Yielding of glass under shear: A directed percolation transition precedes shear-band formation

Under external mechanical loading, glassy materials, ranging from soft matter systems to metallic alloys, often respond via formation of inhomogeneous flow patterns, during yielding. These inhomogeneities can be precursors to catastrophic failure, implying that a better understanding of their underlying mechanisms could lead to the design of smarter materials. Here, extensive molecular dynamics simulations are used to reveal the emergence of heterogeneous dynamics in a binary Lennard-Jones glass, subjected to a constant strain rate. At a critical strain, this system exhibits for all considered strain rates a transition towards the formation of a percolating cluster of mobile regions. We give evidence that this transition belongs to the universality class of directed percolation. Only at low shear rates, the percolating cluster subsequently evolves into a transient (but long-lived) shear band with a diffusive growth of its width. Finally, the steady state with a homogeneous flow pattern is reached. In the steady state, percolation transitions also do occur constantly, albeit over smaller strain intervals, to maintain the stationary plastic flow in the system.

Figure 1

(a) Stress-strain response of the glass at temperature $T=0.2$ for an age of $t_{\mathrm{w}}={10}^4$ and sheared with a constant strain rate. Data shown for different shear rates: $\dot{\gamma }={10}^{-3},3\times {10}^{-4}$, and ${10}^{-4}$. The inset shows the flow curve. Red solid line shows the fitting with Herschel-Bulkley form ${\sigma }_{xz}^{\mathrm{ss}}\left(\dot{\gamma }\right)=0.3974+2.2963{\dot{\gamma }}^{0.43}$. (b) Variation of $z$ component of the MSD of large particles with strain. Brown dotted line marks ${\mu }_{\mathrm{th}}$. The inset is a zoom into the superdiffusive regime for the three different shear rates, shown in linear scale. (c) Velocity profiles, for $\dot{\gamma }={10}^{-4}$, at five different strain values marked in (a). Green solid line represents the expected linear profile. (d) MSD maps at $\dot{\gamma }t=0.06,0.5$ for $\dot{\gamma }={10}^{-4}$. (e) Maps of local strain corresponding to MSD maps shown in (d). (f) Maps of local mobility corresponding to (d). Mobile regions are marked in blue while immobile regions are marked in green.

Figure 2

Emergence of percolation transition. (a) Variation of $p_{\mathrm{span}}$ with $p$, in the box of dimension $20\times 20\times 80$, for strain rates $\dot{\gamma }={10}^{-2},{10}^{-3},3\times {10}^{-4},{10}^{-4},3\times {10}^{-5}$, and ${10}^{-5}$. (b) Percolating cluster at the critical point for $\dot{\gamma }={10}^{-2},{10}^{-3}$, and ${10}^{-4}$ (left to right) in a $20\times 20\times 80$ system. (c) Percolating cluster at the critical point for $\dot{\gamma }={10}^{-4}$ in a ${50}^3$ system. (d) Variation of $p$ with strain for all strain rates shown in (a). Orange dotted line corresponds to the $\dot{\gamma }t=0.5$. (e) 2D slice of MSD map for trajectories shown in (b), at $\dot{\gamma }t=0.5$.

Figure 3

Occurrence of directed percolation (DP) transition. (a) Variation of $p_{\mathrm{span}}$ with $p$ for five different system sizes $10\times 10\times 40$ (black circles), $20\times 20\times 80$ (magenta squares), $30\times 30\times 120$ (red stars), $40\times 40\times 40$ (green diamonds), $50\times 50\times 50$ (blue triangles) for $\dot{\gamma }={10}^{-4}$. (b) Finite-size scaling of critical points for the systems sizes shown in (a); the red solid line shows the fit according to the scaling law $p_c\left(L_{\mathrm{x}}\right)=0.3034-2.412L_{\mathrm{x}}^{-1/1.106}$ corresponding to DP. Green dash-dotted line shows the same scaling function with parameters corresponding to standard percolation: critical point $p_c\left(\infty \right)=3.116$ and $\nu =0.8765$. The blue dotted line shows the asymptotic $p_c\left(\infty \right)$ for DP which is 0.3034. (c) Cluster size distribution around $p_c$ for the system sizes shown in (a); data show a power-law decay with exponent $-2.39$ which is consistent with the prediction for DP.

Figure 4

(a) 2D projections, in $xz$ plane, of MSD maps for $20\times 20\times 80$ showing the evolution of shear band with time, for imposed $\dot{\gamma }={10}^{-4}$, at strains of $\dot{\gamma }t=0.2,0.5,1.0,3.0$. (b) Variation of width of the band ${\xi }_b$, with $\dot{\gamma }={10}^{-3},{10}^{-4},3\times {10}^{-5}$. Also shown, using solid green squares, is the growth of ${\xi }_b$ in a ${50}^3$ system, for $\dot{\gamma }={10}^{-4}$. The zone between the solid red lines marks the regime of $t^{1/2}$ growth. Inset shows the dependence of time scale for fluidization, ${\tau }_f$ (defined in the text), with changing $\dot{\gamma }$; the red line is a fit with ${\dot{\gamma }}^{-1.28}$. (c) 2D projection of MSD map for a ${50}^3$ system at $\dot{\gamma }t=0.5$ sheared with $\dot{\gamma }={10}^{-4}$. (d) Corresponding evolution of spatial profile of mobility, ${\mathrm{\Delta }}^2\left(z\right)$, with strain, in the ${50}^3$ system.

Figure 5

Evolution of flow patterns in different samples. The upper and middle panel show data for ${50}^3$ system at density 1.2 and temperature $T=0.2$. Lower panel shows the similar data for ${53}^3$ system at density 1.3 and temperature $T=0.1$. All the three samples are sheared with constant strain rate $\dot{\gamma }={10}^{-4}$. (a), (e), and (i) show MSD map around percolation transition, (b), (f), and (j) show percolating cluster near the transition point. (c), (g), and (k) show MSD map at large strain while (d), (h), and (l) show strain map at the same strain.

Figure 6

Percolation transition in the steady-state regime for the $20\times 20\times 80$ system. (a) shows evolution of MSD with strain interval $\mathrm{\Delta }{\gamma }_c$, for strain origins $\dot{\gamma }{t}_0=0.0,6.0,7.0,8.0\text{,}\text{and}9.0$, after shear is imposed. (b) Layerwise MSD. Simulation box is divided into eight layers and average MSD for each layer is plotted for a strain origin of $\dot{\gamma }{t}_0=9.0$. Inset shows the same layerwise MSD for time origin taken at the start of the shear, i.e., $\dot{\gamma }{t}_0=0.0$. (c) Percolation transition with respect to the time origins mentioned in (a). (d) Variation of $p$ with $\mathrm{\Delta }{\gamma }_c$.