continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...

, Eshelby's inclusion problem refers to a set of problems involving

ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...

elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...

inclusions in an infinite elastic body. Analytical solutions to these problems were first devised by John D. Eshelby in 1957.Eshelby (1957).Eshelby (1959). Eshelby started with a thought experiment on the possible

stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...

strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...

, and

displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...

fields in a linear elastic body containing an inclusion. In particular, he considered the situation in which the inclusion has undergone a transformation (such as twinning or localized thermal expansion) 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 (also called the eigenstrain), has spawned a large amount of work in the

mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...

of composites. The results find their applications in the effective medium theory for heterogeneous elastic materials.

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Bibliography

* * {{Citation, last=Eshelby, first=J.D., year=1959, title=The elastic field outside an ellipsoidal inclusion, journal=

Proceedings of the Royal Society A ''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life s ...

, volume=252, number=1271, pages=561–569, doi=10.1098/rspa.1959.0173, bibcode=1959RSPSA.252..561E, s2cid=119853168

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See also