Au cours des dernières décennies, le développement de nouveaux matériaux a progressé pour les applications liées à la mécanique. De nouvelles générations de composites ont été développées, qui peut offrir des avantages par rapport aux tapis unidirectionnels renforcés de fibres couramment utilisés les matériaux prennent alors le nom de woven fabrics. Le comportement de ce matériau est fortement influencé par la micro-structure du matériau. Dans la thèse, les modèles mécaniques et les schémas numériques capables de modéliser les comportement des tissus et des matériaux de réseau généraux ont été développés. Le modèle prend en compte la micro-structure au moyen d'une technique d'homogénéisation. Les fibres dans le réseau ont été traités comme des micro-poutres, ayant une rigidité à la fois en extension et en flexion, avec différents types de connexions. La procédure développée a été appliquée pour obtenir les modèles mécaniques homogénéisés pour certains types de réseaux de fibres biaxiaux et quadriaxiaux, simulant soit des réseaux de fibres (en ce cas a été supposé parmi les fibres) ou des tissus avec une interaction négligeable entre les faisceaux de fibres et en empêchant tout glissement relatif (dans ce cas, les connexions ont été simulés au moyen de pivots). Différentes géométries ont été analysées, y compris la cas dans lesquels les fibres ne sont pas orthogonales. On obtient généralement un premier milieu à gradient mais, dans certains cas, la procédure d'homogénéisation lui-même indique qu'un continuum d'ordre supérieur est mieux adapté pour représenter la déformation de la micro-structure. Des résultats spéciaux ont été obtenus dans le cas de fibres reliées par pivots. Dans ce cas, un matériau orthotrope à module de cisaillement nul a été obtenu. Un tel matériau a un tenseur constitutif elliptique, il peut donc conduire à des concentrations de contrainte. Cependant, il a été montré que certaines considérations sur le comportement physique de tels réseaux indiqué que les termes d'ordre supérieur inclus dans l'expansion des forces internes et des déformations, de sorte qu'un matériau de gradient de déformation a été obtenu. Les résultats obtenus peuvent être utilisés pour la conception de matériaux spécifiques nécessitant des propriétés. Bien que le modèle de référence soit un matériau de réseau, les résultats obtenus peuvent être appliqué à d'autres types similaires de microstructures, comme des matériaux pantographiques, des micro-dispositifs composé de micro-poutres, etc. Ils étaient limités à la gamme d'élasticité linéaire, qui est petite déformation et comportement élastique linéaire. Ensuite, les simulations numériques ont été axées sur les tests d'extension et les tests de biais. Le obtenu configurations déformées sont conformes aux tests expérimentaux de la littérature, tant pour tissus équilibrés et non équilibrés. De plus, une comparaison entre les premier et deuxième gradients des prédictions numériques ont été effectuées. Il a été observé que les prédictions de deuxième gradient mieux simuler les preuves expérimentales. / In the past decades there has been an impressive progress in the development of new materials for mechanical related applications. New generations of composites have been developed, that can offer advantages over the unidirectional fibre-reinforced mats commonly used then materials take the name of woven fabrics. The behaviour of this material is strongly influenced by the micro-structure of the material. In the thesis mechanical models and a numerical scheme able to model the mechanical behaviour of woven fabrics and general network materials have been developed. The model takes in to account the micro-structure by means of a homogenization technique. The fibres in the network have been treated like microbeams, having both extensional and bending stiffness, with different types of connection, according to the pattern and detail of the network. The developed procedure was applied for obtaining the homogenized mechanical models for some types of biaxial and quadriaxial networks of fibres, simulating either fibre nets (in this case rigid connection were assumed among the fibres) or tissues with negligible interaction between the fibre bundles, and with relative sliding prevented (in this case the connections were simulated by means of pivots). Different geometries were analysed, including the cases in which the fibres are not orthogonal. A first gradient medium is usually obtained but, in some cases, the homogenization procedure itself indicates that a higher order continuum is better fit to represent the deformation of the micro-structure. Special results were obtained for the case of fibres connected by pivots. In this cases an orthotropic material with zero shear modulus was obtained. Such a material has a not elliptic constitutive tensor, thus it can lead to strain concentrations. However, it was shown that some considerations about the physical behaviour of such networks indicated that higher order terms had to be included in the expansion of the internal forces and deformations, so that a strain gradient material was obtained. The results obtained can be used for the design of specific materials requiring ad-hoc properties. Although the reference model is a network material, the results obtained can be applied to other similar kinds of microstructures, like pantographic materials, micro devices composed by microbeams etc. They have been limited at the range of linear elasticity, that is small deformation and linear elastic behaviour. Then, numerical simulations were focused on extension tests and bias tests. The obtained deformed configurations are consistent with the literature experimental tests, both for balanced and unbalanced tissues. Moreover, a comparison between first and second gradient numerical predictions was performed. It was observed that second gradient predictions better simulate the experimental evidences.

This thesis deals with smooth discretization schemes and inverse problems, the former used in efficient yet accurate numerical solutions to forward models required in turn to solve inverse problems. The aims of the thesis include, (i) development of a stabilization techniques for a class of forward problems plagued by unphysical oscillations in the response due to the presence of jumps/shocks/high gradients, (ii) development of a smooth hybrid discretization scheme that combines certain useful features of Finite Element (FE) and Mesh-Free (MF) methods and alleviates certain destabilizing factors encountered in the construction of shape functions using the polynomial reproduction method and, (iii) a first of its kind attempt at the joint inversion of both static and dynamic source parameters of the 2004 Sumatra-Andaman earthquake using tsunami sea level anomaly data. Following the introduction in Chapter 1 that motivates and puts in perspective the work done in later chapters, the main body of the thesis may be viewed as having two parts, viz., the first part constituting the development and use of smooth discretization schemes in the possible presence of destabilizing factors (Chapters 2 and 3) and the second part involving solution to the inverse problem of tsunami source recovery (Chapter 4). In the context of stability requirements in numerical solutions of practical forward problems, Chapter 2 develops a new stabilization scheme. It is based on a stochastic representation of the discretized field variables, with a view to reduce or even eliminate unphysical oscillations in the MF numerical simulations of systems developing shocks or exhibiting localized bands of extreme plastic deformation in the response. The origin of the stabilization scheme may be traced to nonlinear stochastic filtering and, consistent with a class of such filters, gain-based additive correction terms are applied to the simulated solution of the system, herein achieved through the Element-Free Galerkin (EFG) method, in order to impose a set of constraints that help arresting the spurious oscillations. The method is numerically illustrated through its application to a gradient plasticity model whose response is often characterized by a developing shear band as the external load is gradually increased. The potential of the method in stabilized yet accurate numerical simulations of such systems involving extreme gradient variations in the response is thus brought forth. Chapter 3 develops the MF-based discretization motif by balancing this with the widespread adoption of the FE method. Thus it concentrates on developing a 'hybrid' scheme that aims at the amelioration of certain destabilizing algorithmic issues arising from the necessary condition of moment matrix invertibility en route to the generation of smooth shape functions. It sets forth the hybrid discretization scheme utilizing bivariate simplex splines as kernels in a polynomial reproducing approach adopted over a conventional FE-like domain discretization based on Delaunay triangulation. Careful construction of the simplex spline knotset ensures the success of the polynomial reproduction procedure at all points in the domain of interest, a significant advancement over its precursor, the DMS-FEM. The shape functions in the proposed method inherit the global continuity ( C p 1 ) and local supports of the simplex splines of degree p . In the proposed scheme, the triangles comprising the domain discretization also serve as background cells for numerical integration which here are near-aligned to the supports of the shape functions (and their intersections), thus considerably ameliorating an oft-cited source of inaccuracy in the numerical integration of MF-based weak forms. Numerical experiments establish that the proposed method can work with lower order quadrature rules for accurate evaluation of integrals in the Galerkin weak form, a feature desiderated in solving nonlinear inverse problems that demand cost-effective solvers for the forward models. Numerical demonstrations of optimal convergence rates for a few test cases are given and the hybrid method is also implemented to compute crack-tip fields in a gradient-enhanced elasticity model. Chapter 4 attempts at the joint inversion of earthquake source parameters for the 2004 Sumatra-Andaman event from the tsunami sea level anomaly signals available from satellite altimetry. Usual inversion for earthquake source parameters incorporates subjective elements, e.g. a priori constraints, posing and parameterization, trial-and-error waveform fitting etc. Noisy and possibly insufficient data leads to stability and non-uniqueness issues in common deterministic inversions. A rational accounting of both issues favours a stochastic framework which is employed here, leading naturally to a quantification of the commonly overlooked aspects of uncertainty in the solution. Confluence of some features endows the satellite altimetry for the 2004 Sumatra-Andaman tsunami event with unprecedented value for the inversion of source parameters for the entire rupture duration. A nonlinear joint inversion of the slips, rupture velocities and rise times with minimal a priori constraints is undertaken. Large and hitherto unreported variances in the parameters despite a persistently good waveform fit suggest large propagation of uncertainties and hence the pressing need for better physical models to account for the defect dynamics and massive sediment piles. Chapter 5 concludes the work with pertinent comments on the results obtained and suggestions for future exploration of some of the schemes developed here.