Issue 52

N. Hebbar et alii, Frattura ed Integrità Strutturale, 52 (2020) 230-246; DOI: 10.3221/IGF-ESIS.52.18 231 I NTRODUCTION n recent decades, a new class of composite materials has emerged from a group of researchers at the National Aerospace Laboratory (STA) in Japan, where they have developed a functionally graded materials with characteristics to withstand with thermal and mechanical stresses[1], since the classic materials, despite their advantages of high rigidity, high and low mechanical resistance, do not always meet the required requirements, these functionally graded materials have a microstructure that varies gradually and constantly through the thickness in order to optimize their performances whether mechanical or thermal or both at the same time. Fig.1 shows the microstructure of functionally graded materials [2]. Figure 1: The microstructure of functionally graded materials . Today, structures in advanced composite materials attract several researchers who are immersed in this vast field of research where they have developed several models in order to study the behavior of beams, plates and shells in different applications such as Koizumi [3-5], Karama et al.[6] and Aydogdu et al. [7] where they used a first-order parabolic and the exponential deformation theory to study the free vibration behavior of a functionally graded and simply supported material beam. Bernoulli [8] and Euler [9] have developed a classical theory (CBT) for the analysis of isotropic and anisotropic beams but unfortunately, this theory does not take into account the effect of transverse shear deformation. In order to overcome these limitations, several researchers have introduced the theory of shear deformation which takes into account the effect of transverse shear for the first-order theory and higher-order (HSDT). Timoshenko [10] introduced in his first-order theory the effect of shear deformation but always remains that he must add a correction factor and this is due to the shear stress that is constant across the thickness. These difficulties are eliminated by the introduction of the theory of higher-order shear deformation by researchers Reddy [11] and Touratier [12] they proposed work in bending, buckling and free vibration. Matsunaga [13] investigated the buckling and free vibration of FGM plates using a two- dimensional deformation theory. Vidal et al.[14] carried out an evaluation of the sine model for nonlinear analysis of composite beams. Ş im ş ek [15] used the different higher-order theories to study the dynamic responses of beams in functionally graded materials. Talha et al.[16] studied the static and dynamic response of FGM plates using the theory of higher-order shear strain. Hosseini-Hashemi et al. [17] investigated the free vibration of rectangular-type FGM plates using the first-order shear deformation theory. Hosseini-Hashemi et al. [18] proposed a new analytical approach to study the free vibration of rectangular Reissner-Mindlin plates. Xiang et al. [19] provided a theory of n-shear deformation for the purpose of studying the free vibration of sandwich plates. Thai and Vo [20] have worked on the analysis of the buckling and vibration of beams in functionally graded materials by using higher-order shear deformation theory for beams. Reddy et al. [21] proposed a theory of torque stress depended on the microstructure of functionally graded beams. Eltaher et al. [22] determined the position of the neutral axis and its effect on the eigenfrequencies of functionally graded macros/nano-beams. Li and Batra [23] proposed a relational analysis between the critical buckling loads of a Timoshenko theory of beams and Euler-Bernoulli for different boundary conditions. Nguyen et al. [24] used the theory of first-order shear deformation to analyze the vibration of beams in graded functional materials to obtain an analytical solution I