Constitutive Relations for Shear Fronts in Shear-thickening Suspensions*

When a shear-thickening suspension is submitted to a sudden driving force at a boundary, a front can be generated, which propagates into the bulk and turns the suspension into a solid-like material in its wake. To explain this phenomenon, we generalized the Wyart-Cates model by introducing a characteristic strain scale that controls the crossover from start-up response to steady-state behavior. With the generalized model, we derived a constitutive relation for dense suspensions in the dynamic jamming regime. We found that the relation between stress and velocity is quadratic, as is generally true for inertial effects in liquids, but with a pre-factor that can be much larger than the material density. The prediction of the model matches well with the experimental results obtained with a quasi-one dimensional system. This theoretical framework unifies both transient and steady-state properties of shear-thickening materials.