  In this paper we present a theoretical background of a coupled
analytical-numerical approach to model a crack propagation process in
two-dimensional bounded domains. The goal of the coupled analytical-numerical
approach is to obtain the correct solution behaviour near the crack tip by help
of the analytical solution constructed by using tools of the complex function
theory and couple it continuously with the finite element solution in the
region far from singularity. In this way, crack propagation could be modelled
without using remeshing. Possible directions of crack growth can be calculated
through the minimization of the total energy composed of the potential energy
and the dissipated energy based on the energy release rate. Within this
setting, an analytical solution of a mixed boundary value problem based on
complex analysis and conformal mapping techniques is presented in a circular
region containing an arbitrary crack path. More precisely, the linear elastic
problem is transformed into a Riemann-Hilbert problem in the unit disk for
holomorphic functions. Utilising advantages of the analytical solution in the
region near the crack tip, the total energy could be evaluated within short
computation times for various crack kink angles and lengths leading to a
potentially efficient way of computing the minimization procedure. To this end,
the paper presents a general strategy of the new coupled approach for crack
propagation modelling. Additionally, we also discuss obstacles on the way of
practical realisation of this strategy.
