Free vibration analysis of embedded functionally graded carbon nanotube-reinforced composite conical/cylindrical shells and annular plates using a numerical approach

Buckling and vibration analysis of embedded functionally graded carbon nanotube-reinforced composite annular sector plates under thermal loading

Vibrational analysis of functionally graded carbon nanotube-reinforced composite spherical shells resting on elastic foundation using the variational differential quadrature method

The vibration analysis of functionally graded carbon nanotube-reinforced composite plates with the arbitrarily shaped cutout is presented using a novel numerical approach called variational differential quadrature finite element method (VDQFEM). The governing equations are expressed in matrix form based on Mindlin’s plate theory. In the proposed numerical approach, the space domain of the plate is first transformed into a number of sub-domains known as finite elements. Then, the variational differential quadrature (VDQ) discretization method is used within each element to obtain the mass and stiffness matrices. In order to use the VDQ method, the irregular domain of the element is transformed into a regular one employing the mapping technique. Finally, the assemblage procedure is performed to present total mass and stiffness matrices. The introduced numerical approach can be effectively used for structural analysis of arbitrarily shaped plates. A wide range of comparative and convergence studies are outlined to show the performance of the method. It is observed that the numerical results are rapidly converged and the proposed solution strategy can be successfully applied to examine the vibration of FG-CNTRC plates with different cutouts.

Buckling analysis of axially-loaded functionally graded carbon nanotube-reinforced composite conical panels using a novel numerical variational method

Variational differential quadrature: A technique to simplify numerical analysis of structures

This article presents an attempt to study the nonlinear resonance of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) annular sector plates excited by a uniformly distributed harmonic transverse load. To this purpose, first, the extended rule of mixture including the efficiency parameters is employed to approximately obtain the effective material properties of FG-CNTRC annular sector plates. Then, the focus is on presenting the weak form of discretized mathematical formulation of governing equations based on the variational differential quadrature (VDQ) method and Hamilton’s principle. The geometric nonlinearity and shear deformation effects are considered based on the von Karman assumptions and Reddy’s third-order shear deformation plate theory, respectively. The discretization process is performed via the generalized differential quadrature (GDQ) method together with numerical differential and integral operators. Then, an efficient multi-step numerical scheme is used to obtain the nonlinear dynamic behavior of the FG-CNTRC annular sector plates near their primary resonance as the frequency-response curve. The accuracy of the present results is first verified and then a parametric study is presented to show the impacts of CNT volume fraction, CNT distribution pattern, geometry of annular sector plate and sector angle on the nonlinear frequency-response curve of FG-CNTRC annular sector plates with different edge supports.

Atomistic-continuum modeling of vibrational behavior of carbon nanotubes using the variational differential quadrature method

Eringen’s nonlocal theory (ENT) provides an efficient tool to consider the size dependency for the structural analysis of small-scale structures. Recently, it has been indicated that in some cases, inconsistent results may be obtained using the Eringen differential model (EDM) and Eringen integral model (EIM) should be employed. In this regard, the vibration of nanoplates resting on the elastic medium is investigated on the basis of the EIM using variational differential quadrature (VDQ) method. On the basis of the first-order shear deformation theory (FSDT), the EIM and EDM formulations are given. An effective numerical technique is applied in the framework of the energy method to find the natural frequencies. Comprehensive results are reported to compare the EIM and EDM. The results show that using the proposed integral model, the paradox related to cantilever nanoplates is resolved.

This article deals with the large amplitude free and forced vibration analysis of functionally graded carbon nanotube‐reinforced composite (FG‐CNTRC) annular sector plates based on a numerical approach. The modified rule of mixture is used to estimate the material properties. The equations of motion are developed based on the first‐order shear deformation theory (FSDT) and the von Kármán geometric nonlinearity. First, the discretized form of energy functional of structure is given with the aid of variational differential quadrature method. Then, a time periodic discretization is performed and the frequency response of the nanocomposite plate is determined using the pseudo‐arc length continuation method. After verifying the correctness of the proposed approach, a comprehensive parametric study is presented to investigate the effects of important factors on the nonlinear vibration characteristics of the FG‐CNTRC annular sector plates. The results imply that the volume fraction and distribution type of nanotubes have considerable effects on the fundamental frequency as well as nonlinear frequency response curves. POLYM. COMPOS., 40:E1364–E1377, 2019. © 2018 Society of Plastics Engineers

ABSTRACTEmploying the variational differential quadrature (VDQ) method, the effects of initial thermal loading on the vibrational behavior of embedded single-walled carbon nanotubes (SWCNTs) based on the nonlocal shell model are studied. According to the first-order shear deformation theory and considering Eringen's nonlocal elasticity theory, the energy functionality of the system is presented and discretized using the VDQ method. The effects of thermal loading and elastic foundation are simultaneously taken into account. The use of the numerical discretization technique in the context of variational formulation reduces the order of differentiation in the governing equations and consequently improves the convergence rate. The accuracy of the present model is first checked by comparison with molecular dynamics simulation results and those of other methods. The effects of involved parameters are then investigated on the fundamental frequencies of thermally preloaded embedded SWCNTs. The results imply that the thermal loading has a significant effect on the vibration analysis of embedded SWCNTs.

Numerical study on the buckling and vibration of functionally graded carbon nanotube-reinforced composite conical shells under axial loading

Nonlinear large deformation analysis of shells using the variational differential quadrature method based on the six-parameter shell theory

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

This paper aims to investigate the imperfection sensitivity of the post-buckling behavior and the free vibration response under pre- and post-buckling of nanoplates with various edge supports in the thermal environment. Formulation is based on the higher-order shear deformation plate theory, von Kármán kinematic hypothesis including an initial geometrical imperfection and Gurtin–Murdoch surface stress elasticity theory. The discretized nonlinear coupled in-plane and out-of-plane equations of motion are simultaneously obtained using the variational differential quadrature (VDQ) method and Hamilton’s principle. To this end, the displacement vector and nonlinear strain–displacement relations corresponding to the bulk and surface layers are matricized. Also, the variations of potential strain energies, kinetic energies and external work are obtained in matrix form. Then, the VDQ method is employed to discretize the obtained energy functional on space domain. By Hamilton’s principle, the discretized quadratic form of nonlinear governing equations is derived. The resulting equations are solved employing the pseudo-arc-length technique for the post-buckling problem. Moreover, considering a time-dependent small disturbance around the buckled configuration, the vibrational characteristics of pre- and post-buckled nanoplates are determined. The influences of initial imperfection, thickness, surface residual stress and temperature rise are examined in the numerical results.

A numerical study is performed to investigate the impacts of thermal loading on the vibration and buckling of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) conical shells. Thermo-mechanical properties of constituents are considered to be temperature-dependent. Considering the shear deformation theory, the energy functional is derived, and applying the variational differential quadrature (VDQ) method, the mass and stiffness matrices are obtained. The shear correction factors are accurately calculated by matching the shear strain energy obtained from an exact three-dimensional distribution of the transverse shear stresses and shear strain energy related to the first-order shear deformation theory. Numerical results reveal that considering temperature-dependent material properties plays an important role in predicting the thermally induced vibration of FG-CNTRC conical shells, and neglecting this effect leads to considerable overestimation of the stiffness of the structure.