Rotating Variable-Thickness Inhomogeneous Cylinders: Part I—Analytical Elastic Solutions

In this paper, an analytical solution for the rotation problem of an inhomogeneous hollow cylinder with variable thickness under plane strain assumption is developed. The present cylinder is made of a fiber-reinforced viscoelastic inhomogeneous orthotropic material. The thickness of the cylinder is taken as parabolic function in the radial direction. The elastic properties varies in the same manner as the thickness of the cylinder while the density varies according to an exponential law form. The inner and outer surfaces of the cylinder are considered to have combinations of free and clamped boundary conditions. Analytical solutions are given according to different types of the hollow cylinders. An extension of the present solutions to the viscoelastic ones and some applications are investigated in Part II.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

A. Zenkour, "Rotating Variable-Thickness Inhomogeneous Cylinders: Part I—Analytical Elastic Solutions," Applied Mathematics, Vol. 1 No. 6, 2010, pp. 481-488. doi: 10.4236/am.2010.16063.

 [1] Yu. é. Senitskii, “Stress State of a Rotating Inhomogeneous Anisotropic Cylinder of Variable Density,” International Applied Mechanics, Vol. 28, No. 5, 1992, pp. 28-35. [2] C. O. Horgan and A. M. Chan, “The Pressurized Hollow Cylinder or Disk Problem for Functionally Graded Isotropic Linear Elastic Materials,” Journal of Elasticity, Vol. 55, No. 1, 1999, pp. 43-59. [3] A. T. Vasilenko and N. I. Klimenko, “Stress State of Rotating Inhomogeneous Anisotropic Cylinders,” International Applied Mechanics, Vol. 35, No. 8, 1999, pp. 778-783. [4] F. Rooney and M. Ferrari, “Tension, Bending, and Flexure of Functionally Graded Cylinders,” International Journal of Solids and Structures, Vol. 38, No. 3, 2001, pp. 413-421. [5] Ya. M. Grigorenko and A. T. Vasilenko, “The Effect of Inhomogeneity of Elastic Properties on the Stress State in Composite Cylindrical Panels,” Mechanics of Composite Materials, Vol. 37, No. 2, 2001, pp. 85-90. [6] A. Oral and G. Anlas, “Effects of Radially Varying Moduli on Stress Distribution of Nonhomogeneous Anisotropic Cylindrical Bodies,” International Journal of Solids and Structures, Vol. 42, No. 20, 2005, pp. 5568-5588. [7] E. Pan and A. K. Roy, “A Simple Plane-strain Solution for Functionally Graded Multilayered Isotropic Cylinders,” Structural Engineering and Mechanics, Vol. 24, No. 6, 2006, pp. 727-740. [8] N. Tutuncu, “Stresses in Thick-walled FGM Cylinders with Exponentially-varying Properties,” Engineering Structures, Vol. 29, No. 9, 2007, pp. 2032-2035. [9] E. E. Theotokoglou and I. H. Stampouloglou, “The Radially Nonhomogeneous Elastic Axisymmentric Problem,” International Journal of Solids and Structures, Vol. 45, No. 25-26, 2008, pp. 6535-6552. [10] M. Mohammadi and J. R. Dryden, “Influence of the Spatial Variation of Poisson’s Ratio upon the Elastic Field in Nonhomogeneous Axisymmetric Bodies,” International Journal of Solids and Structures, Vol. 46, No. 3-4, 2009, pp. 788-795. [11] X. F. Li and X. L. Peng, “A Pressurized Functionally Graded Hollow Cylinder with Arbitrarily Varying Material Properties,” Journal of Elasticity, Vol. 96, No. 1, 2009, pp. 81-95. [12] Ya. M. Grigorenko and L. S. Rozhok, “Stress Analysis of Hollow Elliptic Cylinders with Variable Eccentricity and Thickness,” International Applied Mechanics, Vol. 38, No. 8, 2002, pp. 954-966. [13] A. M. Zenkour, “Rotating Variable-thickness Orthotropic Cylinder Containing a Solid Core of Uniform-thickness,” Archive of Applied Mechanics, Vol. 76, No. 1-2, 2006, pp. 89-102. [14] A. M. Zenkour, “Stresses in Cross-ply Laminated Circular Cylinders of Axially Variable Thickness,” Acta Mechanica, Vol. 187, No. 1-4, 2006, pp. 85-102. [15] W. H. Duan and C. G. Koh, “Axisymmetric Transverse Vibrations of Circular Cylindrical Shells with Variable Thickness,” Journal of Sound and Vibration, Vol. 317, No. 3-5, 2008, pp. 1035-1041. [16] G. J. Nie and R. C. Batra, “Static Deformations of Functionally Graded Polar-orthotropic Cylinders with Elliptical Inner and Circular Outer Surfaces,” Composites Science and Technology, Vol. 70, No. 3, 2010, pp. 450-457. [17] M. Abramowitz and A. I. Stegun, “Handbook of Mathematical Functions,” 5th Edition, US Government Printing Office, Washington DC, 1966. [18] H. J. Ding, H. M. Wang and W.Q. Chen, “A Solution of a Non-homogeneous Orthotropic Cylindrical Shell for Axisymmetric Plane Strain Dynamic Thermoelastic Problems,” Journal of Sound and Vibration, Vol. 236, No. 4, 2003, pp. 815-829. [19] L. S. Srinath, “Advanced Mechanics of Solids,” McGraw-Hill, India, 1983.