Three dimensional mechanical behaviors of in-plane functionally graded plates

A semi-analytical solution procedure to investigate the distributions of displacement and stress components in the in-plane functionally graded plates based on the scaled boundary finite element method (SBFEM) in association with the precise integration algorithm (PIA) is developed in this paper. The proposed approach is applicable to conduct the flexural analysis on functionally graded plates with various geometric configurations, boundary conditions, aspect ratios and gradient functions. The elastic material parameters of functionally graded plates discussed here are mathematically formulated as power law, exponential and trigonometric functions varied along with the in-plane directions in a continuous pattern. Only a surface of the plate parallel to the middle plane is required to be discretized with two dimensional high order spectral elements, which contributes to reducing the computational expense. By virtue of the scaled boundary coordinates, the virtual work principle and the internal nodal force vector, the basic equations of elasticity are converted into a first order ordinary differential SBFEM matrix equation. The general solution of the governing equation is analytically expressed as a matrix exponential with respect to the transverse coordinate z. According to the PIA, the stiffness matrix from the matrix exponential can be acquired. Considering that the PIA is a highly accurate method, any desired accuracy of the displacement and stress field can be obtained. The entire derivation process is built on the three dimensional elasticity equations without importing any assumptions on the plate kinematics. Comparisons with numerical solutions available from prevenient researchers are made to validate the high accuracy, efficiency and serviceability of the employed technique. Additionally, circular and perforated examples are provided to highlight the performance of the developed methodology and depict the influences of boundary conditions, thickness-to-length ratios and gradient indexes on the deformable behaviors of in-plane functionally graded plates.