====== EP-GTNB ======

===== Description =====

3D elasto-plastic constitutive combining isotropic and kinematic hardening, anisotropic yield locus and growth of voids. Rupture criterion applied on porous ductile materials (GURSON model).

==== The model ====

  * Mixed hardening and plastic anisotropy.
  * Reproducing of the damage evolution (which is predicted by the experiments) by fitting the parameter q2 (GTNB q2=constant, GTNB2 q2=state variable)
  * All the state variables are integrated implicitly.
  * Analytical computation of the consistent tangent modulus (See IJNME 2011:85:1049-1072).

==== Files ====

Prepro: LGUR3.F \\
Lagamine: GUR3DCLAS/GUR2ACLAS.F

===== Availability =====

|Plane stress state| NO |
|Plane strain state| NO |
|Axisymmetric state| YES (GUR2ACLAS)|
|3D state| YES|
|Generalized plane state| NO |

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

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

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 357 (axi) and 360 (3D)|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (3I5) ^^
|NINTV| = 0 : Not used |
|IKAP| = 0 : Tangent matrix by perturbation (through LOAX3D) |
|:::| = 1 : Tangent matrix by perturbation (calculated within the law) |
|NTYPHP| Type of hardening |
|:::| = 1 : Swift law : $\sigma_Y = K(\varepsilon_0+\bar{\varepsilon}^p)^n$ |
|:::| = 2 : Voce law : $\sigma_Y = \sigma_0 + K[1-\exp(-n.\bar{\varepsilon}^p)]$ |
|:::| = 3 : Ludwik : $\sigma_Y = \sigma_0 + K(\bar{\varepsilon}^p)^n$ |

==== Real parameters ====

^ Line 1 (2G10.0) ^^
|E| YOUNG's elastic modulus |
|ANU| POISSON's ratio |
^ Line 2 (3G10.0) ^^
|SIGO| Coefficient of the hardening law ($K$) |
|DN| Strain hardening exponent ($n$) |
|EPS0| Hardening coefficient ($\varepsilon_0$ or $\sigma_0$) |
^ Line 3 (2G10.0) ^^
|HKIN| First parameter of the kinematic hardening ($C_X.X_{sat}$) |
|HNL| Second parameter of the kinematic hardening ($C_X$) : \[\dot{\underline{X}} = C_X\left(X_{sat}\;\dot{\underline{\varepsilon}}^p-\underline{X}\;\bar{\dot{\varepsilon}}^p\right)\] |
^ Line 4 (3G10.0) ^^
|R0| Lankford coefficient in the direction 0° |
|R45| Lankford coefficient in the direction 45° |
|R90| Lankford coefficient in the direction 90° |
^ Line 5 (4G10.0) ^^
|QUN| Damage parameter ($q_1$) |
|QDEUX| Damage parameter ($q_2$)|
|QTR| Damage parameter ($q_3$)|
|F0| Initial porosity |

===== Stresses =====

==== Number of stresses ====

4 (axi) \\ 6 (3D)

==== Meaning ====

The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates. \\

GUR3DCLAS :
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{zz}$|
|SIG(4)|$\sigma_{xy}$|
|SIG(5)|$\sigma_{yz}$|
|SIG(6)|$\sigma_{xz}$|

GUR2ACLAS :
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{zz}$|
|SIG(4)|$\sigma_{xy}$|

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

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

13 (axi) \\ 16 (3D)

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

GUR3DCLAS :
|Q(1)| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic |
|Q(2)| Equivalent plastic strain in the matrix $\sigma_Y = K(\varepsilon_0+\varepsilon_m^p)^n$ |
|:::| (not $\varepsilon_{eqa}^p$, which is the equivalent plastic strain of the macroscopic medium) |
|Q(3)$\rightarrow$Q(8)| The six components of the macroscopic plastic strain : $\underline{\varepsilon}_{11}^p$, $\underline{\varepsilon}_{22}^p$, $\underline{\varepsilon}_{33}^p$, $\underline{\varepsilon}_{12}^p$, $\underline{\varepsilon}_{13}^p$, $\underline{\varepsilon}_{23}^p$ |
|Q(9)$\rightarrow$Q(14)| The six components of the macroscopic backstress : $X_{11}$, $X_{22}$, $X_{33}$, $X_{12}$, $X_{13}$, $X_{23}$ |
|Q(15)| f : the void porosity fraction |
|Q(16)| T: triaxiality |

Qtrial : anisotropic equivalent shifted stress with HILL criterion calculation.\\

GUR2ACLAS :
|Q(2)| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic |
|Q(3)| Equivalent plastic strain in the matrix $\sigma_Y = K(\varepsilon_0+\varepsilon_m^p)^n$ |
|Q(4)$\rightarrow$Q(7)| The four components of the macroscopic plastic strain : $\underline{\varepsilon}_{11}^p$, $\underline{\varepsilon}_{22}^p$, $\underline{\varepsilon}_{33}^p$, $\underline{\varepsilon}_{12}^p$ |
|Q(8)$\rightarrow$Q(11)| The four components of the macroscopic backstress : $X_{11}$, $X_{22}$, $X_{33}$, $X_{12}$ |
|Q(12)| f : the void porosity fraction |
|Q(13)| T: triaxiality |