Numerical modeling of bending, buckling, and vibration of functionally graded beams by using a higher-order shear deformation theory

The objective of this work is to analyze the behavior beams functionally graded, simply supported, under different conditions such as bending, buckling, and vibration and this by use shear deformation theories a two-dimensional (2D) and quasi-three-dimensional (quasi-3D). The proposed theories take into account a new field of displacement which includes indeterminate whole terms and contains fewer unknowns, compared to other theories of the literature; by taking account of the effects of the transverse shears and the thickness stretching. In this theory, the distribution of the transverse shear stress is hyperbolic and satisfies the boundary conditions on the upper and lower surfaces of the beam without the need for a shear correction factor. In this type of beam the properties of the materials vary according to a distribution of the volume fraction, the Hamilton principle is used to calculate the equations of motion, and in order to check the accuracy of the theory used comparison is made with the studies existing in the literature.


INTRODUCTION
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]. 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][4][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 twodimensional deformation theory. Vidal et al. [14] carried out an evaluation of the sine model for nonlinear analysis of composite beams. Şimşek [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 according to Navier's solution. Hadji et al. [25] have developed a new model of first-order and higher-order shear deformation to analyze the vibration of functionally graded beams. Yaghoobi et al. [26] developed an analytical study on the analysis of nonlinear free vibrations after beam buckling in functionally graded materials resting on a non-linear elastic foundation under thermo mechanical loading using VIM. Rahmani and Pedram [27] they analyzed by modeling the effect of size on the vibration of functionally graded nano-beams based on the nonlocal Timoshenko beam theory. AlKhateeb and Zenkour [28] presented a refined four-variable theory for the analysis of flexion of resting plates on elastic foundations in hygrothermal environments. Vo et al. [29,30] developed a finite element model based on a refined theory of shear deformation in order to study the static and dynamic behavior of beams under different boundary conditions. Meradjah et al. [31] proposed a new theory of shear deformation for the study of beams in functionally graded materials with a consideration of the stretching effect. Vo et al. [32] developed a quasi-3D theory for the study of buckling and vibration of sandwich beams. Zemri, A et al. [33] proposed an unrefined theory of theory for static analysis, buckling and free vibration of beams in nanometrically functionally graded materials. Al-Basyouni et al. [34] analyzed flexion and vibration as a function of the size of functionally graded micro-beams based on the modified theory of torque stress and neutral surface position. Ebrahimi and Dashti [35] explored the effects of linear and non-linear distributions of temperature on the vibration of nano-beams in functionally graded materials. Kar and Panda [36] studied the vibration and nonlinear shear bending of a spherical shaped, shell panel is functionally graded materials. Bourada et al. [37] presented a new simple and refined higher-order trigonometric theory for the analysis of free bending and vibration of beams in functionally graded materials taking into account the effect of stretching the thickness. Celebi et al. [38] proposed a unified method for studying the constraints in a sphere of functionally graded materials with properties that vary exponentially. Boukhari et al. [39] proposed a thermal study on wave propagation in FGM functionally graded materials plates based on the neutral surface position. Ebrahimi and Barati [40] have studied the influence of the environment on the damping vibration of nano-beams in functionally graded materials. Ahouel et al. [41] investigated the size-dependent mechanical behavior of trigonometrically shear functional and trigonometric shear nano-beams, including the concept of the neutral surface position. Shafiei et al. [42] studied the nonlinear vibrations of conical micro-beams in functionally graded imperfect and porous materials based on modified torque constraints and Euler-Bernoulli theories. Raminnea et al. [43] used the non-linear Reddy theory of higher-order for the study of the vibration and instability of embedded pipes carrying a fluid as a function of temperature. Ghumare and Sayyad [44] developed a new theory for the study of fifth-order shear deformation and normal deformation for the analysis of flexion and free vibration of FGM beams. Benadouda et al. [45] proposed a theory of shear deformation for the study of wave propagation in beams in functionally graded materials with porosities. Bellifa et al. [46] used a theory of simple shear deformation as well as the concept of the position of the neutral surface for the analysis of flexion and free vibration of plates made of functionally graded materials. Akbaş [47,48] studied the vibratory response of viscoelastic beams and wave propagation in a beam made of functionally graded materials in thermal environments. Bellifa et al. [49] used the theory of non-local zero-order shear strain for the non-linear post-buckling of nano-beams. Li et al. [50] studied the effect of thickness on the mechanical behavior of nano-beams. Sayyad and Ghugal [51] developed a theory of unified shear deformation for the study of the bending of beams and plates in functionally graded materials. Aldousari [52] studied the bending analysis of different material distributions in a functionally graded beam. Bouafia et al. [53] developed a non-local quasi-3D theory to study the behavior of the free bending of nano-beams in functionally graded materials. Zidi et al. [54] have proposed a new simple two-unknown theory for studying the hyperbolic shear deformation of beams in functionally graded materials. Fouda et al. [55] proposed a porosity model to study shear deformation in the static case, buckling and free vibration of porous beams in functionally graded materials based on the Euler Bernoulli method and finite elements. A study on the free vibration of beams in functionally graded materials is presented by Zaoui et al. [56] where they used a theory of higher-order shear deformation. Mouffoki et al. [57] studied the analysis of the free vibration of nano-beams under a hygro-thermal loading using a new theory of trigonometric beams with two unknowns. Recently, Sayyad and Ghugal [58] studied the bending, buckling and free vibration responses of the hyperbolic shear deformation of FGM beams. Kaci. A. et al. [59] have studied the post-buckling analysis of shear-deformable composite beams using a new, simple two-unknown theory. Dragan et al. [60] developed a new function for the purpose of analyzing plate bending in functionally graded materials. The effect of shear deformation of structures in functionally graded materials requires more investigation. In our study, a theory of 2D and quasi-3D three-variable shear deformation for the analysis of beams in functionally graded materials is presented. The motion equations are derived from the Hamilton principle. Navier's solutions are also presented. Displacements, stresses, critical buckling loads and frequencies obtained using the current beam theory of functionally graded materials in which the properties of materials vary with the power-law (P-FGM) are compared with other results in order to demonstrate the effectiveness of the proposed theory. Numerical examples will be presented for the study of the shear deformation of beams in functionally graded materials in the case of bending, buckling and vibration.  The characteristics of the material can change according to the thickness and the function given in the following equations [24,57,61,62,63]: where: Ec and Em present the property of the upper and lower faces of the beam respectively and p is the exponent which specifies the distribution profile of the material in the thickness. In this work, Young's modulus E and the shear modulus G, change according to the problem case according to Eqn. (1), and the Poisson's ratio  is considered constant. he hypotheses of the present theory are as follows; the origin of the Cartesian coordinate system is taken on the neutral axis of the beam in functionally graded materials; The displacements are small in comparison with the thickness of the beam thus the deformations involved are infinitesimal; Displacements (u) in the x direction consist of extension, bending and shear components.

KINEMATIC AND CONSTITUTIVE EQUATION
n the basis of the assumptions made in the previous section, the displacement field can be presented as follows: where: u0 is the axial displacement in the median plane, and t represents the time.
In this study, f (z) represents the shape function determining the distribution of transverse shear deformation as follows: The deformations associated with displacements in Eqn. (4) are: where: and 0 0 The Navier method is used to solve the integrals defined in the equations [64]: where the coefficient A' is considered according to the type of solution used, in this case via the Navier method. Consequently, A′ and Ka are expressed as follows: The beam in functionally graded materials obeys Hooke's law, so the behavioral relations can be given as follows: with ( x , z , xz ) and ( x  , z  , xz  ) are respectively the stresses and the deformations.
The expressions ij Q depends on the normal deformation z  : In the case of two-dimensional shear deformation (2D) the normal deformation z  = 0, therefore: In the case of quasi-three-dimensional shear deformation (quasi-3D) the normal deformation z   0, therefore:

EQUATION OF MOTION
he Hamilton principle is used in this study to derive the equations of motion; it can be given in the following analytic form [65]: where: δU is the variation of the strain energy,δV is the variation of the kinetic energy and δK is the variation of the potential energy. The variation of the deformation energy of the beam can be defined as follows: with: Nx, Mb, Ms and Q are the resultants of the stress in terms of axial force, bending moment, higher-order moment and shear force, respectively: The variation of the kinetic energy is expressed by: Furthermore, the potential energy of the distributed load is expressed by: Substitute the expressions displacement by deformation as well as stress by deformation which are respectively defined by the Eqns. (16), (18) and (19) in Eqn. (14) and by integrating by parts while putting the coefficients u, v, w and equal to zero. As a result, the governing equations obtained are given as follows: where: (z) is the density and I 0 , I 1 , J 1 , I 2 , J 2 and K 2 are the coefficients of inertia as defined below:

ANALYTICAL SOLUTION
he motion equations admit Navier's solutions for simply supported beams. The variables u0, w0 and ϕcan be written assuming the following variations: with: i = 1  and α = mπ/L The transverse load q is also expressed by the double series of Fourier sine as follows: In which: S 11 = A 11 α 2 , S 12 = -B 11 α 3 , S 13

BENDING ANALYSIS
ab. 2 presents a comparison of dimensionless displacements and stresses of Al/Al 2 O 3 functionally graded materials beams, simply supported and subjected to uniformly distributed loads with different exponent values of the power-law p and for ratios. L/h = 5 and 20. It can be seen through the results obtained that displacements and stresses increase as the power-law index increases and takes a maximum value when p takes the value of one and a minimum value in the case where p takes the value of zero, this interpretation is due to the ductility of the beam since the more the material index is increasing, the more the beam becomes more ductile. The results obtained are compared with other results from the literature such as HSDT of Reddy [11], HSDT of Hadji et al. [25], and HSDT of A.S. Sayyad and Y.M. Ghugal [58]. It can also be noted that the two-dimensional (2D) shear deformation theory is in good agreement with the other theories of shear deformation, whereas the results obtained by the theory of quasi-shear deformation threedimensional (quasi-3D) are slightly larger compared to that of the literature and this is due to the effect of normal transversal deformation which is not neglected (εz 0) compared to other theories where the effect normal transversal deformation is neglected (ε z = 0). Fig. 4 shows the variation of the transverse displacement across the length of the beam made of Al/Al 2 O 3 functionally graded materials, subjected to a uniformly distributed load. It is noted that the transverse displacement increases with the increase of the index of the power-law p and reaches a maximum value. The traced curve takes a parabolic form.      The Fig. 7 illustrates the variation of the axial stress of a beam made of Al/Al 2 O 3 functionally graded materials, subjected to a uniformly distributed load, for different values of the index of the power-law p which takes the values 0, 1, 2, 5 and 10 with a ratio of (L/h = 5). It can be deduced that the upper part is towed and the lower part is compressed and between these two parts the curve takes a parabolic form. The Fig. 8 shows the variation of the transverse shear stress of a beam in Al/Al2O3 functionally graded materials, subjected to a uniformly distributed load for different values of the index of the power-law p which takes the values of 0, 1, 2, 5 and 10 with a ratio of (L/h = 5). It is observed that the plot of the transverse shear stress does not take a parabolic form as in the case of homogeneous beams of metal and ceramic, it can also be noted that the neutral axis is eccentric towards the upper part.

BUCKLING ANALYSIS
n this part we have studied the comparison of dimensionless critical buckling loads ( Ncr ) for Al/Al2O3 type functionally graded beams, simply supported and subjected to axial forces (N 0 ) with respect to different values of the exponent of the law of power p for the ratios L/h= 5 and 10. It is observed that the critical load of dimensionless buckling decreases with the growth of the index of the law of power p and it increases with the increase of the ratio (L/h). The results of the critical buckling load obtained in Tab. 3 are compared with other results of the literature such as HSDT of Reddy [11], FSDT of Touratier [12], HSDT of A.S. Sayyad and Y.M. Ghugal [58], FSDT of Li and Batra [23], and HSDT of Vo et al. [30]. It can also be noted that the theory of two-dimensional shear deformation (2D) is in good agreement with the other theories of shear deformation, whereas the results obtained by the theory of quasi-threedimensional shear deformation (quasi-3D) are slightly larger than those of the literature and this is due to the effect of the normal transversal deformation which is not neglected (εz0) compared to other theories where the effect of the normal transverse deformation is neglected (ε z =0).  Observe that the fundamental non-dimensional frequencies for the first three modes decrease with the growth of the power-law index and they increase with the increase of the ratio (L/h), this is because of an increase in the power-law index p makes the beam more flexible. The results of the fundamental frequencies obtained in Tab. 4 are compared with other results such as HSDT of Reddy [11], HSDT of A.S. Sayyad and Y.M. Ghugal [58], FSDT of Touratier [12], HSDT of Simsek [15], HSDT of Thai and Vo [20], FSDT of Vo et al. [29], HSDT of Vo et al. [30], FSDT of Timoshenko [10], and CBT of Bernoulli-Euler [9]. It can also be noted that the two-dimensional shear deformation theory (2D) is in good agreement with that of the literature, whereas the results obtained by the theory of quasi-three-dimensional shear deformation (quasi-3D) are slightly larger compared to that of the literature and this is due to the effect of normal transversal deformation which is not neglected (ε z 0) compared to other theories where the effect of normal transversal deformation is neglected (ε z =0). Fig. 9 illustrates the variation of the buckling critical load and the dimensionless fundamental natural frequency with respect to the index of the power-law p for different values of the ratio (L/h) by the use of the deformation theory shearing. It can be seen through these plots that the critical load and the frequency decreases with the growth of the index of the law of power p, it is maximum when the law of power p takes the value of zero in this case, the beam is entirely ceramic and is minimal in the case where the index of the power-law p takes the value of one, in this case, the beam is entirely metal, this is due to the increase in the value of the index of the power-law which causes a decrease in the value of the modulus of elasticity. It can also be seen that the ratio (L/h) has a considerable effect on the critical buckling load and the fundamental dimensionless fundamental frequency when it is reduced, the value of the ratio (L/h) decreases. This dependence is related to the effect of shear deformation.
(a) (b) Figure 9: Variation in dimensionless critical buckling loads (a) and natural frequencies (b) with respect to the power-law exponents of exponents of simply supported FG beams.

CONCLUSIONS
he aim of our work is to study the bending, buckling, and vibration of beams functionally graded using a twodimensional (2D) and quasi-three-dimensional shear deformation theory (quasi-3D), without a need for introducing a shear correction factor. These beams are subjected to uniformly distributed loads. The principle of virtual works is used to solve equilibrium equations using the Navier approach with simply supported boundary conditions. The equations of motion are derived from the Hamilton principle. Parametric studies were carried out to examine the influence of the power-law index, and the beam aspect ratio (L/h) on the variation of dimensionless displacements as well as the distribution of dimensionless stresses across the thickness of a beam made of Al/Al2O3 type functionally graded materials. The results obtained are in good agreement with the results of the literature. It can be said that the present theory of shear deformation with the taking into account the stretching effect is not only precise, but also provides an easily feasible approach for the simulation of the mechanical behavior of structures in order to design weak structures which can be used in several fields such as automotive, aeronautical, marine, medical and nuclear. In futuristic work, we envisage the study of these mechanical behaviors with other boundary conditions and different loadings that are mechanical or/and thermal. T