3D elasticity solutions for equilibrium problems of transversely isotropic FGM plates with holes

Abstract

Based on the England–Spencer plate theory, the elastic field in a transversely isotropic functionally graded plate with holes is analyzed by transforming the original three-dimensional (3D) problem into a two-dimensional (2D) one. The general solutions of the governing equations for the 2D problem are expressed by four analytic functions α(ζ), β(ζ), ϕ(ζ) and ψ(ζ) when there are no transverse forces acting on the plate surfaces. In a multiply-connected domain, the four analytic functions are expressed as the sum of logarithmic functions that are multi-valued and holomorphic functions, and should meet the conditions of single-valuedness of displacements. 3D elasticity solutions are then obtained for a transversely isotropic FGM plate with a circular hole subject to loads at infinity based on the power series method. Explicit expressions for concentration factors of the resultant forces are presented. For the degenerated case of homogeneous materials, the present elasticity solutions are exactly the same as those based on plane stress elasticity and classical plate theory. In addition, the elasticity solutions for an infinite FGM plate subject to in-plane point forces (X, Y) and moments (MX, MY) are derived, among which the solutions of point moments are reported for the first time. There are obvious differences between the present solutions of point forces and the existing solutions reported in the literature. In particular, the present solutions for homogeneous materials include a 3D correction term, which depends on the thickness-to-radius ratio, to those of plane stress elasticity. Finally, the elasticity solution for the point moment MZ is discussed.