Natural frequencies of FGM nanoplates embedded in an elastic medium

Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium

Thermal vibration and buckling analysis of two-phase nanobeams embedded in size dependent elastic medium

Vibration analysis of non-uniform and non-homogeneous boron nitride nanorods embedded in an elastic medium under combined loadings using DQM

A size-dependent Kirchhoff micro-plate model resting on elastic medium is developed based on the strain gradient elasticity theory. Three material length scale parameters are introduced in the model, and those parameters may effectively capture the size effect. The model can degenerate into the modified couple stress plate model or the classical plate model by setting two (l 0 and l 1) or all (l 0, l 1 and l 2) of the material length scale parameters to be zero. Analytical solutions for the static bending, buckling and free vibration problems of a rectangular micro-plate with all edges simply supported are obtained. The results predicted by the present model are compared with those predicted by the degraded models. Influences of the elastic medium on the static bending, buckling, and free vibration are discussed. The results show that the present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter. The study may be helpful to guide the design of microplate-based devices resting on elastic medium for a wide range of potential applications.

Frequency domain analysis of nonlocal rods embedded in an elastic medium

In this study, the vibration behavior of annular and circular graphene sheet coupled with temperature change and under in-plane pre-stressed is studied. Influence of the surrounding elastic medium 011 the fundamental frequencies of the single-layered graphene sheets (SLGSs) is investigated. Both Winkler-type and Pasternak- type models are employed to simulate the interaction of the graphene sheets with a surrounding elastic medium. By using the nonlocal elasticity theory the governing equation is derived for SLGSs. The closed-form solution for frequency vibration of circular graphene sheets lias been obtained and nonlocal parameter, inplane pre-stressed, the parameters of elastic medium and temperature change appears into arguments of Bessel functions. The results are subsequently compared with valid result reported in the literature and the molecular dynamics (MD) results. The effects of the small scale, pre-stressed, mode number, temperature change, elastic medium and boundary conditions on natural frequencies are investigated. The non-dimensional frequency decreases at high temperature case with increasing the temperature change for all boundary conditions. The effect of temperature change 011 the frequency vibration becomes the opposite at high temperature case in compression with the low temperature case. The present research work thus reveals that the nonlocal parameter, boundary conditions and temperature change have significant effects on vibration response of the circular nanoplates. The present results can be used for the design of the next generation of nanodevices that make use of the thermal vibration properties of the graphene.

Vibration analysis of fluid-conveying nanotubes embedded in an elastic medium considering surface effects

In this paper, the small size effect on the free vibration behavior of finite length nanotubes embedded in an elastic medium is investigated. The problem is formulated based on the three-dimensional (3D) nonlocal elasticity theory. Since the 3D nonlocal constitutive relations in a cylindrical coordinate system are used, in addition to displacement components, the stress tensor components are chosen as degrees of freedom. The surrounding elastic medium is modeled as the Winkler’s elastic foundation. The differential quadrature method as an efficient and accurate numerical tool in conjunction with the series solution is used to discretize the governing equations. Very fast rate of convergence of the method is demonstrated. The effects of the nonlocal parameter together with the other geometrical parameters and also the stiffness parameter of the elastic medium on the natural frequencies are studied.

In this study, free torsional vibration and forced torsional vibration analysis under the time-dependent exponential and harmonic torsional loadings in single-walled carbon nanotube are investigated. The SWCNT is embedded in an elastic medium. Eringen's theory among the small-scale theories is selected. The nonlocal differential constitutive relation and corresponding boundary condition are derived via Hamilton's principle. Clamped–clamped boundary condition is utilized. The assumed modes method is employed for the dynamic torsional vibration in order to discretize the derived governing equations. The novelty of this work is devoted to the analysis of forced torsional vibration of a carbon nanotube embedded in an elastic medium under the various loadings. The angular displacement for the resonance frequency neglecting the elastic medium is illustrated. For the free analysis, the first three nondimensional natural frequencies with various small-scale parameters and stiffness of the elastic medium are calculated. The results are compared with another study for the first 10 mode numbers. The effects of the nonlocal parameter, length of carbon nanotube, stiffness of the elastic medium, thickness, time constant, and excitation frequency on the nondimensional and dimensional angular displacements are investigated, dynamically. For the greater values of the stiffness of the medium, the nonlocal parameter becomes negligible. When a time-dependent exponential torque is applied to the model, the angular displacement becomes greater and then lower by an increase in the value of the length, but the nondimensional angular displacement decreases continuously by increasing the value of the length under the time-dependent harmonic loading. Moreover, the angular displacement for a determined time becomes lower first and then becomes greater by increasing the time constant.

The oscillations of a rod in an inhomogeneous elastic medium

Thermal–mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

In this research, vibration analysis of a viscoelastic sandwich plate rested on a visco-Pasternak foundation is investigated. Sandwich plate consists of a magnetorheological (MR) fluid core and viscoelastic nanocomposite face sheets which include piezoelectric matrix and functionally graded carbon nanotubes (FG-CNTs) fiber. Core and face sheets are affected by applying magnetic and electric fields, respectively. The elastic medium is simulated by a visco-Pasternak model which is considered in terms of the effects of springiness, shear and damping of foundation. At first, the constitutive equations of sandwich plate are expressed by employing the geometric continuity conditions. Then, the equations of motion are derived by applying the energy method and Hamilton’s principle and an appropriate analytical approach is proposed to solve them. This approach can be considered as the different boundary conditions. Finally, the influences of various parameters such as the type of MR material, core to face sheets thickness ratio, volume fractions and symmetric distributions of CNTs in face sheets, magnetic and electric fields, viscoelastic constant of face sheets and elastic medium on the dimensionless natural frequencies of sandwich plate are studied. The results show that increasing core-to-face sheet thickness ratio leads to a decrease in the natural frequencies because the MR fluid core is softer than nanocomposite face sheets. Thus, the MR fluid core acts like a damper. Also, the loss factor of the MR fluid core decreases with increasing magnetic field intensity, and consequently, the natural frequency grows. In addition, increasing the volume fraction of CNTs causes the stiffness of the structure to increase. The results of this research can be utilized to realize applicable semi-active devices with controllable stiffness.

Optimal design of micron-scale beams as a general case is an important problem for development of micro-electromechanical devices. For various applications, the mechanical parameters such as mass, maximum deflection and stress, natural frequency and buckling load are considered in strategies of micro-manufacturing technologies. However, all parameters are not of equal importance in each operating condition but multi-objective optimization is able to select optimal states of micro-beams which have desirable performances in various micro-electromechanical devices. This paper provides optimal states of design variables including thickness, distribution parameter of functionally graded materials, and aspect ratio in simply supported FG micro-beams resting on the elastic foundation using analytical solutions. The elastic medium is assumed to be as a two-layered foundation including a shear layer and a linear normal layer. Also, the size effect on the mechanical parameters is considered using the modified strain gradient theory and non-dominated sorting genetic algorithm-II is employed to optimization procedure. The target functions are defined such that the maximum deflection, maximum stress and mass must be minimized while natural frequency and critical buckling load must be maximized. The optimum patterns of FG micro-beams are presented for exponential and power-law FGMs and the effect of theory type and elastic foundation discussed in details. Findings indicate that the elastic foundation coefficients and internal length scale parameters of materials have the significant influences on the distribution of design variables. It is seen that the optimum values of inhomogeneity parameter and aspect ratio for E-FG micro-beams predicted by the modified strain gradient theory are larger than those of the classical continuum theory. Also, the multi-objective optimization is able to improve the normalized values of mass, maximum deflection, buckling load and natural frequency of P-FG micro-beams.

A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory

In this scientific work, a new shear deformation theory for free vibration analysis of simply supported rectangular functionally graded plate embedded in an elastic medium is presented. Due to technical problems during the fabrication, porosities can be created in side FGM plate which may lead to reduction in strength of materials. In this investigation the FGM plate are assumed to have a new distribution of porosities according to the thickness of the plate. The elastic medium is modeled as Winkler-Pasternak two parameter models to express the interaction between the FGM plate and elastic foundation. The four unknown shear deformation theory is employed to deduce the equations of motion. The Hamilton’s principle is used to derive the governing equations of motion. The accuracy of this theory is verified by compared the developed results with those obtained using others plate theory. Some examples are performed to demonstrate the effect of changing gradient material, elastic parameters, porosity index, and length to thickness ratios on the fundamental frequency of functionally graded plate.