Isotropic tensors

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See also

Theory: The Walpole basis

Manual: Structured tensors · Elastic moduli · Parametrizations and conversions

Tutorials: Elastic Green's functions · The elastic sphere, solved symbolically · A two-point Green tensor between offset frames · Acoustic and Hill tensors

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The most constrained symmetry class, and the template for every other one: a small closed algebra in which products and inverses are computed on a couple of scalars rather than on components. Implemented by TensISO in src/tens_isotropic.jl, in arbitrary dimension .

The two projectors

With the order-2 identity and    the order-4 identity on symmetric tensors,

extracts the spherical part of a symmetric tensor and the deviatoric one. They are complementary orthogonal projectors:

Their Frobenius norms are the dimensions of the subspaces they project onto:

In TensND the triple is iso_projectors / iso_projectors, and individually tens_J4, tens_K4, tens_Id4, tens_Id2.

The algebra

Every isotropic minor-symmetric order-4 tensor is

and since and are orthogonal idempotents the algebra is that of a pair of independent scalars:

In elasticity   and   in 3-D. TensISO{4} stores exactly the pair ; TensISO{2} stores the single scalar of .

Isotropization

The closest isotropic tensor to an arbitrary , for the Frobenius distance, is its orthogonal projection onto . Because the two projectors are orthogonal with known norms, the normal equations are diagonal and the answer is closed-form (Bornert et al., 2001):

the general- denominator being  . This is isotropify, and it is the simplest instance of the general machinery of Projection onto a symmetry class: no orientation to optimize, because an isotropic tensor has none.

Isotropization does not commute with inversion

Projecting a stiffness and projecting the corresponding compliance give different isotropic materials. This is not a defect of the implementation but of the Euclidean distance itself, which is not invariant under inversion. Distances that are — log-Euclidean, power-Euclidean, arctan-Euclidean — are constructed and compared by Morin et al. (2020); TensND implements the Euclidean one, so the choice of which of or to project is the user's and must be stated when a result is reported.

Storage

Storage	Coefficients	Case
TensISO{2,dim,T,1}
TensISO{4,dim,T,2}

One shape per order, invariant under the whole orthogonal group: an isotropic tensor is its own transpose, and the products and inverses above stay in the shape, so there is no wider one as there is for the transversely isotropic and orthotropic classes.

Relation to the wider classes

Isotropy is the intersection of every other class, so an isotropic tensor satisfies all three predicates:

isisotropic, istransverselyisotropic and isorthotropic reflect this — all three return true on . Conversion up the chain is explicit: iso_to_ti re-expresses an isotropic tensor on the Walpole basis about a chosen axis, and iso_to_ortho on an orthotropic frame; the coefficients that result are computed on Walpole basis and Orthotropy.