Free vibration analysis of functionally graded sandwich beams based on a higher order shear deformation theory

FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED SANDWICH BEAMS BASED ON A HIGHER-ORDER SHEAR DEFORMATION THEORY Nguyen Ba Duy 1 , Nguyen Trung Kien 2,* 1 Faculty of Civil Engineering,

FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED SANDWICH BEAMS BASED ON A HIGHER-ORDER SHEAR

DEFORMATION THEORY Nguyen Ba Duy 1 , Nguyen Trung Kien 2,*

1

Faculty of Civil Engineering, Thu Dau Mot University, 06 Tran Van On Street, Phu Hoa District,

Thu Dau Mot City, Binh Duong Province, Viet Nam

2

Faculty of Civil Engineering and Applied Mechanics, University of Technical Education Ho Chi

Minh City, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Vietnam

*

Email: ntkien@hcmute.edu.vn

Received: 13 November, 2013; Accepted for publication: 07 September, 2014

ABSTRACT

This paper presents free vibration analysis of functionally graded sandwich beams using a higher-order shear deformation theory The face layers of sandwich beam are assumed to have isotropic metal-ceramic material distribution, and Young’s modulus and mass density are assumed to vary according to power-law form in terms of the volume fractions of the constituents The core layer is still homogeneous and made of an isotropic metallic material Governing equations of motion are derived from the Hamilton’s principle Navier-type solution for simply-supported beams is developed to solve the problem Numerical results are obtained for sandwich beams with homogeneous softcore to investigate effects of the power-law index, span-to-height ratio and ratio of layer thickness on the natural frequencies

Keywords: functionally graded sandwich beams, free vibration, beam theory

1 INTRODUCTION

Increase of the application of sandwich structures in aerospace, marine, civil construction led to the development of functionally graded (FG) sandwich structures to overcome the material discontinuity found in classical sandwich materials With the wide application of FG sandwich structures, understanding vibration of FG sandwich structures becomes an important task Thanks to the advantage that no shear correction factor is needed, the higher-order shear deformation theory is widely used for vibration of sandwich plates Based on this theory, though many works on these problems are available in the open literature [1, 2, 3], only representative samples are cited here, while detailed discussions can be found in [4] Although there are several studies of the behavior of FG sandwich plates, research on vibration of FG sandwich beams is a few in number Bhangale and Ganesan [5] studied vibration and buckling analysis of a FG sandwich beam having constrained viscoelastic layer in thermal environment by using finite

natural frequencies of sandwich beams with inhomogeneous FG core using a truly meshfree radial point interpolation method Vo et al [8] proposed a finite element model for vibration and buckling analysis of functionally graded sandwich beams based on a refined shear deformation theory

This paper aims to present free vibration analysis of functionally graded sandwich beams using a higher-order shear deformation theory The face layers of sandwich beam are assumed to have isotropic metal-ceramic material distribution, and Young’s modulus and mass density are assumed to vary according to power-law form in terms of the volume fractions of the constituents The core layer is still homogeneous and made of an isotropic metallic material Governing equations of motion are derived from the Hamilton’s principle Navier-type solution for simply-supported beams is developed to solve the problem Numerical results are obtained for sandwich beams with homogeneous softcore to investigate effects of the power-law index, span-to-height ratio and ratio of layer thickness on the natural frequencies

2 KINEMATICS

Consider a FG sandwich beam composed of three layers as shown in Fig 1 The x-, y-, and z-axes are taken along the length (L), width (b), and height (h) of the beam, respectively The

face layers of the sandwich beam are made of an isotropic material with material properties

varying smoothly in the z-direction only The core layer is made of an isotropic homogeneous

metallic material The vertical positions of the bottom and top, and of the two interfaces between

the layers are denoted by h 0 = − h/2 , h 1 , h 2 , h 3 = h/2, respectively For the brevity, the ratio of

the thickness of each layer from bottom to top is denoted by the combination of three numbers, i.e “1-0-1”, “1-2-1”, “2-1-2”, and so on

Figure 1 Geometry of the sandwich beam with functionally graded faces

The displacement field of the present theory can be obtained as ([9]):

( , ) ( , ) ( , , ) ( , ) w x t b ( ) w x t s

U x z t u x t z f z

( , , ) b( , ) s( , )

where u is the axial displacement, w bandw bare the bending and shear components of transverse displacement along the mid-plane of the beam The non-zero strains are given by:

U

∂

∈ = =∈ + +

g

γ =∂ +∂ = γ

where

2 4 3

z

h

 

=  

2

1 df 1 4 z

g

 

and ∈0x, γxz0 , κx b and κx s are the axial strain, shear strain and curvatures in the beam, respectively, defined as:

0 ,

x u x

0 ,

xz w s x

,

b

x w b xx

κ = −

(4c) ,

s

x w s xx

κ = −

(4d) where the comma indicates the differentiation with respect to the subscript that follows

3 VARIATIONAL FORMULATION

In order to derive the equations of motion, Hamilton’s principle is used:

2

1

t

t dt

where ℧ and ℜdenote the strain energy and kinetic energy, respectively The variation of the strain energy can be stated as:

1

3 / 2 ( ) ( ) / 2

1 0

n

n

l

x x xz xz

b h n

dz dxdy

−

℧

0l(N xδ x M x bδκb x M x sδκx s Q xδγxz)dx

where N x , b

x

M , s

x

M and Q x are the axial force, bending moments and shear force, respectively, defined as:

1

3 ( ) 1

n

n

h n

x h x n

−

=

1

3 ( ) 1

n

n

h

x h x n

−

=

3

3 ( ) 1

n

n

h n

x h xz n

−

=

The variation of the kinetic energy is obtained as:

1

3 / 2 ( )

0 / 2

1

n

n

l b h n

b h n

U U W W dz dxdy

−

−

=

0l[δu I u( I w b x I w f s x) δw I w b ( b w s) δw b x( I u I w b x I w fz s x)

2

s b s s x f fz b x f s x

w I w w w I u I w I w dx

where the differentiation with respect to the time t is denoted by dot-superscript convention;

( )n

ρ is the mass density of the each layer and I0, I1, I2, I , f I and fz I f2 are the inertia coefficients, defined by:

2

1

3

0 1 2

1

( , , , , , ) n (1, , , , , )

n

h n

f fz f h

n

I I I I I I ρ z z f fz f bdz

−

=

By substituting Eqs (6) and (8) into Eq (5), and integrating by parts versus both space and time variables, and collecting the coefficients of δu, δw b, and δw s, the following equations

of motion of the functionally graded sandwich beam are obtained:

b x xx b s x b xx fz s xx

w M I w w I u I w I w

2

s x xx x x b s f x fz b xx f s xx

4 CONSTITUTIVE EQUATIONS

The effective material properties for each layer, like Young’s modulus E and mass density ρ,

can be expressed as:

c m c m

where P and c P denote the material property of ceramic and metal located at the top and m

bottom surfaces, and at the core, respectively The volume fraction of metal V m( )n through the thickness of the sandwich beam faces follows a simple expression:

(1) 0

1 0

p m

V

= 

−

  for z∈[h h0, 1] (12a)

( ) 2 1

m

(3) 3

2 3

p m

z h V

h h

 − 

−

  for z∈[h h2, 3] (12c) where p is a power-law index, which is positive The metal material distribution through the

beam thickness for various values of p is plotted in Fig 2

Figure 2 Distribution of metallic material through the thickness of (1-2-1) FG sandwich

beams with respect to the power-law index p

The stress-strain relations for FG sandwich beams are given by:

( ) ( ) ( , ) ( ) ( , )

x x z E z x x z

( )

( )

( )

2[1 ( )]

n

z

υ

The constitutive equations for beam forces and beam strains are obtained by using Eqs (2), (7) and (13):

0

0

0 0 0

s

s s s

s

M

A Q

κ κ γ

∈ 

=

(14)

where the components of the stiffnesses of FG sandwich beams are given by:

1

3

2 2 ( ) 1

n

h

h n

−

=

1

3

2 ( ) 1

n

n

h

h n

−

=

By substituting Eqs (4) and (14) into Eq (10), the explicit form of the governing equations

of motion can be expressed as:

s

xx b xxx s xxx b x f s x

Au −Bw −B w =I uɺɺ−I wɺɺ −I wɺɺ (16a)

xxx b xxxx s xxxx b s x b xx fz s xx

Bu −Dw −D w =I wɺɺ +wɺɺ +I uɺɺ −I wɺɺ −I wɺɺ (16b)

B u −D w −H w +A w =I wɺɺ +wɺɺ +I uɺɺ −I wɺɺ −I wɺɺ (16c)

Eq (16) is the most general form for the vibration analysis of FG sandwich beams, and the

dependent variables, u, w bandw sare fully coupled

5 ANALYTICAL SOLUTIONS

The above equations of motion are analytically solved for free vibration problems The Navier solution procedure is used to determine the analytical solutions for a simply-supported sandwich beam The solution is assumed to be of the form:

1

( , ) rcos i t

r

=

1

r

=

1

r

=

where ω is the natural frequency, i= −1the imaginary unit, α =rπ /L, u r,w brandw srare wave amplitudes to be determined Substituting Eqs (17a - 17c) into Eq (16), the following eigenvalue problem is obtained:

11 12 13 11 12 13

2

12 22 23 12 22 23

13 23 33 13 23 33

0 0 0

r br sr

u

ω

(18)

Equation (18) can be rewritten under a compact form ( 2 )

r

ω

components of the stiffness matrix K and the mass matrix M associated with

r = u w w r br sr

U are explicitly given as follows:

2

The solution of this eigenvalue problem will allow to calculate the natural frequencies and vibration modes of FG sandwich beams

6 NUMERICAL RESULTS

This section gives numerical examples to investigate the natural frequencies of simply-supported FG sandwich beams Unless mentioned otherwise, Al/Al2O3 FG sandwich beams with

homogeneous softcore for two values of span-to-height ratio, L/h = 5 and 20, are considered

Young’s modulus and mass density of aluminum are E m =70GPa,νm=0.3 and

3

2702 /

m kg m

ρ = and E c =380GPa,νc =0.3and ρc =3960kg m/ 3for alumina at the top and bottom surfaces For convenience, the following non-dimensional natural frequencies is used:

m m

L

ρ ω

Figure 3: Effect of the power-law index and span-to-height ratio on the fundamental frequency of

Al/Al2O3 FG beams with homogeneous softcore

Table 1 The first three non-dimensional natural frequencies of simply-supported Al/Al2O3 beams

5 1 Present 5.1527 4.4102 3.9904 3.6264 3.4012 3.2816

Nguyen et al [10] 5.1525 4.4075 3.9902 3.6344 3.4312 3.3135 Simsek [11] 5.1525 4.4083 3.9902 3.6344 3.4312 3.3134 Thai and Vo (TBT) [12] 5.1527 4.4107 3.9904 3.6264 3.4012 3.2816

2 Present 17.8813 15.4567 14.0100 12.6406 11.5432 11.0241 Nguyen et al [10] 17.8711 15.4250 14.0030 12.7120 11.8157 11.3073 Thai and Vo (TBT) [12] 17.8812 15.4588 14.0100 12.6405 11.5431 11.0240

3 Present 34.2100 29.8331 27.0981 24.3154 21.7162 20.5565 Nguyen et al [10] 34.1449 29.7146 27.0525 24.4970 22.4642 21.3219 Thai and Vo (TBT) [12] 34.2097 29.8382 27.0979 24.3152 21.7158 20.5561

20 1 Present 5.4603 4.6506 4.2051 3.8361 3.6485 3.5390

Nguyen et al [10] 5.4603 4.6504 4.2051 3.8368 3.6509 3.5416 Simsek [11] 5.4603 4.6514 4.2051 3.8368 3.6509 3.5416 Thai and Vo (TBT) [12] 5.4603 4.6511 4.2051 3.8361 3.6485 3.5390

2 Present 21.5732 18.3942 16.6344 15.1619 14.3746 13.9263 Nguyen et al [10] 21.5732 18.3912 16.6344 15.1715 14.4110 13.9653 Thai and Vo (TBT) [12] 21.5732 18.3962 16.6344 15.1619 14.3746 13.9263

3 Present 47.5930 40.6480 36.7679 33.4689 31.5781 30.5370 Nguyen et al [10] 47.5921 40.6335 36.7673 33.5135 31.7473 30.7176 Thai and Vo (TBT) [12] 47.5930 40.6526 36.7679 33.4689 31.5780 30.5369

For verification purpose, the first three natural frequencies of FG beams with different values of span-to-height ratio and power-law index are given in Table 1 The results obtained from the present theory are compared with those of Nguyen et al [10] based on first-order shear deformation beam theory (FSBT), Simsek [11] and Thai and Vo [12] based on higher-order shear deformation beam theory (TBT) It can be seen that the present model is in excellent agreement with earlier works As expected, an increase of the power-law index makes FG beams more flexible, which leads to a reduction in natural frequencies This holds irrespective of the consideration of shear effects Moreover, in order to verify the efficiency of the present study in predicting the vibration responses of FG sandwich beams, Table 2 presents the comparison of the fundamental frequency of Al/Al2O3 sandwich beams for various values of the power-law index and four cases of thickness ratio of layers It can be seen from this table that the present solutions are similar to those of [8] based on a finite element model, except some minor

differences for p = 0.5

Table 2 The non-dimensional fundamental frequencies of simply-supported Al/Al2O3

sandwich beams with homogeneous softcore

L/h

1-0-1 1-1-1 1-2-1 2-1-2

5 0 Present 2.6773 2.6773 2.6773 2.6773

Vo et al (TBT) [8] 2.6773 2.6773 2.6773 2.6773 0.5 Present 4.4443 4.1850 3.9930 4.3059

Vo et al (TBT) [8] 4.4427 4.1839 3.9921 4.3046

1 Present 4.8525 4.5858 4.3663 4.7178

Vo et al (TBT) [8] 4.8525 4.5858 4.3663 4.7178

2 Present 5.0945 4.8740 4.6459 4.9970

Vo et al (TBT) [8] 5.0945 4.8740 4.6459 4.9970

5 Present 5.1880 5.0703 4.8564 5.1603

Vo et al (TBT) [8] 5.1880 5.0703 4.8564 5.1603

10 Present 5.1848 5.1301 4.9326 5.1966

Vo et al (TBT) [8] 5.1848 5.1301 4.9326 5.1966

20 0 Present 2.8371 2.8371 2.8371 2.8371

Vo et al (TBT) [8] 2.8371 2.8371 2.8371 2.8371 0.5 Present 4.8595 4.6306 4.4169 4.7473

Vo et al (TBT) [8] 4.8579 4.6294 4.4160 4.7460

1 Present 5.2990 5.1160 4.8938 5.2216

Vo et al (TBT) [8] 5.2990 5.1160 4.8938 5.2217

2 Present 5.5239 5.4410 5.2445 5.5113

Vo et al (TBT) [8] 5.5239 5.4410 5.2445 5.5113

5 Present 5.5645 5.6242 5.4843 5.6382

Vo et al (TBT) [8] 5.5645 5.6242 5.4843 5.6382

10 Present 5.5302 5.6621 5.5575 5.6452

Vo et al (TBT) [8] 5.5302 5.6621 5.5575 5.6452

Furthermore, in order to investigate the effects of the power-law index and span-to-height ratio on the natural frequencies, different types of symmetric FG sandwich beams are considered Numerical results are tabulated in table 3 and plotted in Fig 3 In general, as the power-law index increases, the natural frequencies increase for sandwich beams with homogeneous softcore This is due to the fact that higher values of power-law index correspond to high portion of ceramic in comparison with the metal part for homogeneous softcore

Table 3 The nondimensional natural frequencies of Al/Al2O3 sandwich beams with

homogeneous softcore

L/h Mode Ratio of layer

thickness

p

5 1 1-0-1 2.6773 4.4443 4.8525 5.0945 5.1880 5.1848

1-1-1 2.6773 4.1850 4.5858 4.8740 5.0703 5.1301 1-2-1 2.6773 3.9930 4.3663 4.6459 4.8564 4.9326 2-1-2 2.6773 4.3059 4.7178 4.9970 5.1603 5.1966

2 1-0-1 9.2909 14.5621 15.9330 16.9348 17.5884 17.7648

1-1-1 9.2909 13.4460 14.5406 15.4382 16.2202 16.5445 1-2-1 9.2909 12.8355 13.7464 14.4701 15.1156 15.4047 2-1-2 9.2909 13.9147 15.1484 16.1356 16.9250 17.2150

3 1-0-1 17.7751 26.6188 29.1639 31.2837 32.9978 33.6204

1-1-1 17.7751 24.2347 25.9687 27.5509 29.1401 29.8897 1-2-1 17.7751 23.1416 24.4327 25.5391 26.6663 27.2351 2-1-2 17.7751 25.1824 27.2875 29.1783 30.9506 31.7167

20 1 1-0-1 2.8371 4.8595 5.2990 5.5239 5.5645 5.5302

1-1-1 2.8371 4.6306 5.1160 5.4410 5.6242 5.6621 1-2-1 2.8371 4.4169 4.8938 5.2445 5.4843 5.5575 2-1-2 2.8371 4.7473 5.2216 5.5113 5.6382 5.6452

2 1-0-1 11.2093 19.0614 20.7919 21.7104 21.9248 21.8172

1-1-1 11.2093 18.1125 19.9718 21.2371 21.9860 22.1607 1-2-1 11.2093 17.2778 19.0826 20.4145 21.3459 21.6432 2-1-2 11.2093 18.5851 20.4231 21.5740 22.1202 22.1774

3 1-0-1 24.7289 41.5970 45.3940 47.5166 48.1669 48.0232

1-1-1 24.7289 39.3631 43.2784 46.0103 47.7391 48.2030 1-2-1 24.7289 37.5526 41.2842 44.0553 46.0589 46.7381 2-1-2 24.7289 40.4406 44.3799 46.9385 48.2846 48.5068

8 CONCLUSIONS

This paper presented free vibration analysis of functionally graded sandwich beams using a higher-order shear deformation theory The face layers of sandwich beam are assumed to have isotropic metal-ceramic material distribution, and Young’s modulus and mass density are assumed to vary according to power-law form in terms of the volume fractions of the constituents The core layer is still homogeneous and made of an isotropic metallic material Governing equations of motion are derived from the Hamilton’s principle Navier-type solution for simply-supported beams is developed to solve the problem Numerical results are obtained for sandwich beams with homogeneous softcore to investigate effects of the power-law index, span-to-height ratio and thickness ratio of layers on the natural frequencies The present model is found to be appropriate and efficient in analyzing vibration of FG sandwich beams

Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology

Development (NAFOSTED) under Grant No 107.02-2012.07.

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