Griffith vs. Barenblatt

Mechanical folklore has it that, for a given brittle sample, cohesive models à la Barenblatt will behave asymptotically like Griffith’s model as the internal length shrinks to 0. By internal length, we mean the ratio between the fracture toughness and the yield stress; see e.g. Figure 2.3 in Subsection 2.6. Provided that cohesive forces are only triggered near the crack tip, similar views were already espoused by (Griffith, 1920), page 166 : “it may therefore be said that the application of the mathematical theory of elasticity on the basis that the crack is assumed to be a traction-free surface, must give the stresses correctly at all points of the body, with the exception of those near the ends of the crack. In a sufficiently large crack the error in the strain energy so calculated must be negligible.”

Our purpose in what follows is to quantify this within our framework of choice, the variational framework. We visit this issue in the context of global minimality and report on Giacomini’s significant contribution (Giacomini, 2005b). We forego a general investigation of local minimality because of the current lack of any kind of meaningful results, but refer the reader to (Marigo and Truskinovsky, 2004) in the case of a pull-out problem, or to Section 9 in the context of fatigue.