Derivatives of the principal invariants of a second-order tensor
\textit{D. E. Carlson} and \textit{A. Hoger} [ibid. 16, 221-224 (1986; Zbl 0585.73006)] recently gave an elegant basis-free derivation of the following expression for the derivatives of the principal invariants \(I_ 1,I_ 2,...,I_ n\) of a second-order tensor \({\mathbb{A}}\) in n- dimensional space: \[ (1)\quad \frac{\partial I_ r}{\partial {\mathbb{A}}}=[\sum^{r-1}_{\beta =0}I_{\beta}(-{\mathbb{A}})^{r-\beta - 1}]^ T,\quad r=1,2,...,n, \] where \(I_ 0=1\) and \({\mathbb{A}}^ 0={\mathbb{I}}\). Representation (1) generalizes (basis-free) derivations given by e.g.: \textit{M. E. Gurtin} [An introduction to continuum mechanics (1981; Zbl 0559.73001)] in which \({\mathbb{A}}\) is taken to be invertible, and (componential) derivations by \textit{J. L. Ericksen} [Tensor fields. Encyclopedia of physics, Vol. III/1 (1959; Zbl 0102.405)] in which \({\mathbb{A}}\) is taken to be symmetric. The approach of Carlson and Hoger, however, is indirect in the sense that they use the expressions for the derivatives of moments given by \textit{C. Truesdell} and \textit{W. Noll} [The nonlinear field theories of mechanics. Encyclopedia of physics Vol. III/3 (1965; Zbl 0137.195)] and the expressions for principal invariants in terms of the moments given by \textit{F. R. Gantmacher} [The theory of matrices. Vol. I (1959); Russian original (1953; Zbl 0050.248)]. The purpose of this note is to prove (1) directly in two different ways, without recourse to the results of Truesdell and Noll and of Gantmacher.
scientific article; zbMATH DE number 4143530