====== LILYCQ ======
===== Description =====
Ilyushin's type constitutive law for 3d shell ([[elements:coqj4|COQJ4]]) element. Use the resultant stresses directly.

==== The model ====
This law is only used for mechanical analysis of elasto-anisotropic plastic with linear anisotropic or non-linear isotropic hardening. The resultants stresses are used directly.
==== Files ====
Prepro: LILYCQ.F \\
Lagamine: CQ4ILY.F
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| = 34 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (3I5) ^^
|IND|Index of the thickness variable|
|:::|= 0 the thickness is variable during deformation|
|:::|= 1 it is constant during deformation.|
|NPO|Number of points to define the uniaxial constitutive law ($\geq$ 0)|
|NINTV|Number of sub-interval step for integrating the constitutive law ($\geq$ 0)|
==== Real parameters ====
**__Global Data__**
^Line 1 (3G10.0)^^
|E|Young Modulus |
|ANU|Poisson's coefficient |
|$D_{NM}$|the parameter for the couple of normal effort and moment.|
|:::|= 0, it is decoupled|
|:::|= 1, it is coupled|

\[
f = \frac{1}{h^2}Q_{NN} + D_{NM} \frac{4}{\sqrt{3}h^3}Q_{NM} + \frac{16}{h^4}Q_{MM} - \sigma_y^2
\]
Where
\[
Q_NN = \alpha_{11}N_x^2 + \alpha_{22}N_y^2 + \alpha_{33}N_{xy}^2 - 2 \alpha_{12}N_xN_y
\]
\[
Q_MM = \alpha_{11}M_x^2 + \alpha_{22}M_y^2 + \alpha_{33}M_{xy}^2 - 2 \alpha_{12}M_xM_y
\]
\[
Q_NM = \alpha_{11}N_xM_x + \alpha_{22}N_yM_y + \alpha_{33}N_{xy}M_{xy} - \alpha_{12}N_xM_y + N_yM_x
\]
**__Uniaxial Constitutive Law__** \\
**For NPO = 0**
^Line 1 (2G10.0)^^
|SY11|Initial yield stress of tension or compression in 1 – direction.|
|ET11|The tangent modulus in this direction|
^Line 2 (2G10.0)^^
|SY22|Initial yield stress in tension or compression in 2 – direction|
|ET22|The tangent modulus in this direction|
^Line 3 (2G10.0)^^
|SY33|Initial yield stress in tension or compression in 3 – direction|
|ET33|The tangent modulus in this direction|
^Line 4 (2G10.0)^^
|SY12|Initial yield shear stress  in 1 - 2 plan|
|ET12|The tangent modulus in this plan|

REMARK: \\
It is possible to use the linear anisotropic or nonlinear isotropic hardening law for the model. For linear anisotropic hardening, it is only valuable for the case: NPO = 0 \\

**For NPO > 0**\\
^Line 1 (4G10.0)^^
|SY11|
|SY22|
|SY33|
|SY12|
^Line 2 - repeated N times (2G10.0)^^
|$\sigma_i$|the value of stree for the referent directional considered point|
|$\varepsilon_i$|the value of strain for this direction at the considered point|

===== Results =====
==== Stresses ====
|SIG(1)|$N_x$, normal effort in the local X‑direction|
|SIG(2)|$N_y$, normal effort in the local Y‑direction|
|SIG(3)|$N_{xy}$, normal effort in the local X-Y plan|
|SIG(4)|$M_x$, moment associated to the local X‑direction|
|SIG(5)|$M_y$, moment associated to the local Y‑direction|
|SIG(6)|$M_{xy}$, moment associated to the local X-Y plan|

==== State variables ====
|Q(1)|thick, the actual thickness for this IP.|
|Q(2)|$\sigma_n^e$, the equivalent VM type stress of the membrane part for this IP.|
|Q(3)|$\sigma_m^e$, the equivalent VM type stress of the flexion part for this IP.|
|Q(4)|$\varepsilon_{ep}^p$, equivalent plastic strain at this IP.|
|Q(5)|$\sigma_y$, the current yield stress for this IP.|
|Q(6)|$\alpha_{11}$, the anisotropic parameter for this IP.|
|Q(7)|$\alpha_{12}$, idem.|
|Q(8)|$\alpha_{22}$, idem.|
|Q(9)|$\alpha_{33}$, idem.|