Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Transient temperature and elastic response of a space-based mirror in the radiation-conduction environment

Author: Bejoy K. Choudhury
Journal: Quart. Appl. Math. 64 (2006), 201-228
MSC (2000): Primary 74B05, 74F05, 42B05
DOI: https://doi.org/10.1090/S0033-569X-06-00970-7
Published electronically: May 3, 2006
MathSciNet review: 2243860
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of the one-dimensional heat equation with radiation boundary conditions is studied. Steady state and transient solutions for a space-based mirror are obtained under a broad class of boundary specification. Associated elastic stress and displacement are investigated. Time-varying heat flux at the boundary is also considered. The method requires solving equations of the type $ \tan\lambda=\left(Bi_1+Bi_2\right)\lambda/\left(\lambda^2-Bi_1Bi_2\right)$ for the eigenvalues $ \lambda$, for which complete asymptotic solutions are given. Selected transient response results for temperature and stresses are presented.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover reprint, New York, 1970. See tables 4-19, 4-20 for the eigenvalues of (A.1), (A.2).
  • 2. R. C. F. Bartels and R. V. Churchill, Resolution of Boundary Problems by the use of a Generalized Convolution, Bull. Amer. Math. Soc. 48, 276-282 (1942). MR 0005994 (3:243g)
  • 3. M. A. Biot, Thermoelasticity and Irreversible Thermodynamics, J. Appl. Phys. 27, 240-253 (1956). MR 0077441 (17:1035e)
  • 4. G. Birkhoff and G-C. Rota, Ordinary Differential Equations, 4th ed, John Wiley, New York, 1989. MR 0972977 (90h:34001)
  • 5. L. M. K. Boelter, V. H. Cherry, H. A. Johnson, R. C. Martinelli, Heat Transfer Notes, McGraw-Hill, New York, 1965.
  • 6. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, Mineola (Dover reprint), New York, 1997.
  • 7. D. E. Carlson, Linear Thermoelasticity, in Handbuch der Physik, Bd. VIa/2, Springer-Verlag, Berlin, 1972.
  • 8. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed, Clarendon Press, Oxford, 1973; Reprint of 2nd ed., 1988. MR 0959730 (89f:80004)
  • 9. W. A. Day, Justification of the Uncoupled and Quasi-Static Approximations in a Problem of Dynamic Thermoelasticity, Arch. Rational Mech. Anal. 77, 387-396 (1981). MR 0642554 (83b:73091)
  • 10. W. A. Day, The Status of the Heat Equation, in Rational Thermodynamics, 2nd ed, (Ed. C. Truesdell), Springer-Verlag, New York, 1984. Also, Heat Conduction within Linear Thermoelasticity, Springer Tracts in Natural Philosophy, vol. 30, Springer-Verlag, Berlin, 1985. MR 0804043 (87c:73001)
  • 11. N. G. de Bruijn, Asymptotic Methods in Analysis, 2nd ed, North-Holland, Amsterdam, 1961. MR 0177247 (31:1510)
  • 12. G. H. Hardy, Collected Papers, vol. III, Clarendon Press, Oxford, 1969. MR 0255362 (41:24)
  • 13. G. H. Hardy and W. W. Rogosinski, Fourier Series, Cambridge Tracts No. 38, 3rd ed, Cambridge University, Cambridge, 1950. MR 0044660 (13:457b)
  • 14. M. P. Heisler, Temperature Charts for Induction and Constant-Temperature Heating, ASME Trans. 69, 227-236 (1947).
  • 15. M. P. Heisler, Transient Thermal Stresses in Slabs and Circular Pressure Vessel, J. Appl. Mech. 20, 261-269 (1953).
  • 16. J. H. Hlinka, H. G. Landau, and V. Paschkis, Charts on Elastic Thermal Stresses in Heating and Cooling of Slabs and Cylinders, ASME paper 57-A-238, 1-19 (1957).
  • 17. E. Landau, Foundations of Analysis, 3rd ed, (English translation by F. Steinhardt), Chelsea/AMS reprint, 2000. MR 0038404 (12:397m)
  • 18. A. V. Luikov, Analytical Heat Diffusion Theory, Academic Press, London, 1968.
  • 19. N. Noda, R. B. Hetnarski, and Y. Tanigawa, Thermal Stresses, 2nd ed, Taylor and Francis, New York, 2003.
  • 20. F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974; Reprint AK Peters, Ltd., Wellesley, MA, 1997. MR 0435697 (55:8655); MR 1429619 (97i:41001)
  • 21. M. N. Özisik, Heat Conduction, 2nd ed, Wiley-Interscience, New York, 1993.
  • 22. P. J. Schneider, Temperature Response Charts, John Wiley, New York, 1963. [A total of 120 uniformly sized charts (mostly 10.5 cm $ \times$ 21 cm) were specially prepared for this book.]
  • 23. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed, McGraw-Hill, New York, 1970.
  • 24. E. C. Titchmarsh, The Theory of Functions, 2nd ed, Clarendon Press, Oxford, 1985.
  • 25. G. P. Tolstov, Fourier Series (English translation by Richard. A. Silverman), Dover reprint, New York, 1962. MR 0425474 (54:13429)
  • 26. J. H. Weiner, A Uniqueness Theorem for the Coupled Thermoelastic Problem, Quart. Appl. Math. 15, 102-105 (1957). MR 0088216 (19:484i)
  • 27. R. D. S. Zerkle and J. E. Sunderland, The transient temperature distribution in a slab subject to thermal radiation, J. Heat Transfer, Trans. ASME. 87, 117-133 (1965).
  • 28. A. Zygmund, Trigonometric Series, 2nd ed, Cambridge University Press, Cambridge, 1988. MR 0933759 (89c:42001)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 74B05, 74F05, 42B05

Retrieve articles in all journals with MSC (2000): 74B05, 74F05, 42B05

Additional Information

Bejoy K. Choudhury
Affiliation: Lockheed Martin Space Systems, Sunnyvale, California 94089
Email: bejoy.choudhury@lmco.com

DOI: https://doi.org/10.1090/S0033-569X-06-00970-7
Keywords: Heat conduction, thermoelasticity, Fourier series
Received by editor(s): December 10, 2004
Published electronically: May 3, 2006
Article copyright: © Copyright 2006 Brown University
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society