An Axisymmetric Torsion Problem of an Elastic Layer on a Rigid Circular Base

Journal of Solid Mechanics

مقاله 14، دوره 12، شماره 1، خرداد 2020، صفحه 204-218   

نوع مقاله: Research Paper

شناسه دیجیتال (DOI): 10.22034/jsm.2019.1868197.1441

نویسندگان

B Kebli^{* 1}؛ S Berkane²؛ F Guerrache¹

¹Laboratoire de Génie Mécanique et Développement, Département de Génie Mécanique, Ecole Nationale Polytechnique El-Harrach Algiers 16200, Algeria

²Department of Computer Science and Engineering, University of Québec in Outaouais, Gatineau, Québec, Canada

چکیده

A solution is presented to a doubly mixed boundary value problem of the torsion of an elastic layer, partially resting on a rigid circular base by a circular rigid punch attached to its surface. This problem is reduced to a system of dual integral equations using the Boussinesq stress functions and the Hankel integral transforms. With the help of the Gegenbauer expansion formula of the Bessel function we get an infinite algebraic system of simultaneous equations for calculating the unknown function of the problem. Both the two contact stresses under the punch and on the lower face of the layer are expressed as appropriate Chebyshev series. The effects of the radius of the disc with the rigid base and the layer thickness on the displacements, contact stresses as well as the shear stress and the stress singularity factor are discussed. A numerical application is also considered with some concluding results.

کلیدواژه‌ها

Elastic torsion؛ Doubly mixed boundary value problem؛ Dual integral equations؛ Infinite algebraic system؛ Stress singularity factor

مراجع

[1] Reissner E., Sagoci H.F., 1944, Forced torsion oscillation of an half-space-I, Journal of Applied Physics 15: 652-654.
[2] Lebedev N.N., Ufliand I.S., 1958, Axisymmetric contact problem for an elastic layer, PMM 22: 320-326.
[3] Florence A.L., 1961, Two contact problems for an elastic layer, The Quarterly Journal of Mechanics and Applied Mathematics 14: 453-459.
[4] Smelyanskaya L.M., Tokar A.S., 1971, Strength of a composite elastic layer weakened by a plane circular crack, Soviet Applied Mechanics 7: 1119-1125.
[5] Low R.D., 1972, On the torsion of elastic half spaces with embedded penny shaped flaws, Journal of Applied Mechanics 72: 786-790.
[6] Sih G.C., Chen E.P., 1972, Torsion of a laminar composite de bonded over a penny- shaped area, Journal of the Franklin Institute 293: 251-261.
[7] Dhawan G.K., 1974, Torsion of elastic half-space with penny shaped crack, Defence Science Journal 24: 15-22.
[8] Tamate O., Saito T., 1975,On the twisting of two elastic layers bonded to a rigid foundation, Translations of the Japan Society of Mechanical Engineers 341: 33-40.
[9] Singh B.M., Dhaliwal R.S., 1977, Torsion of a elastic layer by two circular dies, International Journal of Engineering Science 15: 171-175.
[10] Gazetas G., 1981, Torsional displacements and stresses in non-homogeneous soil, Geotechnique 31: 487-496.
[11] Hara T., Akiyama T., Shibuya T., Koizumi T., 1989, An axisymmetric torsion problem of an elastic layer on a rigid foundation with a cylindrical hole, Translations of the Japan Society of Mechanical Engineers 55: 1339-1346 .
[12] Erguven M.E., 1991, Torsion of two bonded layers by a rigid disc, Meccanica 26: 117-123.
[13] Pak R.Y.S., Saphores J.D.M., 1991, Torsion of a rigid disc in a half-space, International Journal of Engineering Science 29: 1-12 .
[14] Bacci A., Bennati S., 1996, An approximate explicit solution for the local torsion of an elastic layer, Mechanics of Structures and Machines 24: 21-38.
[15] Li C., Zou Z., 1998, Local stress field for torsion of penny-shaped crack in a functionally graded material, International Journal of Fracture 91: 17-22.
[16] Sakamoto M., 2003, An elastic layer with a penny-shaped crack subjected to internal pressure, SME International Journal Series A solid Mechanics and Material Engineering 46: 10-14.
[17] Matysiak S.J., Kulchytsky-Zhyhailo R., Perkowski D.M., 2011, Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space, Mechanics Research Communications 38: 320-325.
[18] Madani F., Kebli B., 2017, Axisymmetric torsion of an internally cracked elastic medium by two embedded rigid discs, Mechanics and Mechanical Engineering 21: 363-377.
[19] Duffy D.G., 2008, Mixed Boundary Value Problems, Boca Raton, Chapman Hall/CRC.
[20] Gradshteyn I.S., Ryzhik I.M., 2007, Table of Integrals- Series and Products, Academic Press, New York.