论文标题
$ {\ bf u} = \ Mathbf {c}^{1/2} $及其不变性的$ \ bf c $及其不变性
${\bf U} = \mathbf{C}^{1/2}$ and its invariants in terms of $\bf C$ and its invariants
论文作者
论文摘要
我们考虑$ n \ times n $张量，$ n = 3,4,5,6 $。在这种情况下，在三个主要不变式$ i_1，i_2，i_3 $ $ {\ bf c} = {\ bf u}^2 $的情况下连接$i_α$和$i_α$的方程是通过取消分解\ [λ^2 {\ bf i} - {\ bf c} =（λ{λ{\ bf i} - {\ bf i} - {\ bf i}）（λ{\ bf u}）（λ{λ}+ coeffer coeft）获得的方程。在消除$ i_2 $时，我们获得了一个具有系数的四分之一方程式，具体仅取决于最大根为$ i_1 $的$i_α$。同样，我们可能会获得一个四分之一的方程，其最大根为$ i_2 $。对于$ n = 4 $，我们发现$ i_2 $再次是四分之一方程的最大根，因此所有$i_α$均以$i_α$表示。然后，$ \ bf u $和$ {\ bf u}^{ - 1} $仅以$ \ bf c $表示，如$ n = 3 $。对于$ n = 5 $，我们发现，但不展示，第20级多项式，其中$ i_1 $是最大的根，它具有四个虚假零。我们无法以$ n = 5 $表示$i_α$的$i_α$。尽管如此，$ \ bf u $和$ {\ bf u}^{ - 1} $以$ \ bf c $的功率表示，现在取决于$i_α$。对于$ n = 6 $，我们发现但不展示，这是一个32度多项式，最大的根$ i_1^2 $。这些根中有16个是相关的，但我们展示的其他16根是虚假的。 $ \ bf u $和$ {\ bf u}^{ - 1} $用$ \ bf c $的功率表示。讨论了$ n> 6 $的案例。 关键字：连续力学，极地分解，张量正方根，主要不变性，立方方程，四分之一方程，度量16
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
