$\mathbf{U}={\mathbf{C}}^{1/2}$ and its invariants in terms of $\mathbf{C}$ 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 $\mathbf{U}$ in terms of the three principal invariants $I_1,I_2,I_3$ of $\mathbf{C}={\mathbf{U}}^2$. Equations connecting the $i_{\alpha }$ and $I_{\alpha }$ are obtained by taking determinants of the factorisation \[λ^2{\bf I}- {\bf C} = (λ{\bf I}- {\bf U}) (λ{\bf I}+ {\bf U})\] 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 $\mathbf{U}$ and ${\mathbf{U}}^{-1}$ are expressed solely in terms of $\mathbf{C}$, 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, $\mathbf{U}$ and ${\mathbf{U}}^{-1}$ are expressed in terms of powers of $\mathbf{C}$ 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. $\mathbf{U}$ and ${\mathbf{U}}^{-1}$ are expressed in terms of powers of $\mathbf{C}$. The cases $N>6$ are discussed. Keywords: Continuum mechanics, polar decomposition, tensor square roots, principal invariants, cubic equations, quartic equations, equations of degree 16

19 pages, 0 figures. Journal of Elasticity (2020)

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