• DOI:10.1007/S10659-020-09780-X
• Corpus ID: 216056205

U = C 1 / 2 ${\mathbf{U}} = \mathbf{C}^{1/2}$ and Its Invariants in Terms of C $\mathbf{C}$ and Its Invariants

@article{Scott2020U,
  title={
U
=
C
1
/
2
\$\{\mathbf\{U\}\} = \mathbf\{C\}^\{1/2\}\$
 and Its Invariants in Terms of 
C
\$\mathbf\{C\}\$
 and Its Invariants},
  author={N. H. Scott},
  journal={Journal of Elasticity},
  year={2020},
  volume={141},
  pages={363-379}
}

• N. Scott
• Published 22 April 2020
• Mathematics
• Journal of Elasticity

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