Anomalous softness in amorphous matter in the reversible plastic regime

Abstract

We study an integer automaton elasto-plastic model of an amorphous solid subject to cyclic shear of amplitude $Γ$. We focus on the reversible plastic regime at intermediate $Γ_0<Γ<Γ_y$, where, after a transient, the system settles into a periodic limit cycle with hysteretic, dissipative plastic events which repeat after an integer number of cycles. We study the plastic strain rate, $\frac{dε}{dγ}$, (where $γ$ is the applied strain and $ε$ is the plastic strain) during the terminal limit cycles and show that it consists of a creeping regime at low $γ$ with very low $\frac{dε}{dγ}$ followed by a sharp transition at a characteristic strain, $γ_*$, and stress, $σ_*$, to a flowing regime with higher $\frac{dε}{dγ}$. We show that while increasing $Γ$ above $Γ_0$ results in lower terminal ground state energy, $U_{\text{min}}$, and a correspondingly narrower distribution of stresses, it, surprisingly, results in lower $γ_*$, and $σ_*$. The stress distribution, $P(σ)$, also becomes skewed for $Γ>Γ_0$. That is, the systems in the RPR are anomalously soft and mechanically polarized. We relate this to an emergent characteristic feature in the stress distribution, $P(σ)$, at a value, $σ_0$, which is independent of $Γ$ and show that $σ_0$ implies a relation between the $Γ$ dependence of $σ_*$, $γ_*$, and the amplitude of plastic strain, $ε_p$. We show that the onset of hysteresis is characterized by a power-law scaling, indicative of a second order transition with $ε_p\propto (Γ-Γ_0)^{1.2\pm0.1}$. We argue that $σ_0$ and, correspondingly, the onset of the RPR at $Γ=Γ_0$, is simply set by the so-called Eshelby-stress. Furthermore, we show that cycling at $Γ_0$ results in a maximally hardened state.