Abstract
For the treatment of plane elasticity problems the use of complex functions has turned out to be an elegant and effective method. The complex formulation of stresses and displacements resulted from the introduction of a real stress function which has to satisfy the 2-dimensional biharmonic equation. It can be expressed therefore with the aid of complex functions. In this paper the fundamental idea of characterizing the elasticity problem in the case of zero body forces by a biharmonic stress function represented by complex valued functions is extended to 3-dimensional problems. The complex formulas are derived in such a way that the Muskhelishvili formulation for plane strain is included as a special case. As in the plane case, arbitrary complex valued functions can be used to ensure the satisfaction of the governing equations. Within the solution of an analytical example some advantages of the presented method are illustrated.
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Piltner, R. The use of complex valued functions for the solution of three-dimensional elasticity problems. J Elasticity 18, 191–225 (1987). https://doi.org/10.1007/BF00044194
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DOI: https://doi.org/10.1007/BF00044194