of structures undergoing large displacement using Topology Optimization Method (TOM). The TOM is a numerical method capable of synthesizing the basic layout of a mechanical structure accomplishing to a given design requirement, for example the maximum strain energy allowed in the structure. Since its introduction nearly three decades, TOM has gained widespread popularity in academia and industry. So far, most papers dealing with the method have been concerned with the optimization of structures with linear geometric and material behavior. Even now a small number of works and articles have been concerned with the modeling and topology optimization of structures undergoing nonlinear effects. This work proposes to compile the formulations described in the literature and adding new techniques to improve the robustness for obtaining results of OT under geometric nonlinearity. The TOM for geometric nonlinear behavior in this work is implemented with Solid Isotropic Microstructure with Penalization (SIMP) material model. The geometrically nonlinear behavior of the structures is modeled using a Lagrangean description for hyperelastic constitutive models for Saint Venant-Kirchhoff and neo-Hookean. Both constitutive models are implemented using the Finite Element Method (FEM) and the static equilibrium of the structure is obtained using an incremental and iterative Full-Newton Method considering all elements and internal force of the design domain (elements called \"voids\"). The sensitivity of the objective function is derived using the adjoint method and the optimization problem is solved using the Optimality Criteria (OC) method and Method of Moving Asymptotes (MMA) together with a Relaxation Function proposed to stabilize the TO nonlinear solution. The nonlinear projection function in conjunction with the Continuation Method is used to obtain checkerboard-free and mesh-independent designs and to improve the convergence results. The objective function of end-compliance is tested, by minimizing it for a fixed load. In this work, some examples show that differences in stiffness of optimized structures using linear and nonlinear modeling are generally small, however they can be large in certain cases involving buckling or bifurcation point, that degenerate the solution obtained.

La simulation de la mise en forme des films polymères de décoration par le procédé de thermoformage a plusieurs objectifs. Elle permet de déterminer la faisabilité, ou les conditions de cette faisabilité et surtout elle permet de résoudre les nombreuses problématiques concernant la maîtrise des propriétés finales du film de décoration, telles que sa distribution d'épaisseur. Les simulations évitent les coûteuses études expérimentales par essais-erreurs. Le travail présenté dans ce document concerne les deux étapes principales de la mise en forme des films polymères par thermoformage, à savoir, l’étape de chauffage thermique par infrarouge et l’étape dite de formage. Les apports de ce travail sont les suivants : le développement d’une loi de comportement viscoélastique isotrope non isotherme, permettant de décrire le comportement mécanique du film polymère au cours de sa déformation. La simulation de l’étape de chauffage infrarouge permettant la mise en forme des films polymères à la température calculée par la prise en compte de la loi de comportement mécanique proposée. Enfin, un ensemble de simulations de mise en forme ont été réalisés et validés par comparaison avec des essais expérimentaux. / The simulation of the forming process of polymer films for thermoforming process has several objectives. It allows to determine feasibility or the conditions of this feasibility and above all it allows to know the thickness distribution of the deformed film of decoration. Simulations avoid the expensive experimental studies by test-errors. The work presented in this document relates to the two steps of the thermoforming of polymer films. The first step consists in heating the sheet using infrared lamps and the second step consists in forming the sheet into a mold. The contributions of this work are as follows: development of numerical modelling of the thermoforming process for non-isothermal viscoelastic sheet under large strains. Simulation of the heating step with taken into account the heterogeneous radiative heat transfer due to the shape of the tools. Finally, a set of simulations of forming processes was realized and the results of the simulations are compared to the results of experiments.

La thèse présente une méthodologie pour construire des éléments finis de type « solide-coque » avec intégration réduite en vue des applications à la simulation de la mise en forme tel que l’emboutissage des tôles où ces éléments finis doivent présenter de bonnes aptitudes à modéliser la flexion mais également les situations de laminage de la tôle. A partir des éléments volumiques à 8 nœuds et 3 degrés de liberté par nœud (les 3 composantes du déplacement), un neuvième nœud est rajouté au centre de l’élément. Ce neuvième nœud n’est pourvu que d’un seul degré de liberté, le déplacement le long de la direction de l’épaisseur. Cette direction privilégiée a un nombre de points d’intégration supérieur ou égal à 3 mais l’intégration est réduite au centre de l’élément diminuant très sensiblement les temps CPU par rapport à une intégration complète. Un soin particulier a été pris pour contrôler tous les modes à énergie nulle dus à l’intégration réduite. Ce nœud supplémentaire permet une distribution linéaire de la déformation normale. Avec les lois de comportement complètement 3D ces nouveaux éléments solide-coque donnent des résultats similaires en flexion à ceux obtenus avec des éléments coques et état plan de contrainte. Le neuvième nœud joue le rôle d’un paramètre supplémentaire pour l’interpolation quadratique du déplacement dans la direction de l’épaisseur. Ce degré de liberté a une signification physique et un effort équivalent à une pression normale peut être prescrit. Dans les situations de pression normale et dans le cas du contact, la contrainte normale obtenue est physique ce qui n’est pas le cas de nombreux éléments solide-coque de la littérature. Le pincement ou le laminage des tôles est correctement modélisé. Pour valider ces éléments, un module d’emboutissage en U avec passage et laminage de la bande de tôle sur des rouleaux a été construit au laboratoire. La comparaison entre les efforts d’emboutissage calculés et mesurés est très bonne ainsi que la géométrie des bandes de tôle obtenue après retour élastique. / This thesis presents a methodology for developing under-integrated “solid-shell” finite elements for sheet forming simulations like deep drawing where these elements must offer a bending capability and sheet thinning conditions as well. Starting from 8-node elements endowing three degrees of freedom per node (three displacement components), a ninth node is added at the centre of the element. This extra node has just one degree of freedom: a displacement along the ‘thickness’ direction. Several integration points are distributed along this privileged direction (5 points, generally) but the in-plane reduced integration at the centre of the element decreases CPU costs compared to a full integration. A special care has been taken to control all zero-energy modes due to the reduced integration. This additional node allows a linear distribution of the normal strain. With fully-3D constitutive laws, these new solid-shell elements give similar bending results as those obtained with shell elements and a plane stress state hypothesis. This ninth node acts as an additional parameter for the quadratic interpolation of the displacement in the ‘thickness’ direction. The corresponding degree of freedom has a physical meaning and a force, equivalent to a normal pressure for instance, may be prescribed. In situations of a normal pressure and in the case of contact, the obtained normal stress is physically defined, which is not the case for many solid-shell elements found in the literature. The pinching (or the thinning) of sheets is properly modelled. To validate these elements, an apparatus for U-drawing tests with ironing or thinning on strip sheets has been built in the laboratory. The comparison between numerical and experimental punch force during sheet forming is pretty good as well as the geometry of blank after springback.

The research presented in this work addresses open questions regarding (i) the fundamental understanding of powder compaction, and (ii) the complex mechanical response of particle-binder composites under large deformations. This work thus benefits a broad range of industries, from the pharmaceutical industry and its recent efforts on continuous manufacturing of solid tablets, to the defense and energy industries and the recurrent need to predict the performance of energetic materials. Powder compacts and particle-binder composites are essentially confined particulate systems with significant heterogeneity at the meso (particle) scale. While particle mechanics strategies for modeling evolution of mesoscale microstructure during powder compaction depend on the employed contact formulation to accurately predict macroscopic quantities like punch and die wall pressures, modeling of highly nonlinear, strain-path dependent macroscopic response without a distinctive yield surface, typical of particle-binder composites, requires proper constitutive modeling of these complex deformation mechanisms. Moreover, continued loading of particle-binder composites over their operational life may introduce significant undesirable changes to their microstructure and mechanical properties. These challenges are addressed with a combined effort on theoretical, modeling and experimental fronts, namely, (a) novel contact formulations for elasto-plastic particles under high levels of confinement, (b) a multi-scale experimental procedure for assessing changes in microstructure and mechanical behavior of particle-binder composites due to cyclic loading and time-recovery, and (c) a finite strain nonlinear elastic, endochronic plastic constitutive formulation for particle-binder composites.

In this thesis a theory for large deformation of plates is presented. Herein aspects of the common 3D-theory for large deformation with the Kirchhoff hypothesis for reducing the dimension from 3D to 2D is combined. Even though the Kirchhoff assumption was developed for small strain and linear material laws, the deformation of thin plates made of isotropic non-linear material was investigated in a numerical experiment. Finally a heavily deformed shell without any change in thickness arises. This way of modeling leads to a two-dimensional strain tensor essentially depending on the first two fundamental forms of the deformed mid surface. Minimizing the resulting deformation energy one ends up with a nonlinear equation system defining the unknown displacement vector U. The aim of this thesis was to apply the incremental Newton technique with a conformal, C^1-continuous finite element discretization. For this the computation of the second derivative of the energy functional is the key difficulty and the most time consuming part of the algorithm. The practicability and fast convergence are demonstrated by different numerical experiments.:1 Introduction 2 The deformation problem in the three-dimensional space 2.1 General differential geometry of deformation in the three-dimensional space 2.2 Equilibrium of forces 2.3 Material laws 2.4 The weak formulation 3 Newton’s method 3.1 The modified Newton algorithm 3.2 Second linearization of the energy functional 4 Differential geometry of shells 4.1 The initial mid surface 4.2 The initial shell 4.3 The plate as an exception of a shell 4.4 Kirchhoff assumption and the deformed shell 4.4.1 Differential geometry of the deformed shell 4.4.2 The Lagrangian strain tensor of the deformed plate 5 Shell energy and boundary conditions 5.1 The resulting Kirchhoff deformation energy 5.2 Boundary conditions 5.3 The resulting weak formulation 6 Newton’s method and implementation 6.1 Newton algorithm 6.2 Finite Element Method (FEM) 6.2.1 Bogner-Fox-Schmidt (BFS) elements 6.2.2 Hsiegh-Clough-Tocher (HCT) elements 6.3 Efficient solution of the linear systems of equation 6.4 Implementation 6.5 Newton’s method and mesh refinement 7 Numerical examples 7.1 Plate deflection 7.1.1 Approximation with FEM using BFS-elements 7.1.2 Approximation with FEM using reduced HCT-elements 7.2 Bending-dominated deformation 7.2.1 Approximation with FEM using BFS-elements 7.2.1.1 1st example: Cylinder 7.2.1.2 2nd example: Cylinder with further rotated edge normals 7.2.1.3 3rd example: Möbiusstrip 7.2.1.4 4th example: Plate with twisted edge 7.2.2 Approximation with FEM using reduced HCT-elements 7.2.2.1 1st example: Partly divided annular octagonal plate 7.2.2.2 2nd example: Divided annulus with rotated edge normals 8 Outlook and open questions Bibliography Notation Theses List of Figures List of Tables

Afin de pouvoir prédire le comportement des renforts de composites 3D interlock au cours d'un procédé de mise en forme, il est nécessaire de connaitre la position des mèches dans le renfort durant la phase de préformage du procédé. Les travaux présentés ici traitent de la simulation du préformage de renforts 3D épais à l'aide d'un élément fini hexaédrique semi-discret spécifique. En utilisant le principe des travaux virtuels, on distingue le travail interne virtuel dû à la tension des mèches des autres travaux virtuels. La raideur due aux tensions de mèches, qui constitue la contribution principale de la rigidité du matériau, est prise en compte à l'aide de barres incluses dans les éléments. Les rigidités dues aux autres sollicitations, comme la compression transverse, les cisaillements ou les frottements inter-mèches, sont décrites par un matériau continu additionnel. La combinaison de ce modèle discret du premier ordre et d'un matériau continu hyperélastique anisotrope dit du second ordre, pour formuler le comportement du matériau va permettre la simulation du préformage des renforts tissés épais. Conjointement aux travaux sur la simulation, des travaux expérimentaux pour l'identification des paramètres matériau de la loi de comportement ont été définis et réalisés. Ces paramètres concernent les deux parties de la formulation du comportement. / In order to simulate 3D interlock composite reinforcement behavior during forming process, it is necessary to predict yarns positions in the fabric during the preforming stage of the process. The present work deals with thick 3D interlock fabric forming simulation using specific hexahedral semi-discrete finite elements. Using the virtual work principle, we distinguish the virtual internal work due to tensions in yarns from other internal virtual works. The stiffness relative to yarns tension which is the main part of the rigidity is described by bars within the elements. The other rigidities - like transverse compression, shears or friction between yarns - are depicted by a continuous additional material. A combination of this "first order" discrete model and a continuous orthotropic hyperelastic "second order" material formulation will enable us to simulate the interlock preforming process. Jointly to the simulation work, we also had to specify and perform experimental testing identification of material parameters. These parameters concern both parts of the model.

Cette étude a pour objectifs la caractérisation expérimentale des frottements saccadés dans les matériaux granulaires modèles constitués des billes de verre monodisperses en compression triaxiale drainée et l'identification des paramètres de contrôle. Cinq paramètres macroscopiques caractérisent ces frottements saccadés : la chute de déviateur et la contraction volumique, l'intermittence de déformation axiale, le module d'Young et le coefficient de Poisson. Les frottements saccadés affectent simultanément le déviateur et la déformation volumique. Le comportement macroscopique est globalement contractant tandis que le matériau tend vers un état limite critique en grandes déformations, à la manière des sables lâches. Cependant il présente localement, dans les phases de blocage qui suivent immédiatement les ruptures temporaires, le comportement dilatant des sables denses, qui obéit à une relation contrainte-dilatance linéaire et unique. Les frottements saccadés disparaissent au-delà d'une vitesse critique d'écrasement axial, qui dépend de la contrainte de confinement et de la taille des grains. Le module d'Young dynamique par propagation d'ondes varie avec la contrainte de confinement selon une loi de puissance. Le module d'Young quasi-élastique au départ des phases de blocage est constant à l'intérieur du domaine élastique, de même que le coefficient de Poisson. Le suivi par granulométrie laser et par analyse d'images des matériaux après un ou plusieurs essais triaxiaux permet de suivre l'évolution de la taille moyenne et de la forme des grains. Tandis que les instabilités par saccade disparaissent suite à un certain nombre d'essais, on observe, simultanément à une légère diminution du volume moyen, l'apparition progressive de populations d'objets non sphériques par une fusion des grains analogue au frittage. / The objectives of this study are the experimental characterisatino of the stick-slip instabilities in a model granular material and the identification of relevant control parameters. As monodisperse glass beads are subjected to drained triaxial compression tests, five macroscopic parameters characterize the stick-slip phenomenon: the deviator drop, the jump in volumetric contraction, the intermittence of the axial strain, Young’s modulus and Poisson’s ratio. The stick-slip events simultaneously affect the deviator stress and the volumetric strain. While the global material behavior is that of a loose sand, gradually contracting and hardening as it approaches its large strain critical state, its response in the “stick” phases immediately following the “slip” instabilities is similar to that of dense, dilatant sands, with a unique, linear stress-dilatancy relationship. Stick-slip events disappear beyond a critical axial strain rate, depending on the confining stress and on the grain diameter. The Young modulus associated to wave propagation varies with the confining stress according to a power law. The quasi-elastic modulus measured at the beginning of the stick phase is constant inside the elastic domain, as well as the Poisson ratio. The evolution of grain size and shape after one or several triaxial tests is monitored by laser granulometry and image analysis. The gradual vanishing of stick-slip events, on repeating the tests, is likely related to the global decreasing trend of average particle volume and to the formation of non-spherical objects, apparently by some phenomenon analogous to sintering.

L'évaluation de la résistance à l'ouverture des mailles dans les filets de pêche est un enjeu important pour l'évaluation de la sélectivité des chaluts, et plus largement des engins de pêche. Les objectifs de cette thèse sont de développer et d'évaluer une méthodologie pour l'évaluation de la résistance à l'ouverture des mailles dans les filets. Différentes méthodes existent déjà (Sala et al., 2007, Priour and Cognard, 2011, Balash, 2012, De la Prada and Gonzales, 2013). Notre objectif est de proposer une méthode expérimentale simple et ne nécessitant pas d'équipements coûteux pour pouvoir être déployée facilement dans les laboratoires et dans l'industrie de la pêche, combinée avec un modèle numérique capable de représenter le comportement mécanique non-linéaire d'un filet. Le filet, constitué de fils tressés ou toronnés, en Polyéthylène ou Polyamide, présente une structure complexe. De plus, les filets peuvent être soumis à de grandes déformations. Afin d'étudier la réponse mécanique des filets à différents types de sollicitations et de créer une base de données expérimentale, de nombreux essais ont été effectués sur un large éventail de filets de pêche. Plusieurs méthodes numériques pour l'évaluation de la résistance à l'ouverture des mailles à partir de données expérimentales ont été développées. Ces travaux offrent une avancée scientifique pour l'évaluation de la résistance à l'ouverture des mailles : malgré le comportement mécanique visco-élasto-plastique des échantillons de filet, une méthodologie plus simple et plus précise que celles existantes, basée sur un dispositif expérimental simple et un modèle éléments finis libre de droits, est présentée. / The evaluation of the mesh resistance to opening in fishing nets is an important issue to assess the selectivity of trawls, and more broadly of fishing gear. The objectives of this thesis are to develop and to assess a methodology for the evaluation of the mesh resistance to opening in netting structures. Several methods are already proposed (Sala et al., 2007, Priour and Cognard, 2011, Balash, 2012, De la Prada and Gonzales, 2013). This thesis aims at proposing a simple experimental test that does not require expensive devices to be easily spread in laboratories and in the fishing industry, a simple test combined with a numerical model able to represent the non-linear mechanical behaviour of a tested netting panel. The netting structure, constituted of braided or stranded twines, made of Polyethylene or Polyamide, is complex. Moreover, the fishing nets can be subjected to large deformation. In order to study the mechanical response of netting samples to different types of solicitations and to obtain experimental data, numerous experimental tests were performed on a large range of netting samples. Several numerical methods for the evaluation of the mesh resistance to opening using experimental data were developed. This work offers scientific advance for the evaluation of the mesh resistance to opening: despite the visco-elasto-plastic mechanical behaviour of netting samples, one more accurate and simpler methodology than the existing ones, based on a simple experimental set up and on a free of rights finite element model, is presented.