Nonlinear free vibration of functionally graded polymer composite beams reinforced with graphene nanoplatelets (GPLs)

Engineering Structures 140 (2017) 110–119

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Nonlinear free vibration of functionally graded polymer composite beams reinforced with graphene nanoplatelets (GPLs) Chuang Feng a, Sritawat Kitipornchai b, Jie Yang a,⇑ a b

School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083, Australia School of Civil Engineering, The University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia

a r t i c l e

i n f o

Article history: Received 5 October 2016 Revised 20 February 2017 Accepted 22 February 2017

Keywords: Graphene platelet Nonlinear vibration Non-uniform distribution Polymer nanocomposites

a b s t r a c t This paper studies the nonlinear free vibration of a multi-layer polymer nanocomposite beam reinforced by graphene platelets (GPLs) non-uniformly distributed along the thickness direction. Theoretical formulations are based on Hamilton’s principle, Timoshenko beam theory, and von Kármán nonlinear straindisplacement relationship. The effective Young’s modulus of the GPL/polymer composites is estimated by Halpin-Tsai micromechanics model to account for the effects of GPL geometry and dimensions. The vibration frequencies and amplitude of the beam are obtained numerically by employing Ritz method. The influences of the distribution pattern, weight fraction, geometry and size of GPL nanofillers, the total number of layers together with the vibration amplitude and boundary conditions on the nonlinear free vibration behavior are investigated. The results show that adding a very small amount of GPLs into polymer matrix as reinforcements significantly increases the natural frequencies of the beam. Using larger sized GPLs with fewer single graphene layers and placing more GPLs near the top and bottom surfaces of the beam are the most effective ways to strengthen the beam stiffness and increase the linear and nonlinear natural frequencies. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Dispersing carbon-based nanomaterials into polymer matrix as reinforcements to produce high-performance composite structures has been attracting huge research attention over the last decades. Carbon nanotube (CNT) is one of the most promising reinforcements as it is strong and lightweight. However, the development of CNT reinforced composite structures is hindered by several challenges, such as high manufacturing costs, difficulty to achieve uniform dispersion and highly anisotropic attributes of CNTs [1–3]. Recently, incorporating graphene and its derivatives into polymer matrix has demonstrated great potential due to their excellent mechanical properties, improved reinforcing effects and moderate cost. For example, the Young’s modulus and ultimate strength of graphene and its derivatives can reach up to 1 TPa and 130 GPa, respectively [3]. It has been experimentally observed that with the same loading of reinforcing fillers, i.e. 0.1% weight fraction (w.t.), the effective Young’s modulus of graphene based epoxy nanocomposites is 31% higher than that of pristine epoxy while only an increase of 3% can be achieved by using CNT reinforcements [4]. This is mainly attributed to the extremely high specific ⇑ Corresponding author. E-mail address: [email protected] (J. Yang). http://dx.doi.org/10.1016/j.engstruct.2017.02.052 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

surface area of graphene and its derivatives that provides excellent load transfer between the reinforcements and the matrix [4–9]. Due to its two-dimensional attributes, the dispersion of graphene reinforcements in polymer matrix is also greatly improved with less agglomeration compared to the one-dimensional anisotropic CNTs. In addition, the excellent mechanical properties, abundance in nature, and mass production with relatively low cost of graphene’s derivatives such as graphene platelet (GPLs) and graphene oxide (GO) make graphene based polymers perfect nanocomposites in developing light weight structures in a wide range of engineering applications [10]. Liang et al. [11] successfully prepared GO/poly(vinyl alcohol) nanocomposites and found that the tensile strength and Young’s modulus were increased by 76% and 62%, respectively, by adding 0.7 w.t.% of GO into the matrix. Lee et al. [12] synthesized functionalized graphene sheet (FGS)/epoxy nanocomposites for cryotank application and observed a significant increase in strength and toughness. The experiments conducted by Tang et al. [8] showed that the graphene/epoxy nanocomposites with highly dispersed graphene have higher strength and fracture toughness. Rahman and Haque [13] developed a molecular modeling technique to determine the mechanical properties of graphene/epoxy nanocomposites and observed a significant improvement in Young’s and shear moduli of the composites. Ji et al. [14] and Spanos et al.

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C. Feng et al. / Engineering Structures 140 (2017) 110–119

[15] used micromechanics model to obtain the effective elastic moduli of graphene reinforce polymer composites. Cho et al. [16] and Shokrieh et al. [5] predicted the mechanical properties of GPL/epoxy nanocomposites through a combined molecular dynamic (MD) simulation and micromechanics analysis. Although extensive theoretical and experimental work has been done on graphene based polymer composites, the majority of them are focused on the manufacturing and characterization of material properties. Research work concerning the mechanical behavior of structures made of graphene nanocomposites is still very limited. It should also be noted that in all of the previous studies, graphene and its derivatives are uniformly distributed in the matrix only. Functionally graded materials (FGMs) are inhomogeneous composites characterized by a continuous and smooth change in both material composition and mechanical properties along one or more direction(s). By using FGMs, the possible interfacial failure due to property mismatch at the interface between two distinct materials [17] can be effectively alleviated or eliminated. Furthermore, the multifunctional requirements can also be met simultaneously, which is not possible in conventional homogeneous composites. Due to their unique advantages, FGMs and their structures have received extensive interests from both research and industry communities since it was first developed in 1980s [18–28]. In particular, composites structures reinforced with non-uniformly distributed CNTs have been explored extensively. Shen [29] studied nonlinear bending behavior of functionally graded CNT-reinforced nanocomposite plates subjected to transverse uniform/sinusoidal load in thermal environments and found the bending behaviours were significantly improved as a result of non-uniform distribution of the reinforcements. Ke et al. [30,31] investigated the nonlinear free vibration and dynamic stability of functionally graded CNT reinforced composite beam and discussed the effect of CNT volume fraction and distribution on the nonlinear vibration. Kwon et al. [32] successfully manufactured functionally graded CNT reinforced aluminum matrix composites that demonstrated great improvement in hardness. Shen and Zhang [33] analyzed the thermal buckling and postbuckling behavior of CNT reinforced composite plates and showed remarkable increase in buckling temperature and strength of the plates with functionally graded reinforcements. Rafiee et al. [21] examined the large amplitude free vibration of CNT reinforced composite beams with piezoelectric layers and confirmed that a functionally graded reinforcement could significantly influence the nonlinear vibration behaviours of the beams. Using finite element analysis, Zhu et al. [34] analyzed static bending and vibration behavior of functionally graded CNT reinforced composite plates. More work on studying the mechanical behaviours of functionally graded CNT reinforced composite structures can be found in the review by Liew et al. [35]. Compared to the numerous works on CNT reinforced functionally graded composite structures, relatively limited work has been found on graphene and its derivatives reinforced functionally graded structures despite of their excellent mechanical properties and moderate cost. Given the challenge and the constraint of current manufacturing techniques to fabricate graphene based composite structures with continuous variation of material properties, alternatively, this paper proposes a composites beam consisting of N layers stacked up in the thickness direction as shown in Fig. 1 to form FGM beams. GPLs are uniformly dispersed in the polymer matrix within each individual layer while the w.t.% of GPL varies from layer to layer according to a specific distribution pattern. Such a multi-layer beam will be an excellent approximation of a continuous FGM structure if the total number of layers used is sufficiently large. Following our most recent studies on functionally graded graphene reinforced polymer nanocomposite beams and plates [36–40], this paper investigates the nonlinear

Fig. 1. Schematic configuration of a multi-layer GPL reinforced nanocomposite beam.

Fig. 2. GPL distribution patterns along the thickness direction.

free vibration of this multi-layer GPL/Polymer nanocomposite beam with four GPL distribution patterns in Fig. 2. The effective Young’s modulus of each individual homogeneous layer is determined by Halpin-Tsai micromechanics model while the governing equation for vibration is established based on Timoshenko beam theory and von Kármán strain-displacement relationship. Parametric study on the influences of distribution pattern, concentration, geometry and size of GPLs, and total number of layers on the vibration of the GPL/polymer composite beam will be conducted by numerically solving the governing equation with Ritz method. 2. Theoretical formulation Shown in Fig. 1 is a multilayer GPL reinforced nanocomposite beam with length L, width b and thickness h, consisting of N individual layers of the same thickness Δh = h/N. GPLs are uniformly distributed within each layer with the weight fraction varying from layer to layer along the thickness direction according to one of the four linear patterns shown in Fig. 2 where darker color represents a higher GPL weight fraction. Pattern 1 is a special case that corresponds to an isotropic homogeneous beam with GPLs uniformly dispersed over the thickness. Patterns 2 and 3 are symmetrical distributions with GPL weight fraction increasing/decreasing linearly from the top and bottom layers to the middle layer, respectively. Consequently, GPL weight fraction in the middle layer is the maximum in Pattern 2 but the minimum in Pattern 3. Pattern 4 is an asymmetrical distribution in which with GPL weight fraction increasing linearly from the top to the bottom layers. Denote the total weight fraction of GPLs dispersed in the beam as fGPL and assume that Patterns 2, 3 and 4 have the same minimum and maximum weight fractions, i.e. fmin and fmax. GPL weight fraction in the ith layer can be expressed as

fi ¼ (

f GPL N

ðPattern 1Þ

f min Þ i 6 N2 f i ¼ f min þ ði  1Þ 2ðf max N2
N  2ðf max f min Þ f i ¼ f max  i  2  1 i > N2 N2

ð1aÞ

ðPattern 2Þ

ð1bÞ

112

(

C. Feng et al. / Engineering Structures 140 (2017) 110–119

f min Þ f i ¼ f max  ði  1Þ 2ðf max i 6 N2 N2
N  2ðf max f min Þ f i ¼ f min þ i  2  1 i > N2 N2

f  f min f i ¼ f min þ ði  1Þ max N1

ðPattern 3Þ

ðPattern 4Þ

ð1cÞ

ð1dÞ

2.1. Effective mechanical properties of GPL/polymer nanocomposites The material properties of each layer, i.e. the effective Young’s modulus, mass density and Poisson’s ratio, need to be determined before the governing equations for the nonlinear free vibration of the nanocomposite beam are established. Assuming GPLs as effective rectangular fillers perfectly bonded to the matrix without slipping between them, the effective Young’s modulus of each individual layer can be predicted by Halpin-Tsai micromechanics model as [41–43]

relationship and considering the nonlinearity due to the stretching of neutral axis, the normal strain exx and shear strain cxz are

exx ¼

 2 @u 1 @w @u @w þz þ ; cxz ¼ u þ @x 2 @x @x @x

The linear stress-strain relationship gives normal stress shear stress sixz in the ith layer as

rixx ¼ 1Eim2

h

@u @x




i

þ 12


@w2 @x

Ei @w sixz ¼ 2ð1þ mi Þ u þ @x



ð8Þ

rixx and

i þ z @@xu

ð9Þ

where Ei and mi are the effective Young’s modulus and Poisson’s ratio of the ith layer. The strain energy can then be written as

V¼

i¼N Z h=2ði1ÞDh LX

Z

1 2

0

h=2iDh

i¼1

ðrixx exx þ sixz exz Þdzdx ¼ V L þ V NL

ð10Þ

where the linear strain energy is

3ð1 þ nL gL V GPL Þ 5ð1 þ nW gW V GPL Þ EC ¼  EM þ  EM 8ð1  gL V GPL Þ 8ð1  gW V GPL Þ

ð2Þ

where

gL ¼

ðEGPL =EM Þ  1 ðEGPL =EM Þ  1 ; gW ¼ ðEGPL =EM Þ þ nL ðEGPL =EM Þ þ nW

ð3Þ

and EC, EM, EGPL are Young’s moduli of the GPL/polymer nanocomposite, polymer matrix, and GPLs, respectively, and VGPL is GPL volume fraction. The effects of the geometry and the size of GPL reinforcements are taken into consideration in this model via two parameters [42]

nL ¼ 2ðlGPL =hGPL Þ; nW ¼ 2ðwGPL =hGPL Þ

Vi ¼

fi f i þ ðqGPL =qM Þð1  f i Þ

ð5Þ

Then mass density and the Poisson’s ratio of the ith layer can be obtained by using the rule of mixture as

qi ¼ qGPL V i þ qM ð1  V i Þ

ð6aÞ

mi ¼ mGPL V i þ mM ð1  V i Þ

ð6bÞ

It should be noted that the accuracy of Halpin-Tsai micromechanics model for the prediction of the effective Young’s modulus of GPL/polymer composites can be verified through a direct comparison against the experimental data at GPL loading of 0.1 w.t.% reported by Rafiee et al. [4]. The Young’s modulus predicted by Eq. (2) is just 2.7% higher than the experimental result.

Z

L

A11

ð11Þ

and nonlinear strain energy is

V NL ¼

1 2

Z

L

( A11

0

"    2 #  2 ) 4 1 @w @u @w @ u @w þ B11 dx þ 4 @x @x @x @x @x ð12Þ

ð4Þ

in which lGPL, wGPL and hGPL denote the length, width and thickness of GPLs. Given the mass densities of GPLs and the polymer matrix, qGPL and qM, the volume fraction of GPL nanofillers in the ith layer can be calculated based on its GPL weight fraction fi by

(

 2  2 @u @u @ u @u þ 2B11 þ A55 u2 þ D11 @x @x @x @x 0  2 ) @w @w þ A55 dx þ2A55 u @x @x

1 VL ¼ 2

The stiffness components are defined as

fA11 ; A55 ; B11 ; D11 g ¼

i¼N Z X i¼1

h=2ði1ÞDh

h=2iDh

  Ei 1  mi 2 dz 1; k ; z; z s 2 1  m2i ð13Þ

where the shear correction factor ks = 5/6. Neglecting the rotational inertia due to its limited effect on the transverse vibration of the beam, the kinetic energy of the beam is

Z K¼

i¼N Z h=2ði1ÞDh LX

0

i¼1

h=2iDh

qi

" 2  2 # @u @u @w dzdx þz þ @t @t @t

ð14Þ

which can be further expressed as

Z K¼ 0

L

"    2 #  2 2 @u @u @ u @u @w I1 þ 2I2 þ I1 dx þ 2I3 @t @t @t @t @t

ð15Þ

in which the inertial components are defined as

fI1 ; I2 ; I3 g ¼

i¼N Z X i¼1

h=2ði1ÞDh

h=2iDh

qi f1; z; z2 gdz

ð16Þ

2.2. Governing equation

Based on Hamilton’s principle, the total energy of the beam structures is

According to Timoshenko beam theory, the displacement fields in the x- and z-directions are

Y

~ ðx; z; tÞ ¼ uðx; tÞ þ zuðx; tÞ u ~ z; tÞ ¼ wðx; tÞ wðx;

ð7Þ

where u(x,t) and w(x,t) are displacement components on the midplane of the beam, u(x, t) represents the cross section rotation, and t is time. Eq. (7) reduces to classical Euler-Bernoulli beam when uðx; tÞ ¼  @w . Based on von Kármán nonlinear strain-displacement @x

¼ V L þ V NL þ K

ð17Þ

When subjected to harmonic vibration, the dynamic displacement components of the beam take the form of

u ¼ u0 eiXt ; w ¼ w0 eiXt ; u ¼ u0 eiXt

ð18Þ

where X is the vibration frequency of the beam. Introducing the following dimensionless quantities

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C. Feng et al. / Engineering Structures 140 (2017) 110–119

qffiffiffiffiffiffiffi

qffiffiffiffiffiffiffi

110 10 fU; Wg ¼ fu0 h;w0 g ; X ¼ xL ; w ¼ u0 ; g ¼ hL ; s ¼ Lt AI110 ; x ¼ XL AI110 Q  n o Y f ;V L ;V NL ;Kge2ixs ; fI1 ; I2 ; I3 g ¼ II101 ; I10I2h ; I I3h2 f ; V L ; V NL ; K  g ¼ A110 h2 =L 10 n o 11 55 11 fa11 ; a55 ; b11 ; d11 g ¼ AA110 ; AA110 ; AB110 ; D11 h A h2 110

ð19Þ where I0 and A110 are the values of I0 and A11 of an isotropic homogeneous beam without GPL reinforcements. The strain energy and kinetic energy can now be written in dimensionless form as

"

 2  2 @U @w @U @w þ 2b11 þ g2 a55 w2 þ d11 @X @X @X @X 0  2 # @W @W þ a55 dX þ2ga55 w @X @X

V L ¼

V NL

1 2

Z

1 ¼ 2

1

1

0

ð20Þ

"

 4  2  2 # a11 @W a11 @U @W 1 @w @W dX þ þ b11 4g2 @X g @X @X g @X @X ð21Þ

K  ¼ x2

Z

1

ðI1 U 2 þ I1 W 2 þ 2I2 Uw þ I3 w2 ÞdX

ð22Þ

0

The dimensionless total energy of the beam is  Y

¼ V L þ V NL þ K 

ð23Þ

3. Solution method The present study is focused on the flexural vibration behavior of the nanocomposite beam. Ritz method is employed in this paper to obtain the vibration frequencies and the free vibration amplitude of the beam. Three typical boundary conditions, i.e. clamped–clamped (C–C), clamped-hinged (C–H) and hinged– hinged (H–H), are considered, for which the following trial polynomials satisfying all displacement conditions are used for the dimensionless displacement components

8 n X > > aj X j ð1  XÞ UðXÞ ¼ > > > > j¼1 > > > > n < X bj X j ð1  XÞ C—C beam : WðXÞ ¼ > > j¼1 > > > n > X > > > cj X j ð1  XÞ > : wðXÞ ¼

Q Q Q @ @ @ ¼ 0; ¼ 0; ¼0 @ aj @bj @ cj

ð27Þ

from which the governing equation for nonlinear vibration of the nanocomposite beam is derived in matrix form

a11

Z

where n is the number of polynomial terms, aj, bj and cj are unknown coefficients to be determined. It should be mentioned that the shear stress obtained from Timoshenko beam theory based formulations does not satisfy the traction free boundary conditions on both the top and bottom surfaces of the beam which can be fully met in higher-order beam theories [44–48]. Substituting the above trial functions into Eqs. (20)–(23) then minimizing the dimensionless total energy with respect to the unknown coefficients, one has

  1 1 KL þ KNL1 þ KNL2 d  x2 Md ¼ 0 2 3

ð28Þ

where vector d = {{aj}T {bj}T {cj}T}T, M is the mass matrix, KL is the linear stiffness matrix, KNL1 and KNL2 are nonlinear stiffness matrices that are linearly and quadratically dependent on unknown vector d, respectively. The elements of the stiffness matrices are listed in the Appendix A. The nonlinear stiffness matrix KNL1 is associated with the bending-stretching coupling effect due to the asymmetric GPL distribution in Pattern 4 and vanishes for beams with symmetric GPL distributions such as Patterns 1, 2 and 3, or for beams with C–C end supports while KNL2 is associated with the elongation of the beam when subjected to large amplitude vibration. Because of the bending-stretching coupling effects, C–H and H–H beams with GPL distribution Pattern 4 have different half-cycle frequencies at positive and negative amplitudes, i.e. x+ – x–. In such circumstances, the nonlinear Eq. (30) needs to be solved by employing an iterative method described in Ref. [30] to find the period T+ at the positive deflection cycle and its counterpart T- at the negative deflection cycle due to the fact that the energy required in each deflection cycle is same

Tþ ¼

p p ; T ¼  xþ x

ð29Þ

Then the nonlinear frequency of the functionally graded GPL reinforced beam is obtained as

x¼

2xþ x xþ þ x

ð30Þ

ð24Þ 4. Results and discussion 4.1. Convergence and validation study

j¼1

8 n X > > aj X j ð1  XÞ > UðXÞ ¼ > > > j¼1 > > > > n < X bj X j ð1  XÞ C—H beam : WðXÞ ¼ > > j¼1 > > > n > X > > > cj X j > : wðXÞ ¼

ð25Þ

j¼1

8 n X > > > aj X j ð1  XÞ UðXÞ ¼ > > > j¼1 > > > > n < X bj X j ð1  XÞ H—H beam : WðXÞ ¼ > > j¼1 > > > n > X > > > cj X j1 > : wðXÞ ¼ j¼1

Table 1 lists the dimensionless linear fundamental natural frequencies xl of C–C, C–H, and H–H multilayer GPL/epoxy nanocomposite beams with varying numbers of polynomial terms to check the convergence of the present analysis. Epoxy is chosen as the polymer matrix material. The material properties of GPLs and epoxy and the dimensions of GPL nanofillers are the same as those in [4], i.e. EM = 2.85 GPa, qM = 1.2 g/cm3, lGPL = 2.5 lm, wGPL = 1.5 lm, hGPL = 1.5 nm, qGPL = 1.06 g/cm3, EGPL = 1.01 TPa. Table 1 Dimensionless linear fundamental natural frequency of multilayer GPL/epoxy nanocomposite beams (fGPL = 0.5%, N = 10, Pattern 3, L/h = 20).

ð26Þ

n

C–C

C–H

H–H

2 3 4 5

1.28878 0.58463 0.58463 0.58463

0.67273 0.40397 0.40397 0.40397

0.40138 0.25853 0.25853 0.25853

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C. Feng et al. / Engineering Structures 140 (2017) 110–119

The dimensions of the beam are chosen as L = 2 m and h = 0.1 m with length-to-thickness ratio being 20 unless stated otherwise. As can be seen, convergent results can be obtained at n = 4 for all boundary conditions. Table 2 compares the nonlinear frequency ratio xnl/xl of an H– H isotropic homogeneous beam with those given by Sing et al. [49] and Lestari and Hanagud [50] where xl and xnl stand for the dimensionless linear and nonlinear fundamental frequencies of the beam. The dimensionless vibration amplitude wmax/H varies pffiffiffiffiffiffiffi from 1 to 5, where H ¼ I=A with A and I representing the cross-section area and area moment of inertia of the beam, respectively. Good agreement is achieved. Table 3 compares the dimensionless linear fundamental pffiffiffiffiffiffiffiffiffiffiffiffiffiffi frequency xl ¼ X D0 =I10 obtained from the present study with the previous results for FGM beams with L/h = 20. Here D0 ¼ D110  B2110 =A110 with D110, B110 and A110 being the corresponding values of D11, B11 and A11 for an isotropic homogeneous beam. Young’s modulus, mass density and Poisson’s ratio on the top surface of the FGM beam are E1 = 70 GPa, q1 = 2780 kg/m3 and m1 = 0.33, respectively. The material properties of the beam vary in the thickness direction following an exponential function as described in [51]. E2 in the table denotes the Young’s modulus at the bottom surface of the FGM beam. As a special case, E2/E1 = 1 represents an isotropic homogeneous beam. Again, the present results agree well with those by Yang and Chen [51]. It is known that the accuracy of the Ritz method is highly dependent on the assumed displacement functions. Table 4 compares the linear natural frequencies of H–H nanocomposite beams obtained by using polynomials in Eqs. (24)–(26) and the following trigonometric functions [52,53]:

UðXÞ ¼ ¼

n X

n X bj sinðjpXÞ; wðXÞ

j¼1

j¼1

aj sinðjpXÞ; WðXÞ ¼

n X

cj cosðjpXÞ

ð30Þ

j¼1

Table 2 Comparison of fundamental frequency ratio xnl/xl of an H–H isotropic homogeneous beam (h = 0.1 m, L/h = 20). Wmax/H

Present

Singh et al. [49]

Lestari and Hanagud [50]

1.0 2.0 3.0 4.0 5.0

1.0684 1.3136 1.6435 2.0097 2.4001

1.0897 1.3299 1.6394 2.0000 2.3848

1.0892 1.3178 1.6257 1.9760 2.3502

Table 3 Comparison of dimensionless linear fundamental natural frequency of FGM beams. E2/E1

0.2 1.0 5.0

C–C

H–H

Present

Yang and Chen [51]

Present

Yang and Chen [51]

5.2903 5.5933 5.3003

5.25 5.59 5.25

2.5205 2.4674 2.5205

2.51 2.47 2.51

Table 4 Comparisons of dimensionless linear fundamental natural frequency of H–H GPL/epoxy nanocomposites beams using different displacement functions (fGPL = 0.5%, N = 10, Pattern 3, L/h = 20).

As can be seen, the results by using these two different sets of trial functions are almost identical when n = 3. The same observations can be found for C–C and C–H beams as well. Together with the results in Tables 2 and 3, it is evident that the assumed polynomial functions are capable of producing results with good accuracy. 4.2. Linear free vibration analysis Table 5 tabulates the dimensionless linear natural frequencies of GPL/epoxy nanocomposite beams for the first three modes. It is found that among all distribution patterns considered, GPL distribution Pattern 3 gives the highest natural frequencies while Pattern 2 results in the lowest natural frequencies. With the same GPL weight fraction fGPL = 0.5%, the fundamental natural frequency with Pattern 3 is 18.6% (C–C), 19.4% (C–H) and 20% (H–H) higher than that of the beams where GPLs are uniformly distributed (Pattern 1), indicating that the best reinforcing effect can be achieved by an appropriate non-uniform distribution of GPLs. Fig. 3 presents the effect of GPL weight fraction on the percentage frequency change of the beam, defined as the percentage ratio between the fundamental frequencies with and without GPLs, by taking C–C GPL/epoxy nanocomposite beam as an example. The frequency is significantly increased through the addition of a very small amount of GPLs into the polymer matrix. For example, the natural frequency can be increased by 25.9% at a very low GPL concentration fGPL = 0.1% with Pattern 3 and can be considerably further increased as fGPL increases. The frequency change is also highly dependent on GPL distribution pattern. At fGPL = 0.5%, the fundamental frequency is almost doubled with Pattern 3 but is increased by 45% only with pattern 2. This clearly indicates that both GPL weight fraction and distribution pattern play an important role in enhancing the beam stiffness and placing more GPLs near the top and bottom surfaces of the beam (Pattern 3) is the most effective way to increase natural frequencies. The effects of GPL geometry and size, in the form of length-tothickness ratio lGPL/hGPL and length-to-width ratio lGPL/wGPL, on the frequency change ratio of GPL/epoxy nanocomposite beams are investigated in Fig. 4 where lGPL remains constant. Note that lGPL/wGPL = 1 and – 1 correspond to a square shaped and a rectangular shaped GPL, respectively, and GPLs with a lower lGPL/wGPL and a bigger lGPL/hGPL have a larger surface area and fewer single graphene layers. A significant increase in natural frequency is observed as lGPL/hGPL increases all the way up to 1000–1500 beyond which the frequency remains almost unaffected regardless of the further increase in lGPL/hGPL. The natural frequency also becomes lower at a bigger lGPL/wGPL ratio. This is because with the same amount of GPLs, a larger surface contact area between the polymer matrix and GPLs provides better load transfer thus

Table 5 Dimensionless linear natural frequency of GPL/epoxy nanocomposite beams (fGPL = 0.5%, N = 10, L/h = 20). Mode

Pattern 1

Pattern 2

Pattern 3

Pattern 4

C–C 1 2 3

0.49288 1.32817 2.53250

0.41449 1.12519 2.16504

0.58463 1.55394 2.91701

0.46755 1.26195 2.41114

0.33846 1.07855 2.19867

0.28360 0.90928 1.86924

0.40397 1.27227 2.55433

0.32305 1.02519 2.09090

0.21542 0.85226 1.88292

0.17992 0.71526 1.59220

0.25853 1.01309 2.20666

0.21180 0.80705 1.79436

n

Polynomial

Trigonometric

C–H 1 2 3

2 3 4 5

0.40138 0.25853 0.25853 0.25853

0.25852 0.25852 0.25852 0.25852

H–H 1 2 3

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C. Feng et al. / Engineering Structures 140 (2017) 110–119

100

120

fGPL = 0.10%

N = 10, fGPL = 0.5%

90

Frequency change ratio (%)

Frequency change ratio (%)

fGPL = 0.25% fGPL = 0.50%

N = 10 60

30

Pattern 1 Pattern 2 Pattern 3 Pattern 4

80

60

40

5

10

15

20

25

30

35

40

45

50

Slenderness ratio L/h

0 Pattern 1

Pattern 2

Pattenr 3

Pattern 4

Fig. 3. Effect of weight fraction of GPL on frequency change ratio of C–C GPL/epoxy nanocomposite beams.

Fig. 6. Variation of frequency change ratio with slenderness ratio of C–C GPL/Epoxy nanocomposite beams.

200

Pattern 1 75

Pattern 4 Pattern 2

50

lGPL/wGPL= 1:

Pattern 1

Pattern 2

lGPL/wGPL= 3:

Pattern 3 Pattern 1

Pattern 4 Pattern 2

Pattern 3

Pattern 4

25

150

1

2

3

4

50

0

5

lGPL/hGPL (×1000) Fig. 4. Effect of GPL geometry and size on linear natural frequency of C–C GPL/ epoxy nanocomposite beams.

Frequency change ratio (%)

80

N=4 N=6 N=8 N = 10 fGPL = 0.5%

60 40 20 0

Pattern 2

Pattern 1

Pattern 2

Pattern 3

Pattern 4

Fig. 7. Effect of boundary conditions on frequency change ratio of GPL/Epoxy nanocomposite beams.

Table 6 Dimensionless nonlinear natural frequency of GPL/Epoxy nanocomposite beams (fGPL = 0.5%, N = 10, L/h = 20).

120 100

H-H

100

0 0

N = 10, fGPL = 0.5%

C-H

Pattern 3

Frequency change ratio (%)

Frequency change ratio (%)

C-C

N = 10, fGPL = 0.5%

100

Pattern 3

Mode

Pattern 1

Pattern 2

Pattern 3

Pattern 4

C–C 1 2 3

0.58302 1.69755 3.30280

0.50870 1.55319 3.11394

0.65875 1.87276 3.57584

0.51877 1.65045 3.21529

C–H 1 2 3

0.43263 1.45787 3.07798

0.39101 1.35461 2.80241

0.49152 1.59764 3.16084

0.41111 1.41459 2.90949

H–H 1 2 3

0.27259 1.07270 2.33122

0.26043 0.98768 2.09158

0.31973 1.20509 2.54097

0.26139 1.04504 2.22756

Pattern 4

Fig. 5. Effect of total number of layers on frequency change ratio of C–C GPL/Epoxy nanocomposite beams.

leading to higher structural stiffness. In other words, GPLs with a larger surface area and fewer single graphene layers are better reinforcing nanofillers than their counterparts with a smaller surface area and more single graphene layers. This observation on

the effects of GPL geometry and dimensions agrees with the arguments in previous studies [54,55]. The fabrication of an ideal functionally graded structure with a continuous and smooth variation of GPLs across the thickness is extremely difficult due to the constraint of manufacture technology. A functionally graded GPL reinforced multilayer nanocomposite structure in which each individual layer is reinforced by

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C. Feng et al. / Engineering Structures 140 (2017) 110–119

uniformly distributed GPLs while GPL concentration incrementally varying layer by layer is much easier to fabricate. Such a multilayer structure is an excellent approximation if a sufficient number of layers are used. Fig. 5 demonstrates the effect of the total number of layers on the frequency change ratio of C–C GPL/epoxy nanocomposite beams. As can be seen, as the total number of layers increases from 4 to 10, the frequency change ratio increases for pattern 2 and pattern 4 while it decreases for pattern 3. The difference between the results with N = 8 and N = 10 becomes very small and negligible (less than 3% for Pattern 2 and less than 1% for Patterns 3 and 4). This suggests that a multilayer structure with 10 layers would be accurate enough to approximate the ideal FGM structure.

N = 10, fCNT = 0.5% Pattern 1 Pattern 2 Pattern 3 Pattern 4

Frequency ratio (ωnl/ωl )

1.6

1.4

1.2

(a) C-C 1.0 -1.0

-0.5

0.0

0.5

1.0

Dimensionless amplitude (wmax/h)

Frequency ratio (ωnl/ωl )

2.00

N = 10, fCNT = 0.5% Pattern 1 Pattern 2 Pattern 3 Pattern 4

1.75

1.50

1.25

(b) C-H 1.00 -1.0

-0.5

0.0

0.5

1.0

Fig. 6 plots the frequency change ratio versus the slenderness ratio curves for C–C GPL/epoxy nanocomposite beams with four different GPL distribution patterns. Results indicate that the frequency change ratio is independent of slenderness ratio of the beams. The same trend is also observed for C–H and H–H beams for which the results are omitted for brevity. Fig. 7 depicts the frequency change ratio results for beams with different boundary conditions. As can be seen, the frequency change ratio is the highest for the H–H beam with GPL distribution Pattern 3 and the lowest for the C–C beam with Pattern 2. 4.3. Nonlinear vibration Table 6 lists the first three dimensionless nonlinear natural frequencies xnl for GPL/epoxy nanocomposite beams at vibration amplitude (wmax/h = 0.5). Similar to the observations in linear free vibration analysis, beams with GPL distribution Pattern 3 have the highest nonlinear natural frequency. Fig. 8 examines the effect of vibration amplitude on the frequency ratio, xnl/xl, for GPL/epoxy nanocomposite beams where 0.5 w.t.% GPLs are dispersed in the matrix. The frequency ratio exhibits a typical ‘‘hard spring” behavior, i.e. increases as the vibration amplitude increases. Due to the bending-stretching coupling effect represented by the third term in Eq. (12), C–H and H–H beams with asymmetrical GPL distribution pattern 4 have different frequency ratios at positive and negative vibration amplitudes of the same magnitude hence their curves are asymmetric. In contrast, the frequency ratio curves of C–C beams and those with symmetrical GPL distribution Patterns 1, 2 and 3 are symmetric due to the fact that the bending-stretching coupling effect vanishes in these beams. To quantify the nonlinearity effects at higher vibration modes, Fig. 9 plots the variation of nonlinear frequency ratio xnl/xl against the maximum beam deflection for the first four vibration modes. It is demonstrated that the frequency ratio becomes more sensitive to the nonlinear effects as vibration amplitude increases, which is attributed to the increasing effect of the stretching of neutral axis at larger deformation. At the same vibration amplitude, the effect of nonlinear deformation on the frequency ratio becomes more pronounced for higher vibration modes. Fig. 10 displays the free vibration response of GPL/epoxy nanocomposite beams with different GPL distribution patterns. It _ is assumed that the initial conditions are Wð0Þ ¼ 1 and Wð0Þ ¼ 0. Apparently, the distance between two consecutive response peaks which virtually represents the period is the smallest for the beam

Dimensionless amplitude (wmax/h) 1.8

2.25 N = 10, fCNT = 0.5%

2.00

Frequency ratio (ωnl /ωl )

Frequency ratio (ωnl/ωl )

N = 10, fCNT = 0.5% Pattern 1 Pattern 2 Pattern 3 Pattern 4

1.75 1.50 1.25

st

1 mode nd 2 mode rd 3 mode th 4 mode

1.6

1.4

1.2

(c) H-H 1.00 -1.0

-0.5

0.0

0.5

1.0

Dimensionless amplitude (wmax/h) Fig. 8. Frequency ratio curves of GPL/epoxy nanocomposite beams with different GPL distribution patterns and boundary conditions: (a) C–C; (b) C–H; (c) H–H.

C-C 1.0 -1.0

-0.5

0.0

0.5

1.0

Dimensionless amplitude (wmax/h) Fig. 9. Nonlinear frequency ratio curves of GPL/epoxy nanocomposite beams for the first four vibration modes (GPL distribution pattern 3, C–C).

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C. Feng et al. / Engineering Structures 140 (2017) 110–119

1.5

Dimensionless amplitude w/h

Pattern 1

Pattern 2

Pattern 3

Pattern 4

1.0 0.5 0.0 -0.5 -1.0

(a) C-C -1.5

0

5

10

15

20

25

30

Dimensionless time τ 1.5

Dimensionless amplitude w/h

Pattern 1

Pattern 2

Pattern 3

Pattern 4

1.0 0.5 0.0 -0.5

Acknowledgements -1.0

(b) C-H -1.5

0

5

10

15

20

25

30

Dimensionless time τ 1.5 Pattern 1

Dimensionless amplitude w/h

comprehensively investigated within the framework of Timoshenko beam theory and von Kármán nonlinear straindisplacement relationship. The results and findings of this study clearly demonstrate the great potential of using GPLs in achieving novel engineering structures with significantly improved structural performance. It is found from numerical results that adding a very small amount of GPL into polymer matrix can significantly increase the natural frequency of the composite beam. At the same weight fraction, GPLs with a larger surface area but fewer single graphene layers exhibit better reinforcing effects. The comparisons among the four GPL distribution patterns indicate that dispersing more GPL reinforcements near the top and bottom surfaces of the beam is the most effective way to increase the beam stiffness for higher natural frequencies. It is also confirmed that a multilayer structure with 10 individual layers stacked up is accurate enough to obtain an ideal functionally graded structures with continuous and smooth variation in material properties. Furthermore, C–H and H–H beams where GPLs are non-symmetrically distributed according to Pattern 4 have different half cycles at negative and positive amplitudes at the same magnitude. It should be pointed out that the present analysis is based on the assumption of perfect bonding with no slipping between GPLs and polymer matrix. In case that there exists imperfect bonding that may deteriorate the mechanical properties of the nanocomposite under larger deformation [56], a cohesive law that accounts for the bonding state needs to be incorporated into the micromechanics model.

Pattern 2

Pattern 3

Pattern 4

1.0

The work described in this paper is fully funded by a research grant from the Australian Research Council under Discovery Project scheme (DP160101978). The authors are grateful for this financial support. Dr. Chuang Feng is also grateful for the support from the Australian Research Council under Discovery Early Career Researcher Award (DECRA) scheme (DE160100086). Appendix A

0.5

The trial functions for various end supports are expressed in following general forms

0.0

UðXÞ ¼ -0.5

n X

n X

j¼1

j¼1

aj H1j ; WðXÞ ¼

bj H1j ; wðXÞ ¼

n X

cj H2j

j¼1

Elements for linear stiffness matrix KL

Z

-1.0

(c) H-H -1.5

0

5

10

15

20

25

a11

@H1j @H1m dX; @X @X

b11

@H1j @H2m dX @X @X

1

KL ðj; mÞ ¼ 0

30

Dimensionless time τ Fig. 10. Free vibration response of GPL/epoxy nanocomposite beams with different GPL distribution patterns and boundary conditions: (a) H–H; (b) C–H; (c) H–H.

Z

1

¼ 0

Z

1

KL ðn þ j; n þ mÞ ¼

a55 0

with GPL distribution Pattern 3 and the biggest for its counterpart with asymmetric GPL distribution Pattern 4. This is consistent with the results for natural frequencies discussed above. In addition, from Fig. 10b and c it can be seen that C–H and H–H beams with GPL distribution according to Pattern 4 have smaller negative amplitude and different half-cycles at positive and negative amplitudes.

Z ¼

Z KL ð2n þ j; 2n þ mÞ ¼ 0

1

@H1j @H2m dX @X @X

  @H2j @H2m d11 þ a55 g2 H2j H2m dX @X @X

Elements for symmetric mass matrix M 1

Mðj; mÞ ¼

I1 @H1j @H1m dX; Mðj; n þ mÞ ¼ 0; Mðj; 2n þ mÞ

0

Z Linear and nonlinear free vibration behaviours of nanocomposite beams reinforced with non-uniformly distributed GPLs are

@H1j @H1m dX; KL ðn þ j; 2n þ mÞ @X @X

a55 g

0

Z 5. Conclusions

1

KL ðj; n þ mÞ ¼ 0; KL ðj; 2n þ mÞ

¼ 0

1

I2 @H1j @H2m dX

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C. Feng et al. / Engineering Structures 140 (2017) 110–119

Z

1

Mðn þ j; n þ mÞ ¼

I1 @H1j @H1m dX; Mðn þ j; 2n þ mÞ

0

Z

1

¼ 0; Mð2n þ j; 2n þ mÞ ¼

I3 @H2j @H2m dX

0

Elements for nonlinear stiffness matrix KNL1

KNL1 ðj; mÞ ¼ 0; KNL1 ðj; n þ mÞ Z 1 a11 @W @H1j @H1m dX; KNL1 ðj; 2n þ mÞ ¼ 0 ¼ g @X @X @X 0 Z KNL1 ðn þ j; n þ mÞ ¼ 0

Z KNL1 ðn þ j; 2n þ mÞ ¼ 0

1

1

  a11 @U b11 @w @H1j @H1m þ dX; g @X g @X @X @X

b11 @W @H1j @H2m dX; KNL1 ð2n þ j; 2n þ mÞ ¼ 0 g @X @X @X

Elements for nonlinear stiffness matrix KNL1

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