Scaling theory for steady-state plastic flows in amorphous solids

Strongly correlated amorphous solids are a class of glass formers whose interparticle potential admits an approximate inverse power-law form in a relevant range of interparticle distances. We study the steady-state plastic flow of such systems, first in the athermal quasistatic limit and second at finite temperatures and strain rates. In all cases we demonstrate the usefulness of scaling concepts to reduce the data to universal scaling functions where the scaling exponents are determined a priori from the interparticle potential. In particular we show that the steady plastic flow at finite temperatures with efficient heat extraction is uniquely characterized by two scaled variables; equivalently, the steady-state displays an equation of state that relates one scaled variable to the other two. We discuss the range of applicability of the scaling theory, and the connection to density scaling in supercooled liquid dynamics. We explain that the description of transient states calls for additional state variables whose identity is still far from obvious.

Figure 1

(Color online) The different pairwise potentials discussed in this work.

Figure 2

(Color online) $r^{-1}\frac{\partial U\left(r\right)}{\partial r}$ in the range of $r_0/\lambda ∊\left[{\rho }_{\text{max}}^{-1/2},{\rho }_{\text{min}}^{-1/2}\right]$ for $k=8$ in green asterisks, and for $k=10$ in blue circles. The line through the points represents the scaling laws Eq. (5)

Figure 3

(Color online) Stress-strain curves averaged over 20 independent runs for an athermal system with $N=4096$, $k=8$ (left panel) and $k=10$ (right panel) as a function of the density, with the density increasing from bottom to top.

Figure 4

(Color online) The same stress-strain curves as in Fig. 3 but with the stress rescaled by ${\rho }^{\nu }$, with $\nu =4.80$ for $k=8$ (top panel) and $\nu =5.87$ for $k=10$ (bottom panel). The insets demonstrate the density dependence of ${\sigma }_Y$ and $\mu$ according to ${\rho }^{\nu }$.

Figure 5

(Color online) Left panels: stress normalized by ${\rho }^{\nu }$ vs strain for the $x$ values $x=0.001$, $x=0.01$, $x=0.1$, and $x=0.2$, increasing from top to bottom. Right panels: log-log plots of the steady-state flow stress as a function of density, for the same corresponding values of $x$.

Figure 6

(Color online) The function $\mathcal{S}\left(x;y_0\right)$. Data is displayed for $\rho =1.15$ (blue circles) over a wide range of $x=\frac{T}{ϵ{\rho }^{\nu -1}}$ values, and for the densities of Fig. 5 over the $x$ values $x=0.001$, $x=0.01$, $x=0.1$, and $x=0.2$. The value of $y_0=\frac{\dot{\gamma }}{{\tau }_{\star }^{-1}{\rho }^{\nu /2}}$ is $1.658\times {10}^{-5}$ for all simulated systems.

Figure 7

(Color online) The scaling function $\mathcal{S}\left(x_0,y\right)$ normalized by the values $\mathcal{S}\left(x_0,y=2.5\times {10}^{-6}\right)$, for various values of $y$.

Figure 8

(Color online) Left panel: stress-strain curves for the potential Eq. (2) which has a repulsive and an attractive part for all densities increasing from bottom to top. Right upper panel: demonstration of the failure of rescaling of the stress-strain curves. Lower panels: ${\sigma }_Y$ and $⟨\mu ⟩$ as a function of the density. Note that predictability is regained only for higher densities, the straight line is ${\rho }^7$.

Figure 9

(Color online) The pure number $\Omega$ as a function of the density for the three potentials discussed in the text. Note that $\Omega$ appears to increase with the exponent of the repulsive part of the potential whenever scaling prevails.