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

Full-text PDF Free Access

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.

**[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

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73G05,
73C50

Retrieve articles in all journals with MSC: 73G05, 73C50

Additional Information

DOI:
https://doi.org/10.1090/qam/1162279

Article copyright:
© Copyright 1992
American Mathematical Society