====== MINTY3_KI ======

===== Description =====

Anisotropic elasto-plastic law based on texture for solid elements at constant temperature combined with the microstructure hardening model of C. TEODOSIU.\\

__M__icroscopic __INT__erpolated __Y__ield locus __3__D __KI__nematic

==== The model ====

This law is used for mechanical analysis of elasto-plastic anisotropic solids undergoing large strains. Isotropic or Isotropic & Kinematic (Teodosiu) hardening are assumed.

==== Files ====

Prepro: LMINTY_KI.F \\
Lagamine: MINTY3_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 \\ Texture data being read as metallurgical data, this constitutive law MUST BE THE FIRST one (IL=1)|
|ITYPE| 511|
|COMMNT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (8I5) ^^
|NINTV| Number of sub-steps used to integrate numerically the constitutive equation in a time step |
|NCRI| Number of crystals for each integration point \\ (to be in accordance with *.MET) |
|NNLP| Maximum number of steps without re-actualisation of the yield locus |
|NTEMPO| Number of TEMPO variables per integration point \\ (NTEMPO=100 for IMETH=5, 7, 9 or 11 and NTEMPO=109 for IMETH=6, 8, 10 or 12) |
|IMETH| Type of anisotropic yield surface to be used |
|:::| = -1 : No updated HILL with no discretized yield locus |
|:::| = +1 : No updated HILL with discretized yield locus |
|:::| = -2 : No updated J. WINTERS with no discretized yield locus (6 order series in stresses space) |
|:::| = +2 : No updated B. van BAEL with discretized yield locus (6 order series in strains space) |
|:::| = +3 : No updated U.L.G. yield locus |
|:::| = +4 : Updated U.L.G. yield locus |
|:::| = +5 : Taylor model without updating |
|:::| = +6 : Taylor model with updating |
|:::| = +7 : Bishop-Hill model without updating |
|:::| = +8 : Bishop-Hill model with updating |
|:::| = +9 : Visco-plastic Taylor model without updating |
|:::| = +10 : Visco-plastic Taylor model with updating |
|:::| = +11 : ALAMEL model without updating |
|:::| = +12 : ALAMEL model with updating |
|IKAP| = 0 : Analytical compliance matrix |
|:::| = 1 : Perturbation compliance matrix |
|MAXIT| Maximum number of iterations during stress integration
|NINTEPS| Number of sub-intervals per unit of delta epsilon |
|VOCE| |

Thus, the number of sub-steps = MAX(NINTV; NINTEPS*DELTA EPSILON).

^ Line 1 (1I5) ^^
|NTEO| = 0 : Classic hardening |
|:::| = 1 : Teodosiu hardening |

__If NTEO = 1 :__
^ Line 1 (3I5) ^^
|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 |
|:::| = 2 : Automatically performed in Lagamine |
|IOPTEO| = 10 : Hofferlin type. $S_L$ only activated if a path change with an elastic transition occurs |
|:::| = 20 : Alves type. $S_L$ 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 $S_L$ activated only if a sufficiently strong path change occurs $\rvert\Delta\hat{\varepsilon}_{n+1}^p-\Delta\hat{\varepsilon}_n^p\rvert\geq prec$, where $n$ and $n+1$ are two successive steps and $\Delta\hat{\varepsilon}^p$ is the increment of the local plastic strain |

__Only if KREAD=1 :__
^ Line 1 (1G10.0) ^^
|COEFK| Value to correct Teodosiu's parameters |

==== Real parameters ====

^ Line 1 (3G10.0) ^^
|E| YOUNG's elastic modulus |
|ANU| POISSON's ratio |
|THETA$\phi$| Angle between stress nodes for the yield locus interpolation |

__If NTEO=1__, the 3 following parameters can be equal to 0 :
^ Line 1 (3G10.0) ^^
|CK| Hardening factor K (see below) |
|GAMMA$\phi$| Hardening GAMMA$\phi$ coefficient ($\Gamma^{\circ}$) (see below) |
|CN| Hardening exponent (see below) |

__If IMETH=9 or 10__ : 
^ Line 1 (1G10.0) ^^
|H| Visco-plastic parameter \\ (=1/$m$ with $m$ the strain rate sensitivity parameter) |

__Only if NTEO=1 :__ Theodosius hardening parameters :
^ Line 1 (2G10.0) ^^
|CP| Polarity saturation rate |
|NP| Polarity exponent |
^ Line 1 (5G10.0) ^^
|CSD| Orientation saturation rate for $S_D$ |
|CSL| Orientation saturation rate for $\underline{\underline{S_L}}$ |
|SSAT0| Initial orientation saturation value |
|NL| Orientation exponent |
|R0| Initial yield limit |
^ Line 1 (2G10.0) ^^
|CX| Back stress saturation rate |
|XSAT0| Initial back stress saturation value |
^ Line 1 (2G10.0) ^^
|M| Influence of $\underline{\underline{S}} on kinematic ($m=0$) - isotropic ($m=1$) hardening |
|Q| $S_D-\underline{\underline{S_L}}$ balance on $X_{sat}$ |
^ Line 1 (2G10.0) ^^
|CR| Isotropic hardening : saturation rate |
|RSAT| Isotropic hardening : saturation value |

__Only if IMETH=$\pm$1__ (HILL) :
^ Line 1 : HILL's coefficient ^^
|ALPHA12| $\alpha_{12}$ |
|ALPHA13| $\alpha_{13}$ |
|ALPHA23| $\alpha_{23}$ |
|ALPHA44| $\alpha_{44}$ |
|ALPHA55| $\alpha_{55}$ |
|ALPHA66| $\alpha_{66}$ |
^ Line 2 ^^
|SIG$\phi$| Uniaxial yield stress in a reference direction |

===== 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 =====

==== Number of state variables ====

= 7 if NTEO=0 \\ = 62 if NTEO=1

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

__If NTEO=0 : __
|Q(1)| Accumulated plastic strains |
|Q(2)| GAMMA hardening parameter ($\Gamma$) |
|Q(3)$\rightarrow$Q(7)| 5 components of the plastic strain rate direction |

__If NTEO=1 :__
|Q(1)| IYIELD |
|:::| = 0 : Elastic |
|:::| = 1 : Plastic |
|Q(2)| pd (for printing) |
|Q(3:7)| 5 components of the plastic strain rate direction |
|Q(8:13)| 6 components back stress X |
|Q(20)| SD (for printing) |
|Q(21:56)| 36 components of $S_L$ |
|Q(57)| $\rvert$S$\rvert$ (for printing) |
|Q(58)| R |
|Q(59:62)| Internal parameters (for printing) |

==== Hardening form ====

__NTEO=0 :__
- For IMETH$\neq$1 : in this case, the yield surface is scaled by the critical resolved shear stress $\tau$ : \[\tau=K(\Gamma^{\circ}+\Gamma)^N\]
- For IMETH=1 : in this case, the yield surface is scaled by $\sigma_{eq}$ : \[\sigma = K(\varepsilon_0+\varepsilon)^n\]

__NTEO=1 :__
Depicted in internal report\\

File *.MET for META=4, IMETH=5 to 12.\\

For each material :
^ Line 1 (1I5) ^^
|NGLI| Number of slip system in the crystal |
^ Line 2-3-4-... (NGLI lines) (8G10.0) ^^
|NORMAL(3)| Vector $\perp$ to the slip plane |
|DIRECTION(3)| Vector direction of the slip |
|CRSS+| Critical resolved shear stress + if slip in positive direction |
|CRSS-| Critical resolved shear stress - otherwise |
^ Line 2 + NGLI and following ones (NCRI lines) (I5,4(1F13.5) ^^
|ICRI| Crystal number (increasing order from 1 to NCRI) |
|WEIGHT| Crystal weight |
|EULER| Euler angles to define crystal orientation \\ Phi_1 / Phi / Phi _2 |