Predicting Plastic Flow Events in Athermal Shear-Strained Amorphous Solids

We propose a method to predict the value of the external strain where a generic amorphous solid will fail by a plastic response (i.e., an irreversible deformation), solely on the basis of measurements of the nonlinear elastic moduli. While usually considered fundamentally different, with the elastic properties describing reversible phenomena and plastic failure epitomizing irreversible behavior, we show that the knowledge of some nonlinear elastic moduli is enough to predict where plasticity sets in.

Figure 1

A typical stress vs strain curve in a system of 4096 particles in two dimensions obtained in the athermal limit. Every elastic (reversible) increase in stress is followed by a sudden plastic (irreversible) drop in stress. The aim of this Letter is to predict the value of the strain at which the next plastic drop will take place.

Figure 2

Upper panel: Straining from an elastic branch of the elastoplastic steady state in a system of $N=484$ particles. Shown are the stress vs strain curve up to the first plastic event (blue rhombi) and the value of ${\gamma }_P$ as predicted by Eq. (13) for every value of $\gamma$ (green squares). In addition we show the prediction of the simplistic Taylor expansion (16) (red circles). Lower panel: The same measurements starting from the equilibrium isotropic stressless state. Note the region where the even derivatives build up.

Figure 3

The first three nonlinear derivatives $B_n$ as a function of $\gamma$, displaying the huge variation in value due to large nonaffine reversible transformation before diving to $-\infty$ because of a plastic event.

Figure 4

Log-log plot of the range of strain $|\gamma -{\gamma }_P|$ where $\delta {\gamma }_P/\Delta {\gamma }_P\le 0.15$ as a function of the system size in the elastoplastic steady state. The slope of the continuous fit line is $\eta =-0.61$. The inset shows how the measurement is done, $(i)/(ii)\le 0.15$.