Graded Carbon Nanotube Research Articles

Nonlinear buckling and post-buckling of functionally graded CNTs reinforced composite truncated conical shells subjected to axial load

This study deals with the nonlinear static analysis of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) truncated conical shells subjected to axial load based on the classical shell theory. Detailed studies for both nonlinear buckling and post-buckling behavior of truncated conical shells. The truncated conical shells are reinforced by single-walled carbon nanotubes which alter according to linear functions of the shell thickness. The nonlinear equations are solved by both the Airy stress function and Galerkin method based on the classical shell theory. In numerical results, the influences of various types of distribution and volume fractions of carbon nanotubes, geometrical parameters, elastic foundations on the nonlinear buckling and post-buckling behavior of FG-CNTRC truncated conical shells are presented. The proposed results are validated by comparing with other authors.

Flexural Analysis of Functionally Graded CNT-Reinforced Doubly Curved Singly Ruled Composite Truncated Cone

AbstractThe flexural analysis of functionally graded carbon nanotube (CNT)-reinforced doubly curved singly ruled composite truncated cone is studied. The mathematical model contains the expansion o...

Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect

Abstract The present paper is aimed to study the buckling and postbuckling response of functionally graded carbon nanotube (FG-CNT)- magnesium (Mg) nanocomposite plate with interphase effect. Interphase zone is characterized by employing a cohesive zone model for its elastic modulus and thickness. An equivalent solid fiber (ESF) of CNT and interphase is modeled and dispersed into the matrix material by utilizing random sequential adsorption (RSA) technique. The effective elastic properties of the nanocomposite are computed by finite element method (FEM) based numerical homogenization technique. The obtained elastic properties of nanocomposite are utilized to investigate the buckling and post-buckling behaviour of different functionally graded (i.e., FG) nanocomposite plates modeled by varying the volume fraction of CNT/ESF along thickness direction, under in-plane compressive loads. The non-linear formulation is based on first-order shear deformation theory and von Karman’s assumptions. It is found that considering the interphase between CNT and Mg matrix would result in decrease in buckling load and postbuckling strength of FG-CNT-reinforced nanocomposite plate as compared to nanocomposite without interphase. It is also reported that the higher volume fraction of CNTs near top and bottom surfaces than the middle portion of nanocomposite plate provide better resistance to buckling and postbuckling.

Nonlinear bending analysis of FG-CNTRC annular plates with variable thickness on elastic foundation

As a first endeavor, the nonlinear bending behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) annular plates with variable thickness on an elastic foundation is scrutinized in this research. The nonlinear governing equations are established according to the third-order shear deformation theory (TSDT) in conjunction with the nonlinear von Karman strain field. A Pasternak elastic foundation is assumed to be in contact with the annular plate during deformation. To discretize the nonlinear governing equations and boundary conditions, the generalized differential quadrature method (GDQM) is adopted and the nonlinear system of equations is solved via the Newton-Raphson iterative method. Through the numerical results, effects of the elastic foundation, geometrical parameters, boundary conditions, CNT dispersion and volume fraction, thickness-to-radius ratio and thickness profile are investigated in detail. It is shown that CNT volume fraction and dispersion have a remarkable effect on improving bending behavior of the annular plate. It is found that thickness profile plays an essential role in the bending response of the annular plate.

Numerical Study on Low Velocity Impact Response of Functionally Graded Carbon Nanotube Reinforced Composite Beams

Numerical Study on Low Velocity Impact Response of Functionally Graded Carbon Nanotube Reinforced Composite Beams

Bending analysis of functionally graded CNT reinforced doubly curved singly ruled truncated rhombic cone

The bending analysis of functionally graded carbon nanotube (CNT) reinforced doubly curved singly ruled truncated rhombic cone is investigated. In this study, a simple C0 isoparametric finite element formulation based on third order shear deformation theory is presented. To characterize the membrane-flexure behavior observed in a CNT reinforced truncated rhombic cone, a displacement field involving higher-order terms in in-plane fields is considered. The proposed kinematics field incorporates for transverse shear deformation and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the CNT reinforced truncated rhombic cone in an accurate sense. The material properties of the CNT reinforced truncated rhombic cone are estimated according to the rule of mixture. The present model eliminates the need of shear correction factor and imposed zero-transverse shear strain at upper and lower surface of the truncated rhombic cone. The new feature in present model is simultaneous inclusion of twist curvature in strain field as well as curvature in displacement field that makes it suitable for moderately thick and deep truncated rhombic cone. The proposed new mathematical model is implemented in finite element code written in FORTRAN. The proposed model has been validated with analytical, experimental, and finite element results from the literature. This is first attempt to study bending response of CNT reinforced doubly curved singly ruled truncated rhombic cone. The effect of CNT distribution, boundary condition, loading pattern, and other geometric parameters are also examined.

Dynamic Stability of Rotating FG-CNTRC Cylindrical Shells under Combined Static and Periodic Axial Loads

The dynamic stability behavior of rotating functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells under combined static and periodic axial forces is investigated. The governing equations are derived based on the first-order shear deformation theory (FSDT) of shells. The initial mechanical stresses due to the steady state rotation of the shell are evaluated by solving the dynamic equilibrium equations. The equations of motion under different boundary conditions are discretized in the spatial domain and transformed into a system of Mathieu–Hill type equations using the differential quadrature method (DQM) together with the trigonometric series. The influences of both the initial mechanical stresses and Coriolis acceleration are considered. Then, the parametric resonance is analyzed and the dynamic instability regions are determined by employing the Bolotin’s first approximation. After validating the approach, the effects of rotational speed, Coriolis acceleration, carbon nanotubes (CNTs) distribution in the thickness direction, CNTs volume fraction, length and thickness-to-mean radius ratios on the principal dynamic instability regions are examined in detail.

Geometrically nonlinear dynamic analysis of FG-CNTRC plates subjected to blast loads using the weak form quadrature element method

Geometrically nonlinear dynamic analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular plates subjected to blast loads is conducted based on Reddy’s higher-order shear deformation theory using the weak form quadrature element method. The von-Kármán strain terms are introduced to consider the geometrically nonlinear effects. The polymer composite plate is reinforced by single-walled carbon nanotubes (SWCNTs) with the uniform and functionally graded distribution in the plate thickness direction. The effective material properties of the FG-CNTRC plates are estimated with the extended rule of mixture. The Newmark-β time integration scheme and the Newton–Raphson iteration technique are adopted to solve the nonlinear incremental dynamic equilibrium equation in temporal domain. Comparative and convergence studies are carried out to validate the accuracy, efficiency and numerical stability of the presented weak form quadrature element formulation. The effects of the carbon nanotube distribution and volume fraction, plate width-to-thickness ratio, plate aspect ratio, load type and boundary condition on the dynamic response of the FG-CNTRC plates under blast loads are systematically investigated through the parametric studies.

Dynamic stability of fluid-conveying thin-walled rotating pipes reinforced with functionally graded carbon nanotubes

In this study, vibration and dynamic stability of fluid-conveying thin-walled rotating pipes reinforced with functionally graded carbon nanotubes are studied. The pipe is modeled based on thin-walled Timoshenko beam theory and reinforced by single-walled carbon nanotubes with uniform distribution as well as three types of functionally graded distribution patterns. The governing equations of motion and the associated boundary conditions are derived via Hamilton’s principle. The governing equations of motion are discretized via the Galerkin method, and the eigenfrequency and the stability region of the pipe are found using the eigenvalue analysis. Some numerical examples are presented to study the effects of length–radius ratio, carbon nanotubes distribution, volume fraction of carbon nanotubes, rotational speed and mass ratio on the non-dimensional eigenfrequency and critical flutter velocity of the thin-walled rotating pipe conveying fluid. The results show that the carbon nanotubes distribution has a significant effect on the non-dimensional eigenfrequency and critical flutter velocity. Also, it is found that the rotational speed has a stabilizing effect on the dynamic behavior of the system.

Closed-form solution for nonlinear buckling analysis of FG-CNTRC cylindrical shells with initial geometric imperfections

The circular cylindrical shells have been widely used in modern engineering structures, especially in the aerospace industry such as the oil pipeline, the missile, spacecraft hull, storage tanks. In recent years, functionally graded carbon nanotube composites (FG-CNTRCs) have emerged, as a promising type of composites. Due to the increasing demands for high structures performance, this research paper proposes a closed-form solution to investigate the nonlinear buckling behavior of the FG-CNTRC cylindrical shells subjected to compressive load. The small initial imperfections of the FG-CNTRC cylindrical shells are also considered through analytical modeling. Effective properties of materials of the shells reinforced by single-walled carbon nanotubes (SWCNTs) are estimated through a micro-mechanical model based on the extended rule of mixtures. The Donnell shell theory and von-Karman nonlinear kinematics are used for nonlinear equilibrium equations. The novelty of this work is to exploit an exact solution via Galerkin procedure and term of the Airy stress function in order to reveal the impacts of the imperfection parameter, different types of CNTs distribution, the volume fraction of CNTs on nonlinear behavior and compressive equilibrium paths of FG-CNTRC cylindrical shells.

Free vibration analysis of functionally graded carbon nanotube reinforced composite truncated conical panels with general boundary conditions

This paper presents the free vibration analysis of functionally graded carbon nanotube reinforced composite truncated conical panels with general boundary conditions for the first time. Based on the modified Fourier series method for the field variables, the Ritz method is employed to obtain the frequency parameters associated with the mode shapes. The truncated conical panels are reinforced by single-walled carbon nanotubes (SWCNTs) which are assumed to be graded through the thickness direction with different types of distributions. The effective material properties of the FG-CNTRC truncated conical panels are estimated through a micromechanical model based on the extended rule of mixture. In the present study, the artificial spring boundary technique is adopted here to implement the general boundary condition. Several examples are given to demonstrate the convergence, accuracy and flexibility of the present method. The effects of the volume fractions of CNTs, distribution type of CNTs, boundary restraint parameters and geometrical parameters on the vibration behavior of the FG-CNTRC truncated conical panels are presented.

A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates

This paper presents a model which is effective and simple based upon Second Order Shear Deformation Theory (SSDT) to comprehensive study of Functionally Graded Carbon Nanotube Reinforced Composite (FG-CNTRC) plates including size effects for the first time. Also it is the first time that the size-dependent static, stability and dynamic response of FG-CNTRC plates considering Winkler–Pasternak elastic foundation effects are investigated. It is assumed that the material properties of FG-CNTRC are varied through the thickness direction using four different distributions of carbon nanotubes (CNTs). In order to include the size effects in our modeling, the nonlocal elasticity theory presented by Eringen is utilized. The influences of different parameters such as volume fraction of carbon nanotubes, nonlocality and elastic foundation effects on the response of FG-CNTRC plates are studied. Numerical results prove high accuracy and reliability of the present method in comparison with other available numerical or analytical methods.

Three-dimensional nonlinear bending analysis of FG-CNTs reinforced composite plates using the element-free Galerkin method based on the S-R decomposition theorem

A three-dimensional element-free Galerkin method (EFG) based on the Strain-Rotation (S-R) decomposition theorem, named 3D-SR-EFG, is developed to investigate the nonlinear bending behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. Due to its overcoming the deficiencies of classic finite deformation theories, S-R decomposition theorem can provide a reliable theoretical support for the geometrically nonlinear simulation. The incremental variational formulation based on the S-R decomposition theorem for three-dimensional static large deformation problems is derived from the updated co-moving coordinate formulation and principle of potential energy rate. Global weak-form EFG is adopted to obtain the discrete form of the formulation. Convergence and comparison studies are conducted to validate the numerical stability and accuracy of the proposed 3D-SR-EFG. The influences of volume fraction and distributions of CNTs, plate’s aspect ratio and width-to-thickness ratio, boundary conditions on the nonlinear bending response of the CNTRC plates are numerically analyzed and discussed in parametric studies. Results demonstrate that the 3D-SR-EFG can effectively predict the nonlinear bending behavior of the CNTRC plates. This work also further extends the applications of the S-R decomposition theorem.

An analytical study of sound transmission through stiffened double laminated composite sandwich plates

The main objective of this research work is focused on sound transmission loss analysis of stiffened double laminated composite sandwich plate structures subjected to plane sound wave excitation, wherein the laminated composite plates are composed of perfectly bonded functionally graded carbon nanotubes (CNTs) reinforced composite layers, and in each layer, the carbon nanotubes is uniform or functionally graded along with the thickness direction of structures, and three types of the CNTs distributions are studied. The extended rule of mixture is employed to determine the properties of the composite material. The compatibility of displacements on the interface between the plate and the stiffeners is employed to derive the governing equation of each stiffener. Then fluid–structure coupling is considered by imposing velocity continuity condition at fluid–structure interfaces. By using the space harmonic approach and virtual work principle, the sound transmission loss is described analytically. Since no existing results of sound insulation can be found for such composite material plate structure, comparison studies can only be made with the isotropic and laminated case. Good agreement is found from these comparison studies. Based on the developed theoretical mode, the influences of the volume fractions of CNTs, distribution type of CNTs, structural damping, lamination angle and the number of layers on sound transmission loss are subsequently presented.

Mechanical behavior of laminated functionally graded carbon nanotube reinforced composite plates resting on elastic foundations in thermal environments

The present study is concerned with static and free vibration analyses of laminated functionally graded carbon nanotube reinforced composite rectangular plates on elastic foundation based on nth-order shear deformation theory. Four types of carbon nanotubes distributions along the plate thickness are considered, which include uniformly distributed and three other functionally graded distributions. Governing differential equations are derived by means of Hamilton’s principle. The differential quadrature method is developed to formulate the problem, and rapid convergence is observed in this study. A numerical comparison with available results in the literature is carried out to show the validity of the proposed theory. Furthermore, effects of the carbon nanotubes volume fraction, thickness side ratio, aspect ratio, foundation parameters, different thermal environments, the number of layers, lamination angle, boundary condition, and carbon nanotubes distribution types on the static response of laminated functionally graded carbon nanotube reinforced composite plates are also investigated.

Thermal and mechanical buckling analysis of FG carbon nanotube reinforced composite plates using modified couple stress theory and isogeometric approach

Abstract This paper investigates the thermal and mechanical buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates by using isogeometric analysis (IGA) based on modified couple stress theory (MCST). A refined hyperbolic shear deformation theory is used for buckling analysis, which satisfies free transverse shear stress conditions on the top and bottom surfaces of plate without a need for shear correction factor. The material properties of carbon nanotube reinforced composite plates are assumed to be temperature-dependent. For numerical analysis the IGA method using B-Spline or Non-Uniform Rational B-Spline (NURBS) functions is employed. The obtained results are compared with those available in the literature. Also, the influence of different parameters on mechanical and thermal buckling analysis is investigated. These parameters include material length scale parameter, boundary conditions, aspect and length-to-thickness ratios of plate, different types of FG-CNTRC distribution, volume fraction of CNTs and temperature dependency.

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.

Thermal buckling analysis of cylindrical shells of carbon nanotubes reinforced composites

The purpose of this study is to investigate the thermal buckling analysis of cylindrical shells of carbon nanotubes reinforced composites (CNTRC). In this study, a hybrid laminate composite plate consisting of N layers which are of thickness t is designed. N is the number of layers and the thickness of each functionally graded carbon nanotube reinforced composite (CNTR-FG) layer is h, and three types of the distribution of carbon nanotubes (CNT) are considered for each layer. CNTR-FG are multi-layered plates which have the length "a", width "b", thickness "t", and an arbitrary combination of boundary conditions along the four edges of the layer. The distribution of carbon nanotubes along the thickness of each layer of CNTR-FG is considered to be given. Implementation is done by using Abaqus software. In the investigation of the thermal buckling of cylindrical shells of CNTRC, in three different types of distribution of carbon nanotubes, the behavior of buckling of a cylindrical shell is investigated. In each case, the amount of buckling generated in the shell is calculated and analyzed. The results show that the behaviors of these three types of distributions are close, and the nanotubes with uniform distribution against thermal buckling loads perform better than the other two distributions.

Free vibration study of composite conical panels reinforced with FG-CNTs

This work analyses the natural frequencies of composite conical panels made of a polymeric matrix reinforced with uniform or functionally graded carbon nanotubes (FG-CNTs). The mechanical properties of the composite conical panels rely on a refined rule of mixtures, including some efficiency parameters, whereas the kinematics of the problem is here tackled with the first order shear deformation shell theory (FSDT) and the Donnell’s theory. The expressions of the virtual strain and kinetic energies of the conical panels are substituted into the Hamilton’s principle. A matrix formulation of the free vibration problem is achieved using the conventional Ritz method, with the shape functions stemming from the Gram-Schmidt process. Based on a parametric investigation, we assess the key role of the CNT volume fractions and patterns, and check for their effect on the free vibrations of the composite conical panels, as useful for practical applications.

Analysis of the low velocity impact response of functionally graded carbon nanotubes reinforced composite spherical shells

The low velocity impact analysis of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) spherical shells including the contact force and indentation of the shell surface center was presented. The material properties were set according to the rule of mixture. The fibers and polymeric matrix were temperature-dependent. Timoshenko-Midlin assumption was used to establish the dimensionless nonlinear governing equations of the axis-symmetric transverse isotropy spherical shells. The Hertz contact theory was established to obtain the force and displacement between the spherical shell and impactor. Seven species of comparison analyses including grading profiles and volume fraction were mentioned to investigate the influence on the low velocity impact response of the FG-CNTRC spherical shells. The results revealed that the contact force was determined by the material properties of both contact surface and noncontact surface collectively. Furthermore, the increase of volume fraction can result in the larger contact force and shorter contact time.