Generalized Differential Quadrature Method Research Articles

Static Torsion of Bi-Directional Functionally Graded Microtube Based on the Couple Stress Theory Under Magnetic Field

This paper presents is a solution for static torsion in a microtube made of bi-directional functionally graded materials (BDFGMs). The material properties are assumed to vary according to the arbitrary function along radius and length of microtube. As for the torque effect of the axial magnetic field, the well-known Maxwell’s relation is used. Couple stress theory is employed to study the influence of small-scale on static torsion of microtube. The Navier equation and boundary conditions of the size-dependent BDFG microtube were derived by the minimum potential energy. These equations were solved by employing the generalized differential quadrature method (GDQM). Comparison between the results of this work with the analytical method for static torsion of microtube made of one direction FG material reveals the accuracy of this study. Finally, numerical results are presented to study the small-scale effect and heterogeneity constants on the static torsion of the bi-directional functionally graded microtube.

RETRACTED ARTICLE: A Mathematical Model for Vibration Behavior Analysis of DNA and Using a Resonant Frequency of DNA for Genome Engineering

The DNA molecule is the most evolved and most complex molecule created by nature. The primary role of DNA in medicine is long-term storage of genetic information. Genetic modifying is one of the most critical challenges that scientists face. On the other hand, it is said that under the influence of acoustic, electromagnetic, and scalar waves, the genetic code of DNA can be read or rewritten. In this article, the most accurate and comprehensive dynamic model will be presented for DNA. Each of the two strands is modeled with an out of plane curved beam and then by doubling this two strands with springs, consider the hydrogen bond strength between this two strands. Beams are traditionally descriptions of mechanical engineering structural elements or building. However, any structure such as automotive automobile frames, aircraft components, machine frames, and other mechanical or structural systems contain beam structures that are designed to carry lateral loads are analyzed similarly. Also, in this model, the mass of the nucleobases in the DNA structure, the effects of the fluid surrounding the DNA (nucleoplasm) and the effects of temperature changes are also considered. Finally, by deriving governing equations from Hamilton’s principle method and solving these equations with the generalized differential quadrature method (GDQM), the frequency and mode shape of the DNA is obtained for the first time. In the end, validation of the obtained results from solving the governing equations of mathematical model compared to the obtained results from the COMSOL software is confirmed. By the help of these results, a conceptual idea for controlling cancer with using the DNA resonance frequency is presented. This idea will be presented to stop the cancerous cell’s protein synthesis and modifying DNA sequence and genetic manipulation of the cell. On the other hand, by the presented DNA model and by obtaining DNA frequency, experimental studies of the effects of waves on DNA such as phantom effect or DNA teleportation can also be studied scientifically and precisely.

Frequency characteristics of FG-GPLRC viscoelastic thick annular plate with the aid of GDQM

Abstract This is the first research on the free vibration analysis of functionally graded graphene platelets reinforced composite (FG-GPLRC) viscoelastic annular plate resting on the visco-Pasternak foundation and subjected to the nonlinear temperature gradient and mechanical loading within the framework of higher-order shear deformation theory (HSDT). Hamilton's principle is employed to establish governing equations within the framework of HSDT. In this paper, viscoelastic properties are modeled according to Kelvin-Voigt viscoelasticity. The deflection as the function of time can be solved by the fourth-order Runge-Kutta numerical method. Generalized differential quadrature method (GDQM) is applied to obtain a numerical solution. Numerical results are compared with those published in the literature to examine the accuracy and validity of the applied approach. A comprehensive parametric study is accomplished to reveal the influence of the stiffness of the substrate, patterns of temperature rise, axial load, damper and viscoelasticity coefficient, weight fraction and distribution patterns of GPLs and geometric dimensions of GPLs on the frequency response of the structure. The results revealed that applying sinusoidal temperature rise and locating more square-shaped GPLs in the vicinity of the top and bottom surfaces have important effect of the highest natural frequency and buckling load of the FG-GPLRC viscoelastic structure.

On the nonlinear dynamics of a multi-scale hybrid nanocomposite disk

This is the first research on the nonlinear frequency analysis of a multi-scale hybrid nanocomposite (MHC) disk (MHCD) resting on an elastic foundation subjected to nonlinear temperature gradient and mechanical loading is investigated. The matrix material is reinforced with carbon nanotubes (CNTs) or carbon fibers (CF) at the nano- or macroscale, respectively. We present a modified Halpin–Tsai model to predict the effective properties of the MHCD. The displacement–strain of nonlinear vibration of multi-scale laminated disk via third-order shear deformation theory (TSDT) and using Von Karman nonlinear shell theory is obtained. Hamilton’s principle is employed to establish the governing equations of motion, which is finally solved by generalized differential quadrature method (GDQM) and perturbation method (PM). Finally, the results show that FG patterns, different orientation angle of the fiber, the VF and WCNT parameters, axial load, nonlinear temperature gradient, and applied temperature of the top surface play an essential impact on the linear and nonlinear dynamic responses of the MHCD. The more significant outcome of this research is that the effects of the VF, WCNT, $$\theta$$, and β parameters on the nonlinear frequency of the MHCD can be considered at the higher value of the large deflection parameter and the effect of negative axial load on the dynamic responses of the structure is more intensive. As an applicable result show that the best functionally graded (FG) pattern for serving the highest nonlinear dynamic response of an MHC reinforced annular plate is FG-A.

Frequency Characteristics of Multiscale Hybrid Nanocomposite Annular Plate Based on a Halpin–Tsai Homogenization Model with the Aid of GDQM

In this article, we study the vibration performance of multiscale hybrid nanocomposite (MHC) annular plates (MHCAP) resting on Winkler–Pasternak substrates exposed to nonlinear temperature gradients. The matrix material is reinforced with carbon nanotubes (CNTs) or carbon fibers (CF) at the nano- or macroscale, respectively. The annular plate is modeled based on higher-order shear deformation theory (HSDT). We present a modified Halpin–Tsai model to predict the effective properties of the MHCAP. Hamilton’s principle was employed to establish the governing equations of motion, which is finally solved by the generalized differential quadrature method (GDQM). In order to validate the approach, numerical results were compared with available results from the literature. Subsequently, a comprehensive parameter study was carried out to quantify the influence of different parameters such as stiffness of the substrate, patterns of temperature increase, outer temperature, volume fraction and orientation angle of the CFs, weight fraction and distribution patterns of CNTs, outer radius to inner radius ratio, and inner radius to thickness ratio on the response of the plate. The results show that applying a sinusoidal temperature rise and locating more CNTs in the vicinity of the bottom surface yielded the highest natural frequency.

Viscoelastic dynamics and static responses of a graphene nanoplatelets-reinforced composite cylindrical microshell

In this study, a cylindrical microshell stability reinforced by graphene nanoplatelets is investigated while an axial load is applied uniformly. In addition, viscoelastic foundation covers the composite nanostructure. Therefore, the impacts of the small scale parameter are studied while nonlocal strain gradient theory (NSGT) is considered. The present research deals for the first time with the consideration of viscoelastic, strain–stress size-dependent parameters along with taking into account of various boundary conditions (BCs), especially C-F ones put into effect on the proposed theory. The governing equations (G.Eqs) and BCs have been obtained utilizing energy method and solved with assistance of the generalized differential quadrature method (GDQM). Besides, for the validation of the results, the results of the current model are compared to the results acquired from analytical method. The results show that, viscoelastic foundation, patterns of GPL distribution, nonlocal parameter, length scale parameter, layers’ numbers, boundary condition as well as GPL weight function have considerable impact on the stability of the GPLRC cylindrical microshell. Another significant result is that; for C-F B.Cs, at the higher value of the parameter, the influence of the on the dimensionless buckling load of the structure is much more remarkable in comparison with the lower one which in results section is investigated in detail. The results of the present investigation would be informative for design and fabrication of the microactuators and microsensors.

Nonlinear vibration behavior of functionally graded porous cylindrical panels

The paper presents large amplitude free vibration response of functionally graded porous (FGP) cylindrical panels considering different shell theories and boundary conditions. Nonlinear governing equations are obtained based on two shell theories, first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT). The von Karman geometrical nonlinearity along with the Hamilton principle is utilized. Mechanical properties of the open-cell foam are assumed to vary continuously through the thickness. This graded porosity offers a smooth stress distribution along the thickness of the panel. Generalized differential quadrature method (GDQM) is utilized to discretize the nonlinear dynamic governing equations along with three different boundary conditions. To solve the set of equations that include highly nonlinear parameters, the harmonic balance method along with the direct iterative approach is used. The results present the influence of geometrical parameters, vibration amplitude, porosity distribution, shell theories and boundary conditions on the nonlinear frequencies. It is found that both porosity distribution and porosity coefficient have a remarkable effect on the nonlinear natural frequencies of FGP cylindrical panel. To enhance the dynamic response of the cylindrical panel, porosity should be avoided near the panels’ surfaces.

Vibration analysis of FG-GPLRC annular plate in a thermal environment

In this article, the thermal vibration of functionally graded graphene platelets reinforced composite (FG-GPLRC) annular plate resting on an elastic foundation under the mechanical load framework of higher order shear deformation theory (HSDT) is analyzed. Governing equations and boundary conditions are established by employing Hamilton’s principle. A generalized differential quadrature method (GDQM) is applied to obtain a numerical solution. Numerical results are compared with those published in the literature to examine the accuracy and validity of the applied approach. A comprehensive parametric study is accomplished to reveal the influence of stiffness of the substrate, patterns of temperature rise, temperature gradient, axial load, weight fraction and distribution patterns of GPLs, outer radius to inner radius ratio, inner radius to thickness ratio of the annular plate, and geometric dimensions of GPLs on the response of the structure. The results revealed that applying sinusoidal temperature rise and locating more square-shaped GPLs in the vicinity of the top and bottom surface result in the highest natural frequency.

A semi‐analytical solution for time‐varying latent heat thermal energy storage problems

Latent heat thermal energy storage (LHTES) problems include a lot of boundary conditions that could not be solved by exact solution, so new approaches to solving such problems could revolutionize the advanced energy storage devices. This paper focuses on reformulating the generalized differential quadrature method (GDQM) for a one-dimensional solidification/melting Stefan problem as a fundamental LHTES problem and solves some practical cases. Convergence and comparisons demonstrate that the proposed approach is sufficiently reliable. By checking the accuracy of the proposed approach for the LHTES problem (where Stefan number is below 0.2), it was demonstrated that for all Stefan numbers, the maximum error is less than 3.81% for temperatures. As the usual range of thermal energy storages, for Stefan numbers up to 0.2 the solution yields errors less than 0.2%. Then, the proposed approach is very ideal for such applications. In comparison, GDQM has a more accurate response than an integral solution for Stefan numbers less than 0.2. When this priority of GDQM comes with its low computational cost, it would undoubtedly be preferable.

Free vibration of FG-GPLRC spherical shell on two parameter elastic foundation

In the present research, the free vibration analysis of functionally graded (FG) nanocomposite deep spherical shells reinforced by graphene platelets (GPLs) on elastic foundation is performed. The elastic foundation is assumed to be Winkler-Past ernak-type. It is also assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the nanocomposite shell. Volume fraction of the graphene platelets as nanofillers may be different in the layers. The modified HalpinTsai model is used to approximate the effective mechanical properties of the multilayer nanocomposite. With the aid of the first order shear deformation shell theory and implementing Hamilton\'s principle, motion equations are derived. Afterwards, the generalized differential quadrature method (GDQM) is utilized to study the free vibration characteristics of FG-GPLRC spherical shell. To assess the validity and accuracy of the presented method, the results are compared with the available researches. Finally, the natural frequencies and corresponding mode shapes are provided for different boundary conditions, GPLs volume fraction, types of functionally graded, elastic foundation coefficients, opening angles of shell, and thickness-to-radius ratio.

Buckling and Free Vibration Analysis of Fiber Metal-laminated Plates Resting on Partial Elastic Foundation

This research presents, buckling and free vibration analysis of fiber metal-laminated (FML) plates on a total and partial elastic foundation using the generalized differential quadrature method (GDQM). The partial foundation consists of multi-section Winkler and Pasternak type elastic foundation. Taking into consideration the first-order shear deformation theory (FSDT), FML plate is modeled and its equations of motion and boundary conditions are derived using Hamilton's principle. The formulations include Heaviside function effects due to the nonhomogeneous foundation. The novelty of this study is considering the effects of partial foundation and in-plane loading, in addition to considering the various boundary conditions of FML plate. A computer program is written using the present formulation for calculating the natural frequencies and buckling loadings of composite plates without contacting with elastic foundation and composite plates resting on partial foundations. The validation is done by comparison of continuous element model with available results in the literature. The results show that the constant of total or partial spring, elastic foundation parameter, thickness ratio, frequency mode number and boundary conditions play an important role on the critical buckling load and natural frequency of the FML plate resting on partial foundation under in-plane force.

Thermo-electro-elastic nonlinear stability analysis of viscoelastic double-piezo nanoplates under magnetic field

The nonlinear thermo-electro-elastic buckling behavior of viscoelastic nanoplates under magnetic field is investigated based on nonlocal elasticity theory. Employing nonlinear strain-displacement relations, the geometrical nonlinearity is modeled while governing equations are derived through Hamilton\'s principle and they are solved applying semi-analytical generalized differential quadrature (GDQ) method. Eringen\'s nonlocal elasticity theory considers the effect of small size, which enables the present model to become effective in the analysis and design of nano-sensors and nano actuators. Based on Kelvin-Voigt model, the influence of the viscoelastic coefficient is also discussed. It is demonstrated that the GDQ method has high precision and computational efficiency in the buckling analysis of viscoelastic nanoplates. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as electric voltage, small scale effects, elastomeric medium, magnetic field, temperature effects, the viscidity and aspect ratio of the nanoplate on its nonlinear buckling characteristics. It is explicitly shown that the thermo-electro-elastic nonlinear buckling behavior of viscoelastic nanoplates is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of viscoelastic nanoplates as fundamental elements in nanoelectromechanical systems.

Vibration analysis of FG porous rectangular plates reinforced by graphene platelets

The aim of this study is to investigate free vibration of functionally graded porous nanocomposite rectangular plates where the internal pores and graphene platelets (GPLs) are distributed in the matrix either uniformly or non-uniformly according to three different patterns. The elastic properties of the nanocomposite are obtained by employing Halpin-Tsai micromechanics model. The GPL-reinforced plate is modeled using a semi-analytic approach composed of generalized differential quadrature method (GDQM) and series solution adopted to solve the equations of motion. The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and those reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. New results reveal the importance of porosity coefficient, porosity distribution, graphene platelets (GPLs) distribution, geometrical and boundary conditions on vibration behavior of porous nanocomposite plates. It is observed that the maximum vibration frequency obtained in the case of symmetric porosity and GPL distribution, while the minimum vibration frequency is obtained using uniform porosity distribution.

Vibration behavior of functionally graded sandwich beam with porous core and nanocomposite layers

This paper presents the influence of carbon nanotubes (CNTs) waviness, aspect ratio, internal pores and graphene platelets (GPLs) on the vibrational behavior of functionally graded nanocomposite sandwich beams resting on two-parameter elastic foundations. The distributions of CNTs are considered functionally graded (FG) or uniform along the thickness of upper and bottom layers of the sandwich beam and their mechanical properties are estimated by an extended rule of mixture. In this study, the classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes\' effective aspect ratio as the filler content increases. The core of structure is porous and the internal pores and graphene platelets (GPLs) are distributed in the matrix of core either uniformly or non-uniformly according to three different patterns. The elastic properties of the nanocomposite are obtained by employing Halpin-Tsai micromechanics model. The equations of motion are derived based on Timoshenko beam theory and employing Hamilton\'s principle. The problem is modeled using a semi-analytical approach composed of generalized differential quadrature method (GDQM) and series solution adopted to solve the equations of motion. Detailed parametric studies are carried out to investigate carbon nanotubes (CNTs) waviness, CNT aspect ratio, porosity coefficient, porosity distribution, graphene platelets (GPLs) distribution, Winkler foundation modulus, shear elastic foundation modulus and geometrical conditions on the vibrational behavior of the sandwich structure.

Effect of porosity on buckling and vibrational characteristics of the imperfect GPLRC composite nanoshell

Due to rapid development of manufacturing process, composite materials with porosity have attracted commercially notices in advanced engineering applications. For this regard, buckling and vibrational characteristics of a porous composite cylindrical nanoshell reinforced with GPLs is investigated in this paper. The material properties of piece-wise graphene-reinforced composites (GPLRC) are assumed to be graded in the thickness direction of a cylindrical nanoshell and are estimated using a nanomechanical model. The novelty of our work is including the effects of porosity and GPLRC on natural frequency, critical axial load and critical temperature of the GPLRC cylindrical nanoshell. The governing equations (GEs) and boundary conditions (BCs) have been developed using Hamilton's principle and have been solved with assistance of the generalized differential quadrature method (GDQM). Besides, for the validation of the results, the results of current model are compared to the results acquired from analytical method. The results show that porosity, GPL distribution pattern, length scale parameter, number of layers and GPL weight function have significant influence on natural frequency, critical load and critical temperature of the porous GPLRC cylindrical nanoshell. The results of the current study are useful suggestions for design of nanoactuators and nanosensors.

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

In this paper, vibration and buckling of two-phase nanobeams embedded in size dependent elastic medium and under thermal load are analyzed. Due to paradoxes of common differential nonlocal elasticity, such as neglecting the size effect of uniform loads, failure to satisfy the constitutive boundary conditions, associated to transformation of integral nonlocal equation to differential one, and incompatibility between the results of differential nonlocal with those of integral nonlocal, the size dependent effects of nanobeam, elastic medium and thermal load are taken into account simultaneously by using two-phase local/nonlocal Eringen's elasticity, for the first time. Governing equations and corresponding boundary conditions are derived using Hamilton's principle. To obtain natural vibration frequencies as well as critical buckling temperature, three different methods of solution are presented, i.e. exact solution, Generalized Differential Quadrature Method (GDQM) and Finite Element Method (FEM) which is based on the integral form of two-phase elasticity. Several comparison studies are conducted to examine the validity of the present formulation and results. The effects of applying two-phase elasticity on elastic medium and thermal load in vibration and buckling of nanobeams with different boundary conditions are investigated in details. Differences appeared in present results, especially in the cases with higher temperature and nonlocality as well as stiffer elastic medium, reveal that the size dependency of elastic medium and uniform thermal load, which is neglected by differential nonlocal, must be considered by employing two-phase elasticity.

Aeroelastic analysis of cantilever non-symmetric FG sandwich plates under yawed supersonic flow

In this paper, a numerical solution is presented for supersonic flutter analysis of cantilever non-symmetric functionally graded (FG) sandwich plates. The plate is considered to be composed of two different functionally graded face sheets and an isotropic homogeneous core made of ceramic. Based on the first order shear deformation theory (FSDT) and linear piston theory, the set of governing equations and boundary conditions are derived. Dimensionless form of the governing equations and boundary conditions are derived and solved numerically using generalized differential quadrature method (GDQM) and critical velocity and flutter frequencies are calculated. For various values of the yaw angle, effect of different parameters like aspect ratio, thickness of the plate, power law indices and thickness of the core on the flutter boundaries are investigated. Numerical examples show that wings and tail fins with larger length and shorter width are more stable in supersonic flights. It is concluded for FG sandwich plates made of Al-Al2O3 that increase in volume fraction of ceramic (Al2O3) increases aeroelastic stability of the plate. Presented study confirms that improvement of aeroelastic behavior and weight of wings and tail fins of aircrafts are not consistent items. It is shown that value of the critical yaw angle depends on aspect ratio of the plate and other parameters including thickness and variation of properties have no considerable effect on it. Results of this paper can be used in design and analysis of wing and tail fin of supersonic airplanes.

Buckling and Frequency Responses of a Graphene Nanoplatelet Reinforced Composite Microdisk

This is the first research on the vibration and buckling analysis of a graphene nanoplatelet composite (GPLRC) microdisk in the framework of a numerical based generalized differential quadrature method (GDQM). The stresses and strains are obtained using the higher-order shear deformable theory (HOSDT). Rule of the mixture is employed to obtain varying mass density, thermal expansion, and Poisson’s ratio, while the module of elasticity is computed by modified Halpin–Tsai model. Governing equations and boundary conditions of the GPLRC microdisk are obtained by implementing Extended Hamilton’s principle. The results show that outer to inner ratios of the radius ([Formula: see text], ratios of length scale and nonlocal to thickness [Formula: see text] and [Formula: see text], and GPL weight fraction [Formula: see text] have a significant influence on the frequency and buckling characteristics of the GPLRC microdisk. Another necessary consequence is that by increasing the value of the [Formula: see text], the distribution of the displacement field extends from radial to tangent direction, especially in the lower mode numbers, this phenomenon appears much more remarkable. A useful suggestion of this research is that, for designing the GPLRC microdisk at the low value of the [Formula: see text], more attention should be paid to the [Formula: see text] and [Formula: see text], simultaneously.

On aeroelastic stability of a piezo-MRE sandwich plate in supersonic airflow

In this paper, aeroelastic characteristics of a sandwich plate subjected to a supersonic flow are studied. The material of face layers has piezoelectric effect and the core is a magnetorheological elastomer (MRE). The aerodynamic pressure duo to supersonic flow is considered based on linear piston theory. The kinematic assumption of classical plate theory for face layers is considered and strain components of MRE core are written based on geometric relations of displacement field of face layers. The equations of motion are obtained in according to Hamilton’s principle. The generalized differential quadrature method (GDQM) is used as a solution technique of governing equations. In order to verify the validity of obtained results, a comparison of the first five natural frequencies is performed with those presented in the literature. The effects of magnetic field, type of piezoelectric material of face layers, boundary conditions, and geometric parameters on the supersonic aeroelastic stability of sandwich plate are investigated. The findings of this study indicate that the magnetic field has a beneficial effect on the maximization of stability boundary of sandwich plate. Also, by increasing the thickness of MRE core the critical aerodynamic pressure of the sandwich plate decreases.

Nonlinear transient dynamic analysis of laminated composite parabolic panels of revolution with variable thickness resting on elastic foundation

In this article, nonlinear transient behavior of laminated composite Parabolic Panels of Revolution Structure(s) (PPRS) with variable thickness resting on elastic foundation is investigated using Generalized Differential Quadrature (GDQ) method. Winkler-Pasternak model is used to represent elastic foundation. Linear, arch, sine and cosine thickness functions are used to express variable thickness. In transient analyses, First Order Shear Deformation Theory (FSDT) is used to consider the transverse shear effects. Nonlinearity is taken into account using Green-Lagrange nonlinear strain-displacement relations considering deepness effect. Virtual work principle is used to derive the equations of motion. Partial derivatives in the equation of motion are expressed with GDQ method and time integration is carried out using Newmark average acceleration method. Several problems are solved and compared with finite element results in order to validate the proposed method. After validation, effects of thickness functions, thickness variation parameter, geometric characteristic parameter of PPRS, boundary conditions, elastic foundation parameters as well as composite lamination scheme on nonlinear transient dynamic behaviour of PPRS are investigated.