Abstract
In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory of two-temperature generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating.
Similar content being viewed by others
References
Anisimov, S. I., Kapeliovich, B.L., and Perelman, T.L., (1974), “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP, 39, 375–377.
Bargmann, H., (1974), “Recent developments in the field of thermally induced waves and vibrations,” Nucl. Eng. Des., 27, 372.
Boley, B. A., (1980), In: Thermal Stresses, Edited by D. P. H. Hasselman and R. A. Heller, Plenum Press, New York, pp. 1–11.
Boley, B. A., and Tolins, I. S., (1962), “Transient coupled thermoelastic boundary value problems in the half-space,” J. Appl. Mech., 29, 637–646.
Chandrasekhariah, D. S., (1986), “Thermoelasticity with second sound A-review,” Appl. Mech. Rev., 39, 355.
Chen, P. J., and Gurtin, M. E., (1968), “On a theory of heat conduction involving two temperatures,” Z. Angew. Math. Phys., 19, 614–627.
Chen, P. J., Gurtin, M. E., and Williams, W. O., (1969), “On the thermodynamics of non-simple elastic materials with two temperatures,” Z. Angew. Math. Phys., 20, 107–112.
Chen, P. J., Gurtin, M. E., and Williams, W. O., (1968), “A note on non-simple heat conduction,” Z. Angew. Math. Phys., 19, 969–970.
Chen, J. K., Beraun, J. E., and Tham, C. L., (2004), “Ultrafast thermoelasticity for short-pulse laser heating,” Int. J. Eng. Sci., 42, 793–807.
Churchill, R. V., (1972), Operational Mathematics, Third edition, McGraw-Hill Book Company, New York.
El-Maghraby, N. M., and Youssef, H., (2004), “State space approach to generalized thermoelastic problem with thermo-mechanical shock,” Appl. Math. Comput., 156, p. 577.
Green, A. E. and Lindsay, K. A., (1972), “Thermoelasticity,” J. Elasticity, 2, No. 1.
Hanig, G., and Hirdes, U., (1984), “A method for the numerical inversion of Laplace Transform,” J. Comput. Appl. Math., 10, 113–132.
Lord, H., and Shulman, Y., (1967), “A generalized dynamical theory of thermo-elasticity,” J. Mech. Phys. Solids, 15, p. 299.
Qiu, T.Q., and Tien, C.L., (1993), “Heat transfer mechanism during short-pulse laser heating of metals,” ASME J. Heat Transf., 115, 835–841.
Sneddon, N., (1995), Fourier Transforms, Dover Publications Inc., New York.
Tzou, D. Y., (1997), Macro-to Microscale Heat Transfer: the Lagging Behavior, Taylor & Francis, Washington, DC.
Warren, W. E., and Chen, P. J., (1973), “Wave propagation in the two-temperature theory of thermoelasticity,” Acta Mech., 16, 21–33.
Youssef, H., (2005-a), “The dependence of the modulus of elasticity and the thermal conductivity on the reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity,” J. Appl. Math. Mech., 26, No. 4, 470–475.
Youssef, H., (2006-a), “Two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating,” Eur. J. Mech. A/Solids, 25, 745–763.
Youssef, H., (2006-b), “Problem of generalized thermoelastic infinite medium with cylindrical cavity subjected to a ramp-type heating and loading,” Arch. Appl. Mech., 75, 8–9, 553–565.
Youssef, H., (2006-c), “Theory of two-temperature generalized thermoelasticity,” J. Appl. Math., 71, No. 3, 383–390.
Rights and permissions
About this article
Cite this article
Youssef, H.M. Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating. Comput Math Model 19, 201–216 (2008). https://doi.org/10.1007/s10598-008-0014-7
Issue Date:
DOI: https://doi.org/10.1007/s10598-008-0014-7