Three-Dimensional Modeling for Thin Plate-Like Structures Including Surface Effects by Using State Space Method

A three-dimensional (3-D) approach based on the state space method is proposed to study size-dependent mechanical properties of ultra-thin plate-like elastic structures considering surface effects. The structure is modeled as a laminate composed of a bulk bounded with upper and bottom surface layers, which are allowed to have different material properties from the bulk layer. State equations, including the surface properties of the structure, can be established on the basis of 3-D fundamental elasticity to analyze the size-dependent static characteristics of the thin plate-like structure. Compared with two-dimensional plate theories based size-dependent models for thin film structures in literature, the present 3-D approach is exact, which can provide benchmark results to assess the accuracy of 2-D plate theories and various numerical approaches. To show the feasibility of the proposed approach, a 3-D analytical solution for a simply supported plate-like thin structure including surface layers is derived. An algorithm is proposed for the calculation of the state equations obtained to ensure that the numerical results can reveal the surface effects clearly even for extremely thin surface layers. Numerical examples are carried out to exhibit the surface effects and some discussions are provided based on the results obtained.