${\bf U} = \mathbf{C}^{1/2}$及其不变量根据$\bf C$及其不变量
${\bf U} = \mathbf{C}^{1/2}$ and its invariants in terms of $\bf C$ and its invariants
摘要：
我们考虑$N= 3,4,5,6$的$N\times N$张量。在$N=3$的情况下，期望根据${\bf C}={\bf U}^2$的三个主要不变量$I_1, I_2, I_3$来找到$\bf U$的三个主要不变量$i_1, i_2, i_3$。通过取因子分解\[λ^2{\bf I}- {\bf C} = (λ{\bf I}- {\bf U}) (λ{\bf I}+ {\bf U})\]的行列式并比较系数，获得连接$i_α$和$I_α$的方程。在消除$i_2$，我们得到一个四次方程的系数完全取决于$I_α$，其最大根是$i_1$。同样，我们可以得到一个四次方程，其最大根是$i_2$。对于$N=4$，我们发现$i_2$再次是四次方程的最大根，因此所有的$i_α$都用$I_α$表示。然后$\bf U$和${\bf U}^{-1}$仅用$\bf C$表示，与$N=3$相同。对于$N= 5$，我们发现，但不展示，二十度多项式的$i_1$是最大的根，并有四个虚假的零。我们无法用$N=5$的$I_α$来表达$i_α$。然而，$\bf U$和${\bf U}^{-1}$是用$\bf C$的幂表示的，其系数现在取决于$i_α$。对于$N=6$，我们发现，但不展示，一个32度多项式，其中有最大的根$i_1^2$。这些根中有16个是相关的，但我们展示的其他16个是虚假的。$\bf U$和${\bf U}^{-1}$以$\bf C$的幂表示。的情况下$N>6$进行了讨论。保留字：连续统力学，极分解，张量平方根，主不变量，三次方程，四次方程，16次方程
英文摘要：
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 $i_α$ and $I_α$ 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 $I_α$ 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 $i_α$ are expressed in terms of the $I_α$. 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 $i_α$ in terms of the $I_α$ 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 $i_α$. 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
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