Sidebar
Table of Contents
EP-ELLI
Description
Elliptic elasto-plastic constitutive law for solid elements at constant temperature
The model
Mechanical analysis of elasto-plastic isotropic porous media undergoing large strains. Dilatancy is included through an elliptic yield surface relating the mean stress and the von MISES equivalent stress. Isotropic hardening is assumed.
Files
Prepro: LELLI.F
Lagamine: ELLI2EA.F, ELLI3D.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 | 73 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
Integer parameters
| Line 1 (9I5) | |
|---|---|
| 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 > 1 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 |
| ICBIF | Computation indice of bifurcation criterion = 0: non computed = 1: computed (plane strain state only) |
| IECROU | = 2 Volumetric strain hardening (including softening) |
| 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 (7G10.0) | |
|---|---|
| E_PAR1 | First elastic parameter |
| E_PAR2 | Second elastic parameter |
| E_PAR3 | Third elastic parameter |
| E_PAR4 | Fourth elastic parameter |
| HARD | Hardening parameter |
| PCONS0 | Preconsolidation pressure (If IPCONS=0) |
| OCR | Over Consolidation Ratio (If IPCONS<>0 , see IPCONS parmeter) |
| Line 2 (7G10.0) | |
| AI1MIN | Minimum value of $I_{\sigma}$ for non-linear elasticity |
| PHIC | Coulomb's angle (in degrees) for compressive paths |
| PHIE | Coulomb's angle (in degrees) for extensive paths |
| AN | Van Eekelen exponent (default value=-0.229) |
| COH | Cohesion value |
| POROS | initial soil porosity ($n_{0}$) |
| RHO | Specific mass |
| DIV | parameter for the computation of NINTV in the law (for NINTV = 0 only). |
Stresses
Number of stresses
6 for the 3-D 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
27: for 2D plane strain analysis with bifurcation criterion (ICBIF=1)
15: 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) | actualized specific mass |
| Q(3) | 0 if the current state is elastic 1 if the current state is elasto-plastic |
| Q(4) | plastic work per unit volume ($W^{p}$) |
| Q(5) | actualised value of porosity |
| Q(6) | equivalent strain n$\circ$1 $\varepsilon_{eq1} = \int {\Delta \dot{\varepsilon}_{eq} \Delta t}$ |
| Q(7) | Updated value of preconsolidation pressure $p_{0}$ |
| Q(8) | equivalent strain indicator n$\circ$1 (Villote n$\circ$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) | number of sub-intervals used for the integration |
| Q(15) | memory of localisation calculated during the re-meshing |
| Q(16) $\rightarrow$ Q(27) | reserved for bifurcation |
Hardening forms
ITYLA = 2: volumic strain hardening
\[Dp_{0}=- ECRO p_{0} \varepsilon_{\nu}^{p} \]
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 ration |
| 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, $\phi$) = $\Big[ \frac{-II_{\hat{\sigma}}^{2}}{m^{2}(I_{\sigma} - \frac{3c}{tg\phi}}-I_{\sigma} \Big]$ /3
