Linear Isotropic Elastic Models#
We implement isotropic linear elasticity both with and without a reference stress and strain state. With a linear elastic material we often compute the deformation relative to an unknown initial stress and strain state. An initial undeformed configuration with zero stress and strain corresponds to reference stress and strain of zero.
Without a reference stress and strain state, we have
(44)#\[\sigma_{ij} = \lambda \epsilon_{kk} \delta{ij} + 2 \mu \epsilon_{ij}\]
and with a reference stress and strain state, we have
(45)#\[\sigma_{ij} = \sigma_{ij}^{ref} + \lambda\left(\epsilon_{kk} - \epsilon_{kk}^{ref}\right) \delta{ij} + 2 \mu \left(\epsilon_{ij} - \epsilon_{ij}^{ref}\right).\]
The mean stress is
(46)#\[\begin{split}\begin{gathered}
\sigma^{mean} = \frac{1}{3} \sigma_{kk}, \\
\sigma^{mean} = \frac{1}{3} \sigma_{kk}^{ref} + K\left(\epsilon_{kk} - \epsilon_{kk}^{ref}\right), \\
\end{gathered}\end{split}\]
where \(K = \lambda + 2 \mu/3\) is the bulk modulus. If the reference stress and reference strain are both zero, then this reduces to
(47)#\[\sigma^{mean} = K \epsilon_{kk}.\]
The deviatoric stress is
\[\begin{split}\begin{gathered}
\sigma_{ij}^{dev} = \sigma_{ij} - \sigma^{mean} \delta_{ij}, \\
\sigma_{ij}^{dev} = \sigma_{ij}^{ref} + \lambda \left(\epsilon_{kk} -
\epsilon_{kk}^{ref}\right) \delta_{ij} + 2 \mu \left(\epsilon_{ij} -
\epsilon_{ij}^{ref}\right) - \left[\frac{1}{3}\sigma_{kk}^{ref} +
\left(\lambda + \frac{2}{3}\mu\right)\left(\epsilon_{kk}-
\epsilon_{kk}^{ref}\right)\right] \delta_{ij}, \\
\sigma_{ij}^{dev} = \sigma_{ij}^{ref} - \frac{1}{3}\sigma_{kk}^{ref}\delta_{ij} + 2 \mu \left(\epsilon_{ij} - \epsilon_{ij}^{ref}\right) - \frac{2}{3} \mu\left(\epsilon_{kk}- \epsilon_{kk}^{ref}\right) \delta_{ij}. \\
\end{gathered}\end{split}\]
For isotropic linear elasticity, all components of \(C_{ikjl}\) are zero except for:
(48)#\[\begin{split}\begin{gathered}
C_{1111} = C_{2222} = C_{3333} = \lambda + 2 \mu, \\
C_{1122} = C_{1133} = C_{2233} = \lambda, \\
C_{1212} = C_{2323} = C_{1313} = \mu. \\
\end{gathered}\end{split}\]
The deviatoric elastic constants are:
(49)#\[\begin{split}\begin{gathered}
C_{1111}^{dev} = C_{2222}^{dev} = C_{3333}^{dev} = \frac{4}{3} \mu, \\
C_{1122}^{dev} = C_{1133}^{dev} = C_{2233}^{dev} = -\frac{2}{3} \mu, \\
C_{1212}^{dev} = C_{2323}^{dev} = C_{1313}^{dev} = \mu. \\
\end{gathered}\end{split}\]