Subcritical crack growth: the microscopic origin of Paris' law

We investigate the origin of Paris' law, which states that the velocity of a crack at subcritical load grows like a power law, da/dt ~ (DeltaK)(m), where DeltaK is the stress-intensity-factor amplitude. Starting from a damage-accumulation function proportional to (Deltasigma)(gamma), Deltasigma being the stress amplitude, we show analytically that the asymptotic exponent m can be expressed as a piecewise-linear function of the exponent gamma, namely, m=6-2gamma for gamma or =gamma(c), reflecting the existence of a critical value gamma(c)=2. We perform numerical simulations to confirm this result for finite sizes. Finally, we introduce bounded disorder in the breaking thresholds and find that below gamma(c) disorder is relevant, i.e., the exponent m is changed, while above gamma(c) disorder is irrelevant.