Ellipsoidal inclusions

The ellipsoid is the one shape for which the strain is uniform inside the inclusion, which is what makes a Hill tensor exist at all (The Eshelby inclusion problem). An Ellipsoid is built from its semi-axes, in any order — the constructor sorts them decreasing and permutes the local frame to match, so    always holds downstream.

using MeanFieldHomogenization, TensND
E, ν = 210e3, 0.3
λ = E*ν/((1+ν)*(1-2ν)); μ = E/(2*(1+ν))
C₀ = TensISO{3}(3*(λ+2μ/3), 2μ)

# Sphere
hill_tensor(Ellipsoid(1.0), C₀)

# Prolate spheroid
hill_tensor(Ellipsoid(3.0, 1.0, 1.0), C₀)

# 2D ellipse
hill_tensor(Ellipsoid(1.0, 0.5), TensISO{2}(3*(λ+2μ/3), 2μ))

The two aspect ratios that recur everywhere are the in-plane   and the flatness  ; both are read off the stored semi-axes:

ell = Ellipsoid(3.0, 1.5, 0.8)
a, b, c = ell.semi_axes
(a = a, b = b, c = c, η = b / a, ω = c / a, shape = MeanFieldHomogenization.shape_trait(ell))

(a = 3.0, b = 1.5, c = 0.8, η = 0.5, ω = 0.26666666666666666, shape = Triaxial)

plotly_scene(shape_traces(ell); uid = "man-ellipsoid", height = 440,
    title = "Ellipsoid(3.0, 1.5, 0.8) — dashed guides are the principal semi-axes")

Every other spheroid and the degenerate limits below are drawn side by side in The inclusion zoo.

Degenerate limits

When an Ellipsoid constructor receives a real semi-axis equal to Inf or 0, it returns the appropriate dedicated type:

Call	Returned type	See
Ellipsoid(Inf, b, c) with b, c > 0	Cylinder	cylindrical inclusions
Ellipsoid(a, b, 0) with a, b > 0	EllipticCrack	cracks
Ellipsoid(Inf, b, 0) with b > 0	RibbonCrack	cracks
Ellipsoid(Inf, Inf, c)	ArgumentError (slab, out of scope)
Ellipsoid(a, 0, 0)	ArgumentError (needle, out of scope)

The detection is active only for real element types; with symbolic types (SymPy.Sym, Symbolics.Num) call the dedicated constructor (Cylinder, EllipticCrack, RibbonCrack) explicitly.