Abstract
The nonaxisymmetric problem of natural vibrations of a hollow sphere made of functionally gradient piezoelectric material is solved based on 3D electroelasticity. The properties of the material change continuously along a radial coordinate according to an exponential law. The external surface of the sphere is free of tractions and either insulated or short-circuited by electrodes. After separation of variables and representation of the components of the displacements and of the stress tensor in terms of spherical functions, the initially three-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed by ordinary differential equations. This problem is solved by a stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Moreover, a numerical investigation is performed based on the algorithm used for solving the problem. In particular, we investigate the influence of the geometric and electric parameters on the frequency spectrum at the nonaxisymmetry of natural vibrations of an inhomogeneous piezoceramic thick-walled sphere.
Similar content being viewed by others
References
Bailey T., Hubbard E.: Modeling and analysis of functionally graded materials and structures. J. Guid. Control. Dyn. 8(5), 605–611 (1985)
Baz A., Poh S.: Performance on an active control system with piezoelectric actuators. J. Sound. Vib. 126(2), 327–343 (1988)
Berlincourt, D.: Piezoelectric crystals and ceramics. In: Mattiat, O.E. (ed.) Ultrasonic Transducer Materials. Plenum Press, New York, ch. 2, pp. 62–124 (1971)
Birman V., Byrd L.W.: Modeling and analysis of functionally graded materials and structures. ASME Appl. Mech. Rev. 60, 195–216 (2007)
Boriseiko V.A., Grinchenko V.T., Ulitko A.F.: Relations of electroelasticity for piezoceramic shells of revolution. Sov. Appl. Mech. 12(2), 126–131 (1971)
Chen W.T.: On some problems in spherically isotropic elastic materials. Trans. ASME Ser. E 33(3), 347–355 (1966)
Chen W.Q., Ding H.J.: On free vibrations of a functionally graded piezoelectric rectangular plate. Acta Mech. 153, 207–216 (2002)
Chen W.Q., Wang L.Z., Lu Y.: Free vibrations of functionally graded piezoceramic hollow spheres with radial polarization. J. Sound Vib. 251(1), 103–114 (2002)
Chen W.Q., Lu Y., Ye J.R., Cai J.B.: 3D electroelastic fields in a functionally graded piezoceramic hollow sphere under mechanical and electric loading. Arch. Appl. Mech. 72, 39–51 (2002)
Chree C.: On the longitudinal vibrations of aeolotropic bars with one axis of material symmetry. Q. J. Math. 24, 340–358 (1890)
Crawly, E.F., de Luis, J.: Use of piezoelectric actuators as elements of intelligent structures. AIAA J. 25(10), 1373–1385 (1987)
Dai H.L., Wang X.: Transient wave propagation in piezoelectric hollow spheres subjected to thermal shock and electric excitation. Struct. Eng. Mech. 19(4), 441–457 (2005)
Dai H.L., Fu Y.M., Yang J.H.: Electromagnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere. Acta Mech. Sinica 23, 55–63 (2007)
Grigorenko, Ya.M., Vasilenko, A.T, Pankratova, N.D.: Statics of Anisotropic Shells. Kiev, Vyshch. Shkola (in Russian) (1985)
Grigorenko, Ya.M., Grigorenko, A.Ya., Vlaikov, G.G.: Problems of Mechanics for Anisotropic Inhomogeneous Shells on Basic of Different Models. Kiev, Akademperiodika, p. 549 (2009)
Grigorenko A., Müller W.H., Wille R., Yaremchenko S.: Numerical solution of stress-strain state in hollow cylinder by Means of spline approximation. J. Math. Sci. 180(2), 135–145 (2012a)
Grigorenko A., Müller W.H., Wille R., Loza I.: Nonaxisymmetric vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials. Continuum Mech. Thermodyn. 24(4-6), 515–524 (2012b)
Jaerisch P.: Über die elastischen Schwingungen einer isotropen Kugel. J. Math. (Crelle) 88, 131–145 (1880)
Lamb H.: On the vibration of an elastic sphere. Proc. Lond. Math. Soc. 13, 189–212 (1882)
Love, A.E.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, reprint in USA (1952)
Loza I.A.: Nonaxisymmetric vibrations of hollow inhomogeneous sphere with piezoceramic layers. Rep. NAN Ukraine (in Russian) 11, 137–143 (2011)
Loza I.A., Shul’ga N.A.: Axisymmetric vibrations of a hollow piezoceramic sphere with radial polarization. Int. Appl. Mech. 20(2), 113–117 (1984)
Loza I.A., Shul’ga N.A.: Forced axisymmetric vibrations of a hollow piezoceramic sphere with an electrical method of excitation. Int. Appl. Mech. 26(6), 818–821 (1990)
Morse F.M., Feshbach G.: Methods of Theoretical Physics. McCraw-Hill, NY (1953)
Petrashen’ G.I.: Vibrations of an isotropic elastic sphere. Dokl. Akad. Nauk SSSR (in Russian) 47(3), 177–181 (1947)
Poisson, S.D.: Memoire sur l’equilibre et le movement des corps elastique. L’Académie Royale des Sciences, VIII, pp. 357–570 (1829)
Sato Y., Usami T.: Basic study on the oscillation on a homogeneous elastic sphere. Geophys. Mag. Tokyo 31, 15–62 (1962)
Shul’ga N.A.: Electroelastic oscillations of a piezoceramic sphere with radial polarization. Int. Appl. Mech. 22(5), 497–500 (1986)
Shul’ga N.A., Grigorenko A.Ya., Efimova T.L.: Free non-axisymmetric oscillations of a thick-walled nonhomogeneous transversally isotropic hollow sphere. Int. Appl. Mech. 24(5), 439–444 (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Rights and permissions
About this article
Cite this article
Grigorenko, A.Y., Müller, W.H., Wille, R. et al. Nonaxisymmetric electroelastic vibrations of a hollow sphere made of functionally gradient piezoelectric material. Continuum Mech. Thermodyn. 26, 771–781 (2014). https://doi.org/10.1007/s00161-014-0337-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-014-0337-x