A Problem of a Semi-Infinite Medium Subjected to Exponential Heating Using a Dual-Phase-Lag Thermoelastic Model

Author(s)
Ahmed E. Abouelregal

ABSTRACT

The problem of a semi-infinite medium subjected to thermal shock on its plane boundary is solved in the context of the dual-phase-lag thermoelastic model. The expressions for temperature, displacement and stress are presented. The governing equations are expressed in Laplace transform domain and solved in that domain. The solution of the problem in the physical domain is obtained by using a numerical method for the inversion of the Laplace transforms based on Fourier series expansions. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The results obtained are presented graphically to show the effect phase-lag of the heat flux and a phase-lag of temperature gradient on displacement, temperature, stress.

The problem of a semi-infinite medium subjected to thermal shock on its plane boundary is solved in the context of the dual-phase-lag thermoelastic model. The expressions for temperature, displacement and stress are presented. The governing equations are expressed in Laplace transform domain and solved in that domain. The solution of the problem in the physical domain is obtained by using a numerical method for the inversion of the Laplace transforms based on Fourier series expansions. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The results obtained are presented graphically to show the effect phase-lag of the heat flux and a phase-lag of temperature gradient on displacement, temperature, stress.

KEYWORDS

Generalized Thermoelasticity, Dual-Phase-Lag Model, Semi-Infinite Medium, Laplace Transform

Generalized Thermoelasticity, Dual-Phase-Lag Model, Semi-Infinite Medium, Laplace Transform

Cite this paper

nullA. Abouelregal, "A Problem of a Semi-Infinite Medium Subjected to Exponential Heating Using a Dual-Phase-Lag Thermoelastic Model,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 619-624. doi: 10.4236/am.2011.25082.

nullA. Abouelregal, "A Problem of a Semi-Infinite Medium Subjected to Exponential Heating Using a Dual-Phase-Lag Thermoelastic Model,"

References

[1] M. Biot, “Thermoelasticity and Irreversible ThermodyNamics,” Journal of Applied Physics, Vol. 27, No. 3, 1956, pp. 240-253. doi:10.1063/1.1722351

[2] H. Lord and Y. Shulman, “A Generalized Dynamical Theory of Thermoelasticity,” Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5

[3] R. Dhaliwal and H. Sherief, “Generalized Thermoelasticity for Anisotropic Media,” Quarterly of Applied Mathematics, Vol. 33, 1980, pp. l-8.

[4] A. E. Green and N. Laws, “On the Entropy Production Inequality,” Archive for Rational Mechanics and Analysis, Vol. 45, No. 1, 1972, pp. 47-53. doi:10.1007/BF00253395

[5] A. E. Green and K. A. Lindsay, “Thermoelasticity,” Journal of Elasticity, Vol. 2, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689

[6] A. E. Green and P. M. Naghdi, “Thermoelasticity Without Energy Dissipation,” Journal of Elasticity, Vol. 31, No. 3, 1993, pp. 189-208. doi:10.1007/BF00044969

[7] D. Y. Tzou, “Macro- to Microscale Heat Transfer: The Lagging Behavior,” 1st Edition, Taylor & Francis, Wa- shington, 1996.

[8] D. Y. Tzou, “A Unified Approach for Heat Conduction From Macro- to Micro- Scales,” Journal of Heat Transfer, Vol. 117, No. 1, 1995, pp. 8-16. doi:10.1115/1.2822329

[9] D. Y. Tzou, “Experimental Support for the Lagging Behavior in Heat Propagation,” Journal of Thermophysics and Heat Transfer, Vol. 9, 1995, pp. 686-693. doi:10.2514/3.725

[10] V. Danilovskaya, “Thermal Stresses in an Elastic Half- space Due to Sudden Heating of Its Boundary,” Prikl Mat. Mekh., In Russian, Vol. 14, 1950, pp. 316-324.

[11] D. S. Chandrasekharaiah and K. S. Srinath, “One-Dimensional Waves in a Thermoelastic Half-Space Without Energy Dissipation,” International Journal of Engineering Science, Vol. 34, No. 13, 1996, pp. 1447-1455. doi:10.1016/0020-7225(96)00034-1

[12] S. K. Roychoudhuri and P. S. Dutta, “Thermoelastic Interaction Without Energy Dissipation in an Infinite Solid with Distributed Periodically Varying Heat Sources,” International Journal of Solids Structures, Vol. 42, 2005, pp. 4192-4203.

[13] H. Sherief, and R. Dhaliwal, “Generalized One-Dimen- sional Thermal Shock Problem for Small Times,” Journal of Thermal Stresses, Vol. 4, No. 3-4, 1981, pp. 407-420. doi:10.1080/01495738108909976

[14] M. N. Allam, K. A. Elsibai and A. E. Abouelregal, “Magneto-Thermoelasticity for an Infinite Body with a Spherical Cavity and Variable Material Properties Without Energy Dissipation,” International Journal of Solids and Structures, Vol. 47, No. 20, 2010, pp. 2631-2638. doi:10.1016/j.ijsolstr.2010.04.021

[15] G. Honig and U. Hirdes, “A Method for the Numerical Inversion of the Laplace Transform,” Journal of Computational and Applied Mathematics, Vol. 10, No. 1, 1984, pp. 113-132. doi:10.1016/0377-0427(84)90075-X

[16] H. Youssef, “Thermomechanical Shock Problem of Generalized Thermoelastic Infinite Body with a Cylindrical Cavity and Material Properties Depends on the Reference Temperature,” Journal of Thermal Stresses, Vol. 28, No. 5, 2005, pp. 521-532. doi:10.1080/01495730590925029

[1] M. Biot, “Thermoelasticity and Irreversible ThermodyNamics,” Journal of Applied Physics, Vol. 27, No. 3, 1956, pp. 240-253. doi:10.1063/1.1722351

[2] H. Lord and Y. Shulman, “A Generalized Dynamical Theory of Thermoelasticity,” Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5

[3] R. Dhaliwal and H. Sherief, “Generalized Thermoelasticity for Anisotropic Media,” Quarterly of Applied Mathematics, Vol. 33, 1980, pp. l-8.

[4] A. E. Green and N. Laws, “On the Entropy Production Inequality,” Archive for Rational Mechanics and Analysis, Vol. 45, No. 1, 1972, pp. 47-53. doi:10.1007/BF00253395

[5] A. E. Green and K. A. Lindsay, “Thermoelasticity,” Journal of Elasticity, Vol. 2, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689

[6] A. E. Green and P. M. Naghdi, “Thermoelasticity Without Energy Dissipation,” Journal of Elasticity, Vol. 31, No. 3, 1993, pp. 189-208. doi:10.1007/BF00044969

[7] D. Y. Tzou, “Macro- to Microscale Heat Transfer: The Lagging Behavior,” 1st Edition, Taylor & Francis, Wa- shington, 1996.

[8] D. Y. Tzou, “A Unified Approach for Heat Conduction From Macro- to Micro- Scales,” Journal of Heat Transfer, Vol. 117, No. 1, 1995, pp. 8-16. doi:10.1115/1.2822329

[9] D. Y. Tzou, “Experimental Support for the Lagging Behavior in Heat Propagation,” Journal of Thermophysics and Heat Transfer, Vol. 9, 1995, pp. 686-693. doi:10.2514/3.725

[10] V. Danilovskaya, “Thermal Stresses in an Elastic Half- space Due to Sudden Heating of Its Boundary,” Prikl Mat. Mekh., In Russian, Vol. 14, 1950, pp. 316-324.

[11] D. S. Chandrasekharaiah and K. S. Srinath, “One-Dimensional Waves in a Thermoelastic Half-Space Without Energy Dissipation,” International Journal of Engineering Science, Vol. 34, No. 13, 1996, pp. 1447-1455. doi:10.1016/0020-7225(96)00034-1

[12] S. K. Roychoudhuri and P. S. Dutta, “Thermoelastic Interaction Without Energy Dissipation in an Infinite Solid with Distributed Periodically Varying Heat Sources,” International Journal of Solids Structures, Vol. 42, 2005, pp. 4192-4203.

[13] H. Sherief, and R. Dhaliwal, “Generalized One-Dimen- sional Thermal Shock Problem for Small Times,” Journal of Thermal Stresses, Vol. 4, No. 3-4, 1981, pp. 407-420. doi:10.1080/01495738108909976

[14] M. N. Allam, K. A. Elsibai and A. E. Abouelregal, “Magneto-Thermoelasticity for an Infinite Body with a Spherical Cavity and Variable Material Properties Without Energy Dissipation,” International Journal of Solids and Structures, Vol. 47, No. 20, 2010, pp. 2631-2638. doi:10.1016/j.ijsolstr.2010.04.021

[15] G. Honig and U. Hirdes, “A Method for the Numerical Inversion of the Laplace Transform,” Journal of Computational and Applied Mathematics, Vol. 10, No. 1, 1984, pp. 113-132. doi:10.1016/0377-0427(84)90075-X

[16] H. Youssef, “Thermomechanical Shock Problem of Generalized Thermoelastic Infinite Body with a Cylindrical Cavity and Material Properties Depends on the Reference Temperature,” Journal of Thermal Stresses, Vol. 28, No. 5, 2005, pp. 521-532. doi:10.1080/01495730590925029