### An Axisymmetric Contact Problem of a Thermoelastic Layer on a Rigid Circular Base

Document Type: Research Paper

Authors

Department of Mechanical Engineering, Ecole Nationale Polytechnique, Algiers, Algeria

10.22034/jsm.2019.668619

Abstract

We study the thermoelastic deformation of an elastic layer. The upper surface of the medium is subjected to a uniform thermal field along a circular area while the layer is resting on a rigid smooth circular base. The doubly mixed boundary value problem is reduced to a pair of systems of dual integral equations. The both system of the heat conduction and the mechanical problems are calculated by solving a dual integral equation systems which are reduced to an infinite algebraic one using a Gegenbauer’s formulas.  The stresses and displacements are then obtained as Bessel function series. To get the unknown coefficients, the infinite systems are solved by the truncation method. A closed form solution is given for the displacements, stresses and the stress singularity factors. The effects of the radius of the punch with the rigid base and the layer thickness on the stress field are discussed. A numerical application is also considered with some concluding results.

Keywords

[1] Dhaliwal R.S., 1966, Mixed boundary value problem of heat conduction for infinite slab, Applied Scientific Research 16: 226-240.
[2] Dhaliwal R.S., 1967, An axisymmetric mixed boundary value problem for a thick slab, SIAM Journal on Applied Mathematics 15: 98-106.
[3] Mehta B.R.C., Bose T.K., 1983, Temperature distribution in a large circular plate heated by a disk heat source, International Journal of Heat and Mass Transfer 26: 1093-1095.
[4] Lebedev N.N., Ufliand IA.S., 1958, Axisymmetric contact problem for an elastic layer, PMM 22: 320-326.
[5] Zakorko V.N., 1974, The axisymmetric strain of an elastic layer with a circular line of separation of the boundary conditions on both faces, PMM 38: 131-138.
[6] Dhaliwal R.S., Singh B.M., 1977, Axisymmetric contact problem for an elastic layer on a rigid foundation with a cylindrical hole, International Journal of Engineering Science 15: 421-428.
[7] Wood D.M., 1984, Circular load on elastic layer, International Journal for Numerical and Analytical Methods in Geomechanics 8: 503-509.
[8] Toshiaki H., Takao A., Toshiakaru S., Takashi K., 1990, An axisymmetric contact problem of an elastic layer on a rigid base with a cylindrical hole, JSME International Journal 33: 461- 466.
[9] Sakamoto M., Hara T., Shibuya T., Koizumi T., 1990, Indentation of a penny-shaped crack by a disc-shaped rigid inclusion in an elastic layer, JSME International Journal 33: 425- 430.
[10] Sakamoto M., Kobayashi K., 2004, The axisymmetric contact problem of an elastic layer subjected to a tensile stress applied over a circular region, Theoretical and Applied Mechanics Japan 53: 27-36.
[11] Sakamoto M., Kobayashi K., 2005, Axisymmetric indentation of an elastic layer on a rigid foundation with a circular hole, WIT Transactions on Engineering Sciences 49: 279-286.
[12] Kebli B., Berkane S., Guerrache F., 2018, An axisymmetric contact problem of an elastic layer on a rigid circular base, Mechanics and Mechanical Engineering 22: 215-231.
[13] Dhaliwal R.S., 1971, The steady-state thermoelastic mixed boundary-value problem for the elastic layer, IMA Journal of Applied Mathematics 7: 295-302.
[14] Wadhawan M.C., 1973, Steady state thermal stresses in an elastic layer, Pure and Applied Geophysics 104: 513- 522.
[15] Negus K.J., Yovanovich M., Thompson J.C., 1988, Constriction resistance of circular contacts on coated surfaces: Effect of boundary conditions, Journal of Thermophysics and Heat Transfer 2: 158-164.
[16] Lemczyk T.F., Yovanovich M.M., 1988, Thermal constriction resistance with convective boundary conditions-1 Half-space contacts, International Journal of Heat and Mass Transfer 31: 1861-1872.
[17] Lemczyk T.F., Yovanovich M.M., 1988, Thermal constriction resistance with convective boundary conditions-2 Half-space contacts, International Journal of Heat and Mass Transfer 31: 1873-1988.
[18] Rao T.V., 2004, Effect of surface layers on the constriction resistance of an isothermal spot. Part I: Reduction to an integral equation and numerical results, Heat and Mass Transfer 40: 439- 453.
[19] Rao T.V., 2004, Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers, Heat and Mass Transfer 40: 455- 466.
[20] Abdel-Halim A.A., Elfalaky A., 2005, An internal penny-shaped crack problem in an infinite thermoelastic solid, Journal of Applied Sciences Research 1: 325-334.
[21] Elfalaky A., Abdel-Halim A.A., 2006, Mode I crack problem for an infinite space in thermoelasticity, Journal of Applied Sciences 6: 598-606.
[22] Hetnarski R.B., Eslami M.R., 2009, Thermal Stresses-Advanced Theory and Applications, Springer, Dordrecht.
[23] Hayek S.I., 2001, Advanced Mathematical Methods in Science and Engineering, Marcel Dekker, New York.
[24] Duffy D.G., 2008, Mixed Boundary Value Problems, Boca Raton, Chapman Hall/CRC.
[25] Gradshteyn I.S., Ryzhik I. M., 2007, Table of Integrals- Series and Products, Academic Press, New York.