A new higher order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beam

A new higher-order sheardeformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams Abstract This paper presents a new higher-order shear de

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A new higher-order shear

deformation theory for

static, buckling and free

vibration analysis of

functionally graded

sandwich beams

Abstract

This paper presents a new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams In this theory, the axial displacement accounts for a third-order and inverse trigonometric distribution, and the transverse shear stress satisfies the traction-free boundary conditions on the top and bottom surfaces of the beams Governing equations of motion are derived from the Hamilton’s principle for sandwich beams with homogeneous hardcore and softcore Navier-type solution for simply-supported beams is developed to solve the problem Numerical results are obtained to investigate effects of the power-law index, span-to-height ratio and thickness ratio of layers on the displacements, stresses, critical buckling load and frequencies

Keywords

Functionally graded sandwich beams, static, buckling, vibration, beam theory

Introduction

Increase of the application of sandwich structures in aerospace, marine, civil con-struction led to the development of functionally graded (FG) sandwich structures

Journal of Sandwich Structures and Materials

0(00) 1–19

! The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1099636215589237

jsm.sagepub.com

1

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

2 Faculty of Civil Engineering, Thu Dau Mot University, Thu Dau Mot City, Binh Duong Province, Viet Nam

Corresponding author:

Trung-Kien Nguyen, 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: kiennt@hcmute.edu.vn

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to overcome the material discontinuity found in classical sandwich structures With the wide application of FG sandwich structures, understanding behaviors of FG sandwich beams becomes an important task

In general, the behavior of FG beams can be predicted using either three-dimen-sional (3D) elasticity theory or equivalent single-layer beam theories such as classical beam theory (CBT), first-order shear deformation beam theory (FSBT), third-order shear deformation beam theory (TSBT) and higher-order shear deformation beam theory (HSBT) Based on the 3D elasticity theory, Sankar [1] derived the exact solutions for bending analysis of FG beams subjected to trans-verse loads Zhong and Yu [2] also used 3D elasticity theory to predict the bending responses of cantilever FG beams under concentrated and uniformly distributed loads The bending responses of FG beams were investigated by Benatta et al [3], Ben-Oumrane et al [4] and Thai and Vo [5] using various equivalent single-layer beam theories Kapuria et al [6] presented a finite element model for static and free vibration responses of layered FG beams using third-order zigzag theory and validated against experiments for two different FGM systems under various boundary conditions Using a unified formulation, Giunta et al [7] presented sev-eral beam theories for the static analysis of FG beams Chakraborty et al [8] developed a new beam finite element based on the FSBT to study static, free vibration and wave propagation problems in bi-material beams fused with FGM layer Li [9] presented a new unified approach for analyzing the static and dynamic behaviors of FG beams with the rotary inertia and shear deformation included Li and Batra [10] derived analytical relations between the critical buckling load of a

FG Timoshenko beam and that of the corresponding homogeneous Euler-Bernoulli beam subjected to axial compressive load Kadoli et al [11] adopted the TSBT to develop a beam finite element to study the static behavior of FG beams under uniformly distributed loads Li et al [12] derived analytical solutions for static and dynamic analysis of FG beams using TSBT Based on the FSBT, Nguyen et al [13] recently proposed the static and free vibration analysis of axially loaded FG beams in which an improved transverse shear stiffness has been intro-duced It should be noted that the CBT is applicable to slender beams only For moderately deep beams, it underestimates deflection and overestimates buckling load and natural frequencies due to ignoring the shear deformation effect The FSBT accounts for the shear deformation effect, but requires a shear correction factor Alternatively, the HSBT considers the shear deformation effect without requiring any shear correction factors However the efficiency of the HSBT depends on the appropriate choice of displacement field which is an interesting subject that attracted many research (see [4, 11, 14–23]) Although there are many works on the FG beams, the studies on behaviors of FG sandwich beams are still limited Bhangale and Ganesan [24] studied vibration and buckling behaviors of an

FG sandwich beam having constrained viscoelastic layer in thermal environment

by using ﬁnite element formulation Amirani et al [25] used the element-free Galerkin method for free vibration analysis of sandwich beam with FG core Bui et al [26] investigated transient responses and natural frequencies of sandwich

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beams with inhomogeneous FG core using a truly mesh-free radial point interpol-ation method Vo et al [27] studied free vibrinterpol-ation and buckling behaviors of

FG sandwich beams by a ﬁnite element model using the TSBT

This paper aims to present the static, buckling and free vibration analysis of

FG sandwich beams using a new higher-order shear deformation theory In this theory, the transverse shear stress accounts for a new hyperbolic distribution and satisﬁes the traction-free boundary conditions on the top and bottom surfaces of the beams Governing equations of motion are derived from the Hamilton’s principle for two core types of sandwich beams Navier-type solution for simply-supported beams

is developed to solve the problem Numerical results are obtained for FG sandwich beams to investigate eﬀects of the power-law index, span-to-height ratio and thick-ness ratio of layers on the deﬂection, stresses, critical buckling load and frequencies

Kinematics

Consider an FG sandwich beam composed of three layers as shown in Figure 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 a mixture of isotropic materials with properties varying smoothly in the z-direction only The core layer is made of an isotropic homogeneous material The vertical pos-itions of the bottom and top, and of the two interfaces between the layers are denoted by h0¼ h2, h1, h2, h3¼h2, 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’’, ’’2-1-2’’ and so on

The displacement ﬁeld of the present theory is given by

uðx, z, tÞ ¼ u0ðx, tÞ zwb,xf ðzÞws,x ð1aÞ

wðx, z, tÞ ¼ wbðx, tÞ þ wsðx, tÞ ð1bÞ

where the comma indicates the diﬀerentiation with respect to the subscript that follows, u0 is the axial displacement, wb and ws are the bending and shear

Figure 1 Geometry of the sandwich beam with functionally graded faces

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components of transverse displacement along the mid-plane of the beam The function f(z) is given by

f ðzÞ ¼ z þ 8rz

3 3h2 ffiffiffiffiffiffiffiffiffiffiffiffiffi

r2þ4

p sinh1 rz

h

The non-zero strains are given by

xx¼0xxþzbxxþfsxx ð3aÞ

where gðzÞ ¼ 1 f,xis a shape function determining the distribution of the trans-verse shear strain and shear stress through the depth of the beam This function is chosen to satisfy the stress-free boundary conditions on the top and bottom surfaces of the beam 0xx, xz0, bxx and sxx are the axial strain, shear strain and curvatures in the beam, respectively, deﬁned as

0xx¼u0,x, bxx¼ wb,xx, sxx¼ ws,xx ð4aÞ

Variational formulation

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

ZT 0

where U, V and K denote the strain energy, work done and kinetic energy, respect-ively The variation of the strain energy can be stated as

U ¼

Z l 0

Zb=2

b=2

X3 n¼1

Zh n

h n1

ðxxðnÞxxþxzðnÞxzÞdz

dx dy

¼Rl

0ðNxx0xxþMbxxbxxþMsxxsxxþQxxz0Þdx

ð6Þ

where Nxx, Mb

xx, Ms

xxand Qxare the axial force, bending moments and shear force, respectively, deﬁned as

Nxx¼X3 n¼1

Z h n

h n1

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Mbxx¼X3 n¼1

Z h n

h n1

Msxx¼X3 n¼1

Zh n

h n1

Qx¼X3 n¼1

Zh n

h n1

The variation of the work done by in-plane and transverse load is given by

V ¼

Zl 0

N0xxðwb,xxþws,xxÞðwbþwsÞbdx

Zl 0 qðwbþwsÞbdx ð8Þ

The variation of the kinetic energy is obtained as

K ¼

Zl

0

Zb=2

b=2

X3 n¼1

Zh n

h n1

ðnÞðu_ u þ w _wÞdz

dx dy

¼

Zl

0

½u_0ðI0u_0I1w_b,xJ1w_s,xÞ þ _wbI0ðw_bþw_sÞ þ _wb,xðI1u_0þI2w_b,xþJ2w_s,xÞ

þ _wsI0ðw_bþw_sÞ þ _ws,xðJ1u_0þJ2w_b,xþK2w_s,xÞdx ð9Þ

where the differentiation with respect to the time t is denoted by dot-superscript convention; ðnÞis the mass density of each layer and I0, I1, I2, J1, J2and K2are the inertia coefficients defined by

ðI0, I1, I2, J1, J2, K2Þ ¼X3

n¼1

Zh n

h n1

ðnÞð1, z, z2, f, fz, f2Þbdz ð10Þ

By substituting equations (6), (8) and (9) into equation (5), and integrating by parts versus both space and time variables, and collecting the coeﬃcients of u0, wb and ws, the following equations of motion of the FG sandwich beam are obtained

u0: Nxx,x¼I0u€0I1w€b,xJ1w€s,x ð11aÞ

wb: Mbxx,xxþq þ N0xxðwb,xxþws,xxÞ ¼I0ðw€bþw€sÞ þI1u€0,xI2w€b,xxJ2w€s,xx

ð11bÞ

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ws: Msxx,xxþQx,xþq þ N0xxðwb,xxþws,xxÞ

¼I0ðw€bþw€sÞ þJ1u€0,xJ2w€b,xxK2w€s,xx ð11cÞ

Constitutive equations

The eﬀective material properties of FG sandwich beams according to the power-law form can be expressed by

Pð j ÞðzÞ ¼ ðPbPtÞVðbj ÞðzÞ þ Pt ð12Þ

where Ptand Pbare the Young’s moduli (E), Poisson’s ratio (), mass densities ()

of materials located at the top and bottom surfaces, and at the core, respectively The volume fraction function Vðbj Þdeﬁned by the power-law form as follows

Vð1Þb ðzÞ ¼ z h0

h1h0

for z 2 ½h0, h1

Vð2Þb ðzÞ ¼1 for z 2 ½h1, h2

Vð3Þb ðzÞ ¼ zh3

h 2 h 3

for z 2 ½h2, h3

8

>

ð13Þ

where p is a power-law index which is positive Distribution of material with Vb through the depth of (1-2-1) FG sandwich beam is displayed in Figure 2 The stress–strain relations for FG sandwich beams are given by

ðnÞxxðx, zÞ ¼ EðnÞðzÞxxðx, zÞ ð14aÞ

ðnÞxzðx, zÞ ¼ E

ðnÞðzÞ 2½1 þ ðnÞðzÞxzðx, zÞ ¼ G

ðnÞðzÞxzðx, zÞ ð14bÞ

The relation between the stress resultants and strains are obtained by using equations (7) and (14)

Nxx

Mb xx

Msxx

Qx

8

>

<

>

:

9

>

=

>

;

¼

Bs Ds Hs 0

2 6 6

3 7 7

0 xx

b xx

s xx

xz0

8

>

<

>

:

9

>

=

>

;

ð15Þ

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

ðA, B, D, Bs, Ds, HsÞ ¼X3

n¼1

Zh n

h n1

ð1, z, z2, f, zf, f2ÞEðnÞðzÞbdz ð16aÞ

As¼X3 n¼1

Zh n

h n1

By substituting equations (4) and (15) into equations (11), the explicit form of the governing equations of motion can be expressed as

Au0,xxBwb,xxxBsws,xxx ¼I0u€0I1w€b,xJ1w€s,x ð17aÞ

−0.5

−0.4

−0.3

−0.2

−0.1

0 0.1

0.2

0.3

0.4

0.5

Volume fraction function

p=0.5 p=1 p=2 p=5 p=10

Figure 2 Distribution of material with Vbthrough the depth of (1-2-1) FG sandwich beams with respect to the power-law index p

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Bu0,xxxDwb,xxxxDsws,xxxxþq þ N0xxðwb,xxþws,xxÞ

¼I1u€0,xþI0ðw€bþw€sÞ I2w€b,xxJ2w€s,xx

ð17bÞ

Bsu0,xxxDswb,xxxxHsws,xxxxþAsws,xxþq þ N0xxðwb,xxþws,xxÞ

¼J1u€0,xþI0ðw€bþw€sÞ J2w€b,xxK2w€s,xx

ð17cÞ

Equation (17) is the general form for the static, buckling and vibration analysis

of FG sandwich beams, and the dependent variables, u0, wb and ws are fully coupled

Table 1 Non-dimensional fundamental frequency ( !) of Al/Al2O3sandwich beams with homogeneous hardcore

L/h p Theory 1-0-1 2-1-2 2-1-1 1-1-1 2-2-1 1-2-1

5 0 Present 5.1528 5.1528 5.1528 5.1528 5.1528 5.1528

Vo et al [27] (TSBT) 5.1528 5.1528 5.1528 5.1528 5.1528 5.1528 0.5 Present 4.1254 4.2340 4.2943 4.3294 4.4045 4.4791

Vo et al [27] (TSBT) 4.1268 4.2351 4.2945 4.3303 4.4051 4.4798

1 Present 3.5735 3.7298 3.8206 3.8756 3.9911 4.1105

Vo et al [27] (TSBT) 3.5735 3.7298 3.8187 3.8755 3.9896 4.1105

2 Present 3.0680 3.2365 3.3546 3.4190 3.5718 3.7334

Vo et al [27] (TSBT) 3.0680 3.2365 3.3514 3.4190 3.5692 3.7334

5 Present 2.7448 2.8440 2.9789 3.0181 3.1965 3.3771

Vo et al [27] (TSBT) 2.7446 2.8439 2.9746 3.0181 3.1928 3.3771

10 Present 2.6934 2.7356 2.8715 2.8809 3.0629 3.2357

Vo et al [27] (TSBT) 2.6932 2.7355 2.8669 2.8808 3.0588 3.2356

20 0 Present 5.4603 5.4603 5.4603 5.4603 5.4603 5.4603

Vo et al [27] (TSBT) 4.3148 4.4290 4.4970 4.5324 4.6170 4.6979

1 Present 3.7147 3.8768 3.9775 4.0328 4.1603 4.2889

Vo et al [27] (TSBT) 3.7147 3.8768 3.9774 4.0328 4.1602 4.2889

2 Present 3.1764 3.3465 3.4756 3.5389 3.7051 3.8769

Vo et al [27] (TSBT) 3.1764 3.3465 3.4754 3.5389 3.7049 3.8769

5 Present 2.8440 2.9311 3.0776 3.1111 3.3030 3.4921

Vo et al [27] (TSBT) 2.8439 2.9310 3.0773 3.1111 3.3028 3.4921

10 Present 2.8042 2.8188 2.9665 2.9662 3.1616 3.3406

Vo et al [27] (TSBT) 2.8041 2.8188 2.9662 2.9662 3.1613 3.3406

TSBT: third-order shear deformation beam theory.

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Analytical solutions

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

u0ðx, tÞ ¼X1

m¼1

wbðx, tÞ ¼X1

m¼1

Table 2 Non-dimensional fundamental frequency ( !) of Al/Al2O3sandwich beams with homogeneous softcore

L/h p Theory 1-0-1 2-1-2 2-1-1 1-1-1 2-2-1 1-2-1

5 0 Present 2.6773 2.6773 2.6773 2.6773 2.6773 2.6773

Vo et al [27] (TSBT) 4.4427 4.3046 4.1960 4.1839 4.0504 3.9921

1 Present 4.8519 4.7168 4.5947 4.5848 4.4305 4.3656

Vo et al [27] (TSBT) 4.8525 4.7178 4.5916 4.5858 4.4270 4.3663

2 Present 5.0940 4.9958 4.8697 4.8725 4.7083 4.6447

Vo et al [27] (TSBT) 5.0945 4.9970 4.8668 4.8740 4.7047 4.6459

5 Present 5.1876 5.1592 5.0422 5.0687 4.9070 4.8547

Vo et al [27] (TSBT) 5.1880 5.1603 5.0399 5.0703 4.9038 4.8564

10 Present 5.1846 5.1957 5.0885 5.1286 4.9730 4.9307

Vo et al [27] (TSBT) 5.1848 5.1966 5.0866 5.1301 4.9700 4.9326

20 0 Present 2.8371 2.8371 2.8371 2.8371 2.8371 2.8371

Vo et al [27] (TSBT) 4.8579 4.7460 4.6050 4.6294 4.4611 4.4160

1 Present 5.2990 5.2216 5.0544 5.1159 4.9124 4.8937

Vo et al [27] (TSBT) 5.2990 5.2217 5.0541 5.1160 4.9121 4.8938

2 Present 5.5239 5.5111 5.3392 5.4409 5.2245 5.2444

Vo et al [27] (TSBT) 5.5239 5.5113 5.3390 5.4410 5.2242 5.2445

5 Present 5.5645 5.6381 5.4836 5.6241 5.4169 5.4841

Vo et al [27] (TSBT) 5.5645 5.6382 5.4834 5.6242 5.4166 5.4843

10 Present 5.5302 5.6451 5.5074 5.6620 5.4670 5.5573

Vo et al [27] (TSBT) 5.5302 5.6452 5.5073 5.6621 5.4667 5.5575

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wsðx, tÞ ¼X1

m¼1

where x is the natural frequency, ﬃﬃ

i

p

¼ transverse load q(x) is also expressed as

qðxÞ ¼X1 m¼1

Table 3 Non-dimensional critical buckling load ( Ncr) of Al/Al2O3sandwich beams with homogeneous hardcore

L/h p Theory 1-0-1 2-1-2 2-1-1 1-1-1 2-2-1 1-2-1

5 0 Present 48.5960 48.5960 48.5960 48.5960 48.5960 48.5960

Vo et al [27] (TSBT) 27.8574 30.0301 31.0728 31.8784 33.2536 34.7653

1 Present 19.6531 22.2113 23.5246 24.5598 26.3609 28.4444

Vo et al [27] (TSBT) 19.6525 22.2108 23.5246 24.5596 26.3611 28.4447

2 Present 13.5808 15.9158 17.3248 18.3591 20.3748 22.7862

Vo et al [27] (TSBT) 13.5801 15.9152 17.3249 18.3587 20.3750 22.7863

5 Present 10.1473 11.6685 13.0272 13.7218 15.7307 18.0914

Vo et al [27] (TSBT) 10.1460 11.6676 13.0270 13.7212 15.7307 18.0914

10 Present 9.4526 10.5356 11.8372 12.2611 14.1995 16.3787

Vo et al [27] (TSBT) 9.4515 10.5348 11.8370 12.2605 14.1995 16.3783

20 0 Present 53.2364 53.2364 53.2364 53.2364 53.2364 53.2364

Vo et al [27] (TSBT) 29.7175 32.2629 33.2376 34.0862 35.6405 37.3159

1 Present 20.7213 23.4212 24.8793 25.9588 27.9537 30.2307

Vo et al [27] (TSBT) 20.7212 23.4211 24.8796 25.9588 27.9540 30.2307

2 Present 14.1973 16.6050 18.1400 19.1999 21.3923 23.9899

Vo et al [27] (TSBT) 14.1973 16.6050 18.1404 19.3116 21.3927 23.9900

5 Present 10.6175 12.0885 13.5519 14.2285 16.3829 18.8874

Vo et al [27] (TSBT) 10.6171 12.0883 13.5523 14.2284 16.3834 18.8874

10 Present 9.9849 10.9074 12.3080 12.6819 14.7520 17.0445

Vo et al [27] (TSBT) 9.9847 10.9075 12.3084 12.6819 14.7525 17.0443

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