This study concerns the development of the peridynamic strain energy density function for a Neo-Hookean type membrane under equibiaxial, planar and uniaxial loading conditions. The material parameters for each loading case are determined by equating the peridynamic strain energy to those of the classical continuum mechanics. Therefore, the peridynamic equations of motion are derived based on the Neo-Hookean model under the assumption of incompressibility. Numerical results concern the deformation of a membrane without and with a defect in the form of a hole, an inclusion and a crack under equibiaxial, planar and uniaxial loading conditions. As part of the verification process, the peridynamic predictions are compared with those of finite element analysis. For all defect types and loading conditions, the comparisons indicate excellent agreement.

Le prolapsus des organes pelviens (vessie, rectum, utérus, vagin) est un problème de santé qui touche de plus en plus de femmes. Ce trouble, dont la fréquence augmente avec le vieillissement de la population, altère inévitablement la qualité de vie des malades. Pour autant, les causes de cette pathologie sont mal connues et les pratiques chirurgicales demeurent mal évaluées. La réalisation d’un simulateur du comportement dynamique des organes pelviens permettant au chirurgien d’estimer l’impact fonctionnel de son geste avant sa réalisation est donc un besoin identifié. Ce travail concerne ainsi le développement, par la méthode des éléments finis, d’un modèle numérique du mouvement des organes pelviens et de leurs interactions. Un modèle est construit à partir d’une segmentation de l’IRM des patiente, permettant de générer la géométrie des organes pelviens. Des lois hyperélastiques sont ensuite adoptées pour modéliser le comportement mécanique des organes. Des résultats qualitatifs sont obtenus, permettant de comprendre les causes de certaines formes de prolapsus et d’estimer l’effet virtuel des interactions entre les organes. / Pelvic organ prolapse is a health problem that occurs only in women and becomes more common. These disorders whose frequency increases with the aging of the population affect the patients’ quality of life. However, the causes of these diseases are poorly understood and the surgical practices remain poorly evaluated. The realization of a simulator will allow surgeon to estimate the functional impact of his actions before implementation, to perform the surgery in a more controlled and reliable way. This work concerns the development of a numerical model of pelvic organs’ movement and their interactions based on the finite element methods. A first model is constructed from patients MRI images, allowing the generation of the organ geometries. Hyperelastic modeling of the organs behaviors were considered. Qualitative results could help to understand the reasons for the prolapse and to estimate the potential effect of organs interactions.

An aim of the given thesis is creating of a computational model of strain-stress behaviour of shock absorbers made of plastic and hyperelastic material. First step is get ecquaint with theory of large strains, theory of hyperelasticity and plasticity. Next step is get ecquaint with options of commercial programs based on MKP. Based on this knowledge, models of simple shock absorber are made. With created models, crash tests are simulated and property of shock absorbers are compared.

In order to validate the appropriate functional characteristics of cartilage, we need to systematically study and understand what constitutes normality and degradation in cartilage. This thesis provides an important step in this direction. To understand the mechanical repercussions of disruption to the matrix properties, cartilage is often artificially degraded using common enzymes. Although the process of artificial degradation does not provide an accurate representation of osteoarthritis, it can provide insight into the biomechanical properties of single matrix components by examining the behaviour of the tissue following its removal. Through histological analysis utilising the optical absorbance measurements of Safranin O stain, this work has demonstrated that for a given time and enzyme concentration, the action of Trypsin on proteoglycans is highly variable and is dependent on: * The initial distribution and concentration of proteoglycans at different depths * The intrinsic sample depth * The location in the joint space, and * The medium type. These findings provide initial data towards a mathematical model which researchers can use to optimise Trypsin treatment of articular cartilage, and therefore model degeneration in vitro with a better degree of certainty. The variability noted in the distribution and concentration of proteoglycans, and most likely the collagen network, creates a large variation in the compressive and tensile stiffness of all samples, and total failure strain energy. The average values for each of these tests indicate that a loss of proteoglycan through Trypsin treatment results in decreased compressive stiffness, increased tensile stiffness, and little change to the failure strains or total failure strain energy. Conversely, disruption to the collagen network shows increased compressive and tensile stiffness, as well as failure strain and total failure strain energy. Due to the large variation in the results for each treatment group, the average values for the treated samples fall within the range of results for normal cartilage. These values cannot therefore be used as dependable parameters to benchmark cartilage, since the parameters for artificially degraded cartilage are within the normal levels. The Yeoh and Polynomial hyperelastic laws were found to best represent the material characteristics of cartilage across the range of tested samples, regardless of differences in health and strength. The results presented here provide important insight into the biomechanical outcomes of artificial degradation and provide direction for future research in this area.

This diploma thesis deals with the creation of computational model of arterial aneurysm, which respects its real geometry and constitutive behaviour. Subsequently there is explored the risk of rupture, comparing the values of the highest main stress in aneurysm´s and intact aorta´s wall. This thesis also offers the comprehensive summary and mutual comparision of current constitutive (hyperelastic) models, which are for simplicity isochoric considered. The main task of this thesis is to formulate the method for finding of „unloaded“ = „outstressed“ = „primary“ = reduced geometry, which is generally unavailable. Models of aneurysm´s geometry, used so far, present already deformed configurations from blood pressure, axial prestrech and rezidual strain.

Smooth muscle contraction regulates the size of the blood vessel lumen which directly affects the mechanical response of the vessel. Folding in arteries has been observed in arteries during excessive contraction, known as a coronary artery spasm. The interplay of muscle contraction, geometry, and material responses and their effects on stability can be understood through mathematical models. Here, we consider a three-layer cross-sectional model of a coronary artery with anisotropic properties and intimal thickening, and perform a linear stability analysis to investigate the circumferential folding patterns that emerge due to muscle contraction. Our model shows that a critical level of contractile activity yields a uniform strain distribution across the arterial wall. When the muscle is contracted above this critical level, the tissue behaves isotropically and it is more prone to circumferential instability. This theoretical framework could serve as a valuable tool to understand the relationship between arterial lumen morphology and wall contraction in health and disease.

In the literature, there are various material models proposed so as to model the constitutive behavior of hyperelastic materials for example, St. Venant-Kirchho_ model, Mooney-Rivlin model etc. The stability of such material models under various states of deformation is of important concern, and generally stability analysis is conducted in homogeneous states of deformation. Within hyperelasticity, instabilities can be broadly classified as geometrical and material types. Geometrical instabilities such as buckling, symmetric bifurcation etc. are of physical origin, and lead to multiple solutions at critical stretch. Material instability is a aw in the material model and leads to unphysical solutions at the onset. It is required that the constitutive model should be materially stable i.e., should not give unphysical results, and be able to predict correctly the onset of geometrical instabilities. Certain constitutive restrictions proposed in the literature are inadequate to characterize such instabilities. In the work, we propose stability criteria which will characterize geometrical as well as material instabilities. A new elasticity tensor is defined, which is found to characterize material instability adequately. In order to investigate the validity of proposed stability criteria, three important constitutive models of hyperelasticity viz., St. Venant-Kirchho_, compressible Mooney-Rivlin and compressible Ogden models are investigated for stability.

An understanding of the mechanical properties of viral capsids (protein assemblies forming shell containers) has become necessary as their perceived use as nano-materials for targeted drug delivery. In this thesis, a heterogeneous, spatially detailed model of the viral capsid is considered. This model takes into account the increased degrees of freedom between the capsomers (capsid sub-structures) and the interactions between them to better reflect their deformation properties. A spatially realistic finite element multi-domain decomposition of viral capsid shells is also generated from atomistic PDB (Protein Data Bank) information, and non-linear continuum elastic simulations are performed. These results are compared to homogeneous shell simulation re- sults to bring out the importance of non-homogenous material properties in determining the deformation of the capsid. Finally, multiscale methods in structural analysis are reviewed to study their potential application to the study of nanoindentation of viral capsids. / text

Dans le cadre d’une collaboration avec la société PHYTOREM, nous avons élaboré un prototype de distribution autorégulé afin d’épandre des Eaux Usées après un simple dégrillage et via la Phytorémédiation (dépollution par les plantes).La première approche du projet de thèse a été de comprendre les comportements rhéologiques des effluents, mis à disposition par Phytorem, et mécaniques du matériau élastomère de type EPDM. Nous avons exposé les différentes façons de retrouver leurs propriétés rhéologiques et mécaniques par le biais de divers tests de rhéométrie, concernant les effluents, et de traction uni-, bi- et équibi-axiale, pour la partie matériau. Ceci nous a permis d’obtenir d’une part, la viscosité de nos effluents, et d’autre part, la loi de comportement la mieux adaptée à notre matériau.La deuxième et dernière approche porte sur les interactions entre un fluide et une membrane hyperélastique ayant pour fonction de réguler un écoulement. Le comportement de la membrane contrainte par la pression a été simulé sous Abaqus. Ces résultats ont permis de modéliser l’écoulement (code CFD commercial) lorsque la membrane est déformée et de déterminer numériquement la loi débit/pression du dispositif. Ces développements numériques s’appuient sur la méthode des éléments finis et un couplage partitionné simple en une étape pour une première approche entre le fluide, la membrane et la structure. Les modèles numériques sont validés expérimentalement. Ces travaux participent à l’élaboration d’un prototype de distributeur auto-régulé. / In collaboration with PHYTOREM, we have developed a prototype of self-regulated drip emitter to spread the Wastewater after a simple screening using phytoremediation (remediation by plants).The first approach of the thesis project was to understand the rheological behaviour of waste provided by PHYTOREM, and mechanical properties behaviour of EPDM elastomer type. We have explained the different ways to find their rheological and mechanical properties through various rheometry tests on waste, and tension uni-, biand equibi-axiale, for the material part. This allowed us to obtain first, the viscosity of our waste, and secondly, the behaviour law of best suited to our material.The second and final approach focuses on the interactions between a fluid and a hyperelastic membrane whose function is to regulate flow. The membrane behaviour under pressure stress was simulated using Abaqus. These results were used to model the flow (commercial CFD) when the membrane is distorted and to determine numerically its flow versus pressure law. These developments are relying on numerical finite element method and partitionned into a single coupling step for a first approach between fluid, membrane and structure. The numerical models are validated experimentally. This work contributes to the development of a prototype of self-regulated drip emitter.

Les modèles d'endommagement à gradient, aussi dénommés modèles à champs de phases, sont désormais largement utilisés pour modéliser la rupture fragile et ductile, depuis l'initiation de l'endommagement jusqu'à la propagation d'une fissure. Cependant, la majorité des études disponibles dans la littérature ne concerne que le cadre des petites déformations, et très peu d'études poussées ont été menées afin d'étudier leur pertinence dans un contexte de grandes déformations. Ce serait pourtant d'un intérêt primordial, notamment pour l'industrie pneumatique, qui deviendrait alors capable de prédire plus précisément l'initiation de l'endommagement dans ses structures.Dans la première partie de ce travail, nous établissons des solutions analytiques d'évolution de l'endommagement (homogène et localisée) pour des matériaux visqueux, en petites et en grandes déformations. En petites déformations, les modèles rhéologiques de Maxwell et Poynting-Thomson sont étudiés, et en grandes déformations, les modèles de Maxwell et Zener sont choisis. Une étude sur l'évolution de l'endommagement dans un cas purement hyperélastique est aussi menée.A cette première partie analytique succède une partie numérique, qui détaille l'implémentation des modèles d'endommagement à gradient dans des codes éléments finis en grandes déformations. De même qu'en petites déformations, une stratégie de minimisation alternée est adoptée pour résoudre successivement les problèmes d'endommagement et de déplacement. Le matériau suit une loi de Mooney-Rivlin quasi-incompressible, et une méthode mixte en déplacement-pression est utilisée. Des tests en 2D et 3D sont effectués, qui mettent en évidence la capacité des modèles à initier de l'endommagement en grandes déformations.Les modèles d'endommagement utilisés pour la seconde partie ne sont cependant capables d'initier de l'endommagement que dans les zones où la déformation est importante, c'est-à-dire dans les zones de forte contrainte déviatorique. Il a toutefois été montré que certains matériaux polymères, quasi-incompressibles, s'endommagent dans les zones de forte pression hydrostatique. Par conséquent, la recherche et l'étude d'un modèle d'endommagement capable d'initier de l'endommagement dans les zones de forte pression, pour des matériaux quasi-incompressibles lorsqu'ils sont sains, fait l'objet d'une troisième partie.Enfin, la croissance brusque de cavités dans un matériau hyperélastique, appelée phénomène de cavitation, est étudiée, ainsi que son interaction avec l'endommagement. Dans un premier temps, nous considérons la cavitation comme une simple bifurcation hyperélastique d'un matériau néo-hookéen compressible isotrope, et déterminons l'expression analytique de l'élongation critique pour laquelle la cavitation fait son apparition. Dans un second temps, nous montrons qu'il y a une compétition entre la cavitation et l'endommagement, et qu'en fonction de la valeur du ratio des élongations critiques respectives pour chaque phénomène, deux types de rupture apparaissent. / Gradient damage models, also known as phase-field models, are now widely used to model brittle and ductile fracture, from the onset of damage to the propagation of a crack in various materials. Yet, they have been mainly studied in the framework of small deformation, and very few studies aims at proving their relevance in a finite deformation framework. This would be more helpful for the tyre industry that deals with very large deformation problems, and has to gain insight into the prediction of the initiation of damage in its structures.The first part of this work places emphasis on finding analytical solutions to unidimensional problems of damaging viscous materials in small and large deformation.In all the cases, the evolution of damage is studied, both in the homogeneous and localised cases. Having such solutions gives a suitable basis to implement these models and validate the numerical results.A numerical part naturally follows the first one, that details the specificities of the numerical implementation of these non local models in large deformation. In order to solve the displacement and damage problems, the strategy of alternate minimisation (or staggered algorithm) is used. When solved on the reference configuration, the damage problem is the same as in small deformation, and consists in a bound constraint minimisation. The displacement problem is non linear, and a mixed finite element method is used to solve a displacement-pressure problem. A quasi-incompressible Mooney-Rivlin law is used to model the behaviour of the hyperelastic material. Various tests in 2D and 3D are performed to show that gradient damage models are perfectly able to initiate damage in sound, quasi-incompressible structures, in large deformation.In the simulations depicted above, it should be noted that the damage laws combined to the hyperelastic potential results in an initiation of damage that takes place in zones of high deformation, or in other words, in zones of high deviatoric stress. However, in some polymer materials, that are known to be quasi-incompressible, it has been shown that the initiation of damage can take place in zones of high hydrostatic pressure. This is why an important aspect of the work consists in establishing a damage law such that the material be incompressible when there is no damage, and the pressure play a role in the damage criterion. Such a model is exposed in the third part.Finally, the last part focuses on the cavitation phenomenon, that can be understood as the sudden growth of a cavity. We first study it as a purely hyperelastic bifurcation, in order to get the analytical value of the critical elongation for which cavitation occurs, in the case of a compressible isotropic neo-hookean material submitted to a radial displacement. We show that there is a competition between the cavitation phenomenon and the damage, and that depending on the ratio of the critical elongation for damage and the critical elongation for cavitation, different rupture patterns can appear.