The authors consider the plane-strain straightening of annular cylindrical sectors composed of isotropic, compressible, nonlinearly elastic solids. For zero-body forces and boundary conditions of place, the existence and uniqueness of solutions are established under the assumption that the material satisfies the tension-extension condition. The result is exemplified by considering a compressible neo-Hookean material and a generalized Blatz-Ko material for which closed-form solutions are displayed. Also discussed are certain cases of nonexistence and nonuniqueness of solutions.
Get full access to this article
View all access options for this article.
References
1.
[1] Ericksen, J. L.: Deformations possible in every isotropic incompressible perfectly elastic body. Z. Angew. Math. Phys., 5, 466-486 (1954).
2.
[2] Ericksen, J. L.: Deformations possible in every compressible, isotropic, perfectly elastic material. J. Math. Phys., 34, 126-128 (1955).
3.
[3] Truesdell, C. and Toupin, R.: The classical field theories, in Handbuch der Physik, 11111, ed., S. Fluigge, Springer Verlag, Berlin, 1960.
4.
[4] Truesdell, C. and Noll, W.: The non-linear field theories of mechanics, in Handbuch der Physik, III/3, ed., S. FlUgge, Springer-Verlag, Berlin, 1965.
5.
[5] Wang, Y.: Finite deformations of a generalized Blatz-Ko material, Ph.D. thesis, University of Plymouth, 1996.
6.
[6] Carroll, M. M. and Horgan, C. O.: Finite strain solutions for a compressible elastic solid. Quart. Appl. Math., 48, 767-780 (1990).
7.
[7] Aron, M. and Wang, Y.: Remarks concerning the flexure of a compressible nonlinearly elastic rectangular block. J. Elasticity, 40, 99-106 (1995).
8.
[8] Levinson, M. and Burgess, W.: A comparison of some simple constitutive relations for slightly compressible rubberlike materials. Int. J. Mech. Sci., 13, 563-572 (1971).
9.
[9] Blatz, P. J.: Finite elastic deformation of a plane strain wedge-shaped radial crack in a compressible cylinder, in Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, pp. 145-161, eds., M. Reiner and D. Abir, Pergamon, Oxford, 1964.
10.
[10] Ogden, R. W.: On non-uniqueness in the traction boundary-value problem for a compressible elastic solid. Quart. Appl. Math., 41, 337-344 (1984).
11.
[11] Ball, J. M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. Lon., A306, 557-611 (1982).
12.
[12] Sivaloganathan, J.: The generalized Hamilton-Jacobi inequality and stability of equilibria in non-linear elasticity. Arch. Ration. Mech. Anal., 107, 347-379 (1989).
13.
[13] Hill, R.: Aspects of invariance in solid mechanics, in Advances in Applied Mechanics, Vol. 18, pp. 1-75, ed., C.-S. Yih, Academic Press, New York, 1978.
14.
[14] Blatz, P. J. and Ko, W. L.: Application of finite elasticity theory to the deformation of rubbery materials. Trans. Soc. Rheology, 6, 223-251 (1962).
15.
[15] Silling, S. A.: Creasing singularities in compressible elastic materials. J. Appl. Mech. Trans. ASME, 58, 70-74 (1991).
16.
[16] Wang, Y. and Aron, M.: A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J. Elasticity, 44, 89-96 (1996).
17.
[17] 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).
18.
[18] Aron, M.: On a class of plane radial deformations of compressible non-linearly elastic solids. I.M.A.J. Appl. Maths., 52, 289-296 (1994).
