appendices:a7
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appendices:a7 [2024/06/17 16:23] frederic |
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| ====== Appendix 7: Notion of effective stress and signification of parameter ISOL in elements CSOL2 and MWAT2 ====== | ====== 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). | ||
| + | |||
| - | \[ \sigma = \begin{cases} \sigma'-b & \quad \text{if } p \geq 0 \\ | + | The total stress σ is split into an effective stress σ' in the matrix and a pressure p_f in the fluid. |
| - | \sigma' & \quad \text{if } p < 0 | + | |
| - | \end{cases} | + | |
| - | \] | + | |
| - | ^ISOL = 4^^ | + | ^ISOL = 1 - All coupled elements^^ |
| - | $\sigma = \sigma' - bp , \forall p$ with ISEMI=1 or 2 for the problems where the negative pressure represents effectively the suction. | + | This case corresponds to Terzaghi's postulate. |
| - | ^ISOL = 5^^ | ||
| - | $\sigma = \sigma' - b\theta(S_r) , \forall p$ with ISEMI = 1 or 2 \\ | + | \[ \sigma = \begin{cases} \sigma'- b p_w & \quad \text{if } p_w \geq 0 \\ |
| - | with $\theta(S_r)$ = Bishop's coefficient, depending on the material saturation, and included between 0 and 1: | + | \sigma' & \quad \text{if } p_w < 0 |
| - | \[ \theta(S_r) = \begin{cases} S_r = 1 & \quad \text{if } p \geq 0 \\ | + | |
| - | S_r = \frac{n}{n_0}=\frac{S}{S_0} & \quad \text{if } p < 0 | + | |
| \end{cases} | \end{cases} | ||
| \] | \] | ||
| - | 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^^ | + | 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) \\ | $\sigma = \sigma^* -p_a$ (Alonso's net stress) \\ | ||
| + | |||
| with $p_a$ is equal to 0 in CSOL2 and equal to air pressure in MWAT2 | 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^^ | ^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: | with: | ||
| * $S_a$ air saturation | * $S_a$ air saturation | ||
| Line 51: | Line 54: | ||
| * $b$ is the Biot coefficient | * $b$ is the Biot coefficient | ||
| * $\pi$ is the generalized pore pressure (Coussy-Danglat) | * $\pi$ 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^^ | ^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: \[b_{ij}=\delta_{ij}-\frac{C^e_{ijkk}}{3K_s}\] | 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: \[b_{ij}=R_{ik}R_{jl}b'_{kl}\] | 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. \\ \\ | 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 $p = 0$ | ||
appendices/a7.1718634198.txt.gz · Last modified: 2024/06/17 16:23 by frederic
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