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