Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Kármán geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part I: Axially-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to axial compression in thermal environments. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially-loaded, perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling load as well as postbuckling strength of the shell under axial compression. The results reveal that the CNT volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells.     

Vibration damping characteristics of carbon nanotubes-based thin hybrid composite spherical shell structures

The present work deals with the evaluation of elastic properties and dynamic analyses of thin hybrid composite shell structures, which consist of conventional carbon fiber as the reinforcing phase and multiwalled carbon nanotubes-based polymer as the matrix phase. The Mori-Tanaka and strength of material method has been implemented to determine the elastic properties of such hybrid composite structures without and with considering agglomerations. An eight-noded shell element, which considers stress resultant-type Koiter's shell theory and transverse shear effect as per Mindlin's hypothesis having five degrees of freedom at each node, has been utilized for discretizing and analysis of such hybrid shell structures. The Rayleigh damping model has been implemented in order to study the effect of carbon nanotubes (CNTs) on damping capacity of such hybrid composite shell structures. Different types of spherical shell panels have been analyzed in order to study the time and frequency responses. Results show that the elastic properties as well as damping properties of such hybrid composite structures improved with the addition of CNTs as compared to conventional carbon fiber reinforced composites laminates; effects of some important parameters on the vibration characteristic of such hybrid composite shell structures have also been presented. The effects of agglomeration parameters on the elastic properties and their influences on the dynamic responses considering different layers and stacking sequences have also been investigated.     

Nonlinear vibration of nanotube-reinforced composite plates in thermal environments

This paper deals with the large amplitude vibration of nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation in thermal environments. The SWCNTs are assumed aligned, straight and a uniform layout. Two kinds of carbon nanotube-reinforced composite (CNTRC) plates, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC plates are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The motion equations are based on a higher-order shear deformation plate theory that includes plate-foundation interaction. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies of CNTRC plates. Numerical results reveal that the natural frequencies as well as the nonlinear to linear frequency ratios are increased by increasing the CNT volume fraction. The results also show that the natural frequencies are reduced but the nonlinear to linear frequency ratios are increased by increasing the temperature rise or by decreasing the foundation stiffness. The results confirm that a functionally graded reinforcement has a significant effect on the nonlinear vibration characteristics of CNTRC plates.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part II: Pressure-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to lateral or hydrostatic pressure in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under external pressure is proposed and a singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. Numerical results for pressure-loaded, perfect and imperfect, FG-CNTRC cylindrical shells are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling pressure as well as postbuckling strength of the shell under external pressure. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling pressure and postbuckling behavior of CNTRC shells.     

Vibration damping characteristics of carbon fiber-reinforced composites containing multi-walled carbon nanotubes

Vibration damping characteristic of nanocomposites and carbon fiber reinforced polymer composites (CFRPs) containing multiwall carbon nanotubes (CNTs) have been studied using the free and forced vibration tests. Several vibration parameters are varied to characterize the damping behavior in different amplitudes, natural frequencies and vibration modes. The damping ratio of the hybrid composites is enhanced with the addition of CNTs, which is attributed to sliding at the CNT-matrix interfaces. The damping ratio is dependent on the amplitude as a result of the random orientation of CNTs in the epoxy matrix. The natural frequency shows negligible influence on the damping properties. The forced vibration test indicates that the damping ratios of the CFRP composites increase with increasing CNT content in both the 1st and 2nd vibration modes. The CNT-epoxy nanocomposites also show similar increasing trends of damping ratio with CNT content, indicating the enhanced damping property of CFRPs arising mainly from the improved damping property of the modified matrix. The dynamic mechanical analysis further confirms that the CNTs have a strong influence on the composites damping properties. Both the dynamic loss modulus and loss factor of the nanocomposites and the corresponding CFRPs show consistent increases with the addition of CNTs, an indication of enhanced damping performance.     

Buckling and Postbuckling Behavior of Functionally Graded Nanotube-Reinforced Composite Plates in Thermal Environments

This paper investigates the buckling and postbuckling of simply supported, nanocomposite plates with functionally graded nanotube reinforcements subjected to uniaxial compression in thermal environments. The nanocomposite plates are assumed to be functionally graded in the thickness direction using single-walled carbon nanotubes (SWCNTs) serving as reinforcements and the plates' effective material properties are estimated through a micromechanical model. The higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity is used to model the composite plates and a two-step perturbation technique is performed to determine the buckling loads and postbuckling equilibrium paths. Numerical results for perfect and imperfect, geometrically mid-plane symmetric functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC plate, which is a special case in the present study, are compared with those of the FG-CNTRC plate. The results show that the buckling loads as well as postbuckling strength of the plate can be significantly increased as a result of a functionally graded nanotube reinforcement. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC plates.     

Vibration Damping of Carbon Nanotube Assembly Materials

Vibration reduction is of great importance in various engineering applications, and a material that exhibits good vibration damping along with high strength and modulus has become more and more vital. Owing to the superior mechanical property of carbon nanotube (CNT), new types of vibration damping material can be developed. This paper presents recent advancements, including our progresses, in the development of high‐damping macroscopic CNT assembly materials, such as forests, gels, films, and fibers. In these assemblies, structural deformation of CNTs, zipping and unzipping at CNT connection nodes, strengthening and welding of the nodes, and sliding between CNTs or CNT bundles are playing important roles in determining the viscoelasticity, and elasticity as well. Toward the damping enhancement, strategies for micro‐structure and interface design are also discussed.

     

Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates

Thermal buckling and postbuckling behavior is presented for functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) subjected to in-plane temperature variation. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, geometrically mid-plane symmetric FG-CNTRC plates and uniformly distributed CNTRC plates under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the plate can be increased as a result of a functionally graded reinforcement. It is found that in some cases the CNTRC plate with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.     

Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes

In this paper, the mechanical buckling of a functionally graded nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs) subjected to uniaxial and biaxial in-plane loadings is investigated. The material properties of the nanocomposite plate are assumed to be graded in the thickness direction and vary continuously and smoothly according to two types of the symmetric carbon nanotubes volume fraction profiles. The material properties of SWCNT are determined according to molecular dynamics (MDs), and then the effective material properties at a point are estimated by either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. The equilibrium and stability equations are derived using the Mindlin plate theory considering the first-order shear deformation (FSDT) effect and variational approach. The results for nanocomposite plate with uniformly distributed CNTs, which is a special case in the present study, are compared with those of the symmetric profiles of the CNTs volume fraction. A numerical study is performed to investigate the influences of the different types of compressive in-plane loadings, CNTs volume fractions, various types of CNTs volume fraction profiles, geometrical parameters and different types of estimation of effective material properties on the critical mechanical buckling load of functionally graded nanocomposite plates.     

Vibration of functionally graded carbon nanotube reinforced quadrilateral plates using geometric transformation discrete singular convolution method

Free vibration characteristics of thick skew plates reinforced by functionally graded carbon nanotubes (CNTs) reinforced composite are presented. Discrete singular convolution (DSC) method is used for the numerical solution of vibration problems via geometric mapping technique. Using the geometric transformation via a four-node element, the straight-sided quadrilateral physical domain is mapped into a square domain in the computational space. Then the method of discrete singular convolution with some singular kernels such as Regularized Shannon's delta (RSD) and Lagrange's delta kernels (LDK) have been used for spatial discretizing of the resulting governing equation of motion. Calculated results have been presented in order to show the effects of volume fraction of CNT, skew angles, CNT distribution types, plate aspect ratio and length-to-thickness ratio on the frequency of CNT reinforced skew plate. The current results are compared with the related results available in the literature and obtained by different methods. It is shown that reasonable accurate results are obtained for free vibration of nanocomposite plates with less computational effort for higher modes. Several test examples for different plate have been selected to demonstrate the convergence properties, accuracy, and simplicity in numerical implementation of DSC procedures. This approach has verified the accuracy and applicability of DSC method to the class of problem considered in this study. Furthermore, in the numerical examples in the scope of the study, the results obtained with DSC method using a coarser grid are more accurate than the values obtained by finite elements and differential quadrature (DQ) methods. It is also revealed that the method of discrete singular convolution is a promising and potential approach for computational mechanics of nonrectangular plates with nanocomposite reinforced.     

Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels

In this paper, natural frequencies characteristics of a continuously graded carbon nanotube-reinforced (CGCNTR) cylindrical panels based on the Eshelby–Mori–Tanaka approach is considered. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The CGCNTR shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The novelty of the present work is to exploit Eshelby–Mori–Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various mid radius-to-thickness ratio, shell angle, length-to-mean radius ratio and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR cylindrical panels. The interesting and new results show that continuously graded oriented CNTs volume fractions can be utilized for the management of vibrational behavior of structures so that the frequency parameters of structures made of such material can be considerably improved than that of the nanocomposites reinforced with uniformly distributed CNTs.     

3D free vibration analysis of elastically supported thick nanocomposite curved panels with finite length and different boundary conditions via 2D GDQ method

Based on the three-dimensional theory of elasticity, free vibration characteristics of nanocomposite panels reinforced by randomly oriented, straight, single-walled carbon nanotubes (SWCNTs) are considered. The volume fractions of randomly oriented agglomerated SWCNTs are assumed to be graded not only in the radial direction but also in the axial direction of the curved panel. This study presents a 2D six-parameter power-law distribution for CNTs volume fraction of 2D continuously graded nanocomposite that gives designers a powerful tool for flexible designing of structures under multi-functional requirements. The material properties are determined in terms of local volume fractions and material properties by the Mori-Tanaka scheme.     

Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to combined axial and radial mechanical loads in thermal environment. Two types of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation shell theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A boundary layer theory and associated singular perturbation technique are employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, FG-CNTRC cylindrical shells under combined action of external pressure and axial compression for different values of load-proportional parameters. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell.     

压电集成碳纳米管CNT增强功能梯度结构非线性建模与仿真

将碳纳米管(carbon nanotube,CNT)以梯度形式分布与基体材料结合,形成功能梯度(functionally graded,FG)结构。为了实现FG-CNT增强复合板在发生大变形时的准确计算,考虑四种典型的CNT分布形式,均匀分布、V型分布、O型分布和X型分布,建立基于Reissner-Mindlin板壳假设的大变形几何非线性有限元模型。该模型不仅包含了几何全非线性应变位移关系,还考虑了薄板结构法向发生大转角的情形。该研究与文献结果进行比较,验证模型的准确性;对FG-CNT增强复合板进行几何大变形非线性计算,分析CNT分布形式、CNT增强角度等因素对FG-CNT增强复合板刚度的影响。研究结果表明,CNT分布形式及增强角度对FG-CNT复合板的力学特征有显著的影响。     

Nonlinear vibration and instability analysis of functionally graded CNT-reinforced cylindrical shells conveying viscous fluid resting on orthotropic Pasternak medium

In this study, nonlinear vibration and instability of embedded temperature-dependent cylindrical shell conveying viscous fluid resting on temperature-dependent orthotropic Pasternak medium are investigated. The equivalent material properties of nanocomposites are estimated using rule of mixture. Both cases of uniform distribution and functionally graded distribution patterns of reinforcements are considered. Based on orthotropic Mindlin shell theory, the governing equations are derived. Generalized differential quadrature method is applied for obtaining the frequency and critical fluid velocity of a system. The effects of different parameters, such as distribution type of single-walled carbon nanotubes (SWCNTs), volume fractions of SWCNTs, and Pasternak medium are discussed.     

Static response and free vibration of two-dimensional functionally graded metal/ceramic open cylindrical shells under various boundary conditions

The free vibration and static response of a two-dimensional functionally graded (2-D FGM) metal/ceramic open cylindrical shell are analyzed using 2-D generalized differential quadrature method. The open cylindrical shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. This paper presents a novel 2-D power-law distribution for ceramic volume fraction of 2-D FGM that gives designers a powerful tool for flexible designing of structures under multifunctional requirements. Various material profiles in two radial and axial directions are illustrated using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori–Tanaka scheme. The 2-D generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated, and to validate the results, comparisons are made with the available solutions for FGM cylindrical shells. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the mechanical stresses and natural frequency than conventional 1-D FGM. The achieved results confirm that natural frequency and mechanical stress distribution can be modified to a required manner by selecting an appropriate volume fraction profile in two directions.     

热环境下纤维增强树脂复合薄壁圆柱壳的强迫振动响应

研究了热环境下纤维增强树脂复合薄壁圆柱壳强迫振动响应的理论求解方法。考虑了基础激励载荷的影响,并依托板壳理论、复弹性模量等方法,在确立了受热后材料与结构的本构关系、物理方程及能量方程基础上,建立了该类型纤维增强树脂复合薄壁圆柱壳在热环境下的理论模型。利用双向梁函数法表示振型函数,在引入比例阻尼的基础上,通过Ritz法和振型叠加法成功求解获得了频域振动响应。以TC300碳纤维/环氧树脂复合薄壁圆柱壳为研究对象,测试获得前3阶振动响应曲线。研究发现,相对于测试获得的前3阶共振响应结果,理论计算获得的共振响应误差最大不超过14.8%,验证了所提出的分析方法的正确性和有效性。      

热环境中功能梯度圆柱壳声辐射特性研究

基于经典壳体理论,推导了热环境下,流场中功能梯度圆柱壳的受迫振动方程,研究了不同温度分布方式、壳体材料参数以及周围流场参数对功能梯度圆柱壳声辐射特性的影响。研究结果表明:温度变化会引起功能梯度圆柱壳材料特性的改变,在壳体内产生热应力,影响流场中壳体的声辐射特性。温度变化对不同频段壳体辐射声功率的影响不同。重流体中,温度变化对声功率的影响随着频率的增高而增大;轻流体中,温度升高使频率较低时的声功率峰值向低频方向移动,且峰值减小,当接近环频率时,随着温度升高,辐射声功率下降。均匀升温对壳体声辐射特性的影响高于非均匀升温。非均匀升温时,重流体中体积分数指数对辐射声功率的影响比轻流体中更明显,且主要体现在对峰值位置的影响。     

The effects of carbon nanotube orientation and aggregation on vibrational behavior of functionally graded nanocomposite cylinders by a mesh-free method

In this paper, axisymmetric natural frequencies of nanocomposite cylinders reinforced by straight single-walled carbon nanotubes are presented based on a mesh-free method. The straight carbon nanotubes (CNTs) are oriented, aligned or randomly or locally aggregated into some clusters. Volume fractions of the CNTs and clusters are assumed to be functionally graded along the thickness, so material properties of the carbon nanotube reinforced composite cylinders are variable and are estimated based on the Eshelby–Mori–Tanaka approach. In the mesh-free analysis, moving least squares shape functions are used for an approximation of the displacement field in the weak form of motion equation, and the transformation method is used for the imposition of essential boundary conditions. The effects of orientation and aggregation of the functionally graded CNT are studied. It is observed that kind of distributions, aggregation or even randomly orientations of CNTs has significant effect on the effective stiffness and frequency parameter.