A phase-field crack model based on a directional strain decomposition and a stress-driven Crack-Opening Indicator

We present a phase-field model for pre-cracked solids with a directional strain decomposition and a stress-driven crack-opening indicator (COI). We first introduce the COI to distinguish the opening and closure of pre-cracks and propose a novel criterion for solids based on the stress component in the direction of the crack surface normal. The stress-driven COI can correctly detect crack opening when principal strains have different signs, which cannot be achieved by most existing phase-field models. We then split the strain tensor into an effective part that induces stress and a crack-induced part that does not result in stress, and suggest a mapping tensor to transform the total strain tensor to the effective strain tensor. The mapping tensor is based on the crack surface normal, the phase variable, and the COI, thus leading to a directional decomposition of the strain. Eventually, we obtain the fourth-order material tangent mapping an arbitrary strain tensor to its stress tensor in the way that the shear stress along the crack surface vanishes and the normal stress in the crack surface normal direction is eliminated when the crack is open. Besides, owing to the directional strain decomposition, the direction where the stiffness decreases is invariant to the current strain once a diffusive crack is generated, which is physically more reasonable. More importantly, the decomposition together with the COI ensures that when the crack is open the spurious Poisson effect associated with the crack-induced strain is eliminated. Numerical examples are provided and the results have demonstrated the capability of the proposed model in simulating material behaviours in the post-fracture stage. Particularly, in a shear test of a squared plate with an initial crack, our model that simulates both the pre-crack and crack growth using the phase-field approach is able to reproduce the crack evolution pattern that has only been obtained in phase-field modelling by considering the initial crack as a geometric notch.