Mechanism of delayed elastic state absorption versus fatigue failure in cyclic shear of amorphous materials

Abstract

We study cyclic shear in a lattice elastoplastic model of amorphous materials. The model displays a threshold shear amplitude $γ_{\rm c}$, which increases with sample annealing. For imposed amplitudes $γ_0\lesssimγ_{\rm c}$, it shows slow progression to an absorbing elastic state. For $γ_0\gtrsim γ_{\rm c}$, it shows slow fatigue and delayed material failure. Our key contribution is to elucidate a mechanistic understanding of this behaviour. We show that progression to an absorbing state maps to single element diffusion on a finite interval with two absorbing boundaries. We derive expressions for the decay rate of the fraction of active elements, and the survival distribution. The diffusing variable is the local elemental strain, which takes two jumps stroboscopically per cycle. Once the value at any site diffuses below its yielding threshold, the element behaves elastically. This leads to a gradual accumulation of strains just below threshold. For $γ_0\lesssim γ_{\rm c}$ this diffusive behaviour is preceded by many cycles in which cooperativity between elements via stress propagation is also important. For $γ_0 > γ_{\rm c}$, stress propagation dominates and precipitates failure. For $γ_0\gtrsim γ_{\rm c}$, the same accumulation of elements just below local threshold that led to an elastic state for $γ_0\lesssimγ_{\rm c}$ accounts for the fatigue that brings the system to the brink of failure. We evidence this via simulations in which the strain amplitude artificially exceeds $γ_0 $ in each half cycle, before reverting to the system's actual state at $\pmγ_0$ and resuming the next half cycle. During fatigue, the stress-strain slope $dσ/dγ$ evolves towards unity at $|γ|=γ_0^-$, signifying perfect elasticity within cycle, yet towards zero at $|γ|=γ_0^+$, precipitating failure.