Elastic Anisotropy of Cubic Crystals

Material Preset

Stiffness Constants (GPa)

C₁₁ 522

C₁₂ 204

C₄₄ (shear) 160

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Zener ratio A

1.01

E<100>

411 GPa

E<111>

414 GPa

E<110>

412 GPa

How it works

In a cubic single crystal, stiffness is not the same in every direction — pulling along the cube edge <100> stretches the lattice differently than pulling along the body diagonal <111>. The directional Young's modulus follows from the crystal's compliance tensor S_ij (the inverse of the stiffness tensor C_ij):

1/E(n) = S11 − 2·(S11 − S12 − S44/2)·Γ
Γ = l1²l2² + l2²l3² + l3²l1²   (direction cosines l1,l2,l3)

S11 = (C11+C12) / [(C11−C12)(C11+2C12)]
S12 = −C12 / [(C11−C12)(C11+2C12)]
S44 = 1 / C44

Zener anisotropy ratio: A = 2·C44 / (C11 − C12)

Γ = 0 exactly along the <100> cube axes and reaches its maximum of 1/3 along the <111> body diagonals — so the sign of (S11 − S12 − S44/2) decides whether a crystal is stiffest along its edges or its diagonals.

A = 1 — elastically isotropic (no direction dependence). Tungsten sits almost exactly here.
A > 1 — stiffer along <111> than <100> (most cubic metals: Cu, Fe, Ni, Al).
A < 1 — stiffer along <100> than <111> (many ionic crystals, e.g. NaCl).

The lobed surface is the 3D polar plot of E(θ,φ): its radius at every direction is the Young's modulus you would measure pulling a rod cut along that crystallographic direction. Drag to rotate; the wireframe cube marks the <100> crystal axes.

Tungsten (nearly isotropic)

Stiff direction (high E) Compliant direction (low E)

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