====== EP-GTNPHY ======

===== Description =====

3D elasto-plastic constitutive combining isotropic and kinematic hardening, anisotropic yield locus, nucleation growth and coalescence of voids. Physically-based models. Applied on porous ductile materials.

==== The model ====

  * Mixed hardening and plastic anisotropy.
  * $q_2$ is a state variable.
  * Different physically-based nucleation, growth and coalescence of voids.
  * Classical nucleation and coalescence modeling by Tvergaard and Needleman (GTN model).
  * Consistent tangent matrix calculated using the perturbation method.

==== Files ====

Prepro: LGUR3.F \\
Lagamine: GTNB3DPHY.F, GUR3DBF.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| 363|
|COMMENT| 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 |
|NINTEPS| Number of sub-intervals per unit of delta epsilon |
|:::| Number of sub-steps = MAX(NINTV; NINTEPS*DELTA EPSILON) |
|MAXIT| Maximum number of iterations in the N-R scheme |
|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 : $\sigma_Y = \sigma_0 + K(\bar{\varepsilon}^p)^n$ |
|NTYCOA| Void nucleation model |
|:::| = 1 : Classic GTN |
|:::| = 2 : Bouaziz, Maire approach |
|:::| = 3 : Landron approach |
|NTYGRO| Void growth model |
|:::| = 1 : Classic GTN |
|:::| = 2 : Bouaziz, Maire approach |
|NTYCOA| Void coalescence model |
|:::| = 1 : Classic GTN |
|:::| = 2 : Brown and Embury (not ready) |
|:::| = 3 : Thomason |

==== Real parameters ====

^ Line 1 (2G10.0) ^^
|E| YOUNG's elastic modulus |
|ANU| POISSON's ratio |
^ Line 2 (3G10.0) ^^
|CK| Coefficient of the hardening law ($K$) |
|CN| Strain hardening exponent ($n$) |
|CW| Hardening coefficient ($\varepsilon_0$ or $\sigma_0$) |
^ Line 3 (2G10.0) ^^
|CX| First parameter of the kinematic hardening ($C_X.X_{sat}$) |
|XSAT| 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| Initial damage parameter ($q_1$) |
|QDEUX| Initial damage parameter ($q_2$)|
|QTR| Initial damage parameter ($q_3$)|
|F0| Initial porosity ($f_0$)|
^Void nucleation parameters (3G10.0)^^
|**If NTYNUC = 1** ||
|FNUC| Nucleation parameter ($f_N$) |
|SNUC| Nucleation parameter ($S_N$) |
|ENUC| Nucleation parameter ($\varepsilon_N$) |
|**If NTYNUC = 2**||
|AA0| $A$ : Number of nucleated voids per mm$^3$ |
|EPSN0| $\varepsilon_{N0}$ : Critical strain for pure shear loading |
|**If NTYNUC = 3** ||
|BB0| Material parameter ($B$) |
|NN0| Initial number of voids per unit volume ($N_0$) |
|SCRITM| Critical shear stress ($\sigma_C$) |
^Void growth parameters (3G10.0)^^
|**If NTYGRO = 2 or NTYGRO = 3**||
|RRi0| = $R_0^i$ : Initial mean radius |
^Void coalescence parameters (3G10.0)^^
|**If NTYCOA = 1** ||
|FCRIT| Coalescence parameter ($f_c$) |
|FFAIL| Coalescence parameter ($f_U$) |

===== 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)|$\sigma_{yz}$|
|SIG(6)|$\sigma_{xz}$|


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

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

25

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

|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)| $f^*$ : (Effective) void volume fraction |
|Q(16)| $T$ : triaxiality (without backstress) |
|Q(17)| $\omega$ : Lode angle (Nahshon and Hutchinson, 2008) (not used) |
|Q(18)| $\varepsilon^p_{eqa}$ : Equivalent macroscopic plastic strain |
|Q(19)| $q$ : Effective eq. macroscopic stress |
|Q(20)| $p$ : Hydrostatic stress |
|Q(21)| $f$ : Void volume fraction |
|Q(22)| $N$ : Number of nucleated voids |
|Q(23)| $f_N$ : Porosity (Nucleation contribution) |
|Q(24)| $R$ : The updated void radius |
|Q(25)| $\ln(R_T/T_{T0})$ where $R_T$ and $R_{T0}$ are are the current and initial radius of the single equivalent porosity |
|Q(26)| $R_T$ : Radius of the single void cavity at the end of the time increment |
|Q(27)| $q_2$ | 
|Q(28)| $q_1=1.5\; q_2$ | 
|Q(29)| $q_3 = q_1^2$ |

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