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--- appendices:a7 [2024/06/17 16:00]
frederic
+++ appendices:a7 [2024/06/20 16:25] (current)
frederic
@@ Line 1: / Line 1: @@
 ====== Appendix 7: Notion of effective stress and signification of parameter ISOL in elements CSOL2 and MWAT2 ======
-Two coupling methods are possible with the 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.
-  * 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.
-^ISOL = 1 (ISEMI = 1 or 2)^^
+__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^^
-\[ \sigma =  \begin{cases}   \sigma'-b  & \quad \text{if } p \geq 0 \\
+This case corresponds to Terzaghi's postulate.
-    \sigma'  & \quad \text{if } p < 0
+\[ \sigma =  \begin{cases}   \sigma'- b p_w  & \quad \text{if } p_w \geq 0 \\
+    \sigma'  & \quad \text{if } p_w < 0
   \end{cases}
 \]
-===== ISOL = 2 (ISEMI = 1) - NOT USED =====
+b is the Biot's coefficient
-\[ \sigma = \sigma' = b (p_A+\Delta p) \\ =b (p_A+\chi_w \varepsilon_v)\\ =b (p_A+\chi_w \dot{\varepsilon}_v\Delta t)\]
-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).
-===== ISOL = 3 - NOT USED =====
+^ISOL = 4  - CSOL2 element^^
-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
-^ISOL = 4^^
+$\sigma = \sigma' - b p_w $, whatever the sign of $p_w$,  for the problems where the negative pressure represents effectively the suction.
-$\sigma = \sigma' - bp , \forall p$ with ISEMI=1 or 2 for the problems where the negative pressure represents effectively the suction.
+^ISOL = 5 - CSOL2 element^^
-^ISOL = 5^^
+$\sigma = \sigma' - b S_r p_w, \forall p_w$  \\
-$\sigma = \sigma' - b\theta(S_r) , \forall p$ with ISEMI = 1 or 2 \\
+with $S_r$ = the water saturation degree, ranging between 0 and 1.
-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 $p = 0$
-^ISOL = 6^^
+^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^^
-\[ \sigma = \sigma' - b\left((1-S_w)p_a+S_w p_w\right) \\ = \sigma' - b(S_a p_a + S_w p_w)\]
+\[ \sigma = \sigma' - b \left((1-S_w)p_a+ S_w p_w\right) \\ = \sigma' - b(S_a p_a + S_w p_w)\]
 with:
   * $S_a$ air saturation
@@ Line 64: / Line 54: @@
   * $b$ 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).
 ^ISOL = 9 : Only for CSOL2 element and ORTHOPLA mechanical law^^
-$\sigma_{ij} = \sigma'_{ij} - b_{ij}\theta(S_r)p , \forall p$ with ISEM = 1 or 2 \\
+$\sigma_{ij} = \sigma'_{ij} - b_{ij} S_r p_w , \forall p_w$  \\
 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. \\ \\
-$\theta(S_r)$ 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 $p = 0$

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