Deviatoric stress

Given a mean (normal) stress defined as \(\sigma_m = \frac{1}{3}(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})\), the deviatoric stress tensor, \(\underline{\underline{\tau}}\), is defined as \(\underline{\underline{\tau}}=\underline{\underline{\sigma}}-\sigma_m\underline{\underline{I}}\) such that:

\[\begin{split} \underline{\underline{\tau}} = \begin{bmatrix} \sigma_{xx}-\sigma_m & \tau_{xy} &\tau_{xz} \\ \tau_{yx} & \sigma_{yy}-\sigma_m & \tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{zz}-\sigma_m \end{bmatrix} \end{split}\]

Note that only the normal stresses \(\sigma_{ij}\) differ between \(\underline{\underline{\tau}}\) and \(\underline{\underline{\sigma}}\) – shear stresses \(\tau_{ij}\) are the same.

\(\sigma_m\) is the component of the stress that does not vary with direction (i.e. its isotropic). It tries to change the volume of our material. Removing it from the total stress \(\underline{\underline{\sigma}}\) leaves only the component of stress that does vary with direction, \(\underline{\underline{\tau}}\), which tries to change the shape of our material.

\(\underline{\underline{\tau}}\) is particularly useful for glaciology because we can often assume that ice is incompressible, so to a first approximation, \(\sigma_m\) does not affect flow, whereas \(\underline{\underline{\tau}}\) does. It is \(\underline{\underline{\tau}}\) which appears in ice flow laws.