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laws:orthoplatra
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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.
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.
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 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 |
laws/orthoplatra.txt · Last modified: 2020/08/25 15:46 (external edit)
