<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>