Fig. 4. Marginal stability and partial irreversiblity.

(A) Difference between ZFC and FC stresses (rescaled by p, $\tilde{\mathrm{\sigma }}=\mathrm{\sigma }/\mathit{p}$) as a function of the volume strain ϵ, for ϕ_g = 0.655 and a few differently fixed γ. The Gardner threshold ϵ_G(ϕ_g; γ) is determined as the point where this difference exceeds 0.0006 (horizontal dashed line). For example, the vertical dashed line marks ϵ_G(ϕ_g = 0.655; γ = 0.06) ≈ −0.036. (B) Distribution of Δ_AB over samples and realizations, at fixed volume strain ϵ = −0.036 (or ϕ = 0.68), for different γ. From the data, we estimate γ_G(ϕ_g = 0.655; ϵ = −0.036) ≈ 0.06, where P(Δ_AB) becomes non-Gaussian (green points), consisting with (A). (C) The stress drops $\mathrm{\delta }{\widehat{\mathrm{\sigma }}}_1$ and $\mathrm{\delta }{\widehat{\mathrm{\sigma }}}_2$ measured in single cycle shear simulations are plotted as functions of γ at fixed ϵ = −0.036 (see the text for the definitions). The vertical dashed line represents γ_G(ϕ_g = 0.655; ϵ = −0.036) ≈ 0.06 estimated in (A) and (B). Note that at ϵ = −0.036, the system jams under constant volume shear (see Fig. 2) so that we can exclude any irreversibility caused by yielding.