====== EP-ELLI======
===== Description =====
Elliptic elasto-plastic constitutive law for solid elements at constant temperature 
 ==== The model ====
Mechanical analysis of elasto-plastic isotropic porous media undergoing large strains.  Dilatancy is included through an elliptic yield surface relating the mean stress and the von MISES equivalent stress.  Isotropic hardening is assumed.
==== Files ====
Prepro: LELLI.F  \\
Lagamine: ELLI2EA.F, ELLI3D.F
===== Availability =====
|Plane stress state| NO |
|Plane strain state| YES|
|Axisymmetric state| YES |
|3D state| NO (?)|
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 73|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^Line 1 (9I5)^^
|NINTV| > 0: number of sub-steps used to integrate numerically the constitutive equation in a time step. \\ = 0: NINTV will be calculated in the law with DIV=$5*10^{-3}$|
|ISOL| = 0: use of total stresses in the constitutive law \\ ≠ 0: use of effective stresses in the constitutive law. See [[appendices:a8|Appendix 8]]|
|IELA| = 0 Linear elasticity \\ > 1 Non linear elasticity|
|ILODEF| Shape of the yield surface in the deviatoric plane \\ = 1: circle in the deviatoric plane \\ = 2: smoothed irregular hexagon in the deviatoric plane|
|ILODEG| Not used : Associated plasticity|
|ICBIF|Computation indice of bifurcation criterion\\ = 0: non computed \\ = 1: computed (plane strain state only)|
|IECROU| = 2 Volumetric strain hardening (including softening) |
|KMETH| = 2 Actualised VGRAD integration \\ = 3 Mean VGRAD integration (Default value)|
|IPCONS| = 0 Definition of pre-consolidation pressure \\ ≠ 0 Definition of OCR|

==== Real parameters ====
^Line 1 (7G10.0)^^
|E_PAR1| First elastic parameter |
|E_PAR2| Second elastic parameter|
|E_PAR3| Third elastic parameter|
|E_PAR4| Fourth elastic parameter|
|HARD| Hardening parameter|
|PCONS0| Preconsolidation pressure (If IPCONS=0)|
|OCR| Over Consolidation Ratio (If IPCONS<>0 , see [[laws:EPELLI# IPCONS parameter |IPCONS parmeter]])|
^Line 2 (7G10.0)^^
| AI1MIN| Minimum value of $I_{\sigma}$ for non-linear elasticity|
| PHIC| Coulomb's angle (in degrees) for compressive paths|
| PHIE| Coulomb's angle (in degrees) for extensive paths|
|AN| Van Eekelen exponent (default value=-0.229)|
|COH| Cohesion value|
|POROS| initial soil porosity ($n_{0}$)|
|RHO| Specific mass|
|DIV|parameter for the computation of NINTV in the law (for NINTV = 0 only). |

===== Stresses =====
==== Number of stresses ====
6 for the 3-D state \\
4 for the other cases
==== Meaning ====
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates. \\
For the other cases:
|SIG(1)|$\sigma_{XX}$|
|SIG(2)|$\sigma_{YY}$|
|SIG(3)|$\sigma_{XY}$|
|SIG(4)|$\sigma_{ZZ}$|

===== State variables =====
==== Number of state variables ====
27: for 2D plane strain analysis with bifurcation criterion (ICBIF=1) \\
15: in all the other cases
==== List of state variables ====
|Q(1)| 1 in plane strain state \\ circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state |
|Q(2)| actualized specific mass|
|Q(3)| 0  if the current state is elastic \\ 1  if the current state is elasto-plastic|
|Q(4)| plastic work per unit volume ($W^{p}$)|
|Q(5)| actualised value of porosity|
|Q(6)| equivalent strain n$\circ$1 $\varepsilon_{eq1} = \int {\Delta \dot{\varepsilon}_{eq} \Delta t}$|
|Q(7)| Updated value of preconsolidation pressure $p_{0}$ |
|Q(8)| equivalent strain indicator n$\circ$1 (Villote n$\circ$1) $\alpha_{1} = (\Delta \dot{\varepsilon}_{eq} \Delta t )/ \varepsilon_{eq1} $|
|Q(9)| X deformation|
|Q(10)| Y deformation|
|Q(11)| Z deformation|
|Q(12)| XY deformation|
|Q(13)| Volumetric strain|
|Q(14)| number of sub-intervals used for the integration|
|Q(15)| memory of localisation calculated during the re-meshing|
|Q(16) $\rightarrow$ Q(27)| reserved for bifurcation|

==== Hardening forms ====
__ITYLA = 2: volumic strain hardening__
\[Dp_{0}=- ECRO p_{0} \varepsilon_{\nu}^{p} \]
Sign dependent on the consolidation stress.\\
Softening is possible.\\
==== Elastic forms====
__IELA = 0: linear elasticity__\\
|E_PAR1|E : Young's Elastic modulus|
|E_PAR2 |ANU : Poisson's ratio|
|E_PAR3 |not used|
|E_PAR4 |not used|
|HARD |ECRO : Hardening parameter|

__IELA = 1: non linear elasticity__\\
|E_PAR1|KAPPA : Elastic slope in oedometer path|
|E_PAR2|ANU : Poisson's ratio|
|E_PAR3|not used|
|E_PAR4 |not used|
|HARD |LAMBDA : Plastic slope in oedometer path|
|ECRO |$\frac{1+e_{0}}{\lambda - \kappa}$|

__IELA = 2: non linear elasticity__\\
|E_PAR1|KAPPA : Elastic slope in oedometer path|
|E_PAR2 |G0: Shear modulus|
|E_PAR3|not used|
|E_PAR4|not used|
|HARD|LAMBDA : Plastic slope in oedometer path|
|ECRO|$\frac{1+e_{0}}{\lambda - \kappa}$|

__IELA = 3: non linear elasticity__\\
|E_PAR1 |KAPPA : Elastic slope in oedometer path|
|E_PAR2 |K0 : Minimum value of the bulk modulus|
|E_PAR3 |G0 : Shear modulus|
|E_PAR4|ALPHA2|
|HARD|LAMBDA : Plastic slope in oedometer path|
|ECRO|$\frac{1+e_{0}}{\lambda - \kappa}$|

__IELA = 4: non linear elasticity__\\
|E_PAR1|K0: Minimum value of the bulk modulus|
|E_PAR2|n: n parameter|
|E_PAR3|G0: Shear modulus|
|E_PAR4 |Patm : Atmospheric pressure|
|HARD| |
|ECRO|HARD|

__IELA = 5: non linear elasticity__\\
|E_PAR1 |$\nu$: Poisson’s ration|
|E_PAR2 |n: n parameter|
|E_PAR3| G0: Shear modulus|
|E_PAR4 |Patm: Atmospheric pressure|
|HARD||
ECRO = HARD

==== IPCONS parameter ====
__IPCONS = 0__:$p_{0}=$PCONS0

__IPCONS = 1__: $p_{0}=\sigma_{v}.OCR$

__IPCONS = 2__: $p_{0}=p_{0}(\sigma, cohesion, \phi). OCR$


Where $ p_{0}$($\sigma$, cohesion, $\phi$) = $\Big[ \frac{-II_{\hat{\sigma}}^{2}}{m^{2}(I_{\sigma} - \frac{3c}{tg\phi}}-I_{\sigma} \Big]$ /3