The influence of third shear stress invariant dependence in the isotropic viscous relation on the reduced model for ice-sheet flow

Abstract

Ice-sheet flow solutions commonly adopt a simple isotropic viscous law, in which the deviatoric stress is coaxial with the strain rate, and the single response function depends on only one invariant, the second principal invariant of the deviatoric stress. This can be correlated with single-stress component tests, which cannot, however, verify the validity of the simplification. Morland (2007) has shown how combined compression and shear tests can verify and correlate a general quadratic isotropic viscous relation, or simply third invariant dependence, and there is evidence that at least third principal invariant dependence is required. Morland (2007) showed that a significant quadratic term changes the relative stress magnitudes in the reduced model for ice-sheet flow, that crucial simplifications are not achieved and formally noted that dependence on the third invariant in the coaxial relation also prevented the simplification. It is now shown that, provided the third invariant dependence does not dominate the isotropic relation, further asymptotic expansion does yield the reduced model simplification, and the influence of different weightings of third invariant dependence is illustrated by a comparison of steady radial flow solutions over a flat bed. It is found that the third invariant dependence does not have a large influence in these examples.