Issue 50

H. Saidi et alii, Frattura ed Integrità Strutturale, 50 (2019) 286-299; DOI: 10.3221/IGF-ESIS.50.24 287 However, in FGM fabrication, micro voids or porosities can occur within the materials during the process of sintering. This is because of the large difference in solidification temperatures between material constituents [12]. Wattanasakulpong [13], also gave the discussion on porosities happening inside FGM samples fabricated by a multi-step sequential infiltration technique. Therefore, it is important to take into account the porosity effect when designing FGM structures subjected to dynamic loadings [14]. Currently, many functionally graded (FG) plate structures which have been employed for engineering fields led to the development of various plate models to study the static, buckling and vibration responses of FG structures [15-19]. The classical plate theory (CPT) is based on the supposition that straight lines which are normal to the neutral surface before deformation remain straight and normal to the neutral surface after deformation. Since the transverse shear deformation is neglected [20-23], it cannot be suitable for the investigating of moderately thick or thick plates in which transverse shear deformation effects are more important. For FG thick and moderately thick plates; the first-order shear deformation theory (FSDT) has been employed [24-27]. In such formulation, the displacements are linearly varied within the thickness and need a shear correction coefficient to correct the unrealistic distribution of the transverse shear stresses and shear strains across the thickness. To avoid the use of the shear correction coefficient, higher-order shear deformation plate theories (HSDTs) have been developed [28-40]. The purpose of this work to propose a new higher-order shears deformation theory for free vibration response of FG plates with porosity embedded in elastic medium. In this investigation the FGM plate are assumed to have a new distribution of porosity according to the thickness of the plate. The elastic medium is modeled as Winkler-Pasternak two parameter models to express the interaction between the FGM plate and elastic foundation. The four unknown shear deformation theory is employed to deduce the equations of motion from Hamilton’s principle. The Hamilton’s principle is used to derive the governing equations of motion. The accuracy of this theory is verified by compared the developed results with those obtained using others plate theory. Some examples are performed to demonstrate the effect of changing gradient material, elastic parameters, porosity index, and length to thickness ratios on the fundamental frequency of functionally graded plate. Figure 1 : Schematic representation of a rectangular FG plate resting on elastic foundation. M ATHEMATICAL FORMULATION n the current work, a FG simply supported rectangular plate with length, width and uniform thickness equal to a, b and h respectively is considered. The geometry of the plate and coordinate system are illustrated in Fig. 1. The material characteristics of FG plate are considered to vary continuously within the thickness of the plate in according to the power law distribution as follows       ( /2) 1 ( ) 1 2 k m c m c m z E z E E E E E e h                (1a)     ( /2) 1 ( ) (1 ) 2 k m c m c m z z E E e h                    (1b) I