Torsion and circular shear coupling in nonlinear-elastic hollow cylinder

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Abstract

Combined torsional and circular shear of an incompressible nonlinear-elastic right-circular hollow cylinder is studied. A solution to the problem is obtained for an arbitrary elastic potential depending on the first invariant of the left Cauchy – Green deformation tensor solely (generalized neo-Hookean solid). For the Gent material, an analytical solution in closed form is obtained. A rotary damper design based on the obtained solution is proposed. Formulas for the dissipation of kinetic energy due to friction on the cylindrical surfaces of the pipe are given. For a strain softening material, a numerical solution is obtained, which is compared with experimental results.

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About the authors

G. M. Sevastyanov

Institute of Machinery and Metallurgy FEB RAS

Author for correspondence.
Email: akela.86@mail.ru
Russian Federation, Komsomolsk-on-Amur

O. N. Komarov

Institute of Machinery and Metallurgy FEB RAS

Email: olegnikolaevitsch@rambler.ru
Russian Federation, Komsomolsk-on-Amur

A. V. Popov

Institute of Machinery and Metallurgy FEB RAS

Email: popov.av@live.com
Russian Federation, Komsomolsk-on-Amur

References

  1. Mioduchowski A., Haddow J.B. Combined torsional and telescopic shear of a compressible hyperelastic tube // J Appl Mech-T Asme. 1979. V. 46. № 1. P. 223–226. https://doi.org/10.1115/1.3424509
  2. Fosdick R.L., MacSithigh G. Helical shear of an elastic, circular tube with a non-convex stored energy // Arch. Ration Mech. AN. 1983. V. 84. P. 31–53. https://doi.org/10.1007/BF00251548
  3. Tao L., Rajagopal K.R., Wineman A.S. Circular shearing and torsion of generalized neo-Hookean materials // IMA J Appl Math. 1992. V. 48. № 1. P. 23–37. https://doi.org/10.1093/imamat/48.1.23
  4. Zidi M. Azimuthal shearing and torsion of a compressible hyperelastic and prestressed tube // Int. J Nonlin Mech. 2000. V. 35. № 2. P. 201–209. https://doi.org/10.1016/S0020-7462(99)00008-6
  5. Zidi M. Combined torsion, circular and axial shearing of a compressible hyperelastic and prestressed tube // J Appl. Mech-T Asme. 2000. V. 67. № 1. P. 33–40. https://doi.org/10.1115/1.321149
  6. Zidi M. Finite torsion and shearing of a compressible and anisotropic tube // Int. J Nonlin Mech. 2000. V. 35. № 6. P. 1115–1126. https://doi.org/10.1016/S0020-7462(99)00083-9
  7. Horgan C.O., Saccomandi G. Helical shear for hardening generalized neo-Hookean elastic materials // Math. Mech. Solids. 2003. V. 5. № 5. P. 539–559. https://doi.org/10.1177/10812865030085007
  8. Saravanan U., Rajagopal K.R. Inflation, extension, torsion and shearing of an inhomogeneous compressible elastic right circular annular cylinder // Math. Mech. Solids. 2005. V. 10. № 6. P. 603–650. https://doi.org/10.1177/1081286505036422
  9. Zhukov B.A. Nonlinear interaction of finite longitudinal shear with finite torsion of a rubber-like bushing // Mech. Solids. 2015. V. 50. P. 337–344. https://doi.org/10.3103/S0025654415030097
  10. Sevastyanov G.M. Torsion with circular shear of a Mooney – Rivlin solid // Mech. Solids. 2020. V. 55. P. 273–276. https://doi.org/10.3103/S0025654420020156
  11. Rivlin R.S. Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure // Philos. TR R Soc. S-A. 1949. V. 242. № 845. P. 173–195. https://doi.org/10.1098/rsta.1949.0009
  12. Ericksen J.L. Deformations possible in every isotropic, incompressible, perfectly elastic body // Z Angew Math. Phys. 1954. V. 5. № 6. P. 466–489. https://doi.org/10.1007/BF01601214
  13. Knowles J.K. The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids // Int. J Fracture. 1977. V. 13. P. 611–639. https://doi.org/10.1007/BF00017296
  14. Gent A.N. A new constitutive relation for rubber // Rubber Chem. Technol. 1996. V. 69. № 1. P. 59–61. https://doi.org/10.5254/1.3538357
  15. Horgan C.O., Saccomandi G. Constitutive modelling of rubber-like and biological materials with limiting chain extensibility // Math. Mech. Solids. 2002. V. 7. № 4. P. 353–371. https://doi.org/10.1177/108128028477
  16. Horgan C.O., Saccomandi G. Pure azimuthal shear of isotropic, incompressible hyperelastic materials with limiting chain extensibility // Int. J Nonlin. Mech. 2001. V. 36. № 3. P. 465–475. https://doi.org/10.1016/S0020-7462(00)00048-2
  17. Anssari-Benam A., Horgan C.O. Extension and torsion of rubber-like hollow and solid circular cylinders for incompressible isotropic hyperelastic materials with limiting chain extensibility // Eur. J Mech. A-Solids. 2022. V. 92. P. 104443. https://doi.org/10.1016/j.euromechsol.2021.104443
  18. Begun A.S., Burenin A.A., Kovtanyuk L.V. Helical viscoplastic flow in a gap between rigid cylinders // Mech. Solids. 2017. V. 52. P. 640–652. https://doi.org/10.3103/S0025654417060048
  19. Lurie A.I. Nonlinear theory of elasticity. North-Holland, New York, 1990.
  20. Polyanin A.D., Zaitsev V.F., Moussiaux A. Handbook of first order partial differential equations. Taylor & Francis, London, 2002.
  21. Kut S., Ryzińska G. Modeling elastomer compression: Exploring ten constitutive equations // Materials. 2023. V. 16. № 1. P. 4121. https://doi.org/10.3390/ma161141
  22. Mullins L., Tobin N.R. Stress softening in rubber vulcanizates. Part I. Use of a strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber // J Appl. Polym. Sci. 1965. V. 9. P. 2993–3009. https://doi.org/10.1002/app.1965.070090906

Supplementary files

Supplementary Files
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2. Fig. 1. Loading scheme of the deformable element of the damper.

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3. Fig. 2. Design of the rotary damper with friction: 1 – central sleeve, 2 – outer cage, 3 – elastic element, 4 – grippers.

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4. Fig. 3. Structural elements of a rotary damper with friction: 1 – central sleeve, 2 – outer cage, 3 – elastic element, 4 – grippers.

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5. Fig. 4. Simple tension, s11 in MPa. The markers are experimental data for polyurethane A83 (engineering stresses). The Ghent model with parameters E = 3m = 17 MPa, Jm = 16 (data for polyurethane A90 according to [21]). Model (3.2) with parameters E0 = 17 MPa, E1 = 1 MPa, b = 0.29, I1a – 3 = 0.107.

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6. Fig. 5. Torque M [N · m] when loading the rotary damper with the angle of rotation a (in degrees). Markers are experimental data. The solid line is a numerical and analytical calculation based on the elastic model (3.2). The dotted line is an analytical calculation for the Gent material with parameters E = 3m = 17 MPa, Jm = 16.

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