Functionally Graded Carbon Nanotube Research Articles

Large-amplitude nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes

The present study discusses the nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes (FG-CNTs). Mechanical properties of the plate have been continuously changed along the thickness of the plate by considering four different distribution patterns of CNTs. Based on the first-order shear deformation theory (FSDT) and von Ḱarḿan’s geometrical nonlinearity, the governing equations of the plate are derived using Hamilton’s principle. Galerkin discretization technique is employed to convert the partial differential equations of motion to ordinary differential equations. Afterward, the nonlinear time-dependent equations of the plate were analytically solved using the multiple time scales method. Eventually, the effect of CNT parameters, the geometrical ratios and rotational speed on the nonlinear frequency of the plate have been studied. For verification purposes, the present numerical results have been compared with those available in the literature with evident agreement. The present research may give benchmark solutions and some guidelines for designing FG-CNTs rotating plates.

A nonlocal higher-order shear deformation approach for nonline ar static analysis of magneto-electro-elastic sandwich Micro/Nano-plates with FG-CNT core in hygrothermal environment

In this study, the nonlocal static analysis of Magneto-Electro-Elastic (MEE) sandwich micro/nano-plates with Functionally Graded Carbon Nanotubes (FG-CNTs) core in a hygrothermal environment is studied. The nonlocal static formulation is developed based on Reddy's High-order Shear Deformation theory (HSDT). The von Kármán nonlinear strains are used and the governing equations of the HSDT are derived accounting for Eringen's nonlocal stress-gradient model. The governing equations are solved using the Galerkin method, which involves expanding the displacement field based on orthogonal functions. The variation of the volume fractions through the thickness of FG-CNTs core layer has been computed using two different homogenization techniques: the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the characteristics of the FG-CNTs core, nonlocal parameters, thermal and moisture changes, types of CNT reinforcement and volume fraction, electric and magnetic potentials, and geometric parameters. The results are compared with the first-order shear deformation theory (FSDT) and classical plate theory to show the accuracy of nonlocal High-order Shear Deformation theory(NHSDT) formulation.

Free vibration analysis of functionally graded carbon nanotubes reinforced double plates

The specific objective of this study is to analyze the dynamics of functionally graded carbon nanotubes (FGCNT) reinforced double plates. Connected via an elastic layer, the plates have simply supported boundary conditions. In the current study, three carbon nanotubes functionally graded patterns, varying in the thickness direction are considered, including uniformly distributed, functionally graded O-pattern, and functionally graded X-pattern. Following the development of the coupled equations of motion using the Hamilton principle while considering the influences of the elastic layer, the equations are subsequently solved utilizing a two-spatial-variable modal decomposition method. For verification purposes, the equations developed are compared to simplified configurations provided in the existing studies. The solution methodology is verified through comparing against numerical results of simplified configurations of plates obtained from the development of the finite element method and existing studies. Both verifications have shown very good agreement. Influences of plates’ dimensions, carbon nanotubes reinforcement, and the stiffness of elastic layer are analyzed and provided in this study. The transverse-motion natural frequencies of the double plates are also identified, and they follow a decreasing trend as the aspect ratio increases for all the cases. The fundamental lateral-motion and axial-motion natural frequency also follows a similar trend as the aspect ratio increases. The reinforcement effect of carbon nanotubes on the transverse-motion natural frequencies is less obvious for thinner plates. An increase in the elastic layer stiffness increases the second series transverse-motion natural frequencies of the double-plate system. Among the considered functionally graded patterns, the functionally graded X-pattern reinforcement provides the largest increase in the transverse-motion natural frequencies.

Modified couple stress theory (MCST) and refined zigzag theory (RZT) for wave propagation of embedded smart sandwich plates with magnetorheological fluid core and viscoelastic FG-CNTRC magnetostrictive face sheets

This paper discusses the wave propagation of a magnetorheological fluid (MRF) micro sandwich plate with magnetostrictive face sheets embedded in the Kerr foundation. The face sheets are reinforced with functionally graded carbon nanotubes (FG-CNT). It was hypothesized that the distribution of carbon nanotubes would be functionally graded along the thickness of face sheets in various patterns such as uniform distribution (UD), and three type of functionally graded (FG), that is, FG-O, FG-X, and FG-AV. The properties of the magnetostrictive layer are considered viscoelastic based on the Kelvin-Voigt model. The refined zigzag theory (RZT) is utilized to simulate displacements of the plate due to the sandwich nature of the system. The modified couple stress theory (MCST) is utilized to predict the influence of small-scale parameter. The equations of motion for the micro sandwich plate are derived by Hamilton's principle, and analytical solutions are utilized to acquire the phase speed, escape, and cut-off frequencies. After model verification, extensive analytical results are presented in detail. They demonstrated that the wave characteristics of the system are influenced by the micro sandwich plate parameters. Consequently, the dimensionless phase speed rises 66% via increasing the magnetic field magnitude from 12.5 to 17.5 G.

Exploring the benefits of functionally graded carbon nanotubes (FG-CNTs) as a platform for targeted drug delivery systems

Background and ObjectiveModern therapeutic systems have benefited from the use of functionally graded carbon nanotubes (FG-CNTs) to enhance their efficiency. Various studies have shown that the study of dynamic response and stability of fluid-conveying FG-nanotubes can be improved by considering a Multiphysics framework for the modeling of such a complex biological environment. However, despite noticing important aspects in modeling, the previous studies have drawbacks such as underrepresenting the effect of varying composition of the nanotube on magnetic drug release in drug delivery systems. The present work has the novelty of studying the combined effects of fluid flow, magnetic field, small-scale parameters, and functionally graded material on the performance of FG-CNTs for drug delivery applications. Additionally, the lack of an inclusive parametric study is resolved in the present study by evaluating the significance of different geometrical and physical parameters. As such, the achievements support the development of an efficient drug delivery treatment. MethodsThe Euler–Bernoulli beam theory is implemented to model the nanotube and Hamilton's principle based on Eringen's nonlocal elasticity theory is used to derive the constitutive equations of motion. To add the effect of slip velocity on the CNT's wall, a correction factor is applied to velocity based on the Beskok–Karniadakis model. Resultsdemonstrate that the dimensionless critical flow velocity increases by 227% as the magnetic field intensity increases from 0 to 20 T, and improves the system stability. On the contrary, drug loading on the CNT has the opposite effect, as the critical velocity decreases from 10.1 to 8.38 using a linear function for drug loading, and it decreases to 7.95 using an exponential function. By employing a hybrid load distribution, an optimum material distribution can be achieved. ConclusionsTo benefit from the potential of CNTs in drug delivery systems while minimizing the instability problems, a suitable design for the drug loading is required prior to the clinical implementation of the nanotube.

A strategy resisting wrinkling of sandwich structures reinforced using functionally-graded carbon nanotubes

Sandwich structures have been widely applied in the wing and the horizontal tail of the aircraft, so face sheets of such structure might occur wrinkling deformation in the process of service, which will largely decrease capability of sustaining loads. As a result, this paper aims at proposing a reasonable strategy resisting wrinkling deformation of sandwich structures. To this end, an enhanced higher-order model has been proposed for wrinkling analysis of sandwich structures. Buckling behaviors of a five-layer sandwich plate are firstly analyzed, which is utilized to assess performance of the proposed model. Subsequently, wrinkling behaviors of four sandwich plates are further investigated by utilizing present model, which have been evaluated by using quasi three-dimensional (3D) elasticity solutions, 3D Finite Element Method (3D-FEM) results and experimental datum. Finally, the present model is utilized to study the buckling and the wrinkling behaviors of sandwich plates reinforced by Carbon Nano Tubes (CNTs). In addition, influence of distribution profile of CNTs on wrinkling behaviors has been analyzed, and a typical distribution profile of CNTs has been chosen to resist wrinkling deformation. Without increase of additional weight, the present strategy can effectively resist wrinkling deformation of sandwich plates, which is rarely reported in published literature.

Optimization Design of Laminated Functionally Carbon Nanotube-Reinforced Composite Plates Using Deep Neural Networks and Differential Evolution

This study presents a new approach as an integration of deep neural networks (DNN) into differential evolution (DE) to give the so-called DNN-DE for frequency optimization of laminated functionally graded carbon nanotube (FG-CNT)-reinforced composite quadrilateral plates under free vibration. In the presented approach, the DNN is applied to predict the objective and constraints during the optimization process instead of using the time-consuming finite element analysis (FEA) procedures while the DE is used as an optimizer for solving the optimization problem. Several numerical examples are performed to illustrate the performance of the proposed method. Optimal results obtained by the DNN-DE are compared with those achieved by other methods in order to show the reliability and effectiveness of the proposed methodology. Additionally, the influence of various parameters such as the boundary condition, the carbon nanotube (CNT) volume fraction, the CNT distribution on the optimal results is also investigated. The obtained results indicate that the proposed DNN-DE is an effective and promising method in solving optimization problems of engineering structures.

An analysis of vibration and buckling behaviors of nano-composite beams reinforced with agglomerated carbon nanotubes via differential quadrature finite element method

This article aims to investigate the buckling and free vibration behaviors of nanocomposite beams reinforced with functionally graded (FG) agglomerated carbon nanotubes (CNTs) in polymeric beams matrix. In order to estimate the material properties of the FG CNTs reinforced beam, the Eshelby–Mori–Tanaka rule on the basis of equivalent fiber is employed. Moreover, the Lagrange’s principle is used to achieve the governing equations under supposition of the refined higher order beam theory without need of any shear correction factor. The differential quadrature finite element method (DQFEM) as a robust and precise numerical procedure is utilized to achieve the natural frequencies and buckling loads with various boundary supports. Then, the assessment of the effects of different factors such as of CNTs agglomeration parameters, different patterns of CNTs distribution, length to thickness quotients, and mixed boundary conditions on the vibrational and buckling responses of agglomerated CNTs reinforced beam are explored. It is concluded that the current numerical solution based DQFEM is efficient and accurate for predicting the mechanical behavior of the nanocomposite structures. Moreover, the stability and vibration responses of the NC beams can be significantly affected by tailoring the agglomeration parameters.

Nonlocal Strain Gradient Theory for the Bending of Functionally Graded Porous Nanoplates.

Many investigators have become interested in nanostructures due to their outstanding mechanical, chemical, and electrical properties. Two-dimensional nanoplates with higher mechanical properties compared with traditional structural applications are a common structure of nanosystems. Nanoplates have a wide range of uses in various sectors due to their unique properties. This paper focused on the static analysis of functionally graded (FG) nanoplates with porosities. The nonlocal strain gradient theory is combined with four-variable shear deformation theory to model the nanoplate. The proposed model captures both nonlocal and strain gradient impacts on FG nanoplate structures by incorporating the nonlocal and strain gradient factors into the FG plate's elastic constants. Two different templates of porosity distributions are taken into account. The FG porous nanoplate solutions are compared with previously published ones. The impact of nonlocal and strain gradient parameters, side-to-thickness ratio, aspect ratio, and porosity parameter, are analyzed in detail numerically. This paper presents benchmark solutions for the bending analysis of FG porous nanoplates. Moreover, the current combination of the nonlocal strain gradient theory and the four-variable shear deformation theory can be adapted for various nanostructured materials such as anisotropic, laminated composites, FG carbon nanotube reinforced composites, and so on.

Uncertain frequency responses of CNT – reinforced polymeric graded structure using fuzzified elastic properties – fuzzy finite element approach

A novel fuzzy finite element computational model is prepared to predict the frequency responses of the functionally graded carbon nanotube (FG-CNT) reinforced composite panel structure. The nanotube distributions in the matrix have an affinity towards the agglomeration and affect final properties. The model incurred the fuzzification of elastic properties (E 11, E 22, G 12, and υ 12) considering a ±20% as the degree of uncertainty via Mamdani’s fuzzy inference system (FIS). The extended rule of the mixture has been adopted to count the CNT properties for different grading patterns to achieve the realistic one in the framework of the higher-order polynomial model. Further, a set of fuzzy rules have been adopted to count the uncertainty in elastic properties with the help of the design of the experiment (Taguchi-L25 orthogonal array). The equation of motion is derived for the CNT-reinforced composite in association with Hamilton’s principle for the computation of natural frequency. An in-house computer code is derived in MATLAB with the help of a higher-order fuzzified finite element model (FFEM) for the computation. The suitability of the proposed model has been tested by solving a series of numerical examples and the output is discussed in detail.

Magneto-hygro-thermal vibration analysis of the viscoelastic nanobeams reinforcedwith carbon nanotubes resting on Kerr’s elastic foundation based on NSGT

This paper studies the vibration analysis of the viscoelastic nanobeams reinforced with functionally graded carbon nanotubes (FG-CNTs), considering the thickness stretching effect subjected to magneto-hygro-thermo loading based on a novel higher-order shear deformation theory. It is assumed that the nanosize structure is resting on Kerr’s elastic foundation. The effective material properties of FG-CNTs are changed through the thickness direction based on five diverse distributions of CNTs. All of the effective material properties of the nanobeam are temperature-dependent. Nonlocal strain gradient theory is applied to consider the size effect. A higher-order shear deformation beam theory is utilized to consider the shear impacts. Three kinds of relations are employed to study the effects of the hygro-thermo loading comprehensively on the frequency of the nanobeams. An analytical solution technique is utilized to solve the nonlocal governing equations achieved from Hamilton’s principle for several boundary conditions. The influences of size effect parameters, elastic foundation factors, distributions of CNTs, hygrothermal environment, and viscoelastic parameter on the frequency of nanobeams are investigated.

A size-dependent differential quadrature element model for vibration analysis of FG CNT reinforced composite microrods based on the higher order Love-Bishop rod model and the nonlocal strain gradient theory

In the current paper, a size-dependent differential quadrature element model using the higher-order Love-Bishop rod theory in conjunction with the nonlocal strain gradient theory for axial vibration analysis in the embedded functionally graded (FG) carbon nanotube (CNT) reinforced composite microrods with varying cross-section is presented for the first time. The CNT agglomeration effect is considered applying the two-parameter Eshelby-Mori-Tanaka homogenization scheme. The differential equation form of nonlocal strain gradient theory is employed to take into account the size-dependent effects. Based on the Hamilton's principle, the governing differential equation of the model is derived and solved using differential quadrature element method (DQEM). The higher-order variationally consistent boundary conditions are obtained using the weighted residual approach. The numerical results indicate that the present model is simple and it can be successfully employed for analysis of the small-sized structures. Moreover, the effects of small-scale parameters, the CNT distribution's patterns, the CNT agglomeration parameters, the non-uniformity of the cross section and the stiffness of the surrounding medium on the frequency are studied and discussed in details.

Quasi-3D nonlinear flexural response of isogeometric functionally graded CNT-reinforced plates with various shapes with variable thicknesses

This exploration deals with the nonlinear large deflection behavior of nanocomposite plates reinforced with functionally graded carbon nanotube (FG-CNT) having various shapes with variable thickness. To accomplish this end, a hybrid quasi-3D plate model is constructed by incorporating trigonometric normal and sinusoidal transverse shear functions. The applied plate formulations have the capability to model the nanocomposite plate with only four independent variables which results in a significant reduction in the computational cost of computational calculations. Four different dispersion schemes for the CNT reinforcement are considered, the effective mechanical properties of each one are achieved via an extension of the rule of mixture. In addition, plate thickness is assumed to vary in convex, concave and linear patterns. For solving nonlinear problem, isogeometric technique using non-uniform rational B-spline basis functions is employed to integrate an accurate geometric description. It is revealed that at deeper parts of flexural response, the role of CNT dispersion pattern on the nonlinear flexural stiffness of FG-CNT reinforced composite plates gets lower importance. Moreover, it is demonstrated that by substituting linear thickness variation scheme with convex one, the influence of CNT dispersion pattern reduces, but by changing from the linear type to the concave one, this influence increases.

Carrera unified formulation for bending and free vibration analysis of sandwich plate with FG-CNT faces considering the both soft and stiff cores

In this article, the bending and free vibration behavior of sandwich plates reinforced with functionally graded carbon nanotube (FG-CNT) were investigated based on the Carrera unified formulation(CUF). CUF is a technique which can handle a large category of plate models in a unified way. For FG-CNT, variable kinematic method was used to apply the properties of functionally graded CNTs, which reduce the degrees of freedom and increase the accuracy of the results. Accuracy and efficiency of the presented solution procedure are demonstrated by comparing the obtained results for the sandwich panel with the solution are available. It can be seen that the CUF is the robust method for modeling the sandwich plate with every configuration of CNTs. It must be mentioned, one of the capability of the CUF is that allows us to study the soft core considering the flexibility of the core which not reported in the literature. Therefore, results indicate that the CUF is a suitable model for analysis of the FG-CNT sandwich panels with both stiff and soft cores. A detailed numerical study is performed to examine the influence of soft and hard cores, different plate thickness ratios, different volume fractions of the CNTs, core to face sheet thickness, and also different types of nanotube configurations FG-X, FG-O, FG-VΛ, and FG-ΛV.

Vibration of FG-CNT and FG-GNP sandwich composite coupled Conical-Cylindrical-Conical shell

In this paper, free vibration analysis of sandwich composite joined conical-cylindrical-conical shells is implemented using First-order Shear Deformation Theory (FSDT). The sandwich structure is made of three layers including two face sheets and one core. These layers are reinforced with functionally graded carbon nanotubes (FG-CNTs) and functionally graded graphene nanoplatelets (FG-GNPs). Eight patterns are considered for distribution of CNTs and GNPs throughout the thickness of layers. To obtain the equivalent mechanical properties of each layer, the well-known rule of mixture is utilized. The coefficients of material matrix is calculated using the Equivalent Single Layer (ESL) theory. Moreover, the Donnell's approach is employed for obtaining the equations of motion associated to the conical and cylindrical shells. The principle of Hamilton is used for achieve the governing system of partial differential equation (PDE) of joined shells. Afterwards, this system of PDE is solved by using an efficient semi-analytical solution method, namely Generalized Differential Quadrature Method (GDQM). Different cases of boundary conditions are investigated in this study. By comparing the obtained results with the reference solutions, the correctness and efficiency of the proposed formulation are proved. Furthermore, several new, complicated and applicable examples are considered to study the effect of different parameters of geometric, material and boundary conditions on the natural frequency of the joined shells as well.

Flutter of geometrical imperfect functionally graded carbon nanotubes doubly curved shells

In this paper, flutter of geometrical imperfect functionally graded carbon nanotubes (FG-CNTs) doubly curved shell subjected to a supersonic flow is investigated. For this purpose, the imperfect doubly curved shell is reinforced by four different carbon nanotubes (CNTs) distributions in the thickness direction. The Hill’s elastic moduli are used to obtain the effective material properties of FG-CNTs doubly curved shell. The piston aerodynamic theory and von Karman geometrical nonlinearity terms are used to study large deflection flutter analysis of FG-CNTs imperfect shell. The effect of different parameters including large deflection, geometrical imperfection and CNTs volume fraction on critical flutter pressure of doubly curved shells are studied. According to the results, distributions with poor CNTs in the middle of thickness are more efficient in improving critical flutter pressure. Additionally, the effect of shell imperfection on flutter pressure is more considerable than CNTs volume fraction and distributions.

Static and dynamic instability analysis of tapered CNTRC sandwich plates under uniform and non-uniform in-plane loadings using spline finite strip method

This paper presents the buckling analysis of rectangular sandwich plates with pure polymeric tapered cores and functionally graded carbon nanotube (FG-CNT) reinforced composite face sheets under static and harmonic dynamic loads. One uniform and four linearly-varying patterns are considered for the applied in-plane loads. The effective material properties of face sheets, in which the polymeric matrix is reinforced with aligned single-walled carbon nanotubes (SWCNT), are estimated using the extended rule of mixture. A higher-order zigzag shear deformation theory, which has the same number of dependent unknowns as the first-order theory, is employed to express the plate's displacement field. The governing equations are derived using Hamilton's principle and discretized by the spline finite strip method. The dynamic instability regions of plates with uniform and variable thickness are obtained by employing Bolotin's method. The accuracy of the present method is demonstrated by considering the problems for which solutions are available. A detailed numerical study is conducted to examine and compare the effects of different parameters, including the taper angle, type of the applied in-plane load, static and dynamic load factors, length-to-width ratios, the volume fraction of CNT, distribution pattern of CNTs along the thickness direction, temperature, and boundary conditions on the behavior of sandwich plate.

Stress distribution of a sector of cylindrical sandwich shell with FG-CNT core and piezoelectric face sheets subjected to blast pressure

ABSTRACT A novel layerwise theory along with higher-order shear deformation theory is studied to determine the stress distribution in a three-layer simply supported sector of cylindrical sandwich shell with piezoelectric face sheets and functionally graded carbon nanotube (FG-CNT) core. The out-of-shell displacement of the sandwich shell at each layer is assumed to be a quadratic polynomial function of the radial component in addition to a function of the coordinate components within the shell. The sandwich shell is subjected to the internal blast pressure so that the positive and negative phases of the pressure are considered. The 19 equations of motion have been derived by Hamilton’s principle and Maxwell’s static equations. The results indicate that the thickness, length and opening angle of the cylindrical shell have more effect on the stress distribution. In addition, the type of FG-CNT core has an effect on stress components in terms of time.

Time-depended stress analysis of a sector of the spherical sandwich shell with piezoelectric face sheets and FG-CNT core subjected to blast pressure

A novel Layerwise theory (LT) along with the higher-order shear deformation theory (HSDT) is investigated to determine the stress distribution in a three-layer sector of the spherical sandwich shell with piezoelectric face sheets and functionally graded carbon nanotube (FG-CNT) core. The simply-supported sandwich shell is subjected to the internal blast pressure so that the positive and the negative phases of the pressure are considered. The blast pressure is generated by a 1 Kg TNT hemispherical surface burst detonated. The out-of-shell displacement of the sandwich spherical shell at each layer is assumed to be a quadratic polynomial function of the radial component in addition to a function of the coordinate components within the shell. The nineteen equations of motion are derived by Hamilton's principle and Maxwell's static equations. By solving these nineteen equations, the stress distribution is calculated in the spherical sandwich shell in terms of time.

Nonlinear Vibration Analysis of Functionally Graded Carbon Nanotubes Sandwich Cylindrical Panels

In this research, we investigate the nonlinear vibration of functionally graded carbon nanotubes (FG-CNTs) for simply supported sandwich cylindrical panels. The sandwich consisting of three layers formed of (FG-CNTs) and isotropic material as (CNT, ALMINUME, CNT).  Mechanical properties of the sandwich media are acquired according to a refined rule of blend approach. The governing equations were derived using a first-order deformation theory (FOSDT). Four kinds of carbon nanotubes of sandwich cylindrical panels were analyzed. The volume fraction of CNTs is varied. The properties of nonlinear responses and free vibration are studied. The numerical approach employs the fourth-order Runge-Kutta and Galerkine procedure. Which conducted for the dynamic analysis of the panels to present the natural frequencies and non-linear dynamic response expression. The results show that; the natural frequencies and the nonlinear vibration amplitude decrease with the volume fraction and thickness ratio increase. The nonlinear vibration amplitude response increases when increasing the excitation force. The initial imperfection and the elastic foundation have a minor impact on the nonlinear vibration response of the panel. The Pasternak Foundation has a larger impact than the Winkler foundation. The structure formed of FG-CNT present an excellent choice for high-performance of engineering applications.