Mathematical and Physical Analyses of Middle/Neutral Surfaces Formulations for Static Response of Bi-Directional FG Plates with Movable/Immovable Boundary Conditions

Abstract

This article is prompted by the existing confusion about correctness of responses of beams and plates produced by middle surface (MS) and neutral surface (NS) formulations. This study mathematically analyzes both formulations in the context of the bending of bi-directional functionally graded (BDFG) plates and discusses where the misconceptions are. The relation between in-plane displacement field variables on NS and on MS are derived. These relations are utilized to define a modified set of boundary conditions (BCs) for immovable simply supported plates that enables either formulation to apply fixation conditions on the refence plane of the other formulation. A four-variable higher order shear deformation theory is adopted to present the displacement fields of BDFG plates. A 2D plane stress constitution is used to govern stress–strain relations. Based on MS and NS, Hamilton’s principles are exploited to derive the equilibrium equations which are described by variable coefficient partial differential equations. The governing equations in terms of stress resultants are discretized by the differential quadrature method (DQM). In addition, analytical expressions that relate rigidity terms and stress resultants associated with the two formulations are proved. Both the theoretical analysis and the numerical results demonstrate that the responses of BDFG plates based on MS and NS formulations are identical in the cases of clamped BCs and movable simply supported BCs. However, the difference in responses of immovable simply supported BCs is expected since each formulation assumes plate fixation at different planes. Further, numerical results show that the responses of immovable simply supported BDFG plates obtained using the NS formulation are identical to those obtained by the MS formulation if the transferred boundary condition (from NS- to MS-planes) are applied. Theoretical and numerical results demonstrate also that both MS and NS formulations are correct even for immovable simply supported BCs if fixation constraints at different planes are treated properly.