Deformation of an elliptical inclusion in two-dimensional incompressible power-law viscous flow

An iterative, semi-analytical solution is derived for deformation of an elliptical (in cross-section) power-law viscous inclusion within an infinite linear viscous matrix undergoing a general 2D incompressible flow. Finite-element numerical models are used to extend the analysis to that of a power-law viscous matrix. The general behaviour of a deformable elliptical inclusion is not dramatically changed by power-law viscous rheology, but the effective viscosity is now a function of the orientation and axial ratio of the inclusion. Overall, the effect is similar to a markedly increased viscosity ratio for a stronger inclusion, or a decreased ratio for a weaker inclusion, when compared to the linear viscous case. As a result, rather low reference state viscosity ratios between inclusion and matrix (e.g., 2 to 3, determined at the same effective strain rate for both materials) can produce marked differences in behaviour for the range of power-law stress exponents established experimentally for many minerals and rocks (typically 3–6). Even for very high strain within a shear zone (γ > 100), initially nearly circular inclusions (R < 2) can maintain low axial ratios (R < 2–3) and widely variable orientations. These inclusions deform internally and are not rigid, but continue to rotate or oscillate without strong elongation.