Carbon Nanotube-reinforced Composite Beams Research Articles

Static, buckling, and free vibration responses of functionally graded carbon nanotube-reinforced composite beams with elastic foundation in non-polynomial framework

In this work, the static, buckling, and free vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam resting on a Pasternak elastic foundation are studied. The secant function-based shear deformation theory (SFSDT) is used for this analysis. This theory fulfills the traction-free boundary conditions at the top and bottom surfaces of the beam, hence there is no need for a shear correction factor. Hamilton’s principle is used to determine the governing differential equations and boundary conditions whereas Navier’s solution technique is used for determining the closed-form solution. The analytical approach is used to examine the deflection, stresses, critical buckling load, and natural frequencies of the FG-CNTRC beam resting on the Pasternak elastic foundation including a shear layer and Winkler springs. To determine the material characteristics of FG-CNTRC beams, the Rule of the mixture is used. Uniform distribution (UD-beam), FG-X beam, FG-O beam, and FG-V beam are the different forms of CNT reinforcement distribution that are used in this study. Considering different span thickness ratios, the volume fraction and distribution of CNT, the Winkler spring, and the shear layer constant factors, all the structural responses are predicted. It is also observed that the present theory predicts the structural responses of the FG-CNTRC beam accurately when compared to other existing theories. A few new results are also included as the benchmark solutions for the new research.

1/2 and 1/3 Subharmonic Resonance Behaviors of a Rotating FG-CNTRC Beam with Geometric Imperfections

This paper aims at investigating 1/2 and 1/3 subharmonic resonances of a rotating functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam in the presence of geometric imperfections. A general function is introduced to denote three types of imperfections containing sine, global, and local modes. At first, based on the von-Kármán geometric nonlinearity assumption, the forced vibration equation is set up. Then, the Galerkin method and multiple scale method are utilized to study subharmonic resonance behaviors. Finally, coupled effects of FG-CNTRCs, rotating motion, and geometric imperfections on subharmonic resonance responses are investigated. It is found that a specific range of the excitation amplitude and frequency motivates the subharmonic resonance behavior. The coupling of geometric imperfections and geometric nonlinearities generates the 1/2 resonance phenomena, and geometric nonlinearities result in the 1/3 resonance behavior. Both the imperfection mode and amplitude can affect the resonance response and resonance regions. Moreover, in contrast with the 1/3 subharmonic resonance, geometric imperfections produce greater influence on the 1/2 subharmonic resonance.

Nonlinear dynamical characteristics of carbon nanotube-reinforced composite beams with piezoelectric actuators and elastically restrained ends under thermo-electro-mechanical loads

Nonlinear free vibration and dynamical responses of carbon nanotube (CNT) reinforced composite beams with surface-bonded piezoelectric layers and tangentially restrained ends under thermo-electro-mechanical loads are investigated in this paper. The properties of constitutive materials are assumed to be temperature-dependent and effective properties of nanocomposite are estimated using an extended rule of mixture. Unlike previous studies, the present work considers the effects of tangentially elastic constraints of two ends on the nonlinear dynamic characteristics of hybrid beams. Motion equation is established within the framework of Euler-Bernoulli beam theory taking into account von Kármán nonlinearity. Analytical solution is assumed to satisfy simply supported boundary conditions and Galerkin procedure is employed to obtain a time ordinary differential equation including both quadratic and cubic nonlinear terms. This differential equation is numerically solved employing fourth-order Runge-Kutta scheme to determine the frequencies of nonlinear free vibration and nonlinear transient response. Parametric studies are executed to examine numerous influences on the nonlinear dynamical characteristics of hybrid nanocomposite beams. The study reveals that tangential constraints of ends substantially effect the frequencies and dynamic response of the beam, especially at elevated temperatures. The results also indicate that nonlinear dynamic responses can be controlled effectively by means of piezoelectric actuators and elasticity of tangential constraints of ends should be considered in design of piezo-CNTRC beams.

Free vibration of functionally graded carbon nanotube-reinforced composite beam

The purpose of this paper is to investigate the free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam. The carbon nanotubes (CNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the FG-CNTRC beam are estimated by using the rule of mixture. The trigonometric shear deformation beam theory is employed to deal with the problem. The mathematical models provided in this paper are numerically validated by comparison with some available results. New results of free vibration analyses of FG-CNTRC beam are presented and discussed in detail. The effects of the length to thickness ratio and CNT volume fraction are taken into investigation.

Thermo-electrical postbuckling behavior of carbon nanotubes-reinforced composite beams with piezoelectric layers and tangentially restrained ends

An analytical investigation on the buckling and postbuckling behavior of carbon nanotube-reinforced composite beams integrated with surface-bonded piezoelectric layers under uniform temperature rise is presented in this paper. Carbon nanotubes (CNTs) are reinforced into isotropic matrix through uniform distribution and functionally graded distributions. The properties of material constituents are assumed to be temperature-dependent and effective properties of CNT-reinforced composite are estimated using an extended rule of mixture. Equilibrium equations of the beams are established based on Euler-Bernoulli theory including von Kármán nonlinearity and solved using analytical solutions and Galerkin method. Critical temperatures and postbuckling load-deflection paths are determined using an iteration algorithm. Parametric studies are performed to examine the influences of CNT distribution and volume fraction, applied voltage, in-plane and out-of-plane conditions of the ends, slenderness, and thickness ratio of layers on the critical loads and postbuckling load carrying capacity of beams. Results reveal that CNT volume fraction and degree of in-plane ends constraint have slight and significant influences on the critical temperatures and thermal postbuckling paths, respectively. The study also finds that negative and positive voltages increase and decrease the thermal buckling temperatures of piezoelectric CNT-reinforced composite beams.

On Vibration Responses of Advanced Functionally Graded Carbon Nanotubes Reinforced Composite Nanobeams

This article presents an analytical approach to explore the free vibration behaviour of new functionally graded carbon nanotube-reinforced composite beams (FG-CNTRC) based on a two-variable higher-order shear deformation theory and nonlocal strain gradient theory. The beams resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The kinematic relations of the shaft are proposed according to novel trigonometric functions. The vibrated nanobeam’s motion equations are obtained via the classical Hamilton’s principle and solved using Navier’s steps. A comparative evaluation of results against predictions from literature demonstrates the accuracy of the proposed analytical model. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanobeams to nonlocal length scale, strain gradient microstructure scale, material distribution, constant spring factors, and geometry. The current work presents the free vibration problem of supported (FG-CNTRC) beams reinforced by different patterns of carbon nanotube (CNT) distributions in the polymeric matrix.

Free Vibration Analysis of Functionally Graded Carbon Nanotube-Reinforced Higher Order Refined Composite Beams Using Differential Quadrature Finite Element Method

Present paper deals on the free vibration investigation of carbon nanotube-reinforced composite (CNTs) beams, based on refined third order shear deformation finite element beam theory. The particularity of this model is that, it can capture shear deformation effect without using of any shear correction factor by satisfying shear stress free at free edges. The carbon nanotubes are supposed to be immersed in a polymeric matrix with functionally graded pattern across the thickness direction of the beam, and their material properties are evaluated using the rule of mixture. The differential equations of motion and related boundary conditions are extracted using Lagrange’s principle and solved employing a robust numerical tool called, Differential Quadrature Finite Element Method (DQFEM) for the first time, with high convergence speed, fast calculus performance as well as a good numerical stability. The obtained results have been validated with those available in literature, in order to show the correctness of the present model. Afterwards, a deep parametric study is performed to examine the effects of various geometrical and material parameters on the vibration behavior of FG-CNTs beams.

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.

Influence of the boundary relaxation on free vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections

Influence of the boundary relaxation on free vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections

Static bending of functionally graded single-walled carbon nanotube conjunction with modified couple stress theory

This paper focused on studying the transeverse deflection behavior of three functionally graded microbeam models: material, porous material, and functionally graded single-walled carbon nanotube-reinforced composite (FG-SWCNTRC) microb beam. To calculate micro size effects of the non-classical beam model, the modified couple stress theory (MCST) uses just length material scale parameters. A numerical solution to the static deflection equation is the Lagrange multiplier method. The characteristics studied included length, material parameter ratio, volume fraction of material, porosity and carbon nanotube, SWCNT distribution types, boundary conditions, and aspect ratio (length/thickness). The static behavior of FG-micro beams demonstrates that the theory of modified couple stress (MCST) produces more accurate results than classical beams. This is especially true if the beam thickness is close to the length scale parameter. As the CNT volume fraction grows and the porosity volume fraction falls, the microbeam FG structure deflects less.

Free vibration and modal stress analysis of FG-CNTRC beams under hygrothermal conditions using zigzag theory

The present study aims to carry out free vibration analysis of functionally graded single-walled carbon nanotube-reinforced composite (FG-CNTRC) beams under hygro-thermal conditions. The study is carried out using the recently proposed C-0 finite element-based higher-order zigzag theory. The study is carried out on five different CNTRC graded beams. Temperature and moisture-dependent material properties are used. Constant temperature or moisture distribution across the thickness of the beam is taken. Hamilton's principle has been used for defining the governing differential equations. Modal stresses are also studied for the first six mode shapes. Stresses at the higher mode of vibration are found to be more affected by temperature or moisture concentration as compared to the stresses observed for the fundamental mode of vibration. It is observed that the nature of stress distribution across the beam is widely affected by the gradation law along with moisture or temperature values to which the beam is subjected. FG-O beam is found to be least sensitive under thermal conditions whereas FG-X beam is found to be the most.

Study on the detailed and homogenized models for functionally graded carbon nanotube-reinforced composite beams

Study on the detailed and homogenized models for functionally graded carbon nanotube-reinforced composite beams

Bifurcation and Chaos of Functionally Graded Carbon Nanotube Reinforced Composite Beam with Piezoelectric Layer

Bifurcation and Chaos of Functionally Graded Carbon Nanotube Reinforced Composite Beam with Piezoelectric Layer

Forced Vibration Analysis of Composite Beams Reinforced by Carbon Nanotubes.

This paper presents forced vibration analysis of a simply supported beam made of carbon nanotube-reinforced composite material subjected to a harmonic point load at the midpoint of beam. The composite beam is made of a polymeric matrix and reinforced the single-walled carbon nanotubes with their various distributions. In the beam kinematics, the first-order shear deformation beam theory was used. The governing equations of problem were derived by using the Lagrange procedure. In the solution of the problem, the Ritz method was used, and algebraic polynomials were employed with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method was applied in the time history. In the numerical examples, the effects of carbon nanotube volume fraction, aspect ratio, and dynamic parameters on the forced vibration response of carbon nanotube-reinforced composite beams are investigated. In addition, some comparison studies were performed, with special results of published papers to validate the using formulations.

Finite element model for carbon nanotube-reinforced and graphene nanoplatelet-reinforced composite beams

This paper investigates bending, vibration and buckling analysis of carbon nanotube-reinforced composite (CNTRC) and graphene nanoplatelet-reinforced composite (GPLRC) beams. Both normal and shear effects are included in the formulation of the governing equations of motion of the beams. Finite element model is employed to determine displacements, critical buckling loads and natural frequencies of the beams with different boundary conditions. Several important factors in parametric studies including distribution pattern, volume fraction of CNTs, GPL weight fraction, slenderness ratio, boundary condition, etc. are investigated. Some new results are given as benchmark for further validation.

Theoretical solutions for auxetic laminated beam subjected to a sudden load

In the current work, the concept of auxetic structures design is introduced into carbon nanotube-reinforced composite (CNTRC) laminate. Larger values of negative Poisson’s ratio (NPR) of CNTRC laminate are obtained by setting the optimal orientation. Meanwhile, a theoretical model is presented for the nonlinear dynamic response of functionally graded (FG) CNTRC beam with NPR.Using Reddy’s higher order shear deformation plate theory with von Karman's strain–displacement relations, the motion equations are obtained A two-perturbation approach and the fourth-order Runge Kutta method are then used to obtain the time-deflection curve of the auxetic FG-CNTRC beam. The effects of factors such as the CNT distribution, thermal field and foundation stiffness on the dynamic characteristic of the auxetic FG-CNTRC beam are studied in details. Numerical results indicate that the thermal–mechanical behavior of FG-CNTRC beams might be significantly improved by determining a proper distribution of CNTs. Furthermore, the results show that the change of temperature and foundation stiffness have a significant effect on the dynamic characteristic of FG-CNTRC laminated beams with NPR.

On size-dependent nonlinear free vibration of carbon nanotube-reinforced beams based on the nonlocal elasticity theory: Perturbation technique

Based on the first-order shear deformation (FSD) model and nonlocal elasticity theory, the simultaneous effects of shear and small scale on the nonlinear vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams are investigated for the first time. To this end, the governing equations of bending and stretching with von Kármán geometric nonlinearity are decoupled into one fourth-order partial differential equation in terms of transverse deflection. A closed-form solution of the nonlinear natural frequency, which can be used in conceptual design and optimization algorithms of FG- CNTRC beams with different boundary conditions, is developed using a hybrid method of Galerkin and perturbation technique. First, the decoupled equation is reduced to a nonlinear ordinary one with respect to time by implementing the Galerkin method. Next, multiple scales perturbation technique is used to replace this nonlinear ordinary differential equation with a series of linear equations which can be solved analytically. Finally, numerical results are presented to compare with the existing ones in the literature as well as to study the effects of CNTs distribution, boundary conditions and nonlocal parameter on the frequency ratio. It is seen that the influence of CNTs distribution becomes more significant when the nonlocal parameter increases. Also, the difference between results obtained from the classical and nonlocal elasticity theories increases as the amplitude of vibration increases. Therefore, it is concluded that the classical elasticity theory is inadequate to predict the nonlinear vibration behavior of FG-CNTRC beams with large deformation.

An analytical study on the nonlinear forced vibration of functionally graded carbon nanotube-reinforced composite beams on nonlinear viscoelastic foundation

This paper deals with the nonlinear forced vibration of nanocomposite beams resting on a nonlinear viscoelastic foundation and subjected to a transverse periodic excitation. It is considered that the functionally graded carbon nanotubereinforced composite (FG-CNTRC) beam is made of an isotropic matrix reinforced by either aligned- or randomly oriented-straight single-walled carbon nanotubes (SWCNTs) with four types of distributions through the thickness direction of the beam. Both the Eshelby–Mori–Tanaka approach and extended rule of mixtures are used to predict the effective material properties of the FG-CNTRC beams. The mathematical model of the beam is developed based on the Euler–Bernoulli beam theory together with von Karman assumptions. Subsequently, the accurate analytical solutions of the governing equation are obtained through applying the variational iteration method (VIM). Several examples are verified to have higher accuracy than those available in the literature. In addition, a comprehensive investigation into the effect of carbon nanotubes (CNTs) distribution, CNTs volume fraction, end supports, vibration amplitude, and foundation coefficients on the vibrational characteristics of the nanocomposite beam is performed and some new results are presented.

FREE VIBRATION OF TAPERED FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE BEAMS USING A HIERARCHICAL BEAM ELEMENT

Free vibration of tapered functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams is investigated. The beams with four types of carbon nanotube distribution in the thickness, namely the uniform (UD-CNT), X-type (FGX-CNT), A-type (FGA-CNT) and O-type (FGO-CNT), are assumed to be linearly tapered in longitudinal direction by three different taper cases. Based on the first-order shear deformation theory, equations of motion with variable coefficients are derived from Hamilton’s principle. Using hierarchical functions to interpolate the displacement field, a two-node beam element with nine degrees of freedom is formulated and employed to compute frequencies of the beams. The accuracy of the derived formulation is confirmed by comparing frequencies obtained in the present work with the published data. The effects of the total CNT volume fraction, CNT distribution type, taper cases, taper ratio, aspect ratio, boundary conditions, etc., on the vibration characteristics of the beams are examined and discussed.

FREE VIBRATION OF TAPERED FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE BEAMS USING A HIERARCHICAL BEAM ELEMENT

Free vibration of tapered functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams is investigated. The beams with four types of carbon nanotube distribution in the thickness, namely the uniform (UD-CNT), X-type (FGX-CNT), A-type (FGA-CNT) and O-type (FGO-CNT), are assumed to be linearly tapered in longitudinal direction by three different taper cases. Based on the first-order shear deformation theory, equations of motion with variable coefficients are derived from Hamilton’s principle. Using hierarchical functions to interpolate the displacement field, a two-node beam element with nine degrees of freedom is formulated and employed to compute frequencies of the beams. The accuracy of the derived formulation is confirmed by comparing frequencies obtained in the present work with the published data. The effects of the total CNT volume fraction, CNT distribution type, taper cases, taper ratio, aspect ratio, boundary conditions, etc., on the vibration characteristics of the beams are examined and discussed.