### Wednesday, 13 October 2004 - 2:15 PM

### This presentation is part of : Micromechanics and Physics of Materials

### The Translation and Rotation of a Nominally Spherical Inclusion in an Elastic Medium

**M. Rahman**, Daimler-Chrysler Corporation, Daimler-Chrysler Corporation, 800 Chrysler Dr., CIMS 484-14-01, Auburn Hills, MI 48326
Elasticity literature abounds in solutions of problems for solids containing inclusions or cavities possessing a high degree of symmetry, e.g. those of spherical and ellipsoidal configurations. Although solutions of this type are important for synthesizing those for more complicated cases, their applicability is however somewhat restricted, because inclusions and cavities of such canonical shapes are uncommon of the irregular shapes usually encountered in practical applications. As a simple model of such asymmetrical shapes, we consider the example of a rigid inclusion in an elastic solid, whose shape deviates slightly from that of a perfect sphere, hereafter called the reference sphere. The inclusion is given a small translation and a rotation in some arbitrary directions. The problem then consists in determining the resulting elastic field in the medium. In terms of the small parameter characterizing the boundary perturbation, we reduce the problem to an infinite set of problems, each satisfying the equilibrium equations and some appropriate conditions at the surface of the reference sphere. We then use Love's representation for the elastic equilibrium equations in terms of three sets of spherical harmonics, which, in conjunction with Brenner’s methodology, furnishes elegant solutions to these problems. To the first order in the small parameter, we deduce explicit expressions that relate the displacement field as well as the total force and torque required to oppose the translation and rotation of the inclusion to its geometry.

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