The Walpole basis

Transverse isotropy about an axis is the symmetry class where a well-chosen basis pays off most: in the Walpole basis [6] the double contraction becomes a   matrix product plus two scalar products, and inversion becomes a   matrix inverse plus two reciprocals. TensND stores such tensors as their coefficients on this basis (TensTI, src/tens_anisotropic.jl) rather than as 81 components.

The six tensors

From the axial and transverse projectors

build [6], [7]:

and any transversely isotropic order-4 tensor decomposes as

They are tens_W1 … tens_W6, collectively walpole_basis.

Kelvin–Mandel matrices

In the frame where  , with the ordering of Kelvin–Mandel representation:

Note that and are transposes of each other and are not major-symmetric individually; every other is.

The synthetic triplet

Gather the first four coefficients into a   matrix and keep the last two apart:

The Walpole basis is closed under double contraction, and in this notation the algebra is [6]

The reason is the multiplication table, which identifies with the matrix units of   matrices, while and are idempotents spanning two independent one-dimensional ideals. Every non-vanishing product is:

(row , column is  ; the table is not symmetric.)

Orthogonality and norms

For the Frobenius scalar product the basis is orthogonal but not orthonormal:

This single fact is what makes every transversely isotropic projection closed-form: the normal equations of Projection onto a symmetry class are diagonal, so each coefficient is an independent quotient  .

The isotropic tensors on this basis

The identity, spherical and deviatoric tensors of Isotropic tensors are transversely isotropic about any axis, so they have Walpole coefficients — and these are worth recording because they are easy to get wrong:

or, in the synthetic notation,

Both and have singular blocks ( ), as they must, being projectors.    is immediate on the table. Converting an isotropic tensor to this representation is fromISO.

\mathbb{I} is not the sum of the six

 : the sum overshoots by  , whose norm is . Likewise    and     . The three correct identities above are pinned by tests (Testing and conventions).

Major symmetry and storage

A major-symmetric transversely isotropic tensor — an elastic stiffness, for instance — therefore needs five coefficients, a general one six:

Storage	Coefficients	Case
TensTI{4,T,5}		major-symmetric,
TensTI{4,T,6}		general
TensTI{4,T,8}		full axially-invariant space, see The extended Walpole algebra

The widening is not cosmetic: the product of two major-symmetric Walpole tensors is generally not major-symmetric, because     unless and commute. TensTI{4,T,5} ⊡ TensTI{4,T,5} therefore returns an N=6 container.

The symmetrized basis sometimes met in the literature merges the pair,   , and relabels  ,  ; it spans exactly the five-dimensional major-symmetric subspace and is walpole_basis_sym.

Order-2 transverse isotropy

The same construction one order down: a transversely isotropic order-2 tensor is

with the transverse and the axial coefficient, stored as TensTI{2,T,2}. Products and inverses are again termwise, since and are complementary orthogonal projectors. The tensor is isotropic exactly when  . A third generator exists and is discussed on The extended Walpole algebra.