Based on the third-order shear deformation theory (TSDT), the investigation of the free vibration response of a continuously graded carbon nanotube-reinforced (CGCNTR) cylindrical shell is presented. The volume fractions of randomly oriented straight single-walled carbon nanotubes are assumed to be graded in the thickness direction. An embedded carbon nanotube in a polymer matrix and its surrounding inter-phase is replaced with an equivalent fiber for predicting the mechanical properties of the carbon nanotube/polymer composite. The Mori–Tanaka scheme as an accurate micromechanics model is used for estimating the homogenized material properties of nanocomposites reinforced with equivalent fibers. The equations of motion and the associated boundary conditions are derived using the Hamilton’s principle based on TSDT. The discretization of the system by means of the Generalized Differential Quadrature Method leads to a standard linear eigenvalue problem. Detailed parametric studies have been carried out to study the impacts of the various types of equivalent fiber distribution, different boundary conditions and geometrical parameters on the vibration characteristics of CGCNTR cylindrical shells. The interesting finding of the present study is that the graded CNT volume fractions with symmetric distribution through the shell thickness have high capabilities to reduce or increase the natural frequency in comparison with uniformly and asymmetric CNT distribution.
CallahanJBaruhH. A closed-form solution procedure for circular cylindrical shell vibrations. Int J Solids Struct1999; 36: 2973–3013.
2.
LeissaAW. Vibration of shells. NASA SP288. Washington, DC: US Government Printing Office, 1973 [Republished 1993, Acoustical Society of America].
3.
ReddyJN. Mechanics of laminated composite plates and shells: theory and analysis. 2nd ed.New York: CRC Press, 2004.
4.
SoedelW. Simplified equations and solutions for the vibration of orthotropic cylindrical shells. J Sound Vib1983; 87: 555–566.
5.
ZhangLXiangY. Vibration of open circular cylindrical shells with intermediate ring supports. Int J Solids Struct2006; 43: 3705–3722.
6.
SalvetatDRubioA. Mechanical properties of carbon nanotubes: a fiber digest for beginners. Carbon2002; 40: 1729–1734.
7.
EndoMHayashiTKimYA. Applications of carbon nanotubes in the twenty-first century. Phil Trans R Soc Lond A2004; 362: 2223–2238.
8.
WernikJMMeguidSA. Multiscale modeling of the nonlinear response of nano-reinforced polymers. Acta Mech2011; 217: 1–16.
9.
ThostensonETRenZFChouT-W. Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol2001; 61: 1899–1912.
ValterBRamMKNicoliniC. Synthesis of multiwalled carbon nanotubes and poly (o-anisidine) nanocomposite material: fabrication and characterization of its Langmuir-Schaefer films. Langmuir2002; 18: 1535–1541.
12.
QianDDickeyECAndrewsR. Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites. Appl Phys Lett2000; 76: 2868–2870.
13.
YokozekiTIwahoriYIshiwataS. Matrix cracking behaviors in carbon fiber/epoxy laminates filled with cup-stacked carbon nanotubes (CSCNTs). Compos Part A2007; 38: 917–924.
14.
HuNFukunagaHLuC. Prediction of elastic properties of carbon nanotube reinforced composites. Proc R Soc London Ser A2005; 461: 1685–910.
15.
FidelusJDWieselEGojnyFH. Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites. Compos Part A2005; 36: 1555–1561.
16.
BonnetPSireudeDGarnierB. Thermal properties and percolation in carbon nanotube–polymer composites. J Appl Phys2007; 91: 2019–2030.
17.
HanYElliottJ. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci2007; 39: 315–323.
JinYYuongFG. Simulation of elastic properties of single-walled carbon nanotubes. Compos Sci Technol2003; 63: 1507–1515.
20.
ChangTGaoH. Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J Mech Phys Solid2003; 51: 1059–1074.
21.
GriebelMHamaekersJ. Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites. Mech Eng2004; 193: 1773–1788.
22.
WuiteJAdaliS. Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis. Compos Struct2005; 71: 388–396.
23.
ShokriehMMRafieeR. Investigation of nanotube length effect on the reinforcement efficiency in carbon nanotube based composites. Compos Struct2010; 92: 2415–2420.
24.
ShenH-S. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, part I: axially-loaded shells. Compos Struct2011; 93: 2096–2108.
25.
SeidelGDLagoudasDC. Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites. Mech Mater2006; 38: 884–907.
26.
Sobhani AraghBHedayatiHBorzabadi FarahaniE. A novel 2-D six-parameter power-law distribution for free vibration and vibrational displacements of two-dimensional functionally graded fiber-reinforced curved panels. Eur J Mech A Solids2011; 30: 865–883.
27.
YasMHSobhani AraghB. Three-dimensional analysis for thermo elastic response of functionally graded fiber reinforced cylindrical panel. Compos Struct2010; 92: 2391–2399.
28.
BatraRCJinJ. Natural frequencies of a functionally graded anisotropic rectangular plate. J Sound Vib2005; 282: 509–516.
29.
Sobhani AraghBYasMH. Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced of cylindrical panels. Int J Mech Sci2010; 52: 1047–1063.
30.
VelSS. Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells. Compos Struct2010; 92: 2712–2727.
31.
Sobhani AraghBYasMH. Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution. Acta Mech2010; 215: 155–173.
32.
ReddyJNChengZQ. Three-dimensional thermomechanical deformations of functionally graded rectangular plates. Eur J Mech A Solids2001; 20: 841–855.
33.
MatsunagaH. Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Compos Struct2008; 82: 499–512.
34.
JabbariMBahtuiAEslamiMR. Axisymmetric mechanical and thermal stresses in thick long FGM cylinders. J Therm Stresses2006; 29: 643–663.
35.
ShenH-S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Compos Struct2009; 91: 9–19.
36.
ShenH-SZhuZH. Buckling and postbuckling behavior of functionally graded nanotube-reinforced composite plates in thermal environments. Comput Mater Continua2010; 18: 155–182.
37.
ShenH-SZhangC-L. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates. Mater Des2010; 31: 3403–3411.
WangZ-XShenH-S. Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets. Composites Part B2012; 43: 411–421.
40.
ShuC. Differential quadrature and its application in engineering. Berlin: Springer, 2000.
41.
ShuCRichardsBE. Application of generalized differential quadrature to solve two-dimensional incompressible Navier Stokes equations. Int J Numer Meth Fluid1992; 15: 791–798.
42.
BertCWMalikM. Differential quadrature method in computational mechanics, a review. Appl Mech Rev1996; 49: 1–28.
43.
MalekzadehP. Differential quadrature large amplitude free vibration analysis of laminated skew plates based on FSDT. Compos Struct2008; 83: 89–200.
44.
ShiDLFengXQHuangYY. The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. J Eng Mater Technol2004; 126: 250–257.
ShokriehMMRafieeR. Prediction of mechanical properties of an embedded carbon nanotube in polymer matrix based on developing an equivalent long fiber. Mech Res Commun2010; 37: 235–240.
49.
MoriTTanakaK. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall1973; 21: 571–574.
EshelbyJD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc London Ser A1957; 241: 376–396.
52.
MuraT. Micromechanics of defects in solids. The Hague: Martinus Nijhoff, 1982.
53.
HillR. A self-consistent mechanics of composite materials. J Mech Phys Solids1965; 13: 213–222.
54.
WangZ-XShenH-S. Nonlinear vibration of nanotube-reinforced composite plates in thermal environments. Comput Mater Sci2011; 50: 2319–2330.
55.
ReddyJN. Theory and Analysis of elastic plates and shells. 2nd ed.Boca Raton, FL: CRC Press, 2007.
56.
QatuMS. Vibration of laminated shells and plates. Oxford: Elsevier Academic Press, 2004.
57.
SoldatosKPHadjigeorgiouVP. Three-dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels. J Sound Vib1990; 137: 369–384.
58.
MalekzadehPFaridMZahedinejadP. Three-dimensional free vibration analysis of thick cylindrical shells resting on two-parameter elastic supports. J Sound Vib2008; 313: 655–675.
59.
MatsunagaH. Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory. J Compos Struct2009; 88: 519–531.
