An explicit representation of the rotation tensor is given in terms of the deformation gradient. The derivation does not require the a priori determination of the stretch tensor. Instead, it relies on systematic use of the Cayley-Hamilton theorem and basic properties of proper orthogonal tensors.
Get full access to this article
View all access options for this article.
References
1.
[1] Truesdell, C. and Toupin, R. A.The classical field theories, in Handbuch der Physik III/I, pp. 226-793, ed., S. FIligge, Springer-Verlag, Berlin, 1960.
2.
[2] Marsden, J. E. and Hughes, T.J.R.Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs, NJ, 1983.
3.
[3] Hoger, A. and Carlson, D. E.Determination of the stretch and rotation in the polar decomposition of the deformation gradient. Quarterly of Applied Mathematics, 42, 113-117 (1984).
4.
[4] Ting, T.C.T.Determination of C 1/2, C - 1/2 and more general isotropic tensor functions of C. Journal of Elasticity, 15, 319-323 (1985).
5.
[5] Horn, R. A. and Johnson, C. R.Matrix Analysis, Cambridge University Press, Cambridge, 1985.
6.
[6] Horn, B.K.P.Closed-form solution of absolute orientation using unit quaternions. Journal of the Optical Society of America A, 4, 629-642 (1987).
[10] Abramowitz, M. and Stegun, L. A., eds. Handbook of Mathematical Functions; with formulas, graphs, and mathematical tables, Dover, NY, 1970.
11.
[11] Rivlin, R. S.Further remarks on the stress-deformation relations for isotropic materials. Journal of Rational Mechanics and Analysis, 4, 681-702 (1955).
