Going variational

In this section, the starting premise is Griffith’s model for crack evolution, as presented in his celebrated paper (Griffith, 1920). Of course, continuum mechanics has seen and weathered many storms in eighty seven years and it would make little sense to present fracture exactly as in (Griffith, 1920). The reader will find below what we believe to be a very classical introduction to brittle fracture within a rational mechanical framework. Whether this strictly conforms to the tenet of Rational Mechanics is a matter best left to the experts in the field.

Our starting assumptions are two-fold. First, as mentioned in the introduction, we do not wish to contribute at this point to the hesitant field of dynamic fracture, thereby restricting our focus to quasi-static evolution. At each time, the investigated sample is in static equilibrium with the loads that are applied to it at that time. We use the blanket label “loads” for both hard devices (displacement type boundary conditions) and soft devices (traction type boundary conditions and/or body forces). In the former case, we often refer to those boundary conditions as “displacement loads”. Then, we do not concern ourselves with changes in temperature, implicitly assuming that those will not impact upon the mechanics of the evolution: in particular, thermal expansion is not an option in this model, at least to the extent that it couples thermal and mechanical effects. However, thermal stresses induced by a known temperature field fall squarely within the scope of the forthcoming analysis.