Free and forced vibration analyses of functionally graded hollow cylinders and spheres are performed and analytical benchmark solutions are presented. The material is assumed to be graded in the radial direction according to a simple power law. The Laplace transform method is used, and the inversion into the time domain is performed exactly using calculus of residues. The Complex Laplace parameter in the free vibration equation has directly given natural frequencies, and the results are given in tabular form. On the inner surface, various axisymmetric dynamic pressures are applied, and radial displacement and hoop stress are presented in the form of graphs. The exponent in the power law, called the inhomogeneity parameter, essentially refers to the degree of inhomogeneity. Increasing the inhomogeneity parameter provides a stress-shielding effect. Closed-form solutions obtained in the present paper are tractable, and they allow for further parametric studies. The inhomogeneity constant is a useful parameter from a design point of view in that it can be tailored for specific applications to control the stress distribution.
Issue Section:
Research Papers
References
1.
Tranter
, C. J.
, 1942, “The Applications of the Laplace Transformation to a Problem on Elastic Vibrations
,” Philos. Mag.
, 33
, pp. 614
–622
.2.
Mirsky
, I.
, 1964, “Axisymmetric Vibrations of Orthotropic Cylinders
,” J. Acoust. Soc. Am.
, 36
(11
), pp. 2106
–2112
. 3.
Klosner
, J. M.
, and Dym
, C. L.
, 1965, “Axisymmetric, Plane-Strain Dynamic Response of a Thick Orthotropic Shell
,” J. Acoust. Soc. Am.
, 39
(1
), pp. 1
–7
. 4.
Ahmed
, N.
, 1965, “Axisymmetric Plane-Strain Vibrations of a Thick-Layered Orthotropic Cylindrical Shell
,” J. Acoust. Soc. Am.
, 40
(6
), pp. 1509
–1516
. 5.
Ghosh
, A. K.
, 1995, “Axisymmetric Vibration of a Long Cylinder
,” J. Sound Vib.
, 186
(5
), pp. 711
–721
. 6.
Tarn
, J. Q.
, 2001, “Exact Solutions for Functionally Graded Anisotropic Cylinders Subjected to Thermal and Mechanical Loads
,” Int. J. Solids Struct.
, 38
(46–47
), pp. 8189
–8206
. 7.
Erdogan
, F.
, 1995, “Fracture Mechanics of Functionally Graded Materials
,” Composites Eng.
, 5
(7
), pp. 753
–770
. 8.
Horgan
, C. O.
, and Chan
, A. M.
, 1999, “The Pressurized Hollow Cylinder or Disk Problem for Functionally Graded Isotropic Linearly Elastic Materials
,” J. Elasticity
, 55
(1
), pp. 43
–59
. 9.
Jabbari
, M.
, Sohrabpour
, S.
, and Eslami
, M. R.
, 2002, “Mechanical and Thermal Stresses in a Functionally Graded Hollow Cylinder due to Radially Symmetric Loads
,” Int. J. Pressure Vessels Piping
, 79
(7
), pp. 493
–497
. 10.
Jabbari
, M.
, Sohrabpour
, S.
, and Eslami
, M. R.
, 2003, “General Solution for Mechanical and Thermal Stresses in a Functionally Graded Hollow Cylinder due to Nonaxisymmetric Steady-State Loads
,” ASME J. Appl. Mech.
, 70
(1
), pp. 111
–118
. 11.
Shao
, Z. S.
, 2005, “Mechanical and Thermal Stresses of a Functionally Graded Circular Hollow Cylinder with Finite Length
,” Int. J. Pressure Vessels Piping
, 82
(3
), pp. 155
–163
. 12.
Dai
, H. L.
, Fu
, Y. M.
, and Dong
, Z. M.
, 2006, “Exact Solutions for Functionally Graded Pressure Vessels in a Uniform Magnetic Field
,” Int. J. Solids Struct.
, 41
(18–19
), pp. 5570
–5580
. 13.
Jabbari
, M.
, Bahtui
, A.
, and Eslami
, M. R.
, 2006, “Axisymmetric Mechanical and Thermal Stresses in Thick Long FGM Cylinders
,” J. Therm. Stresses
, 29
(7
), pp. 643
–663
. 14.
Loy
, C. T.
, Lam
, K. Y.
, and Reddy
, J. N.
, 1999, “Vibration of Functionally Graded Cylindrical Shells
,” Int. J. Mech. Sci.
, 41
(3
), pp. 309
–324
. 15.
Pradhan
, S. C.
, Loy
, C. T.
, Lam
, K. Y.
, and Reddy
, J. N.
, 2000, “Vibration Characteristics of Functionally Graded Cylindrical Shells Under Various Boundary Vonditions
,” Appl. Acoust.
, 61
(1
), pp. 111
–129
. 16.
Zhou
, D.
, Cheung
, Y. K.
, Lo
, S. H.
, and Au
, F. T. K.
, 2003, “3D Vibration Analysis of Solid and Hollow Cylinders via Chebyshev-Ritz Method
,” Comput. Methods Appl. Mech. Eng.
, 192
(13–14
), pp. 1575
–1589
. 17.
Han
, X.
, Liu
, G. R.
, Xi
, Z. C.
, and Lam
, K. Y.
, 2001, “Transient Waves in a Functionally Graded Cylinder
,” Int. J. Solids Struct.
, 38
(17
), pp. 3021
–3037
. 18.
Ng
, T. Y.
, Lam
, K. Y.
, Liew
, K. M.
, and Reddy
, J. N.
, 2001, “Dynamic Stability Analysis of Functionally Graded Cylindrical Shells Under Periodic Axial Loading
,” Int. J. Solids Struct.
, 38
(8
), pp. 1295
–1309
. 19.
Shakeri
, M.
, Akhlaghi
, M.
, and Hoseini
, S. M.
, 2006, “Vibration and Radial Wave Propagation Velocity in Functionally Graded Thick Hollow Cylinder
,” Compos. Struct.
, 76
(1–2
), pp. 174
–181
. 20.
Nelson
, R. B.
, 1973, “Natural Vibrations of Laminated Orthotropic Spheres
,” Int. J. Solids Struct.
, 9
(3
), pp. 305
–311
. 21.
Gautham
, P.
, and Ganesan
, N.
, 1992, “Axisymmetric Vibration of Layered Orthotropic Spherical Shells of Variable Thickness
,” Comput. Struct.
, 45
(5–6
), pp. 893
–900
. 22.
Jiang
, H.
, Young
, P. G.
, and Dickinson
, S. M.
, 1996, “Natural Frequencies of Vibration of Layered Hollow Spheres Using Exact Three-Dimensional Elastic Equations
,” J. Sound Vib.
, 195
(1
), pp. 155
–162
. 23.
Heyliger
, P. R.
, and Jilani
, A.
, 1992, “The Free Vibrations of Inhomogeneous Elastic Cylinders and Spheres
,” Int. J. Solids Struct.
, 29
(22
), pp. 2689
–2708
. 24.
Chen
, W. Q.
, 1999, “Effect of Radial Inhomogeneity on Natural Frequencies of an Anisotropic Hollow Sphere
,” J. Sound Vib.
, 226
(4
), pp. 787
–794
. 25.
Chen
, W. Q.
, Wang
, X.
, and Ding
, H. J.
, 1999, “Free Vibration of a Fluid-Filled Hollow Sphere of a Functionally Graded Material with Spherical Isotropy
,” J. Acoust. Soc. Am.
, 106
(5
), pp. 2588
–2594
. 26.
Chen
, W. Q.
, Wang
, L. Z.
, and Lu
, Y.
, 2002, “Free Vibrations of Functionally Graded Piezoceramic Hollow Spheres with Radial Polarization
,” J. Sound Vib.
, 251
(1
), pp. 103
–114
. 27.
Chiroiu
, V.
, and Munteanu
, L.
, 2007, “On the Free Vibrations of a Piezoceramic Hollow Sphere
,” Mech. Res. Commun.
, 34
(2
), pp. 123
–129
. 28.
Sheehan
, J. P.
, and Debnath
L.
, 1972, “Transient Vibrations of an Isotropic Elastic Sphere
,” Pure Appl. Geophys.
, 99
(1
), pp. 37
–48
. 29.
Ghosh
, A. K.
, and Agrawal
, M. K.
, 1994, “Radial Vibrations of Spheres
,” J. Sound Vib.
, 171
(3
), pp. 315
–322
. 30.
Ding
, H. J.
, Wang
, H. M.
, and Chen
, W. Q.
, 2002, “Analytical Thermo-Elastodaynamic Solutions for a Nonhomogeneous Transversely Isotropic Hollow Sphere
,” Arch. Appl. Mech.
, 72
(8
), pp. 545
–553
. 31.
Chen
, W. Q.
, Lu
, Y.
, Ye
, G. R.
, and Cai
, J. B.
, 2002, “3D Electroelastic Field in a Functionally Graded Piezoceramic Hollow Sphere Under Mechanical and Electric Loadings
,” Arch. Appl. Mech.
, 72
(1
), pp. 39
–51
. 32.
Ding
, H. J.
, Wang
, H. M.
, and Chen
, W. Q.
, 2003, “Dynamic Responses of a Functionally Graded Pyroelectric Hollow Sphere for Spherically Symmetric Problems
,” Int. J. Mech. Sci.
, 45
(6–7
), pp. 1029
–1051
. 33.
Wang
, H. M.
, and Ding
, H. J.
, 2006, “Spherically Symmetric Transient Responses of Functionally Graded Magneto-Electro-Elastic Hollow Sphere
,” Struct. Engin. Mech.
, 23
(5
), pp. 525
–542
.34.
Dai
, H. L.
, and Fu
, Y. M.
, 2007, “Magnetothermoelastic Interactions in Hollow Structures of Functionally Graded Material Subjected to Mechanical Loads
,” Int. J. Pressure Vessels Piping
, 84
(3
), pp. 132
–138
. 35.
Manolis
, G. D.
, and Beskos
, D. E.
, 1981, “Dynamic Stress Concentration Studies by Boundary Integrals and Laplace Transform
,” Int. J. Numer. Methods Eng.
, 17
(4
), pp. 573
–599
. 36.
Watson
, G. N.
, 1996, A Treatise on the Theory of Bessel Functions
, Cambridge University Press
, Cambridge, UK
.Copyright © 2011
by American Association of Physics Teachers
You do not currently have access to this content.
