Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading

The response of amorphous materials to an applied strain can be continuous or instead discontinuous if the initial configuration is very stable. We study theoretically how such a stress drop emerges as the system's initial stability is increased. We show that this emergence is well reproduced by elastoplastic models and is predicted by a mean field approximation, where it corresponds to a continuous transition. In the mean field, failure can be forecasted from the avalanche statistics. We show that this is not the case for very stable materials in finite dimensions due to rare weak regions where a shear band nucleates. To understand the nucleation, we build an analogy with fracture mechanics predicting that the critical nucleation radius of a shear band follows $a_c\sim {(\mathrm{\Sigma }-{\mathrm{\Sigma }}_b)}^{-2}$, where $\mathrm{\Sigma }$ is the stress and ${\mathrm{\Sigma }}_b$ is the stress that a shear band can carry.

Figure 1

An elastoplastic model can be continuous without (1) and with (2) a stress overshoot or be discontinuous (3), depending on the preparation. (a) Schematic of the initial distribution of stabilities, $P_0\left(x\right)$, representing the three system preparations. (b) Stress versus strain curves for these three different $P_0\left(x\right)$ together with snapshots of the spatial distribution of plastic strain ${\epsilon }_p$. (c) Snapshots of the plastic strain taken at the positions indicated by red circles on the stress versus strain curves. (1) If the system preparation is not very stable, strain remains homogeneous and there is no stress overshoot. (2) As the stability of the preparation is increased a shear band is formed. (3) For a very stable preparation, a sharp shear band is formed during the macroscopic stress drop.

Figure 2

(a) Stress versus plastic strain curve consists of alternating elastic stress increases $\mathrm{\Delta }\mathrm{\Sigma }$ and stress relaxations by an avalanche of plastic events $\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathrm{avalanche}}=-\mu \mathrm{\Delta }{\epsilon }_p$. (b) When the slope of the macroscopic stress versus plastic strain curve reaches $-\mu$, an extensive avalanche occurs. (c) Stress versus strain curve corresponding to the stress versus plastic strain curve from panel (b) during a strain-controlled loading.

Figure 3

(a) Initial stability distributions $P_0\left(x\right)\sim (1-\alpha )exp[-{(x-1)}^2/\left(2s_P^2\right)]+\alpha x(2-x)$ with $s_P=0.05$ we use to find (b) stress versus strain curves in the Hebraud-Lequeux model, showing the onset of macroscopic stress drop as the initial stability distribution $P_0\left(x\right)$ is narrowed.

Figure 4

(a) Stress versus strain curve in an elastoplastic model with $N={706}^2$ in which a defect of varying size $a_s$ (as indicated in color) was inserted. (b) Maximal stress reached ${\mathrm{\Sigma }}_{\max }$ as a function of the defect length $a_s$. (c) When no defects are inserted, ${\mathrm{\Sigma }}_{\max }$ decreases very slowly with $N$, consistent with our prediction ${\mathrm{\Sigma }}_{\max }-{\mathrm{\Sigma }}_b\sim 1/\sqrt{lnN}$.