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This paper demonstrated the coupled surface effects of thermal Casimir force and squeeze film damping (SFD) on size-dependent electromechanical stability and bifurcation of torsion micromirror actuator. The governing equations of micromirror system are derived, and the pull-in voltage and critical tilting angle are obtained. Also, the twisting deformation of torsion nanobeam can be tuned by functionally graded carbon nanotubes reinforced composites (FG-CNTRC). A finite element analysis (FEA) model is established on the COMSOL Multiphysics platform, and the simulation of the effect of thermal Casimir force on pull-in instability is utilized to verify the present analytical model. The results indicate that the numerical results well agree with the theoretical results in this work and experimental data in the literature. Further, the influences of volume fraction and geometrical distribution of CNTs, thermal Casimir force, nonlocal parameter, and squeeze film damping on electrically actuated instability and free-standing behavior are detailedly discussed. Besides, the evolution of equilibrium states of micromirror system is investigated, and bifurcation diagrams and phase portraits including the periodic, homoclinic, and heteroclinic orbits are described as well. The results demonstrated that the amplitude of the tilting angle for FGX-CNTRC type micromirror attenuates slower than for FGO-CNTRC type, and the increment of CNTs volume ratio slows down the attenuation due to the stiffening effect. When considering squeeze film damping, the stable center point evolves into one focus point with homoclinic orbits, and the dynamic system maintains two unstable saddle points with the heteroclinic orbits due to the effect of thermal Casimir force.

A scale-dependent analytical model is presented to solve the nonlinear pull-in instability of functionally graded carbon nanotubes (CNTs) reinforced nano-actuator with piezoelectric layer considering high order-corrected electrostatic pressure and finite temperature and conductivity corrections of Casimir force. Based on Eringen’s nonlocal elasticity theory considering the long range forces among atoms, and geometrical nonlinearity, the electro-thermo-mechanical coupling governing equation of nano-actuator is derived, and solved by utilizing natural mode function and Galerkin’s decomposition method. The higher-order corrected model of electrostatic force with fringing field effect accounting for large gap and geometrical nonlinearity is employed. The results indicate that pull-in voltage decreases with increase of positive piezoelectric effect but increases with increment of negative piezoelectric effect. Pull-in voltage declines as the piezoelectric layer thickness of nano-actuator increases. Casimir force appears in more significant effect on the pull-in voltage of nano-actuator than that of van der Waals force, which shows that the analysis of nanoscale devices cannot neglect the influences of intermolecular forces within sub-micron separations. Finally, the coupling influences of van der Waals force and Capillary force on pull-in voltage are compared.

This paper aims to investigate the coupling influences of thermal loading and surface effects on pull-in instability of electrically actuated circular nanoplate based on Eringen's nonlocal elasticity theory, where the electrostatic force and thermally corrected Casimir force are considered. By utilizing the Kirchhoff plate theory, the nonlinear equilibrium equation of axisymmetric circular nanoplate with variable coefficients and clamped boundary conditions is derived and analytically solved. The results describe the influences of surface effect and thermal loading on pull-in displacements and pull-in voltages of nanoplate under thermal corrected Casimir force. It is seen that the surface effect becomes significant at the pull-in state with the decrease of nanoplate thicknesses, and the residual surface tension exerts a greater influence on the pull-in behavior compared to the surface elastic modulus. In addition, it is found that temperature change plays a great role in the pull-in phenomenon; when the temperature change grows, the circular nanoplate without applied voltage is also led to collapse.

A size-dependent model of functionally graded carbon nanotubes (FGCNTs) reinforced microbeam with piezoelectric layer is presented based on unifying nonlocal stress and strain gradient framework. Nonlinear dynamic characteristics of the microbeam arise from electrostatic, piezoelectric actuation and thermal loading, with consideration of quantum and thermal fluctuations induced Casimir force. The nonlinearly dynamic frequency and pull-in instability of FGCNTs reinforced microbeam with damping effect are investigated by perturbation technique and verified by numerical method, where the coupling effects of nonlocal stress gradient and strain gradient parameters on fundamental frequency are described. The results show that the frequency of piezoelectric laminated microbeam-damping system declines with the growth of nonlocal stress gradient parameter while increases with the increment of strain gradient parameter. The pull-in voltage of micro- piezoelectric laminated beam can be tuned by excitation voltage exerted on piezoelectric layer, and the frequency and pull-in voltage decrease with the increase of the excitation voltage exerted on piezoelectric layer. In addition, dissipative effects originating from viscous and structural damping are evaluated. It is found that the pull-in voltage of micro-structures with damping system is higher than that of undamped system.

The paper reports a result of investigation on the dynamic pull-in instability of functionally graded carbon nanotubes (FGCNTs) reinforced nano-actuator considering damping behavior. Here, the nonlocal stress gradient and strain gradient theories are jointly utilized to capture the size effects of nanoscale structures. The material properties of FGCNTs reinforced nano-actuator are temperature-dependent, and the influences of geometrical nonlinearity and intermolecular forces such as van der Waals interaction and Casimir force are also considered. The results indicate that the fundamental frequency decreases with the increase of initial amplitude and electrostatic voltage until drops to zero when the dynamic pull-in occurs. It is shown that the frequencies of nano-actuator decline with the increment of nonlocal stress gradient parameter whereas increasing the strain gradient parameters enlarges the frequencies. In addition, it is observed from the time history and phase portrait that the dynamic pull-in voltage of system with damping behavior is larger than that without damping system, since the dissipative effect of damping requires more energy injected into the system.

An analytical model of investigating the pull-in characteristics of CNTs reinforced nano-actuator with temperature-dependency subjected to coupled electrostatic loading, dispersion forces as van der Waals and thermally corrected Casimir force, and surface stress are derived based on von Karman׳s geometric nonlinearity and surface elasticity. The results illustrate that increment of volume fraction of CNTs enhances the structural stiffness and leads to the increases of pull-in voltage of CNTs reinforced nano-actuator; the growth of temperature increases the axial compression stress and thus decreases the pull-in voltage; and free-standing behavior depends on the characteristic scale of the CNTs reinforced nano-actuator. The doubly-clamped and doubly-supported configuration of nano-actuator are considered and the pull-in voltage of doubly-clamped nano-actuator is larger than that of doubly-supported type. Casimir force with thermal correction significantly depends on the temperature and initial gap between electrodes. The results show the difference between Casimir force with and without thermal correction is small within couples of nanometers gap but the effect of Casimir force on pull-in instability of CNTs reinforced nano-actuator cannot be neglected.

In this paper, an analytical method is presented to investigate the effect of surface characteristic and temperature change on the pull-in instability of electrically actuated nano-switches reinforced by carbon nanotubes (CNTs) based on Eringen's nonlocal beam theory. An extremely nonlinear fourth-order governing equation for the doubly clamped nano-switches made of CNTs/Si composites nanobeam is derived and solved by using the principle of virtual work, where Van der Waals force as atomic interactions and Casimir force as macro effects of quantum field fluctuation of vacuum are combined as an electrostatic force with fringing field effects. The results show that both the pull-in voltage and pull-in deflection of CNTs/Si composite nanobeam increase with the increase of CNTs volume ratio but decrease with the increase of temperature change. The coupling influences of small scale parameter, geometric behavior, surface characteristic and thermal effect on the pull-in instability of electrostatically actuated CNTs/Si nanobeam are detailedly discussed.

This paper proposes a scale-dependent model to investigate the dynamic-pull-in characteristics of a functionally graded carbon nanotubes (FGCNTs) reinforced nanodevice with a piezoelectric ...

The electromechanical coupling characteristics of carbon nanotubes (CNTs) reinforced cantilever nano-actuator are investigated by considering surface effect, nonlocal scale effect containing the long-range forces among atoms, van der Waals force as molecular interaction, and Casimir force as macro effect of quantum field fluctuation. The extremely nonlinear governing equation is derived by utilizing energy methods based on Eringen's nonlocal elasticity theory and Young–Laplace's surface effect model. Some useful finding and results show that the pull-in voltage and deflection of nano-actuator increase with the increase of CNTs volume ratio, the increase of the nonlocal scale parameter enhances the pull-in voltage, but declines the pull-in deflection, and the surface effect becomes gradually significant as the thickness of CNTs reinforced nano-actuator decreases. In addition, it is found that van der Waals force and Casimir force could lead to the collapse of nano-actuator without applied voltage, where the influence of Casimir force is more significant than that of van der Waals force on the electromechanical coupling behavior of CNTs reinforced nano-actuator.

In this study, we report a new energy dissipation mechanism of nanomechanical resonators due to the Casimir effect originating from quantum fluctuation of the vacuum electromagnetic field at the nanoscale. An analytical study on the evaluation of the Casimir effect-induced energy loss in nanobeam resonators undergoing in-plane flexural vibration is presented. Two-dimensional elastic wave theory is employed to determine the energy transmission from the vibrating resonator to the substrate. Fourier transform and Green's function technique are adopted to solve the problem of wave motions on the surface of the substrate excited by the Casimir force. Analytical expressions of the Casimir effect-induced energy loss in terms of the quality factor, taking into account both pressure wave propagation in the noncontact substrate and shear wave propagation in the supporting substrate, as well as linear and nonlinear terms of time-varying Casimir force, have been derived. Effects of beam geometry, initial separation gap, and structural boundary conditions on energy loss are examined. Results of the present study demonstrate that the Casimir effect-induced energy loss plays an important role in the dissipation of the nanobeam resonators, in which the influence of shear wave propagation is remarkable. Also, as reflected by our results, the influence of nonlinear terms of time-varying Casimir force on the energy dissipation cannot be neglected for large-amplitude vibration, which is obviously a feature of nonlinear damping. Furthermore, we propose a possible way to experimentally measure the Casimir force by using the energy dissipation mechanism due to the Casimir force.

An analytical method is presented to investigate the pull-in instability of a carbon nanotube (CNT)-reinforced variable cross-section nanoswitch with a piezoelectric effect. Governing equations with variable coefficients are derived based on the nonlocal beam model with geometrical nonlinearity and are solved using the shooting method. All the nonlinear effects of the piezoelectric voltage, van der Waals force, Casimir force, CNT volume fraction, nonlocal parameters and width ratio on the pull-in instability are investigated. The pull-in electrostatic voltage increases with the increment of nonlocal parameters, which exhibits the significant scale-dependent behavior of nanostructures. The results show that the variable cross-section improves the flexural rigidity of the cantilever-type nanoswitch effectively, and that the piezoelectric effect of the piezoelectric layer is utilized to control the electrostatic force induced by the voltage exerted on the elastic layer, owing to piezoelectric materials' advantages of rapid response, light weight and low energy consumption.