Elasticity Solutions For Bending Problems Of Functionally Graded Plates With Transverse Isotropy

Since the functionally graded materials (FGMs) were first introduced as a new type of design materials, a number of investigations have been conducted to deal with a variety of typical structures made of FGMs in all aspects, among which bending of FGM plates is one kind of the most fundamental problems. Material inhomogeneity inherent in FGMs leads to great difficulties for mechanics analysis since the governing partial differential equations have variable coefficients. In the three-dimensional (3D) analysis available in literature, the material constants are assumed to be certain special functions (e.g. the exponential function) only in order to obtain the elasticity solutions. This can not reflect the impact of diversity of material inhomogeneity on the mechanical responses of FGM structures.In this work, the bending problems of FGM plates with transverse isotropy are investigated based on 3D elasticity theory. In view of the current limitation, the Main & Spencer's theory is extended in two aspects. First, the material is assumed to be transversely isotropic, rather than isotropic. Second, the tractions-free conditions on the top and bottom surfaces are replaced by the conditions of uniform loads applied on the surfaces. The governing equations are derived rigorously based on 3D elasticity theory, which are expressed in terms of three components of the mid-plane displacement.England's theory is also extended to functionally graded plates of materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the 3D theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable. In the above-mentioned extensions, the hypothesis that the material parameters can vary in the thickness direction in an arbitrary fashion is retained. The solutions satisfy exactly the boundary conditions at the top and bottom surfaces, but approximately those at the four edges in the same sense as CPT. Therefore, the analytical solutions obtained in this work belong to 3D elasticity solutions, which can serve as benchmark for accessing the validity of various approximate plate theories or numerical methods that may be used in the analysis of such plates.