Sage Journals: Discover world-class research

Abstract

The determination of exact deformations that apply to all homogeneous, isotropic, incompressible, hyperelastic materials is referred to in the literature as Ericksen's problem. For this class of materials, there remains only the case of constant strain invariants to be examined. The authors consider, in spherical polar coordinates, a radially symmetric deformation for which the constancy of the strain invariants arises as a natural part of the equilibrium conditions. In addition, we show that the remaining equilibrium equations admit two exact first integrals, which can be exploited to reduce the order of the problem. Finally, we show in certain complex coordinates (Z, Z) that the determination of further solutions of Ericksen's problem hinges on finding a real function P(Z, Z) that must satisfy two nonlinear partial differential equations. One of these partial differential equations admits linearization, and separable solutions may be determined involving associated Legendre functions. However, the remaining partial differential equation does not simplify under this transformation, and the authors have been unable to proceed further. Although the authors have been unable to determine any new solutions as such, the present analysis constitutes a major simplification of one of the remaining outstanding problems of nonlinear continuum mechanics.

Get full access to this article

View all access options for this article.

References

[1] Ericksen, J. L. : Deformations possible in every isotropic incompressible perfectly elastic body. J. Appl. Math. Phys. (ZAMP), 5, 466-486 (1954).

[2] Beatty, M. F. : Introduction to nonlinear elasticity, in Nonlinear Effects in Fluids and Solids, pp. 13-112, eds., M. M. Carroll and M. A. Hayes , Plenum Press, New York, 1996.

[3] Martins, L. C. and Duda, F. P. : Constrained elastic bodies and universal solutions in the class of plane deformations with uniform transverse stretch. Math. Mech. Solids, 3, 91-106 (1998).

[4] Marris, A. and Shiau, J. : Universal deformations in isotropic incompressible hyperelastic materials when the deformation tensor has equal proper values. Arch. Rat. Mech. Anal., 36, 135-160 (1970). Corrigenda, Arch. Rat. Mech. Anal., 38, 406 (1970).

[5] Kafadar, C. B. : On Ericksen's problem. Arch. Rat. Mech. Anal., 47, 15-27 (1972).

[6] Marris, A. W. : Universal deformations in incompressible isotropic elastic materials. J. Elasticity, 5, 111-128 (1975).

[7] Marris, A. W. : Two new theorems on Ericksen's problem. Arch. Rat. Mech. Anal., 79, 131-173 (1982).

[8] Yin, W-L. : Two families of finite deformations with constant strain invariants. Mech. Research Communications, 10, 127-132 (1983).

[9] Fosdick, R. L. and Schuler, K. W. : On Ericksen's problem for plane deformations with uniform transverse stretch. Int. J. Eng. Sci., 7, 217-233 (1969).

10.

[10] Fosdick, R. L. : Statically possible radially symmetric deformations in isotropic incompressible elastic solids. J. Appl. Math. Phys. (ZAMP), 22, 590-607 (1971).

11.

[11] Hill, J. M.: Some partial solutions of finite elasticity. Ph.D. thesis, University of Queensland, 1972.

12.

[12] Hill, J. M. : On static similarity deformations for isotropic materials. Q. J. Appl. Math, 40, 287-292 (1982).

13.

[13] Hill, J. M. and Arrigo, D. J. : New families of exact solutions for finitely deformed incompressible elastic materials. IMA J. Appl. Math, 54, 109-123 (1995).

14.

[14] Hill, J. M. and Arrigo, D. J. : Transformations and equation reductions in finite elasticity I: Plane strain deformations. Math. Mech. Solids, 1, 155-175 (1996).

15.

[15] Arrigo, D. J. and Hill, J. M. : Transformations and equation reductions in finite elasticity II: Plane stress and axially symmetric deformations. Math. Mech. Solids, 1, 177-192 (1996).

16.

[16] Hill, J. M. and Arrigo, D. J. : Transformations and equation reductions in finite elasticity III: A general integral for plane strain deformations. Math. Mech. Solids, 4, 3-15 (1999).

17.

[17] Hill, J. M. : Partial solutions of finite elasticity: Three dimensional deformations. J. Appl. Math. Phys. (ZAMP), 24, 609-618 (1973).

A Note on Ericksen's Problem for Radially Symmetric Deformations

Abstract

Get full access to this article

References