Orthotropic viscous response of polar ice

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. A simple macroscopic description is that these material glide planes are rotated towards planes normal to an axis of compression, and away from planes normal to an axis of extension, inducing an instantaneous orthotropic viscous response with reflexional symmetries in the planes orthogonal to the current principal stretch axes. An associated orthotropic viscous law expresses the stress in terms of the strain-rate, strain, and three structure tensors based on the principal stretch axes. The fabric induced during differential stretchings along fixed principal axes, and the subsequent instantaneous viscous shear response in different planes due to the frozen fabric when the axial stress and strain-rate are removed, define a set of instantaneous directional viscosities in terms of the frozen principal stretches and the material response coefficients. Various inequalities and equalities between these viscosities are derived from the original rotation concepts, which, together with observed enhancement factors at large stretch and shearing, impose restrictions on the permitted response coefficients. It is shown how a simple viscous law can meet all these requirements, and such a law is illustrated for continued axial stretchings and shearing.