Variational Differential Quadrature Research Articles

Analyzing primary resonant dynamics of functionally graded nanoplates based on a surface third-order shear deformation model

In the context of Gurtin-Murdoch (GM) surface elasticity theory, a size-dependent third-order shear deformable plate model is developed herein in order to study the nonlinear forced vibration behavior of rectangular nanoplates with considering surface stress effect. Nanoplates are assumed to be made of functionally graded materials (FGMs) whose properties are graded in the thickness direction based on a power-law distribution. First, the constitutive relations of GM model are matricized. Then, Hamilton’s principle is used to derive the governing equations. The variational differential quadrature, a numerical Galerkin, time periodic discretization, and pseudo arc-length methods are also employed for numerical solution of the geometrically nonlinear forced vibration problem. The frequency-response curves of rectangular nanoplates with different boundary conditions are investigated for different values of thickness, power-law index, surface constants and side length-to-thickness ratio. The results reveal that the surface stress has an important influence on the frequency-response curve of nanoplates at very small scales.

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

The present work is concerned with the application of the variational differential quadrature (VDQ) method (Faghih Shojaei and Ansari, 2017), in the area of computational mechanics, to the nonlinear large deformation analysis of shell-type structures. To this end, based on the six-parameter shell model, the functional of energy in quadratic form is derived based on Hamilton’s principle which is then directly discretized by the VDQ technique. The formulation of article is presented in a general form so that it can be readily used for different structures such as beams, annular plates, cylindrical shells and hemispherical shells under various loading conditions. In order to reveal the accuracy of developed solution strategy, it is tested in several popular benchmark problems for the geometric nonlinear analysis of shells. The results show that the present numerical method is capable of yielding highly accurate solution in the nonlinear large deformation analysis of shells. It is also easy to implement due to its compact and explicit matrix formulation.

Numerical study on the thermal buckling analysis of CNT-reinforced composite plates with different shapes based on the higher-order shear deformation theory

In the present study, based on the higher-order shear deformation plate theory, the unified numerical formulation is developed in variational framework to investigate the thermal buckling of different shapes of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. Since the thermal environment has considerable effects on the material properties of carbon nanotubes (CNTs), the temperature-dependent (TD) thermo-mechanical material properties are taken into account. In order to present the governing equations, the quadratic form of the energy functional of the plate structure is derived and its discretized counterparts are presented employing the variational differential quadrature (VDQ) approach. The discretized equations of motion are finally obtained based on Hamilton's principle. In order to convenient application of differential quadrature numerical operators in irregular physical domain, the mapping procedure is considered in accordance to the conventional finite element formulation. Some comparison and convergence studies are performed to show validity and efficiency of the proposed approach. A wide range of numerical results are also reported to analyze the thermal buckling behavior of different shaped FG-CNTRC plates.

Imperfection Sensitivity of Post-Buckling Behavior and Vibration Response in Pre- and Post-Buckled Regions of Nanoscale Plates Considering Surface Effects

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.

Geometrically nonlinear resonance of higher-order shear deformable functionally graded carbon-nanotube-reinforced composite annular sector plates excited by harmonic transverse loading

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.

Thermally induced mechanical analysis of temperature-dependent FG-CNTRC conical shells

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.

Nonlinear Forced Vibration Analysis of FG-CNTRC Cylindrical Shells Under Thermal Loading Using a Numerical Strategy

Employing an efficient numerical strategy, the nonlinear forced vibration analysis of composite cylindrical shells reinforced with single-walled carbon nanotubes (CNTs) is carried out. It is assumed that the distribution of CNTs along the thickness direction of the shell is uniform or functionally graded and the temperature dependency of the material properties is accounted. The governing equations are presented based on the first-order shear deformation theory along with von-Karman nonlinear strain-displacement relations. The vectorized form of energy functional is derived and directly discretized using numerical differential and integral operators. By the use of variational differential quadrature (VDQ) method, discretized nonlinear governing equations are obtained. Then, the time periodic differential operators are applied to perform the discretization procedure in time domain. Finally, the pseudo-arc length continuation method is employed to solve the nonlinear governing equations and trace the frequency response curve of the nanocomposite cylindrical shell. A comparison study is first presented to verify the efficiency and validity of the proposed numerical method. Comprehensive numerical results are then given to investigate the effects of the involved factors on the nonlinear forced vibration characteristics of the structure. The results show that the changes of fundamental vibrational mode shape have considerable effects on the frequency response curves of composite cylindrical shells reinforced with CNTs.

Nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite rectangular plates

The geometrically nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite (FG-GPLRC) rectangular plates with different edge conditions is examined. The considered plate with NL-layers is made from a mixture of an isotropic polymer matrix and graphene platelets (GPLs) in each layer. The weight fraction of GPLs changes in a layer-wise manner. The modified Halpin-Tsai model and rule of mixture are utilized to compute the effective material properties of FG-GPLRCs. To mathematically model the vibrations of FG-GPLRC plates, the displacement field, strain tensor and constitutive relations as well as the energy functional of system including strain and kinetic energies and external work are represented in matrix forms as a function of the displacement components. Then, by simultaneous use of Hamilton’s principle and an efficient numerical scheme namely, the variational differential quadrature (VDQ) technique, the weak form of discretized nonlinear equations of motion is obtained. The present model includes the influences of geometric nonlinearity, rotary inertia and transverse shear deformation. Furthermore, a multistep numerical approach based on the Galerkin method, time periodic discretization method and pseudo arc-length continuation technique in conjunction with the modified Newton-Raphson method is employed to solve the problem of nonlinear harmonically excited vibration of FG-GPLRC rectangular plates. Results are plotted in the form of frequency-response and force-response curves to indicate the effect of various parameters such as GPL distribution pattern, weight fraction, geometry of GPL nanofillers and boundary constraints of FG-GPLRC plates.

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

A numerical approach is adopted for the multiscale analysis of vibrations of single-walled carbon nanotubes (SWCNTs). The SWCNT is modeled by a hyperelastic membrane whose kinematics is described using the higher-order Cauchy-Born rule. The constitutive model is formulated exclusively in terms of the interatomic potential, so, it inherits the atomistic information and involves no other phenomenological input. The variational differential quadrature (VDQ) method is employed in which the continuum model is discretized using DQ, and a weak form of equation of motion is obtained via a variational approach. VDQ is computationally advantageous since it has a fast rate of convergence and can reproduce the results of molecular dynamics simulations. Detailed investigations into frequencies and mode shapes of SWCNTs with different geometrical parameters, boundary conditions and chiralities are carried out. It is found that short nanotubes display a coupling between the axial/torsional and bending modes. Also, as the tube diameter or length increases, mode transitions are made at several critical points. If the edge supports are more flexible and tube length is longer, the critical diameters are larger. Eventually, the vibration characteristics of axially strained nanotubes are analyzed, and it is concluded that SWCNTs with smaller radii have higher strain sensitivity.

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.

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

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.

An efficient numerical method for analyzing the thermal effects on the vibration of embedded single-walled carbon nanotubes based on the nonlocal shell model

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.

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

• Proposing a novel solution methodology called VDQ. • Presenting a highly accurate integral operator using Taylor series and DQ rule. • Presenting a matrix representation for the nonlinear Green-St Venant strain tensor. • Presenting symmetric formulation for equations arising from the fully nonlinear elasticity theory. • Unified and standard formulating the nonlinear discretized equations independent of material type, structure type and even selected theory. A new solution method in the area of computational mechanics is developed in this article, which is called variational differential quadrature (VDQ). The main idea of this method is based on the accurate and direct discretization of the energy functional in the structural mechanics. In the VDQ method, through developing an efficient matrix formulation and using an accurate integral operator, the discretized governing equations are derived directly from the weak form of the equations with no need for the analytical derivation of the strong form. This technique provides an alternative way to discretize the energy functional, which avoids the local interpolation and the assembly process of the methods of this kind. We first implement the VDQ method for the nonlinear elasticity theory considering the Green-St. Venant strain tensor; then we simplify the formulation further for the first-order shear deformable beam and plate theories. The final formulation of these cases demonstrates the simplicity of the implementation for the VDQ method in the numerical analysis of the structures, which is a major goal for this article. Using these examples, one can easily learn and apply this technique to other structures. To assess the performance of the VDQ method, we compare it with the generalized differential quadrature (GDQ) method and finite element method (FEM) in the case of bending analysis of Mindlin plates. It is indicated that computational cost of VDQ is less than that of GDQ, and the convergence rate of VDQ is faster than that of FEM.

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

Abstract The main objective of this paper is to present the buckling and vibration analysis of thermally pre-stressed functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) annular sector plates resting on the elastic foundation via the variational differential quadrature (VDQ) method. The material properties of nanocomposite plate are considered to continuously vary across the thickness and are estimated according to the modified rule of mixture. The governing equations are derived on the basis of first order shear deformation theory. Applying two-dimensional generalized differential quadrature (GDQ) method, the energy functional of the structure is discretized. Then, based on Hamilton's principle and the VDQ method, the reduced forms of mass and stiffness matrices are obtained. After verifying the accuracy of the present method, comprehensive numerical results are presented to examine the effects of important parameters on the stability and vibrational behavior of the nanotube-reinforced composite annular sector plates. The results indicate that functionally graded distributions of CNTs in the thickness direction and the increase of elastic foundation coefficients can improve the stability of the structure.

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

Buckling analysis of axially-compressed functionally graded carbon nanotube-reinforced composite (FG-CNTRC) conical panels is presented employing the variational differential quadrature (VDQ) method. The material properties of nanocomposite conical panel are assumed to be graded along the thickness direction and are estimated through the micromechanical model. To present the energy functional of the structure, the first-order shear deformation theory is utilized. Applying the generalized differential quadrature (GDQ) method in axial and circumferential directions, the discretized form of energy functional is obtained. Then, based on Hamilton’s principle and matrix relations, the reduced form of stiffness matrices is derived. A comparison between the obtained results and those given in the literature shows the accuracy of the present approach. Numerical results indicate that volume fractions and distribution patterns of CNTs have significant effects on the buckling load of FG-CNTRC conical panels.

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

Abstract In this paper, the free vibration characteristics of embedded functionally graded carbon nanotube-reinforced composite (FG-CNTRC) spherical shells are studied based on a numerical approach. The elastic foundation is considered to be Pasternak-type. Moreover, the extended rule of mixture is used so as to obtain the material properties of FG-CNTRC. The shell is also modeled according to the first-order shear deformation shell theory. The energy functional of the structure is obtained first. Using differential operators, the discretized form of the energy functional is derived. By means of the variational differential quadrature (VDQ) method, the reduced forms of mass and stiffness matrices are then obtained. Selected numerical results are given to investigate the effects of different parameters such as elastic foundation coefficients, boundary conditions, CNT volume fraction, thickness-to-radius ratio and type of distribution of CNT on the vibrations of FG-CNTRC spherical shells.

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

Free vibration analysis of embedded functionally graded carbon nanotube-reinforced composite (FG-CNTRC) conical, cylindrical shells and annular plates is carried out using the variational differential quadrature (VDQ) method. Pasternak-type elastic foundation is taken into consideration. It is assumed that the functionally graded nanocomposite materials have the continuous material properties defined according to extended rule of mixture. Based on the first-order shear deformation theory, the energy functional of the structure is calculated. Applying the generalized differential quadrature method and periodic differential operators in axial and circumferential directions, respectively, the discretized form of the energy functional is derived. Based on Hamilton’s principle and using the VDQ method, the reduced forms of mass and stiffness matrices are obtained. The comparison and convergence studies of the present numerical method are first performed and then various numerical results are presented. It is found that the volume fractions and functionally grading of carbon nanotubes play important roles in the vibrational characteristics of FG-CNTRC cylindrical, conical shells and annular plates.

Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method

By developing a nonlinear microstructure-dependent third-order shear deformable beam model based on the most general form of Mindlin’s strain gradient elasticity theory (SGT) and the von Kármán hypothesis, the size-dependent nonlinear mechanical behavior of a microbeam made of functionally graded materials (FGMs) are described. The matrix representations of classical and non-classical kinematic and constitutive relations are obtained first. Then, the variation of energy functional is obtained in a matrix form. Afterwards, the variational differential quadrature (VDQ) rule is utilized to directly derive the discretized form of nonlinear governing equations of motion on the space domain. The periodic time differential operators and the pseudo arc-length continuation method are also used to solve the nonlinear problems of microbeams including the nonlinear free and forced vibration as well as nonlinear bending and postbuckling. The effects of length-scale parameter and boundary conditions on the microstructure-dependent nonlinear mechanical characteristics of FGM microbeams are investigated. The present model accommodates the simple forms of microstructure-dependent formulations based on the modified strain gradient theory (MSGT) and modified couple stress theory (MCST) for some specific values of the material length scale parameters. The results predicted by MSGT, MCST and classical theory are compared.

Three-dimensional bending and vibration analysis of functionally graded nanoplates by a novel differential quadrature-based approach

In this paper, the three dimensional (3D) nonlocal bending and vibration analyses of functionally graded (FG) nanoplates are presented using a novel numerical solution method which is called variational differential quadrature (VDQ) due to its numerical essence and the framework of implementation. Through this approach, a quadratic weak formulation of 3D nonlocal elasticity for the considered phenomena is presented. Two types of the distribution of functionally graded materials (FGMs) namely power law distribution and exponentially varied along the thickness of the plate are considered. The energy quadratic representation of the problems is first obtained based on the 3D theory of elasticity. A weak form of local governing equations is then derived from this representation by a variational approach. To incorporate the effects of small size into the local model, a size-dependent energy functional based on the nonlocal elasticity theory is developed. By introducing this functional into Hamilton’s principle, the discretized equations of motion including size effects are derived. By the VDQ method, the need for derivation of strong statement of the problems through minimizing the energy functional in the differential quadrature formulation is bypassed. In several numerical examples, the obtained results are compared with the available solutions in the literature, and the validity and high accuracy as well as fast convergence rate of the VDQ are indicated. It is also found that the small scale has a decreasing effect on the stiffness of nanoplates.