Functionally Graded Carbon Nanotube Reinforced Composite Research Articles

Four-node quadrilateral C 0-element based on cell-based smoothed strains strategy and third-order shear deformation theory for functionally graded carbon nanotube reinforced composite plates

This study indicates the analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates using a four-node quadrilateral element related to the C0-type of Reddy’s third-order shear deformation theory (C0 HSDT) and cell-based smoothed strains (CS) strategy. Reddy’s theory is surely taking the advantages and desirable properties of the third-order shear deformation theory. Besides, FG-CNTRC plates with advanced material properties are changed from the bottom to top surface with four kinds of carbon nanotube (CNTs). Numerical results and comparison with other reference solutions suggest that the benefits of the present element are accuracy and efficiency in analysis of FG-CNTRC plates.

Piezoelectric Patches for Deflection Control of Functionally Graded Carbon Nanotube-Reinforced Composite Plates

In the present work, a smart structure is being investigated, where a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate is equipped with piezoelectric actuators to provide vibration control. Due to their high mechanical properties coupled with lightweight, FG-CNTRCs are mainly used in the aerospace industry and in advanced engineering applications. The CNTs have a linear and non-linear distribution along the thickness of the plate and are distributed according to five configurations, namely: UD, FG-X, FG-O, FG-A and FG-V. The first order shear deformation (FOSD) theory is considered in the formulation of a 9-node quadratic finite element with 5 degrees-of-freedom per node, and an additional degree of freedom is provided for the piezoelectric layer. The model developed in this study assesses the free vibration behavior and controls the nanocomposite plate deflection through the electromechanical coupling factor piezoelectric. In addition, it investigates: (i) the effect of the plate configuration, (ii) the CNT volume fraction, (iii) the CNT destruction patterns, (iv) the linear and nonlinear distribution of CNTs, (v) the number of CNTRC ply, (vi) the boundary conditions and (vii) the dimensions with different locations of actuators. The results obtained show the first natural frequencies for all configurations, which are considered to be in good agreement with those available in the literature and illustrate that the effective stiffness of the nanocomposite plates can be improved further when the reinforcement is dispersed according to the FG-X pattern. In addition, for the case of the deflection control analysis, results indicate that the distributed piezoelectric layers (actuators) attenuate the deflection of the CNTRC to the desired tolerance. It is noted that patches with partial coverage compared to the case of total coverage of piezoelectric layers require more electrical power to reach the same level of attenuation. The developed numerical model is intended to be used in a variety of potential advanced engineering applications.

Effect of the electric field on the sound transmission loss of double-walled electro-rheological fluid sandwich plates with functionally graded carbon nanotube reinforced composite facesheets

Numerous studies have been conducted to examine the vibroacoustic characteristics of lightweight double-walled structures due to the anticipated applications of these structures in noise reduction engineering. In this study, third-order shear deformation theory (TSDT) is used to develop a theoretical model that predicts sound transmission loss (STL) across double-walled electro-rheological fluid (ERF) sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) facesheets. The extended rule of mixture is utilized to evaluate the properties of the FG-CNTRC material in the thickness. Depending on the four FG models, the CNT volume fraction varies. A sufficient displacement continuity condition is taken into account between layers. Furthermore, it should be noted that changing the electric field affects the pre-yield zone’s ERF features. The vibroacoustic equations are derived using Hamilton’s principle and solved utilizing weighted residual (Galerkin) technique considering simply supported and clamped boundary conditions. Several studies are done to compare the results of the suggested model with other results found in the literature. An extensive numerical study is conducted to examine the dependence of STL on several parameters, including electric field strength, volume percentage of the CNTs, CNTs distribution, depth of acoustic cavity.

Postbuckled vibration behaviour of skew sandwich plates with metal foam core under arbitrary edge compressive loads using isogeometric approach

In the present work, nonuniform rational B-spline (NURBS) based isogeometric formulation in conjunction with refined higher-order theory is used to investigate the linear buckling, post-buckling, and post-buckled vibration behaviour of initially imperfect skew sandwich plates. The face sheets are functionally graded carbon nanotube-reinforced composite (FGCNTRC), and the core layer is made up of aluminium foam. The effects of three types of CNT distributions (uniform, FGX and FGO) in the face sheets, two types (uniform, symmetric) of porosity distribution functions for the core layer and five types of in-plane compressive loads are examined in the present investigation. The pre-buckling stresses are calculated using static analysis to evaluate accurate, critical loads. The post-buckling paths are traced using the modified Riks method. The accuracy of the present results is ascertained by comparing the results for critical loads and post-buckling paths with those available in the literature. Subsequently, the influence of CNT distribution functions, porosity functions, compressive loads, skew angle and the side-to-thickness ratio is studied on the nonlinear stability and free vibration behaviour of the post-buckled skew sandwich plates. The obtained results highlight that the buckling strength can be improved by increasing the skew angle, increasing the concentration of CNTs towards the surface of the face sheets and by using a metal foam core with nonuniform porosity distribution.

Nonlinear buckling analysis of higher-order shear deformable FG-CNTRC plates stiffened by oblique FG-CNTRC stiffeners

The nonlinear buckling analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates subjected to axial compression load is analytically examined in this paper. Assuming that the FG-CNTRC plates are stiffened by an oblique FG-CNTRC stiffener system. Reddy’s higher-order shear deformation plate theory (HSDPT) with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Moreover, the smeared stiffener technique is successfully improved for the higher-order shear deformable anisotropic oblique stiffener system by using a homogeneous model of the anisotropic beam. Galerkin’s method is used to achieve the expressions of critical buckling loads and postbuckling load-deflection curves in explicit form. The numerical values display the influences of FG-CNTRC stiffeners, material, and geometrical properties on the nonlinear buckling response of plates.

Hybrid control of laminated FG-CNTRC shell structures using an advanced smoothed finite element approach based on zig-zag theory

This study proposes an advanced cell-based smoothed discrete shear gap method (CS-DSG3) using zig-zag theory integrated with a hybrid control mechanism for analysis of smart damping control of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) shell structures. The two-layer smart constrained layer damping (SCLD) treatment patches are attached to the surface of shells in which the constraining layer is made of 1–3 piezoelectric composite material to enhance the shear deformation of the constrained viscoelastic layer. This study hence contributes two new developments including: (1) this is the first study that applies a hybrid control mechanism for smart damping control of the laminated FG-CNTRC shell structures; (2) this study incorporates successfully the CS-DSG3 with the zig-zag theory to give an effective global–local numerical approach which not only improves the accuracy of global numerical solutions and reduce the computational cost but also considers the important local behaviors of laminated FG-CNTRC shell structures. Various numerical examples are studied to investigate the attenuation of amplitude of vibrations on the laminated FG-CNTRC shell structures. In addition, the influence of CNTs distribution and nanotube volume fraction on the damping behavior of laminated FG-CNTRC shell structures are also investigated in detail.

Size Dependent Buckling Analysis of a FG-CNTRC Microplate of Variable Thickness under Non-Uniform Biaxial Compression

This paper combines third-order shear deformation theory (TSDT) and modified couple stress theory (MCST) with the principle of total potential energy to analyze the size-dependent buckling behavior of a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) rectangular microplate of variable thickness subject to non-uniform biaxial compression when resting on an elastic medium. To determine the thickness qualities of the material, the extended rule of mixture was applied. In the context of microplate buckling in the presence of small length scale effects, the three kinds of Carbon Nanotube (CNT) distribution—(a) UD, (b) FG-O, and (c) FG-X—were used and compared. The equations governing various combinations of simply supported or clamped boundary conditions have been solved using the differential quadrature method (DQM). The correctness and precision of the solutions have been compared to another study. A numerical study was conducted to examine the dependence of buckling load on several parameters, including percentage change of thickness, length scale parameter, nonuniform edge loads, boundary conditions, volume percentage of the CNTs, CNT distribution, and elastic medium parameter. The results of their effects are presented in this paper.

An energy method for buckling behavior analysis of functionally graded carbon nanotube-reinforced composite sandwich structures

The present work aims to investigate the buckling performance of sandwich structure of functionally graded carbon nanotube-reinforced composite (FG-CNTRC). Through first-order shear deformation theory, an analytical model for the sandwich structure of FG-CNTRC was established. The governing equation for the prediction of the buckling performance of the sandwich structure of FG-CNTRC was obtained through energy method. There was analytical solution that can satisfy both boundary conditions. The theoretical model and method were verified by literature analysis, and the influence of each parameter on the buckling performance was evaluated and performed on the basis of the corroborated model. The findings can lay a solid foundation of the design and application of the sandwich structure of FG-CNTRC.

Free vibration analysis of a nanoscale FG-CNTRCs sandwich beam with flexible core: Implementing an extended high order approach

The present work uses an extended high-order technique to examine free vibration analysis of Nanoscale CNTRCs sandwich beams. In this study, face sheets are made of PMMA reinforced with single-wall carbon nanotubes, and the honeycomb structure is selected for the flexible core. First-order shear deformation theory is utilized for the skins. The so-called high-order sandwich panel theory (HSAPT) is used for the core to consider the transverse shear strains in theoretical formulations. Then, skins and cores are subjected to the modified couple stress theory (MCST). For obtaining equations and related boundary conditions, the Hamilton principle is used. Additionally, three forms of beam support are examined: simply supported on both sides (SS), simply supported on one side and clamped supported on the other (SC), and clamped supported on both sides (CC). The material properties in skins are simulated to be graded in the thickness direction. The shooting method, as the numerical solution, is adopted to solve the governing differential equations to determine the natural frequencies of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams in which boundary conditions are converted into initial conditions. Compared to other techniques, this method is a reliable way to solve complicated equations and boundary conditions; therefore, its applicability, convergence, and accuracy are verified by comparing obtained results with those existing ones in the literature. Furthermore, the influences of distribution pattern and volume fraction of carbon nanotubes, boundary conditions, the size dependency parameter, core thickness, and slenderness ratio on vibration responses are discussed. It is concluded that natural frequencies rise with an increase in skin thicknesses and length scale parameter values while decreasing when beams’ length and core thicknesses increase.

STATIC INVESTIGATION OF A FUNCTIONALLY GRADED CARBON NANOTUBES REINFORCED COMPOSITE CYLINDRICAL SHELL, DOUBLE-ENDED CLAMPED SUBJECTED TO EXTERNAL PRESSURE LOADS

In this article, static analysis of clamped-clamped (C-C) functionally graded carbon nanotubes reinforced composite (FG-CNTRC) cylindrical shell subjected to external pressure was conducted. The governing equations were established by using higher-order shear deformation theory (HSDT) taking transverse normal stress effect into account. In this theory, the transverse displacement w is not a constant but rather is the second order polynomial of the coordinate along the thickness direction. Distribution of carbon nanotubes (CNT) across the shell thickness is assumed to be uniform (UD) or functionally graded in four types: FG-Ʌ, FG-V, FG-O, and FG-X. Effective material properties of FG-CNTRC cylindrical shells were estimated by the rule of mixture. An analytical solution using the simple trigonometric series and the Laplace transformation to solve governing equations of shell with clamped boundary condition at the both ends is presented. The validation of the applied approach was examined by comparing the results based on 3D exact model. The effects of the CNT distribution, the CNT volume fraction, and the geometrical parameters on the static behaviour of cylindrical shells subjected to external pressure were investigated. The result is remarkable that the stress components near the outer or inner surface vary most strongly, and in the case of a short or/and thick shell, the geometric parameter greatly affects the stress of shell.

An analytical solution for vibration response of CNT/GPL/fibre/polymer hybrid composite micro/nanoplates

In the present article, a closed-form solution is carried out for the vibration response of CNT/GPL/fiber/polymer hybrid composite macro/micro/nanoplates resting on elastic support. Multi-layered fiber/polymer hybrid plates with either functionally graded carbon nanotube reinforced composite (FG-CNTRC) material or functionally graded graphene platelets reinforced composite (FG-GPLRC) material are two different types of plate considered here. The distribution of reinforcing nanocomposites can be uniform or functionally graded through the plate layers. The governing vibration equations are developed in the framework of the nonlocal elasticity theory of Eringen and using the first-order shear deformation plate theory. In the analytical solution procedure, the partial differential equations of motion are simplified to a series of homogeneous ordinary differential equations using a displacement field for Levy-type of the boundary conditions. The general solution of the obtained equations is expressed in terms of exponential functions. Using that the displacements should satisfy the remaining boundary conditions, the vibrational frequencies are derived. Finally, the influences of different parameters including geometry and material properties such as nonlocal parameter, number and angle of layers, volume fraction, and type of distribution of reinforcing nanocomposites are investigated and discussed. Natural frequencies for various boundary conditions are illustrated. The numerical results illustrate that the effects of the nonlocal parameter on the frequency decrease when one of the boundaries is free, and these effects increase when one of the boundaries is clamped for both FG-CNTRC and FG-GPLRC plates.

Using Laurent’s series in the theoretical solution to estimate the stress resultants of FG-CNTRC plates weakened by a central cutout at different temperatures

Conventional ordinary materials have many weaknesses when exposed to extreme loadings. Hence, scientists develop novel engineered materials to address these weaknesses. Functionally Graded Carbon Nanotube Reinforced Composites (FG-CNTRCs) are a modern group of materials that have recently flourished thanks to their excellent and unique mechanical properties. One common loading condition is subjecting FG-CNTRC plates containing cutouts to in-plane loadings. The presence of an opening disturbs the stress field, especially in the proximity of the cutout, and creates stress concentration. This study estimates the stress and moment resultants on the edge of elliptical cutouts in asymmetric FG-CNTRC plates under various loading conditions using a new analysis based on Lekhnitskii’s complex variables method, mapping function, and Laurent series. Unlike previous studies on perforated asymmetric plates, which were based on numerical methods or the Schwartz formulations, this study presents a new solution using Laurent’s series to represent the holomorphic function. In calculating the stress and moment components around the opening, the effect of determining variables is studied. This approach can be generalized to solve different anisotropic body problems (FG_CNTRC, FGM, Laminate composites). Thus, stress and moment components in perforated anisotropic plates can be determined simply and systematically using this method.

Influence of cut-out shape on the dynamic behavior of carbon nanotubes reinforced functionally graded plates

This research investigates the effects of cut-out on the dynamic behaviour of functionally graded carbon nanotube reinforced composite (FGCNTRC) plates. Finite element method (FEM) is utilized and SHELL181 type element of the ANSYS Workbench 17.2 program is utilized. To understand the vibration characteristics of the FG plates, various shapes of cut-outs such as triangular, square, rectangular, hexagonal, circular etc are considered. For comparison, solid FGCNTRC plate is modeled and natural frequencies of plates integrated with cut-outs of different shapes are determined using free vibration study. The mode superposition method is used to address the harmonic response analysis problem keeping the damping ratio constant. The stress, phase changes and displacement responses for definite frequency ranges have been used to interpret the findings. To learn more about the effects of cut-outs on the dynamic response, the plates are designed considering various grading patterns. The findings show that the shape of cut-outs significantly affects the harmonic responses as well as the fundamental frequency of the FG plates and are also depending on the grading patterns of these plates.

Dynamic response of FG-CNTRC beams subjected to a moving mass

This article presents the forced vibration of composite beams reinforced by single-walled carbon nanotubes (SWCNTs) and subjected to a moving mass. Considering the distribution of carbon nanotubes such as uniform (UD-CNT), functionally graded Λ (FGΛ-CNT) and X (FGX-CNT), three different beams are studied. Based on a third-order shear deformation theory (TSDT), the motion equations of the beams are derived using Hamilton's principle. Including mass interaction forces, the motion equations are transformed into a finite element equation in which a two-node beam element with eight degrees of freedom is utilized. To improve the efficiency of the beam element, the transverse shear rotation is employed as an independent variable in the derivation of the beam element. The vibration characteristics, including the dynamic magnification factors and the time histories for mid-span deflections are computed by using the Newmark method. Numerical result reveal that the vibration of the beams is clearly influenced of the CNT reinforcement, and the dynamic magnification is significantly decreased by increasing the CNT volume fraction. It is also shown that the FGX-CNT beam is the best in dynamic resistance in terms of the lowest dynamic deflection and dynamic magnification factors. The effects of the total volume fraction and the moving load velocity on the dynamic behaviour of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) beams are examined in detail and highlighted.

An extensible double director 3D shell formulation for FGM-CNTRC shell bending analysis

In this paper, the continuum based three-dimensional shell element for static linear analysis of Functionally Graded Carbon Nanotubes Reinforced Composite (FG-CNTRC) shell structures is discussed. The present finite element formulation used the extensible double directors theory including the upgrade parameter of thickness stretching. Four new elements are developed, in the present formulation, revealing firstly the enhancement of the thickness strain with one or four parameters and then the enrichment of the membrane strains with four added parameters by means of the Enhanced Assumed Strain (EAS) method. To further illustrate the applicability and efficiency of the developed methodology, it is worth to consider its application to plate and shell structures for isotropic and FG-CNTRC materials. Results of deflections and stresses of the studied shell structures are examined and depicted for various volume fractions and configurations of CNTs, different geometrical parameters and upgrading load values in order to highlight the impact of these parameters on static behavior of FG-CNTRC shells.

Low-Velocity Impact Behavior of Sandwich Plates with FG-CNTRC Face Sheets and Negative Poisson’s Ratio Auxetic Honeycombs Core

The combination of auxetic honeycomb and CNT reinforcement composite is expected to further improve the impact protection performance of sandwich structures. This paper studies the low-velocity impact response of sandwich plates with functionally graded carbon nanotubes reinforced composite (FG-CNTRC) face sheets and negative Poisson’s ratio (NPR) auxetic honeycomb core. The material properties of FG-CNTRC were obtained by the rule of mixture theory. The auxetic honeycomb core is made of Ti-6Al-4V. The governing equations are derived based on the first-order shear deformation theory and Hamilton’s principle. The nonlinear Hertz contact law is used to calculate the impact parameters. The Ritz method with Newmark’s time integration schemes is used to solve the response of the sandwich plates. The (20/−20/20)s, (45/−45/45)s and (70/−70/70)s stacking sequences of FG-CNTRC are considered. The effects of the gradient forms of FG-CNTRC surfaces, volume fractions of CNTs, impact velocities, temperatures, ratio of plate length, width and thickness of surface layers on the value of the plate center displacement, the recovery time of deformation, contact force and contact time of low-velocity impact were analyzed in detail.

Investigation of Mechanical Behaviors of Functionally Graded CNT-Reinforced Composite Plates.

In this paper, the mechanical behavior of a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate is numerically investigated. According to the concept of a hierarchical model, the displacement is decomposed into the in-field functions and the assumed thickness-wise monomial. The former is defined on the plate midsurface and is approximated by the 2-D meshfree natural element method (NEM). The FG-CNTRC plate is modeled as a homogenized orthotropic body, and its effective elastic properties are determined by referring to MD simulation and the linear rule of mixtures. Regarding the thickness-wise distribution of CNTs, one uniform and three functionally gradient distributions are taken. Through comparative numerical experiments, the reliability of the numerical method is justified with the maximum relative difference of 6.12%. The effects of the volume fraction and vertical distribution of CNTs, the plate width-thickness and aspect ratios, and the boundary conditions on the bending, free vibration, and buckling behaviors of FG-CNTRC plates are examined. It is found that the mechanical behavior of FG-CNTRC plates is significantly dependent of these major parameters.

Postbuckling and postbuckled vibration behaviour of imperfect trapezoidal sandwich plates with FG-CNTRC face sheets under nonuniform loadings

The present work investigates the postbuckling, and postbuckled vibration behaviour of initially imperfect trapezoidal sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets and FG porous metal foam core under the influence of non-uniform edge compression. The plate's kinematic assumptions are based on a refined higher order theory and the strain-displacement relations include von Karman assumptions for geometrical nonlinearity. The weak form of governing equations derived using Hamilton's principle is transformed into a discretized form of algebraic equations using the element free Galerkin (EFG) method in conjunction with moving kriging (MK) interpolation functions. The pre-buckling stresses are determined using static analysis to evaluate accurate critical buckling loads. Modified Riks technique is used to trace nonlinear equilibrium paths. Parametric studies include the effect of CNT distribution in face sheets, porosity distribution in the core layer and edge loading conditions on the nonlinear stability and vibration behaviour of sandwich plates. New results on trapezoidal sandwich plates with initial imperfections, hitherto not found in the literature, are presented for the first time, which can be used as benchmark solutions for further research.

Maximization of the fundamental frequency of the FG-CNTRC quadrilateral plates using a new hybrid PSOG algorithm

This research work presents the fundamental frequency optimization of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) quadrilateral plates using an improved hybrid algorithm of the particle swarm optimization and genetic (PSOG) algorithm. The objective function is to maximize the fundamental frequency of the FG-CNTRC quadrilateral plates and the design variables are a set of the discrete values of the fiber orientations in the layers. The natural frequency of the plates is obtained by using the cell-based smoothed discrete shear gap method. The several comparative numerical examples are investigated to show the efficiency of the proposed hybrid PSOG algorithm. It is demonstrated in the numerical examples that this design method can provide with the accurate optimum solutions for the stacking sequences of the vibrating FG-CNTRC quadrilateral plates.

Nonlinear bending analysis of FG-CNTRC plate resting on elastic foundation by natural element method

A nonlinear numerical analysis is addressed for a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate on elastic foundation. The large deflection bending problem is formulated according to the von Kármán nonlinear theory and the (1,1,0)* hierarchical model. And, it is numerically solved by 2-D (two-dimensional) natural element method (NEM) which is characterized by a high accuracy even for coarse grid. The derived nonlinear matrix equations are iteratively calculated by combining the load increment scheme and the Newton-Raphson iteration method. The present method was verified through the benchmark test, where the present method shows excellent agreement with the reference solutions. The nonlinear behavior of FG-CNTRC plate on elastic foundation is evaluated in terms of the non-dimensional deflection and bending moment. These non-dimensional quantities are examined with respect to the foundation stiffness, the volume fraction and gradient pattern of CNT, the aspect and width-thickness ratios of plate and the boundary condition. From the parametric experiments, it is found that both non-dimensional quantities are significantly influenced by such parameters.