====== 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.

{{  :laws:epmohr1.png?400  |}}

=== 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).

{{  :laws:epmohr2.png?600  |}}

==== 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 [[appendices:a8|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)| 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 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 |