Abstract
The axisymmetric torsion of a layer of incompressible Mooney–Rivlin material placed between two rigid coaxial cylindrical surfaces is considered. It is assumed that the ends of the cylinder are fixed to avoid axial displacement and the cross-sections orthogonal to the axis of symmetry do not distort during deformation. In this case, the material points move along arcs of circles with an angular velocity which in the general case may depend on both the axial and the radial coordinates. Closed-form solution of the equilibrium equation is obtained. The well-known Rivlin solution is a particular case of this solution. Like in the classical solution, the stress tensor is a function of only the radial coordinate and the twist angle of the material points is a linear function of the axial coordinate. It is shown that torsion that is symmetric about a plane orthogonal to the axis of rotation can be realized only within the framework of the Rivlin solution. The obtained exact solution can be realized in the presence of friction on cylindrical surfaces. The initial-boundary-value problem described by this solution is considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
J. L. Ericksen, “Deformations possible in every isotropic, incompressible, perfectly elastic body,” ZAMP 5 (6), 466–489 (1954).
R. S. Rivlin, “Large elastic deformations of isotropic materials. VI. further results in the theory of torsion, shear and flexure,” Phil. Trans. R. Soc. Lond. A. 242 (845), 173–195 (1949).
N. Kh. Arutyunyan and B. L. Abramyan, Torsion of Elastic Bodies (Fizmatgiz, Moscow, 1963) [in Russian].
N.Kh. Arutyunyan and Yu.N. Radayev, “Elastoplastic torsion of a cylindrical rod for finite deformations,” J. Appl. Math. Mech. 53 (6), 804–811 (1989).
G. M. Sevastyanov and A. A. Burenin, “Finite strain upon elastic–plastic torsion of an incompressible circular cylinder,” Dokl. Phys.63, 393–395 (2018).
D. A. Warne and P. G. Warne, “Torsion in nonlinearly elastic incompressible circular cylinders,” Int. J. Non-Linear Mech.86, 158–166 (2016).
R. L. Fosdick and G. MacSithigh, “Helical shear of an elastic, circular tube with a non-convex stored energy,” in The Breadth and Depth of Continuum Mechanics: A Collection of Papers Dedicated to J. L. Ericksen on His Sixtieth Birthday (Springer, Berlin, Heidelberg, 1986), pp. 31–53.
C. O. Horgan and G. Saccomandi, “Helical shear for hardening generalized neo-Hookean elastic materials,” Math. & Mech. Sol. 8 (5), 539–559 (2003).
B. A. Zhukov, “Nonlinear interaction of finite longitudinal shear with finite torsion of a rubber-like bushing,” Mech. Solids 50 (3), 337–344 (2015).
M. Mooney, “A theory of large elastic deformation,” J. Appl. Phys. 11, 582–592 (1940).
A. I. Lurie, Nonlinear Theory of Elasticity (Nauka, Moscow, 1980; North-Holland, Amsterdam, 1990).
A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (Chapman & Hall/CRC Press, Boca Raton-London, 2012).
Funding
This study was carried out within the framework of state contract of Khabarovsk Federal Research Center of the Far Eastern Branch of the Russian Academy of Sciences.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by A.A. Borimova
About this article
Cite this article
Sevastyanov, G.M. Torsion with Circular Shear of a Mooney–Rivlin Solid. Mech. Solids 55, 273–276 (2020). https://doi.org/10.3103/S0025654420020156
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654420020156