eshelby

eshelby - EN224: Linear Elasticity Division of Engineering...

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EN224: Linear Elasticity Division of Engineering 3.6 Eshelby Inclusion Problems Eshelby found an important application of the results outlined in the preceding section. Consider an infinite, homogeneous, isotropic, linear elastic solid. Suppose we introduce a uniform eigenstrain in the ellipsoidal region We can use the procedure outlined in the preceding section to compute the fields in the solid. For our present application, the body force is zero everywhere except on the surface of the ellipsoid, where the body force is singular Where n denotes the normal to the ellipsoid. The integrals for the potentials cannot be evaluated exactly (except for the special case of a spherical region) but they can be reduced to elliptic integrals. Generated by www.PDFonFly.com at 1/10/2010 9:47:22 PM URL: http://www.engin.brown.edu/courses/en224/eshelby/eshelby.html
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Remarkably, it turns out that the strain and stress fields inside the ellipsoid are uniform . Outside the ellipsoid, the fields are more complex, but can still be reduced to expressions involving a small number of elliptic integrals. We will give results only for fields within the ellipsoidal region here. The total strain is usually expressed as where is a function of the elastic properties of the solid and the semi-axes of the ellipsoid, and is known as the Eshelby tensor. Its components are comlicated, but here they are (at least for an isotropic solid) The remaining components may be computed by the cyclic permutation of (1,2,3). Any components that cannot be obtained in this way are zero: thus Assume that . Then, the
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eshelby - EN224: Linear Elasticity Division of Engineering...

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