Contributed Talk

Monogenic series expansions and an application to linear elasticity

Sebastian Bock
Bauhaus-Universitaet Weimar, Germany


In the talk we shall give a brief overview of some interesting characteristics of recently studied orthogonal Appell bases [1,2] of solid spherical monogenics in IR3. In this context, a compact closed form representation of the Appell basis elements in terms of classical spherical harmonics is shown. Based on a recently developed spatial generalization of the Kolosov-Muskhelishvili formulae in terms of a monogenic and an anti-monogenic function [3], the Appell basis is used to construct a polynomial basis of solutions to the Lamé-Navier equation from linear elasticity. Finally, a hypercomplex version of the classical Kelvin solution which describes the elastic displacements of a concentrated force acting at the origin of an infinite body is discussed.

[1] Bock, S.: On a three dimensional analogue to the holomorphic z-powers: Power series and recurrence formulae, Complex Var. Elliptic Eqns, Volume 57, Issue 12, pp. 1271-1287, 2012.
[2] Bock, S.; Gürlebeck, K.; Lavicka, R.; Soucek, V.: Gelfand-Tsetlin bases for spherical monogenics in dimension 3, Revista Matemática Iberoamericana, Volume 28, Issue 4, pages 1165-1192, 2012.
[3] Weisz-Patrault, D.; Bock, S. and Gürlebeck, K.: Three-dimensional elasticity based on quaternion-valued potentials, International Journal of Solids and Structures, Volume 51, Issues 19/20, pages 3422-3430, 2014.

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