Elasticity with Infinitesimal Strain and Prescribed Slip on Faults#
For each fault, which is an internal interface, we add a boundary condition to the elasticity equation prescribing the jump in the displacement field across the fault,
where \(\vec{u}^+\) is the displacement vector on the “positive” side of the fault, \(\vec{u}^-\) is the displacement vector on the “negative” side of the fault, \(\vec{d}\) is the slip vector on the fault, and \(\vec{n}\) is the fault normal which points from the negative side of the fault to the positive side of the fault. We enforce the jump in displacements across the fault using a Lagrange multiplier corresponding to equal and opposite tractions on the two sides of the fault.
We apply conservation of momemtum,
to a fault interface \(\Omega_f\) with boundaries \(\Gamma_{f^+}\) and \(\Gamma_{f^-}\). For a fault interface, the body force is zero, \(\vec{f}(\vec{x},t) = \vec{0}\). The tractions on the positive and negative fault faces are
where \(\vec{\lambda}\) is the Lagrange multiplier that corresponds to the fault traction generating the prescribed slip and \(\boldsymbol{\sigma}^+\) and \(\boldsymbol{\sigma}^-\) are the stresses in the domain at the positive and negative sides of the fault. Thus, for a fault interface, we have
Category |
Symbol |
Description |
---|---|---|
Unknowns |
\(\vec{u}\) |
Displacement field |
\(\vec{v}\) |
Velocity field |
|
\(\vec{\lambda}\) |
Lagrange multiplier field |
|
Derived quantities |
\(\boldsymbol{\sigma}\) |
Cauchy stress tensor |
\(\boldsymbol{\epsilon}\) |
Cauchy strain tensor |
|
Common constitutive parameters |
\(\rho\) |
Density |
\(\mu\) |
Shear modulus |
|
\(K\) |
Bulk modulus |
|
Source terms |
\(\vec{f}\) |
Body force per unit volume, for example \(\rho \vec{g}\) |
\(\vec{d}\) |
Slip vector field on the fault corresponding to a jump in the displacement field across the fault |