Elliptic elasto-plastic constitutive law for solid elements at constant temperature
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.
Prepro: LELLI.F
Lagamine: ELLI2EA.F, ELLI3D.F
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | NO (?) |
| Generalized plane state | NO |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 73 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| 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 |
| 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). |
6 for the 3-D state
4 for the other cases
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) | $\sigma_{ZZ}$ |
27: for 2D plane strain analysis with bifurcation criterion (ICBIF=1)
15: in all the other cases
| 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 |
ITYLA = 2: volumic strain hardening
\[Dp_{0}=- ECRO p_{0} \varepsilon_{\nu}^{p} \]
Sign dependent on the consolidation stress.
Softening is possible.
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 = 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