Application of biparametric perturbation method to functionally graded thin plates with different moduli in tension and compression

Abstract

In this study, a biparametric perturbation method is used for the solution of the large‐deflection bending problem of a functionally graded thin plate with different moduli in tension and compression. First, the Föppl‐von Kármán equations for the bimodular functionally graded thin plate are established in rectangular coordinates system, thus obtaining the axisymmetric simplified form in polar coordinates system. By adopting two groups of perturbation parameters, one group is gradient index and central deflection, another is gradient index and load, the biparametric perturbation solution for the established governing equations are obtained, respectively, under different boundary constrains. The result indicates that the two groups of parameter selections are valid, and the biparametric perturbation solution is also consistent to single‐parameter perturbation solution. The bimodular effects on stiffness and deformation of thin plates are also discussed via biparametric perturbation solutions obtained. The study indicates that the dominant factor influencing the stiffness is the relative magnitudes relation among the tensile modulus, the neutral layer modulus and the compressive modulus.

Variational Solution and Numerical Simulation of Bimodular Functionally Graded Thin Circular Plates under Large Deformation

“…Table 3 lists the central deflection values under different load magnitudes (from q = 10 kPa to q = 200 kPa, with an interval of 10 kPa). For an effective comparison, in Table 3, we also list two other groups of value from different theoretical solutions, the variational solution in this study and the perturbation solution in our previous study [26], in which the results from the variational solution are obtained via Equation (52) in this study; the results from the perturbation solution are based on Equation (104) in [26]. By comparing the values of central deflection in Table 3, it is easily found that the values from the three solutions are basically consistent, but there still exist small differences between them.…”

“…Table 3 lists the central deflection values under different load magnitudes (from q = 10 kPa to q = 200 kPa, with an interval of 10 kPa). For an effective comparison, in Table 3, we also list two other groups of value from different theoretical solutions, the variational solution in this study and the perturbation solution in our previous study [26], in which the results from the variational solution are obtained via Equation (52) in this study; the (c) q = 60 kPa; (d) q = 80 kPa; (e) q = 100 kPa; and (f) q = 120 kPa.…”

“…Recently, the bimodular effect of materials was further introduced into the analysis of functionally graded materials, and some works finally emerged, including bimodular FGM beams [24] and bimodular FGM plates [25][26][27][28]. Aiming at the bimodular FGM plates, a simplified theory on the neutral layer under small deflection was established in [25]; thereafter, the governing equations of the large-deflection problem of bimodular FGM thin circular plates was derived in [26]. For large-deformation problems, both the deflection and rotation angle will increase with the increase in external loads.…”

Application of Multi-Parameter Perturbation Method to Functionally-Graded, Thin, Circular Piezoelectric Plates

“…(1) 6 are integral constants of the first-order perturbation to D 31 . By using boundary conditions (38) and (39), these constants are determined as…”

“…He et al [36] further studied the application of the biparametric perturbation method to beam problems with the height difference of end supports under various boundary conditions. More recently, progress has been made in the application of this multi-parameter perturbation to bimodular plates, mainly including the combined loads problem [37], the parameter selection problem in a single load [38], and the parameter selection problem in a functionally-graded material (FGM) plate [39]. More recently, Fallah et al [40] reviewed the perturbation method in mechanical, thermal, and thermo-mechanical loadings of cylindrical bendings of FGM plates, both clamped and simply supported, in which one-and two-parameter perturbations were used.…”