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====== ORTHOPLATRA ======

===== Description =====

Elasto-plastic constitutive law for solid elements at constant temperature (non-associated) with non-linear anisotropic elasticity. Isotropic hardening/softening of friction angle and cohesion is possible. Tensile criterion for desiccation cracking included. Isotropic hardening/softening of tensile strength is possible. \\

This law is a variation of ORTHOPLA including a tensile failure criterion.

==== The model ====

This law is used for mechanical analysis of elasto-plastic anisotropic porous media undergoing drying (large strain). It is primarily used to predict desiccation cracks onset (can be used for other tensile loading configurations).

=== Tensile criterion ===

\[f = II_2 +\frac{1}{(3.\cos\beta-\sqrt{3}\sin\beta)}.(I_1-3.\sigma'_t)\]
where $I_1$, $II_2$ are respectively the first invariant of the stress tensor and the second invariant of the deviatoric stress tensor. $\beta$ is Lode's angle and $\sigma_t$ is the material tensile strength. \\

{{  :laws:orthoplatra1.png?600  |}}

=== Peron's formulation for tensile strength variation with suction ===

\[\sigma_t'=\sigma_t'^{sat}-\left(k_2*\left(1-\exp\left(\frac{-k_1.s}{k_{20}}\right)\right)\right)\]
where $\sigma_t'$ is the effective tensile strength, $\sigma_t'^{sat}$ is the saturated effective tensile strength, $k_1$ and $k_2$ are model parameters. $k_1$ controls the variation speed and $k_2$ is the maximum delta.

{{  :laws:orthoplatra2.png?400  |}}
==== Files ====

Prepro: LORTHOPLATRA.F \\

===== Availability =====

|Plane stress state| NO |
|Plane strain state| YES|
|Axisymmetric state| YES|
|3D state| NO |
|Generalized plane state| NO |

===== Input file =====

==== Parameters defining the type of constitutive law ====

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 619|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (12I5) ^^
|NINTV| $\neq$ 0 : 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 |
|:::| $\neq$ 0 : Use of effective stresses in the constitutive law (See [[appendices:a8|Appendix 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) |
|IREDUC| = 0 : nothing |
|:::| = 1 : Phi-C reduction method |
|ICOCA| = 0 : nothing |
|:::| = 1 : Capillary cohesion formulation ($c = c_0+AK1.s+AK2.s^2$) |
|:::| = 2 : $c=c_0+AK1.\log(s)+AK2$ |
|:::| = 3 : Linear dependence of the elastic modulus to the suction $E_i=(1+AK2.s).E_{0,i}$ |
|:::| = 4 : Dependence of the elastic modulus on the suction through an exponential relation $E_i=E_{0,i}+EK_i.(1+AK2.s).E_{0,i}$ |
|:::| = 5 : Dependence of the elastic modulus on the mean stress state $E_i=E_{0,i}+EK_i.(\frac{p}{AK1})^{AK2}$ |
|IBEDDING| Bedding orientation (normal to the bedding plane) |
|:::| = 1 : bedding plane in $e_1e_2$ anisotropic plane (normal $e_3$) |
|:::| = 2 : bedding plane in $e_1e_3$ anisotropic plane (normal $e_2$) |
|:::| = 3 : bedding plane in $e_2e_3$ anisotropic plane (normal $e_1$) |
|IANISO| = 0 : Anisotropy of cohesion with major principal stress orientation relative to bedding |
|:::| = 1 : Anisotropy of cohesion by microstructure fabric tensor |
|IVISCO| = 0 : nothing |
|:::| = 1 to 3 : viscoplastic model |

==== Real parameters ====

^ Line 1 (3G10.0) ^^
|ALPHA| Angle of rotation of the anisotropic axis around Z axis (see figure above) |
|THETA| Angle of rotation of the anisotropic axis around $e_1$ axis (see figure above) |
|PHI| Angle of rotation of the anisotropic axis around $e_2$ axis (see figure above) |
^ Line 2 (6G10.0) ^^
|E1| Elastic Young modulus E($e_1$) |
|E2| Elastic Young modulus E($e_2$) |
|E3| Elastic Young modulus E($e_3$) |
|G12| Elastic shear modulus G($e_1e_2$) |
|G13| Elastic shear modulus G($e_1e_3$) |
|G23| Elastic shear modulus G($e_2e_3$) |
^ Line 3 (5G10.0) ^^
|ANU12| Poisson ratio NU($e_1e_2$) |
|ANU13| Poisson ratio NU($e_1e_3$) |
|ANU23| Poisson ratio NU($e_2e_3$) |
|RHO| Specific mass |
|DIV| Size of sub-steps for computation of NINTV (only if NINTV=0, Default value=$5.D-3$) |
^ Line 4 (7G10.0) ^^
|PSIC| Dilatancy angle (in degrees) for compressive paths |
|PSIE| Dilatancy angle (in degrees) for extensive paths (iff ILODEG=2) |
|PHMPS| Constant value for definition of (????!!!!) |
|BIOPT| Bifurcation computation parameter |
|AK1| Capillary cohesion first parameter |
|AK2| Capillary cohesion second parameter |
|DECCOH| Cohesion hardening shifting |
^ Line 5 (7G10.0) ^^
|PHICF| Final Coulomb's angle (in degrees) for compressive paths |
|PHIEF| Final Coulomb's angle (in degrees) for extensive paths (iff ILODEF=2) |
|RAYPHIC| Ratio between initial and residual friction angle for compressive paths |
|BPHI| Only if there is hardening/softening |
|AN| Van Eekelen exponent (default value = -0.229) |
|DECPHI| Coulomb's angle hardening shifting |
|RAYPHIE| Ratio between initial and residual friction angle for extensive paths (iff ILODEF=2) |
^ Line 6 (6G10.0) ^^
|COHF0| Residual value of cohesion for major principal stress 1 perpendicular to the bedding plane (parallel to the normal to the bedding plane) (if IANISO=0) |
|:::| = COHC0 = $c_0$ (if IANISO=1) (See paragraph 7.4) |
|COHFMIN| Minimal residual value of cohesion (if IANISO=0) |
|:::| = COHAISO = $A_{11}$ (if IANISO=1) (See paragraph 7.4) |
|COHF90| Residual value of cohesion for major principal stress 1 parallel to the bedding plane (perpendicular to the normal to the bedding plane) (if IANISO=0) |
|:::| = COHB1 = $b_1$ (if IANISO=1) (See paragraph 7.4) |
|ANGLEMIN| Angle between the normal to the bedding plane and the major principal stress 1 for which the cohesion is minimum (if IANISO=0) |
|:::| = COHB2 = $b_2$ (if IANISO=1) (See paragraph 7.4) |
|RAYCOH| Ratio between initial and residual cohesion |
|BCOH| Only if there is hardening/softening |
^ Line 7 (7G10.0) ^^
|SIGMAT0| Initial value of the uniaxial tensile strength |
|SIGMATF| Final value of the uniaxial tensile strength (for hardening/softening) |
|AKSIGMAT1| First parameter of Peron's formulation to account for the tensile strength evolution with suction |
|AKSIGMAT20| Initial value of the second parameter of Peron's formulation to account for the tensile strength evolution with suction |
|AKSIGMAT2F| Final value of the second parameter of Peron's formulation to account for the tensile strength evolution with suction (for hardening/softening) |
|BSIGMAT| Only if there is hardening/softening |
|PSI2| Dilatancy angle for the tensile criterion |
^ Line 8 (3G10.0) ^^
|EK1| Model parameter for non-linear elasticity (depending on ICOCA) |
|EK2| Model parameter for non-linear elasticity (depending on ICOCA) |
|EK3| Model parameter for non-linear elasticity (depending on ICOCA) |


===== Stresses =====

==== Number of stresses ====

4 for 2D 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)|$\sigma_{zz}$|

===== State variables =====
==== Number of state variables ====

54 for 2D plane strain analysis with bifurcation criterion (ICBIF=1) \\ 42 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)| Reduced deviatoric stress (varies from 0 to 1) |
|Q(4)| = 0 if the current state is elastic |
|:::| = 1 if the current state is elasto-plastic |
|Q(5)| Equivalent viscoplastic shear strain, i.e. the generalised plastic distortion, which increment is $\dot{\gamma}_{vp}=\sqrt{\frac{2}{3}\dot{e}_{ij}^{vp}\dot{e}^{vp}_{ij}}$) (see PLASOL) |
|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)| $=\varepsilon_{xx}$ |
|Q(9)| $=\varepsilon_{yy}$ |
|Q(10)| $=\varepsilon_{zz}$ |
|Q(11)| $=\gamma_{xy}=2.\varepsilon_{xy}$ |
|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 $\varepsilon^p_{eq}$ |
|Q(15)| Actualised value of cohesion for bedding perpendicular to 1st principal stress |
|:::| = COHF0 (if IANISO=0) or COHC0 (if IANISO=1) |
|Q(16)| Actualised value of cohesion $c$ |
|Q(17)| Actualised value of Coulomb's friction angle for compressive paths $\phi_C$ |
|Q(18)| Actualised value of Coulomb's friction angle for extensive paths $\phi_E$ |
|Q(19)| = 0 if the stress state is not at the criterion apex |
|:::| = 1 if the stress state is at the criterion apex |
|Q(20)| Number of sub-intervals used for the integration |
|Q(21)| Memory of localisation calculated during the re-meshing |
|Q(22)| ? |
|Q(23)| ? |
|Q(24)| ORIENTBED |
|Q(25)| Dilatancy angle in compression |
|Q(26)| Dilatancy angle in extension |
|Q(27)| Damage variable |
|Q(28)| x plastic deformation |
|Q(29)| y plastic deformation |
|Q(30)| z plastic deformation |
|Q(31)| xy plastic deformation |
|Q(32)| Saturated effective material tensile strength |
|Q(33)| First invariant of the strength sensor |
|Q(34)| Second invariant of the deviatoric strength sensor |
|Q(35)| Lode's angle |
|Q(36)| Effective material tensile strength |
|Q(37)| Volumic plastic strain generated by the tensile failure criterion |
|Q(38)| E1 |
|Q(39)| E2 |
|Q(40)| E3 |
|Q(41)| Saturation degree * suction |
|Q(42)| Second model parameter for Peron's formulation |
|Q(43)$\rightarrow$Q(54)| Reserved for bifurcation |
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