Numerical study of the three-dimensional wave equation using the mesh-free kp-Ritz method

This paper deals with numerical modeling of three-dimensional linear wave propagation based on the mesh-free kp-Ritz method. The mesh-free kernel particle estimate is employed to approximate the 3D displacement field. A system of discrete equations is obtained through application of the Ritz minimization procedure to the energy expressions. Convergence analysis and error estimates of the kp-Ritz method for three-dimensional wave equation are also presented in the paper. From the error analysis, we found that the error bound between the numerical and the exact solution is directly related to the radii of weight functions and the time step length. Effectiveness of the kp-Ritz method for three-dimensional wave equation is investigated by three numerical examples.     

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.     

Geometrically nonlinear analysis of cylindrical shells using the element-free kp-Ritz method

In this paper, the geometrically nonlinear analysis of cylindrical shells is carried out using the element-free kp-Ritz method. The first-order shear deformation shell theory, which can cater for both thin and relatively thick shells, is utilized in the present study. Meshfree kernel particle functions are employed to approximate the two-dimensional displacement field. The nonlinear equilibrium equations are formulated by applying the Ritz procedure to the energy functional of shells. The Newton–Raphson method and the arc length technique are used to determine the load–displacement path. To validate the accuracy and stability of this method, convergence studies based on the support size and number of nodes were performed. Comparisons were also made with the existing results available in the open literature, and good agreement is obtained.     

Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method

In this paper, dynamic analysis of nanocomposite cylinders reinforced by single-walled carbon nanotubes (SWCNTs) subjected to an impact load was carried out by a mesh-free method. Free vibration and stress wave propagation analysis of carbon nanotube reinforced composite (CNTRC) cylinders are presented. In this simulation, an axisymmetric model is used. Four types of distributions of the aligned carbon nanotubes (CNTs) are considered; uniform and three kinds of functionally graded (FG) distributions along the radial direction of cylinder. Material properties are estimated by a micro mechanical model. In the mesh-free analysis, moving least squares (MLSs) shape functions are used for approximation of displacement field in the weak form of motion equation and the transformation method was used for the imposition of essential boundary conditions. Effects of the kind of distribution and volume fractions of carbon nanotubes and cylinder thickness on the natural frequencies and stress wave propagation of CNTRC cylinders are investigated. Results obtained for this analysis were compared with FEM and previous published work and good agreement was seen between them.     

Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory

This paper mainly presents bending and free vibration analyses of thin-to-moderately thick composite plates reinforced by single-walled carbon nanotubes using the finite element method based on the first order shear deformation plate theory. Four types of distributions of the uniaxially aligned reinforcement material are considered, that is, uniform and three kinds of functionally graded distributions of carbon nanotubes along the thickness direction of plates. The effective material properties of the nanocomposite plates are estimated according to the rule of mixture. Detailed parametric studies have been carried out to reveal the influences of the volume fractions of carbon nanotubes and the edge-to-thickness ratios on the bending responses, natural frequencies and mode shapes of the plates. In addition, the effects of different boundary conditions are also examined. Numerical examples are computed by an in-house finite element code and the results show good agreement with the solutions obtained by the FE commercial package ANSYS.     

Elastic wave propagation in a functionally graded nanocomposite reinforced by carbon nanotubes employing meshless local integral equations (LIEs)

The transient dynamic analysis of displacement field and elastic wave propagation in finite length functionally graded nanocomposite reinforced by carbon nanotubes are carried out using local integral equations (LIEs) based on meshless local Petrov–Galerkin (MLPG) method. The distribution of the aligned carbon nanotubes (CNTs) is assumed to vary as three kinds of functionally graded distributions as well as uniform distribution (UD) through radial direction of axisymmetric reinforced cylindrical composites. The mechanical properties are simulated using a micro-mechanical model in volume fraction form. A unit step function is used as a test function in the local weak form, which leads to local integral equations (LIEs). The analyzed domain is divided into small subdomains with a circular shape. The radial basis functions are used for approximation of the spatial variation of field variables. For treatment of time variations, the Laplace-transform technique is utilized. The 2D propagation of elastic waves through 2D domain is illustrated for various kinds of carbon nanotubes distributions. The time histories of displacement fields are studied in detail for various kinds of carbon nanotube distributions in reinforced cylindrical composites.     

Postbuckling responses of functionally graded cylindrical shells under axial compression and thermal loads

This paper presents a postbuckling analysis of functionally graded cylindrical shells under axial compression and thermal loads using the element-free kp-Ritz method. The formulation is developed to handle problems of small strains and moderate rotations, based on the first-order shear deformation shell theory and von Kármán strains. The effective material properties of the shells are assumed to be continuous along their thickness direction, and are obtained using a power-law distribution of the volume fractions of the constituents. The approximations of the two-dimensional displacement fields are expressed in terms of a set of mesh-free kernel particle functions. The system bending stiffness is evaluated using a stabilized conforming nodal integration method and the membrane and shear terms are estimated using direct nodal integration to eliminate shear locking. The postbuckling path is traced using a combination of the arc-length and mesh-free kp-Ritz methods. The proposed formulation is validated by comparing the results of the proposed method with those in the literature. The postbuckling responses of two types of functionally graded conical shells, one composed of Al/ZrO₂ and the other of SUS304/Si₃N₄, are investigated and the effects of volume fraction, boundary condition, and length-to-thickness ratio on postbuckling behavior are discussed in detail.     

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.     

Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach

The buckling behavior of functionally graded carbon nanotube (FG-CNT) reinforced composite thick skew plates is studied. The CNTs are reinforced uniaxially aligned in the axial direction. Material properties of the nanocomposites are assumed to be graded in the thickness direction. The element-free IMLS-Ritz method is employed for the numerical analysis. The theoretical formulation has incorporated the effects of transverse shear deformation and rotary inertia through employing the first-order shear deformation theory (FSDT). A few numerical examples are chosen to demonstrate the numerical stability and accuracy of the IMLS-Ritz method. The validity of the IMLS-Ritz results is examined by comparing them with those of the known data in the literature. Parametric studies are conducted for various types of CNTs distributions, CNT ratios, skew plates, aspect ratios and thickness-to-height ratios under different boundary conditions. Some conclusions are drawn on the parametric studies with respect to the buckling characteristics.     

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.     

Analysis of Metal Matrix Laminated Cylindrical Panels Subjected to Nonuniform Temperature Loading

The moderately large deflection response of metal matrix composite (MMC) cylindrical panels to nonuniform thermal loading is addressed. Temperature dependence of both elastic and viscoplastic properties of the metallic matrix is taken into account; this suggests that a nonuniformly heated panel should be considered as a nonhomogeneous structure. At each loading increment, a micromechanical analysis is performed to establish the instantaneous thermoinelastic constitutive law at each point of the panel. This is followed by a structural analysis that yields quasi-static thermal response of antisymmetrically laminated MMC panels. Results are presented for simply supported and clamped graphite-aluminum panels and for two types of in-plane boundary conditions. The effects of boundary conditions, lamination angle, panel's curvature, length-to-thickness ratio, and different types of spatial temperature distributions are illustrated. Comparisons with results obtained using an approach that treats the effect of temperature-dependent material properties in a simplified manner are shown. Comparisons with the corresponding elastic solutions (which neglect the inelastic effects in the metallic matrix) are given.     

Magneto-thermo-mechanical dynamic buckling analysis of a FG-CNTs-reinforced curved microbeam with different boundary conditions using strain gradient theory

In this article, dynamic buckling analysis of an embedded curved microbeam reinforced by functionally graded carbon nanotubes is carried out. The structure is subjected to thermal, magnetic and harmonic mechanical loads. Timoshenko beam theory is employed to simulate the structure. Furthermore, the temperature-dependent surrounding elastic foundation is modeled by normal springs and a shear layer. Using strain gradient theory, the small scale effects are taken into account. The extended rule of mixture is employed to estimate the equivalent properties of the composite material. The governing equations and different boundary conditions are derived based on the energy method and Hamilton’s principle. Dynamic stability regions of the system are obtained using differential quadrature method. The aim of this paper is to investigate the influence of different parameters such as small scale effect, boundary conditions, elastic foundation, volume fraction and distribution types of carbon nanotubes, magnetic field, temperature and central angle of the curved microbeam on the dynamic stability region of the system. The results indicate that by increasing the volume fraction of CNTs, the frequency of the system increases and thus the dynamic stability region occurs at higher frequencies.     

A mesh-free method for analysis of the thermal and mechanical buckling of functionally graded cylindrical shell panels

The buckling response of functionally graded ceramic-metal cylindrical shell panels under axial compression and thermal load is presented here. The formulation is based on the first-order shear deformation shell theory and element-free kp-Ritz method. The material properties of shell panels are assumed to vary through their thickness direction according to a power-law distribution of the volume fraction of constituents. Approximations of the displacement field are expressed in terms of a set of mesh-free kernel particle functions. A stabilized conforming nodal integration approach is employed to estimate the bending stiffness, and the shear and membrane terms are evaluated using a direct nodal integration technique to eliminate membrane and shear locking for very thin shells. The mechanical and thermal buckling responses of functionally graded shell panels are investigated, and the influences of the volume fraction exponent, boundary conditions, and temperature distribution on their buckling strengths are also examined.     

Global/local analysis of composite plates with cutouts

 The study focuses on the development of a simple and accurate global/local method for calculating the static response of stepped, simply-supported, isotropic and composite plates with circular and elliptical cutouts. The approach primarily involves two steps. In the first step a global approach, the Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the cutout are augmented with a perturbation function, which is accurate for uniform thickness plates only, to account for the cutout. The Ritz solution does not accurately satisfy the natural boundary conditions at the cut-out boundary, nor does it accurately model the discontinuities caused by abrupt thickness changes. Therefore, a second step, local in nature is taken in which a small area in the vicinity of the hole and encompassing other points of singularities is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The chosen perturbation function is reliable for circular cutout in uniform plates, therefore elliptical cutouts were suitably transformed to circular shapes using conformal mapping. The methodology is then applied to the analysis of composite plates, and its usefulness successfully proved in such cases. The proposed approach resulted in accurate prediction of stresses, with considerable savings in CPU time and data storage for composite flat panels.     

用团簇法计算单层纳米碳管的电子结构

用分子团簇法计算了三种边界条件下单层纳米碳管的电子结构,依靠得到的态密度、能级、分子轨道和结合能,对合成纳米尺度器件时可能出现的结果进行了预测。     

Fabrication of carbon-based nanocomposite films by spin-coating process: An experimental and modeling study of the film thickness

Spin-coating is used for the fabrication of nanocomposite thin films, consisting of carbon nanoparticles embedded in epoxy matrix, on Mylar substrate. The final thickness of the heat-cured film was measured as a function of the spinning speed and nanoparticle concentration. Multi-walled carbon nanotubes with carboxyl functionalization (MWCNT-COOH) or exfoliated graphite nanoplatelets (xGnP) were used as fillers. Experimental results were in good agreement with the predictions from a model that considered the rheology and flow behavior of the reinforced resin fluids on a rotating disk. The model was differentiated for Newtonian and non-Newtonian regime of the spinning polymer fluid. In case of non-Newtonian behavior of the epoxy resin at high particle concentrations, a semi-empirical approach was used to determine the model constants from rheology measurements. Results from this analysis also indicate how rheological and wetting properties of the nano-reinforced polymer fluids depend on the aspect ratio of the graphene nanoplatelets.     

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.     

拉应力下碳纳米管增强高分子基复合材料的应力分布EI北大核心CSCD

采用基于剪切滞后模型的数值计算和有限元仿真结合的研究方法,通过构建由碳纳米管增强的高分子复合材料的圆柱形代表性体积元模型,分析在一定拉伸应力下不同碳纳米管的层数、长径比、含量以及环氧树脂、尼龙和聚甲基丙烯酸甲酯3种基体材料对碳纳米管内各层应力分布的影响。结果表明:在一定的拉伸应力下,层数和长径比对碳纳米管中各层的应力分布影响很大。碳纳米管的饱和应力值随着层数增加而减小,其值与层数存在一定的相关性,在对碳纳米管本身性能的利用率上,单壁碳纳米管表现最好;长径比的增大能有效提升碳纳米管的有效长度;随着碳纳米管含量的减少,其饱和应力值明显增大,有效长度不断减小;不同的高分子基体材料对碳纳米管的应力分布影响并不明显。     

Numerical analysis and experimental correlation of composite shell buckling and post-buckling

This paper deals with the buckling and post-buckling behaviour of carbon fibre reinforced plastic cylindrical shells under axial compression. The finite element analysis is used to investigate this problem and three different types of analysis are compared: eigenvalue analysis, non-linear Riks method and dynamic analysis. The effect of geometric imperfection shape and amplitude on critical loads is discussed. A numerical–experimental correlation is performed, using the results of experimental buckling tests. The geometric imperfections measured on the real specimens are accounted for in the finite element model. The results show the reliability of the method to follow the evolution of the cylinder shape from the buckling to the post-buckling field and good accuracy in reproducing the experimental post-buckling behaviour.     

高速旋转薄壁圆柱壳的行波共振特性研究

基于传递矩阵法研究了不同边界条件下高速旋转薄壁圆柱壳的行波共振特性。首先,基于Love 壳体理论,考虑离心力、科氏力和惯性力的影响,建立了旋转态薄壁圆柱壳的振动微分方程;然后,引入传递矩阵方法,根据壳体子段间的状态向量表达式,推导了结构的整体传递矩阵;最后,通过高精度的精细积分法进行求解,得到了两端简支、两端固支和固支-自由边界条件下的共振特性。算例结果表明,传递矩阵方法适合于求解高速旋转薄壁圆柱壳的行波共振特性,在三种边界条件下以周向模态的振动为主;在工作转速和1倍频激振力作用下,共振裕度小于10%的共振转速点仅有一个,而在其它倍频激振下的共振转速点不在安全裕值范围内。