On Itskov (2024) counterexample to the functional basis of isotropic vector and tensor functions by Shariff (2023)

Abstract

In the paper Itskov (Mechanics of Soft Materials 6:1, 20242), he gave a counterexample to show that elements of the functional basis by Shariff (Q. J. Mech. Appl. Math. 76, 143–161, 2023) do not represent isotropic invariants of the vector and tensor arguments and cannot thus be referred to as the functional basis. In this paper, we prove that his counterexample is incorrect.

Appendix

In view of Eq. 4, we can write

$$\begin{aligned} f(\varvec{A}_1,\varvec{A}_2) = g(\lambda _1,\lambda _2,\lambda _3, \varvec{v}_1,\varvec{v}_2,\varvec{v}_3,\varvec{A}_2) \,, \end{aligned}$$

(A1)

with the symmetry property

$$\begin{aligned} g(\lambda _1,\lambda _2,\lambda _3, \varvec{v}_1,\varvec{v}_2,\varvec{v}_3,\varvec{A}_2)=g(\lambda _2,\lambda _1,\lambda _3, \varvec{v}_2,\varvec{v}_1,\varvec{v}_3,\varvec{A}_2)=g(\lambda _3,\lambda _2,\lambda _1, \varvec{v}_3,\varvec{v}_2,\varvec{v}_1,\varvec{A}_2) =\text {etc.} \,. \end{aligned}$$

(A2)

Since g is an isotropic scalar function, we have,

$$\begin{aligned} g(\lambda _1,\lambda _2,\lambda _3, \varvec{v}_1,\varvec{v}_2,\varvec{v}_3,\varvec{A}_2) = g(\lambda _1,\lambda _2,\lambda _3, \varvec{Q}\varvec{v}_1,\varvec{Q}\varvec{v}_2,\varvec{Q}\varvec{v}_3,\varvec{Q}\varvec{A}_2\varvec{Q}^T) \,, { }\forall \varvec{Q}\in Orth^3 \,. \end{aligned}$$

(A3)

Since $\varvec{v}_i\cdot \varvec{v}_j = \delta _{ij}$ (the Kronecker delta), we then have, the basis containing the spectral elements

$$\begin{aligned} \lambda _i \,, { } {A}^{(2)}_{ij} \, \end{aligned}$$

(A4)

that are isotropic functions.