====== EP-GTNB2 ======

===== Description =====

3D elasto-plastic constitutive combining isotropic and kinematic hardening, anisotropic yield locus, nucleation 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=internal variable)
  * Homogenization procedure to incorporate Bouaziz’s experimental observations of void nucleation and growth. It is based on the replacement of the all voids by one equivalent void with the same volume embedded in the matrix.
  * It finds a global (nucleated and Growth) value of the void volume fraction.

==== Files ====

Prepro: LGUR3.F \\
Lagamine: GUR3DANI/GUR2DANI.F

===== Availability =====

|Plane stress state| NO |
|Plane strain state| YES |
|Axisymmetric state| YES (GUR2DANI)|
|3D state| YES|
|Generalized plane state| NO |

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

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

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 358 (axi) and 361 (3D)|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (I5) ^^
|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 (2G10.0/3G10.0/2G10.0/3G10.0/7G10.0/2G10.0) ^^
|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 ($C_X$) : \[\dot{\underline{X}} = C_X\left(s\;\dot{\underline{\varepsilon}}^p-\underline{X}\;\bar{\dot{\varepsilon}}^p\right)\] |
^ 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 (7G10.0) ^^
|QUN| Damage parameter ($q_1$) |
|QDEUX| Damage parameter ($q_2$)|
|QTR| Damage parameter ($q_3$)|
|F0| Initial porosity |
|EPSN0| Related to Bouaziz equation of void nucleation : $\varepsilon_n=\varepsilon_{n0}\;\exp(-T)$ |
|AA0| Constant equal to 5000 voids/mm$^3$ for calculation of N (number of nucleated voids per mm$^3$) |
|RR0| Initial radius of a singular void |
^ Line 6 (2G10.0) ^^
|AB| Thomason parameter (not used) |
|BETA| Thomason parameter (not used) |

===== Stresses =====

==== Number of stresses ====

4 (axi) \\ 6 (3D)

==== Meaning ====

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

GUR3DANI :
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{zz}$|
|SIG(4)|$\sigma_{xy}$|
|SIG(5)|$\sigma_{yz}$|
|SIG(6)|$\sigma_{xz}$|

GUR2DANI :
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{zz}$|
|SIG(4)|$\sigma_{xy}$|

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

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

22 (axi) \\ 25 (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 |
|Q(17)| The equivalent macroscopic plastic strain |
|Q(18)| = N : Number of nucleated voids |
|Q(19)| The volume of the equivalent single void |
|Q(20)| The updated void radius computed by the integration equation |
|Q(21)| $q_2$ |
|Q(22)| $q_1=1.5\;q_2$ |
|Q(23)| $q_3=(1.5\;q_2)^2$ |
|Q(24)| ln=(RT/RT0) where RT and RT0 are are the current and initial radius of the single equivalent porosity |
|Q(25)| RT radius of the single void cavity at the end of the time increment |

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 |
|Q(14)| The equivalent macroscopic plastic strain |
|Q(15)| N : Number of nucleated voids |
|Q(16)| The volume of the equivalent single void |
|Q(17)| The updated void radius computed by the integration equation |
|Q(18)| $q_2$ |
|Q(19)| $q_1=1.5\;q_2$ |
|Q(20)| $q_3=(1.5\;q_2)^2$ |
|Q(21)| ln=(RT/RT0). where RT and RT0 are are the current and initial radius of the single equivalent porosity |
|Q(22)| RT radius of the single void cavity at the end of the time increment |

Qtrial : anisotropic equivalent shifted stress with HILL criterion calculation.