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====== HILL3D_KI ======
===== Description =====
Anisotropic elasto-plastic law based on HILL48 yield locus for solid elements at constant temperature.
==== The model ====
Mechanical analysis of elasto-plastic anisotropic solids undergoing large strains. \\ Classic isotropic and kinematic hardening is available. \\ Teodosiu isotropic and kinematic hardening is available.
==== Files ====
Prepro: LHIL3D_KI.F  \\
Lagamine:  HILL3D_KI.F
===== Availability =====
|Plane stress state| NO |
|Plane strain state| NO|
|Axisymmetric state| NO |
|3D state| YES |
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 66|
|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|
|NINTEPS| Number of sub-intervals per unit of delta epsilon |
|$\Rightarrow$  number of sub-steps = MAX(NINTV; NINTEPS*DELTA EPSILON)||
|IKAP| 0 = Analytical compliance matrix **not working**\\ 1 = Perturbation compliance matrix|
|MAXIT|Maximum number of iterations during stress integration|
|NTYPHP| Type of hardening law (see [[laws:HILL3D_KI#Hardening form|Hardening form]]), no action if NTEO = 1\\ = 1 (Swift  hardening) \\ = 2 (Voce hardening) \\ = 3 (Ludwick hardening) |
|INDAM| 0   (no fatigue damage computation)\\ >  1   (fatigue damage computation)|
^Line 2 (4I5)^^
|NTE0|0 Classic hardening\\ 1 Teodosiu hardening\\ 2 Teodosiu + Jauman hardening \\ 3 Ziegler hardening \\ 4 Armstrong-Frederick hardening with 2 terms|
|//if nteo = 1 or nteo= 2, then//||
|NREAD| = 1 Read the 58 state variables in .f72  (in column)|
|KREAD|= 1 Changes hardening parameters if fitted with a Von Mises yield locus and shear tests|
|IOPTEO|= 10 Hofferlin type. **S**<sub>L</sub> only activated if a path change with an elastic transition occurs.\\ = 20 Alves type. **S**<sub>L</sub> activated for any path change (continuous path change or even local path change, for example simple shear where material frame rotates)\\ = 21 Alves type but **S**<sub>L</sub> activated only if a sufficiently strong path change occurs $|\Delta\mathbf{\hat{\varepsilon}}_{n+1}^{p} - \Delta\mathbf{\hat{\varepsilon}}_{n}^{p}| \geq$ prec, where n and n+1 are two successive steps and $\Delta\mathbf{\hat{\varepsilon}}^{p}$ is the increment of the local plastic strain.|
==== Real parameters ====
^ Line 1 (6G10.0) ^^
|$E_{1}$|YOUNG's orthotropic elastic moduli|
|$E_{2}$|:::|
|$E_{3}$|:::|
|$\mbox{ANU}_{12}$|Orthotropic POISSON's ratios|
|$\mbox{ANU}_{13}$|:::|
|$\mbox{ANU}_{23}$|:::|
^Line 2 (3G10.0)^^
|$G_{12}| COULOMB's orthotropic elastic moduli|
|$G_{13}|:::|
|$G_{23}|:::|

The inverse of the orthotropic elastic matrix is defined:
$$
\begin{pmatrix} \varepsilon_{11} \\  \varepsilon_{22}\\  \varepsilon_{33}\\  \varepsilon_{12} \\ \varepsilon_{13} \\ \varepsilon_{23} \end{pmatrix} = \begin{pmatrix}  
      \frac{1}{E_{1}} & \frac{-\nu_{12}}{E_{1}} & \frac{-\nu_{13}}{E_{1}} & 0 & 0 & 0\\
      \frac{-\nu_{12}}{E_{1}} & \frac{-\nu_{12}}{E_{2}} & \frac{1}{E_{2}} & 0 & 0 & 0\\ 
      \frac{-\nu_{13}}{E_{1}} & \frac{-\nu_{23}}{E_{2}} & \frac{1}{E_{3}} & 0 & 0 & 0\\ 
      0 & 0 & 0 & \frac{1}{2G_{12}} & 0 & 0 \\
      0 & 0 & 0 & 0 & \frac{1}{2G_{13}} & 0 \\
      0 & 0 & 0  & 0 & 0 & \frac{1}{2G_{23}} 
      \end{pmatrix}
      \begin{pmatrix}
      \sigma_{11}\\
      \sigma_{22}\\
      \sigma_{33}\\
      \sigma_{12}\\
      \sigma_{13}\\
      \sigma_{23}\\
      \end{pmatrix}
$$

^Line 3 (6G10.0)^^
|F|Hill’s coefficients defining the yield locus, // param(16, ilaw) //|
|G| Hill’s coefficients defining the yield locus, // param(17, ilaw) //|
|H| Hill’s coefficients defining the yield locus, // param(18, ilaw) //|
|N| Hill’s coefficients defining the yield locus, // param(19, ilaw) //|
|L| Hill’s coefficients defining the yield locus, // param(20, ilaw) //|
|M| Hill’s coefficients defining the yield locus, // param(21, ilaw) //|
The yield locus is defined:\\

  - Classic: $f = \frac{1}{2} \underline{\sigma}^{T} \underline{\underline{H}} \ \underline{\sigma} - \sigma_{F}^{2} = 0 $\\
  - Teodosiu: $f = \sqrt{\underline{\sigma}^{T} \underline{\underline{H}} \ \underline{\sigma}^{T}} - \sigma_{F}$\\
Be careful! $\underline{\underline{H}}$ is divided by two for Teodosiu model

\[\underline{\underline{H}} = \begin{pmatrix} G+H &-H&-G&0&0&0\\-H&H+F&-F&0&0&0\\-G&-F&F+G&0&0&0\\0&0&0&2N&0&0\\0&0&0&0&2L&0\\0&0&0&0&0&2M\\ \end{pmatrix}\]

The 6 parameters of this matrix can be computed:\\
$\mathbf{\rightarrow}$ from the yield stress limits: 

$\sigma_{XX}^{yield} = \sigma_{F}^{initial}$ = initial value of $\sigma_{F}$ (see [[laws:HILL3D_KI#Hardening form|Hardening form]])\\
$\sigma_{YY}^{yield} = \sqrt{\frac{2}{H+F}}\sigma_{F}^{initial}$\\
$\sigma_{ZZ}^{yield} = \sqrt{\frac{2}{G+F}}\sigma_{F}^{initial}$\\
$\sigma_{XY}^{yield} = \sqrt{\frac{1}{N}}\sigma_{F}^{initial}$\\
$\sigma_{XZ}^{yield} = \sqrt{\frac{1}{L}}\sigma_{F}^{initial}$\\
$\sigma_{YZ}^{yield} = \sqrt{\frac{1}{M}}\sigma_{F}^{initial}$\\
with the additional condition:
  - Classic : H + G = 2\\
  - Teodosiu: H + G = 1\\

$\mathbf{\rightarrow}$  for sheet metal fitting, it is more convenient to perform 3 tensile tests and use the following fitting through the Lankford coefficients:\\
$r_{0} = \frac{H}{G}$; $r_{90} = \frac{H}{F}$; $r_{45} = \frac{2N-F-G}{2(F+G)}$\\
with these additional conditions:
  -  $H + G = 2$; $N=L=M$
  -  $H + G =1$; $N=L=M$ 
And $\sigma_{F}$ is computed as above\\ \\
$\rightarrow$ The 3 conditions to compute $N$, $L$ and $M$ with the yield stress limits can be replaced by the following using the tensile test along the $45^{\circ}$ direction:  \\
$\sigma_{45^{\circ}}^{yield} = \sqrt{\frac{8}{G+F+2N}}  \sigma_{F}^{initial}$ with the additional conditions $N=L=M$\\
It should be noticed that these 3 fittings cannot generally be fulfilled simultaneously. Indeed, each one gives enough equations to compute all the Hill coefficients.\\
However, for an isotropic material, the Von Mises criterion is obtained. The condition on the Lankford coefficients gives: \\
\[\left. \begin{array}{lll} r_{0} = 1\\ r_{90} = 1\\ r_{45} = 1 \end{array} \right\} \begin{array} 
11. & NTEO=0, NTEO=3 & \rightarrow & H=G=F=1 & N=L=M=3 \\ 
2. & NTEO=1 & \rightarrow & H=G=F=0.5 & N=L=M=1.5 \end{array}\] 
And the condition on the yield stresses gives :

\[\left. \begin{array}{ll} \sigma_{YY}^{yield} = \sigma_{ZZ}^{yield} = \sigma_{XX}^{yield} \\
\sigma_{45^{\circ} }^{yield} = \sigma_{XX}^{yield}\\ 
\end{array} 
\right\} \begin{array} 
11. & NTEO=0, NTEO=3 & \rightarrow & H=G=F=1 & N=L=M=3 \\ 
2. & NTEO=1 & \rightarrow & H=G=F=0.5 & N=L=M=1.5 \end{array}\] 
__**1. Classic or Ziegler hardening parameters (NTEO=0, 3 or 4)**__

^Line 1 (3G10.0)^^
|CK| hardening factor K (see 6.3), //param(28,ilaw)//|
|CW| hardening coefficient W<sub>0</sub>  or $\varepsilon_{0}$ or $\sigma_{0}$(see [[laws:HILL3D_KI#Hardening form|section hardening form]]), //param(29,ilaw)//|
|CN| hardening exponent n (see 6.3), //param(30,ilaw)//|
__NTEO = 0__\\
^Line 1 (2G10.0)^^
|CX|kinematic hardening saturation rate, //param(25,ilaw)//|
|XSAT|kinematic hardening saturation value, //param(26,ilaw)//|

__NTEO = 4__\\
^Line 1 (2G10.0)^^
|CX1|kinematic hardening saturation rate 1, //param(31,ilaw)//|
|XSAT1|kinematic hardening saturation value 1, //param(32,ilaw)//|
|CX2|kinematic hardening saturation rate 2, //param(33,ilaw)//|
|XSAT2|kinematic hardening saturation value 2, //param(34,ilaw)//|

__NTEO = 3__ 
^Line 1 (2G10.0)^^
|$C_{A}$| initial kinematic hardening modulus, //param(25,ilaw)//|
|$G_{A}$| rate at which the kinematic hardening modulus decrease with increasing plastic deformation, //param(26,ilaw)//|
^Line 2 (1G10.0)^^
|m|kinematic-isotropic hardening balance : **__0 or 1__** \\(m = 1 : isotropic/mixed, m = 0 : full kinematic), //param(27, ilaw)// \\ **Always** use m = 1 when NTEO=0 or NTEO=3.|

Amstrong-Frederick (classic) : $\underline{\dot{X}} = C_{X}(X_{sat} \underline{\dot{\varepsilon}}^{plastic} \ - \overline{\dot{\varepsilon}}^{plastic}. \ \underline{X})$ \\ or \\
Ziegler kinematic hardening : $\underline{\dot{X}} = C_{A}\frac{1}{\sigma_{0}} (\underline{\sigma} - \underline{X}).\dot{\overline{\varepsilon}}^{plastic} – G_{A}. \underline{X}. \dot{\overline{\varepsilon}}^{plastic}$\\
\\
**(If NTEO = 4)**: __X__ = __X__1 + __X__2\\
\\
**Parameters of fatigue law** ONLY IF INDAM > 1
^Line 1 (10G10.0)^^
|SIGL| fatigue limit|
|SIGU|stress fracture stress|
|BETA|fatigue damage parameter law|
|ADAM|fatigue damage parameter law|
|bdam|fatigue damage parameter law|
|cdam|fatigue damage parameter law|
|dlim|damage critic value|
|am0|fatigue damage parameter law|
|aprim|fatigue damage parameter law|
|a|fatigue damage parameter law|

**__2. Teodosiu hardening parameters (NTEO=1 or 2)__**
^Line 1(2G10.0)^^
|CP|polarity saturation rate, //param (22, ilaw)//|
|NP|polarity exponent, // param(23,ilaw)//|
^Line 2 (5G10.0)^^
|CSD| orientation saturation rate for $S_{D}$, //param(24,ilaw)//|
|CSL| orientation saturation rate for$ \underline{\underline{S_{L}}}$, //param(34, ilaw)//| 
|SSAT0| initial orientation saturation value, //param(28, ilaw)//|
|NL|orientation exponent, //param(29,ilaw)//|
|R0| initial yield limit, //param(31,ilaw)//|
^Line 3 (2G10.0)^^
|CX|back stress saturation rate, //param(25, ilaw)//|
|XSAT0| initial back stress saturation value, //param(26, ilaw)//|
^Line 4 (2G10.0)^^
|M|influence of __S__ on kinematic(m = 0)-isotropic(m = 1) hardening, //param(27, ilaw)//|
|Q| $S_{D}$ - ____S<sub>L</sub> balance on X<sub>sat</sub>, //param(30, ilaw)//|
^Line 5 (2G10.0)^^
|CR|isotropic hardening : saturation rate, //param(32, ilaw)//|
|RSAT| isotropic hardening : saturation value, //param(33, ilaw)//|


===== Stresses =====
==== Number of stresses ====
6
==== Meaning ====
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates. \\
|SIG(1)|$\sigma_{XX}$|
|SIG(2)|$\sigma_{YY}$|
|SIG(3)|$\sigma_{ZZ}$|
|SIG(4)|$\sigma_{XY}$|
|SIG(5)|$\sigma_{XZ}$|
|SIG(6)|$\sigma_{YZ}$|

===== State variables =====
__**1. Classic hardening parameters (NTEO = 0, 3 or 4)**__
==== Number of state variables ====
9 if NTEO = 0 or 3\\
21 if NTEO = 4
==== List of state variables (if NTEO = 0 or 3)====
|Q(1)| Yield criterion = 0 : the previous step was elastic \\= 1: the previous step was elasto-plastic|
|Q(2)| Accumulated plastic work ($W^{pl}$)|
|Q(3):Q(8)|Back stress (__X__)|
|Q(9)| Accumulated plastic equivalent strain divided by $\sqrt{2}$ ($\frac{\underline{\varepsilon^{pl}}}{\sqrt{2}}$ )|
|Q(10):Q(15)|Plastic strain tensor $\underline{\varepsilon}^p$ (6 components)|
|Q(16):Q(18)|Principal strains $\varepsilon_{I}$, $\varepsilon_{II}$, $\varepsilon_{III}$|
==== List of state variables (if NTEO = 4)====
|Q(1)| Yield criterion = 0 : the previous step was elastic \\= 1: the previous step was elasto-plastic|
|Q(2)| Accumulated plastic work ($W^{pl}$)|
|Q(3):Q(8)|Total back stress (__X__)|
|Q(9):Q(14)|First back stress (__X__1)|
|Q(15):Q(20)|Second back stress (__X__2)|
|Q(21)| Accumulated plastic equivalent strain divided by $\sqrt{2}$ ($\frac{\underline{\varepsilon^{pl}}}{\sqrt{2}}$ )|
**__State variables of fatigue law (used only if nteo = 0, 3 or 4)__**\\
__IF **INDAM.GT.1** ONLY__\\
__Number of state variables__ = 32\\
|QBCRO(1)|maximum x-component of deviatoric stress over a fatigue cycle|
|QBCRO(2)| maximum y-component of deviatoric stress over a fatigue cycle |
|QBCRO(3) |maximum z-component of deviatoric stress over a fatigue cycle |       
|QBCRO(4)|maximum xy-component of deviatoric stress over a fatigue cycle|
|QBCRO(5)|maximum xz-component of deviatoric stress over a fatigue cycle|            
|QBCRO(6)| maximum yz-component of deviatoric stress over a fatigue cycle|
|QBCRO(7)| minimum x-component of deviatoric stress over a fatigue cycle|
|QBCRO(8)| minimum y-component of deviatoric stress over a fatigue cycle |
|QBCRO(9) | minimum z-component of deviatoric stress over a fatigue cycle| 
|QBCRO(10)|minimum xy-component of deviatoric stress over a fatigue cycle|          
|QBCRO(11)|minimum xz-component of deviatoric stress over a fatigue cycle | 
|QBCRO(12)|minimum yz-component of deviatoric stress over a fatigue cycle |    
|QBCRO(13)| maximum hydrostatic stress over a fatigue cycle |
|QBCRO(14) |maximum of V.M stress over a fatigue cycle|
|QBCRO(15) | number of fatigue cycles  |
|QBCRO(16)| cumul of time increment|
|QBCRO(17) | mean stress (3D case) over a fatigue cycle|
|QBCRO(18)| damage variable|
|QBCRO(19)|number of cycles|
|QBCRO(20)|minimum hydrostatic stress over a fatigue cycle |
|QBCRO(21) |sines criterion |
|QBCRO(22)|variable used for the damage computation (multi-blocks case)|
|QBCRO(23)| normalised damage variable|
|QBCRO(24)| minimum triaxiality factor over a fatigue cycle |
|QBCRO(25)| maximum triaxiality factor over a fatigue cycle|
|QBCRO(26)|equivalent stress amplitude (AIIa) over a fatigue cycle|
|QBCRO(27)|maximum triaxiality function over a fatigue cycle|
|QBCRO(28)|maximum damage equivalent stress over a fatigue cycle|
|QBCRO(29)| number of cycle to failure for the block loading |
|QBCRO(30) |x-component of the gradient of equivalent stress amplitude (AIIa)|
|QBCRO(31) |y-component of the gradient of equivalent stress amplitude (AIIa)|
| QBCRO(32)| z-component of the gradient of equivalent stress amplitude (AIIa)|

==== Hardening form ====
**if (NTYPHP = 0)**, The user’s parameters CK, CW and CN are respectively $K^{W}$ , $W_{0}$ and $n^{W}$ of the hardening equation:\\
$\sigma_{F} = K^{W} (W_{0} + W^{pl})^{n^{W}}$\\
$ K^{W}$ = //param(22,ilaw)// \\
$ W_{0}$ = //param(23,ilaw)// \\
$ n^{W}$ = //param(24,ilaw)// \\

**if  (NTYPHP = 1)**, The user’s parameters CK, CW and CN are respectively K, $\varepsilon_{0}$ and n of the hardening equation:\\
$\sigma_{F} = K(\varepsilon_{0} +m. \varepsilon^{pl})^{n}$\\
K = //param(24,ilaw)// \\
$\varepsilon_{0}$ = //param(29,ilaw)// \\
n = //param(30,ilaw)// \\
m = //param(27,ilaw)// \\

**if (NTYPHP = 2)**, The user’s parameters CK, CW and CN are respectively K, $\sigma_{0}$ and n of the hardening equation:\\
$\sigma_{F} = \sigma_{0} + K(1-exp(-n.\varepsilon^{pl}.m))$\\
K = //param(24,ilaw)// \\
$\sigma_{0}$ = //param(29,ilaw)// \\
n = //param(30,ilaw)// \\

**if (NTYPHP = 3)**, The user’s parameters CK, CW and CN are respectively K, $\sigma_{0}$ and n of the hardening equation:\\
$\sigma_{F} = \sigma_{0} + K.(\varepsilon^{pl})^{n}$\\
K = //param(24,ilaw)// \\
$\sigma_{0}$ = //param(29,ilaw)// \\
n = //param(30,ilaw)// \\

**__2. Teodosiu hardening state variables (NTEO = 1)__** \\
For details see internal report\\
==== Number of state variables ====
54
==== List of state variables ====
|Q(1)| Yield criterion \\ = 0: the previous step was elastic \\ = 1: the previous step was elasto-plastic|
|Q(2)| Accumulated plastic work ($W^{pl}$)|
|Q(3):Q(8)|Back stress (__X__)|
|Q(9)| Accumulated plastic equivalent strain ($\underline{\varepsilon}^{pl}$)|
|Q(10:15)| Polarity vector (__P__)|
|Q(16)| Orientation (S<sub>D</sub>)|
|Q(17:52)|Orientation matrix ($\underline{\underline{S}}_{D}$)|
|Q(53)|Current elastic limit ($\sigma_{F}$)|
|Q(54)|Isotropic hardening (R )|
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