Stress and strain-rate formulations for fabric evolution in polar ice

Abstract

Re-orientation of individual crystal glide planes, as isotropic surface ice is deformed during its passage to depth in an ice sheet, creates a fabric and associated anisotropy. We re-examine an orthotropic viscous law which was developed to reflect the induced anisotropy arising from the mean rotation of crystal axes during deformation. This expresses the deviatoric stress, the stress formulation, in terms of the strain-rate, strain, and three structure tensors based on the principal stretch axes, and involves two fabric response coefficient functions which determine the strength of the anisotropy. A validity condition implicitly relates the two response functions, so the model law has only one independent fabric response function. A modified formulation is now presented in which the two fabric response coefficients are expressed as functions of different invariant arguments, and the validity condition becomes an explicit algebraic relation between the two functions. The response can therefore be described explicitly in terms of a single fabric response function. An analogous orthotropic viscous law for the strain-rate, the strain-rate formulation, akin to the conventional “flow law” for isotropic ice, expressed in terms of the deviatoric stresss, strain and the three structure tensors, is also constructed. Correlations with complete (idealised) uni-axial compression and shearing responses are made for the stress formulation, to determine the fabric response function which would yield these responses.