Sharif Digital Repository

In this study, dynamic response of a micro- and nanobeams under electrostatic actuation is investigated using strain gradient theory. To solve the governing sixth-order partial differential equation, mode shapes and natural frequencies of beam using Euler–Bernoulli and strain gradient theories are derived and then compared with classical theory. Galerkin projection is utilized to convert the partial differential equation to ordinary differential equations representing the system mode shapes. Accuracy of proposed one degree of freedom model is verified by comparing the dynamic response of the electrostatically actuated micro-beam with analogue equation and differential quadrature methods.... 

Since the classical continuum theory is neither able to evaluate the accurate stiffness nor able to justify the size-dependency of micro-scale structures, the non-classical continuum theories such as the modified couple stress theory have been developed. In this paper, a new comprehensive Timoshenko beam element has been developed on the basis of the modified couple stress theory. The shape functions of the new element are derived by solving the governing equations of modified couple stress Timoshenko beams. Subsequently, the mass and stiffness matrices are developed using energy approach and Hamilton's principle. The formulations of the modified couple stress Euler-Bernoulli beam element... 

The modified couple stress theory is employed in this paper in order to formulate the size-dependent strain energy release rate of Euler-Bernoulli and Timoshenko notched beams. As a case study, the normalized energy release rate of the aforementioned beams has been developed for three-point bending as a function of the ratio of material length scale parameter to the beam depth. The results of the current model are compared to the experimental data. The good agreement between the present results and the experimental values indicates that the current model can be successfully applied to evaluate size dependent energy release rate of structures  

This paper investigates the dynamic behavior of microcantilevers under suddenly applied DC voltage based on the modified couple stress theory. The cantilever is modeled based on the Euler-Bernoulli beam theory and equation of motion is derived using Hamilton's principle. Both analytical and numerical methods are utilized to predict the dynamic behavior of the microbeam. Multiple scales method is used for analytical analysis and the numerical approach is based on a hybrid finite element/finite difference method. The results of the modified couple stress theory are compared with those from the literature as well as the results predicted by the classical theory. It is shown that the modified... 

In recent years, extensive experiments have shown that classical continuum theory cannot predict the behavior of mechanical microstructures with small size. To accurately design and analyze micro- and nano-electro-mechanical systems, size-dependent continuum theories should be used. These theories model micro- and nano-electro-mechanical systems with higher accuracy because they include size-dependent parameters. In this paper, polysilicon nano-beam is modeled using modified couple stress and Eringen's nonlocal elasticity theories. First, partial differential equations governing the vibration of nano-beams are converted to a one D.O.F. differential equations using Galerkin method, resulting... 

Modified couple stress theory is a size-dependent theorem capturing the micro/nanoscale effects influencing the mechanical behaviors of the micro- and nanostructures. In this paper, it is applied to investigate the nonlinear vibration of carbon nanotubes under step DC voltage. The vibration, natural frequencies and dynamic pull-in characteristics of the carbon nanotubes are studied in detail. Moreover, the effects of various boundary conditions and geometries are scrutinized on the dynamic characteristics. The results reveal that application of this theory leads to the higher values of the natural frequencies and dynamic pull-in voltages  

A size-dependent, explicit formulation for coupled thermoelasticity addressing a Timoshenko microbeam is derived in this study. This novel model combines modified couple stresses and non-Fourier heat conduction to capture size effects in the microscale. To this purpose, a length-scale parameter as square root of the ratio of curvature modulus to shear modulus and a thermal relaxation time as the phase lag of heat flux vector are considered for predicting the thermomechanical behavior in a microscale device accurately. Governing equations and boundary conditions of motion are obtained simultaneously through variational formulation based on Hamilton's principle. As for case study, the model is... 

In this paper, a size-dependent non-classical yield criterion is introduced on the basis of the modified couple stress theory in order to capture the size-dependency of the micro-scale structure yielding loads where the attempts of the classical yield criteria such as the von-Mises have been in vain. In order to develop the new yield criterion, the deviatoric part of the micro-scale structure strain energy density, including both classical and non-classical parts, is equated to the deviatoric strain energy density of a macro-size tensile-test sample at the yielding point. For bending of microbeams and torsion of microbars, the size-dependent yielding moments have been determined based on the... 

The paper presents the effects of fluid flow on the static and dynamic properties of carbon nanotubes that convey a viscous fluid. The mathematical model is based on the modified couple stress theory. The effects of various fluid parameters and boundary conditions on the pull-in voltages are investigated in detail. The applicability of the proposed system as nanovalves or nanosensors in nanoscale fluidic systems is elaborated. The results confirm that the nanoscale system studied in this paper can be properly applied for these purposes  

The paper presents size-dependant mechanical behaviors of carbon nanotubes under electrostatic actuation using modified couple stress theory. The behaviors of the carbon nanotubes with different geometries and boundary conditions are studied in detail. The results reveal that application of this theory results in higher pull-in voltages for both cantilever and doubly clamped boundary conditions  

In this paper, a size-dependent formulation is presented for mechanical analyses of inhomogeneous micro-plates based on the modified couple stress theory. The plate properties can arbitrarily vary through the thickness. The governing differential equations of motion are derived for functionally graded (FG) plates with arbitrary shapes utilizing a variational approach. Moreover, the boundary conditions are provided at smooth parts of the plate periphery and also at the sharp corners of the periphery. Utilizing the derived formulation, the free-vibration behavior as well as the static response of a rectangular FG micro-plate is investigated  

In this paper size-dependent static and dynamic behavior of nonlinear Euler-Bernoulli beams made of functionally graded materials (FGMs) is investigated on the basis of the strain gradient theory. The volume fraction of the material constituents is assumed to be varying through the thickness of the beam based on a power law. As a consequence, the material properties of the microbeam (including length scales) are varying in the direction of the beam thickness. To develop the model, the usual simplifying assumption which considers the length scale parameter to be constant through the thickness is avoided and equivalent length scale parameters are introduced for functionally graded microbeams... 

The classical continuum theory is neither able to accurately model the mechanical behavior of micro/nano-scale structures nor capable of justifying the size-dependent behavior observed in these structures; so the non-classical continuum theories such as the strain gradient theory have been emerged and developed. In order to enable the finite element method (FEM) to more accurately deal with the problems in micro/nano-scale structures, a size-dependent Euler-Bernoulli beam element is developed based on the strain gradient theory. Compared to the classical Euler-Bernoulli beam element, the nodal displacement vector of the new Euler-Bernoulli beam element has an additional component, i.e. the... 

In this paper, static behavior and pull-in of micro tweezers is studied. The micro tweezer is modelled as two cantilever beams. Static behavior of the micro tweezer under the effect of electrostatic actuation is modelled using the Euler-Bernoulli beam theory. In order to capture size effects on the behavior of micro tweezers, modified couple stress theory is utilized. It is shown when the voltage between two electrodes increased from some specific value, micro beams adhere to each other and it is observed that the pull-in voltage predicted by the modified couple stress theory considerably differs with that of the classical theory of elasticity. Results of this paper can be used for accurate... 

In this work an analytical solution is presented for a viscoelastic micro-beam based on the modified couple stress theory which is a non-classical theory in continuum mechanics. The modified couple stress theory has the ability to consider small size effects in micro-structures. It is strongly emphasized that without considering these effects in such structures the solution will be wrong and not suitable for designing systems in micro-scales. In this study correspondence principle is used for deriving constitutive equations for viscoelastic material based on the modified couple stress theory. Governing equilibrium equations are obtained by considering an element of micro-beam. Closedform... 

In this paper an analytical solution for size-dependent response of cantilever micro-beams is presented. Using the modified couple stress theory, the small scale effects are accounted for. Employing the Modified Variational Iteration Method, efficient and accurate analytical expressions for the deflection of the micro-beam are presented. Very good agreement is observed between the present work results and available experimental data. This study may be helpful to characterize the size-dependent mechanical properties of MEMS. Consequently, the proposed analytical solution can be used as an efficient tool for studying the effects of the material or geometrical parameters on small scale devices... 

In this paper, the nonlinear forced-vibration of Euler-Bernoulli beams with large deflections is investigated based on the modified couple stress theory, a non-classical theory capable of capturing size effects. The classical theory is unable to predict the size effects. In systems with the dimensions in order of microns and sub-microns the size effects are very significant. For some specific beams subjected to a concentrated force at its middle as the harmonic exciter, the size-dependent responses are investigated for primary, super-harmonic and sub-harmonic resonances. The results show that the frequency-responses of the system are highly size-dependent  

The couple stress theory is a non-classical continuum theory which is capable to capture size effects in small-scale structures. This property makes it appropriate for modeling the structures in micron and sub-micron scales. The purpose of this paper is the derivation of the governing motion equations and boundary conditions for the geometrically nonlinear micro-plates with arbitrary shapes based on the modified version of the couple stress theory. The consistent boundary conditions are provided at smooth parts of the plate periphery and also at the sharp corners of the periphery using variational approach  

This paper investigates the deflection and static pull-in of microbridges based on the modified couple stress theory, a non-classic continuum theory able to predict the size effects for structures in micron and sub-micron scales. The beam is modeled using EulerBernoulli beam theory and the nonlinearities caused by mid-plane stretching have been considered. It is shown that modified couple stress theory predicts size dependent normalized deflection and pull-in voltage for microbeams while according to classical theory the normalized behavior of microbeams is independent of the size of the beam. According to results, when the thickness of the beam is in order of length scale of the beam... 

In this paper, the modified couple stress theory is employed to develop a size-dependent beam element able to predict the size-dependency observed in microbeams. The stiffness matrix is obtained for the aforementioned beam element. As an example, the deflection of a microcantilever is evaluated using the proposed beam elements and the results of the finite element method are compared to the analytical results obtained by the classical beam theory. The maximum deflection of the beam is depicted versus the ratio of the beam thickness to the material length scale parameter, the parameter appearing in non-classical continuum theories. The results show that when the characteristic size of the...