The Walpole basis

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See also

Theory: Transversely isotropic parametrizations · Isotropic tensors

Manual: Structured tensors · Parametrizations and conversions

Tutorials: The Walpole basis end to end · Acoustic and Hill tensors · Elastic Green's functions · A two-point Green tensor between offset frames

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Transverse isotropy about an axis is the symmetry class where a well-chosen basis pays off most: in the Walpole basis (Walpole, 1984) 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 (Walpole, 1981; Walpole, 1984):

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 reads (Walpole, 1984)

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 iso_to_ti.

\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.