Validating a Unified Effective Stress Equation for Soil

Effective stress ${\sigma }^{\prime }$ is the stress acting on a soil’s skeleton due to the total stress $\sigma$, pore water pressure (Terzaghi 1943), interparticle capillary stress (Bishop 1959), and adsorptive stress (Lu and Likos 2006). A general effective stress is proposed (Lu and Likos 2006), using the suction stress ${\sigma }^{\mathrm{s}}$ as a function of water content $w$, including capillary stress ${\sigma }_{\mathrm{cap}}^s(w)$ and adsorptive stress ${\sigma }_{\mathrm{ads}}^s(w)$

An explicit form of suction stress and its dependence on soil water content was proposed and experimentally validated (Lu et al. 2010) for all types of soils under matric suction $<1.5\mathrm{MPa}$where $w_r$ = residual water content for adsorption; $w_{\mathrm{s}}$ = saturated water content; ${\alpha }^{\mathrm{SWR}}$ is related to the inverse of the air-entry pressure; and $n^{\mathrm{SWR}}$ is related to the pore size distribution for soil water retention (SWR). Recently, Lu and Zhang (2019) showed that defining matric suction ${\psi }_m$ as the air pressure $u_a$ and pore water pressure $u_w$ difference (i.e., $u_{\mathrm{a}}–u_{\mathrm{w}}$) is incomplete, and a general definition including both adsorption and capillarity iswhere $x$ = statistical distance to a particle surface; and ${\psi }_{\mathrm{ads}}(w)$ = soil sorptive potential (SSP). In light of the general matric suction and SSP concepts, suction stress equations more general than Eq. (2) have been proposed under a full range of water content [Fig. 1(a)] (Zhang and Lu 2020)where ${\alpha }^{SS}$ is related to the inverse of the average capillary suction stress; $n^{SS}$ is related to the pore size distribution; $w_{\mathrm{amax}}^{SS}$ is related to maximum adsorptive water content; ${\sigma }_{\mathrm{dry}}^s$ = suction stress at the oven-dry state; and $\beta$ = strength of the adsorptive suction stress. Capillary suction stress [Eq. (4a) and Fig. 1(a)] is due to capillary pressure $(u_a-u_w)$ and varies nonmonotonically with water content, whereas adsorptive suction stress [Eq. (4b) and Fig. 1(a)] is due to SSP and varies monotonically at low water content.