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

\$\{\mathbf\{U\}\} = \mathbf\{C\}^\{1/2\}\$
 and Its Invariants in Terms of 
 and Its Invariants},
  author={N. H. Scott},
  journal={Journal of Elasticity},
  • 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… 
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