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where u is the local velocity of the front of size a(x, t) at time t, k is the local growth rate of the damage, β is the activation energy, A is the loading intensity and α is a positive parameter which controls the length-scale of the damage domain under consideration, d is the average crack advance at time t, a is the front size, and a is the front size. To reproduce a crack that is wider than the observation window, the front size a is chosen to be much larger than the crack size c (2 mm vs. 1 μm). We set a=20mm and c=2mm, so that the crack can be observed for a time window of 10 s.
The front is modelled as a heterogeneous mixture of n elements with d u, and is thus characterized by its fractal dimensiond and the mean front advanced at time t (see Fig. 1). To reproduce the experimental conditions, we set n=3.2 × 105 and d=1.4. Each element is randomly selected (see Fig. 1) from the local velocity profile at time t with the probability of selection being proportional to the actual local velocity, u. This is motivated by the fact that the crack is well localized, and so most elements are ‘almost’ in a steady state, with a local velocity u(x, t) close to u(x, t+d/u).
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