PSI - Issue 26

Merdaci Slimane et al. / Procedia Structural Integrity 26 (2020) 35–45 Slimane et al. / Structural Integrity Procedia 00 (2020) 000 – 000

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1. Introduction

Functionally graded materials (FGMs) are a future engineered material wherein material properties are continually varied through the thickness direction by mixing two different materials for example ceramic and metal. As a result, internal boundaries does’t occurs and overcomes the stress concentration setup in composite laminates. The functionally graded materials have been widely employed in the various applications such as constructions, aerospace, nuclear, civil, automotive, barrier for ceramic engines, electrical devices, energy transformation, biomedical engineering, and optics, etc (Akbaş (2015), Bennai et al.(2015), Arefi (2015), Ait Atmane et al. (2015), Ebrahimi et al. (2015), Darılmaz (2015), Ebrahimi et al. (2016), Kar et al. (2016), Moradi-Dastjerdi (2016) , Trinh, T.H. et al. (2016) and A. Hadj Mostefa et al. (2018)). The concept of functionally graded material was first considered in Japan in 1984 during a space plane project. Such kind material is produced by mixing two or more materials by a graded distribution of the volume fractions of the constituents Koizumi (2017). It is should be noted that the importance of including transverse shear deformation effects comes from the fact that composite materials have very high ratios of in- plane Young’s moduli to transverse shear moduli. Using assumptions similar to those of Shimpi (2002), many theories of shear deformation with four unknowns have been developed using different shape functions. For example, Merdaci et al. (2011) and Ameur et al. (2011) developed a four-unknown HSDT for FG sandwich plates and FG plates using the sinusoidal function. To the best of the authors’ knowledge, a few works has been done on mechanical beh avior of porous structures. Rezaei et al. (2016) investigated the effect of porosity on natural frequency of thick porous cellular plates by using Carrere unified formulation. The effect of deformation coupling between solid and fluid on the free vibration characteristics of isotropic rigid porous rectangular plates under undrained condition is studied by the same authors in Rezaei et al. (2017). Kamranfard et al. (2017) presented an analytical approach for vibration and buckling of annular sectorial porous plates under in-plane uniform compressive loading. Benachour et al. (2011) and Belabed et al. (2014) have developed a model for the free vibration of FGM plates with an arbitrary coefficient. Approximate solutions are obtained using the Navier solution, the fundamental frequencies are found by solving a problem of eigenvalues. Recently Wagih et al. (2019) studied the effect of contact with an elastoplastic FG substrate and a rigid spherical indenture with the help of FEM. Merdaci et al. (2019) investigated rectangular porous thick FGPs by applying higher order theory for bending response. 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 Neves et al. (2017). In the manufacture of FGMs, micro-porosities or voids can occur in the materials during the sintering process. This is due to the large difference in solidification temperature between the material constituents Zhu et al. (2001). Wattanasakulpong et al. (2012) also gave a discussion of the porosities that occur within FGM specimens made by a sequential multi-step infiltration technique. Therefore, it is important to take into account the effect of porosity in the design of FGM structures subjected to static Merdaci et al. (2019) and Merdaci (2018) and dynamic loads (Wattanasakulpong et al. (2014), Mouaici et al. (2016), Merdaci (2018), Merdaci et al. (2019), Merdaci (2019), Saidi et al. (2019) and Reddy (2002)). Consequently, studies devoted to understanding the static and dynamic behavior of FGM material structures have been given more and more attention in recent years. This present research focuses on the free vibration analysis of simply supported porous functionally graded plates (FGP) for even porosity distribution perfect and imperfect based on the higher-order shear deformation plates theory. This theory satisfies the nullity of the stresses at the upper and lower surfaces of the plate without using the shear correction factor contrary to other theories. In this investigation the FGM plate are assumed to have a distribution of porosity according to the thickness direction of the plate are supposed even porosity. The equations of motion of the P- FGP are determined by applying Hamilton’s principle and a Navier -type analytical solution. 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 parameters such as the material index, porosity factor, and length to thickness ratios on the fundamental frequency of functionally graded plate of simply supported P-FGP.

2. General formulations

Consider a thick rectangular plate FG of length a, width b and thickness h made of functionally graded material as shown in Fig.1 together with the adopted coordinate system. As can be observed in Fig. 2, the embedded FG plate