Three-dimensional elastic solutions for functionally graded circular plates

Roberta Sburlati

DICAT, Faculty of Engineering, University of Genova, Via Montallegro, 1––16145, Genova, Italy

Lorenzo Bardella

DICATA, Faculty of Engineering, University of Brescia, Via Branze, 43––25123, Brescia, Italy

Abstract

Three-dimensional elastic solutions are obtained for a functionally graded thick circular plate subject to axisymmetric conditions. We consider a isotropic material where the Young modulus depends exponentially on the position along the thickness, while the Poisson ratio is constant. The solution method utilises a Plevako's representation form which reduces the problem to the construction of a potential function satisfying a linear fourth-order partial differential equation. We write this potential function in terms of Bessel functions and we pointwise assign mixed boundary conditions. The analytic solution is obtained in a general form and explicitly presented by assuming transversal load on the upper face and zero displacements on the mantle; this is done by superposing the solutions of problems with suitably imposed radial displacement. We validate the solution by means of a finite element approach; in this way, we highlight the effects of the material inhomogeneity and the limits of the employed numerical method near the mantle, where the solution shows a large sensitivity to the boundary conditions.

Author Keywords: Functionally graded plates; Boundary value problems; Finite element method