DAMAGE DUE TO SELF-SIMILAR MICROCRACKING

We analyze the effect of scale invariant spatial correlations in a mic ro-crack array on the elastic properties of a solid, in the dilute lim it where the distance between defects is larger than the size of indiv idual cracks. When they form a fractal set, both the long-range spatia l correlations and the mutual long-range elastic interactions are to b e taken into account. A real-space renormalization scheme is developed to address this problem, based on an influence function which is buil t on an ultrametric distance. We show that the collective behavior of the micro-crack set induces a far-field stress perturbation which is s imply proportional to the number of cracks, for all fractal dimensions , but the proportionality constant differs from the individual crack c haracteristics by the effect of elastic interactions. The infinite siz e behavior of deterministic hierarchical fractal sets is computed anal ytically, as well as the generalisation to random fractal sets. The le ading finite size effect correction is shown to be a power-law of the system size, and the prefactor is computed analytically. A formal exte nsion to continuous dilation symmetry is introduced and the previous r esults are extended to this case. Finally, some numerical simulation r esults are presented.