Nucleation of fracture: The first-octant evidence against classical variational phase-field models

As a companion work to [1], this Note presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on ``tension-dominated'' problems where all principal stresses are non-negative, that is, problems taking place entirely within the first octant in the space of principal stresses.