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

Abstract

We investigate the origin of Paris's law, which states that the velocity of a crack at subcritical load grows like a power law, $da/dt \sim (ΔK)^{m}$, where $ΔK$ is the stress intensity factor amplitude. Starting from a damage accumulation function proportional to $(Δσ)^γ$, $Δσ$ 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 $γ$, namely, $m=6-2γ$ for $γ< γ_{c}$, and $m=γ$ for $γ\ge γ_{c}$, reflecting the existence of a critical value $γ_{c}=2$. %In this way, here we discover the existence of a critical %value $γ_{c}=2$ characterized by a scaling law with a critical %exponent separating two regimes of different linear functions $m %(γ)$. We performed numerical simulations to confirm this result for finite sizes. Finally, we introduce bounded disorder in the breaking thresholds and find that below $γ_{c}$ disorder is relevant, i.e., the exponent $m$ is changed, while above $γ_{c}$ disorder is irrelevant.