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.A subclass of these x(sigma) have flow curves that touch the (.)gamma=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) approximately ln 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.