## Analysis on the Establishment of Angular Displacement and Stresses for Pure Azimuthal Deformation in an Incompressible Hollow Cylindrical Gent Material

Okujagu Tamunobarachueye Edward

Department of Mathematics, Federal University of Technology, Owerri, Nigeria.

Nwagwu Isaac Ogazie *

Department of Mathematics, Federal University of Technology, Owerri, Nigeria.

Julius Bassey Effiong

Department of Mathematics, Federal University of Technology, Owerri, Nigeria.

*Author to whom correspondence should be addressed.

### Abstract

This project work is concerned with the study of an Isotropic, Incompressible hollow cylinder deforming under pure azimuthal shear loading. The analysis resulted into a non-linear second order ordinary differential equation for the determination of the angular displacement. The resulting solution of the boundary value problem gave an equation which needed the wolfram Alpha software for the determination of the analytic solution of the angular displacement. Relationship between the displacement, stress components and the material parameter for hardening and softening is shown and the effect of change on the material parameter on the angular displacement is illustrated.

Keywords: Angular displacement, deformed radius, shear strain, displacement gradient, shear stresses

**How to Cite**

*Asian Journal of Pure and Applied Mathematics*,

*5*(1), 157–169. Retrieved from https://globalpresshub.com/index.php/AJPAM/article/view/1810

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### References

Gent AN. Rubber elasticity: basic concepts and behaviour. OH: The University of Akron; 2005.

Horgan O, Saccomandi G. A molecular-statistical basis for the gent constitutive model of rubber elasticity. J Elast. 2003;68:167-76.

Abo-El-Nou NA, Fatimah A. The dispersion relation of flexural waves in a magnetoelastic anisotropic circular cylinder. Adv Phys Theor Appl. 2013;13:20-33.

Haddow JB, Erbay HA. Some aspects of finite amplitude transverse waves in a compressible hyperelastic solid. Q J Mech Appl Math. 2002;55(1):17-28.

Haddow JB, Jiang X. Finite amplitude azimuthal shear waves in an incompressible hyperelastic solids, ibid. 2002;68:145-52.

Horgan CO, Polignome DA. Pure azimuthal shear of compressible nonlinear elastic tubes. Q Appl Math. 1997;52:11-3.

Beatty MF, Jiang Q. On Compressible Matherials capable of sustaining Axisymmetric shear Deformations. Part 2: rotation shear of isotropic hyperelastic material, Q. J Mech. 1997;50:211-37.

Jiang X, Ogden RW. Azimuthal shear of a circular cylindrical tube of compressible elastic, Q uarterly. J Mech Appl Math. 1998;51:143-58.

Akbarov SA, Guliev MS. Axisymmetric Longitudinal wave propagation in a finite prestretched compound circular cylinder made of an incompressible material. Int Appl Mech. 2009;45:1141-51.

Selim MM. Torsional Waves Propagation in an initially stressed dissipative cylinder. Appl Math Sci. 2009;1:1419-27.

Horgan CO. The remarkable Gent constitutive model for hyperelastic materials. Int J Non Linear Mech. 2015;68:9-16.

Shearer T, Abrahams ID, Parnell WJ, Daros CH. Torsional wave propagation in a pre- stressed hyperelastic annular circular cylinder. Q J Mech Appl Math. 2013;66(4):465-87.

Moreira DC, Nunes LCS. Comparison of simple and pure shear for an incompressible isotropic hyperelastic material under large deformation. Polym Test. 2013;32(2):240-8.

Erumaka EN. Finite deformation of a class of Ogden solid under anti-plane shear. J Math Sci. 2003;18:19-24.

Ogden RW. Non-Linear elastic deformations. chichester: Ellis Horwood; 1984.

Anani Y, Rahimi G. Stress analysis of thick pressure vessel composed of incompressible hyperelastic materials. IJMECH. 2015;4(3):19-37.

Merodio J, Ogden RW. Extension, inflation and torsion of a residually stressed circular cylindrical tube, continnum mechanics thermodynamics. 2015;28:157-74.

M. and Darijani H. New polynomial strain energy function; application to rubbery circular cylinders under finite extension and torsion. J Appl Polym Sci. 2015;132.

Mangan R, Destrade M, Saccomandi G. Strain energy function for isotropic non-linear elastic incompressible solids with linear finite strain response in shear and torsion. Extreme Mech Lett. 2016;9:204-6.

Erumaka EN, Onugha EE, Nwagwu IO. Axial shear wave in an incompressible cylindrical solid. J Niger Assoc Math Math Phys. 2017;42:137-42.

Erumaka EN, Onugha EE, Nwagwu IO. Combined axial and azimuthal shear wave in an incompressible hollow circular cylinder. Trans Niger Math Phys. 2018;7:41-6.