Mean-Field Description of Plastic Flow in Amorphous Solids

Failure and flow of amorphous materials are central to various phenomena including earthquakes and landslides. There is accumulating evidence that the yielding transition between a flowing and an arrested phase is a critical phenomenon, but the associated exponents are not understood, even at a mean-field level where the validity of popular models is debated. Here, we solve a mean-field model that captures the broad distribution of the mechanical noise generated by plasticity, whose behavior is related to biased Lévy flights near an absorbing boundary. We compute the exponent $\theta$ characterizing the density of shear transformation $P(x)\sim x^{\theta }$, where $x$ is the stress increment beyond which they yield. We find that after an isotropic thermal quench, $\theta =1/2$. However, $\theta$ depends continuously on the applied shear stress; this dependence is not monotonic, and its value at the yield stress is not universal. The model rationalizes previously unexplained observations and captures reasonably well the value of exponents in three dimensions. Values of exponents in four dimensions are accurately predicted. These results support the fact that it is the true mean-field model that applies in large dimensions, and they raise fundamental questions about the nature of the yielding transition.

Popular Summary

Amorphous solids such as emulsions, foams, and granular materials can fail and flow as a fluid if a sufficiently large shear stress is applied. This yielding process, which plays an important role in various phenomena including landslides and earthquakes, is poorly understood; there is no accepted microscopic description of this problem, which is central to material science, physics, and geophysics. Recent experiments and numerical simulations have revealed very rich dynamics near the threshold stress where flow stops. Here, we introduce a class of mean-field models that take into account that the mechanical noise generated in flow is very broad in amplitude. We solve the mean-field mode analytically, and we compute the pseudogap exponent $\theta$ characterizing the solid’s stability.

These mean-field models have an elegant formulation in terms of biased Levy flights near an absorbing boundary, a mathematical problem that has not been answered in the past. Using this analogy, we recover previous puzzling empirical observations that were hitherto unexplained, even at a qualitative level. We find that some key exponents describing the density of elementary excitations (shear transformations) are nonuniversal, underlining the curious nature of this transition and calling into question previous theoretical approaches. We also find that our ability to predict $\theta$ increases with spatial dimension; our results are quite satisfactory in three dimensions and very accurate in four dimensions.

We expect that our findings can be used to address long-standing questions in other fields such as electron and spin glasses where the density of excitations is also known to be singular.

Figure 1

Mean-field model: An unstable site (red line) returns to $x_i=1$ from $x_i<0$ after a time ${\tau }_r$. Concomitantly, other sites implement a drift toward negative $x$ to conserve the mean stress and a random jump of symmetric distribution $w(\xi )$. The dashed line represents $P(x)$, assuming $\mathrm{\Sigma }>0$. A pseudogap is represented at $x=0$.

Figure 2

Theoretical prediction of the pseudogap exponent $\theta$ vs the Lévy index of the random kicks $\mu$. Green line: $\theta$ at the yield stress ${\mathrm{\Sigma }}_c$. Red-dotted line: $\theta$ after a quench, $\mathrm{\Sigma }=0$, overlapping with the green line for $\mu >1$. Blue dashed line: Marginal values of $\theta$, overlapping with the green line for $\mu >2$, and $0<\mu <1$. Below the blue line, the system is unstable and forbidden dynamically [16]. The data points are the measured value of $\theta$ at ${\mathrm{\Sigma }}_c$ in the simulated mean-field model: For $\mu =4/3$, $A=0.15$ (circle), $A=0.35$ (square); $\mu =1$, $A=0.15$ (circle), $A=0.35$ (square), and the yellow stars are the corresponding theoretical values indicated when $\mu =1$.

Figure 3

$P(x)$ for $\mu =4/3$ (a) and $\mu =0.8$ (b) for $N=L^2$ as shown in the legend. (a) The dashed line is the theoretical value $\theta =0.667$; the measured values are $\theta =0.61\pm 0.04$ ($A=0.15$) and $\theta =0.62\pm 0.03$ ($A=0.35$). (b) The dashed line is the fits of Eq. (15), which are $P(x)=5.273+6.506x^{0.2}$ ($A=0.15$) and $P(x)=1.655+6.618x^{0.2}$ ($A=0.35$).

Figure 4

(a) $P(x)$ for $\mu =1$, $N={512}^2$ with different $A$. (b) Dependence of $\theta$ on $A$ for $\mu =1$. The blue dots are the numerical values extracted from (a), and the black line is the theoretical prediction Eq. (19) using the five measured values of $v$ (see inset) and a fifth-order polynominal interpolation to guide the eye.

Figure 5

$\theta$ as a function of the relative stress $\mathrm{\Sigma }/{\mathrm{\Sigma }}_c$, where ${\mathrm{\Sigma }}_c$ is the yield stress for $\mu =1$, $A=0.3$, and $N={1024}^2$. Right after the quench, $\theta =0.53\pm 0.03$. As $\mathrm{\Sigma }$ is increased, a sharp drop to a lower value occurs. Inset (a): The stress vs plastic strain curves. Zooming in on a finite-size system reveals a staircase. Inset (b): The bias $v$’s dependence on $\mathrm{\Sigma }/{\mathrm{\Sigma }}_c$, from which we get the black dashed line in the main panel as the theoretical prediction using Eq. (20).

Figure 6

(a) $P(x)$ at ${\mathrm{\Sigma }}_c$ for the shuffled Eshelby kernel for $d=2$ and $d=3$. We find ${\theta }_{2\mathrm{D}}^{\mathrm{SF}}=0.39\pm 0.02$ and ${\theta }_{3\mathrm{D}}^{\mathrm{SF}}=0.29\pm 0.02$. (b) Direct measurements of the distribution $P(\mathcal{G})$ of the Eshelby kernel corresponding to a stress drop of unity at the origin, from which we extract $A_0=0.7\pm 0.2$ ($d=2$) and $A_0=0.4\pm 0.1$ ($d=3$).

Figure 7

Comparison between the measurements of the pseudogap exponent $\theta$ in finite-dimensional elastoplastic models (circle) and its mean-field prediction (square) as a function of the spatial dimension $d$. As $d$ increases, the difference becomes smaller and undetectable for $d=4$.

Figure 8

Numerical calculation of $w_{\gamma }(\xi )$ at $\gamma =0.1$, 0.01, 0.001, $\mu =1$. The colorful dashed lines are the theoretical prediction, for ${\gamma }^{1/\mu }\ll \xi \ll {\xi }_m$. The black dashed line is at ${\xi }_m$, where the upper cutoff begins to play a role.

Figure 9

(a) Direct measurements of $A_0$ as extracted from the distribution $P(\mathcal{G})$ of the Eshelby kernel in $d=4$. We extract $A_0=0.29\pm 0.4$ and predict ${\theta }_{4\mathrm{D}}^{\mathrm{MF}}=0.24\pm 0.04$. (b) $P(x)$ at ${\mathrm{\Sigma }}_c$ using extremal dynamics for the shuffled Eshelby kernel for $d=4$, where ${\theta }_{4\mathrm{D}}^{\mathrm{SF}}=0.24\pm 0.01$. (c) Finite-dimensions measurement of $P(x)$ in $d=4$ at fixed stress ${\mathrm{\Sigma }}_c=0.5025$, where ${\theta }_{4\mathrm{D}}=0.25\pm 0.01$.