Generalized Thermo Elasticity in an Infinite Nonhomogeneous Solid Having a Spherical Cavity Using DPL Model

Author(s)
Ahmed Elsayed Abouelrega

ABSTRACT

The induced temperature, displacement, and stress fields in an infinite nonhomogeneous elastic medium having a spherical cavity are obtained in the context dual-phase-lag model. The surface of the cavity is stress free and is subjected to a thermal shock. The material is elastic and has an in¬homogeneity in the radial direction. The type of non homogeneity is such that the elastic constants, thermal conductivity and density are propor¬tional to the nth power of the radial distance. The solutions are obtained analytically employing the Laplace transform technique. The numerical inversion of the transforms is carried out using Fourier series expansions. The stresses, temperature and displacement are computed and presented graphically. A comparison of the results for different theories is presented.

The induced temperature, displacement, and stress fields in an infinite nonhomogeneous elastic medium having a spherical cavity are obtained in the context dual-phase-lag model. The surface of the cavity is stress free and is subjected to a thermal shock. The material is elastic and has an in¬homogeneity in the radial direction. The type of non homogeneity is such that the elastic constants, thermal conductivity and density are propor¬tional to the nth power of the radial distance. The solutions are obtained analytically employing the Laplace transform technique. The numerical inversion of the transforms is carried out using Fourier series expansions. The stresses, temperature and displacement are computed and presented graphically. A comparison of the results for different theories is presented.

KEYWORDS

Generalized Thermo Elasticity, Nonhomogeneous, Functionally Graded Material (FGM), Laplace Transform, Three-Phase-Lag Model

Generalized Thermo Elasticity, Nonhomogeneous, Functionally Graded Material (FGM), Laplace Transform, Three-Phase-Lag Model

Cite this paper

nullA. Abouelrega, "Generalized Thermo Elasticity in an Infinite Nonhomogeneous Solid Having a Spherical Cavity Using DPL Model,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 625-632. doi: 10.4236/am.2011.25083.

nullA. Abouelrega, "Generalized Thermo Elasticity in an Infinite Nonhomogeneous Solid Having a Spherical Cavity Using DPL Model,"

References

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

[4] 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

[5] D. Y. Tzou, “Macro- to Microscale Heat Transfer: the Lagging Behavior,” 1st Edition, Taylor & Francis, Washington, 1996.

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

[7] D. Y. Tzou, “Experimental Support for the Lagging Behavior in Heat Propagation,” Journal of Thermophysics and Heat Transfer, Vol. 9, No. 4, 1995, pp. 686-693. doi:10.2514/3.725

[8] D. S. Chandrasekharaiah, “Thermoelasticity with Second Sound,” Applied Mechanics Reviews, Vol. 39, No. 3, 1986, pp. 354-376.

[9] G. Honig and V. 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

[10] S. B. Sinha and K. A. Elsibai, “Thermal Stresses for an Infinite Body with Spherical Cavity with Tow Relaxation Times,” Journal of Thermal Stresses, Vol. 19, No. 5, 1996, pp. 745-759. doi:10.1080/01495739608946190

[11] M. N. Allam, K. A. Elsibai and A. E. Abouelergal, “Thermal Stresses in a Harmonic Field for an Infinite Body with a Circular Cylindrical Hole Without Energy Dissipation,” Journal of Thermal Stresses, Vol. 25, No. 1, 2002, pp. 57-68. doi:10.1080/014957302753305871

[12] S. Mukhopadhyay, “Thermoelastic Interactions without Energy Dissipation in an Unbounded Body with a Spherical Cavity Subjected to Harmonically Varying Temperature,” Mechanics Research Communications, Vol. 31, No. 1, 2004, pp. 81-89. doi:10.1016/S0093-6413(03)00082-X

[13] S. Mukhopadhyay and R. Kumar, “A Study of Generalized Thermoelastic Interactions in an Unbounded Medium with a Spherical Cavity,” Computers and Mathematics with Applications, Vol. 56, No. 9, 2008, pp. 2329- 2339.doi:10.1016/j.camwa.2008.05.031

[14] S. K. Roychoudhuri, “One-Dimensional Thermoelastic Waves in Elastic Half-Space with Dual-Phase-Lag Effects,” Materials and Structures Journal, Vol. 2, No. 1, 2007, pp. 489-503. doi:10.2140/jomms.2007.2.489

[15] S. Suresh and A. Mortensen, “Fundamentals of Functionally Graded Materials,” Institute of Materials Communications Ltd., London, 1998.

[16] R. C. Wetherhold and S. S. Wang, “The Use of Functionally Graded Materials to Eliminate or Control Thermal Deformation,” Composites Science and Technology, Vol. 56, No. 9, 1996, pp. 1099-1104. doi:10.1016/0266-3538(96)00075-9

[17] R. B. Hetnarski and J. Ignaczak, “Generalized Thermoelasticity,” Journal of Thermal Stresses, Vol. 22, No. 4, 1999, pp. 451-476. doi:10.1080/014957399280832

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

[4] 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

[5] D. Y. Tzou, “Macro- to Microscale Heat Transfer: the Lagging Behavior,” 1st Edition, Taylor & Francis, Washington, 1996.

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

[7] D. Y. Tzou, “Experimental Support for the Lagging Behavior in Heat Propagation,” Journal of Thermophysics and Heat Transfer, Vol. 9, No. 4, 1995, pp. 686-693. doi:10.2514/3.725

[8] D. S. Chandrasekharaiah, “Thermoelasticity with Second Sound,” Applied Mechanics Reviews, Vol. 39, No. 3, 1986, pp. 354-376.

[9] G. Honig and V. 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

[10] S. B. Sinha and K. A. Elsibai, “Thermal Stresses for an Infinite Body with Spherical Cavity with Tow Relaxation Times,” Journal of Thermal Stresses, Vol. 19, No. 5, 1996, pp. 745-759. doi:10.1080/01495739608946190

[11] M. N. Allam, K. A. Elsibai and A. E. Abouelergal, “Thermal Stresses in a Harmonic Field for an Infinite Body with a Circular Cylindrical Hole Without Energy Dissipation,” Journal of Thermal Stresses, Vol. 25, No. 1, 2002, pp. 57-68. doi:10.1080/014957302753305871

[12] S. Mukhopadhyay, “Thermoelastic Interactions without Energy Dissipation in an Unbounded Body with a Spherical Cavity Subjected to Harmonically Varying Temperature,” Mechanics Research Communications, Vol. 31, No. 1, 2004, pp. 81-89. doi:10.1016/S0093-6413(03)00082-X

[13] S. Mukhopadhyay and R. Kumar, “A Study of Generalized Thermoelastic Interactions in an Unbounded Medium with a Spherical Cavity,” Computers and Mathematics with Applications, Vol. 56, No. 9, 2008, pp. 2329- 2339.doi:10.1016/j.camwa.2008.05.031

[14] S. K. Roychoudhuri, “One-Dimensional Thermoelastic Waves in Elastic Half-Space with Dual-Phase-Lag Effects,” Materials and Structures Journal, Vol. 2, No. 1, 2007, pp. 489-503. doi:10.2140/jomms.2007.2.489

[15] S. Suresh and A. Mortensen, “Fundamentals of Functionally Graded Materials,” Institute of Materials Communications Ltd., London, 1998.

[16] R. C. Wetherhold and S. S. Wang, “The Use of Functionally Graded Materials to Eliminate or Control Thermal Deformation,” Composites Science and Technology, Vol. 56, No. 9, 1996, pp. 1099-1104. doi:10.1016/0266-3538(96)00075-9

[17] R. B. Hetnarski and J. Ignaczak, “Generalized Thermoelasticity,” Journal of Thermal Stresses, Vol. 22, No. 4, 1999, pp. 451-476. doi:10.1080/014957399280832