====== EP-TSOIL ======

===== Description =====

Cap model : elasto-plastic constitutive law for solid elements at constant temperature with thermoplasticity (A thermomechanical model of clays, CUI et al., 2000).

==== The model ====

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

==== Files ====

Prepro: LTSOIL.F \\
Lagamine: TSOIL2EA.F, TSOIL3D.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| 169|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (14I5) ^^
|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=$1.10^{-5}$ |
|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 |
|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 |
|ILY| = 0 : Evolution of the pre-consolidation pressure with temperature (ENPC, LY curve) \[p'_{cT} = p'_{c_0T_0}\exp(-\alpha_0\Delta T)\] |
|:::| = 1 : Evolution of the pre)consolidation pressure with temperature (ACMEG-T) \[p'_{cT} = p'_{c_0T_0}\left[1-\gamma_T\log\left(\frac{T}{T_0}\right)\right]\] |

==== 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 PCONS0=0) |
|OCR| Over Consolidation Ratio (if PCONS0$\neq$0, see section 6.5) |
|AI1MIN| Minimum value of I$_{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 ITRACT$\neq$0) |
^ Line 5 (4G10.0) ^^
|POROS| Initial soil porosity ($n_0$) |
|RHO| Specific mass |
|DIV| Parameter for the computation of NINTV in the law (for NINTV = 0 only) |
|BIOPT| Bifurcation computation parameter |
^ Line 6 (7G10.0) ^^
|ALPHA2| Volumetric thermal expansion coefficient of the solid |
|ALPHA0| Shape parameter for LY curve (ILY=0 : formulation ENPC, for ACMEG-T see below) |
|BETA0| Hardening parameter for TY curve (1/Pa) |
|TC| Critical temperature (TY curve) (°K) |
|T0| Initial temperature (TY curve) (°K) \[TY \equiv T_{cT} - \left[(T_c-T_0)\exp(-\beta p')+T_0\right] = 0\] |
^ Line 7 (4G10.0) ^^
|C1| Parameter for HC curve |
|C2| Parameter for HC curve |
|A parameter| Parameter for the definition of the thermal volumetric plastic strain |
|GAMA| Shape parameter for LY curve according ACMEG-T (ILY = 1) \[HC\equiv p'-c_1 p'_{c0}\exp(c_2\Delta T) = 0\] |

===== 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 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 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 $p_0$ |
|Q(8)| Equivalent strain indicator n°1 (Villote n°1) : $\alpha_1 = \dfrac{\Delta\dot{\varepsilon}_{eq}\;\Delta t}{\varepsilon_{eq1}}$ |
|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 ALPHAP |
|Q(21)| Actualised value of temperature |
|Q(22)| ITESTALPHA : Variable for the calculation of ALPHAP |
|Q(23)| Actualised value of the pre-consolidation pressure as a function of the temperature (Pa) |
|Q(24)| Number of sub-intervals used for the integration |
|Q(25)| KSUMMITER |
|Q(26)| K (Pa) |
|Q(27)| G (Pa) |
|Q(28)$\rightarrow$Q(39)| Reserved for bifurcation |

==== 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 = 2__ : $p_0$ = $p_0$($\sigma$, cohesion, $\phi$) . OCR\\

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