====== DAFA2 ======
===== Description =====
Full law name : EVP-DAFALIAS-KALIAKIN \\

DAFALIAS-KALIAKIN elasto-visco-plastic constitutive law for isotropic cohesive soils.

==== The model ====

This law is used for mechanical analysis of elasto-visco-plastic isotropic porous media undergoing large strains according to DAFALIAS-KALIAKIN "bounding surface model".\\

Use ISTRA(3) = -1 in the loading file.

==== Files ====

Prepro: LDAFA.F \\
Lagamine: INT2DA.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| 89|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (4I5) ^^
|NINTV| Number of sub steps used to integrate numerically the  constitutive equation in a time step |
|ISOL| = 0 : Use of total stresses in the constitutive law |
|:::| $\neq$ 0 : Use of effective stresses in the constitutive law (See annex 8) |
|ICBIF| = 0 : Nothing |
|:::| = 1 : Compute the bifurcation criterion |
|ISTRAIN| = 0 : Use of "CAUCHY" strains |
|:::| = 1 : Use of natural strains |

==== Real parameters ====

^ Line 1 (7G10.0) ^^
|$\lambda$| Slope of the consolidation line in a plot of $e$ vs. $\ln(p)$ |
|$\kappa$| Slope of the swell/recompression line in a plot of $e$ vs. $\ln(p)$ |
|$\phi_c$| Frictional angle in triaxial compression path |
|$\phi_e$| Frictional angle in triaxial extension path |
|G| Elastic shear modulus (if $\nu > 0$, G is useless in the simulation) |
|$\gamma$| Without signification for the moment (= 0) |
|$I_l$| Non-zero limiting value of $I$ and $I_0$ below which the relation between $I$ (or $I_0$) and void ratio changes continuously from logarithmic to linear (usually $I_l$ = $P_a$) |
^ Line 2 (7G10.0) ^^
|$P_a$| Atmospheric pressure |
|$R_c$| Parameter defining the shape of the bounding surface for ellipse 1 (corresponding to a state of triaxial compression, it may assume any value in the range $1.0\leq R\leq\infty$. Typical values : 2.00-3.00) |
|$A_c$| Parameter defining the shape of the hyperbolic portion of the bounding surface (corresponding to a state of triaxial compression. In theory, $0$\leq$A_c<\infty$, experimentally : 0.02-0.2) |
|t| Parameter defining the shape of the ellipse 2 portion of the bounding surface ($0.05\leq t\leq 0.95$) |
|$R_e/R_c$| Ratio of the values of the shape parameter associated with ellipse 1 in extension ($R_e$) and in compression ($R_c$) |
|$A_e/A_c$| Ratio of the values of the shape parameter associated with hyperbola in extension ($A_e$) and in compression ($A_c$) |
|c| Projection center parameter ($0.0\leq c < 1.0$) |
^ Line 3 (7G10.0) ^^
|$S_p$| Parameter controlling the size of elastic nucleus associated with plastic strains |
|$H_c$| Primary hardening parameter $H$ in a state of triaxial compression (typical values: 5-50) |
|m| Secondary hardening parameters which applies to both extension and compression (m=0.02 is recommended) |
|$H_e/H_c$| Ratio of the values of the primary hardening parameter $H$ in extension ($H_e$) and in compression ($H_c$) |
|s| Hardening parameter used only for the single ellipse version of the bounding surface (not available now) |
|w| Hardening parameter used only for the single ellipse version of the bounding surface (not available now) |
|$S_v$| Parameter controlling the size of elastic nucleus associated with visco-plastic response |
|:::| < 0 : The effect of viscosity will not be considered |
^ Line 4 (7G10.0) ^^
|V| Visco-plastic parameter (typical values : $10^7-10^8$ kPa-min) |
|n| Visco-plastic parameter (typical values : 0.7-10) |
|$e_{in}$| Initial void ratio |
|$\nu$| Poisson's ratio (if $\nu$ < 0, G is specified explicitly) |
|$p_c$| Initial size of the bounding surface ($p_c$ is the effective pre-consolidation pressure) |
|OCR| Over consolidation ratio (if OCR = 0 : we give $p_c$) |
|RHO| Specific mass |

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

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

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

22 (11 for axisymmetric state)

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

|Q(1)| Element thickness (t) in plane stress state |
|:::| = 1 : Plane strain state |
|:::| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state |
|:::| = 0 : 3D state |
|:::| Element thickness (t) in generalized plane state |
|Q(2)| e : Void ratio |
|Q(3)| $I_0$ : Intersection of the bounding surface with the positive $I$-axis |
|Q(4)| $\varepsilon_{kk}$ : Accumulated volumetric strain |
|Q(5)| $I$ : First invariant of the effective stress (corresponding to the current state) |
|Q(6)| J : Square root of the second deviatoric stress invariant (corresponding to the current state) |
|Q(7)| An index to projection zone |
|:::| = 1 : Indicates that the "image" state is on the ellipse 1 |
|:::| = 2 : Indicates that the "image" state is on the hyperbola |
|:::| = 3 : Indicates that the "image" state is on the ellipse 2 |
|Q(8)| $\sin(3\alpha)$ where $\alpha$ is the "Lode" angle |
|Q(9)| b : Projection parameter (b$\geq$1, if b=1 the current state is on the bounding surface) |
|Q(10)| Sum of $\dot{\varepsilon}_V \;dt$ |
|Q(11)| Actualised specific mass |
|Q(12)$\rightarrow$Q(23)| Modulus for the analysis of bifurcation |

{{ :laws:dafa2.png?600 |}}