====== EP-ARB ======

===== Description =====

Elasto-plastic constitutive law for solid elements at constant temperature.

==== The model ====

This law is used for a mechanical analysis of elasto-plastic isotropic solids undergoing large strains. Mixed hardening is assumed.

==== Files ====

Prepro: LARB.F \\

===== Availability =====

|Plane stress state| YES |
|Plane strain state| YES|
|Axisymmetric state| YES|
|3D state| YES|
|Generalized plane state| YES|

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

==== Parameters defining the type of constitutive law ====

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 50|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (6I5) ^^
|NINTV| number of sub-steps used to integrate numerically the constitutive equation in a time step.|
|NPOINT| = 0 if the law is described by Figure 1 (bilinear or one parabola)|
|:::| = NN if the law is described by points (a lot of linear segments) (figure 2)|
|:::| = -1 if the law is described by $\sigma=C.\varepsilon^n$ (figure 3)|
|ICBIF| = 1 if there is a comp. bifurcation criterion (plane strain analysis only) |
|:::| = 0 otherwise (no comp.)|
|ITHICK| = 1 if the initial thickness $\neq$ ONE|
|:::| = 0 if the initial thickness = ONE, to be introduced for plane stress state only| 
|NIANA| Type of problem |
|INDAM| $\geq$ 1 if fatigue damage computes (see section 4.4)|
|:::| = 0 if there is no damage|


==== Real parameters ====
=== If NPOINT=0 ===
^ Line 1 (7G10.0) ^^
|E| YOUNG's elastic modulus|
|ANU| POISSON's ratio|
|SIGY1| lower yield limit ($\sigma_{y1}$)|
|SIGY2| upper yield limit ($\sigma_{y2}$) \\ If SIGY2$\leq$SIGY1 : bilinear law)|
|EPS2| upper yield strain ($\varepsilon_2$)|
|ET| elasto-plastic tangent modulus (E_t)|
|ECROU| = 0 for isotropic hardening|
|:::| = 1 for kinematic hardening|
|:::| $\in$[0,1] for mixed hardening|
{{  :laws:eparb1.png?400  |}}

=== If NPOINT>0 ===
^ Line 1 (2G10.0) ^^
|ANU| POISSON's ratio|
|ECROU| = 0 for isotropic hardening |
|:::| = 1 for kinematic hardening |
|:::| $\in$[0,1] for mixed hardening |
^Line 2 (2G10.0) - Repeat NPOINT times^^
|EPS| true strain [-] |
|SIG| true stress [MPa] |
{{  :laws:eparb2.png?400  |}}

=== If NPOINT=-1 ===
 
^ Line 1 (5G10.0) ^^
|E| YOUNG's elastic modulus |
|ANU| POISSON's ratio |
|ECROU| = 0 for isotropic hardening |
|:::| = 1 for kinematic hardening |
|:::| $\in$[0,1] for mixed hardening |
|PARAC| parameter C in $\sigma=C.\varepsilon^n$ |
|PARAN| parameter n in $\sigma=C.\varepsilon^n$ |
{{  :laws:eparb3.png?400  |}}
**N.B.** : Possibility of a ($\bar{\sigma},\bar{\varepsilon}$) curve increasing or decreasing. \\

=== If ITHICK = 1 ===

^1 Line (G10.0) ^^
|THICK| Initial thickness \\ :!: Doesn't work with [[elements:jet2d|JET2D]] element |


==== Fatigue Continuum damage parameters ====

^ Line 1 (Only if INDAM$\neq$0 : 8G10.0) ^^
|SIGL| fatigue damage yield stress |
|SIGU| ultimate tensile stress |
|BETA| fatigue damage parameter |
|ADAM| fatigue damage parameter |
|BDAM| fatigue damage parameter |
|M0| fatigue damage parameter |
|DLIM| crack limit damage |
|PERIO| period of loading |

About INDAM : there is currently three ways to compute fatigue damage. The choice of one or another method depends on INDAM parameter value. \\

  * If INDAM = 1, fatigue is computed in __real time__ during Lagamine simulation : after each period of PERIO sec., the DAM state variable (= Q(55) ) is incremented. Calculus of fatigue increment is based on stresses SIG «imposed» (!!! cycle !!!), in each element, during last period of time. This way is interesting only if laws coupled with damage or often varying loading are used. For a large number of steady cycles without coupling (like in H.C.F.), see method n°2 or 3.
  * If INDAM = 2 or 3, fatigue is computed by __post-processing__. After each cycle of PERIO sec., where fatigue increment has been calculated as usual, the program automatically computes damage evolution for N cycles (just identical to the last one). If simulation has not ended, it is then possible to impose another cycle and to restart the damage computation from its preceding value. If many cycles are imposed (INDAM= 2), we must create an ASCII file : //namdat.hcf//, where the $i^{th}$ line gives the number $N_i$ (format I10) of times to repeat the cycle between t = (i-1)*PERIO and i*PERIO sec. It is also possible to impose only one cycle (INDAM= 3) and to directly compute life duration $N_F$ (Q(56)) for each element, just as if cycle were maintained until rupture: file //namdat.hcf// is then no more necessary. A direct analytical method is then used to calculate life duration. To conclude : method n°2 is fast and appropriate for a not too long sequence of steady states (=> cycle n°i), but is useless for only one state. Method n°3 is the best for a unique cycle whose life duration is sought. //Remark : Method n°2 is still in development and cannot be certificated to be 100 percent operational.// 


===== 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 ====


20 for the 3D state \\ 30 for EP \\ 18 for the other cases


==== List of state variables ====


|Q(1)|THICK | Element thickness (t) in plane stress state |
|:::|:::| = 1 in plane strain state |
|:::|:::| circumferential strain rate ($\dot{\varepsilon_{\theta}}$) in axysimmetrical state |
|:::|:::| = 0 in 3D state |
|:::|:::| element thickness (t) in generalized plane state |
|Q(2)|AK2| current yield limit in tension \\ its initial value is $\sigma_{y1}$ |
|Q(3)|YIELD| = 0 if the current state is elastic |
|:::|:::| = 1 if the current state is elasto-plastic |
|Q(4)|EPSB| equivalent plastic strain ($\bar{\varepsilon}^p$) |
|Q(5)|DISSIP| plastic work per unit volume |
|Q(6)|TWORK| total strain energy per unit volume (elastic + plastic) |
|Q(7) $\rightarrow$ Q(12)|| fracture criteria |
|Q(13)|AKE| equivalent stress for mixed hardening |
|Q(14)|AF1I| back stress for kinematic and mixed hardening |
|Q(15)|AF2I | ::: |
|Q(16)|AF3I | ::: |
|Q(17)|AF12I | ::: |
|Q(18)|| equivalent stress ($\bar{\sigma}$) |
|Q(18)|AF13I | only for 3D state |
|Q(19)|AF23I | ::: |
|Q(20)|| equivalent stress ($\bar{\sigma}$) |
|Q(19) $\rightarrow$ Q(30)|| modulus for the analysis of bifurcation (only for EP) |
|Q(31)|| indicator type VILOTTE n°1 |
|Q(32)|| unity of $\varepsilon_{eq}$ |
|Q(33)|| age of the point of the mesh |
|:::|| = 1 if the element must be re-meshed|
|:::|| = 0 otherwise |
|Q(37)|SMAX1| maximum of deviator $S_{xx}$ |
|Q(38)|SMAX2 | maximum of deviator $S_{yy}$ |
|Q(39)|SMAX3| maximum of deviator $S_{zz}$ |
|Q(40)|SMAX12| maximum of deviator $S_{xy}$ |
|Q(41)|SMAX13 | maximum of deviator $S_{xz}$ |
|Q(42)|SMAX23 | maximum of deviator $S_{yz}$ |
|Q(43)|SMIN1 | minimum of deviator $S_{xx}$ |
|Q(44)|SMIN2 | minimum of deviator $S_{yy}$ |
|Q(45)|SMIN3 | minimum of deviator $S_{zz}$ |
|Q(46)|SMIN12 | minimum of deviator $S_{xy}$ |
|Q(47)|SMIN13 | minimum of deviator $S_{xz}$ |
|Q(48)|SMIN23 | minimum of deviator $S_{yz}$ |
|Q(49)|CUMTR | sum of stress trace |
|Q(50)|NCAL| sum of number of step |
|Q(51)|SIGEQM | maximum equivalent stress |
|Q(52)|CYCI | number of cycle |
|Q(53)|TIME | time |
|Q(54)|DAM | damage variable |
|Q(55)| NFAT | number of fatigue cycles before failure (initialized to 0 and calculated when DAM reaches the value of 1) |
|Q(56)|SIGHM | maximum of hydrostatic stress |