When does elasticity
select a shape?

The Bitter–Crum theorem.

For a coherent two-phase system with spatially uniform, isotropic linear elastic moduli and purely dilatational phase eigenstrains, in an infinite or periodic medium, the elastic energy at fixed phase volumes is independent of inclusion size, shape and arrangement. The stress-free strain in each phase is held fixed. Uniform applied stress does not introduce shape selection under these homogeneous-modulus assumptions.

This does not mean the elastic energy or local stress vanishes. It means rearranging a fixed amount of each phase cannot lower the elastic energy under these conditions. Elastic anisotropy, modulus contrast, nondilatational misfit or boundaries that permit different relaxation can remove that restriction.

Reference: Fratzl, Penrose & Lebowitz, Modeling of Phase Separation in Alloys with Coherent Elastic Misfit, Theorem 2 and its stated assumptions.

See the result through the kernel.

For the scalar homogeneous isotropic dilatational model, B(n) becomes a direction-independent constant B₀. Under the stress-free mean condition used in our lesson, Parseval’s theorem gives:

For two fixed composition values cα and cβ and a β volume fraction φ, the variance is φ(1−φ)(cβ−cα)². Energy then depends on phase fraction, not morphology. For a diffuse field, fixed mean alone is insufficient: changing the distribution of composition values changes its variance and energy.

This is the scalar-kernel connection derived for this teaching example. Its 2D coefficient should not be substituted for a 3D material coefficient.

Eshelby’s ellipsoidal inclusion theorem.

A uniform eigenstrain prescribed within an ellipsoid embedded in an infinite, homogeneous, isotropic, linearly elastic medium produces a uniform constrained strain inside the ellipsoid. The Eshelby tensor maps eigenstrain to that interior strain:

S depends on ellipsoid shape and the elastic constants. Uniformity inside does not imply a uniform exterior field or equality between constrained strain and eigenstrain. This classic statement concerns a single inclusion; arbitrary interacting precipitates and finite boundaries require additional treatment.

Original source: J. D. Eshelby (1957), The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proceedings of the Royal Society A 241, 376–396.

The applied-stress result also called Eshelby’s theorem.

In coherent-phase-separation discussions, “Eshelby’s theorem” can instead refer to the result that, with homogeneous stiffness and uniform external stress, the external-stress contribution depends on phase volumes and fixed phase eigenstrains, not inclusion shape or arrangement. That statement concerns the external loading contribution, not the entire self-energy. See Theorem 1 in the Fratzl–Penrose–Lebowitz treatment.

Use the theorems as solver checks.

1. Isotropic limit: set C₄₄ = (C₁₁ − C₁₂)/2. For dilatational eigenstrain, the scalar kernel should lose its angular dependence.
2. Arrangement test: rearrange the same composition values on a periodic grid. A constant-kernel calculation should preserve elastic energy, even though its chemical gradient energy can change.
3. Anisotropic case: restore C₁₁ = 4, C₁₂ = 2, C₄₄ = 3. B depends on direction, so morphology selection is allowed.
4. Inclusion benchmark: compare a separate, sufficiently resolved isolated ellipsoidal inclusion calculation against Eshelby’s interior strain. A small periodic cell is not an infinite-medium benchmark.

The movie uses anisotropic stiffness: its Zener ratio 2C₄₄/(C₁₁−C₁₂) is 3. Its directional patterns therefore do not contradict the isotropic Bitter–Crum theorem. The movie solves a scalar composition model, not Eshelby’s isolated-inclusion problem.

Derive the Fourier kernel and elastic driving force →