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