Effect of fiber transverse isotropy on effective thermal conductivity of metal matrix composites reinforced by randomly distributed fibers

Accepted Manuscript Effect of fiber transverse isotropy on effective thermal conductivity of metal matrix composites reinforced by randomly distributed fibers Wenlong Tian, Lehua Qi, Changqing Su, Jian Liu, Jiming Zhou PII: DOI: Reference:

S0263-8223(16)30692-4 http://dx.doi.org/10.1016/j.compstruct.2016.05.070 COST 7483

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

31 January 2016 16 May 2016 23 May 2016

Please cite this article as: Tian, W., Qi, L., Su, C., Liu, J., Zhou, J., Effect of fiber transverse isotropy on effective thermal conductivity of metal matrix composites reinforced by randomly distributed fibers, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.05.070

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Effect of fiber transverse isotropy on effective thermal conductivity of metal matrix composites reinforced by randomly distributed fibers Wenlong Tiana,b, Lehua Qia, ∗, Changqing Sua, Jian Liuc, Jiming Zhoua

aSchool

of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072,

P.R.China bDepartment

of Civil and Environmental Engineering, Northwestern University, Evanston

60208, Illinois, USA cInstitute

of Printing and Packaging Engineering, Xi'an University of Technology, Xi'an

710048, P.R.China *Corresponding author. Tel.: +86-29-88460447, Fax: +86-29-88491982, E-mail address: [email protected] (Lehua Qi)

Abstract

The effective thermal conductivity of metal matrix composites (MMCs) reinforced by the randomly distributed transversely isotropic fibers is evaluated using the representative volume element (RVE) based finite element (FE) homogenization method. The modified random sequential absorption (RSA) algorithm is proposed to generate the periodic RVEs of MMCs and thus the periodic boundary conditions are introduced. The transversely isotropic thermal conductivity of fiber is considered and its numerical implementation in the FE analysis is detailed. The RVE based FE homogenization method to predict the effective thermal conductivity of MMCs reinforced by the randomly distributed

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fibers is verified by comparing against the experiment and the Hashin–Shtrikman bounds. The simulation results show that the effect of the transverse isotropy of fiber thermal conductivity on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers can be neglected owing to the large matrix/fiber thermal conductivity ratio. The effective thermal conductivity of MMCs reinforced by the randomly distributed fibers decreases with the increase of fiber volume fraction, while the effect of fiber aspect ratio on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers is not significant. Keywords: Fiber transverse isotropy; Finite element homogenization; Metal matrix composites (MMCs); Short-fiber composites; Thermal conductivity

1. Introduction

Metal matrix composites (MMCs) are widely used in the aviation, aerospace, automobile industries and other related fields [1, 2], because of the low weight, superior material properties [3, 4]. Different types of reinforcements including fibers, particles, whiskers and various metal matrix materials such as aluminum, copper and magnesium alloys have been used to fabricate these heterogeneous composites for the different applications [5, 6]. There exists many applications of MMCs relating to the heat transfer, which in turn requires modelling the effective thermal conductivity of MMCs.. Over the last decades, the theoretical predictive models ranging from the

2

simple weighted average of the constituent thermal conductivities (for example the rule of mixtures) to the complicated formulations that account for the spatial and orientation variation of inclusions have been developed to evaluate the effective thermal conductivity of composites. For composites with the simple microstructures, the effective thermal conductivity can be modelled using the existing fundamental structural models, such as Series model [7], Parallel model [7], Maxwell-Eucken model (two forms) [8], Effective Medium Theory model [9] and Mori–Tanaka model [10, 11]. However, these fundamental structural models are not appropriate any more for composites with the complicated microstructures. An alternative method to predict the effective thermal conductivity of composites with the complicated microstructures is to introduce the empirical models which are generally derived by modifying or combining the fundamental structural models. Wang et al. [12, 13] proposed the procedures to combine the fundamental structural models based on the inclusion volume fraction and by using the simple combinatory rules to evaluate the effective thermal conductivity of composites. The combined models possessed advantages that they were not dependent on any empirical parameter and that they could provide the reasonable accuracy. However, it does not appear to exist one single empirical model applicable for all types of composites. Meanwhile, the empirical models do not have the clear physical explanations nor provide the detailed heat flux and temperature fields.

3

Therefore, the numerical homogenization method following the direct finite element (FE) computation of the boundary-value problem at a representative volume element (RVE [14, 15]) (entitled the RVE based FE homogenization method) comes into sight [16, 17], which has been well documented for the determination of the effective thermal conductivity of composites. Tu et al. [18] adopted the FE method to investigate the effective thermal conductivity of PTFE composites reinforced by the randomly distributed short fibers with the aspect ratio l / d of 7. The method was verified by the comparison with the experiment results. Considering the interface thermal conductance, Marcos-Gómez et al. [19] introduced the FE method based on the concept of RVE and equivalent inclusion method to predict the thermal conductivity of a composite. The simulation results showed the most important parameters controlling the thermal conductivity were the interface thermal conductance and the preferred orientation of the fibers. Duschlbauer et al. [20] used the FE homogenization of the unit cells to validate the proposed replacement scheme which combined the orientation average and multi-inclusion Mori-Tanaka model to study the thermal conduction responses of composites reinforced by randomly oriented short fibers at non-dilute volume fractions. In this paper, the effective thermal conductivity of MMCs reinforced by randomly distributed transversely isotropic fibers is evaluated by using the RVE based FE homogenization method. To generate the periodic RVEs of MMCs reinforced by the randomly distributed fibers, the modified random sequential

4

absorption (RSA) algorithm is proposed and implemented, and thus the periodic boundary conditions are introduced. The transversely isotropic thermal conductivity of fiber is considered and the numerical implementation in the FE analysis is detailed. The effects of the transverse isotropy of fiber thermal conductivity, the fiber volume fraction and the fiber aspect ratio on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers are investigated.

2. Representative volume element

To predict the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers by using the RVE based FE homogenization method, the first step should be to generate the periodic RVEs of MMCs reinforced by the randomly distributed fibers. During the generation of RVE, the fiber intersection is not acceptable [21]. Considering any two fibers P1(x) and P2 (x) , the shortest distance between any two points x1 and x2 located on the surfaces of fibers P1(x) and P2 (x) , respectively, must be greater than 0,

d(x1, x2 ) > 0 x1 ∈P1 and x2 ∈P2

(1)

The detailed algorithm and implementation of RVE generation are given in the previous publication [22]. Note that a minimum separation distance

dmin (e.g. 5%

of the fiber radius) between the surfaces of any two fibers and between the surfaces of any fiber and the cubic RVE is necessary to avoid the finite elements becoming distorted [23].

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To guarantee the periodicity of a RVE, the fibers penetrating the surfaces of the RVE are slit into the appropriate number of parts which then are translated to the opposite surfaces (Figure.1) [22, 24]. The formulation is given as,

PN (xi ) = P(xi ) − κ i L i = 1,2 or 3

(2)

i

where PN ( x ) are the translated fiber parts, x i are the translated coordinates in the corresponding dimension,

κ i ∈{−1,0or1}

is the coefficient depending on

the specified penetrated surface of the RVE. Note that up to three surfaces of a RVE can be penetrated by one single fiber. The RSA algorithm [24, 25] combining the above conditions (entitled modified RSA algorithm) is used to generate the periodic RVEs of MMCs reinforced by the randomly distributed fibers. Here, the identical fibers are considered only and a generated periodic RVE and mesh of MMCs reinforced by the randomly distributed fibers with the aspect ratio of 15 and fiber volume fraction of 10% are illustrated in Figures. 2.

3. Governing equations

The boundary conditions and the presence of internal heat source may strongly affect the temperature and heat flux fields in the RVEs. The thermal conductivity, however, is intrinsic to materials and should not depend on such external effects. Therefore, here the steady state heat transfer of a RVE Ω with the prescribed temperature boundary condition is considered,

div ( k ⋅∇T ) = 0 with T = T on ∂ΩT

6

(3)

where

∂ΩT is the temperature boundary. For the isotropic matrix, the

components of the thermal conductivity tensor k m have the following m

m

m

m

m

m

m

expressions: k11 = k22 = k33 and k12 = k21 = k23 =⋅⋅⋅ = k31 = 0 . However, the short fibers are considered to be transversely isotropic so that for the short fibers the principal axis directions should be assigned by establishing the element coordinate system (local coordinate system attached to a fiber), and only the f

f

f

components k11 , k22 , k33 of the thermal conductivity tensor in three principal axis directions are required, which will be discussed in Section 5. Based on Eq. (3) with the prescribed temperature boundary condition, one can get the temperature and heat flux fields by the FE analysis. Then, the effective heat flux vector

q

is related to temperature gradient vector ∇T by the

Fourier's law,

q = − k ⋅∇T

(4)

where ∇T = (∇T1 , ∇T2 , ∇T3 )T is the temperature gradient vector imposed on as the

boundary

conditions,

and

the

effective

heat

flux

vector

q = ( q 1 , q 2 , q 3 )T can be obtained by the post-processing of the FE analysis results by,

qi =

1 VRVE

 neint  1 V qi ( yI ) ⋅W ( yI )  = ∑e e  ∑  I =1  VRVE

nint

∑ q ( y ) ⋅ IVOL( y ) i

I

I

(5)

I =1

where neint and nint are the numbers of the integration point in the element e and in the whole RVE, respectively. W ( yI ) is the integration point weight at an integration point positioned at yI in the element e , whose volume is Ve .

IVOL ( yI ) = Ve ⋅W ( yI ) , in which IVOL( yI ) is the integration point volume at an

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integration point positioned at yI .

4. Periodic boundary condition

The periodic boundary conditions [24, 26] are introduced to predict the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers. Consequently, the boundaries ∂Ω of a RVE (domain Ω ) are decomposed into two opposite parts: ∂ΩT + and ∂ΩT − such that ∂Ω = ∂ΩT + ∪ ∂ΩT − and

∅ = ∂ΩT + ∩ ∂ΩT − . Each node x + on the boundaries ∂ΩT + is associated with a unique node x − on the opposite boundaries ∂ΩT − . Then, the periodic boundary conditions are defined by (no Einstein’s summation), Tx+ − Tx− = ∇Ti ⋅ ( xi+ − xi− ) with i = 1, 2 and 3 i

(6)

i

where Tx+ and Tx− are the applied temperature for each pair of the nodes xi+ and i

i

xi− located on the parallel boundary surfaces, edges and corner vertices on the

RVE, respectively. ∇Ti is the temperature gradient. Clearly, in the RVE xi+ − xi− is constant for each pair of the nodes located on the parallel boundary surfaces, edges and corner vertices so that ∇Ti ( xi+ − xi− ) becomes constant. Therefore, the periodic boundary condition is implemented in the FE analysis as a nodal temperature constraint,

Tx+ − Tx− = ∆Ti i

with i = 1, 2 and 3

(7)

i

The nodes on the RVE boundaries are categorized into three different sets, as shown in Figure. 3:


Set I: Inner surface nodes ( S Top , S Bottom , S Front , S Back , S Left and S Right ) are

8

the nodes that belong to one surface excluding the nodes on the common edges and vertices;


Set II: Inner edge nodes ( E AB , E BC , E CD , E DA , E A1B1 , E B1C1 , E C1D1 , E D1 A1 , E AA1 , E BB1 , E CC1 and E DD1 ) are the nodes that belong to an edge

excluding the end nodes which resembles the vertices;


Set III: Vertical nodes ( V A , V B , V C , V D , V A1 , V B1 , V C1 and V D1 ) are the eight vertical nodes. A linear multi-point constraint [27] is performed to set up the periodic

boundary conditions in Eq. (7),

Tk + − Tk − − TRP = 0 i

(8)

i

where ki+ and ki− are subscripts representing the relative periodic nodes on the opposite surfaces noted. The temperature difference is introduced to the system of equation through a Reference Point, which is not linked to any element within the model but simply to provide it with the necessary degrees of freedom (DOFs) through which the thermal conduction behaviors of RVE are controlled. Note that in ABAQUS [27], once a DOF has been used in a constraint equation, it cannot be used in another constraint equation, because this DOF has been eliminated by the previous constraint equation.

To determine the components kij of the effective thermal conductivity tensor k in Eq. (4), a RVE has to be numerically simulated three times for ki1 ,

ki 2 and ki3 ( i = 1, 2 and 3 ), respectively, by using the different periodic boundary conditions.

9

5. Fiber transversely isotropic thermal conductivity

The thermal conductivity of fiber is considered as transversely isotropic along its axis. To define the transverse isotropy of fiber in the FE analysis, a local coordinate system o L ( x1L )i ( x2L )i ( x3L )i is required and attached to the fiber i , because that the fibers are distributed randomly in the RVEs of MMCs reinforced by the randomly distributed fibers which means the principal axes of fibers do not coincide with the axes x1G , x2G and x3G of the global coordinate system o G x1G x2G x3G attached to the RVE.

In the FE analysis, the thermal conductivity tensor of a fiber requires to be transformed from the local coordinate system to the global coordinate system. For a spatially oriented fiber in the RVE, its orientation can be described by the Euler angles θ , ϕ and φ as depicted in Figure. 4. Denote u Lj and ulG as the unit basis vectors of the local and global coordinate systems, respectively. The relation between these two unit basis vectors is given by,

ulG = R jl ⋅ u Lj

(9)

where the matrix R jl is the transformation matrix mapping the j th axis in the local coordinate system o L ( x1L )i ( x2L )i ( x3L )i to the l th axis in the global coordinate system o G x1G x2G x3G and is defined as [28],

nq mq   p  R jl =  qt −mpt + ms −mpt − ns   −qs nps + mt mps − nt 

(10)

q = sin θ , s = cos ω and t = sin ω . Note

with m = cos ϕ , n = sin ϕ , p = cos θ ,

that for simplification, the Euler angle ω can be selected as 0° . Then, the

10

transformation of the thermal conductivity tensor kiL for the fiber i from the local coordinate system o L ( x1L )i ( x2L )i ( x3L )i to the global coordinate system o G x1G x2G x3G can be performed through the tensor transformation law, and the

resulting thermal conductivity tensor for the fiber i is denoted by kiG ,

(k )

G i mn

( )

= Rm′m ⊗ Rn′n ⋅ kiL

m ′n′

.

(11)

In Abaqus [27], the “MATERIAL ORIENTATION” option is activated to transform the thermal conductivity tensor of the fibers from their local coordinate systems to the global coordinate system.

6. Results and discussion

MMCs reinforced by the randomly distributed fibers analyzed here are Csf/Mg composites, consisting of AZ91D magnesium alloy matrix with the isotropic material properties and spatially randomly distributed T300 short carbon fibers with the transversely isotropic material properties. The thermal conductivity of the matrix is 72 W ⋅ m −1 ⋅ K −1 , while the thermal conductivities of the fibers are 7.81 W ⋅ m−1 ⋅ K −1 and 0.675 W ⋅ m −1 ⋅ K −1 in the longitudinal (axial) and transverse directions, respectively. Note that in the case of isotropy, the thermal conductivity of the fibers is 7.81 W ⋅ m−1 ⋅ K −1 . The perfect bonded interfaces between the matrix and fibers in the Csf/Mg composites are hypothesized [16, 17].

6.1 Critical size of representative volume element

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Because of the complex structures of the matrix and irregular shape of the fibers, both the matrix and fibers are meshed using the tetrahedral elements (see Figures. 2). It is well known that the tetrahedral elements have relatively unfavorable convergence properties so that the mesh size of RVE might affect the predicted thermal conductivities of MMCs reinforced by the randomly distributed fibers. Thus, before the numerical homogenization of the effective thermal conductivities of MMCs reinforced by the randomly distributed fibers, the convergence studies in terms of mesh size of RVE are performed. The results demonstrate that the mesh size of RVE has the neglectable effect on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers, resulting from that the heat transfer due to the thermal conduction is actually linear. Thus, all RVEs hereafter are meshed using the mesh size L / le = 45 ( L is the length of RVE and le is the length of the element located on the surfaces of RVE). The RVE is a statistical representation of composites to effectively include a sampling of all microstructural heterogeneities in the composites such that the resulting geometrical RVE is independent of the properties of composites. On the other hand, it requires to take care of the minimum size of the RVE for saving the computational resource and time under the precondition of sufficiently accurately characterizing the constitutive responses of composites. In this paper, the different size RVEs of Csf/Mg composites are generated, in which the fiber aspect ratios are selected as 5 , 10 and 15 (the fiber diameter d is 7 µ m and

12

the fiber length l is 35µ m , 70µ m and 105µ m , respectively) and the fiber volume fraction is 10%. Figure. 5 illustrates the variation of the effective thermal conductivity of Csf/Mg composites with the fiber volume fraction of 10% and with the fiber aspect ratio of 5, 10 and 15, respectively, regarding the change of the size of RVE. Here onwards in all the graphs, the legend represents the error of the effective thermal conductivity around the mean value, which are obtained from the ensemble averages of the effective thermal conductivity of three different RVE samples at each length. From Figure. 5(a), we can observe that it is sufficient to consider the size of RVE as cell/fiber length ratio L / l = 3.0 ( L is the length of cubic RVE and l is the length of identical fiber) when the fiber aspect ratio l / d is 5 by taking into account both the relative error (less than 0.069% in the case of L / l = 3.5 , see Table. 1) and the variation of effective thermal conductivity around the mean value. However, Figure. 5(b) indicates that the critical size of RVEs with the larger fiber aspect ratios l / d = 10 and 15 can be considered as L / l = 2.0 (relative error being less than 0.13% in the case of L / l = 2.5 , see Table. 1). The phenomenon is explained that under the condition of the same size L / l , the RVEs with the larger fiber aspect ratios contain more fibers than those with the smaller fiber aspect ratios. So when the fibers with the smaller aspect ratios are used to generate RVEs of composites, the size of RVE should be chosen to be larger to contain more fibers such that the resulting material properties of composites are more accurate and are not of much variation. The results agree

13

well with the previous studies [24] on the critical size for RVEs, which are used to predict the elastic properties of composites with the randomly distributed fibers. Thus, the critical size of RVE of MMCs reinforced by the randomly distributed fibers is selected as L / l = 3.0 in the case of the fiber aspect ratio l / d < 10 , while in the case of the fiber aspect ratio l / d ≥ 10 the critical size of RVE is selected as L / l = 2.0 .

6.2 Validation of numerical homogenization

The MMCs used to validate the proposed RVE based FE homogenization method here are the short potassium titanate fibers reinforced the AlSi10Mg aluminum alloy (abbreviated as PTisf/Al composites), in which the fiber volume fraction is 25%. The thermal conductivities of the matrix and fibers are 155 W ⋅ m −1 ⋅ K −1 and 1.7 W ⋅ m −1 ⋅ K −1 , respectively. The average length and radium of

the fiber are 65 µ m and 6.5 µ m , respectively. The microstructures of PTisf/Al composites indicate that the potassium titanate fibers are oriented in a random configuration [29]. The effective thermal conductivity of PTisf/Al composites predicted by the RVE based FE homogenization method is compared against the standard Hashin–Shtrikman bounds [30], which are given by,  3 ⋅ (1 − v f ) ⋅ ( k m − k f )  klower = k f ⋅ 1 + f m f   3 ⋅ k + v f ⋅ ( k − k ) 

and

  3 ⋅ v f ⋅ (k f − k m ) kupper = k m ⋅ 1 + m f m   3 ⋅ k + (1 − v f ) ⋅ ( k − k ) 

(12) The calculated thermal conductivity (100.5041 W ⋅ m −1 ⋅ K −1 ) of PTisf/Al

14

composites with the fiber volume fraction of

25% falls within the

Hashin–Shtrikman bounds ( klower = 15.20 W ⋅ m −1 ⋅ K −1 , and kupper = 104.09 W ⋅ m −1 ⋅ K −1 ).

The RVEs of PTisf/Al composites with the fiber volume fraction of 25% are generated, the sizes of which are selected as cell/fiber length ratio L / l = 3.0 due to the fiber aspect ratio l / d of 5. The temperature distribution along the

x1 direction in the matrix and fibers is given in Figure. 6 and the corresponding heat flux fields on the opposite surfaces ABCD and A1B1C1D1 of RVE is shown in Figure. 7. The effective thermal conductivities of PTisf/Al composites predicted by the RVE based FE homogenization method and measured from the thermal conduction tests are listed in Table. 2, respectively. Note that the effective thermal conductivities of PTisf/Al composites parallel to the pressed plane and perpendicular to the pressed plane are given in Ref. [29], respectively. Here, the average of the effective thermal conductivities of PTisf/Al composites parallel to the pressed plane and perpendicular to the pressed plane is selected as the effective isotropic thermal conductivity of PTisf/Al composites. The effective thermal conductivity of PTisf/Al composites with the fiber volume fraction of 25% predicted by the RVE based FE homogenization method agreeing well with that measured from the thermal conduction experiments indicates the validation of the RVE based FE homogenization method on the prediction of the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers.

6.3 Effect of fiber transverse isotropy

15

Here, the effect of the transverse isotropy of fiber thermal conductivity on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers is emphasized. Table. 3 gives the effective thermal conductivities of Csf/Mg composites (fiber volume fraction of 10%) with the transversely isotropic and isotropic fibers, respectively. The effective thermal conductivities of Csf/Mg composites with the isotropic fibers are compared against the Hashin–Shtrikman bounds ( klower = 53.16 W ⋅ m−1 ⋅ K −1 and kupper = 63.24 W ⋅ m−1 ⋅ K −1 ). It is found that the numerical estimates for the effective conductivity of Csf/Mg composites with the isotropic fibers exceed the upper Hashin–Shtrikman bound by a relatively small margin of 1.5%, which can be considered as acceptable. From Table. 3, we can find the differences between the effective thermal conductivities of Csf/Mg composites with the transversely isotropic and isotropic fibers are less than 2.5%. Therefore, the effect of the transverse isotropy of fiber thermal conductivity on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers is not of significance. The reasons are stated as the random distribution of fibers and the large matrix/fiber thermal conductivity ratio, which is explained as that the matrix contributes more than the fibers to the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers. The homogenized effective thermal conductivity matrices kT and k I of Csf/Mg composites reinforced by the transversely isotropic and isotropic fibers

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with the fiber volume fraction of 10% and with the aspect ratio of 15 are presented as follows (in W ⋅ m −1 ⋅ K −1 ), respectively,

62.58 0.00 0.00   64.01 0.00 0.00    kT =  0.00 62.60 0.00  and kI =  0.00 63.99 0.00   0.00 0.00 62.50  0.00 0.00 63.91 Since the computed values of the off-diagonal entries in the matrices kT and k I are three or four orders of magnitude smaller than the diagonal entries and, thus, is considered as null. To investigate the isotropy of the effective thermal conductivity matrices kT and k I , the following isotropic parameter is defined,

Iii =

min kii , kΣ max kii , kΣ

with k Σ = ( k11 + k22 + k33 ) / 3

and

i = 1, 2 and 3

(13)

where Iii takes the value 1 for the composites with the isotropic thermal conductivity. Table. 4 gives the values of Iii by substituting the diagonal entries in the thermal conductivity matrices kT and k I , and it is illustrated that the effective thermal conductivities of MMCs reinforced by the randomly distributed transversely isotropic and isotropic fibers are perfectly approximated to that of an isotropic material, resulting from the spatially random fiber distribution [21].

6.4 Effects of fiber volume fraction and fiber aspect ratio

The effective material properties of composites are affected by the fiber volume fraction and the fiber aspect ratio, besides the material properties of the constituents. In this section, the effects of the fiber volume fraction and fiber aspect ratio on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers are investigated.

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The effective thermal conductivities of Csf/Mg composites with the fiber aspect ratios of 5, 10 and 15 for a range of the fiber volume fractions are plotted in Figure. 8, which shows that the effective thermal conductivity of Csf/Mg composites decreases approximately linearly with the increase of fiber volume fraction (the nonlinear behavior still exists). The fibers have the smaller thermal conductivities than the metal matrix, resulting that the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers decreases with the increase of fiber volume fraction. Figure. 9 gives the effective thermal conductivity of Csf/Mg composites with the fiber volume fraction of 10% regarding the change of fiber aspect ratio. It can be found that the effect of the fiber aspect ratio on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers can be neglected.

7. Conclusions

The effective thermal conductivity of MMCs reinforced by randomly distributed transversely isotropic fibers is investigated by using the RVE based FE homogenization method. To generate the periodic RVEs of MMCs reinforced by the randomly distributed fibers, the modified RSA algorithm is proposed. The thus required periodic boundary conditions are introduced. The transversely isotropic thermal conductivity of fiber is considered and the detailed numerical implementation in the FE analysis is described. Thus, the following conclusions are obtained:

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(1). The RVE based FE homogenization method is verified to predict the effective thermal conductivity of MMCs reinforced with the randomly distributed fibers by comparing against the experiments and the Hashin–Shtrikman bounds; (2). The effect of the transverse isotropy of fiber thermal conductivity on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers can be neglected owing to the large cell/fiber thermal conductivity ratio; (3). Due to the spatially random distribution of fibers, the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers can be viewed as isotropic; (4). The effective thermal conductivity of MMCs reinforced by the randomly distributed fibers decreases with the increase of fiber volume fraction and the effect of fiber aspect ratio on the effective thermal conductivity of MMCs reinforced by the randomly distributed fibers is not of significance.

Acknowledgement

This work is done under the financial support from the National Nature Science Foundation of China (Nos.51472203, 51521061 and 51432008), National

High

Tech

(2015AA8011004B),

Research Excellent

and

Development

Doctorate

Foundation

Polytechnical University and China Scholarship Council.

References

19

Program of

of

China

Northwestern

[1] I. Alfonso, O. Navarro, J. Vargas, A. Beltrán, C. Aguilar, G. González, I. Figueroa, FEA evaluation of the Al4C3 formation effect on the young’s modulus of carbon nanotube reinforced aluminum matrix composites, Composite Structures 127 (2015) 420 - 425. [2] L. Qi, J. Liu, J. Guan, J. Zhou, H. Li, Tensile properties and damage behaviors of Csf/Mg composite at elevated temperature and containing a small fraction of liquid, Composites Science and Technology 72 (14) (2012) 1774 - 1780. [3] I. Kientzl, I. Orbulov, J. Dobranszky, A. Nemeth, Mechanical behaviour Al-matrix composite wires in double composite structures, Advances in Science and Technology 50 (2006) 147 - 150. [4] H. Mechakra, A. Nour, S. Lecheb, A. Chellil, Mechanical characterizations of composite material with short Alfa fibers reinforcement, Composite Structures 124 (2015) 152 - 162. [5] S. Uddin, T. Mahmud, C. Wolf, C. Glanz, I. Kolaric, C. Volkmer, H. Höller, U. Wienecke, S. Roth, H. Fecht, Effect of size and shape of metal particles to improve hardness and electrical properties of carbon nanotube reinforced copper and copper alloy composites, 70 (2010) 2253 – 2257. [6] M. Barmouz, P. Asadi, M. Besharati Givi, M. Taherishargh, Investigation of mechanical properties of Cu/SiC composite fabricated by FSP: Effect of SiC particles’ size and volume fraction, 528 (2011) 1740 – 1749. [7] R. Progelhof, J. Throne, R. Ruetsch, Methods for predicting the thermal conductivity of composite systems: A review, Polymer Engineering & Science 16

20

(9) (1976) 615 – 625. [8] J. Maxwell, A Treatise on Electricity and Magnetism, third ed., Dover Publications Inc., New York, reprinted 1954 (Chapter 9). [9] R. Landauer, The electrical resistance of binary metallic mixtures, Journal of Applied Physics 23 (7) (1952) 779 – 784. [10] D. Duschlbauer, H. Böhm, H. Pettermann, Computational simulation of composites reinforced by planar random fibers: homogenization and localization by unit cell and mean field approaches, Journal of Composite Materials 40 (2006) 2217 - 2234. [11] B. Mortazavi, M. Baniassadi, J. Bardon, S. Ahzi, Modeling of two-phase random composite materials by finite element, Mori–Tanaka and strong contrast methods, Composites Part B: Engineering, 45 (2013) 1117 – 1125. [12] J. Wang, J. Carson, M. North, D. Cleland, A new approach to modelling the effective thermal conductivity of heterogeneous materials, International Journal of Heat and Mass Transfer 49 (17 - 18) (2006) 3075 – 3083. [13] J. Wang, J. Carson, M. North, D. Cleland, A new structural model of effective thermal conductivity for heterogeneous materials with co-continuous phases, International Journal of Heat and Mass Transfer 51 (9 - 10) (2008) 2389 – 2397. [14] D. Trias, J. Costa, A. Turon, J. Hurtado, Determination of the critical size of a statistical representative volume element (SRVE) for carbon reinforced polymers, Acta Materialia 54 (13) (2006) 3471 – 3484. [15] S. Dhiman, P. Potluri, C. Silva, Influence of binder configuration on 3D woven

21

composites, Composite Structures 134 (2015) 862 – 868. [16] J. Gou, H. Zhang, Y. Dai, S. Li, W. Tao, Numerical prediction of effective thermal conductivities of 3D four-directional braided composites, Composite Structures 125 (2015) 499 – 508. [17] H. Yu, D. Heider, S. Advani, Prediction of effective through-thickness thermal conductivity of woven fabric reinforced composites with embedded particles, Composite Structures 127 (2015) 132 – 140. [18] S. Tu, W. Cai, Y. Yi, X. Ling, Numerical simulation of saturation behavior of physical properties in composites with randomly distributed second-phase, Journal of Composite Materials 39 (2005) 617 - 631. [19] D. Marcos-Gómez, J. Ching-Lloyd, M. Elizalde, W. Clegg, J. Molina-Aldareguia, Predicting the thermal conductivity of composite materials with imperfect interfaces, Composites Science and Technology 70 (2010) 2276 – 2283. [20] D. Duschlbauer, H. Böhm, H. Pettermann, Modelling interfacial effects on the thermal conduction behavior of short fiber reinforced composites, International Journal of Materials Research 6 (2011) 717 - 728. [21] G. Catalanotti, On the generation of RVE-based models of composites reinforced with long fibres or spherical particles, Composite Structures 138 (2016) 84 – 95. [22] W. Tian, L. Qi, J. Zhou, J. Liang, Y. Ma, Representative volume element for composites reinforced by spatially randomly distributed discontinuous fibers and its applications, Composite Structures 131 (2015) 366 – 373.

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[23] H. Böhm, A. Eckschlager, W. Han, Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements, Computational Materials Science 25 (2002) 42 – 53. [24] L. Qi, W. Tian, J. Zhou, Numerical evaluation of effective elastic properties of composites reinforced by spatially randomly distributed short fibers with certain aspect ratio, Composite Structures 131 (2015) 843 – 851. [25] C. Dandekar, Y. Shin, Multi-step 3-D finite element modeling of subsurface damage in machining particulate reinforced metal matrix composites, Composites Part A: Applied Science and Manufacturing 40 (2009) 1231 – 1239. [26] S. Jacques, I. D. Baere, W. V. Paepegem, Application of periodic boundary conditions on multiple part finite element meshes for the meso-scale homogenization of textile fabric composites, Composites Science and Technology 92 (2014) 41 - 54. [27] ABAQUS/Standard, ABAQUS scripting/analysis user’s manual, Version 6.12-3, RI: ABAQUS Inc., 2012. [28] B. Jiang, C. Liu, C. Zhang, B. Wang, Z. Wang, The effect of non-symmetric distribution of fiber orientation and aspect ratio on elastic properties of composites, Composites Part B: Engineering 38 (1) (2007) 24 - 34. [29] K. Asano, H. Yoneda, Y. Agari, Microstructure and thermal properties of squeeze cast aluminum alloy composite reinforced with short potassium titanate fiber, Materials Transactions 49 (11) (2008) 2664 - 2669. [30] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective

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magnetic permeability of multiphase materials, Journal of applied Physics, 33 (10) (1962) 3125 - 3131.

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Figures:

Figure 1: Illustration of fibers penetrating the surfaces of the RVEs which are slit into the appropriate number of parts and then are translated to the opposite surfaces

Figure 2: RVE and mesh of composites reinforced by the randomly distributed fibers with the aspect ratio of 15 and fiber volume fraction of 10%: (a) RVE, (b) Fibers

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Figure 3: Categorized three different sets of nodes on the RVE boundaries: Set I: Inner surface nodes, Set II: Inner edge nodes and Set III: Vertical nodes

Figure 4: Two Euler orientation angles θ ∈ [0, π ] and φ ∈ [0, 2π ] of fiber with respect to the global Cartesian coordinate system o G x1G x2G x3G and the local Cartesian coordinate system o L ( x1L )i ( x2L )i ( x3L )i

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Figure 5: Effective thermal conductivities of Csf/Mg composites with the fiber volume fraction of 10% predicted by the RVEs with the different size: (a) Fiber aspect ratio of 5, (b) Fiber aspect ratios of 10 and 15

Figure 6: Temperature distribution in the RVE of PTisf/Al composites obtained by using the corresponding periodic boundary condition: (a) Matrix, (b) Fibers

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Figure 7: Corresponding heat flux fields in the opposite surfaces of RVE of PTisf/Al composites

Figure 8: Effective thermal conductivity of Csf/Mg composites reinforced by the randomly distributed transversely isotropic fibers with the different fiber volume fractions: (a) Fiber aspect ratio of 5, (b) Fiber aspect ratio of 10 and (c) Fiber aspect ratio of 15

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Figure 9: Variation of effective thermal conductivity of Csf/Mg composites reinforced by the randomly distributed transversely isotropic fibers regarding the change in the fiber aspect ratio

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Tables: Relative error

Fiber aspect ratio

1.0 (2.0)

1.5 (2.5)

2.0 (3.0)

2.5 (3.5)

3.0 (4.0)

l / d =5

#

0.96%

0.75%

0.069%

0.044%

l / d = 10

#

1.56%

0.33%

0.13%

0.08%

l / d = 15

#

0.62%

0.10%

0.11%

0.07%

Table 1: Relative errors of effective thermal conductivity of Csf/Mg composites predicted by the RVEs with the different size (Note that in the case of l / d = 5 ,

L / l are 2.0, 2.5, 3.0, 3.5 and 4.0, while l / d = 10 or 15, L / l are 1.0, 1.5, 2.0, 2.5 and 3.0)

Effective thermal conductivity −1

Numerical homogenization

Experimental test

Error 1

100.50

97.11

3.38%

−1

(W ⋅ m ⋅ K )

Table 2: Comparison of the effective thermal conductivities of PTisf/Al composites with the fiber volume fraction of 25% predicted by the RVE based FE homogenization method and measured experiments.

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from the thermal

conduction

l / d = 10

Thermal conductivity −1

l / d = 15

Transversel y isotropic

Isotropic

Error #1

Transversel y isotropic

Isotropic

Error #2

62.57

63.99

2.27%

62.56

63.97

2.25%

−1

(W ⋅ m ⋅ K )

Table 3: Effective thermal conductivities of Csf/Mg composites reinforced by the randomly distributed transversely isotropic and isotropic fibers, respectively

Transversely isotropic

Isotropic

Isotropic parameter

Value

I11

I 22

I33

I11

I 22

I33

0.9999

0.9995

0.9990

0.9997

1.0000

0.9988

Table 4: Isotropic parameters of the effective thermal conductivity of Csf/Mg composites with the randomly distributed transversely isotropic and isotropic fibers, respectively

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