Pure shear and simple shear deformation of inhomogeneous viscous fluids. 1. Theory

Abstract

This paper is an analysis of deformation of inhomogeneous materials by simple and pure shear. The model adopted is that of a homogeneous Newtonian fluid matrix in which are embedded small, circular or elliptical particles. The particles are also viscous bodies but they may differ from the matrix in coefficient of viscosity. The ratio of the coefficient of the viscosity of a particle to the coefficient of viscosity of the matrix is called the viscosity ratio — it is an important factor which controls the change in particle shape during the deformation. First, the change in shape of a single elliptical particle embedded in a viscous matrix during pure shear deformation of the particle-matrix system is examined. An equation is derived which relates the change in shape of the particle, the viscosity ratio and the total finite strain experienced by the system, if the particle is aligned with its axes parallel to the strain axes. If the particle does not have its axes parallel to the strain axes, it experiences both rotation and change in shape. Deformation paths illustrating this change in orientation and shape, with increasing pure shear, are obtained numerically. The simple shear deformation of a circular particle in a viscous matrix is also examined numerically and deformation paths relating the change in particle shape, the orientation of the particle major axis, the viscosity ratio and the amount of simple shear are calculated. The single particle results are then applied to multiparticle systems. The development of preferred orientations during deformation is discussed and an equation is derived to calculate the change in viscosity ratio with increasing concentration of particles in the system. The theoretical results are checked experimentally using ethyl cellulose-benzyl alcohol solutions to represent both particles and matrix.