<p>The rapid growth of fields such as metamaterials, composites, architected materials, porous solids, and micro/nano materials, along with the continuing advancements in design and fabrication procedures have led to the synthesis of complex structures having intricate material distributions and non-trivial geometries. These materials find important applications including biomedical implants and devices, aerospace and naval structures, and micro/nano-electromechanical devices. Theoretical and experimental evidences have shown that these structures exhibit size-dependent (or, nonlocal) effects. This implies that the response of a point within the solid is affected by a collection of points; ultimately a manifestation of the multiscale deformation process. Broadly speaking, at a continuum level, the mathematical description of these multiscale phenomena leads to integral constitutive models, that account for the long-range interactions via nonlocal kernels. </p> <p><br></p> <p>Despite receiving considerable attention, the existing class of approaches to nonlocal elasticity are predominantly phenomenological in nature, following from their definition of the material parameters of the nonlocal kernel based on 'representative volume element' (RVE)-based statistical homogenization of the heterogeneous microstructure. The size of the RVE required for practical simulation, does not achieve a full-resolution of the intricate heterogeneous microstructure, and also implicitly enforces the use of symmetric nonlocal kernels to achieve thermodynamic consistency and mathematically well-posedness. The latter restriction directly limits the application of existing approaches only to the linear deformation analysis of either periodic or isotropic nonlocal structures. Additionally, the lack of a consistent characterization of the nonlocal effects, often results in inconsistent (also labeled as 'paradoxical') predictions depending on the nature of the external loading. In order to address these fundamental theoretical gaps, this dissertation develops a fractional-order kinematic approach to nonlocal elasticity by leveraging cutting-edge mathematical operators derived from the field of fractional calculus.</p> <p><br></p> <p>In contrast to the class of existing class of approaches that adopt an integral stress-strain constitutive relation derived from the equilibrium of the RVE, the fractional-order approach is predicated on a differ-integral (fractional-order) strain-displacement relation. The latter relation is derived from a fractional-order deformation-gradient mapping between deformed and undeformed configurations, and this approach naturally localizes and captures the effect of nonlocality at the root of the deformation phenomena. The most remarkable consequence of this reformulation consists in its ability to achieve thermodynamic and mathematical consistency, irrespective of the nature of the nonlocal kernel. The convex and positive-definite nature of the formulation enabled the use of variational principles to formulate well-posed governing equations, the incorporation of nonlinear effects, and enabled the development of accurate finite element simulation methods. The aforementioned features, when combined with a variable-order extension of the fractional-order continuum theory, enabled the physically consistent application of the nonlocal formulation to general continua exhibiting asymmetric interactions; ultimately a manifestation of material heterogeneity. Indeed, a rigorous theoretical analysis was conducted to demonstrate the natural ability of the variable-order in capturing the role of microstructure in the deformation of heterogeneous porous solids. These advantages allowed the application of the fractional-order kinematic approach to accurately and efficiently model the response of porous beams and plates, with random microstructural descriptions. Results derived from multiphysical loading conditions, as well as nonlinear deformation regimes, are used to demonstrate the causal relation between the kinematics-based fractional-order characterization of nonlocal effects and the natural role of microstructure in determining the macroscopic response of heterogeneous solids. The potential implications of the developed formalism on scientific discovery of material laws are examined in-depth, and different areas for further research are identified.</p>

Au cours de cette thèse, nous avons développé un outil numérique, basé sur une formulation intégrale en éléments de frontière, qui permet une analyse classique du comportement d'une fissure 3D soumise à des sollicitations mécaniques complexes. Cet outil industriel est destiné à être intégré dans un code de calcul à usage industriel. Dans le but d'appréhender l'impact de la microstructure sur le comportement en fissuration fragile, nous nous sommes intéressés aux modèles de comportement non local. Nous avons commencé par adopter le modèle de comportement élastique non local de Eringen, qui permet de décrire plus finement le comportement élastique au voisinage de la fissure en prenant en compte les interactions à longue distance au sein du matériau. Cette modélisation du comportement conduit, contrairement à l'approche classique, à un un champ de contrainte fini sur le front de la fissure et localement maximal en avant du front. Ces résultats montrent qu'il est possible de prévoir la stabilité et la direction de propagation de la fissure à l'aide d'un critère plus simple et plus naturel, basé sur les variations du champ de contrainte au voisinage du front de la fissure. La stratégie numérique adoptée permet de traiter indifféremment des cas de fissure en traction, compression, cisaillement ou sollicitation mixte. L'intérêt de l'approche non-locale étant clairement démontré, nous avons considéré la version améliorée du modèle de Eringen telle que proposée par Polizzotto. Cette modélisation est la plus appropriée pour les milieux finis et requiert une mise en oeuvre numérique particulière. Les bases d'une méthodologie numérique, initiée par R. Kouitat ont été établies. Cette méthode est fondée sur un couplage des éléments de frontière avec une méthode de collocation par points d'équations aux dérivées partielles. Les premiers résultats obtenus dans ce cadre sont très encourageants et montrent qu'il sera effectivement possible de traiter le phénomène irréversible de fissuration de la même façon que les problèmes de plasticité / In this thesis, we have developed a numerical tool, based on a classical boundary elements method, which allows a conventional analysis of a stationary crack in a 3D specimen under complex mechanical loading. In order to assess the impact of the microstructure on the brittle fracture, we were interested in non local models of behavior. First, we have adopted the non local elastic model due to Eringen. This refined constitutive equation allows to account for long range interactions in the description of the elastic behavior in the vicinity of the crack front. Unlike the traditional approach, this type of model leads to a finite stress field at the crack front. Moreover, the stress is locally maximal ahead of the front. These interesting results indicate that it is possible to predict the stability and direction of crack propagation in a simple and more naturel way by using stress based criteria. The implemented numerical strategy can handle cases of crack in tension or compression, under shear stress or mixed loadings. Having clearly highlighted the interest of non local models, we have adopted the improved version of Eringen elastic model as proposed by Polizzotto. This elastic model is applicable to finite domains and requires a specific numerical treatment. The basis of such a numerical strategy initiated by R. Kouitat has been established. The method couples the conventional boundary element method with local point interpolation of a strong form differential equation. Promising results are obtained and show that with such modeling of material behavior, it is possible to describe the irreversible process of fracturing in a similar way as plasticity

<p dir="ltr">The continuous advancements in material science and manufacturing engineering have revolutionized the material design and fabrication techniques therefore drastically accelerating the development of complex structured materials. These novel materials, such as micro/nano-structures, composites, porous media, and metamaterials, have found important applications in the most diverse fields including, but not limited to, micro/nano-electromechanical devices, aerospace structures, and even biological implants. Experimental and theoretical investigations have uncovered that as a result of structural and architectural complexity, many of the above-mentioned material classes exhibit non-negligible nonlocal effects (where the response of a point within the solid is affected by a collection of other distant points), that are distributed across dissimilar material scales.</p><p dir="ltr">The recognition that nonlocality can arise within various physical systems leads to a challenging scenario in solid mechanics, where the occurrence and interaction of nonlocal elastic effects need to be taken into account. Despite the rapidly growing popularity of nonlocal elasticity, existing modeling approaches primarily been concerned with the most simplified form of nonlocality (such as low-dimensional, isotropic, and homogeneous nonlocal problems), which are often inadequate to identify the nonlocal phenomena characterizing real-world problems. Further limitations of existing approaches also include the inability to achieve a mathematically well-posed theoretical and physically consistent framework for nonlocal elasticity, as well as the absence of numerical approaches to achieving efficient and accurate nonlocal simulations. </p><p dir="ltr">The above discussion identifies the significance of developing theoretical and numerical methodologies capable of capturing the effect of nonlocal elastic behavior. In order to address these technical limitations, this dissertation develops an advanced continuum mechanics-based approach to nonlocal elasticity by using fractional calculus - the calculus of integrals and derivatives of arbitrary real or even complex order. Owing to the differ-integral definition, fractional operators automatically possess unusual characteristics such as memory effects, nonlocality, and multiscale capabilities, that make fractional operators mathematically advantageous and also physically interpretable to develop advanced nonlocal elasticity theories. In an effort to leverage the unique nonlocal features and the mathematical properties of fractional operators, this dissertation develops a generalized theoretical framework for fractional-order nonlocal elasticity by implementing force-flux-based fractional-order nonlocal constitutive relations. In contrast to the class of existing nonlocal approaches, the proposed fractional-order approach exhibits significant modeling advantages in both mathematical and physical perspectives: on the one hand, the mathematical framework only involves nonlocal formulations in stress-strain constitutive relationships, hence allowing extensions (by incorporating advanced fractional operator definitions) to model more complex physical processes, such as, for example, anisotropic and heterogeneous nonlocal effects. On the other hand, the nonlocal effects characterized by force-flux fractional-order formulations can be physically interpreted as long-range elastic spring forces. These advantages grant the fractional-order nonlocal elasticity theory the ability not only to capture complex nonlocal effects, but more remarkably, to bridge gaps between mathematical formulations and nonlocal physics in real-world problems.</p><p>An efficient nonlocal multimesh finite element method is then developed to solve partial integro-differential governing equations in the fractional-order nonlocal elasticity to further enable nonlocal simulations as well as practical applications. The most remarkable consequence of this numerical method is the mesh-decoupling technique. By separating the numerical discretization and approximation between the weak-form integral and nonlocal integral, this approach surpasses the limitations of existing nonlocal algorithms and achieves both accurate and efficient finite element solutions. Several applications are conducted to verify the effectiveness of the proposed fractional-order nonlocal theory and the associated multimesh finite element method in simulating nonlocal problems. By considering problems with increasing complexity ranging from one-dimensional to three-dimensional problems, from isotropic to anisotropic problems, and from homogeneous to heterogeneous nonlocality, these applications have demonstrated the effectiveness and robustness of the theory and numerical approach, and further highlighted their potential to effectively model a wider range of nonlocal problems encountered in real-world applications.</p>

<p>У овој тези проучене су осцилације и стабилност запремински оптерећене правоугаоне<br />нано-плоче уз коришћење Ерингенове теорије еластичности. Запреминско оптерећење<br />је константно са правцем који је у равни плоче. Гранични услови су моделовани као<br />покретна укљештења. Класична теорија плоча и Карманова теорија плоча, које су<br />надограђене Ерингеновом теоријом еластичности, искоришћене су за формирање<br />диференцијалне једначине стабилности и осциловања нано-плоче. Галеркиновом<br />методом одређене су сопствене фреквенције трансверзалних осцилација нано-плоче у<br />зависности од ефеката запреминског оптерећења и нелокалности. Одређене су<br />критичне вредности параметра запреминског оптерећења при којима нано-плоча губи<br />стабилност. Приказан је утицај ефеката запреминског оптерећења и нелокалности на<br />неколико облика осциловања. Верификација резултата извршена је помоћу методе<br />диференцијалних квадратура.</p> / <p>U ovoj tezi proučene su oscilacije i stabilnost zapreminski opterećene pravougaone<br />nano-ploče uz korišćenje Eringenove teorije elastičnosti. Zapreminsko opterećenje<br />je konstantno sa pravcem koji je u ravni ploče. Granični uslovi su modelovani kao<br />pokretna uklještenja. Klasična teorija ploča i Karmanova teorija ploča, koje su<br />nadograđene Eringenovom teorijom elastičnosti, iskorišćene su za formiranje<br />diferencijalne jednačine stabilnosti i oscilovanja nano-ploče. Galerkinovom<br />metodom određene su sopstvene frekvencije transverzalnih oscilacija nano-ploče u<br />zavisnosti od efekata zapreminskog opterećenja i nelokalnosti. Određene su<br />kritične vrednosti parametra zapreminskog opterećenja pri kojima nano-ploča gubi<br />stabilnost. Prikazan je uticaj efekata zapreminskog opterećenja i nelokalnosti na<br />nekoliko oblika oscilovanja. Verifikacija rezultata izvršena je pomoću metode<br />diferencijalnih kvadratura.</p> / <p>In this thesis, the problem of stability and vibration of a rectangular single-layer graphene<br />sheet under body force is studied using Eringen&rsquo;s theory. The body force is constant and<br />parallel with the plate. The boundary conditions correspond to the dynamical model of a<br />nanoplate clamped at all its sides. Classical plate theory and von K&aacute;rm&aacute;n plate theory,<br />upgraded with nonlocal elasticity theory, is used to formulate the differential equation of<br />stability and vibration of the nanoplate. Natural frequencies of transverse vibrations,<br />depending on the effects of body load and nonlocality, are obtained using Galerkin&rsquo;s method.<br />Critical values of the body load parameter, i.e., the values of the body load parameter when<br />the plate loses its stability, are determined for different values of nonlocality parameter. The<br />mode shapes of nanoplate under influences of body load and nonlocality are presented as<br />well. Differential quadrature method is used for verification of obtained results.</p>