Invariants of C$^{1/2}$ in terms of the invariants of C

TL;DR

This paper derives simple, symmetric polynomial expressions relating the invariants of a tensor C and its square root C^{1/2}, facilitating easier computation and understanding of their mathematical relationship.

Contribution

It presents new, explicit polynomial formulas connecting the invariants of C and C^{1/2}, highlighting symmetries and simplifying their computation.

Findings

Derived a bivariate function relating I_1, I_2 to i_1, i_2

Expressed invariants using a single function call

Highlighted symmetries among the invariants

Abstract

The three invariants of C$^{1/2}$ are key to expressing this tensor and its inverse as a polynomial in C. Simple and symmetric expressions are presented connecting the two sets of invariants $I_1, I_2,I_3$ and $i_1, i_2,i_3 $ of C and C$^{1/2}$, respectively. The first result is a bivariate function relating $I_1, I_2$ to $i_1, i_2$. The functional form of $i_1$ is the same as that of $i_2$ when the roles of the C-invariants are reversed. The second result expresses the invariants using a single call to a single function. The two sets of expressions emphasize symmetries in the relations among these four invariants.