Abstract
AbstractThe response of ice to applied stress on ice-sheet flow timescales is commonly described by a non-linear incompressible viscous fluid, for which the deviatoric stress has a quadratic relation in the strain rate with two response coefficient functions depending on two principal strain-rate invariants I2 and I3. Commonly, a coaxial (linear) relation between the deviatoric stress and strain rate, with dependence on one strain-rate invariant I2 in a stress formulation, equivalently dependence on one deviatoric stress invariant in a strain-rate formulation, is adopted. Glen's uni-axial stress experiments determined such a coaxial law for a strain-rate formulation. The criterion for both uni-axial and shear data to determine the same relation is determined. Here, we apply Steinemann's uni-axial stress and torsion data to determine the two stress response coefficients in a quadratic relation with dependence on a single invariant I2. There is a non-negligible quadratic term for some ranges of I2; that is, a coaxial relation with dependence on one invariant is not valid. The data does not, however, rule out a coaxial relation with dependence on two invariants.
Highlights
Ice-sheet flow plays a significant role in climate change, and flow solutions require a constitutive law for the stress dependence on ice deformation and strain rate, and on temperature
We correlate the two responses with a quadratic relation involving two response coefficient functions, both depending on only one, the second principal strain-rate invariant, which shows that a nontrivial quadratic term arises at both high and low stress level ranges; that is, the coaxial form with dependence on one invariant is not valid at all stress levels, including shear stress levels arising in ice-sheet flows which is where the viscous law is required
We have examined three different particular stress formulations of the general quadratic viscous fluid relation: coaxial with dependence on one strain-rate invariant, coaxial with dependence on two strain-rate invariants, and quadratic with dependence on one strain-rate invariant
Summary
Ice-sheet flow plays a significant role in climate change, and flow solutions require a constitutive law for the stress dependence on ice deformation and strain rate, and on temperature. Glen (1958) (acknowledging Fritz Ursell) presented this general quadratic viscous relation for the strain rate, but adopted the simple coaxial form proposed by Nye (1953) with dependence on one, the second principal, deviatoric stress invariant, which is a measure of the shear stress magnitude His pioneering uni-axial stress experiments at Cambridge (Glen, 1952, 1953, 1955, 1958) provided data to determine the one response coefficient function dependence on the one, the second principal, deviatoric stress invariant, for which he assumed a power law. We correlate the two responses with a quadratic relation involving two response coefficient functions, both depending on only one, the second principal strain-rate invariant, which shows that a nontrivial quadratic term arises at both high and low stress level ranges; that is, the coaxial form with dependence on one invariant is not valid at all stress levels, including shear stress levels arising in ice-sheet flows which is where the viscous law is required. This background of Steinemann’s largely ignored work is presented in the Preface of the book by Hutter (2020)
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