Effect of Initial Stress on the Propagation Characteristics of Waves in Fiber-Reinforced Transversely Isotropic Thermoelastic Material under an Inviscid Liquid Layer
The present investigation deals with the propagation of waves in fiber-reinforced transversely isotropic thermoelastic solid half space with initial stresses under a layer of inviscid liquid. The secular equation for surface equation in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficients of plane waves are studied numerically for a particular model. Effects of initial stress and thickness of the layer on the phase velocity, attenuation coefficient, and specific loss of energy are predicted graphically in the certain model. A particular case of Rayleigh wave has been discussed and the dispersion curves of the phase velocity and attenuation coefficients have also been presented graphically. Some other particular cases are also deduced from the present investigation. 1. Introduction Fiber-reinforced materials are widely used in engineering structures due to their superiority over the structural materials in applications requiring high strength and stiffness in lightweight components. Consequently, characterization of their mechanical behavior is of particular importance for structural design using these materials. Fibers are assumed an inherent material property, rather than some form of inclusion in models as Spencer [1]. In the case of an elastic solid reinforced by a series of parallel fibers it is usual to assume transverse isotropy. The idea of continuous self-reinforcement at every point of an elastic solid was introduced by Belfield et al. [2]. The characteristic property of reinforced concrete member is that its components, namely, concrete and steel, act together as a single anisotropic unit as main long as they remain in the elastic condition; that is, the two components are bound together so that there can be no relative displacement between them. The dynamical interaction between the thermal and mechanical fields in solids has great practical applications in modern aeronautics, astronautics, nuclear reactors, and high energy particle accelerators. The analysis of stress and deformation of fiber-reinforced composite materials has been an important subject of solid mechanics for last three decades. Pipkin [3] did pioneer works on the subject. Sengupta and Nath [4] discussed the problem of surface waves in fiber-reinforced anisotropic elastic media. Lord and Shulman [5] introduced a theory of generalized thermoelasticity with one relaxation time for an isotropic body. The theory was extended for anisotropic body by Dhaliwal and Sherief [6]. In this theory, a modified law of heat
References
[1]
A. J. M. Spencer, Deformation of Fibre-reinforced Materials, University of Oxford, Clarendon, Va, USA, 1941.
[2]
A. J. Belfield, T. G. Rogers, and A. J. M. Spencer, “Stress in elastic plates reinforced by fibres lying in concentric circles,” Journal of the Mechanics and Physics of Solids, vol. 31, no. 1, pp. 25–54, 1983.
[3]
A. C. Pipkin, “Finite deformations of ideal fiber-reinforced composites,” in Composites Materials, G. P. Sendeckyj, Ed., vol. 2 of Mechanics of Composite Materials, pp. 251–308, Academic Press, New York, NY, USA, 1973.
[4]
P. R. Sengupta and S. Nath, “Surface waves in fibre-reinforced anisotropic elastic media,” Indian Academy of Sciences, Sadhana Proceedings, vol. 26, pp. 363–370, 2001.
[5]
H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” Journal of the Mechanics and Physics of Solids, vol. 15, no. 5, pp. 299–309, 1967.
[6]
R. S. Dhaliwal and H. H. Sherief, “Generalized thermoelasticity for anisotropic media,” Quarterly of Applied Mathematics, vol. 38, no. 1, pp. 1–8, 1980/81.
[7]
A. U. Erdem, “Heat Conduction in fiber-reinforced rigid bodies,” 10 Ulusal Ist Bilimi ve Tekmgi Kongrest (ULIBTK), 6–8 Eylul, Ankara, Turkey, 1995.
[8]
R. Kumar and R. R. Gupta, “Study of wave motion in an anisotropic fiber-reinforced thermoelastic solid,” Journal of Solid Mechanics, vol. 2, no. 1, pp. 91–100, 2010.
[9]
P. Chadwick and L. T. C. Seet, “Wave propagation in a transversely isotropic heat-conducting elastic material,” Mathematika, vol. 17, pp. 255–274, 1970.
[10]
H. Singh and J. N. Sharma, “Generalized thermoelastic waves in transversely isotropic media,” Journal of the Acoustical Society of America, vol. 77, pp. 1046–1053, 1985.
[11]
B. Singh, “Wave propagation in an anisotropic generalized thermoelastic solid,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 10, pp. 1479–1485, 2003.
[12]
P. Chadwick, Progress in Solid Mechanics, vol. 1, North-Holland, Amsterdam, The Netherlands, 1960, edited by R. Hill, I. N. Sneddon.
[13]
P. Chadwick and D. W. Windle, “Propagation of Rayleigh waves along isothermal and insulated boundaries,” Proceedings of the Royal Society of America, vol. 280, pp. 47–71, 1964.
[14]
J. N. Sharma and H. Singh, “Thermoelastic surface waves in a transversely isotropic half space with thermal relaxations,” Indian Journal of Pure and Applied Mathematics, vol. 16, no. 10, pp. 1202–1219, 1985.
[15]
A. Montanaro, “On singular surfaces in isotropic linear thermoelasticity with initial stress,” Journal of the Acoustical Society of America, vol. 106, no. 3, pp. 1586–1588, 1999.
[16]
J. Wang and P. Slattery, “Thermoelasticity without energy dissipation for initially stressed bodies,” International Journal of Mathematics and Mathematical Sciences, vol. 31, no. 6, pp. 329–337, 2002.
[17]
D. Ie？an, “A theory of prestressed thermoelastic Cosserat continua,” Journal of Applied Mathematics and Mechanics, vol. 88, no. 4, pp. 306–319, 2008.
[18]
K. Ames and B. Straughan, “Continuous dependence results for initially prestressed thermoelastic bodies,” International Journal of Engineering Science, vol. 30, no. 1, pp. 7–13, 1992.
[19]
J. Wang, R. S. Dhaliwal, and S. R. Majumdar, “Some theorems in the generalized theory of thermoelasticity for prestressed bodies,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 2, pp. 267–276, 1997.
[20]
M. Marin and C. Marinescu, “Thermoelasticity of initially stressed bodies, asymptotic equipartition of energies,” International Journal of Engineering Science, vol. 36, no. 1, pp. 73–86, 1998.
[21]
V. V. Kalinchuk, T. I. Belyankova, Y. E. Puzanoff, and I. A. Zaitseva, “Some dynamic properties of the nonhomogeneous thermoelastic prestressed media,” The Journal of the Acoustical Society of America, vol. 105, no. 2, p. 1342, 1999.
[22]
M. I. A. Othman and Y. Song, “Reflection of plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation,” International Journal of Solids and Structures, vol. 44, no. 17, pp. 5651–5664, 2007.
[23]
B. Singh, “Wave propagation in an initially stressed transversely isotropic thermoelastic solid half-space,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 705–715, 2010.
[24]
M. I. A. Othman, S. M. Said, and N. Sarker, “Effect of hydrostatic initial stress on a fiber-reinforced thermoelastic medium with fractional derivative heat transfer,” Multidiscipline Modeling in Materials and Structures, vol. 9, no. 3, pp. 410–426, 2013.
[25]
A. M. Abd-Alla, S. M. Abo-Dahab, and A. Al-Mullise, “effects of rotation and gravity field on surface waves in fibre-reinforced thermoela stic media under four theories,” Journal of Applied Mathematics, vol. 2013, Article ID 562369, 10 pages, 2013.
[26]
M. I. A. Othman and S. Y. Atwa, “Effect of rotation on a fiber-reinforced thermo-elastic under Green-Naghdi theory and influence of gravity,” Meccanica, vol. 49, no. 1, pp. 23–36, 2014.
[27]
I. A. Abbas, “Generalized magneto-thermoelastic interaction in a fiber-reinforced anisotropic hollow cylinder,” International Journal of Thermophysics, vol. 33, no. 3, pp. 567–579, 2012.
[28]
I. A. Abbas, “A GN model for thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a circular hole,” Applied Mathematics Letters, vol. 26, no. 2, pp. 232–239, 2013.
[29]
I. A. Abbas and A. Zenkour, “Two-temperature generalized thermoelastic interaction in an infinite fiber-reinforced anisotropic plate containing a circular cavity with two relaxation times,” Journal of Computational and Theoretical Nanoscience, vol. 11, no. 1, pp. 1–7, 2014.
[30]
I. A. Abbas and M. I. A. Othman, “Generalized thermoelastic interaction in a fiber-reinforced anisotropic half-space under hydrostatic initial stress,” Journal of Vibration and Control, vol. 18, no. 2, pp. 175–182, 2012.
[31]
W. M. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York, NY, USA, 1957.
[32]
H. Kolsky, Stress Waves in Solids, Clarendon Press, Dover, New York, NY, USA, 1963.
[33]
M. A. Ezzat, “Fundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect conductor cylindrical region,” International Journal of Engineering Science, vol. 42, no. 13-14, pp. 1503–1519, 2004.
