Dynamical Behaviours of Size Dependant and Functionally Graded Beams via Numerical Solutions

Abstract

Nanostructures have made significant advancements in the fields of science, and engineering because of their unique mechanical, electrical, and electronic properties. Due to these characteristics, nanomaterials such as nanowires, nanoparticles, nanoribbons, and nanotubes, etc., play a critical role in a variety of nanoelectromechanical systems, such as nanoprobes, nanotube resonators, and nanoactuators. In nanotechnology, non-classical models and nonlocal elasticity theories are essential due to the presence of small-scale effects at the nano or micro scale. Unlike classical theories, the nonlocal theories contain internal material length scale parameters that can capture size effects at the nano or micro scale and can predict the behavior of the nano-sized structures accurately. Hence, accurate prediction of their dynamical behaviors becomes essential for engineering design and manufacturing. On the other hand, Functionally Graded Materials (FGM) have gained enormous attention as heat-shielding advanced structural materials in various engineering applications and industrial sectors, such as aerospace, nuclear power, automobiles, aviation, space vehicles, biomedical, and steel. These are often inhomogeneous materials composed of ceramic-metal composites, with the composition or volume of constituents varying continuously in one or more specified directions. As a result, their properties vary continuously along one interface to the next in a predetermined mathematical pattern. The ceramic component withstands high temperatures due to its low thermal conductivity, making it appropriate for usage in high-temperature environments including nuclear reactors, chemical plants, and the production of high-speed vessels. The ductile metal component prevents fracture caused by high temperature gradient-induced strains. As a result, the mechanical strength of FGM substantially increases while reducing the weight of structure. As said above, the functionally graded materials have a variety of positive aspects as per requirements which enables the material to be more adaptable. Combining FGM concept with nano scaled effect produce materials with much-enhanced functionality specially in the development of devices and equipments, viz. nano-electro-mechanical systems, including thin shape memory alloy, atomic force microscopy, etc. Modelling and analysis of FG-Nano beams are challenging as nanoscale devices are built exploiting the characteristics of nanotubes, nanobeams, nanomembranes, and nanosheets. Further, the uncertainties or randomness of the material properties of structural components are of serious concern. Structural analysis is usually done by taking deterministic or crisp parameters, but the truth is quite diverse. The primary causes of the spread of uncertainty or randomness are defects in atomic configurations, measurement errors, environmental conditions, etc., which affect the behaviour of dynamical structures. As a matter of fact, these structural anomalies indicate that the materials may not have the capability to demonstrate their normal mechanical behaviors. The influence of uncertainties may become much more profound in case of nano and micro structures due to the small-scale effects. In fact, several nanoscale experiments and molecular dynamics study also support the claim of possible inclusion of randomness in various parameters. In view of the above, the objective of this thesis has been to develop dynamical models as well as study dynamical characteristics of nanobeams, microbeams, functionally graded beams, functionally graded nanobeams, and functionally graded microbeams considering various boundary conditions, complicating effects as well as with material or geometrical uncertainties by employing different efficient numerical or analytical methods where appropriate. In order to capture small scale effects of nano or microstructures, various non-classical continuum theories such as Eringen's nonlocal elasticity theory, nonlocal strain gradient theory, conformable fractional nonlocal elasticity theory, nonlocal elasticity theory with bi Helmholtz operator, modified couple stress theory, and a new nonlocal elasticity theory have been discussed. Several beam theories, such as Euler-Bernoulli beam theory, Timoshenko beam theory, one variable first-order shear deformation theory, and refined higher-order shear deformation theory, are considered. Further, various complicating effects, viz. Winkler-Pasternak elastic foundation, variable elastic foundation, Kerr elastic foundation, longitudinal magnetic field, electromagnetic field, hygroscopic environment, linear and nonlinear thermal environment, surface energy, surface residual stresses, porosity, etc., are taken into this investigation. Fuzzy concepts such as triangular fuzzy number, symmetric Gaussian fuzzy number, single and double parametric forms, etc., have also been attributed to deal with material uncertainties and their propagations. As such, differential quadrature method, differential transform method, Rayleigh-Ritz Method, Hermite-Ritz Method, shifted Chebyshev polynomials-based Rayleigh-Ritz methods, Navier's method, Galerkin weighted residual method, Monte Carlo simulation technique, and wavelet-based methods such as Haar wavelet and higher-order Haar wavelet methods have also been applied to solve the problems. Additionally, comprehensive studies have been conducted on all the scaling parameters to determine their effect on frequencies and buckling loads.