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.

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

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

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

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

$\sigma = \sigma^* -p_a$ (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)

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