Viscous response of polar ice with evolving fabric

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, deformation, and three structure tensors based on the principal stretch axes. General frame indifferent forms are analysed to determine the fabric induced during differential stretchings along fixed principal axes. Then, freezing the fabric by removal of the stress, and hence strain-rate, the instantaneous simple shear responses in different planes are determined and compared. The corresponding instantaneous viscosities are expressed in terms of the response coefficients of the constitutive model. A simple model is adopted to illustrate the evolution of the viscosities during axial stretchings and shearing.