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,

(108)#\[\begin{gathered} \vec{u}^+ - \vec{u}^- - \vec{d}(\vec{x},t) = \vec{0} \text{ on }\Gamma_f, \end{gathered}\]

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,

(109)#\[\begin{equation} \int_\Omega \rho(\vec{x}) \frac{\partial \vec{v}}{\partial t} \, d\Omega = \int_\Omega \vec{f}(\vec{x},t) \, d\Omega + \int_\Gamma \vec{\tau}(\vec{x},t) \, d\Gamma, \end{equation}\]

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

(110)#\[\begin{gather} \tau^+(\vec{x},t) = \boldsymbol{\sigma}^+ \cdot \vec{n} + \vec{\lambda} \\ \tau^-(\vec{x},t) = \boldsymbol{\sigma}^- \cdot \vec{n} - \vec{\lambda}, \end{gather}\]

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

(111)#\[\begin{equation} \int_{\Omega_f} \rho(\vec{x}) \frac{\partial \vec{v}}{\partial t} \, d\Omega = \int_{\Gamma_{f^+}} \boldsymbol{\sigma} \cdot \vec{n} + \vec{\lambda} \, d\Gamma + \int_{\Gamma_{f^-}} \boldsymbol{\sigma} \cdot \vec{n} - \vec{\lambda} \, d\Gamma. \end{equation}\]

Table 4 Mathematical notation for elasticity equation with infinitesimal strain and prescribed slip on faults.#
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