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====== EP-GURSHEAR ======

===== Description =====

Law name : GTNB \\

3D elasto-plastic constitutive combining isotropic and kinematic hardening, anisotropic yield locus and nucleation, growth and coalescence of voids. Extended to shear loads. Applied on porous ductile materials.

==== The model ====

  * Mixed hardening and plastic anisotropy.
  * All the state variables are integrated implicitly.
  * Two shear extensions available.
  * Analytical computation of the consistent tangent modulus (See IJNME 2011:85:1049-1072) available.

==== Files ====

Prepro: LGUR3.F \\
Lagamine: GUR2DEXT.F, GUR3DEXT.F

===== Availability =====

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

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

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

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 364 (3D) and 365 (axi)|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (7I5) ^^
|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) |
|:::| = 2 : Analytical tangent matrix |
|NTYPHP| Type of hardening |
|:::| = 1 : Swift law : $\sigma_Y = K(\varepsilon_0+\varepsilon_M^P)^n$ |
|:::| = 2 : Voce law : $\sigma_Y = \sigma_0 + K[1-\exp(-n.\varepsilon_M^P)]$ |
|:::| = 3 : Ludwik : $\sigma_Y = \sigma_0 + K(\varepsilon_M^P)^n$ (not working)|
|NTYPEX| = 0 : Classic Gurson (only void growth) |
|:::| = 1 : GTN model (Nucleation+growth+coalescence) |
|:::| = 2 : Shear (Nahshon and Hutchinson, 2008) |
|:::| = 3 : GTN+Shear (Nahshon and Hutchinson, 2008) |
|:::| = 4 : GTN+Shear (Xue, 2008) |
|NTYCOA| = 1 : Coalescence criterion (Tvergaard & Needemen) |
|:::| = 2 : Coalescence criterion Thomason (Zhang et al. 2000) |


==== Real parameters ====

^ Line 1 (2G10.0) ^^^
|E| YOUNG's elastic modulus | = param(1,ilaw) |
|ANU| POISSON's ratio | = param(2,ilaw) |
^ Line 2 (3G10.0) ^^^
|CK| Coefficient of the hardening law ($K$) | = param(3,ilaw) |
|CW| Hardening coefficient ($\varepsilon_0$ or $\sigma_0$) | = param(5,ilaw) |
|CN| Strain hardening exponent ($n$) | = param(4,ilaw) |
^ Line 3 (2G10.0) ^^^
|CX| Parameter of the kinematic hardening ($C_X$) | = param(6,ilaw) |
|XSAT| Parameter of the kinematic hardening ($X_{sat}) \[\dot{\underline{X}} = C_X\left(X_{sat}\;\dot{\underline{\varepsilon}}^p-\underline{X}\;\dot{\bar{\varepsilon}}^p\right)\] | = param(7,ilaw) |
^ Line 4 (6G10.0) ^^^
|F| Hill's coefficients | = param(8,ilaw) |
|G| ::: | = param(9,ilaw) |
|H| ::: | = param(10,ilaw) |
|L| ::: | = param(11,ilaw) |
|M| ::: | = param(12,ilaw) |
|N| ::: | = param(13,ilaw) |
^ Line 5 (7G10.0) ^^^
|QUN| Damage parameter ($q_1$) | = param(15,ilaw) |
|QDEUX| Damage parameter ($q_2$) | = param(16,ilaw) |
|QTR| Damage parameter ($q_3$) | = param(17,ilaw) |
|F0| Initial porosity ($f_0$) | = param(18,ilaw) |
|**If NTYPEX=2,3 (Nahshon and Hutchinson, 2008)**|||
|KOMEGA| Shear parameter ($k_{\omega}$) | = param(19,ilaw) |
|TR1| Shear parameter ($T_1$) | = param(26,ilaw) |
|TR2| Shear parameter ($T_2$) | = param(27,ilaw) |
|**If NTYPEX=4 (Xue, 2008)**|||
|QQ3| Shear parameter ($k_g$) | = param(19,ilaw) |
|QQ4| Shear parameter ($q_4$) \[\dot{D}_{shear} = k_g\;f^{q_4}\;g_{\theta}\;\varepsilon_{eq}\;\dot{\varepsilon}_{eq}\]| = param(20,ilaw) |
^If NTYPEX=1,3,4^^^
^ Line 6 (5G10.0) ^^^
|FNUC| Nucleation parameter ($f_N$) | = param(21,ilaw) |
|SNUC| Nucleation parameter ($S_N$) | = param(22,ilaw) |
|ENUC| Nucleation parameter ($\varepsilon_N$) | = param(23,ilaw) |
|FCRIT| Coalescence parameter ($f_c$) | = param(24,ilaw) |
|FFAIL| Coalescence parameter ($f_U$) | = param(25,ilaw) |

Note : If FCRIT=FFAIL=0, there is no coalescence.

===== Stresses =====

==== Number of stresses ====

4 (axi) \\ 6 (3D)

==== Meaning====

__GUR3DEXT__ : \\

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)|$\sigma_{xz}$|

__GUR2DEXT__ : \\

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

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

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

27 (axi) \\ 30 (3D)

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

__GUR3DEXT__ :\\

|Q(1)| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic |
|Q(2)| Equivalent plastic strain in the matrix |
|:::| (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 $f^*$) void volume fraction |
|Q(16)| $T$ : triaxiality (without backstress) |
|Q(17)| If NTYPEX=2,3 (Nahshon and Hutchinson, 2008) :\\ = $\omega$ : Lode parameter corrected (Nielsen and Tvergaard, 2010) |
|:::| If NTYPEX=4 (Xue, 2008) : \\ = $g_{\theta}$ : Lode parameter |
|Q(18)| = $\varepsilon^p_{eqa}$ : Equivalent macroscopic plastic strain |
|Q(19)| $q$ : Effective eq. macroscopic stress |
|Q(20)| $p$ : Effective hydrostatic stress |
|Q(21)| $f$ : Void volume fraction |
|Q(22)| Porosity (Nucleation contribution) |
|Q(23)| Porosity (Growth contribution) |
|Q(24)| Porosity (Shear contribution) |
|Q(25)| $D$ : Damage variable |
|Q(26)| $\mu_{\sigma}$ : Lode parameter (Lode, 1926) |
|Q(27)| $X$ : Lode parameter (Wierzbicki et al., 2005) |
|Q(28)| $\bar{\theta}$ : Lode parameter (Bai and Wierzbicki, 2008) |
|Q(29)| $\omega$ : Lode parameter (Nahshon and Hutchinson, 2008) |
|Q(30)| $\theta_V$ : Lode parameter (Voyiadjis, 2012) |
|Q(31)| $\beta_{coal}$ if (.GE.0) then coalescence |
|Q(32)| Fcr computed by Thomason criterion |

__GUR2DEXT__ :\\

|Q(2)| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic |
|Q(3)| Equivalent plastic strain in the matrix |
|:::| (not $\varepsilon_{eqa}^p$, which is the equivalent plastic strain of the macroscopic medium) |
|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)| $f$ : (Effective $f^*$) void volume fraction |
|Q(13)| $T$ : triaxiality (without backstress) |
|Q(14)| If NTYPEX=2,3 (Nahshon and Hutchinson, 2008) :\\ = $\omega$ : Lode parameter corrected (Nielsen and Tvergaard, 2010) |
|:::| If NTYPEX=4 (Xue, 2008) : \\ = $g_{\theta}$ : Lode parameter |
|Q(15)| $\varepsilon^p_{eqa}$ : Equivalent macroscopic plastic strain |
|Q(16)| $q$ : Effective eq. macroscopic stress |
|Q(17)| $p$ : Effective hydrostatic stress |
|Q(18)| $f$ : Void volume fraction |
|Q(19)| Porosity (Nucleation contribution) |
|Q(20)| Porosity (Growth contribution) |
|Q(21)| Porosity (Shear contribution) |
|Q(22)| $D$ : Damage variable |
|Q(23)| $\mu_{\sigma}$ : Lode parameter (Lode, 1926) |
|Q(24)| $X$ : Lode parameter (Wierzbicki et al., 2005) |
|Q(25)| $\bar{\theta}$ : Lode parameter (Bai and Wierzbicki, 2008) |
|Q(26)| $\omega$ : Lode parameter (Nahshon and Hutchinson, 2008) |
|Q(27)| $\theta_V$ : Lode parameter (Voyiadjis, 2012) |
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