Study of Torsional Vibrations of Composite Poroelastic Spherical Shell-Biot’s Extension Theory
Subject Areas : Engineering
R
Gurijala
1
(Department of Mathematics, Kakatiya University, Warangal, Telangana, India)
M
Reddy Perati
2
(Department of Mathematics, Kakatiya University, Warangal, Telangana, India)
Keywords: Frequency equation, Attenuation, Torsional vibrations, Composite spherical shell, Phase velocity,
Abstract :
Torsional vibrations of composite poroelastic dissipative spherical shell are investigated in the framework of Biot’s extension theory.Here composite poroelastic spherical shell consists of two spherical shells, one is placed on other, and both are made of different poroelastic materials. Consideration of the stress-free boundaries of outer surface and the perfect bonding between two shells leads to complex valued frequency equation. Limiting case when the ratio of thickness to inner radius is very small is investigated numerically. In this case, thick walled composite spherical shell reduces to thin composite spherical shell. For illustration purpose, four composite materials, namely, Berea sandstone saturated with water and kerosene, Shale rock saturated with water and kerosene are employed. The particular cases of a poroelastic solid spherical shell and poroelastic thick walled hollow spherical shell are discussed. If the shear viscosity of fluid is neglected, then the problem reduces to that of classical Biot’s theory. Phase velocity and attenuation are computed and the results are presented graphically. Comparison is made between the results of Biot’s extension theory and that of classical Biot’s theory. It is conclude that shear viscosity of fluid is causing the discrepancy of the numerical results.
[1] Richard Rand H., 1968, Torsional vibrations of elastic prolate spheroids, Journal of the Acoustical Society of America 44 (3): 749-751.
[2] Heyliger P.R., Pan E., 2016, Free vibrations of layered magneto electro elastic spheres, Journal of the Acoustical Society of America 140(2): 988-999.
[3] Biot M.A., 1956, The theory of propagation of elastic waves in fluid-saturated porous solid, Journal of the Acoustical Society of America 28: 168-178.
[4] Shah S.A., Tajuddin M., 2011, Torsional vibrations of poroelastic prolate spheroids, International Journal of Applied Mechanics and Engineering 16: 521-529.
[5] Shanker B., Nageswara Nath C., Ahmed Shah S., Manoj Kumar J., 2013, Vibration analysis of a poroelastic composite hollow sphere, Acta Mechanica 224: 327-341.
[6] Rajitha G., Sandhya Rani B., Malla Reddy P., 2012, Vibrations in a plane angular sector of poroelastic elliptic cone, Special Topics and Reviews in Porous Media 3(2): 157-168.
[7] Rajitha G., Malla Reddy P., 2014, Investigation of flexural vibrations in poroelastic elliptic cone, Proceedings of International Conference on Mathematical Sciences.
[8] Rajitha G., Malla Reddy P., 2014, Axially symmetric vibrations of composite poroelastic spherical shell, International Journal of Engineering Mathematics 2014: 416406.
[9] Shah A., Nageswaranath Ch., Ramesh M., Ramanamurthy M.V., 2017, Torsional vibrations of coated hollow poroelastic spheres, Open Journal of Acoustics 7: 18-26.
[10] Sahay P.N., 1996, Elasto dynamics of deformable porous media, Proceedings of the Royal Society of London A 452: 1517-1529.
[11] Solorza S., Sahay P.N., 2009, On extensional waves in a poroelastic cylinder within the framework of viscosity-extended Biot theory: The case of traction-free open-pore cylindrical surface, Geophysics Journal International 179: 1679-1702.
[12] Malla Reddy P., Rajitha G., 2015, Investigation of torsional vibrations of thick-walled hollow poroelastic cylinder using Biot's extension theory, Indian Academy of Sciences 40(6): 1925-1935.
[13] Rajitha G., Malla Reddy P., 2018, Analysis of radial vibrations in thick walled hollow poroelastic cylinder in the framework of Biot’s extension theory, Multidiscipline Modeling in Materials and Structures 14(5): 970-983.
[14] Milton A., Irene A.S., 1964, Handbook of Mathematical Functions, Dover Publications, National Bureau of Standard Applied Mathematics Series 55.
[15] Sandhya Rani B., Anand Rao J., Malla Reddy P., 2018, Study of radial vibrations in an infinitely long fluid-filled transversely isotropic thick-walled hollow composite poroelastic cylinders, Journal of Theoretical and Applied Mechanics 48(3): 31-44.