The Shear Box

Push the top half of the box sideways under a fixed normal load. The work you do goes into sliding against friction and into lifting the load as the sample rises (Taylor's balance): τ/σ′ = tan φ′_cs + dy/dx. A dense sample has to climb; a loose one sinks. Both end at the critical state.

What you are seeing

• The box. The bottom half is fixed and the top half is pushed sideways by the force T (blue) under the normal load N. The darker band is the shear zone: it leans as the top half moves, and thickens as the sample dilates (orange) or thins as it contracts (blue). The rise is exaggerated three times.
• Strength and rise. Top: the stress ratio τ/σ′ (black) against the shear displacement x, and friction alone, tan φ′_cs (blue dashed). The shaded strip between them is dy/dx: the work of lifting the load. Bottom: the rise y of the top half, orange if the sample ends up higher, blue if lower. The dashed line and dots mark the current displacement.

The model

Peak dilatancy follows Bolton's (1986) relative dilatancy index, I_R = I_D(10 − ln p′) − 1, with φ′_peak − φ′_cs ≈ 5I_R degrees in plane strain. The shape of the curves is schematic.