Dynamic characteristic analysis of an electrostatically-actuated circular nanoplate subject to surface effects

This study investigates the influence of surface effect on the nonlinear behavior of an electrostatically actuated circular nanoplate. The Casimir force, surface effects, and the electrostatic force are modelled. In performing the analysis, the nonlinear governing equation of a circular nanoplate is solved using a hybrid computational scheme combining a differential transformation and finite differences. The method is used to model systems found in previous literature using different methods, producing consistent results, thus verifying that it is suitable for treatment of the nonlinear electrostatic coupling phenomenon. The obtained results from numerical methods demonstrated that the relationship between the thickness, radius, and gap size of a circular nanoplate, and its pull-in voltage, is scale-dependent. The model exhibits size-dependent behavior, showing that surface effects significantly influence the dynamic response of circular nanoplate actuators. Moreover, the influence of surface stress on the pull-in voltage of circular nanoplate is found to be more significant than the influence of surface elastic modulus. Finally, the effects of actuation voltage, excitation frequency, and surface effects on the dynamic behavior of the nanoplate are examined through use of phase portraits. Overall, the results show that the using hybrid method here presented is a suitable technique for analyzing nonlinear behavior characteristic of circular nanoplates.     

Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force

The pull-in instability and the vibration for a prestressed circular electrostatically actuated microplate are investigated in consideration of the Casimir force. Based on von Kármán’s nonlinear bending theory of thin plates, the governing equations for the whole analysis are decomposed into two two-point boundary value problems. For static deformation of the plate, the geometric nonlinearity is involved and the pull-in parameters are obtained by using the shooting method through taking the applied voltage or Casimir parameter as an unknown. This algorithm is also used to study the small amplitude free vibration about the predeformed bending configuration following an assumed harmonic time mode, and the variation of the prestress and Casimir parameters dependent fundamental natural frequency with the applied voltage is presented. Several case studies are compared with available published simulations to confirm the proposed method. The influences of various parameters, such as the initial gap-thickness ratio, Casimir effect, prestress on the pull-in instability behavior and the natural frequency are examined.     

Size-dependent dynamic analysis of rectangular nanoplates in the presence of electrostatic,Casimir and thermal forces

In the present study by considering the small-scale effects, the dynamic instability of fully clamped and simply supported nanoplates is studied in the attendance of electrostatic, Casimir as well as thermal forces. To this end, by applying the nonlocal elasticity theory of Eringen along with the classical plate theory, the dynamic equilibrium equation of nanoplates is obtained by incorporating the in-plane thermal and transverse intermolecular distributed loads. The solution of the obtained nonlinear governing equation is done using the Galerkin method and the dynamic pull-in instability voltage of the nanoplates is compared with the available experimental results. Finally, the simultaneous effects of thermal force as well as nonlocal parameter on the dynamic response of nanoplates are examined in the presence of Casimir force in detail.     

Pull-in parameters of cantilever type nanomechanical switches in presence of Casimir force

In this paper, the effect of the Casimir force on pull-in parameters of cantilever type nanomechanical switches is investigated by using a distributed parameter model. In modeling of the electrostatic force, the fringing field effect is taken into account. The model is nonlinear due to the inherent nonlinearity of the Casimir and electrostatic forces. The nonlinear differential equation of the model is transformed into the integral form by using the Green’s function of the cantilever beam. The integral equation is solved analytically by assuming an appropriate shape function for the beam deflection. The pull-in parameters of the switch are computed in three cases including nanoactuators without applied voltages, microswitches, and the general case of nanocantilevers. Nanoactuators without applied voltages are studied to determine the detachment length and the minimum initial gap of freestanding nanocantilevers, which are the basic design parameters for NEMS switches. The pull-in parameters of microswitches are investigated as a special case of our study by neglecting the Casimir effect and the results are verified through comparison with other works published in the literature. The general case of nanocantilevers is studied considering coexistence of the electrostatic and Casimir forces. The results of the distributed parameter model are compared with the lumped parameter model.     

Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory

This paper studies the electro-mechanical shear buckling analysis of piezoelectric nanoplate using modified couple stress theory with various boundary conditions.In order to be taken electric effects into account, an external electric voltage is applied on the piezoelectric nanoplate. The simplified first order shear deformation theory (S-FSDT) has been employed and the governing differential equations have been obtained using Hamilton's principle and nonlinear strains of Von-Karman. The modified couple stress theory has been applied to considering small scale effects. An analytical approach was developing to obtain exact results with various boundary conditions. After all, results have been presented by change in some parameters, such as; aspect ratio, effect of various boundary conditions, electric voltage and length scale parameter influences. At the end, results showed that the effect of external electric voltage on the critical shear load occurring on the piezoelectric nanoplate is insignificant.     

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic circular cylindrical shells

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic (FGMETE) circular cylindrical shell are carried out in the present work. The Hamilton principle, higher order shear deformation theory, constitutive equation considering coupling effect between mechanical, electric, magnetic, thermal are considered to derive the equations of motion and distribution of electrical potential, magnetic potential along the thickness direction of FGMETE circular cylindrical shell. The influences of various external loads, such as axis force, temperature difference between the bottom and top surface of shell, surface electric voltage and magnetic voltage, on the buckling response of FGMETE circular cylindrical shell are investigated. The natural frequency obtained by present method is compared with results in open literature and a good agreement is obtained.     

Three-dimensional exact solution of layered two-dimensional quasicrystal simply supported nanoplates with size-dependent effects

A size-dependent plate model is developed to investigate the elastic responses of the multilayered two-dimensional quasicrystal nanoplates based on the nonlocal strain gradient theory for the first time. A nonlocal stress field parameter and a length scale parameter are taken into account in the new model to capture both stiffness-softening and stiffness-hardening size effects. The exact solution for a single-layer two-dimensional quasicrystal simply supported nanoplate is derived by utilizing the pseudo-Stroh formalism in conjunction with the nonlocal strain gradient theory. Afterward, a dual variable and position method is used to deal with the multilayered case. Numerical examples are presented to study the dependence of size-dependent effect on nanoplate length and the influences of scale parameters on the quasicrystal nanoplate subjected to a z-direction mechanical load on its top surface. The proposed model should be useful to verify various nanoplate theories and other numerical methods.     

Casimir effects on Riemann surfaces

The Casimir effect whose existence was first predicted by Casimir in 1948 is considered as a manifestation of macroscopic quantum field theory. This force is evaluated theoretically by using the value of the Riemann zeta function at −3. The aim of the present paper is to introduce a similar Casimir energy for a Riemann surface, and to express it by a special value of the Mellin transform of a theta series arising from the heat kernel and also by a weighted integral of the logarithm of the Selberg zeta function.     

Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes

Stability of a functionally graded (FG) micro-beam, based on modified couple stress theory (MCST), subjected to nonlinear electrostatic pressure and thermal changes regarding convection and radiation, is the main purpose of this paper. It is assumed that the functionally graded beam, made of metal and ceramic, follows the volume fraction definition and law of mixtures, and its properties change as an exponential function through its thickness. By changing the ceramic constituent percent of the bottom surface, five different types of the micro-beams are investigated. The static pull-in voltages in presence of temperature changes are obtained by using step-by-step linearization method (SSLM) and, by adapting Runge–Kutta approach, the dynamic pull-in voltages are obtained numerically. Though the temperature distribution through the thickness of FG micro-beam (due to its too small measurement) is considered uniform, owing to the different thermal expansions of layers, temperature changes cause deflection in the micro-beam, and consequently affect pull-in values. Hence the profound effects of different material constituent over the pull-in voltages are illustrated and it is graphically displayed that how in some cases neglecting components of the couple stress leads to inaccurate results.     

Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory

A variety of micro-scale experiments have demonstrated that the mechanical property of some metals and polymers on the order of micron scale are size dependence. Taking into account the size effect on the mechanical property of materials and the inherent nonlinear property of electrostatic force, the static pull-in behavior of an electrostatically actuated Bernoulli–Euler microbeam is analyzed on the basis of a modified couple stress theory. The approximate analytical solutions to the pull-in voltage and pull-in displacement of the microbeam are derived by using the Rayleigh–Ritz method. The results show that the normalized pull-in voltage of the microbeam increases by a factor of 3.1 as the microbeam thickness equals to the material length scale parameter and exhibits size effect remarkably. However, the size effect on the pull-in voltage is almost diminishing as the microbeam thickness is far greater than the material length scale parameter. The normalized pull-in displacement of the microbeam exhibits size independence and equals to 0.448 and 0.398 for the cantilever beam and clamped–clamped beam, respectively.     

Three-dimensional exact thermo-elastic analysis of multilayered two-dimensional quasicrystal nanoplates

A three-dimensional thermo-elastic analytical solution for two-dimensional quasicrystal simply supported nanoplates subjected to a temperature change on their top surface is presented. The nonlocal theory and pseudo-Stroh formalism are used to obtain the exact solution for a homogeneous two-dimensional decagonal quasicrystal nanoplate with its thickness direction as a quasi-periodic direction. The propagator matrix method is introduced to deal with the corresponding multilayered nanoplates. Comprehensive numerical results show that nonlocal parameters, stress-temperature coefficients, stacking sequences have great influence on the stress, displacement components and heat fluxes of the nanoplates. In addition, the stacking sequences also influence the temperature and heat fluxes of the nanoplate. The exact thermo-elastic solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and multilayered plates. The mechanical behaviors of the nanoplates in numerical results can also serve as benchmarks to verify various thin-plate theories or other numerical methods.     

Quantum Backreaction (Casimir) Effect II. Scalar and Electromagnetic Fields

Casimir effect in most general terms may be understood as a backreaction of a quantum system causing an adiabatic change of the external conditions under which it is placed. This paper is the second installment of a work scrutinizing this effect with the use of algebraic methods in quantum theory. The general scheme worked out in the first part is applied here to the discussion of particular models. We consider models of the quantum scalar field subject to external interaction with “softened” Dirichlet or Neumann boundary conditions on two parallel planes. We show that the case of electromagnetic field with softened perfect conductor conditions on the planes may be reduced to the other two. The “softening” is implemented on the level of the dynamics, and is not imposed ad hoc, as is usual in most treatments, on the level of observables. We calculate formulas for the backreaction energy in these models. We find that the common belief that for electromagnetic field the backreaction force tends to the strict Casimir formula in the limit of “removed cutoff” is not confirmed by our strict analysis. The formula is model dependent and the Casimir value is merely a term in the asymptotic expansion of the formula in inverse powers of the distance of the planes. Typical behaviour of the energy for large separation of the plates in the class of models considered is a quadratic fall-of. Depending on the details of the “softening” of the boundary conditions the backreaction force may become repulsive for large separations. Communicated by Klaus Fredenhagen submitted 9/09/04, accepted 1/07/05     

变温度荷载作用下半无限成层饱和介质的热固结分析

对半无限成层饱和多孔介质作用随时间变化的温度荷载的热固结问题进行解析求解．其中,热-水-力耦合线性弹性控制方程考虑了热渗效应和等温热流效应的影响．先采用Laplace变换求其在变换域上的解,然后用数值方法求逆变换．对半无限体表面作用呈指数衰减热荷载的双层体系进行研究,分析了两层介质热固结系数、弹性模量等的差异性对热固结特征的影响．研究表明:位移场和应力场对温度场的耦合作用可以忽略,而热渗效应对温度和孔压有显著影响．     

A moving thermal dielectric crack in piezoelectric ceramics with a shearing force applied on its surface

This article conducts an exact analysis of a thermal dielectric crack moving in piezoelectric materials. Self-generating thermal and electric loadings by the crack interior are exerted on the crack surfaces as well as various external loadings including a shearing force. Fundamental solutions of the thermal and electro-elastic coupling fields are given by determining a temperature function and a harmonic function with eigenvalues properties due to material properties considered. Analytical expressions are obtained benefiting evaluation of key parameters. Numerical analysis is done and some interesting observations are found. There is a critical crack velocity within and beyond which the electric loading exerts different influences on the thermal flux of crack interior and the thermal stress intensity factor.     

Fluctuation forces in a three-layer medium with rough boundaries. IV. Calculations in the second order of perturbation theory (Casimir range)

The perturbation theory developed in the previous paper of the series is used to calculate the correction to the Casimir force due to the surface roughness.Scientific Council on the Problem Complex Kibernetika, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 113–118, July, 1992.     

Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies

A modified continuum model of electrically actuated nanobeams is presented by incorporating surface elasticity in this paper. The classical beam theory is adopted to model the bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. On the basis of this modified beam theory the governing equation of an electrically actuated nanobeam is derived and a powerful technology, analog equation method (AEM) is applied to solve this complex problem. Beams made from two materials: aluminum and silicon are chosen as examples. The numerical results show that the pull-in phenomena in electrically actuated nanobeams are size-dependent. The effects of the surface energies on the static and dynamic responses, pull-in voltage and pull-in time are discussed.     

横向非均升温下弹性梁的热过屈曲

基于轴向可伸长梁的几何非线性理论和打靶法,研究了两端不可移简支弹性梁在横向非均匀分布升温场作用下的热弹性屈曲响应。着重分析了横向升温变化对热过屈曲变形的影响,给出了相应的特性曲线。数值结果表明,由于横向温度改变会产生热弯曲内力,因此过屈曲平衡路径与有初始变形梁的过屈曲平衡路径相似。     

Thermal instability of a layered viscoelastic rectangular prism under high-frequency shear loading

Within the framework of a coupled problem of thermoviscoelasticity, using the method of numerical modeling, we have studied the thermal instability of a rectangular prism, composed of copper and polyethylene or polymethylmethacrylate layers, in the course of its dissipative heating. The prism is subjected to high-frequency force or kinematic shear. We have established that, in the case of polyethylene, thermal instability occurs under conditions of force loading and is absent under a kinematic one. However, for polymethylmethacrylate, thermal instability takes place for both cases of loading. This effect is attributable to the existence of temperature intervals where the shear and volume loss compliances for each of the polymers increases with the temperature. We have also revealed that the critical values of thermal instability for a prism with metal layers are substantially higher than those for a homogeneous prism. In addition to thermal instability, thermal resonance instability is also possible. It is caused by the jump of the thermal state from the low- to high-temperature branch of the soft-type resonance characteristic.     

Energy cycle of brushless DC motor chaotic system

The vector field of the brushless DC motor (BLDCM) chaotic system is regarded as the force field of a pure mechanical system via the transformation of Kolmogorov system. The BLDCM force field is decomposed into four types of torque: inertial, internal, dissipative, and generalized external torque. The forcing effect of each term in the force field is identified via the analogue of the electrical and mechanical system. The BLDCM energy transformation of four forms of energy—kinetic, potential, dissipative, and generalized external is investigated. The physical interpretation of force decomposition and energy exchange is given. The rate of change of the Casimir energy is equivalent to the power exchanged between the dissipative energy and the energy supplied to the motor, and it governs the different dynamic modes. A simple and optimal supremum bound for the chaotic attractor is proposed using the Casimir function and optimization.     

Analysis of surface mechanical properties effects on the dynamic characteristics of a circular micro-diaphragm with a distributed mass

The influences of surface mechanical properties effects on the dynamic characteristics of a circular micro-diaphragm with a partially uniformly distributed mass are investigated. Surface mechanical properties effects on two sides are considered to be different as one side of the diaphragm contacts with biochemical media. A size-dependent analytical model is developed based on the thin plate theory and surface elasticity theory for the micro-diaphragm with a partially uniformly distributed mass. The Galerkin procedure is used to solve the governing equation. The size-dependence of the natural frequency and mass sensitivity of the micro-diaphragm with different surface mechanical properties effects is discussed. Results show that the influences of different surface mechanical properties effects on the natural frequency and mass sensitivity are more significant for thinner micro-diaphragms. The influences depend on the differences in the surface mechanical properties effects of the two surfaces.