Fatigue in disordered media

TL;DR

This paper models fatigue crack growth in disordered materials using a fuse network, revealing how microscopic damage accumulation influences macroscopic crack propagation laws and material lifetime.

Contribution

It introduces a model linking microscopic damage exponents to the macroscopic Paris law in disordered media, supported by analytical and simulation results.

Findings

Crack jump and waiting time scale as power laws with crack length.

Disorder can decrease the Paris law exponent, increasing material lifetime.

Analytical results confirm simulation findings for zero damage exponent.

Abstract

We obtain the Paris law of fatigue crack propagation in a disordered solid using 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 in 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 $a$, $<\Delta a> \propto a^{\alpha}$ and $<\Delta t>/<t_r> \propto a^{-\beta}$, 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.