Mechanical failure in amorphous solids: Scale-free spinodal criticality

The mechanical failure of amorphous media is a ubiquitous phenomenon from material engineering to geology. It has been noticed for a long time that the phenomenon is “scale-free,” indicating some type of criticality. In spite of attempts to invoke “Self-Organized Criticality,” the physical origin of this criticality, and also its universal nature, being quite insensitive to the nature of microscopic interactions, remained elusive. Recently we proposed that the precise nature of this critical behavior is manifested by a spinodal point of a thermodynamic phase transition. Demonstrating this requires the introduction of an “order parameter” that is suitable for distinguishing between disordered amorphous systems. At the spinodal point there exists a divergent correlation length which is associated with the system-spanning instabilities (known also as shear bands) which are typical to the mechanical yield. The theory, the order parameter used and the correlation functions which exhibit the divergent correlation length are universal in nature and can be applied to any amorphous solid that undergoes mechanical yield. The phenomenon is seen at its sharpest in athermal systems, as is explained below; in this paper we extend the discussion also to thermal systems, showing that at sufficiently high temperatures the spinodal phenomenon is destroyed by thermal fluctuations.

Figure 1

A typical stress-vs.-strain curve resulting from a shear loading of an amorphous solids using an AQS protocol. Similar transitions between a regime in which the stress rises on the average as a function of strain to a second regime after a yield point ${\gamma }_{{}_Y}$ have been observed in countless experiments and simulations, requiring an explanation using a generic theory that is insensitive to microscopic details.

Figure 2

The averaged order parameter $\overline{\langle Q_{12}\rangle }$ as a function of $\gamma$ (left scale) and the averaged stress as a function of $\gamma$ (right scale). The averaging is over all the patches for this systems size $N=10000$. Note the phase transition that occurs near the yield stress. The transition gets sharper with the system size, see Fig. 6 and the associated discussion below.

Figure 3

The probability distribution function $\overline{P_{\gamma }\left(Q_{12}\right)}$ at ${\gamma }_{{}_Y}=0.088$ averaged over 100 initial configurations each of which has 500 different realizations to obtain $\overline{P_{\gamma }\left(Q_{12}\right)}$. At this value of the strain the pdf has two peaks of equal heights. We identify this value of $\gamma$ as the point of the phase transition.

Figure 4

The probability distribution function $\overline{P_{\gamma }\left(Q_{12}\right)}$ in the vicinity of the critical point ${\gamma }_{{}_Y}=0.088$.

Figure 5

The number of configurations which pass below the threshold value $Q_{12}=0.8$ of the overlap order parameter as a function of the strain $\gamma$ for $N=4000$. In the inset we show the same test for $N=500$. The conclusion is that all the configurations lose the mutual overlap in the vicinity of the yield point ${\gamma }_{{}_Y}$.

Figure 6

Demonstrating the sharpening of the transition with the system size.

Figure 7

Panel (a): A typical graph of $\overline{\langle Q_{12}\rangle }$ as a function of $\gamma$ (here for $N=4000$ (right scale) and the slope of the same function (left scale). Panel (b): The maximal slope of the function $\overline{\langle Q_{12}\rangle \left(\gamma \right)}$ as a function of system size.

Figure 8

Panel (a): The pdf's $P({\gamma }_c,N)$ for different systems sizes from $N=500$ to $N=10000$. Panel (b): Data collapse on rescaling the pdf's $P({\gamma }_c,N)$ according to Eq. (6) to obtain a scaling function $\tilde{P}\left(x\right)$ with $x=\left[({\gamma }_c-{\gamma }^{*})\sqrt{N}\right]$.

Figure 9

A three-dimensional projection of the three correlation function as a function of $x,y$.

Figure 10

The susceptibilities ${\chi }_{{}_{{\mathrm{\Gamma }}_2}}$ (a), and ${\chi }_{{}_{G_R}}$ (b), and ${\chi }_{{}_{G_L}}$ (c) as a function of $\gamma$ for the three systems sizes available. Superimposed are the stress-vs.-strain curves for comparison. The color code is violet for $N=1000$, red for 4000 and green for $10000$.

Figure 11

The system size dependence of the maxima ${\chi }_{{}_{{\mathrm{\Gamma }}_2}}^{\max }$ indicating a dependence on $\sqrt{N}$ in correspondence to the results in Fig. 7.

Figure 12

The function $G_R(x=0,y;\gamma )$ for various values of $\gamma$ from $5\times {10}^{-5}$ to 0.09405. Note the increase in the over all amplitude of the correlator as well as the increase in the correlation length. The lines through the data are the fit function Eq. (16).

Figure 13

The $\gamma$ dependence of the correlation length $\xi \left(\gamma \right)$ (a), the amplitude $A\left(\gamma \right)$ (b) and the constant $C\left(\gamma \right)$ (c) in the best fit to the function $G_R(x=0,y;\gamma )$, cf. Eq. (16).

Figure 14

The difference between $\int dy{G}_R(x=0,y)$ and $C\times L$.

Figure 15

The full $y$ dependence of $G_R(x=0,y)$. The region fitted by Eq. (10) in the paper is shown. Note that the minimum in the function resides in higher values of $y$ for larger system sizes.

Figure 16

Panel (a): The correlation length $\xi$ read from an exponential fit to the $x$ and $y$ projection of the correlation function ${\mathrm{\Gamma }}_2(x,y)$ for three values of the system size. Panel (b): The dependence of the correlation length $\xi$ on $\gamma -{\gamma }_c$.

Figure 17

Panel (a): Average stress vs. strain in quasistatic straining at finite temperature. The average is computed over all the configurations in a patch and then over the patches. Panel (b): The averaged overlap order parameter computed at the designated temperatures as a function of the strain $\gamma$. Here $N=10000$.

Figure 18

The dependence of $\overline{P_{\gamma }(Q_{12},T)}$ on $\gamma$ for three different temperatures $T=0,0.1,0.2$.