Functionally Graded Carbon Nanotube-reinforced Research Articles

Vibration of FG-CNTRC annular sector plates resting on the Winkler-Pasternak elastic foundation under a periodic radial compressive load

The vibration and stability of functionally graded carbon nanotube-reinforced (FG-CNTRC) composite annular sector plates resting on Winkler-Pasternak elastic foundation subjected to a periodic radial compressive load are investigated for various boundary conditions. To this end, a shear deformable plate model is established according to a parabolic theory which can interpret the shear deformation and rotary inertia effects without using any shear correction factor. The modified micromechanical technique is employed to compute the effective material properties of the FG-CNTRCs. The discretized form of higher-order governing equations is directly accessed by employing Hamilton’s principle and the variational differential quadrature (VDQ) method. In addition, by considering the applied radial compressive force as a periodic function, the discretized equations are expressed as the Mathieu–Hill equations, and then the instability regions are determined via the Bolotin’s scheme. In numerical results, the effects of the CNT distribution pattern and volume fraction, geometry parameters, sector angle, elastic foundation and static load factor on the stability of FG-CNTRC annular sector plates subject to arbitrary edge conditions are thoroughly discussed.

Geometrically nonlinear buckling analysis of functionally graded carbon nanotube reinforced cylindrical panels resting on Winkler–Pasternak elastic foundation

This paper deals with geometrically nonlinear buckling analysis of functionally graded carbon nanotube reinforced (FG-CNTR) cylindrical panels. The FG-CNTR cylindrical panel is assumed to be rested on the Winkler–Pasternak elastic foundation and subjected to uniform pressure. In the FG-CNTR cylindrical panel model, uniform and three distributions of carbon nanotubes, which are graded in the thickness direction of the panel, are considered. Effective properties of materials of the panels reinforced by single-walled carbon nanotubes are estimated through a micromechanical model based on the extended rule of mixtures. Governing equilibrium equations of the FG-CNTRC cylindrical panel are obtained based on the classical shell theory and considering the von Karman geometrically nonlinearity and initial geometric imperfection. A closed form of the resulting stability equations is established via the Galekin procedure to obtain the buckling load–deflection relations in case of simply supported boundary condition. In the numerical results section, the exactness of formulation is validated by comparing the obtained results with those reported in the open database. Then, a comprehensive investigation into the influence of carbon nanotube volume fraction, carbon nanotube distribution rule, imperfection parameter, elastic foundation as well as the geometry parameters on the nonlinear buckling behaviors of the FG-CNTRC cylindrical panels is discussed in detail.

Computation of elastodynamic behavior of a hybrid laminated plate containing CNTR-FG layers and FRC layers under dynamic loading

This paper studies the elastodynamic behavior of a hybrid laminated plate composed of carbon nanotube-reinforced functionally graded (CNTR-FG) layers and conventional fiber reinforced composite (FRC) layers. Based on first order shear deformation theory (FSDT), the element-free kp-Ritz method was used in this paper. Four types of functionally graded distributions of single walled carbon nanotubes (SWCNTs) along the thickness direction of layers are considered. The extended rule of mixture was used to calculate the effective material properties of the resulting nanocomposites. Two-dimensional displacement fields of the plates are obtained by using the mesh-free kernel particle functions. The Ritz procedure is used to obtain the discretized governing equations and the Newmark-β method is applied to perform numerical time integration. The stability and accuracy of the present element-free method are verified through convergence studies by considering the effect of support size and number of nodes. Detail parametric studies are conducted to investigate the influence of carbon nanotube volume fraction, plate width-to-thickness ratio, plate aspect ratio, boundary condition and distribution type of carbon nanotubes on the dynamic responses of hybrid laminated plates containing CNTR-FG layers and FRC layers.

Characterizing buckling behavior of matrix-cracked hybrid plates containing CNTR-FG layers

In this paper, the effect of matrix cracks on the buckling of a hybrid laminated plate is investigated. The plate is composed of carbon nanotube reinforced functionally graded (CNTR-FG) layers and conventional fiber reinforced composite (FRC) layers. Different distributions of single walled carbon nanotubes (SWCNTs) through the thickness of layers are considered. The cracks are modeled as aligned slit cracks across the ply thickness and transverse to the laminate plane, and the distribution of cracks is assumed statistically homogeneous corresponding to an average crack density. The first-order shear deformation theory (FSDT) is employed to incorporate the effects of rotary inertia and transverse shear deformation, and the meshless kp-Ritz method is used to obtain the buckling solutions. Detailed parametric studies are conducted to investigate the effects of matrix crack density, CNTs distributions, CNT volume fraction, plate aspect ratio and plate length-to-thickness ratio, boundary conditions and number of layers on buckling behaviors of hybrid laminated plates containing CNTR-FG layers.

Modeling large amplitude vibration of matrix cracked hybrid laminated plates containing CNTR-FG layers

• A numerical framework for large amplitude vibration analysis of matrix cracked hybrid laminated plates. • Matrix crack is considered based on self-consistent model. • We show that the crack density of matrix cracks plays an important role in vibration of hybrid laminated plates. • Parametric studies are conducted to investigate effects of various parameters on nonlinear vibration responses. This paper presents the mathematical modeling of the nonlinear vibration behavior of a hybrid laminated plate composed of carbon nanotube reinforced functionally graded (CNTR-FG) layers and conventional fiber reinforced composite (FRC) layers. Three type symmetric distributions of single walled carbon nanotubes (SWCNTs) through the thickness of layers are considered. The cracks are modeled as aligned slit cracks across the ply thickness and transverse to the laminate plane. The distribution of cracks is assumed to be statistically homogeneous corresponding to an average crack density. The obtained partial differential equations are solved by the element-free kp-Ritz method, and the iteration process is dealt with using the linearized updated mode method. Detailed parametric studies are conducted investigate the effects of matrix crack density, CNTs distributions, CNT volume fraction, plate aspect ratio and plate length-to-thickness ratio, boundary conditions and number of layers on the frequency-amplitude responses of hybrid laminated plates containing CNTR-FG layers.

Bending and vibration behaviors of matrix cracked hybrid laminated plates containing CNTR-FG layers and FRC layers

The influence of matrix cracks on bending and vibration behaviors of hybrid laminated plates is investigated based on an element-free numerical framework. Carbon nanotube reinforced functionally graded (CNTR-FG) layers and conventional graphite fiber reinforced composite (FRC) layers are selected as layer elements for the considered hybrid laminated plates. To illustrate the degraded stiffness, a matrix-cracked model, namely self-consistent model (SCM), is employed. Based on the first-order shear deformation theory (FSDT), we can obtain the governing equation. An element-free numerical framework is proposed to solve the governing equation for obtaining the bending and vibration solutions. In addition, some selected calculations are furnished to illustrate the influences of matrix crack densities, boundary conditions, material parameters and geometric parameters on the bending and vibration behavior characteristics.

Meshless modeling of geometrically nonlinear behavior of CNT-reinforced functionally graded composite laminated plates

A geometrically nonlinear analysis of carbon nanotube reinforced functionally graded (CNTR-FG) composite laminated plates is presented. Single-walled carbon nanotubes (SWCNTs) are selected as reinforcement and the effective material properties of CNTR-FG plates are assumed to be graded through the thickness direction in each layer. The two-dimensional displacement fields of the plates are approximated by a set of meshless kernel particle functions. For the purpose of eliminating shear locking, a stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness, and the membrane and shear terms are calculated by the direct nodal integration method. Parametric studies are conducted to investigate the effect of various types of CNT distribution, CNT volume fraction, plate aspect ratio and boundary conditions on the nonlinear responses of CNTR-FG laminated plates. Moreover, the effects of the number of layers and lamination angle are also investigated.

Buckling analysis of functionally graded carbon nanotube-reinforced curved panels under axial compression and shear

Abstract The need for high-performance lightweight materials for the design of aerospace structures predicts a remarkable future role of carbon nanotubes (CNTs). In particular, thin curved fuselage panels are widely employed. However, the theoretical difficulties in the underlying differential problem induced by the curvature, as well as a still under development technology of CNT-based composites, the number of studies dealing with buckling behavior of this structural typology is limited. The present study aims to provide some insight into the linear buckling analysis of functionally graded carbon-nanotube reinforced (FG-CNTRC) cylindrical curved panels under compressive and shear loading. Effective properties of materials of the panels reinforced by single-walled carbon nanotubes (SWCNTs) are estimated through a micromechanical model based on either the Eshelby-Mori-Tanaka approach or the extended rule of mixtures. A series of numerical simulations have been carried out to inspect the influence of curvature, panel aspect ratio, the distribution profile of reinforcements (uniform and three non-uniform distributions) and CNTs orientation angle on the buckling critical load under compressive and shear loading in uniform thermal environments. Results demonstrate that the change of fiber orientation, CNTs distribution, panel aspect ratio, loading condition and temperature have noticeable effects on the buckling strength and buckling modes of FG-CNTRC curved panels.

Buckling analysis of CNT reinforced functionally graded laminated composite plates

Buckling behavior of carbon nanotube reinforced functionally graded (CNTR-FG) composite laminated plate is studied. The first-order shear deformation theory (FSDT) is employed to incorporate the effects of rotary inertia and transverse shear deformation, and the meshless kp-Ritz method is used to obtain the buckling solutions. Using the kernel particle approximation in the field variables and minimizing the energy functional via the Ritz procedure, a discretized eigenvalue equation of the problem is derived. The numerical stability and accuracy of the kp-Ritz method is validated through convergence and comparison studies. Besides, parametric studies are conducted for various types of CNTs distributions, CNT volume fraction, plate aspect ratio and plate length-to-thickness ratio under different boundary conditions. Moreover, the effects of number of layers and lamination angle are also investigated.

Analysis of laminated CNT reinforced functionally graded plates using the element-free kp-Ritz method

This paper presents the well known bending responses of laminated carbon nanotube-reinforced functionally graded composite plates. The laminated plate is composed of perfectly bonded carbon nanotube-reinforced functionally graded (CNTR-FG) layers. Within each CNTR-FG layer, different distributions of single walled carbon nanotubes (SWCNTs) through the thickness of the layers are considered. Effective material properties of the laminated nanocomposite plates are estimated using the extended rule of mixture. The shear deformation is considered using the first-order shear deformation plate theory. In the solution process, the governing differential equations of the bending behavior of laminated CNTR-FG plates are solved by the element-free kp-Ritz method. Effects of the volume fractions of carbon nanotubes, plate width-to-thickness ratio, plate aspect ratio, distribution type of carbon nanotubes, boundary conditions, number of layers and lamination angle on the bending responses of the laminated nanocomposite plates are examined in detail.

Elastodynamic analysis of carbon nanotube-reinforced functionally graded plates

In this paper, elastodynamic problem of carbon nanotube-reinforced functionally graded (CNTR-FG) plates is studied using the element-free kp-Ritz method based on first order shear deformation theory (FSDT). The plates are reinforced by single-walled carbon nanotubes (SWCNTs) with different types of distributions, i.e. uniform and three kinds of functionally graded distributions of carbon nanotubes along thickness direction of the panels. Extended rule of mixture is employed to estimate effective material properties of the resulting nanocomposites. Two-dimensional displacement fields of the plates are approximated by a set of mesh-free kernel particle functions. The discretized governing equations are developed via the Ritz procedure, and numerical time integration is performed through the Newmark-β method. Convergence studies are carried out by considering the influence of support size and number of nodes to verify the numerical stability of the present element-free method. Numerical simulations are used to study the effect of carbon nanotube volume fraction, plate width-to-thickness ratio, plate aspect ratio, boundary condition, load type and distribution type of carbon nanotubes on the dynamic responses of CNTR-FG plates.

Stress analysis of functionally graded open cylindrical shell reinforced by agglomerated carbon nanotubes

In this paper, stresses due to bending behavior of functionally graded carbon nanotube-reinforced (FGCNTR) open cylindrical shells subjected to mechanical loads is studied. The material properties of FGCNTR shells are assumed to be graded in the thickness direction, and are estimated using a two-parameter micromechanics model in which Eshelby–Mori–Tanaka approach is employed. The primary bending formulation is based on the linear, small-strain, three-dimensional elasticity theory. In addition, the cylindrical shells are analyzed using the third-order shear deformation theory (TSDT). In order to discretize the governing equations, the two-dimensional generalized differential quadrature method (2-D GDQM) in the thickness and longitudinal directions and the trigonometric functions in tangential direction are used. The effects of agglomeration parameters, CNTs volume fraction, and CNTs distribution through the thickness on the bending behavior of FGCNTR open cylindrical shells are studied. In addition, the mechanical stresses obtained from 3-D elasticity are compared with those obtained using TSDT for a different range of geometric and agglomeration parameters.

Dynamic stability analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the element-free kp-Ritz method

In this paper, a first-known dynamic stability analysis of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under static and periodic axial force by using the mesh-free kp-Ritz method is presented. The cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) with different types of distributions, i.e. uniform and three kinds of functionally graded distributions of carbon nanotubes along thickness direction of the panels. Eshelby–Mori–Tanaka approach is employed to estimate effective material properties of the resulting nanocomposite panels. By applying the Ritz minimization procedure to the energy expressions, a system of Mathieu–Hill equations is formulated. Then the principal instability regions are analyzed through Bolotin’s first approximation. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and radius-to-thickness ratio. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined in detail.

Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels

A large deflection geometrically nonlinear behavior of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under uniform point transverse mechanical loading is studied. The analysis is carried out using the kp-Ritz method with kernel particle function is employed to construct the shape functions for the two-dimensional displacement approximations. Based on the first-order shear deformation shell theory, nonlinear governing equations are developed with geometric nonlinearity taking the form of von Kármán strains. It is assumed that carbon nanotubes are uniaxially aligned in the axial direction and are functionally graded in thickness direction of the cylindrical panels. The effective material properties of resulting CNTR-FG panels are estimated by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. A stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness and the membrane as well as shear terms are calculated by the direct nodal integration method to eliminate shear locking, for a very thin cylindrical panel. Several numerical example problems are examined to reveal the influences of volume fraction of carbon nanotubes, span angle, edge-to-radius ratio and thickness on nonlinear responses of the CNTR-FG panels. Moreover, effects of different boundary conditions and distribution type of carbon nanotubes are also investigated.

Eshelby-Mori-Tanaka approach for vibrational behavior of functionally graded carbon nanotube-reinforced plate resting on elastic foundation

In this study, based on the three-dimensional theory of elasticity, free vibration characteristics of functionally graded (FG) nanocomposite plates reinforced by randomly-oriented straight single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation are considered. Material properties are graded in the thickness direction of the plate according to the volume fraction power law distribution. An embedded carbon nanotube (CNT) in a polymer matrix and its surrounding inter-phase which is perfectly bonded to surrounding resin is replaced with an equivalent fiber to predict the mechanical properties of the carbon nanotube/polymer composite. The Mori-Tanaka approach is employed to calculate the effective elastic moduli of the plate. The natural frequencies of the plate are obtained by means of the generalized differential quadrature (GDQ) method. Detailed parametric studies have been carried out to investigate the influences of the CNT volume fraction, Winkler foundation modulus, shear elastic foundation modulus and various geometrical parameters on the vibration behavior of the functionally graded carbon nanotube-reinforced (FG-CNTR) plates.

Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach

This paper presents a postbuckling analysis of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under axial compression. Based on kernel particle approximations for the field variables, the Ritz method is employed to obtain the discretized governing equations. The cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) which are assumed to be graded through the thickness direction with different types of distributions. The effective material properties of CNTR-FG cylindrical panels are estimated through a micromechanical model based on the extended rule of mixture. To eliminate shear locking for a very thin cylindrical panel, the system’s bending stiffness is evaluated by a stabilized conforming nodal integration scheme and the membrane as well as shear terms are calculated by the direct nodal integration method. In the present study, the arc-length method combined with the modified Newton–Raphson method is used to trace the postbuckling path. Detailed parametric studies are carried out to investigate effects of various parameters on postbuckling behaviors of CNTR-FG cylindrical panels and results for uniformly distributed (UD) CNTR-FG cylindrical panel are provided for comparison.

Free Vibration Analysis of Nanocomposite Plates Reinforced by Graded Carbon Nanotubes Based on First-Order Shear Deformation Plate Theory

AbstractBased on the Mindlin’s first-order shear deformation plate theory this paper focuses on the free vibration behavior of functionally graded nanocomposite plates reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs). The material properties of simply supported functionally graded carbon nanotube-reinforced (FGCNTR) plates are assumed to be graded in the thickness direction. The effective material properties at a point are estimated by either the Eshelby-Mori-Tanaka approach or the extended rule of mixture. Two types of symmetric carbon nanotubes (CNTs) volume fraction profiles are presented in this paper. The equations of motion and related boundary conditions are derived using the Hamilton’s principle. A semi-analytical solution composed of generalized differential quadrature (GDQ) method, as an efficient and accurate numerical method, and series solution is adopted to solve the equations of motions. The primary contribution of the present work is to provide a comparative study of the natural frequencies obtained by extended rule of mixture and Eshelby-Mori-Tanaka method. The detailed parametric studies are carried out to study the influences various types of the CNTs volume fraction profiles, geometrical parameters and CNTs volume fraction on the free vibration characteristics of FGCNTR plates. The results reveal that the prediction methods of effective material properties have an insignificant influence of the variation of the frequency parameters with the plate aspect ratio and the CNTs volume fraction.