evaluation of effective thermal conductivity of fiber-reinforced composites

International Journal of Architecture, Engineering and ConstructionVol.1, No. 1, March 2012, pp. 14-29

Evaluation of Effective Thermal Conductivity of

Fiber-Reinforced Composites

Changyong Cao1, Aibing Yu2, Qing-Hua Qin1,∗

1Research School of Engineering, Australian National University, Canberra, ACT 0200, Australia2School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia

Abstract: In this paper, effective thermal conductivity of fiber-reinforced composites are estimated by thenewly developed hybrid finite element method (FEM). In the hybrid FEM, foundational solutions are employedto approximate the intra-element displacement field in any given element, while the polynomial shape functionsused in traditional FEM are utilized to interpolate the frame field. The homogenization procedures using therepresentative volume element are integrated with the hybrid fundamental solution based finite element method(HFS-FEM) to estimate the effective thermal conductivity of the composites and to investigate the effect of fibervolume fraction and fiber arrangement pattern on the effective thermal conductivity. A special element with aninclusion is constructed by means of related special fundamental solutions. Due to the fact that the proposedspecial element exactly satisfy its boundary conditions along the fibre-matrix interface, only element boundaryintegrals are involved and significant mesh reduction can be achieved. Mesh regeneration may be avoided aswell when the fiber volume fraction is slightly changed. The accuracy of the numerical results obtained bythe proposed method is verified against with that obtained from commercial software package ABAQUS. Theresults indicate that the proposed method is efficient and accurate in analyzing the micromechanical thermalbehavior of fiber-composites and has the potential to be scaled up to macro-scale modeling of practical problemsof interest.

Keywords: Thermal conductivity, fiber-reinforced composites, representative volume element, hybrid finiteelement method, fundamental solutions, special element

1 INTRODUCTION

Fiber-reinforced composites are structural materialsthat consist of fiber reinforcing phase and matrix phasein which the fiber is embedded at a macroscopic level topossibly experience a range of mechanical, thermal andchemical environment during their service life (Chung1994). Due to the superiority of their physical proper-ties over the single matrix, such as high thermal andelectrical conductivity, high stiffness and strength etc.,fiber-reinforced composites have been widely used inengineering applications. The determination of effec-tive properties of composite materials is of paramountimportance in engineering design and application ofcomposite materials. The effective thermal conductiv-ity and other thermo-physical properties of compositeshave attracted considerable interest in theoretical, nu-merical and experimental researchers during the lastseveral decades (Brennan and Walrath 2009; Chen and

Cheng 1967; Farooqi and Sheikh 2006; Kachanov andSevostianov 2005; Landis et al. 2000; Li et al. 2011;Tsukrov and Novak 2002). It is expected to save mucheffort, time and expense if the properties of the newreinforced composites could be predicted accurately ordesigned from micro-structural properties of its con-stituents.In literatures, the averaged or homogenized method

using representative volume element (RVE) is usuallyemployed in micro mechanical modeling of compos-ites (Miehe 2003; Zohdi and Wriggers 2008). Theboundary value problems defined on the RVEs can beanalyzed by proper numerical methods. In the pasttwo decades, considerable attention has been given tothe determination of the thermal or mechanical prop-erties of composites by using finite element method(FEM) (Islam and Pramila 1999). The reason for us-ing FEM is that by employing it we can easily sim-ulate the effect of various possible defects to the me-

*Corresponding author. Email: [email protected]

14

mailto:[email protected]

Cao et al./International Journal of Architecture, Engineering and Construction 1 (2012) 14-29

chanical properties, i.e., the effect of improper bondbetween fiber and matrix and the effect of cracks hav-ing different orientations. However, the drawback ofthis method is that refined meshes near the defects(cracks, holes or inclusions) are usually required toachieve the desired accuracy. This is not practical foranalysis of composites whose fiber distribution mightchange repeatedly. Unlike FEM, the boundary elementmethod (BEM) simply requires division on boundariesof the domain under consideration, reducing the di-mensionality of the problem by one. This approach hasbeen successfully applied to steady-state and transientheat conduction (Ma et al. 2008), interface perfor-mance (Chen and Papathanasiou 2004), and thermoe-lastic behavior (Henry et al. 2007) of fiber-reinforcedcomposites. Singular or hyper-singular integrals, how-ever, are unavoidable in BEM. Moreover, the BEM so-lution process becomes extremely complex for imposingcontinuity conditions across the interface between fiberand matrix when solving multi-material problems likefiber-reinforced composites (Gao and Davies 2002).To overcome these difficulties, hybrid Trefftz

FEM(HT-FEM) was developed based on two indepen-dent fields: an intra-element field and an auxiliaryframe field (Qin 2000; Qin and Wang 2008). Inter-element continuity is enforced by using a modified vari-ational principle, which is used to construct the stiff-ness equation and to establish relationship of the framefield and the internal field of the element. This ap-proach involves the element boundary integrals only,which inherits the advantages of both conventionalFEM and BEM, and has been successfully applied tovarious engineering problems (Qin 1994; Qin 1995; Qin1996; Qin 2003; Qin and Wang 2008). The propertyof non-singular element boundary integral appearing inHT-FEM enables us to construct arbitrary shaped ele-ments conveniently; however, the terms of truncated T-complete functions must be carefully selected to achiev-ing the desired results, and it is difficult to generate T-complete functions for certain complex or new physicalproblems. Further, in the HT-FEM a coordinate trans-formation is required to keep the system matrix stable,and the necessary variational functional is somewhatcomplex for practical use.As an alternative to HT-FEM, a novel hybrid finite

formulation based on the fundamental solutions, calledHFS-FEM, was developed for solving two-dimensionallinear heat conduction problems (Wang and Qin 2009;Wang and Qin 2011a), and isotropic elastic (Wang andQin 2011a), functionally graded elastic (Wang and Qin2012) and piezoelectric problems (Cao et al. 2012). Inthe approach, a linear combination of the fundamen-tal solution at different source points located outsideof the elements is used to approximate the unknownfield within the element instead of the truncated T-complete functions used in HT-FEM. The proposedHFS-FEM inherits all the advantages of HT-FEM andobviates the difficulties that occur in HT-FEM. If only

the general hybrid element is used, however, the pre-sented HFS-FEM has no significant improvement inefficiency for handling heterogeneous composites.In the present work, the formulations of HFS-FEM

for the heat conduction problem is presented to modelheterogeneous fiber-reinforced composites and to inves-tigate the applicability of the new method in predictingthe effective thermal conductivity of ideal fiber rein-forced composites. Both of general element and specialelement for circular fiber inclusions are proposed basedon the relevant fundamental solutions. The special ele-ment is based on a special fundamental solution whichanalytically satisfies the continuity of temperature andheat flux on the interface between fiber and matrixand is constructed to reduce the mesh refine effort inmodeling heterogeneous composites. Then, indepen-dent intra-element and frame fields as well as a mod-ified variational functional are constructed to derivefinal stiffness equations and determine the unknowns.The RVE is utilized for estimating the effective ther-mal property of the composites. Two examples arepresented in order to get insight into the influence offiber volume fraction and fiber arrangement pattern tothe effective thermal conductivity. The accuracy of thenumerical results obtained by the proposed method isverified against by those calculated by commercial soft-ware package ABAQUS. The results indicate that theproposed method is efficient and accurate in analyzingthe thermal behavior of fiber-composites and has thepotential to be scaled up to macro-scale modeling oflarge-scale practical problems of considerable interest.The paper is organized as follows: the governing

equations of the heat conduction problem and the basicconcepts of RVE and homogenization procedures arefirst introduced in Section 2. Then, the fundamentalsolutions for interpolation of temperature field withinelement domain are given out in Section 3. Further,the detailed solution procedures for the derivation ofthe HFS-FEM are presented in Section 4. Finally, twonumerical examples for RVEs with simple and complexfiber patterns are considered in Section 5, and someconcluding remarks are presented in Section 6.

2 GOVERNING EQUATIONS AND HO-MOGENIZATION

In this section, a brief review of the basic equationsand concepts used in micromechanical analysis of theheat conduction problem is presented to introduce thenotation and provide a common source for reference inlate sections.

2.1 Governing Equations

The thermal equilibrium governing equation for thetemperature field can be expressed as (Temizer andWriggers 2010):

∇q = 0 (1)

15


Figure 1. Periodic fiber-reinforce composites and RVE

with the boundary conditions:

θ = θ onΓθ, h = q · n = h onΓk (2)

where ∇ = [∂/∂x1, ∂/∂x2] is the divergence operator;θ = T − T0 is unknown temperature change, where Tis the current absolute temperature and T0 is a refer-ence absolute temperature; g = ∇θ is the temperaturegradient; h is the normal heat flux; n is the outwardnormal vector to the boundary Γ = Γθ ∪ Γk; and θand h are specified values on the related boundaries,respectively. The space derivatives are denoted by acomma, i.e., θ,i = ∂θ/∂xi, and the subscript i takesvalues 1 and 2 in our analysis.The constitutive law for flux vector q = [q1, q2]T fol-

lows the Fourier’s law as (Temizer and Wriggers 2010):

q = −k · g (3)

where k is the thermal conductivity tensor, which issymmetry and positive-definite and depends on the lo-cal materials in the heterogeneous composites. Thematrix and fiber are herein assumed to be locallyisotropic and homogeneous, so k11 = k22 = km in thematrix and equal to kf in the fiber area, respectively.

2.2 Representative Volume Elements(RVE)

The microstructure of the composite, such as theshape, size distribution, spatial distribution, and orien-tation distribution of the reinforcing inclusions in thematrix, has significant influence of the overall (effec-tive) properties of the heterogeneous materials (Nemat-Nasser and Hori 1999). Although most composites pos-sess inclusions of random distributions, great insight ofthe effect of microstructure on the effective propertiescan be gained from investigation of composites with pe-riodic structures. For a composite with periodic fibres,it is usually sufficient to draw conclusions for the wholestructure by considering only a unit cell, so called RVE,which is shown in Figure 1 (Zohdi and Wriggers 2008).

For simplification, following assumptions for an idealfiber-reinforced composites will be applied (Islam andPramila 1999): (1) the composites are macroscopicallyhomogeneous, (2) locally both the matrix and the fiberare homogeneous and isotropic, (3) the thermal contactresistance between the fibre and the matrix is negligi-ble, (4) the composite is free of voids, (5) the problem istwo dimensional, and (6) reinforced fibers are arrangedin a square periodic array, i.e., they are uniformly dis-tributed in the matrix. This last assumption impliesthat fibers are equal and uniform in shape and size andare symmetrical about the x1- and x2- directions.The macroscopic overall properties of the heteroge-

neous materials are primarily extracted from the RVEsin an averaged or homogenized method. In the presentanalysis, a RVE consisting of matrix material and fiberphase, as shown in Figure 1, is chosen to be statisti-cally representative of the two-phase composite. It isnoted that the characteristic size of the heterogeneitiesis supposed to be much smaller than the dimension ofRVE, which in turn is supposed to be small comparedto the characteristic length of the macroscopic struc-ture. As a result, a RVE is regarded just as a point witha homogenized constitutive law on the macro-level.

2.3 Homogenization for RVE

The macroscopic constitutive formulation must be de-termined by materials testing on RVE, from which theheterogeneous conduction problem is solved with spec-ified boundary conditions. The macroscopic flux andthe macroscopic gradient fields are identified as the vol-ume averages of the microscopic counterparts and theyare related to each other by the macroscopic constitu-tive formulations.The thermal constitutive law that governs each mate-

rial or phase in a RVE is given by the standard Fourier’slaw as shown in Eq. (3). To evaluate the effectivethermal conductivity of the microscopically heteroge-neous fiber-reinforce composites, the effective flux q

16


and effective temperature gradient g are defined as thevolume average values of the respective fields in theRVE (Zohdi and Wriggers 2008).

q = 〈q〉 =1Ω

∫

Ω

qdΩ =1Ω

∫

Γ

hXdΓ (4)

g = 〈g〉 =1Ω

∫

Ω

∇θdΩ =1Ω

∫

Γ

θndΓ (5)

where Ω is the volume of RVE. It can be seen fromEqs. (4) and (5) that the volume average gradient andflux are related only to the flux on the boundary ofRVE.For the isotropic case, the thermal tensor k∗ can be

expressed as k∗ = k∗I, where k∗ is the conductivity co-efficient and I is the identity tensor. Thus, accordingto Fourier’s law, the effective thermal conductivity canbe calculated by:

k∗ =‖q∗‖‖g∗‖ =

‖q‖‖g‖ (6)

It should be noticed that the material test for RVEmust be conducted under steady-state conditions andthere should be no external heat supply in order toeliminate any external influents or rate effects (Tem-izer and Wriggers 2010).

2.4 Boundary Conditions for RVE

Using the RVE model described above, three typesof boundary condition are usually employed to eval-uate the overall/effective thermal properties of hetero-geneous materials: (1) Uniform flux boundary condi-tion, (2) Linear temperature gradient boundary condi-tion and (3) periodic boundary condition (Temizer andWriggers 2010; Zohdi and Wriggers 2008). The threeboundary conditions are listed as follows:

1. Uniform flux boundary condition (UF-BC):

h = Q · n (7)

so that 〈q〉 = Q.

2. Linear temperature boundary condition (LT-BC):

θ = G ·X (8)

so that 〈g〉 = G.

3. Periodic boundary condition (PR-BC):

θ+ − θ− = G · (X+ −X−) (9)

and

h+ = −h− (10)

so that 〈q〉 = Q, where Q and G are controlledconstant vectors.

Previous investigations have found that the PR-BC is much more accurate in micromechanical anal-ysis of composite materials for both periodic materials

and random materials (Hazanov and Huet 1994; Huet1990). Consequently, PR-BC will be employed for theRVE models in the present research.From the above homogenization procedures, we can

see that the flux and temperature on the boundary ofthe RVE are sufficient to calculate the effective ther-mal conductivity of the composites. This feature willenable us to easily employ the special element intro-duced in Section 5 to evaluate the effective parametersof RVE.

3 FUNDAMENTAL SOLUTIONS

For the proposed hybrid FEM, it is essential to findthe fundamental solutions of the plane heat conduc-tion problems to interpolate the intra-element approx-imation fields. The Green’s function of the two-dimensional heat transfer problem in an infinite do-main can be defined by:

k∇2G(x,y) + δ(x,y) = 0 (11)

where δ(x,y) is the Dirac delta function, x = (x1, x2)denotes the field point where response is calculatedand y = (y1, y2) denotes the source point where unitconcentrated heat applied. For plane heat conductionproblems of fiber-reinforced composites two kinds offundamental solutions will be used: one is a generalfundamental solution for homogeneous materials with-out any inclusions, and the other is a special funda-mental solution satisfying the interfacial conditions be-tween the circular inclusion and the matrix.

3.1 General Fundamental Solution

Consider a unit heat source applied at the source pointz0 = x10 + ix20 in the infinite homogeneous domainΩm, the temperature response Gm at any field point zis given in the form (Chao and Shen 1997):

Gm(z, z0) = − 12πkm

Re ln(z − z0) (12)

where Re denotes the real part of the bracketed ex-pression, z = x1 + ix2 and i =

√−1 is imaginary num-ber (Wang and Qin 2011b).

3.2 Special Fundamental Solution

For the case that a central circular inclusionis embed-ded in an infinite domain Ωm, if a unit heat source isapplied at the source point z0 in Ωm, as shown in Fig-ure 2, the temperature responses Gm and Gf at anyfield point z in matrix or fiber regions can be obtainedby using the complex potential theory by consideringthe interface (| z | = R) continuity conditions betweeninclusion and matrix as (Chao and Shen 1997):

17


Figure 2. Fundamental solutions for plane heat con-duction problems in fiber-reinforced composites

Gm =− 12πkm

(Re

[ln(z − z0)

]+

km − kf

km + kfRe

[ln(z − z0)

]) z ∈ Ωm

Gf =− 1(km + kf )π

·Re[ln(z − z0)]

z ∈ Ωf

(13)

Similarly, the induced temperature Gm in the matrixshows a proper singular behavior at the source pointz0, while Gf in the fiber is regular because the sourcepoint z0 is outside the fiber.

4 FORMULATIONS OF THE HFS-FEM

The main idea of the proposed hybrid FEM is origi-nated from the HT-FEM, which utilizes two indepen-dent approximate fields: intra-element field and an in-dependent frame field along element boundary. Butunlike the HT-FEM, the intra-element fields in theHFS-FEM are constructed based on the fundamentalsolutions, rather than a truncated T-complete functionset. In this section, the solution procedures of the hy-brid finite element model with the fundamental solu-tions as interior approximation functions are describedto solving the linear heat transfer problems of compos-ite materials.

4.1 Intra-Element Field

For a particular element e, which occupies sub-domainΩe, we assume that the temperature field defined inthe element domain is approximated by a linear com-bination of foundational solutions at different sourcepoints located outside of the element domain (see Fig-ure 3, Wang and Qin 2010).

θe(x) =ns∑

j=1

N(x,ysj)cj = Ne(x)ce

∀x ∈ Ωe,ysj 6∈ Ωe

(14)

where ns is the number of source points outside the ele-ment domain, which is equal to the number of nodes ofan element in the present research based on the gener-ation approach of the source points; ce is an unknowncoefficient vector (not nodal temperatures); and Ne isthe fundamental solution matrix which can be writtenas:

Ne =[G∗1(x,ys1) G∗2(x,ys2) . . . G∗ns(x,ysns)

](15)

ce = [c1 c2 . . . cns]T (16)

where Gi(x,ysj) represents the corresponding funda-mental solution.

Gi(x,ysj) =

Gm(x,ysj) x ∈ Ωm

Gf (x,ysj) x ∈ Ωf(17)

It is noted that since the fundamental solutions al-ready include the presence of interface between thefillers and matrix, it is not necessary to model the tem-perature and heat flux continuity conditions on theinterface and then the analysis will become simpler.One of the advantages in the presented HFS-FEM isto reduce computation effort by using special purposeelements. In addition, due to use of two groups of inde-pendent interpolation functions in the HFS-FEM, wecan construct arbitrarily shaped elements to be usedfor analysis, as shown in Figure 3.For a particular element as shown in Figure 3, we

can use the nodes of the element to generate relatedsource points for simplicity, so that the singular sourceslocate on the pseudo boundary to achieve certain nu-merical stability. The source point ysj(j = 1, 2, . . . , ns)can be practically generated by means of the followingmethod (Wang and Qin 2010):

ys = x0 + γ(x0 − xc) (18)

Figure 3. Intra-element field, frame field in a partic-ular element in HFS-FEM for a particular element

18


where γ is a dimensionless coefficient and x0 is thepoint on the element boundary (the nodal point in thiswork) and xc is the geometrical centroid of the element.

The corresponding outward normal derivative of ue

on Γe is:

he = −k∂θe

∂n= Qece (19)

where

Qe = −k∂Ne

∂n= −knTe (20)

with

Te =[∂Ne

∂x1

∂Ne

∂x2

]T(21)

4.2 Auxiliary Frame Field

To enforce conformity on the field variable θ, for in-stance, θe = θf on Γe ∩ Γf of any two neighboring el-ements e and f , an auxiliary inter-element frame fieldθ is used and expressed in terms of nodal displacementvector, d, as used in conventional FEM. In this case, θis confined to the whole element boundary, that is

θe(x) = Ne(x)de x ∈ Γe (22)

which is independently assumed along the elementboundary in terms of nodal degree of freedom (DOF)de, where Ne represents the conventional finite elementinterpolating functions. For example, a simple interpo-lation of the frame field on the side with three nodes ofa particular element as shown in Figure 4 can be givenin the form:

θ = N1θ1 + N2θ2 + N3θ3 (23)

where Ni(i = 1, 2, 3) stands for shape functions interms of natural coordinate ξ defined in Figure 4.

Figure 4. Typical quadratic interpolation for theframe field

4.3 Hybrid Variational Functional

For the boundary value problem defined in Eqs. (1)and (2), since the stationary conditions of the tra-ditional potential or complementary variational func-tional can not guarantee the required inter-elementcontinuity condition in the proposed hybrid finite el-ement model, a modified potential functional is devel-oped as follows:

∏m

=∑

e

∏me

(24)

with∏me

=− 12

∫

Ωe

kθ,iθ,idΩ−∫

Γqe

hθdΓ+

∫

Γe

h(θ − θ)dΓ(25)

in which the governing equation, i.e., Eq. (1), is as-sumed to be satisfied a priori. The boundary Γe of aparticular element consists of the following parts:

Γe = Γθe ∪ Γhe ∪ ΓIe (26)

where ΓIe represents the inter-element boundary of theelement e.Applying the divergence theorem, we can eliminate

the domain integral and obtain the final form of thefunctional for the HFS-FE model.

∏e

= −12

∫

Γe

hθdΓ−∫

Γqe

hθdΓ +∫

Γe

hθdΓ (27)

Finally, substituting Eqs. (14), (19) and (22) into thefunctional Eq. (27) produces:

∏e

= −12cT

e Hece − dTe ge + cT

e Gede (28)

where He =∫Γe

QTe NedΓ, Ge =

∫Γe

QTe NedΓ and

ge =∫Γqe

NTe hdΓ.

To enforce inter-element continuity on the commonelement boundary, the unknown vector ce should beexpressed in terms of nodal DOF de. Minimization ofthe functional

∏e with respect to ce and de, respec-

tively, yields:

∂∏

e

∂cTe

=−Hece + Gede = 0

∂∏

e

∂dTe

=GTe ce − ge = 0

(29)

from which the optional relationship between ce andde, and the stiffness equation can be produced as:

ce = H−1e Gede (30)

and

Kede = ge (31)

where Ke = GTe H−1

e Ge stands for the symmetric ele-ment stiffness matrix. It is worth pointing out that theevaluation of the vector in Eq. (31) is the same as thatin the conventional FEM, which is obviously convenient

19


for the implementation of HFS-FEM into the existingFEM program. The matrices He, Ge and vector ge

can be calculated by the commonly used Gaussian nu-merical integration.

4.4 Recovery of Rigid-Body Motion

Considering the physical definition of the fundamentalsolution, it is necessary to recover the missing rigid-body motion modes from the above results. Followingthe method presented in (Qin 2000), the missing rigid-body motion can be recovered by writing the internalpotential field of a particular element e as:

θe(x) = Ne(x)ce + c0 (32)

where the undetermined rigid-body motion parameterc0 can be calculated using the least square matching ofθe and θe at element nodes.∫ n

i=1

[θe(x)− θe(x)

]2 = min (33)

which finally gives:

c0 =1n

∫ n

i=1

M θei (34)

in which M θei = (θe−Nece)|node i and n is the numberof element nodes.Once the nodal field is determined bysolving the final stiffness equation, the coefficient vec-tor ce can be evaluated from Eq. (30), and then c0 isevaluated from Eq. (34). Finally, the temperature fieldθ at any internal point in an element can be determinedby means of Eq. (32).

5 NUMERICAL EXAMPLE AND DIS-CUSSION

In this section, numerical experiments are conducted todemonstrate the performance and efficiency of the hy-brid algorithm with special element and also to studythe micro-thermal behavior of composites. To evalu-ate the effective thermal conductivity of the compositesimplies a homogenization procedure that requires theapplication of several independent loading conditionson the RVE of the heterogeneous materials (Zohdi andWriggers 2008). In the present work, temperature gra-dient controlled method is employed to conduct thenecessary tests. Resolving the heat flux field in hetero-geneous materials through a steady-state heat transferanalysis allows for the calculation of the average fluxfield components q. Since the average temperature gra-dient g is imposed, the effective thermal conductivitycoefficient k∗ for the equivalent homogenous materialcan be directly calculated from Eq. (6).In the following two examples, effective properties

of two-phase composite materials are investigated bymeans of the HFS-FEM. It is assumed that both fiberand matrix are linear thermal conduction and that theyare perfectly bonded at their interface. However, com-plex inclusion geometry is not included in the RVEs

and the problem on how to generate realistic RVEmicro-level geometry for numerical analysis is not dis-cussed in the present work.

5.1 Example 1: RVE with One ReinforcedFiber

In the first example, a square RVE with only one rein-forced fiber, as shown in Figure 5, is investigated by theproposed method. The thermal conductivities of thefiber and matrix are respectively assumed as k2 = 20,and k1 = 1. The mismatch ratio between fiber and ma-trix m = k2/k1 = 20 will be maintained in this analysisunless otherwise specified. For verification and com-parison purpose, the numerical results from traditionalFEM (here we employ ABAQUS for the analysis) arealso given in the analysis as well. It should be pointedout that a constant temperature gradient G = [1, 0]T

or G = [0, 1]T is applied in the analysis and the PR-BC illustrated in Section 2 are employed for both theHFS-FEM and ABAQUS modeling.

Figure 5. Schematic of RVE containing one centralfiber

Convergence Verification

Before we employ the proposed method for heat trans-fer analysis of composite materials, the convergencestudy is first carried out by checking against the con-cerned variables. Three different meshes: Mesh 1 (128-node elements and one 16-node special elements with atotal of 60 nodes), Mesh 2 (328-node elements and one16-node special elements with a total of 128 nodes),and Mesh 3 (618-node elements and one 16-node spe-cial element with a total of 220 nodes) are employed.The results calculated by ABAQUS are employed asa reference benchmark for comparison, which are ob-tained using a very fine mesh (as shown in Figure 6,1940 3D8R element with 5941 nodes).Figure 7 presents the effective thermal conductivities

k∗ versus the number of element meshes or nodes. Itis shown that the results obtained from HFS-FEM are

20


(a) HFS-FEM with special element (b) ABAQUS

Figure 6. Mesh configurations for RVE by (a) for HFS-FEM and by (b) for ABAQUS

converging to the benchmark values with the increaseof number of DOF. It is also noted that HFS-FEM isable to give similar accuracy, when using 128 nodesonly, to that from ABAQUS using thousands of nodes.

Effect of Fiber Volume Fraction on the EffectiveThermal Properties

Fiber volume fraction (FVF)has significant influence tothe overall thermal conductivity of the heterogeneouscomposites. In this section, the effect of FVF on themacroscopic effective thermal conductivities k∗ of is in-vestigated. The FVF is changed from 3.14% to 63.62%by varying the radius of the fiber. The predicted effec-tive thermal conductivities from HFS-FEM are graph-ically shown in Figure 8. It can be observed, as wasexpected, that effective thermal conductivities k∗ rises

along an increase in the FVF. However, the increas-ing rate becomes larger along with an increase in theFVF. When the FVF increases to 63.62%, the effectivethermal conductivity k∗ of the composite reaches up tomore than four times of that of the matrix. It is alsoobserved from Figure 8 that, for all the cases investi-gated here, better accuracy can be achieved by HFS-FEM with special elements than that from ABAQUS.Compared to the results calculated from ABAQUS,

which are achieved by refined meshes around andwithin the fiber, the specially purposed inclusion el-ements can achieve very good accuracy. This con-clusion can also be confirmed by Figures 9 and 10,which presents the variations of the effective thermalconductivities k∗ varies with the materials mismatchratio m = k2/k1 or 1/m = k1/k2.

Figure 7. RVE with one central fiber: Convergence of the effective thermal conductivity

21


Figure 8. Effective thermal conductivities k∗ of composite for different fiber volume factions

Effect of Material Mismatch Ratio on the EffectiveThermal Properties

Figure 9 shows the effective thermal conductivity k∗

of the heterogeneous composite for different materialmismatch ratios when the thermal conductivity k2 ofthe fiber is larger than the thermal conductivity k1

of the matrix, i.e., a conducting fiber embedded intoan insulated matrix. The FVFis constant at 19.63%,i.e., R = 2.5. It can be seen from Figure 9 that fora given matrix the effective thermal conductivity in-creases with the increasing conductivity of the fiberand the heat conduction performance can be dramat-ically improved when adding the conducting fiber but

the effect will become weaker after the thermal conduc-tivity of fiber larger than 20 times of that of the ma-trix. For comparison, the predicted results obtainedfrom ABAQUS are also presented in Figure 9. Thesame conclusion can be obtained from the results andboth methods provide equivalent accuracy.Figure 10 presents the variation of the effective ther-

mal conductivity k∗ for the different material mismatchratios when the thermal conductivity k2 of the fiberis smaller than the thermal conductivity k1 of matrix,i.e., an insulated fiber embedded into a conducting ma-trix. The FVF also keeps constant at 19.63%. Com-pared with Figure 9, it is obvious that the nearly linearrelationship between effective thermal conductivity k∗

Figure 9. Effective thermal conductivity k∗ of the composite for different mismatch ratios when k2 > k1

22



Figure 11. Contour plots of the temperature and heat flux distribution in the RVE (m = 20, FVF=19.63%)

Figure 12. Contour plots of the temperature and heat flux distribution in the RVE (1/m = 20, FVF=19.63%))

23


and the matrix thermal conductivity k1 can be found,which is different from the nonlinearity shown in Fig-ure 9.Contour plots of the temperature and heat flux dis-

tributions in the RVE under constant temperature gra-dient and periodic boundary conditions are shown inFigures 11 and 12, in which the FVF is kept as 19.63%.It can be seen from Figure 11 that for higher conduc-tion fiber the heat flux are mainly pass through fiberwhile for the insulated fiber with lower conductivitythan that of matrix, most flux will carry out by thematrix (see Figure 12).

5.2 Example 2: RVE with Multi-Fibers

Two different fiber patterns as shown in Figure 13 areinvestigated to reveal the effect of fiber configurationon effective thermal properties of heterogeneous com-posites. The geometry and dimensions of the RVEsare given in Figure 13, in which L = 10 and R = 1. Inthe two cases, the FVF is kept as a constant at 15.71%.Mesh configuration of the RVEs for HFS-FEM is shownin Figure 14, in which the 16-node special element (i.e.,8-edges) for fibers are employed. Mesh configurationsof the RVEs for ABAQUS are shown in Figure 15.The predicted effective thermal conductivities k∗ of

the two-phase composites by the HFS-FEM for the twodifferent patterns are given in Figures 16 and 17, re-spectively. For the case of conducted fibres embed-ded in an insulated matrix, i.e., k2 > k1, it can beseen from Figure 16 that for both patterns the effec-tive thermal conductivity k∗ are increasing with theincreasing of fiber thermal conductivity k2 and the in-creasing rates are dramatic first between 1 and 20, thentends to be slightly smooth. The relationship betweeneffective thermal conductivities k∗ and mismatch ratiom are nonlinear. When the thermal conductivities k2

of the fiber is 20 times of that of matrix, the effectivethermal conductivities k∗ of the composites, comparedwith the pure matrix materials, can increase by 33.3%for pattern 1 and 35.2% for pattern 2, respectively. Itis obvious from Figure 16 that the effective thermalconductivities k∗ of pattern 2 is slightly better thanthose of pattern 1, and the beneficial influence will be-come clearer when the mismatch ratio becomes larger(m > 20).For the case of insulated fiber embedded in a conduc-

tor matrix, i.e., k2 < k1, it can be seen from Figure 17that the effective thermal conductivity k∗ for both pat-terns are linearly increasing with the matrix thermalconductivity k1 if fixed k2 as a constant, which is com-pletely different from the former case of nonlinearity.For such case, it is found that the thermal propertyof the matrix has a significant influence on the overallmaterial prosperity. The matrix conductivity reducesby 26.80% for pattern 1 and 28.02% for pattern 2, re-spectively, when the thermal conductivity k1 of matrixis 100 times of that of fiber. It can be observed from

Figure 17 that the effective thermal conductivities k∗

of pattern 1 is slightly better than those of pattern 2,in other words the conductivity reducing effect of pat-tern 2 is slight better than that of pattern 1, as theincreasing effect for the case of k2 > k1.Figure 18 presents the contour plots of the tempera-

ture distribution and heat flux components across theRVE when m = 20 and the FVF is 15.71%. It is obvi-ous that the fiber inclusions has a significant influencefor the heat flux distribution in the RVE, and that theheat fluxes are concentrated through the fibers whenthere is fiber appearing at a cross section. It is indi-cated that most of the heat are carried out through thefibers in the composites when the thermal conductivityof the fiber is higher than that of the matrix. Oppo-sitely, it can be found from Figure 19 that the insulatedfibers expel the heat flux to the matrix and most flux gothrough the matrix, not the fiber. Comparing the twopatterns, it can be seen that this phenomenon is muchmore significant for pattern 2. Numerical results ob-tained above also show that the results from HFS-FEMhave a good agreement with those from ABAQUS al-though much less meshes are employed by HFS-FEM.It can be concluded that the proposed method is ac-curate and efficient in analyzing micro heat transferproblems.

6 CONCLUSIONS

A new hybrid FEM based on the fundamental so-lutions is proposed and successfully applied to heattransfer problems of heterogeneous composites. Thetwo independent intra-element and frame fields facili-tate the construction of arbitrary-shaped elements andthe modified variational functional for stiffness ma-trix derivation involves element boundary integral only.Based on the special fundamental solution, a type ofspecial element with inclusion(fiber) involved is pro-posed for mesh reduction in analyzing heterogeneouscomposites.The effective thermal conductivity of composites is

evaluated through the RVEs with single or multi-ple fibers using the homogenization technique. Themethod is used to investigate the effect of the FVFand the mismatch ratio between fiber and matrixin the composites. Numerical results obtained fromABAQUS are also presented for comparison and veri-fication. It is shown that the numerical solutions ob-tained by HFS-FEM coincided with the results calcu-lated by ABAQUS with fine element mesh. It is foundthat the effective thermal conductivity rises with theincreasing FVF. The employment of special inclusionelement can significantly reduce model meshing effortand computing cost, and simultaneously avoid meshregeneration when the FVF is slightly changed.It may be concluded that the proposed micromechan-

ical models based on HFS-FEM have the potential tomodel fiber-reinforced composites and to be further de-

24


(a) Pattern 1 (b) Pattern 2

Figure 13. RVEs of the composite with two different fiber configurations


Figure 14. Mesh configurations of the RVEs for HFS-FEM


Figure 15. Mesh configurations of the RVEs for ABAQUS

25




26


Figure 18. Contour plots of the temperature and heat flux distribution in the RVE (m = 20, FVF=15.71%)

Figure 19. Contour plots of the temperature and heat flux distribution in the RVE (1/m = 20, FVF=15.71%)

27


veloped for considering the defects such as the cracksand pore voids in microstructures and then for multi-scale simulation in the future work.

REFERENCES

Brennan, K. P., and Walrath, D. E. (2009). “A self-contained module for predicting micro-scale materialproperties during fiber-reinforced polymer process-ing.” Journal of Thermoplastic Composite Materials,22, 727–751.

Cao, C., Qin, Q. H., and Yu, A. (2012). “Hybridfundamental-solution-based FEM for piezoelectricmaterials.” Computational Mechanics (in press).

Chao, C., and Shen, M. (1997). “On bonded circularinclusions in plane thermoelasticity.” Journal of ap-plied mechanics, 64, 1000–1004.

Chen, C., and Cheng, S. (1967). “Mechanical propertiesof fiber reinforced composites.” Journal of CompositeMaterials, 1, 30–41.

Chen, X., and Papathanasiou, T. (2004). “Interfacestress distributions in transversely loaded continuousfiber composites: parallel computation in multi-fiberRVEs using the boundary element method.” Com-posites science and technology, 64, 1101–1114.

Chung, D. D. L. (1994). Carbon Fiber Compos-ites. Butterworth-Heinemann, Newton, MA, UnitedStates.

Farooqi, J. K., and Sheikh, M. A. (2006). “Finite ele-ment modelling of thermal transport in ceramic ma-trix composites.” Computational Materials Science,37, 361–373.

Gao, X. W., and Davies, T. G. (2002). Boundary El-ement Programming in Mechanics. Cambridge Uni-versity Press, Cambridge, United Kingdom.

Hazanov, S., and Huet, C. (1994). “Order relationshipsfor boundary-conditions effect in heterogeneous bod-ies smaller than the representative volume.” Journalof the Mechanics and Physics of Solids, 42, 1995–2011.

Henry, D., Ma, F., Chatterjee, J., and Banerjee, P.(2007). “Steady state thermoelastic analysis of 3dsolids with fiber inclusions by boundary elementmethod.” Computer Methods in Applied Mechanicsand Engineering, 197, 294–307.

Huet, C. (1990). “Application of variational concepts tosize effects in elastic heterogeneous bodies.” Journalof the Mechanics and Physics of Solids, 38, 813–841.

Islam, M. R., and Pramila, A. (1999). “Thermal con-ductivity of fiber reinforced composites by the FEM.”Journal of Composite Materials, 33, 1699–1715.

Kachanov, M., and Sevostianov, I. (2005). “On quanti-tative characterization of microstructures and effec-tive properties.” International Journal of Solids andStructures, 42, 309–336.

Landis, C. M., Beyerlein, I. J., and McMeeking, R. M.(2000). “Micromechanical simulation of the failure offiber reinforced composites.” Journal of the Mechan-

ics and Physics of Solids, 48, 621–648.Li, H., Li, S., and Wang, Y. (2011). “Prediction of effec-tive thermal conductivities of woven fabric compos-ites using unit cells at multiple length scales.” Jour-nal of Materials Research, 26, 384–394.

Ma, F., Chatterjee, J., Henry, D., and Banerjee, P.(2008). “Transient heat conduction analysis of 3dsolids with fiber inclusions using the boundary ele-ment method.” International Journal for NumericalMethods in Engineering, 73, 1113–1136.

Miehe, C. (2003). “Computational micro-to-macrotransitions for discretized micro-structures of hetero-geneous materials at finite strains based on the mini-mization of averaged incremental energy.” ComputerMethods in Applied Mechanics and Engineering, 192,559–591.

Nemat-Nasser, S., and Hori, H. (1999). Micromechan-ics: Overall Properties of Heterogeneous Materials.Elsevier Science, Amsterdam, Netherlands.

Qin, Q. H. (1994). “Hybrid trefftz finite-element ap-proach for plate-bending on an elastic-foundation.”Applied Mathematical Modelling, 18, 334–339.

Qin, Q. H. (1995). “Hybrid-trefftz finite-elementmethod for reissner plates on an elastic-foundation.”Computer Methods in Applied Mechanics and Engi-neering, 122, 379–392.

Qin, Q. H. (1996). “Transient plate bending analysis byhybrid trefftz element approach.” Communicationsin Numerical Methods in Engineering, 12, 609–616.

Qin, Q. H. (2000). The Trefftz Finite and BoundaryElement Method. WIT Press, Southampton, Unitedkingdom.

Qin, Q. H. (2003). “Solving anti-plane problems ofpiezoelectric materials by the trefftz finite elementapproach.” Computational Mechanics, 31, 461–468.

Qin, Q. H., and Wang, H. (2008). Matlab and C Pro-gramming for Trefftz Finite Element Methods. CRCPress, New York, United States.

Temizer, I., and Wriggers, P. (2010). “A micromechani-cally motivated higher-order continuum formulationof linear thermal conduction.” Journal of AppliedMathematics and Mechanics/ZAMM, 90, 768–782.

Tsukrov, I., and Novak, J. (2002). “Effective elasticproperties of solids with defects of irregular shapes.”International Journal of Solids and Structures, 39,1539–1555.

Wang, H., and Qin, Q. H. (2009). “Hybrid fem withfundamental solutions as trial functions for heat con-duction simulation.” Acta Mechanica Solida Sinica,22, 487–498.

Wang, H., and Qin, Q. H. (2010). “Fundamental-solution-based finite element model for plane or-thotropic elastic bodies.” European Journal of Me-chanics - A/Solids, 29, 801–809.

Wang, H., and Qin, Q. H. (2011a). “Fundamental-solution-based hybrid FEM for plane elasticity withspecial elements.” Computational Mechanics, 48,515–528.

28


Wang, H., and Qin, Q. H. (2011b). “Special fiber el-ements for thermal analysis of fiber-reinforced com-posites.” Engineering Computations, 28, 1079–1097.

Wang, H., and Qin, Q. H. (2012). “Boundary inte-gral based graded element for elastic analysis of 2d

functionally graded plates.” European Journal of Me-chanics - A/Solids, 33, 12–33.

Zohdi, T. I., and Wriggers, P. (2008). An Introductionto Computational Micromechanics. Springer, Heidel-berg, Germany.

29

evaluation of effective thermal conductivity of fiber-reinforced composites

Documents

given element

element boundaryintegrals

proposedspecial element

fiber volume fraction

finite element methodhfsfem

hybrid fem

fiber arrangement pattern

fiber reinforcing phase