On Theory, Modeling And Solutions For Statics And Dynamics Of A Nanobeam And Nanobeam-like Structure Based On Nonlocal Elasticity Theory

Nanoscale materials and structures have recently received considerable attention as these small-scale structures offer great potential for different applications in several areas. With rapid development of nanotechnology, miniaturized structures with nanoscale features can be precisely manufactured and applied in nano-electromechanical systems (NEMS). The widespread availability of NEMS such as atomic force microscopes has resulted in renewed interest in nano-systems for chemical, physical and biological sensors and devices as nano-systems facilitate higher precision in sensors and higher frequency in mixers, and the performance of these devices is extremely dependent on properties of their nanobeam-like elements. These properties need to be well characterized in order to control their functionality.In comparison with macro materials and structures, some effects not observable at macro-size are clear at nanoscale and, therefore, classical mechanics have failed to work in nanostructures. For instance, according to the classical continuum theory, the stress is singular at a crack tip despite the weak external load. However, each material has limited fatigue strength and in fact, atomic simulation and experiments have proved nonsingularity of stress at the crack tip. Currently, three main approaches are being used to investigate nanomechanics: experiment, molecular dynamic (MD) simulation and the continuum theory. Due to the complexity of instruments, equipment and technology, precise experiments at nanoscale are extremely difficult to be conducted. As MD simulation considers each individual molecule and its multiple mechanical or chemical web-interactions, its use requires extremely fast computing facilities and thence it is largely confined to relatively restricted systems with a limited number of molecules. That is why many researchers focus on the new continuum theory in nanomechanics.In this study, static and dynamic behaviors of nanobeams and nanobeam-like structures were investigated by the method based on the nonlocal elasticity theory. Transverse, longitudinal and torsional mechanics were considered. Particularly, subminiature belt is a common component in nanoscale devices which involves transverse vibration of axially moving nanostructures. Therefore, this work also concerns the dynamics of axially moving nanobeam-like structures based on the nonlocal theory. The nonlocal elasticity is based on atomic theory of lattice dynamics and experimental observations of phonon dispersion. In this theory, stress at a reference point is considered to be a function of strain fields at every point in the body. Such dependence of nonlocal stress on strains over the deformed body is observed in lattice dynamics and can also be inferred from the continuum mechanical model of carbon nanotubes. The nonlocal theory contains information about forces between atoms, and the internal length scale is introduced into the constitutive equations as a material parameter. However, it is strange to see that there are two nonlocal models, termed the traditional nonlocal model and the effective nonlocal model, that offer different predictions, i.e. stiffness is enhanced in one and is weakened in the other. The two theoretical models are compared with MD simulation and they are above and below MD results, respectively.In order to verify the two models, a semi-continuum model with discrete atomic layers in the thickness direction was developed to investigate the bending behaviors of ultra-thin beams of nanoscale thickness. The relaxation effect was considered in one atomic layer on each of the upper and lower surfaces and comparison with the two nonlocal models are inconclusive due to different material parameters. The long range attractive or repulsive interaction, or, the looser or tighter atomic lattice near surface than the bulk results in two different nonlocal models.The main aim of this project is to develop a new, exact nonlocal stress model for analytical solutions of nanobeams subjected to an axial load with simple support boundary conditions. It also concerns the dynamic response of a pre-tensioned nanobeam when precise internal axial force is considered. In addition, free vibration and stability of a nanobeam subjected to a variable axial load, and dynamic behaviors of axially moving nanobeams are also analyzed. Similarly, an effective torsion nonlocal model is proposed, based on the variational principle and static twist, torsional and longitudinal vibrations of nanobeam-like structures (such as nanorods), as well as dynamics of an axially moving nanorod were investigated in detail. Both analytical and numerical solutions are presented in this thesis. The results conclude that deflection decreases, or natural frequency (or structure stiffness) increases, with increase in nonlocal effects based on the effective nonlocal model.The results reported in this study are expected to be useful for theoretical predicting and designing NEMS or some other nanoscale devices using nanobeam-like structures.