====== LLEVCQ ======
===== Description =====
An elasto-viscoplastic simplified constitutive law - LEV for 3D shell ([[elements:coqj4|COQJ4]]) element, numerically integrated on the thickness.

==== The model ====
This law is only used for mechanical analysis of elasto-viscoplastic with numerical integrations on the thickness.
==== Files ====
Prepro: LLEVCQ.F \\
Lagamine: CQ4LEV.F
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| = 137 |
|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.|
|METH|the method used to calculate the IP on thickness|
|:::|= 0 → GAUSS|
|:::|= 1 → LOBATTO|
|:::|= 2 → NEWTON-COTE|
|MLAW|= the method used to calculate the increments of stress|
|:::|= 0 → Backward Euler method|
|:::|= 1 → Radial Return method|
|:::|= 2 → Decomposed mode method|
|MANA|= 0 → The tangent matrix obtained by setting $\dot{\lambda} = 0$|   
|:::|= 1 → The tangent matrix obtained by setting $\dot{\lambda}_{eq} \Rightarrow \dot{\sigma}_{eq}$| 
|:::|= 2 → The tangent matrix obtained by setting $\dot{\lambda}_{eq} \Rightarrow \dot{\varepsilon}_{eq}$|
|:::|= 3 → The tangent matrix obtained by the consistent condition|
==== Real parameters ====
^Line 1 (4G10.0)^^
|E|Young Modulus |
|ANU|Poisson's coefficient |
|C|Material parameter for $\sigma-\varepsilon$ relation|
|M|idem|
**Remark :**
\[
\sigma_{eq} = c \left( \dot{\varepsilon}_{eq}\right)^m
\]
===== 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)| $\sigma_{yt}$, equivalent stress 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 $3 + NPI*6$