On the numerical implementation of special solutions to the homogeneous Riemann–Hilbert problem in two-dimensional elasticity

Abstract

The special solutions proposed by Muskhelishvili for a particular kind of homogeneous Riemann–Hilbert problems are very important in the mechanical analysis of cracked materials. The numerical implementation of these special solutions relies on specific choices of the arguments of relevant parameters. We establish here a unified principle to specify the admissible arguments of relevant parameters in the numerical implementation of these special solutions. We show that the use of the conventional argument branch (such as $\left( {-\pi ,\pi } \right]$ or $\left[ {0,2\pi } \right) )$ may lead to erroneous evaluation of these special solutions when the corresponding crack is arc-shaped. In the context of our principle, we provide also the consistent asymptotic expressions of these special solutions in the neighborhood of a certain point or in the remote region.