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We study an integer automaton elasto-plastic model of an amorphous solidsubject to cyclic shear of amplitude $\Gamma$. We focus on the reversibleplastic regime at intermediate $\Gamma_0<\Gamma<\Gamma_y$, where, after atransient, the system settles into a periodic limit cycle with hysteretic,dissipative plastic events which repeat after an integer number of cycles. Westudy the plastic strain rate, $\frac{d\epsilon}{d\gamma}$, (where $\gamma$ isthe applied strain and $\epsilon$ is the plastic strain) during the terminallimit cycles and show that it consists of a creeping regime at low $\gamma$with very low $\frac{d\epsilon}{d\gamma}$ followed by a sharp transition at acharacteristic strain, $\gamma_*$, and stress, $\sigma_*$, to a flowing regimewith higher $\frac{d\epsilon}{d\gamma}$. We show that while increasing $\Gamma$above $\Gamma_0$ results in lower terminal ground state energy,$U_{\text{min}}$, and a correspondingly narrower distribution of stresses, it,surprisingly, results in lower $\gamma_*$, and $\sigma_*$. The stressdistribution, $P(\sigma)$, also becomes skewed for $\Gamma>\Gamma_0$. That is,the systems in the RPR are anomalously soft and mechanically polarized. Werelate this to an emergent characteristic feature in the stress distribution,$P(\sigma)$, at a value, $\sigma_0$, which is independent of $\Gamma$ and showthat $\sigma_0$ implies a relation between the $\Gamma$ dependence of$\sigma_*$, $\gamma_*$, and the amplitude of plastic strain, $\epsilon_p$. Weshow that the onset of hysteresis is characterized by a power-law scaling,indicative of a second order transition with $\epsilon_p\propto(\Gamma-\Gamma_0)^{1.2\pm0.1}$. We argue that $\sigma_0$ and, correspondingly,the onset of the RPR at $\Gamma=\Gamma_0$, is simply set by the so-calledEshelby-stress. Furthermore, we show that cycling at $\Gamma_0$ results in amaximally hardened state.

Anomalous softness in amorphous matter in the reversible plastic regime