Eshelby's inclusion


In continuum mechanics, Eshelby's inclusion problem refers to a set of problems involving ellipsoidal elastic inclusions in an infinite elastic body. Analytical solutions to these problems were first devised by John D. Eshelby in 1957.
Eshelby started with a thought experiment on the possible stress, strain, and displacement fields in a linear elastic body containing an inclusion. In particular, he considered the situation in which the inclusion has undergone a transformation but its change in shape and size are restricted because of the surrounding material. In that situation, the inclusion and the surrounding material remains in a stressed state. Also the strain states in the body and the inclusion are potentially inhomogeneous and complicated.
Eshelby found that the resulting elastic field can be found using a "sequence of imaginary cutting, straining and welding operations." Eshelby's finding that the strain and stress field inside the ellipsoidal inclusion is uniform and has a closed-form solution, regardless of the material properties and initial transformation strain, has spawned a large amount of work in the mechanics of composites.
The results find their applications in the effective medium theory for heterogeneous elastic materials.