Stationarity versus local or global minimality – A comparison

Before embarking on the variational journey, we wish to explore the ramifications of minimality in the context of both Griffith and cohesive fracture. The adopted setting, or rather settings, for such an analysis are designed so that the “crack path” is not at stake. Nor is irreversibility a concern here because the monotonicity of the loads combined with the geometry of the problems result in an increase of both the measure of the discontinuity set and the magnitude of the discontinuities on that set. The focus is squarely on minimality, although, at times energy balance (Eb) will come to the rescue.The two settings are

• 1. A 1d-traction experiment under a hard or a soft device;

• 2. A 2d-tearing experiment.

In the first setting, cracks are merely points of discontinuity along the bar; in the second setting, symmetry of the geometry and of the loads suggests a straight crack path in mode III. In both settings, we assess the potential existence of weak variational evolutions satisfying unilateral stationarity (Ust), unilateral minimality (Ulm), or still unilateral global minimality (Ugm), together with energy balance (Eb), this for both Griffith, or cohesive fracture energies. The resulting picture is a dizzying labyrinth, but maybe it is because we have “realized that [fracture] and the labyrinth were one and the same”.