====== INFM2 ======
Infinite plane or axisymmetric element.
===== Description =====
This infinite element is based on a transformation of coordinates between a classic reference element and a real element. For instance, the reference element can be similar to the classic [[elements:plxls|PLXLS]] element (coordinates ξ and η vary between [-1, 1]). However the real element INFM2 has some of its nodes to infinite coordinates. It is therefore an open element. The infinite nodes are not used in the formulation. \\
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The element is defined by 4 or 6 nodes (NNODX) defining the geometry, specified in NODES according to the order indicated in the figure. The direction of the path along the element is always from node 1 to node NNODX. \\
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The displacement field is defined by 6 or 2 nodes (NNODD).
{{  :elements:infm2.png?600  |}}
Nodes 2 and 3 (in the case of the linear element NNODX = 4) give the direction of semi-infinite sides and regulate the speed of variation of coordinates between the finite border and the infinite border of the element. The positions of these intermediary nodes have an important role in the value of the element thus created. No displacement is associated to these 2 intermediary nodes. \\
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Nodes 2, 3 and 4 (in the case of the parabolic element NNODX = 6) have the same role as nodes 2 and 3 in the linear element. However, in this case, 2 displacements (degrees of freedom) are associated to each of these nodes. \\
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__For the parabolic element defined by NNODX = 6 and NNODD = 6: __ \\
The geometry is defined by: $x=\displaystyle\sum_{l=1}^{NNODX}(\phi_l^xx_l)$ with: 
\[\left \{ \begin{array}{r c l}
 \phi_1^x & = & \displaystyle\frac{2\xi}{1-\xi}\frac{\eta(1-\eta)}{2} \\
 \phi_2^x & = & -\displaystyle\frac{1+\xi}{1-\xi}\frac{\eta(1-\eta)}{2} \\ 
 \phi_3^x & = & \displaystyle\frac{1+\xi}{1-\xi}(1+\eta^2) \\
 \phi_4^x & = & \displaystyle\frac{1+\xi}{1-\xi}\frac{\eta(1+\eta)}{2} \\ 
 \phi_5^x & = & -\displaystyle\frac{2\xi}{1-\xi}\frac{\eta(1+\eta)}{2} \\
 \phi_6^x & = & -\displaystyle\frac{2\xi}{1-\xi}(1+\eta^2)
\end{array} \right . \]
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The displacement field is defined by: $u=\displaystyle\sum_{m=1}^{NNODD}(\phi_m^du_m)$ with: 
\[\left \{ \begin{array}{r c l}
 \phi_1^d & = & \displaystyle\frac{1}{4}\xi(1-\xi)\eta(1-\eta) \\
 \phi_2^d & = & -\displaystyle\frac{1}{2}(1-\xi^2)\eta(1-\eta) \\
 \phi_3^d & = & (1-\xi^2)(1-\eta^2) \\
 \phi_4^d & = & \displaystyle\frac{1}{2}(1-\xi^2)\eta(1+\eta) \\
 \phi_5^d & = & -\displaystyle\frac{1}{4}\xi(1-\xi)\eta(1+\eta) \\
 \phi_6^d & = & -\displaystyle\frac{1}{2}\xi(1-\xi)(1-\eta^2) \\
\end{array} \right . \]
These interpolation functions are identical to those of a classic 9-nodes element: the 3 interpolation function relating to the infinite node ($\xi=1$) do not play a part as the displacements of these nodes are considered equal to 0. \\
__For the linear element defined by NNODX = 4 and NNODD = 2: __ \\
The geometry is defined by: $x=\displaystyle\sum_{l=1}^{NNODX}(\phi_l^xx_l)$ with: 
\[\left \{ \begin{array}{r c l}
 \phi_1^x & = & -\displaystyle\frac{2\xi}{1-\xi}\frac{(1-\eta)}{2} \\
 \phi_2^x & = & \displaystyle\frac{1+\xi}{1-\xi}\frac{(1-\eta)}{2} \\ 
 \phi_3^x & = & \displaystyle\frac{1+\xi}{1-\xi}\frac{(1+\eta)}{2} \\ 
 \phi_4^x & = & -\displaystyle\frac{2\xi}{1-\xi}\frac{(1+\eta)}{2} \\
\end{array} \right . \]
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The displacement field is defined by: $u=\displaystyle\sum_{m=1}^{NNODD}(\phi_m^du_m)$ with: 
\[\left \{ \begin{array}{r c l}
 \phi_1^d & = & \displaystyle\frac{1}{4}(1-\xi)(1-\eta) \\
 \phi_2^d & = & -\displaystyle\frac{1}{4}(1-\xi)(1+\eta) \\
\end{array} \right . \]
These interpolation functions are identical to those of a classic 4-nodes element: the 2 interpolation function relating to the infinite node ($\xi=1$) do not play a part as the displacements of these nodes are considered equal to 0. \\
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As can be seen, the infinite element is not isoparametric. \\
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For the axisymmetric element, the axis of symmetry is necessarily the Y axis.
Type: 62 \\
Implemented by: J.P. Radu, 1999
==== Files ====
Prepro: INFM2A.F \\
Lagamine: INFM2B.F

===== Input file =====
^Title (A5)^^
|TITLE|"INFM2" in the first 5 columns|
^Control data (3I5)^^
|NELEM|Number of elements|
|INSIG|= 0 if no initial stress \\ = 1 or 2 if initial stresses|
|IDES| Drawing index of the element: \\ = 0 drawing only of the "finite" border of the element (1 line) \\ = 1 more complete drawing of the element|
^Initial stresses - Only if INSIG = 1 or 2^ (3G10.0)^
|If INSIG = 1 → $\sigma_y=\sigma_{y0}+yd\sigma_y$ \\ If INSIG = 2 → $\sigma_y = \min(\sigma_{y0}+yd\sigma_y, \phi*D\phi)$ ||
|SIGY0|$\sigma_{y0}$ effective stress $\sigma_y$ at the axis origin|
|DSIGY|Gradient of effective stress along Y axis|
|AK0|$k_0$ ratio $\sigma_x/\sigma_y$|
|In the case where this element is used for an analysis in effective stress (cf ISOL parameter in the constitutive law), on needs: \\  - To define a pressure $p$ through the DoF n°3 \\  - Fix this DoF in the FIXED section of the *.lag file \\  - Use control parameter NTANA = 4 \\ The computation of SIGY0 and DSIGY must then be performed from effective stresses, from which the water pressure is substracted: $\sigma'=\sigma - p$ \\ In addition, the specific density to input in the mechanical law is the apparent specific density $\rho_a=(1-n)\rho_s+n\rho_w$ and not $\rho_s$ as usually done.||
^Definition of the elements (3I5/8I5)^^
|NNODX|Number of nodes defining the geometry: 4 or 6|
|NINTE|Number of integration points: 1, 4, or 9|
|LMATE|Material law|
|NODES(NNODX)|List of nodes defining the geometry|

===== Results =====
Cauchy stresses in global axes $\sigma_x,\sigma_y,\tau_{xy},\sigma_{z}$