The Eshelby inclusion problem

Everything in this section rests on one result. This page states it, and defines the three tensors it produces — $\mathbb{P}$, $\mathbb{S}$, $\mathbb{Q}$. Their closed forms are the subject of the next page, Hill polarization tensors.

The problem

Fill $\mathbb{R}^3$ with a homogeneous linear elastic medium of stiffness $\mathbb{C}$, and single out an ellipsoid $\mathcal{E}_{\boldsymbol{A}}$ centered at the origin (shape tensor $\boldsymbol{A}$, semi-axes $a\ge b\ge c$; see Notation). Prescribe a uniform polarization stress $\boldsymbol{\tau}$ inside the ellipsoid and zero outside, with no remote loading.

[3] 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.

\[\forall\,\underline{x}\in\mathcal{E}_{\boldsymbol{A}},\qquad \boldsymbol{\varepsilon}(\underline{x}) = -\,\mathbb{P}:\boldsymbol{\tau}.\]

$\mathbb{P} = \mathbb{P}(\boldsymbol{A},\mathbb{C})$ is the Hill polarization tensor [4], [5]. It depends only on the ellipsoid's shape and orientation (through $\boldsymbol{A}$) and on the reference medium (through $\mathbb{C}$) — 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 $\mathbb{S}$. Introduce the equivalent eigenstrain (stress-free strain) $\boldsymbol{\varepsilon}^{\star} = -\mathbb{C}^{-1}:\boldsymbol{\tau}$. Then

\[\boldsymbol{\varepsilon}(\underline{x}) = \mathbb{S}:\boldsymbol{\varepsilon}^{\star}, \qquad \boxed{\;\mathbb{S} = \mathbb{P}:\mathbb{C}\;}\]

which is Eshelby's original form. $\mathbb{S}$ is dimensionless; $\mathbb{P}$ has the dimension of a compliance.

Second Hill tensor $\mathbb{Q}$. Asking for the stress inside the inclusion rather than the strain gives the dual statement

\[\boldsymbol{\sigma}(\underline{x}) = -\,\mathbb{Q}:\boldsymbol{\varepsilon}^{\star}, \qquad \boxed{\;\mathbb{Q} = \mathbb{C} - \mathbb{C}:\mathbb{P}:\mathbb{C}\;}\]

$\mathbb{Q}$ 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 $\mathbb{P}$ — see Crack opening displacement.

In MeanFieldHom, $\mathbb{P}$ and $\mathbb{S}$ are hill_tensor and eshelby_tensor. There is no public accessor for $\mathbb{Q}$: assemble it from $\mathbb{P}$ 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 $T$, the flux is $\underline{q} = -\boldsymbol{K}\cdot\nabla T$, and the reference property is an order-2 conductivity $\boldsymbol{K}$. Eshelby's uniformity result holds unchanged: a uniform polarization flux inside the ellipsoid produces a uniform gradient inside it, and an order-2 Hill tensor $\boldsymbol{P}(\boldsymbol{A},\boldsymbol{K})$ plays the role of $\mathbb{P}$ [5]:

\[\nabla T(\underline{x}) = -\,\boldsymbol{P}\cdot\underline{\tau}_q, \qquad \boldsymbol{s} = \boldsymbol{P}\cdot\boldsymbol{K}, \qquad \boldsymbol{Q} = \boldsymbol{K} - \boldsymbol{K}\cdot\boldsymbol{P}\cdot\boldsymbol{K}.\]

The two problems are handled by the same functions in MeanFieldHom, which dispatch on the order of the property tensor passed in: an order-4 $\mathbb{C}$ selects the elastic path, an order-2 $\boldsymbol{K}$ the transport one.

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. The step from this idealized problem 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 $\mathbb{P}$ 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).