Toughness or strength? Regularization in phase-field fracture explained by the coupled criterion

In recent years the phase-field method and the coupled energy and stress-based criterion have attracted much attention due to their adaptability in modeling fractures. Both approaches have been successfully used to determine crack initiation and have compared well with real-life experiments. The phase-field method diffuses the crack surface into the volume of the solid, thus making the solution viable through variational techniques. The diffusion is controlled by an internal length scale, which is primarily considered to be a numerical aid without any real physical meaning. In this paper, we question the consideration that the internal length is only a numerical parameter, and assess its mechanical significance with the help of the coupled criterion. Through elaborate benchmark examples, the correlation between the two methods is demonstrated based on the critical loading, the crack topology, and the crack arrest length. We reveal that independently of the chosen aspect, the phase-field approach and the coupled criterion present excellent correspondence. We show that the correlation between tensile strength and length scale is unique for the standard phase-field formulation. Interestingly, we find that both stress and energy criteria are satisfied in the phase-field fracture, and this is explained by demonstrating the alteration in global energy release rate due to the regularization introduced by the smeared model.