On Axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells

ABSTRACT

Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hollow poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially symmetric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervious and an impervious surface. Results of previous works are obtained as a particular case of the present study.

Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hollow poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially symmetric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervious and an impervious surface. Results of previous works are obtained as a particular case of the present study.

KEYWORDS

Biot’s Theory, Axially Symmetric Vibrations, Radial Vibrations, Rotatory Vibrations, Spherical Shell, Elastic Fluid, Pervious Surface, Impervious Surface, Frequency

Biot’s Theory, Axially Symmetric Vibrations, Radial Vibrations, Rotatory Vibrations, Spherical Shell, Elastic Fluid, Pervious Surface, Impervious Surface, Frequency

Cite this paper

nullS. Shah and M. Tajuddin, "On Axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells,"*Open Journal of Acoustics*, Vol. 1 No. 2, 2011, pp. 15-26. doi: 10.4236/oja.2011.12003.

nullS. Shah and M. Tajuddin, "On Axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells,"

References

[1] R. Kumar, “Axially Symmetric Vibrations of a Fluid- Filled Spherical Shell,” Acustica, Vol. 21, 1969, pp. 143- 149.

[2] R. Rand and F. DiMaggio, “Vibrations of Fluid Filled Spherical and Spheroidal Shells,” Journal of the Acoustical Society of America, Vol. 42, No. 6, 1967, pp. 1278- 1286. doi:10.1121/1.1910717

[3] M. A. Biot, “Theory of Propagation of Elastic Waves in Fluid-Saturated Porous Solid,” Journal of the Acoustical Society of America, Vol. 28, 1956, pp. 168-178. doi:10.1121/1.1908239

[4] S. Paul, “A Note on the Radial Vibrations of a Sphere of Poroelastic Material,” Indian Journal of Pure and Applied Mathematics, Vol. 7, 1976, pp. 469-475.

[5] G. Chao, D. M. J. Smeulders and M. E. H. van Dongen, “Sock-Induced Borehole Waves in Porous Formations: Theory and Experiments,” Journal of the Acoustical Society of America, Vol. 116, No. 2, 2004, pp. 693-702. doi:10.1121/1.1765197

[6] S. Ahmed Shah, “Axially Symmetric Vibrations of Fluid- Filled Poroelastic Circular Cylindrical Shells,” Journal of Sound and Vibration, Vol. 318, No. 1-2, 2008, pp. 389- 405. doi:10.1016/j.jsv.2008.04.012

[7] J. N. Sharma and N. Sharma, “Three Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermoelastic Sphere,” Journal of Applied Mechanics - Transactions of the ASME, Vol. 77, No. 2, 2010, p. 021004.

[8] S. Ahmed Shah and M. Tajuddin, “Torsional Vibrations of Poroelastic Prolate Spheroids,” International Journal of Applied Mechanics and Engineering, Vol. 16, 2011, pp. 521-529.

[9] A. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” National Bureau of Standards, Wa- shington, 1965.

[10] I. Fatt, “The Biot-Willis Elastic Coefficients for a Sandstone,” Journal of Applied Mechanics, Vol. 26, 1959, pp. 296-296.

[11] C. H. Yew, and P. N. Jogi, “Study of Wave Motions in Fluid-Saturated Porous Rocks,” Journal of the Acoustical Society of America, Vol. 60, 1976, pp. 2-8. doi:10.1121/1.381045

[1] R. Kumar, “Axially Symmetric Vibrations of a Fluid- Filled Spherical Shell,” Acustica, Vol. 21, 1969, pp. 143- 149.

[2] R. Rand and F. DiMaggio, “Vibrations of Fluid Filled Spherical and Spheroidal Shells,” Journal of the Acoustical Society of America, Vol. 42, No. 6, 1967, pp. 1278- 1286. doi:10.1121/1.1910717

[3] M. A. Biot, “Theory of Propagation of Elastic Waves in Fluid-Saturated Porous Solid,” Journal of the Acoustical Society of America, Vol. 28, 1956, pp. 168-178. doi:10.1121/1.1908239

[4] S. Paul, “A Note on the Radial Vibrations of a Sphere of Poroelastic Material,” Indian Journal of Pure and Applied Mathematics, Vol. 7, 1976, pp. 469-475.

[5] G. Chao, D. M. J. Smeulders and M. E. H. van Dongen, “Sock-Induced Borehole Waves in Porous Formations: Theory and Experiments,” Journal of the Acoustical Society of America, Vol. 116, No. 2, 2004, pp. 693-702. doi:10.1121/1.1765197

[6] S. Ahmed Shah, “Axially Symmetric Vibrations of Fluid- Filled Poroelastic Circular Cylindrical Shells,” Journal of Sound and Vibration, Vol. 318, No. 1-2, 2008, pp. 389- 405. doi:10.1016/j.jsv.2008.04.012

[7] J. N. Sharma and N. Sharma, “Three Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermoelastic Sphere,” Journal of Applied Mechanics - Transactions of the ASME, Vol. 77, No. 2, 2010, p. 021004.

[8] S. Ahmed Shah and M. Tajuddin, “Torsional Vibrations of Poroelastic Prolate Spheroids,” International Journal of Applied Mechanics and Engineering, Vol. 16, 2011, pp. 521-529.

[9] A. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” National Bureau of Standards, Wa- shington, 1965.

[10] I. Fatt, “The Biot-Willis Elastic Coefficients for a Sandstone,” Journal of Applied Mechanics, Vol. 26, 1959, pp. 296-296.

[11] C. H. Yew, and P. N. Jogi, “Study of Wave Motions in Fluid-Saturated Porous Rocks,” Journal of the Acoustical Society of America, Vol. 60, 1976, pp. 2-8. doi:10.1121/1.381045