26. Plane wave propagation in a 3D anisotropic half‑space under Green-Naghdi theory II

S. Chakraborty1, S. C. Mandal2, A. K. Das3, N. Sarkar4, A. Lahiri5

1, 2, 5Department of Mathematics, Jadavpur University, Kolkata, 700032, India

3Department of Geology, Asutosh College, 92, S. P. Mukherjee Road, Kolkata, 700026, India

4Department of Applied Mathematics, University of Calcutta, Kolkata, 700009, India

4Corresponding author

E-mail: 1csanjukjta1977@gmail.com, 2scmandal@math.jdvu.ac.in, 3akdas.geology@gmail.com, 4nsarkarindian@gmail.com, 5lahiriabhijit2000@yahoo.com

Received 26 August 2016; accepted 27 October 2016

DOI https://doi.org/10.21595/mme.2016.17621

Abstract. In this paper, the theory of coupled thermoelasticity in three dimension is employed for triclinic half-space, subjected to time dependent heat source on the boundary of the space which is traction free and is considered in the context of Green-Naghdi model of type II (thermoelasticity without energy dissipation) of generalized thermoelasticity. Normal mode analysis is used to the non-dimensional coupled equations. Finally, the resulting equations are written in the form of a vector-matrix differential equation which is then solved by eigenvalue approach. Numerical results for the temperature, thermal stresses, and displacements are presented graphically and analyzed. Mathematical results shown in thermoelastic curves were supplemented by tectonic movements of elastic lithospheric plates.

Keywords: anisotropic solid, Green-Naghdi model II, normal mode analysis, eigenvalue approach, lithostatic pressure and lithospheric plates.

References

[1]        Biot M. A. Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics, Vol. 27, 1956, p. 240‑253.

[2]        Lord H. W., Shulman Y. A generalized dynamical theory of thermoelasticity. Journal of Mechanics and Physics of Solids, Vol. 15, 1967, p. 299‑309.

[3]        Dhaliwal R. S., Sherief H. H. Generalized thermoelasticity for anisotropic media. Quarterly of Applied Mathematics, Vol. 33, 1980, p. 1‑8.

[4]        Ignaczak J. Uniqueness in generalized thermoelasticity. Journal of Thermal Stresses, Vol. 2, 1979, p. 171‑175.

[5]        Ignaczak J. A note on uniqueness in thermoelasticity with one relaxation time. Journal of Thermal Stresses, Vol. 5, 1982, p. 257‑263.

[6]        Green A. E., Lindsay K. A. Thermoelasticity. Journal of Elasticity, Vol. 2, 1972, p. 1‑7.

[7]        Green A. E., Naghdi P. M. A re-examination of the basic postulate of thermomechanics. Proceedings of Royal Society of London, Vol. 432, 1991, p. 171‑194.

[8]        Green A. E., Naghdi P. M. On undamped heat waves in an elastic solid. Journal of Thermal Stresses, Vol. 15, 1992, p. 253‑264.

[9]        Green A. E., Naghdi P. M. Thermoelasticity without energy dissipation. Journal of Elasticity, Vol. 31, 1993, p. 189‑208.

[10]     Roychoudhuri S. K., Bandyopadhyay N. Thermoelastic wave propagation in a rotating elastic medium without energy dissipation. International Journal of Mathematics and Mathematical Sciences, Vol. 1, 2004, p. 99‑107.

[11]     Roychoudhuri S. K., Dutta P. S. Thermoelastic interaction without energy dissipation in an infinite solid with distributed periodically varying heat sources. International Journal of Solids and Structures, Vol. 42, 2005, p. 4192‑4203.

[12]     Sharma J. N., Chouhan R. S. On the problem of body forces and heat sources in thermoelasticity without energy dissipation. Indian Journal of Pure and Applied Mathematics, Vol. 30, 1999, p. 595‑610.

[13]     Chandrasekharaiah D. S., Srinath K. S. Thermoelastic plane waves without energy dissipation in a rotating body. Mechanics Research Communications, Vol. 24, 2000, p. 551‑560.

[14]     Sarkar N., Lahiri A. A three dimensional thermoelastic problem for a half Space without energy dissipation. International Journal of Engineering Science, Vol. 51, 2012, p. 310‑325.

[15]     Bachher M., Sarkar N., Lahiri A. State-space approach to 3D generalized thermoviscoelasticity under Green and Naghdi theory II. Mathematical Models in Engineering, Vol. 1, 2015, p. 111‑124.

[16]     Othman M. I. A., Lotfy K. H., Farouk R. K. Transient disturbance in a half-space using generalized magneto-thermoelasticity with internal heat source. Acta Physica Polonica A, Vol. 116, 2009, p. 185‑192.

[17]     Pal P. K., Acharya D. Effects of inhomogeneity on the surface waves in anisotropic media. Sadhana, Vol. 23, 1998, p. 247‑258.

[18]     Pal P. C., et al. Wave propagation in an inhomogeneous anosotropic generalized thermoelastic solid. Journal of Thermal Stresses, Vol. 37, 2014, p. 817‑831.

[19]     Achenbach J. D. Wave Propagation in Elastic Solids. North Holland, New York, 1976.

[20]     Nayfeh A. H. Wave Propagation in Layered Anisotropic Media with Application to Composites. Elsevier, Amsterdam, 1995.

[21]     Pal A. K., Chattopadhyay A. The reflection phenomena of plane waves at a free boundary in a pre‑stressed elastic half-space. Journal of the Acoustical Society of America, Vol. 76, 1984, p. 924‑925.

[22]     Chattopadhyay A., Rogerson G. A. Wave reflection in slightly compressible, finitely deformed elastic media. Archive of Applied Mechanics, Vol. 71, 2001, p. 307‑316.

[23]     Sharma M. D. 3-D wave propagation in a general anisotropic poroelastic medium: reflection and transmission at an interface with fluid. Geophysical Journal International, Vol. 157, 2004, p. 947‑958.

[24]     Chattopadhyay A., Kumari P., Sharma V. K. Reflection and refraction at the interface between distinct generally anisotropic half spaces for three-dimensional plane quasi-P waves. Journal of Vibration and Control, Vol. 21, 2015, p. 493‑508.

[25]     Mensch T., Rasolofosaon P. Elastic-wave velocities in anisotropic media of arbitrary symmetry generalization of Thomsonís parameters ε, δ and γ. Geophysical Journal International, Vol. 128, 1997, p. 43‑64.

[26]     Abbas I. A., Othman M. I. A. Generalized thermoelastic interaction in a fiber-reinforced anisotropic half-space under hydrostatic initial stress. Journal of Vibration and Control, Vol. 18, 2011, p. 175‑182.

[27]     Verma K. L. Thermoelastic waves in anisotropic plates using normal mode expansion method with thermal relaxation time. World Academy of Science, Engineering and Technology, Vol. 2, 2008, p. 19‑26.

[28]     Zhou B., Greenhalgh S. On the computation of elastic wave group velocities for a general anisotropic medium. Journal of Geophysics and Engineering, Vol. 1, 2004, p. 205‑215.

[29]     Grechka V. Shear-wave group-velocity surfaces in low-symmetry anisotropic media. Geophysics, Vol. 80, 2015, p. 1‑7.

[30]     Vernik L., Liu X. Velocity anisotropy in shale: a petrophysical study. Geophysics, Vol. 62, 1997, p. 521‑532.

[31]     Kumar R., Gupta V. Plane wave propagation in an anisotropic thermoelastic medium with fractional order derivative and void. Journal of Thermoelasticity, Vol. 1, 2013, p. 21‑34.

[32]     Othman M. I. A., Atwa S. Y., Elwan A. W. Two and three dimensions of generalized thermoelastic medium without energy dissipation under the effect of rotation. Applied Mathematics, Vol. 6, 2015, p. 793‑805.

[33]     Zhu A., Wiens D. A. Thermoelastic stress in oceanic lithosphere due to hotspot reheating. Journal of Geophysical Research, Vol. 96, 1991, p. 18323‑18334.

[34]     Othman M. I. A., Song Y. Effect of rotation on plane waves of generalized electro‑magneto‑thermoviscoelasticity with two relaxation times. Applied Mathematical Modelling, Vol. 32, 2008, p. 811‑825.

[35]     Santra S., Das N. C., Kumar R., Lahiri A. Three dimensional fractional order generalized thermoelastic problem under effect of rotation in a half space. Journal of Thermal Stresses, Vol. 38, 2015, p. 309‑324.

[36]     Lahiri A., Das N. C., Sarkar S., Das M. Matrix method of solution of coupled differential equations and its application to generalized thermoelasticity. Bulletin of the Calcutta Mathematical Society, Vol. 101, 2009, p. 571‑590.

[37]     Irwin G. R. Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics, Vol. 24, 1957, p. 361‑364.

[38]     Turcotte D. L. Are transform faults thermal contraction cracks? Journal of Geophysical Research, Vol. 79, 1974, p. 2573‑2577.

[39]     Wilson J. T. A possible origin of the Hawaiian Islands. Canadian Journal of Physics, Vol. 41, 1963, p. 863‑870.

[40]     Koyanagi R. Y., Endo E. T., Ebisu J. S. Reawakening of Mauna Loa volcano, Hawaii: a preliminary evaluation of seismic evidence. Geophysical Research Letters, Vol. 2, 1975, p. 405‑408.

[41]     Ando M. The Hawaiian earthquake of November 29, 1975: Low dip angle faulting due to forceful injection of magma. Journal of Geophysical Research, Vol. 84, 1979, p. 7616‑7626.

[42]     Stewart G. S., Helmberger D. V. The Bermuda earthquake of March 24, 1978; a significant oceanic intraplate event. Journal of Geophysical Research, Vol. 86, 1981, p. 7020‑7036.

[43]     Nishenko S. P., Kafka A. L. Earthquake focal mechanisms and the intraplate setting of the Bermuda Rise. Journal of Geophysical Research, Vol. 87, 1982, p. 3929‑3941.

[44]     Bergman E. A. Intraplate earthquakes and the state of stress in oceanic lithosphere. Tectonophysics, Vol. 132, 1986, p. 1‑35.

[45]     Anderson T. L. Fracture Mechanics: Fundamentals and Applications. Taylor and Francis, Boca Raton, FL, 2005.

Cite this article

Chakraborty S., Mandal S. C., Das A. K., Sarkar N., Lahiri A. Plane wave propagation in a 3D anisotropic half‑space under Green‑Naghdi theory II. Mathematical Models in Engineering, Vol. 2, Issue 2, 2016, p. 114‑134.

 

Mathematical Models in Engineering. December 2016, Volume 2, Issue 2

© JVE International Ltd. ISSN Print 2351-5279, ISSN Online 2424-4627, Kaunas, Lithuania