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.

Many people have readily suggested different types of hyperelastic models for high strain materials and biotissues since the 1940’s without validating them. But, there is no agreement for those models and no model is better than the other because of the ambiguity. The existence of ambiguity is because the error analysis has not been done yet (Criscione, 2003). The error analysis is motivated by the fact that no physical quantity can be measured without having some degree of uncertainties. Inelastic behavior is inevitable for the high strain materials and biotissues, and validity of the model should be justified by understanding the uncertainty due to it. We applied the fundamental statistical theory to the data obtained by randomized uniaxial stretch-controlled tests. The goodness-of-fit test (2R) and test of significance (t-test) were also employed. We initially presumed the factors that give rise to the inelastic deviation are time spent testing, stretch-rate, and stretch history. We found that these factors characterize the inelastic deviation in a systematic way. A huge amount of inelastic deviation was found at the stretch ratio of 1.1 for both specimens. The significance of this fact is that the inelastic uncertainties in the low stretch ranges of the rubber-like materials and biotissues are primarily related to the entropy. This is why the strain energy can hardly be determined by the experimentation at low strain ranges and there has been a deficiency in the understanding of the exclusive nature of the strain energy function at low strain ranges of the rubber-like materials and biotissues (Criscione, 2003). We also found the answers for the significance, effectiveness, and differences of the presumed factors above. Lastly, we checked the predictive capability by comparing the unused deviation data to the predicted deviation. To check if we have missed any variables for the prediction, we newly defined the prediction deviation which is the difference between the observed deviation and the point forecasting deviation. We found that the prediction deviation is off in a random way and what we have missed is random which means we didn’t miss any factors to predict the degree of inelastic deviation in our fitting.

Very little work has been done to study and understand the internal mechanisms that provide goose down with its resiliency under repeated compression. We have employed low magnification optical microscopy to identify some of these important mechanisms. Microscopy showed that a small tertiary structure exists on most goose down fibers and creates an important point of contact when two fibers interact. This tertiary contact mechanism has been coupled with fiber orientation and incorporated into a unique strain-energy function. The principal stresses for an initial compression cycle can be determined from this strain-energy function according to the hyperelastic constitutive theory. Irreversible deformation and hysteresis necessitate another means to determine the stresses during unloading and reloading. For these stages, the framework used by Beatty et al. (2002) for an ideal Mullins material will be utilized in conjunction with a shift in the stress-free state to determine the principal stresses. The proposed model is then evaluated for uniaxial compression and shown to capture the general behavior of goose down in compression including the irreversible deformation and hysteresis.

Many people have readily suggested different types of hyperelastic models for high strain materials and biotissues since the 1940’s without validating them. But, there is no agreement for those models and no model is better than the other because of the ambiguity. The existence of ambiguity is because the error analysis has not been done yet (Criscione, 2003). The error analysis is motivated by the fact that no physical quantity can be measured without having some degree of uncertainties. Inelastic behavior is inevitable for the high strain materials and biotissues, and validity of the model should be justified by understanding the uncertainty due to it. We applied the fundamental statistical theory to the data obtained by randomized uniaxial stretch-controlled tests. The goodness-of-fit test (2R) and test of significance (t-test) were also employed. We initially presumed the factors that give rise to the inelastic deviation are time spent testing, stretch-rate, and stretch history. We found that these factors characterize the inelastic deviation in a systematic way. A huge amount of inelastic deviation was found at the stretch ratio of 1.1 for both specimens. The significance of this fact is that the inelastic uncertainties in the low stretch ranges of the rubber-like materials and biotissues are primarily related to the entropy. This is why the strain energy can hardly be determined by the experimentation at low strain ranges and there has been a deficiency in the understanding of the exclusive nature of the strain energy function at low strain ranges of the rubber-like materials and biotissues (Criscione, 2003). We also found the answers for the significance, effectiveness, and differences of the presumed factors above. Lastly, we checked the predictive capability by comparing the unused deviation data to the predicted deviation. To check if we have missed any variables for the prediction, we newly defined the prediction deviation which is the difference between the observed deviation and the point forecasting deviation. We found that the prediction deviation is off in a random way and what we have missed is random which means we didn’t miss any factors to predict the degree of inelastic deviation in our fitting.

Mooney-Rivlin hyperelasticity equations are nonlinear coupled partial differential equations (PDEs) that are used to model various elastic materials. These models have been extended to account for fiber reinforced solids with applications in modeling biological materials. As such, it is important to obtain solutions to these physical systems. One approach is to study the admitted Lie symmetries of the PDE system, which allows one to seek invariant solutions by the invariant form method. Furthermore, knowledge of conservation laws for a PDE provides insight into conserved physical quantities, and can be used in the development of stable numerical methods. The current Thesis is dedicated to presenting the methodology of Lie symmetry and conservation law analysis, as well as applying it to fiber reinforced Mooney-Rivlin models. In particular, an outline of Lie symmetry and conservation law analysis is provided, and the partial differential equations describing the dynamics of a hyperelastic solid are presented. A detailed example of Lie symmetry and conservation law analysis is done for the PDE system describing plane strain in a Mooney-Rivlin solid. Lastly, Lie symmetries and conservation laws are studied in one and two dimensional models of fiber reinforced Mooney-Rivlin materials.

Articular cartilage is the translucent, heterogeneous three-component biological load processing gel that overlays the end of the articulating bones of mammalian joints. Normally, healthy intact articular cartilage performs two biomechanical functions very effectively. These are (i) redistribution of stresses due to loads acting on the joint; (ii) act as a near-frictionless interface between contacting bone ends. These principal functions are enabled by its highly elastic properties. Under normal physiological conditions, these essential biomechanical functions are provided over the lifetime of a mammalian joint with little or no degenerative changes. However, certain levels of physiological and traumatic loads and degenerative processes induced by activities such as running, walking, extreme sport, and aging can alter the composition and structure of the tissue, leading to changes in its biomechanical properties. This, inturn, influences its functional characteristics. The most common degenerative change in articular cartilage is osteoarthritis and the management and treatment of this disease is pivotal to all research targeted toward articular cartilage. Several scientific groups around the world have developed models of articular cartilage to predict its fundamental and functional responses to load and altered biochemical conditions through both in vivo and in vitro studies. The most predominant of these models are the biphasic and triphasic models, which are based on the conceptualisation of articular cartilage as a dispersed mixture of its three main components namely collagen fibrils proteoglycan aggregates and water. The triphasic model is an extension of the biphasic model and incorporates swelling as a separate identifiable component of the tissue's biomechanical response. While these models are capable of predicting the elastic and viscoelastic behaviour and certain aspects of the swelling characteristics of articular cartilage, they are incapable of accounting for its short-term responses where the fluid component is the main carrier of the applied pressure. The hydrostatic and swelling components of the fluid content determine the manner of stress-sharing and hence transient load processing within the matrix as stress is transmitted to the underlying structure. Furthermore, the understanding of the nature of this stress-sharing between fluid and solid components of the tissue is fundamental to the comprehension of the nature of degeneration and its biomechanical consequence in the function of the articulating joint. The inability of the biphasic and triphasic theories to predict, in accordance with experimental results, the transient behaviour of the loaded matrix fluid requires a more representative model. This imperative therefore forms the basis for the research work presented in this thesis. In this thesis, a new mathematical model of articular cartilage load carriage is presented which can predict the transient load-induced responses. The model is based on a continuum framework invoking the principle of mechanical consolidation of fluid-saturated, swollen porous elastic materials. The cartilage matrix is conceptualised as a heterogeneous anisotropic fluid-saturated porous material in which its solid component responds to load as a hyperelastic material and whose interaction with the swelling component produces a partially distributed time-varying permeability. In accordance with the principle of consolidation, a phenomenological approach is adopted for developing both analogue/engineering models and mathematical models for the tissue. The models are then used to predict both bulk matrix responses and the properties of the hypothetical layers of the tissue when subjected to physiological loading conditions. Ultimately, the generalized mathematical model is used to analyse the effect of superficial layer laceration on the stress-processing or stress-sharing characteristic of normal healthy articular cartilage. Finally, predicted results are shown to compare with experimental data demonstrating that the new models for swelling deformation, the hyperelastic law for solid skeletal structure and the distributed, time-dependent permeability are representative of the articular cartilage.

[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.

Le projet de thèse concerne la construction, la génération et l'identification de modèles continus stochastiques, pour des milieux hétérogènes exhibant des comportements non linéaires. Le domaine d'application principal visé est la biomécanique, notamment au travers du développement d'outils de modélisation multi-échelles et stochastiques, afin de quantifier les grandes incertitudes exhibées par les tissus mous. Deux aspects sont particulièrement mis en exergue. Le premier point a trait à la prise en compte des incertitudes en mécanique non linéaire, et leurs incidences sur les prédictions des quantités d'intérêt. Le second aspect concerne la construction, la génération (en grandes dimensions) et l'identification multi-échelle de représentations continues à partir de résultats expérimentaux limités / This work is concerned with the construction, generation and identification of stochastic continuum models, for heterogeneous materials exhibiting nonlinear behaviors. The main covered domains of applications are biomechanics, through the development of multiscale methods and stochastic models, in order to quantify the great variabilities exhibited by soft tissues. Two aspects are particularly highlighted. The first one is related to the uncertainty quantification in non linear mechanics, and its implications on the quantities of interest. The second aspect is concerned with the construction, the generation in high dimension and multiscale identification based on limited experimental data

Le présent travail étudie la caractérisation expérimentale et la modélisation de l'anisotropie induite par effet Mullins, i.e., la perte de raideur après les premiers cycles de chargement, très souvent observée dans les matériaux de type élastomère. Après une description des caractéristiques mécaniques du matériau silicone utilisé dans notre étude, des essais expérimentaux originaux sont développés pour créer des historiques de chargement complexes. D'une part, des successions d'essais de traction uniaxiale classiques sont réalisées, avec changement de directions de chargement. D'autre part, des états hétérogènes de contrainte et déformation obtenus lors d'essais de gonflement de membrane circulaire ont été complètement caractérisés grâce à des mesures de champs cinématiques réalisées par la méthode de corrélations d'images 3D ; les chargements effectués sont alors de type traction biaxiale-traction simple. Les paramètres clés pour la modélisation de l'effet Mullins ont ainsi pu être mis en évidence, avec notamment ses parts isotrope et anisotrope. Un modèle a ainsi été développé à partir d'une théorie de double réseau prenant en compte des critères expérimentalement motivés. Une version adaptée à une implantation simple dans un code de calculs éléments finis est finalement développée pour la réalisation de calculs de structures. / The present work studies the experimental characterization and modeling of the anisotropy induced by Mullins effect, i.e., the loss of stiffness in the first loading cycles, often observed in rubber-like materials. After a description of the mechanical characteristics of the particular silicone material used in our study, experimental tests are developed to create original and complex loading histories. First, successions of conventional uniaxial tensile tests are performed with changing directions of loading. Second, the state of heterogeneous stress and strain obtained in circular membrane swelling tests was completely characterized by means of kinematic field measurements made by the 3D image correlation method, and the loadings are then biaxial tension followed by uniaxial traction. The key parameters for modeling the Mullins effect were able to be identified, including its isotropic and anisotropic parts. A model was thus developed based on the double-network theory taking into account the experimentally motivated criteria. A suitable version with simple implementation in a finite element computer code was finally developed to allow the calculation of a structural part.

Análises numéricas confiáveis do comportamento mecânico de materiais como borrachas, dependem muito de uma calibração precisa do modelo constitutivo hiperelástico utilizado. Estes modelos são calibrados ajustando as curvas teóricas aos dados experimentais, obtidos por meio de ensaios usuais. Em muitos casos as amostras de matéria prima desses elastômeros já não se encontram disponíveis ou é impossível fabricar os corpos de prova requeridos. O objetivo deste trabalho é verificar a possibilidade de encontrar constantes constitutivas testando o próprio componente, ao invés dos usuais ensaios de tração, compressão e cisalhamento. A abordagem proposta consiste em criar uma rotina de programação associada a uma função custo onde, a partir de uma estimativa inicial de constantes constitutivas, sejam realizados processos iterativos de otimização buscando aproximar as curvas de força × deslocamento teórica e experimental. Um componente automotivo será utilizado nos estudos e dois modelos constitutivos hiperelásticos serão testados. As equações de tensões nominais dos modelos hiperelásticos serão utilizadas para predizer o comportamento teórico dos ensaios usuais, de forma a verificar a qualidade das constantes obtidas. Conclui-se que é possível utilizar o ensaio da própria peça para caracterizar o material hiperelástico, com resultados comparáveis aos que seriam obtidos com os ensaios típicos para esta aplicação. / The reliable numerical analysis of the mechanical behavior of rubber-like materials depends strongly on accurately calibrated hyperelastic constitutive models. Such models are calibrated by fitting theoretical curves against experimental data obtained in well known tests. In many cases samples of the original elastomer are no longer available or it is impossible to manufacture the specimens required by the standard tests. The aim of this work is verify the possibility of finding the constitutive constants by testing the actual component instead of the usual tensile, compression and shear tests. The proposed approach consists in creating a programming routine with a cost function that, starting from an initial estimate of the constitutive constants, iterate through an optimization algorithm in order to fit the theoretical force × displacement curves to the experimental ones. An automotive component will be used during the studies and two hyperelastic constitutive models will be tested. The nominal stress equations for the hyperelastic models are used to predict the standard tests behavior, to assess the quality of the constants obtained. The results shown that is possible to characterize an hyperelastic material by testing the actual component, with results comparable to those which would be obtained with standard tests.