====== ARBTH ======
===== Description =====
Elasto-plastic constitutive law with thermal effects for solid elements at variable temperature \\
Developed from 1985 to 2005 by AM Habraken, F. Libon, De Montleau - 3D version by C. Lequesne (2005) \\
Project: 
==== Use ====
Coupled thermo-mechanical analysis of elasto-plastic solids undergoing large strains
==== Files ====
Prepro: LARBTN.F \\
Lagamine: ARBC2N.F (plane strain, axissymetric, generalized plane state)\\
ARB3C (3D state)
==== Subroutines ====
^File^Subroutine^Description^
|ARBC2N.F| ARBC2N |Main subroutine of the law for plane strain, axissymetric, or generalized plane state|
|ARB3C.F| ARB3C |Main subroutine of the law for 3D state|
|CALMAT.F| CALAMAT | Linear interpolation of parameters at a given temperature|
|CALDER.F| CALDER | Computation of $\frac{dE}{dT}$, $\frac{d\nu}{dT}$, and $\frac{d\alpha}{dT}$|
|CALSYT.F| CALSYT | Computes actualized plastic limit |
|CONCAT.F| CONCAT | Concatenation of 2 vectors |
|CALMU2.F| CALMU2 | Computation of the plastic modulus|

===== Availability =====
|Plane stress state| NO |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| YES |
|Generalized plane state| YES |

===== Input file =====

==== Parameters defining the type of constitutive law ====
^ 1 line (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 250|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ 1 line (6I5) ^^
|NTEMP| number of temperatures at which material data are given |
|NINTV| number of sub-steps used to integrate numerically the constitutive equation in a time step \\ if NINTV <= 0 : number of sub-steps is based on the norm of the deformation increment and on DIV=1.D-04 |
|IENTH| = 0 to use the classical formulation for $\alpha$|
|:::| = 1 to use $\int \alpha dT$ |
|NPOINT| number of points (eps,sig) to define the law|
|:::|=0 if parabolic or bilinear law|
|IZENER| =0 if $\sigma$-$\varepsilon$ curves do not depend on strain rate|
|:::| = 1 if $\sigma$-$\varepsilon$ curves depend on strain rate (Not available)|
|IDYN|= 1 if recrystallisation|
|:::| = 0 else|

==== Real parameters ====
^ Line 1 (2G10.0) ^^
|ACTIVE| energy activation (not used)|
|EPSRATE| epsilon rate (not used)|
//Note: This first line was implemented in the prepro but these parameters are not used in the law - probably a development that was never completed...//

=== If NPOINT=0 - repeat NTEMP times===
//Not available in ARBC2N ?//
^ Line 1 (9G10.0) ^^
|T| temperature|
|E| Young's elastic modulus at temperature T|
|Nu| Poisson's ratio at temperature T|
|ALPHA| Thermal expansion coefficient $\alpha$ at temperature T|
|SIGY1|Lower yield limit ($\sigma_{y1}$) at temperature T|
|SIGY2|Upper yield limit ($\sigma_{y2}$) at temperature T (SIGY2<SIGY1 bilinear case) \\ (parabolic case wrong very often AMH - better use bilinear case or npoint>0) |
|EPS2 | upper yield strain ($\varepsilon_2$) at temperature T|
|ET| Elasto-plastic tangent modulus (Et) at temperature T|
|COEFTQ| TAYLOR-QUINNEY's coefficient (q) at temperature T|

=== If NPOINT>0 ===
^ Lines 1:NPOINT - (G10.0)^^
|EPS| Strain for which stress will be given at each temperature|
//Repeat NTEMP times//
^Line 1 (3G10.0)^^
|T| temperature|
|Nu| Poisson's ratio at temperature T|
|ALPHA| Thermal expansion coefficient $\alpha$ at temperature T|
^Line 2:NPOINT+1^^
|SIGY|Stress for strain defined here above at temperature T|
^Line NPOINT+2^^
|COEFTQ| TAYLOR-QUINNEY's coefficient (q) at temperature T|

=== If IDYN=1 ===
Recrystallisation function $\varepsilon = C_1 * atan((ln(Z)-C_2)*C_3)+C_4$
^Line 1 (4G10.0)^^
|C1| for $\varepsilon_C$ |
|C2| :::|
|C3| :::|
|C4| :::|
^Line 2 (4G10.0)^^
|C1| for $\varepsilon_S$ |
|C2| :::|
|C3| :::|
|C4| :::|

===== 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 ====
13
==== List of state variables ====
|Q(1)|= element thickness (t) in plane stress state and generalized plane state|
|:::| =1 in plane strain state |
|:::| = circumferential strain rate ($\dot{\varepsilon}_\theta$) in axisymmetric state|
|:::| = 0 in 3D state |
|Q(2)|Current yield limit in tension; its initial value is $\sigma_{yl}$|
|Q(3)|= 0 if the current state is elastic|
|:::| = 1 if the current state is elasto-plastic|
|Q(4)| Equivalent plastic strain |
|Q(5)| Instantaneous thermal flow at the end of the step|
|Q(6)| adiabatic temperature increase due to plastic dissipation \\ since the beginning (coupled analysis) \\ since the preceding thermomechanical meeting (semi-coupled analysis) |
|Q(7)| Initial temperature|
|Q(8)| Capacity $\rho C$ (used in semi-coupled analysis)|
|Q(9)| epsilon rate for idyn=1|
|Q(10)| Ln(Z) Zener parameter for idyn=1|
|Q(11)| $T_{eq}$ equivalent temperature for izener=1|
|Q(12)| Xdyn = recrystallisation fraction for all step|
|Q(13)| Xdy2 = recrystallisation fraction for actual step|