Abstract
In order to describe stress and displacement fields in the neightborhood of singularities in fracture mechanics the so-called theory of complex stress function based on Kolosov and Muskhelishvili can be used. The relation between linear elasticity and complex function theory is based on the Theorem of Goursat. In this paper a discrete version of the Theorem is proved.
Similar content being viewed by others
References
G.W. Kolosov, Über eine Anwendung der Theorie der Funktion einer komplexen Veränderlichen auf das ebene Problem der mathematischen Elastizitätstheorie (Russian). Ph. D. Thesis.Yuriew (Dorpat) 1909.
Kolosov G.W.: Über einige Eigenschaften des ebenen Problems der Elastizitätstheorie. Zeitschrift fü r Mathematik und Physik 62, 383–409 (1914)
Muskhelishvili N.I.: Einige Grundaufgaben der mathematischen Elastizitätstheorie. VEB Fachbuchverlag, Leipzig (1971)
Meleshko V.V.: Selected topics in the history of the two-dimensional biharmonic problem. Journal of Applied Mechanics 56((1), 33–85 (2003)
E. Goursat, Sur l’équation Δ Δu = 0. . Bulletin de la Société Mathématique de France 26 (1898), 236.
S. Bock, K. Gü rlebeck, On a Spatial Generalization of the Kolosov-Muskhelishvili Formulae. Math. Meth. Appl. Sciences 32 (2) (2009), 223-240
A.A. Samarskij, Theorie der Differenzenverfahren. Akademische Verlagsgesellschaft Geest und Portig K.-G. Leipzig 1984.
Duffin R.J.: Discrete potential theory. Duke Math. J. 20, 233–251 (1953)
Duffin R.J.: Basic properties of discrete analytic functions. Duke Math. J. 23, 335–363 (1956)
J. Ferrand, Fonctions preharmonique et fonctions preholomorphes. Bulletin des Sciences Mathematique, sec. series.
N. Faustino, U. Kähler, F. Sommen, Dirac Operators in Clifford Analysis. Adv. appl. Clifford alg. Birkhäuser Verlag Basel, doi:10.1007/s00006-007-0041-z
H. Ridder, Schepper H, Kähler U, Sommen F. Discrete function theory based on skew Weyl relations. Proceedings of the American Mathematical Society, 2009
A. Hommel, Fundamentallösungen partieller Differenzenoperatoren und die Lösung diskreter Randwertprobleme mit Hilfe von Differenzenpotentialen. Dissertation, Bauhaus Universität Weimar 1998.
F Brackx, H. Schepper, F. Sommen, L.V. de Voorde, Discrete Clifford analysis: an overview. Cubo 11 (2009), 55-71.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hommel, A. A Discrete Theorem of Goursat. Adv. Appl. Clifford Algebras 24, 1039–1045 (2014). https://doi.org/10.1007/s00006-014-0500-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-014-0500-2