====== PERSOL ======

===== Description =====

Non-associated PERZYNA type visco-plastic constitutive law with non-linear elasticity. Isotropic hardening/softening for friction angle, cohesion and pre-consolidation pressure. For solid elements at constant temperature.

==== The model ====

This law is used for mechanical analysis of visco-plastic isotropic porous media undergoing large strains.

==== Files ====

Prepro: LPERSOL.F \\
Lagamine: PERZINT.F

===== Availability =====

|Plane stress state| NO |
|Plane strain state| YES |
|Axisymmetric state| YES (no bifurcation analysis) |
|3D state| NO |
|Generalized plane state| NO |

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

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

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 69|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

LPARAM(5) to LPARAM(14) :
^ Line 1 (11I5) ^^
|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 |
|:::| = 1 : Use of effective stresses in the constitutive law (See [[appendices:a8|appendix 8]]) |
|IELA| = 0 : Linear elasticity |
|:::| > 1 : Non-linear elasticity for the moment MAX=4 |
|ILODEF| Deviatoric section shape of the loading surface in {$\sigma_1 ; \sigma_2 ; \sigma_3$} |
|:::| = 1 : Circle in the deviatoric plane |
|:::| = 2 : Smoothed irregular hexagon in the deviatoric plane |
|ILODEG| Deviatoric section shape of the potential surface in {$\sigma_1 ; \sigma_2 ; \sigma_3$} |
|:::| = 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 by the Taylor rule : PHMPS=$\phi_C-\Psi_C=\phi_E-\Psi_E$ |
|IECROUC| = 1 : no hardening in cap |
|:::| = 2 : Hardening in cap |
|IECROUD| = 1 : No hardening in failure |
|:::| = 2 : Hardening in failure |
|KMETH| = 3 : Mean vgrad method |
|:::| = 2 : Actualised vgrad method (=KJaum ?) |
|ICBIF| = 0 : Nothing |
|:::| = 1 : Rice bifurcation criterion is computed (only for 2D plane strain analysis) |
|IUNDIM| ??? |

==== Real parameters ====

^ Line 1 (6G10.0) ^^
|E_PAR1| First elastic parameter |
|E_PAR2| Second elastic parameter | 
|E_PAR3| Third elastic parameter |
|E_PAR4| Fourth elastic parameter |
|ECRO|Hardening parameter - **TO BE CHECKED**|
|H1| |
^Line 2 (6G10.0)^^
|PCONS0|Pre-consolidation pressure\\ If PCONSO0=0, OCR is used|
|OCR|Over Consolidation Ratio |
|AI1MIN| Minimum value of $I_{\sigma}$ for non-linear elasticity |
|PSIC | $\psi_C$ \\ IECPS = 1 : not used \\ IECPS = 0 :  PSIC = const = PARAM(32) |
|PSIE| $\psi_E$ \\ IECPS = 1 : not used \\ IECPS = 0 :  PSIE = const = PARAM(33) |
|PHMPS| TAYLOR constant, used if IECPS=1 \\ then PSIC = PHIC-PHMPS \\ PSIE = PHIE - PHMPS |
^Line 3 (6G10.0)^^
| PHIC0 | $\phi_{C0}$ ($\phi_{C}$ is the Coulomb angle in degrees for compression)|
| PHICF | $\phi_{Cf}$ ($\phi_{C}$ is the Coulomb angle in degrees for compression)|
| BPHI | $B_p$ |
| PHIE0 | $\phi_{E0}$ ($\phi_{E}$ is the Coulomb angle in degrees for extension)|
| PHIEF | $\phi_{Ef}$ ($\phi_{E}$ is the Coulomb angle in degrees for extension)|
|AN| Van Eekelen exponent (default value=-0.229) |
^Line 4 (5G10.0)^^
| COH0 | $c_0$ (COH is the cohesion value)|
| COHF | $c_f$ |
| BCOH | $B_c$ |
|POROS0| Initial soil porosity ($n_o$) |
|RHO| Specific mass |
^Line 5 (4G10.0)^^
| ALPHAC | Visco-plastic parameter for $\Phi_c=\left(\frac{f_c}{p_0}\right).\alpha_c$ |
| OMEGA | Viscosity parameter for $\gamma_c$ |
| AIOT | Viscosity parameter for $\gamma_c$ |
| PATM | Atmospheric pressure, defined two times for IELA=4 |
^Line 6 (4G10.0)^^
| A2D in degree ($\phi$) | Parameter $a_2$ for $\gamma_d=a_2\gamma_c$ |
| ALPHAD | Parameter $\alpha_d$ for $\Phi_d=\left(\frac{f_d}{p_0}\right)^{\alpha_d}$ |
|DIV| Parameter for the computation of NINTV (if NINTV=0)|
|BIOPT| |


| | IELA = 0 | IELA = 1 | IELA = 2 | IELA = 3 | IELA = 4 |
|E_PAR1| E | $\kappa$ | $\kappa$ | $\kappa$ | $\kappa$ |
|E_PAR2| $\nu$ | $\nu$ | $G_0$ | $G_0$ | $G_0$ |
|E_PAR3| | | | $K_0$ | Exp_N |
|E_PAR4| | | | ALPHA_2 | PATM |
|ECRO| ECRO | $\lambda$ \[ECRO=\frac{1+e_0}{\lambda-\kappa}\] | $\lambda$ \[ECRO=\frac{1+e_0}{\lambda-\kappa}\] | $\lambda$ \[ECRO=\frac{1+e_0}{\lambda-\kappa}\] | $\lambda$ \[ECRO=\frac{1+e_0}{\lambda-\kappa}\] |

Where : 
|$K_0$| Minimum value of the bulk modulus |
|Exp_N| $n$ parameter |
|$G_0$| Shear modulus |
|PATM| Atmospheric pressure |
|$\lambda$| Plastic slope in oedometer path |
|$\kappa$| Elastic slope in oedometer path |
|ECRO| Hardening parameter (TO BE CHECKED) |


===== Stresses =====

==== Number of stresses ====

6 for 3D state \\ 4 for the other cases

==== Meaning ====

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)|$\sigma_{zz}$|
|SIG(4)|$\sigma_{xy}$|

===== State variables =====

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

= 32 for 2D plane strain analysis with bifurcation criterion (ICBIF=1) \\ = 20 in all the other cases

==== List of state variables ====

|Q(1)| = 1 : Plane strain state |
|:::| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state |
|Q(2)| Specific mass $\rho$ actualised in the element routine |
|Q(3)| = 0 : Current state is elastic |
|:::| = 1,2 : Current state is inelastic CAP or D |
|Q(4)| Plastic work per unit volume ($W^p$) |
|Q(5)| Porosity : $n=\frac{e}{1-e}=\frac{\Omega_{voids}}{\Omega_{total}}$ |
|Q(6)| = $\sin(3\beta)$ where $-\frac{\pi}{6}\leq\beta\leq\frac{\pi}{6}$ is a Lode's angle |
|Q(7)| Pre-consolidation pressure $p_0$ |
|Q(8)| First stress invariant $I_{\sigma}$ |
|Q(9)| Stress deviator second invariant $II_s$ |
|Q(10)| |
|Q(11)| |
|Q(12)| |
|Q(13)| Actualised value of inelastic volume strain in CAP : $\varepsilon_v^c$ |
|Q(14)| Actualised value of equivalent plastic strain in DEV. : $\bar{e}_d$ |
|Q(15)| Actualised value of cohesion $c$ |
|Q(16)| Actualised value of Coulomb’s friction angle for compressive paths $\phi_C$ |
|Q(17)| Actualised value of Coulomb’s friction angle for extensive paths $\phi_E$ |
|Q(18)| Not used |
|Q(19)| Number of sub-intervals used for the integration NINTV |
|Q(20)| Memory of localisation calculated during the re-meshing |
|Q(21)$\rightarrow$Q(32)| Reserved for bifurcation |

===== Formulation =====

==== Loading and potential surfaces ====

The stresses and stress invariants are : 
\[I_{\sigma} = \sigma_{ij}\quad ; \quad \hat{\sigma}_{ij}=\sigma_{ij}-\frac{I_{\sigma}}{3}\delta_{ij} \]
\[II_{\hat{\sigma}}=\sqrt{\frac{1}{2}\hat{\sigma}_{ij}\hat{\sigma}_{ij}}\quad ;\quad III_{\hat{\sigma}} = \frac{1}{3}\hat{\sigma}_{ij}\hat{\sigma}_{jk}\hat{\sigma}_{ki}\]
\[\beta =-\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}}{2}\frac{III_{\hat{\sigma}}}{II^3_{\hat{\sigma}}}\right)\]

=== Criterion with friction angle different from 0 (Drücker Prager or Van Eekelen) ===

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 $\psi$ )

=== Criterion with friction angle equal to 0 (Von Mises criterion) ===

\[f=II_{\hat{\sigma}}-\frac{2c}{\sqrt{3}} = 0\]

==== 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 : \[\varepsilon_{eq}^p=\sqrt{\frac{2}{3}\hat{\varepsilon}_{ij}^p\hat{\varepsilon}_{ij}^p}\]

Hyperbolic functions are used :
  - If ILODE = 1 or 2 : \[\phi_C=\phi_{C0}+\frac{(\phi_{Cf}-\phi_{C0})\varepsilon_{eq}^p}{B_p+\varepsilon_{eq}^p}\]\[c=c_0+\frac{(c_f-c_0)\varepsilon_{eq}^p}{B_c+\varepsilon_{eq}^p}\]
  - Only if ILODE = 2 : \[\phi_E=\phi_{E0}+\frac{(\phi_{Ef}-\phi_{E0})\varepsilon_{eq}^p}{B_p+\varepsilon_{eq}^p}\] 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 below).

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

===== Structure of PARAM =====
|PARAM(1,ILAW)| ZERO | |
|PARAM(2,ILAW)| E_PAR1 | First elastic parameter |
|PARAM(3,ILAW)| E_PAR2 | Second elastic parameter |
|PARAM(4,ILAW)| E_PAR3 | Third elastic parameter |
|PARAM(5,ILAW)| E_PAR4 | Fourth elastic parameter |
|PARAM(6,ILAW)| COH | Cohesion value |
|PARAM(7,ILAW)| PCONS0 | Pre-consolidation pressure\\ If PCONSO0=0, OCR is used) |
|PARAM(8,ILAW)| AI1MIN | Minimum value of $I_{\sigma}$ for non-linear elasticity |
|PARAM(9,ILAW)| PHIC | Coulomb's angle (in degrees) for compression |
|PARAM(10,ILAW)| PHIE | Coulomb's angle (in degrees) for extension |
|PARAM(11,ILAW)| AN | Van Eekelen exponent (default value=-0.229) |
|PARAM(12,ILAW)| POROS0 | Initial soil porosity ($n_o$) |
|PARAM(13,ILAW)| ECRO | |
|PARAM(14,ILAW)| DIV | Parameter for the computation of NINTV (if NINTV=0) |
|PARAM(15,ILAW)| RHO | Specific mass |
|PARAM(16,ILAW)| BIOPT | ??? |
|PARAM(17,ILAW)| OCR | |
^ Visco-parameters for CAP case -"c" ^^^
|PARAM(18,ILAW)| ALPHAC | Visco-plastic parameter for $\alpha_c=\left(\frac{f_c}{p_0}\right).\alpha_c$ |
|PARAM(19,ILAW)| OMEGA | Viscosity parameter for $\gamma_c$ |
|PARAM(20,ILAW)| AIOT | Viscosity parameter for $\gamma_c$ |
|PARAM(21,ILAW)| PATM | Atmospheric pressure, defined two times for IELA=4 |
^ Visco-parameters for DEVIATORIC case -"d" ^^^
|PARAM(22,ILAW)| A2D in degree ($\phi$) | Parameter $a_2$ for $\gamma_d=a_2\gamma_c$ |
|PARAM(23,ILAW)| ALPHAD | Parameter $\alpha_d$ for $\Phi_d=\left(\frac{f_d}{p_0}\right)^{\alpha_d}$ |
^ Parameters for failure hardening laws ^^^
|PARAM(24,ILAW)| PHIC0 | $\phi_{C0}$ |
|PARAM(25,ILAW)| PHICF | $\phi_{Cf}$ |
|PARAM(26,ILAW)| BPHI | $B_p$ |
|PARAM(27,ILAW)| PHIE0 | $\phi_{E0}$ |
|PARAM(28,ILAW)| PHIEF | $\phi_{Ef}$ |
|PARAM(29,ILAW)| COH0 | $c_0$ |
|PARAM(30,ILAW)| COHF | $c_f$ |
|PARAM(31,ILAW)| BCOH | $B_c$ |
|PARAM(32,ILAW)| PSIC | $\psi_C$ \\ IECPS = 1 : not used \\ IECPS = 0 :  PSIC = const=PARAM(32) |
|PARAM(33,ILAW)| PSIE | $\psi_E$ \\ IECPS = 1 : not used \\ IECPS = 0 :  PSIE = const=PARAM(33) |
|PARAM(34,ILAW)| PHMPS | TAYLOR constant, used if IECPS=1 \\ then PSIC = PHIC-PHMPS \\ PSIE = PHIE - PHMPS |