====== EP-DAM ======
===== Description =====
Elasto-plastic isotropic constitutive law with damage for solid elements at constant temperature endochronic 

==== The model ====
This law is used for mechanical analysis of elasto‑plastic isotropic solids undergoing large strains, taking account of internal damage generated by plastic strains. Plastic isotropic hardening is assumed.
==== Files ====
Prepro: LEPDAM.F \\
Lagamine: ENDO2A.F
===== Availability =====
|Plane stress state| NO |
|Plane strain state| NO |
|Axisymmetric state| YES |
|3D state| NO |
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 220 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (I5) ^^
|NINTV| number of sub-steps used to integrate numerically the constitutive equation in a time step. |
==== Real parameters ====
^ Line 1 (7G10.0) ^^
|E| YOUNG’s elastic modulus |
|ANU| Poisson ratio |
|TAU| ratio of volumetric damage to deviatoric damage $(=\tau)$ |
|AG| rate of deviatoric damage $(=a_G)$ |
|RE| initial yield limit $(=R_e)$ |
|AK| hardening coefficient $(=k)$ |
|AN| hardening exponent $(=n)$ |

===== 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 ====
= 9
==== List of state variables ====
|Q(1)| = element thickness (t) in plane stress state |
|:::| = 1 in plane strain state|
|:::| = circumferential strain rate ($\dot{\varepsilon_{\theta}}$)  in axisymmetrical state|
|:::| = 0 in 3‑D state | 
|:::| = element thickness (t) in generalized plane state| 
|Q(2)| = 0  if the current state is elastic |
|:::| = 1  if the current state is elasto‑plastic| 
|Q(3)|equivalent plastic strain $(\varepsilon_p)$ |
|Q(4)| amount of deviatoric damage (= d) |
|Q(5)| amount of volumetric damage (= `U) |
|Q(6)| thermodynamic reaction conjugated to deviatoric damage $(=Y_d)$|
|Q(7)| thermodynamic reaction conjugated to deviatoric damage $(=Y_{\delta})$ |
|Q(8)| hardening level (R) |
|Q(9)| damage level (B)  |