The extended Walpole algebra

The classical Walpole basis of The Walpole basis spans the tensors that are transversely isotropic in the usual sense. That space is not the full space of tensors invariant under rotations about the axis: two generators are missing. This page derives the eight-dimensional space TensND actually implements (TensTI{4,T,8}, src/tens_anisotropic.jl), which has no counterpart in the Echoes manual or, as far as we are aware, in the standard references.

Why eight and not six

Fix a unit axis and let act by rotations about it. The space of minor-symmetric order-4 tensors is six-dimensional in the Kelvin–Mandel picture, and the tensors invariant under the action form the commutant of that representation. Decomposing the -dimensional Kelvin–Mandel space into isotypic components of :

Harmonic	Subspace	Dimension	Commutant	Parameters
	axial and in-plane spherical	2	real matrices
	axial shears	2
	in-plane deviatoric	2

By Schur's lemma the commutant of each isotypic block is a division algebra — for the trivial ( ) block, for each two-dimensional real irreducible — giving     real parameters. The classical Walpole basis captures the real parts , and drops the imaginary parts , .

The two extra generators

Let be the in-plane rotation generator,

which is antisymmetric and odd in . Then

In the Kelvin–Mandel frame with  :

Both matrices are antisymmetric, so

they are major-antisymmetric. They are therefore invisible to any major-symmetric description — which is exactly why they are absent from the classical basis, whose target was the elastic stiffness.

The algebra

The eight-dimensional space is a commutant, hence an algebra: closed under double contraction and under inversion. In terms of the block and the two complex numbers,

with an ordinary   matrix product and inverse, and ordinary complex products and inverses. Setting    recovers the real rules of The Walpole basis exactly.

What the extra generators do to order-2 contraction

and annihilate precisely the order-2 tensors that are themselves axially invariant:

They do not annihilate a general symmetric tensor

A common shortcut is to assume and never matter for order-4 : order-2 contraction. They do. maps the axial-shear pair onto itself rotated by a quarter turn, and does the same for the in-plane deviatoric pair   . Both results are symmetric and generally nonzero.

What survives is the weaker statement above, and it is enough for the rules that involve only and — which is why the machinery carries over unchanged to  . The   contraction methods do account for , and are checked against the generic dense route.

Order 2: three generators, not two

The same argument one order down. The space of order-2 tensors invariant under rotations about is three-dimensional:

the third generator being the antisymmetric in-plane rotation that a symmetric parametrization cannot represent. With  :

This is TensTI{2,T,3}; the familiar symmetric case is TensTI{2,T,2} with  .

When this matters

For an elastic stiffness or compliance — major-symmetric by construction — the extra coefficients vanish and the classical five- or six-parameter description is complete. They become essential for objects that are not major-symmetric, the archetype being a strain-concentration tensor

whose exact average over rotations about an axis lands in the full eight-dimensional space. Projecting such an object onto the five-parameter major-symmetric subspace — which is what a best-fit projection does, see Projection onto a symmetry class — silently discards the   split together with and . Averaging and projecting are different operations, and this space is where the difference lives.

Accessors: get_ℓ returns the six classical coefficients (dropping ), get_ℓ8 always returns the full eight-tuple, padding with zeros for N=5 and N=6 inputs.