The definition of effective stress is mandatory when using CSOL2, MWAT2, CSOL3, MWAT3, FAIL2, FAIL3, FAIF2, FAIF3, FAIN2, FAIN3, SGRC2, SGRT2:

The total stress σ is split into an effective stress σ' in the matrix and a pressure p_f in the fluid.

This case corresponds to Terzaghi's postulate.

\[ \sigma = \begin{cases} \sigma'- b p_w & \quad \text{if } p \geq 0 \\ \sigma' & \quad \text{if } p < 0 \end{cases} \]

b is the Biot's coefficient

$\sigma = \sigma' - bp , \forall p$ with ISEMI=1 or 2 for the problems where the negative pressure represents effectively the suction.

$\sigma = \sigma' - b\theta(S_r) , \forall p$ with ISEMI = 1 or 2
with $\theta(S_r)$ = Bishop's coefficient, depending on the material saturation, and included between 0 and 1: \[ \theta(S_r) = \begin{cases} S_r = 1 & \quad \text{if } p \geq 0 \\ S_r = \frac{n}{n_0}=\frac{S}{S_0} & \quad \text{if } p < 0 \end{cases} \] with:

• $p$ the pore pressure in CSOL2 and water pressure in MWAT2
• $n$ the soil porosity
• $S$ the accumulated fluid volume
• $S_0$: $S$ in $p = 0$

$\sigma = \sigma^* -p_a$ (Alonso's net stress)
with $p_a$ is equal to 0 in CSOL2 and equal to air pressure in MWAT2

\[ \sigma = \sigma' - b\left((1-S_w)p_a+S_w p_w\right) \\ = \sigma' - b(S_a p_a + S_w p_w)\] with:

• $S_a$ air saturation
• $S_w$ water saturation
• $p_a$ air pressure
• $p_w$ water pressure

\[\sigma = \sigma' - b\pi , \forall p\] where:

• $b$ is the Biot coefficient
• $\pi$ is the generalized pore pressure (Coussy-Danglat)

Remark:
For other elements than CSOL2 or MWAT2, the parameter ISOL can only be equal to 0 or 1.
For PLXLS : ISOL < 0 ⇒ for non saturated soil (suction effect and considered in the mechanic law) (for the Alonso's law).

$\sigma_{ij} = \sigma'_{ij} - b_{ij}\theta(S_r)p , \forall p$ with ISEM = 1 or 2
with $b_{ij}$ the anisotropic Biot’s coefficient. In the orthotropic axes: \[b_{ij}=\delta_{ij}-\frac{C^e_{ijkk}}{3K_s}\] In case of orthotropic axes rotation, it is transposed in the global axes as follows: \[b_{ij}=R_{ik}R_{jl}b'_{kl}\] where $R_{ij}$ is the rotation matrix. More details about this anisotropy are available in the definition of element CSOL2 and orthotropic law ORTHOPLA.

$\theta(S_r)$ is the Bishop's coefficient, depending on the material saturation, and included between 0 and 1: \[ \theta(S_r) = \begin{cases} S_r = 1 & \quad \text{if } p \geq 0 \\ S_r = \frac{n}{n_0}=\frac{S}{S_0} & \quad \text{if } p < 0 \end{cases} \] with:

• $p$ the pore pressure in CSOL2
• $n$ the soil porosity
• $S$ the accumulated fluid volume
• $S_0$: $S$ in $p = 0$