Analytical solutions for deformable elliptical inclusions in general shear

Abstract

Using Muskhelishvili's method, we present closed-form analytical solutions for an isolated elliptical inclusion in general shear far-field flows. The inclusion is either perfectly bonded to the matrix or, as in the case of a circular inclusion, to a possible intermediate layer. The solutions are valid for incompressible all-elastic or all-viscous systems. The actual values of the shear mondulus or viscosity in the inclusion, mantle and matrix can be different and no limits are imposed on the possible material property contrasts. The solutions presented are complete 2-D solutions and the parameters that can be analysed include all kinematic (e.g. stream functions, velocities, strain rates, strains) and dynamic parameters (e.g. pressure, maximum shear stress, etc.). We refrain from giving the tedious derivation of the presented solutions, and instead we focus on how to use the solutions, how to extract the parameters of interest and how to apply and verify them. In order to demonstrate the usefulness of Muskhelishvili's method for slow viscous flow problems, we apply our results to a clast in a shear zone and obtain important new insights, even for simple, circular inclusions. Another important application is the benchmarking of numerical codes for which the presented solutions are most suitable due to the infinite range of viscosity contrasts and the strong local gradients of properties and results. To stimulate a broader use of Muskhelishvili's method, all solutions are implemented in MATLAB and downloadable from the web.