====== LVMCQ4 ======
===== Description =====
Von Mises type constitutive law for 3D shell ([[elements:coqj4|COQJ4]]) element numerically integrated on the thickness.

==== The model ====
This law is only used for mechanical analysis of elastoplastic isotropic hardening. Numerical integration on the thickness is used.
==== Files ====
Prepro: LVMCQ4.F \\
Lagamine: CQ4VMS.F
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| = 31 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (5I5) ^^
|NPI|number of integration points across the thickness ( $\leq$ 10 )|
|IND|index of the thickness variable|
|:::|= 0  the thickness is variable during deformation|
|:::|= 1  it is constant during deformation|
|:::|= -1  incompressible|
|NPO|number of points to define the uniaxial constitutive law ($\geq$ 0)|
|:::|= 0 -> bilinear|
|:::|= n -> multilinear|
|METH|=	the method used to calculate the IP on thickness|
|:::|= 0 -> GAUSS|
|:::|= 1 -> LOBATTO|
|:::|= 2 -> NEWTON-COTE|
|NINTV|number of sub-interval used to integrate the law|
==== Real parameters ====
__**Global Data**__
^Line 1 (2G10.0)^^
|E|Young Modulus|
|ANU|Poisson's coefficient|

**__Uniaxial Constitutive Law__** \\
**For NPO = 0**
^Line 2 (2G10.0)^^
|SY| Initial yield stress|
|ET|The tangent modulus (must be different than E)|

**For NPO $\neq$ 0** 
^Line 2 - Repeated N times (2G10.0)^^
|$\sigma_i$|the value of stress at the considered point|
|$\varepsilon$|	the value of strain 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)| $\sigma_y^e$, the equivalent VM type stress integrated through the thickness of this IP|
|Q(3)|Yield, index of stress state for this IP|
|:::|= 0, the stress state in this IP is elastic.|
|:::|= 1, it is elastoplastic.|
^Repeat NPI times ($K = 3 + (IPI-1)*6$)^^
|Q(K+1)| $\sigma_y$, the yield stress at this IP in the thickness.|
|Q(K+2)| Yield, index of stress state for this IP in the thickness.|
|Q(K+3)| $\varepsilon_{ep}^p$, equivalent plastic strain at this IP in the thickness.|
|Q(K+4)| $\sigma_{11}$, the local stress at this IP in the thickness.|
|Q(K+5)| $\sigma_{22}$, idem.|
|Q(K+6)| $\sigma_{12}$, idem.|
the total number of state variables is equal to $3 + NPI*6$