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Modelling the elastic and thermoelastic properties of short fibre composites with anisotropic phases
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 66 (2006) 69–79 www.elsevier.com/locate/compscitech
Modelling the elastic and thermoelastic properties of short fibre composites with anisotropic phases C.D. Price a, P.J. Hine a
a,*
, B. Whiteside b, A.M. Cunha c, I.M. Ward
a
IRC in Polymer Science and Technology, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK b IRC in Polymer Science and Technology, University of Bradford, Bradford, UK c Department of Polymer Engineering, Universidada Do Minho, Campus Azurem 4800-058, Portugal Received 15 February 2005; received in revised form 23 May 2005; accepted 25 May 2005 Available online 14 July 2005
Abstract An analytical procedure is proposed and validated for predicting the elastic anisotropy and thermal expansion behaviour of a short fibre polymer composite, where both the fibre and the polymer matrix possess anisotropic material properties. The modelling strategy is to consider the fibre composite as an aggregate of units of structure, with an averaging scheme which takes into account the state of orientation. Validation of this strategy required the accurate determination of the unit properties and a satisfactory orientation averaging procedure. First, the unit properties were determined for very highly aligned samples with either an isotropic or an anisotropic fibre (glass or carbon) and either an isotropic or an anisotropic matrix (Nylon or liquid crystalline polymer). The fibre orientation and length distribution were determined by image analysis together with measurements of their elastic and thermoelastic properties. In combination with finite element calculations developed by Gusev, these results provided the basis for validation of appropriate analytical schemes to predict the unit properties. The orientation averaging procedures were validated by measurements on a model ribbed box component manufactured from a carbon fibre filled liquid crystalline polymer (anisotropic fibre and anisotropic matrix). Measurements of YoungÕs modulus and the coefficient of thermal expansion were combined with the determination of fibre orientation by image analysis. The key result was that if the fibre orientation level was high, the best prediction was to assume that the matrix orientation was identical to that of the fibres. For a lower degree of alignment, a better prediction was obtained by assuming that the matrix was isotropic. 2005 Elsevier Ltd. All rights reserved. Keywords: Modelling; Anisotropic elasticity; Short fibre composites
1. Introduction To use composite materials for sophisticated applications such as those incorporating optical fibres, it is necessary to make products which show minimal coefficient of thermal expansion in specific directions in addition to excellent mechanical stiffness and strength. To meet such requirements in a short fibre polymer composite, where the product can be manufactured in large quantities by injection moulding, presented a challenging design prob*
Corresponding author. Tel.: +44 113 343 3827; fax: +44 113 343 3846. E-mail address: [email protected] (P.J. Hine). 0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.05.024
lem. In our recent research, this was resolved by choosing carbon fibres as the reinforcing phase and a liquid crystalline polymer as the matrix phase. This system extends already very extensive studies of fibre composites to a system where both the fibre and the matrix are anisotropic. To validate the thermoelastic behaviour of these materials it was therefore necessary to combine and extend existing analytical procedures for predicting the elastic anisotropy and thermal expansion behaviour of anisotropic short fibre composites. This paper presents a strategy for dealing with this complex situation, and confirms its validity by analysing results for a range of injection moulded short fibre composites.
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The modelling approach adopted assumes that the oriented material can be regarded as an aggregate of units of structure, each of which consists of an ideal perfectly aligned short fibre composite. This strategy has previously been shown to be applicable to both oriented polymers [1,2] and polymer composites [3–6]. The calculations require three inputs: (1) the properties of the unit of structure; (2) orientation functions which describe the state of orientation; (3) averaging procedures to obtain the aggregate properties. Research over many years has addressed the several issues which this approach raises, for example, the best model for predicting unit properties, the most appropriate average fibre length to represent a distribution of fibre lengths and the best assumptions for undertaking the averaging of the unit properties. In recent research, this analytical modelling strategy has been validated on a range of injection moulded samples with an isotropic matrix and isotropic or anisotropic fibres, and underpinned by the novel finite element methodology developed by Gusev and colleagues at ETH Zurich [6– 10]. It has been concluded that: • The best model for the prediction of the unit properties is that based on the original ideas of Eshelby and Mori and Tanaka as modified by Tandon and Weng [11] for isotropic phases and as reported by Qui and Weng [12] for discontinuous anisotropic fibre reinforcement. This agrees with the conclusions of many authors as summarised in the recent review paper by Tucker and Liang [13]. • The best approach for predicting the coefficient of thermal expansion of the unit uses the explicit treatment of Levin [14] which is identical to that of Christensen [15] and of Rosen and Hashin [16]. • The best measure of the fibre length is the number average fibre length [7]. • The best method for predicting the properties of a partially aligned fibre composite is to use a tensor averaging approach [3,17] and a constant strain prediction [5]. This is a more general approach than that developed previously (for example Cox [18]and Halpin [19,20]) where the composite is considered to consist of discrete layers of different fibre orientation and the stiffness constants of these layers is averaged using standard laminate analysis (e.g. [21]). Our recent work with Gusev [10] confirms that the constant strain prediction is valid for all orientation states and for all elastic and thermoelastic constants provided the volume fraction is greater than 5% and the fibre aspect ratio is greater than 10. The paper of Camacho and Tucker [17] is a particularly useful reference for this calculation as it details the tensor averaging for both elastic and thermoelastic properties.
For validation of the unit predictions, dumbbellshaped specimens were made by the University of Minho from a range of materials with different combinations of component isotropy: carbon fibre filled Nylon, pure liquid crystalline polymer (LCP), glass fibre filled LCP and carbon fibre filled LCP. These samples were considered appropriate because in the central shaft section, the fibres were measured to be very highly aligned, allowing the orientation averaging contribution to be minimised and the unit predictions to be assessed. Fully aligned samples could, however, not be produced experimentally, so direct assessment of the unit predictions was made by comparing with numerical finite element predictions of perfectly aligned samples made by the group of Gusev, ETH, Zurich. For all the composite materials, characterisation of their fibre orientation and fibre length distributions was carried out using an in-house developed image analysis system [22]. For an assessment of the orientation averaging scheme, a demonstrator ribbed box component was manufactured at the University of Bradford, from the carbon fibre filled LCP material (that is both phases anisotropic). The ribs of this component proved ideal structures for evaluating the proposed scheme, being convenient for mechanical measurement and also partially aligned in terms of the fibres. Mechanical measurements (bending modulus and coefficient of thermal expansion) were made on these ribs, and their orientation structure was measured using image analysis. Comparison of the measured properties (YoungÕs modulus and coefficient of thermal expansion) with the model predictions for the carbon/LCP samples (both phases anisotropic) showed that there were two possible choices: either assume the matrix had the same orientation as the fibres, or assume the matrix was randomly oriented. A randomly oriented LCP matrix can itself be considered to be composed of randomly arranged fully aligned polymeric units. The aggregate approach was therefore used to determine the properties of the randomly oriented LCP matrix, based on the properties of the fully aligned LCP material and using orientation averaging with the constant stress (Reuss) bound, which from previous work has been found to be the most appropriate choice in this situation [2]. 2. Injection moulded dumbbells 2.1. Experimental 2.1.1. Materials A range of injection moulded dumbbell samples was produced at The University of Minho, Portugal, by the group of Professor Cunha. Fig. 1 shows a picture of the dumbbell sample, which was composed of a cylindrical central shaft (20 mm long and 1.5 mm in diameter),
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Fig. 1. Injection moulded dumbbell sample.
gradually opening out to a 10 mm · 3.5 mm rectangular cross section and 10 mm square end tags. Samples were made from three filled materials of different combinations of isotropy and one unfilled material: details of these materials are given in Table 1. The samples were injection moulded using the processing conditions specified by the material manufacturers. 2.1.2. Measurement of the elastic properties of the unfilled matrix materials Prediction of the composite properties requires a knowledge of the unfilled matrix materials, namely Nylon and liquid crystalline polymer (LCP). For the isotropic aromatic Nylon polymer, only two elastic constants were required, the YoungÕs modulus E and the PoissonÕs ratio m. A sample of the unfilled aromatic Nylon was not available, so the pertinent values were taken from manufacturersÕ data sheets: 2 GPa and 0.4 for E and m, respectively. The LCP material when fully aligned is transversely isotropic, thus requiring five elastic constants to specify its elastic anisotropy, E11, E22, m21, m23 and G12, where the 1-axis is the direction of preferred orientation (the dumbbell axis). For modelling the composite ÔunitÕ, these properties should be those of a perfectly aligned matrix sample. Injection moulded dumbbell samples, produced from the unfilled pure LCP matrix (Hoechst Celanese grade B950), showed a very high degree of molecular orientation in the central shaft as measured by wide angle X-ray scattering (WAXS). For this reason the central shaft of these samples could be use to determine the five elastic constants of the LCP perfectly aligned unit. The axial modulus, E11, was measured in tension (up to a strain of 0.5%) using an RDP servo-mechanical test machine. The sample strain was measured using a Messphysik video extensometer and the tests were carried out at a nominal strain rate of 103 s1. To check these mea-
surements, three point bending tests were made and excellent agreement was found between these two methods: the determined value was 25.6 GPa. The transverse modulus, E22, was measured by the method of transverse compression [23] and gave a value of 1.45 GPa. The shear modulus, G12, was measured (on a separate piece of moulded LCP machined to size) using a rectangular torsion test on a Rheometric RDA2 torsion rheometer. An issue with rectangular torsion tests is the effect of the sample aspect ratio on the shear modulus. Three different aspect ratios (sample length/width) were measured (4, 6 and 12;1) and all gave a very similar value for the shear modulus of 1.13 GPa. Attempts were made to measure the PoissonÕs ratio of the samples using a tensile test and monitoring the transverse strain, but these results proved scattered. As an alternative, the values determined from a previous study on an LCP material, where an ultrasonic technique was employed [24], were used. It is considered that using an ultrasonic value is valid, as PoissonÕs ratio is less frequency-dependent than the other elastic constants, being given by ratios of compliances: the previously measured values were 0.48 for m21 and 0.71 for m23. 2.1.3. Measurement of the coefficient of thermal expansion of the unfilled matrix materials The value for the coefficient of thermal expansion, a, of the aromatic Nylon, was taken from the literature and was 70 · 106 K1. The anisotropic LCP polymer required two values, the axial coefficient of thermal expansion a1 and the transverse coefficient of thermal expansion a2. These were measured using a custom built facility [25], where the temperature was cycled between 15 and 25 C over a 25 min cycle. The average value at 20 C was calculated from the maximum slope of the plot of displacement versus temperature. The axial coefficient of thermal expansion was measured from 6 mm long pieces taken from the dumbbell shaft: these mea-
Table 1 Details of the materials used
1 2 3 4
Description
Grade
Fibre
Matrix
Weight fraction (%)
Carbon fibre filled Nylon Unfilled LCP Glass fibre filled LCP Carbon fibre filled LCP
Solvay IXEF 300/6/9019 Hoechst Celanese Ticona B950 Hoechst Celanese Ticona B130 Hoechst Celanese Ticona B230
Anisotropic – Isotropic Anisotropic
Isotropic Anisotropic Anisotropic Anisotropic
30 – 30 30
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surements gave a value of 5 · 106 K1. Initial experiments measuring the transverse coefficient of thermal expansion across the diameter of the central shaft section gave quite scattered results. Much more consistent results were obtained by measuring across the end tabs sections of the dumbbells. This was considered representative of the transverse expansion of the fully aligned polymer, as our previous work on both polymers and composites (e.g. [26]) has shown that the transverse elastic and thermoelastic properties are little affected by molecular orientation. This measurement gave a value of 80 · 106 K1. In summary, the elastic and thermoelastic properties of the matrix materials (at 20 C) are shown in Table 2. 2.1.4. The elastic properties of the fibre phase The elastic properties of the fibres were taken from a combination of published literature and manufacturerÕs data sheets. The values for isotropic ÔEÕ glass fibres are well known with quoted values in good agreement: the values used here were a YoungÕs modulus E of 72.5 GPa, a PoissonÕs ratio m of 0.2 and a coefficient of thermal expansion a of 5 · 106 K1. For the transversely isotropic carbon fibres the values, particularly the transverse constants, are more difficult to obtain. The filled LCP materials used a carbon fibre with a quoted axial modulus, E11, of 230 GPa and an axial coefficient of thermal expansion, a1, of 0.4 · 106 K1. Previous work by a number of authors, including Smith [27], has shown that there is a relationship between the axial YoungÕs modulus and the other elastic constants. Using the data of Smith, and others, the other elastic constants of the carbon fibre used in our materials could be estimated: this does, however, introduce some degree of uncertainty in the final predictions, in that these constants have to estimated and are not measured directly. The values for both reinforcements are shown in Table 3. 2.1.5. The fractions of the two phases The fractions of the phases in each composite were specified by the manufacturers but they were also checked by direct density measurement (and the known Table 2 Matrix elastic properties Aromatic Nylon
E m a
2 0.4 70
Data sheet Data sheet Data sheet
LCP (B950)
E11 E22 m21 m23 G12 a1 a2
25.6 1.45 0.48 0.71 1.13 5 80
Measured statically Measured statically Measured-ultrasonics Measured-ultrasonics Measured statically Measured statically Measured statically
E and G in GPa, a · 106 K1.
Table 3 Reinforcement elastic properties Glass fibres (E glass)
E m a
Carbon fibres
E11 E22 m21 m23 G12 a1 a2
72.5 0.2 5
Literature Literature Literature
230 14 0.26 0.38 17.5 0.4 26
Manufacturer Estimated Estimated Estimated Estimated Manufacturer Literature
E and G in GPa, a · 106 K1. Table 4 Fractions of the various materials
Carbon filled Nylon Glass filled LCP Carbon filled LCP
Grade
Weight fraction (%)
Volume fraction (%)
Solvay grade IXEF 300/6/9019 Ticona grade B130 Ticona grade B230
30
21
30 30
19 24
densities of the two phases) by density column and density bottle tests. The weight and volume fractions for the three materials are shown in Table 4. Density measurements were also used to check for internal voiding. Samples that were more than 0.5% below the average density (also usually accompanied by sink marks), were discarded. 2.1.6. Characterisation of the fibre orientation and length distributions of the composite materials The fibre orientation and length distributions of the injection moulded composite samples were measured using an in-house image analysis facility developed at the University of Leeds [22]. For fibre orientation investigations, the chosen method is one of optical reflection microscopy of polished 2D sections taken from the areas of interest of the composite. Each fibre that meets the 2D section is seen as an elliptical footprint (see Fig. 2(a)), and measuring the ellipticity of these images allows the two polar angles, h and /, that specify the orientation of each fibre to be determined: h is the angle the fibre makes with the sectioned surface normal (1) and / is the angle the fibre makes with the 2-axis when projected into the 23 plane (Fig. 2(b)). An XY stage allows a large area to be scanned, allowing a determination of fibre orientation over a large area. For further details of the system and the analysis routines see [28]. Fibre length characterisation is achieved by taking a chosen section of the sample, removing the matrix material (by placing into a furnace set at 450 C for 6 h) and then spreading the resulting fibres thinly on a glass plate. To aid dispersion of the fibre residue, a solution of water and a mild surfactant are also placed into the dish and
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of the dumbbell shaft. The measured values for the three injection moulded dumbbells are shown in Table 5. 2.2. Comparison between theoretical predictions and experimental measurements
Fig. 2. (a) A typical image frame. (b) The definition of h and /.
then left to dry. For the length determination, the fibres are viewed in transmitted light and image analysis routines are used to determine the length of each fibre shadow image. The glass fibre/LCP sample was measured in this way and gave a value of the fibre aspect ratio (based on the number average fibre length) of 25. The two carbon fibre filled materials were also measured in this way and both gave a very similar value for the fibre aspect ratio of 33. The fibre orientation structure of the central shaft of the injection moulded dumbbells was found, as might be expected, to be both highly aligned and transversely isotropic, that is no preferred / orientation. The fibre orientation could, therefore, be specified by only two orientation averages, hcos2hi and hcos4hi, where h is the angle between a fibre and the longitudinal (1) axis
2.2.1. Anisotropic fibre/isotropic matrix – carbon fibre filled Nylon As described earlier, the modelling strategy is to undertake the prediction of properties in two stages: first the prediction of the properties of the fully aligned ÔunitÕ and secondly the effects of misorientation using tensor averaging. For the carbon fibre filled Nylon, that is an anisotropic fibre in an isotropic matrix, the scheme previously validated in the former studies was used: the modification of Qui and Weng for the elastic properties of the unit: the treatment of Levin for the determination of the coefficients of thermal expansion of the unit: the aggregate model and constant strain bound for the composite properties. Using this procedure, theoretical predictions were determined based on the phase properties and the measured fibre microstructure detailed earlier. For comparison with these predictions, measurements were carried out on the composite samples of four properties: the axial modulus E11, the transverse modulus E22, the axial coefficient of thermal expansion a1 and the transverse coefficient of thermal expansion a2. As with the tests on the unfilled polymers, E11 was measured in bending and tension in the central shaft section, E22 by transverse compression of this cylindrical section, a1 by a measurement along the axial direction of the central shaft and a2 from a transverse measurement of the end tab section. Table 6 shows a comparison of these measured values (fifth column) with the theoretical predictions (fourth column). It is seen that the agreement is good, considering the uncertainty in the carbon fibre elastic constants. For interest, the table also shows the properties for various other material states. The second column shows the properties of a fully aligned composite calculated assuming the fibres are infinitely long, while the third column shows the properties again for a fully aligned composite, but where the fibres have the same aspect ratio as the measured material. It is of interest to note that for this combination of materials, the correction for finite length is more important than that for misalignment. Table 5 Measured fibre orientation averages of the various materials
Carbon filled Nylon Glass filled LCP Carbon filled LCP
Grade
hcos2hi
hcos4hi
Solvay grade IXEF 300/6/9019 Ticona grade B130 Ticona grade B230
0.856
0.830
0.904 0.929
0.865 0.870
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Table 6 A comparison of the measured and predicted properties for the carbon fibre/Nylon dumbbells Theoretical predictions
E11 E22 a1 a2
Measured for dumbbell
Fully aligned fibres infinite fibre length
Fully aligned fibres finite fibre length (aspect ratio = 33:1)
Actual fibre alignment finite fibre length (aspect ratio = 33:1)
49.9 3.24 1.92 80.7
31.1 3.22 3.62 80.0
26.2 4.38 6.09 65.4
2.2.2. Anisotropic matrix composites For modelling the glass/LCP and carbon/LCP materials, the modelling strategy is identical to that described above when the matrix is isotropic, apart from the requirement for the calculation of the Eshelby tensor for an anisotropic material, which has been determined by Mura [29]. The prediction of the coefficients of thermal expansion of the unit follows directly from Levin, which just requires the correct unit elastic constants and the coefficients of thermal expansion of the component phases. As an additional step in the validation, the analytical predictions obtained in this way (Mura and Qui; Weng and Levin) for the composite unit were compared with a finite element prediction determined using the ETH Zurich PALMYRA package, which in previous work has been found to give excellent agreement with experimentally measured values. This comparison is shown in Table 7. It is seen that the analytical predictions of the unit properties are very close to those obtained by FE calculations, confirming the usefulness of the analytical approach. Experimental measurements of the axial and transverse YoungÕs Modulus (E1 and E2) and the axial and transverse coefficients of thermal expansion (a1 and a2) were made on the glass fibre and carbon fibre LCP dumbbells for comparison with the theoretical predictions: the results of this comparison are shown in Tables 8 and 9, respectively. The aggregate modelling is identiTable 7 Comparison of prediction of properties of fully aligned unit based on the best analytical route and finite element calculation (PALMYRAETH Zurich) Glass/LCP
Carbon/LCP
Analytical
FE
Analytical
FE
E11 E22 m21 m23 G12
33.9 2.01 0.427 0.686 1.63
33.8 2.34 0.421 0.642 1.90
62.7 2.09 0.443 0.682 1.76
64.2 2.53 0.419 0.636 2.13
a1 a2
1.04 63.2
0.90 61.9
1.90 61.0
1.50 57.0
E in GPa, a · 106 K1.
23.9 3.6 5.5 60.0
cal to that for a fibre reinforced material with an isotropic matrix, although this makes the implicit assumption that the level of orientation in the two phases (fibres and matrix) is the same. Several points in this comparison are of interest. First, it is of interest to examine the magnitude of the corrections for fibre length and fibre misorientation in comparison with the results of the carbon fibre/aromatic Nylon dumbbell described earlier (Table 6). There appears to be a correlation between the degree of anisotropy between the two phases and the effect of the length correction. This is seen particularly for E11, which is most affected when the fibres are aligned, in that the difference is largest for carbon/Nylon (49.9 GPa down to 25.8 GPa) and only very small for glass fibres in LCP (34.5 GPa down to 33.9 GPa). The effect of the correction for fibre misorientation does depend on which property is considered: in general, it is observed as having a larger percentage effect on the transverse properties (E22 and a2) compared to the longitudinal properties. A second point of interest is that both of the filled LCP materials have axial coefficients of thermal expansion (both measured and predicted) close to zero, which could be a valuable material property. In general terms, the agreement between theoretical predictions and measurements is reasonable, but not as good as seen with the isotropic aromatic Nylon matrix (Table 6). One possible reason for this discrepancy is the limitation of the aggregate modelling strategy described above in that the tensor averaging approach assumes that the degree of orientation of the matrix and fibre phases, in the misaligned composite, is identical: of course this may not be exactly the case. The development of a model able to handle different levels of orientation of the two phases would require a substantial amount of further work, and would also require characterisation of the orientation of the LCP matrix phase using wide angle X-ray scattering (WAXS), both of which were outside the scope of the current research project. As will be seen from the results presented in the next section, this discrepancy increases as the level of preferred orientation decreases, and a new strategy is proposed for this situation.
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Table 8 A comparison of the measured and predicted properties for the glass fibre/LCP dumbbells Theoretical predictions
E11 E22 a1 a2
Measured for dumbbell
Fully aligned fibres infinite fibre length
Fully aligned fibres finite fibre length (25:1)
Actual fibre alignment finite fibre length (25:1)
34.5 2.00 0.78 63.0
33.9 2.01 1.04 63.2
29.1 2.86 1.20 55.0
28.0 1.95 0.43 60.0
E in GPa, a · 106 K1.
Table 9 A comparison of the measured and predicted properties for carbon fibre/ LCP dumbbells Theoretical predictions
E11 E22 a1 a2
Measured for dumbbell
Fully aligned fibres infinite fibre length
Fully aligned fibres finite fibre length (33:1)
Actual fibre alignment finite fibre length (33:1)
74.6 2.09 1.49 60.8
65.9 2.09 1.77 61.0
54.3 2.55 4.89 58.1
3. Model injection moulded ribbed box 3.1. Experimental 3.1.1. Component details The aim of this second part of the project was to further test the modelling scheme outlined above through the study of a more complicated component, in this case a ribbed box, using the material where both phases were anisotropic, (carbon fibres in LCP). The demonstrator component, of external dimensions 22 mm · 76 mm · 6 mm, was injection moulded at The University of Bradford, UK: a picture of this is shown in Fig. 3(a). The injection gates were located at the end of the component at positions between the ribs (between 1 and 2, and 2 and 3) to stop any jetting down the rib: the positions are shown by the arrows on Fig. 3(a). The processing conditions used were as specified by the manufacturer, and are shown in Table 10. The near zero coefficient of thermal expansion of this material, as measured in the previous section of the study, made the extraction of this component from the mould a real challenge, so that the machine had to be operated in manual mode and not automatically. 3.1.2. Measurement of fibre orientation The three main ribs on the component were chosen as prime candidates for structures on which the mechanical properties (YoungÕs modulus and coefficient of thermal expansion) could easily be measured experimentally. For this reason, it was decided to measure the fibre orientation at four positions along each of the three ribs, denoted A, B, C and D, located 10, 20, 30 and 40 mm
43.2 2.45 1.00 49.0
from the end face. Details of these positions are shown schematically on Fig. 3(b). 2D sections were taken across the rib at each chosen position: the 1-axis here is defined as parallel to the ribs, the 2-axis perpendicular to the ribs but in the plane of the component, and the 3-axis as out of the plane of the component: the sections across the ribs were therefore taken in the 23 plane. With fibre orientation data (FOD) there are a number of ways of presenting the results. A helpful method of visualising the data is to recreate the scanned area in terms of the elliptical images that were identified and fitted during analysis. Fig. 4(b) shows such a reconstruction, taken from Rib 2 at position C (location shown in Fig. 4(a)). The picture shows both the rib and the plate onto which it is connected: the junction between the rib and the plate is shown by the dotted line: the left and right boundaries of the diagram below the dotted line are the saw cuts severing the specimen from the rest of the box. It is seen that there is a region at the outer edges of the rib where the fibres are aligned preferentially with the flow direction (1-axis) and a central portion where the orientation is more in the 23 plane. In terms of producing a theoretical prediction of a rib, the most suitable measure is the average orientation over the scanned area. This is most appropriate as the experimental measurements were for the YoungÕs modulus and coefficient of thermal expansion in the 1 direction (i.e., averaged properties across the cross section). The orientation averages calculated in this way from the measured image analysis data showed that the fibre average orientation, when averaged over the whole section (rib plus plate underneath) indicated that the fibres were still predominantly aligned with
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Fig. 3. (a) Details of the injection moulded ribbed box (the arrows show the positions of the injection gates). (b) Details of image analysis section positions on the three ribs chosen for analysis: Positions A, B, C and D: 10, 20, 30 and 40 mm from the gate end, respectively.
respect to in the 1 direction, and the pattern in the three ribs was very similar. The level of preferred orientation increased significantly between positions A and B, and then was measured to be reasonably constant for positions B, C and D. In light of these measurements, it was decided to measure YoungÕs modulus and coefficient of thermal expansion for the three ribs between positions B and C, where the average value of orientation with respect to rib axis (1 direction) was hcos2h1i = 0.675 ± 0.029. In addition to the FOD, the prediction of mechanical properties also requires a measurement of the number average fibre length. This was carried out using the procedure described in an earlier section and gave an average fibre length of 252 lm, which together with an average measured diameter of 6.27 lm, gave an average fibre aspect ratio of 40.2. This is a higher value than was seen for the injection moulded dumbbells (33) and it is likely that the higher degree of fibre attrition seen in the moulded dumbbells is due to the much smaller Table 10 Processing conditions used for demonstrator ribbed box component Injection speed Shot size Screw speed Back pressure Cooling time Melt temperature Mould temperature
15 mm/s 29 mm 150 rpm 50 bar 10 s 290 C 90 C
mould cavities and channels in this mould, leading to significantly higher shear forces during processing. 3.1.3. Measurement of mechanical properties of the ribs From the image analysis measurements, the decision was made to measure the mechanical properties (YoungÕs modulus and coefficient of thermal expansion) of ribs taken from the component, between positions B and C on all three ribs, as the fibre orientation averages are reasonably consistent between these positions. The ribs were removed from the component, with the plate underneath still attached, therefore corresponding to the measured image analysis sections, and calculated orientation averages. The YoungÕs modulus was measured in flexure, and the rib/plate samples were tested with either the 3-axis vertical (upright) or the 2-axis vertical (on-side). As a result of the material being anisotropic, and the low value of the bending span to width ratio (L/W) which had to be used for testing these samples (particularly for the upright measurement), a correction had to be made to the results to correct for shear deformation. After correction, the values measured for E11 in flexure were 24.3 ± 1.7 GPa with the sample upright and 25.9 ± 1.7 GPa with the sample on its side. In view of the good agreement, these two values were combined to give an average value of E11 of 25 ± 2 GPa. The coefficient of thermal expansion of the component was measured with respect to both the 1 and 3 axes
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Table 11 A comparison of measured and theoretical predictions Measured
E11 (GPa) a1 (·106 K1) a3 (· 106 K1)
Fig. 4. (a) Location of analysis region as shown below. (b) Reconstructed image scan from Rib 2 at position C as shown above.
on samples cut from between positions B and C. The average values measured were (0.9 ± 1.8) · 106 K1 for a1 and (52 ± 3) · 106 K1 for a3. 3.2. Comparison between theoretical predictions and experimental measurements The final part of this study was to generate analytical predictions for the ribs at the chosen positions to compare with the experimental measurements, based on the modelling scheme developed for the injection moulded dumbbells. The properties of the carbon fibres and LCP matrix have already been reported earlier (Tables 1 and 2). Table 11 shows a comparison of the measured (column 2) and predicted (column 3) mechanical properties for Ribs 1, 2 and 3 at positions B and C. The agreement
25.0 ± 2.0 0.9 ± 1.8 52.0 ± 3.0
Best analytical Aligned matrix
Isotropic matrix
29.7 ± 1.7 5.02 ± 0.25 37.5 ± 1.9
24.2 ± 1.4 1.25 ± 0.1 31.9 ± 1.6
is reasonable considering that this composite material is the most challenging system to model, as both phases are oriented and could potentially have a different level of orientation: in addition there is uncertainty in some of the carbon fibre properties. Although at the current time an analytical model is not available for modelling a system where the orientation is different in the two phases (although an FE approach could model such a system) a second prediction can be made by assuming the matrix is randomly oriented, and therefore isotropic in terms of elastic properties. Previous studies (e.g. [30]) have shown that in the presence of fibres, the level of matrix orientation can be much lower than seen in a pure matrix sample for the same geometry. For this reason an assumption of a randomly oriented matrix could be a good one in the situation where the fibres are partially aligned. To carry out this prediction, the properties of the randomly oriented LCP matrix were first calculated, using the aggregate model again. In this case, previous work has shown [2] that the constant stress (Reuss) bound is the most appropriate, based on the fully oriented LCP properties (Table 1). The results of this calculation are shown in Table 12. Using these properties for the LCP matrix, the predictions of the properties of the rib were recalculated, and the results of this are shown in the final column of Table 11. The predictions of E11 and a1 are in better agreement with the measurements based on an isotropic matrix, whereas the prediction of the transverse coefficient of thermal expansion a2 is farther away. At the present time the reason for this final discrepancy is unknown. Two alternate strategies can therefore be proposed for predicting the properties of a system where both the fibres and matrix are anisotropic: (1) if the degree of fibre orientation is high (hcos2hi > 0.8) then a good prediction can be produced by assuming the orientation in the two phases is the same: (2) if the degree of fibre orientation is lower (hcos2hi < 0.8) then a good predic-
Table 12 Calculated properties for an ÔisotropicÕ LCP matrix LCP isotropic (Reuss bound) E (GPa) m a (·106 K1)
3.42 0.292 51.7
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Table 13 Modelling schemes for different combinations of fibre and matrix anisotropy Fibre
Matrix
Unit
Aggregate
Elastic Isotropic glass Anisotropic carbon Isotropic glass Anisotropic carbon
Isotropic Isotropic Nylon Anisotropic LCP Anisotropic LCP
Qui Qui Qui Qui
and and and and
Weng Weng Weng Weng
Eshelby Eshelby Mura Mura
tion can be produced by assuming the orientation of the matrix is isotropic.
4. Conclusions By combining careful microstructural characterisation, together with detailed mechanical measurements, an analytical modelling scheme, which is able to give good predictions for short fibre composites where both phases are anisotropic, has been validated. The chosen modelling scheme is as follows: • Use the modification of Qui and Weng (for an anisotropic fibre) to determine the elastic properties of the unit based on the number average fibre length. • If the matrix is anisotropic, determine the Eshelby tensor from the work of Mura. • Use the treatment of Levin to determine the coefficients of thermal expansion of the unit, based on the mechanical properties determined above. • Determine the properties of the misaligned composite using tensor averaging and a constant strain prediction. If both phases are isotropic this scheme gives excellent predictions. For anisotropic carbon fibres in an aromatic Nylon isotropic matrix the predictions were also excellent. For isotropic glass fibres in an anisotropic LCP matrix the predictions are reasonable. For anisotropic carbon fibres in an anisotropic LCP matrix the predictions are reasonable. Two suggestions have been made for the poorer predictions of the LCP filled materials: (1) the uncertainty in the carbon fibre elastic and thermoelastic constants and (2) the degree of orientation in the LCP matrix. As the predictions of the carbon fibre/Nylon material were in excellent agreement with measurements, this suggests that it is the unknown level of orientation in the LCP matrix that is the most likely cause of the discrepancy. At present there are only two possible choices, either assume the orientation is the same as the matrix, or assume that the LCP matrix is randomly oriented, giving isotropic properties. Interestingly which of these
Coefficient of thermal expansion
Elastic
Coefficient of thermal expansion
Christensen Christensen Christensen Christensen
Camacho + Tucker Camacho + Tucker Camacho + Tucker Camacho + Tucker
Camacho + Tucker Camacho + Tucker Camacho + Tucker Camacho + Tucker
to choose appears to depend on the level of fibre orientation. For high fibre alignment (the central shaft of the dumbbells) the former gives a better prediction while for lower fibre alignment (the ribbed component) the randomly oriented matrix gave a slightly better prediction. The suggested strategy is therefore if the average value of the principal second order orientation tensor is greater than 0.8, then assume fibre and matrix have the same orientation, and if the average value is less than 0.8, then assume the matrix is randomly oriented. The proposed modelling schemes, for different combinations of anisotropy, are finally summarised in Table 13. It is the conclusion of this work that this combination gives the best predictions of the elastic and thermoelastic properties available at present without recourse to finite element procedures.
Acknowledgements This project was undertaken with the financial support of Marconi UK and Moldflow. The authors thank Ticona for providing the raw composite materials. The authors also thank Andrei Gusev at ETH, Zurich, for carrying out the finite element predictions of the fully aligned carbon fibre filled LCP material.
References [1] Ward IM. Optical and mechanical anisotropy in crystalline polymers. Proc Phys Soc London 1962;80:1176–88. [2] Troughton MJ, Davies GR, Ward IM. Dynamic mechanicalproperties of random copolyesters of 4-hydroxybenzoic acid and 2-hydroxy 6-naphthoic acid. Polymer 1989;30(1):58–62. [3] Brody H, Ward IM. Modulus of short carbon and glass fiber reinforced composites. Polym Eng Sci 1971;11:139–51. [4] Advani SG, Tucker III CL. The use of tensors to describe and predict fibre orientation in short fibre reinforced composites. J Rheol 1987;31:751–84. [5] Hine PJ, Duckett RA, Ward IM, Allan PS, Bevis MJ. A comparison of short glass fiber reinforced polypropylene plates made by conventional injection molding and using shear controlled injection molding. Polym Compos 1996;17(3):400–7. [6] Lusti HR, Hine PJ, Gusev AA. Direct numerical predictions for the elastic and thermoelastic properties of short fibre composites. Compos Sci Technol 2002;62(15):1927–34.
C.D. Price et al. / Composites Science and Technology 66 (2006) 69–79 [7] Hine PJ, Lusti HR, Gusev AA. Numerical simulation of the effects of volume fraction, aspect ratio and fibre length distribution on the elastic and thermoelastic properties of short fibre composites. Compos Sci Technol 2002;62(10–11):1445–53. [8] Gusev AA, Lusti HR, Hine PJ. Stiffness and thermal expansion of short fiber composites with fully aligned fibers. Adv Eng Mater 2002;4(12):927–31. [9] Gusev A, Heggli M, Lusti HR, Hine PJ. Orientation averaging for stiffness and thermal expansion of short fiber composites. Adv Eng Mater 2002;4(12):931–3. [10] Hine PJ, Lusti HR, Gusev A. On the possibility of reduced variable predictions for the thermoelastic properties of short fibre composites. Compos Sci Technol 2004;64: 1081–8. [11] Tandon GP, Weng GJ. The effect of aspect ratio of inclusions on the elastic properties on unidirectionally aligned composites. Polym Compos 1984;5:327–33. [12] Qiu YP, Weng GJ. On the application of Mori–TanakaÕs theory involving transversely isotropic spheroidal inclusions. Int J Eng Sci 1990;28:1121–37. [13] Tucker III CL, Liang E. Stiffness predictions for unidrectional short-fiber composites: review and evaluation. Compos Sci Technol 1999;59:655–71. [14] Levin VM. Mech Tverd tela 1968;88:25. [15] Christensen RM. Mechanics of composites materials. Malabar, FL: Kreiger Publishing Company; 1991. [16] Rosen BW, Hashin Z. Effective thermal expansion coefficients and specific heats of composite materials. Int J Eng Sci 1970;8:157–73. [17] Camacho CW, Tucker CL. Stiffness and thermal expansion predictions for hybrid short fiber composites. Polym Compos 1990;11:229–39. [18] Cox HL. The elasticity and strength of paper and other fibrous materials. Brit J Appl Phys 1952;3(MAR):72–9.
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[19] Halpin JC, Pagano NJ. Laminate approximation for randomly oriented fibrous composites. J Compos Mater 1969;3:720–4. [20] Halpin JC, Jerine K, Whitney JM. Laminate analogy for 2 and 3 dimensional composite materials. J Compos Mater 1971;5(JAN):36–49. [21] Tsai SW, Hahn HT. Introduction to composite materials. West Park, CT: Technomic; 1969. [22] Hine PJ, Davidson N, Duckett RA, Ward IM. Measuring the fiber orientation and modeling the elastic properties of injectionmolded long-glass-fiber-reinforced nylon. Compos Sci Technol 1995;53(2):125–31. [23] Kotani T, Sweeney J, Ward IM. The measurement of transverse mechanical-properties of polymer fibers. J Mater Sci 1994;29(21):5551–8. [24] Hine PJ, Ward IM. Measuring the elastic properties of highmodulus fibres. J Mater Sci 1996;31(2):371–9. [25] Orchard GAJ, Davies GR, Ward IM. The thermal-expansion behavior of highly oriented polyethylene. Polymer 1984;25(8):1203–10. [26] Hine PJ, Wire S, Duckett RA, Ward IM. Hydrostatically extruded glass fiber reinforced polyoxymethylene. 2. Modeling the elastic properties. Polym Compos 1997;18(5):634–41. [27] Smith RE. Ultrasonic elastic constants of carbon fibers and their composites. J Appl Phys 1972;43:2555–611. [28] Hine PJ, Davidson N, Duckett RA, Clarke AR, Ward IM. Hydrostatically extruded glass-fiber-reinforced polyoxymethylene. I: the development of fiber and matrix orientation. Polym Compos 1996;17:720–9. [29] Mura T. Mechanics of elastic and inelastic solids. The Hague: Nijhoff; 1982. [30] Cooke AH. Orientation studies in short glass fibre reinforced polyproplyene composites, in Physics Department. Leeds: University of Leeds; 1997.
Modelling the elastic and thermoelastic properties of short fibre composites with anisotropic phases C.D. Price a, P.J. Hine a
a,*
, B. Whiteside b, A.M. Cunha c, I.M. Ward
a
IRC in Polymer Science and Technology, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK b IRC in Polymer Science and Technology, University of Bradford, Bradford, UK c Department of Polymer Engineering, Universidada Do Minho, Campus Azurem 4800-058, Portugal Received 15 February 2005; received in revised form 23 May 2005; accepted 25 May 2005 Available online 14 July 2005
Abstract An analytical procedure is proposed and validated for predicting the elastic anisotropy and thermal expansion behaviour of a short fibre polymer composite, where both the fibre and the polymer matrix possess anisotropic material properties. The modelling strategy is to consider the fibre composite as an aggregate of units of structure, with an averaging scheme which takes into account the state of orientation. Validation of this strategy required the accurate determination of the unit properties and a satisfactory orientation averaging procedure. First, the unit properties were determined for very highly aligned samples with either an isotropic or an anisotropic fibre (glass or carbon) and either an isotropic or an anisotropic matrix (Nylon or liquid crystalline polymer). The fibre orientation and length distribution were determined by image analysis together with measurements of their elastic and thermoelastic properties. In combination with finite element calculations developed by Gusev, these results provided the basis for validation of appropriate analytical schemes to predict the unit properties. The orientation averaging procedures were validated by measurements on a model ribbed box component manufactured from a carbon fibre filled liquid crystalline polymer (anisotropic fibre and anisotropic matrix). Measurements of YoungÕs modulus and the coefficient of thermal expansion were combined with the determination of fibre orientation by image analysis. The key result was that if the fibre orientation level was high, the best prediction was to assume that the matrix orientation was identical to that of the fibres. For a lower degree of alignment, a better prediction was obtained by assuming that the matrix was isotropic. 2005 Elsevier Ltd. All rights reserved. Keywords: Modelling; Anisotropic elasticity; Short fibre composites
1. Introduction To use composite materials for sophisticated applications such as those incorporating optical fibres, it is necessary to make products which show minimal coefficient of thermal expansion in specific directions in addition to excellent mechanical stiffness and strength. To meet such requirements in a short fibre polymer composite, where the product can be manufactured in large quantities by injection moulding, presented a challenging design prob*
Corresponding author. Tel.: +44 113 343 3827; fax: +44 113 343 3846. E-mail address: [email protected] (P.J. Hine). 0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.05.024
lem. In our recent research, this was resolved by choosing carbon fibres as the reinforcing phase and a liquid crystalline polymer as the matrix phase. This system extends already very extensive studies of fibre composites to a system where both the fibre and the matrix are anisotropic. To validate the thermoelastic behaviour of these materials it was therefore necessary to combine and extend existing analytical procedures for predicting the elastic anisotropy and thermal expansion behaviour of anisotropic short fibre composites. This paper presents a strategy for dealing with this complex situation, and confirms its validity by analysing results for a range of injection moulded short fibre composites.
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The modelling approach adopted assumes that the oriented material can be regarded as an aggregate of units of structure, each of which consists of an ideal perfectly aligned short fibre composite. This strategy has previously been shown to be applicable to both oriented polymers [1,2] and polymer composites [3–6]. The calculations require three inputs: (1) the properties of the unit of structure; (2) orientation functions which describe the state of orientation; (3) averaging procedures to obtain the aggregate properties. Research over many years has addressed the several issues which this approach raises, for example, the best model for predicting unit properties, the most appropriate average fibre length to represent a distribution of fibre lengths and the best assumptions for undertaking the averaging of the unit properties. In recent research, this analytical modelling strategy has been validated on a range of injection moulded samples with an isotropic matrix and isotropic or anisotropic fibres, and underpinned by the novel finite element methodology developed by Gusev and colleagues at ETH Zurich [6– 10]. It has been concluded that: • The best model for the prediction of the unit properties is that based on the original ideas of Eshelby and Mori and Tanaka as modified by Tandon and Weng [11] for isotropic phases and as reported by Qui and Weng [12] for discontinuous anisotropic fibre reinforcement. This agrees with the conclusions of many authors as summarised in the recent review paper by Tucker and Liang [13]. • The best approach for predicting the coefficient of thermal expansion of the unit uses the explicit treatment of Levin [14] which is identical to that of Christensen [15] and of Rosen and Hashin [16]. • The best measure of the fibre length is the number average fibre length [7]. • The best method for predicting the properties of a partially aligned fibre composite is to use a tensor averaging approach [3,17] and a constant strain prediction [5]. This is a more general approach than that developed previously (for example Cox [18]and Halpin [19,20]) where the composite is considered to consist of discrete layers of different fibre orientation and the stiffness constants of these layers is averaged using standard laminate analysis (e.g. [21]). Our recent work with Gusev [10] confirms that the constant strain prediction is valid for all orientation states and for all elastic and thermoelastic constants provided the volume fraction is greater than 5% and the fibre aspect ratio is greater than 10. The paper of Camacho and Tucker [17] is a particularly useful reference for this calculation as it details the tensor averaging for both elastic and thermoelastic properties.
For validation of the unit predictions, dumbbellshaped specimens were made by the University of Minho from a range of materials with different combinations of component isotropy: carbon fibre filled Nylon, pure liquid crystalline polymer (LCP), glass fibre filled LCP and carbon fibre filled LCP. These samples were considered appropriate because in the central shaft section, the fibres were measured to be very highly aligned, allowing the orientation averaging contribution to be minimised and the unit predictions to be assessed. Fully aligned samples could, however, not be produced experimentally, so direct assessment of the unit predictions was made by comparing with numerical finite element predictions of perfectly aligned samples made by the group of Gusev, ETH, Zurich. For all the composite materials, characterisation of their fibre orientation and fibre length distributions was carried out using an in-house developed image analysis system [22]. For an assessment of the orientation averaging scheme, a demonstrator ribbed box component was manufactured at the University of Bradford, from the carbon fibre filled LCP material (that is both phases anisotropic). The ribs of this component proved ideal structures for evaluating the proposed scheme, being convenient for mechanical measurement and also partially aligned in terms of the fibres. Mechanical measurements (bending modulus and coefficient of thermal expansion) were made on these ribs, and their orientation structure was measured using image analysis. Comparison of the measured properties (YoungÕs modulus and coefficient of thermal expansion) with the model predictions for the carbon/LCP samples (both phases anisotropic) showed that there were two possible choices: either assume the matrix had the same orientation as the fibres, or assume the matrix was randomly oriented. A randomly oriented LCP matrix can itself be considered to be composed of randomly arranged fully aligned polymeric units. The aggregate approach was therefore used to determine the properties of the randomly oriented LCP matrix, based on the properties of the fully aligned LCP material and using orientation averaging with the constant stress (Reuss) bound, which from previous work has been found to be the most appropriate choice in this situation [2]. 2. Injection moulded dumbbells 2.1. Experimental 2.1.1. Materials A range of injection moulded dumbbell samples was produced at The University of Minho, Portugal, by the group of Professor Cunha. Fig. 1 shows a picture of the dumbbell sample, which was composed of a cylindrical central shaft (20 mm long and 1.5 mm in diameter),
C.D. Price et al. / Composites Science and Technology 66 (2006) 69–79
71
Fig. 1. Injection moulded dumbbell sample.
gradually opening out to a 10 mm · 3.5 mm rectangular cross section and 10 mm square end tags. Samples were made from three filled materials of different combinations of isotropy and one unfilled material: details of these materials are given in Table 1. The samples were injection moulded using the processing conditions specified by the material manufacturers. 2.1.2. Measurement of the elastic properties of the unfilled matrix materials Prediction of the composite properties requires a knowledge of the unfilled matrix materials, namely Nylon and liquid crystalline polymer (LCP). For the isotropic aromatic Nylon polymer, only two elastic constants were required, the YoungÕs modulus E and the PoissonÕs ratio m. A sample of the unfilled aromatic Nylon was not available, so the pertinent values were taken from manufacturersÕ data sheets: 2 GPa and 0.4 for E and m, respectively. The LCP material when fully aligned is transversely isotropic, thus requiring five elastic constants to specify its elastic anisotropy, E11, E22, m21, m23 and G12, where the 1-axis is the direction of preferred orientation (the dumbbell axis). For modelling the composite ÔunitÕ, these properties should be those of a perfectly aligned matrix sample. Injection moulded dumbbell samples, produced from the unfilled pure LCP matrix (Hoechst Celanese grade B950), showed a very high degree of molecular orientation in the central shaft as measured by wide angle X-ray scattering (WAXS). For this reason the central shaft of these samples could be use to determine the five elastic constants of the LCP perfectly aligned unit. The axial modulus, E11, was measured in tension (up to a strain of 0.5%) using an RDP servo-mechanical test machine. The sample strain was measured using a Messphysik video extensometer and the tests were carried out at a nominal strain rate of 103 s1. To check these mea-
surements, three point bending tests were made and excellent agreement was found between these two methods: the determined value was 25.6 GPa. The transverse modulus, E22, was measured by the method of transverse compression [23] and gave a value of 1.45 GPa. The shear modulus, G12, was measured (on a separate piece of moulded LCP machined to size) using a rectangular torsion test on a Rheometric RDA2 torsion rheometer. An issue with rectangular torsion tests is the effect of the sample aspect ratio on the shear modulus. Three different aspect ratios (sample length/width) were measured (4, 6 and 12;1) and all gave a very similar value for the shear modulus of 1.13 GPa. Attempts were made to measure the PoissonÕs ratio of the samples using a tensile test and monitoring the transverse strain, but these results proved scattered. As an alternative, the values determined from a previous study on an LCP material, where an ultrasonic technique was employed [24], were used. It is considered that using an ultrasonic value is valid, as PoissonÕs ratio is less frequency-dependent than the other elastic constants, being given by ratios of compliances: the previously measured values were 0.48 for m21 and 0.71 for m23. 2.1.3. Measurement of the coefficient of thermal expansion of the unfilled matrix materials The value for the coefficient of thermal expansion, a, of the aromatic Nylon, was taken from the literature and was 70 · 106 K1. The anisotropic LCP polymer required two values, the axial coefficient of thermal expansion a1 and the transverse coefficient of thermal expansion a2. These were measured using a custom built facility [25], where the temperature was cycled between 15 and 25 C over a 25 min cycle. The average value at 20 C was calculated from the maximum slope of the plot of displacement versus temperature. The axial coefficient of thermal expansion was measured from 6 mm long pieces taken from the dumbbell shaft: these mea-
Table 1 Details of the materials used
1 2 3 4
Description
Grade
Fibre
Matrix
Weight fraction (%)
Carbon fibre filled Nylon Unfilled LCP Glass fibre filled LCP Carbon fibre filled LCP
Solvay IXEF 300/6/9019 Hoechst Celanese Ticona B950 Hoechst Celanese Ticona B130 Hoechst Celanese Ticona B230
Anisotropic – Isotropic Anisotropic
Isotropic Anisotropic Anisotropic Anisotropic
30 – 30 30
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surements gave a value of 5 · 106 K1. Initial experiments measuring the transverse coefficient of thermal expansion across the diameter of the central shaft section gave quite scattered results. Much more consistent results were obtained by measuring across the end tabs sections of the dumbbells. This was considered representative of the transverse expansion of the fully aligned polymer, as our previous work on both polymers and composites (e.g. [26]) has shown that the transverse elastic and thermoelastic properties are little affected by molecular orientation. This measurement gave a value of 80 · 106 K1. In summary, the elastic and thermoelastic properties of the matrix materials (at 20 C) are shown in Table 2. 2.1.4. The elastic properties of the fibre phase The elastic properties of the fibres were taken from a combination of published literature and manufacturerÕs data sheets. The values for isotropic ÔEÕ glass fibres are well known with quoted values in good agreement: the values used here were a YoungÕs modulus E of 72.5 GPa, a PoissonÕs ratio m of 0.2 and a coefficient of thermal expansion a of 5 · 106 K1. For the transversely isotropic carbon fibres the values, particularly the transverse constants, are more difficult to obtain. The filled LCP materials used a carbon fibre with a quoted axial modulus, E11, of 230 GPa and an axial coefficient of thermal expansion, a1, of 0.4 · 106 K1. Previous work by a number of authors, including Smith [27], has shown that there is a relationship between the axial YoungÕs modulus and the other elastic constants. Using the data of Smith, and others, the other elastic constants of the carbon fibre used in our materials could be estimated: this does, however, introduce some degree of uncertainty in the final predictions, in that these constants have to estimated and are not measured directly. The values for both reinforcements are shown in Table 3. 2.1.5. The fractions of the two phases The fractions of the phases in each composite were specified by the manufacturers but they were also checked by direct density measurement (and the known Table 2 Matrix elastic properties Aromatic Nylon
E m a
2 0.4 70
Data sheet Data sheet Data sheet
LCP (B950)
E11 E22 m21 m23 G12 a1 a2
25.6 1.45 0.48 0.71 1.13 5 80
Measured statically Measured statically Measured-ultrasonics Measured-ultrasonics Measured statically Measured statically Measured statically
E and G in GPa, a · 106 K1.
Table 3 Reinforcement elastic properties Glass fibres (E glass)
E m a
Carbon fibres
E11 E22 m21 m23 G12 a1 a2
72.5 0.2 5
Literature Literature Literature
230 14 0.26 0.38 17.5 0.4 26
Manufacturer Estimated Estimated Estimated Estimated Manufacturer Literature
E and G in GPa, a · 106 K1. Table 4 Fractions of the various materials
Carbon filled Nylon Glass filled LCP Carbon filled LCP
Grade
Weight fraction (%)
Volume fraction (%)
Solvay grade IXEF 300/6/9019 Ticona grade B130 Ticona grade B230
30
21
30 30
19 24
densities of the two phases) by density column and density bottle tests. The weight and volume fractions for the three materials are shown in Table 4. Density measurements were also used to check for internal voiding. Samples that were more than 0.5% below the average density (also usually accompanied by sink marks), were discarded. 2.1.6. Characterisation of the fibre orientation and length distributions of the composite materials The fibre orientation and length distributions of the injection moulded composite samples were measured using an in-house image analysis facility developed at the University of Leeds [22]. For fibre orientation investigations, the chosen method is one of optical reflection microscopy of polished 2D sections taken from the areas of interest of the composite. Each fibre that meets the 2D section is seen as an elliptical footprint (see Fig. 2(a)), and measuring the ellipticity of these images allows the two polar angles, h and /, that specify the orientation of each fibre to be determined: h is the angle the fibre makes with the sectioned surface normal (1) and / is the angle the fibre makes with the 2-axis when projected into the 23 plane (Fig. 2(b)). An XY stage allows a large area to be scanned, allowing a determination of fibre orientation over a large area. For further details of the system and the analysis routines see [28]. Fibre length characterisation is achieved by taking a chosen section of the sample, removing the matrix material (by placing into a furnace set at 450 C for 6 h) and then spreading the resulting fibres thinly on a glass plate. To aid dispersion of the fibre residue, a solution of water and a mild surfactant are also placed into the dish and
C.D. Price et al. / Composites Science and Technology 66 (2006) 69–79
73
of the dumbbell shaft. The measured values for the three injection moulded dumbbells are shown in Table 5. 2.2. Comparison between theoretical predictions and experimental measurements
Fig. 2. (a) A typical image frame. (b) The definition of h and /.
then left to dry. For the length determination, the fibres are viewed in transmitted light and image analysis routines are used to determine the length of each fibre shadow image. The glass fibre/LCP sample was measured in this way and gave a value of the fibre aspect ratio (based on the number average fibre length) of 25. The two carbon fibre filled materials were also measured in this way and both gave a very similar value for the fibre aspect ratio of 33. The fibre orientation structure of the central shaft of the injection moulded dumbbells was found, as might be expected, to be both highly aligned and transversely isotropic, that is no preferred / orientation. The fibre orientation could, therefore, be specified by only two orientation averages, hcos2hi and hcos4hi, where h is the angle between a fibre and the longitudinal (1) axis
2.2.1. Anisotropic fibre/isotropic matrix – carbon fibre filled Nylon As described earlier, the modelling strategy is to undertake the prediction of properties in two stages: first the prediction of the properties of the fully aligned ÔunitÕ and secondly the effects of misorientation using tensor averaging. For the carbon fibre filled Nylon, that is an anisotropic fibre in an isotropic matrix, the scheme previously validated in the former studies was used: the modification of Qui and Weng for the elastic properties of the unit: the treatment of Levin for the determination of the coefficients of thermal expansion of the unit: the aggregate model and constant strain bound for the composite properties. Using this procedure, theoretical predictions were determined based on the phase properties and the measured fibre microstructure detailed earlier. For comparison with these predictions, measurements were carried out on the composite samples of four properties: the axial modulus E11, the transverse modulus E22, the axial coefficient of thermal expansion a1 and the transverse coefficient of thermal expansion a2. As with the tests on the unfilled polymers, E11 was measured in bending and tension in the central shaft section, E22 by transverse compression of this cylindrical section, a1 by a measurement along the axial direction of the central shaft and a2 from a transverse measurement of the end tab section. Table 6 shows a comparison of these measured values (fifth column) with the theoretical predictions (fourth column). It is seen that the agreement is good, considering the uncertainty in the carbon fibre elastic constants. For interest, the table also shows the properties for various other material states. The second column shows the properties of a fully aligned composite calculated assuming the fibres are infinitely long, while the third column shows the properties again for a fully aligned composite, but where the fibres have the same aspect ratio as the measured material. It is of interest to note that for this combination of materials, the correction for finite length is more important than that for misalignment. Table 5 Measured fibre orientation averages of the various materials
Carbon filled Nylon Glass filled LCP Carbon filled LCP
Grade
hcos2hi
hcos4hi
Solvay grade IXEF 300/6/9019 Ticona grade B130 Ticona grade B230
0.856
0.830
0.904 0.929
0.865 0.870
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Table 6 A comparison of the measured and predicted properties for the carbon fibre/Nylon dumbbells Theoretical predictions
E11 E22 a1 a2
Measured for dumbbell
Fully aligned fibres infinite fibre length
Fully aligned fibres finite fibre length (aspect ratio = 33:1)
Actual fibre alignment finite fibre length (aspect ratio = 33:1)
49.9 3.24 1.92 80.7
31.1 3.22 3.62 80.0
26.2 4.38 6.09 65.4
2.2.2. Anisotropic matrix composites For modelling the glass/LCP and carbon/LCP materials, the modelling strategy is identical to that described above when the matrix is isotropic, apart from the requirement for the calculation of the Eshelby tensor for an anisotropic material, which has been determined by Mura [29]. The prediction of the coefficients of thermal expansion of the unit follows directly from Levin, which just requires the correct unit elastic constants and the coefficients of thermal expansion of the component phases. As an additional step in the validation, the analytical predictions obtained in this way (Mura and Qui; Weng and Levin) for the composite unit were compared with a finite element prediction determined using the ETH Zurich PALMYRA package, which in previous work has been found to give excellent agreement with experimentally measured values. This comparison is shown in Table 7. It is seen that the analytical predictions of the unit properties are very close to those obtained by FE calculations, confirming the usefulness of the analytical approach. Experimental measurements of the axial and transverse YoungÕs Modulus (E1 and E2) and the axial and transverse coefficients of thermal expansion (a1 and a2) were made on the glass fibre and carbon fibre LCP dumbbells for comparison with the theoretical predictions: the results of this comparison are shown in Tables 8 and 9, respectively. The aggregate modelling is identiTable 7 Comparison of prediction of properties of fully aligned unit based on the best analytical route and finite element calculation (PALMYRAETH Zurich) Glass/LCP
Carbon/LCP
Analytical
FE
Analytical
FE
E11 E22 m21 m23 G12
33.9 2.01 0.427 0.686 1.63
33.8 2.34 0.421 0.642 1.90
62.7 2.09 0.443 0.682 1.76
64.2 2.53 0.419 0.636 2.13
a1 a2
1.04 63.2
0.90 61.9
1.90 61.0
1.50 57.0
E in GPa, a · 106 K1.
23.9 3.6 5.5 60.0
cal to that for a fibre reinforced material with an isotropic matrix, although this makes the implicit assumption that the level of orientation in the two phases (fibres and matrix) is the same. Several points in this comparison are of interest. First, it is of interest to examine the magnitude of the corrections for fibre length and fibre misorientation in comparison with the results of the carbon fibre/aromatic Nylon dumbbell described earlier (Table 6). There appears to be a correlation between the degree of anisotropy between the two phases and the effect of the length correction. This is seen particularly for E11, which is most affected when the fibres are aligned, in that the difference is largest for carbon/Nylon (49.9 GPa down to 25.8 GPa) and only very small for glass fibres in LCP (34.5 GPa down to 33.9 GPa). The effect of the correction for fibre misorientation does depend on which property is considered: in general, it is observed as having a larger percentage effect on the transverse properties (E22 and a2) compared to the longitudinal properties. A second point of interest is that both of the filled LCP materials have axial coefficients of thermal expansion (both measured and predicted) close to zero, which could be a valuable material property. In general terms, the agreement between theoretical predictions and measurements is reasonable, but not as good as seen with the isotropic aromatic Nylon matrix (Table 6). One possible reason for this discrepancy is the limitation of the aggregate modelling strategy described above in that the tensor averaging approach assumes that the degree of orientation of the matrix and fibre phases, in the misaligned composite, is identical: of course this may not be exactly the case. The development of a model able to handle different levels of orientation of the two phases would require a substantial amount of further work, and would also require characterisation of the orientation of the LCP matrix phase using wide angle X-ray scattering (WAXS), both of which were outside the scope of the current research project. As will be seen from the results presented in the next section, this discrepancy increases as the level of preferred orientation decreases, and a new strategy is proposed for this situation.
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Table 8 A comparison of the measured and predicted properties for the glass fibre/LCP dumbbells Theoretical predictions
E11 E22 a1 a2
Measured for dumbbell
Fully aligned fibres infinite fibre length
Fully aligned fibres finite fibre length (25:1)
Actual fibre alignment finite fibre length (25:1)
34.5 2.00 0.78 63.0
33.9 2.01 1.04 63.2
29.1 2.86 1.20 55.0
28.0 1.95 0.43 60.0
E in GPa, a · 106 K1.
Table 9 A comparison of the measured and predicted properties for carbon fibre/ LCP dumbbells Theoretical predictions
E11 E22 a1 a2
Measured for dumbbell
Fully aligned fibres infinite fibre length
Fully aligned fibres finite fibre length (33:1)
Actual fibre alignment finite fibre length (33:1)
74.6 2.09 1.49 60.8
65.9 2.09 1.77 61.0
54.3 2.55 4.89 58.1
3. Model injection moulded ribbed box 3.1. Experimental 3.1.1. Component details The aim of this second part of the project was to further test the modelling scheme outlined above through the study of a more complicated component, in this case a ribbed box, using the material where both phases were anisotropic, (carbon fibres in LCP). The demonstrator component, of external dimensions 22 mm · 76 mm · 6 mm, was injection moulded at The University of Bradford, UK: a picture of this is shown in Fig. 3(a). The injection gates were located at the end of the component at positions between the ribs (between 1 and 2, and 2 and 3) to stop any jetting down the rib: the positions are shown by the arrows on Fig. 3(a). The processing conditions used were as specified by the manufacturer, and are shown in Table 10. The near zero coefficient of thermal expansion of this material, as measured in the previous section of the study, made the extraction of this component from the mould a real challenge, so that the machine had to be operated in manual mode and not automatically. 3.1.2. Measurement of fibre orientation The three main ribs on the component were chosen as prime candidates for structures on which the mechanical properties (YoungÕs modulus and coefficient of thermal expansion) could easily be measured experimentally. For this reason, it was decided to measure the fibre orientation at four positions along each of the three ribs, denoted A, B, C and D, located 10, 20, 30 and 40 mm
43.2 2.45 1.00 49.0
from the end face. Details of these positions are shown schematically on Fig. 3(b). 2D sections were taken across the rib at each chosen position: the 1-axis here is defined as parallel to the ribs, the 2-axis perpendicular to the ribs but in the plane of the component, and the 3-axis as out of the plane of the component: the sections across the ribs were therefore taken in the 23 plane. With fibre orientation data (FOD) there are a number of ways of presenting the results. A helpful method of visualising the data is to recreate the scanned area in terms of the elliptical images that were identified and fitted during analysis. Fig. 4(b) shows such a reconstruction, taken from Rib 2 at position C (location shown in Fig. 4(a)). The picture shows both the rib and the plate onto which it is connected: the junction between the rib and the plate is shown by the dotted line: the left and right boundaries of the diagram below the dotted line are the saw cuts severing the specimen from the rest of the box. It is seen that there is a region at the outer edges of the rib where the fibres are aligned preferentially with the flow direction (1-axis) and a central portion where the orientation is more in the 23 plane. In terms of producing a theoretical prediction of a rib, the most suitable measure is the average orientation over the scanned area. This is most appropriate as the experimental measurements were for the YoungÕs modulus and coefficient of thermal expansion in the 1 direction (i.e., averaged properties across the cross section). The orientation averages calculated in this way from the measured image analysis data showed that the fibre average orientation, when averaged over the whole section (rib plus plate underneath) indicated that the fibres were still predominantly aligned with
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Fig. 3. (a) Details of the injection moulded ribbed box (the arrows show the positions of the injection gates). (b) Details of image analysis section positions on the three ribs chosen for analysis: Positions A, B, C and D: 10, 20, 30 and 40 mm from the gate end, respectively.
respect to in the 1 direction, and the pattern in the three ribs was very similar. The level of preferred orientation increased significantly between positions A and B, and then was measured to be reasonably constant for positions B, C and D. In light of these measurements, it was decided to measure YoungÕs modulus and coefficient of thermal expansion for the three ribs between positions B and C, where the average value of orientation with respect to rib axis (1 direction) was hcos2h1i = 0.675 ± 0.029. In addition to the FOD, the prediction of mechanical properties also requires a measurement of the number average fibre length. This was carried out using the procedure described in an earlier section and gave an average fibre length of 252 lm, which together with an average measured diameter of 6.27 lm, gave an average fibre aspect ratio of 40.2. This is a higher value than was seen for the injection moulded dumbbells (33) and it is likely that the higher degree of fibre attrition seen in the moulded dumbbells is due to the much smaller Table 10 Processing conditions used for demonstrator ribbed box component Injection speed Shot size Screw speed Back pressure Cooling time Melt temperature Mould temperature
15 mm/s 29 mm 150 rpm 50 bar 10 s 290 C 90 C
mould cavities and channels in this mould, leading to significantly higher shear forces during processing. 3.1.3. Measurement of mechanical properties of the ribs From the image analysis measurements, the decision was made to measure the mechanical properties (YoungÕs modulus and coefficient of thermal expansion) of ribs taken from the component, between positions B and C on all three ribs, as the fibre orientation averages are reasonably consistent between these positions. The ribs were removed from the component, with the plate underneath still attached, therefore corresponding to the measured image analysis sections, and calculated orientation averages. The YoungÕs modulus was measured in flexure, and the rib/plate samples were tested with either the 3-axis vertical (upright) or the 2-axis vertical (on-side). As a result of the material being anisotropic, and the low value of the bending span to width ratio (L/W) which had to be used for testing these samples (particularly for the upright measurement), a correction had to be made to the results to correct for shear deformation. After correction, the values measured for E11 in flexure were 24.3 ± 1.7 GPa with the sample upright and 25.9 ± 1.7 GPa with the sample on its side. In view of the good agreement, these two values were combined to give an average value of E11 of 25 ± 2 GPa. The coefficient of thermal expansion of the component was measured with respect to both the 1 and 3 axes
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Table 11 A comparison of measured and theoretical predictions Measured
E11 (GPa) a1 (·106 K1) a3 (· 106 K1)
Fig. 4. (a) Location of analysis region as shown below. (b) Reconstructed image scan from Rib 2 at position C as shown above.
on samples cut from between positions B and C. The average values measured were (0.9 ± 1.8) · 106 K1 for a1 and (52 ± 3) · 106 K1 for a3. 3.2. Comparison between theoretical predictions and experimental measurements The final part of this study was to generate analytical predictions for the ribs at the chosen positions to compare with the experimental measurements, based on the modelling scheme developed for the injection moulded dumbbells. The properties of the carbon fibres and LCP matrix have already been reported earlier (Tables 1 and 2). Table 11 shows a comparison of the measured (column 2) and predicted (column 3) mechanical properties for Ribs 1, 2 and 3 at positions B and C. The agreement
25.0 ± 2.0 0.9 ± 1.8 52.0 ± 3.0
Best analytical Aligned matrix
Isotropic matrix
29.7 ± 1.7 5.02 ± 0.25 37.5 ± 1.9
24.2 ± 1.4 1.25 ± 0.1 31.9 ± 1.6
is reasonable considering that this composite material is the most challenging system to model, as both phases are oriented and could potentially have a different level of orientation: in addition there is uncertainty in some of the carbon fibre properties. Although at the current time an analytical model is not available for modelling a system where the orientation is different in the two phases (although an FE approach could model such a system) a second prediction can be made by assuming the matrix is randomly oriented, and therefore isotropic in terms of elastic properties. Previous studies (e.g. [30]) have shown that in the presence of fibres, the level of matrix orientation can be much lower than seen in a pure matrix sample for the same geometry. For this reason an assumption of a randomly oriented matrix could be a good one in the situation where the fibres are partially aligned. To carry out this prediction, the properties of the randomly oriented LCP matrix were first calculated, using the aggregate model again. In this case, previous work has shown [2] that the constant stress (Reuss) bound is the most appropriate, based on the fully oriented LCP properties (Table 1). The results of this calculation are shown in Table 12. Using these properties for the LCP matrix, the predictions of the properties of the rib were recalculated, and the results of this are shown in the final column of Table 11. The predictions of E11 and a1 are in better agreement with the measurements based on an isotropic matrix, whereas the prediction of the transverse coefficient of thermal expansion a2 is farther away. At the present time the reason for this final discrepancy is unknown. Two alternate strategies can therefore be proposed for predicting the properties of a system where both the fibres and matrix are anisotropic: (1) if the degree of fibre orientation is high (hcos2hi > 0.8) then a good prediction can be produced by assuming the orientation in the two phases is the same: (2) if the degree of fibre orientation is lower (hcos2hi < 0.8) then a good predic-
Table 12 Calculated properties for an ÔisotropicÕ LCP matrix LCP isotropic (Reuss bound) E (GPa) m a (·106 K1)
3.42 0.292 51.7
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Table 13 Modelling schemes for different combinations of fibre and matrix anisotropy Fibre
Matrix
Unit
Aggregate
Elastic Isotropic glass Anisotropic carbon Isotropic glass Anisotropic carbon
Isotropic Isotropic Nylon Anisotropic LCP Anisotropic LCP
Qui Qui Qui Qui
and and and and
Weng Weng Weng Weng
Eshelby Eshelby Mura Mura
tion can be produced by assuming the orientation of the matrix is isotropic.
4. Conclusions By combining careful microstructural characterisation, together with detailed mechanical measurements, an analytical modelling scheme, which is able to give good predictions for short fibre composites where both phases are anisotropic, has been validated. The chosen modelling scheme is as follows: • Use the modification of Qui and Weng (for an anisotropic fibre) to determine the elastic properties of the unit based on the number average fibre length. • If the matrix is anisotropic, determine the Eshelby tensor from the work of Mura. • Use the treatment of Levin to determine the coefficients of thermal expansion of the unit, based on the mechanical properties determined above. • Determine the properties of the misaligned composite using tensor averaging and a constant strain prediction. If both phases are isotropic this scheme gives excellent predictions. For anisotropic carbon fibres in an aromatic Nylon isotropic matrix the predictions were also excellent. For isotropic glass fibres in an anisotropic LCP matrix the predictions are reasonable. For anisotropic carbon fibres in an anisotropic LCP matrix the predictions are reasonable. Two suggestions have been made for the poorer predictions of the LCP filled materials: (1) the uncertainty in the carbon fibre elastic and thermoelastic constants and (2) the degree of orientation in the LCP matrix. As the predictions of the carbon fibre/Nylon material were in excellent agreement with measurements, this suggests that it is the unknown level of orientation in the LCP matrix that is the most likely cause of the discrepancy. At present there are only two possible choices, either assume the orientation is the same as the matrix, or assume that the LCP matrix is randomly oriented, giving isotropic properties. Interestingly which of these
Coefficient of thermal expansion
Elastic
Coefficient of thermal expansion
Christensen Christensen Christensen Christensen
Camacho + Tucker Camacho + Tucker Camacho + Tucker Camacho + Tucker
Camacho + Tucker Camacho + Tucker Camacho + Tucker Camacho + Tucker
to choose appears to depend on the level of fibre orientation. For high fibre alignment (the central shaft of the dumbbells) the former gives a better prediction while for lower fibre alignment (the ribbed component) the randomly oriented matrix gave a slightly better prediction. The suggested strategy is therefore if the average value of the principal second order orientation tensor is greater than 0.8, then assume fibre and matrix have the same orientation, and if the average value is less than 0.8, then assume the matrix is randomly oriented. The proposed modelling schemes, for different combinations of anisotropy, are finally summarised in Table 13. It is the conclusion of this work that this combination gives the best predictions of the elastic and thermoelastic properties available at present without recourse to finite element procedures.
Acknowledgements This project was undertaken with the financial support of Marconi UK and Moldflow. The authors thank Ticona for providing the raw composite materials. The authors also thank Andrei Gusev at ETH, Zurich, for carrying out the finite element predictions of the fully aligned carbon fibre filled LCP material.
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