Different universality classes at the yielding transition of amorphous systems

We study the yielding transition of a two-dimensional amorphous system under shear by using a mesoscopic elasto-plastic model. The model combines a full (tensorial) description of the elastic interactions in the system and the possibility of structural reaccommodations that are responsible for the plastic behavior. The possible structural reaccommodations are encoded in the form of a “plastic disorder” potential, which is chosen independently at each position of the sample to account for local heterogeneities. We observe that the stress must exceed a critical value ${\sigma }_c$ in order for the system to yield. In addition, when the system yields a flow curve (relating stress $\sigma$ and strain rate $\dot{\gamma }$) of the form $\dot{\gamma }\sim {(\sigma -{\sigma }_c)}^{\beta }$ is obtained. Remarkably, we observe the value of $\beta$ to depend on some details of the plastic disorder potential. For smooth potentials a value of $\beta \simeq 2.0$ is obtained, whereas for potentials obtained as a concatenation of smooth pieces a value $\beta \simeq 1.5$ is observed in the simulations. This indicates a dependence of critical behavior on details of the plastic behavior. In addition, by integrating out nonessential, harmonic degrees of freedom, we derive a simplified scalar version of the model that represents a collection of interacting Prandtl-Tomlinson particles. A mean-field treatment of this interaction reproduces the difference of $\beta$ exponents for the two classes of plastic disorder potentials and provides values of $\beta$ that compare favorably with those found in the full simulations.

Figure 1

(a) Definition of the three elementary distortions $e_1$, $e_2$, $e_3$ that describe the elastic state of the system at each spatial position. (b–c) Sketch of the state of a sample under an applied shear. (b) Corresponds to the case of a system formed by identical elements, and (c) the case in which each element has its own energy potential and energy minima.

Figure 2

Sketch of the local free energy depending on the strain state of a cell. (a) Perfectly elastic case. (b) Plastic case. In this case, other minima appear as the strain is increased further.

Figure 3

Strain rate vs stress curves, for systems with different values of $B/\mu$, for smooth and parabolic potentials. System size is $256\times 256$. (a) Linear scale. (b) Logarithmic scale with the value of ${\sigma }_c$ subtracted. Dotted lines are drawn for reference.

Figure 4

Examples of the evolution of stress in the system, under the quasistatic protocol described in the text. The left part corresponds to smooth potentials, and the right part to parabolic potentials. In (a) we see the stress-strain plot, and in (b) the stress-time one. The strain rate is zero in the gray regions [when $Z$, shown in panel (c), is larger than a threshold value $z_0$], whereas it is a fixed, small $\dot{\gamma }$ outside these periods. $T$ and $S$ measure the duration and size of the avalanches.

Figure 5

Histogram of avalanche size distribution, in systems of different sizes, for (a) parabolic and (b) smooth potentials. The straight lines show some reference slopes.

Figure 6

The cutoff avalanche size $S_{\max }$ as a function of system size, for the smooth and parabolic potential cases. A dependence close to $S_{\max }\sim L$ is observed in both cases.

Figure 7

Avalanche duration vs avalanche size, for both kinds of potential, in a system of 256 $\times$ 256. Black dots are the results of individual avalanches; red (parabolic) and blue (smooth) curves are the average of $T$ in successive $S$ slices. Black lines are shown to display the overall behavior.

Figure 8

Evolution of the variance of the strain in the system (${\mathrm{\Sigma }}^2=\overline{e_2^2}-{\overline{e_2}}^2$) as a function of the average strain $\overline{e_2}=\dot{\gamma }t$. Different curves were obtained for different values of $\dot{\gamma }$, as indicated. System size is $64\times 64$.

Figure 9

Flow curves for the iterative implementation of the self-consistently driven PT model, for parabolic and smooth potentials. The first iteration gives the results of the standard PT model, and the second one corresponds to the stochastically driven PT model. After a few iterations the flow curves converge to a limiting curve with an intermediate value of the $\beta$ exponent.

Figure 10

The data from the previous figure corresponding to the first and second iteration, and the average from iterations 10 to 20, plotted as a power law around $\sigma ={\sigma }_c$. Dotted lines are guides to the eye, drawn with the indicated slope.

Figure 11

Typical plastic potentials that are generated for the parabolic case (a) and the smooth case (b). Note that the curvature of the potentials at all minima is the same.

Figure 12

Results for the standard ($\nu =0$) and stochastically driven ($\nu =1$) PT model [Eqs. (B1) and (B2)], for the case of parabolic and smooth potentials. Panels (a) and (b) are the results in linear scale, whereas (c) and (d) are in logarithmic scale, with $\sigma$ shifted in each case by the numerically determined ${\sigma }_c$. The asymptotic forms (dotted lines) display the exponents predicted by the analytical treatment. The numerical data tend to match the analytical behavior in the small $\dot{\gamma }$ limit.