On a Hypercomplex Version of the Kelvin Solution in Linear Elasticity

  • Sebastian BockEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


The article gives an overview about recently developed spatial generalizations of the Kolosov-Muskhelishvili formulae using the framework of hypercomplex function theory. Based on these results, a hypercomplex version of the classical Kelvin solution is obtained. For this purpose a new class of monogenic functions with (logarithmic) line singularities is studied and an associated two step recurrence formula is proved. Finally, a connection of the function system to the Cauchy-kernel function is established.


Recurrence formulae Generalized Kolosov-Muskhelishvili formulae Kelvin solution 

Mathematics Subject Classification (2010)

Primary 30G35; Secondary 74B05 


  1. 1.
    C. Álvarez-Peña, Contragenic Functions and Appell Bases for Monogenic Functions of Three Variables. Ph.D. thesis. Centro de Investigacion y de Estudios Avanzados del I.P.N., Mexico (2013)Google Scholar
  2. 2.
    C. Álvarez-Peña, R.M. Porter, Contragenic functions of three variables. Compl. Anal. Oper. Theory 8, 409–427 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L.C. Andrews, Special Functions of Mathematics for Engineers (SPIE Optical Engineering Press, Bellingham; Oxford University Press, Oxford, 1998)Google Scholar
  4. 4.
    J.R. Barber, Elasticity. Solid Mechanics and Its Applications, vol. 172, 3rd rev. edn. (Springer, New York, 2010)Google Scholar
  5. 5.
    S. Bock, K. Gürlebeck, On a spatial generalization of the Kolosov-Muskhelishvili formulae. Math. Methods Appl. Sci. 32, 223–240 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    S. Bock, On a three dimensional analogue to the holomorphic z-powers: Laurent series expansions. Compl. Var. Elliptic Equ. 57(12), 1271–1287 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Bock, On a three dimensional analogue to the holomorphic z-powers: power series and recurrence formulae. Compl. Var. Elliptic Equ. 57(12), 1349–1370 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S. Bock, On monogenic series expansions with applications to linear elasticity. Adv. Appl. Clifford Algebr. 24(4), 931–943 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis. Pitman Research Notes Math. Ser. 76, Pitman, London etc. (1982)Google Scholar
  10. 10.
    K. Gürlebeck, H.R. Malonek, A hypercomplex derivative of monogenic functions in \(\mathbb {R}^{n+1}\) and its applications. Compl. Var. 39, 199–228 (1999)Google Scholar
  11. 11.
    K. Gürlebeck, K. Habetha, W. Sprößig, Holomorphic functions in the plane and n-dimensional space, in A Birkhäuser Book (2008), ISBN: 978-3-7643-8271-1Google Scholar
  12. 12.
    B. Klein Obbink, On the Solutions of D n D m F. Reports on Applied and Numerical Analysis (Eindhoven University of Technology, Department of Mathematics and Computing Science, 1993)Google Scholar
  13. 13.
    G.W. Kolosov, Über einige Eigenschaften des ebenen Problems der Elastizitätstheorie, Z. Math. Phys. 62, 383–409 (1914)Google Scholar
  14. 14.
    M.E. Luna-Elizarrarás, M. Shapiro, A survey on the (hyper-) derivates in complex, quaternionic and Clifford analysis. Millan J. Math. 79, 521–542 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    H.R. Malonek, Zum Holomorphiebegriff in höheren Dimensionen, Habilitationsschrift. Pädagogische Hochschule Halle (1987)Google Scholar
  16. 16.
    J. Morais, M.H. Nguyen, K.I. Kou, On 3D orthogonal prolate spheroidal monogenics. Math. Methods Appl. Sci. 39(4), 635–648 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity,(translated from the Russian by J.R.M. Radok.) (Noordhoff International Publishing, Leyden, 1977)Google Scholar
  18. 18.
    H. Neuber, Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie; der Hohlkegel unter Einzellast als Beispiel. Z. Angew. Math. Mech. 14, 203–212 (1934)CrossRefzbMATHGoogle Scholar
  19. 19.
    P. Papkovic, Solution générale des équations différentielles fondamentales de l’élasticité, exprimée par un vecteur et un scalaire harmonique (Russisch), in Bull. Acad. Sc. Leningrad (1932), pp. 1425–1435Google Scholar
  20. 20.
    J.R. Rice, Mathematical analysis in the mechanics of fracture, in Fracture, An Advanced Treatise, ed. by H. Liebowitz. Mathematical Fundamentals, vol. 2 (Academic Press, New York, 1968), pp. 191–311Google Scholar
  21. 21.
    D. Weisz-Patrault, S. Bock, K. Gürlebeck, Three-dimensional elasticity based on quaternion-valued potentials. Int. J. Solids Struct. 51(19), 3422–3430 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics/PhysicsBauhaus-Universität WeimarWeimarGermany

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