# 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…
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## References

SHOWING 1-10 OF 12 REFERENCES
Determination of the stretch and rotation in the polar decomposition of the deformation gradient
• Mathematics
• 1984
On montre qu'en appliquant le theoreme de Cayley-Hamilton on peut calculer directement le tenseur d'etirement a droite U a partir du tenseur de deformation de Cauchy-Green a droite sans avoir recours
Derivatives of the Rotation and Stretch Tensors
Previous work on representing the rotation and stretch tensors, their time derivatives and their gradients with respect to the deformation gradient tensor is reviewed and some new results are
On the Explicit Determination of the Polar Decomposition in n-Dimensional Vector Spaces
A method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is
Nonlinear Fluid-Structure Interactions in Flapping Wing Systems
Title of dissertation: NONLINEAR FLUID–STRUCTURE INTERACTIONS IN FLAPPING WING SYSTEMS Timothy Fitzgerald, Doctor of Philosophy, 2013 Dissertation directed by: Professor Balakumar Balachandran
Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient
• Mathematics
• 2005
We develop a new effective method of determining the rotation R and the stretches U and V in the polar decomposition F = RU = VR of the deformation gradient. The method is based on a minimum property
Invariants of C1∕2 in terms of the invariants of C
The three invariants of C$^{1/2}$ are key to expressing this tensor and its inverse as a polynomial in C. Simple and symmetric expressions are presented connecting the two sets of invariants \$I_1,