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--- elements:csol2 [2019/09/04 17:45]
helene created
+++ elements:csol2 [2020/08/25 15:46] (current)
@@ Line 4: / Line 4: @@
 {{  :elements:plxnt.png?400|}}
 Coupled mechanical-flow analysis in large deformations. \\
+\\
+Type: 203 \\
 \\
 The element is defined by 3, 4, 6, 8, 15 or 25 nodes indicated in NODES in the order indicated in the figure. \\
@@ Line 84: / Line 86: @@
 \end{bmatrix}\] with $det=\dfrac{1-\nu_{31}\nu_{13}-\nu_{21}\nu_{12}-\nu_{32}\nu_{23}-2\nu_{31}\nu_{12}\nu_{23}}{E_1E_2E_3}$ \\
 Adopting the micro-homogeneity and micro-isotropy assumptions, in which the bulk modulus of the solid phase Ks is isotropic, the Biot’s coefficient can be expressed as:
-\[b_{11}=1-\frac{C^e_{1111}+C^e_{1122}+C^e_{1133}}{3K_s}=1-\frac{1-\nu_{23}\nu_{32}+\nu_{21}+\nu_{31}\nu_{23}+\nu_{21}\nu_{32}+\nu_{31}}{E_2E_3det3K_s} \]
+\[b_{11}=1-\frac{C^e_{1111}+C^e_{1122}+C^e_{1133}}{3K_s}=1-\frac{1-\nu_{23}\nu_{32}+\nu_{21}+\nu_{31}\nu_{23}+\nu_{21}\nu_{32}+\nu_{31}}{E_2E_3\det{3K_s}} \\
+b_{22}=1-\frac{C^e_{2211}+C^e_{2222}+C^e_{2233}}{3K_s}=1-\frac{\nu_{12}+\nu_{13}\nu_{32}+1-\nu_{13}\nu_{31}+\nu_{32}+\nu_{31}\nu_{12}}{E_1E_3\det{3K_s}} \\
+b_{33}=1-\frac{C^e_{3311}+C^e_{3322}+C^e_{3333}}{3K_s}=1-\frac{\nu_{13}+\nu_{23}\nu_{12}+\nu_{23}+\nu_{21}\nu_{13}+1-\nu_{21}\nu_{12}}{E_1E_2\det{3K_s}} \]
+For cross anisotropy, let us consider ($e_1$,$e_2$) as the isotropic plane (bedding plane for sedimentary rocks) and $e_3$ the normal to this plane. The subscripts ${\parallel}$ and $\perp$ indicates, respectively, the direction parallel to bedding and perpendicular to bedding.
+\[{E_1=E_2=E_{\parallel}}\quad , \quad {E_3=E_{\perp}}\]
+\[\frac{\nu_{\perp\parallel}}{E_{\perp}}=\frac{\nu_{\parallel\perp}}{E_{\parallel}}\]
+Biot's coefficients become:
+\[b_{11}=b_{22}=b_{\parallel\parallel}=1-\frac{1+\nu_{\parallel\parallel}+\nu_{\parallel\parallel}\nu_{\perp\parallel}+\nu_{\perp\parallel}}{E_{\parallel}E_{\perp}\det{3K_s}} \\
+b_{33}=b_{\perp\perp}=1-\frac{1-\nu_{\parallel\parallel}^2+2\nu_{\parallel\perp}+2\nu_{\parallel\perp}\nu_{\parallel\parallel}}{E_{\parallel}E_{\parallel}\det 3K_s} \\
+\det=\frac{1-\nu_{\parallel\parallel}^2-2\nu_{\perp\parallel}\nu_{\parallel\perp}(1+\nu_{\parallel\parallel})}{E_{\parallel}E_{\parallel}E_{\perp}} \]
+with identical value in the isotropic plane. For isotropy, Biot’s coefficients reduce to $b_{ij}=b\delta_{ij}$
+leading to $b_{11}=b_{22}=b_{33}=b$. In that case, use INBIO = 1. \\ \\
+__Remark:__
+  * The anisotropy of Biot’s coefficients can only be used with ISOL=9 (see appendix) and [[laws:orthopla|ORTHOPLA]].
+  * When variation of the solid density and of the material porosity is taken into account (see flow law), they depend on the mean effective stress $\sigma'=\sigma+b\theta(S_r)p$, which is expressed as a function of a generalized Biot’s coefficient $b=\frac{b_{ii}}{3}$ . The latter is linked to a generalized drained bulk modulus $K_0=\frac{C^e_{iijj}}{9}$ by the relation $b=1-\frac{K_0}{K_s}$.
+^Definition of the elements (6I5/16I5(/9I5)) ^^
+|NNODM|Number of nodes for the mechancial description: 3, 4, 6, 8, 15, or 25|
+|NINTM|Number of integration point (1, 3, 4, 7, 9, 12, or 16) for the mechanical description|
+|LMAT1|Mechanical material|
+|NNODP|Number of nodes for the flow description: 3, 4, 6, 8, 15, or 25|
+|NINTP|Number of integration points (1, 3, 4, 7, 9, 12, or 16) for the flow description \\ Must be equal to NINTM|
+|LMAT2|Flow material|
+|NODES(NNODEM)|List of nodes|
+===== Results =====
+  * Stresses (global axes): $\sigma_x$, $\sigma_y$, $\sigma_{xy}$, $f_x$, $f_y$, $f_{stored}$, 0
+  * Internal variables: Internal variables of the mechanical law + internal variables of the flow law

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