#### Vol. 2, No. 9, 2007

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

### Andrew N. Norris

Vol. 2 (2007), No. 9, 1805–1812
##### 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 $\left\{{I}_{1},{I}_{2},{I}_{3}\right\}$ and $\left\{{i}_{1},{i}_{2},{i}_{3}\right\}$ 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 function call. The two sets of expressions emphasize symmetries in the relations among these four invariants.

##### Keywords
invariants, finite elasticity, stretch tensors, polar decomposition