Linear Isotropic

We assume linear elasticity for the solid phase, so we have the following formulation for the stress tensor:

(201)\[\begin{equation} \boldsymbol{\sigma}(\vec{u},p) = \boldsymbol{C}:\boldsymbol{\epsilon} - \alpha p \boldsymbol{I} = \lambda \boldsymbol{I} \epsilon_{v} + 2 \mu \boldsymbol{\epsilon} - \alpha \boldsymbol{I} p \end{equation}\]

where \(\lambda\) and \(\mu\) are Lamé’s parameters, \(\lambda = K_{d} - \frac{2 \mu}{3}\), \(\mu\) is the shear modulus, and the volumetric strain is defined as \(\epsilon_{v} = \nabla \cdot \vec{u}\). For isotropic linear elasticity, all components of \(C_{ikjl}\) are zero except for:

\[\begin{split}\begin{gathered} C_{1111} = C_{2222} = C_{3333} = \lambda + 2 \mu, \\ C_{1122} = C_{1133} = C_{2233} = \lambda, \\ C_{1212} = C_{2323} = C_{1313} = \mu. \\ \end{gathered}\end{split}\]

The deviatoric elastic constants are:

\[\begin{split}\begin{gathered} C_{1111}^{dev} = C_{2222}^{dev} = C_{3333}^{dev} = \frac{4}{3} \mu, \\ C_{1122}^{dev} = C_{1133}^{dev} = C_{2233}^{dev} = -\frac{2}{3} \mu, \\ C_{1212}^{dev} = C_{2323}^{dev} = C_{1313}^{dev} = \mu. \\ \end{gathered}\end{split}\]