14. State-space approach to 3D generalized thermoviscoelasticity under Green and Naghdi theory II
M. Bachher1, N. Sarkar2, A. Lahiri3
1, 3Department of Mathematics, Jadavpur University, Kolkata-700 032, India
2Department of Applied Mathematics, University of Calcutta, Kolkata-700 009, India
E-mail: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
(Received 23 November 2015; accepted 27 December 2015)
Abstract. The present paper is aimed at studying the effects of viscosity on thermoelastic interactions in a three-dimensional homogeneous isotropic half-space solid medium whose surface is subjected to a thermal shock and is assumed to be stress free. The formulation is applied to the generalized thermoelasticity based on the GN model without energy dissipation (GN II model). The normal mode analysis together with state-space approach is used to obtain the exact analytical expressions for the field variables considered. Numerical computations are performed for a specific material and the results obtained are represented graphically. Comparisons are made within the theory in the presence and absence of viscosity effects.
Keywords: generalized thermoviscoelasticity, GN model, energy dissipation, normal mode analysis, state-space approach.
 Lord H. W., Shulman Y. A. Generalized dynamical theory of thermoelasticity. Journal of Mechanics and Physics of Solids, Vol. 15, 1967, p. 299‑309.
 Biot M. Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics, Vol. 27, 1956, p. 240‑253.
 Green A. E., Lindsay K. A. Thermoelasticity. Journal of Elasticity, Vol. 2, 1972, p. 1‑7.
 Green A. E., Naghdi P. M. A re-examination of the basic postulate of thermo-mechanics. Proceedings of Royal Society of London, Vol. 432, 1991, p. 171‑194.
 Green A. E., Naghdi P. M. Thermoelasticity without energy dissipation. Journal of Elasticity, Vol. 31, 1993, p. 189‑208.
 Green A. E., Naghdi P. M. An unbounded heat wave in an elastic solid. Journal of Thermal Stresses, Vol. 15, 1992, p. 253‑264.
 Othman M. I. A., Zidan M. E. M., Hilal M. I. M. The influence of gravitational field and Rotation on thermoelastic solid with voids under Green-Naghdi theory. Journal of Physics, Vol. 2, 2013, p. 22‑34.
 Othman M. I. A., Zidan M. E. M., Hilal M. I. M. Effect of rotation on thermoelastic material with voids and temperature dependent properties of type III. Journal of Thermoelasticity, Vol. 1, 2013, p. 1‑11.
 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.
 Gupta R. R. Propagation of waves in the transversely isotropic thermoelastic Green-Naghdi type II and type III medium. Journal of Applied Mechanics and Technical Physics, Vol. 52, 2011, p. 825‑833.
 Lazzari B., Nibbi R. On the exponential decay in thermoelasticity without energy dissipation an of type III in presence of an absorbing boundary. Journal of Mathematical Analysis and Applications, Vol. 338, 2008, p. 317‑329.
 Roychoudhuri S. K., Bandyopadhyay N. Interactions due to body forces in generalized thermos‑elasticity III. Computers and Mathematics with Applications, Vol. 54, 2007, p. 1341‑1352.
 Verma K. L. Generalized thermoelastic vibrations in heat conducting plates without energy dissipation. Tamkung Journal of Science and Engineering, Vol. 10, 2007, p. 1‑9.
 Leseduarte M. C., Quintanilla R. Thermal stresses in type III thermoelastic plates. Journal of Thermal Stresses, Vol. 29, 2006, p. 485‑503.
 Taheria H., Fariborza S. J., Eslamia M. R. Thermoelastic analysis of an annulus using the Green‑Naghdi model. Journal of Thermal Stresses, Vol. 28, 2005, p. 911‑927.
 Ilioushin A. A., Pobedria B. E. Fundamentals of the Mathematical Theory of Thermal Viscoelasticity. Nauka, Moscow, 1970.
 Tanner R. I. Engineering Rheology. Oxford University Press, Oxford, 1988.
 Huilgol R., Phan-Thien N. Fluid Mechanics of Viscoelasticity. Elsevier, Amsterdam, 1997.
 Ezzat M. A., Othman M. I. A., El-Karamany A. S. State-space approach to two-dimensional generalized thermoviscoelasticity with two relaxation times. International Journal of Engineering Science, Vol. 40, 2002, p. 1251‑1274.
 Ezzat M. A., El-Karamany A. S. The uniqueness and reciprocity theorems for generalizedthermo‑viscoelasticity with two relaxation times. International Journal of Engineering Science, Vol. 40, 2002, p. 1275‑1284.
 Abd‑Alla A. N., Yahia A. A., Abo-Dahab S. M. On the reflection of the generalized magneto‑thermoviscoelastic plane waves. Chaos Solitons Fractals, Vol. 16, 2003, p. 211‑231.
 Othman M. I. A. Generalized electromagneto-thermoviscoelastic in case of 2-D thermal shock problem in a finite conducting half-space with one relaxation time. Acta Mechanica, Vol. 169, 2004, p. 37‑51.
 Abd‑Alla A. N., Abo-Dahab S. M. Time-harmonic sources in a generalized magneto‑thermoviscoelastic continuum with and without energy dissipation. Applied Mathematical Modelling, Vol. 33, 2009, p. 2388‑2402.
 Kanoria M., Mallik S. H. Generalized thermoviscoelastic interaction due to periodically varying heat source with three-phase-lag effect. European Journal of Mechanics – A/Solid, Vol. 29, 2010, p. 695‑703.
 Deswal S., Kalkal K. A two-dimensional generalized electro-magneto-thermoviscoelastic problem for a half-space with diffusion. International Journal of Thermal Science, Vol. 50, 2011, p. 749‑759.
 Sarkar N. Analysis of magneto-thermoelastic response in a fiber-reinforced elastic solid due to hydrostatic initial stress and gravity field. Journal of Thermal Stresses, Vol. 37, 2014, p. 1‑18.
 Bahar L., Hetnarski R. State space approach to thermoelasticity. Journal of Thermal Stresses, Vol. 1, 1978, p. 135‑145.
 Anwar M. A., Sherief H. H. State space approach to generalized thermoelasticity. Journal of Thermal Stresses, Vol. 11, 1988, p. 353‑365.
 Sherief H. H., Anwar M. A. State space approach to two-dimensional generalized thermoelasticity problems. Journal of Thermal Stresses, Vol. 17, 1994, p. 567‑590.
 Youssef H. M. A two-temperature generalized thermoelastic medium subjected to a moving heat source and ramp-type heating: a state-space approach. Journal of Mechanics of. Materials Structure, Vol. 4, 2009, p. 1637‑1649.
 Simmons G. F. Introduction to Topology and Modern Analysis. Krieger Publishing Company, U.S.A., 2003.
Cite this article
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, Issue 2, 2015, p. 111‑124.
Mathematical Models in Engineering. December 2015, Volume 1, Issue 2
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