$K$-core analysis of shear-thickening suspensions

Shear thickening of suspensions is studied by discrete-particle simulation, accounting for hydrodynamic, repulsive, and contact forces. The contact forces, including friction, are activated when the imposed shear stress $\sigma$ is able to overcome the repulsive force. The simulation method captures strong continuous and discontinuous shear thickening (CST and DST) in the range of solid volume fraction $0.54 \le \phi\le 0.56$ studied here. This work presents characteristics of the contact force network developed in the suspension under shear. The number of frictional contacts per particle $Z$ is shown to have a one-to-one relationship with the suspension stress, and the conditions for simple percolation of frictional contacts are found to deviate strongly from those of a random network model. The stress is shown to have important correlations with topological invariant metrics of the contact network known as $k$-cores; the $k$-cores are maximal subgraphs (`clusters') in which all member particles have $k$ or more frictional contacts to other members of the same subgraph. Only $k\le 3$ is found in this work at solid volume fractions $\phi \le 0.56$. Distinct relationships between the suspension rheology and the $k$-cores are found. One is that the stress susceptibility, defined as $\partial \sigma/\partial \dot{\gamma}$ where $\dot{\gamma}$ is the shear rate, is found to peak at the condition of onset of the $3$-core, regardless of whether the system exhibits CST or DST. A second is that the stress per particle within cores of different $k$ increases sharply with increase of $k$ at the onset of DST; in CST, the difference is mild.