The Eshelby inclusion problem

Everything in this section rests on one result. This page states it, and defines the three tensors it produces — , , . Their closed forms are the subject of the next page, Hill polarization tensors.

The problem

Fill with a homogeneous linear elastic medium of stiffness , and single out an ellipsoid centered at the origin (shape tensor , semi-axes   ; see Notation). Prescribe a uniform polarization stress inside the ellipsoid and zero outside, with no remote loading.

[6] showed that the resulting strain field is uniform inside the ellipsoid. This is the whole reason mean-field homogenization works, and it is specific to the ellipsoid: no other shape has it.

  is the Hill polarization tensor [7], [8]. It depends only on the ellipsoid's shape and orientation (through ) and on the reference medium (through ) — in particular it is independent of the ellipsoid's size.

Three tensors, one contraction apart

The same solution is written three ways in the literature. Knowing which is which avoids most confusion when comparing formulas across papers.

Eshelby tensor . Introduce the equivalent eigenstrain (stress-free strain)    . Then

which is Eshelby's original form. is dimensionless; has the dimension of a compliance.

Second Hill tensor . Asking for the stress inside the inclusion rather than the strain gives the dual statement

is what degenerates in a controlled way when the inclusion becomes flat, which is why the crack theory is built on it rather than on — see Crack opening displacement.

In MeanFieldHomogenization, and are hill_tensor and eshelby_tensor. There is no public accessor for : assemble it from when you need it,

P = hill_tensor(inclusion, C₀)
Q = C₀ - C₀ ⊡ P ⊡ C₀

which is all the crack machinery does internally before taking the flat limit.

The transport counterpart

Replace elasticity by a scalar diffusion problem — heat conduction, mass diffusion, electric conduction, Darcy flow. The unknown is a scalar potential , and the reference property is an order-2 conductivity . Fourier's law carries a minus sign that Hooke's does not, so the stress analog is taken to be minus the flux,

the convention fixed in Elasticity and transport: one set of formulas —   is then, in both theories, what the exterior transmits to the interior across a surface. With that substitution Eshelby's uniformity result holds literally: a uniform polarization inside the ellipsoid produces a uniform gradient inside it, and an order-2 Hill tensor plays the role of [8]:

which is the elastic     above with replaced by — no sign changed.

The two problems are handled by the same functions in MeanFieldHomogenization, which dispatch on the order of the property tensor passed in: an order-4 selects the elastic path, an order-2 the transport one. That single implementation is possible because of the convention above.

See Conduction and diffusion for the call.

Why this matters for a real material

A real heterogeneous material is not one ellipsoid in an infinite medium. What comes closest is the inhomogeneity problem — an ellipsoid of a different stiffness , loaded remotely by    — and it reduces to the inclusion problem above by the equivalent-polarization argument of Localization:

The step from there to an estimate of effective properties is the subject of the next two pages, and it has two parts:

1. each inclusion is treated as if it were alone in an infinite reference medium — this is what makes usable, and it is exactly the approximation that distinguishes one mean-field scheme from another (Homogenization schemes);

2. the choice of that reference medium is the scheme: the matrix itself (Mori–Tanaka), the effective medium being sought (self-consistent), or something in between (Localization).