====== Appendix 7: Notion of effective stress and signification of parameter ISOL in elements CSOL2 and MWAT2 ======

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

__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).
 

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

^ISOL = 1 - All coupled elements^^

This case corresponds to Terzaghi's postulate.


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

b is the Biot's coefficient

^ISOL = 4  - CSOL2 element^^

$\sigma = \sigma' - b p_w $, whatever the sign of $p_w$,  for the problems where the negative pressure represents effectively the suction.

^ISOL = 5 - CSOL2 element^^

$\sigma = \sigma' - b S_r p_w, \forall p_w$  \\

with $S_r$ = the water saturation degree, ranging between 0 and 1.


^ISOL = 6 - CSOL2 and MWAT2^^

$\sigma = \sigma^* -p_a$ (Alonso's net stress) \\

with $p_a$ is equal to 0 in CSOL2 and equal to air pressure in MWAT2

^ISOL = 7 - only for element MWAT2: Bishop's model^^

\[ \sigma = \sigma' - b \left((1-S_w)p_a+ S_w p_w\right) \\ = \sigma' - b(S_a p_a + S_w p_w)\]
with:
  * $S_a$ air saturation
  * $S_w$ water saturation
  * $p_a$ air pressure
  * $p_w$ water pressure

^ISOL = 8 - CSOL2 element^^

\[\sigma = \sigma' - b\pi , \forall p\]
where:
  * $b$ is the Biot coefficient
  * $\pi$ is the generalized pore pressure (Coussy-Danglat)

^ISOL = 9 : Only for CSOL2 element and ORTHOPLA mechanical law^^

$\sigma_{ij} = \sigma'_{ij} - b_{ij} S_r p_w , \forall p_w$  \\
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. \\ \\