Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes


Authors: Debra A. Polignone and Cornelius O. Horgan
Journal: Quart. Appl. Math. 50 (1992), 323-341
MSC: Primary 73G05; Secondary 73C50
DOI: https://doi.org/10.1090/qam/1162279
MathSciNet review: MR1162279
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Abstract | References | Similar Articles | Additional Information

Abstract: The axial shear problem for a hollow circular cylinder, composed of homogeneous isotropic compressible nonlinearly elastic material, is described. The inner surface of the tube is bonded to a rigid cylinder while the outer surface is subjected to a uniformly distributed axial shear traction and the radial traction is zero. For an arbitrary compressible material, the cylinder will undergo both a radial and axial deformation. These axisymmetric fields are governed by a coupled pair of nonlinear ordinary differential equations, one of which is second-order and the other first-order. The class of materials for which axisymmetric anti-plane shear (i.e., a deformation with zero radial displacement) is possible is described. The corresponding axial displacement and stresses are determined explicitly. Specific material models are used to illustrate the results.


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  • [1] J. E. Adkins, Some generalizations of the shear problem for isotropic incompressible materials, Proc. Cambridge Philos. Soc. 50 (1954), 334–345. MR 0060991
  • [2] A. E. Green and J. E. Adkins, Large elastic deformations, Second edition, revised by A. E. Green, Clarendon Press, Oxford, 1970. MR 0269158
  • [3] A. E. Green and W. Zerna, Theoretical elasticity, Second edition, Clarendon Press, Oxford, 1968. MR 0245245
  • [4] James K. Knowles, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids, Internat. J. Fracture 13 (1977), no. 5, 611–639 (English, with French summary). MR 0462075, https://doi.org/10.1007/BF00017296
  • [5] R. Abeyaratne and C. O. Horgan, Bounds on stress concentration factors in finite anti-plane shear, J. Elasticity 13 (1983), no. 1, 49–61. MR 708842, https://doi.org/10.1007/BF00041313
  • [6] A. H. Jafari, C. O. Horgan, and R. Abeyaratne, Finite anti-plane shear of an infinite slab with a traction-free elliptical cavity: bounds on the stress concentration factor, Internat. J. Non-Linear Mech. 19, 431-443 (1984)
  • [7] C. O. Horgan and S. A. Silling, Stress concentration factors in finite antiplane shear: numerical calculations and analytical estimates, J. Elasticity 18 (1987), no. 1, 83–91. MR 899418, https://doi.org/10.1007/BF00155438
  • [8] James K. Knowles, On finite anti-plane shear for incompressible elastic materials, J. Austral. Math. Soc. Ser. B 19 (1975/76), no. 4, 400–415. MR 0475116, https://doi.org/10.1017/S0334270000001272
    James K. Knowles, A note on anti-plane shear for compressible materials in finite elastostatics, J. Austral. Math. Soc. Ser. B 20 (1977/78), no. 1, 1–7. MR 0475117, https://doi.org/10.1017/S0334270000001399
  • [9] James K. Knowles, On finite anti-plane shear for incompressible elastic materials, J. Austral. Math. Soc. Ser. B 19 (1975/76), no. 4, 400–415. MR 0475116, https://doi.org/10.1017/S0334270000001272
    James K. Knowles, A note on anti-plane shear for compressible materials in finite elastostatics, J. Austral. Math. Soc. Ser. B 20 (1977/78), no. 1, 1–7. MR 0475117, https://doi.org/10.1017/S0334270000001399
  • [10] R. W. Ogden, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984
  • [11] A. Mioduchowski and J. B. Haddow, Finite telescopic shear of a compressible hyperelastic tube, Internat. J. Non-Linear Mech. 9, 209-220 (1974)
  • [12] M. G. Faulkner, Compressibility effects for nearly incompressible elastic solids, Appl. Sci. Res. 25, 328-336 (1972)
  • [13] V. K. Agarwal, On finite anti-plane shear for compressible elastic circular tube, J. Elasticity 9, 311-319 (1979)
  • [14] Debra A. Polignone and Cornelius O. Horgan, Pure torsion of compressible non-linearly elastic circular cylinders, Quart. Appl. Math. 49 (1991), no. 3, 591–607. MR 1121689, https://doi.org/10.1090/qam/1121689
  • [15] A. Ertepinar and G. Erarslanoglu, Finite anti-plane shear of compressible hyperelastic tubes, Internat. J. Engrg. Sci. 28, 399-406 (1990)
  • [16] 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
  • [17] Fritz John, Plane elastic waves of finite amplitude. Hadamard materials and harmonic materials, Comm. Pure Appl. Math. 19 (1966), 309–341. MR 0201113, https://doi.org/10.1002/cpa.3160190306
  • [18] P. J. Blatz, Application of finite elastic theory in predicting the performance of solid propeliane rocket motors, Calif. Inst. of Tech., GALCIT SM60-25, 1960
  • [19] P. J. Blatz, Application of large deformation theory to the thermo-mechanical behavior of rubberlike polymers--porous, unfilled, and filled, Rheology, Theory and Applications, vol. 5 (F. R. Eirich, ed.), Academic Press, New York, 1969, pp. 1-55
  • [20] R. W. Ogden, Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids, Proc. Roy. Soc. London Ser. A 328, 567-583 (1972)
  • [21] J. B. Haddow and R. W. Ogden, Compression of bonded elastic bodies, J. Mech. Phys. Solids 36 (1988), no. 5, 551–579. MR 964175, https://doi.org/10.1016/0022-5096(88)90010-5
  • [22] R. M. Christensen, A two material constant, nonlinear elastic stress constitutive equation including the effect of compressibility, Mech. of Materials 7, 155-162 (1988)
  • [23] P. J. Blatz and W. L. Ko, Application of finite elasticity to the deformation of rubbery materials, Transactions of the Society of Rheology 6, 223-251 (1962)
  • [24] M. Levinson and I. W. Burgess, A comparison of some simple constitutive relations for slightly compressible rubber-like materials, Internat. J. Mech. Sci. 13, 563-572 (1971)
  • [25] Y. C. Fung, Biomechanics. Mechanical properties of living tissues, Springer-Verlag, Berlin, 1981
  • [26] M. F. Beatty, Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues-- with examples, Appl. Mech. Reviews 40, 1699-1734 (1987)
  • [27] P. F. Chu, Strain energy function for biological tissues, J. Biomechanics 3, 547-550 (1970)
  • [28] Qing Jiang and James K. Knowles, A class of compressible elastic materials capable of sustaining finite anti-plane shear, J. Elasticity 25 (1991), no. 3, 193–201. MR 1115552, https://doi.org/10.1007/BF00040926

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DOI: https://doi.org/10.1090/qam/1162279
Article copyright: © Copyright 1992 American Mathematical Society


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