Minimum energy multiple crack propagation. Part I: Theory and state of the art review

Abstract

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith’s theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. The key contributions of Part-I of this three-part paper are given as follows. (1) Formulation of the total energy functional governing multiple crack behaviour. (2) Three solution methods to the problem of competing crack growth for different fracture front stabilities, e.g. stable, unstable, or partially stable crack tip configurations; we compare our approach to Budyn et al. (2004) and demonstrate via example cases that the latter approach of resolving competing crack growth is not energy minimal in some cases. Finally, (3), the minimum energy criterion for a set of crack tip extensions is posed as the condition of vanishing rotational dissipation rates with respect to the extension angles. The proposed formulation lends itself to a straightforward application within a discrete framework involving multiple finite-length crack tip extensions. The open-source Matlab code, documentation, benchmark/example cases are included as supplementary material.