Equilibrium of transversely isotropic FGM plates with an elliptical hole: 3D elasticity solutions

A transversely isotropic plate of functionally graded material (FGM) with an elliptical hole subjected to non-lateral loads is analyzed by employing England-Spencer plate theory. The problem is analytically solved by determining the four analytic functions in the general solution to the governing equations when there are no transverse forces acting on the plate surfaces. Three kinds of basic problems are classified according to the boundary conditions. For the first kind problem, which is studied in detail in the paper, the boundary conditions expressed by four real functions are rewritten as two complex function equations. The problem is eventually transformed to a complex function theory problem. Three-dimensional elasticity solutions are obtained for a transversely isotropic FGM plate containing an elliptical hole subject to loads at infinity or on the hole boundary. The conformal mapping technology is used along with the Cauchy integral method. Explicit expressions for the concentration factors of resultant forces are also presented. The present elasticity solutions are exactly the same as those available in the literature if the FGM degenerates to the homogeneous material. When the elliptical hole becomes a circular one, the present elasticity solutions are also found consistent with those obtained in the authors' previous work.