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EP-MOHR
Description
Elasto-plastic constitutive law for solid elements at constant temperature (non-associated) with linear elasticity. Isotropic hardening/softening of friction angle and cohesion is possible and the Mohr Coulomb yield surface is represented.
The model
This model is used for mechanical analysis of elasto-plastic isotropic porous media undergoing large strains, with a Mohr Coulomb yield surface.
Integration is performed using an implicit backward Euler scheme with a return mapping normal to the flow surface $g$. This law can take into account the influence of:
- the first stress invariant, i.e. the yield surface is either parallel to the pressure axis in the $p-q$ plane,
- the third stress invariant, i.e. the Lode angle : the trace in the deviatoric plane is an irregular hexagon.
Yield and flow surfaces
The stresses and stress invariants are: $$ I_{\sigma} = \sigma_{ij} \quad ; \quad \hat{\sigma}_{ij}=\sigma_{ij}-\frac{I_{\sigma}}{3}\delta_{ij} $$ $$ II_{\sigma} = \sqrt{\frac{1}{2}\hat{\sigma}_{ij}\hat{\sigma}_{ij}} \quad ; \quad III_{\sigma} = \frac{1}{3}\hat{\sigma}_{ij}\hat{\sigma}_{jk}\hat{\sigma}_{ki} $$ $$ \beta = -\frac{1}{3} \;\sin^{-1}\left(\frac{3\sqrt{3}}{2} \frac{III_{\sigma}}{II_{\sigma}^3}\right) $$
The Mohr-Coulomb failure criterion is an intrinsic curve criterion. It expresses a linear relationship between the shear stress $\tau$ and the normal stress $\sigma_N$ acting on a failure plane. $$ \tau = c + \sigma_N \;\tan\phi $$ where $c$ is the cohesion and $\phi$ the friction angle. This criterion can be expressed in a more general fashion in term of principal stresses by the relation $$ f=\frac{I_{\sigma}}{3}\sin\phi + II_{\sigma}\cos\beta -\frac{II_{\sigma}}{\sqrt{3}}\sin\beta\sin\phi-c\cos\phi = 0 $$
The criterion predicts identical friction angles under triaxial compression paths and triaxial extension paths.
Hardening/softening
Hardening/softening is assumed to be represented by the evolution of friction angles and/or cohesion as a function of the Von Mises equivalent plastic strain $$ \varepsilon_{eq}^p = \sqrt{\frac{2}{3}\hat{\varepsilon}^p_{ij}\hat{\varepsilon}^p_{ij}}$$
Hyperbolic functions are used $$ \phi = \phi_0+\frac{(\phi_f-\phi_0)\varepsilon_{eq}^p}{B_p+\varepsilon_{eq}^p}\quad , \quad c=c_0+\frac{(c_f-c_0)\varepsilon_{eq}^p}{B_C+\varepsilon_{eq}^p} $$ where coefficients $B_p$ and $B_c$ are respectively the values of equivalent plastic strain for which half of the hardening/softening on friction angles and cohesion is achieved (see figure 2).
Files
Prepro: LMOHR.F
Lagamine: MOHR2EA.F
Availability
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalised plane state | NO |
Input file
Parameters defining the type of constitutive law
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 591 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
Integer parameters
| Line 1 (5I5) | |
|---|---|
| NINTV | number of sub-steps used to integrate numerically the constitutive equation in a time step. If NINTV=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 |
| $\neq 0$ : use of effective stresses in the constitutive lax. See Appendix 8. | |
| ICBIF | = 0 : nothing |
| = 1 : Rice bifurcation criterion is computed (only for 2D plane strain analysis) | |
| 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) | |
Real parameters
| Line 1 (6G10.0) | |
|---|---|
| E | Young elastic modulus |
| ANU | Poisson ratio |
| PSI | Dilatancy angle (in degrees) |
| 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) | |
| PHI0 | Initial Coulomb's angle (in degrees) |
| PHIF | Final Coulomb's angle (in degrees) |
| BPHI | Only if there is hardening/softening |
| Line 3 (3G10.0) | |
| COH0 | Initial value of cohesion |
| COHF | Final value of cohesion |
| BCOH | Only if there is hardening/softening |
Stresses
Number of stresses
4 for 2D analysis
6 for 3D analysis
Meaning
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.
For 2D analysis :
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{xy}$ |
| SIG(4) | $\sigma_{zz}$ |
For 3D analysis :
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{zz}$ |
| SIG(4) | $\sigma_{xy}$ |
| SIG(5) | $\sigma_{xz}$ |
| SIG(6) | $\sigma_{yz}$ |
State variables
Number of state variables
37 for the 2D plane strain analysis with bifurcation criterion (ICBIF=1)
25 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) | actualised specific mass |
| Q(3) | = 0 i the current state is elastic |
| = 1 if the current state is elasto-plastic | |
| Q(4) | plastic work per unit volume ($W^p$) |
| Q(5) | volume 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) | |
| 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 friction angle |
| Q(17) | = 0 if the stress state is not at the criterion apex |
| = 1 if the stress state is at the criterion apex | |
| Q(18) | number of sub-intervals used for the integration |
| Q(19) | memory of localisation calculated during the re-meshing |
| Q(20) | … |
| Q(21) | principal stress n°1 |
| Q(22) | principal stress n°2 |
| Q(23) | principal stress n°3 |
| Q(24) | ICRITF |
| Q(25) $\rightarrow$ Q(37) | reserved for bifurcation |
