In this paper, free transverse vibration and buckling analyses of a nanobeam are presented by coupling the Euler-Bernoulli beam (EBT) theory and Eringen’s nonlocal elasticity theory. The nanobeam is embedded in the Pasternak foundation. Hamilton’s energy principle is used to derive governing differential equations. The Lagrange polynomial-based differential quadrature method (PDQM) and a harmonic differential quadrature method (HDQM) are used to convert the governing differential equation and boundary conditions into a set of linear algebraic equations. The first three frequencies and the lowest critical buckling loads for clamped-clamped, clamped-simply supported, and simply supported-simply supported boundary conditions are obtained by implementing the bisection method through a computer program written in C++. The impacts of nonlocal Eringen’s parameter (scaling effect parameter), boundary conditions, axial force, and elastic foundation moduli on frequencies are examined. The effects of nonlocal Eringen’s parameter, boundary conditions, and elastic foundation moduli on critical buckling load are also studied. A convergence study of both versions of DQM is conducted to validate the present analysis. A comparison of frequencies and critical buckling loads with those available in the literature is presented.

In this study, the vibration behavior of functional graded (FG) circular and annular nanoplate embedded in a Visco-Pasternak foundation and coupled with temperature change is studied. It is assumed that a uniform radial magnetic field acts on the top surface of the plate and the magnetic permeability coefficient of the plate along its thickness are assumed to vary according to the volume distribution function. The effect of in-plane pre-load and temperature change are investigated on the vibration frequencies of FG circular and annular nanoplate. To obtain the vibration frequencies of the FG circular and annular nanoplate, two different size dependent theories are utilized. The material properties of the FGM nanoplates are assumed to vary in the thickness direction and are estimated through the Mori– Tanaka homogenization technique. The FG circular and annular nanoplate is coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak foundation. By using the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST), the governing equation is derived for FG circular and annular nanoplate. The differential quadrature method (DQM) and the Galerkin method (GM) are utilized to solve the governing equation to obtain the frequency vibration of FG circular and annular nanoplate. The results are subsequently compared with valid result reported in the literature. The effects of the size dependent, the in-plane pre-load, the temperature change, the magnetic field, the power index parameter, the elastic medium and the boundary conditions on the natural frequencies are investigated. The results show that the size dependent parameter has an increasing effect on the vibration response of circular and annular nanoplate. Also, the temperature change play important role in the mechanical behavior of the FG circular and annular nanoplate. According to the results, the application of radial magnetic field to the top surface of the plate changes the state of stress in both tangential and radial direction. The present analysis results can be used for the design of the next generation of nano-devices that make use of the thermal vibration properties of the nanoplate.

The present work introduces the thermoelastic vibrations of nonlocal nanobeams resting on a two-parameter foundation. The governing equations are formulated for linear Winkler– Pasternak foundation type based on the generalized dualphase-lag heat conduction and nonlocal beams theories. The nanobeam is subjected to a temperature ramping function. The coupled equations of the problem are formulated and solved by Laplace transform technique. The effects of the nonlocal parameter and different foundation parameters on the field variables are illustrated graphically and discussed. The results obtained are consistent with previous analytical and numerical results.

This article proposes a refined higher order nonlocal strain gradient theory for stresses and deflections of new model of functionally graded (FG) sandwich nanoplates resting on Pasternak elastic foundation. Material properties of the FG layers are supposed to vary continuously through-the-thickness according to a power function or a sigmoid function in terms of the volume fractions of the constituents. The face layers are made of FG material while the core layer is homogeneous and made of ceramic. In this study, an analytical approach is proposed using the higher-order shear deformation plate theory and nonlocal strain gradient theory with combination of various boundary conditions. Numerical outcomes are reported to display the impact of the material distribution, boundary conditions, elastic foundation parameters and the sandwich nanoplate geometry on the deflections and stresses of FG sandwich nanoplates. The exactness of this theory is determined by comparing it to other published outcomes.

his study analyzes the free vibration, forced vibration, resonance, and stress wave propagation of orthotropic sandwich plates made of functionally graded materials (FGMs). Dynamic analyses are conducted using a mesh-free method based on first-order shear deformation theory and the shape functions constructed using moving least squares approximation. The sandwich plates are rested on a Pasternak elastic foundation and subjected to periodic or impact loading and essential boundary conditions, which are imposed through a transfer function method. The sandwich plates are assumed to be composed of a homogeneous orthotropic core and two orthotropic FGM face sheets made of two orthotropic materials. The volume fractions of the materials are varied smoothly along the thickness of the face sheets. The convergence and accuracy of the applied method are demonstrated, after which numerical analyses are conducted to investigate the effects of elastic foundation coefficients, material distributions, geometrical dimensions, time-dependent loading, and boundary conditions on the vibrational and dynamic characteristics of the orthotropic FGM sandwich plates.

In this paper, the free vibration of the beam on elastic foundation is studied using finite element method. To this end, a two- node Timoshenko element is used for beam modeling. Each node in this element has a rotational and translational degree of freedom, which encompasses all four degrees of freedom. The displacement and rotational fields of this beam is selected from the third and the second order, respectively. Moreover, the shear strain of the element is assumed as a constant value. Interpolation functions for displacement field and beam rotation are explicitly calculated by employing total beam energy and it's stationary with respect to shear strain. Also, two-parameter elastic foundation model is used. In this method, the soil is modeled as a layer of Winkler springs with a shear layer on it. Next, by utilizing the interpolation functions, the stiffness matrices of beam and foundation, as well as their mass matrices are introduced; hence, the free vibration analysis on the elastic foundation is carried out. Finally, after conducting several tests, the high efficiency and accuracy of the proposed element is demonstrated.

In this study, the vibration behavior of circular and annular graphene sheet embedded in a Visco-Pasternak foundation and coupled with temperature change and under in-plane pre-load is studied. The single-layered annular graphene sheet is coupled by an enclosing viscoelastic medium which is simulated as a Visco- Pasternak foundation. By using the nonlocal elasticity theory and classical plate theory, the governing equation is derived for single-layered graphene sheets (SLGSs). The closed-form solution for frequency vibration of circular graphene sheets has been obtained and nonlocal parameter, in-plane pre-load, the parameters of elastic medium and temperature change appears into arguments of Bessel functions. To verify the accuracy of the present results, the new version differential quadrature method (DQM) is also developed. Closed-form results are successfully verified with those of the DQM results. The results are subsequently compared with valid result reported in the literature. The effects of the small scale, pre-load, 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 on the non-dimensional 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, temperature change and initial pre-load have significant effects on vibration response of the circular nanoplates. The present analysis results can be used for the design of the next generation of Nano devices that make use of the thermal vibration properties of the graphene.