Zirconium is a widely applied material in nuclear industry and characterized by highly anisotropic mechanical behaviour. With the development of the nuclear industry, zirconium alloys have proven to be very important structural material used as pressure tubes in heavy water reactors, and thin-walled tubing in light water reactors due to improved corrosion resistance. To study the special property of zirconium, this thesis will evaluate different numerical models and will present an overview about the deformation mechanisms for zirconium alloys. Twinning and De-twinning (TDT) and Predominant Twin Reorientation (PTR) models are both implemented in the EVPSC polycrystal model. Various self-consistent models as well as TDT and PTR twinning models are evaluated by studying large strain behavior of zirconium alloys slab under different deformation processes. The material parameters for those models are determined by virtue of the experimental uniaxial tension and compression along the normal direction (ND) as well as tension along the rolling direction (RD). The predictability of those models is assumed through comparison of macroscopic deformation behavior (stress strain curves and R values) and microscopic mechanical response (texture coefficients and lattice strain) between the predicted and experimental data. Numerical results reveal that, among the various models examined, the VPSC-TDT model with the Affine self-consistent scheme gives the best overall performance for zirconium alloys slab, TDT twinning model gives the better overall performance than the PTR twinning model. / Thesis / Master of Applied Science (MASc)

This thesis is concerned with large strain soil consolidation and a large portion of the work is devoted to the theoretical development in modelling this phenomenon. An experimental programme based on a specially designed oedometer is also included. The governing equations are formulated in Chapter 2, where both the material and the space coordinates are considered. In space coordinate the problem is a moving boundary problem, and special numerical techniques for the idealized case of a thin soil layer are developed in Chapter 3. In Chapter 4 the consolidation of a normally consolidated stratum and a dredged fill are considered with a linear soil model, and the consolidation of a deposit undergoing continuous sedimentation is considered in Chapter 5. The material coordinate is used in the analysis which also includes the effect of self weight of soil. These problems are again considered in Chapter 6 and Chapter 7 with a nonlinear soil model. A specially designed oedometer for testing very soft soil is described in Chapter 8, and the experimental results are discussed in Chapter 9. These are compared with the theoretical solutions developed in Chapter 3, and good agreement has been obtained.

[pt] É apresentado um modelo não-linear para estruturas unidimensionais em equilíbrio, onde não são feitas restrições de caráter geométrico. Este modelo é capaz de descrever movimentos que envolvam flexão, torção, cilhamento e extensão. As configurações de referência e deformada têm sua geometria descrita através da posição espacial de uma curva e da orientação de uma base ortonormal associada a cada ponto dessa curva. O uso dos ângulos de Euler na descrição das rotações, o que pode implicar em instabilidades numéricas, é evitado através do uso de uma nova parametrização para o problema. O problema de equilíbrio que envolve o comportamento não-linear de uma estrutura unidimensional é formulado de diferentes maneiras. São apresentados dois métodos numéricos para a solução desse problema. Um deles é baseado numa decomposição via lagrangeano aumentado e outro é um método de Newton não convencional. São discutidos detalhes acerca da implementação computacional desses métodos. A validade das formulações é atestada através de alguns exemplos numéricos. Em particular. São analisadas algumas aplicações relacionadas com operações de cabos umbilicais em prospecção petrolífera, que envolvem carregamentos estáticos complicados como aqueles ocasionados por flutuadores e pela atração gravitacional. / [en] It is presented a model of the static geometrically non-linear behavior of an elastic rod which considers flexion, torsion, shear and tension. The geometry of the body, in the reference and deformed configurations, is described given the position of the centerline and the geometry of a rigid frame attached to each point of the line. A particular parametrization that avoids the difficulties associated with the use of Euler angles is employed simplifying the numerical treatment. The equilibrium problem for a nonlinear rod is formulated in several different ways and two numerical methods for solution of these problems are presented. One is based on augmented Lagrangian splitting and the second is a non-standard Newton’s method. Details pertaining to the implementation of that method are discussed. A number of numerical simulations have been documented to demonstrate the robustness of the formulations. In particular, some applications in connection with Off shore pipe lines operations, which involves complicated static loading conditions that includes floaters and gravitational forces, are analysed.

The mechanical properties of pure Aluminium, Al-0.2%Cu and Al-0.4%Cu at large strains were studied by channel-die compression at three different temperatures: 77K, 200K and 300K. The results were interpreted in terms of work hardening rate versus stress (0/r) diagrams. The evolution of the structure was studied on a range of scales from macroscopic to microscopic, by optical study of slip lines, X-ray diffraction for texture measurements, Electron Back-Scattering Kikuchi Patterns for local texture measurements, and by Transmission Electron Microscopy for microstructural information. Intense shear banding was observed at large strains in all alloys at all temperatures. The texture evolution was shown to be consistent with this change in deformation mode. At low temperatures, stage HI of deformation was shown to be represented by a straight line in the 6lr diagram. Increasing the temperature lead to a dramatic decrease in work hardening rate and to an increasing concavity of the 0lr plots. The addition of solutes to pure Aluminium was shown to result in an increase of the work hardening rate, which could be represented by a simple translation of the 0/r plots on the stress axis. At large strains, all three materials experienced a stage (stage IV) of constant work hardening at low rate. The stage IV work hardening rate decreased with increasing temperature, and was not influenced by solute content. The stage Ill-Stage IV transition was very sharp at 77K and smoother at higher testing temperatures. Phenomenological models were developed for the prediction of the influence of temperature and solute content on work hardening. Moderate strains were modelled taking into account the evolution of the dislocation density into two different populations during the deformation. The influence of solutes on work hardening was modelled by considering how segregation of solute atoms at the dislocation cores influences dynamic recovery. Stage IV work hardening was considered to arise from the accumulation of dislocation debris resulting from the dynamic recovery events. / Thesis / Master of Engineering (ME)

Durant ce travail, nous avons étudié les propriétés mécaniques et de fracture d'élastomères à réseaux multiples synthétisés par des étapes successives de gonflement/polymérisation inspirées de l'architecture moléculaire développée par Gong pour les doubles réseaux hydrogels. Une méthode de synthèse plus versatile a été utilisée pour varier de façon continue le pré-étirement isotrope du premier réseau λ0, qui contrôle le module d'Young et le durcissement. Dans le cas d'une dilution importante du premier réseau (<10%), une scission moléculaire apparaît à grande déformation dans le réseau pré-étiré sans rompre le matériau. Le taux de dilution contrôle la quantité d’endommagement et donc la pente de la courbe contrainte-déformation. Finalement, pour les systèmes les plus dilués (<3%), une striction est observée au-dessus d’un seuil de contrainte. Changer le taux de réticulant du premier réseau ou les monomères utilisés ont conduit par ailleurs à l’obtention de comportements mécaniques similaires. L’énergie de fracture Γ est une fonction croissante de λ0. Des techniques de visualisation locale comme la Corrélation d’Image Numérique et l’intégration de molécules méchanoluminescentes ont été utilisées pour décrire une zone d’endommagement en tête de fissure dont la taille augmente avec λ0. Enfin, le mécanisme de renforcement des élastomères à réseaux multiples a pu être partiellement décrit dans le contexte du modèle de Brown sur les doubles réseaux. / We investigated systematically the mechanical and fracture properties of multiple network elastomers synthesized by successive swelling/polymerization steps inspired by the molecular architecture of Gong’s double network gels. A more versatile synthesis method was used to vary continuously the isotropic degree of prestretching λ0 of the first network resulting in a wider range of mechanical behaviours, where λ0 controls the Young’s modulus at small strain and the strain hardening at large strain. If the first network is diluted enough (<10%) molecular bond breakage occurs in this prestretched network at high strain while avoiding sample failure. The degree of dilution controls the amount of damage and therefore the slope of the stress-strain curve. Finally, for the most diluted systems (<3%), a yield stress and a necking phenomenon was observed. Changing the degree of crosslinking of the first network or the monomers used led to the same qualitative mechanical behaviour. The fracture energy Γ was shown to be an increasing function of λ0 however different regimes could be distinguished with macroscopic fracture occurring before or after bulk damage was detected. Visualisation techniques such as Digital Image Correlation and embedded mechanoluminescent molecules were used to map a damage zone in front of the crack tip, the size of which increased with λ0. Finally, the toughening mechanism of the multiple network elastomers could be understood in a nearly quantitative way within the framework of Brown's model of fracture of double network gels.

Les travaux de cette thèse ont porté sur le développement d’une nouvelle loi de comportement hyperélastique, isotrope et incompressible permettant de modéliser les matériaux caoutchouteux en grande déformation et en grand déplacement. Cette nouvelle loi combine une approche moléculaire et une approche phénoménologique, ce qui permet de couvrir un spectre large de sollicitations. Elle est constituée par la superposition de quatre termes :– un terme lié à la contrainte d’entrelacement des chaînes macromoléculaires observée avec le phénomène de cristallisation. Ce terme est modélisé par une fonction logarithmique provenant de l’énergie phénoménologique de Gent-Thomas,– un terme lié à l’hypothèse des déformations affines observées avec le raidissement final de certaines chaînes macromoléculaires des élastomères. Ce terme provient de la probabilité non-Gaussienne de Langevin. Nous l‘avons modélisé par la loi moléculaire 8-chaines d’Arruda-Boyce avec un aménagement qui consiste à utiliser une approximation originale de la fonction de Langevin inverse,– un terme lié à la contrainte des chaînes ayant des déformations non-affines. Ce terme est modélisé par une fonction Gaussienne sous forme intégrale. Il s’agit de l’une des contributions originale de ce travail de thèse,– une partie volumique standard permettant de prendre en compte l’incompressibilité du matériau.Les deux principales originalités de la thèse concernent donc l’élaboration d’une approximation inédite de la fonction de Langevin inverse ainsi que la construction d’une nouvelle densité d’énergie hyperélastique isotrope, incompressible et hybride.Afin d’étudier la pertinence du modèle proposé, des comparaisons ont été réalisées avec plusieurs jeux de données expérimentales disponibles dans la littérature. Ces comparaisons ayant été couronnées de succès, l’implémentation numérique du modèle que nous proposons a été effectuée dans le code universitaire aux éléments finis FER. / This thesis concerns the development of a new incompressible isotropic hyperelastic behavior law allowing the modeling of rubber materials with large strain and large displacement. This new law mixes a molecular approach with a phenomenological one and therefore covers a wide range of loading. It has been built by a sum over four terms:– a term related to the interleaving macromolecular chains observed with the crystallization phenomenon. This term is modeled by a logarithmic function coming from the phenomenological energy of Gent-Thomas,– a term related to the assumption of affine deformations observed with the final stiffening of a part of macromolecular elastomeric chains. This term comes from the non Gaussian probability of Langevin. We have modeled it by the 8-chains molecular law of Arruda-Boyce but with an original approximation of the inverse of the Langevin function,– a term related to the stress occurring with non affine strains. This term has been modeled by a Gaussian function adopting an integral form. This is one of the original contribution of this thesis work,– a classical volumetric term taking into account the incompressibility of the material.The two main originalities of the thesis are therefore the introduction of a new approximation of the inverse of the Langevin function and the development of a new hyperelastic energy density which is isotropic, incompressible and hybrid.In order to study the efficiency of the proposed model, comparisons were made with several experimental data available in the literature. These comparisons have been successful and we have implemented our model in the university finite element software FER.

We present an implicit a-posteriori finite element procedure to compute bounds for functional outputs of finite element solutions in large strain elasticity. The method proposed relies on the existence of a potential energy functional whose local minima, over a space of suitably chosen continuous functions, corresponds to the problem solution. The output of interest is cast as a constrained minimization problem over an enlarged discontinuous finite element space. A Lagrangian is formed were the multipliers are an adjoint solution, which enforces equilibrium, and hybrid fluxes, which constrain the solution to be continuous. By computing approximate values for the multipliers on a coarse mesh, strict upper and lower bounds for the output of interest on a suitably refined mesh, are obtained. This requires a minimization over a discontinuous space, which can be carried out locally at low cost. The computed bounds are uniformly valid regardless of the size of the underlying coarse discretization. The method is demonstrated with two applications involving large strain plane stress incompressible neo-hookean hyperelasticity. / Singapore-MIT Alliance (SMA)

[pt] Apresenta-se aqui uma aplicação do método de continuação, baseado no algoritmo de Euler-Quase Newton, a problemas de equilíbrio de materiais hiperelásticos incompressíveis sujeitos a grandes deformações. Discretiza-se o problema misto estado deformado-campo de pressão pela utilização do método dos elementos finitos, prevendo-se a compatibilidade LBB entre os espaços envolvidos. Propõe-se a utilização de uma função densidade de energia de deformação para o material de Mooney-Rivlin distinta daquela apresentada na literatura clássica, devido ao mal comportamento do Hessiano associado à formulação original. / [en] The application of a continuation method based on the Euler-Chord algorithm to equilibrium problems of incompressible, hyperelastic materials subjected to large deformations is here presented. The mixed strained state-pressure field problem is discretized by means of the finite element method, taking into account the LBB compatibility condition between the involved spaces. The utilization of a strain energy density function diverse from the one presented in the classical literature, is proposed, due to the ill behavior of the Hessian associated with the original formulation.

[pt] Neste trabalho analisa-se o problema de contato entre dois corpos bidimensionais ou tridimensionais com comportamento elasto-plástico submetidos a grandes deformações através de duas metodologias de solução implementadas. O procedimento de solução numérica inclui o tratamento conjunto das não-linearidades geométricas e físicas como também da não-linearidade das condições de contato baseando- se numa estratégia incremental-iterativa. Na modelagem dos corpos utiliza-se o método dos elementos finitos em conjunto com uma formulação Lagrangeana Atualizada e relações constitutivas para grandes deformações elasto- plásticas. Os corpos são considerados ambos deformáveis, supondo a superfície de contato sem atrito. Os métodos implementados foram: o método da Penalidade onde as restrições de contato são satisfeitas de forma aproximada através do emprego do parâmetro de penalidade e o método da Programação Linear Complementar que formula um problema de programação matemática para cada configuração de equilíbrio incremental onde as condições de Kuhn-Tucker são resolvidas pelo esquema de pivoteamento de Lemke [43]. Foi realizada uma análise comparativa entre os métodos e também um estudo paramétrico verificando a influência de parâmetros relevantes para o problema. Foi implementado um algoritmo para consideração da geometria de contato onde são supostas condições genéricas de contato: nó-superfície e onde as relações cinemáticas são dadas em termos de uma função diferenciável da distância.Foram utilizados elementos lineares formulados em deslocamentos, hexaédricos de 8 nós e elementos com uma formulação híbrida - Enhanced Assumed Strain - EAS, que possuem, além de oito nós, três parâmetros internos de deformação, com um campo de deformações trilinear completo, minorando os efeitos de -locking- presentes nos elementos lineares formulados em deslocamentos. O emprego desses dois tipos de elementos teve como finalidade comparar as suas performances em problemas de contato. / [en] In this theses the frictionless contact problem between two- dimensional or three-dimensional deformable bodies with elasto-plastic behavior under finite strains is studied. The numeric solution procedure is based on an incremental- iterative strategy and includes the treatment of geometric and physical nonlinearities as well as those which arise from contact conditions. In the model of body the finite element method together with an updated Lagrangian and elasto-plastic constitutive relations for finite strains is used.For the solution of the contact problem two methods were investigated. The methods were the Penalty method, where the contact restritions are imposed in an aproximated manner with the use of a penalty parameter and the Linear Complementarity Programming method,which is based on the formulation of a mathematic programming problem for each configuration of equilibrium where the Kuhn-Tucker omplementarity conditions corresponding to the contact conditions are solved by the Lemke pivoting scheme [43]. A algorithm was implemented for consideration for the contact geometry where generic conditions of contact are assumed and the kinematic relation is expressed through a differentiable gap function.Displacement based elements with linear interpolations - hexahedral 8 node elements were employed for the space discretization. In order to diminish the locking effects present in these elements, a hibrid formulation - Enhanced Assumed Strain - EAS, with three aditional internal parameters of deformation with an complete trilinear deformation field was employed.

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