Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended Finite Element Method

In computational fracture mechanics as applied, for example, to damage tolerance assessment, it has been common practice to determine the onset of fracture growth and the growth direction by post-processing the solution of the linear elastostatics problem, at a particular instance in time. For mixed mode loading the available analytically derived criteria that can be used for determining the onset of crack growth typically rely on the assumptions of an idealized geometry e.g. a single crack subjected to remote loading [1, 2] and that the kink angle of the infinitesimal crack increment is quite small [3]. Moreover, the growth direction given by a criterion that is based on an instantaneous local crack tip field can only be valid for infinitesimally small crack growth increments. Consequently, the maximum hoop stress criterion [4] and other similar criteria [5] disregard the changes in the solution that take place as fractures advance over a finite size propagation. Hence, due to the error committed in time-integration, fractures may no longer follow the most energetically favorable paths that theoretically could be achieved for a specific discrete problem. In our approach, we investigate multiple fracture evolution under quasistatic conditions in an isotropic linear elastic solid based on the principle of minimum potential elastic energy, which can help circumvent the aforementioned difficulties. The technique enables a minimization of the potential energy with respect to all crack increment directions minding their relative interactions. The directions are optimized (in the energy sense) by considering virtual crack rotations to find the energy release rates and its first derivatives in order to determine, via an iterative process, the directions that yield zero energy release rates with respect to all virtual rotations [6]. We use the extended finite element method (XFEM) [7, 8] for discretization of a 2D continua in order to model an elaborate crack evolution over time, similar in principle to [9], although we consider here