Lifting a buried object: Reverse hopper theory

•Formulae predict force to push up an object buried in granular material.•The formulae are for (1) axi-symmetric motion and (2) two-dimensional motion.•Axi-symmetric motion implies granule movement within a cone, reverse hopper flow.•Two-dimensional motion implies granule movement within a triangular prism.•The theoretical push-up forces agree well with published experimental data.

A theory is given to predict the upward force, F, to lift an object buried at depth H in non-cohesive granular material. Explicit formulae give F in terms of the material friction coefficient of the granular material and the geometric parameters.The lifted object is either (1) a horizontal disc of diameter D or (2) a horizontal plate of width B and length L  , where L⪢BL⪢B. In case (1), the lifted disc is assumed to cause axi-symmetric upward particle motion, reverse hopper flow, within an inverted cone. Active   failure is assumed: the vertical stress, σ2σ2, is K×(horizontalstressσ1); here K=(1+sinϕ)/(1−sinϕ), ϕϕ being the angle of friction for the granular material. This gives the vertical stress, σ20σ20, on the disc. An additional lift force is needed to overcome the frictional stress, ττ, at the conical interface between stationary and upward moving particles: it is assumed that τ=μσ1τ=μσ1, μμ being the internal friction coefficient. For consolidated granules, μ=tanϕ, but for the sheared material, μ