Two-inclusion interaction tensors

Before this page

Concentration and contribution tensors, whose one-site picture this page extends to a pair of inclusions, and the sign of the Green operator fixed in the Conventions, which this page follows.

Every one-site scheme of the package needs one object: the Hill tensor of a single inclusion in a reference medium. The two N-body schemes need one more — the tensor measuring the field one inclusion induces in another. This page defines it, gives the closed forms, and states why they are exact.

Notation follows the Nomenclature: underline for vectors, bold for order 2, blackboard bold for order 4. Two symbols are introduced here.

Symbol	Object	Elasticity	Conduction
/	Green operator of the reference	order 4	order 2
/	interaction tensor of two inclusions	order 4	order 2

Molinari and El Mouden (1996) write the interaction tensor and Brisard et al. (2014) write it ; the letter is kept here because a Greek capital carries no order in this typeface convention.

Everything below is written once and read twice, in elasticity and in transport, using the dictionary of Elasticity and transport: one set of formulas:  ,  ,   . Order-2 objects keep the bold typeface (, , ); no sign changes between the two columns.

1. The Lippmann-Schwinger equation

With a homogeneous reference and the polarization    , the local problem is equivalent to the integral equation of (Zeller and Dederichs, 1973), written here in the sign convention of (Brisard et al., 2023):

whose kernel is the Green operator of the reference medium, built from the Green function by

the brackets denoting symmetrization on and on . In conduction the same object is minus the Hessian of the scalar Green function ,

The leading minus is what makes this convention the coherent one. As a distribution the conduction kernel splits into the regular part above and a Dirac self term,

and the transform of the whole is

which is the convention of Brisard et al. (2023): a positive semi-definite operator. The regular part averages to zero over any ball centered on the source, so the whole interior average is the Dirac term — which is exactly of a sphere. Plus the Hill tensor, positive definite as is: the identity tying this page to the rest of the package. (The functions of the package evaluate the regular part; the Dirac term is what self_interaction_tensor carries.)

2. Definition

For two inclusions (receiver) and (source) whose centers are separated by ,

so that   is the average field induced in by a uniform polarization carried by . Over a single inclusion the same integral gives

so the one-inclusion case is exactly the     of the Eshelby problem. See self_interaction_tensor and the Mori-Tanaka limit on the cluster-model page.

Two sign conventions exist

The package follows Brisard et al. (2023): the Green operator maps a polarization onto minus the induced field, its Fourier symbol   is positive, and   , coherent with the Hill tensor.

Molinari and El Mouden (1996) and Berveiller et al. (1987) write instead     , so their is the opposite of and their self term is   . Any formula transcribed from them — their Appendix A component table in particular — must be flipped before it is compared with anything on this page. The closed form below is their table with the sign already flipped, which is why its is positive where theirs is negative.

3. Exact closed forms for balls and disks

The solid mean-value expansion of a smooth field over a ball of radius reads

and applying it once for the source and once for the receiver gives

The elastic Green function is biharmonic, so   away from the source and the expansion terminates: the formula is exact for two non-overlapping balls at any separation. In conduction is harmonic,  , and the interaction is exactly the point-dipole field.

Elasticity, 3D

Let   be the unit vector along the line of centers,  , and

— note the sign of : Molinari and El Mouden (1996) write  , and the flip to this package's convention is applied here, once. In the frame whose third axis is , after their Appendix A and Berveiller et al. (1987):

This set is transversely isotropic about and satisfies    identically, so it is carried exactly by five Walpole coefficients rather than by an 81-component array — the storage described in Conventions:

Equivalently, basis-free, with    and   :

Conduction

The kernel is harmonic, so the dipole field is exact and the receiver radius drops out entirely:

The two-dimensional form is taken literally from (Brisard et al., 2023) — same sign, same prefactor — and the test suite checks it component by component. It is the sharpest available statement that the package and its normative reference share one convention.

Elasticity, plane strain

being the plane deviatoric projector. Note that does not depend on the reference Poisson ratio, and that both terms of the bracket above carry the same overall convention: and are negated together, never one without the other.

4. The vanishing isotropic part

For any two distinct inclusions,

i.e. the interaction tensor has no isotropic part (in conduction, it is traceless). The decomposition into isotropic and anisotropic parts being linear, any sum of interaction tensors inherits the property. The consequence is sharp: for an arrangement of cubic symmetry the effective bulk modulus is blind to the spatial distribution and coincides exactly with the Mori-Tanaka estimate. Only the shear response sees the arrangement.

5. General ellipsoids

Beyond balls the series does not terminate, and (Brisard et al., 2014), §4.2, expand the regular part of the kernel about the line of centers. In moment form, with the normalized second moment of a region about its centroid — for an ellipsoid of semi-axes ,    in the principal frame —

Truncating there is method = :multipole, the default for non-spherical geometries. It is asymptotic in (inclusion size / center distance) and reduces to the exact ball formula when both moments are isotropic. method = :quadrature integrates the definition directly by a product rule — geometry-agnostic, far slower, and the oracle the closed forms are validated against.

6. Anisotropic reference media

Everything above assumes an isotropic reference, whose Green operator is a closed form. That is not a restriction of the method, only of the kernel, and it is lifted by the Barnett line integral of Barnett (1972) and Willis (1975): for an arbitrary anisotropic stiffness the displacement Green function is

being the acoustic (Christoffel) tensor and the contour the unit circle in the plane perpendicular to  . The integrand is smooth and periodic, so a Gauss-Legendre rule converges fast, and the whole expression is homogeneous of degree in .

The Green operator takes two more derivatives of this. They are obtained by differentiating the quadrature itself with forward-mode AD rather than by differentiating the line integral by hand — exact, and reusing one verified expression instead of introducing a second. See green_function_aniso and green_operator_aniso; the dispatcher green_operator keeps the closed form whenever the reference is isotropic, which matters because the anisotropic route is some three orders of magnitude dearer.

In conduction no quadrature is needed at all — the anisotropic scalar Green function is elementary,

and its Hessian is written out directly.

Two consequences worth keeping in mind when reading an anisotropic result:

• the closed forms of the previous sections no longer apply, not even for a ball pair. Their exactness rested on the isotropic Green function being biharmonic, which a general anisotropic one is not, so the series does not terminate and every pair goes through the truncated multipole expansion;

• the isotropic part no longer vanishes.    is a property of the isotropic kernel, so with an anisotropic reference the bulk response does see the spatial arrangement. The statement "a cubic array keeps the Mori-Tanaka bulk modulus exactly" holds for an isotropic matrix only.

The one case still open is plane-strain elasticity with an anisotropic reference, whose Green function needs the Stroh formalism rather than the Barnett integral; green_operator raises rather than returning an isotropic approximation.

Why this matters for multiscale chaining

This is not a corner case. A cluster or equivalent-inclusion estimate on a cubic array is anisotropic — its two shear constants differ — so using one N-body result as the reference medium of another scale requires exactly this kernel. Chaining the two schemes across scales was impossible without it; see the manual and scripts/92.

7. Periodic images

Under a periodic boundary treatment each source carries a family of images and the interaction becomes a lattice sum, truncated to a cluster of radius :

Summed over all of the series is only conditionally convergent and the summation order has to be prescribed — the difficulty Brisard et al. (2023) meet, where a naive real-space lattice sum needs a heuristic correction. The cluster truncation above does not have that problem, and Appendix B of Molinari and El Mouden (1996) says why: the kernel integrates to zero over the exterior of a sphere centered on the receiver,

so the neglected images contribute a vanishing amount as grows. Summing over a sphere of images rather than a box is therefore not a detail but the reason the truncation is legitimate.

Where to go next

The next chapter returns to a single inclusion, in its flat limit: Crack opening displacement and compliance describes a crack by how much it opens rather than by a volume fraction and a stiffness. The two schemes built on come at the end of the chapter on schemes: The cluster model solves for the mean strain of every inclusion inside a cluster of neighbors, and The equivalent inclusion method poses the same problem as a Galerkin discretization that also yields a rigorous bound. The functions evaluating are documented in API — Interactions.