Surface energy

Since the surface energy term scales with c and the strain energy term scales with c², U_tot has to go through a maximum at a certain critical crack size c_crit(Fig. F c). This is an important result since it implies that extending a crack that is smaller than c_crit consumes rather than liberates energy and is thus stable. In contrast, flaws that are longer than c_crit are unstable since extending them releases more energy than is consumed. Note that increasing the applied stress (Fig. F d) will result in failure at smaller critical flaw sizes. For instance, a solid for which the size of the largest flaw lies somewhere between those shown in Fig. F c and d will not fail at the stress shown in Fig. F c, but will fail if that stress is increased (Fig. F d).

   The location of the maximum is determined by differentiating Eq. (7) and equating it to zero. Carrying out the differentiation, replacing ?_app by ?_f, and rearranging terms, one can show that the condition for failure is