## 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

• In this paper we present a theoretical background for 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 complex function theory and couple it continuouslyIn this paper we present a theoretical background for 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 complex function theory and couple it continuously with the finite element solution in the region far from the 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 in the way of practical realisation of this strategy.  Document Type: Article Dr. Dmitrii LegatiukORCiDGND, Daniel Weisz-Patrault https://doi.org/10.1007/s40315-021-00403-7Cite-Link https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20210805-44763Cite-Link https://link.springer.com/article/10.1007/s40315-021-00403-7 Computational Methods and Function Theory Springer Heidelberg English 2021/08/02 2021/07/01 2021/08/05 Bauhaus-Universität Weimar Fakultät Bauingenieurwesen / Professur Angewandte Mathematik 2021 23 1 23 coupling; crack propagation; energetic approach; fracture mechanics Angewandte Mathematik; Finite-Elemente-Methode; Rissausbreitung; Modellierung; Bruchmechanik 600 Technik, Medizin, angewandte Wissenschaften / 620 Ingenieurwissenschaften 50 Technik allgemein / 50.31 Technische Mechanik Creative Commons 4.0 - Namensnennung (CC BY 4.0)