====== EP-CAPSOL ======
===== Description =====
CAP MODEL : Elasto-plastic constitutive law for solid elements at constant temperature. \\
==== The model ====
This law is used for mechanical analysis of elasto-plastic isotropic porous media undergoing large strains.
==== Files ====
Prepro: LCAP.F \\
Lagamine: CAP2EA.F, CAP3D.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| 79|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (10I5) ^^
|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]] |
|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|
|ITRACT| = 0 : No traction limitation|
|:::|≠ 0 : Traction stresses limitation|
|IECPS| = 0 : $\psi$ is defined with PSIC and PSIE - Not used|
|:::| = 1 : $\psi$ is defined with PHMPS - Not used|
|ICBIF|Computation indice of bifurcation criterion|
|:::| = 0 : non computed|
|:::| = 1 : computed (plane strain state only|
|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 (5G10.0) ^^
|E_PAR1| First elastic parameter |
|E_PAR2| Second elastic parameter |
|E_PAR3| Third elastic parameter |
|E_PAR4| Fourth elastic parameter |
|HARD| Hardening parameter |
^ Line 2 (6G10.0) ^^
|PCONS0| Preconsolidation pressure (If IPCONS0=0) |
|OCR| Over Consolidation Ratio (If IPCONS0<>0, see section 6.5 |
|AI1MIN| Minimum value of $I_\sigma$ for non-linear elasticity |
|PSIC| Coulomb's angle (in degrees) for compressive paths -Not used. |
|PSIE| Coulomb's angle (in degrees) for extensive paths -Not used. |
|PHMPS| Van Eekelen exponent (default value=-0.229) - Not used |
^ Line 3 (6G10.0) ^^
|PHIC0| Initial Coulomb’s angle (in degrees) for compressive paths  |
|PHICF| Final Coulomb’s angle (in degrees) for compressive paths |
|BPHI|Only if there is hardening/softening|
|PHIE0| Initial Coulomb’s angle (in degrees) for extensive paths  |
|PHIEF| Final Coulomb’s angle (in degrees) for extensive paths (psi ILODEF = 2)|
|AN| Van Eekelen exponent (default value=-0.229) |
^ Line 4 (4G10.0) ^^
|COH0| Initial value of cohesion |
|COHF| Final value of cohesion |
|BCOH| Only if there is hardening/softening |
|TRACTION| |
^ Line 5 (3G10.0) ^^
|POROS| Initial soil porosity ($n_0$)|
|RHO| Specific mass|
|DIV| Parameter for the computation of NINTV in the law (for NINTV=0 only).|

=== Hardening forms ===

__ITYLA = 2__: Volumetric strain hardening \\
$dp_0$ = ECRO $p_0\varepsilon_v^p$\\
Sign depedent 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 ratio \\
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 = 0:__ $p_0 = p_0(\sigma,\text{cohesion}, \phi) . OCR$\\

Where $p_0(\sigma,\text{cohesion},\phi) = \left[ \frac{-II_{\widehat{\sigma}}^2}{m^2(I_{\sigma}-\frac{3c}{tg\phi})} - I_{\sigma} \right] / 3$
===== Stresses =====
==== Number of stresses ====
= 6 : for 3D 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 ====
= 34 : for 2D plane strain analysis with bifurcation criterion (ICBIF=1)\\
= 22 : 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)| actualised specific mass |
|Q(3)| = 0 if the current state is elastic |
|:::|= 1 if the current state is elasto-plastic: PLASOL |
|:::|= 2 if the current state is elasto-plastic: ELLIPSE |
|:::|= 3 if the current state is elasto-plastic: TRACTION |
|:::|= 4 if the current state is elasto-plastic: ELL + PLASOL |
|:::|= 5 if the current state is elasto-plastic: PLASOL + TRACTION |
|Q(4)| plastic work per unit volume ($W^p$) |
|Q(5)| Actualised value of porosity |
|Q(6)| equivalent strain $n^o$1 $\varepsilon_{eq1} = \int \Delta \dot{\varepsilon}_{eq}\Delta t$ |
|Q(7)| Updated value of preconsolidation pressure $p_0$ |
|Q(8)| equivalent strain indicator $n^o 1$ (Villote $n^o 1$) $\alpha_1 = (\Delta\dot{\varepsilon}_{eq}\Delta t ) / \varepsilon_{eq1}$ |
|Q(9)| X deformation |
|Q(10)| Y deformation |
|Q(11)| Z deformation |
|Q(12)| XY deformation |
|Q(13)| Plastic volumetric strain |
|Q(14)| Plastic equivalent strain |
|Q(15)| Cohesion |
|Q(16)| Frictional angle in compression |
|Q(17)| Frictional angle in extension |
|Q(18)| APEX |
|Q(19)| number of sub-intervals used for the integration |
|Q(20)| Cubic modulus |
|Q(21)| Shear modulus |
|Q(22)$\rightarrow$ Q(34)| reserved for bifurcation |