Fig. 4: Statistics of yielding, emergence of viscoelasticity and plastic turbulence.

a Within the intermittent phase, bursts of activity having duration τ₁ occur amidst quiescent states having duration τ₂, for f = 1.0. b, c The distributions of the periods of intermittent bursts [shown in b] and quiescence [shown in c] can be fit to a power law with an exponential cut-off, \(P({\tau }_{1}) \sim {\tau }_{1}^{-\alpha }\exp (-{\tau }_{1}/{\tau }_{10})\) and \(P({\tau }_{2}) \sim {\tau }_{2}^{-\beta }\exp (-{\tau }_{2}/{\tau }_{20})\), with the corresponding fit functions shown as lines. Data shown for f = 0.8 (red), 1.0 (green), 1.2 (cyan) and 1.4 (blue); the exponents vary with f and are measured to be α = 2.97, 2.30, 2.04 and 2.07 in b, β = 0.57, 0.57, 0.72 and 0.94 in (c). The inset shows the variation of the cut-offs τ₁₀ and τ₂₀ with f. We note that the cut-off for τ₁, moves to larger times as f → f^*(τ_p) from below, whereas τ₂₀ increases as f → f_c(τ_p) from above, giving rise to a power-law behaviour at the two limits. d The intermittent yielding events involve the non-affine displacement of a finite fraction of particles, as seen in this plot of n_c, the number of particles that show a non-affine displacement within a time window Δt = 10⁴, versus total particle number N, where the dashed line has slope 1. e The scale-free intermittency close to the phase boundary f^*(τ_p), is associated with plastic turbulence as seen in the spectrum of the energy density E(k) that shows an inverse cascade from an injection scale, shown by the arrow. Data shown for f = 1.6, τ_p = 10⁴. The crossover from a steep spectrum k⁻⁵ to the Kolmogorov spectrum k^−5/3 at lower k, is set by the scale of the vorticity.