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

Vieira, André P.; Andrade Jr., José S.; Herrmann, Hans J.

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

Authors

Vieira, André P.

Andrade Jr., José S.

Herrmann, Hans J.

Description

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.
4 pages, 4 figures