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In this paper we present a theoretical background of a coupledanalytical-numerical approach to model a crack propagation process intwo-dimensional bounded domains. The goal of the coupled analytical-numericalapproach is to obtain the correct solution behaviour near the crack tip by helpof the analytical solution constructed by using tools of the complex functiontheory and couple it continuously with the finite element solution in theregion far from singularity. In this way, crack propagation could be modelledwithout using remeshing. Possible directions of crack growth can be calculatedthrough the minimization of the total energy composed of the potential energyand the dissipated energy based on the energy release rate. Within thissetting, an analytical solution of a mixed boundary value problem based oncomplex analysis and conformal mapping techniques is presented in a circularregion containing an arbitrary crack path. More precisely, the linear elasticproblem is transformed into a Riemann-Hilbert problem in the unit disk forholomorphic functions. Utilising advantages of the analytical solution in theregion near the crack tip, the total energy could be evaluated within shortcomputation times for various crack kink angles and lengths leading to apotentially efficient way of computing the minimization procedure. To this end,the paper presents a general strategy of the new coupled approach for crackpropagation modelling. Additionally, we also discuss obstacles on the way ofpractical realisation of this strategy.

computational methods and function theory

Coupling of complex function theory and finite element method for crack  propagation through energetic formulation: conformal mapping approach and  reduction to a Riemann-Hilbert problem