On the functional basis of isotropic vector and tensor functions by Shariff (2023)

Abstract

In the paper Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) a functional basis of a system of vectors and symmetric tensors is proposed. The functional basis is expressed in terms of eigenvalues and eigenvectors of the first tensor and includes a smaller number of terms in comparison to the classical irreducible representation (see, e.g., Boehler, J. Appl. Math. Mech. 57:323–327, 1977; Pennisi and Trovato, Int. J. Eng. Sci. 25:1059–1065, 1987). In the present contribution, we 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. To this end, a counterexample with two symmetric tensors is considered. Under an arbitrary orthogonal transformation the functional basis (Shariff, Q. J. Mech. Appl. Math. 76:143–161, 2023) of these two tensors should remain constant but it does change in contrast to the classical representation.