Abstract
In this paper the representation of three-dimensional displacement fields in linear elasticity in terms of six complex valued functions is considered. The representation includes the complex Muskhelishvili formulation for plane strain as a special case. The completeness of the complex representation for regular solutions is shown and a relationship to the Neuber/Papkovich solutions is given.
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References
R. Piltner: The use of complex valued functions for the solution of three-dimensional elasticity problems. J. Elasticity 18 (1987) 191–225.
N.I. Muskhelishvili: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1963).
M.E. Gurtin: The linear theory of elasticity. In: S. Flügge (ed.), Handbuch der Physik, Vol. VIa/2 (ed. by C. Truesdell), Springer-Verlag, Berlin (1972) pp. 1–295.
R.A. Eubanks and E. Sternberg: On the completeness of the Boussinesq-Papkovich stress functions. J. Rat. Mech. Anal. 5 (1956) 735–746.
M. Stippes: Completeness of the Papkovich potentials. Q. Appl. Math. 26 (1969) 477–483.
T. Tran Cong and G.P. Steven: On the representation of elastic displacement fields in terms of three harmonic functions. J. Elasticity 9 (1979) 325–333.
K. Hackl and U. Zastrow: On the existence, uniqueness and completeness of displacement and stress functions in linear elasticity. J. Elasticity 19 (1988) 3–23.
R.F. Millar: On the completeness of the Papkovich potentials. Q. Appl. Math. 41 (1984) 385–393.
D.E. Carlson: A note on the Beltrami stress functions. Z. angew. Math. Mech. 47 (1967) 206–207.
R.D. Mindlin: Note on the Galerkin and Papkovich stress functions. Bull. Am. Math. Soc. 42 (1936) 373–376.
U. Zastrow: Zur Lösung von Halbraumproblemen nach der Spannungsfunktionenmethode — neue und alte Ansätze für Verschiebungs-und Spannungsfunktionen in der räumlichen Elastizitätstheorie (habilitation thesis). VDI-Verlag, Düsseldorf (1986).
P.F. Papkovich: Solution générale des équations différentielles fondamentales d'élasticité, exprimée par trois fonctions harmoniques. C. R. Acad. Sci. Paris 195 (1932) 513–515.
H. Neuber: Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel. Z. angew. Math. Mech. 14 (1934) 203–212.
H. Neuber: Kerbspannungslehre. Springer-Verlag, Berlin (1958).
E.T. Whittaker and G. N. Watson: A Course of Modern Analysis. Cambridge University Press (1927).
E.T. Copson: Partial Differential Equations. Cambridge University Press (1974).
E.T. Whittaker: On the partial differential equations of mathematical physics. Math. Ann. 57 (1903) 333–355.
H. Bateman: Partial Differential Equations of Mathematical Physics. Cambridge University Press (1932).
E.W. Hobson: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press (1931).
E.T. Whittaker: On Lamé's differential equation and ellipsoidal harmonics. Proc. London Math. Soc., 2. Ser. 14 (1915) 260–268.
G. Temple. Whittaker's work on the integral representation of harmonic functions. Proc. Edinburgh Math. Soc. (1958) 11–24.
G.B. Jeffrey: On spheroidal harmonics and allied functions. Proc. Edinburgh Math. Soc. 33 (1915) 118–121.
G.B. Jeffrey: Bipolar and toroidal harmonics. Proc. Edinburgh Math. Soc. 34 (1916) 102–108.
G.B. Jeffrey: Transformations of axes for Whittaker's solution of Laplace's equations. Proc. Edinburgh Math. Soc. 35 (1917) 32–37.
S. Bergmann: Zur Theorie der ein-und mehrwertigen harmonischen Funktionen des dreidimensionalen Raumes. Math. Z. 24 (1926) 641–669.
S. Bergmann: Zur Theorie der algebraischen Potentialfunktionen des dreidimensionalen Raumes (part I, II). Math. Ann. 99 (1928) 629–659 and 101 (1929) 534–558.
S. Bergmann: A class of harmonic functions in three variables and their properties. Trans. Amer. Math. Soc. 59 (1946) 216–247.
E. Kreyszig: On singularities of solutions of partial differential equations in three variables. Arch. Rat. Mech. Anal. 2 (1958/59) 151–159.
E. Kreyszig: On regular and singular harmonic functions of three variables. Arch. Rat. Mech. Anal. 4 (1959/60) 352–370.
Ph.M. Morse and H. Feshbach: Methods of Theoretical Physics, Part II. McGraw-Hill, New York (1953) pp. 1252–1309.
H.G. Hahn: Elastizitätstheorie. Teubner-Verlag, Stuttgart (1985).
H. Leipholz: Einführung in die Elastizitätstheorie. Braun-Verlag, Karlsruhe (1968).
Added in proof
R. Piltner: The representation of three-dimensional elastic displacement fields with the aid of complex valued functions for several curvilinear coordinates. Mechanics Research Communications 15 (2) (1988) 79–85.
R. Piltner: The application of a comples 3-dimensional elasticity solution representation for the analysis of a thick rectangular plate. Acta Mechanica 75 (1988) 77–91.
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Piltner, R. On the representation of three-dimensional elasticity solutions with the aid of complex valued functions. J Elasticity 22, 45–55 (1989). https://doi.org/10.1007/BF00055333
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DOI: https://doi.org/10.1007/BF00055333