Abstract
The dynamic solution of a multilayered spherically isotropic piezoelectric hollow sphere subjected to radial dynamic loads is obtained. By the method of superposition, the solution is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static part is derived by the state-space method, and the dynamic part is obtained by the method of separation of variables coupled with the initial parameter method as well as the orthogonal expansion technique. By using the quasi-static and dynamic parts, the electric boundary conditions as well as the electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully by means of the interpolation method. The displacements, stresses and electric potentials are finally obtained. The present method is suitable for a multilayered spherically isotropic piezoelectric hollow sphere consisting of arbitrary layers and subjected to arbitrary spherically symmetric dynamic loads. Finally, numerical results are presented and discussed.
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References
Loza, I.A., Shul’ga, N.A.: Axisymmetric vibrations of a hollow piezoceramic sphere with radial polarization. Sov Appl Mech 20, 113–117 (1984)
Loza, I.A., Shul’ga, N.A.: Forced axisymmetric vibrations of a hollow piezoceramic sphere with an electrical method of excitation. Sov Appl Mech 26, 818–822 (1990)
Shul’ga, N.A.: Electroelastic oscillation of a piezoceramic sphere with radial polarization. Sov Appl Mech 22, 497–500 (1986)
Shul’ga, N.A.: Radial electroelastic vibrations of a hollow piezoceramic sphere. Sov Appl Mech 22, 731–734 (1990)
Shul’ga, N.A.: Harmonic electroelastic oscillations of spherical bodies. Sov Appl Mech 29, 812–817 (1993)
Cai, J.B., Chen, W.Q., Ye, G.R., Ding, H.J.: Nature frequencies of submerged piezoceramic hollow spheres. Acta Mech Sin 16, 55–62 (2000)
Chen, W.Q., Ding, H.J., Xu, R.Q.: Three-dimensional free vibration analysis of a fluid-filled piezoelectric hollow sphere. Comput Struct 79, 653–663 (2001)
Borisyuk, A.I., Kirichok, I.F.: Steady-state radial vibrations of piezoceramic spheres in compressible fluid. Sov Appl Mech 15, 936–940 (1979)
Ding, H.J., Wang, H.M., Chen, W.Q.: Transient responses in a piezoelectric spherically isotropic hollow sphere for symmetric problems. ASME J Appl Mech 70, 436–445 (2003)
Liu, G.R., Tani, J.: Surface waves in functionaly gradient piezoelectric plates. J Vib Acoust 116, 440–448 (1994)
Ray, M.C., Bhattacharya, R., Samanta, B.: Exact solutions for dynamic analysis of composite plates with distributed piezoelectric layers. Comput Struct 66, 737–743 (1998)
Heyliger, P.R., Ramirez, G.: Free vibration of laminated circular piezoelectric plates and discs. J Sound Vib 229, 935–956 (2000)
Liu, G.R., Dai, K.Y., Han, X., Ohyoshi, T: Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates. J Sound Vib 268, 131–147 (2003)
Siao, J.C.T., Dong S.B., Song J.: Frequency spectra of laminated piezoelectric cylinders. J Vib Acoust 116, 364–370 (1994)
Kharouf, N., Heyliger P.R.: Axisymmetric free vibrations of homogeneous and laminated piezoelectric cylinders. J Sound Vib 174, 539–561 (1994)
Li, H.Y., Lin, Q.R., Liu, Z.X., Wang, C.: Free vibration of piezoelectric laminated cylindrical shells under hydrostatic pressure. Int J Solids Struct 38, 7571–7585 (2001)
Heyliger, P., Wu, Y.C.: Electroelastic fields in layered piezoelectric spheres. Int J Eng Sci 37, 143–161 (1999)
Chen, W.Q., Free vibration analysis of laminated piezoelectric hollow sphere. J Acoust Soc Am 109, 41–50 (2001)
Li, H.Y., Liu, Z.X., Lin, Q.R.: Spherical-symmetric steady-state response of piezoelectric spherical shell under external excitation. Appl Math Mech 21, 947–956 (2000)
Wang, X., Lu, G., Guillow, S.R.: Stress wave propagation in orthotropic laminated thick-walled spherical shells. Int J Solids Struct 39, 4027–4037 (2002)
Ding, H.J., Wang, H.M., Chen, W.Q.: Elastodynamic solution for spherically symmetric problems of a multilayered hollow sphere. Arch Appl Mech 73, 753–768 (2004)
Ding, H.J., Wang, H.M., Chen W.Q.: Dynamic responses of a functionally graded pyroelectric hollow sphere for spherically symmetric problems. Int J Mech Sci 45, 1029–1051 (2003)
Ding, H.J., Wang, H.M., Chen W.Q.: New numerical method for Volterra integral equation of the second kind in piezoelectric dynamic problems. Appl Math Mech 25, 16–23 (2004)
Berry, J.G., Naghdi, P.M.: On the vibration of elastic bodies having time-dependent boundary conditions. Quart Appl Math 14, 43–50 (1956)
Yin, X.C., Yue, Z.Q.: Transient plane-strain response of multilayered elastic cylinders to axisymmetric impulse. ASME J Appl Mech 69, 825–835 (2002)
Kress, R.: Linear integral equations, applied mathematical sciences, volume 82. Springer, Berlin Heidelberg New York, 1989
Adelman, N.T., Stavsky, Y., Segal, E.: Axisymmetric vibration of radially polarized piezoelectric ceramic cylinders. J Sound Vib 38, 245–254 (1975)
Kharouf, N., Heyliger, P.R.: Axisymmetric free vibrations of homogeneous and laminated piezoelectric cylinders. J Sound Vib 174, 539–561 (1994)
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Wang, H., Ding, H. & Chen, Y. Transient responses of a multilayered spherically isotropic piezoelectric hollow sphere. Arch Appl Mech 74, 581–599 (2005). https://doi.org/10.1007/s00419-005-0374-9
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DOI: https://doi.org/10.1007/s00419-005-0374-9