Mean-Field Scenario for the Athermal Creep Dynamics of Yield-Stress Fluids

We develop a theoretical description based on an existent mean-field model for the transient dynamics prior to the steady flow of yielding materials. The mean-field model not only reproduces the experimentally observed nonlinear time dependence of the shear-rate response to an external stress, but also allows for the determination of the different physical processes involved in the onset of the reacceleration phase after the initial slowing down and a distinct fluidization phase. The fluidization time displays a power-law dependence on the distance of the applied stress to an age-dependent yield stress, which is not universal but strongly dependent on initial conditions.

Figure 1

Creep curves. Strain-rate time series $\dot{\gamma }(t)$ and corresponding strain time series $\gamma (t)$ (inset); the dashed linear line serves as a guide for the eye. (a) Creep behavior for different applied stresses for a fixed initial aging level $s_d=0.28$ with $\mathrm{\Delta }\sigma =0.18,0.2,\dots ,0.28$. (b) Creep behavior for a fixed applied stress $\mathrm{\Delta }\sigma =0.18$ for different initial aging levels $s_d=0.28,0.3,\dots ,0.36$.

Figure 2

Creep behavior. (a) Fluidization time ${\tau }_f$ versus $\mathrm{\Delta }\sigma =\sigma -{\sigma }_y$ for $\alpha =0.3$ and $s_d=0.22,0.24,\dots ,0.34$. Inset: $\beta$ versus $s_d$ for $\alpha =0.4$ (triangle), 0.3 (square), and 0.2 (circle). (b) Comparison between ${\dot{\gamma }}_q(t)$ (dashed curve) generated by Eq. (4) and $\dot{\gamma }(t)$ (solid curve) generated by Eq. (1). (c) ${\tau }_f$ versus $\mathrm{\Delta }{\sigma }_s$ for $\alpha =0.3$. Different symbols for different values of $s_d$; the correspondence is the same as in (a). (d) The difference between static and dynamic yield stress ${\sigma }_y^S-{\sigma }_y$ against the normalized initial relaxation $s_d/({\sigma }_c-{\sigma }_y)$. Inset: ${\beta }_s$ versus $s_d/({\sigma }_c-{\sigma }_y).\alpha =0.4$ (triangle), 0.3 (square), and 0.2 (circle).