====== ZIZOU ======
===== Description =====
Cap-model elasto-plastic + cap-model visco-plastic constitutive laws for solid elements at constant temperature with influence of lode angle and suction
==== The model ====
This law is used for mechanical analysis of elastoplastic‑viscoplastic isotropic porous media undergoing large strains. 
==== Files ====
Prepro: LZIZOU.F \\
Lagamine: ZIZOU.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| 46 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (12I5) ^^
|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|
|:::| > 0 : Non linear elasticity|
|IELAS| = 0: Constant KAPPAS|
|:::| > 0: Variable KAPPAS|
|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|
|:::| = 1 : $\psi$ is defined with PHMPS|
|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|
|ILC| = 0: Barcelona LC curve|
|:::| ≠ 0: Pasachalk LC curve|
==== Real parameters ====
**__Parameters and internal variables for elasto-plastic model__**
^ 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_EP|Hardening parameter |
^ Line 2 (3G10.0) ^^
|PCONS0_EP|Preconsolidation pressure (If IPCONS0=0) |
|OCR_EP|Over Consolidation Ratio (If IPCONS0<>0, see section 6.5 |
|AI1MIN_EP|Minimum value of $I_\sigma$ for non-linear elasticity |
^ Line 3 (3G10.0) ^^
|PSIC_EP|Coulomb's angle (in degrees) for compressive paths|
|PSIE_EP|Coulomb's angle (in degrees) for extensive paths|
|PHMPS_EP|Van Eekelen exponent (default value=-0.229)|
^ Line 4 (5G10.0) ^^
|PHIC0_EP|Initial Coulomb’s angle (in degrees) for compressive paths  |
|PHICF_EP|Final Coulomb’s angle (in degrees) for compressive paths |
|BPHI_EP| Only if there is hardening/softening|
|PHIE0_EP|Initial Coulomb’s angle (in degrees) for extensive paths  |
|PHIEF_EP|Final Coulomb’s angle (in degrees) for extensive paths (psi ILODEF = 2)|
^ Line 5 (5G10.0) ^^
|AN_EP|Van Eekelen exponent (default value=-0.229) |
|COH0_EP|Initial value of cohesion |
|COHF_EP|Final value of cohesion |
|BCOH_EP|Only if there is hardening/softening |
|TRACTION_EP|Limit of the traction stress (Only if ITRACT <>0 ) |
^ Line 6 (4G10.0) ^^
|POROS_EP|Initial soil porosity ($n_o$)|
|RHO_EP|Specific mass|
|DIV_EP|Parameter for the computation of NINTV in the law (for NINTV=0 only)|
|BIOPT_EP|Bifurcation computation parameter|
^ Line 7 (5G10.0) ^^
|S0_EP|Yield limit in term of suction (SI curve)|
|PCRATIO_EP|Relative Reference pressure PCONS0/PC for the definition of the LC curve|
|RRATIO_EP|Max soil stiffness|
|BETA_EP|Beta soil stiffness parameter|
|LAMBDA-S_EP|Plastic suction coefficient|
^ Line 8 (5G10.0) ^^
|KAPPA-S_EP|Elastic suction coefficient|
|PATM_EP|Atmospheric pressure|
|k_EP| Evolution of cohesion with suction ($c(s) = c(0) + k.s$)|
|AKAPPAS1_EP|First parameter of KAPPAS formulation|
|AKAPPAS2_EP|Second parameter of KAPPAS formulation|

**__Parameters and internal variables for visco-plastic model__**
^ 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_VP|Hardening parameter |
^ Line 2 (3G10.0) ^^
|PCONS0_VP|Preconsolidation pressure (If IPCONS0=0) |
|OCR_VP|Over Consolidation Ratio (If IPCONS0<>0, see section 6.5 |
|AI1MIN_VP| = Minimum value of $I_\sigma$ for non-linear elasticity |
^Line 3 (3G10.0)^^
|PSIC_VP|Coulomb's angle (in degrees) for compressive paths|
|PSIE_VP|Coulomb's angle (in degrees) for extensive paths|
|PHMPS_VP|Van Eekelen exponent (default value=-0.229)|
^Line 4 (5G10.0)^^
|PHIC0_VP|Initial Coulomb’s angle (in degrees) for compressive paths|
|PHICF_VP|Final Coulomb’s angle (in degrees) for compressive paths|
|BPHI_VP|Only if there is hardening/softening|
|PHIE0_VP|Initial Coulomb’s angle (in degrees) for extensive paths|
|PHIEF_VP|Final Coulomb’s angle (in degrees) for extensive paths (psi ILODEF = 2)|
^Line 5 (5G10.0)^^
|AN_VP|Van Eekelen exponent (default value=-0.229)|
|COH0_VP|Initial value of cohesion|
|COHF_VP|Final value of cohesion|
|BCOH_VP|Only if there is hardening/softening|
|TRACTION_VP|Limit of the traction stress (Only if ITRACT <>0 )|
^Line 6 (4G10.0)^^	
|POROS_VP|Initial soil porosity ($n_o$)|
|RHO_VP|Specific mass|
|DIV_VP|Parameter for the computation of NINTV in the law (for NINTV = 0 only)|
|BIOPT_VP|Bifurcation computation parameter|
^Line 7 (5G10.0)^^
|S0_VP|Yield limit in term of suction (SI curve)|
|PCRATIO_VP|Relative Reference pressure PCONS0/PC for the definition of the LC curve|
|RRATIO_VP|Max soil stiffness|
|BETA_VP|Beta soil stiffness parameter|
|LAMBDA-S_VP|Plastic suction coefficient|
^Line 8 (5G10.0)^^
|KAPPA-S_VP| Elastic suction coefficient|
|PATM_VP|Atmospheric pressure|
|k_VP|Evolution of cohesion with suction ($c(s) = c(0) + k.s$)|
|AKAPPAS1_VP|First parameter of KAPPAS formulation|
|AKAPPAS2_VP|Second parameter of KAPPAS formulation|
^Line 9 - Visco_parameters for Cap (3G10.0)^^
|ALPHAC|Viscoplastic parameter for $\Phi_C = ( f_c /p())* * \alpha_c$|
|OMEGA|Viscosity parameter for $\gamma_c$|
|AIOT|Viscosity parameter for  $\gamma_c$|
^Line 10 - Visco_parameters for friction (2G10.0)^^
|ALPHAD|parameter $\alpha_d$ for $\Phi_d = \left( \frac{f_d}{p_0} \right)^{\alpha_d}$|
|A2D|parameter $\alpha_d$ for $\gamma_d = a_2 \gamma_c$|
===== 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 ====
= 68 : for 2D plane strain analysis with bifurcation criterion (ICBIF=1)\\
= 57 : 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|
|:::| = THICK_EP|
|Q(2)|actualised specific mass = RHO_EP |
|Q(3)| = 0 if the current state is elastic |
|:::|= 1 if the current state is elasto-plastic (Friction mechanism) |
|:::|= 2 if the current state is elasto-plastic (Pore collapse mechanism) |
|:::|= 3 if the current state is elasto-plastic (Traction mechanism) |
|:::|= 4 if the current state is elasto-plastic (Friction + pore mechanisms) |
|:::|= 5 if the current state is elasto-plastic (Friction + traction mechanisms) |
|Q(4)|plastic work per unit volume ($W^p$) |
|Q(5)|Actualised value of porosity = PORO0_EP |
|Q(6)|equivalent strain $n^o$1 $\varepsilon_{eq1} = \int \Delta \dot{\varepsilon}_{eq}\Delta t$ |
|Q(7)|Updated value of preconsolidation pressure $p_0$ |
|:::|PCONS0_EP|
|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)|Volumetric strain |
|Q(14)|Deviatoric strain |
|Q(15)|Actualised value of cohesion = COH0_EP|
|Q(16)|Actualised value of frictional angle in compression path ($\phi_C$) |
|:::|PHIC0_EP|
|Q(17)|Actualised value of frictional angle in compression path ($\phi_E$) |
|:::|PHIE0_EP|
|Q(18)|APEX criterion |
|Q(19)|Actualised value of ALAMBDAS = ALAMBDAS_EP |
|Q(20)|Actualised value of AKAPPAS = AKAPPAS_EP |
|Q(21)|Actualised value of $S_0$ = S0_EP |
|Q(22)|Absolute value of reference pressure $P_C$ |
|Q(23)|PCONS0 |
|Q(24)|number of sub-intervals used for the integration |
|Q(25)|number of interation used for the integration |
|Q(26)|Cubic modulus |
|Q(27)|Shear modulus |
|Q(28)|OVERS |
|Q(29)|THICK_VP |
|Q(30)|RHO_VP|
|…|:::|
|Q(33)|POROS0_VP|
|…|:::|
|Q(35)|PCONS0_VP|
|…|:::|
|Q(43)|COH0_VP|
|Q(44)|PHIC0_VP|
|Q(45)|PHIE0_VP|
|…|:::|
|Q(47)|ALAMBDAS_VP|
|Q(48)|AKAPPAS_VP|
|Q(49)|S0_EP|
|…|:::|
|Q(59)$\rightarrow$Q(68)|reserved for bifurcation|

===== Hardening forms =====

__ITYLA = 2__ : Volumetric strain hardening \\
$dp_0$ = -ECRO $p_0\varepsilon_v^p$\\
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 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 $p_0(\sigma,\text{cohesion},\phi) = \left[ \frac{-II_{\widehat{\sigma}}^2}{m^2(I_{\sigma}-\frac{3c}{tg\phi})} - I_{\sigma} \right] / 3$