On the completeness of the Papkovich-Neuber solution
Author:
Ton Tran Cong
Journal:
Quart. Appl. Math. 47 (1989), 645-659
MSC:
Primary 73C05
DOI:
https://doi.org/10.1090/qam/1031682
MathSciNet review:
MR1031682
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P. F. Papkovich, The representation of the general integral of the theory of elasticity in terms of harmonic functions (in Russian), Izvest. Akad. Nauk. SSSR. No. 10, 1425–1435 (1932), also in Solution générale des équations differentielles fondamentales d’élasticité exprimeé par trois fonctions harmoniques, Comptes Rendus, Acad. de Sci., Paris, 513–515, Sept. (1932)
H. Neuber, Kerbspannungslere, Springer, Berlin, 1937
- 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
- M. E. Gurtin, On Helmholtz’s theorem and the completeness of the Papkovich-Neuber stress functions for infinite domains, Arch. Rational Mech. Anal. 9 (1962), 225–233. MR 187467, DOI https://doi.org/10.1007/BF00253346
M. E. Gurtin, The linear theory of elasticity, Handbuch der Physik, VI a/2, Springer-Verlag, Berlin-Heidelberg-New York, 1972
- R. A. Eubanks and E. Sternberg, On the completeness of the Boussinesq-Papkovich stress functions, J. Rational Mech. Anal. 5 (1956), 735–746. MR 79897, DOI https://doi.org/10.1512/iumj.1956.5.55027
- M. G. Slobodyanskiĭ, General forms of solutions, expressed by harmonic functions, of the equations of elasticity for simply connected and multiply connected regions, Akad. Nauk SSSR. Prikl. Mat. Meh. 18 (1954), 55–74 (Russian). MR 0064605
- I. S. Sokolnikoff, Mathematical theory of elasticity, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. 2d ed. MR 0075755
- M. Stippes, Completeness of the Papkovich potentials, Quart. Appl. Math. 26 (1968/69), 477–483. MR 239801, DOI https://doi.org/10.1090/S0033-569X-1969-0239801-4
- S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, Oxford-New York-Paris, 1965. Translated from the Russian by W. J. A. Whyte; Translation edited by I. N. Sneddon. MR 0185399
- T. Tran Cong and G. P. Steven, On the representation of elastic displacement fields in terms of three harmonic functions, J. Elasticity 9 (1979), no. 3, 325–333. MR 547720, DOI https://doi.org/10.1007/BF00041103
- 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
P. F. Papkovich, The representation of the general integral of the theory of elasticity in terms of harmonic functions (in Russian), Izvest. Akad. Nauk. SSSR. No. 10, 1425–1435 (1932), also in Solution générale des équations differentielles fondamentales d’élasticité exprimeé par trois fonctions harmoniques, Comptes Rendus, Acad. de Sci., Paris, 513–515, Sept. (1932)
H. Neuber, Kerbspannungslere, Springer, Berlin, 1937
R. D. Mindlin, Note on the Galerkin and Papkovitch stress functions, Bull. Amer. Math. Soc. 42, 373–376 (1936)
M. E. Gurtin, On Helmholtz’s theorem and the completeness of the Papkovitch-Neuber stress functions for infinite domains, Arch. Rational Mech. Anal. 9, 225–233 (1962)
M. E. Gurtin, The linear theory of elasticity, Handbuch der Physik, VI a/2, Springer-Verlag, Berlin-Heidelberg-New York, 1972
R. A. Eubanks and E. Sternberg, On the completeness of the Boussinesq-Papkovich stress functions, J. Rational Mech. Anal. 5, 735–746 (1956)
M. G. Slobodyansky, General forms of solutions, expressed by harmonic functions, of the equations of elasticity for simply connected and multiply connected regions, Akad. Nauk SSSR Prikl. Mat. Mekh. 18, 54–78 (1954)
I. S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd ed., McGraw-Hill, New York, 1956
M. Stippes, Completeness of the Papkovich potentials, Quart. Appl. Math. 26, 477–483 (1969)
S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, New York, 1953
Ton Tran-Cong and G. P. Steven, On the representation of elastic displacement fields in terms of three harmonic functions, J. Elasticity 9, 325–333 (1979)
O. D. Kellog, Foundations of Potential Theory, Springer, 1929, also Dover, New York, 1953
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© Copyright 1989
American Mathematical Society