Free vibration analysis of doubly convex/concave functionally graded sandwich beams
Ashraf M. Zenkour^{1} , Ibrahim A. Abbas^{2}
^{1}Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
^{1}Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr ElSheikh 33516, Egypt
^{2}Department of Mathematics, Faculty of Science and ArtsKhulais, King Abdulaziz University, Jeddah, Saudi Arabia
^{2}Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
^{1}Corresponding author
Journal of Vibroengineering, Vol. 16, Issue 6, 2014, p. 27472755.
Received 11 May 2014; received in revised form 30 June 2014; accepted 22 August 2014; published 30 September 2014
JVE Conferences
This paper presents the highly accurate analytical investigation of the natural frequencies for doubly convex/concave sandwich beams with simplysupported or clampedsupported boundary conditions. The present sandwich beam is made of a functionally graded material composed of metal and ceramic. The properties are graded in the thickness direction of the two faces according to a volume fraction powerlaw distribution. The bottom surface of the bottom face and the top surface of the top face are both metalrich material. The core is made of a fully ceramic material. The thickness of the sandwich beam varies along its length according to a quadraticlaw distribution. Two types of configuration with doubly convex and doubly concave thickness variations are presented. The governing equation and boundary conditions are derived using the dynamic version of the principle of minimum of the total energy. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effect of configurations of the constituent materials on the frequencies. Natural vibration frequencies of sandwich beams versus many parameters are graphically presented and remarking conclusions are made.
Keywords: sandwich beam, thickness variations, free vibration, functionally graded.
1. Introduction
Functionally graded materials (FGMs) are nonhomogeneous but isotropic material which properties vary gradually and continuously with location within the material. FGMs are developed for military, automotive, biomedical application, semiconductor industry and general structural element in high thermal environments. Many weaknesses of composites can be improved by functionally grading the material to have a smooth spatial variation of material composition, with ceramicrich material placed at the hightemperature locations and metalrich material in regions where mechanical properties need to be high.
Natural vibration frequencies of plates are important in the design and analysis of engineering structures in diverse fields as aerospace, ocean and nuclear engineering, electronics, and oil refineries. Loy et al. [1] have studied the vibration of stainless steel and nickel graded cylindrical shells under simply supported ends by using Love’s theory and RayRitz method. Pradhan et al. [2] have presented the vibration of a FG cylindrical shell. The effects of boundary condition and volume fractions on the natural frequencies are studied. Reddy and Cheng [3] have studied the harmonic vibration problem of FG plates by means of a threedimensional asymptotic theory formulated in terms of transfer matrix. Vel and Batra [4] have studied the threedimensional exact solution for the vibration of FG rectangular plates. Based on the higherorder shear deformation theory, Chen [5] has analyzed the nonlinear vibration of a shear deformable FG plate including the effects of transverse shear deformable and rotary inertia. Giunta et al. [6] have presented the free vibration analysis of FG beams via several axiomatic refined theories.
Sandwich constructions have been developed and utilized for almost many years because of its outstanding bending rigidity, low specific weight, superior isolating qualities, good fatigue properties, and excellent natural vibration frequencies. The last characteristic is the major reason why sandwich structures are used more often in aerospace vehicles. Chan and Cheung [7] have carried out a dynamic analysis of multilayered sandwich plates using linear elastic theory. Reddy and Kuppusamy [8] have studied the free vibration of laminated and sandwich rectangular plates using 3D elasticity equations and the associated finite element model. Kanetmatsu et al. [9], Wang [10] and Lee and Fan [11] have proposed the study of the bending and vibration of sandwich plates. The free vibration analysis of fiber reinforced plastic composite sandwich plates have been presented by Meunier and Shenoi [12]. Pandya and Kant [13] have developed a simple finite element formulation for flexural analysis of multilayer symmetric sandwich plates. A study on the eigenfrequencies of sandwich plates has been presented by Kant and Swaminathan [14].
Recently, sandwich construction becomes even more attractive due to the introduction of advanced composite and FGMs. Sandwich structures made from FGMs have increasing use because of smooth variation of material properties along some preferred direction. Zenkour [15] has studied the buckling and free vibration of FG sandwich plates. Studies on vibration of doubly convex/concave sandwich beams made of FGMs have not been seen in the literature. Compared with FG plates and shells, studies for FG beams are relatively less (Sankar [16], Wu et al. [17], Aydogdu and Taskin [18]). Bhangale and Ganesan [19] have presented the buckling and vibration behavior of a FG sandwich beam having constrained viscoelastic layer in thermal environment by using finite element formulation.
In the present article, free vibration analysis of a variablethickness FG sandwich beam is presented. The governing differential equation is exactly satisfied at every point of the beam. The boundary conditions at the end edges of the beam are also exactly satisfied. The core layer of the beam is made from an isotropic ceramic material. The bottom face is made of a FG metalceramic material, which components vary smoothly in the thickness direction from metal to ceramic ones. The top face is made of a FG ceramicmetal material, which constituents vary smoothly in the thickness direction from ceramic to metal ones. The effective material properties of the present FG sandwich beam are determined using a simple power law distribution. Natural frequencies are presented for metal/ceramic/metal FG sandwich beam with variable thickness. Results obtained are tabulated for future comparison with other investigators. Additional results are plotted to show the effects of vibrations in the geometric and lamination parameters.
2. Geometrical preliminaries
The present study considers FGMs composed of metal and ceramic. The grading is accounted for only across the thickness of the sandwich beam. The present approach adopts the smooth and continuous variation of the volume fraction of either ceramic or metal based on the power law index. The bottom face of the present sandwich beam is made from a metalceramic FG material, the core layer is still homogeneous and made from a ceramic material, and the top face is made from a ceramicmetal FG material (see Zenkour [15, 2024]). The present beam is assumed to have length $L$, width $b$ and variable thickness $h$, as shown in Figs. 1 and 2. Rectangular Cartesian coordinates $\left(x,y,z\right)$ are used and the midplane is defined by $z=\text{0}$ and its bounding planes are defined by $z=\pm 1/2h\left(x\right)$.
Fig. 1. Plot of the FG sandwich beam with a doubly convex thickness variation ($\text{0.5}\le \lambda \le \text{0}$)
Fig. 2. Plot of the FG sandwich beam with a doubly concave thickness variation ($\text{0}\le \lambda \le \text{0.5}$)
A simple power lawtype definition for the volume fraction of the metal across the thickness direction of the sandwich beam is assumed. This is defined as:
where ${h}_{0}$ is the thickness of the core layer and the volume fraction index $k$ represents the material variation profile through the faces thickness, which is always greater than or equal to zero. The value of $k$ equal to zero represents a fully ceramic beam while the value of $k$ tends to infinity represents fully metal faces with a ceramic core. Based on the volume fraction definition, the effective material property definition follows:
Note that $P\left(x,z\right)$ represents the effective material property for each interval while ${P}_{m}$ and ${P}_{c}$ represent, respectively, the corresponding properties of the metal and ceramic of the FG sandwich beam. The superscript ‘($n$)’ stands for the layer number ($n=\text{1}$ means bottom face, $n=\text{2}$ means core, and $n=\text{3}$ means top face). Generally, this study assumes that Young’s modulus $E$ and material density $\rho $ of the FGM change continuously through the thickness direction of the beam and obey the gradation relation given in Eq. (2). It should be noted that the material properties of the considered beam are metalrich at the bottom and top surfaces ($z=\pm {h}_{0}/2$) of the beam and ceramicrich at the interfaces ($z=\mp {h}_{0}/2$).
The total thickness of the beam accounts for doubly convexity/concavity variation in the $x$ direction. It reads:
where ${h}_{1}$ is the constant reference thickness value located at the beam center ${\text{(}h}_{1}>{h}_{0}\text{)}$, $\lambda $ is a small thickness parameter show the convexity and the concavity of the thickness variation, and $f\left(x\right)$ describes the convex/concave thickness variation of the sandwich beam:
The present sandwich beam has a doubly convex thickness variation when $\text{0.5}<\lambda <\text{0}$ (see Fig. 1), and it is a doubly concave thickness variation when $\text{0}<\lambda <\text{0.5}$ (see Fig. 2). It is to be noted here that the ends of the beam is $\text{(}1+\lambda \text{)}$ times thicker (thinner) than the thickness at the center of the beam, $x=L/2$.
3. Basic equations
The dynamic version of the principle of minimum of the total energy is used to derive the governing equation and associated boundary conditions. It is given in terms of the deflection $w$ and the transverse distributed load $q$ is as:
where:
in which $E\left(x,z\right)$ and $\rho \left(x,z\right)$ are given according to Eq. (2). The associated boundary conditions are given as follows:
1) $w$ is specified or ${I}_{2}\left(x\right)\frac{d\ddot{w}}{dx}+\frac{d}{dx}\left({I}_{e}\left(x\right)\frac{{d}^{2}w}{d{x}^{2}}\right)=0$.
2) $\frac{dw}{dx}$ is specified or $\left({I}_{e}\left(x\right)\frac{{d}^{2}w}{d{x}^{2}}\right)=0$.
The moment and shear force are given by:
4. Free vibration of sandwich beams
The edges of the considered beam $\text{(}x=0,L\text{)}$ have two combinations of simply supported (S) and clamped (C) boundary conditions. The displacement $w$ is presented as products of determined function of the axial coordinate and unknown function of time. For free vibration, the load force $q$ is vanished and the displacement is given by:
where ${\omega}_{m}$ denotes the eigenfrequency associated with the $m$ eigenmode, ${W}_{m}$ is arbitrary parameter and $\xi (\equiv x/L)$ is the dimensionless axial variable. The function $X\left(\xi \right)$ depends on the boundary conditions on the beam edges as follows:
SS:
CS:
${\mu}_{m}=\left(m+\frac{1}{4}\right)\pi ,{\eta}_{m}=\frac{\mathrm{cos}{\mu}_{m}+\mathrm{cosh}{\mu}_{m}}{\mathrm{sin}{\mu}_{m}+\mathrm{sinh}{\mu}_{m}}.$
Using Eq. (8) into the governing equation, Eq. (5), and setting $q=0$, one obtains:
where:
and
is the frequency parameter of the natural vibration. After imposing the boundary conditions of the problem, Eq. (11) can be solved directly by numerical computation to obtain the positive root of the frequency parameter $\mathrm{\Omega}$ according to the SS and CS boundary conditions.
5. Numerical examples and discussion
The FGM can be obtained combining two distinct materials such as a metal and a ceramic. The isotropic FG sandwich beam considered in the examples is assumed to be composed of metal (Ti6Al4V) and ceramic (zirconiaZrO_{2}). The relevant material properties for the constituent materials are listed in Table 1 (Reddy and Chin [25]).
Table 1. Material properties of Ti6Al4V/ZrO_{2} sandwich beam
Ti6Al4V

ZrO_{2}


$E\mathrm{}\mathrm{}$(GPa)

105.802

168.4

$\nu $

0.2982

0.2979

$\rho \mathrm{}\mathrm{}\text{(}{\text{10}}^{\text{3}}\text{kg/}{\text{m}}^{\text{3}}\text{)}$

8.9

2.37

The dimensionless fundamental frequencies $\text{(}m=\text{1)}$ for a doubly convex FG sandwich beam with ${h}_{1}/{h}_{0}=\text{2}$ and $L/{h}_{0}=\text{10}$ at the center of sandwich beam are presented in Table 2 for different values of the volume fraction index $k$ and the thickness parameter $\lambda $. Similar results for the doubly concave sandwich beam are presented in Table 3. It is to be noted that, frequencies increase as $k$ increases and $\lambda $ decreases and this irrespective of the boundary conditions. As the volume fraction index $k$ tends to infinity (full ceramic beam), the frequencies are the same for the two shapes of the beam and are independent on the thickness parameter $\lambda $.
Table 2. Fundamental frequencies for a doubly convex FG sandwich beam $\text{(}{h}_{1}/{h}_{0}=\text{2,}L/{h}_{0}=\text{10)}$
$\lambda $

BC

$k=\text{0}$

$k=\text{0.5}$

$k=\text{1.5}$

$k=\text{3.5}$

$k=\text{5.5}$

$k=\text{7.5}$

$k=\text{9.5}$

$k\to \mathrm{\infty}$

$\frac{1}{2}$

$\mathrm{S}\mathrm{S}$

4.33140

5.11231

5.99776

6.81849

7.21183

7.44358

7.59652

2.01050

CS

5.80357

6.84438

8.02089

9.10980

9.63159

9.93907

10.14204

3.13970


$\frac{1}{3}$

$\mathrm{S}\mathrm{S}$

3.92943

4.63588

5.43469

6.17336

6.52683

6.73492

6.87220

2.01050

CS

5.43421

6.40635

7.50256

8.51496

8.99941

9.28470

9.47296

3.13970


$\frac{1}{4}$

$\mathrm{S}\mathrm{S}$

3.71216

4.37826

5.13004

5.82405

6.15580

6.35101

6.47975

2.01050

CS

5.23977

6.17570

7.22947

8.20137

8.66605

8.93958

9.12002

3.13970


$\frac{1}{5}$

$\mathrm{S}\mathrm{S}$

3.57546

4.21615

4.93823

5.60403

5.92204

6.10910

6.23243

2.01050

CS

5.11956

6.03308

7.06055

8.00733

8.45972

8.72595

8.90155

3.13970

The dimensionless natural frequencies for a doubly convex $\text{(}\lambda =\text{0.5)}$ and a doubly concave $\text{(}\lambda =\text{0.5)}$ FG sandwich beams with ${h}_{1}/{h}_{0}=\text{2}$, $L/{h}_{0}=\text{10}$ and ${h}_{1}/{h}_{0}=\text{2}$ and $\xi =\text{0.5}$ are given, respectively, in Tables 4 and 5. Once again, the frequencies increase as $k$ increases and the eigenmode $m$ increases. Some plots are presented for natural frequencies $\text{(}m=\text{5)}$ of FG sandwich beam $\text{(}k=\text{5.5)}$ with doubly convex/concave thickness variations $\text{(}\left\lambda \right=\text{1/3)}$. All plots shown henceforth are obtained for natural frequencies versus the thickness ratio ${h}_{1}/{h}_{0}$ at $L/{h}_{0}=\text{10,15,20}$ and 25. Figs. 3 and 4 show that the frequencies for beams subjected to SS and CS boundary conditions are stable and increasing with the increase of ${h}_{1}/{h}_{0}$ and $L/{h}_{0}$ ratios.
Table 3. Fundamental frequencies for a doubly concave FG sandwich beam $\text{(}{h}_{1}/{h}_{0}=\text{2,}L/{h}_{0}=\text{10)}$
$\lambda $

BC

$k=\text{0}$

$k=\text{0.5}$

$k=\text{1.5}$

$k=\text{3.5}$

$k=\text{5.5}$

$k=\text{7.5}$

$k=\text{9.5}$

$k\to \mathrm{\infty}$

$\frac{1}{2}$

$\mathrm{S}\mathrm{S}$

1.07824

1.31547

1.62791

1.94863

2.11093

2.20886

2.27437

2.01050

CS

2.96251

3.46706

4.00711

4.48159

4.70096

4.82791

4.91078

3.13970


$\frac{1}{3}$

$\mathrm{S}\mathrm{S}$

1.46901

1.70702

1.94605

2.14111

2.22600

2.27345

2.30374

2.01050

CS

3.59546

4.22193

4.90950

5.52889

5.82041

5.99067

6.10245

3.13970


$\frac{1}{4}$

$\mathrm{S}\mathrm{S}$

1.95405

2.28825

2.64683

2.96228

3.10815

3.19256

3.24764

2.01050

CS

3.87333

4.55267

5.30343

5.98420

6.30604

6.49444

6.61831

3.13970


$\frac{1}{5}$

$\mathrm{S}\mathrm{S}$

2.19419

2.57476

2.98946

3.35999

3.53326

3.63410

3.70016

2.01050

CS

4.03088

4.74005

5.52633

6.24146

6.58024

6.77876

6.90937

3.13970

Table 4. Natural frequencies for a doubly convex FG sandwich beam $\text{(}{h}_{1}/{h}_{0}=\text{2,}L/{h}_{0}=\text{10,}\lambda =\text{0.5)}$
$m$

BC

$k=\text{0}$

$k=\text{0.5}$

$k=\text{1.5}$

$k=\text{3.5}$

$k=\text{5.5}$

$k=\text{7.5}$

$k=\text{9.5}$

$k\to \mathrm{\infty}$

3

$\mathrm{S}\mathrm{S}$

24.2850

28.5355

33.4461

38.1951

40.5746

42.0165

42.9866

17.8051

CS

27.5301

32.3342

37.9065

43.3339

46.0722

47.7387

48.8631

20.8278


5

$\mathrm{S}\mathrm{S}$

52.5505

61.5724

72.2832

83.1402

88.8323

92.3806

94.8139

47.9605

CS

56.2863

65.9319

77.4150

89.1082

95.2656

99.1149

101.7595

52.6267


7

$\mathrm{S}\mathrm{S}$

82.6627

96.6864

113.6417

131.3489

140.8997

146.9604

151.1679

90.0591

CS

86.4410

101.0933

118.8311

137.4100

147.4576

153.8380

158.2753

96.0230


9

$\mathrm{S}\mathrm{S}$

112.7417

131.7330

154.9500

179.6297

193.1659

201.8473

207.9186

141.3344

CS

116.3723

136.1056

160.1347

185.5152

199.6253

208.5695

214.7331

148.2557

Table 5. Natural frequencies for a doubly concave FG sandwich beam $\text{(}{h}_{1}/{h}_{0}=\text{2,}L/{h}_{0}=\text{10,}\lambda =\text{0.5)}$
$m$

BC

$k=\text{0}$

$k=\text{0.5}$

$k=\text{1.5}$

$k=\text{3.5}$

$k=\text{5.5}$

$k=\text{7.5}$

$k=\text{9.5}$

$k\to \mathrm{\infty}$

3

$\mathrm{S}\mathrm{S}$

21.4001

25.1182

29.3835

33.4843

35.5320

36.7710

37.6037

17.8051

CS

24.6967

28.9792

33.9168

38.7024

41.1101

42.5735

43.5603

20.8278


5

$\mathrm{S}\mathrm{S}$

50.2223

58.8216

69.0062

79.3111

84.7087

88.0721

90.3782

47.9605

CS

54.0201

63.2549

74.2266

85.3776

91.2479

94.9146

97.4334

52.6267


7

$\mathrm{S}\mathrm{S}$

80.7744

94.4591

110.9847

128.2286

137.5257

143.4245

147.5193

90.0591

CS

84.6029

98.9551

116.2173

134.3278

144.1806

150.3806

154.7344

96.0230


9

$\mathrm{S}\mathrm{S}$

111.1768

129.8889

152.7484

177.0363

190.3547

198.8957

204.8686

141.3344

CS

115.5447

135.1096

158.7798

183.2037

198.1876

207.450

213.3862

148.2557

Finally, the dimensionless fundamental frequency is plotted through the length of the FG sandwich beam according to different parameters. Figs. 5 and 6 show plots of the fundamental frequency of a SS doubly convex/concave FG sandwich beam with the volume fraction index $k=\text{5}$, ${h}_{1}/{h}_{0}=\text{3}$, $L/{h}_{0}=\text{10}$ and for different values of $\lambda $. However, Figs. 7 and 8 show plots of the fundamental frequency of a SS doubly convex $\text{(}\lambda =\text{0.5)}$ or concave $\text{(}\lambda =\text{0.5)}$ FG sandwich beam with the volume fraction index $k=\text{3.5}$, ${h}_{1}/{h}_{0}=\text{3}$ and for different values of $L/{h}_{0}$ ratio. In addition, Figs. 9 and 10 shows fundamental frequencies of doubly convex/concave SS sandwich beam with fully metallic faces and ceramic core ($k$ tends to infinity).
As shown in Figs. 510, results are symmetric about the center of the beam. Frequencies are maximum at the edges of the doubly convex beam and at the center of the doubly concave beam. Once again, frequencies increase as $\lambda $ decreases. However, the value of $\lambda $ has a very little effect on frequencies at the center of beam with fully metallic faces and ceramic core.
Fig. 3. Natural frequency $\mathrm{\Omega}$ vs thickness ratio ${h}_{1}/{h}_{0}$ for a SS FG sandwich beam with doubly convex/concave thickness variation
Fig. 4. Natural frequency $\mathrm{\Omega}$ vs thickness ratio ${h}_{1}/{h}_{0}$ for a CS FG sandwich beam with doubly convex/concave thickness variation
Fig. 5. Fundamental frequency $\mathrm{\Omega}$ through the length of a doubly convex SS FG sandwich beam for different values of $\xi $$\text{(}k=\text{5,}{h}_{1}/{h}_{0}=\text{3,}L/{h}_{0}=\text{10)}$
Fig. 6. Fundamental frequency $\mathrm{\Omega}$ through the length of a doubly concave SS FG sandwich beam for different values of $\xi $$\text{(}k=\text{5,}{h}_{1}/{h}_{0}=\text{3,}L/{h}_{0}=\text{10)}$
Fig. 7. Fundamental frequency $\mathrm{\Omega}$ through the length of a doubly convex SS FG sandwich beam for different values of $L/{h}_{0}$$\text{(}k=\text{3.5,}{h}_{1}/{h}_{0}=\text{3)}$
Fig. 8. Fundamental frequency $\mathrm{\Omega}$ through the length of a doubly concave SS FG sandwich beam for different values of $L/{h}_{0}$$\text{(}k=\text{3.5,}{h}_{1}/{h}_{0}=\text{3)}$
6. Conclusions
This article focuses on the derivation of natural vibration frequencies of variablethickness FG sandwich beams subjected to various boundary conditions. The core layer is composed of a homogeneous ceramic material while the faces are made of a symmetric FG metalceramic material. The material properties such as Young’s modulus and material density can vary through the axial and thickness directions of the beam according to a mixed powerlaw type distributions. Some vibration frequencies for metalceramic/ceramic/ceramicmetal sandwich beam with a doubly convex/concave variable thickness are tabulated for future comparisons. The effects of many parameters such as thickness ratio, lengthtocore thickness ratio, thickness parameter and the volume fraction index on frequencies are investigated. The results show that the fundamenta1 frequencies are similar to that observed for homogeneous convex/concave beams and the natural frequencies are affected by the thickness variation and the constituent volume fractions and the configurations of the constituent materials.
Fig. 9. Fundamental frequency $\mathrm{\Omega}$ through the length of a doubly convex SS sandwich beam of fully metallic faces and a ceramic core $\text{(}k\to \infty \text{,}{h}_{1}/{h}_{0}=\text{3,}L/{h}_{0}=\text{10)}$
Fig. 10. Dimensionless fundamental frequency $\mathrm{\Omega}$ through the length of a doubly concave SS sandwich beam of fully metallic faces and a ceramic core $\text{(}k\to \infty \text{,}{h}_{1}/{h}_{0}=\text{3,}L/{h}_{0}=\text{10)}$
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks the DSR technical and financial support.
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