Crystallography & Symmetry
patterns derived from their groups
Concept

Compatibility — where it appears

The condition that two differently deformed regions can meet across a plane without the material tearing. It requires the two deformations to differ by a rank-one term, which happens exactly when a certain symmetric matrix has one positive eigenvalue, one negative and one zero.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

3m → 1: the two directions a wall between domains may take. The difference between the strains of two domains, sampled around a circle of directions: the first colour where that direction is stretched, the second where it is compressed. The two solid lines are the directions where it is neither, and those are the only orientations a straight wall between the two domains can take without straining itself — Sapriel's condition, one dimension down from the planes it is usually written for. There are exactly two, and that is not luck: the two domains are images of one another under the parent group, so their strains have the same area change and their difference changes no area at all. A form that changes no area takes both signs, and its zero set is a pair of directions.

The walls a strain permits

Two domains of different shape can only meet along a line neither of them stretches. That condition is a quadratic in a direction, so a pair of domains has exactly two permissible walls — and the reason there are always two rather than sometimes none is that their strains differ by no area at all.

applied · Domains Three variants, and not one undistorted plane. The three tetragonal variants a cubic parent produces, with the principal stretches of each. Every one of them has the same three numbers in a different order, and the middle one is not one — so none of the three leaves any plane undistorted, and none of them can meet the parent phase across an interface. That is the difficulty the whole of the crystallographic theory of martensite exists to resolve, and it is visible in one column.

The plane a deformation leaves alone

Two differently deformed regions can meet across a plane only if that plane is deformed identically from both sides — which forces the two deformations to differ by a rank-one term. Multiplying each side by its own transpose removes the rotation and leaves a condition on a signature: one positive eigenvalue, one negative, one exactly zero. In that form the classical rule that the middle principal stretch must be one is not quoted but derived, and it says that no single variant of a cubic-to-tetragonal transition can meet its parent at all.

applied · Domains Where the laminate's middle eigenvalue crosses zero. The middle eigenvalue of FᵀF − I for the average deformation of a twinned laminate, against the volume fraction of one variant. At both ends the laminate is a single variant and the value is well away from zero; in between it crosses, twice, and each crossing is a volume fraction at which the laminate can meet the parent phase across a plane. The two roots are complementary, which is the same plate with the two variants exchanged.

The plate that only fits when it is twinned

No single variant of a cubic-to-tetragonal transition can meet its parent across a plane. A fine mixture of two variants can, because its average deformation carries a free parameter — the volume fraction — and that parameter passes through the compatibility condition twice. Sweeping it gives the two fractions, the two habit planes, and one inequality: the plate exists exactly when the two principal stretches satisfy η₁² + η₃² ≤ 2.

applied · Domains

Named alongside it

The objects these essays reach for when they reach for this one.

Domain wallStrainTwinDomain stateEigenvalueFerroelasticityHabit planeSapriel conditionSpontaneous strainStrain tensorTwin law

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