====== MIPAY3 ======

===== Description =====

Anisotropic elasto-plastic law based on texture for solid elements at constant temperature.\\

__MI__croscopic __PA__rt __Y__ield law __3__D

==== The model ====

This law is used for mechanical analysis of elasto-plastic anisotropic solids undergoing large strains. Isotropic hardening is assumed.

==== Files ====

Prepro: LMIPAY.F \\
Lagamine: MIPAY3.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| 501|
|COMMNT| 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 |
|NCRI| Number of crystals for each integration point (400 or 1600) (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 (=65) |
|IMETH| Type of anisotropic yield surface to be used |
|:::| = -1 : No updated HILL without discretized yield locus |
|:::| = +1 : No updated HILL with discretized yield locus |
|:::| = -2 : No updated J. WINTERS no discretized yield locus (6 order series in stresses space) |
|:::| = +2 : No updated B. van BAEL 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 plasticity with no yield locus |
|IKAP| = 0 : Analytical compliance matrix (available only for IMETH=-2) |
|:::| = 1 : Perturbation compliance matrix (general case) |
|:::| = 2 : Compliance matrix always the elastic one (available only for IMETH =-5) |

==== Real parameters ====

^ Line 1 (5G10.0) ^^
|E| YOUNG's elastic modulus |
|ANU| POISSON's ratio |
|CK| Hardening factor K (see below) |
|GAMMA$\phi$| Hardening GAMMA$\phi$ coefficient ($\Gamma^{\circ}$) (see below) |
|CN| Hardening exponent (see below) |
^ Line 2 (6G10.0) ^^
|THETA$\phi$| = 10 |
|THETAMAX| = 85 |
|CMIN| = 0,2 |
|FLIMIT| = 0,05 |
|SLIMIT| = 0,025 |
|TOL| = 1.10-7 |

The last 6 parameters are temporary values. See report 11 of research Micro-Macro (RW 2748).

__Only if IMETH=1__ (HILL) :
^ Line 1 (6G10.0) : HILL's coefficient ^^
|ALPHA12| $\alpha_{12}$ |
|ALPHA13| $\alpha_{13}$ |
|ALPHA23| $\alpha_{23}$ |
|ALPHA44| $\alpha_{44}$ |
|ALPHA55| $\alpha_{55}$ |
|ALPHA66| $\alpha_{66}$ |
^ Line 2 (1G10.0) ^^
|SIG$\phi$| Uniaxial yield stress in a reference direction |

===== Stresses =====

==== Number of stresses ====

6 for 3D state

==== 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)|$\sigma_{yz}$|

===== State variables =====

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

2

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

|Q(1)| Yield indicator |
|:::| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic |
|Q(2)| GAMMA hardening parameter ($\Gamma$) |

==== Hardening form ====

- 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 $\sigma_{eq}$ : \[\sigma = K(\varepsilon_0+\varepsilon)^n\]

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

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 |