====== EP-CHEMSOL ======

===== Description =====

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

==== The model ====

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

==== Files ====

Prepro: LCHEM.F \\
Lagamine: CHEM.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| 570|
|COMMNT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (11I5) ^^
|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 annex 8) |
|IELA| = 0 : Linear elasticity |
|:::| > 0 : Non-linear elasticity |
|IELAS| = 0 : Constant BETA |
|:::| > 0 : Variable BETA |
|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 |
|:::| $\neq$ 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 |
|:::| $\neq$ 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 Ratio (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 (ssi 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 ITRACT$\neq$0) |
^ Line 5 (3G10.0) ^^
|POROS| Initial soil porosity ($n_o$) |
|RHO| Specific mass |
|DIV| Parameter for the computation of NINTV in the law (for NINTV = 0 only) |
^ Line 6 (3G10.0) ^^
|C0| Yield limit in term of concentration. This is not used at present in the model, so C0 should be given a large value (>1) so that yielding never takes place |
|A| Chemical softening parameter |
|BETA| Chemo-elastic expansion coefficient BETA if it is constant (IELAS=0) |
^ Line 7 (3G10.0) ^^
|k| Evolution of cohesion with concentration ($c(c) = c(0) + k.c$) |
|BETA0| First parameter of chemo-elastic expansion coefficient BETA |
|F0| Second parameter of chemo-elastic expansion coefficient BETA |

===== 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 ====

= 36 for 2D plane strain analysis with bifurcation criterion (ICBIF=1) \\ = 24 in all the other cases

==== List of state variables ====

|Q(1)| =1 : Plane strain state |
|:::| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state |
|Q(2)| Actualised specific mass |
|Q(3)| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic (Friction mechanism) |
|:::| = 2 : Current state is elasto-plastic (Pore collapse mechanism) |
|:::| = 3 : Current state is elasto-plastic (Traction mechanism) |
|:::| = 4 : Current state is elasto-plastic (Friction + pore mechanisms) |
|:::| = 5 : Current state is elasto-plastic (Friction + traction mechanisms) |
|Q(4)| Plastic work per unit 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 $p_c$ |
|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 BETA |
|Q(20)| Actualised value of C0 (NOT USED AT PRESENT) |
|Q(21)| Updated value of pre-consolidation pressure at zero concentration $p_c^*$ |
|Q(22)| Number of sub-intervals used for the integration |
|Q(23)| Number of iteration used for the integration |
|Q(24)| Memory of localisation calculated during the re-meshing |
|Q(25)$\rightarrow$Q(36)| Reserved for bifurcation |

==== Hardening forms ====

__ITYLA=2 :__ Volumetric strain hardening : \[dp_0=-ECRO\;p_0\;\varepsilon_v^p\] where the sign is 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 parameters ====

__IPCONS = 0 :__ \[p_0 = PCONS0\]

__IPCONS = 1 :__ \[p_0=\sigma_v\;OCR\]

__IPCONS = 2 :__ \[p_0=p_0(\sigma,cohesion,\phi)\;OCR\] where \[p_0(\sigma,cohesion,\phi) = \left[\frac{-II_{\hat{\sigma}}^2}{m^2\left(I_{\sigma}-\frac{3c}{\tan\phi}\right)}-I_{\sigma}\right]/3\]