Smoothness of the scalar coefficients in the representation \(H(A)= \alpha I+ \beta A+ \gamma A^ 2\) for isotropic tensor functions of class \(C^ r\) (Q1905671)

An isotropic tensor function can often be expressed in different ways that conform to the same representation formula. An arbitrary choice would likely result in a representation with terms which are discontinuous, even if the initial isotropic tensor function is smooth. The author deals with the aforementioned problems for a simple representation formula, namely the Rivlin-Ericksen formula for isotropic functions that map symmetric tensors onto symmetric tensors. The restrictive assumption of polynomial dependence is replaced by the more general and natural requirement of smoothness. Representations in terms of repeated eigenvalues as well as in terms of principal invariants are given which can find useful applications in continuum mechanics and nonlinear elasticity. The two-dimensional case is considered separately.