Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method

Accepted Manuscript Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method Y. Yang, C.C. Lam, K.P. Kou, V.P. Iu PII: DOI: Reference:

S0263-8223(14)00285-2 http://dx.doi.org/10.1016/j.compstruct.2014.06.016 COST 5749

To appear in:

Composite Structures

Please cite this article as: Yang, Y., Lam, C.C., Kou, K.P., Iu, V.P., Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method, Composite Structures (2014), doi: http:// dx.doi.org/10.1016/j.compstruct.2014.06.016

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Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method Y. Yang1*, C.C. Lam1, K.P. Kou1, V.P. Iu1 1

Department of Civil Engineering, Faculty of Science and Technology, University of Macau, Macau, China *Corresponding author. E-mail addresses: [email protected]

Abstract Free vibration of the functionally graded (FG) sandwich beams are studied by a meshfree boundary-domain integral equation method. Two sandwich beams, namely, FG core with homogeneous face sheets and homogeneous core with FG face sheets are considered in the study. Based on the two-dimensional elasticity theory, the boundary-domain integral equations are derived by applying the elastostatic fundamental solutions. Radial integration method is employed to transform the domain integrals related to material nonhomogeneity and inertia effect into boundary integrals, hence a meshfree scheme involving boundary integrals only is achieved. Each layer in the sandwich beam is modelled separately and the equilibrium and compatibility conditions are enforced at the interfaces to derive the equations for the whole sandwich beam. Extensive parametric study is presented to examine the influence of material composition, material gradient, layer thickness proportion, thickness to length ratio and boundary conditions on the free vibration of FG sandwich beams. Keywords: FG sandwich beams, free vibration, boundary-domain integral equations, meshfree method, boundary element method 1. Introduction Sandwich construction has being used extensively in a variety of engineering fields including aerospace, nuclear, civil and mechanical engineering for its high bending stiffness with low specific weight. Availability of a wide selection of face sheet and core materials makes it possible to obtain multifunctional benefits. For example, use of a viscoelastic layer to a structural sandwich to control vibration and noise propagation. Double-sided coatings for cutting tools help reducing friction and hence extending the edge life for the high-speed, dry and micro machining. Sandwich construction can produce structural elements with high strength and stiffness-to-weight ratio, low weight, good corrosion resistance; good fatigue resistance and functional benefit. These attractive properties have led to a great 1

amount of research efforts in studying sandwich structures. Analysis of sandwich structures is well documented in the books of Plantema [1] and Allen [2]. However, sandwich structures have not been fully exploited in structural applications due to the damage tolerance concerns. Such as core/face sheet delamination, face sheet damage and shear cracking of the core, as a consequence of the thermal expansion mismatch and stress concentration resulted from stiffness discontinuity between layers. It represents a critical design concern since it can susceptible induce failures. As the improvements to the abovementioned deleterious effect, functionally graded materials (FGMs) are believed to alleviate this weak point. The reason is that FGMs are characterized by smooth and continuous variation in the properties from one surface to another, and also can achieve the desirable functionalities. Although a lot of new technologies of researching the FG sandwich structures are at their infancy, their potential advantages appear great. Venkataraman and Sankar [3] analyzed the elasticity solution for stresses in a sandwich beam with functionally graded core, in which materials were assumed with an exponential variation of the elastic stiffness coefficients across the thickness. The results demonstrated the significant reduction in shear stresses at the face-sheet-core interface achieved by functionally grading the core properties. Deflections and stresses of simply supported FGM sandwich plates under sinusoidal loading were investigated by Zenkour [4] according to different plate theories. A power-law distribution in terms of the volume fraction of the constituents was used to simulate the FG face sheet. He concluded that the gradients in material properties play an important role in determining the response of the FGM plates. Neves et al. [5] also did the static analysis of the power-law FG sandwich plates according to a hyperbolic theory considering Zig-Zag and warping effects. The results leaded to the conclusion that the deflection of the simply supported sandwich plate with FG core or FG skins increases as the power-law exponent of the material increases. An exact thermoelasticity solution for a two-dimensional thick FGM sandwich structures were presented by Shodja et al. [6], they also indicated that a graded core could effectively reduce the face sheet/core interfacial shear stress. Apetre et al. [7] addressed the low-speed impact response of sandwich beams with FG core. The results revealed that FG cores could be used effectively to mitigate or completely prevent impact damage in sandwich composites. Model sandwich structures comprising of graded core with bilinear variation of volume fraction of hollow microballoons were considered for experimental and numerical simulations by Kirugulige et al. [8]. They showed that a decrease of stress intensity factors was obtained by using a FGM core. Etemadi et al. [9] conducted three-dimensional finite element simulations for analyzing low velocity 2

impact behavior of sandwich beams with a FG core, the results revealed that the maximum contact force increased and the maximum strain decreased compared to those of sandwich beams with a homogeneous core. Buckling analysis of the power-law sandwich plates with FG skins using a new quasi-3D hyperbolic sine shear deformation theory and collocation with radial basis functions was presented by Neves et al. [10]. They concluded that the critical buckling loads decrease as the power-law exponent increases. As the core to plate total thickness ratio increases, the buckling loads increases as well. Vibration analysis always plays an important role to exploit the mechanics world of the novel constructions. Chehel Amirani et al. [11] analyzed the free vibration of sandwich beam with FG core by using the element free Galerkin method. The Mori-Tanaka method was used as the mircomechanics technique to determine the effective properties of FG core. Zenkour [12] studied the free vibration of the simply supported FG sandwich plate by the sinusoidal shear deformation plate theory. Li et al. [13] presented three-dimensional vibration analytical solutions for multi-layer FGM sandwich plates based on Ritz method in conjunction with Chebhyshev polynomial series. Neves et al. conducted the free vibration of the FG plates and FG sandwich plates by a thickness-stretching sinusoidal shear deformation theory [14], a new hyperbolic sine shear deformation theory [15], and a thickness-stretching higher-order shear deformation theory [16]. All of them assumed that the properties of the FGMs follow the power law volume fraction function. Furthermore, the three-dimensional models may be computationally involved and expensive. Hence, there is a growing appreciation of the importance of applying two-dimensional theories for the evolution of the accurate structural analysis. More effective numerical solution scheme is also necessary for predicting the vibration behaviors of FG sandwich structures, especially for the free vibration behaviors of the exponentially FG sandwich beams which are rarely studied in the literature. In this paper, a meshfree boundary-domain integral equation is presented for studying free vibration of sandwich beams composed of FGM. Based on the two-dimensional elasticity theory, the boundary-domain integral equations for each layer of the FG sandwich beams are derived initially by using the elastostatic fundamental solutions. The resulting domain integrals are due to the material nonhomogeneity and the inertial effect. By applying the radial integration method, these domain integrals are transformed into boundary integrals which can be evaluated by a meshfree scheme. Then, enforcing the equilibrium and compatibility conditions at the interfaces in the sandwich beam, an eigenvalue system involving the system matrices with only boundary integrals for free vibration of the sandwich beams are obtained. Numerical 3

study is carried out to evaluate the effect of various important parameters on the free vibration of the FG sandwich beams. 2. FG Sandwich Beam In this study, two types of the FG sandwich beams are studied, namely, a non-symmetrical FG core with homogeneous face sheets and a homogeneous core with FG face sheets as depicted in Fig. 1. By assuming that the layers are perfectly bonded to each other. The total length and height of these sandwich beams are denoted by L and ht. In the first type of sandwich beam, the top and bottom homogeneous face sheets are pure ceramic and pure steel, respectively, while the FG core consists of material with continuous variation of steel from the bottom face to ceramic on the top face. For the other type of beam, the core is ceramic with face sheet properties varying inwardly from steel to ceramic. The material parameters of the ceramic and the steel are described in Table 1. For the FG layer, the Young’s modulus and the mass density are varying gradually in the transverse direction according to an exponential function described in Eqs (1) and (2), while the Poisson’s ratio is a constant. This functional form of property variation has been recognized convenient in solving elasticity problems by many researchers [17, 18, 19]. E 1 E ( x2 ) = Eb e β x2 ， where β = ln( t ) ， (1) Tm Eb

ρ ( x2 ) = ρb eγ x ， 2

where

γ=

ρ 1 ln( t ) ， Tm ρb

(2)

where Et, ρt are the Young’s modulus and mass density for the top face constituent of the FG layer, and Eb, ρb are for the bottom face constituent. β and γ represent the FG gradation parameters for the Young’s modulus and mass density respectively. x2 denotes the Cartesian coordinates variable in the transverse direction and Tm is the thickness of the FG layer. The through thickness variation of the Young’s modulus for these two type of sandwich beams are shown in Fig. 2. 3. Problem formulation Based on the two-dimensional elasticity theory, the governing differential equations of the undamped steady-state elastodynamics for each single layer of the considered FG sandwich beams can be expressed as [17]

σ ij , j ( x ) + ω 2 ρ ( x )ui ( x ) = 0 ,

(3)

where σij, ρ and ui is the stress tensor, mass density and displacements and ω is the vibration frequency. A comma after a quantity represents spatial derivatives and repeated indexes denote summation. 4

The elasticity tensor cijkl for nonhomogeneous isotropic material is described in the form of 0 cijkl ( x ) = μ ( x )cijkl ,

0 = cijkl

where

2v δ ijδ kl + δ ik δ jl + δ ilδ jk , 1 − 2v

(4a, b)

0 where cijkl is the elastic tensor of a “fictitious” reference homogeneous material with

μ=1. The shear modulus μ(x) is related to E(x) as μ ( x ) = E ( x ) / 2(1 + v) . μ(x) varies continuously with the coordinates for the FG layers , while it keeps a constant for the homogeneous layer. δij is the Kronecker delta. By taking the elastostatic displacement fundamental solutions Uij(x, y) as the weight function, the weak-form of the equilibrium Eq. (3) can be obtained as

∫

Ω

[σ jk ,k + ρω 2u j ] ⋅ U ij d Ω = 0 .

(5)

Substitution of the generalized Hooke’s law

0 σ ij = cijkl uk ,l = μ ( x )cijkl u k ,l

and

application of the Gauss’s divergence theorem, the boundary-domain integral equations yield as ui ( y ) = ∫ U ij ( x , y )t j ( x )d Γ − ∫ Tij ( x , y )u j ( x )d Γ + ∫ Vij ( x , y )u j ( x )d Ω Γ

Γ

Ω

ρ ( x) U ij ( x, y )u j ( x )d Ω Ω μ ( x)

+ω 2 ∫

,

(6)

where ti = σ ij n j is the traction vector and nj is the components of the outward unit normal to the boundary Γ of the considered layer domain Ω. u~i is the normalized displacement vector relating with the normalized shear modulus μ~ , which are

defined by [19] ui ( x ) = μ ( x )ui ( x ) ，

μ ( x ) = ln[ μ ( x )] .

(7a, b)

The fundamental solutions arising in equation (6) are presented in the following, where Uij(x, y) and Tij(x, y) are chosen as the elastostatic displacement fundamental solutions for homogeneous, isotropic and linear elastic solids with µ=1 [20]. −1 [(3 − 4v )δ ij ln(r ) − r,i r, j ] , U ij = 8π (1 − v) 0 Σijl = crsjl U ir , s =

Tij = Σijl nl =

−1 [(1 − 2v)(δ il r, j + δ ij r,l − δ jl r,i ) + 2r,i r, j r,l ] , 4π (1 − v)r

−1 [(1 − 2v)(ni r, j − n j r,i ) + ((1 − 2v)δ ij + 2r,i r, j )r,l nl ] , 4π (1 − v)r 5

(8) (9) (10)

Vij = Σijl μ ,l =

−1 [(1 − 2v)( μ ,i r, j − μ , j r,i ) + ((1 − 2v)δ ij + 2r,i r, j )r,l μ ,l ] , 4π (1 − v)r

(11)

where r=|x-y| is the distance from the field point x to the source point y. Boundary-domain integral equations for boundary points can be obtained by letting y to the boundary Γ in Eq. (6). It can be readily observed that, the two domain integrals in Eq. (6) are related to the material nonhomogeneity and the inertial effect. For homogeneous layer the shear modulus and density are constant, the boundary-domain integral equation is simplified with only one domain integral as ui ( y ) = ∫ U ij ( x , y )t j ( x )d Γ − ∫ Tij ( x , y )u j ( x )d Γ + ω 2 Γ

Γ

ρh μh

∫

Ω

U ij ( x , y )u j ( x )d Ω .

(12)

For the FG layer with elastic properties following an exponential law, μ ,l in Eq. (7b) is constant and Vij(x,y) thus becomes very simple for integration. The two domain integrals are further transformed to boundary integrals by the radial integration method (RIM) proposed by Gao [21]. In the RIM, u~i in the domain integrals are approximated by a combination of the radial basis function and the polynomials of the global coordinates as ui ( x ) = ∑ α iAφ A ( R ) + aik xk + ai0 , A

where

∑α

A i

= 0,

A

∑α

A i

x Aj = 0 .

(13a, b, c)

A

R 2 R 3 R 4 ⎧ 0 ≤ R ≤ dA, ⎪1 - 6( ) + 8( ) - 3( ) ， dA dA dA . φ ( R) = ⎨ ⎪0, R ≥ d , A ⎩ A

(13d)

The 4th order spline-type radial basis function is chosen for φ A (R) [17], where the application point A (Nt=Nb+Ni) is constituted by all the boundary nodes (Nb) and internal nodes (Ni) as depicted in Fig 3. By substituting the coordinates of the field pointes (xk) and the application point A ( x Aj ) into Eqs. (13), if with on two coincide nodes, the unknown coefficient vectors α can be calculated by a set of linear algebraic equations as u = φ ⋅ α ,

and

α = φ −1 ⋅ u .

(14a, b)

After determining the coefficients α iA , aik and ai0 , substitute Eq. (13a) into the domain integrals and apply the RIM, the two domain integrals of Eq. (6) are transformed into the boundary integrals in the form of [17] r,k ∂r 12 1 ∂r A 2 1 ∂r 02 A k k 0 ∫Ω Vij u j d Ω = α j ∫Γ r ∂n Fij d Γ + a j ∫Γ r ∂n Fij d Γ + (a j yk + a j )∫Γ r ∂n Fij d Γ , (15) 6

ω2 ∫

r ∂r 12 ρ 1 ∂r A 2 ρ U ij u j d Ω = ω 2 b e(γ − β ) y [α jA ∫ Pij d Γ + a kj ∫ ,k Pij d Γ Γ Γ μ μb r ∂n r ∂n 2

Ω

1 ∂r 02 + ( a yk + a ) ∫ P d Γ] Γ r ∂n ij k j

,

(16)

0 j

where the relation xi=yi+r,ir is used to relate x with r. By rewriting Eq. (11) with Vij = Vij r , the integral functions in Eqs. (15) and (16) can be expressed as r

r

r

0

0

0

FijA 2 = ∫ rVijφ A dr = Vij ∫ φ A dr , Fij12 = ∫ r 2Vij dr = r

PijA 2 = ∫ rU ijφ Ae

( γ − β ) r,2 r

0

r

dr , Pij12 = ∫ r 2U ij e

( γ − β ) r,2 r

0

r 1 2 r Vij , Fij02 = ∫ rVij dr = rVij ,(17a,b,c) 0 2 r

dr , Pij02 = ∫ rU ij e 0

( γ − β ) r,2 r

dr .

(18a,b,c)

Since r,i in the above radial integrals is constant, then Eqs. (17b, c) can be evaluated analytically and other integrals are calculated by standard Gaussian quadrature formula [17]. Therefore the displacement boundary integral equations with only boundary integrals are obtained as

r ∂r 12 1 ∂r A 2 Fij d Γ + a kj ∫ ,k Fij d Γ Γ r ∂n r ∂n r ∂r 12 ρ 1 ∂r 02 1 ∂r A 2 +(a kj yk + a 0j ) ∫ Fij d Γ] + ω 2 0 e(γ − β ) y2 [α jA ∫ Pij d Γ + a kj ∫ ,k P d Γ . (19) Γ r ∂n Γ r ∂n Γ r ∂n ij μ0

cij u j = ∫ U ij t j d Γ − ∫ Tij u j d Γ + [α jA ∫ Γ

Γ

+(a kj yk + a 0j ) ∫

Γ

Γ

1 ∂r 02 Pij d Γ] r ∂n

Discretizing of each layer boundary into quadratic boundary elements with Nb boundary nodes and collocating the resulting boundary integral equations at the Nt boundary and internal nodes, two sets of discretized boundary integral equations are obtained, in the matrix forms as follows, for the boundary nodes

([ H

b

⎧u ⎫ 0] − [Vb ])2 N ×2 N ⎨ b ⎬ − ω 2 [ Pb ]2 N ×2 N {u}2 N ×1 = [Gb ]2 N ×2 N {tb }2 N ×1 , (20a) t b b t b b b t  u ⎩ i ⎭2 Nt ×1

for the internal nodes

([ H

i

⎧u ⎫ I ] − [Vi ])2 N ×2 N ⎨ b ⎬ − ω 2 [ Pi ]2 N ×2 N {u}2 N ×1 = [Gi ]2 N ×2 N {tb }2 N ×1 , t b i t i b i t  u ⎩ i ⎭2 Nt ×1

(20b)

where I is the identity matrix with the size of 2Ni x 2Ni. { ub }, { ui } and { u } are the displacement vectors of the boundary nodes, internal nodes and applications points respectively. By considering the boundary conditions, the sub-columns of the coefficient matrices [Hb], [Hi] respect to the known displacement nodes should be interchanged with that of [Gb], [Gi] respected to the tractions, so do the displacement 7

vectors { ub } and the traction vectors {tb}. Meanwhile, it is noticed that the sub-columns of the matrices [Vb], [Pb] and [Vi], [Pi] corresponding to the known boundary displacement nodes should be taken as zero [17]. Then Eqs. (20) lead to the following system of linear algebraic equations

⎧x ⎫

[ Ab ] ⎨ ub ⎬ − ω 2 [ Pb ]{u} = [ Bb ]{ yb } ⎩

i

for boundary nodes,

⎭

⎧x ⎫

[ Ai ] ⎨ ub ⎬ − ω 2 [ Pi ]{u} = [ Bi ]{ yb } ⎩

i

(21a)

for internal nodes.

⎭

(21b)

where { yb } is the traction vector for the boundary nodes. Pre-multiplying [Bb]-1, Eqs. (21) become

⎧x ⎫ ⎡⎣ Ab ⎤⎦ ⎨ b ⎬ − ω 2 ⎡⎣ Pb ⎤⎦ {u} = { yb } ⎩ ui ⎭

for boundary nodes,

(22a)

⎧x ⎫ ⎡⎣ Ai ⎤⎦ ⎨ b ⎬ − ω 2 ⎡⎣ Pi ⎤⎦ {u} = 0 ⎩ ui ⎭

for internal nodes,

(22b)

where

⎡⎣ Ab ⎤⎦ = [ Bb ]

−1

[ Ab ] ,

⎡⎣ Pb ⎤⎦ = [ Bb ]

−1

⎡⎣ Ai ⎤⎦ = [ Ai ] − [ Bi ][ Bb ]

−1

[ Ab ] ,

[ Pb ] ,

(23a, b)

⎡⎣ Pi ⎤⎦ = [ Pi ] − [ Bi ][ Bb ]

−1

[ Pb ] .

(23c, d)

The nodes for each layer in the sandwich beam are classified into three groups as shown in Fig 4. Namely, the ‘s’ group for the boundary nodes solely associated with a single layer, ‘c’ group for the remaining boundary nodes on the interfaces between the layers and ‘i’ group for the internal nodes. Then Eqs. (22) can be rewritten in terms of these three groups of nodes as

⎡ Abss ⎢ ⎢ Abcs ⎢A ⎣ is

Absc Abcc Aic

Absi ⎤ ⎧ xbs ⎫ ⎡ Pbss Pbsc ⎥⎪ ⎪ 2 ⎢ Abci ⎥ ⎨ ubc ⎬ − ω ⎢ Pbcs Pbcc ⎪ ⎪ ⎢P Aii ⎥⎦ ⎩ u i ⎭ ⎣ is Pic

Pbsi ⎤ ⎧ xbs ⎫ ⎧ ybs ⎫ ⎥⎪ ⎪ ⎪ ⎪ Pbci ⎥ ⎨ ubc ⎬ = ⎨ tbc ⎬ . Pii ⎥⎦ ⎩⎪ ui ⎭⎪ ⎩⎪ 0 ⎭⎪

(24)

In Eq. (24), xbs contains the unknown normalized displacements and the unknown traction vectors for the ’s’ group nodes, while ybs contains all the corresponding known vectors. In the free vibration analysis, only the homogeneous system of the linear algebraic equations is needed, which can be obtained by taking the vectors ybs containing the known normalized boundary displacements as well as the known 8

boundary traction to be zero. However, the normalized displacements u bc and traction tbc for the ‘c’ group nodes are all unknown. In order to construct the total stiffness matrix and mass matrix for the whole FG sandwich beams, the isolated system linear algebraic equations for each single layer will be assembled together by employing the equilibrium and compatibility conditions for the displacements and tractions of the nodes which are shared by the interfaces. The divided boundaries of the whole FG sandwich beams are described in Fig. 5. The boundary of the top and bottom face sheet is discretized into two sub-boundaries Γ1, Γ2 and Γ7, Γ8. While, the edges of core layer are departed on their own, those are Γ3,

Γ4, Γ5 and Γ6. Among them Γ2 and Γ3, Γ5 and Γ7 are two common interface pairs and they are composed by the same common nodes but in an opposite direction. Let u m and tm denote the nodal displacements and the traction vectors on boundary Γm respectively, the boundary integral equations of the three layers expressed in the matrix form as

⎡ Ab11 ⎢ ⎢ Ab 21 ⎢A ⎣ i11

Ab12 Ab 22 Ai12

⎡ Ab 33 Ab 34 ⎢ ⎢ Ab 43 Ab 44 ⎢A Ab 54 ⎢ b 53 ⎢ Ab 63 Ab 64 ⎢ ⎢⎣ Ai 23 Ai 24

Ab1i ⎤ ⎧ xb1 ⎫ ⎡ Pb11 Pb12 Pb1i ⎤ ⎧ xb1 ⎫ ⎧ 0 ⎫ ⎥⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ 2 ⎢ Ab 2i ⎥ ⎨ub 2 ⎬ − ω ⎢ Pb 21 Pb 22 Pb 2i ⎥ ⎨ub 2 ⎬ = ⎨tb 2 ⎬ forΩ1, m=1, 2, (25a) ⎪ ⎪ ⎢P ⎥⎪  ⎪ ⎪ ⎪ Ai1 ⎥⎦ ⎩ ui1 ⎭ ⎣ i11 Pi12 Pi1 ⎦ ⎩ ui1 ⎭ ⎩ 0 ⎭ Ab 35

Ab 36

Ab 45

Ab 46

Ab 55

Ab 56

Ab 65

Ab 66

Ai 25

Ai 26

Ab 3i ⎤ ⎧ u ⎫ ⎡ Pb 33 Pb 34 ⎥ ⎪ b3 ⎪ ⎢ Ab 4i ⎥ xb 4 ⎢ Pb 43 Pb 44 ⎪⎪ ⎪⎪ 2 Ab 5i ⎥ ⎨ub 5 ⎬ − ω ⎢ Pb 53 Pb 54 ⎥ ⎢ ⎪ ⎪ ⎢ Pb 63 Pb 64 Ab 6i ⎥ ⎪ xb 6 ⎪ ⎥ ⎢ Ai 2 ⎥⎦ ⎪⎩ui 2 ⎪⎭ ⎢⎣ Pi 23 Pi 24

Pb 35

Pb 36

Pb 45

Pb 46

Pb 55

Pb 56

Pb 65

Pb 66

Pi 25

Pi 26

Pb 3i ⎤ ⎧ u b 3 ⎫ ⎧t ⎫ b2 ⎥ Pb 4i ⎥ ⎪ xb 4 ⎪ ⎪ 0 ⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ Pb 5i ⎥ ⎨ub 5 ⎬ = ⎨ tb 5 ⎬ ⎥ Pb 6i ⎥ ⎪⎪ xb 6 ⎪⎪ ⎪⎪0 ⎪⎪ ⎥ Pi 2 ⎥⎦ ⎪⎩ui 2 ⎪⎭ ⎪⎩0 ⎪⎭

for Ω2, m=3, 4, 5, 6,

⎡ Ab 77 ⎢ ⎢ Ab87 ⎢A ⎣ i 37

Ab 78 Ab88 Ai 38

Ab 7 i ⎤ ⎧ xb 7 ⎫ ⎡ Pb 77 ⎥⎪ ⎪ 2 ⎢ Ab8i ⎥ ⎨ ub8 ⎬ − ω ⎢ Pb87 ⎪ ⎪ ⎢P Ai 3 ⎥⎦ ⎩ u i 3 ⎭ ⎣ i 37

(25b)

Pb 7 i ⎤ ⎧ xb 7 ⎫ ⎧tb 7 ⎫ ⎥⎪ ⎪ ⎪ ⎪ Pb8i ⎥ ⎨ ub8 ⎬ = ⎨ 0 ⎬ forΩ3, m=7, 8. (25c) Pi 3 ⎥⎦ ⎪⎩ u i 3 ⎪⎭ ⎪⎩ 0 ⎪⎭

Pb 78 Pb88 Pi 38

Considering the equilibrium and compatibility conditions at the two interfaces,

tb 2 = − tb 3 , tb 5 = − tb 7 ,

ub 2 = u b 3 , ub 5 = u b 7 ,

between Ω1 and Ω2, between Ω2 and Ω3.

(26a) (26b)

Finally, Eqs. (25) yields a generalized eigenvalue system with the eigenvector containing the unknowns at the boundary nodes and internal nodes of the FG 9

sandwich beam,

[ K ]{X } = ω 2 [ M ]{X } ,

(27)

where

⎡ Ab11 ⎢ ⎢ Ab 21 ⎢0 ⎢ ⎢0 ⎢ [K ] = ⎢ 0 ⎢ ⎢0 ⎢A ⎢ i11 ⎢0 ⎢ ⎢⎣ 0 ⎡ Pb11 ⎢ ⎢ Pb 21 ⎢0 ⎢ ⎢0 ⎢ [M ] = ⎢ 0 ⎢0 ⎢ ⎢ Pi11 ⎢ ⎢0 ⎢ ⎣0

{X } = {xb1

Ab12

0

0

0

0

Ab1i

0

Ab 22 + Ab 33

Ab 34

Ab 35

Ab 36

0

Ab 2i

Ab 3i

Ab 43

Ab 44

Ab 45

Ab 46

0

0

Ab 4i

Ab 53

Ab 54

Ab 55 + Ab 77

Ab 56

Ab 78

0

Ab 5i

Ab 63

Ab 64

Ab 65

Ab 66

0

0

Ab 6i

0

0

Ab87

0

Ab88

0

0

Ai12

0

0

0

0

Ai1

0

Ai 23

Ai 24

Ai 25

Ai 26

0

0

Ai 2

0

0

Ai 37

0

Ai 38

0

0

Pb12

0

0

0

0

Pb1i

0

Pb 22 + Pb 33

Pb 34

Pb 35

Pb 36

0

Pb 2i

Pb 3i

Pb 43

Pb 44

Pb 45

Pb 46

0

0

Pb 4i

Pb 53

Pb 54

Pb 55 + Pb 77

Pb 56

Pb 78

0

Pb 5i

Pb 63

Pb 64

Pb 65

Pb 66

0

0

Pb 6i

0

0

Pb87

0

Pb88

0

0

Pi12

0

0

0

0

Pi1

0

Pi 23

Pi 24

Pi 25

Pi 26

0

0

Pi 2

0

0

Pi 37

0

Pi 38

0

0

ui 2

ui 3 } .

ub 2

xb 4

ub 5

xb 6

ub8

ui1

T

⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ Ab 7 i ⎥ ⎥ 0 ⎥, ⎥ Ab8i ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ Ai 3 ⎥⎦

(28a)

0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ Pb 7 i ⎥ ⎥ 0 ⎥, Pb8i ⎥⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ Pi 3 ⎦

(28b)

0

(28c)

The natural frequencies and vibration modes of FG sandwich beams can be determined by resolving this general eigenvalue equation.

4. Numerical analysis and discussion Three-noded quadratic boundary elements are applied in the analysis and plane-strain condition is considered. Three basic support conditions of simply supported (S), fixed (C) and free (F) ends [17] are imposed and shown in Fig. 6. The natural frequency is normalized by

ϖ = ω ht ρ Al E Al .

(29)

10

To verify the accuracy and convergence of the present method, the first type FG sandwich beam composed by the pure ceramic face-sheet, ceramic-steel FG core, and the pure steel face-sheet from the bottom to the top are considered in the analysis. For the sake of convenience, this considered kind of the FG sandwich beam is defined by CFS sandwich beam. Assume that the two face-sheet have the same thickness which are denote by hf, and the thickness of the core layer is presented by hc. The CFS sandwich beams with five thickness proportions (hf/hc=0.1, 0.2, 0.3, 0.4 and 0.5) are analyzed by the present meshfree boundary-domain integral equation method and the results are compared with which calculated by the software of the finite element method (FEM). First ten normalized natural frequencies of the fixed-fixed CFS sandwich beams with aspect ratio L/ht=5 are listed in Table 2. It can be seen that, compared to the results obtained from the FEM with tens of thousands quadratic elements, the results evaluated by the present method with fewer boundary and internal nodes provided higher precision degree, including the high frequencies. A comprehensive parametric study is considered to investigate the free vibration of the FG sandwich beams. For the homogeneous face-sheet with non-symmetrical FG core sandwich beam, the abovementioned CFS sandwich beam represents the kind of which with a softer base face-sheet, while a harder top face-sheet. Another property is defined by SFC sandwich beam which is made up in a reversed composition orders as CFS. Similarly, there are two different layering for the FG face-sheet homogeneous core sandwich beam, namely, SCCS sandwich beam represents a soft core composed by ceramic and CSSC is the sandwich beam with a harder core of steel, respectively. For all of these four kinds of FG sandwich beams, five layer thickness proportion (hf/hc=0.1, 0.2, 0.3, 0.4, and 0.5), four important boundary conditions, namely, the cantilever beam (C-F), fixed beam (C-C), fixed-simply supported beam (C-S) and the simply supported beam (S-S) as well as three beam aspect ratios (L/ht=5, 10 and 20) are applied to investigate the dynamic response of the FG sandwich beams. Fig. 7 shows the effect of layer thickness ratio on the frequency for these four types of sandwich beams with various boundary conditions. SCCS sandwich beams give higher frequencies while CSSC beams show lower frequencies. This is obvious as SCCS beams are softer than CSSC beams. For C-S and S-S support conditions, SFC sandwich beams exhibit much higher frequencies than CFS while for C-C and C-F support condition, both give almost the same frequencies. It can be observed that the frequency increases with layer thickness ratio for SCCS beams and decrease for the other three types. Based on this result, it can be concluded that, the natural frequency of the FG sandwich beams may be controlled by the stiffness of the whole structure. Stiffening the FG sandwich beam could reduce the structural natural frequency. 11

First four vibration modes of these four kinds of FG sandwich beams with hf/hc=0.2, and L/ht=5 are plotted in Fig. 8 to Fig. 11 for different support conditions. For each kind of FG sandwich beam, the transverse vibration displacements at the first, second interface and the central line are presented respectively. Similar displacements at the three layer are observed for CSSC and SCCS beams, while CFS and SFC show variation of displacements which indicate thickness stretching during vibration. Thickness stretching phenomenon cannot be observed in vibration analysis by using the classical and higher order beam theories as they assume constant transverse displacement in the thickness direction. From the figures, it is shown that the displacements increased along the transverse direction for the CFS and decreased for the SFC sandwich beam. Therefore, it could be concluded that the thickness stretching phenomenon of the FG sandwich beam expanded with increased the layer stiffness. The effect of length to thickness ratio of vibration frequency is also investigated by the present method. Taking SCCS sandwich fixed beam with L/ht=5, 10 and 20 as the numerical examples, the normalized natural frequency versus the layer thickness proportion is plotted in Fig 12. It can be seen that increasing the length to thickness ratio, the normalized natural frequency decreases, which is consistent with the characteristic of the homogeneous FG sandwich beam. Figure 13 indicates the effect of the important boundary conditions on the response of the FG sandwich beams. As an example, for the SCCS sandwich beams with L/ht=5, the C-C sandwich beams hold the highest frequency, while the C-F sandwich beam has the smallest frequency and the natural frequency of the other two support conditions beams are between them. Among which, the natural frequency of the C-S sandwich beams is larger than that of the S-S sandwich beams.

5. Conclusion In this paper, a developed meshfree boundary-domain integral equation method is applied to investigate the free vibration behavior of the FG sandwich beams. Two classical type of the FG sandwich beams are considered in the analysis, and the material properties of the FG layer is assumed to vary along the transverse direction in an exponential law. The material composition, layer thickness proportion, length to thickness ratio as well as the important boundary conditions are investigated in a great detail. The results of these numerical analyses demonstrate that the current developed method is not only convenient to implement, but also demonstrated higher accuracy, convergence and efficiency.

Reference 12

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Figure Captions Figure 1 FG sandwich beam configurations Figure 2 Young’s modulus variation of the sandwich beam configuration Figure 3 Boundary nodes, internal nodes and application points Figure 4 Node grouping Figure 5 Boundary discretization of the FG sandwich beam Figure 6 Different supports for the sandwich beams (a) simply supported; (b) fixed; (c) free Figure 7 Effect of layer thickness ratio for different support conditions Figure 8 Vibration modes of the C-C FG sandwich beams (hf/hc=0.2, L/ht=5) Figure 9 Vibration modes of the C-F FG sandwich beams (hf/hc=0.2, L/ht=5) Figure 10 Vibration modes of the C-S FG sandwich beams (hf/hc=0.2, L/ht=5) 14

Figure 11 Vibration modes of the S-S FG sandwich beams (hf/hc=0.2, L/ht=5) Figure 12 Normalized fundamental frequency for C-C SCCS sandwich beam with (L/ht=5, 10, 20) Figure 13 Normalized fundamental frequency for C-C, C-F, C-S and S-S SCCS sandwich beam with L/ht=5

Table Captions Table 1 Material properties of ceramic and steel Table 2 First ten normalized frequencies of the C-C CFS sandwich beams results of the present methods compared with the FEM (L/ht=5)

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