GENERAL SOLUTION OF ELASTICITY PROBLEMS IN TWO DIMENSIONAL POLAR COORDINATES USING MELLIN TRANSFORM

Ike, Charles

doi:10.22059/jcamech.2019.278288.370

GENERAL SOLUTION OF ELASTICITY PROBLEMS IN TWO DIMENSIONAL POLAR COORDINATES USING MELLIN TRANSFORM

Document Type : Research Paper

Author

Charles Ike

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

10.22059/jcamech.2019.278288.370

Abstract

Abstract
In this work, the Mellin transform method was used to obtain solutions for the stress field components in two dimensional (2D) elasticity problems in terms of plane polar coordinates. the Mellin transformation was applied to the biharmonic stress compatibility equation expressed in terms of the Airy stress potential function, and the boundary value problem transformed to an algebraic problem which was solved to obtain the Mellin transformed Airy stress potential function. The Mellin transform was similarly used to obtain the Mellin transformed stress field components. The use of Mellin transform inversion formula yielded the solutions to the 2D elasticity problem in the physical space domain variables. Specific illustration was considered of the solution by using the Mellin transform method for the Flamant problem and the Mellin transform solutions found to agree with solutions from the literature.

Keywords

Mellin transform method

Mellin transform inversion formula

biharmonic stress compatibility equation

Airy stress potential function

two dimensional (2D) elasticity problem

Subjects

Inverse problems

[1] S.K. Borg. Fundamentals of Engineering Elasticity Second Edition. World Scientific Publishing Co. Ltd. London, 1970.

[2] J. Blaauwendraad. Theory of Elasticity ct 5141 Direct Methods. DelftUniversity of Technology, Faculty of Civil Engineering and Sciences. June 2003.

[3] A. Szekrenyes. Introduction to plane problems subject Application of plane stress, plane strain and revolution symmetric (axisymmetric) models. www.mm.bme.hu/../11_fejezetes_bevezetes_sikfeladatok_lektorait_kovrigait_VA_eng.pdf.

[4] I.S. Sokolnikoff. Mathematical Theory of Elasticity Second Edition. Tata McGraw-Hill Publishing Company Ltd, Bombay, New Delhi 1956.

[5] R.J. Atkin and N. Fox. An introduction of the theory of elasticity. Longman Group Ltd, London 1980.

[6] T.G. Sitharam and L. Govinda Reju. Applied Elasticity for Engineers Module: Elastic Solutions and Applications in Geomechanics. 14.139.172.204/npte/1/ CSE/web/105108070/module 8/lecture 17.pdf.

[7] H.R. Hamidzadeh and R.N. Jazar. Vibrations of thick cylindrical structures. Springer Science Business Media p. 15 – 26, 2010.

[8] A. Hazel. MATH 350211: Elasticity www.maths.manchester.ac.uk/~ahazel/ MATHS Nov 30 2015.

[9] D. Palaniappian. A general solution of equations of equilibrium in linear elasticity. Applied Mathematical Modelling 35 (2011). Pp. 5494 – 5499. Elsevier, 2011.

[10] S.P. Timoshenko and J.N. Goodier. Theory of Elasticity, Third Edition. McGraw Hill, New York 1970.

[11] C. Ramadas. 2D Theory of Elasticity R&DE (Engineers) DRDO. http://imechanica.org/files/2D theory of elasticity.pdf.

[12] O. Joubert. The Mellin Transform. October 2011. math.sun.ac.za/wp-content/uploads/2013/02/Hons_Projek.pdf.

[13] D. Zagier. Appendix. The Mellin Transform and Related Analytic Techniques. people.mpim-bonn.mpg.de/zagier/files/text/Mellin Transform/fulltext.pdf

[14] T. Cindy. Mellin Transform and Riemann Zeta Function. www.math.clemson.edu/~kevja/COURSES/math952/MTHSC 952 - 2013 - SPRING PRESENTATIONS/Cindy Tagaris Mellin Transforms.pdf.

[15] B.M. Das. Advanced Soil Mechanics Third Edition. Taylor and Francis, New York, 2008.

[16] H.N. Onah, N.N. Osadebe, C.C. ike, C.U. Nwoji. Determination of stresses caused by infinitely long line loads on semi-infinite elastic soils using Fourier transform method. Nigerian Journal of Technology NIJOTECH Vol 35 No 4 October 2016 pp. 726 – 731.