Pure torsion of compressible non-linearly elastic circular cylinders

Polignone, Debra A.; Horgan, Cornelius O.

doi:10.1090/qam/1121689

Authors: Debra A. Polignone and Cornelius O. Horgan
Journal: Quart. Appl. Math. 49 (1991), 591-607
MSC: Primary 73G05; Secondary 73C50, 73K05
DOI: https://doi.org/10.1090/qam/1121689
MathSciNet review: MR1121689
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Abstract: The large deformation torsion problem of an elastic circular cylinder, composed of homogeneous isotropic compressible nonlinearly elastic material and subjected to twisting moments at its ends, is described. The problem is formulated as a two-point boundary-value problem for a second-order nonlinear ordinary differential equation in the radial deformation field. The class of materials for which pure torsion (i.e., a deformation with zero radial displacement) is possible is described. Specific material models are used to illustrate the results.

R. S. Rivlin, Large elastic deformations of isotropic materials. IV. Further developments of the general theory, Philos. Trans. Roy. Soc. London Ser. A 241 (1948), 379–397. MR 27674, DOI https://doi.org/10.1098/rsta.1948.0024
R. S. Rivlin, A note on the torsion of an incompressible highly-elastic cylinder, Proc. Cambridge Philos. Soc. 45 (1949), 485–487. MR 29695
J. L. Ericksen, Deformations possible in every compressible, isotropic, perfectly elastic material, J. Math. and Phys. 34 (1955), 126–128. MR 70397, DOI https://doi.org/10.1002/sapm1955341126
A. E. Green, Finite elastic deformation of compressible isotropic bodies, Proc. Roy. Soc. London Ser. A 227 (1955), 271–278. MR 67671, DOI https://doi.org/10.1098/rspa.1955.0010
A. E. Green and J. E. Adkins, Large elastic deformations, Clarendon Press, Oxford, 1970. Second edition, revised by A. E. Green. MR 0269158

R. W. Ogden, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984

A. E. Green and E. W. Wilkes, A note on the finite extension and torsion of a circular cylinder of compressible elastic isotropic material, Quart. J. Mech. Appl. Math. 6 (1953), 240–249. MR 57131, DOI https://doi.org/10.1093/qjmam/6.2.240
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, Springer-Verlag, Berlin, 1965, pp. 1–602. MR 0193816

R. T. Shield, An energy method for certain second-order effects with application to torsion of elastic bars under tension, J. Appl. Mech. 47, 75–81 (1980)

M. Levinson, Finite torsion of slightly compressible rubberlike circular cylinders, Internat. J. Non-Linear Mech. 7, 445–463 (1972)

M. G. Faulkner and J. B. Haddow, Nearly isochoric finite torsion of a compressible isotropic elastic circular cylinder, Acta Mech. 13, 245–253 (1972)

R. W. Ogden, Nearly isochoric elastic deformations: application to rubberlike solids, J. Mech. Phys. Solids 26 (1978), no. 1, 37–57. MR 505905, DOI https://doi.org/10.1016/0022-5096%2878%2990012-1

P. K. Currie and M. Hayes, On non-universal finite elastic deformations, Finite Elasticity (D. E. Carlson and R. T. Shield, eds.), Proceedings of IUTAM Symposium, Martinus Nijhoff, The Hague, 1982, pp. 143–150

P. J. Blatz and W. L. Ko, Application of finite elasticity to the deformation of rubbery materials, Trans. Soc. Rheol. 6, 223–251 (1962)

M. F. Beatty, Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with examples, Appl. Mech. Rev. 40, 1699–1734 (1987)

M. M. Carroll and C. O. Horgan, Finite strain solutions for a compressible elastic solid, Quart. Appl. Math. 48 (1990), no. 4, 767–780. MR 1079919, DOI https://doi.org/10.1090/qam/1079919
J. K. Knowles and Eli Sternberg, On the ellipticity of the equations of nonlinear elastostatics for a special material, J. Elasticity 5 (1975), no. 3-4, 341–361. Special issue dedicated to A. E. Green. MR 475115, DOI https://doi.org/10.1007/BF00126996
R. Abeyaratne and C. O. Horgan, Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium, J. Elasticity 15 (1985), no. 3, 243–256. MR 804497, DOI https://doi.org/10.1007/BF00041423
C. O. Horgan and R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid, J. Elasticity 16 (1986), no. 2, 189–200. MR 849671, DOI https://doi.org/10.1007/BF00043585

D.-T. Chung, C. O. Horgan, and R. Abeyaratne, The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials, Internat. J. Solids and Structures 22, 1557–1570 (1986)

Cornelius O. Horgan, Some remarks on axisymmetric solutions in finite elastostatics for compressible materials, Proc. Roy. Irish Acad. Sect. A 89 (1989), no. 2, 185–193. MR 1051392

J. S. K. Wong and R. T. Shield, The stability of a cylindrical elastic membrane of biological tissue and the effect of internal fluid flow, Ing.-Archiv 49, 393–412 (1980)

Y. C. Fung, Biomechanics. Mechanical Properties of Living Tissues, Springer-Verlag, Berlin, 1981

M. M. Carroll, Finite strain solutions in compressible isotropic elasticity, J. Elasticity 20 (1988), no. 1, 65–92. MR 962367, DOI https://doi.org/10.1007/BF00042141
Fritz John, Plane elastic waves of finite amplitude. Hadamard materials and harmonic materials, Comm. Pure Appl. Math. 19 (1966), 309–341. MR 201113, DOI https://doi.org/10.1002/cpa.3160190306
D. M. Haughton, Inflation and bifurcation of thick-walled compressible elastic spherical shells, IMA J. Appl. Math. 39 (1987), no. 3, 259–272. MR 983745, DOI https://doi.org/10.1093/imamat/39.3.259
O. H. Varga, Stress-strain behavior of elastic materials. Selected problems of large deformations, Polymer Reviews, Vol. 15, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0205523
Debra A. Polignone and Cornelius O. Horgan, Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes, Quart. Appl. Math. 50 (1992), no. 2, 323–341. MR 1162279, DOI https://doi.org/10.1090/qam/1162279