Statistical and renewal results for the random sequential adsorption model applied to a unidirectional multicracking problem

We work out a stationary process on the real line to represent the positions of the multiple cracks which are observed in some composites materials submitted to a fixed unidirectional stress ɛ . Our model is the one-dimensional random sequential adsorption. We calculate the intensity of the process and the distribution of the inter-crack distance in the Palm sense. Moreover, the successive crack positions of the one-sided process (denoted by X i ɛ , i ⩾ 1 ) are described. We prove that the sequence { ( X i ɛ , Y i ɛ ) , 1 ⩽ i ⩽ n } is a “conditional renewal process”, where Y i ɛ is the value of the stress at which X i ɛ forms. The approaches “in the Palm sense” and “one-sided process” merge when n → + ∞ . The saturation case ( ɛ =+ ∞ ) is also investigated.