====== INTME2/INTME3 ======

===== Description =====

Constitutive law for mechanical contact for interface elements ([[elements:fail2|FAIL2B]]/[[elements:fain2|FAIN2B]]/[[elements:faif2|FAIF2B]] or [[elements:fail3|FAIL3B]]/[[elements:fain3|FAIN3B]]/[[elements:faif3|FAIF3B]]).

==== The model ====
This law is similar to the Coulomb's Law in 2D/3D and is used in mechanical analysis of problems involving unilateral contact between two bodies. Coulomb dry friction law is used. The contact condition is enforced via a penalty method or augmented Lagrangian method according to ISTRA(4). \\

The fault behaviour can be expressed according to two formulations: 
  * **Classical formulation** (IFRAC = 0) : \[\Delta \sigma = K_p\;\Delta V\]
  * **Goodman formulation** (IFRAC = 1) : \[\Delta\sigma = \frac{K_p}{\left(1+\frac{V}{D_0}\right)^{\gamma}}\Delta V \leftrightarrow V = D_0\left[\sqrt[1-\gamma]{\left(\frac{(1-\gamma)}{D_0\;K_n}\sigma'+1\right)}-1\right] \quad\text{and}\quad d-V=D_0\]

Thus: 
  * if contact pressure $\sigma'=0$ : Hydraulic aperture = $d = D_0$ and Fault closure = $V=0$
  * if contact pressure $\sigma'=-\infty$ : Hydraulic aperture = $d = 0$ and Fault closure = $V = -D_0$ (negative value)

{{:  laws:intme.png?700|  }}

==== Files ====

Prepro: LINTME.F \\
Lagamine: INTME2.F, INTME3.F

===== Availability =====

|Plane stress state| YES|
|Plane strain state| YES|
|Axisymmetric state| YES|
|3D state| YES|
|Generalized plane state| YES|

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

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

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 86|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (6I5) ^^
|ISOL| = 0 : use of total stresses in the constitutive law |
|:::| $\neq$ 0 : use of effective stresses in the constitutive law. See [[appendices:a8|Appendix 8]] |
|IFRAC| = 0 : Classical formulation of fault behaviour |
|:::| = 1 : Goodman formulation of fault behaviour | 
|NINTV| Only in 3D : number of sub-steps used to integrate numerically the constitutive equation in a time step (useful only in dynamic) |
|INDIC| Only in 3D : 0 or 1 to define the outside pressure used in case of no contact (see "Use" paragraph) |
|IREDUC| = 1 : Phi-C reduction method |
|:::| = 0 : nothing |
|ITYPEL| = 0 : for DAIL3 element (Nsig_meca = 6) |
|:::| $\neq$ 0 : for [[elements:fain3|FAIN3]] or [[elements:faif3|FAIF3]] element (Nsig_meca = 4) |

==== Real parameters ====

^ Line 1 (6G10.0) ^^
|AKP| Penalty coefficient on the contact pressure $K_p$ |
|AKTAU| Penalty coefficient on the shear frictional stress $K_{\tau}$Â |
|PHI| Coulomb's friction coefficient $\tan\phi$ |
|B| Cohesion |
|TAUMAX| Maximum contact friction (only for 2D state) (default value = $10^{20}$) |
|PRESID| Residual pressure |
^ Line 2 (3G10.0) ^^
|GAMMA| Exponent value of Goodman formulation (useful if IFRAC = 1 ) |
|D0| Maximal fault closure in absolute value for $\sigma'=-\infty$ (useful if IFRAC = 1) |
|RHO| Specific mass (useful if element is FAIF2 and IENTH = 1 in INTFL2 law) |

===== Stresses =====

==== Number of stresses ====

4 for both the 2-D and 3-D states \\

==== Meaning ====

The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates. \\

For the 2-D state:

|SIG(1)| normal contact pressure (> 0 if contact and < 0 if no contact) |
|SIG(2)| tangent contact stress |
|SIG(3)| meaningless |
|SIG(4)| meaningless |

For the 3-D state:

|SIG(1)| normal contact pressure |
|SIG(2)| tangent contact stress in the $\xi$ direction |
|SIG(3)| tangent contact stress in the $\eta$ direction |
|SIG(4)| reduced deviatoric stress = $\frac{||\tau||}{p'}$ if $p' > 0$ |

The $\xi$ and $\eta$ correspond to the intrinsic co-ordinates of the contact element FAIL3.\\

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

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

4 (+6 for 2D state, +4 for [[elements:fail3|FAIL3]], +2 for [[elements:fain3|FAIN3]]/[[elements:faif3|FAIF3B]]) \\

These are the 4 state variables related to the law, they are the first ones printed. After them, you find the state variables related to the contact geometry, 6 for the [[elements:fail2|FAIL2B]]/[[elements:fain2|FAIN2B]]/[[elements:faif2|FAIF2B]], 4 for the [[elements:fail3|FAIL3B]] and 2 for [[elements:fain3|FAIN3B]]/[[elements:faif3|FAIF3B]] their meaning are explained in the element section.


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

|Q(1)| = 0 if the current state is elastic (no sliding) |
|:::| = 1 if the current state is elastoplastic (sliding at contact) |
|:::| = -1 if no mechanical, nor thermal contact |
|Q(2)| amount of mechanical energy dissipated per unit area, due to friction |
|Q(3)| Fault closure (V : < 0 for closed fracture, > 0 for opened fracture) (useful if IFRAC = 1) |
|Q(4)| Hydraulic aperture $d$ (>0) (useful if IFRAC = 1) |

For [[elements:fail2|FAIL2B]]/[[elements:fain2|FAIN2B]]/[[elements:faif2|FAIF2B]] element :
|Q(5)| Number of foundation segment |
|Q(6)| Relative interpenetration distance $\Delta V = V-V_{ini}$ (<0 for compression cases and >0 for extension cases) |
|Q(7)| Jacobian |
|Q(8)| NOCO contact indicator given by CALFON subroutine |
|Q(9)| Relative tangential speed |
|Q(10)| Relative sliding |

For [[elements:fail3|FAIL3]] element :
|Q(5)| Number of foundation segment |
|Q(6)| Interpenetration distance $\lambda_c=\Delta V$ |
|Q(7)| Relative tangential speed n°1 |
|Q(8)| Relative tangential speed n°2 |

For [[elements:fain3|FAIN3]] element :
|Q(5)| Number of foundation segment |
|Q(6)| Interpenetration distance $\lambda_c=\Delta V$ |