Elasto-plastic constitutive law for solid elements at constant temperature (non-associated) with linear elasticity. Isotropic softening of cohesion is possible.
This law is used for mechanical analysis of elasto-plastic isotropic porous media undergoing large strains.
Prepro: LMOMA.F
Lagamine: MOMA2EA.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 | 586 |
| COMMNT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (7I5) | |
|---|---|
| NINTV | Number of sub-steps used to integrate numerically the constitutive equation in a time step |
| = 0 : Number of sub-steps is based on the norm of the deformation increment and on DIV | |
| ISOL | = 0 : Use of total stresses in the constitutive law |
| ≠ 0 : Use of effective stresses in the constitutive law (See annex 8) | |
| ICBIF | = 0 : Nothing |
| = 1 : Rice bifurcation criterion is computed (only for 2D plane strain analysis) | |
| 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 | Shape of the flow surface in the deviatoric plane |
| = 1 : Circle in the deviatoric plane | |
| = 2 : Smoothed irregular hexagon in the deviatoric plane | |
| IECPS | = 0 : $\psi$ is defined with PSIC and PSIE |
| = 1 : $\psi$ is defined with PHMPS | |
| KMETH | = 2 : Actualised VGRAD integration |
| = 3 : Mean VGRAD integration (Default value) | |
| Line 1 (7G10.0) | |
|---|---|
| E | Young's elastic modulus |
| ANU | Poisson's ratio |
| PSIC | Dilatancy angle (in degrees) for compressive paths |
| PSIE | Dilatancy angle (in degrees) for extensive paths (iff ILODEG=2) |
| RHO | Specific mass |
| DIV | Size of sub-steps for computation of NINTV (only if NINTV=0; Default value=5.D-3) |
| PHMPS | Constant value for definition of |
| Line 2 (3G10.0) | |
| 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) |
| Line 3 (4G10.0) | |
| COH0 | Initial value of cohesion |
| ALPHA | Coefficient in the softening relationship |
| GAMMARP | |
| BIOPT | |
6 for 3D state
4 for the other cases
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) | $\sigma_{zz}$ |
= 36 for 2D plane strain analysis with bifurcation criterion (ICBIF=1)
= 24 in all the other cases
| Q(1) | = 1 : Plane strain state |
| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state | |
| Q(2) | Actualised specific mass |
| Q(3) | = 0 : Current state is elastic |
| = 1 : Current state is elasto-plastic | |
| Q(4) | Plastic work per unit volume ($W^p$) |
| Q(5) | Volumic variation |
| Q(6) | Equivalent strain n°1 : $\varepsilon_{eq1}=\int\Delta\dot{\varepsilon}_{eq}\;\Delta t$ |
| Q(7) | Equivalent strain indicator n°1 (Villote n°1) : $\alpha_1=\frac{\Delta\dot{\varepsilon}_{eq}\;\Delta t}{\varepsilon_{eq1}}$ |
| Q(8) | X deformation |
| Q(9) | Y deformation |
| Q(10) | Z deformation |
| Q(11) | XY deformation |
| Q(12) | Equivalent strain n°2 : $\varepsilon_{eq2}=\int\Delta\varepsilon_{eq}$ |
| Q(13) | Equivalent strain indicator n°2 (Villote n°2) $\alpha_2=\frac{\Delta\varepsilon_{eq}}{\varepsilon_{eq2}}$ |
| Q(14) | Actualised value of equivalent plastic strain $\varepsilon_{eq}^p$ |
| Q(15) | Actualised value of cohesion $c$ |
| Q(16) | Actualised value of Coulomb's frictional angle in compressive path ($\phi_C$) |
| Q(17) | Actualised value of Coulomb's frictional angle in extensive path ($\phi_E$) |
| Q(18) | = 0 : If the stress state is not at the criterion apex |
| = 1 : If the stress state is at the criterion apex | |
| Q(19) | Number of sub-intervals used for the integration |
| Q(20) | |
| Q(21) | |
| Q(22) | Actualised value of volumetric plastic deformations |
| Q(23) | Second deviatoric strain increment invariant |
| Q(24) | Plastic loading index |
| Q(25) | Memory of localisation calculated during the re-meshing |
| Q(26)$\rightarrow$Q(36) | Reserved for bifurcation |
The stresses and stress invariants are : \[I_{\sigma} = \sigma_{ij}\quad ; \quad \hat{\sigma}_{ij}=\sigma_{ij}-\frac{I_{\sigma}}{3}\delta_{ij} \] \[II_{\hat{\sigma}}=\sqrt{\frac{1}{2}\hat{\sigma}_{ij}\hat{\sigma}_{ij}}\quad ;\quad III_{\hat{\sigma}} = \frac{1}{3}\hat{\sigma}_{ij}\hat{\sigma}_{jk}\hat{\sigma}_{ki}\] \[\beta =-\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}}{2}\frac{III_{\hat{\sigma}}}{II^3_{\hat{\sigma}}}\right)\]
The regular criterion is used if $I_{\sigma}-m'II_{\hat{\sigma}}<\frac{3c}{\tan\phi_c}$ : \[f=II_{\hat{\sigma}}+m\left(I_{\sigma}-\frac{3c}{\tan\phi_c}\right)=0\] with :
- Drücker Prager : $m = \frac{2\sin\phi_c}{\sqrt{3}(3-\sin\phi_c)}$
- Van Eekelen : $m=a(1+b\sin 3\beta)^n$ where $a$ and $b$ are functions of $\phi_C$, $\phi_E$ and $n$.
The apex criterion is used if $I_{\sigma}-m'II_{\hat{\sigma}}\geq\frac{3c}{\tan\phi_C}$ : \[f=I_{\sigma}-\frac{3c}{\tan\phi_c}=0\] where $m'$ is the equivalent of $m$ but for the flow surface (i.e. $\phi$ is replaced by $\psi$ )
Softening is assumed to be represented by the evolution of cohesion as a function of the Von Mises equivalent plastic strain : \[\gamma^p=\sqrt{\hat{\varepsilon}_{ij}^p\hat{\varepsilon}_{ij}^p}\] where $\hat{\varepsilon}_{ij} = \varepsilon_{ij}-\frac{I_{\varepsilon}}{3}\delta_{ij}$ is the deviatoric strain tensor.
The following function is used : \begin{align*}f(\gamma^p)&=\left(1-(1-\alpha)\frac{\gamma^p}{\gamma_R^p}\right)^2\;quad\text{if}\quad 0<\gamma^p<\gamma^p_R \\ &= \alpha^2 \quad\text{if}\quad \gamma^p\geq \gamma_R^p \end{align*} where $\alpha$ and $\gamma_R^p$ are two models parameters.