Eigenvalue Approach on Generalized Thermoelastic Interactions of a Layer

Nilanjana Gangopadhyaya; Surath Roy; Manasi Sahoo; Suparna Roychowdhury

doi:https://doi.org/10.14445/22315373/IJMTT-V54P527

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 54 | Number 3 | Year 2018 | Article Id. IJMTT-V54P527 | DOI : https://doi.org/10.14445/22315373/IJMTT-V54P527

Eigenvalue Approach on Generalized Thermoelastic Interactions of a Layer

Nilanjana Gangopadhyaya, Surath Roy, Manasi Sahoo, Suparna Roychowdhury

Citation :

Nilanjana Gangopadhyaya, Surath Roy, Manasi Sahoo, Suparna Roychowdhury, "Eigenvalue Approach on Generalized Thermoelastic Interactions of a Layer," International Journal of Mathematics Trends and Technology (IJMTT), vol. 54, no. 3, pp. 240-252, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V54P527

Abstract

The unified form of the basic equations of generalized thermoelastic interactions for Lord Shulman (LS) and Green Lindsay (GL) models for a layer have been written in the form of a vector matrix differential equation and solved by the eigen value approach technique in the Laplace transform domain in a closed form. The inversions of the physical variables from the transformed domain have been made by using Zakian algorithm for the numerical inversion from the Laplace transform domain. Graphs for the physical variables have been presented for different cases and the results are compared with the existing literature.

Keywords

Generalized thermoelasticity, relaxation time, unified form of equation, eigen value approach, Zakian algorithm.

References

[1] M. A. Biot, “Thermoelasticity and irreversible thermodynamics”, J. Appld. Phys, vol.27, pp.240-253, 1956
[2] H. W. Lord, Y. Shulman, “A generalized dynamical theory of thermoelasticity”, J. Mech. Phys. Solids, vol. 15(5), pp.299-309, 1967.
[3] R.B. Hetnarski, M.R. Eslami, Thermal Stresses- Advanced Theory and Applications, Springer, New York, 2009.
[4] A. E. Green, K.A. Lindsay, “Thermoelasticity”, J. Elasticity, vol. 2(1), pp.1-7, 1972.
[5] A. E. Green, P.M. Naghdi, “A re-examination of the basic postulates of thermomechanics”, Proc. R .Soc. London.A, vol.432, pp.171-194, 1991.
[6] D. S. Chandrasekharaiah, “Thermoelasticity with second sound: A review”, Appld .Mech .Rev., vol. 39(3), pp.355-376, 1986.
[7] D. S. Chandrasekharaiah, “Hyperbolic thermoelasticity: A review of recent literature”, Appl. Mech .Rev., vol.51(12), pp.705-729, 1998.
[8] H.Taheri, S. Fariborz, & M.R. Eslami, “Thermoelasticity solution of a layer using the Green-Naghdi model”, J. Therm. Stresses, vol.27(9), pp.795-809, 2004.
[9] A. Bagri, H. Taheri, M.R. Eslami & S. Fariborz, “Generalized coupled thermoelasticity of a layer”, J. Therm. Stresses, vol.29(4), pp.359-370, 2006
[10] M.A.Ezzat, “State space approach to generalized magneto-thermoelasticity with two relaxation times in a medium of perfect conductivity”, Int. J. Engg. Sc., vol.35(8), pp.741-752,1997.
[11] M.A.Ezzat, H.M.Youssef, “Generalized magneto-thermoelasticity in a perfectly conducting medium”, Int. J. Solid Struct., vol.42, pp.6319-6334, 2005.
[12] R.B. Hetnarski, J. Ignaczak, “On solution like thermoelastic waves”, Appld .Anal., vol.65, pp.183-204, 1997.
[13] S.A. Hosseini Kordkheili, R. Naghdabadi, “Thermoelastic analysis of functionally graded cylinders under axial loading”, J. Therm Stresses, vol.31(1), pp.1-17, 2007.
[14] J. Ignaczak, M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, OUP, Oxford, 413 pages, 2010.
[15] M. Islam, M. Kanoria, “One dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity”, Math. Mech.Solids, vol. 19(6), pp.672-693, 2014.
[16] A. Kar, M. Kanoria, “Analysis of thermoelastic response in a fiber reinforced thin annular disc with three-phase-lag effect”, Eur. J. Pure & Appld. Maths., vol.4(3), pp.304-321 ,2011.
[17] R. Kumar, S. Mukhopadhyay, “Effects of thermal relaxation time on plane wave propagation under two-temperature thermoelasticity”, Int . J .Engg. Sci., vol.48(2), pp.128-139, 2010.
[18] N.C. Das, A. Lahiri, S. Sarkar, S. Basu, “Reflection of generalized thermoelastic waves from isothermal and insulated boundaries of a half space ", Comp. & Maths. with Applications , vol.56(11), pp.2795-2805 , 2008.
[19] N. Das Gupta, N.C. Das, “Eigenvalue approach to fractional order generalized thermoelasticity with line heat source in an infinite medium”, J. Therm. Stresses, vol.39 (8), pp.977-990, 2016.
[20] M.B. Bera, N.C. Das & A. Lahiri, “Eigenvalue approach to two temperature generalized thermoelastic interaction in an annular disk”, J.Therm. Stresses, vol.38 (11), pp.1308-1322, 2015.
[21] J. Ignaczak, “Soliton-like solutions in a nonlinear dynamic coupled thermoelasticity”. J. Therm. Stresses, vol.13 (1), pp.73-98, 1990.
[22] R.B. Hetnarski, J. Ignaczak, “Soliton like waves in a low temperature nonlinear thermoelastics solid”, Int .J. Engg .Sc, vol.34 (15), pp.1767-1787, 1996.
[23] Y. Kiani, M.R. Eslami, “Nonlinear generalized thermoelasticity of an isotropic layer based on Lord-Shulman theory”, Euro. J. Mech./A Solids.vol. 61, pp.245-253, 2017.
[24] I.A. Abbas, H.M. Youssef, “A nonlinear generalized thermoelasticity model of temperature dependent materials using finite element method”, Int .J. Thermophys, vol.33(7), pp.1302-1313, 2012
[25] M. Bateni, M.R. Eslami, “Thermally nonlinear generalized thermoelasticity of a layer”. J. Therm. Stresses, vol.40 (10), pp.1320-1338, 2017.
[26] A. Bagri, M.R. Eslami, “A unified generalized thermoelasticity formulation; Application to thick functionally graded cylinders”, J. Therm. Stresses, vol.30 (9-10), pp.911-930, 2007.
[27] X.G. Tian, Y.P. Shen, C.Q. Chen, T.H. He, “A direct finite element method to study of generalized thermoelastic problems”, Int. J. Solids, Struct., vol.43(7), pp.2050-2063, 2006.
[28] M. Shariyat, “Nonlinear transient stress and wave propagation analysis of the FGM thick cylinders employing a unified generalized thermoelasticity theory”, Int .J. Mech .Sc., vol.65(1), pp.24-37, 2012.
[29] V. Zakian, “Numerical inversion of Laplace transform”, Electronics letters, vol.5(6), pp.120-121,1969.
[30] N. Noda, R.B. Hetnarski, & Y. Tanigawa, Thermal Stresses, 2nd ed., Taylor & Francis, New York, 2003.
[31] A. Lahiri, N.C. Das, S. Sarkar & M. Das, “Matrix method of solution of coupled differential equations and its applications in generalized thermoelasticity”, Bull .Cal. Math. Soc. vol.101 (6), pp.571-590, 2009.