Energetically motivated crack orientation vector for phase-field fracture with a directional split

Abstract

The realistic approximation of structural behavior in a post fracture state by the phase-field method requires information about the spatial orientation of the crack surface at the material point level. For the directional phase-field split, this orientation is specified by the crack orientation vector, that is defined perpendicular to the crack surface. An alternative approach to the determination of the orientation based on standard fracture mechanical arguments, i.e. in alignment with the direction of the largest principle tensile strain or stress, is investigated by considering the amount of dissipated strain energy density during crack evolution. In contrast to the application of gradient methods, the analytical approach enables the determination of all local maxima of strain energy density dissipation and, in consequence, the identification of the global maximum, that is assumed to govern the orientation of an evolving crack. Furthermore, the evaluation of the local maxima provides a novel aspect in the discussion of the phenomenon of crack branching. As the directional split differentiates into crack driving contributions of tension and shear stresses on the crack surface, a consistent relation to Mode I and Mode II fracture is available and a mode dependent fracture toughness can be considered. Consequently, the realistic simulation of rock-like fracture is demonstrated. In addition, a numerical investigation of $$\Gamma $$ Γ -convergence for an AT-2 type crack surface density is presented in a two-dimensional setup. For the directional split, also the issues internal locking as well as lateral phase-field evolution are addressed.

A phase‐field formulation based on an extended F‐criterion for rock fracture

“…To enforce crack irreversibility, a history variable  is used in Equations ( 35) and (37), which is defined as…”

“…To accurately simulate material behaviors in the post-fracture stage, Luo et al 36 presented a phase-field model with a directional strain decomposition to consider the crack surface normal and a stress-driven crack opening indicator to distinguish the opening and closure of pre-cracks. Steinke et al 37 discussed in detail the spatial orientation of the potential crack surface and the issue of internal locking and lateral phase-field evolution due to local orientation misalignment.…”

The origin of the energy split in phase‐field fracture and eigenfracture

“…The solution of this challenging optimisation problem is studied in ref. [18] for simple examples. A generalised, robust and efficient solution scheme for the energetically motivated crack normal is part of ongoing research activities.…”