Vibration Of Nano-Structures With Elastic Boundary Conditions

Nanostructures have attracted considerable attention for their outstanding mechanical,electrical,chemical and physical properties.Hence,nanostructures hold exciting promise as nanoresonators.At the nanoscale,the interactions between nanostructures are mainly van der Waals(vd W)forces,rather than direct contact.Because the atoms of these nanostructures at the edge cannot be completely fixed,it is difficult to realize the fully simply or clamped supported boundary conditions.The vd W force at the boundary can be described by the elastic boundary conditions.Boundary conditions have very important influence on the vibrational behaviours of the nanostructures,which lead to complex dynamic behaviors.This dissertation focuses on the dynamics of carbon nanotubes,nonlocal nanobeams,and molybdenum disulfide.Continuum mechanic models were used to study the vibrations of single-walled carbon nanotubes(SWCNTs)and double-walled carbon nanotubes(DWCNTs)suspended on a trench,the thermal vibration of stress gradient and strain gradient beams with elastic boundary conditions,the vibration of single-layered molybdenum disulfide(SLMo S2)suspended on a hole,and the vibration of SLMo S2 suspended on a trench.The dissertation begins with a brief survey on the mechanics problems of the nanostructures in Chapter 1.There follows the main body of the dissertation,from Chapters 2 to 7.It terminates with a few concluding remarks in Chapter 8.The results and the main contributions of the dissertation are as following.Chapter 2 presents the vibration of SWCNTs suspended on a trench by using three-segment Timoshenko beam(TSTB)model and a one-segment Timoshenko beam(OSTB)model with elastic boundaries together with molecular dynamics(MD)simulation.The boundary elastic constants of the SWCNTs suspended on the silicon trench are obtained by fitting the bending curve of SWCNTs subjected to a static uniformly distributed lateral load simulated via the MD method.Explicit formulas are derived for the vd W interaction coefficients between the SWCNTs and silicon substrates.An analytically modified Fourier series method(MFSM)is proposed to analyse the free vibration of the Timoshenko beam models with elastic boundary conditions.The MD simulations show that both the TSTB model and the OSTB model with elastic boundaries have a relatively good ability to predict the vibrational behaviours of SWCNTs suspended on a silicon trench.Chapter 3 proposes a continuum three-segment double Timoshenko beam(TSDTB)model to investigate the vibrational analysis of DWCNTs suspended on a trench.Accurate explicit formulas of the vd W interactions between each tube as well as between the tubes and a substrate are derived.An analytically MFSM is developed for the vibrational analysis of the TSDTB model.Numerical results show that the suspended DWCNTs as nanomechanical resonators could reach very high frequencies.The overlapped lengths between DWCNTs and the substrate at both ends have almost no effects on the lower natural frequencies of the TSDTB model when the overlapped lengths are long enough.Moreover,the natural frequencies of the TSDTB model are much less than those of one-segment double Timoshenko beam(OSDTB)model with clamped supported boundary condition,whereas the TSDTB model can be equivalent to the simply supported OSDTB model under certain condition.The intertube vd W interaction can raise the natural frequencies of the DWCNTs for the antiphase vibration.The amplitudes of the inner tubes of the DWCNTs are larger than those of the outer tubes.The vd W interaction between the inner tube and silicon substrate has little effect on the vibration of the DWCNTs.However,the vd W interactions between the outer tube and silicon substrate have important effects on the vibration of the DWCNTs.Chapter 4 presents a nonlocal Euler beam model with second-order gradient of stress taken into consideration to study the thermal vibration of beams with elastic boundary conditions.An analytical solution is proposed to investigate the free vibration of stress gradient Euler beams subjected to axial thermal stress.The effects of the nonlocal parameter,thermal stress and stiffness of boundary constraint on the vibration behaviors of beams are revealed.The results show that natural frequencies including the thermal stress are lower than those without the thermal stress when temperature rises.The temperature frequency ratios decrease with the increase of dimensionless translational spring stiffness.However,the temperature frequency ratios increase with the increase of dimensionless rotational spring stiffness.Chapter 5 presents a strain gradient Euler beam model described by a sixth-order differential equation to investigate the thermal vibrations of beams made of strain gradient elastic materials.The sixth-order differential equation of motion and elastic boundary conditions are determined simultaneously by a variation formulation based on Hamilton's principle.Analytical solutions for the free vibration of the elastic constraint strain gradient beams subjected to axial thermal stress are obtained.The effects of the thermal stress,nonlocal effect parameter,and boundary spring stiffness on the vibration behaviors of the strain gradient beams are investigated.The results show that the natural frequencies obtained by the strain gradient Euler beam model with the thermal stress decrease while the temperature is rising.The temperature frequency ratios decrease with the increase of dimensionless translational spring stiffness.But,the temperature frequency ratios increase with the increase of dimensionless rotational spring stiffness.Chapter 6 investigates the vibrational behaviors of SLMo S2 suspended over a circular hole by using a continuum model of two-segment circular Kirchhoff plate(TSCKP)together with MD simulation.Explicit formula of the vd W interaction coefficient between the SLMo S2 and silicon substrate is derived theoretically.An analytical solution is proposed to analyze the free vibrations of the TSCKP model with radial initial stresses.The effects of the overlapped width,vd W force,the radius of the circular hole and radial initial stresses on the vibrational behaviors of the circular single-layered Mo S2(CSLMo S2)are investigated.Numerical results show that the natural frequencies of the CSLMo S2 s change obviously as the overlapped width increases,when the overlapped width is less than 1.5 nm.As the overlapped width between the CSLMo S2 and silicon substrate is large enough,the overlapped width and radial initial stresses of the outer-segment circular plate have almost no effects on the natural frequencies of the TSCKP model.However,the radial initial stresse of the inner-segment circular plate has important effect on the natural frequencies of the TSCKP model.In addition,the natural frequencies of the TSCKP model are smaller than of those of the one-segment circular Kirchhoff plate(OSCKP)with clamped supported boundary condition.The TSCKP model can also be used to predict the vibrational behaviors of the arbitrary shape SLMo S2 suspended over a circular hole,when the overlapped width between the SLMo S2 and the substrate is large enough.Chapter 7 presents the vibrational behaviors of rectangular single-layered Mo S2(RSLMo S2)suspended over a trench by using a continuum three-segment rectangular Kirchhoff plate(TSRKP)model.An analytically MFSM and a 4-node 12-degree of freedom Kirchhoff plate element are used to analyze the free vibrations of the TSRKP model with initial stresses.Numerical results show that the natural frequencies of the RSLMo S2 change obviously as the overlapped width increases,when the overlapped length is less than 1 nm.As the overlapped length between the RSLMo S2 and silicon substrate is large enough,the overlapped length,the initial stresses of the rectangular Kirchhoff plate that overlaps over substrate have almost no effects on the natural frequencies of the TSRKP model.However,the natural frequencies of the TSRKP model increase as the initial stresses of the rectangular Kirchhoff plate that does not overlap over substrate increase.In addition,the natural frequencies of the TSRKP model are much less than of those of the one-segment rectangular Kirchhoff plate(OSRKP)model with two opposite edges clamped supported and the other two free,but higher than the OSRKP model with two opposite edges simply supported and the other two free.