The principal invariants of the stretch tensors in the polar decomposition of the deformation gradient stand in one-to-one relation to the principal invariants of their squares, the Cauchy—Green deformation tensors. These relations are used to obtain the derivatives of the first set of invariants with respect to the deformation gradient. The latter generate expressions for the stretch and rotation factors in the polar decomposition as explicit functions of the deformation gradient, and lead to new formulae for the stress and elastic moduli for isotropic elastic materials.
Get full access to this article
View all access options for this article.
References
1.
[1] John, F.: Plane strain problems for a perfectly elastic material of harmonic type. Commun. Pure Appl. Math., 13, 239-296 (1960).
2.
[2] Abeyaratne, R. and Horgan, C. O.: The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials. Int. J. Solids Struct., 20, 715-723 (1984).
3.
[3] Wheeler, L. T.: Finite deformations of a harmonic elastic medium containing an ellipsoidal cavity. Int. J. Solids Struct., 21, 799-804 (1985).
4.
[4] Carroll, M.M.: Finite strain solutions in compressible isotropic elasticity. J. Elasticity, 20, 65-92 (1988).
5.
[5] Horgan, C. O.: Equilibrium solutions for compressible nonlinearly elastic materials. In Nonlinear Elasticity: Theory and Applications, London Mathematical Society Lecture Note Series 283 (Y.B. Fu and R. W. Ogden eds.) Cambridge University Press, Cambridge, 2001, pp. 135-159.
6.
[6] Sawyers, K.: Comments on the paper ‘Determination of the stretch and rotation in the polar decomposition of the deformation gradient’ by A. Hoger and D.E. Carlson. Q. Appl. Math., 44, 309-311 (1986).
7.
[7] Cohen, H. and Wang, C.-C.: A note on hyperelasticity. Arch. Ration. Mech. Anal., 85, 213-236 (1984).
[9] Lu, J. and Papadopoulos, P.: On the direct determination of the rotation tensor from the deformation gradient. Math. Mech. Solids, 2, 17-26 (1997).
10.
[10] Ting, T.C.T.: Determination of C½, C−½ and more general isotropic functions of C. J. Elasticity, 15, 319-323 (1985).
11.
[11] Rivlin, R.S. and Smith, G.F.: A note on material frame indifference. Int. J. Solids Struct., 23, 1639-1643 (1987).
12.
[12] Wheeler, L.T.: On the derivatives of the stretch and rotation with respect to the deformation gradient. J. Elasticity, 24, 129-133 (1990).
13.
[13] Beatty, M.F. and Hayes, M.A.: Deformations of an elastic, internally constrained material. Part 1: Homogeneous deformations. J. Elasticity, 29, 1-84 (1992).
14.
[14] Beatty, M.F. and Hayes, M.A.: Deformations of an elastic, internally constrained material. Part 2: Nonhomogeneous deformations. Q. J. Mech. Appl. Math., 45, 663-709 (1992).
15.
[15] Ogden, R.W.: Non-Linear Elastic DeformationsDover Publications, New York, 1997.
16.
[16] Knowles, J.K. and Sternberg, E.: On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Ration. Mech. Anal., 63, 321-336 (1977).
17.
[17] Haughton, D.M.: Elastic membranes. In Nonlinear Elasticity: Theory and Applications, London Mathematical Society Lecture Note Series 283 (Y.B. Fu and R.W. Ogden eds.), Cambridge University Press, Cambridge, 2001, pp. 233-267.
