====== EP-GUR ======

===== Description =====

Elasto-plastic constitutive law for porous ductile metals at constant temperature, using the GURSON model.

==== The model ====

This law is used for mechanical analysis of elasto-plastic isotropic porous ductile solids undergoing large strains. Combined isotropic and kinematic hardening is assumed.

==== Files ====

Prepro: LGUR.F \\

===== Availability =====

|Plane stress state| NO |
|Plane strain state| YES|
|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| 80|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (2I5) ^^
|NINTV| = 0 : NINTV will be calculated in the law |
|:::| > 0 : Number of sub-steps used to integrate numerically the constitutive equation in a time step |
|ICBIF| = 0 : Nothing |
|:::| = 1 : RICE bifurcation criterion |

==== Real parameters ====

^ Line 1 (6G10.0) ^^
|E| YOUNG's elastic modulus |
|ANU| POISSON's ratio |
|SIGY1| Initial yield limit ($\sigma_{y1}$) |
|BB| Parameter of hardening (0$\leq$BB$\leq$1) |
|:::| = 0 : Pure kinematic hardening |
|:::| = 1 : Pure isotropic hardening |
|DN| > 0 : Strain hardening exponent ($n$) for piecewise power law |
|:::| $\leq$ 0 : Plastic tangent modulus ($-E_t$) for bi-linear law |
|FO| Initial void volume fraction |
^ Line 2 (7G10.0) ^^
|FC| Parameter of material ($f_C$) |
|FF| Parameter of material ($f_F$) |
|RFS| Parameter of material ($f_F^*/f_F$) |
|$Q_{UN}$| Parameter of material ($Q_1$) |
|$Q_{DEUX}$| Parameter of material ($Q_2$) |
|$Q_{TR}$| Parameter of material ($Q_3$) |
|$\alpha_n$| Parameter for calculation of NINTV (if NINTV=0 and $\alpha_n$=0 : its value will be $1.0\times10^{-4}$) |
^ Line 3 (7G10.0) ^^
|EPN| Mean strain for nucleation ($\varepsilon_N$) |
|SPN| Mean stress for nucleation ($\sigma_N$) |
|FN| Volume fraction of void nucleating particles ($f_N$) |
|SEP| Corresponding standard deviation for strain ($S_{\varepsilon}$) |
|SSIG| Corresponding standard deviation for stress ($S_{\sigma}$) |
|DETEP| Failure parameter of material ($\Delta_{\varepsilon}$) |
|DNCEPS| Central parameter of material for nucleation and failure |
|:::| = 0 : Nucleation of new voids and failure of material are not considered |
|:::| = $\pm$1 : Nucleation is controlled by the plastic strain |
|:::| = $\pm$2 : Nucleation is controlled by the maximum normal stress |
|:::| = $\pm$3 : Both strain controlled and stress controlled nucleation take place |
|:::| = -4 : Nucleation of new voids is not considered |
|:::| > 0 : Failure of material is not considered |

===== Stresses =====

==== Number of stresses ====

==== 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)|$\sigma_{xy}$|
|SIG(4)|$\sigma_{zz}$|

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

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

26

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

|Q(1)| Element thickness (t) in plane stress state |
|:::| = 1 : Plane strain state |
|:::| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state |
|:::| = 0 : 3-D state |
|:::| Element thickness (t) in generalized plane state |
|Q(2)| Current yield limit in tension (its initial value is $\sigma_{y1}$) | 
|Q(3)| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic |
|Q(4)| = $\left(\varepsilon_M^p\right)_{max}$ : Current maximum value of equivalent plastic strain of matrix material (its initial value is $\varepsilon_N$) |
|Q(5)| Current maximum value of $\sigma_m+\sigma_K^K/3$ (its initial value is $f_N$) |
|Q(6)| Current void volume fraction (f) (its initial value is $f_o$) |
|Q(7)| Equivalent plastic strain of matrix material ($\varepsilon_M^P$) |
|Q(8)| A |
|Q(9)| B |
|Q(10)| C |
|Q(11)| Yield surface centre for $\sigma_{xx}$ (its initial value is 0) |
|Q(12)| Idem for $\sigma_{yy}$ |
|Q(13)| Idem for $\sigma_{zz}$ |
|Q(14)| Idem for $\sigma_{xy}$ |
|Q(15)$\rightarrow$Q(26)| Modules for the analysis of bifurcation |