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====== BBM ======
Barcelona Basic Model
===== Description =====

Full law name : ALONSO-GENS BBM \\

An elasto-plastic constitutive law for saturated or partially saturated soils based on the CAMCLAY-type model.\\

It can take into account:
  * the influence of the LODE angle;
  * linear or non-linear elasticity;
  * non-associated plasticity;
  * the elasto-plasticity induced by the suction.

==== The model ====

This law is used for mechanical analysis of saturated and partially saturated elasto-plastic isotropic porous media undergoing suction changes.
=== Hardening forms ===

__Elasticity__
|IELAP| Expression | Parameters |
|0| \[\kappa=\kappa_0\] | $\kappa_0$ = KAPPA0 |
|1| \[\kappa = \kappa_0\left[1+\alpha_1.s+\alpha_2.\ln\left(\frac{s+u_{atm}}{u_{atm}}\right)\right]\] | $\kappa_0$ = KAPPA0 \\ $\alpha_1$ = KAPPA1 \\ $\alpha_2$ = KAPPA2 |

|IELAS| Expression | Parameters |
|0| \[\kappa_s=\kappa_{s0}\] | $\kappa_{s0}$ = KAPPAS0 |
|1| \[\kappa_s = \kappa_{s0}\left[1+\alpha_p.\ln\left(\frac{p}{u_{atm}}\right)\right].\exp(\alpha_s.s)\] | $\kappa_{s0}$ = KAPPAS0 \\ $\alpha_p$ = KAPPAS1 \\ $\alpha_s$ = KAPPAS2 \\ $u_{atm}$ = PATM |
|2|  Not defined  ||
|3| \[\kappa_s = \kappa_{s0}.(1-\alpha_s.s)\] | $\kappa_{s0}$ = KAPPAS0 \\ $\alpha_s$ = KAPPAS2 |
|4| \[\kappa_s = \kappa_{s0}.\exp(-\alpha_p.p)\] | $\kappa_{s0}$ = KAPPAS0 \\ $\alpha_p$ = KAPPAS1 |
|5| \[\kappa_s = \begin{cases} \kappa_{s0} & \quad \text{if } s>s^* \\ \kappa_{res} & \quad \text{if } s\leq s^*\end{cases}\] | $\kappa_{s0}$ = KAPPAS0 \\ $\kappa_{res}$ = KAPPAS1 \\ $s^*$ = KAPPAS2 |
|6| \[\kappa_s = \begin{cases} \kappa_{s0} & \quad \text{if } S_r\leq S_r^* \\ \kappa_{s0}(1-S_r)^{\gamma_{\kappa_s}}& \quad \text{if } S_r>S_r^*\end{cases}\] | $\kappa_{s0}$ = KAPPAS0 \\ $\gamma_{\kappa_s}$ = KAPPAS1 \\ $S_r^*$ = KAPPAS2 |
|7| \[\kappa_s = \begin{cases} \kappa_{s0} & \quad \text{if }  S_r\leq S_r^* \\ \kappa_{res} & \quad \text{if } S_r>S_r^*\end{cases}\] | $\kappa_{s0}$ = KAPPAS0 \\ $\kappa_{res}$ = KAPPAS1 \\ $S_r^*$ = KAPPAS2 |

__Yield surfaces__

{{ :laws:bbm.png?400 |}}

==== Files ====

Prepro: LBBM.F \\
Lagamine: BBMPIL.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| 812|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (16I5) ^^
|NINTV| > 0 : Number of sub-steps used to integrate numerically the constitutive equation in a time step |
|:::| = 0 : NINTV will be calculated in the law with DIV |
|ISOL| = 0 : Use of total stresses $\sigma_{t,ij}$ in the constitutive law |
|:::| = 6 : Use of net stresses $\sigma_{ij}$ in the constitutive law. The net stresses are defined as \[\sigma_{ij}=\sigma_{t,ij}-\max(u_a,u_w)\;\delta_{ij}\] with $u_a$ and $u_w$ the air and water pressures and $\delta_{ij}$ the Kronecker delta |
|IELA\\ (used in BBMELA)| = 0 : Non-linear elasticity \[K = \frac{1+e}{\kappa}.p\quad\text{and}\quad G=\frac{3}{2}.\frac{1-2\nu}{1+\nu}.K\] |
|:::| = 1 : Non-linear elasticity \[K = \frac{1+e}{\kappa}.p\quad\text{and}\quad G=G_0\] |
|IELAP\\ (used in BBMELA)| = 0 : Constant KAPPA ($\kappa$) |
|:::| > 0 : Variable KAPPA ($\kappa$) |
|:::| = 1 : $\kappa = \kappa_0\left[1+\alpha_1. s + \alpha_2. \ln\left(\frac{s+u_{atm}}{u_{atm}}\right)\right]$ |
|IELAS\\ (used in BBMINT)| = 0 : Constant KAPPAS ($\kappa_s$) |
|:::| > 0 : Variable KAPPAS ($\kappa_s$) |
|:::| = 1 : $\kappa_s = \kappa_{s0}\left[1+\alpha_p.\ln\left(\frac{p}{u_{atm}}\right)\right].\exp(\alpha_s.s)$ |
|:::| = 2 : Not defined |
|:::| = 3 : $\kappa_s = \kappa_{s0}.(1-\alpha_s.s)$ |
|:::| = 4 : $\kappa_s = \kappa_{s0}.\exp(-\alpha_p.p)$ |
|:::| = 5 : if $s\leq s^*$ : $\kappa_s = \kappa_{res}$, else : $\kappa_s=\kappa_{s0}$ |
|:::| = 6 : if $S_r>S_r^*$ : $\kappa_s=\kappa_{s0}(1-S_r)^{\gamma_{\kappa_s}}$, else : $\kappa_s=\kappa_{s0}$ |
|:::| = 7 : if $S_r>S_r^*$ : $\kappa_s=\kappa_{res}$, else : $\kappa_s=\kappa_{s0}$ |
|IVOID| = 0 : Initial void ratio |
|:::| = 1 : Updated void ratio |
|ILC| Shape of the LC curve in the $p-s$ plane |
|:::| = 0 : Original formulation (Alonso et al., 1990) |
|:::| = 1 : Modified version for $S_r^*$ (Dieudonné, 2016) |
|IDEV| Shape of the LC curve in the $p–q$ plane |
|:::| = 0 : Original formulation (Alonso et al., 1990) |
|ICS| Increase in cohesion with suction |
|:::| = 0 : Original formulation (Alonso et al., 1990) |
|IASSOC| = 0 : Non-associated plasticity (Alonso et al., 1990) |
|:::| = 1 : Associated plasticity |
|ILODEF| Shape of the yield surface in the deviatoric plane |
|:::| = 0 : Circle in the deviatoric plane |
|:::| = 1 : Smoothed irregular hexagon in the deviatoric plane |
|ILODEG| Shape of the flow surface in the deviatoric plane (meaningful if IASSOC$\neq$1) |
|:::| = 0 : Circle in the deviatoric plane |
|:::| = 1 : Smoothed irregular hexagon in the deviatoric plane |

==== Real parameters ====

^ Line 1 (3G10) ^^
|POROS0| Initial porosity |
|RHO| Solid specific mass |
|DIV| Parameter for the computation of NINTV in the law (for NINTV=0 only) |
^ Line 2 (5G10) ^^
|KAPPA0| 1st elastic parameter (relative to changes in stress) |
|KAPPA1| 2nd elastic parameter (relative to changes in stress) |
|KAPPA2| 3rd elastic parameter (relative to changes in stress) |
|NUG| Poisson’s ratio $\nu$ (if IELA = 0) |
|:::| Shear modulus G (if IELA = 1) |
|AI1MIN| Minimum value of $I_{\sigma}$ for non-linear elasticity |
^ Line 3 (6G10) ^^
|COH| Value of cohesion in saturated conditions |
|COH_PAR1| 1st parameter for the evolution of cohesion with suction |
|COH_PAR2| 2nd parameter for the evolution of cohesion with suction |
|PHIC| Friction angle (in degrees) for compressive paths |
|PHIE| Friction angle (in degrees) for extensive paths (only if ILODEF=2) |
|AN| Van Eekelen exponent (default value = -0.229) |
^ Line 4 (6G10) ^^
|LAMBDA0| Plastic coefficient in saturated conditions |
|P0ST| Pre-consolidation pressure in saturated conditions |
|PCrel| Relative reference pressure P0ST/PC for the definition of the LC curve |
|LC_PAR1| 1st parameter of the LC curve |
|LC_PAR2| 2nd parameter of the LC curve |
|LC_PAR3| 3rd parameter of the LC curve |
^ Line 5 (6G10) ^^
|KAPPAS0| 1st elastic parameter (relative to changes in suction) |
|KAPPAS1| 2nd elastic parameter (relative to changes in suction) |
|KAPPAS2| 3rd elastic parameter (relative to changes in suction) |
|LAMBDAS| Plastic suction coefficient |
|PATM| Atmospheric pressure |
|S0| Yield limit in term of suction (SI curve) |

===== 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 other cases:
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{xy}$|
|SIG(4)|$\sigma_{zz}$|

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

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

25 in all the other cases

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

|Q(1)| = 1 : Plane strain state |
|:::| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetric state |
|Q(2)| Specific mass |
|Q(3)| Porosity |
|Q(4)| Suction |
|Q(5)| = 0 if the current state is elastic |
|:::| = 1 if the current state is elasto-plastic (LC) |
|:::| = 2 if the current state is elasto-plastic (SI) |
|:::| = 12 if the current state is elasto-plastic (LC + SI) |
|Q(6)| Pre-consolidation pressure $p_0$ in saturated conditions |
|Q(7)| Current pre-consolidation pressure $p_0$ |
|Q(8)| Maximum suction $s_0$ |
|Q(9)| Apparent cohesion |
|Q(10)| Apparent resistance in extension |
|Q(11)| Elastic slope relative to changes in pressure ($\kappa$) |
|Q(12)| Cubic modulus (K) |
|Q(13)| Shear modulus (G) |
|Q(14)| Elastic slope relative to changes in pressure ($\kappa_s$) |
|Q(15)| Plastic slope relative to changes in pressure ($\lambda(s)$) |
|Q(16)| X deformation |
|Q(17)| Y deformation |
|Q(18)| Z deformation |
|Q(19)| XY deformation |
|Q(20)| Volumetric strain |
|Q(21)| Deviatoric strain |
|Q(22)| Number of sub-intervals used for the integration |
|Q(23)| Number of iteration used for the integration |
|Q(24)| Equivalent strain indicator $\alpha = \Delta\dot{\varepsilon}_{vol}\;\Delta t/\varepsilon_{vol}$ |
|Q(25)| = 0 if elastic, 1 if softening, 2 if hardening |
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