Introduction

Kerid R., Bourouina H., Yahiaoui R., Bounekhla M., Aissat A.

In this paper, we investigate the magnetic field, thermal loads and small scale effects on the dynamics vibration of a nanobeam structure composed of a rectangular configuration perforated with periodic square holes network and subjected to axial magnetic field based on Euler–Bernoulli beam model (EBM) and Timoshenko beam model (TBM). The developed resonance frequencies expressions are derived by modifying the standard equations of dynamics beam vibration. The small scale effect is adopted via the Eringen's nonlocal theory while the coupled governing equations are obtained and solved using analytical solution method in order to determine the resonance frequency of perforated nanobeam. It is found that the resonance frequency change, the magnetic field intensity , the thermal loads and small scale effects are in dependence with geometrical parameters such as size and number of holes. Therefore, these results are discussed for the investigation of the structure dynamic deformation and compared with literature results where new remarks are deduced and presented with detail for a proper design of M/NEMS structures. Geometry and coordinates of perforated nanobeam structure with periodic square holes network. • Resonance frequencies of perforated beams are remarkably affected on geometrical parameters. • Magnetic field, thermal loads and small scale effects reduce the resonance frequencies. • Magnetic field and thermal loads effects on the frequency ratios vary as the resonance frequency change.

Park W., Han S.

Buckling analysis of nonlocal magneto-electro-elastic nano-plate is investigated based on the higher-order shear deformation theory. The in-plane magnetic and electric fields can be ignored for magneto-electro-elastic nano-plates. According to magneto-electric boundary condition and Maxwell equation, the variation of magnetic and electric potentials along the thickness direction of the magneto-electro-elastic plate is determined. To reformulate the elastic theory of magneto-electro-elastic nano-plate, the nonlocal differential constitutive relations of Eringen is applied. Using the variational principle, the governing equations of the nonlocal theory are derived. The relations between local and nonlocal theories are studied by numerical results. Also, the effects of nonlocal parameters, in-plane load directions, and aspect ratio on buckling response are investigated. Numerical results show the effects of the electric and magnetic potentials. These numerical results can be useful in the design and analysis of advanced structures constructed from magneto-electro-elastic materials.

Arefi M., Soltan Arani A.H.

Alibeigi B., Tadi Beni Y., Mehralian F.

In this paper, the buckling response of nanobeams on the basis of the Euler-Bernoulli beam model with the von Kármán geometrical nonlinearity using the modified couple stress theory is investigated under various types of thermal loading and electrical and magnetic fields. The modified couple stress theory, used in this paper, is capable to consider the higher-order electro-mechanical coupling effects besides size effects. The governing equations and boundary conditions are derived using minimum potential energy principle. The nanobeam is assumed to be under two types of thermal loading, uniform and linear, along thickness direction. The buckling response of nanobeams is studied using the Galerkin method and the effects of different parameters, such as size effect, length and thickness, on the critical buckling temperature are shown. The buckling behavior of nanobeam is illustrated significantly size-dependent particularly with an increase in thickness and decrease in length.

Barati M.R.

Magneto-hygro-thermal vibration analysis of double-layered nanoplates made of functionally graded materials is presented based on higher order refined plate theory. For the first time, a double-layered nanoplate is modeled via nonlocal strain gradient theory in which both stiffness-softening and stiffness-hardening effects are incorporated. Another novelty of this paper is that the effects of magnetic and hygro-thermal fields on inhomogeneous double-layered nanoplates are considered to study their behavior under different physical fields. The gradation of material properties is considered using power-law model. The governing equations and related classical and non-classical boundary conditions are derived based on Hamilton’s principle. These equations are solved for hinged nanoplates via Galerkin’s method. It is indicated that type of vibration, moisture rise, temperature rise, nonlocal parameter, strain gradient parameter, material gradation, elastic foundation and side-to-thickness have a remarkable influence on vibration behavior of double-layered nanoscale plates.

Karličić D., Cajić M., Adhikari S., Kozić P., Murmu T.

The recent development in nanotechnology resulted in growing of various nanoplate like structures. High attention was devoted to graphene sheet nanostructure, which enforced the scientist to start developing various theoretical models to investigate its physical properties. Magnetic field effects on nanoplates, especially graphene sheets, have also attracted a considerable attention of the scientific community. Here, by using the nonlocal theory, we examine the influence of in-plane magnetic field on the viscoelastic orthotropic multi-nanoplate system (VOMNPS) embedded in a viscoelastic medium. We derive the system of m partial differential equations describing the free transverse vibration of VOMNPS under the uniaxial in-plane magnetic field using the Eringen's nonlocal elasticity and Kirchhoff's plate theory considering the viscoelastic and orthotropic material properties of nanoplates. Closed form solutions for complex natural frequencies are derived by applying the Navier's and trigonometric method for the case of simply supported nanoplates. The results obtained with analytical method are validated with the results obtained by using the numerical method. In addition, numerical examples are given to show the effects of nonlocal parameter, internal damping, damping and stiffness of viscoelastic medium, rotary inertia and uniaxial in-plane magnetic force on the real and the imaginary parts of complex natural frequencies of VOMNPS. This study can be useful as a starting point for the research and design of nanoelectromechanical devices based on graphene sheets.

Nickell R.E., Sackman J.L.

Several variational principles are derived for the initial-boundary-value problem of fully coupled linear thermoelasticity for an inhomogeneous, anisotropic continuum. A consistent set of field variables is employed and a method based on the Laplace transform is used to incorporate the initial conditions explicitly into the formulation. These principles lend themselves readily to numerical solutions based on an extended Ritz method.

Awrejcewicz J., Krysko-Jr. V.A., Yakovleva T.V., Krysko V.A.

In this work mathematical models of temporal part of chaos at chosen spatial locations of a plate locally reinforced by ribs taking into account an interplay of their interactions are derived and studied numerically for the most relevant dynamical parameters. In addition, an influence of the additive external noise on chaotic vibrations of multi-layer beam–plate structures coupled only by boundary conditions is investigated. We illustrate and discuss novel nonlinear phenomena of the temporal regular and chaotic contact/no-contact dynamics with the help of Morlet wavelets and Fourier analysis. We show how the additive white noise cancels deterministic chaos close to the boundary of chaotic region in the space of parameters, and we present windows of on/off switching of the frequencies during the contact dynamics between structural members. In order to solve the mentioned design type nonlinear problem we apply methods of qualitative theory of differential equations, the Bubnov–Galerkin method in higher approximations, the Runge–Kutta methods of 4th, 6th and 8th order, as well as the computation and analysis of the largest Lyapunov exponent (Benettin׳s and Wolf׳s algorithms are used). The agreement of outcomes of all applied qualitatively different numerical approaches validate our simulation results. In particular, we have illustrated that the Fourier analysis of the studied mechanical structures may yield erroneous results, and hence the wavelet-based analysis is used to investigate chaotic dynamics in the system parameter space.

Ghadiri M., Hosseini S.H., Shafiei N.

ABSTRACTIn this article, the equation of motion for a rotating nanocantilever has been developed based on the Euler–Bernoulli beam model, which includes the effect of temperature, small scale effect, and centrifugal force. A power series method has been employed to obtain the exact solution of the natural frequencies. The results also compared with other solutions of exact and approximate differential quadrature method. The effects of temperature, angular velocity, and small scale in the vibration characteristics of a rotating nanocantilever beam are investigated. It is shown that the effect of temperature plays a significant role in the behavior of the vibration of a rotating nanocantilever. Nondimensional frequency increases in the first mode with increasing the nonlocal parameter while it is inverse for the second and third modes of vibration.

Vatankhah R., Kahrobaiyan M.H.

Hosseini M., Jamalpoor A.

In this article, based on the nonlocal elasticity theory of Eringen, dynamic characteristics of a double-FGM viscoelastic nanoplates-system subjected to temperature change with considering surface effects (surface elasticity, tension and density) is studied. Two Kirchhoff nanoplates are coupled by an internal Kelvin–Voigt viscoelastic medium and also are limited to the external Pasternak elastic foundation. The material properties of the simply supported functionally graded nanoplates are assumed to follow power law distribution in the thickness direction. The governing equations of motion for three cases (out-of-phase vibration, in-phase vibration and one nanoplate fixed) are derived from Hamilton's principle. The analytical approach is employed to determine explicit closed-form expression for complex natural frequencies of the system. Numerical results are presented to show variations of the frequency of double-FGM viscoelastic nanoplates corresponding to various values of the nonlocal parameter, temper...

Ansari R., Faghih Shojaei M., Ebrahimi F., Rouhi H., Bazdid-Vahdati M.

Based on Mindlin’s strain gradient elasticity and Euler–Bernoulli beam theory, a non-classical beam element capable of considering micro-structure effects is developed. To accomplish this aim, the higher-order tensors of energy pairs in the energy functional are vectorized and written in the quadratic representation, from which the stiffness and mass matrices of the element are obtained. In comparison with the classical Euler–Bernoulli beam element, the new element needs one additional nodal degree of freedom (DOF) which results in a total of three DOFs per node. The formulation of the paper is general so that it can be reduced to that based on the modified couple stress theory, the modified strain gradient theory, and the classical elasticity theory. To show the reliability of the proposed element, the bending and free vibration problems of microbeams under different kinds of end conditions are addressed. It is revealed that the present finite element results are in excellent agreement with the ones achieved through analytical solutions.

Karličić D., Kozić P., Adhikari S., Cajić M., Murmu T., Lazarević M.

Nano-materials such as graphene sheets have a great opportunity to be applied in development of a new generation of nanomechanical sensors and devices due to their unique physical properties. Based on the nonlocal continuum theory and vibration analysis, the single-layered graphene sheet with attached nanoparticles affected by in-plane magnetic field is proposed as a new type of the mass-nanosensor. The nonlocal Kirchhoff-Love plate theory is adopted to describe mechanical behavior of single-layered graphene sheet as an orthotropic nanoplate. The equation of motion of a simply supported orthotropic nanoplate is derived, where the influence of Lorentz magnetic force is introduced through classical Maxwell׳s equations. Complex natural frequencies, damped frequency shifts and relative shift of damping ratio for nanoplate with attached nanoparticles are obtained in the explicit form. The influences of the nonlocal and magnetic field parameter, different mass weights and positions of attached nanoparticles and damping coefficients on the relative damped frequency shift and relative shift of damping ratio are examined. The presented results can be useful in the analysis and design of nanosensors applied in the presence of strong magnetic field. Our results show that magnetic field could be successfully used to improve sensibility performances of nanomechanical sensors.

Ansari R., Pourashraf T., Gholami R., Sahmani S., Ashrafi M.A.

In the present study, the resonant frequency and flexural sensitivity of atomic force microscope (AFM) microcantilevers are predicted incorporating size effects. To this end, the modified strain gradient elasticity theory is applied to the classical Euler-Bernoulli beam theory to develop a non-classical beam model which has the capability to capture size-dependent behavior of microcantilevers. On the basis of Hamilton's principle, the size-dependent analytical expressions corresponding to the frequency response and sensitivity of AFM cantilevers are derived. It is observed that by increasing the contact stiffness, the resonant frequencies of AFM cantilevers firstly increase and then tend to remain constant at an especial value. Moreover, the resonant frequencies of AFM cantilevers obtained via the developed non-classical model is higher than those of the classical beam theory, especially for the values of beam thickness close to the internal material length scale parameter.

Karličić D., Jovanović D., Kozić P., Cajić M.