Variational Differential Quadrature Research Articles

An integral nonlocal model for the free vibration analysis of Mindlin nanoplates using the VDQ 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.

Vibration analysis of FG-CNTRC plates with an arbitrarily shaped cutout based on the variational differential quadrature finite element 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.

Vibration analysis of graphene sheets resting on Winkler/Pasternak foundation: A multiscale approach

In the present article, an atomistic-continuum multiscale model is developed to study the free-vibration response of single-layered graphene sheets (SLGSs) embedded in an elastic medium based upon the higher-order Cauchy-Born (HCB) rule. In order to take both transverse shear stress and normal pressure into account, the elastic foundation is considered to be of Winkler-Pasternak type. The governing equations are derived within a variational formulation using a newly proposed method called Variational Differential Quadrature (VDQ). Using the VDQ approach together with the Generalized Differential Quadrature (GDQ) technique, the variational form of the governing equation is discretized in a computationally efficient manner. Finally, a generalized eigenvalue problem is solved to calculate the frequencies of SLGSs. The convergence and correctness of the presented numerical solutions are examined firstly. Then, a number of numerical examples are given to study the effects of boundary conditions, elastic medium and arrangement of atoms on the vibrational response of SLGSs. The present model does not involve any additional phenomenological input, and it considers size effect and material nonlinearity due to atomic interactions.

A new numerical approach for low velocity impact response of multiscale-reinforced nanocomposite plates

In this paper, the low velocity impact analysis of carbon nanotube (CNT)/carbon fiber (CF)-reinforced hybrid nanocomposite plates is presented using variational differential quadrature (VDQ) method due to its numerical essence and the framework of implementation. The hybrid nanocomposite plate deformation is formulated based on classical plate theory and the contact force between the plate and projectile is estimated using Hertzian contact law. Also, a new micromechanics approach is presented to calculate the effective mechanical properties of the CNT/CF polymer hybrid nanocomposites. Five important factors including, random orientation and random distribution of CNTs, CNT/polymer interphase region, waviness and transversely isotropic behavior of CNT are incorporated in the micromechanical analysis. The accuracy of the present approach is verified with the available open literature results showing a clear agreement. The effects of various factors such as volume fraction and non-straight shape of CNT, CNT/polymer interphase region, CF volume fraction, random and regular arrangement of CFs, plate geometrical parameters and impactor velocity on the low velocity impact behavior of the CNT/CF-reinforced hybrid nanocomposite plates are studied.

Nonlinear vibration analysis of graphene sheets resting on Winkler–Pasternak elastic foundation using an atomistic-continuum multiscale model

Based on the higher-order Cauchy–Born (HCB) rule, an atomistic-continuum multiscale model is proposed to address the large-amplitude vibration problem of graphene sheets (GSs) embedded in an elastic medium under various kinds of boundary conditions. By HCB, a linkage is established between the deformation of the atomic structure and macroscopical deformation gradients without any parameter fitting. The elastic foundation is formulated according to the Winkler–Pasternak model which considers both normal pressure and transverse shear stress effects. The weak form of nonlinear governing equations is derived via a variational approach, namely based on the variational differential quadrature (VDQ) method and Hamilton’s principle. In order to solve the obtained equations, a numerical scheme is adopted in which the generalized differential quadrature (GDQ) method together with a numerical Galerkin technique is utilized for discretization in the space domain, and the time-periodic discretization method is used to discretize in the time domain. The effects of the arrangement of atoms, the Winkler and Pasternak coefficients of the elastic foundation, and boundary conditions on the frequency–response curves of GSs are illustrated. It is revealed that the nonlinear effects on the response of GSs with larger size in armchair direction are less important.

A numerical approach to the elastic/plastic axisymmetric buckling analysis of circular and annular plates resting on elastic foundation

Presented herein is the elastic/plastic axisymmetric buckling analysis of circular and annular plates resting on elastic foundation under radial loading based on a variational numerical method named as variational differential quadrature. To accomplish this aim, a first-order shear deformable plate model is developed in the context of incremental theory of plasticity (IT) (with the Prandtl-Reuss constitutive equations) and the deformation theory of plasticity (DT) (with the Hencky constitutive equations). It is considered that the material of plates exhibits strain hardening characterized by the Ramberg-Osgood relation. Also, the Winkler and Pasternak models are employed in order to formulate the elastic foundation. To implement the variational differential quadrature method, the matrix formulations of strain rates and constitutive relations are first derived. Then, based upon Hamilton's principle and using the variational differential quadrature derivative and integral operators, the discretized energy functional of the problem is directly obtained. Selected numerical results are presented to study the effects of various parameters including thickness-to-radius ratio, elastic modulus-to-nominal yield stress ratio, power of the Ramberg-Osgood relation and parameters of elastic foundation on the elastic/plastic buckling of circular and annular plates subject to different boundary conditions. Moreover, several comparisons are provided between the results of two plasticity theories, i.e. IT and DT. The effect of transverse shear deformation is also highlighted.

Semi-analytical postbuckling analysis of polymer nanocomposite cylindrical shells reinforced with functionally graded graphene platelets

In this study, the postbuckling analysis of functionally graded graphene platelets reinforced composite (FG-GPLRC) cylindrical shells is presented. The continuous uniform and functionally graded distribution of GPLs is considered through the thickness direction of the shell. In order to present the effective mechanical properties of GPL-reinforced composites the modified Halpin-Tsai micromechanical model is taken into account. On the basis of first-order shear deformation shell theory and von-Karman geometrically nonlinear relations, the nonlinear governing equations are present in the context of the variational formulation. Employing the Fourier series and variational differential quadrature (VDQ) numerical approach, the semi-analytical solution methodology is further presented. The pseud arc-length continuation method in conjunction with a load disturbance approach was utilized to solve the nonlinear governing equations and trace the postbuckling path. Note that based on the proposed formulation, the mode changes and secondary buckling can be considered through the postbuckling path. The results indicate that in addition to geometrical parameters, distribution patterns and weight fractions of GPLs have significant effects on the buckling and postbuckling characteristics of FG-GPLRC cylindrical shells.

Large deformation analysis of 2D hyperelastic bodies based on the compressible nonlinear elasticity: A numerical variational method

A numerical solution technique named as variational differential quadrature (VDQ) is adopted herein for the compressible nonlinear elasticity problems. The governing equations are obtained based on the virtual work principle by considering displacement as the unknown field. The neo-Hookean model is also considered for the hyperelastic behavior of material. In the solution method, an efficient vector–matrix formulation is developed from which the discretized governing equations are achieved from the weak form of equations in a direct approach. Simplicity in implementation and accuracy are among the features of the proposed approach. Moreover, it does not suffer from the locking problem and unphysical instabilities. Fast convergence rate and computational efficiency are other advantages of this method. A number of numerical examples are given to reveal the good performance of VDQ in the large deformation analysis of compressible and nearly-incompressible bodies.

Elastoplastic postbuckling analysis of moderately thick rectangular plates using the variational differential quadrature method

In this research, the elastoplastic postbuckling response of moderately thick rectangular plates subjected to in-plane loadings is analyzed by a novel numerical approach. The influence of transverse shear deformation is taken into account via the first-order shear deformation theory (FSDT). Also, the elastoplastic behavior is captured based on two theories of plasticity including the incremental theory (IT) (with the Prandtl-Reuss constitutive relations) and the deformation theory (DT) (with the Hencky constitutive relation). Moreover, it is assumed that the material of plate obeys the Ramberg-Osgood (RO) elastoplastic stress-strain relation. First, the matrix formulations of strain rates and constitutive relations are derived. In the next step, according to Hamilton's principle, the weak form of governing equations is derived which is then directly discretized using the variational differential quadrature (VDQ) technique. The discretization process is performed by accurate matrix derivative and integral operators of VDQ. Plates with various boundary conditions under uniaxial and equibiaxial compressions are considered. It is first indicated that the present results are in excellent agreement with the analytical solutions existing in the open literature. Thereafter, the influences of geometrical properties, boundary conditions, elastic modulus-to-nominal yield stress ratio and value of power c in the RO relation on the elastoplastic postbuckling paths of plates are studied. Furthermore, several comparisons are made between the predictions of IT and DT.

Nonlinear vibration response of higher-order shear deformable FG-CNTRC conical shells

In this work, the large-amplitude free and forced vibrations of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shells are investigated based on the higher-order shear deformation theory. In order to obtain the governing equations, the metricized energy functional of the structure is presented based on the higher-order shear deformation theory (HSDT) and von-Karman geometric nonlinearity. Then, the variational differential quadrature (VDQ) method is adopted to present the discretized energy functional. The numerical time differential operators together with the arc-length continuation scheme are utilized to solve the governing equations and find the free and forced vibration responses. In order to evaluate the influences of reinforcement factors and geometrical parameters on the nonlinear vibration behavior, various numerical results are reported. The numerical results reveal that the increase of semi-vertex angle of the shell increases the nonlinear to linear frequency ratio.

Nonlinear bending analysis of nanoplates made of FGMs based on the most general strain gradient model and 3D elasticity theory

In this article, the nonlinear bending behavior of rectangular nanoplates made of functionally graded materials (FGMs) is studied in the context of a variational formulation. To capture size effects, the most general form of strain gradient theory is employed. The three-dimensional (3D) elasticity theory is used for modeling the nanoplate. The governing equations are also derived in the discretized weak form using the variational differential quadrature (VDQ) method. Finally, the solution of the nonlinear bending problem is obtained by the pseudo arc-length continuation algorithm. In the numerical results, the effects of thickness-to-length scale ratio, side length-to-thickness ratio and material gradient index on the nonlinear bending response of nanoplates subject to different types of boundary conditions are analyzed. Moreover, a comparison is provided between the predictions of various strain gradient-based theories.

A VDQ‐based multifield approach to the 2D compressible nonlinear elasticity

SummaryA numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu‐Washizu principle, considering displacement, displacement gradient, and the first Piola‐Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix‐vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ‐based approach. Some well‐known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.

Nonlinear Pull-In Instability of Strain Gradient Microplates Made of Functionally Graded Materials

In this paper, the size-dependent nonlinear pull-in behavior of rectangular microplates made from functionally graded materials (FGMs) subjected to electrostatic actuation is numerically studied using a novel approach. The small scale effects are taken into account according to Mindlin’s first-order strain gradient theory (SGT). The plate model is formulated based on the first-order shear deformation theory (FSDT) using the virtual work principle. The size-dependent relations are derived in general form, which can be reduced to those based on different elasticity theories, including the modified strain gradient, modified couple stress and classical theories (MSGT, MCST and CT). The solution of the problem is arrived at by employing an efficient matrix-based method called the variational differential quadrature (VDQ). First, the quadratic form of the energy functional including the size effects is obtained. Then, it is discretized by the VDQ method using a set of matrix differential and integral operators. Finally, the achieved discretized nonlinear equations are solved by the pseudo arc-length continuation method. In the numerical results, the effects of material length scale parameters, side length-to-thickness ratio and FGM’s material gradient index on the nonlinear pull-in instability of microplates with different boundary conditions are investigated. A comparison is also made between the predictions by the MSGT, MCST and CT.

Vibration analysis of pressurized sandwich FG-CNTRC cylindrical shells based on the higher-order shear deformation theory

The present study deals with the vibration analysis of sandwich cylindrical shells with the functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets resting on elastic medium under internal pressure. Two FG-CNTRC face sheets along with homogeneous core are considered as the sandwich cylindrical shell. The overall mechanical properties of CNT-reinforced composites are presented in accordance to the refined rule of mixture. Based on the higher-order shear deformation theory (HSDT) and in the context of the variational differential quadrature (VDQ) method, the discretized version of governing equations is provided. Validation of the proposed model is demonstrated. Several numerical results are also represented to study the impacts of various material and geometrical factors on the vibration analysis of sandwich cylindrical shells. The results reveal that in the case of constant total thickness, increasing the core-to-face sheet thickness ratio decreases the dimensionless frequencies.

Studying nonlinear thermomechanical wave propagation in a viscoelastic layer based upon the Lord-Shulman theory

Based on the Lord–Shulman (L-S) theory, the thermally nonlinear coupled thermo-viscoelasticity of a layer is analyzed using a numerical approach. Using the Kelvin-Voigt theory of viscoelasticity in conjunction with the L-S theory, the coupled governing equations are first derived in variational form. Then, the variational differential quadrature (VDQ) method is applied to solve the problem numerically. Parametric studies are presented in order to reveal the effects of viscoelastic parameter, intensity of heat flux and relaxation time on the propagation of displacement, temperature and stress waves. Furthermore, the influence of considering nonlinearity in the energy equation is analyzed.

Geometrically nonlinear free vibration analysis of shear deformable magneto-electro-elastic plates considering thermal effects based on a novel variational approach

In this article, a variational numerical method is utilized to investigate the nonlinear free vibrations of magneto-electro-elastic (MEE) plates under thermal environment. To this end, first, the basic equations are written based on the first-order shear deformation theory and von Kármán's geometric nonlinearity. Next, the constitutive equations are represented in matrix form. In the context of Hamilton's principle, the quadratic and matricized form of energy functional is derived which is then directly discretized on space domain by a method called Variational Differential Quadrature (VDQ). Periodic time differential operators are also constructed for discretizing on time domain. The final solution is obtained by the pseudo arc-length continuation algorithm. The effects of applied electric voltage, applied magnetic potential, temperature change and geometrical parameters on the response curves of MEE plates with different types of boundary conditions are studied.

Nonlinear free vibration analysis of shell-type structures by the variational differential quadrature method in the context of six-parameter shell theory

In the present paper, an attempt is made to adopt the variational differential quadrature (VDQ) technique for the large-amplitude vibration analysis of shell-type structures based on the six-parameter shell theory. The functional of energy in quadratic form is derived based on Hamilton's principle which is then directly discretized by the VDQ method. Although the derived formulation is general, the focus of paper is on the cylindrical and spherical shells. The nonlinear vibration problem is solved by means of the time periodic discretization method. The results reveal that the present numerical method can solve the problem accurately. It is also easy to implement due to its compact and explicit matrix formulation. Comprehensive numerical results are presented to study the effects of geometrical properties and boundary conditions on the frequency-response curves of cylindrical and spherical shells. Moreover, comparison studies are presented between the results of the six-parameter shell theory and the first-order shear deformation shell theory.

Nonlinear free and forced vibration analysis of Timoshenko nanobeams based on Mindlin's second strain gradient theory

The size-dependent geometrically nonlinear free and forced vibration behaviors of nanoscale beams are studied in this article using a numerical approach. The size effects are captured using Mindlin's second strain gradient theory (SSGT) in which the second- and third-order derivatives of displacement components are taken into account in the strain energy density. The basic relations are first derived using the Timoshenko beam theory and SSGT. Then, the variational differential quadrature (VDQ) method is adopted to solve the obtained governing equations in the context of variational formulation. Comprehensive numerical results are given to investigate the influences thickness-to-lattice parameter ratio on the nonlinear free and forced vibrations of nanobeams under different types of end conditions. Also, comparisons are made between the predictions of SSGT and the first strain gradient theory as well as the classical elasticity theory.

Vibration of FG-CNTRC annular sector plates resting on the Winkler-Pasternak elastic foundation under a periodic radial compressive load

The vibration and stability of functionally graded carbon nanotube-reinforced (FG-CNTRC) composite annular sector plates resting on Winkler-Pasternak elastic foundation subjected to a periodic radial compressive load are investigated for various boundary conditions. To this end, a shear deformable plate model is established according to a parabolic theory which can interpret the shear deformation and rotary inertia effects without using any shear correction factor. The modified micromechanical technique is employed to compute the effective material properties of the FG-CNTRCs. The discretized form of higher-order governing equations is directly accessed by employing Hamilton’s principle and the variational differential quadrature (VDQ) method. In addition, by considering the applied radial compressive force as a periodic function, the discretized equations are expressed as the Mathieu–Hill equations, and then the instability regions are determined via the Bolotin’s scheme. In numerical results, the effects of the CNT distribution pattern and volume fraction, geometry parameters, sector angle, elastic foundation and static load factor on the stability of FG-CNTRC annular sector plates subject to arbitrary edge conditions are thoroughly discussed.

Nonlinear stability and vibration of pre/post-buckled multilayer FG-GPLRPC rectangular plates

• Proposing the nonlinear equations of parabolic shear deformable FG-GPLRPC plates. • Prediction of nonlinear stability and vibration of pre- and post-buckled FG-GPLRPC plates. • Investigating the impacts of GPL distribution scheme and weight fraction. • Exploring the impacts of the GPLs’ and plate's geometry and boundary conditions. The present study examines the nonlinear stability and free vibration features of multilayer functionally graded graphene platelet-reinforced polymer composite (FG-GPLRPC) rectangular plates under compressive in-plane mechanical loads in pre/post buckling regimes. The GPL weight fractions layer-wisely vary across the lateral direction. Furthermore, GPLs are uniformly dispersed in the polymer matrix of each layer. The effective Young's modulus of GPL-reinforced nanocomposite is assessed via the modified Halpin–Tsai technique, while the effective mass density and Poisson's ratio are attained by the rule of mixture. Taking the von Kármán-type nonlinearity into account for the large deflection of the FG-GPLRPC plate, as well as utilizing the variational differential quadrature (VDQ) method and Lagrange equation, the system of discretized coupled nonlinear equations of motions is directly achieved based upon a parabolic shear deformation plate theory; taking into account the impacts of geometric nonlinearity, in-plane loading, rotary inertia and transverse shear deformation. Afterwards, first, by neglecting the inertia terms, the pseudo-arc length approach is used in order to plot the equilibrium postbuckling path of FG-GPLRPC plates. Then, supposing a time-dependent disturbance about the postbuckling equilibrium status, the frequency responses of pre/post-buckled FG-GPLRC plate are obtained in terms of the compressive in-plane load. The influences of various vital design parameters are discussed through various parametric studies.