Self-similar crack patterns induced by spatial stress fluctuations

A mechanism of the formation of multiscale self-similar crack structures in brittle materials is proposed based on the action of spatial random stress fluctuations (random stress fields) with vanishing mathematical expectation. The presence of self-equilibrating stress fluctuations is, in many cases, characteristic of heterogeneous brittle materials even in the absence of external loading. Some examples are given by residual stresses-stresses caused by phase transformations or internal microscopic heat sources. If the stress fluctuations are strong enough to cause cracking, the cracks will be originated in regions with and perpendicular to the highest local tension. As a legacy of this special location, additional local tractions opening the crack in its centre are developed even in self-equilibrating stress fields. This mechanism is modelled by disklike cracks opened by a pair of concentrated forces applied at the centre. Two cases are considered: (i) the crack growth in the moving equilibrium conditions caused by the self-equilibrating stress fluctuations of increasing amplitude; and (ii) the subcritical crack growth under the stress fluctuations of either sustained (creep) or oscillating (fatigue) amplitudes. A's the cracks grow, the interaction between them leads to a separation of crack sizes. The effects of the interaction are modelled by the differential self-consistent method that is shown to become asymptotically accurate as the ratio between the sizes of the interacting cracks increases. The interaction results in the crack distribution tending to a self-similar one with the distribution function proportional to the inverse fourth power of the crack radius.