Two coupling methods are possible with the elements CSOL2 and MWAT2:

• Fully coupled method: ISEMI = 2
• Semi-coupled method: ISEMI = 1

The total stress σ is split in an effective stress σ' in the matrix and a pressure p in the fluid, according to Terzaghi's law.

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

\[ \sigma = \sigma' = b (p_A+\Delta p) \\ =b (p_A+\chi_w \varepsilon_v)\\ =b (p_A+\chi_w \dot{\varepsilon}_v\Delta t)\] with:

• $p_A$, the pressure at the beginning of the step;
• $\varepsilon_v$, the pores volume variation;
• $\chi_w$, the bulk modulus of water;
• $\Delta t$, the time step.

In the semi-coupled method (ISEMI=1), one replaces favorably in the Terzaghi's principle, the pressure (ISOL=1) by the effect of water on mechanics (ISOL=2).

This case is reserved for the semi-coupled analysis (ISEMI=1) with the law CLOE: \[\sigma=\sigma'-bQ_B(20)\] with $Q_B(20) = Q_A(20) - \chi_w \dot{\varepsilon}_v\Delta t$, state variable containing the pore pressure in the case of non drained analysis

$\sigma = \sigma' - bp , \forall p$ 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 $p = 0$