====== EP-ELL======
===== 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: LELL.F  \\
Lagamine: ELL2EA.F, ELL3D.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| 70|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^Line 1 (4I5)^^
|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 \\ $\neq$ 0 : use of effective stresses in the constitutive law. See [[appendices:a8|Appendix 8]]|
|ITYLA| = 0 : unlimited work hardening \\ = 1: hyperbolic work hardening\\ = 2: volumic strain hardening (including softening)|
|ICBIF|Computation indice of bifurcation criterion\\ = 0 : non computed \\ = 1 : computed (plane strain state only)|
==== Real parameters ====

^Line 1 (7G10.0)^^
|E| YOUNG's elastic modulus |
|ANU| POISSON's ratio|
|COULB|tangent of COULOMB's angle (= ratio between the short diameter and the long diameter of the ellipse = m) |
|ECRO1|hardening rate (= d2a/dg) voir ci-dessous|
|ECRO2|hardening rate (= d2a/dg) voir ci-dessous|
|DEUSA|initial value of the long diameter of the ellipse (=2$a_{0}$)|
|POROS| initial soil porosity ($n_{0}$)|
^Line 2 (2G10.0)^^
|DIV|parameter for the computation of NINTV in the law (for NINTV = 0 only). |
|RHO| Specific mass|

===== 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. \\
{{  :laws:ep-ell_stresses.png?400|}}
For the 3-D state:
|SIG(1)|$\sigma_{XX}$|
|SIG(2)|$\sigma_{YY}$|
|SIG(3)|$\sigma_{ZZ}$|
|SIG(4)|$\sigma_{XY}$|
|SIG(5)|$\sigma_{XZ}$|
|SIG(6)|$\sigma_{YZ}$|
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 ====
19 (for EP$\varepsilon$)\\
6 (for the other cases)
==== List of state variables ====
|Q(1)| 1 in plane strain state \\ circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state \\ 0 in 3-D state|
|Q(2)| current value of the long diameter of the ellipse \\ its initial value is 2$a_{0}$|
|Q(3)| 0  if the current state is elastic \\ 1  if the current state is elasto-plastic|
|Q(4)| plastic work per unit volume ($\varepsilon_{\nu}^{p}$ si ITYLA = 2)|
|Q(5)| current value of soil porosity \\ its initial value is $n_{0}$)|
|Q(6)| somme des $\dot{\varepsilon}_{\nu} dt$|
|Q(7)| PSNIV|
|Q(8)| actualised specific mass|
|Q(9) $\rightarrow$ Q(20)| modulus for the analysis of bifurcation (only for EP$\varepsilon$)|

==== Hardening forms ====
__ITYLA = 0 : monotonic work hardening __
{{  :laws:ityla0_ep-ell.png?300  |}}
\[\frac{d2a}{d\lambda} = ECRO1\]
\[$\lambda$] = $[\sigma]^{-1}\]

__ITYLA = 1: hyperbolic work hardening (with asymptotic maximum)__
{{ :laws:ityla1_ep-ell.png?300 |}}
\[2a = 2a_{0} + \frac{ECRO1 * 1}{ECRO2*\lambda +1}\]

\[\frac{d2a}{d\lambda} = \frac{ECRO1}{(ECRO2*\lambda +1)^{2}}\]

\[2 a_{max} = 2 a_{0} + \frac{ECRO1}{ECRO2}\]

__ITYLA = 2: volumic strain hardening__
\[da = ECRO1*d\varepsilon_{\nu}^{p} \]
Sign dependent on the consolidation stress.\\
Softening is possible.\\