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Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures

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Abstract

This article deals with the vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shell structures. The material properties of an FG-CNTRC shell are graded smoothly through the thickness direction of the shell according to uniform distribution and some other functionally graded (FG) distributions (such as FG-X, FG-V, FG-O and FG-\({\Lambda}\)) of the volume fraction of the carbon nanotube (CNT), and the effective material properties are estimated by employing the extended rule of mixture. An eight-noded shell element considering transverse shear effect according to Mindlin’s hypothesis has been employed for the finite element modelling and analysis of the composite shell structures. The formulation of the shell midsurface in an arbitrary curvilinear coordinate system based on the tensorial notation is also presented. The Rayleigh dam** model has been implemented in order to study the effects of carbon nanotubes (CNTs) on the dam** capacity of such shell structures. Different types of shell panels have been analyzed in order to study the impulse and frequency responses. The influences of CNT volume fraction, CNT distribution, geometry of the shell and material distributions on the dynamic behavior of FG-CNTRC shell structures have also been presented and discussed. Various types of FG-CNTRC shell structures (such as spherical, ellipsoidal, doubly curved and cylindrical) have been analyzed and discussed in order to compare studies in terms of settling time, first resonant frequency and absolute amplitude corresponding to first resonant frequency based on the impulse and frequency responses, and the effects of CNTs on vibration responses of such shell structures are also presented. The results show that the CNT distribution and volume fraction of CNT have a significant effect on vibration and dam** characteristics of the structure.

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Abbreviations

\({{V_{{\rm cnt}}, V_{\rm m}}}\) :

Carbon nanotube and matrix volume fraction

\({{E_{11}^{{\rm cnt}}}}\) and \({{{E}_{22}^{{\rm cnt}}}}\) :

Young’s moduli of CNT in longitudinal and transverse directions

\({{{G}_{12}^{{\rm cnt}}}}\) :

Shear modulus of CNT

\({{{E}_{\rm m}}}\) and \({{{G}_{\rm m}}}\) :

Young’s moduli and shear modulus of the isotropic matrix.

\({{\eta _1, \eta _2}}\) and \({{\eta _3}}\) :

CNT efficiency parameters

\({{\upsilon_{12}^{{\rm cnt}}}}\) and \({{\upsilon_{\rm m}}}\) :

Poisson’s ratio of CNT and matrix

h :

Thickness

\({{{\rm nd}}}\) :

Number of nodes in an element

\({{N_i}}\) :

Shape function corresponding to the i node

\({{\alpha _1,\alpha _2 }}\) :

Curvilinear coordinates

\({{u_{0i}, v_{0i}}}\) and \({{w_{0i}}}\) :

Deflection of midsurface at ith node in \({{\alpha _1,\alpha _2 }}\) and z directions

\({{\theta _{1i}, \theta _{2i}}}\) :

Rotation of normal at ith node about \({{\alpha _2}}\) axis and \({{\alpha _1}}\) axis, respectively

\({{\varepsilon _{xx}^0, \varepsilon _{yy}^0 \, {\rm and} \, \gamma _{xy}^0}}\) :

In-plane strains of the midsurface in the Cartesian coordinate system

\({{k_{xx}, k_{yy} \, {\rm and} \, k_{xy}}}\) :

Bending strains (curvatures) of the midsurface in the Cartesian coordinates system

\({{A_{ij}, B_{ij}, D_{ij}}}\) and \({{A_{ij}^s}}\) :

Extensional, flexural-extensional coupling, bending and transverse shear stiffness, respectively

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  1. Department of Mechanical Engineering, National Institute of Technology, Rourkela, Rourkela, Odisha, 769008, India

    Benedict Thomas & Tarapada Roy

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Thomas, B., Roy, T. Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures. Acta Mech 227, 581–599 (2016). https://doi.org/10.1007/s00707-015-1479-z

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