Elzaki transform method for finding solutions to two-dimensional elasticity problems in polar coordinates formulated using Airy stress functions

Document Type : Research Paper

Author

Dept of Civil Engineering, Enugu State University of Science and Technology, Enugu State, Nigeria

Abstract

In this paper, the Elzaki transform method is used for solving two-dimensional (2D) elasticity problems in plane polar coordinates. Airy stress function was used to express the stress compatibility equation as a biharmonic equation. Elzaki transform was applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified to a fourth order ordinary differential equation (ODE). The general solution for the Airy stress potential function in the Elzaki transform space was obtained by solving the ODE. By inversion, the general solution for the Airy stress potential function was obtained in the physical domain space variables in terms of four unknown integration constants. Normal stresses and shear stress fields were also determined for the general case of 2D elasticity problems. The Flamant problem was solved as a particular illustration of 2D elasticity problems. The stress boundary conditions and the requirement of equilibrium of the internal stress resultants and the external forces were used simultaneously to determine the four constants of integration. The Airy stress potential function and the normal and shear stress fields were thus completely determined. The principle of superposition was used to obtain the elasticity solutions for the stress fields in the elastic half plane due to strip load of infinite extent, and solutions for horizontal stresses on smooth rigid retaining walls due to strip loads and parallel line load of infinite extent acting on the elastic half plane.

Keywords

[1]   C.C. Ike. First principles derivation of a stress function for axially symmetric elasticity problems, and application to Boussinesq problem. Nigerian Journal of Technology, Vol 36 No 3, pp. 767-772, July 2017. http://dx.doi.org/10.4314/nijt.v36i3.15 [Cross Ref]
[2]   C.C. Ike, H.N. Onah, C.U. Nwoji. Bessel functions for axisymmetric elasticity problems of the elastic half space soil, a potential function method. Nigerian Journal of Technology, Vol 36, No 3, pp. 773-781, July 2017. http://dx.doi.org/10.4314/nijt.v36i3.16        [Cross Ref]
[3]   C.U. Nwoji, H.N. Onah, B.O. Mama, C.C Ike. Solution of the Boussinesq problem of half space using Green and Zerna displacement potential function method. Electronic Journal of Geotechnical Engineering (EJGE) 22.11 pp 4305-4314, 2017. Available at ejge.com.
[4]   C.C. Ike, B.O. Mama, H.N. Onah, C.U. Nwoji. Trefftz harmonic function method for solving Boussinesq problem. Electronic Journal of Geotechnical Engineering (EJGE), 22:12 pp 4589-4601, 2017. Available at ejge.com
[5]   H.N. Onah, C.C. Ike, C.U. Nwoji, B.O. Mama. Theory of elasticity solution for stress fields in semi-infinite linear elastic soil due to distributed load on the boundary using the Fourier transform method. Electronic Journal of Geotechnical Engineering (EJGE) 22.13 pp 4945-4962, 2017. Available at ejge.com.
[6]   C.U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike. Solution of elastic half space problem using Boussinesq displacement potential functions. Asian Journal of Applied Sciences (AJAS), Vol 5 Issue 5, pp. 1100-1106, October 2017.
[7]   H.N. Onah, B.O. Mama, C.U. Nwoji, C.C. Ike. Boussineq displacement potential functions method for finding vertical stresses and displacement fields due to distributed load on elastic half space. Electronic Journal of Geotechnical Engineering (EJGE) 22.15 pp 5687-5709, 2017. Available at ejge.com.
[8]   C.C. Ike. Exponential Fourier integral transform method for stress analysis of boundary load on soil. Mathematical Modelling of Engineering Problems (MMEP) 5(1), pp. 33-39, March 2018. https//doi.org/10.18280/mmep.050105
[9]   C.C. Ike. Hankel transform method for solving axisymmetric elasticity problems of circular foundation on semi-infinite soils. International Journal of Engineering and Technology (IJET) Vol 10 No 2, pp 549-564, Apr-May 2018. doi: 10.21817/ijet/2018/v10i2/181002111
[10] C.C. Ike. General solutions for axisymmetric elasticity problems of elastic half space using Hankel transform method. International Journal of Engineering and Technology (IJET) Vol 10 No 2, pp 565-580,  Apr – May 2018. https//doi.org/10.21817/ijet/2018/v10i2/181002112
[11] C.C. Ike. Fourier sine transform method for solving the Cerrutti problem of elastic half plane in plane strain. Mathematical Modelling in Civil Engineering (MMCE) Vol 14 No 1, pp 1-11, 2018. DOI: 10.2478/mmce-2018-0001              [Cross Ref]
[12] C.C. Ike. Solution of elasticity problems in two dimensional polar coordinates using Mellin transform. Journal of Computational Applied Mechanics, JCAMECH, Vol 50 Issue 1, pp. 174-181, June 2019. DOI: 10.22059/jcamech.2019.278288.370
[13] C.C. Ike. On Maxwell’s stress functions for solving three dimensional elasticity problems in the theory of elasticity. Journal of Computational Applied Mechanics (JCAMECH) Vol 49 Issue 2, pp. 342-350, December 2018. DOI: 10.22059/jcamech.2018.266787.330
[14] C.C. Ike. Hankel transform method for solving the Westergard problem for point, line and distributed loads on elastic half space. Latin American Journal of Solids and Structures (LAJSS) Vol 16 No 1, pp. 1-19, 2019. e155 DOI: http://dx.doi.org/10.1590/1679-78255313
[15] C.C. Ike. Fourier-Bessel transform method for finding vertical stress fields in axisymmetric elasticity problems of elastic half-space involving circular foundation areas. Advances in Modelling and Analysis A, AMA_A Vol 55 No. 4, pp. 207-216, December 2018. https://doi.org//10.18280/ama_a.550405.
[16] C.C. Ike. Love stress function method for solving axisymmetric elasticity problems of the elastic half-space. Electronic Journal of Geotechnical Engineering (EJGE) Vol 24 No. 3, pp. 663-706, 2019. Available at ejge.com.
[17] A.P. Boresi, K.P. Chong, J.D. Lee. Plane Elasticity in Polar Coordinates, In Elasticity in Engineering Mechanics, Third Edition, 2010. https//doi.org/10/1002/9780470950005.ch6
[18] R.T. Fenner. Mechanics of Solids. CRC Press, Washington D.C., 1999. ISBN: 0-632-02018-0.
[19] D.R. Kavati. Airy stress function for two dimensional inclusion problems. Master of Science in Mechanical Engineering thesis, The University of Texas at Arlington, December 2005.
[20] Theory of Elasticity – NTNU https://www.ntnu.no.wiki.
[21] M. Kenderova, F. Trebuna, P. Frankovsky. Verification of stress components determined by experimental methods using Airy stress functions. Procedia Engineering, 48(2012) pp. 295-303, 2012. doi.10.1016/j.proeng.2012.09.517.
[22] P.J.G. Schreurs. Applied Elasticity in Engineering Toegepaste Elasticiteitsleer Lecture notes -course 4A 450 Eindhoven University of Technology, Department of Mechanical Engineering, Materials Technology, May 8, 2013. www.mate.tue.nl>ela>pdf
[23] M. Madhavan. A novel approach for two-dimensional inclusion/hole problems using the Airy stress function method. Thesis for Master of Science in Mechanical Engineering, The University of Texas at Arlington, May 2013, 76pp.
[24] A. Maceri. Theory of Elasticity. Springer, New York, 2010. eISBN 978-3-642-11392-5, ISBN 978-3-642-11391-8, doi: 10.1007/978-3-642-11392-5
[25] E.I. Starovoitov, F.B. Nagiyev. Foundations of the theory of elasticity, plasticity and viscoelasticity. Apple Academic Press, CRC Press Taylor & Francis Group, New Jersey, 2013. ISBN 978-1-926895-11-6
[26] T.M. Atanackovic, A. Guran. Theory of Elasticity for Scientists and Engineers. Springer Science Business Media, LLC, New York, 2000. ISBN 978-1-4612-1330-7(ebook), ISBN 978-1-4612-7097-3, doi: 10.1007/978-1-4612-1330-7
[27] M. Danesh, A. Farajpour, M. Mohammadi. Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Research Communications, Vol. 39, No 1, pp 23-27, January 2012. DOI: 10.1016/j.mechrescom.2011.09.004
[28] M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. Rastgoo. Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads. European Journal of Mechanics – A / Solids, Vol. 77, Sept-Oct 2019, 103793, https://doi.org/10.1016/j.euromechsol.2019.05.008
[29] M. Mohammadi, A. Rastgoo. Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core. Structural Engineering and Mechanics, Vol. 69 No. 2, pp 131-143, 2019. DOI: 10.12989/semi2019.69.2.131
[30] M. Mohammadi, A. Rastgoo. Primary and secondary resonance analysis of FG / lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium. Mechanics of Advanced Materials and Structures, pp 1-22, Dec 2019. https://doi.org/10.1080/15376494.2018.1525453
[31] M. Safarabadi, M. Mohammadi, A. Farajpour, M. Goodarzi. Effect of surface energy on the vibration analysis of rotating nanobeam. Journal of Solid Mechanics, Vol. 7 No. 3, pp 299-311, 2015.
[32] A. Barati, MM. Adeli, A. Hadi. Static Torsion of Bi-Directional Functionally Graded Microtube Based on the Couple Stress Theory Under Magnetic Field. International Journal of Applied Mechanics, Vol. 12, No 2, pp 2050021, 2020.
[33] T.M. Elzaki. The new integral transform “Elzaki Transform”. Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Number 1, pp 57-64, 2011.
[34] T.M. Elzaki, S.M. Elzaki. On Elzaki transform and ordinary differential equation with variable coefficients. Advances in Theoretical and Applied Mathematics. ISSN 0973-4554, Vol 6 Number 1, pp. 13-18, 2011.
[35] T.M. Elzaki, S.M. Elzaki. On the new integral transform “Elzaki transform”: Fundamental principles, investigation and applications. Global Journal of Mathematical Sciences. 4(1) pp 1-13, 2012.
[36] A. Devi, P. Roy, V. Gill. Solution of ordinary differential equations with variable coefficients using Elzaki transform. Asian Journal of Applied Science and Technology (AJAST) Vol 1, Issue 9, pp 186-194, 2017.
Volume 51, Issue 2
December 2020
Pages 302-310
  • Receive Date: 16 January 2020
  • Revise Date: 05 February 2020
  • Accept Date: 12 February 2020