Abstract
We obtain the Paris law of fatigue crack propagation in a fuse network
model where the accumulated damage in each resistor increases with time
as a power law of the local current amplitude. When a resistor reaches
its fatigue threshold, it burns irreversibly. Over time, this drives
cracks to grow until the system is fractured into two parts. We study
the relation between the macroscopic exponent of the crack-growth rate
-entering the phenomenological Paris law-and the microscopic damage
accumulation exponent, gamma, under the influence of disorder. The way
the jumps of the growing crack, Delta a, and the waiting time between
successive breaks, Delta t, depend on the type of material, via gamma,
are also investigated. We find that the averages of these quantities,
<Delta a > and <Delta t >/< t(r)>, scale as power laws of the crack
length a, <Delta a > proportional to a(alpha) and <Delta t >/< t(r)>
proportional to a(-beta), where < t(r)> is the average rupture time.
Strikingly, our results show, for small values of gamma, a decrease in
the exponent of the Paris law in comparison with the homogeneous case,
leading to an increase in the lifetime of breaking materials. For the particular case of gamma = 0, when fatigue is exclusively ruled by
disorder, an analytical treatment confirms the results obtained by
simulation. Copyright (C) EPLA, 2012
