====== DAM-EP-TH ======
===== Description =====
Elastic-(visco)-plastic constitutive law fully coupled with damage for solid elements at variable temperature
==== The model ====
Mechanical analysis of thermo-elasto-(visco)-plastic-damage isotropic solids undergoing large strains, plastic mixed hardening and damage isotropic hardening are assumed.\\
:!: Error in mechanical computation if elastic strain is not negligible compared to the plasticity. Small example at the end. 
==== Files ====
Prepro: LZDMGC.F \\
===== Availability =====
|Plane stress state| YES |
|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| 215|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing.|
==== Integer parameters ====
^ Line 1 (6I5) ^^
|NTEMP|number of temperatures at which material date are given|
|NINTV| number of sub-steps used to integrate numerically the constitutive equation in a time step|
|MNINTV|Max. of number of sub-steps (0$\rightarrow$100)|
|MITERA|number of sub-iteration (0$\rightarrow$10)|
|MUTIP|number of multiplicator for sub-steps (0$\rightarrow$2)|
|IVISC| = 0 (EP LAW)\\ = 1 (EVP LAW)|
==== Real parameters ====
^Line 1 (6G10.0)^^
|TAU| ratio of volumetric damage to deviatoric damage|
|ECROU| = 0 for isotropic hardening\\ = 1 for kinematic hardening\\ ∈ [0,1] for mixed hardening|
|COEFQ| TAYLOR-QINNEY's coefficient (q)|
|DNMAX| 0 for EP without damage\\ (0,1) $rightarrow$ Max. damage value at initial fracture \\ otherwise $\rightarrow$ 0.95 limit damage value|
|PROC|precision of iteration\\ (0 $\rightarrow$1.D-3)|
|THICK|thickness for plane state|
==== Real parameters====
__Repeated NTEMP times__
^Line 1 (7G10.0)^^
|T| temperature|
|E| YOUNG's elastic modulus at temperature T|
|ANU| POISSON's ratio at temperature T|
|ALPHA| thermal expansion coefficient ($\alpha$) at temperature T|
|RP$\Phi$| lower yield limit $\sigma_0$  at temperature T|
|ET|  elasto-plastic tangent modulus ($E_t$) at temperature T1|
|RD$\Phi$| initial damage limit at temperature T|
|DTG|damage tangent modulus at temperature T|
^Line 2 (G10.0)^^
|VISCO| visco fluid parameter (unit : time)|
===== Stresses =====
==== Number of stresses ====
6 for 3D 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 ====
28 for 3D state\\ 26 for the other cases
==== 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 axisymmetric 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)|= 0  if the current state is not damage\\ = 1  if the current state is damage |
|Q(4)| generalized plastic strain ($\alpha$) |
|Q(5)| amount of current deviatoric damage (D)|
|Q(6)| plastic hardening level (R)|
|Q(7)| damage hardening level (B)|
|Q(8)| back stresses for kinematic and mixed hardening|
|Q(N)| (N = 14 for 3-D state, N = 12 for other cases)|
|Q(N+1)| equivalent plastic strain|
|Q(N+2)| equivalent stress|
|Q(N+3)| thermodynamic reaction conjugated to deviatoric damage ($Y_{d}$)|
|Q(N+4)| thermodynamic reaction conjugated to volumetric damage ($Y_{*}$)|
|Q(N+5)| plastic work per unit volume|
|Q(N+6)| damage work per unit volume|
|Q(N+7)| total strain energy per unit volume (elastic + plastic + damage)|
|Q(N+8)| part of the dissipated power converted into heat|
|Q(N+9)|current temperature|
|Q(N+10|fracture criteria|
|Q(N+15)||

Example of the error that might occur:
{{ :laws:dam-ep-th.png?700 |}}