Pure azimuthal shear of compressible non-linearly elastic circular tubes
Authors:
Debra A. Polignone and Cornelius O. Horgan
Journal:
Quart. Appl. Math. 52 (1994), 113-131
MSC:
Primary 73G05; Secondary 73C50, 73K05
DOI:
https://doi.org/10.1090/qam/1262323
MathSciNet review:
MR1262323
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Abstract: The azimuthal (or circular) 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. The deformation may be achieved either by applying a uniformly distributed azimuthal shear traction on the outer surface together with zero radial traction (Problem 1) or by subjecting the outer surface to a prescribed angular displacement, with zero radial displacement (Problem 2). For an arbitrary compressible material, the cylinder will undergo both a radial and angular 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 pure azimuthal shear (i.e., a deformation with zero radial displacement) is possible is described. The corresponding angular displacement and stresses are determined explicitly. Specific material models are used to illustrate the results.
D. A. Polignone and C. O. Horgan, Pure torsion of compressible nonlinearly elastic circular cylinders, Quart. Appl. Math. 49, 591–607 (1991)
D. A. Polignone and C. O. Horgan, Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes, Quart. Appl. Math. 50, 323–341 (1992)
J. L. Ericksen, Deformations possible in every compressible, isotropic, perfectly elastic material, J. Math. Phys. 34, 126–128 (1955)
P. K. Currie and M. Hayes, On non-universal finite elastic deformations, Finite Elasticity, (D. E. Carlson and R. T. Shield, eds.), Proc. IUTAM Sympos., 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)
F. John, Plane elastic waves of finite amplitude: Hadamard materials and harmonic materials, Comm. Pure Appl. Math. 19, 309–341 (1966)
R. S. Rivlin, Large elastic deformations of isotropic materials. VI: Further results in the theory of torsion, shear and flexure, Philos. Trans. Roy. Soc. London Ser. A 242, 173–195 (1949)
A. E. Green and W. Zerna, Theoretical Elasticity, Oxford Univ. Press, London, 1968
R. W. Ogden, P. Chadwick, and E. W. Haddon, Combined axial and torsional shear of a tube of incompressible isotropic elastic material, Quart. J. Mech. Appl. Math. 26, 34–41 (1973)
R. W. Ogden, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984
J. E. Adkins, Some general results in the theory of large elastic deformations, Proc. Roy. Soc. London Ser. A 231, 75–99 (1955)
R. W. Ogden and D. A. Isherwood, Solution of some finite plane-strain problems for compressible elastic solids, Quart. J. Mech. Appl. Math. 31, 219–249 (1978)
M. M. Carroll and C. O. Horgan, Finite strain solutions for a compressible elastic solid, Quart. Appl. Math. 48, 767–780 (1990)
J. G. Simmonds and P. Warne, Azimuthal shear of compressible or incompressible, rubber-like, polar-orthotropic tubes of infinite extent, Internat. J. Nonlinear Mechanics 27, 447–464 (1992)
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik III/3, (S. Flugge, ed.), Springer, Berlin, 1965
A. Ertepinar, Finite deformations of compressible hyperelastic tubes subjected to circumferential shear, Internat. J. Engr. Sci. 23, 1187–1195 (1985)
A. Ertepinar, On the finite circumferential shearing of compressible hyperelastic tubes, Internat. J. Engr. Sci. 28, 889–896 (1990)
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)
Y. C. Fung, Biomechanics, Mechanical Properties of Living Tissues, Springer-Verlag, Berlin, 1981
M. F. Beatty, Topics in finite elasticity. Hyperelasticity of rubber, elastomers, and biological tissues-with examples, Appl. Mech. Reviews 40, 1699–1734 (1987)
D. M. Haughton, Circular shearing of compressible elastic cylinders, University of Glasgow, Department of Mathematics, preprint (1991)
D. A. Polignone and C. O. Horgan, Pure torsion of compressible nonlinearly elastic circular cylinders, Quart. Appl. Math. 49, 591–607 (1991)
D. A. Polignone and C. O. Horgan, Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes, Quart. Appl. Math. 50, 323–341 (1992)
J. L. Ericksen, Deformations possible in every compressible, isotropic, perfectly elastic material, J. Math. Phys. 34, 126–128 (1955)
P. K. Currie and M. Hayes, On non-universal finite elastic deformations, Finite Elasticity, (D. E. Carlson and R. T. Shield, eds.), Proc. IUTAM Sympos., 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)
F. John, Plane elastic waves of finite amplitude: Hadamard materials and harmonic materials, Comm. Pure Appl. Math. 19, 309–341 (1966)
R. S. Rivlin, Large elastic deformations of isotropic materials. VI: Further results in the theory of torsion, shear and flexure, Philos. Trans. Roy. Soc. London Ser. A 242, 173–195 (1949)
A. E. Green and W. Zerna, Theoretical Elasticity, Oxford Univ. Press, London, 1968
R. W. Ogden, P. Chadwick, and E. W. Haddon, Combined axial and torsional shear of a tube of incompressible isotropic elastic material, Quart. J. Mech. Appl. Math. 26, 34–41 (1973)
R. W. Ogden, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984
J. E. Adkins, Some general results in the theory of large elastic deformations, Proc. Roy. Soc. London Ser. A 231, 75–99 (1955)
R. W. Ogden and D. A. Isherwood, Solution of some finite plane-strain problems for compressible elastic solids, Quart. J. Mech. Appl. Math. 31, 219–249 (1978)
M. M. Carroll and C. O. Horgan, Finite strain solutions for a compressible elastic solid, Quart. Appl. Math. 48, 767–780 (1990)
J. G. Simmonds and P. Warne, Azimuthal shear of compressible or incompressible, rubber-like, polar-orthotropic tubes of infinite extent, Internat. J. Nonlinear Mechanics 27, 447–464 (1992)
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik III/3, (S. Flugge, ed.), Springer, Berlin, 1965
A. Ertepinar, Finite deformations of compressible hyperelastic tubes subjected to circumferential shear, Internat. J. Engr. Sci. 23, 1187–1195 (1985)
A. Ertepinar, On the finite circumferential shearing of compressible hyperelastic tubes, Internat. J. Engr. Sci. 28, 889–896 (1990)
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)
Y. C. Fung, Biomechanics, Mechanical Properties of Living Tissues, Springer-Verlag, Berlin, 1981
M. F. Beatty, Topics in finite elasticity. Hyperelasticity of rubber, elastomers, and biological tissues-with examples, Appl. Mech. Reviews 40, 1699–1734 (1987)
D. M. Haughton, Circular shearing of compressible elastic cylinders, University of Glasgow, Department of Mathematics, preprint (1991)
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© Copyright 1994
American Mathematical Society
