On the steady-state thermoelastic problem for the half-space
Authors:
E. Sternberg and E. L. McDowell
Journal:
Quart. Appl. Math. 14 (1957), 381-398
MSC:
Primary 73.2X
DOI:
https://doi.org/10.1090/qam/87367
MathSciNet review:
87367
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Abstract: This paper deals with the determination of the steady-state thermal stresses and displacements in a semi-infinite elastic medium which is bounded by a plane. The problem is treated within the classical theory of elasticity and is approached by the method of Green. It is shown that the stress field induced by an arbitrary distribution of surface temperatures is plane and parallel to the boundary. If the surface temperature is prescribed arbitrarily over a bounded “region of exposure” and is otherwise constant, the problem reduces to the determination of Boussinesq’s three-dimensional logarithmic potential for a disk in the shape of the region of exposure, whose mass density is equal to the given temperature. Moreover, it is found that there exists a useful connection between the solutions to Boussinesq’s and to the present problem for the half-space. An exact closed solution, in terms of complete and incomplete elliptic integrals of the first and second kind, is given for a circular region of exposure at uniform temperature. Exact solutions in terms of elementary functions are presented for a hemispherical distribution of temperature over a circular region, as well as for a rectangle at constant temperature.
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J. N. Goodier, On the integration of the thermo-elastic equations, Phil. Mag., 23, 1017 (1937)
C. W. Borchardt, Untersuchungen über die Elasticität fester isotroper Körper unter Berücksichtigung der Wärme, Monatsber. Akad. Wiss. Berlin, 9 (1873)
R. D. Mindlin and D. H. Cheng, Thermoelastic stress in the semi-infinite solid, J. Appl. Phys. 21, 931 (1950)
A. E. H. Love, The stress produced in a semi-infinite solid by pressure on part of the boundary, Trans. Roy. Soc. (London), Series A, 228, 377 (1929)
M. A. Sadowsky, Thermal shock on a circular surface of exposure of an elastic half-space, J. Appl. Mech. 22, 2, 177 (1955).
E. Sternberg and R. A. Eubanks, On the concept of concentrated loads and an extension of the uniqueness theorem in the linear theory of elasticity, J. Rat. Mech. and Anal. 4, 1, 135 (1955)
J. Boussinesq, Application des potentiels à l’étude de l’équilibre et du movement des solides élastiques, Gauthiers-Villars, Paris, 1885
P. F. Papkovich, Solution générale des équations differentielles fondamentales d’élasticité exprimée par trois fonctions harmoniques, C. R., Acad. Sci. Paris 195, 513 (1932)
H. Neuber, Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie, Z. angew. Math. Mech. 14, 203 (1934)
R. D. Mindlin, Note on the Galerkin and Papkovich stress functions, Bull. Am. Math. Soc. 42, 373 (1936)
O. D. Kellogg, Foundations of potential theory, Springer, Berlin, 1929
H. Jeffreys and B. S. Jeffreys, Methods of mathematical physics, University Press, Cambridge, 1950
H. Lamb, On Boussinesq’s problem, Proc. London Math. Soc. 34, 276 (1902)
K. Terazawa, On the elastic equilibrium of a semi-infinite solid, J. Coll. Sci., Univ. Tokyo, 37 art. 7 (1916)
A. E. H. Love, A treatise on the mathematical theory of elasticity, 4th ed., Dover, New York, 1944
M. A. Sadowsky and E. Sternberg, Elliptic integral representation of axially symmetric flows, Quart. Appl. Math. 8, 2, 113 (1950)
E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., University Press, Cambridge, 1935
Eugene Jahnke and Fritz Emde, Tables of functions, 4th ed., Dover, New York, 1945
H. Hertz, Über die Berührung fester elastischer Körper, J. Math. (Crelle), 92, 1881
M. T. Huber, Zur Theorie der Berührung fester elastischer Körper, Ann. Physik 14, 153 (1904)
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Article copyright:
© Copyright 1957
American Mathematical Society