An exponential-trigonometric higher order shear deformation theory (HSDT) for bending, free vibration, and buckling analysis of functionally graded materials (FGMs) plates

Abstract

Two assumptions have been made based on by this proposed theory, which come from recently developed exponential–trigonometric shape function for transverse shear deformation effect and a simple higher order shear deformation theory for plate, based on a constraint between two rotational displacements of axis parallel to the plate midplane, about the axes x, y Cartesian coordinates system, which caused fewer unknown number. For the application of this method, a displacement field extended as only bending membrane for transverse displacement is used, a governing equations of motion as a result are determined according to Hamilton’s principle, and simplified using Navier analytical solutions, as well as the transverse shear stresses effect that satisfied the stress-free boundary conditions on the simply supported plate free faces as a parabolic variation along the thickness are taken into account. A functionally graded materials plates are chosen for the parametric study, where the plates are functionally graded continuously in materials through the plate thickness as a function of power law or exponential form. The aim of this study is to analyze the bending, free vibration as well as the buckling mechanical behaviors, where the results are more focused on the investigation of different parameters such as the volume fraction index, geometric ratios, frequency modes, in-plane compressive load parameters and material properties effects on the deflection, stresses, natural frequencies, and critical buckling load, which are validated in terms of accuracy and efficiency with other plate theories results found in the literature.

Three new hybrid quasi-3D and 2D higher-order shear deformation theories for free vibration analysis of functionally graded material monolayer and sandwich plates with stretching effect

“…results are computed in Table 4, for homogeneous ceramic, stiffer, moderately stiff, softer, thick, moderately thick, thin, square (Al) 1 /Al 2 O 3 P-FGM plates, under the three frequency modes ((m, n) ¼ 1(1,1), 2(1,2), 3(2,2)). The 2D HSDT 1 and 2 results are in very good agreement with all the presented theories solutions, especially with Belkhodja et al 37 as well as Thai and Kim, 23 where the calculated maximum errors are, respectively, 0.0747% as well as 0.6683% and 0.7937%, these are an insignificant values obtained for (p ¼ 0, a/h ¼ 5,(m, n) ¼ 3(2,2)) and (a/h ¼ 20, p ¼ 0.5,(m, n) ¼ 1 (1,1)). The 2D HSDT 1 results are identical to the others theories solutions, for thin and also moderately thin plates.…”

“…1 Thus, the used architectural design is a monolayer (single-layer) and sandwich (symmetric as well as non-symmetric, with hardcore as well as softcore) plates of the dimensions indicated as length (a), width (b) and uniform thickness (h) as shown in Figures 1, 2 and 3, respectively. The thickness evolution follows the z-coordinate (z-axis) perpendicular to the rectangular Cartesian coordinates (0, x, y) defined in the plate median plane, 37 and all in-plane edges of the plate are parallel to x and y axes as well as the bottom and top plate faces are at the plate extremities z ¼ +h/2, with respect to the coordinates h1 and h4.…”

“…The sandwich plates (symmetric as well as nonsymmetric, with hardcore as well as softcore) are composed of three elastic isotropic homogeneous and anisotropic microscopically heterogeneous layers (Figures 2 and 3): Layer 1 is a bottom face layer (z2[h1, h2]); layer 2 is a median core layer (z2[h2, h3]); and layer 3 is a top face layer (z2[h3, h4]), where h2 and h3 are the vertical coordinates of the two median interfaces (Figures 2 and 3). Therefore, the plates were characterized by geometric ratios such as the side-to-thickness ratio (a/h), the aspect ratio (a/b) 37 and the thickness ratio. The ratio (a/h) specifies three plate thicknesses that are a thick plate (with a low ratio values such as (a/h ¼ 2, ffiffiffiffiffi 10 p , 5)), moderately thick plate (a/h % 10) and a thin plate (with high ratio values such as (a/h ¼ 20)).…”

“…It is necessary to know that the transverse shear deformation effect proportionally varies with the plate thickness value. 37 Moreover, the ratios (a/b ¼ 1) and (a/b 6 ¼ 1) define a square plate and a rectangular plate, respectively. 37 The thickness ratio is denoted by the combination of three numbers i-j-k, the sum of the three numbers is X, each number is assumed to be taken from X as ratio as i/X, j/X, k/X, X represent the number of small thicknesses that the plate is divided to, h1 and h2 are known.…”

“…37 Moreover, the ratios (a/b ¼ 1) and (a/b 6 ¼ 1) define a square plate and a rectangular plate, respectively. 37 The thickness ratio is denoted by the combination of three numbers i-j-k, the sum of the three numbers is X, each number is assumed to be taken from X as ratio as i/X, j/X, k/X, X represent the number of small thicknesses that the plate is divided to, h1 and h2 are known. The plate in-plane always must be in the median of the plate, if it is noticed that the in-plane pass in a small thickness of the plate and it divided again, then the other small thicknesses will be divided too, all the new smallest thicknesses must have the same thickness.…”