====== LEV-JET ======
===== Description =====
An elasto‑viscoplastic constitutive law for solid elements at constant temperature ‑ Levi model.

==== The model ====
This law is used for mechanical analysis of elastoplastic isotropic element undergone large deformation. 
==== Files ====
Prepro: LJETV.F \\
Lagamine: JETLEV.F, JT3LEV.F
===== Availability =====
|Plane stress state| NO |
|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| 57 for JET2D; 59 for JET3D |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^Line 1 (2I5)^^
|MLAW|= the method used to calculate the increments of stress|
|:::|= 0 → Radial Return method|
|:::|= 1 → Implicit integration method|
|:::|= 2 → Modified implicit integration method|
|MANA| See explanation below|

__If MLAW = 0__
|MANA|= 0 → The tangent matrix obtained by setting $\dot{\omega} = 0$ |  
|:::|= 1 → The tangent matrix obtained by setting $\dot{\sigma}_{eq} = \dot{\sigma}_{eq}^{Trial}$ | 
|:::|= 2 → The tangent matrix obtained by setting $\dot{\sigma}_{eq} => \dot{\varepsilon}_{eq}$ |  
__Else__
|MANA|= 0 → The tangent matrix obtained by setting $\dot{\lambda}_{vp} = 0$ |
|:::|= 1 → The tangent matrix obtained by setting $\dot{\lambda}_{vp} => \dot{\sigma}_{eq}$ |  
|:::|= 2 → The tangent matrix obtained by setting $\dot{\lambda}_{vp} => \dot{\varepsilon}_{eq}$ |  
==== Real parameters ====
^ Line 1 (4G10.0) ^^
|E|YOUNG's elastic modulus |
|<| POISSON's ratio. |
|$A_c$| parameter for  $\sigma - \dot{\varepsilon}$ relation |
|$A_m$| parameter for  $\sigma - \dot{\varepsilon}$ relation |
\[
\hat{\sigma}_{eq} = A_c \hat{D}_{eq}^{A_m}
\]

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

===== State variables =====
==== Number of state variables ====
= 24 for 2D analysis\\
= 15 for 3D analysis
==== List of state variables ====
__For 2D analysis:__
|Q(1)|current yield limit in tension; its initial value is $R_e$ |
|Q(2)|equivalent plastic strain $(\bar{\varepsilon}^p)$ |
|Q(3)| equivalent VM type stress in this element. |
|Q(4)| $\sigma_{xx}$ in local axes of the element |
|Q(5)| $\sigma_{yy}$ in local axes of the element |
|Q(6)| $\sigma_{zz}$ in local axes of the element |
|Q(7)| $\sigma_{xy}$ in local axes of the element|
|Q(8)| $\sigma_{1}$ anti‑hourglass stress |
|Q(9)| $\sigma_{2}$ anti‑hourglass stress |
|Q(10to13)| x nodal coordinates in local axes of the element |
|Q(14to17)| y nodal coordinates in local axes of the element |
|Q(18)| = 0 in plane strain state |
|:::| = average radius (X coordinate) of the element in axisymmetric state |
|Q(19)|area of the element in the XY plane |
|Q(20)|area of the no deformed element |
|Q(21)|X(4) ‑ X(2) in initial structure |
|Q(22)|X(3) ‑ X(1) in initial structure |
|Q(23)|Y(4) ‑ Y(2) in initial structure |
|Q(24)|Y(3) ‑ Y(1) in initial structure |
|Q(25)| = 0 - if the current stress state is elastic|
|:::| = 1 - if the current stress state is plastic.|

__For 3D analysis:__
|Q(1)| current yield limit tension |
|Q(2)| equivalent plastic strain $(\bar{\varepsilon}^p)$ |
|Q(3)| equivalent VM type stress for this element. |
|Q(4)| $\sigma_{11}$ anti‑hourglass stress |
|Q(5)| $\sigma_{12}$ anti‑hourglass stress |
|Q(6)| $\sigma_{13}$ anti‑hourglass stress |
|Q(7)| $\sigma_{21}$ anti‑hourglass stress|
|Q(8)| $\sigma_{22}$ anti‑hourglass stress |
|Q(9)| $\sigma_{23}$ anti‑hourglass stress |
|Q(10)| $\sigma_{31}$ anti‑hourglass stress |
|Q(11)| $\sigma_{32}$ anti‑hourglass stress |
|Q(12)| $\sigma_{33}$ anti‑hourglass stress |
|Q(13)| $\sigma_{41}$ anti‑hourglass stress |
|Q(14)| $\sigma_{42}$ anti‑hourglass stress |
|Q(15)| $\sigma_{43}$ anti‑hourglass stress |