Hypercomplex functions methods in three-dimensional problems of elasticity
by
Yuri Grigoriev
Yakutsk State University, Russia

For multidimensional problems there exists different theories of hypercomplex functions as analogs of complex functions. For example, the theory of regular quaternionic functions of an incomplete quaternionic variable - the Moisil-Teodorescu system's theory. Effective applications of this theory in three dimensional problems of elasticity theory in star-shaped domains were obtained by the authors in previous papers. In this paper the further outcomes in this area are presented. Three dimensional analog of the Kolosov-Muckhelishvili formula in arbitrary domains is obtained in a quaternionic form. Solutions of the four main problems of an elastic sphere equilibrium are expressed in quadratures by means of Appels hypergeometric functions. For a solution of the first main elasticity problem in a bounded Lyapunov's domain the quaternionic 1-st kind integral equation with a polar kernel (in a real form - a system of integral equations of 1-st kind with polar kernels) is obtained. This equation is constructed similar to the plane case, but it is obtained by means of quaternionic analog of the Kolosov-Muckhelishvili formula. An equivalence of this equation to 2-nd kind one with a completely continuous operator is shown. An other way of reducing of the first main elasticity problem to a Fredholm 2-d kind quaternionic integral equation is presented.

Two new hypercomplex algebras of 4-th rank are introduced. Fundamentals of regular function's theories with values in these algebras are constructed. The first algebra helps to extract a square root from the Lame operator, i.e. the last one is presented as a square of a hypercomplex differential operator of the first order. If there exists some dependences between elastic parameters of a transversely-isotropic elastic medium then general solution of equilibrium equations can be expressed by means of regular hypercomplex functions with values in the second algebra.

Date received: August 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-78.