====== RUBB ======
===== Description =====
Hyperelastic constitutive law for solid elements at constant temperature

==== The model ====
This law is only used for large strains analysis of rubber-like materials
==== Files ====
Prepro: LRUBB.F \\
Lagamine: RUBB2S.F, RUBB2E.F, RUBB2A.F, RUBB3D.F
===== Availability =====
|Plane stress state| YES |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| YES |
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 6 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (2I5) ^^
|MODEL| = 0 : using Mooney-Rivlin model |
|:::| = 1 : using Kilian model |
|MFUN| = 0 : using penalty function : $ln I_3$ |
|:::| = 1 : using penalty function : $I_3 - l$ |

**Remark**: the strain energy density function W for Mooney-Rivlin model :
\[
W = C_l (I_l - 3) + C_2 (I_2 - 3)
\]
For Kilian model :
\[
W = G \{ 2 D_m \left[ \ln\left( l \frac{D_l}{D_m}\right) + \frac{D_l}{D_m} \right] + \frac{2}{3} a D_l^{\frac{2}{3}} \}
\]
\[
D_l = 0.5 (I_l - 3 )
\]
($I_1, I_2, I_3$ are the three strain invariants of the Cauchy-Green deformation tensor)
==== Real parameters ====
__For Mooney-Rivlin model:__ \\
^Line 1 (3G10.0)^
|$C_1$| 
|$C_2$|
|ANU|

__For Kilian model (4G10.0):__ \\
^Line 1 (4G10.0)^
|$G$|
|$D_m$|
|ANU|
|a|

( the range of the Poisson's ratio : $0.489 < ANU < 0.5$ )

===== Stresses =====
==== Number of stresses ====
 = 6 for the 3-D state \\
= 4 for the other cases.
==== Meaning ====
The stresses are the components of CAUCHY stress tensor in global (X, Y, Z) coordinates.
For the 3-D state :
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{zz}$|
|SIG(4)|$\sigma_{xy}$|
|SIG(5)|$\sigma_{xz}$|
|SIG(6)|$\sigma_{yz}$|
For the other cases :
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{xy}$|
|SIG(4)|$\sigma_{zz}$|


===== State variables =====
==== Number of state variables ====
3
==== List of state variables ====
|Q(1)|= element thickness (t) in plane stress state|
|:::|= 1 in plane strain state|
|:::|= circumferential strain rate ($\varepsilon_r$) in axisymmetrical state|
|:::|= 0 in 3‑D state|
|:::|= element thickness (t) in generalized plane state.|
|Q(2)|hydrostatic pressure|
|Q(3)|the third strain invariant of the Cauchy-Green deformation tensor|