Bishop parameter $\chi$ from static equilibrium

For unsaturated soils, Terzaghi’s effective stress σ=σ$^{tot}$−u has been generalized by Bishop to the form σ$^{B}$=σ$^{tot}$−$\chi$u. Factor $\chi$, for example $\chi$(S)≈S, takes into account the degree of saturation S. However, σ$^{B}$ is unable to characterize strength and deformation of soil unless applied with suction. A new stress σ$^{E}$=σ$^{tot}$−$\chi$Eu with $\chi$$^{E}$(S,n) is proposed. It is likewise insufficient if applied without suction, but it has (at least) a clear physical meaning: σ$^{E}$ represents contact forces between grains in the static equilibrium in the same manner as the effective stress does in saturated soils. All grains must be entirely surrounded by free or capillary water.