The completeness of Biot’s solution of the coupled thermoelastic problem
Author:
A. Verruijt
Journal:
Quart. Appl. Math. 26 (1969), 485-490
MSC:
Primary 73.35; Secondary 35.00
DOI:
https://doi.org/10.1090/qam/239802
MathSciNet review:
239802
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Abstract: In one of his papers on the theory of thermoelasticity, M. A. Biot [2] has presented a solution to the differential equations very similar to the well-known Boussinesq—Papcovitch solution in the theory of elasticity. In this note it is proved that this solution is complete, the proof being based upon Mindlin’s theorem of completeness of the Boussinesq—Papcovitch solution in elasticity.
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 0222317
- M. A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27 (1956), 240–253. MR 77441
P. F. Papcovitch, Solution générale des équations différentielles fondamentales d’élastictté, exprimé par trois fonctions harmoniques, C. R. Acad. Sci. Paris 195, 513–516 (1932)
H. Neuber, Ein neuer Ansatz zur Lösung der Grundgleichungen der Elastizitätstheorie, Z. angew. Math. Mech. 14, 203–212 (1934)
J. Boussinesq, Application des potentiels ä l’étude de l’équilibre et du mouvement des solides elastiques, Gauthiera-Villars, Paris, 1885
- R. D. Mindlin, Note on the Galerkin and Papkovitch stress functions, Bull. Amer. Math. Soc. 42 (1936), no. 6, 373–376. MR 1563303, DOI https://doi.org/10.1090/S0002-9904-1936-06304-4
R. A. Eubanka and E. Sternberg, On the completeness of the Boussinesq–Papkovich stress functions, J. Rational Mech. Anal. 5, 735–746 (1956)
H. B. Phillips, Vector analysis, Wiley, New York, 1933
L. R. Ford, Differential equations, McGraw-Hill, New York, 1955
O. D. Kellogg, Foundations of potential theory, Springer, Berlin, 1929
M. A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27, 240–253 (1956)
P. F. Papcovitch, Solution générale des équations différentielles fondamentales d’élastictté, exprimé par trois fonctions harmoniques, C. R. Acad. Sci. Paris 195, 513–516 (1932)
H. Neuber, Ein neuer Ansatz zur Lösung der Grundgleichungen der Elastizitätstheorie, Z. angew. Math. Mech. 14, 203–212 (1934)
J. Boussinesq, Application des potentiels ä l’étude de l’équilibre et du mouvement des solides elastiques, Gauthiera-Villars, Paris, 1885
R. D. Mindlin, Note on the Galerkin and Papkovich stress functions, Bull. Amer. Math. Soc. 42, 373–376 (1936)
R. A. Eubanka and E. Sternberg, On the completeness of the Boussinesq–Papkovich stress functions, J. Rational Mech. Anal. 5, 735–746 (1956)
H. B. Phillips, Vector analysis, Wiley, New York, 1933
L. R. Ford, Differential equations, McGraw-Hill, New York, 1955
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Article copyright:
© Copyright 1969
American Mathematical Society
