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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.
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View Full DocumentRemarkably, 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 semiaxes 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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 Spring '05
 ALLENBOWER

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