Programming of the Complex Logarithm Function in the Solution of the Cracked Anisotropic Plate Loaded by a Point Force.

摘要

In programming solutions of complex function theory, the complex logarithm function is replaced by the complex logarithmic function, introducing a discontinuity along the branch cut into the programmed solution which was not present in the mathematical solution. Recently, Liaw and Kamel presented their solution of the infinite anisotropic centrally cracked plate loaded by an arbitrary point force, which they used as Green's function in a boundary element method intended to evaluate the stress intensity factor at the tip of a crack originating from an elliptical hole. Their solution may be used as Green's function for many more numerical methods involving anisotropic elasticity. In programming applications of Liaw and Kamel's solution, the standard definition of the logarithmic function with the branch cut at the non-positive real axis cannot provide a reliable computation of the displacement field for Liaw and Kamel's solution. Either the branch cut should be redefined outside the domain of the logarithmic function, after proving that the domain is limited to a part of the plane, or the logarithm function should be defined on its Riemann surface. A two dimensional line fractal can provide the link between all mesh points on the plane, essential to evaluate the logarithm function on its Riemann surface. As an example, a two dimensional line fractal is defined for a mesh once used by Erdogan and Arin.