Jamming, hysteresis, and oscillation in scalar models for shear thickening

We investigate shear thickening and jamming within the framework of a family of spatially homogeneous, scalar rheological models. These are based on the "soft glassy rheology" model of Sollich et al. [Phys. Rev. Lett. 78 2020 (1997)], but with an effective temperature x that is a decreasing function of either the global stress sigma or the local strain l. For appropriate x = x(sigma), it is shown that the flow curves include a region of negative slope. around which the stress exhibits hysteresis under a cyclically varying imposed strain rate (gamma )over dot. A subclass of these x(sigma) have flow curves that touch the (gamma )over dot = 0 axis for a finite range of stresses; imposing a stress from this range jams the system, in the sense that the strain gamma creeps only logarithmically with time t, gamma (t) similar to In t. These same systems may produce a finite asymptotic yield stress under an imposed strain, in a manner that depends on the entire stress history of the sample, a phenomenon we refer to as history-dependent jamming. In contrast, when x = x(l) the flow curves are always monotonic, but we show that some x(l) generate an oscillatory strain response for a range of steady imposed stresses. Similar spontaneous oscillations are observed in a simplified model with fewer degrees of freedom. We discuss this result in relation to the temporal instabilities observed in rheological experiments and stick-slip behavior found in other contexts, and comment on the possible relationship with "delay differential equations" that are known to produce oscillations and chaos.