====== EP-SUCSOL ======


===== Description =====

Cap model : elasto-plastic constitutive law for solid elements at constant temperature with effect of suction.

==== The model ====

This law is used for mechanical analysis of elasto-plastic isotropic porous media undergoing large strains.

==== Files ====

Prepro: LSUC.F \\
Lagamine: SUC2EA.F, SUC3D.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| 90|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====


^ Line 1 (11I5) ^^
|NINTV| $\geq 0$ : 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 |
|:::| > 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 |


==== 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| Pre-consolidation pressure (if IPCONS=0) |
|OCR| Over Consolidation Ration (if IPCONS<>0, see section 6.5 !!!!!!!) |
|AI1MIN| Minimum value of $I_{\sigma}$ for non-linear elasticity |
|PSIC| Coulomb's angle (in degrees) for compressive paths |
|PSIE| Coulomb's angle (in degrees) for extensive paths |
|PHMPS| Van Eekelen exponent (default value = -0.229) |
^ 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 (iff 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| Limit of the traction stress (only if TRACT<>0) |
^ 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) |
^ Line 6 (7G10.0) ^^
|S0| Yield limit in term of suction (SI curve) |
|PCrel| Relative Reference pressure PCONS0/PC for the definition of the LC curve |
|RRATIO| |
|BETA| |
|LAMBDA-S| Plastic suction coefficient |
|KAPPA-S| Elastic suction coefficient |
|PATM| Atmospheric pressure |
^ Line 7 (3G10.0) ^^
|k| Evolution of cohesion with suction ($c(s) = c(0)+k.s$)|
|AKAPPAS1| First parameter of KAPPAS formulation |
|AKAPPAS2| Second parameter of KAPPAS formulation |

===== 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 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 ====

39 for 2D plane strain analysis with bifurcation criterion (ICBIF=1) \\ 28 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 (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 unite volume ($W^p$) |
|Q(5)| Actualised value of porosity |
|Q(6)| equivalent strain n°1 : $\varepsilon_{eq1}=\int\Delta\dot{\varepsilon}_{eq}\Delta t$ |
|Q(7)| Updated value of pre-consolidation pressure $p_0$ |
|Q(8)| equivalent strain indicator n°1 (Villote n°1) : $\alpha_1=\frac{\Delta\dot{\varepsilon}_{eq}\Delta t}{\varepsilon_{eq1}}$ |
|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 |
|Q(16)| Actualised value of frictional angle in compression path ($\phi_C$) |
|Q(17)| Actualised value of frictional angle in extension path ($\phi_E$) |
|Q(18)| Apex criterion |
|Q(19)| Actualised value of ALAMBDAS |
|Q(20)| Actualised value of AKAPPAS |
|Q(21)| Actualised value of $S_0$ |
|Q(22)| Absolute value of reference pressure $P_C$ |
|Q(23)| Number of sub-intervals used for the integration |
|Q(24)| Number of iterations used for the integration |
|Q(25)| Cubic modulus |
|Q(26)| Shear modulus |
|Q(27)| Memory of localisation calculated during the re-meshing |
|Q(28)$\rightarrow$Q(39)| 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 $$ ECRO = \frac{1+e_0}{\lambda-\kappa} $$ |
|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 |
|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 |
|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 |
|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 : Hardening parameter $$ ECRO = \frac{1+e_0}{\lambda-\kappa} $$ |
|IELA = 5 | Non-linear elasticity |
|:::| E_PAR1 = $\nu$ (ANU ???) : Poisson's ratio |
|:::| E_PAR2 = n : $n$ parameter |
|:::| E_PAR3 = G0 : Shear modulus |
|:::| E_PAR4 = Patm : Atmospheric pressure |
|:::| HARD = ECRO : Hardening parameter $$ ECRO = \frac{1+e_0}{\lambda-\kappa} $$ |

==== IPCONS Parameters ====

|IPCONS = 0| $p_0$ = PCONS0 |
|IPCONS = 1| $p_0$ = $\sigma_v.$OCR |
|IPCONS = 2| $p_0$ = $p_0(\sigma, \text{cohesion}, \phi).$OCR |

where $p_0(\sigma, \text{cohesion}, \phi) = \left[\frac{-II_{\sigma}^2}{m^2\left(I_{\sigma}-\frac{3c}{\tan\phi)}\right)}-I_{\sigma}\right]/3 $