nonlinear vibration and postbuckling of functionally ...porous beams made of open-cell metal foams,...

Accepted Manuscript

Nonlinear vibration and postbuckling of functionally graded graphene reinforcedporous nanocomposite beams

Da Chen, Jie Yang, Sritawat Kitipornchai

PII: S0266-3538(16)32038-3

DOI: 10.1016/j.compscitech.2017.02.008

Reference: CSTE 6660

To appear in: Composites Science and Technology

Received Date: 23 December 2016

Revised Date: 7 February 2017

Accepted Date: 11 February 2017

Please cite this article as: Chen D, Yang J, Kitipornchai S, Nonlinear vibration and postbuckling offunctionally graded graphene reinforced porous nanocomposite beams, Composites Science andTechnology (2017), doi: 10.1016/j.compscitech.2017.02.008.

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http://dx.doi.org/10.1016/j.compscitech.2017.02.008

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Nonlinear vibration and postbuckling of functionally graded graphene

reinforced porous nanocomposite beams Da Chen a, Jie Yang b,*, Sritawat Kitipornchaia

a School of Civil Engineering, the University of Queensland, Brisbane, St Lucia 4072, Australia b School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083 Australia

*Corresponding author: [email protected] Tel: +61 3 99256169; Fax: +61 3 99256108

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Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams

___________________________________________________________________________

Abstract The nonlinear free vibration and postbuckling behaviors of multilayer functionally graded (FG) porous nanocomposite beams that are made of metal foams reinforced by graphene platelets (GPLs) are investigated in this paper. The internal pores and GPL nanofillers are uniformly dispersed within each layer but both porosity coefficients and GPL weight fraction change from layer to layer, resulting in position-dependent elastic moduli, mass density and Poisson’s ratio along the beam thickness. The mechanical property of closed-cell cellular solids is employed to obtain the relationship between coefficients of porosity and mass density. The effective material properties of the nanocomposite are determined based on the Halpin-Tsai micromechanics model for Young’s modulus and the rule of mixture for mass density and Poisson’s ratio. Timoshenko beam theory and von Kármán type nonlinearity are used to establish the differential governing equations that are solved by Ritz method and a direct iterative algorithm to obtain the nonlinear vibration frequencies and postbuckling equilibrium paths of the beams with different end supports. Special attention is given to the effects of varying porosity coefficients and GPL’s weight fraction, dispersion pattern, geometry and size on the nonlinear behavior of the porous nanocomposite beam. It is found that the addition of a small amount of GPLs can remarkably reinforce the stiffness of the beam, and the its nonlinear vibration and postbuckling performance is significantly influenced by the distribution patterns of both internal pores and GPL nanofillers. Keywords:

Functionally graded beam; porous nanocomposites; graphene platelet; nonlinear free vibration; postbuckling.

1. Introduction

Since being first isolated by Novoselov et al. [1] in Manchester, graphene has become one of the most exciting topics in materials science due to its high Young’s modulus, superior fracture strength and extreme thermal conductivity, etc [2, 3]. Tremendous research efforts have been focused to show the possibility of large scale adoption of graphene in different applications [4-7]. GPLs composed of a few graphene layers are one of the most promising graphene-based nanomaterials with great potential to be an ideal candidate as ultralight

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reinforcements in high-performance composites [8-10]. It is supposed that they can provide even better structural enhancement than other carbon nanomaterials, such as carbon nanotubes (CNTs) [11, 12].

Metal foams are a new class of advanced engineering materials characterised by low density, light weight, good stiffness and excellent energy absorption [13-18]. To optimize the global mechanical properties, graded non-uniform porosities are introduced to produce FG metal foams which have been proved to have better structural response than the normal uniform foams and are investigated by increasing number of researchers in experiments and theoretical works. He et al. [19] prepared FG aluminium foams by coupling the foaming process and the cooling process of the direct foaming method. Hangai et al. [20] fabricated FG aluminium foams with the friction stir processing route and conducted tests to reveal their compressive response. Hassani et al. [21] also successfully produced aluminium foams with graded cell size and density, and measured their mechanical properties. Chen et al. [22-24] obtained the numerical results of buckling, bending, linear and nonlinear vibrations of FG porous beams made of open-cell metal foams, and compared the influences from different porosity distributions. Magnucka-Blandzi and Magnucki [25] performed an effective design of a sandwich beam with an FG metal foam core and calculated the optimal dimensionless parameters to maximize the critical force and minimize the beam mass.

The main drawback of metal foams is that the existence of internal pores actually reduces the structural stiffness, which limits their applications in some engineering fields. Ultralight reinforcements, such as GPLs and CNTs, therefore can be used in the matrix of metal foams to strengthen their mechanical properties and maintain their advantages as low density and light weight, leading to the development of porous nanocomposites supported by the rapidly developed nanotechnology [26-33].

Most of the existing studies about nanocomposites are focused on the mechanical properties and performance of structures reinforced by CNTs. Kim et al. [34] presented a study combining the processing, characterization, and modelling of CNT-reinforced multiscale composites. Yeh et al. [35] conducted tensile tests to measure the mechanical properties of composites with different contents of network multi-walled nanotubes (MWNTs) and dispersed MWNTs. Yas and Samadi [36] investigated the free vibration and buckling of nanocomposite Timoshenko beams reinforced by single-walled CNTs resting on elastic foundation. Wu et al. [37] analysed the nonlinear vibration of shear deformable FG CNT-reinforced composite beams with initial geometric imperfections and considered various distribution patterns of nanofillers. They [38] also examined the imperfection sensitivity of postbuckling behaviour of nanocomposite beams subjected to axial compression. Ansari and Torabi [39] employed a numerical method within the framework of variational formulation to study the buckling and vibration of axially-compressed FG conical shells and took CNTs as reinforcements. However, in comparison with CNT-based nanocomposites, structures reinforced by GPLs are more promising but little researched.

This paper deals with the large amplitude deformations of FG porous beams reinforced by GPLs, including nonlinear free vibration and postbuckling behaviors. The proposed beam is assumed to be made of closed-cell metal foams with graded internal pores characterized by

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uniform and non-uniform porosity distributions. GPL nanofillers are dispersed in the metal matrix with varying volume fractions along the beam thickness. The combined effects of porosity distributions and GPL dispersion patterns are examined by using a shear deformable multilayer beam model constructed with Timoshenko beam theory and von Kármán type nonlinearity. The nonlinear free vibration frequencies and postbuckling loads are computed based on Ritz method and a direct iterative algorithm. The influences from porosities and nanofillers are discussed in details to indicate the best way to improve the nonlinear structural performance of porous nanocomposite beams.

2. Porous nanocomposite beams

A multilayer beam model under current consideration is described in a Cartesian

coordinate system (x, z), where [ ]0,x L∈ is along the mid-plane direction and [ ]/ 2, / 2z h h∈ − is perpendicular to x-axis and points upwards, as shown in Fig. 1. Each layer

is of the same thickness ( /h h n∆ = , n is the total number of layers), but both the porosity coefficient and GPL volume fraction are graded from layer to layer.

Three porosity distributions shown in Fig. 2 are considered in the present study. It is assumed that the size and density of internal pores vary along the beam thickness in non-uniform porosity distributions 1 and 2 while in uniform porosity distribution, the pores are evenly distributed over the whole beam. For beams made of homogeneous materials and with

non-uniform porosity distributions, the maximum value 1E ( 1ρ ) and minimum value 2E ( 2ρ ) of Young’s modulus (mass density) are corresponding to the smallest and largest porosities,

respectively. As depicted in Fig. 2, for symmetric distribution 1, 1E ( 1ρ ) is on the top and

bottom surfaces and 2E ( 2ρ ) is on the mid-plane. While for asymmetric distribution 2, 1E

( 1ρ ) and 2E ( 2ρ ) are on the top and bottom surfaces, respectively. For beams with each porosity distribution, GPLs are dispersed according to three different

patterns as illustrated in Fig. 3 which include symmetric pattern A, asymmetric pattern B and

uniform pattern C. The peak values ijs ( , 1,2,3i j = ) of GPL volume fraction GPLV are

calculated based on the same nanofiller weight fraction where 1,2,3i = are for non-uniform

porosity distributions 1, 2 and uniform porosity distribution, and 1, 2,3j = are for GPL patterns A, B and C, respectively.

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Fig. 1. Configuration of a multilayer beam.

Porosity distribution 1 Uniform distribution Porosity distribution 2

Fig. 2. Porosity distributions.

Pattern A Pattern B Pattern C

Fig. 3. GPL dispersion patterns ( 1, 2,3i = for porosity distributions 1, 2 and uniform porosity distribution).

Young’s modulus and mass density of porous nanocomposite beams are given by

( ) ( )( ) ( )

01

1

c

c m

E z E e z

z e z

ψ

ρ ρ ψ

= −

= − (1)

in which cE and cρ denote the maximum values of ( )E z and ( )zρ , i.e., the material properties of a pure nanocomposite beam without pores, 0e ( 2 11 /E E= − ) and me

( 2 11 /ρ ρ= − ) are porosity coefficient and mass density coefficient. For different porosity

distributions, ( )zψ can be written as

z

/ 2h

/ 2h−

x

1 1( )E ρ

1 1( )E ρ

2 2( )E ρ

11 ( )E ρ

2 2( )E ρ

z

GPLV GPLV GPLV

z z

0

0

1is

1is

2is 3is/ 2h

/ 2h−

123

2n −1n −

n

z

/ 2h

/ 2h−

x0 L

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( )

( )

*

cos Porosity distribution 1

cos Porosity distribution 22 4

Uniform distribution

z

πβπ πψ β

ψ

= +

(2)

where /z hβ = . By using the relationship between Young’s modulus and mass density in Eq. (3) based on the closed-cell Gaussian random field model for cellular solids [40],

( ) ( ) 2.30.121 /1.121

c c

E z z

E

ρρ

= +

(( )

0.15 1c

zρρ

< < ) (3)

me is defined as

( )( )

2.301.121 1 1

m

e ze

z

ψ

ψ

− − = (4)

It is assumed that the beams with uniform and non-uniform porosity distributions have the same mass, which gives

( )2.3

/2*

/20 0

1 1 1d 0.121 /1.121

h

hc

z ze e h

ψ ρρ −

= − +

∫ (5)

The shear modulus of porous nanocomposite beams is

( ) ( )( )2 1E z

G zv z

=+

(6)

in which Poisson’s ratio ( )v z of closed-cell cellular solids is expressed as [41]

( ) ( ) ( ) ( )2

0.342 0.526 0.221 0.132 0.221c c cc c

z zv z v v v

ρ ρρ ρ

= + − + +

(7)

where cv is the Poisson’s ratio of nanocomposites without pores.

The effective Young’s modulus cE is determined by Halpin-Tsai micromechanics model

[11, 26, 35] as

1 13 5

8 1 8 1

GPL GPL GPL GPLL L GPL W W GPL

c m mGPL GPLL GPL W GPL

V VE E E

V V

ξ η ξ ηη η

+ += + − − (8)

in which 2 /GPLL GPL GPLl tξ = , 2 /GPLW GPL GPLw tξ = , mE is the Young’s modulus of metal matrix,

GPLl , GPLt and GPLw are GPL’s average length, thickness and width. GPLLη and

GPLWη can be

computed by

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GPL GPL mL GPL

GPL L m

E E

E Eη

ξ−=

+ (9)

GPL GPL mW GPL

GPL W m

E E

E Eη

ξ−=

+ (10)

The mass density and Poisson’s ratio of the nanocomposite are given by the rule of mixture as [42]

c GPL GPL m mV Vρ ρ ρ= + (11)

c GPL GPL m mv v V v V= + (12)

where GPLρ and mρ are the mass densities of GPLs and metal matrix, GPLv and mv are

Poisson’s ratios of these two components, 1m GPLV V= − is the volume fraction of metal matrix.

As stated before, the volume fraction GPLV of GPLs is assumed to vary along the beam

thickness according to different dispersion patterns in Eq. (13), of which the peak values 1is ,

2is , 3is can be calculated by Eq. (14) based on the weight fraction GPLΛ for the whole beam.

It should be mentioned that ( )zρ in Eq. (14) is related to specific porosity distributions as given in Eqs. (1) and (2).

( )( )

( )

1

2

3

1 cos Pattern A

1 cos Pattern B2 4

Pattern C

i

GPL i

i

s z

V s z

s

πβ

π πβ

− = − +

(13)

( )

( )( ) ( )

( ) ( )

( )

/2

1/2

/2 /2

2/2 /2

/2

3/2

1 cos d

d 1 cos d2 4

d

h

ih c

h hGPL m

ih hGPL m GPL GPL GPL c c

h

ih c

zs z z

z zz s z z

zs z

ρπβ

ρρ ρρ π πβ

ρ ρ ρ ρ ρρ

ρ

−

− −

−

−

Λ = − + Λ + − Λ

∫

∫ ∫

∫

(14)

3. Theories and formulations

3.1 Strain energy, potential energy and work done by external force

According to Timoshenko beam theory, the displacements along the x- and z-axes of an

arbitrary point in the beam are denoted by ( ), ,u x z t% and ( ), ,w x z t% as

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( ) ( ) ( )( ) ( )

, , , ,

, , ,

u x z t u x t z x t

w x z t w x t

φ= +

=

%

% (15)

where ( ),u x t and ( ),w x t are the axial and transverse displacement components on the mid-plane ( 0z= ), ( ),x tφ is the rotation of the beam cross-section, and t represents time. The von Kármán type nonlinear strain-displacement relationship can be expressed as

21

2xx

xz

u wz

x x x

w

x

φε

γ φ

∂ ∂ ∂ = + + ∂ ∂ ∂

∂ = + ∂

(16)

The normal and transverse shear stresses are given by the elastic constitutive law as

( )2( )

1

( )

xx xx

xz xz

E z

v z

G z

σ ε

σ γ

= − =

(17)

Making use of Eqs. (16)-(17), the strain energy U, kinetic energy K, and the work V done by an external axial load P can be obtained as

22 2 2 2

11 55 11 110

1 1 12

2 2 2

L u w w u wU A A B D dx

x x x x x x x x

φ φ φφ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∫

(18)

2 2 2

0 1 20

12

2

L u w uK I I I dx

t t t t t

φ φ ∂ ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ∂ ∫ (19)

2

0

1d

2

L wV P x

x

∂ = ∂ ∫ (20)

in which the stiffness components 11A , 55A , 11B , 11D and inertia items 0I , 1I , 2I of the

multilayer beam are calculated based on the material properties of each layer, and can be written as

( ) ( )( ) ( ){ }

2011 11 11 2

1

1{ , , } 1, ,

1

nc i i

i iii

E h e hhA B D h h

n v h

β ψ ββ β

β=

− =−∑ (21)

( ) ( )( )

055

1

1

2 1

nc i i

ii

E h e hhA k

n v h

β ψ ββ=

− =+

∑ (22)

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( ) ( ) ( ){ }20 1 21

{ , , } 1 1, ,n

c i m i i i

i

hI I I h e h h h

nρ β ψ β β β

=

= − ∑ (23)

where ( )1/ 2 1/ 2 /i n i nβ = + − , i ( 1,2,3,...,n= ) is the layer number, k ( 5 / 6= ) is the shear correction factor. By introducing the following dimensionless quantities

x

Lς = , { } { }* * ,, u wu w

h= , *φ φ= , L

hη = , 1102

10

At

I Lτ = , *

110

PP

A= ,

{ } 5511 11 1111 11 11 55 2110 110 110 110

, , , , , ,AA B D

a b d aA A h A h A

=

, { } 0 1 20 1 2 210 10 10

, , , ,I I I

I I II I h I h

=

(24)

where 110A and 10I are the values of 11A and 0I of a homogeneous metal beam (i.e., 0 0e =

and 0GPLV = ), Eqs. (18)-(20) can be rewritten in dimensionless form as

2* * *

110

2 2 2* * * * *1* *

11 11 11 550

2 4* * * * *1* 11 11 11

20

/

12 d

2

1

2 4

Linear Nonlinear

Linear

Nonlinear

hU U A U U

L

u u wU a b d a

a a bu w w wU

φ φ ηφ ςς ς ς ς ς

φη ς ς η ς η ς

= = +

∂ ∂ ∂ ∂ ∂= + + + + ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂= + + ∂ ∂ ∂ ∂ ∂

∫

∫2

dςς

(25)

2 2 22 * * * * *1*

110 0 1 20

1/ 2 d

2

h u w uK K A I I I

L

φ φ ςτ τ τ τ τ

∂ ∂ ∂ ∂ ∂ = = + + + ∂ ∂ ∂ ∂ ∂ ∫ (26)

22 *1* *

1100

1/ d

2

h wV V A P

Lς

ς

∂= = ∂ ∫ (27)

3.2 Nonlinear vibration analysis

For a beam undergoing harmonic vibration, its dynamic displacements take the form of

{ } { }* * *( , ), ( , ), ( , ) ( ), ( ), ( ) iu w u w eωτς τ ς τ φ ς τ ς ς φ ς′ ′ ′= (28)

where 1i = − , 10 110/L I Aω Ω= is the dimensionless form of natural frequency Ω . Ritz method is used with the following trial functions for beams with different boundary conditions [43]

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1

( ) (1 )N

jj

j

u pς ς ς=

′ = −∑ (29)

1

( ) (1 )N

jj

j

w qς ς ς=

′ = −∑ (30)

1

1

1

1

,

( ) (1 ) ,

,

Nj

j

j

Nj

j

j

Nj

j

j

r

r

r

ς

φ ς ς ς

ς

−

=

=

=

′ = −

∑

∑

∑

Hinged-hinged (H-H) beam

(31) Clamped-clamped (C-C) beam

Clamped-hinged (C-H) beam

in which N is the total number of polynomial items in the trial functions, jp , jq and jr

( 1, 2, ,j N= ⋅⋅ ⋅ ) are the unknown coefficients. Substituting Eqs (29)-(31) into Eqs (25)-(26)

then minimizing the total energy * *U K∏ = + with respect to the unknown coefficients

0jp

∏∂ =∂

, 0jq

∏∂ =∂

, 0jr

∏∂ =∂

, (32)

yields the governing equation in matrix form as

2( ) = 0Linear Nonlinear ω+ −K K M d (33)

where LinearK and NonlinearK are 3 3N N× dimensionless symmetric stiffness matrices, M

denotes the 3 3N N× dimensionless mass matrix, and T T T T{{ } { } { } }j j jp q rd = . Eq. (33) can

be solved by using a direct iterative algorithm [24, 43] to obtain the nonlinear natural

frequencies ω+ and ω− at positive and negative half cycles. Consequently, the nonlinear frequency nlω of porous nanocomposite beams can be obtained as

2nl

ω ωωω ω

+ −

+ −= + (34)

It should be noted that in the free vibration analysis, the maximum deflection ( )0.5maxw w′ ′=for H-H and C-C beams, while ( )0.57maxw w′ ′= for C-H beam.

3.3 Postbuckling analysis

The principle of minimum total potential energy is used for postbuckling analysis, which requires

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( )* * 0U Vδ − = (35) where δ stands for the variational operator. Substituting Eqs. (29)-(31) into the dimensionless strain energy *U and work *V done by external force and employing Eq. (35) lead to the differential equation governing the postbuckling behavior

*( ) = 0Linear Nonlinear P+ −K K V d (36)

in which V is the 3 3N N× dimensionless geometric matrix. An iterative algorithm similar to the one used in the nonlinear free vibration analysis is followed herein to solve Eq. (36) to

determine the postbuckling load nlP under a specific value of mid-span deflection *mw .

4. Numerical results and discussions

In this section, detailed numerical results are presented to examine the effects of porosities and GPL nanofillers on the nonlinear free vibration and postbuckling behaviors of functionally graded porous nanocomposite beams. The material and geometrical parameters

for GPLs are 1.5GPLw = µm, 2.5GPLl = µm, 1.5GPLt = nm, 1.01GPLE = TPa, 1062.5GPLρ =

kg/m3, 0.186GPLv = [11, 44]. As copper based nanocomposites are commonly used in open

literature [26, 28, 29], copper is selected as the metal matrix with 130ME = GPa, 8960Mρ =

kg/m3, 0.34Mv = . It should be noted that according to the range of relative density specified

in Eq. (3), the porosity coefficient is required to be within 00 0.9618e< < . Therefore, in the

following calculations, the employed maximum and minimum values of 0e are 0.6 and 0.2,

respectively.

4.1 Validation and convergence analysis

The free vibration analysis of functionally graded material (FGM) beams with uniform and non-uniform porosity distributions is conducted to validate the proposed formulations.

Young’s modulus ( )E z and mass density ( )zρ of porous FGM beams can be expressed as

FGM-I beam with uniform porosities:

( ) ( ) ( )

( ) ( ) ( )

1

2 2

1

2 2

m

c m m c m

m

c m m c m

zE z E E E E E

h

zz

h

α

αρ ρ ρ ρ ρ ρ

= − + + − +

= − + + − +

(37)

FGM-II beam with non-uniform porosities:

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( ) ( ) ( )

( ) ( ) ( )

211

2 2

211

2 2

m

c m m c m

m

c m m c m

zzE z E E E E E

h h

zzz

h h

α

αρ ρ ρ ρ ρ ρ

= − + + − + −

= − + + − + −

(38)

where α is the porosity volume fraction, 380cE = GPa and 3960cρ = kg/m3 for Alumina as

the ceramic component, 70mE = GPa and 2702mρ = kg/m3 for Aluminium as the metal

component. Poisson’s ratio 0.3v = is fixed through the thickness, which is used to calculate the shear correction factor

10(1 )

12 11sv

kv

+=+

(39)

Table 1 gives the dimensionless fundamental frequency lω of perfect and porous FGM beams with C-C end supports. It is found that the present results become convergent when

10N ≥ and agree very well with the numerical solutions by Wattanasakulpong and Chaikittiratana [45] based on the Chebyshev collocation method. Hence, 10N = is used in the following calculations. Table 1.

Dimensionless fundamental frequency lω of perfect and porous FGM C-C beams ( 0.2α = , 0.2m= ).

Beam type

L / h Present

[45] N = 2 N = 4 N = 6 N = 8 N = 10 N = 12

Perfect FGM

5 2.3415 1.8836 1.8823 1.8823 1.8823 1.8823 1.8823 10 2.0018 1.0904 1.0884 1.0884 1.0884 1.0884 1.0884 20 1.8708 0.5707 0.5691 0.5691 0.5691 0.5691 0.5691

Porous FGM-I

5 2.3945 1.9219 1.9206 1.9205 1.9205 1.9205 1.9205 10 2.0487 1.1112 1.1092 1.1092 1.1092 1.1092 1.1092 20 1.9158 0.5813 0.5797 0.5797 0.5797 0.5797 0.5797

Porous FGM-II

5 2.3782 1.9251 1.9238 1.9238 1.9238 1.9238 1.9238 10 2.0292 1.1185 1.1166 1.1166 1.1166 1.1166 1.1166 20 1.8931 0.5863 0.5847 0.5847 0.5847 0.5847 0.5847

We then look into the nonlinear vibration analysis of porous FGM-I C-C beams with

uniform porosity distributions. The present nonlinear frequency ratios /nl lω ω are compared in Table 2 with those given by Ebrahimi and Zia [46] using Galerkin’s method and the method of multiple scales. Excellent agreement can be obtained with.

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Table 2.

Nonlinear frequency ratio /nl lω ω of porous FGM-I C-C beams

( / 2.8868L h = , 0.2α = ).

m / 0.03maxw L = / 0.06maxw L =

Present [46] Present [46]

0 1.0057 1.0060 1.0227 1.0239 0.2 1.0059 1.0061 1.0233 1.0242 0.5 1.0061 1.0062 1.0242 1.0249 1 1.0065 1.0065 1.0257 1.0261 2 1.0069 1.0068 1.0271 1.0273 3 1.0068 1.0068 1.0269 1.0272 5 1.0064 1.0065 1.0251 1.0260

The postbuckling behavior of unidirectional and cross-ply C-C laminated beams is

depicted in Fig. 4 where comparisons are made between our results and the closed-form solutions derived by Gupta et al. [47] based on the classical plate theory. The parameters used

in this example are 1 155E = GPa, 2 12.1E = GPa, 12 4.4G = GPa, 12 0.248v = , 250L = mm,

1h = mm, and 10b = mm (beam width). Again, our results are in excellent agreement with the existing ones.

0.0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

Gupta et al. [47]: Unidirectional beam Cross-ply beam

Present: Unidirectional beam Cross-ply beam

Po

stbu

cklin

g lo

ad

(N)

Central deflection (mm) Fig. 4. Postbuckling load versus central deflection curves of unidirectional and cross-ply C-C

laminated beams.

It is obvious that a beam with fewer layers is preferred from easy manufacturing and cost

effective point of view. The dimensionless nonlinear frequency and postbuckling load of beams with different total number of layers n are compared in Tables 3 and 4, respectively, in order to find the minimum total number of layers required to approximate an ideal functionally graded beam with enough accuracy yet at minimized fabrication cost. The results

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indicate that a beam with a total of 14 layers (n = 14) is sufficient as results are just 2.0 %

lower than those with 10000n = . Table 3. Dimensionless nonlinear frequency of functionally graded porous nanocomposite beams (Porosity

distribution 1, GPL pattern A, C-C copper-matrix beam, / 20L h = , 0 0.5e = ).

n 0.2maxw′ = 0.4maxw′ = 0.6maxw′ = 0.8maxw′ = 1.0maxw′ =

1 wt.%

2 0.3004 0.3167 0.3418 0.3739 0.4109 6 0.4317 0.4451 0.4665 0.4944 0.5279 10 0.4422 0.4554 0.4765 0.5042 0.5375 14 0.4451 0.4582 0.4793 0.5068 0.5401 18 0.4462 0.4594 0.4804 0.5079 0.5411

10000 0.4480 0.4612 0.4822 0.5096 0.5428

0 wt.% (Pure metal foam)

14 0.3159 0.3269 0.3444 0.3673 0.3945 10000 0.3172 0.3282 0.3457 0.3684 0.3956

Table 4. Dimensionless postbuckling load of functionally graded porous nanocomposite beams (Porosity

distribution 1, GPL pattern A, C-C copper-matrix beam, / 20L h = , 0 0.5e = ).

n * 0.2mw = * 0.4mw =

* 0.6mw = * 0.8mw =

* 1.0mw =

1 wt.%

2 0.00549 0.00614 0.00722 0.00873 0.01067 6 0.01139 0.01216 0.01342 0.01520 0.01749 10 0.01196 0.01273 0.01401 0.01581 0.01812 14 0.01212 0.01289 0.01418 0.01598 0.01830 18 0.01218 0.01296 0.01425 0.01605 0.01837

10000 0.01228 0.01306 0.01435 0.01616 0.01848

0 wt.% (Pure metal foam)

14 0.00655 0.00704 0.00787 0.00902 0.01050 10000 0.00660 0.00710 0.00792 0.00908 0.01057

4.2 Nonlinear free vibration

Figs. 5(A)-5(C) illustrate the effect of GPL weight fraction on the nonlinear frequency ratio versus maximum deflection curves for porous nanocomposite beams. This effect is seen to be different for beams with different GPL distribution patterns. As GPL weight fraction increases, the nonlinear frequency ratio becomes lower for symmetric GPL distribution pattern A but tends to be slightly higher for asymmetric GPL distribution pattern B. For uniform pattern C, however, this effect is negligible.

Fig. 6 depicts the effect of GPLs (pattern A) on the nonlinear frequency of C-C beams with symmetric porosity distribution 1. It is found that the addition of GPL nanofillers improves the effective beam stiffness remarkably as evidenced by the pronounced increase of

GPLΛ

GPLΛ

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nonlinear frequency. It also should be noted that all beams exhibit similar hardening behavior, i.e., their nonlinear frequencies and ratios increase with larger vibration amplitudes.

0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20

1.25 Λ

GPL = 0.0 wt.%

ΛGPL

= 0.5 wt.%

ΛGPL

= 1.0 wt.%

ω nl /

ωl

w'max

0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20

1.25

w'max

ω nl /

ωl

0.80 0.851.18

1.19

1.20

ΛGPL

= 0.0 wt.%

ΛGPL

= 0.5 wt.%

ΛGPL

= 1.0 wt.%

0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20

1.25

w'max

ΛGPL

= 0.0 wt.%

ΛGPL

= 0.5 wt.%

ΛGPL

= 1.0 wt.%

ω nl /

ωl

(A) GPL pattern A (B) GPL pattern B (C) GPL pattern C Fig. 5. Nonlinear frequency ratio versus maximum deflection curves of nanocomposite beams: effect

of GPL weight fraction (porosity distribution 1, C-C beam, 0 0.5e = , / 20L h = ).

-1.0 -0.5 0.0 0.5 1.00.3

0.4

0.5

0.6Porosity 1 (symmetric)GPL A (symmetric)C-C copper-matrix beame

0 = 0.5

L / h = 20

ΛGPL

= 0.0 wt.%

ΛGPL

= 0.5 wt.%

ΛGPL

= 1.0 wt.%

ωnl

w'min w'max Fig. 6. Dimensionless nonlinear frequency versus vibration amplitude curves of nanocomposite beams:

effect of GPL weight fraction (porosity distribution 1, GPL pattern A, C-C beam, 0 0.5e = ,

/ 20L h = ).

Figs. 7(A) and 7(B) show that under the same weight fraction and GPL length GPLl , both

nanofiller geometry and size have important influences on GPL reinforcement effects. The

nonlinear frequency increases remarkably when /GPL GPLl t for square platelets ( / 1GPL GPLl w = )

is increased up to 102 beyond which not much further improvement can be observed. This suggests that platelets with less single graphene layers can enhance the structural stiffness

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more effectively. At a given /GPL GPLl t , GPL with a higher /GPL GPLl w has a lower surface area,

consequently, the nonlinear frequency of the beam is reduced, as shown in Fig. 7 (B).

-1.0 -0.5 0.0 0.5 1.00.40

0.45

0.50

0.55

0.60

w'min w'max

Porosity 1 (symmetric)GPL A (symmetric)C-C copper-matrix beame

0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%

lGPL

/ wGPL

= 1

lGPL

/ tGPL

= 10

lGPL

/ tGPL

= 102

lGPL

/ tGPL

= 103

lGPL

/ tGPL

= 104

ω nl

(A) Effect of /GPL GPLl t

-1.0 -0.5 0.0 0.5 1.00.40

0.45

0.50

0.55


0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%

lGPL

/ tGPL

= 100

w'min w'max

lGPL

/ wGPL

= 1

lGPL

/ wGPL

= 4

lGPL

/ wGPL

= 7

lGPL

/ wGPL

= 10

ω nl

(B) Effect of /GPL GPLl w

Fig. 7. Dimensionless nonlinear frequency versus vibration amplitude curves of nanocomposite beams:

effects of GPL geometry and size (porosity distribution 1, GPL pattern A, C-C beam, 0 0.5e = ,

/ 20L h = , 1.0GPLΛ = wt.%).

The effects of porosity coefficient and distribution on the nonlinear vibration behavior of nanocomposite beams are demonstrated in Fig. 8 where the nonlinear frequency ratio is given and Fig. 9 showing the nonlinear frequency. Since an increase in porosity coefficient corresponds to a larger size and a higher density of the internal pores which weakens beam

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stiffness, the nonlinear frequency drops as the porosity coefficient increases, regardless of the porosity distribution. The results for the nonlinear frequency ratio, however, are seen to be different for the three different porosity distributions. A higher porosity coefficient leads to a lower nonlinear frequency ratio for beams with symmetric porosity distribution 1 and a slightly higher frequency ratio for beams with asymmetric porosity distribution 2, while for beams with uniform porosity distribution, the nonlinear frequency ratio is almost unaffected by the change in porosity coefficient.

0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20

1.25

w'max

e0 = 0.2

e0 = 0.4

e0 = 0.6

ωnl /

ωl

0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20

1.25

w'max

e0 = 0.2

e0 = 0.4

e0 = 0.6

ω nl /

ωl

0.80 0.851.17

1.18

1.19

1.20

0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20

1.25

w'max

e0 = 0.2

e0 = 0.4

e0 = 0.6

ω nl /

ωl

(A) Porosity distribution 1 (B) Porosity distribution 2 (C) Uniform distribution Fig. 8. Nonlinear frequency ratio versus maximum deflection curves of nanocomposite beams: effect

of porosity coefficient (GPL pattern A, C-C beam, / 20L h = , 1.0GPLΛ = wt.%).

-1.0 -0.5 0.0 0.5 1.00.44

0.48

0.52

0.56

w'min w'max

Porosity 1 (symmetric)GPL A (symmetric)C-C copper-matrix beamL / h = 20Λ

GPL = 1.0 wt.%

e0 = 0.2

e0 = 0.4

e0 = 0.6

ω nl

Fig. 9. Dimensionless nonlinear frequency versus vibration amplitude curves of nanocomposite beams:

effect of porosity coefficient (porosity distribution 1, GPL pattern A, C-C beam, / 20L h = , 1.0GPLΛ = wt.%).

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Fig. 10(A) examines the combined effects of porosity distributions and GPL patterns on dimensionless nonlinear frequency versus vibration amplitude curves. The reinforcement effect of GPL nanofillers is seen to be more remarkable with symmetric pattern A, less significant with uniform pattern C, and least pronounced with asymmetric pattern B, which is also confirmed by the percentage frequency increment results presented in Fig. 10(B). The beam with porosity distribution 1 and GPL pattern A has the highest nonlinear frequency, indicating that the non-uniformly symmetric dispersion of internal pores and nanofillers is an effective way to achieve the best reinforcing performance of porous nanocomposite beams from nonlinear free vibration perspective. This is due to the fact that the symmetric pattern with more GPLs distributed in high bending stress regions and more pores in low stress regions around the mid-plane is capable of making the best use of materials. It can also be seen in Fig. 10(B) that as the vibration amplitude increases, the frequency increment tends to be lower for GPL distribution pattern A, becomes higher for pattern B, and stays almost unchanged for pattern C.

-1.0 -0.5 0.0 0.5 1.00.3

0.4

0.5

0.6

w'min w'max

C-C copper-matrix beame

0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%

A: GPL A (symmetric)B: GPL B (asymmetric)C: GPL C (uniform)

Porosity 1 (symmetric) Porosity 2 (asymmetric) Uniform porosity

B

B C

A

A

ω nl

(A) Dimensionless nonlinear frequency versus vibration amplitude

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-1.0 -0.5 0.0 0.5 1.020

30

40

50

w'min w'max

B

C

A



0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%


Non

linea

r fr

eque

ncy

incr

emen

t (%

)

(B) Nonlinear frequency increment versus vibration amplitude

Fig. 10. Comparisons of dimensionless nonlinear frequency and its percentage increment of nanocomposite beams (C-C beam, 0 0.5e = , / 20L h = , 1.0GPLΛ = wt.%)

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

w'min w'max

Porosity 1 (symmetric)e

0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%

H-H

C-H

C-C

GPL A (symmetric) GPL B (asymmetric) GPL C (uniform)

ω nl

Fig. 11. Dimensionless nonlinear frequency versus vibration amplitude curves of nanocomposite

beams: effect of boundary condition (porosity distribution 1, 0 0.5e = , / 20L h = , 1.0GPLΛ = wt.%).

The nonlinear vibration behaviors of C-C, C-H and H-H porous nanocomposite beams are depicted in Fig. 11. As expected, the C-C beam possesses the largest nonlinear frequency. It can also be noted that the curves of C-H and H-H beams with GPL pattern B are asymmetric. This is because pattern B leads to an asymmetric material distribution across the beam thickness, thus the energy balance equation cannot yield equal and opposite roots due to the

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existence of bending-extensional coupling, making the nonlinear frequency dependent on both the magnitude and the sign of the vibration amplitude. For a C-C beam with GPL pattern B, the nonlinear vibration curve remains symmetric since the fully clamped end supports can produce necessary bending moment to eliminate the coupling effects.

The dimensionless nonlinear frequency and the frequency ratio of beams with varying slenderness ratios are tabulated in Tables 5 and 6, respectively. It is worth noting that although the beam with a higher slenderness ratio has a significantly lower nonlinear frequency, its frequency ratio drops just slightly under the same vibration amplitude.

Table 5. Dimensionless nonlinear frequency of porous nanocomposite beams: effect of slenderness ratio

(porosity distribution 1, GPL pattern A, C-C beam, 0 0.5e = , 1.0GPLΛ = wt.%).

maxw′ / 20L h = / 30L h = / 40L h = / 50L h =

0.2 0.4451 0.3010 0.2269 0.1819 0.4 0.4582 0.3098 0.2335 0.1872 0.6 0.4793 0.3238 0.2440 0.1957 0.8 0.5068 0.3423 0.2579 0.2068 1.0 0.5401 0.3646 0.2747 0.2202

Table 6.

Nonlinear frequency ratio /nl lω ω of porous nanocomposite beams: effect of slenderness ratio

(porosity distribution 1, GPL pattern A, C-C beam, 0 0.5e = , 1.0GPLΛ = wt.%).

maxw′ / 20L h = / 30L h = / 40L h = / 50L h =

0.2 1.0102 1.0101 1.0100 1.0100 0.4 1.0401 1.0396 1.0394 1.0393 0.6 1.0879 1.0868 1.0865 1.0863 0.8 1.1504 1.1489 1.1484 1.1482 1.0 1.2258 1.2237 1.2229 1.2226

4.3 Postbuckling

Fig. 12 illustrates the effect of GPL weight fraction on the postbuckling equilibrium paths of porous nanocomposite beams. The results confirm the remarkable reinforcement effects by GPLs.

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0.0 0.2 0.4 0.6 0.8 1.00.005

0.010

0.015


0 = 0.5

L / h = 20

ΛGPL

= 0.0 wt.%

ΛGPL

= 0.5 wt.%

ΛGPL

= 1.0 wt.%

Pn

l

w*m

Fig. 12. Dimensionless postbuckling load versus midspan deflection curves of nanocomposite beams:

effect of GPL weight fraction (porosity distribution 1, GPL pattern A, C-C beam, 0 0.5e = ,

/ 20L h = ).

Figs. 13(A) and 13(B) reveal the effects of GPL geometry and size on the postbuckling behavior of porous nanocomposite beams. Consistent with the observations in nonlinear vibration analysis, it is found that the postbuckling load-carrying capacity can be most effectively improved through the addition of GPLs with fewer single graphene layers and a larger surface area.

0.0 0.2 0.4 0.6 0.8 1.00.010

0.012

0.014

0.016

0.018


0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%

lGPL

/ wGPL

= 5 / 3

lGPL

/ tGPL

= 10

lGPL

/ tGPL

= 102

lGPL

/ tGPL

= 103

lGPL

/ tGPL

= 104

Pn

l

w*m

(A) Effect of /GPL GPLl t

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0.0 0.2 0.4 0.6 0.8 1.00.010

0.012

0.014

0.016

0.018


0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%

lGPL

/ tGPL

= 100

lGPL

/ wGPL

= 1

lGPL

/ wGPL

= 4

lGPL

/ wGPL

= 7

lGPL

/ wGPL

= 10

Pn

l

w*m

(B) Effect of /GPL GPLl w

Fig. 13. Dimensionless postbuckling load versus midspan deflection curves of nanocomposite beams: effects of GPL geometry and size (porosity distribution 1, GPL pattern A, C-C beam, 0 0.5e = ,

/ 20L h = , 1.0GPLΛ = wt.%).

The postbuckling equilibrium path and the percentage increment of the postbuckling load at a given midspan deflection of porous nanocomposite beams are given in Figs. 14(A) and 14(B), respectively. Again, the functionally graded symmetric dispersions of internal pores and GPL nanofillers are proved to be the best distribution patterns to achieve the highest postbuckling load. The percentage increment of the postbuckling load is found to be much higher than that of the nonlinear vibration frequency given in Fig. 10(B). This is because the buckling load is directly proportional to the beam stiffness while the frequency is proportional to the square root of the stiffness.

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0.0 0.2 0.4 0.6 0.8 1.00.006

0.009

0.012

0.015

0.018

0.021

CB

B C

A

A




0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%

Pnl

w*m

(A) Dimensionless postbuckling load versus midspan deflection

0.0 0.2 0.4 0.6 0.8 1.025

50

75

100

C

B

AC-C copper-matrix beame

0 = 0.5

L / h = 20Λ

GPL = 1.0 wt.%



Pos

tbu

cklin

g lo

ad

incr

em

en

t (%

)

w*m

(B) Postbuckling load increment versus midspan deflection

Fig. 14. Comparisons of postbuckling equilibrium path and its percentage change of nanocomposite beams (C-C beam, 0 0.5e = , / 20L h = , 1.0GPLΛ = wt.%).

5. Conclusions

In order to assess GPL’s reinforcing effect on the nonlinear performance of functionally graded porous nanocomposites, the nonlinear vibration and postbuckling analyses on closed-cell copper-matrix beams have been presented. The effects of weight fraction, dispersion pattern, geometry and size of GPL nanofillers on beams with varying porosity distributions, porosity coefficient, boundary conditions and slenderness ratio are investigated through a detailed parametric study. It can be concluded from the results that:

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(1) GPLs can provide remarkable reinforcing effect on porous beams as evidenced by higher

nonlinear vibration frequencies and larger postbuckling loads, both of which decrease with the increasing porosity coefficient.

(2) The effect of GPL weight fraction on the nonlinear frequency ratio tends to be different for beams with different GPL distribution patterns. Similar phenomenon can be observed in the effect of porosity coefficient on the nonlinear frequency ratio of beams with different porosity distributions.

(3) Beams with porosity distribution 1 reinforced by GPL pattern A have the largest nonlinear frequencies and the highest postbuckling loads, which implies that the functionally graded symmetric distributions of both internal pores and GPL nanofillers can achieve the highest beam stiffness.

(4) The reinforcement effects from pattern A decrease with the increasing of nonlinear vibration amplitude and postbuckling midspan deflection, but the results for pattern B show the other way around.

(5) GPLs with fewer single graphene layers and a larger surface area are preferred nanofillers as they offer the best structural performance of porous nanocomposite beams.

Acknowledgements

The work described in this paper was fully funded by an Australian Government Research Training Program Scholarship and three research grants from the Australian Research Council under Discovery Project scheme (DP160101978, LP150100103, LP150101033). The authors are grateful for their financial support.

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nonlinear vibration and postbuckling of functionally ...porous beams made of open-cell metal foams,...

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