Elasticity with Infinitesimal Strain and No Faults#

We begin with the elasticity equation including the inertial term,

(12)#\[\rho \frac{\partial^2\vec{u}}{\partial t^2} - \vec{f}(\vec{x},t) - \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} (\vec{u}) = \vec{0} \text{ in }\Omega,\]

(13)#\[\boldsymbol{\sigma} \cdot \vec{n} = \vec{\tau}(\vec{x},t) \text{ on }\Gamma_\tau,\]

(14)#\[\vec{u} = \vec{u}_0(\vec{x},t) \text{ on }\Gamma_u,\]

where \(\vec{u}\) is the displacement vector, \(\rho\) is the mass density, \(\vec{f}\) is the body force vector, \(\boldsymbol{\sigma}\) is the Cauchy stress tensor, \(\vec{x}\) is the spatial coordinate, and \(t\) is time. We specify tractions \(\vec{\tau}\) on boundary \(\Gamma_\tau\), and displacements \(\vec{u}_0\) on boundary \(\Gamma_u\). Because both \(\vec{\tau}\) and \(\vec{u}\) are vector quantities, there can be some spatial overlap of boundaries \(\Gamma_\tau\) and \(\Gamma_u\); however, a degree of freedom at any location cannot be associated with both prescribed displacements (Dirichlet) and traction (Neumann) boundary conditions simultaneously.

Table 2 Mathematical notation for elasticity equation with infinitesimal strain.#
Category	Symbol	Description
Unknowns	\(\vec{u}\)	Displacement field
	\(\vec{v}\)	Velocity 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 \(\rho \vec{g}\)