The viscous relation for the initial isotropic response of ice

Abstract

Uni-axial compressive stress and simple shear stress experiments on ice determine the corresponding minimum strain-rates following the very small primary elastic strain or viscoelastic creep for a range of applied constant stresses. Assuming these responses are those of a non-linear incompressible simple viscous fluid, they can be related to the two response functions, each depending on two invariants, of the general viscous law, and more specifically to simpler special cases for which the response functions are determined by the uni-axial and shear responses. In particular, the customary co-axial relation with one response function of one invariant argument can only apply if there is an explicit relation between the uni-axial and shear responses. Given that experimental data shows that this relation is not satisfied, then the uni-axial and shear stress responses determine the two response functions, each depending on only one invariant, of a non-co-axial quadratic relation. Single independent strain-rate component responses can determine dependence on only one invariant. However, there is no data for both these responses, and correlations have been made with data from combined uni-axial stress and shear stress tests carried out in laboratories in Tasmania. Expressions for the two general response functions are derived in terms of the responses in combined uni-axial stress and shear stress tests, from which it is found that the quadratic coefficient is very significant at the majority of the data points. This implies that a co-axial relation depending on two invariants, suggested by Steinemann (1954) when his data showed that the customary co-axial relation failed, is not valid. It is also seen that the available data points cover little of the two invariants plane, so dependence on two invariants could not be deduced. The data is used to determine the two response functions of the quadratic, non-co-axial, relation, each with dependence on only one invariant, a measure of the shear strain-rate. Note, though, that the experiments performed in Tasmania do not measure the complete strain-rate response, but instead make an assumption that the longitudinal strain-rate is zero which is not confirmed, so the present construction hinges on the validity of that approximation. The theory shows how accurate data can be used to construct a viscous law.