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Controlling thermal expansion to obtain negative expansivity using laminated composites
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 47–59 www.elsevier.com/locate/compscitech
Controlling thermal expansion to obtain negative expansivity using laminated composites A. Kelly a
a,* ,
L.N. McCartney b, W.J. Clegg a, R.J. Stearn
a
Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, CB2 3QZ, UK b NPL Materials Centre, National Physical Laboratory, Teddington, Middlesex, TW11 0LW, UK Received 31 December 2003; received in revised form 27 May 2004; accepted 7 June 2004 Available online 19 August 2004
Abstract It is confirmed, by considering a wide range of laminated composites containing a variety of fibres and matrices, that negative thermal expansion coefficients may be obtained. These are usually accompanied by a correspondingly large value of the in-plane axial PoissonÕs ratio (PR). By making use of this large PR extremely negative values of expansivity may be obtained – much further negative than for any monolithic materials. The use of laminated composites also overcomes some of the previously reported limitations of a device to control thermal expansion when made with monolithic materials. The use of the device, as a platform to control the expansivity of an optical fibre containing a Bragg grating, is discussed in detail and it is shown that the required negative expansivity of 105 K1 may easily be obtained with a number of composite systems. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Polymer–matrix composites; B. Thermomechanical properties; C. Laminates; C. Optical gratings; C. Thermal compensation
1. Introduction Bragg fibre gratings are used as filters in optical communication networks. The reflected wavelength from the grating should remain well within the wavelength range for a single channel so as to avoid cross talk between channels. At present it is required that the filtering capacity of the grating should maintain the value of the wavelength to which the grating is tuned to within 0.25 pmK1 over a temperature range from 20 to 80 C. Both the refractive index and the grating spacing change with temperature, the former being more sensitive to temperature than the latter. Control may be exercised by straining the grating – in which case the value of the photo-elastic constant comes into play. Speaking approximately, it is neces-
*
Corresponding author. Tel.: +44-1223-363691; fax: +44-122333456. E-mail address: [email protected] (A. Kelly). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.06.003
sary to be able to impose upon the grating an effective thermal expansion coefficient of 105 K1. A recent patent [1] describes a number of ways of achieving this control. By providing a platform to which the fibre may be attached, it is possible to impose upon the grating a uniform negative expansivity of the required amount. One approach, and the one with which this paper is concerned, is to make use of the Poisson contraction in the lateral direction which accompanies an axial extension (in materials with a PoissonÕs ratio having a positive sign).
2. Description of the device The device, which is illustrated in Fig. 1, involves a strip of a material, denoted by S, of relatively low thermal expansion coefficient (a) having an adequately large PoissonÕs ratio m, that is to be coupled with a frame of material, denoted by F, of relatively high thermal expansion coefficient that has adequate stiffness and strength.
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
Fig. 1. The frame and strip device designed to control thermal expansion by use of PoissonÕs ratio.
Consider the strip of a material having a low isotropic thermal expansion coefficient, aS, which is fixed at its ends to the frame made of a material with a high isotropic thermal expansion coefficient, aF, as shown in Fig. 1. As the temperature increases the material F of higher thermal expansivity will increase in length more than the strip S, which has a lower expansivity. Because the frame is joined to the strip, the latter will be placed in tension by an amount depending on the difference in the expansion coefficients of the two members, provided that the geometry of the frame is such that bending cannot occur. This elastic tension will cause the strip, S, to contract laterally due to the Poisson effect (assuming, for the present, that m has its normal positive sign). However, because the temperature has been raised the strip will expand laterally. The effective lateral thermal expansion coefficient of the strip, aeff, is given by aeff ¼ mS ðaC aS Þ þ aS ;
ð1Þ
where mS is PoissonÕs ratio of the strip and aC is a ÔcompositeÕ expansion coefficient that is given by aC ¼
a F AF E F þ a S AS E S ; AF E F þ AS E S
ð2Þ
where A represents area fraction and E is YoungÕs modulus, and where the subscript S indicates the strip and F the frame. The effective axial modulus of the composite, comprising frame and strip, in the axial direction (see Fig. 1) is E C ¼ AF E F þ AS E S :
ð3Þ
Assuming for the moment that the area fraction of the frame is very much greater than that of the strip, we may set aC equal to the expansivity of the frame, aF, in which case the net lateral coefficient of thermal expansion of the strip is given by
aeff ¼ aS þ mS ðaS aF Þ:
ð4Þ
The strip will contract in the lateral direction when the temperature is raised if aF P
ð1 þ mS Þ aS : mS
ð5Þ
It has been demonstrated [2,3] that the device functions as predicted, and can produce effective negative thermal expansion coefficients.
3. Limitation of materials available Clegg and Kelly [2] first address the question of whether or not the device, if constructed of normally available materials, can produce values of the expansivity that are more negative than 105 K1. Firstly, it should be noted that a negative expansivity is favoured by choosing a material for the strip having an expansivity that is as low as possible. From Eq. (4), if aS is zero, then in order to attain a value of aeff of 105 K1 we require that aF has the value 105 K1/mS. For common values of mS of about 1/3 or less, which is an appropriate value for materials of low expansivity, the value of the expansivity of the frame must be as much as 30 · 106 K1. Very few materials, other than plastics, show such large values of thermal expansivity. Some alloys of zinc are a possible exception. Such alloys are not stiff, with Young moduli of around 60–70 GPa. Since the area fraction of the frame may be made very large this would not preclude the use of rather compliant materials for the construction of the frame. However, since the actual size of the device in practice, is likely to be limited, we must look for other materials. It is interesting to note that manganese is a noteworthy anomaly in the pattern that materials of high melting
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
point are also those of greater stiffness and low expansivity. Both a and b Manganese have reported values [4] of expansivity of 22.8–25.6 · 106 K1 – as large as that of aluminium. However, YoungÕs modulus of manganese, 191 GPa, and its melting point 1244 C should be compared with the corresponding values for aluminium of 70 GPa and 660 C, respectively. There are conditions also for the material of the strip, which must be subject to a longitudinal strain that is sufficient to produce the desired value of the Poisson contraction in the lateral direction. The transverse strain in the strip arising from thermally induced mechanical loading in the longitudinal direction, eT, is given, on using Eq. (1), by eT ¼ mðaC aS ÞDT;
ð6Þ
where, of course, the quantity in brackets times the temperature range DT over which the device is designed to operate, represents the longitudinal strain in the strip. We need to assess whether the yield strain of the strip, if composed of a ductile material, or the breaking strain, if the strip is made of brittle material, will be exceeded if the device is used over a temperature range of, say, 100 C. Yield and breaking strains are not usually tabulated, but yield and breaking stresses are. The longitudinal stress in the strip is rS ¼ ES eT =mS ¼ ES ðaC aS ÞDT;
ð7Þ
where ES is YoungÕs modulus of the strip. On using (1) the elastic stress in the strip may be written rS ¼ ðES =mS Þ½aeff aS DT:
ð8Þ
Taking DT = 100 C, and noting that we wish to have a value for aeff of 105 K1 the stress in the strip is 103 (ES/mS), if we assume that aS is zero. For normal values of mS, of about 1/3 or less, these are very large stresses that will normally exceed the yield or breaking stress of most materials. Clegg and Kelly [2] consider this requirement in detail and using the Cambridge Materials Selector [5] as a database, concluding that only a very limited range of materials – some high strength steels and titanium alloys would satisfy the requirements. They also point out that quite modest increases in the assumed value of PoissonÕs ratio would admit the use of many more materials, and cite some examples, one of which is the use of composite materials. These can be designed both to yield large values of m and small values of a. Before describing the use of laminated fibre composites we should remark that very many materials in the form of single crystals can show values of PoissonÕs ratio much larger than one half – which is the upper limit for an isotropic material. We have explored the values possible for a number of crystal systems [6] and the results will be reported elsewhere.
49
4. Laminated fibre composites One of the difficulties of exploring the possibilities presented by composite materials is the enormous range of possible systems that arise because choice of fibre, matrix, volume fraction, and arrangement of the fibres are all parameters that must be considered. Fortunately, it is already well known that laminated fibre composite materials, particularly balanced laminates, can produce values of PoissonÕs ratio that are extremely large (e.g., [7–9]), and can also have negative values. This is relevant if the aim were to increase the in-plane expansivity. Such laminates may also show negative values of the in-plane expansivity [10] and hence appear to be excellent materials to be considered for the strip in the device. For a laminated composite the coefficient of thermal expansion will not be isotropic. To emphasize this we rewrite Eq. (4) as aeff ¼ aS;trans mS;axial ðaF aS;axial Þ; where the subscripts ÔaxialÕ and ÔtransÕ refer to the axial and transverse directions of the strip (see Fig. 1). For a given value of aF we must then minimise the expansivities of the strip in both the axial and transverse directions and maximise the axial PoissonÕs ratio of the strip. We shall not repeat here the detailed theory for a laminated [+h/h]ns composite, which may be found in [3]. The equations are a special case that can be analysed using the NPL computer program LAMPROPS designed for general symmetric laminates [11]. The predictions are robust and consistent with other analytical expressions – this is particularly important for the prediction of the CTE in the through-the-thickness direction [12]. We have compared the predictions accurately with the extensive results on carbon fibres in epoxy resins and with those for Kevlar – see in Section 6. There are two stages in our validation of our computer programs. We calculate the properties of an individual ply from the measured properties of the fibre and matrix and the known volume fraction. The equations and approximations for doing this are listed in [13]. Examples of the ÔaccuracyÕ of such calculations are also given in [13]. We have obtained a similar agreement. An example of the demonstration of the validity of the programs for estimating thermal expansion coefficients is shown in Fig. 2. We have made similar tests using two other systems where both fibre and matrix properties have been measured or estimated and where measurements have been made of the in-plane expansivity of the composite. These systems were HMS (high modulus) carbon fibre in epoxy and HTS (high tensile strength) carbon fibre in epoxy using the extensive experimental results [14–17] for CFRP, and the results for Kevlar in epoxy [18,19]. There are a few published measurements [20,21] of the through-thickness expansivity of a composite.
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59 100
1.6
6
Thermal expansion coefficient x 10 K
-1
1.4
80
60 1
40
0.8
Axial coefficent Through-thickness coefficient Transverse coefficient Experiment Axial Poisson's ratio
20
0.6
Axial Poisson's ratio
1.2
0.4
0 0.2
-20
0
0
10
20
30
40
50
60
70
80
90
Interply angle (degrees) Fig. 2. Predicted values of PoissonÕs ratio and thermal expansion coefficients for a composite of Kevlar 49 fibres in PR 286 (3M) resin. The fibre volume fraction is 0.5. Experimental results from [17].
Fig. 2 demonstrates the excellent agreement between prediction of the axial expansion coefficient and experiment for the case of Kevlar. The measurements of thermal expansion were carried out over a temperature range of 20–150 C for the composites and were closely linear implying a constant value of expansivity over this range. The values assumed in predictions for the resin and fibre are for a temperature of 30 C. It is to be noted from Fig. 2 that the in-plane expansivity curves reflect about h = 45 so that the value of the axial expansivity at an angle h is equal to the transverse expansivity at an angle of (90h). We are not publishing here validation of the computer programs for the prediction of values of PoissonÕs ratio. For HMS carbon in epoxy and for HTS carbon in epoxy, our programs reproduce accurately the values obtained and compared with experiment [8,9]. Similar quantitative agreement, as that shown in Fig. 2 for the values of the expansivity for Kevlar composites, was found for HMS carbon and for HTS, except that in the latter case there was rather poor agreement for the predicted value of the expansivity in the h = 90 direction, i.e. in the transverse direction for a single ply. In subsequent experiments, referred to below, we have found this discrepancy to arise often in other systems. The transverse expansivity of a single ply is dominated by the value of the expansivity of the matrix and when the matrix is a polymer this may be a varying quantity depending on the mode of fabrication of the composite and the value depends significantly on tem-
perature at temperatures close to room temperature. The difficulties of making measurements of the thermal expansion characteristics of fibre laminates have been reported in the literature [21]. Having obtained an accurate predictive capability we must first entertain the idea that the required expansivity and strength for the control of a Bragg fibre grating might be obtained simply from a composite material without the need to use our particular device – see, for example Fig. 2. We have explored a number of composite systems which seem likely to provide large values of PoissonÕs ratio (PR) combined with small values of CTE, and have found the appropriate angle of a balanced laminate so as to maximise/minimise the relevant effect for given values of the thermoelastic constants. The essential reason for a peak in the value of PR shown in Fig. 2, observed also in other systems at an angle of about 30, and the corresponding small and possibly negative value of the axial expansivity (often but not always at an angle slightly less than that at which the PR peaks) is due to the shear deformation in the two orientations of the laminate producing a type of scissoring action. The requirement for the effect to be large is that the axial shear modulus of the component plies be small, with respect to the other elastic constants, particularly the axial stiffness. The value of the axial stiffness of an angle-ply laminate is usually governed by the value of the shear modulus of the matrix. It is not important that the expansivity of the matrix be small – in fact the larger the better, provided the axial expan-
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
sivity of the fibre is a good deal smaller than that of the matrix. This is essentially because the matrix must be squeezed sideways by the fibre as the temperature is raised, so that the scissoring action occurs in the transverse direction and hence reduces the axial expansivity of the composite. In order to illustrate the effect we have run the computer programs to predict the values of PR and expansivity for a number of systems, characterised by being composed of a stiff fibre with a small (lateral) expansivity in a matrix of a much more compliant material. We have studied fibres that are commercially available. The systems examined and the measured properties of fibre and matrix used as inputs are given in Table 1. These data have been used to calculate the properties of single plies of the binary composite for a particular volume fraction, and following this the properties of a [+h/ h]ns laminate. We have also used the experimentally measured properties of single plies of certain combinations of fibre and matrix where these are given in the literature for well-characterised materials.
51
The predictions of thermoelastic constants for a number of systems are shown in Table 2, whence it is apparent that a negative in-plane expansion coefficient is predicted in a number of systems, together with a large value of PoissonÕs ratio. The condition necessary to obtain this negative expansivity is, as emphasised by Ito et al. [10], is to have as large as possible a value of the ratio of the axial stiffness EA to the axial shear modulus lA. The value of lA is, for most volume fractions of interest, dominated by the value of the shear modulus of the matrix, and so this should be chosen to be as low as possible. It is to be noted from Table 2 that the minimum value of the expansivity occurs in all the systems at approximately the same angle h of about 30. However, from the data presented in Table 1, one sees that in order to obtain a negative value in the laminate a small thermal expansion coefficient of the fibre is also a great advantage – compare the values of aA for the isotropic fibres in Table 1. Tungsten and glass have about the same expansivity, but the former is very considerably stiffer. Textron SCS-6 and tungsten do not differ greatly in stiffness but the former attains a lower value of
Table 1 Properties of fibres and matrices used in calculations Material
EA (GPa)
ET (GPa)
mA
mT
lA (GPa)
aA (· 106 K1)
aT (· 106 K1)
Fibre AS 4 Silenka glass Modmor carbon V-tex glass (E-glass) Kevlar 49 SCS6 W-wire SiC
225 74 405 73 124 395 411 483
15 74 8.6 73 6.0 259 411 483
0.2 0.2 0.35 0.2 0.35 0.25 0.28 0.19
0.07 0.2 0.53 0.2 0,35 0.34 0.28 0.19
15 30.8 13.7 30.4 2.2 96.6 160.5 170
0.5 4.9 0.1 5.1 5.7 2.3 4.5 3.3
15 4.9 22.1 5.1 60 23 4.5 3.3
Matrix Epoxy MY Epoxy Ciba LY558 Epoxy PR-286 Polypropylene shell Silicone rubber Rubber Ti–6Al–4V Al
3.35 5.28 3.5 1.3 0.03 0.03 125 70
3.35 5.8 3.5 1.3 0.03 0.03 125 70
0.35 0.35 0.35 0.3 0.495 0.495 0.31 0.34
0.35 0,35 0.35 0.3 0.495 0.495 0.31 0.34
1.24 1.95 1.29 0.5 0.01 0.01 38.5 26.1
58 55 65 90 60 200 9.0 23
58 55 65 90 60 200 9.0 23
Table 2 System comparison System (all at Vf = 0.5)
W wire in epoxy
W wire in PP
SCS 6 in poxy
SCS 6 in PP
E-glass in epoxy
Silenka glass in PP
Minimum value of axial expansivity · 106 (K1) h value for minimum () Maximum value of PoissonÕs ratio EA/ET EA/lT (aT aA) · 106 K1
1.51
5.54
3.91
7.97
5.69
0.299
33 1.93 at 24 20.0 56.8 30.9
33 3.32 at 20 54.6 138 46
33 1.90 at 24.5 19.5 55.3 32.1
33 3.20 at 20 53.5 134 47.1
27 0.73 at 32.5 4.09 11.4 28.1
31 1.26 at 28 10.3 28.0 44.4
52
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
expansivity in the laminate due to the lower expansivity of the fibre. As well as a low value of the shear modulus of the matrix, a correspondingly large value of the expansion coefficient of the matrix will promote a negative expansivity of the laminate. This is reflected in the comparison between the values in Tables 1 and 2 for glass fibres in epoxy and in polypropylene (PP). The physical reason for this follows from the scissoring tendency of the laminate as already pointed out to explain the high values of axial PoissonÕs ratio. In the case of thermal expansion, the individual plies are trying to expand sideways (transverse to the fibres) because of the larger thermal expansion coefficient of the matrix, and this Ôdraws inÕ the material in the axial direction and produces a negative, or small axial expansivity. The final column in Table 2 contains data for the system Silenka glass in PP. A small negative value of expansivity is predicted at angle h of 31. This material system has been investigated experimentally in [10,22]. Negative values have been found which are rather more negative than those predicted in Table 2. In the particular case of this thermosetting resin, the predicted values of the laminate depend sensitively on the assumed properties of the matrix. None of the systems studied in detail show a negative value of the expansivity of the laminate of the desired amount except the system Kevlar in epoxy – see Fig. 2. Here a minimum value of 12.7 · 106 K1 is predicted. This value is in fact slightly less negative than the experimental value of 13.7 · 106 K1 found by Strife and Prewo [18] in an experimental laminate with h equal to approximately 30. It follows that such a system could provide a platform on which a Bragg fibre grating might be mounted so as to control the expansivity of the grating by the required amount. A value of the expansivity of less than 105 K1 is shown to occur over the range of angles from 23.5 to 37.5 – a useful range of angles. The low value is due to the negative expansivity of the fibre 5.7 · 106 K1 and its relatively high stiffness, viz. 124 GPa. We discuss this system a little further in Section 8.
5. The use of laminated composites in the device We have written and validated a computer program that carries out the following calculation. A [h/h]s laminate forms a strip that is attached to a square frame made of an isotropic material having a YoungÕs modulus EF and a linear thermal expansion coefficient aF. The axial Young modulus of the anisotropic strip is deðSÞ noted by EA ðhÞ. The axial thermal expansion of the ðSÞ strip is denoted by aA ðhÞ while the transverse in-plane ðSÞ thermal expansion coefficient is denoted by aT ðhÞ. The (S) axial PoissonÕs ratio is denoted by mA (h).
The effective axial YoungÕs modulus of the strip and frame, regarded as a parallel bar model, is given by the mixtures rule ðSÞ
ECA ðhÞ ¼ AF EF þ AS EA ðhÞ;
ð9Þ
where AF is the area fraction of the frame, and AS = 1AF is the area fraction of the strip. The corresponding axial thermal expansion coefficient is given by the mixtures rule ðSÞ
aCA ðhÞ ¼
ðSÞ
AF EF aF þ AS EA ðhÞaA ðhÞ : ECA ðhÞ
ð10Þ
The effective transverse expansion coefficient of the strip is denoted by aeff T ðhÞ, and is defined by ðSÞ
ðSÞ
ðSÞ
ðSÞ
C aeff T ðhÞ ¼ aT ðhÞ þ mA ðhÞaA ðhÞ mA ðhÞaA ðhÞ:
ð11Þ
A computer program LAMPROPS (version 5) [11] calculates values of aeff T ðhÞ which are tabulated in an output file as a function of h. The values of AF, EF, and aF are input via the control ðSÞ ðSÞ ðSÞ file for the software. The values of EA ðhÞ, aA ðhÞ, aT ðhÞ ðSÞ and mA ðhÞ are calculated using the existing set of routines mentioned above – Section 4. An example of the use of the program is to predict, as shown in Fig. 3, the value of the effective transverse expansion coefficient of the strip for a laminate containing high modulus carbon fibres in epoxy when attached to a steel frame of Young modulus 200 GPa, thermal expansion coefficient of 14 · 106 K1, and with the frame occupying 95% of the cross-sectional area. In order to obtain the largest possible negative value of expansivity when using the device, the value of PoissonÕs ratio appears to be more important than the value of the minimum transverse expansivity of the composite system found when standing alone. For instance the minimum predicted value of transverse expansivity for APC-2–3.08 · 106 K1 when the angle h is 61.5. For HMS carbon the corresponding figures are 3.9 · 106 K1 and 61 which are not greatly different. However, the maximum value of PR for the latter is 1.965 at 29 whereas for the former the maximum PR is 1.41 at 26. When placed in the device under identical conditions, employing a steel frame, the minimum value of the expansivity for the HMS carbon is 15.9 · 106 K 1 at 29, whereas for APC-2 the minimum expansivity is only 7.0 · 106 K1 at 36.
6. Experimental verification We have verified the efficacy of the device by using an aluminium frame with the dimensions shown in Fig. 4. The dimensions of the strip are also shown in this figure. The long axis of the aluminium frame contains a circular tunnel in each arm that accommodates a resistance-
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
53
30
6
Transverse thermal expansivity x 10 K
-1
25 20 15 10 5 0 -5 -10 -15 -20 0
10
20
30
40
50
60
70
80
90
Interply angle (degrees) Fig. 3. Predicted values of the effective transverse expansivity of high modulus carbon fibres in epoxy using a steel frame.
Fig. 4. A photograph of the device as set up to measure transverse expansivity. The dimensions of the strip are given.
heating element. The cross-section of each of the longitudinal arms of the frame was 15.8 · 10.5 mm. The diameter of the tunnel in each arm is 6.0 mm. The area fractions of strip and frame for use in Eqs. (9) and (10) are 0.053 and 0.947, respectively, which we have taken as representing a value of AS of 0.05. We have measured the thermal expansion coefficient of the aluminium frame as 23.34 · 106 K1 and used as a strip a balanced symmetric laminate made of APC-2 material from Fiberite containing a nominal 61% of Hercules
Magnamite AS4 carbon fibre in Peek resin. We have taken YoungÕs modulus of the aluminium frame as 70 GPa. The specimens consisted of 8 layers of pre-preg forming a balanced symmetric laminate with fibres running at ±0, ±10, ±17.5, ±25, ±35 and ±45. The laminates were made by pressing for half an hour at a pressure of 2 MPa, and at a temperature of 400 C. The relative area fractions when mounted in the device are 0.95 for the aluminium frame and 0.05 for the strip of
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
laminated composite. The transverse expansivity of the strip has been measured with a laser micrometer and the axial PoissonÕs ratio found by using resistance strain gauges attached to the specimen. A photograph of the apparatus with a thermocouple attached is shown in Fig. 4. The dimensions of the strip are also shown – the in-plane dimensions being 52.0 mm · 15.4 mm and the thickness 1.01 mm. The distance between the centres of the two holes is 40 mm. The transverse thermal expansion coefficient of the strip was measured before it was attached to the device. Fig. 5 shows an example for the strip containing fibres at an angle h = 45. The measured transverse expansivity of the material of the strip in the free state over the temperature range shown is 1.3 · 106 K1 – the corresponding value calculated using Eq. (11) is found to be 2.7 · 106 K1. For the heating times that we have employed, it is found that the resin strip containing carbon fibre heats up much more slowly than the aluminium frame so a correction has to be applied to take the difference of temperature into account. When mounted in the device the measured value of transverse expansivity at h = 45 is found to be 19.3 · 10 6 K1 and the calculated value from Eq. (11) above is 20.9 · 106 K1. A comparison of the values predicted from Eq. (11) for the transverse expansivity and the measured values for the whole range of angles is given in Fig. 6. We have found that in order to obtain agreement at h = 0 we need to use in Eq. (11), and hence in the com-
puter program, our measured value of the transverse expansivity of the strip, viz. three measurements of 41.7, 42.7 and 43.8 · 106 K1 having a mean value of 42.7 · 106 K1. This measured value approximates more closely to that found by us when using our ply program to calculate the value of aT from the properties of fibre and matrix, viz. AS 4 fibre in Peek resin, than it does to the value given by the manufacturerÕs value for APC-2. For this reason the data in Fig. 6 are designated AS4 Peek rather than APC-2. The qualitative agreement between prediction and experiment for the form of the curve is good. Quantitative uncertainties arise because of a number of factors which are discussed elsewhere [23], but it may be seen that the efficacy of the device is demonstrated, and a negative expansivity of as much as 30.5 · 106 K1 at h = 35 is experimentally obtained. The measured value of PR for APC-2 at 35 was 2.07 (the mean of three measurements), which accounts for the very negative value of aTeff measured at these angles. In the case of APC-2 the predicted range of angles h, over which a value of the transverse expansivity of less than 105 K1 may be obtained with the device, is from 17 to 59. Consequently, in principle the objective of obtaining a platform having an effective expansivity of 105 K1 or less is demonstrated. Of course it is not necessary to consider only the use of a material such as APC-2 in the device. We may show, having demonstrated the validity of the approach, that the systems considered in Section 4 (see Table 2) are also predicted to show negative values of expansivity
15.56
y = 2E-05x + 15.554
Length (mm)
15.55
Clamped Free
15.54
y = -0.0003x + 15.552
15.53
15.52 20
30
40
50
60
70
o
Temperature ( C ) Fig. 5. Measurements of the width of the strip as a function of temperature.
80
90
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
55
40
30
Eqn. (11) Measured
6
-1
Expansivity x 10 (K )
20
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
80
90
Interply angle (degrees) Fig. 6. Comparison of predicted and measured values of the transverse expansivity of a strip of APC-2 (AS-4 in Peek) in the device as a function of the angle h of the specimen.
Table 3 Range of angles h over which aeff <105 K1 for various material systems Material
Min (·106 K1)
Angle ()
Rangea
SCS 6 in PP Ww in epoxy Ww in PP SCS 6 in epoxy E-glass in epoxy
69 31 62 36 2.57
23 29 23 29 46
12–56 18–52 12–53 16–55 Zero
a
Remark
PoissonÕs ratio too small
aeff < 105 K1.
over an appreciable range of angles h. Predictions are collected in Table 3. A frame of aluminium is assumed and the volume fraction of the strip is taken as 5%. The case of E-glass fibre in epoxy demonstrates most clearly the necessity of having a system with a large value of PoissonÕs ratio.
7. Internal strains We have demonstrated that, by designing laminated materials so that they show large values of PoissonÕs ratio, the difficulty outlined in Section 3 of finding a suitable material for the frame may be removed. The second difficulty remarked upon there is that the material of which the strip is made may be insufficiently strong to withstand the stresses placed upon it if the
device is to control expansivity over a sufficiently large range of temperature. In that case we assumed a necessary value of 100 C for the temperature range. In the case of a composite strip there are two types of stresses to consider. The first is, in complete analogy with a monolithic material, whether the strip can withstand the force exerted by the frame upon it. This, of course is directly proportional to temperature difference over which the device is to control the expansivity. Fig. 7 shows the values of the axial stress and strain in the strip made of APC-2 material for a temperature change of 100 C. The stress at h = 0 is 274 MPa and at h = 45 is 34 MPa. The first figure is well within the breaking strength of 2130 MPa for a 0 strip quoted by the manufacturer who also gives a value for a ±45 laminate of 300 MPa, again an order of magnitude greater than the stress imposed on the strip in these experiments. It is to be noted that the axial stress and strain peak at different values of h because, of course, the axial modulus varies as h changes. In addition note that both stress and strain can become negative at large enough inter-ply angles, indicating that the strip is being placed in compression. This occurs at the angle at which the axial expansivity of the strip becomes equal to that of the frame, namely at an angle of 67.5 when the predicted axial expansivity is 23 · 106 K1 – the value for aluminium. At larger angles, the Poisson effect increases the transverse expansivity rather than reducing it. The use of the device to increase expansivity will be considered elsewhere.
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59 300
0.3
0.25
250 Stress
0.2
Strain % 200
0.1 100
Strain %
Stress (MPa)
0.15 150
0.05 50 0 0
-0.05
-50
-0.1 0
10
20
30
40
50
60
70
80
90
Interply angle (degrees)
Fig. 7. Predicted values of the axial stress and strain as a function of angle h for a specimen of APC-2, mounted in the device, if subject to a temperature change of 100 C.
All composite methods of altering the expansivity of a two-component system from that of either component rely upon the generation of internal stresses between the components. We must therefore demonstrate that, when using laminated composites in the device that we are concerned with, the component plies may withstand the stresses between them. In a fibre composite that undergoes a change of temperature, internal stresses arise at the boundary between fibre and matrix due to the differences in thermal expansion coefficient of the two components. We shall not consider these – important though they may be. Our experiments described above show that, for the case of APC-2, the laminated composite exhibits reversible behaviour over a temperature range of at least 80. In the case of [h/h]s laminated fibre composites containing glass with a PP matrix, experiments by Landert [22], working with one of us (AK), showed that the material crept when exposed to temperature changes of this amount. The calculated inter-laminar stresses are expected to lead to this effect. The same trend has been shown by Ito et al. [24] for PP. We have therefore evaluated the inter-laminar stresses in the systems considered here. In the detailed analysis of [h/h]s laminated fibre composites given in [3], expressions for the inter-laminar shear stresses are given. A relatively simple formula is found, viz. s¼
a4 ðhÞ DT; g44 ðhÞ
ð12Þ
where a4 ðhÞ ¼ ½aA aT sin 2h;
ð13Þ
and g44 ¼
1 1 2mA cos2 2h þ þ : sin2 2h þ E A ET E A lA
ð14Þ
We have evaluated Eq. (12) for the relevant systems studied. The value of the shear stress s peaks of course at h = 45 in all systems. For the use of the device, the relevant angle is that at which a minimum in the expansivity occurs, namely close to that at which for the same material removed from the device, large values of axial PoissonÕs ratio occur. PR is usually at a maximum for values of h in the range 20–30. In the experimental case investigated, namely APC-2, the effective transverse expansivity is less than 105 K1 for angles h between 15 and 60 – see Fig. 6. Over this range the predicted value of s varies from 4.82 MPa at 15 through 16.6 MPa at 45 to 11.6 MPa at 60. Such values are much less than the quoted shear strength of the epoxy resin so that internal stresses should not cause the failure of the strip. However, we recognise that the use of Eq. (12) is extremely approximate and completely neglects any edge effects that could be important. Since the work of Ito et al. [24], it is well known that with thermoplastic matrices of low melting point, such as polypropylene, the matrix material creeps under the conditions necessary to obtain negative CTE
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
over a temperature range of more than a few degrees. It was for this reason that we used Peek resin, one of the thermoplastic matrices of highest melting point available. Our experiments show no evidence of creep over the times of our experiments. However, we have not yet shown experimentally that even this material may not creep if used to obtain a negative CTE over an appreciable temperature range for a long period of time.
8. Matching the properties of the optical fibre We have demonstrated that it is possible to make a platform that shows, over an appreciable range of angles in its plane, a negative expansivity of the desired amount to provide temperature compensation for a Bragg fibre grating. The optical fibre containing the grating must be attached to the platform, with an adhesive or by other means, so that it lies in a direction parallel to the transverse direction on the platform. The predicted expansivity of the platform will then be in accordance with the theory just given (If laid at some other angle to the axial direction of a ±h laminate then further calculation of the expansivity in that direction will need to be carried out.). The diameter of the fibre is about 100 l m and the length of the grating about one centimetre. A fibre composite laminate of the required properties can provide a sufficiently wide and robust platform to which the fibre could be attached – say by anchoring it at two points. This is the usual method of attachment.
57
Were it necessary to embed the fibre within the platform an interesting additional point arises. It would be very nice, though it is by no means imperative, for YoungÕs modulus of both optical fibre and platform (in the required direction) to be identical. Where that is the case the strain in fibre and platform will be the same at all points along the fibre direction – provided there is no slippage in the coupling mechanism. If the platform is less stiff than the fibre, then the fibre must be attached to the platform over a rather longer length than that occupied by the grating, so that the required strain in the fibre may be built into it. This distance is not very long and its estimation is a classic and wellknown problem in composite mechanics. Using shearlag theory [25,26], we estimate that, even if the elastic modulus of the platform in the relevant direction is only one half of that of the fibre, the distance required to attain the required strain in the fibre is only 2–3 fibre diameters. For any of the material systems we have discussed above in the device, we have also examined whether one could, within the range of angles over which the desired expansivity of no more than 105 K1 is obtained, concurrently obtain a value of 70 GPa for the transverse YoungÕs modulus of the strip, which is the approximate value for the optical fibre. This does appear to be possible but only with a rather extreme set of conditions. Fig. 8 shows the transverse modulus of a strip composed of APC-2, which is typical of the variation of ET with angle for these laminated composites. Clearly, the value of ET is much less than 70 GPa for angles
160
Transverse Young's modulus (GPa)
140
120
100
80
60
40
20
0 0
10
20
30
40
50
60
70
80
Interply angle (degrees) Fig. 8. Calculated value of the transverse YoungÕs modulus for APC-2 material as function of the angle h.
90
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
below 70. Reference to Fig. 6 shows that the expansivity exceeds 105 K1 for angles greater than about 60 and comparing the two figures, one sees that, over the whole range of angles within which the required expansivity is obtained, ET is much less than 70 GPa. In order to obtain a value of 70 GPa in a fibrous laminate it will be necessary to use a fibre with an appreciable value of ET itself. This precludes carbon or Kevlar (see Table 1). The isotropic fibres are tungsten, SCS-6 and glass. We have found that the required combination of an expansivity of the strip of no more than 105 K1 together with a value of ET of 70 GPa could be obtained with a laminated composite of SCS-6 in epoxy where the required conditions occur at an angle h = 55. However, the fibre content must be very large to obtain this – nearer to 0.8 vol% than 0.75. Such a loading is possible to envisage, particularly with a large diameter fibre such as SCS-6, but is greater than that which can normally be obtained. Such a loading is possible, particularly using a large diameter fibre such as SiC SCS-6. Volume fractions of 0.74, and as high as 0.8, have been obtained using tungsten fibre [27,28]. We return to the case of the system Kevlar in epoxy, which corresponds to a volume fraction of fibre of 50%, where we remarked in Section 4 that the desired expansivity could be obtained without the use of the device. The value of YoungÕs modulus of Kevlar is only 124 GPa so, when used in the device, this system will not attain the required value of ET. However, if used by itself as a platform, the relevant expansivity is not the transverse one but could be that in the axial direction which is the stiff direction for values of h at which the axial expansivity is a minimum. Indeed, as Fig. 2 shows, this system attains the required negative values over a range of angles between 20 and 40. Unfortunately, the axial modulus at h = 30 is only 20 GPa. By greatly increasing the volume of fibre to more than 75% it is possible to predict a value of the axial modulus EA, in this case, of 74.8 GPa at h = 16 where aA is 105 K1. Kevlar is a small diameter fibre and so such fibre packing would be very difficult to obtain.
9. Conclusions We have demonstrated, by using theory and experiment, that laminated composites can be used to make a simple easily manufactured device with which it is possible to produce platforms with extremely negative values of transverse expansivity in a range of directions. These values are much more negative than may be obtained using monolithic materials. We have set ourselves, and achieved, the target of attaining a negative value of 105 K1, which is of interest to those wishing to obtain thermal compensation of Bragg fibre gratings inscribed in optical fibres.
Acknowledgement The contribution to this paper by one of us (L.N.M.) was supported by the Materials Measurement Programme of the Department of Trade and Industry, UK.
References [1] Kelly A, Clegg WJ, Stearn RJ, Vandeperre LJ. International patent application PCT/GB01/04955; 2000. [2] Clegg WJ, Kelly A. Choosing materials for composites with a greater range of properties. Adv Eng Mater 2002;4(6):388–92. [3] Kelly A., McCartney LN, Clyne TW. Development of a device with controllable thermal expansion characteristics incorporating a composite laminate. In: Sol, H., Degrieck, J, editors. Proceedings of the tenth European conference on composite materials – ECCM 10, Paper 154; 2002. [4] Gregoriev IS, Meilikhov EZ, editors. Handbook of physical properties. New York: CRC Press; 1997 [chapter 10, p. 287]. [5] Cambridge Materials Selector. V 2.0 Granta Design Ltd. Grantchester; 1994–1996. [6] Vandeperre LJ, Kelly A. [to be published]. [7] Tsai SW, Hahn HT. Introduction to composite materials. Stamford (CT): Technomic Publishing; 1980. [8] Clarke JF, Duckett RA, Hine PJ, Hutchinson IJ, Ward IM. Negative PoissonÕs ratios in angle-ply laminates; theory and experiment. Composites 1994;9:863–8. [9] Hine PJ, Duckett RA, Ward IM. Negative PoissonÕs ratios in angle-ply laminates. J Mater Sci Lett 1997;16:541–4. [10] Ito T, Suganuma T, Wakashima K. Glass fiber/polypropylene composite laminates with negative coefficients of thermal expansion. J Mater Sci Lett 1999;18:1363–5. [11] Software system LAMPROPS developed at the National Physical Laboratory by one of the authors (L.N.M.). [12] Pagano NJ. Thickness expansion coefficients of composite laminates. J Compos Mater 1974;8:310–2. [13] McCartney LN. Physically based damage models for laminated composites. Proc Inst Mech Eng Part L: J Mater: Des Appl 2003;217:163–99. [14] Rogers KF, Phillips LN, Kingston-Lee DM, Yates B, Overy MJ, Sargent JP, et al. The thermal expansion of carbon fibrereinforced plastics, Part 1: The influence of fibre type and orientation. J Mater Sci 1977;12:718–34. [15] Rogers KF, Phillips LN, Kingston-Lee DM, Yates B, Overy MJ, et al. The thermal expansion of carbon fibre-reinforced plastics, Part 2: The influence of fibre volume fraction. J Mater Sci 1978;13:433–40. [16] Rogers KF, Phillips LN, Kingston-Lee DM, Yates B, Overy MJ, et al. The thermal expansion of carbon fibre-reinforced plastics, Part 3, The influence of resin type. J Mater Sci 1978;13:2217–25. [17] Rogers KF, Phillips LN, Kingston-Lee DM, Yates B, Overy MJ, et al. The thermal expansion of carbon fibre-reinforced plastics, Part 4, Ply multidirectional effects. J Mater Sci 1978;13:2226–32. [18] Strife JR, Prewo KM. Unidirectional and bi-directional Kevlar/ epoxy composites. J Compos Mater 1979;13:264–77. [19] Rojstaczer S, Cohn D, Marom G. Thermal expansion of Kevlar fibres and composites. J Mater Sci Lett 1985;4:1233–6. [20] Barnes JA, Simms IJ, Farrow GJ, Jackson D, Wostenholm G, Yates B. Thermal expansion characteristics of PEEK composites. J Mater Sci 1991;26:2259–71. [21] Romeo G, Frulla G. Analytical and experimental results of the coefficients of thermal expansion of high modulus graphite epoxy materials. J Compos Mater 1995;29:751–65.
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59 [22] Landert M. Controlled thermal expansion of fibre reinforced polymer laminates. Diploma Thesis, ETH, Zurich; 2002. [23] Kelly A, Stearn RJ, McCartney LN. Composite materials of controlled thermal expansion. In: Presented at sixth international seminar on experimental techniques in composite materials, Vicenza, Italy [in press]. [24] Ito T, Suganuma T, Wakashima K. A micro-mechanics based analysis for tailoring glass-fiber-reinforced thermoplastic laminates with near-zero coefficients of thermal expansion. Compos Sci Technol 2000;60:1851–61. [25] Nayfeh AH. Thermomechanically induced interfacial stresses in fibrous composites. Fibre Sci Tech 1977;10:195–209.
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[26] McCartney LN. Analytical models of stress transfer in unidirectional composites and cross-ply laminates, and their application to the prediction of matrix/transverse cracking. In: Reddy JN, Reifsnider KL, editors. Proceedings of the IUTAM symposium on local mechanics concepts for composite material systems, Blacksburg, VA. Berlin, New York, London: Springer; 1991. p. 251–82. [27] Tyson WR, Fibre reinforcement in metals, PhD Thesis, Cambridge; 1964. [28] Dragoi D, Ustandag E, Clausen B, Bourke MAM. Investigation of thermal residual stresses in tungsten-fiber/bulk metallic glass matrix composites. Scripta Mater 2001;45:245–52.
Controlling thermal expansion to obtain negative expansivity using laminated composites A. Kelly a
a,* ,
L.N. McCartney b, W.J. Clegg a, R.J. Stearn
a
Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, CB2 3QZ, UK b NPL Materials Centre, National Physical Laboratory, Teddington, Middlesex, TW11 0LW, UK Received 31 December 2003; received in revised form 27 May 2004; accepted 7 June 2004 Available online 19 August 2004
Abstract It is confirmed, by considering a wide range of laminated composites containing a variety of fibres and matrices, that negative thermal expansion coefficients may be obtained. These are usually accompanied by a correspondingly large value of the in-plane axial PoissonÕs ratio (PR). By making use of this large PR extremely negative values of expansivity may be obtained – much further negative than for any monolithic materials. The use of laminated composites also overcomes some of the previously reported limitations of a device to control thermal expansion when made with monolithic materials. The use of the device, as a platform to control the expansivity of an optical fibre containing a Bragg grating, is discussed in detail and it is shown that the required negative expansivity of 105 K1 may easily be obtained with a number of composite systems. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Polymer–matrix composites; B. Thermomechanical properties; C. Laminates; C. Optical gratings; C. Thermal compensation
1. Introduction Bragg fibre gratings are used as filters in optical communication networks. The reflected wavelength from the grating should remain well within the wavelength range for a single channel so as to avoid cross talk between channels. At present it is required that the filtering capacity of the grating should maintain the value of the wavelength to which the grating is tuned to within 0.25 pmK1 over a temperature range from 20 to 80 C. Both the refractive index and the grating spacing change with temperature, the former being more sensitive to temperature than the latter. Control may be exercised by straining the grating – in which case the value of the photo-elastic constant comes into play. Speaking approximately, it is neces-
*
Corresponding author. Tel.: +44-1223-363691; fax: +44-122333456. E-mail address: [email protected] (A. Kelly). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.06.003
sary to be able to impose upon the grating an effective thermal expansion coefficient of 105 K1. A recent patent [1] describes a number of ways of achieving this control. By providing a platform to which the fibre may be attached, it is possible to impose upon the grating a uniform negative expansivity of the required amount. One approach, and the one with which this paper is concerned, is to make use of the Poisson contraction in the lateral direction which accompanies an axial extension (in materials with a PoissonÕs ratio having a positive sign).
2. Description of the device The device, which is illustrated in Fig. 1, involves a strip of a material, denoted by S, of relatively low thermal expansion coefficient (a) having an adequately large PoissonÕs ratio m, that is to be coupled with a frame of material, denoted by F, of relatively high thermal expansion coefficient that has adequate stiffness and strength.
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
Fig. 1. The frame and strip device designed to control thermal expansion by use of PoissonÕs ratio.
Consider the strip of a material having a low isotropic thermal expansion coefficient, aS, which is fixed at its ends to the frame made of a material with a high isotropic thermal expansion coefficient, aF, as shown in Fig. 1. As the temperature increases the material F of higher thermal expansivity will increase in length more than the strip S, which has a lower expansivity. Because the frame is joined to the strip, the latter will be placed in tension by an amount depending on the difference in the expansion coefficients of the two members, provided that the geometry of the frame is such that bending cannot occur. This elastic tension will cause the strip, S, to contract laterally due to the Poisson effect (assuming, for the present, that m has its normal positive sign). However, because the temperature has been raised the strip will expand laterally. The effective lateral thermal expansion coefficient of the strip, aeff, is given by aeff ¼ mS ðaC aS Þ þ aS ;
ð1Þ
where mS is PoissonÕs ratio of the strip and aC is a ÔcompositeÕ expansion coefficient that is given by aC ¼
a F AF E F þ a S AS E S ; AF E F þ AS E S
ð2Þ
where A represents area fraction and E is YoungÕs modulus, and where the subscript S indicates the strip and F the frame. The effective axial modulus of the composite, comprising frame and strip, in the axial direction (see Fig. 1) is E C ¼ AF E F þ AS E S :
ð3Þ
Assuming for the moment that the area fraction of the frame is very much greater than that of the strip, we may set aC equal to the expansivity of the frame, aF, in which case the net lateral coefficient of thermal expansion of the strip is given by
aeff ¼ aS þ mS ðaS aF Þ:
ð4Þ
The strip will contract in the lateral direction when the temperature is raised if aF P
ð1 þ mS Þ aS : mS
ð5Þ
It has been demonstrated [2,3] that the device functions as predicted, and can produce effective negative thermal expansion coefficients.
3. Limitation of materials available Clegg and Kelly [2] first address the question of whether or not the device, if constructed of normally available materials, can produce values of the expansivity that are more negative than 105 K1. Firstly, it should be noted that a negative expansivity is favoured by choosing a material for the strip having an expansivity that is as low as possible. From Eq. (4), if aS is zero, then in order to attain a value of aeff of 105 K1 we require that aF has the value 105 K1/mS. For common values of mS of about 1/3 or less, which is an appropriate value for materials of low expansivity, the value of the expansivity of the frame must be as much as 30 · 106 K1. Very few materials, other than plastics, show such large values of thermal expansivity. Some alloys of zinc are a possible exception. Such alloys are not stiff, with Young moduli of around 60–70 GPa. Since the area fraction of the frame may be made very large this would not preclude the use of rather compliant materials for the construction of the frame. However, since the actual size of the device in practice, is likely to be limited, we must look for other materials. It is interesting to note that manganese is a noteworthy anomaly in the pattern that materials of high melting
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
point are also those of greater stiffness and low expansivity. Both a and b Manganese have reported values [4] of expansivity of 22.8–25.6 · 106 K1 – as large as that of aluminium. However, YoungÕs modulus of manganese, 191 GPa, and its melting point 1244 C should be compared with the corresponding values for aluminium of 70 GPa and 660 C, respectively. There are conditions also for the material of the strip, which must be subject to a longitudinal strain that is sufficient to produce the desired value of the Poisson contraction in the lateral direction. The transverse strain in the strip arising from thermally induced mechanical loading in the longitudinal direction, eT, is given, on using Eq. (1), by eT ¼ mðaC aS ÞDT;
ð6Þ
where, of course, the quantity in brackets times the temperature range DT over which the device is designed to operate, represents the longitudinal strain in the strip. We need to assess whether the yield strain of the strip, if composed of a ductile material, or the breaking strain, if the strip is made of brittle material, will be exceeded if the device is used over a temperature range of, say, 100 C. Yield and breaking strains are not usually tabulated, but yield and breaking stresses are. The longitudinal stress in the strip is rS ¼ ES eT =mS ¼ ES ðaC aS ÞDT;
ð7Þ
where ES is YoungÕs modulus of the strip. On using (1) the elastic stress in the strip may be written rS ¼ ðES =mS Þ½aeff aS DT:
ð8Þ
Taking DT = 100 C, and noting that we wish to have a value for aeff of 105 K1 the stress in the strip is 103 (ES/mS), if we assume that aS is zero. For normal values of mS, of about 1/3 or less, these are very large stresses that will normally exceed the yield or breaking stress of most materials. Clegg and Kelly [2] consider this requirement in detail and using the Cambridge Materials Selector [5] as a database, concluding that only a very limited range of materials – some high strength steels and titanium alloys would satisfy the requirements. They also point out that quite modest increases in the assumed value of PoissonÕs ratio would admit the use of many more materials, and cite some examples, one of which is the use of composite materials. These can be designed both to yield large values of m and small values of a. Before describing the use of laminated fibre composites we should remark that very many materials in the form of single crystals can show values of PoissonÕs ratio much larger than one half – which is the upper limit for an isotropic material. We have explored the values possible for a number of crystal systems [6] and the results will be reported elsewhere.
49
4. Laminated fibre composites One of the difficulties of exploring the possibilities presented by composite materials is the enormous range of possible systems that arise because choice of fibre, matrix, volume fraction, and arrangement of the fibres are all parameters that must be considered. Fortunately, it is already well known that laminated fibre composite materials, particularly balanced laminates, can produce values of PoissonÕs ratio that are extremely large (e.g., [7–9]), and can also have negative values. This is relevant if the aim were to increase the in-plane expansivity. Such laminates may also show negative values of the in-plane expansivity [10] and hence appear to be excellent materials to be considered for the strip in the device. For a laminated composite the coefficient of thermal expansion will not be isotropic. To emphasize this we rewrite Eq. (4) as aeff ¼ aS;trans mS;axial ðaF aS;axial Þ; where the subscripts ÔaxialÕ and ÔtransÕ refer to the axial and transverse directions of the strip (see Fig. 1). For a given value of aF we must then minimise the expansivities of the strip in both the axial and transverse directions and maximise the axial PoissonÕs ratio of the strip. We shall not repeat here the detailed theory for a laminated [+h/h]ns composite, which may be found in [3]. The equations are a special case that can be analysed using the NPL computer program LAMPROPS designed for general symmetric laminates [11]. The predictions are robust and consistent with other analytical expressions – this is particularly important for the prediction of the CTE in the through-the-thickness direction [12]. We have compared the predictions accurately with the extensive results on carbon fibres in epoxy resins and with those for Kevlar – see in Section 6. There are two stages in our validation of our computer programs. We calculate the properties of an individual ply from the measured properties of the fibre and matrix and the known volume fraction. The equations and approximations for doing this are listed in [13]. Examples of the ÔaccuracyÕ of such calculations are also given in [13]. We have obtained a similar agreement. An example of the demonstration of the validity of the programs for estimating thermal expansion coefficients is shown in Fig. 2. We have made similar tests using two other systems where both fibre and matrix properties have been measured or estimated and where measurements have been made of the in-plane expansivity of the composite. These systems were HMS (high modulus) carbon fibre in epoxy and HTS (high tensile strength) carbon fibre in epoxy using the extensive experimental results [14–17] for CFRP, and the results for Kevlar in epoxy [18,19]. There are a few published measurements [20,21] of the through-thickness expansivity of a composite.
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59 100
1.6
6
Thermal expansion coefficient x 10 K
-1
1.4
80
60 1
40
0.8
Axial coefficent Through-thickness coefficient Transverse coefficient Experiment Axial Poisson's ratio
20
0.6
Axial Poisson's ratio
1.2
0.4
0 0.2
-20
0
0
10
20
30
40
50
60
70
80
90
Interply angle (degrees) Fig. 2. Predicted values of PoissonÕs ratio and thermal expansion coefficients for a composite of Kevlar 49 fibres in PR 286 (3M) resin. The fibre volume fraction is 0.5. Experimental results from [17].
Fig. 2 demonstrates the excellent agreement between prediction of the axial expansion coefficient and experiment for the case of Kevlar. The measurements of thermal expansion were carried out over a temperature range of 20–150 C for the composites and were closely linear implying a constant value of expansivity over this range. The values assumed in predictions for the resin and fibre are for a temperature of 30 C. It is to be noted from Fig. 2 that the in-plane expansivity curves reflect about h = 45 so that the value of the axial expansivity at an angle h is equal to the transverse expansivity at an angle of (90h). We are not publishing here validation of the computer programs for the prediction of values of PoissonÕs ratio. For HMS carbon in epoxy and for HTS carbon in epoxy, our programs reproduce accurately the values obtained and compared with experiment [8,9]. Similar quantitative agreement, as that shown in Fig. 2 for the values of the expansivity for Kevlar composites, was found for HMS carbon and for HTS, except that in the latter case there was rather poor agreement for the predicted value of the expansivity in the h = 90 direction, i.e. in the transverse direction for a single ply. In subsequent experiments, referred to below, we have found this discrepancy to arise often in other systems. The transverse expansivity of a single ply is dominated by the value of the expansivity of the matrix and when the matrix is a polymer this may be a varying quantity depending on the mode of fabrication of the composite and the value depends significantly on tem-
perature at temperatures close to room temperature. The difficulties of making measurements of the thermal expansion characteristics of fibre laminates have been reported in the literature [21]. Having obtained an accurate predictive capability we must first entertain the idea that the required expansivity and strength for the control of a Bragg fibre grating might be obtained simply from a composite material without the need to use our particular device – see, for example Fig. 2. We have explored a number of composite systems which seem likely to provide large values of PoissonÕs ratio (PR) combined with small values of CTE, and have found the appropriate angle of a balanced laminate so as to maximise/minimise the relevant effect for given values of the thermoelastic constants. The essential reason for a peak in the value of PR shown in Fig. 2, observed also in other systems at an angle of about 30, and the corresponding small and possibly negative value of the axial expansivity (often but not always at an angle slightly less than that at which the PR peaks) is due to the shear deformation in the two orientations of the laminate producing a type of scissoring action. The requirement for the effect to be large is that the axial shear modulus of the component plies be small, with respect to the other elastic constants, particularly the axial stiffness. The value of the axial stiffness of an angle-ply laminate is usually governed by the value of the shear modulus of the matrix. It is not important that the expansivity of the matrix be small – in fact the larger the better, provided the axial expan-
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
sivity of the fibre is a good deal smaller than that of the matrix. This is essentially because the matrix must be squeezed sideways by the fibre as the temperature is raised, so that the scissoring action occurs in the transverse direction and hence reduces the axial expansivity of the composite. In order to illustrate the effect we have run the computer programs to predict the values of PR and expansivity for a number of systems, characterised by being composed of a stiff fibre with a small (lateral) expansivity in a matrix of a much more compliant material. We have studied fibres that are commercially available. The systems examined and the measured properties of fibre and matrix used as inputs are given in Table 1. These data have been used to calculate the properties of single plies of the binary composite for a particular volume fraction, and following this the properties of a [+h/ h]ns laminate. We have also used the experimentally measured properties of single plies of certain combinations of fibre and matrix where these are given in the literature for well-characterised materials.
51
The predictions of thermoelastic constants for a number of systems are shown in Table 2, whence it is apparent that a negative in-plane expansion coefficient is predicted in a number of systems, together with a large value of PoissonÕs ratio. The condition necessary to obtain this negative expansivity is, as emphasised by Ito et al. [10], is to have as large as possible a value of the ratio of the axial stiffness EA to the axial shear modulus lA. The value of lA is, for most volume fractions of interest, dominated by the value of the shear modulus of the matrix, and so this should be chosen to be as low as possible. It is to be noted from Table 2 that the minimum value of the expansivity occurs in all the systems at approximately the same angle h of about 30. However, from the data presented in Table 1, one sees that in order to obtain a negative value in the laminate a small thermal expansion coefficient of the fibre is also a great advantage – compare the values of aA for the isotropic fibres in Table 1. Tungsten and glass have about the same expansivity, but the former is very considerably stiffer. Textron SCS-6 and tungsten do not differ greatly in stiffness but the former attains a lower value of
Table 1 Properties of fibres and matrices used in calculations Material
EA (GPa)
ET (GPa)
mA
mT
lA (GPa)
aA (· 106 K1)
aT (· 106 K1)
Fibre AS 4 Silenka glass Modmor carbon V-tex glass (E-glass) Kevlar 49 SCS6 W-wire SiC
225 74 405 73 124 395 411 483
15 74 8.6 73 6.0 259 411 483
0.2 0.2 0.35 0.2 0.35 0.25 0.28 0.19
0.07 0.2 0.53 0.2 0,35 0.34 0.28 0.19
15 30.8 13.7 30.4 2.2 96.6 160.5 170
0.5 4.9 0.1 5.1 5.7 2.3 4.5 3.3
15 4.9 22.1 5.1 60 23 4.5 3.3
Matrix Epoxy MY Epoxy Ciba LY558 Epoxy PR-286 Polypropylene shell Silicone rubber Rubber Ti–6Al–4V Al
3.35 5.28 3.5 1.3 0.03 0.03 125 70
3.35 5.8 3.5 1.3 0.03 0.03 125 70
0.35 0.35 0.35 0.3 0.495 0.495 0.31 0.34
0.35 0,35 0.35 0.3 0.495 0.495 0.31 0.34
1.24 1.95 1.29 0.5 0.01 0.01 38.5 26.1
58 55 65 90 60 200 9.0 23
58 55 65 90 60 200 9.0 23
Table 2 System comparison System (all at Vf = 0.5)
W wire in epoxy
W wire in PP
SCS 6 in poxy
SCS 6 in PP
E-glass in epoxy
Silenka glass in PP
Minimum value of axial expansivity · 106 (K1) h value for minimum () Maximum value of PoissonÕs ratio EA/ET EA/lT (aT aA) · 106 K1
1.51
5.54
3.91
7.97
5.69
0.299
33 1.93 at 24 20.0 56.8 30.9
33 3.32 at 20 54.6 138 46
33 1.90 at 24.5 19.5 55.3 32.1
33 3.20 at 20 53.5 134 47.1
27 0.73 at 32.5 4.09 11.4 28.1
31 1.26 at 28 10.3 28.0 44.4
52
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
expansivity in the laminate due to the lower expansivity of the fibre. As well as a low value of the shear modulus of the matrix, a correspondingly large value of the expansion coefficient of the matrix will promote a negative expansivity of the laminate. This is reflected in the comparison between the values in Tables 1 and 2 for glass fibres in epoxy and in polypropylene (PP). The physical reason for this follows from the scissoring tendency of the laminate as already pointed out to explain the high values of axial PoissonÕs ratio. In the case of thermal expansion, the individual plies are trying to expand sideways (transverse to the fibres) because of the larger thermal expansion coefficient of the matrix, and this Ôdraws inÕ the material in the axial direction and produces a negative, or small axial expansivity. The final column in Table 2 contains data for the system Silenka glass in PP. A small negative value of expansivity is predicted at angle h of 31. This material system has been investigated experimentally in [10,22]. Negative values have been found which are rather more negative than those predicted in Table 2. In the particular case of this thermosetting resin, the predicted values of the laminate depend sensitively on the assumed properties of the matrix. None of the systems studied in detail show a negative value of the expansivity of the laminate of the desired amount except the system Kevlar in epoxy – see Fig. 2. Here a minimum value of 12.7 · 106 K1 is predicted. This value is in fact slightly less negative than the experimental value of 13.7 · 106 K1 found by Strife and Prewo [18] in an experimental laminate with h equal to approximately 30. It follows that such a system could provide a platform on which a Bragg fibre grating might be mounted so as to control the expansivity of the grating by the required amount. A value of the expansivity of less than 105 K1 is shown to occur over the range of angles from 23.5 to 37.5 – a useful range of angles. The low value is due to the negative expansivity of the fibre 5.7 · 106 K1 and its relatively high stiffness, viz. 124 GPa. We discuss this system a little further in Section 8.
5. The use of laminated composites in the device We have written and validated a computer program that carries out the following calculation. A [h/h]s laminate forms a strip that is attached to a square frame made of an isotropic material having a YoungÕs modulus EF and a linear thermal expansion coefficient aF. The axial Young modulus of the anisotropic strip is deðSÞ noted by EA ðhÞ. The axial thermal expansion of the ðSÞ strip is denoted by aA ðhÞ while the transverse in-plane ðSÞ thermal expansion coefficient is denoted by aT ðhÞ. The (S) axial PoissonÕs ratio is denoted by mA (h).
The effective axial YoungÕs modulus of the strip and frame, regarded as a parallel bar model, is given by the mixtures rule ðSÞ
ECA ðhÞ ¼ AF EF þ AS EA ðhÞ;
ð9Þ
where AF is the area fraction of the frame, and AS = 1AF is the area fraction of the strip. The corresponding axial thermal expansion coefficient is given by the mixtures rule ðSÞ
aCA ðhÞ ¼
ðSÞ
AF EF aF þ AS EA ðhÞaA ðhÞ : ECA ðhÞ
ð10Þ
The effective transverse expansion coefficient of the strip is denoted by aeff T ðhÞ, and is defined by ðSÞ
ðSÞ
ðSÞ
ðSÞ
C aeff T ðhÞ ¼ aT ðhÞ þ mA ðhÞaA ðhÞ mA ðhÞaA ðhÞ:
ð11Þ
A computer program LAMPROPS (version 5) [11] calculates values of aeff T ðhÞ which are tabulated in an output file as a function of h. The values of AF, EF, and aF are input via the control ðSÞ ðSÞ ðSÞ file for the software. The values of EA ðhÞ, aA ðhÞ, aT ðhÞ ðSÞ and mA ðhÞ are calculated using the existing set of routines mentioned above – Section 4. An example of the use of the program is to predict, as shown in Fig. 3, the value of the effective transverse expansion coefficient of the strip for a laminate containing high modulus carbon fibres in epoxy when attached to a steel frame of Young modulus 200 GPa, thermal expansion coefficient of 14 · 106 K1, and with the frame occupying 95% of the cross-sectional area. In order to obtain the largest possible negative value of expansivity when using the device, the value of PoissonÕs ratio appears to be more important than the value of the minimum transverse expansivity of the composite system found when standing alone. For instance the minimum predicted value of transverse expansivity for APC-2–3.08 · 106 K1 when the angle h is 61.5. For HMS carbon the corresponding figures are 3.9 · 106 K1 and 61 which are not greatly different. However, the maximum value of PR for the latter is 1.965 at 29 whereas for the former the maximum PR is 1.41 at 26. When placed in the device under identical conditions, employing a steel frame, the minimum value of the expansivity for the HMS carbon is 15.9 · 106 K 1 at 29, whereas for APC-2 the minimum expansivity is only 7.0 · 106 K1 at 36.
6. Experimental verification We have verified the efficacy of the device by using an aluminium frame with the dimensions shown in Fig. 4. The dimensions of the strip are also shown in this figure. The long axis of the aluminium frame contains a circular tunnel in each arm that accommodates a resistance-
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
53
30
6
Transverse thermal expansivity x 10 K
-1
25 20 15 10 5 0 -5 -10 -15 -20 0
10
20
30
40
50
60
70
80
90
Interply angle (degrees) Fig. 3. Predicted values of the effective transverse expansivity of high modulus carbon fibres in epoxy using a steel frame.
Fig. 4. A photograph of the device as set up to measure transverse expansivity. The dimensions of the strip are given.
heating element. The cross-section of each of the longitudinal arms of the frame was 15.8 · 10.5 mm. The diameter of the tunnel in each arm is 6.0 mm. The area fractions of strip and frame for use in Eqs. (9) and (10) are 0.053 and 0.947, respectively, which we have taken as representing a value of AS of 0.05. We have measured the thermal expansion coefficient of the aluminium frame as 23.34 · 106 K1 and used as a strip a balanced symmetric laminate made of APC-2 material from Fiberite containing a nominal 61% of Hercules
Magnamite AS4 carbon fibre in Peek resin. We have taken YoungÕs modulus of the aluminium frame as 70 GPa. The specimens consisted of 8 layers of pre-preg forming a balanced symmetric laminate with fibres running at ±0, ±10, ±17.5, ±25, ±35 and ±45. The laminates were made by pressing for half an hour at a pressure of 2 MPa, and at a temperature of 400 C. The relative area fractions when mounted in the device are 0.95 for the aluminium frame and 0.05 for the strip of
54
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
laminated composite. The transverse expansivity of the strip has been measured with a laser micrometer and the axial PoissonÕs ratio found by using resistance strain gauges attached to the specimen. A photograph of the apparatus with a thermocouple attached is shown in Fig. 4. The dimensions of the strip are also shown – the in-plane dimensions being 52.0 mm · 15.4 mm and the thickness 1.01 mm. The distance between the centres of the two holes is 40 mm. The transverse thermal expansion coefficient of the strip was measured before it was attached to the device. Fig. 5 shows an example for the strip containing fibres at an angle h = 45. The measured transverse expansivity of the material of the strip in the free state over the temperature range shown is 1.3 · 106 K1 – the corresponding value calculated using Eq. (11) is found to be 2.7 · 106 K1. For the heating times that we have employed, it is found that the resin strip containing carbon fibre heats up much more slowly than the aluminium frame so a correction has to be applied to take the difference of temperature into account. When mounted in the device the measured value of transverse expansivity at h = 45 is found to be 19.3 · 10 6 K1 and the calculated value from Eq. (11) above is 20.9 · 106 K1. A comparison of the values predicted from Eq. (11) for the transverse expansivity and the measured values for the whole range of angles is given in Fig. 6. We have found that in order to obtain agreement at h = 0 we need to use in Eq. (11), and hence in the com-
puter program, our measured value of the transverse expansivity of the strip, viz. three measurements of 41.7, 42.7 and 43.8 · 106 K1 having a mean value of 42.7 · 106 K1. This measured value approximates more closely to that found by us when using our ply program to calculate the value of aT from the properties of fibre and matrix, viz. AS 4 fibre in Peek resin, than it does to the value given by the manufacturerÕs value for APC-2. For this reason the data in Fig. 6 are designated AS4 Peek rather than APC-2. The qualitative agreement between prediction and experiment for the form of the curve is good. Quantitative uncertainties arise because of a number of factors which are discussed elsewhere [23], but it may be seen that the efficacy of the device is demonstrated, and a negative expansivity of as much as 30.5 · 106 K1 at h = 35 is experimentally obtained. The measured value of PR for APC-2 at 35 was 2.07 (the mean of three measurements), which accounts for the very negative value of aTeff measured at these angles. In the case of APC-2 the predicted range of angles h, over which a value of the transverse expansivity of less than 105 K1 may be obtained with the device, is from 17 to 59. Consequently, in principle the objective of obtaining a platform having an effective expansivity of 105 K1 or less is demonstrated. Of course it is not necessary to consider only the use of a material such as APC-2 in the device. We may show, having demonstrated the validity of the approach, that the systems considered in Section 4 (see Table 2) are also predicted to show negative values of expansivity
15.56
y = 2E-05x + 15.554
Length (mm)
15.55
Clamped Free
15.54
y = -0.0003x + 15.552
15.53
15.52 20
30
40
50
60
70
o
Temperature ( C ) Fig. 5. Measurements of the width of the strip as a function of temperature.
80
90
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
55
40
30
Eqn. (11) Measured
6
-1
Expansivity x 10 (K )
20
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
80
90
Interply angle (degrees) Fig. 6. Comparison of predicted and measured values of the transverse expansivity of a strip of APC-2 (AS-4 in Peek) in the device as a function of the angle h of the specimen.
Table 3 Range of angles h over which aeff <105 K1 for various material systems Material
Min (·106 K1)
Angle ()
Rangea
SCS 6 in PP Ww in epoxy Ww in PP SCS 6 in epoxy E-glass in epoxy
69 31 62 36 2.57
23 29 23 29 46
12–56 18–52 12–53 16–55 Zero
a
Remark
PoissonÕs ratio too small
aeff < 105 K1.
over an appreciable range of angles h. Predictions are collected in Table 3. A frame of aluminium is assumed and the volume fraction of the strip is taken as 5%. The case of E-glass fibre in epoxy demonstrates most clearly the necessity of having a system with a large value of PoissonÕs ratio.
7. Internal strains We have demonstrated that, by designing laminated materials so that they show large values of PoissonÕs ratio, the difficulty outlined in Section 3 of finding a suitable material for the frame may be removed. The second difficulty remarked upon there is that the material of which the strip is made may be insufficiently strong to withstand the stresses placed upon it if the
device is to control expansivity over a sufficiently large range of temperature. In that case we assumed a necessary value of 100 C for the temperature range. In the case of a composite strip there are two types of stresses to consider. The first is, in complete analogy with a monolithic material, whether the strip can withstand the force exerted by the frame upon it. This, of course is directly proportional to temperature difference over which the device is to control the expansivity. Fig. 7 shows the values of the axial stress and strain in the strip made of APC-2 material for a temperature change of 100 C. The stress at h = 0 is 274 MPa and at h = 45 is 34 MPa. The first figure is well within the breaking strength of 2130 MPa for a 0 strip quoted by the manufacturer who also gives a value for a ±45 laminate of 300 MPa, again an order of magnitude greater than the stress imposed on the strip in these experiments. It is to be noted that the axial stress and strain peak at different values of h because, of course, the axial modulus varies as h changes. In addition note that both stress and strain can become negative at large enough inter-ply angles, indicating that the strip is being placed in compression. This occurs at the angle at which the axial expansivity of the strip becomes equal to that of the frame, namely at an angle of 67.5 when the predicted axial expansivity is 23 · 106 K1 – the value for aluminium. At larger angles, the Poisson effect increases the transverse expansivity rather than reducing it. The use of the device to increase expansivity will be considered elsewhere.
56
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59 300
0.3
0.25
250 Stress
0.2
Strain % 200
0.1 100
Strain %
Stress (MPa)
0.15 150
0.05 50 0 0
-0.05
-50
-0.1 0
10
20
30
40
50
60
70
80
90
Interply angle (degrees)
Fig. 7. Predicted values of the axial stress and strain as a function of angle h for a specimen of APC-2, mounted in the device, if subject to a temperature change of 100 C.
All composite methods of altering the expansivity of a two-component system from that of either component rely upon the generation of internal stresses between the components. We must therefore demonstrate that, when using laminated composites in the device that we are concerned with, the component plies may withstand the stresses between them. In a fibre composite that undergoes a change of temperature, internal stresses arise at the boundary between fibre and matrix due to the differences in thermal expansion coefficient of the two components. We shall not consider these – important though they may be. Our experiments described above show that, for the case of APC-2, the laminated composite exhibits reversible behaviour over a temperature range of at least 80. In the case of [h/h]s laminated fibre composites containing glass with a PP matrix, experiments by Landert [22], working with one of us (AK), showed that the material crept when exposed to temperature changes of this amount. The calculated inter-laminar stresses are expected to lead to this effect. The same trend has been shown by Ito et al. [24] for PP. We have therefore evaluated the inter-laminar stresses in the systems considered here. In the detailed analysis of [h/h]s laminated fibre composites given in [3], expressions for the inter-laminar shear stresses are given. A relatively simple formula is found, viz. s¼
a4 ðhÞ DT; g44 ðhÞ
ð12Þ
where a4 ðhÞ ¼ ½aA aT sin 2h;
ð13Þ
and g44 ¼
1 1 2mA cos2 2h þ þ : sin2 2h þ E A ET E A lA
ð14Þ
We have evaluated Eq. (12) for the relevant systems studied. The value of the shear stress s peaks of course at h = 45 in all systems. For the use of the device, the relevant angle is that at which a minimum in the expansivity occurs, namely close to that at which for the same material removed from the device, large values of axial PoissonÕs ratio occur. PR is usually at a maximum for values of h in the range 20–30. In the experimental case investigated, namely APC-2, the effective transverse expansivity is less than 105 K1 for angles h between 15 and 60 – see Fig. 6. Over this range the predicted value of s varies from 4.82 MPa at 15 through 16.6 MPa at 45 to 11.6 MPa at 60. Such values are much less than the quoted shear strength of the epoxy resin so that internal stresses should not cause the failure of the strip. However, we recognise that the use of Eq. (12) is extremely approximate and completely neglects any edge effects that could be important. Since the work of Ito et al. [24], it is well known that with thermoplastic matrices of low melting point, such as polypropylene, the matrix material creeps under the conditions necessary to obtain negative CTE
A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
over a temperature range of more than a few degrees. It was for this reason that we used Peek resin, one of the thermoplastic matrices of highest melting point available. Our experiments show no evidence of creep over the times of our experiments. However, we have not yet shown experimentally that even this material may not creep if used to obtain a negative CTE over an appreciable temperature range for a long period of time.
8. Matching the properties of the optical fibre We have demonstrated that it is possible to make a platform that shows, over an appreciable range of angles in its plane, a negative expansivity of the desired amount to provide temperature compensation for a Bragg fibre grating. The optical fibre containing the grating must be attached to the platform, with an adhesive or by other means, so that it lies in a direction parallel to the transverse direction on the platform. The predicted expansivity of the platform will then be in accordance with the theory just given (If laid at some other angle to the axial direction of a ±h laminate then further calculation of the expansivity in that direction will need to be carried out.). The diameter of the fibre is about 100 l m and the length of the grating about one centimetre. A fibre composite laminate of the required properties can provide a sufficiently wide and robust platform to which the fibre could be attached – say by anchoring it at two points. This is the usual method of attachment.
57
Were it necessary to embed the fibre within the platform an interesting additional point arises. It would be very nice, though it is by no means imperative, for YoungÕs modulus of both optical fibre and platform (in the required direction) to be identical. Where that is the case the strain in fibre and platform will be the same at all points along the fibre direction – provided there is no slippage in the coupling mechanism. If the platform is less stiff than the fibre, then the fibre must be attached to the platform over a rather longer length than that occupied by the grating, so that the required strain in the fibre may be built into it. This distance is not very long and its estimation is a classic and wellknown problem in composite mechanics. Using shearlag theory [25,26], we estimate that, even if the elastic modulus of the platform in the relevant direction is only one half of that of the fibre, the distance required to attain the required strain in the fibre is only 2–3 fibre diameters. For any of the material systems we have discussed above in the device, we have also examined whether one could, within the range of angles over which the desired expansivity of no more than 105 K1 is obtained, concurrently obtain a value of 70 GPa for the transverse YoungÕs modulus of the strip, which is the approximate value for the optical fibre. This does appear to be possible but only with a rather extreme set of conditions. Fig. 8 shows the transverse modulus of a strip composed of APC-2, which is typical of the variation of ET with angle for these laminated composites. Clearly, the value of ET is much less than 70 GPa for angles
160
Transverse Young's modulus (GPa)
140
120
100
80
60
40
20
0 0
10
20
30
40
50
60
70
80
Interply angle (degrees) Fig. 8. Calculated value of the transverse YoungÕs modulus for APC-2 material as function of the angle h.
90
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A. Kelly et al. / Composites Science and Technology 65 (2005) 47–59
below 70. Reference to Fig. 6 shows that the expansivity exceeds 105 K1 for angles greater than about 60 and comparing the two figures, one sees that, over the whole range of angles within which the required expansivity is obtained, ET is much less than 70 GPa. In order to obtain a value of 70 GPa in a fibrous laminate it will be necessary to use a fibre with an appreciable value of ET itself. This precludes carbon or Kevlar (see Table 1). The isotropic fibres are tungsten, SCS-6 and glass. We have found that the required combination of an expansivity of the strip of no more than 105 K1 together with a value of ET of 70 GPa could be obtained with a laminated composite of SCS-6 in epoxy where the required conditions occur at an angle h = 55. However, the fibre content must be very large to obtain this – nearer to 0.8 vol% than 0.75. Such a loading is possible to envisage, particularly with a large diameter fibre such as SCS-6, but is greater than that which can normally be obtained. Such a loading is possible, particularly using a large diameter fibre such as SiC SCS-6. Volume fractions of 0.74, and as high as 0.8, have been obtained using tungsten fibre [27,28]. We return to the case of the system Kevlar in epoxy, which corresponds to a volume fraction of fibre of 50%, where we remarked in Section 4 that the desired expansivity could be obtained without the use of the device. The value of YoungÕs modulus of Kevlar is only 124 GPa so, when used in the device, this system will not attain the required value of ET. However, if used by itself as a platform, the relevant expansivity is not the transverse one but could be that in the axial direction which is the stiff direction for values of h at which the axial expansivity is a minimum. Indeed, as Fig. 2 shows, this system attains the required negative values over a range of angles between 20 and 40. Unfortunately, the axial modulus at h = 30 is only 20 GPa. By greatly increasing the volume of fibre to more than 75% it is possible to predict a value of the axial modulus EA, in this case, of 74.8 GPa at h = 16 where aA is 105 K1. Kevlar is a small diameter fibre and so such fibre packing would be very difficult to obtain.
9. Conclusions We have demonstrated, by using theory and experiment, that laminated composites can be used to make a simple easily manufactured device with which it is possible to produce platforms with extremely negative values of transverse expansivity in a range of directions. These values are much more negative than may be obtained using monolithic materials. We have set ourselves, and achieved, the target of attaining a negative value of 105 K1, which is of interest to those wishing to obtain thermal compensation of Bragg fibre gratings inscribed in optical fibres.
Acknowledgement The contribution to this paper by one of us (L.N.M.) was supported by the Materials Measurement Programme of the Department of Trade and Industry, UK.
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[26] McCartney LN. Analytical models of stress transfer in unidirectional composites and cross-ply laminates, and their application to the prediction of matrix/transverse cracking. In: Reddy JN, Reifsnider KL, editors. Proceedings of the IUTAM symposium on local mechanics concepts for composite material systems, Blacksburg, VA. Berlin, New York, London: Springer; 1991. p. 251–82. [27] Tyson WR, Fibre reinforcement in metals, PhD Thesis, Cambridge; 1964. [28] Dragoi D, Ustandag E, Clausen B, Bourke MAM. Investigation of thermal residual stresses in tungsten-fiber/bulk metallic glass matrix composites. Scripta Mater 2001;45:245–52.