Statistical physics of the yielding transition in amorphous solids

The art of making structural, polymeric, and metallic glasses is rapidly developing with many applications. A limitation is that under increasing external strain all amorphous solids (like their crystalline counterparts) have a finite yield stress which cannot be exceeded without effecting a plastic response which typically leads to mechanical failure. Understanding this is crucial for assessing the risk of failure of glassy materials under mechanical loads. Here we show that the statistics of the energy barriers $\Delta E$ that need to be surmounted changes from a probability distribution function that goes smoothly to zero as $\Delta E=0$ to a pdf which is finite at $\Delta E=0$. This fundamental change implies a dramatic transition in the mechanical stability properties with respect to external strain. We derive exact results for the scaling exponents that characterize the magnitudes of average energy and stress drops in plastic events as a function of system size.

Figure 1

(Color online) Evolution of the conditional mean energy drops with the loading, for all system sizes simulated, increasing from bottom to top. Superimposed (scale on the right ordinate) is the mean stress vs strain curve for the largest system of $N=20164$. The magenta dots represent equistress states but of highly different stability properties.

Figure 2

(Color online) Panel (a). Mean energy drop $⟨\Delta U⟩$ and mean strain interval $⟨\Delta \gamma ⟩$ in two dimensions as functions of system size, measured in AQS simulations of steady plastic flow of a model glass former, see text. Panel(b). The same for three dimensions. The continuous lines represent the scaling laws (1). The scaling exponents are the same in 2D and 3D.

Figure 3

(Color online) Scaling of energy barriers for various system sizes, see [8] for details. The slope of the continuous lines is 3/2.

Figure 4

(Color online) The measured pdf in two dimension for $N=4096$ in panel (a) and in three dimensions for $N=2000$ in panel (b) for the observed value of $x\equiv \Delta {\gamma }_{\text{iso}}$ for isotropic systems (in red) and of $x\equiv \Delta \gamma$ for the steady state (in blue). The scaling law for the mean shown in the inset indicates a value of $\eta \approx 0.6$ for both two and three dimensions. Note the dramatic change in the power-law tail of the distributions at small values of $\Delta \gamma$: in equilibrium the tail guarantees that the probability to see a zero value of $\Delta E$ is zero. This is not the case in the steady state, and this is the physical hallmark of the yielding transition. The black line through the red dots in panel (b) is the Weibull distribution Eq. (5).

Figure 5

(Color online) Pdf of the observed strain intervals between avalanches compared to the converged pdf of the iterative model, see text for details. Here $q=2/3$. The agreement indicates the robustness of the model and the crucial role of the avalanches in renormalizing the pdf. Here we opted to show the pdf of strain intervals because it is the directly measurable quantity related to the energy drop by Eq. (3).