Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams

Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams

A.S. Sayyad^a*, Y.M. Ghugal^b

^a Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423601, Maharashtra, India

^b Department of Applied Mechanics, Government College of Engineering, Karad-415124, Maharashtra, India

1. Introduction   

Fibrous composite laminated beams are often subjected to delamination and stress concentration problems. This leads to the development of beams made of functionally graded materials (FGMs). A FGM is formed by varying the microstructure from one material to another in a specific gradient. The most commonly used FGMs are ceramic and metal for application in many engineering industries. Some typical practical applications for FGMs are given below.

1)   Nuclear Projects: Fuel pellets and plasma walls of fusion reactors

2)   Aerospace and Aeronautics: Stealth aircraft, rocket components, space plane frames, and space vehicles

3)   Civil Engineering: Building materials, structural elements, and window glass

4)   Defense: Armor plates and bullet-proof vests

5)   Manufacturing: Machine tools, forming and cutting tools, metal casting, and forging processes

6)   Energy Sector: Thermoelectric generators, solar cells and sensors

Detailed descriptions of applications of FGMs in various fields were presented by Koizumi [1, 2], Muller et al. [3], Pompe et al. [4], and Schulz et al. [5]. Increased use of beams, plates, and shells made of FGMs has led to the development of various analytical and numerical models for predicting accurate static bending, elastic buckling, and free vibration responses in these beams. Few studies have addressed the development of elasticity solutions for the analysis of FG beams, but those that have include Sankar [6], Zhong and Yu [7], Daouadji et al. [8], Ding et al. [9], Huang et al. [10], Ying et al. [11], Chu et al. [12], and Xu et al. [13].

Two-dimensional elasticity solutions are analytically difficult and computationally cumbersome. Therefore, various approximate beam theories have been developed to analyze FG beams. Recently, Sayyad and Ghugal [14, 15] presented a comprehensive literature survey on various analytical and numerical methods for the analysis of isotropic and anisotropic beams and plates using displacement-based shear deformation theories. Similarly, Carrera et al. [16] presented recent developments in refined beam theories and related applications.

Since the effect of shear deformation is more pronounced in thick beams made of advanced composite materials, such as FGMs, classical beam theory (CBT)  [17, 18] and first-order shear deformation theory [19] are not suitable for the analysis of thick FG beams. Therefore, higher order shear deformation theories are preferable for accurate analysis of FG thick beams. Various higher order shear deformation theories have been developed by various researchers for the analysis of isotropic and anisotropic beams, such as those by Reddy [20], Soldatos [21], Touratier [22], Karama et al. [23], Mantari et al. [24, 25], Neves et al. [26, 27], Sayyad and Ghugal [28, 29], Sayyad et al. [30, 31], Carrera et al. [32], Zenkour [33], Carrera and Ginuta [34], Carrera et al. [35], and Giunta et al. [36]. Such theories address bending, buckling, and free vibration analyses of beams using Carrera’s unified formulation.

Thai and Vo [37] obtained Navier-type analytical solutions for the bending and vibration of FG beams using various higher order shear deformation theories. Li and Batra [38] obtained the critical buckling load of FG beams in various boundary conditions using the first-order shear deformation theory and CBT. Simsek [39] presented free vibration analysis of FG beams with various boundary conditions based on higher order shear deformation theories. Nguyen et al. [40] presented a Navier-type closed form solution for the static deformation and free vibration of FG beams using the first-order shear deformation theory. Hadji et al. [41, 42] developed new first-order and higher order shear deformation models for static and free vibration analysis of simply supported FG beams. Bourada et al. [43] presented a trigonometric shear and normal deformation theory that considers the effects of transverse shear and normal deformations for the analysis of FG higher order beams. The theory included three unknowns, one of which was the effect of the transverse normal. Vo et al. [44, 45] presented static bending and free vibration analysis based on higher order shear deformation theories using the finite element method. Recently, Sayyad and Ghugal [46] developed a unified shear deformation theory for the bending of FG beams and plates. Hebali et al. [47] developed five variable quasi-three-dimensional hyperbolic shear deformation theories for static and free vibration behavior of FG plates. Bennoun et al. [48] also developed five new variable shear and normal deformation plate theories for free vibration analysis of FG sandwich plates. Beldjelili et al. [49] investigated hygro-thermo-mechanical bending behavior of sigmoid FG plates resting on elastic foundations using four variable trigonometric shear deformation theories. Bouderba et al. [50] developed a simple first-order shear deformation theory for thermal buckling responses of FG sandwich plates to various boundary conditions. Bousahla et al. [51] also developed a four-variable refined plate theory for buckling analysis of FG plates subjected to uniform, linear, and non-linear temperature increases for various thicknesses. Boukhari et al. [52] developed a four-variable refined plate theory for wave propagation analysis of an infinite FG plate in thermal environments. Rahmani and Pedram [53] applied Timoshenko beam theory for free vibration analysis of FG nanobeams. Akgoz and Civalek [54] studied the static bending response of single-walled carbon nanotubes embedded in an elastic medium using higher-order shear deformation microbeams and a modified strain gradient theory. Ebrahimi and Barati [55] obtained a Navier-type solution for free vibration characteristics of FG nanobeams based on the third-order shear deformation beam theory.

In this paper, bending, buckling, and free vibration responses of hyperbolic shear deformable FG higher order beams were studied using the hyperbolic shear deformation theory of Soldatos [21]. Soldatos suggested using the hyperbolic function in the modeling and analysis of composite beams and plates in 1992, recommending that the hyperbolic function yields more accurate predictions of stress, frequencies, and the buckling loads of composite beams and plates. Since then, many researchers have used this function for the analysis of isotropic, laminated, and sandwich beams and plates. However, most research has not focused on the application of this function to evaluate the response of FG beams. Instead, researchers have applied hyperbolic shear deformation theory to bending, buckling, and free vibration analysis of FG beams.

The material properties of FG beams are varied through the thickness of the beam according to power law distribution. Here, the FG beam was made of aluminum (Al)/alumina (Al₂O₃₎, and the hyperbolic cosine distribution of transverse shear stress through the thickness of the beam satisfied the zero traction boundary conditions on the top and bottom surfaces without using the shear correction factor. The variationally consistent governing differential equations and boundary conditions of the theory were obtained using Hamilton’s principle, and an analytical solution for simply supported boundary conditions was obtained using Navier’s solution. The non-dimensional displacements, stress, natural frequencies, and critical buckling loads of FG beams were obtained for various values of the power law exponent. The numerical results were then compared to previously published results and were in excellent agreement with these.

1. 2.      Variational formulation

2.1 Kinematics

Consider a FG beam with length L, width b, and thickness h made of Al/Al₂O₃ as shown in Figure 1. The bottom surface of the FG beam was ceramic-rich and top surface was metal-rich. The beam occupied the region 0≤ x ≤ L; -b/2≤ y ≤ b/2; -h/2≤ z ≤ h/2 in the Cartesian coordinate systems. The x-axis was coincident with the beam neutral axis. The z-axis was assumed to be downward positive, and the beam was assumed to be deformed in the x-z plane only.

The mathematical formulation of the FG beam was based on the following kinematical assumptions.

1)      The axial displacement u consists of the extension, bending, and shear components as

  ,                           (1)

where

 .        (2)

2)      There is no relative motion in the y-direction at any point in the cross section of the beam.

3)      Transverse displacement is assumed to be a function of the x-coordinate only.

                                                                 (3)

4)      The theory applies to the hyperbolic cosine distribution of transverse shear stress through the thickness of the beam and satisfies zero traction boundary conditions on the top and bottom surfaces of the beam.

5)      The axial displacement, u, is such that the resultant axial stress , acting over the cross-section, should result only in a bending moment and should not result in force in the x-direction.

6)      Displacements are small, compared to beam thickness.

7)      One-dimensional constitutive law is used to obtain stress values

Based on these assumptions, the displacement field of the hyperbolic shear deformation theory is given by:

  ,      (4)

where u₀ is the axial displacement of a point on the neutral axis of the beam, w₀ is the transverse displacement of a point on the neutral axis of the beam, and the hyperbolic function is assumed according to the transverse shearing strain distribution across the thickness of the beam (see Figures 2 and 3). The nonzero normal and shear strains at any point of the beam are

    (5)

Figure 1. FG beam under bending conditions in the x-z plane

Figure 2. Through thickness distribution of the transverse shearing strain function

Figure 3. Through thickness distribution of the derivative of transverse shearing strain function

2.2 Constitutive relations

The FG beam was made of Al/Al₂O₃, and the properties of the material varied continuously throughout beam thickness, according to the power law distribution given by Equation (6).

                                               (6)

where E represents the Young’s modulus, G represents the shear modulus, represents the Poisson’s ratio, andrepresents mass density. Subscripts m and c represent the metallic and ceramic constituents, respectively, and p is the power law exponent. The variation of the Young’s modulus E(z) through the thickness z/h of the beam for various values of the power law exponent is shown in Figure 4. The stress–strain relationship at any point of the beam is given by one-dimensional Hooke’s law as follows.

                              (7)

1. 3.      Equations of motion

Equations of motion of hyperbolic shear deformable FG beam are derived using Hamilton’s principle,

 ,                                          (8)

where denotes variations in total strain energy, potential energy, and kinetic energy respectively, and t₁ and t₂ are the lower and upper limits of desired time period, respectively.

Figure 4. Variation in Young’s modulus E(z) through the thickness of the FG beam for various values of the power law exponent (p)

The variation of the strain energy  can be stated as:

 ,    (9)

where are the axial force, bending moment, higher order moment, and shear force resultants, respectively. Additionally,

  , (10)

where

.             (11)

The variation of the potential energy due to transverse and axial loads can be written as

 .                            (12)

The variation of kinetic energy  can be written in following form

 ,    (13)

where is the mass density, and   are the inertia coefficients.

                                         (14)

Substituting Equations (9), (12), and (13) into Equation (8), doing the integrations and setting the coefficients of , , and to equal zero, the following equations of motion are obtained.

           (15)

By substituting the stress resultants from Equation (10) into Equation (15), the following equations of motion can be obtained for unknown displacement variables.

            (16)

1. 4.      Analytical solution

Consider a simply supported FG beam with length ‘L’ and rectangular cross-section ‘b×h’. For simply supported boundary conditions, according to Navier’s solution, the unknown displacement variables are expanded in a Fourier series as given below:

                                             (17)

where , , and  are the unknown coefficients, and is the natural frequency. The uniform transverse load (q) acting on the top surface of the beam was also expanded in the Fourier series as

,                                                    (18)

where q₀ is the maximum intensity of the load at the center of the beam. By substituting Equations (17) and (18) into Equation (16), the analytical solution can be obtained from the following equations.

For bending, ignore time derivatives and axial force.  .       (19)

For buckling, ignore time derivatives and transverse load.

,   (20)

For free vibrations, ignore transverse load and axial force.

.   (21)

1. 5.      Numerical results and discussion

In this section, the accuracy of hyperbolic shear deformation theory for predicting bending, buckling, and vibration responses of FG higher order beams was investigated. The numerical results were obtained using Navier’s solution for simply supported boundary conditions. The beam was made of Al₂O₃ for ceramic ( = 380 GPa, =3960 kg/m³, = 0.3) and Al for metal ( = 70 GPa, =2702 kg/m³, = 0.3). The material properties of the beam were varied across beam thickness according to power law distribution. The bottom surface of the FG beam was ceramic-rich, and the top surface was metal-rich.

5.1 Bending the FG beam

The bending response of the FG beam under a uniform transverse load was investigated. The displacements and stress are presented in the following non-dimensional form.

Axial displacement (u) at x = 0 and z = -h/2: .

Transverse displacement (w) at x = L/2 and z = 0: .

Axial stress ( ) at x = L/2 and z = h/2:

.

Transverse shear stress ( ) at x = 0 and z = 0: .

The numerical results obtained using the present theory were compared to those of other theories, which is shown in Table 1. Comparisons of numerical results are presented in Table 2. Through thickness distribution of displacements and stress are shown in Figure 5 (a–c). The displacements and stress are presented for various values of the power law exponent (p). The transverse shear stress was evaluated directly from constitutive relations. The present results were compared to higher order shear deformation theories of Reddy [20], Touratier [22], and Hadji et al. [42] and the CBT. The present results were in good agreement with those obtained using various shear deformation theories for all values of the power law exponent. Because the effect of transverse shear deformation is not included in the CBT, this theory underestimated displacement and stress. The stress presented by Hadji et al. [42] was higher compared to that obtained using shear deformation theories. The present theory gives a linear variation of axial stress through the thickness for p = 0 and p = ∞; however, for other values of the power law exponent, this is non-linear through the thickness (see Figure 5b). Displacements and stress are increased as the power law exponent increases, creating more flexibility in FG beams.

Table 1. Displacement fields of the present and referred theories

Figure 5(b) shows that an increase in the power law exponent increased the compression zone in the beam, while Figure 5(c) shows the hyperbolic cosine variation of transverse shear stress  that was across the thickness of the beam and that satisfied the traction free conditions at the top and bottom surfaces of the beam. Figure 5(c) also shows an increase in the power law exponent neutral axis that shifted toward the bottom. This was due to ceramic, with which metal has a low elastic modulus.

5.2 Buckling an FG beam

In this section, the buckling response of an FG beam subjected to axial force (N₀) was investigated. A non-dimensional critical buckling load is presented in Table 3. The non-dimensional form of the buckling load was as follows:

.

The critical buckling load was obtained for various values regarding the power law exponent (p) and a length-to-thickness ratio (L/h). Results were compared with those presented by Li and Batra [38], Nguyen et al. [40], and Vo et al. [45]. Table 3 reveals that this study's results agreed with those available in the literature. Specifically, the critical buckling load was higher for a thin, slender beam and lower for a thick beam. However, the critical buckling load was in a non-dimensional form; non-dimensional quantities are reciprocal of dimensional quantities. According to Euler’s buckling theory, critical buckling loads are directly proportional to cross-sections of beams (i.e., moments of inertia). Therefore, it can be noted that the dimensional critical buckling load for the slender beam was actually smaller than the load for the thicker beam.

Table 2. A comparison of the non-dimensional displacements and stress of the FG beams subjected to uniform loads with various power law exponent values

5.3 The free vibrations of FG beams

The free vibration responses of FG beams were investigated. Fundamental frequencies were obtained for various power law exponent values and L/h ratios. The results were compared to those presented by Reddy [20], Simsek [39], Thai and Vo [37], Vo et al. [45], and Timoshenko [19] and those obtained with the CBT. Fundamental frequencies were presented in the following non-dimensional form:

.

Table 4 shows the non-dimensional fundamental frequencies ( ) of simply supported FG beams. The natural frequencies of first three bending modes are presented. Table 4 reveals that the fundamental frequencies obtained using the theory presented in this research were in excellent agreement with those obtained by other researchers. The numerical results showed that all shear deformation theories predicted more or less the same frequencies, whereas the CBT overestimated all frequencies due to a neglect of shear deformation. The effects of a power law exponent, p, on the frequencies of FG beams are shown in Figure 6(b). It was observed that increases in power law exponent values led to reductions of fundamental frequencies. This was because the increases in power law exponent values resulted in decreases in elasticity modulus values. It should be noted that the fundamental frequencies were higher when there were higher modes of vibration.

Figure 5. Through thickness distribution of the non-dimensional (a) axial displacement ( ), (b) the axial stress ( ), and (c) transverse shear stress ( ) simply supported the FG beam under a uniform load throughout various power law exponent values (L/h = 5)

Figure 6. The variations in non-dimensional (a) critical buckling loads and (b) natural frequencies with respect to the power law exponents of simply supported FG beams.

Table 3. A comparison of the non-dimensional critical buckling loads ( ) of the FG beams subjected to axial forces in regards to various power law exponent values

Table 4. A comparison of the first three non-dimensional fundamental frequencies of the FG beams in regards to various power law exponent values

6. Conclusions

A hyperbolic shear deformation theory developed by Soldatos [21] was extended in this paper to conduct bending, buckling, and free vibration analyses of FG beams. With the theory,  hyperbolic cosine variations of transverse shear stress were found at the top and bottom surfaces of the beams. Subsequently, Hamilton’s principle was employed to derive equations of motion. The equations of motion, with the theory, were variationally consistent and allowed the avoidance of a shear correction factor. Then, an analytical solution for a simply supported boundary condition was obtained using Navier’s solution procedure.

The numerical results were compared to those obtained by other researchers to determine the accuracy of the theory. Based on the comparisons and a discussion, it was concluded that the displacements, stress, critical buckling loads, and natural frequencies obtained using the theory were accurate and in agreement with those obtained using other refined shear deformation theories. It was seen that varying material properties had significant effects on the dimensionless stress, frequencies, and buckling loads of the FG beams. Increasing power law exponent values reduced the stiffnesses of the FG beams and consequently led to increases in displacements and reductions of frequencies and buckling loads. Overall, the investigation of the bending, buckling, and free vibration responses of the FG beams confirmed the effects and credibility of the hyperbolic shear deformation theory.

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