====== EVP-NH ======
===== Description =====
ELASTO VISCO PLASTIC CONSTITUTIVE LAW FOR SOLID ELEMENTS AT VARIABLE TEMPERATURE (Norton-Hoff)\\
Implemented by: Pascon F (1998), Charles JF (1997 - 1999)\\
Project: continuous casting research for ARBED (RW2748)\\
==== The model ====
Coupled dynamic recrystallisation-thermo-mechanical analysis of elasto-visco-plastic solids undergoing large strains.\\
JAUMANN stress rate is used\\
__IANA= 2, 3, 5:__\\
See intermediate report RW2748 (1, 8, 17, 24) and intermediate report of April 1998 \\
For details on equations used in analytical compliance matrix computation, see appendix D of April 1998\\
__IANA= 4:__\\
See intermediate report RW2748 (17, 24) 

==== Files ====
Prepro: LNHC2.F \\
Lagamine: NHIC2E.F (IANA= 2, 3 or 5) or NHIC3D.F (IANA= 4)
==== Subroutines ====

^File^Subroutine^Description^
|CALMAT.F | CALMAT|Computes material data at temperature T |
|NHIMAT.F |CALSIGY | |
|:::|MATMSGS2|Used for analytical compliance matrix|
|:::|MATMSGL2|Used for analytical compliance matrix|
|:::|MATMSGS|Used for analytical compliance matrix (3D case)|
|:::|MATMSGL|Used for analytical compliance matrix (3D case)|
|:::|EIGVECT| Computes eigen vectors|
|:::|CMATINV| Inverse complex matrix|
|:::|VGMOYEN |Computes the constant velocities gradient matrix |
|CALPNH.F| CALPNH|Computes $K_0, P_1, P_2, P_3, P_4$ at temperature T |
|RECRYDYN.F|RECRYDYN |Dynamic recrystallization computation |

===== Availability =====
|Plane stress state| NO |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| YES |
|Generalized plane state| YES |
===== Input file =====
^ 1 Line (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 270|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ 1 Line (7I5) ^^
|NINTV| number of sub-steps used to integrate numerically the constitutive equation in a time step. \\ If NINTV < 0 or = 0, then the number of sub-steps will be computed automatically|
|NTEMP| number of temperatures at which material data (E, ANU and ALPHA) are given|
|IDYN| = 1:	if recrystallisation computation|
|:::| = 0:	else|
|ICHP2| = 2:	if parameters $K_0, P_1, P_2, P_3, P_4$ are given at several temperatures|  
|:::| = 1:	if $P_2= e^{-(\frac{T-C_4}{C_5})}.T^{C_6}$	(**__only if nodes temperature in Kelvin !!!__**)|
|:::| any other value if $P_2= (\frac{C_4}{T})^2 - \frac{C_5}{T} + C_6$	(**__only if nodes temperature in Kelvin !!!__**)|
|IALG| = 1:	if enthalpic formulation for ALPHA|
|:::| = 0:	if classical formulation for ALPHA |
|MAXITER| maximum number of iteration in elastic field \\ ≤ 0:	set default value = 50 |
|NTEMP2| number of temperatures at which parameters $K_0, P_1, P_2, P_3, P_4$ are given (only if ICHP2 = 2)|

==== Real parameters ====
^ 1 Line repeated NTEMP times (4G10) \\ Note: parameters introduced by increasing temperature order^^
|T| Temperature|
|E| YOUNG’s elastic modulus at temperature T|
|ANU| POISSON’s ratio at temperature T|
|ALPHA| Thermal expansion coefficient (α) at temperature T.  \\ Even if IALG = 1, ALPHA must be introduced at temperature T. \\ In this case, $\int_0^T\alpha(T).dT$ will be automatically computed |

^ If ICHP2 = 2: 1 Line repeated NTEMP2 times (6G10) \\ Note: parameters introduced by increasing temperature order^^
|T| Temperature|
|$K_0$| See further|
|$P_1$| See "Information about EVP-NH"|
|$P_2$| |
|$P_3$| |
|$P_4$| |

^ If ICHP2 ≠ 2: 2 Lines  (5G10/4G10) (**__only if nodes temperature in Kelvin !!!__**) ^^
|$AK_0$| |
|$C_1$| See further|
|$C_2$| See "Information about EVP-NH"|
|$C_3$| |
|$C_4$| |
|$C_5$| |
|$C_6$| |
|$P_3$| (be careful: 0 < $P_3$ < 1)|
|$P_4$| |

^ 1 Line (4G10) ^^
|TQ| Taylor-Quinney’s coefficient. Absolute value between 0 and 1 :\\ < 0:	when thermal analysis within a semi-coupled analysis\\ > 0:	for other cases (total coupled analysis or mechanical analysis within a semi-coupled analysis)|
|PRECVG| precision in VGMOY calculation (3D state only)\\ ≤ 0:	set default value = $1.10^{-5}$|
|PRECELA| precision in elastic computation\\ ≤ 0:	set default value = $1.10^{-4}$|
|EPSINC| increment of deformation for the automatic computation of NINTV\\ ≤ 0:	set default value = $1.10^{-3}$|

^ If IDYN = 1: 4 Lines  (3I5/4G10.0/4G10.0/2G10.0) ^^
|ICOUPL| =	1: the recrystallisation is coupled |
|:::| = 0:	the recrystallisation is uncoupled |
|ITYPEPS| =	0: the equations defining the beginning and the end of the recryst. have the form : $\varepsilon= Q_1. Q_4^{Q_2} . [LN(Zener)]^{Q_3}$ |
|:::| =	1: the equations defining the beginning and the end of the recryst. have the form : $\varepsilon= Q_1.ATAN[Q_3.[LN(Zener)-Q_2]]+ Q_4$ |
|:::| =	2: the equations defining the beginning and the end of the recryst. have the form : $\varepsilon= Q_1.[LN(Zener)]^{Q_2}+ Q_3.LN(Zener) + Q_4$ |
|NSSMAX| used if ICOUPL = 1: Maximum number of sub-structures\\ The precision on the recryst. fraction is 1/NSSMAX |
|$Q_1$| parameters for the __beginning__ of the recrystallisation: $\varepsilon_c$ |
|$Q_2$| parameters for the __beginning__ of the recrystallisation: $\varepsilon_c$ |
|$Q_3$| parameters for the __beginning__ of the recrystallisation: $\varepsilon_c$ |
|$Q_4$| parameters for the __beginning__ of the recrystallisation: $\varepsilon_c$ |
|$Q_1$| parameters for the __end__ of the recrystallisation: $\varepsilon_s$ |
|$Q_2$| parameters for the __end__ of the recrystallisation: $\varepsilon_s$ |
|$Q_3$| parameters for the __end__ of the recrystallisation: $\varepsilon_s$ |
|$Q_4$| parameters for the __end__ of the recrystallisation: $\varepsilon_s$ |
|ACTIV| Activation energy for Zener computation : $Z=\dot{\varepsilon}.EXP(\frac{ACTIV}{R.T})$ \\ with T the temperature and R the Boltzman gas constant|
|EXPO| Exponent for the AVRAMI law : $X= 1- EXP[-3.(\frac{\bar{\varepsilon}-\varepsilon_c}{\varepsilon_s-\varepsilon_c})^{expo}]$ |

__**NOTE:**__ ISTRA(3) parameter of the execution file:\\ **Units:** \\ = 0:	analytical compliance matrix used (default value) \\ = 1:	perturbation method \\ **Tens:** \\ = 0:	mean velocities gradient (default value) \\ = 1:	initial velocities gradient \\ **Hundreds:** \\ = 0:	yield limit given by intersection between N-H curve and Young’s straight line \\ = 1:	yield limit given by K0 (given parameter – see below) \\

\\ __**Information about EVP-NH:**__ \\
For the 1D case, we have: \\
$\bar{\sigma}= A. K_0. \bar{\varepsilon}^{P_4}. exp(-P_1.\bar{\varepsilon}). P_2. \sqrt{3}. (\sqrt{3}. \bar{\dot{\varepsilon}})^{P_3}$ with $P_1 \geq 0$ \\
The parameters $K_0, P_1, P_2, P_3, P_4$ can be given at several temperatures (ICHP2 = 2) \\
Otherwise, if ICHP2 ≠ 2: (see the law in section: "integer parameters") \\
$P_1= (\frac{T}{C_1})^{C_2} + C_3$ \\
$P_2= f(C_4, C_5, C_6, T)$ \\
$P_3, P_4, K_0= constants$ \\




===== 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 ====
27
==== List of state variables ====
|Q(1)|thickness|
|Q(2)|equivalent stress (effective if icoupl=1)|
|Q(3)|equivalent strain|
|Q(4)|equivalent strain rate|
|Q(5)|instantaneous thermal flow (effective if icoupl=1)|
|Q(6)|plastic dissipation (effective if icoupl=1)|
|Q(7)|ΔT|
|Q(8)|RHOC capacity|
|Q(9)|LN (ZENER)|
|Q(10)|recrystallised fraction since the beginning of the simulation|
|Q(11)|recrystallised fraction on this step|
|Q(12)|elastic part on this step – in percent \\ (>0 : loading ; <0 : unloading) (effective if icoupl=1)|
|Q(13)|number of sub-structures|
|Q(14)|volumic fraction of the unrecrystallised sub-structure|
|Q(15)|effective equivalent strain|
|Q(16)|equivalent strain standard deviation|
|Q(17)|= 0 if always elastic state since the beginning|
|:::| = 1 if any previous step has been performed in visco-plastic domain|
|Q(18)|recrystallised fraction during previous step|
|Q(19)||
|Q(20)||
|Q(21)||
|Q(22)||
|Q(23)||
|Q(24)||
|Q(25)|triaxiality (BLZ2T)|
|Q(26)|shape parameter of the element (BLZ2T)|
|Q(27)|Remeshing parameter (BLZ2T)|