${\bf U} = \mathbf{C}^{1/2}$ and its invariants in terms of $\bf C$ and its invariants

Abstract

We consider $N\times N$ tensors for $N= 3,4,5,6$. In the case $N=3$, it is desired to find the three principal invariants $i_1, i_2, i_3$ of $\bf U$ in terms of the three principal invariants $I_1, I_2, I_3$ of ${\bf C}={\bf U}^2$. Equations connecting the $i_\alpha$ and $I_\alpha$ are obtained by taking determinants of the factorisation \[\lambda^2{\bf I}- {\bf C} = (\lambda{\bf I}- {\bf U}) (\lambda{\bf I}+ {\bf U})\] and comparing coefficients. On eliminating $i_2$ we obtain a quartic equation with coefficients depending solely on the $I_\alpha$ whose largest root is $i_1$. Similarly, we may obtain a quartic equation whose largest root is $i_2$. For $N=4$ we find that $i_2$ is once again the largest root of a quartic equation and so all the $i_\alpha$ are expressed in terms of the $I_\alpha$. Then $\bf U$ and ${\bf U}^{-1}$ are expressed solely in terms of $\bf C$, as for $N=3$. For $N= 5$ we find, but do not exhibit, a twentieth degree polynomial of which $i_1$ is the largest root and which has four spurious zeros. We are unable to express the $i_\alpha$ in terms of the $I_\alpha$ for $N=5$. Nevertheless, $\bf U$ and ${\bf U}^{-1}$ are expressed in terms of powers of $\bf C$ with coefficients now depending on the $i_\alpha$. For $N=6$ we find, but do not exhibit, a 32 degree polynomial which has largest root $i_1^2$. Sixteen of these roots are relevant but the other 16, which we exhibit, are spurious. $\bf U$ and ${\bf U}^{-1}$ are expressed in terms of powers of $\bf C$. The cases $N>6$ are discussed. Keywords: Continuum mechanics, polar decomposition, tensor square roots, principal invariants, cubic equations, quartic equations, equations of degree 16