Indentation into a plastic fluid layer

We study the indentation of a rigid object into a layer of a cohesive or non-cohesive plastic material. Existing approaches to this problem using slip-line theory assume that the penetration depth is relatively small, employing perturbation theory about a flat surface. Here, we use two alternative approaches to account for large penetration depths, and for the consequent spreading and uplift of the surrounding material. For a viscoplastic fluid, which reduces to an ideal plastic under the limit of vanishing viscosity, we adopt a viscoplastic version of lubrication theory. For a Mohr-Coulomb material, we adopt an extension of slip-line theory between two parallel plates to account for arbitrary indenter shapes. We compare the theoretical predictions of penetration and spreading with experiments in which a flat plate, circular cylinder or sphere are indented into layers of Carbopol or glass spheres with successively higher loads. We find reasonable agreement between theory and experiment, though with some discrepancies that are discussed. There is a clear layer-depth dependence of the indentation and uplift for the viscoplastic material. For a cylinder indented into a Mohr-Coulomb material, there is a much weaker dependence on layer depth.