Infinitesimal Strain (Bathe) and No Faults#

In this section we apply a similar approach to the one we use for the elasticity equation to the case of an incompressible material. As the bulk modulus (\(K\)) approaches infinity, the volumetric strain (\(\mathop{\mathrm{Tr}}(\epsilon)\)) approaches zero and the pressure remains finite, \(p = -K \mathop{\mathrm{Tr}}(\epsilon)\). We consider pressure \(p\) as an independent variable and decompose the stress into the pressure and deviatoric components. As a result, we write the stress tensor in terms of both the displacement and pressure fields,

(129)#\[\begin{equation} \boldsymbol{\sigma}(\vec{u},p) = \boldsymbol{\sigma}^\mathit{dev}(\vec{u}) - p\boldsymbol{I}. \end{equation}\]

The strong form is

(130)#\[\begin{gather} % Solution \vec{s}^T = \left( \vec{u} \quad \ p \right)^T, \\ % Elasticity \rho \frac{\partial^2\vec{u}}{\partial t^2} - \vec{f}(\vec{x},t) - \left(\boldsymbol{\sigma}^\mathit{dev}(\vec{u}) - p\boldsymbol{I}\right) = \vec{0} \text{ in }\Omega, % Pressure \vec{\nabla} \cdot \vec{u} + \frac{p}{K} = 0 \text{ in }\Omega, \\ % Neumann \boldsymbol{\sigma} \cdot \vec{n} = \vec{\tau} \text{ on }\Gamma_\tau, \\ % Dirichlet \vec{u} = \vec{u}_0 \text{ on }\Gamma_u, \\ p = p_0 \text{ on }\Gamma_p. \end{gather}\]
Table 5 Mathematical notation for incompressible elasticity with infinitesimal strain#

Category

Symbol

Description

Unknowns

\(\vec{u}\)

Displacement field

\(p\)

Pressure field (\(p>0\) corresponds to negative mean stress)

Derived quantities

\(\boldsymbol{\sigma}\)

Cauchy stress tensor

\(\boldsymbol{\sigma}^\mathit{dev}\)

Cauchy deviatoric 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 \(\rho \vec{g}\)