Induced anisotropy in large ice shields: theory and its homogenization

For polycrystalline ice, an isothermal flow law is derived from microscopic considerations concerning constitutive equations and kinematic assumptions. On the basis of an elasto-plastic decomposition of the deformation gradient on the grain level and by assuming a continuous distribution of different orientated grains in the vicinity of each material point the classical macroscopic field quantities are obtained by calculating the weighted mean values of the associated microscopic quantities. The weighting function is represented by a so called Orientation Distribution Function (ODF). For the general two dimensional (plane and rotationally symmetric) flow regime analytical representations of the ODF are derived under the assumption of a uniform stress distribution over all polycrystals (Sachs-Condition) and a plane or rotationally symmetric orientation distribution. Additionally, the influence of the macroscopic constitutive relations on the microscopic level is restricted to isotropic parts only. Simple examples are used to demonstrate the ability of the ODF to perform the evolving texture. The microscopic constitutive relation for the dissipation potential is assumed to be an objective function of the stress deviator and is expressed as a polynomial law up to the power $n_{max}=4$, as proposed by Lliboutry (1993). A second order structure tensor which depends on the ODF is introduced to consider induced anisotropy. The resulting macro fluidities (inverse viscosities) are then calculated from the analytical representation of the ODF for the case of uniaxial loading underlying linear $n_{max}=1$ and nonlinear $n_{max}=3$ material behaviour.