Abstract
In this paper, we propose a method for the solution of the axisymmetric boundary value problem for a finite elastic cylinder with assigned stress and/or displacements acting on the ends and side. The technique utilizes the Love representation, which allows for reduction of the solution of the elastic problem to the search for a biharmonic function on a cylindrical domain. In the solution method suggested here, we write the Love function with a Bessel expansion and analyze in detail the conditions under which it is possible to differentiate the expansion term by term. We show that this is possible only for a restricted class of elastic solutions. In the general case, we introduce two new auxiliary functions of the z-coordinate. In this way, we obtain the general form of the axisymmetric biharmonic function, which is discussed in relation to certain specific boundary conditions applied on the side and ends of the cylinder. We obtain an exact explicit solution of practical interest for a cylinder with free ends and assigned displacements applied to the side.
Similar content being viewed by others
References
Barber, J.R.: Three-dimensional elasticity problems for the prismatic bar. Proc. R. Soc. A 462, 1877–1896 (2006)
Bloor, M.I., Wilson, M.J.: An approximate analytic solution method for the biharmonic problem. Proc. R. Soc. A 462, 1107–1121 (2006)
Filon, L.N.G.: On the elastic equilibrium of circular cylinder under certain practical systems of load. Philos. Trans. R. Soc. A 198, 147–233 (1902)
Grinchenko, V.T.: The biharmonic problem and progress in the development of analytical methods for the solution of boundary-value problems. J. Eng. Math. 46, 281–297 (2003)
Levinson, M.: The simply supported rectangular plate: an exact, three-dimensional, linear elasticity solution. J. Elast. 7, 283–291 (1985)
Love, A.E.H.: A Treatise of the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1892)
Lur’e, A.I.: Three-Dimensional Problems of the Theory of Elasticity. Interscience, New York (1964)
Meleshko, V.V.: Selected topics in the history of the two-dimensional biharmonic problem. Appl. Mech. Rev. 56(1), 33–85 (2003)
Meleshko, V.V.: Equilibrium of a elastic finite cylinder: Filon’s problem revisited. J. Eng. Math. 46, 355–376 (2003)
Meleshko, V.V., Gomilko, A.M.: Infinite systems for a biharmonic problem in a rectangle: further discussion. Proc. R. Soc. A 460, 807–819 (2004)
Meleshko, V.V., Gomilko, A.M., Gourjii, A.A.: Normal reactions in a clamped elastic rectangular plate. J. Eng. Math. 40, 377–398 (2001)
Nardinocchi, P., Podio Guidugli, P.: Levinson-type benchmarks for slide-clamped and elastically supported plates. J. Elast. 73, 211–220 (2003)
Nicotra, V., Podio-Guidugli, P., Tiero, A.: Exact equilibrium solutions for linearly elastic plate-like bodies. J. Elast. 56, 231–245 (1999)
Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)
Sburlati, R.: An exact solution for the impact law in thick elastic plates. Int. J. Solids Struct. 41, 2539–2550 (2004)
Sburlati, R.: Elastic indentation problems in thin films on substrate systems. J. Mech. Math. Struct. 1(3), 101–117 (2006)
Sburlati, R.: Adhesive elastic contact between a symmetric indenter and an elastic film. Int. J. Solids Struct 46(5), 975–988 (2009)
Watson, G.N.: A Treatise on the Theory of Bessel Function, 2nd edn. Cambridge University Press, Cambridge (1922)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sburlati, R. Three-Dimensional Analytical Solution for an Axisymmetric Biharmonic Problem. J Elasticity 95, 79–97 (2009). https://doi.org/10.1007/s10659-009-9195-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-009-9195-3