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Natural vibration of the three-layered solid sphere with middle layer made of FGM: three-dimensional approach
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 Title & Authors
Natural vibration of the three-layered solid sphere with middle layer made of FGM: three-dimensional approach
Akbarov, Surkay D.; Guliyev, Hatam H.; Yahnioglu, Nazmiye;
 Abstract
The paper studies the natural oscillation of the three-layered solid sphere with a middle layer made of Functionally Graded Material (FGM). It is assumed that the materials of the core and outer layer of the sphere are homogeneous and isotropic elastic. The three-dimensional exact equations and relations of linear elastodynamics are employed for the investigations. The discrete-analytical method proposed by the first author in his earlier works is applied for solution of the corresponding eigenvalue problem. It is assumed that the modulus of elasticity, Poisson`s ratio and density of the middle-layer material vary continuously through the inward radial direction according to power law distribution. Numerical results on the natural frequencies related to the torsional and spheroidal oscillation modes are presented and discussed. In particular, it is established that the increase of the modulus of elasticity (mass density) in the inward radial direction causes an increase (a decrease) in the values of the natural frequencies.
 Keywords
functionally graded material;three-layered solid sphere;natural vibration;natural frequencies;torsional oscillation;spheroidal oscillation;
 Language
English
 Cited by
 References
1.
Akbarov, S.D. (2006), "Frequency response of the axisymmetrically finite pre-stretched slab from incompressible functionally graded material on a rigid foundation", Int. J. Eng. Sci., 44, 484-500. crossref(new window)

2.
Akbarov, S.D. (2015), Dynamics of Pre-Strained Bi-Material Elastic Systems: Linearized Three-Dimensional Approach, Springer-Heidelberg, New York.

3.
Anderson, D.L. (2007), New Theory of the Earth, Cambridge University Press.

4.
Asgari, M. and Akhlagi, M., (2011), "Natural frequency analysis of 2D-FGM thick hollow cylinder based on three-dimensional elasticity equations", Eur. J. Mech. A-Solid, 30, 72-81. crossref(new window)

5.
Asemi, K., Salehi, M. and Sadighi, M. (2014), "Three dimensional static and dynamic analysis of two dimensional functionally graded annular sector plates", Struct. Eng. Mech., 51, 1067-1089. crossref(new window)

6.
Chen, W.Q. and Ding, H.J. (2001), "Free vibration of multi-layered spherically isotropic hollow spheres", Int. J. Mech. Sci., 43, 667-680. crossref(new window)

7.
Chree, C. (1889), "The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solution and applications", Tran. Cambridge Philos. Soc., 14, 250-309.

8.
Eringen, A.C. and Suhubi, E.S. (1975), Elastodynamics. Vol I. Finite motion; Vol II. Linear theory, Academic Press, New York.

9.
Grigorenko, Y.M. and Kilina, T.N. (1989), "Analysis of the frequencies and modes of natural vibration of laminated hollow spheres in three- and two-dimensional formulations", Int. Appl. Mech., 25, 1165-1171.

10.
Guz, A.N. (1985a), "Dynamics of an elastic isotropic sphere of an incompressible material subjected to initial uniform volumetric loading", Int. Appl. Mech., 21(8), 738-746.

11.
Guz, A.N. (1985b), "Dynamics of an elastic isotropic sphere of a compressible material under cubic initial loading", Int. Appl. Mech., 21(12), 1153-1159.

12.
Hasheminejad, S.M. and Mirzaei, Y. (2011), "Exact 3D elasticity solution for free vibrations of an eccentric hollow sphere", J. Sound Vib., 330, 229-244. crossref(new window)

13.
Ilhan, N. and Koc, N. (2015), "Influence of polled direction on the stress distribution in piezoelectric materials", Struct. Eng. Mech., 54, 955-971. crossref(new window)

14.
Ipek, C. (2015), "The dispersion of the flexural waves in a compound hollow cylinder under imperfect contact between layers", Struct. Eng. Mech., 55, 338-348.

15.
Jiang, H., Young, P.G. and Dickinson, S.M. (1996), "Natural frequencies of vibration of layered hollow spheres using exact three-dimensional elasticity equations", J. Sound Vib., 195(1), 155-162. crossref(new window)

16.
Lamb, H. (1882), "On the vibrations of an elastic sphere", Pr. London Math. Soc., 13, 189-212.

17.
Lapwood, E.R. and Usami, T. (1981), Free Oscillations of the Earth, Cambridge University Press.

18.
Love, A.E.H. (1944), A Treatise on the Mathematical Theory of Elasticity, Dover, New York.

19.
Sato, Y. and Usami, T. (1962a), "Basic study on the oscillation of a homogeneous elastic sphere; part I, frequency of the free oscillations", Geoph. Mag., 31, 15-24.

20.
Sato, Y. and Usami, T. (1962b), "Basic study on the oscillation of a homogeneous elastic sphere; part II, distribution of displacement", Geoph. Mag., 31, 25-47.

21.
Sato, Y., Usami, T. and Ewing, M. (1962), "Basic study on the oscillation of a homogeneous elastic sphere, IV. Propagation of disturbances on the sphere", Geoph. Mag., 31, 237-242.

22.
Shah, A.H., Ramakrishnan, C.V. and Datta, S.K. (1969a), "Three dimensional and shell theory analysis of elastic waves in a hollow sphere, Part I. Analytical foundation", J. Appl. Mech., 36, 431-439. crossref(new window)

23.
Shah, A.H., Ramakrishnan, C.V. and Datta, S.K. (1969b), "Three dimensional and shell theory analysis of elastic waves in a hollow sphere, Part II. Numerical results", J. Appl. Mech., 36, 440-444. crossref(new window)

24.
Sharma, J.N., Sharma, D.K. and Dhaliwai, S.S. (2012), "Free vibration analysis of a viscotermoelastic solid sphere", Int. J. Appl. Math. Mech., 8(11), 45-68.

25.
Ye, T., Jin, G. and Su, Z. (2014), "Three-dimensional vibration analysis of laminated functionally graded spherical shells with general boundary conditions", Compos. Struct., 116, 571-588. crossref(new window)

26.
Yun, W., Rongqiao, X. and Haojiang, D. (2010), "Three-dimensional solution of axisymmetric bending of functionally graded circular plates", Compos. Struct., 92, 1683-1693. crossref(new window)