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The deformation of rubber cylinders and tubes by rotation

Published online by Cambridge University Press:  17 February 2009

P. Chadwick ,
C. F. M. Creasy  and
V. G. Hart
P. Chadwick
Affiliation:
School of Mathematics and Physics, University of East Anglia, Norwich, England
C. F. M. Creasy
Affiliation:
Department of Mathematics, University of Queenland, St Lucia, Q. 4067, Australia
V. G. Hart
Affiliation:
Department of Mathematics, University of Queenland, St Lucia, Q. 4067, Australia
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Abstract

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A detailed analytical and numerical study is made of the deformation of highly elastic circular cylinders and tubes produced by steady rotation about the axis of symmetry. Explicit results are obtained through the use of Ogden's strain–energy function for incompressible isotropic elastic materials which, as well as being analytically convenient, is capable of reproducing accurately the observed isothermal behaviour of vulcanized rubber over a wide range of deformations. The three problems of steady rotation considered here concern (i) a tube shrink-fitted to a rigid spindle, (ii)an unconstrained tube, and (iii) a solid cylinder. In each case a set of restictions on the material constans appearing in the strain–energy function is stated which ensures that a tubular of cylindrical shape-preserving deformation exists for all angular spees and that, for problems (i) and (iii), there is no other solution. In connection with problems (ii) and (iii) values of the material constans are also given which correspond to the bifuraction or non-existence of soultions. Enegry consideration are used to determine the local stability of the various solutions obtained.

Type
Research Article
Information
The ANZIAM Journal , Volume 20 , Issue 1 , June 1977 , pp. 62 - 96
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Alexander, H., “Tensile instability of initially spherical balloons”, Int. J. Engng Sci., 9 (1971), 151–162.CrossRefGoogle Scholar
[2]Alexander, H., “The tensile instability of an inflated cylindrical membrane as affected by an axial load”, Int. J. Mech. Sci. 13 (1971), 87–95.CrossRefGoogle Scholar
[3]Chadwick, P., “Aspects of the dynamics of a rubberlike material”, Q. Jl Mech. Appl. Math., 27 (1974), 263–285.CrossRefGoogle Scholar
[4]Chadwick, P. and Creasy, C. F. M., “Some existence-uniqueness results for problems of finite elastic bending”, Q. Jl Mech. Appl. Math. 30 (1977), 187–202.CrossRefGoogle Scholar
[5]Chadwick, P. and Haddon, E. W., “Inflation-extension and eversion of a tube of in-compressible isotropic elastic material”, J. Inst. Maths Applics 10 (1972), 258–278.CrossRefGoogle Scholar
[6]Green, A. E. and Shield, R. T., “Finite elastic deformation of incompressible isotropic bodies”, Proc. Roy. Soc. A 202 (1950), 407–419.Google Scholar
[7]Green, A. E. and Zerna, W., Theoretical Elasticity, Oxford University Press (1st ed., 1954).Google Scholar
[8]Jones, D. F. and Treloar, L. R. G., “The properties of rubber in pure homogeneous strain”, J. Phys. D: Appl. Phys. 8 (1975), 1285–1304.CrossRefGoogle Scholar
[9]Knowles, J. K. and Jakub, M. T., “Finite dynamic deformations of an incompressible elastic medium containing a spherical cavity”, Arch. Ration. Mech. Analysis 18 (1965), 367–378.CrossRefGoogle Scholar
[10]Ogden, R. W., “Large deformation isotropic elasticity—on the correlation of theory and experiment for incompresible rubberlike solids”, Proc. Roy. Soc. A 326 (1972), 565–584.Google Scholar
[11]Ogden, R. W. and Chadwick, P., “On the deformation of solid and tubular cylindrs of incompressible isotropic elastic material”, J. Mech. Phys. Solids 20 (1972), 77–90.CrossRefGoogle Scholar
[12]Ogden, R. W., Chadwick, P. and Haddon, E. W., “Combined axial and torsional shear of a tube of incompressible elastic material”, Q. Jl Mech. Appl. Math. 26 (1973), 23–41.CrossRefGoogle Scholar
[13]Patterson, J. C. and Hill, J. M., “The stability of a solid rotating neo-Hookean cylinder”, Mech. Research Comm. (1977), to appear.CrossRefGoogle Scholar
[14]Treloar, L. R. G., “Stress-strain data for vulcanised rubber under various types of deformation”, Trans. Faraday Soc. 40 (1944), 59–70.CrossRefGoogle Scholar
[15]Treloar, L. R. G., The Physics of Rubber Elasticity Oxford University Press (3rd ed., 1975).Google Scholar