Coulomb.md

User Function Coulomb Friction Law

General Information

• USF Fortran File: USF_Sliding.f90

• USF Name: Friction_Coulomb

• Required Input Variable(s): A Flow Solution in Flow Solution Name, Normal Vector, Stress or the Effective Pressure variable.

General Description

The file USF_Sliding.f90 contains three user functions to apply non-linear friction at the base of glacier.

The first user function (Sliding_Weertman) is a non-linear Weertman-type friction law and is described here. The second user function (Friction_Coulomb) is a non-linear water-pressure-dependent friction law, as proposed by Schoof (2005) and Gagliardini et al. (2007), and is presented in this page. The third user function (Sliding_Budd) is described here and is from Budd et al 1984 (Annals of Glaciology 5, page 29-36).

The friction law in Friction_Coulomb is of the form: tau_b = C.N {[{ {chi . {u_b}^{-n} }/ {(1 + a . chi^q)} }]}^{1/n} . u_b where a = {(q - 1)^{q-1}}/{q^q} and chi = {u_b}/{C^n N^n A_s} The Slip Coefficient in Elmer is then given as C.N {[{ {chi . {u_b}^{-n} }/ {(1 + a . chi^q)} }]}^{1/n} When u_b < u_{t0}, u_b in the previous equation is replaced by u_{t0}.

The parameters to be given are:

• Friction Law Sliding Coefficient → A_s

• Friction Law Post-Peak Exponent → q >= 1

• Friction Law Maximum Value → C ~ max bed slope

• Friction Law Exponent → m = (n Glen's law)

• Friction Law Linear Velocity → u_{t0} The effective pressure is defined as N = -sigma_{nn} -p_w, where sigma_{nn} is the normal Cauchy stress and p_w the water pressure. If a variable Effective Pressure exists, it is used to evaluate directly N. Else, the normal Cauchy stress is estimated from the stress computed at previous timestep. The water pressure is prescribed as an External Pressure (Negative - Compressive convention, and therefore 'External Pressure' should be equal to the opposite of the water pressure in the sif).

SIF contents

The required keywords in the SIF file for this user function are:

!!! Bedrock Boundary Condition 
Boundary Condition 1
  Target Boundaries = 1

  Normal-Tangential Velocity = Logical True
  Flow Force BC = Logical True
  
  !! Water pressure given through the Stokes 'External Pressure' parameter 
  !! (Negative = Compressive)
  External Pressure = Equals Water Pressure
   
  Velocity 1 = Real 0.0
  
  Slip Coefficient 2 =  Variable Coordinate 1
    Real Procedure "ElmerIceUSF" "Friction_Coulomb"
  Slip Coefficient 3 =  Variable Coordinate 1
    Real Procedure "ElmerIceUSF" "Friction_Coulomb"
    
  !! Parameters needed for the Coulomb Friction Law
  Friction Law Sliding Coefficient = Real 4.1613e5  
  Friction Law Post-Peak Exponent  = Real 1.0      !(q=1)
  Friction Law Maximum Value = Real 1.0            !(C=1)
  Friction Law PowerLaw Exponent = Real 3.0        !(m = n = 3 Glen's law) 
  Friction Law Linear Velocity = Real 0.01         
End

Examples

The Coulomb friction law is tested in [ELMER_TRUNK]/elmerice/Tests/Friction_Coulomb with a direct input of the effective pressure and [ELMER_TRUNK]/elmerice/Tests/Friction_Coulomb_Pw with the effective pressure computed from the stress and a prescribed water pressure.

Reference

When this friction law is used, it can be cited using the following reference: Gagliardini O., D. Cohen, P. Råback and T. Zwinger, 2007. Finite-Element Modeling of Subglacial Cavities and Related Friction Law. J. of Geophys. Res., Earth Surface, 112, F02027.