Functionally Graded Carbon Nanotube Reinforced Composite Research Articles

Large deflection of functionally graded carbon nanotube reinforced composite cylindrical shell exposed to internal pressure and thermal gradient

In this paper, the nonlinear bending of functionally graded carbon nanotube‐reinforced composite (FG‐CNTRC) shell exposed to thermomechanical loading is perused. It is assumed that the composite shell is reinforced in the longitudinal axis and is also made from a polymeric matrix. Mechanical features of the constituents are obtained based on the modified rule of mixture, and they are considered to be temperature dependent (TD). Using the first‐order shear deformation shell theory (FSDT) as well as von Kármán type of geometrical nonlinearity, the equilibrium mathematical relations are derived. Utilizing the dynamic relaxation (DR) procedure combined with the central finite difference method, these mathematical relations are solved in diverse boundary conditions. Finally, roles of carbon nanotube (CNT) distributions, boundary conditions, shell radius, thickness‐to‐radius ratios, volume fraction of CNTs, mechanical loads, thermal gradient, and temperature dependency are examined on the results. From the numerical results, it can be inferred that in the shell with the CC boundary condition, the FG‐O distribution of nanotubes has the maximum deflection, and the lowest deflection belongs to the uniform distribution. However, in the SS boundary condition, the highest and lowest values of deflections are related to V and uniform distributions, respectively.

Vibration characteristics of piezoelectric functionally graded carbon nanotube-reinforced composite doubly-curved shells

This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here, the linear distribution of electric potential across the thickness of the piezoelectric layer and five different types of carbon nanotube (CNT) distributions through the thickness direction are considered. Based on the four-variable shear deformation refined shell theory, governing equations are obtained by applying Hamilton’s principle. Navier’s solution for the shell panels with the simply supported boundary condition at all four edges is derived. Several numerical examples validate the accuracy of the presented solution. New parametric studies regarding the effects of different material properties, shell geometric parameters, and electrical boundary conditions on the free vibration responses of the hybrid panels are investigated and discussed in detail.

Dynamic stability analysis for rotating pre-twisted FG-CNTRC beams with geometric imperfections restrained by an elastic root in thermal environment

This paper deals with the problems of free vibration, buckling, and dynamic stability of rotating pre-twisted functionally graded carbon nanotube reinforced composite (FG-CNTRC) imperfect beams in thermal environment. The imperfect beam contains different modes of geometric imperfections such as sine, global, and local modes, and it is restrained by an elastic root. Three types of CNTs distributions including FG-X, UD, and FG-O distributions are considered and the material is temperature-dependent. First, bending–bending coupled governing equations are established through the Hamilton’s principle based on the Euler–Bernoulli beam theory. By setting different parameters, the governing equations can solve the problems of free vibration, buckling, and dynamic stability of the beam. Then, the differential quadrature method (DQM) is employed to get the discrete equations and numerical solutions of the natural frequency, critical buckling load, and instability region. Finally, parametric studies are carried out to present the effects of hub radius, rotating speed, material properties, geometric imperfections, and rigidity of the elastic root on the natural frequencies, critical buckling load, and instability regions. Results show that the elastic root and imperfection mode have obvious influence on the instability regions.

Nonlinear Primary Resonant Characteristics of Higher-Order Shear Deformable FG-CNTRC Circular Cylindrical Panels

In this work, the nonlinear primary resonance of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) circular cylindrical panels is numerically studied. The FG-CNTRC cylindrical panels under the radial harmonic loading are modeled on the basis of the hyperbolic shear deformation shell theory (HSDST). The von Kármán hypothesis is employed to incorporate geometric nonlinearity into mathematical modeling. After representing the kinetic and strain energies and the external work in matrix forms (in terms of displacement vector), the variational differential quadrature (VDQ) method is utilized to obtain the discretized form of the energy functional on the space domain. Then, the nonlinear governing equations are achieved via Hamilton’s principle. In the next step, a multistep numerical solution approach is employed to illustrate the influences of geometrical parameters, subtended angle, CNTs distribution scheme and volume fraction on the primary resonant characteristics of FG-CNTRC cylindrical panels. The results are provided for the panels with clamped and simply-supported boundary conditions (BCs).

Thermoelastic free vibration of rotating pretwisted sandwich conical shell panels with functionally graded carbon nanotube-reinforced composite face sheets using higher-order shear deformation theory

This article presents a numerical investigation on the free vibration characteristics of rotating pretwisted sandwich conical shell panels with two functionally graded carbon nanotube-reinforced composite (FG-CNTRC) face sheets and a homogeneous core in uniform thermal environments. The carbon nanotubes are considered to be aligned with the span length and distributed either uniformly or functionally graded along the thickness of the sandwich conical shell panel. The material properties of FG-CNTRC face sheets and homogenous core are assumed to be temperature-dependent and computed employing micromechanics models. The shallow conical shell is modeled using finite element method within a framework of the higher-order shear deformation theory. Lagrange’s equation of motion is employed to derive the dynamic equilibrium equations of sandwich conical shell panels rotating at moderate rotational speeds wherein Coriolis effect is neglected. Computer codes are developed on the basis of present mathematical formulation to determine the natural frequencies of the sandwich conical panels. Convergence and comparison studies are performed to examine the consistency and accurateness of the present formulation. The numerical results are presented to analyze the effects of various parameters on the natural frequencies of the pretwisted FG-CNTRC sandwich conical shell panels under different thermal conditions.

Dynamic Analysis of Carbon Nanotube-Reinforced Multilayer Composite Plates

The paper studied the analysis of vibrations of rectangular carbon nanotube-reinforced composite plates. To this end, a three-layer nanocomposite plate - two layers with the targeted distribution of carbon nanotubes as FG-X at the top and bottom and a layer without an amplifier in the middle of the plate - were analyzed. The governing equations for this problem are based on First-order Shear Deformation Theory (FSDT). The distribution of nanotubes on these plates is as targeted FG-X. The effect of various types of SWCNTs distributions in the direction of thickness on the vibrational behavior of nanocomposite plates was examined. The effective properties of nanocomposite materials Functionally Graded Carbon Nanotube-Reinforced Composite (FG-CNTRC) were estimated using the rule of mixtures. Detailed parametric studies were performed to determine the effects of the volume fraction of carbon nanotubes and the thickness-to-length ratio of the plate on the natural frequency responses and the shape of the plate mode. The equations obtained in this problem were coded in MATLAB software, the nanocomposite plate was modelled in ABAQUS software, and the comparison of the results obtained from the numerical solution with ABAQUS software showed relatively right consistency with the results obtained from the analytical solution.

Elasticity Solution for Bending and Frequency Behavior of Sandwich Cylindrical Shell with FG-CNTRC Face-Sheets and Polymer Core Under Initial Stresses

This paper explores the high-accuracy analysis of bending and frequency response of the sandwich cylindrical shell with functionally graded (FG) carbon nanotubes reinforced composite (FG-CNTRC) face-sheets and polymeric core under the effect of initial axial stress and various mechanical loading based upon the three-dimensional theory of elasticity for various sets of boundary conditions. The sandwich structure is composed of multilayers with uniformly dispersed carbon nanotubes (CNT) in each fictitious layer of face-sheets, but its weight fraction changes layer-by-layer along the thickness direction. With the aid of compatibility conditions, the sandwich structure with three layers is modeled. Analytical bending and frequency solutions are obtained for simply supported shells. Also, the state-space based differential quadrature method (SS-DQM) is employed to determine the bending and frequency response of the sandwich cylindrical shell by considering various boundary conditions. The bending response of the sandwich cylindrical shell is obtained under the impact of various mechanical loadings. The influences of several parameters, such as initial stress and various mechanical loadings are investigated on the bending and frequency of the structures. The results of the presented study can be served as benchmarks to assess the validity of the conventional two-dimensional theory.

Thermomechanical analysis of large deflection in shear deformable FG‐CNT reinforced composite beams using perturbation technique

Present investigation deals with the geometrically nonlinear bending response of functionally graded carbon nanotube reinforced composite (FG‐CNTRC) beams. It is assumed that beam is resting on a three‐parameter elastic foundation. Uniform thermal field is also included into the formulation and properties are considered as temperature dependent. Distribution of CNTs in the beam may be uniform or functionally graded according to linear distribution of volume fraction of CNTs across the thickness. The equivalent properties of the composite media are obtained with the aid of the refined rule of mixtures approach which is calibrated with the available data based on the molecular dynamics simulations. The governing equations of the beam are obtained based on first‐order, third‐order, and sinusoidal theories. To capture the large deformations, the von Kármán type of kinematic assumptions is included. Two types of boundary conditions are included which are immovable pinned and immovable clamped. The established equations are reformulated and transferred into dimensionless presentation. After that, using the powerful two step perturbation technique, closed form expressions are provided to trace the nonlinear load‐deflection paths of the beam. Results of this study are compared with the available data in the open literature. After that, novel results are provided to discuss the effects of boundary conditions, temperature change, foundation stiffness, slenderness ratio, CNT volume fraction, and distribution profiles. It is shown that graded pattern of CNTs and volume fraction of CNTs are two important factors on the geometrically nonlinear response of the CNTRC beam.

A comparative study of different dynamic condensation techniques applied to multi-damage identification of FGM and FG-CNTRC plates

The dynamic condensation method has been recognized as an effective alternative for structural damage identification using spatially-incomplete modal measurements. However, comparative studies of different dynamic condensation techniques applied to the subject of structural damage identification have been scarcely found, especially for composite structures. In this regard, we conduct a comparative study of six typical dynamic condensation techniques utilized for addressing damage identification problems of composite plates made of functionally graded materials (FGM) and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) materials. Firstly, the six techniques consisting of Guyan’s method, Kidder’s method, Neumann series expansion-based second-order model reduction (NSEMR-II) method, improved reduced system (IRS) method, iterated IRS (IIRS) method, and iterative order reduction (IOR) method are reviewed. Then, their performance for reduced Eigen and optimization-damage identification problems are evaluated by studying two numerical examples of FGM plate and FG-CNTRC plate. For solving the optimization-damage identification problem of plate structures, the article proposes to use a hybrid global–local algorithm, Manta Ray Foraging Optimization—Sequential Quadratic Programming (MRFO-SQP), where the MRFO algorithm is utilized for global exploration and the SQP algorithm is used for the local searching process. The comparative study indicates that the IOR technique is the best dynamic condensation technique and is effective for addressing the structural damage identification problems when comparing with the other five techniques. It is also found that the damage identification approach based on the hybrid MRFO–SQP algorithm combined with the IOR technique can archive the high accuracy and low computational cost for damage localization and quantification.

Free Vibration Analysis of Laminated CNTRC Plates using the pb2-Ritz Method

This paper presents the free vibration analysis of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. The CNTRC layer consists of single-walled carbon nanotubes (SWCNTs) as reinforcement and polymer as a matrix. The material properties are determined according to the extended rule of mixture. Four different patterns of SWCNTs distribution across the thickness of individual layers are considered. Based on the first-order shear deformation theory (FSDT), the equations of motion are derived and then solved by employing the pb2-Ritz method. The accuracy of the present approach is verified by comparing the obtained results with those available in the literature and commercial ANSYS software. The significant influences of CNT volume fractions, CNT distribution patterns, plate aspect ratio, plate width-to-thickness ratio, and boundary conditions on the non-dimensional fundamental frequency of symmetric and anti-symmetric laminated FG-CNTR plates has been proven through the numerical examples. Furthermore, the influence of lamination schemes, CNT fiber orientation and the number of layers is also investigated.

Applying nonlocal strain gradient theory to size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoplates

In this research paper, as initial endeavors, the vibrational responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates taking into account the effect of nonlocal parameter and strain gradient coefficient are investigated. The study aims at developing mathematical modeling via an analytical solution to FG-CNTRC nanoplate structure with allowance for the nonlocal strain gradient effect. The four types of CNT distribution are used and compared in the context of the vibration of nanoplate in the presence of the small length scale effects, namely the (a) UD, (b) FG-V, (c) FG-O, and (d) FG-X. Some theoretical equations based on the first-order shear deformation plate theory (FSDT) are presented to provide a lucid understanding of how the small length-scale influences the FG-CNTRC nanoplate. For the vibrational analysis of a nanoplate, which is simply supported boundary condition, Navier solutions are obtained. Also, in contrast to earlier studies, an analytical approach is used to establish the governing equations of the FG-CNTRC nanoplate. Some specific numerical examples are given and compared with the results presented in the literature. In the section of numerical results, the influence of the nonlocal parameter, strain gradient coefficient, geometric parameters and vibrational modes on the non-dimensional natural frequency are investigated and discussed in detail. These could be useful to analysts and designers to estimate the fundamental natural frequencies in each of the four CNT distributions that the FG-CNTRC nanoplate possesses.

A numerical study to investigate the active vibration attenuation of functionally graded carbon nano-tube reinforced composite shell

Abstract In the present study, active vibration control of functionally graded carbon nano-tubes reinforced composite (FG-CNTRC) cylindrical shell is presented. In this study, straight and aligned carbon nanotubes are used and are functionally graded in the transverse direction. Different distribution configurations are considered for this analysis. First order shear deformation theory (FSDT) is used to derive the kinematic and constitutive equations. Moreover, in order to obtain the equation of motion, the Hamilton’s principle is implemented. For active vibration attenuation application, the FG-CNTRC cylindrical shell is sandwiched between the sensor and actuator layers made of piezoelectric material. The piezoelectric patches occupy the optimal places to obtain effective attenuation of the vibrations. The effect of different volume fractions of CNT and various configurations on natural frequencies is demonstrated numerically. It is observed that with increase in the amount of CNT the vibrational amplitude decreases.

The Linear and Nonlinear Bending Analyses of Functionally Graded Carbon Nanotube-Reinforced Composite Plates Based on the Novel Four-Node Quadrilateral Element

This paper presents the linear and nonlinear analyses of functionally graded carbon nanotube- reinforced composite (FG-CNTRC) plates using a four-node quadrilateral element based on the C0-type of Shi’s third-order shear deformation theory (C0-STSDT). Shi’s theory is taking the advantages and desirable properties of the third-order shear deformation theory. Besides, material properties of FG-CNTRC plates are changed from the bottom to top surface and based on the rule of mixture. Numerical results and comparison with other reference solutions suggest that the advantages of present element are accuracy and efficiency in analysis of FG-CNTRC plates. Some nonlinear numerical results of FG-CNTRC plates are also given in this paper and this contributes to providing additional data for future research work.

Static and dynamic hygrothermal postbuckling analysis of sandwich cylindrical panels with an FG-CNTRC core surrounded by nonlinear viscoelastic foundations

In this article, analytical and semi-analytical methods are used to analyze the nonlinear static and dynamic hygrothermal postbuckling response of sandwich cylindrical panels with functionally graded carbon nanotube-reinforced composite (FG-CNTRC) core. The sandwich cylindrical panels with an FG-CNTRC core are resting on generalized nonlinear viscoelastic foundations which are consisted of a Winkler and Pasternak foundation parameters augmented by a Kelvin-Voigt viscoelastic model and a nonlinear cubic stiffness. The sandwich cylindrical panels with four types of FG-CNTRC core are considered such that the CNTs distribution types consist of the FG-O, FG-Λ, FG-X, and UD. Regarding to the classical shell theory and geometrical nonlinearity in von Kármán-Donnell sense, the governing equation is obtained and discretized utilizing the Galerkin method. The nonlinear dynamic hygrothermal (NDHT) postbuckling is analyzed by means of the fourth-order Runge-Kutta method. The influences of material parameters, various geometrical characteristics, and nonlinear viscoelastic foundation parameters are studied on the nonlinear static hygrothermal (NSHT) postbuckling and NDHT postbuckling analysis of sandwich cylindrical panels with an FG-CNTRC core. The results show that the different types of CNTs distribution and nonlinear viscoelastic foundation parameters have a strong effect on the hygrothermal postbuckling behaviors of sandwich cylindrical panels with an FG-CNTRC core.

Dynamic instability analysis of FG-CNTRC laminated conical shells surrounded by elastic foundations within FSDT

A comprehensive understanding of the dynamic instability of shell structure is critical to avoid resonance damage. On the basis of that, an accurate and analytical method for investigation the dynamic instability of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell surrounded by the elastic foundations is presented in this work based on the first-order shear deformation theory. In the analysis, uniform or functionally graded distributions of reinforcements across the shell thickness are considered and the extended Voigt model is employed to estimate the CNTRC material properties. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton's principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin's method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of volume fractions and types of CNTs, lamination angle, elastic foundations stiffness and lamination angle on the dynamic stability of laminated FG-CNTRC conical shell have been investigated.

Vibration and nonlinear dynamic analysis of sandwich FG-CNTRC plate with porous core layer

This paper focuses on the influence of CNTs, porosity, mechanical and thermal loading on the vibration and dynamic response of the sandwich functionally graded carbon nanotube-reinforced composite (FG-CNTRC) composite plate. The plate is made by three layers in which the core layer is porous FGM materials, bottom and top surfaces are FG-CNTRC. The motion equations are given based on Hamilton’s principle, TSDT, Galerkin method and the fourth-order Runge–Kutta method. The numerical illustration is shown to examine the influence of various parameters such as porosity distribution, CNTs volume fraction, geometrical parameters, elastic foundations, temperature, mechanical loads on the dynamic behaviors of the plate. Highlights The FG-CNTRC plates with porous core layer The vibration and nonlinear dynamic analysis Elastic foundation and temperature Analytical solutions, Hamilton’s principle with the high-order shear deflection theory is used Galerkin method and the fourth-order Runge–Kutta method are applied

Predicting bucking and vibration behaviors of functionally graded carbon nanotube reinforced composite cylindrical panels with three-dimensional flexibilities

A full understanding of the mechanical behaviors of a three-dimensional (3D) functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical panel is important for structural design of engineering composite components. In this paper, the buckling and free vibration studies of FG-CNTRC cylindrical panel with different patterns of CNT distribution are investigated using the three-dimensional theory of elasticity. The cylindrical panels are subjected to axial and circumferential initial stresses, which frequently occur in real engineer structures. The state space technique along the radial direction and the Fourier series expansion along the in-plane coordinate (are) employed to formulate the problem and solve it analytically. A parametric study is carried out to examine the effects of the CNT distribution pattern, CNT volume fraction, length to mid radius ratio, and mid radius to thickness ratio on the buckling and vibration behaviors of FG-CNTRC cylindrical panels. Some interesting findings are observed, which may help design the CNTRC cylindrical panel structures. Besides, the presented studies may serve as benchmarks for researchers to check the validity of their future research works.

Free vibration analysis of hybrid laminated plates containing multilayer functionally graded carbon nanotube-reinforced composite plies using a layer-wise formulation

This paper studies the free vibration behavior of hybrid laminated plates using a layer-wise formulation. The laminated structures are made of graphite fiber-reinforced composite (GFRC) and multilayer functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plies. The internal single-walled carbon nanotubes (SWCNTs) are distributed in layer-wise form along the ply thickness direction either uniformly or functionally graded according to four separate patterns. Both the linear and nonlinear of the CNTs distributions form are considered. Theatrical formulations are based on the first-order shear deformation theory (FSDT), and finite element method is employed to obtain the nondimensional natural frequency. The elastic modulus of both GFRC and CNTRC plies composed the hybrid laminated plate is obtained by a micromechanics model. Initially, the provided results are validated by comparing it with the literature; thereafter, a parametric study is carried out the influences of laminates configuration, number of CNTRC plies layers, CNT volume fraction, CNT distribution patterns, linear and nonlinear distribution of CNTs, plate width-to-thickness ratio, plate aspect ratio, boundary conditions, stacking sequences, and number of plies on the free vibration behavior of hybrid laminated plates. Some final remarks considering the laminated plate configuration and the distribution form of the CNT fillers are presented in order to provide a useful observation on the design criteria of the laminated composite plate structures.

Free vibration analysis of FG-CNTRC conical shell panels using the kernel particle Ritz element-free method

The free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell is carried out by using element-free kp-Ritz method. Based on the first-order shear deformation shell theory, the approximate displacement field is expressed by the shape function of the nuclear particle, and the governing equation is established. The material properties of conical shell plates are determined by rule of mixtures and change with the thickness. In this work, the convergence is studied in terms of the number of nodes, and the accuracy of the proposed method is verified by comparing the current solution with the solution in the literature from different aspects. The effects of volume fraction, boundary conditions, distribution patterns of CNTs, half vertex angle and radius thickness ratio on their free vibration frequency characteristics are discussed in detail.

Thermal vibration and buckling analysis of functionally graded carbon nanotube reinforced composite quadrilateral plate

This paper investigates the thermal vibration and buckling of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) quadrilateral plate using the meshless method. Four distributions of the reinforcements along the thickness direction are considered, which are the uniform distributions (UD), FG-V, FG-O and FG-X. The corresponding effective material properties including Young's modulus, mass density, Poisson's ratio and thermal expansion coefficients are estimated by the rule of mixture with the CNT efficiency parameters accounting for the size-dependence. The first-order shear deformation theory (FSDT) with the consideration of thermal effects is employed. Based on Hamilton's principle and the moving least square (MLS) approximation, the discrete governing equations for the vibration of the FG-CNTRC plate are derived. The free vibration of the regular and irregular plates with and without the temperature effect are considered respectively, and the corresponding natural frequencies and mode shapes are obtained by the eigenvalue problem. The stability analysis of the uniaxial and biaxial mechanical and thermal buckling are also conducted subsequently. The effects of the volume fraction, distribution pattern, geometrical characteristics and temperature on the buckling behaviors of the CNTRC quadrilateral plate are discussed in detail.