## User Function Coulomb Friction Law

### General Informations

• 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 dependant 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:

where

and

The Slip Coefficient in Elmer is then given as

When , in the previous equation is replaced by .

The parameters to be given are:

• `Friction Law Sliding Coefficient`
• `Friction Law Post-Peak Exponent`
• `Friction Law Maximum Value` ~ max bed slope
• `Friction Law Exponent` → m = (n Glen's law)
• `Friction Law Linear Velocity`

The effective pressure is defined as , where is the normal Cauchy stress and the water pressure. If a variable `Effective Pressure` exists, it is used to evaluate directly . 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.