Mechanical analyses of shear deformable functionally graded nanobeams by using nonlocal finite element formulation

In the last 20 years, the design of structures and systems at this scale has gained importance since the developments at the nano/micro scale technology remarked. The mechanical behaviours of nano and micro scaled structures are not understood by using classical elasticity theory due to characteristic internal lengths of structure. Therefore, various higher-order continuum mechanics formulations depending on the size-effect of nano/micro structure are developed. Additionally, it is known that the results of analysis based on the shear effect in the structural elements present more realistic results about the mechanical behavior of structure. With this in motivation, this thesis study has investigated the shear-dependent and atomic size-dependent vibrational behavior of one-dimensional, small-scaled, and functionally graded structures. The nonlocal elasticity theory has been formulated to consider the atomic size dependency. Firstly, nonlocal vibration analysis of functionally graded nanobeams according to Timoshenko beam theory, which is fundamental shear deformable beam theory, has been examined. Then, the longitudinal free vibration of related structural element model has been studied by employing the Love-Bishop rod theory that considers the shear effect in the axial rods. It is assumed that the related structural element model subjected to a thermal and/or elastic environment. In addition to these, free vibration of functionally graded nanoframes whose bending effects are based on fundamental shear deformation have been also investigated. In order to determine the mechanical behavior more accurately, the neutral axis of the functionally graded nanostructure has also been considered. In the whole of mechanical analyses, the governing equations have been obtained based on the variational algebra. Since the large number of parameters affecting the problem complicates the analytical solution, a finite element formulation has been developed and used in the solution of mechanical analysis. The general formulation of finite element analysis has been modified since it gives inappropriate results for some nonlocal mechanical analysis problems. For the numerical applications, free vibration frequencies of related structures have been presented with tables and graphics under different parameters. Detailed discussions of numerical results are given and finally, the most general results are outlined. When the comparisons are performed with analytical solutions and previous literature research, it has been observed that the finite element formulations developed within the scope of thesis has attained very successful results. With this thesis study, it is aimed to understand the mechanical behavior under different effects of one-dimensional composite structures that play a role in the configuration of nano and micro-scale devices and systems by means of a numerical modelling.