Modelling the bulk viscosity of two-phase mixtures in terms of clast shape

Abstract

The rheological behaviour of many rock-types will depend on how mixtures of different minerals and rock clasts behave together. This paper develops two-dimensional modelling of the bulk viscosity of two-phase viscous mixtures with different shape fabrics in pure shearing, based on self consistent mechanics and behaviour of inclusion-matrix systems. An equation is presented for the bulk viscosity in terms of the phase viscosity contrast, phase fractions, and clast shape. Results are given for idealised two-dimensional mixtures of circles, squares or ellipses with a variety of phase viscosity contrasts, in pure shearing parallel/perpendicular to the shape fabric. The normalised bulk viscosities of all these mixtures fall between the theoretical upper and lower viscous bounds given by homogeneous strain-rate and homogeneous stress, respectively, but nearer to the upper bound with increasing shape fabric. Mixtures with a particular geometry are shown to have a constant ‘relative horizontal distance’ between these two bounds, as proposed by P.D. Bons.Using graphs for the normalised bulk viscosities of mixtures with variably elliptical aligned shape fabrics, the results for fabrics orthogonal to pure shearing can be extrapolated to: (a) diagonal fabrics in pure shearing, and (b) orthogonal or diagonal fabrics in simple shearing. These reveal the bulk viscous anisotropy of mixtures with shape fabric, in two dimensions. It is speculated that during a progressive finite deformation, two-phase mixtures might undergo changes in bulk viscosity over time, with steady stiffening during pure shear, and perhaps cyclic stiffening and softening during simple shear. The two-dimensional models provide an approximation to the three-dimensional properties of mixtures with shape fabrics, with potential applications to rocks, including conglomerates.