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We investigate the origin of Paris's law, which states that the velocity of acrack 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 adamage accumulation function proportional to $(\Delta\sigma)^{\gamma}$,$\Delta\sigma$ being the stress amplitude, we show analytically that theasymptotic 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 theexistence of a critical value $\gamma_{c}=2$. %In this way, here we discoverthe existence of a critical %value $\gamma_{c}=2$ characterized by a scalinglaw with a critical %exponent separating two regimes of different linearfunctions $m %(\gamma)$. We performed numerical simulations to confirm thisresult for finite sizes. Finally, we introduce bounded disorder in the breakingthresholds and find that below $\gamma_{c}$ disorder is relevant, i.e., theexponent $m$ is changed, while above $\gamma_{c}$ disorder is irrelevant.

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