Subcritical crack growth: the microscopic origin of Paris's law

We investigate the origin of Paris's law, which states that the velocity of a crack at subcritical load grows like a power law, $da/dt \sim (\Delta K)^{m}$, where $\Delta K$ is the stress intensity factor amplitude. Starting from a damage accumulation function proportional to $(\Delta\sigma)^{\gamma}$, $\Delta\sigma$ being the stress amplitude, we show analytically that the asymptotic exponent $m$ can be expressed as a piecewise-linear function of the %damage accumulation exponent $\gamma$, namely, $m=6-2\gamma$ for $\gamma < \gamma_{c}$, and $m=\gamma$ for $\gamma \ge \gamma_{c}$, reflecting the existence of a critical value $\gamma_{c}=2$. %In this way, here we discover the existence of a critical %value $\gamma_{c}=2$ characterized by a scaling law with a critical %exponent separating two regimes of different linear functions $m$. We performed 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.