Sign conventions

Slip is defined as

\[\boldsymbol{S} = \boldsymbol{u}^- - \boldsymbol{u}^+ \]

Let the orthogonal basis \(\boldsymbol{n},\boldsymbol{d},\boldsymbol{s}\) be given, where normal \(\boldsymbol{n}\) points from the “-“-side to the “+”-side, \(\boldsymbol{d}\) is the dip direction, and \(\boldsymbol{s}\) is the strike direction. The slip-rate vector is defined as

\[\boldsymbol{V} = [\boldsymbol{\dot{S}}\cdot \boldsymbol{d}, \boldsymbol{\dot{S}}\cdot \boldsymbol{s}], \]

the shear traction vector is

\[\boldsymbol{\tau} = [\boldsymbol{d}\cdot \sigma\boldsymbol{n}, \boldsymbol{s}\cdot \sigma\boldsymbol{n}], \]

and the normal stress is given by

\[\sigma_n = \boldsymbol{n}\cdot \sigma \boldsymbol{n}. \]

Note that in 2D we drop the second component of the slip-rate and shear traction vector.

The friction law is given by

\[-(\boldsymbol{\tau}^0 + \boldsymbol{\tau}) = (\sigma_n^0 - \sigma_n) f(|\boldsymbol{V}|,\psi)\frac{\boldsymbol{V}}{|\boldsymbol{V}|} + \eta \boldsymbol{V}, \]

where \(\boldsymbol{\tau}^0\) and \(\sigma_n^0\) are pre-stresses. We take \(\sigma_n^0\) to be positive in compression, thus the sign is different to \(\sigma_n\).