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 Part-II of our three-part paper examines three discrete solution methods for solving fracture mechanics problems based on the principle of minimum total energy. The suitability of each solution approach is determined by the stability property of the fracture configuration at hand. The first method is based on external load-control. It is suitable for stable crack growth and stable fracture configurations. The second method is based on fracture area-control (or length-control in 2D). This method is applicable to stable or unstable fracture growth but the fracture front must be stable. The third solution method is based on a gradient-descent. Although the method is aimed at solving general crack growth problems, its weak point is that the converged solution cannot be guaranteed to be optimal in the particular case of competing crack growth and an unstable fracture front configuration. Nonetheless, the main focus is on the implementation and application of the gradient-descent solution approach within the framework of the extended finite element method. Concerning the aforementioned case of competing crack growth, an alternative solution strategy is pursued to supplement the gradient-descent approach. The proposed method, however, is only a proof of concept since its robustness is assessed by solving fabricated benchmark problems. The open-source Matlab code, documentation and example cases are included as supplementary material.

Link: https://doi.org/10.1016/j.engfracmech.2017.07.029