Table of Contents
EP-ELL
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: LELL.F
Lagamine: ELL2EA.F, ELL3D.F
Availability
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | NO |
Input file
Parameters defining the type of constitutive law
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 70 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
Integer parameters
| Line 1 (4I5) | |
|---|---|
| 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 Appendix 8 |
| ITYLA | = 0 : unlimited work hardening = 1: hyperbolic work hardening = 2: volumic strain hardening (including softening) |
| ICBIF | Computation indice of bifurcation criterion = 0 : non computed = 1 : computed (plane strain state only) |
Real parameters
| Line 1 (7G10.0) | |
|---|---|
| E | YOUNG's elastic modulus |
| ANU | POISSON's ratio |
| COULB | tangent of COULOMB's angle (= ratio between the short diameter and the long diameter of the ellipse = m) |
| ECRO1 | hardening rate (= d2a/dg) voir ci-dessous |
| ECRO2 | hardening rate (= d2a/dg) voir ci-dessous |
| DEUSA | initial value of the long diameter of the ellipse (=2$a_{0}$) |
| POROS | initial soil porosity ($n_{0}$) |
| Line 2 (2G10.0) | |
| DIV | parameter for the computation of NINTV in the law (for NINTV = 0 only). |
| RHO | Specific mass |
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 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
19 (for EP$\varepsilon$)
6 (for the other cases)
List of state variables
| Q(1) | 1 in plane strain state circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state 0 in 3-D state |
| Q(2) | current value of the long diameter of the ellipse its initial value is 2$a_{0}$ |
| Q(3) | 0 if the current state is elastic 1 if the current state is elasto-plastic |
| Q(4) | plastic work per unit volume ($\varepsilon_{\nu}^{p}$ si ITYLA = 2) |
| Q(5) | current value of soil porosity its initial value is $n_{0}$) |
| Q(6) | somme des $\dot{\varepsilon}_{\nu} dt$ |
| Q(7) | PSNIV |
| Q(8) | actualised specific mass |
| Q(9) $\rightarrow$ Q(20) | modulus for the analysis of bifurcation (only for EP$\varepsilon$) |
Hardening forms
ITYLA = 0 : monotonic work hardening
\[\frac{d2a}{d\lambda} = ECRO1\]
\[$\lambda$] = $[\sigma]^{-1}\]
ITYLA = 1: hyperbolic work hardening (with asymptotic maximum)
\[2a = 2a_{0} + \frac{ECRO1 * 1}{ECRO2*\lambda +1}\]
\[\frac{d2a}{d\lambda} = \frac{ECRO1}{(ECRO2*\lambda +1)^{2}}\]
\[2 a_{max} = 2 a_{0} + \frac{ECRO1}{ECRO2}\]
ITYLA = 2: volumic strain hardening
\[da = ECRO1*d\varepsilon_{\nu}^{p} \]
Sign dependent on the consolidation stress.
Softening is possible.