Application of Residual Correction Method on Hyperbolic Thermoelastic Response of Hollow Spherical Medium in Rapid Transient Heat Conduction
In this article, we used the residual correction method
to deal with transient thermoelastic problems with a hollow spherical
region when the continuum medium possesses spherically isotropic
thermoelastic properties. Based on linear thermoelastic theory, the
equations of hyperbolic heat conduction and thermoelastic motion
were combined to establish the thermoelastic dynamic model with
consideration of the deformation acceleration effect and non-Fourier
effect under the condition of transient thermal shock. The approximate
solutions of temperature and displacement distributions are obtained
using the residual correction method based on the maximum principle
in combination with the finite difference method, making it easier and
faster to obtain upper and lower approximations of exact solutions.
The proposed method is found to be an effective numerical method
with satisfactory accuracy. Moreover, the result shows that the effect
of transient thermal shock induced by deformation acceleration is
enhanced by non-Fourier heat conduction with increased peak stress.
The influence on the stress increases with the thermal relaxation time.
[1] Forgac, J.M. and Angus, J.C., “Solidification of metal spheres,”
Metallurgical Transactions, Vol. 12, No. 2, pp 413-416, 1981.
[2] Hata, T., “Thermal shock in a hollow sphere caused by rapid uniform
heating,” ASME Journal of Applied Mechanics, Vol. 58, pp. 64-69, 1991.
[3] Tang, D.W. and Araki, N., “Non-Fourier heat conduction in a finite
medium under periodic surface thermal disturbance,” International
Journal of Heat and Mass Transfer, Vol. 39, No. 8, pp. 1585-1590, 1996.
[4] Biot, M.A., “Thermoelasticity and irreversible thermodynamics,” Journal
of Applied Physics, Vol. 27, No. 3, pp. 240-253, 1956.
[5] Peshkov, V., “Second sound in helium II,” Journal of Physics, Vol. 8, pp.
381-386, 1944.
[6] Joseph, D.D. and Preziosi, L., “Heat waves,” Reviews of Modern Physics,
Vol.61, No. 1, pp. 41-73, 1989.
[7] Maxwell, J.C., “On the dynamic theory of gases,” Philosophical
Transactions of the Royal Society of London, Vol. 157, pp. 49-88 (1867).
[8] Nernst, W., “Die Theoretischen und Experimentellen Grundlagen des
Neuen Warmesatzes,” Knapp, Halle (1918).
[9] Chester, M., “Second sound in solids,” Physical Review, Vol. 131, pp.
2013-2015, 1963.
[10] Cattaneo, C., “A form of heat conduction equation which eliminates the
paradox of instantaneous propagation,” Compte Rendus Acad. Sci., Vol.
247, pp. 431-433 (1958).
[11] Vernotte, P., “Some possible complications in the phenomena of thermal
conduction,” Compt. Rend. Acad. Sci. 252, pp. 2190-2191 (1961).
[12] Chandrasekharaiah, D.S., “Hyperbolic thermoelasticity, a review of
recent literature,” Applied Mechanics Reviews, Vol. 51, 705-729 (1998). [13] Taitel, Y., “On the parabolic, hyperbolic and discrete formulation of the
heat conduction equation, International Journal of Heat and Mass
Transfer,” Vol. 15, pp. 369-371 (1972).
[14] Frankel, J.I., Vick, B. and Ozisik, M.N., “Flux formulation of hyperbolic
heat conduction,” Journal of Applied Physics, Vol. 58, No. 9, pp.
3340-3345 (1985).
[15] Hong, B.S., Su, P.J., Chou, C.Y.; Hung, C.I., “Realization of non-Fourier
phenomena in heat transfer with 2D transfer function,” Applied
Mathematical Modelling, Vol. 35, pp. 4031-4043 (2011).
[16] Kaminiski, W., “Hyperbolic heat conduction equation for materials with a
nonhomogeneous inner structure,” ASME Journal of Heat Transfer, Vol.
112, pp. 555-560 (1990).
[17] Mitra, K., Kumar, S., Vedavarz, A. and Moallemi, M.K., “Experimental
evidence of hyperbolic heat conduction in processed meat,” ASME
Journal of heat transfer, Vol. 117, No. 3, pp. 568-573 (1995).
[18] Lord, H.W. and Shulman Y., “A generalized dynamical theory of
thermoelasticity,” Journal of the Mechanics and Physics of Solids, Vol.
15, No. 5, pp. 299-309 (1967).
[19] Tanigawa, Y., Takeuti, Y. and Ueshima, K., “Transient thermal stresses
of solid and hollow spheres with spherically isotropic thermoelastic
properties,” Archive of applied mechanics, Vol. 54, pp. 259-267 (1984).
[20] Hetnarski, R.B. and Ignaczak, J., “Generalized Thermoelasticity:
Response of semi-space to a short laser pulse,” Journal of Thermal
Stresses, Vol. 17, pp. 377-396 (1994).
[21] Lee, Z.Y., “Coupled problem of thermoelasticity for multilayered spheres
with time-dependent boundary conditions,” Journal of Marine Science
and Technology, Vol. 12, No. 2, pp. 93-101 (2004).
[22] Yu, N., Imatani, S. and Inoue, T., “Title: Hyperbolic Thermoelastic
Analysis due to Pulsed Heat Input by Numerical Simulation,” JSME
International Journal Series A, Vol. 49, No. 2, pp. 180-187 (2006).
[23] Proter, M.H. and Weinberger, H.F., “Maximum Principles in Differential
Equations,” Prentice-Hall (1967).
[24] Lee, Z.Y., Chen, C.K. and Hung, C.I., “Upper and lower bounds of the
solution for an elliptic plate problem using a genetic algorithm,” Acta
Mechanica, Vol. 157, pp. 201-212 (2002).
[25] Su, P.J. and Chen, C.K., “Application of Residual Correction Method on
non-Fourier Heat Transfer for Sphere with Time-Dependent Boundary
Condition,” CMES: Computer Modeling in Engineering & Science, Vol.
91, No. 2, pp. 135-151 (2013).
[1] Forgac, J.M. and Angus, J.C., “Solidification of metal spheres,”
Metallurgical Transactions, Vol. 12, No. 2, pp 413-416, 1981.
[2] Hata, T., “Thermal shock in a hollow sphere caused by rapid uniform
heating,” ASME Journal of Applied Mechanics, Vol. 58, pp. 64-69, 1991.
[3] Tang, D.W. and Araki, N., “Non-Fourier heat conduction in a finite
medium under periodic surface thermal disturbance,” International
Journal of Heat and Mass Transfer, Vol. 39, No. 8, pp. 1585-1590, 1996.
[4] Biot, M.A., “Thermoelasticity and irreversible thermodynamics,” Journal
of Applied Physics, Vol. 27, No. 3, pp. 240-253, 1956.
[5] Peshkov, V., “Second sound in helium II,” Journal of Physics, Vol. 8, pp.
381-386, 1944.
[6] Joseph, D.D. and Preziosi, L., “Heat waves,” Reviews of Modern Physics,
Vol.61, No. 1, pp. 41-73, 1989.
[7] Maxwell, J.C., “On the dynamic theory of gases,” Philosophical
Transactions of the Royal Society of London, Vol. 157, pp. 49-88 (1867).
[8] Nernst, W., “Die Theoretischen und Experimentellen Grundlagen des
Neuen Warmesatzes,” Knapp, Halle (1918).
[9] Chester, M., “Second sound in solids,” Physical Review, Vol. 131, pp.
2013-2015, 1963.
[10] Cattaneo, C., “A form of heat conduction equation which eliminates the
paradox of instantaneous propagation,” Compte Rendus Acad. Sci., Vol.
247, pp. 431-433 (1958).
[11] Vernotte, P., “Some possible complications in the phenomena of thermal
conduction,” Compt. Rend. Acad. Sci. 252, pp. 2190-2191 (1961).
[12] Chandrasekharaiah, D.S., “Hyperbolic thermoelasticity, a review of
recent literature,” Applied Mechanics Reviews, Vol. 51, 705-729 (1998). [13] Taitel, Y., “On the parabolic, hyperbolic and discrete formulation of the
heat conduction equation, International Journal of Heat and Mass
Transfer,” Vol. 15, pp. 369-371 (1972).
[14] Frankel, J.I., Vick, B. and Ozisik, M.N., “Flux formulation of hyperbolic
heat conduction,” Journal of Applied Physics, Vol. 58, No. 9, pp.
3340-3345 (1985).
[15] Hong, B.S., Su, P.J., Chou, C.Y.; Hung, C.I., “Realization of non-Fourier
phenomena in heat transfer with 2D transfer function,” Applied
Mathematical Modelling, Vol. 35, pp. 4031-4043 (2011).
[16] Kaminiski, W., “Hyperbolic heat conduction equation for materials with a
nonhomogeneous inner structure,” ASME Journal of Heat Transfer, Vol.
112, pp. 555-560 (1990).
[17] Mitra, K., Kumar, S., Vedavarz, A. and Moallemi, M.K., “Experimental
evidence of hyperbolic heat conduction in processed meat,” ASME
Journal of heat transfer, Vol. 117, No. 3, pp. 568-573 (1995).
[18] Lord, H.W. and Shulman Y., “A generalized dynamical theory of
thermoelasticity,” Journal of the Mechanics and Physics of Solids, Vol.
15, No. 5, pp. 299-309 (1967).
[19] Tanigawa, Y., Takeuti, Y. and Ueshima, K., “Transient thermal stresses
of solid and hollow spheres with spherically isotropic thermoelastic
properties,” Archive of applied mechanics, Vol. 54, pp. 259-267 (1984).
[20] Hetnarski, R.B. and Ignaczak, J., “Generalized Thermoelasticity:
Response of semi-space to a short laser pulse,” Journal of Thermal
Stresses, Vol. 17, pp. 377-396 (1994).
[21] Lee, Z.Y., “Coupled problem of thermoelasticity for multilayered spheres
with time-dependent boundary conditions,” Journal of Marine Science
and Technology, Vol. 12, No. 2, pp. 93-101 (2004).
[22] Yu, N., Imatani, S. and Inoue, T., “Title: Hyperbolic Thermoelastic
Analysis due to Pulsed Heat Input by Numerical Simulation,” JSME
International Journal Series A, Vol. 49, No. 2, pp. 180-187 (2006).
[23] Proter, M.H. and Weinberger, H.F., “Maximum Principles in Differential
Equations,” Prentice-Hall (1967).
[24] Lee, Z.Y., Chen, C.K. and Hung, C.I., “Upper and lower bounds of the
solution for an elliptic plate problem using a genetic algorithm,” Acta
Mechanica, Vol. 157, pp. 201-212 (2002).
[25] Su, P.J. and Chen, C.K., “Application of Residual Correction Method on
non-Fourier Heat Transfer for Sphere with Time-Dependent Boundary
Condition,” CMES: Computer Modeling in Engineering & Science, Vol.
91, No. 2, pp. 135-151 (2013).
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71304", author = "Po-Jen Su and Huann-Ming Chou", title = "Application of Residual Correction Method on Hyperbolic Thermoelastic Response of Hollow Spherical Medium in Rapid Transient Heat Conduction", abstract = "In this article, we used the residual correction method
to deal with transient thermoelastic problems with a hollow spherical
region when the continuum medium possesses spherically isotropic
thermoelastic properties. Based on linear thermoelastic theory, the
equations of hyperbolic heat conduction and thermoelastic motion
were combined to establish the thermoelastic dynamic model with
consideration of the deformation acceleration effect and non-Fourier
effect under the condition of transient thermal shock. The approximate
solutions of temperature and displacement distributions are obtained
using the residual correction method based on the maximum principle
in combination with the finite difference method, making it easier and
faster to obtain upper and lower approximations of exact solutions.
The proposed method is found to be an effective numerical method
with satisfactory accuracy. Moreover, the result shows that the effect
of transient thermal shock induced by deformation acceleration is
enhanced by non-Fourier heat conduction with increased peak stress.
The influence on the stress increases with the thermal relaxation time.", keywords = "Maximum principle, non-Fourier heat conduction,
residual correction method, thermo-elastic response.", volume = "9", number = "7", pages = "417-8", }
