Transverse thermal conductivity of CFRP laminates: A numerical and experimental validation of approximation formulae

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TRANSVERSE THERMAL CONDUCTIVITY OF CFRP LAMINATES: A NUMERICAL AND EXPERIMENTAL VALIDATION OF APPROXIMATION FORMULAE

R. Rolfes DLR, Institute of Structural Mechanics, Lilienthalplatz 7, D-38108 Braunschweig,

Germany

&

U. Hammerschmidt PTB, Bundesallee 100, D-38112 Braunschweig,

Germany

(Received 18 July 1994: revised version received 27 January 1995; accepted 8 February 1995) tional laminated composite structures.‘-” These are based on the thermal laminate theory, which is the thermal counterpart to the classical laminate theory, or on higher order theories. Reliable thermal conductivity data are an essential input for such an analysis. This paper not only deals with the calculation of the thermal conductivities of CFRP from the properties of the constituents (fibres and matrix), but also with measurement of the thermal conductivity. We restrict ourselves to multidirectionally laminated composites consisting of unidirectional laminae. Fabrics and three-dimensionally reinforced composites are not considered. In a unidirectional lamina (Fig. 1) the thermal conductivity parallel to the fibre direction, Ai, is easily obtained from the rule of mixtures by

Abstract Existing

micromechanical

approximation

formulae

for

the transverse thermal conductivity of unidirectional CFRP laminates from the properties of the constituents provide rather different results. The analytical accuracy of such formulae is examined by comparing with finite-element calculations and yields a pre-selection of two equations. Measurements by two different methods reveal that the self-consistent formula is the most realistic one.

predicting

Keywords: thermal conductivity, CFRP, devices, micromechanics, finite elements

measuring

1 INTRODUCTION With the increasing use of carbon-fibre-reinforced plastics (CFRP) the determination of reliable thermal conductivity data is gaining increasing importance. The field of aircraft construction may serve as an illustrative example. For many years CFRP have been used for certain components of civil aircraft, and in today’s modern airliners they constitute, typically, 20-25% of the total structural mass. This value can be significantly increased only by building safety-critical primary components from lightweight composites. The realization of CFRP wings for example would therefore be a breakthrough from the viewpoint of weight reduction. However, since the mechanical properties of the CFRP matrix strongly depend on temperature, such a wing cannot be safely designed without considering the temperature field during flight and on the ground. Special finite-element methods have been developed for calculating the temperature field in multidirec-

A, = V,Ap+ V,A,

(I)

where V, and A, denote the volume fraction and the thermal conductivity of the matrix while V, and Ar stand for the corresponding properties of the fibres, parallel to their longitudinal axis. Since the crystal lattice of the fibres shows basal planes that are either circumferentially (PAN fibres) or radially (pitch fibres) arranged around its axis (Fig. 2), they are highly anisotropic. While the conductivity in the basal planes is good, the heat flow across the planes is poor. As a consequence there is a remarkable difference in the thermal conductivities in the longitudinal, Ap, and transverse directions, A:, of the fibre. The absolute magnitude of A$’depends strongly on the degree of perfection of the crystal lattice and, hence, large variations may occur. The T300 fibre (PAN, Toray), for example, exhibits a thermal conductivity of about 7 W m-’ K-’ while for the FT700 fibre (pitch, Tonen) 45

46

R. Rolfes,

U. Hammerschmidt

G3 +

// A’

x,1 Fig. 1.

Unidirectional lamina.

a value of 360 W rn-’ K-’ is obtained.5’h The transverse conductivity, A:, varies to a much lesser degree. The corresponding values for the fibres mentioned above are 2.0 and 2.4 W rn-’ K-l.“,7 Thus, A: can be two orders of magnitude higher than A:. Taking into account this anisotropy, eqn (1) provides reliable conductivities, A,. In the transverse direction of the lamina the situation is much more involved. Assuming that the lamina (Fig. 1) is transversely isotropic, the identity AZ= A3

(2) applies and the thermal behaviour can be sufficiently described by the two properties A, and AZ. In the literature there are three different ways of treating the problem of transverse conductivity: numerically by micromechanical calculations; analytically by deriving approximation formulae; and experimentally by measurements. Each approach has its own advantages and contributes like the others towards solving the problem. For practical applications a validated approximation formula is, of course, the most simple and efficient way, which is why there are a lot of equations of this kind.Xm’7However, since they provide rather different results, especially for high fibre volume fractions, the engineer making the calculations needs reliable criteria for selecting the appropriate equation.

circumferentially

orthotropic

Fig. 2.

radially orthotropic

Orthotropic fibres.

All formulae mentioned above are based on analytical assumptions rather than on empirical data, for which reason a numerical micromechanical calculation can serve as a yardstick in order to check the analytical accuracy of the formulae. Numerical results are given by Han and Cosner,” GrovelY and Adams and Doner.*” Adams and Doner have performed finite difference methods for the longitudinal/transverse shear modulus CL.,.. Springer and Tsa? verify the analogy between the governing equations for AZ and GLT. The results of Adams and Doner*” may therefore be directly transferred to transverse conductivity. Like Grove,” they consider a quadratic (or square) packing order of the fibres within the matrix. Grove modifies the numerical values through statistical techniques in order to match random fibre distributions. Han and Cosner’” analyze the quadratic and the hexagonal packing order. None of the authors systematically compares their results with approximation formulae for the corresponding packing order. However, this is essential, since it checks the physical correctness of the approximation formulae. This is done in the following section, where a preselection of two formulae is provided-one for the quadratic and one for the hexagonal array. Subsequently, the results are compared with experimental data. However, this comparison raises severe problems. First, high-quality samples having different fibre volume fractions from 0.45 to 0.75 have to be produced (Section 3). Second, a priori values for A: are needed in order to evaluate the different approximation formulae. Direct measurement is extremely difficult and the calculation of A: from measurements on composite specimens is doubtful. Regarding the general behaviour of the formulae depending on the parameters V, and A; one observes that at low fibre volume fractions (V, 5 0.5) all formulae provide very similar results, which are quite insensitive to variations of A;, and at high volume fractions (V, = 0.7) the discrepancies between the formulae are very significant and their sensitivity with respect to A,’ is high. Thus, the calculation of A, from a measurement on a sample with low V, would allow the use of any formula, but would need extreme accuracy of measurement in order to achieve a reliable value for A:. The use of a specimen with high volume fraction would reduce the accuracy requirement, but would require the preselection of one of the formulae under investigation. We conclude that it makes no sense to calculate A: from any of the measurement data. However, a comparison of experimental data with different approximation formulae is possible if only regarding the best fit. which can be achieved with an arbitrary value for A;. A formula selected by this procedure is qualified for translating conductivities measured at a distinct fibre volume fraction to another volume fraction. This

47

Thermal conductivity of CFRP laminates meets the needs of engineering practice. However, one should not believe the values for A; which are provided by the calculation. Only a few measurements are reported in the This is why we have carried out literature.*’ measurements of our own by two different methods. The first one is the conventional steady-state guarded hot-plate (GHP) method, while the second one is the transient hot-strip (THS) method. 2 MICROMECHANICAL

CALCULATIONS

formulae of In this section, the approximation previous worker?” are compared with micromechanical finite-element calculations.** The formulae in Refs 14-17 were not included in the comparison for different reasons. One equation is only applicable to fabric composites,” two formulae only provide bounds for the conductivity’“.” and the last one delivers values far removed from our numerical solution.” Though based on different theories, all formulae take into account the volume fractions and the thermal conductivities of fibre and matrix, as well as a distinct geometrical arrangement of the fibres within the surrounding matrix. These intrinsic properties are denoted as primary variables. Some formulae try to improve the results by considering additional effects of void content, interfacial barrier resistance between fibres and matrix (which is more important for glass-matrix composites than for CFRP because of the shrinkage of the matrix), or by introducing adjustment parameters in order to match some experimental results. These properties will be denoted as secondary variables. The micromechanical finite-element calculation can only be taken as. a yardstick for comparing different approximation formulae if the finite-element model is uniquely determined by the variables entering the approximation formulae. This does not hold for secondary variables, since the geometrical distribution of voids and the value of an interfacial barrier resistance are unknown and the influence of empirical adjustment parameters on the numerical model is fully undetermined. Therefore only the influence of the primary variables is taken into account. The approximation formulae and the finite-element calculations are based on some general prerequisites: l

l

l

l

l

the fibres have a circular cross-section; all fibres are identical; the fibres are homogeneously distributed within the matrix; the temperature profile is independent of position x (Fig. 1): fibre and matrix are homogeneous and isotropic in the plane under consideration.

The last prerequisite is incorrect, since the fibres are circumferentially or radially orthotropic (see above).

However, the assumption must be made here, since all formulae under consideration are based on it. In the work of Knott and Herakovich’ a formula taking into account the orthotropy is developed, but it uses the ratio of the thermal conductivities in the radial and circumferential directions (A+‘/A:‘), which is normally unknown. The first approach is provided by Hasselman et af.I3 They assume A,’ = A? and A?’ = 2.9 W m ’ K-’ for circumferentially orthotropic fibres which in turn has still to be verified. In the literature many attempts have been made to classify the existing equations for A?. We add another, which is perhaps the simplest. Looking at the basic ideas behind the various models, it turns out that only two are essentially different. These are the electrical resistance analogy and the potential theory approach. Other names like ‘shear analogy’ and ‘elastic moduli analogy’ often appear in the literature. Both categories can be ascribed to the potential theory approach as will be seen subsequently. In 1882 Rayleigh wrote his fundamental work”’ which relies on potential theory. This approach is obvious because the mathematical formulation of the steady-state heat conduction in an isotropic medium is the potential equation V’T = 0

(3

where V’ denotes the Laplacian operator. For the matrix a translational and a dipole potential are considered, while for the fibre only a translational potential has to be taken into account. Both the potentials are given in terms of a trigonometrical series. The resulting streamline shape is physically reasonable. A high fibre content is explicitly allowed. The only disadvantage is the restriction to quadratic arrays. The formula is:

v,

$cp2 111

3v: v’+ v,-p v’IIJ sf

(4a)

with

I+“’ v’ =m

A (4b)

*-$ and S, = 0.03235021 I=’

(4c)

HermansZJ retains only the first element of Rayleigh’s trigonometrical series for calculating the transverse/longitudinal shear modulus. He does not consider a specific packing order, but a special geometry consisting of a circular hbre and two concentric rings of matrix and composite material

48

K. KolJes,

U

Fiber Matrix Composite (homogenized) Fig. 3.

Self-consistent model geometry.

(Fig. 3), which is why self-consistent one:

his method

A2 -=

A: + A, + (A: - A,)Vf

L

/\:+A,-(A:-A,,,)&

is called

a

(5)

This equation probably originates from Polder and van Santen,‘” but because its origin is not fully clear, it will subsequently be referred to as the self-consistent formula. It can also be denoted as a potential theory equation since it is based on Rayleigh’s approach. The widely used relationship of Tsai and Halpin”

(64 with

(6b)

Hammerschmidt

can also be ascribed to the potential theory approach. The kind of inclusions (either spheres, cylinders or rods) and the heat flow direction are taken into consideration by CL+ the influence of the packing order is reflected by $, where V,,, is the maximum packing density. These parameters were introduced in order to fit measurement results, for example CLN = 0.84 was suggested for fibre composites. As mentioned above, we have excluded secondary effects. The value of C,, = 0.5 was therefore used, which applies to heat flow perpendicular to uniaxially oriented fibres. The last candidate in the group of potential theory approaches is the method of Hasselman and Johnson.” They retain the zero and first-order terms of the Rayleigh series and additionally introduce an interfacial barrier resistance between fibres and matrix. Disregarding this secondary effect yields a Rayleigh equation which is only valid for small fibre volume fractions (not to be confused with eqn (4)). Since typical CFRP structures exhibit a rather high fibre content, this equation is not examined further. Another approach besides the potential theory is the electrical resistance analogy. Springer and Tsai” consider a unit cell for a quadratic array (Fig. 4). The total heat flux through the cell is separated into three individual fluxes, two of which pass through the matrix, 4, and q,, only the third one, q2, crossing matrix and fibre in series. By infinitesimally dividing q2 into the heat flux elements dqz, the cross-sectional shape of the fibre is taken into account. Finally the following equation is obtained:

v: vf 1 -+n s

A2 -_=l-2 n

and

“I

log I$= log(a/b)ti

(6~)

is derived from eqn (5). The parameter 5 that depends on the sizes of both the axes of the fibre cross-sectional area (a and 6) is introduced in order to match the numerical results obtained by Adams and Doner’” for circular and elliptic fibres. For circular fibres 5 = 1 and eqn (6) reduces to eqn (5), which again is a potential theory formulation. The formula of Lewis and Nielsen:” A2 AIll

_

1 + 1 -

CLNQ_NV~

DLNW’f

X

@a> where s=2

[

3-l f

1

@b)

(74

with A: --1

(7b) and

*=l+(*)v

(7c)

Fig. 4.

Unit cell according

to Springer

and

Tsai.’

Thermal conductivity of CFRP laminates

it is based on the assumption of a constant heat flow direction, that is not influenced by the fibre. This is a major drawback compared with the equations relying on the potential theory. Thornburgh and Pears’ derive an electrical resistance analogy for fabric composites. Their equation can be applied to unidirectional composites if the amount of heat passing through any fibre-to-fibre contact through the laminate may be neglected. Their formula is then identical with that of Springer and Tsai’ for a quadratic fibre cross-section. The finite-element calculations were carried out on representative cells (Figs 5 and 6) for a quadratic and a hexagonal array using the general purpose code MSC/NASTRAN. The horizontal edges of the cells were considered adiabatically bounded, while the others were kept isothermal at temperatures of 20 and 30°C. At the edges the combined nodal heat fluxes were summed, yielding the total heat flux 9. The transverse conductivity of the composite was then calculated using Fourier’s law

49

However,

where 1 is the length of the cell in the direction of the heat flow (horizontal), and the temperature drop AT equals 10°C. Fibres and matrix were isotropic, values for &‘/A, up to 120 were chosen, and the fibre volume fraction was varied between 0.4 and 0.7. Our numerical results agree well with those of Adams and Doner,“’ Han and Cosner” and Grove.‘” The isothermal plots (Figs 5 and 6) clearly show the strong influence of the fibres on the heat flow direction.

3O.Q 29.3 288 27X!__ 27.2 28.5 25.8~~. 25.1. 24.4 23.7 23.Q 22.3 21.e ZO.Qi_ 20.0 1Q.Q I

Fig. 6. Isothermals

for the hexagonal array.

Figure 7 compares the finite-element results for the quadratic array with analytical results obtained from eqns (4) (5), (7) and (8). The Rayleigh equation matches the numerical results almost perfectly. This is in full agreement with our theoretical considerations. In Fig. 8 the finite-element results for the hexagonal array are compared with those from eqns (5) and (7). The self-consistent eqn (5) represents our results best,

6.5 6

Lew/Niel *Self-Con. Raylelgh Spr/lsai FE. auad

2.5

0.45

Fig. Fig. 5. Isothermals

for the quadratic array.

0,5

0.55

0.6

0.65

”f results versus approximation mulae; quadratic array, A; /A,, = 50.

7. Finite-element

0,7

for-

50

R. Rolfes, U. Hammerschmidt

The aforementioned degree of accuracy can readily be achieved by a steady-state technique, e.g. the absolute measuring GHP method. The GHP is considered the primary referencing instrument but its applicability is limited to poor conducting material. For higher conducting CFRP specimens, a non-steadystate technique was therefore used, the THS method. Although we have successfully applied this technique to a number of other materials, it is still at the development stage. Therefore we performed comparative runs on both the instruments within the overlapping part of the individual measuring ranges. It turned out that the uncertainty and repeatability of the GHP method are somewhat better than those of the THS set-up, while the range of applicability of the latter is wider. From one short and easy experiment the THS method not only provides the thermal conductivity but also the thermal diffusivity and, hence, the volumetric heat capacity. Some basic features of both our instruments are listed in Table 1.

5.5

5

4.5

4

x,

%ll

3.5

3

2.5

2 0.4s

0.5

0.55

0.6

0.65

0.7

“f

Fig. 8. Finite-element results versus approximation formulae; hexagonal array, A:/h,, = 50. which again is not surprising. Conductivity ratios up to 120 showed the same results. It can therefore be concluded that the Rayleigh equations and the self-consistent one are the best equations for the quadratic and the hexagonal array, respectively, from the viewpoint of analytical accuracy. However, the difference between the two is significant, especially for high fibre volume fractions. Measurements were carried out in order to check whether the quadratic or the hexagonal array is more realistic. 3 MEASUREMENTS

The standard instrument for the determination of the thermal transport properties of CFRP is the laser flash apparatus because of its convenience, short experiment times, and large measurement range.2h The latter feature is of great importance because of the extensive variations of CFRP transport properties with temperature and direction. However, the laser flash technique is capable of measuring the thermal diffusivity, a, only. The quantity considered, A, has to be derived through the relationship a = A/(cp) with the further knowledge of the specimen density, p, and specific heat, c. This indirect determination of A leads to an uncertainty that is intrinsically not better than 10%. This is not sufficient for our purposes. To be able to decide that one given array model is preferable the experimental results should be to another, accurate to at least 5%.

3.1 The GHP method In our GHP instrument, as shown in Fig. 9, the disc-shaped specimen is sandwiched between the lower cold plate and the upper heating unit. The heating unit consists of the hot plate, the guard ring, and the guard plate. Prior to a measurement a constant temperature gradient, AT/f, must be established across the sample of cross-sectional. area A and thickness 1. This is done by imposing the unidirectional and homogeneous rate of heat flow @ which equals the electrical power P fed into the resistance heater of the hot plate. A zero temperature difference must be maintained simultaneously across all parts of the heating unit. The thermal conductivity of the sample is then calculated using Fourier’s law, eqn (9), where in this case, 4 = P/A. Analogous to the work of Hemminger and Juge12’ a detailed evaluation procedure leads to a total uncertainty of the GHP instrument of 2.6%. The CFRP specimens are of a circular cylindrical shape, 100 mm in diameter and 2-3 mm thick, and had a unidirectional lay-up. The surfaces have to be plane parallel in order to ensure good thermal contact with both the plates. This was achieved by applying a thin coating of pure resin to the surfaces after finishing the first consolidation process. An epoxy resin (LY556 HT976) with T300-fibre was chosen for testing. Specimens with fibre volume fractions covering a wide range are necessary for verifying the micromechanical equation. However, manufacturing voidless specimens with very low fibre contents turned out to be difficult. After many unsatisfactory attempts, the problem was solved by adding liquid resin to the laminate during the consolidation process. The experimental data are listed in Table 2 and

51

Thermal conductivity of CFRP laminates Table 1. Measuring instruments Range (Wm ’ Km’)

Apparatus

Guarded Transient

hot plate hot strip

Temperature (“C) 20-200 -751000

0.1-5 0.1-400

in Fig. 10 where they are also compared results from the equations in Section 2. self-consistent equation with At/h, (30°C) = 8.5 At/h,,, (120°C) = 13.5 agrees favourably with measurement results.

plotted

with The and the

3.2 The THS method In contrast to the aforementioned steady-state method with its straightforward working eqn (9), in the THS technique the temperature history of the heat source must be analyzed following a complicated mathematical model. This can only be done by means of a computer. However, the experimental set-up is very simple. It consists only of an electrically-heated thin metal foil (the ‘strip’) as the heat source sandwiched between two similar brick-shaped specimen halves (Fig. 11). In order to determine the thermal transport properties h and a from a THS experiment, a constant current is passed through the metal foil. The (temperature-dependent) voltage drop U(t) across the foil is simultaneously recorded pointwise by a fast data acquisition system and then transferred to a PC. It serves as a measure of the foil temperature excursion AT(t). A straight line fit of U = U(f(&, k(A), a, to)) to the data is made using the Levenberg-Marquard

range

details

Test period (min)

Uncertainty (%)

Repeatability (“/) 0.01 2.5

2-3 5

360-600 l-5

algorithm,” which gives an estimate of the unknown thermal properties. Each experimental run and the subsequent analysis can be performed in less than 10 min. The computer program not only provides the CFRP thermal properties along with their individual confidence bands and uncertainties but also useful information on possible improvements to the experiment parameters (e.g. strip width, current, sample dimensions, etc.). The uncertainty does not exceed 5% which has been confirmed by experiment.2y Compared with the costly laser flash technique there is no material blow-off due to effects of the heating pulse on the sample surface. However, the THS method requires greater samples. Several specimens (170 mm X 30 mm X 20 mm) with fibres in the longitudinal direction were prepared for the THS measurements using the mould and autoclave technique. Their thickness of 20mm was achieved in one shot, i.e. no adhesive bonding of thinner plates, which might have effected the conductivity, was needed. However, this complicated the problem of with the voidless manufacturing in comparison cylindrical GHP specimens (see Table 3). If the densities of fibre (pr) and matrix (p,) are known, the void content can be determined from: cp=l-I/,+-l(“rp,-p,)

(10)

Pm

where pc is the density of the composite. The M40 fibre, exhibiting a higher conductivity than the T300 fibre, was used, since a more anisotropic composite better reveals the differences between the micromechanical formulae. Table 3 lists the specimens with void content and measurement results. Since neither the finite-element calculations nor the micromechanical formulae take voids into account, the test results had to be corrected. Belle et al.“” give a relationship for calculating the voidless conductivity of Table 2. Transverse conductivity of carbon/epoxy (LY556HT976/T300) as measured with a GHP apparatus Sample

BAC

Fig. 9. GHP apparatus for a temperature

J

v,

G

range from -75 to 200°C. A, specimen; B, hot plate; C, cold plate; D, guard plate; E, guard ring heater; F, guard ring; G, vessel; H, pressure stamp; I, performances; J, liquid bath.

1 2 3

Pure resin 0447 0643

A(Wm

‘K

‘)

7‘ = 30°C

T = 120°C

0.230 0.484 0.708

0.233 0.528 0.808

52

R.Rolfes,U.”

I

I

I

Table 3. Transverse (LY5564U976/M40)

I

- - - - - Rayleigh --

’ ” nammerscnmm~

Sample

-Self-Con.

conductivity as measured

V,

of carbon/epoxy with a THS set-upm

A (T=20”C)in

cp

Wm

‘K-’

-

- - - - - Rayleigh

Porous material

Voidless material

0.329 0.692 0.854 1.029 1.251

0.329 0.752 0.894 1.085 1.266

-Self-Con. -

0

Meas.

0

Meas.

4 5 6 7 8

L

I

Pure resin 0.493 0.501 0.618 0.692

0 0.055 0.030 0.035 0.008

T=l2O”C-

the approximation is measured by the geometric mean m of the errors in all four data points. For the self-consistent equation m = 0.14 (At/A, = 13) is obtained while Rayleigh’s approach yields m = 0.17 (At/h, = 10.5).

0.4

0.45

0.5

0,SS

0.6

0.65

0.7

”f Fig. 10. Comparison of Rayleigh’s formula (A,’ /Am = 7,s (11,O) at 30°C (120°C)) and the self-consistent formula (A,‘/&, = 8.5 (135) at 30°C (1ZoOC)) with measuresments in a GHP apparatus.

3.3 Results of Pilling et QZ?’ Among the very few measurements that can be found in the literature, those of Pilling et ~1.~’ may be directly compared with our results. Figure 13 shows, that again, the self-consistent equation turns out to be the best fit (m = O-14 (At/A, = 27) for the selfconsistent formula, m = 0.19 (Aklh, = 15) for Rayleigh’s formula).

bulk materials:

4.5

A0= A ’ + Oeh

(11)

2 2

1-q

where A2 is the conductivity of the porous material and cp is the void content. Equation (11) is based on the Maxwell theory of a body with dilute spherical inclusions, the conductivity of which tends to zero. It can be seen from Fig. 12 that again, the self-consistent equation agrees best with the results. The quality of

4 -Self-Con.

35

3

2.5

2 0.45

, Fig.

-IL 2d 11. Schematic

0.55

0.6

0.65

0.7

“f

Y

diagram of set-up.”

0.5

Fig. 12. Comparison the

THS

experimental

and the

of Rayleigh’s formula (At/A,, = 10,5) self-consistent formula (A//A, = 13,0) with measurements in a THS set-up.

Thermal

conductivity

o,f CFRP laminates

53

fraction. However, the value for A:, which is provided by the formula, should not be trusted too much. Furthermore, the work is based on the assumption of a transverse isotropic fibre. Future investigations should also include circumferentially and radially orthotropic fibres. The THS set-up we used for the experimental studies in conjunction with a GHP apparatus turned out to be an adequate instrument to determine thermal properties of composite materials fast and easily. Nevertheless, extensive work has to be done to improve the precision and reproducibility of the set-up until it compares to the steady state GHP apparatus.

6

ACKNOWLEDGEMENTS

op5

0.5

0.55

0.6

0.65

0.7

0.75

”f Fig. 13. Comparison

of Rayleigh’s formula (A,‘/&, = 15,O) and the self-consistent formula (h,l/A, = 27.0) with measurements of Pilling et al.”

4 DISCUSSION

AND CONCLUSION

Micromechanical equations for the transverse conductivity of unidirectional composites were investigated. Theoretical considerations as well as numerical calculations reveal that Rayleigh’s equation best represents the apparent thermal conductivity for a quadratic arrangement of the fibres within the matrix, while the self-consistent equation is the appropriate one for a hexagonal array. This result was achieved by neglecting secondary variables (void content, empirical adjustment parameters, interfacial barrier resistance, etc.), since their influence on the numerical model is not unique. Nevertheless it makes sense to introduce secondary variables into approximation formulae in order to fit experimental data. However, only formulae which correctly describe the influence of the primary variables should be used as starting points: these are the ones selected in the present paper. Comparing relationships with data taken from both literature and experimental results provided by a steady-state and a transient technique gives rise to the conclusion that the self-consistent approach is more realistic than a quadratic array, if porosity is taken into account via eqn (11). The self consistent formula can be used in engineering practice, where the conductivity at a distinct fibre volume fraction is known and shall be translated to another volume

The assistance of A. Pabsch and his team in manufacturing the test specimen, and of H. Kharadi and W. Sabuga in performing the measurements, is gratefully acknowledged. R. R. acknowledges the support of the European programme ‘Human Capital and Mobility’. REFERENCES 1. Rolfes, R., Efficient thermal analysis of anisotropic composite plates using new finite elements. In o,f Developments in the Science and Technology Composite Materials, Proceedings of the Fourth European Conference on Composite Materials (ECCM4). ed. J. Fuller, G. Griininger. K. Schulte, A. R. Bunsell & A. Massiah. Stuttgart, 1900. pp. 743-8. 2. Rolfes, R., Nonlinear transient thermal analysis of composite structures. In Numerical Methods in Engineering Y2, Proceedings of the First European Conference on Numerical Methods in Engineering, ed. Ch. Hirsch. 0. C. Zienkiewicz & E. Onate. Brussels, 1992. pp. 653-Y. 3. ‘Noor. A. K. & Tentik, L. H’.. Stea@~ate nonline& heat transfer in multilayered composite panels. J. Eng. Mech. 118 (1992) 1661-78. 4. Knott, T. W. & Herakovich, C. T.. Effect of fiber orthotropy on effective composite properties. J. Cwnp. Mater., 25 (1991) 732-59. 5. Rolfes. R., WIrmeleitzahlen von UD-Laminaten aus CFK, Berechnung und Messung. DL.R-report IB13191/12, Braunschweig. 1991. 6. The Tonen Corporation, Leaflet for forca fibres. 7. Tonen Corporation (pers. comm.). 8. Springer, G. S. & Tsai, S. W.. Thermal conductivities of unidirectional materials. J. Comp. Marer., 1 (1967) 166-73. 9. Thornburgh. J. D. & Pears, C. D., Prediction of the thermal conductivity of filled and reinforced plastics. ASME Heat Transfer Division. Winter Annual Meeting, 6WA/HT-4, Chicago. IL, 1965. 10. Lord Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium. Phil. Msg., 34 (1882) 481-502. 11. Hasselman, D. P. H. & Johnson, L. F., Effective thermal conductivity of composites with interfacial thermal barrier resistance. J. Camp. Mater., 21 ( 1987) 50X-15. 12. Tsai, S. W. & Halpin, J. C.. Environmental factors in

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13.

14.

IS.

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20.

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