Self-Consistent Theory of Rupture by Progressive Diffuse Damage
  We analyze a self-consistent theory of crack growth controlled by a cumulative damage variable d(t) dependent on stress history. As a function of the damage exponent $m$, which controls the rate of damage dd/dt \propto sigma^m as a function of local stress $\sigma$, we find two regimes. For 0 < m < 2, the model predicts a finite-time singularity. This retrieves previous results by Zobnin for m=1 and by Bradley and Wu for 0 < m < 2. To improve on this self-consistent theory which neglects the dependence of stress on damage, we apply the functional renormalization method of Yukalov and Gluzman and find that divergences are replaced by singularities with exponents in agreement with those found in acoustic emission experiments. For m =2 and m > 2, the rupture dynamics is not defined without the introduction of a regularizing scheme. We investigate three regularization schemes involving respectively a saturation of damage, a minimum distance of approach to the crack tip and a fixed stress maximum. In the first and third schemes, the finite-time singularity is replaced by a crack dynamics defined for all times but which is controlled by either the existence of a microscopic scale at which the stress is regularized or by the maximum sustainable stress. In the second scheme, a finite-time singularity is again found. In the first two schemes within this regime m > 2, the theory has no continuous limit. 
S. Gluzman (Univ. Nice/CNRS) and D. Sornette (Univ. Nice/CNRS and   UCLA)