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Flexural modulus of short fibre-reinforced polymers
CHAPTER 7
Flexural modulus of short fibre-reinforced polymers 7.1 Introduction Short fibre-reinforced polymer (SFRP) composites have many applications as engineering materials. And short fibre-reinforced thermoplastic composites are frequently processed by extrusion compounding and injection moulding techniques (Fu and Lauke, 1997,1998a,b; Ulrych et al., 1993; Takahashi and Choi, 1991; Gupta et al., 1989; Xia et al., 1995; Chin et al., 1988). Injection moulding-grade thermoplastic polymers are usually highviscosity materials and this results in considerable shear-induced fibre breakage during compounding and moulding. This finally results in a fibre length distribution (FLD). During the injection moulding process, progressive and continuous changes in fibre orientation throughout the moulded components take place. As a result, final SFRP composite parts display a fibre orientation distribution (FOD) (Xia et al., 1995; Chin et al., 1988; Hine et al., 1995; Fakirov and Fakirova, 1985; McGrath and Wille, 1995). The mechanical properties of SFRP composites depend not only on the properties and volume fractions of constituent materials but also on the FLD and FOD in the final injection moulded SFRP composite parts. In general the fibre content (volume fraction) and the FLD are regarded explicitly or implicitly as uniform throughout the whole thickness of the product of SFRP composites (Ulrych et al., 1993; Gupta et al., 1989; Xia et al., 1995; Chin et al., 1988: 13; Baily et al., 1989). Unlike the fibre content and the FLD, the FOD may not only be non-uniform but also vary through the thickness of the product of SFRP composites. The FOD can be classified into three major cases. The first case is the uniform fibre alignment in which fibres are distributed uniformly through the thickness. And one limiting case of it is the unidirectional fibre alignment. In injection moulded dumbbell-shaped specimens most fibres are aligned along the mould fill direction (Fu and Lauke, 1998b; Takahashi and Choi, 1991; Ramsteiner and Theysohn, 1979), thus the fibre alignment in these
Science and Engineering of Short Fibre-Reinforced Polymer Composites DOI: https://doi.org/10.1016/B978-0-08-102623-6.00007-X
© 2019 Elsevier Ltd. All rights reserved.
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specimens can be regarded approximately as the unidirectional case. Another limiting case is the random fibre alignment (O’Connell and Duckett, 1991) in which the fibres are distributed uniformly and randomly throughout the specimen. The second case is the continuous fibre alignment in which the fibre orientation changes continuously from the centre to the surface of the specimen. For example the fibre orientation angle with respect to the mould fill direction was observed to increase from the surface to the centre of the plaque specimens of short glass fibre (SGF)reinforced poly(ethylene terephthalate) composites (Fakirov and Fakirova, 1985); in injection moulded plaques of short carbon fibre-reinforced PEEK composites, the fibre orientation averages change continuously across the thickness of the plaque (O’Connell and Duckett, 1991). These can be roughly regarded as examples of the continuous case. In the third case, namely the skincore layered structure in fibre orientation, was also often observed in injection moulded plaque or disc samples (Xia et al., 1995; Karger-Kocsis, 1993; Spahr et al., 1990; Harmia and Friedrich, 1995). In the core layer the fibres have a distinct FOD and are distributed uniformly while in the skin layers the fibres have a different FOD and are also distributed uniformly. The flexural stiffness of an SFRP composite plate with a given fibre orientation and FLD is discussed in this chapter. The flexural modulus of a unidirectional SFRP composite will be evaluated first as this will serve as the basis for the latter cases. Second, we will estimate the flexural stiffness of the SFRP composite with a uniform FOD. Third, we will study the SFRP composite with a continuous FOD. Lastly, we will consider the SFRP composite with a layered structure in fibre orientation. The planar FOD is considered for estimating the flexural modulus of SFRP composites. When there is a three-dimensional FOD, the out-of-plane fibres can be projected onto one plane so that all fibres in SFRP composites lie in the same plane and thus we can perform an approximate estimation of the flexural modulus in the same way.
7.2 Flexural modulus of unidirectional short fibrereinforced polymer composite Here the unidirectional case is considered first as shown in Fig. 7.1, where the short fibres are aligned with an angle θ with the reference direction ‘1’. The flexural modulus of such an SFRP composite plate containing
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191
1
Fibre
θ
3
2
Figure 7.1 Schematic drawing of a unidirectional SFRP composite. P
L
Figure 7.2 Schematic drawing of three-point bend testing for an SFRP composite beam with a span length of L under the applied load, P, at the mid-span.
aligned short fibres is analysed first. The constitutive relations for bending of a plate of thickness h as shown in Fig. 7.2 are of the form 2 3 2 32 3 D11 D12 D16 k1 M1 4 M2 5 5 4 D12 D22 D26 54 k2 5 (7.1) M6 D16 D26 D66 k6 where M1, M2 and M6 are the resultant bending and twisting moments per unit width. k1, k2 and k6 are the bending and twisting curvatures of the plate. Dij are the plate bending stiffnesses and are of the form ð h=2 Dij 5 Qij z2 dz (7.2) 2h=2
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where z is the axis along the thickness direction and varies from 2 h/2 to h/2. Qij (i, j 5 1, 2, 6) are the tensile stiffness constants in the off-axis system (Tsai, 1980). Inverting the momentcurvature relation (Eq. 7.1) one can obtain the following relation in terms of flexural compliance: 2 3 2 32 3 d11 d12 d16 M1 k1 4 k2 5 5 4 d12 d22 d26 54 M2 5 (7.3) k6 d16 d26 d66 M6 where dij are flexural compliances which can be obtained from Dij. d11 is written as follows: 2 2 2 d11 5 ðD26 2 D22 D66 Þ=ðD16 D22 2 2D12 D16 D26 1 D11 D26 2 1 D12 D66 2 D11 D22 D66 Þ
(7.4)
Under simple pure bending of moment M, for example in a threepoint bend test (Fig. 7.2), when a load is applied at the mid-span and the longitudinal direction of the beam is assumed in the reference direction ‘1’ (see Fig. 7.1), for a beam with width b we have M1 5
M b
(7.5)
M2 5 M6 5 0
(7.6)
the resulting momentcurvature relation is k1 5 d11 M1 5
d11 M b
(7.7)
From elementary theory, the rigidity of beam is Eflex I 5
M b 5 k1 d11
(7.8)
where Eflex is the flexural modulus of the SFRP composite plate. I is the moment of inertia of the cross section of plate about the centroidal axis: I5
bh3 12
(7.9)
So Eflex 5
12 h3 d
11
5
1 I d11
(7.10)
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where I 5 I/b. Finally, we obtain the expression for the flexural modulus of the SFRP composite plate by Eflex 5
2 2 2 D16 D22 2 2D12 D16 D26 1 D11 D26 1 D12 D66 2 D11 D22 D66 (7.11) 2 I ðD26 2 D22 D66 Þ
where Dij can be expressed as D11 5 U1 I 1 U2 V1 1 U3 V2
(7.12)
D22 5 U1 I 2 U2 V1 1 U3 V2
(7.13)
D12 5 U4 I 2 U3 V2
(7.14)
D66 5 U5 I 2 U3 V2
(7.15)
1 D16 5 U2 V3 1 U3 V4 2
(7.16)
1 D26 5 U2 V3 2 U3 V4 2
(7.17)
where Vi can be expressed as ð h=2 V1 5 cos ð2θÞ z2 dz 5 I cos ð2θÞ 2h=2
V2 5
V3 5
V4 5
ð h=2 2h=2
ð h=2 2h=2
ð h=2 2h=2
(7.18)
cos ð4θÞ z2 dz 5 I cos ð4θÞ
(7.19)
sin ð2θÞ z2 dz 5 I sin ð2θÞ
(7.20)
sin ð4θÞ z2 dz 5 I sin ð4θÞ
(7.21)
and Ui are functions of the tensile stiffness constants (Qxx, Qyy, Qxy and Qss) in the on-axis system and can be expressed as (Tsai, 1980)
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1 U1 5 ð3Qxx 1 3Qyy 1 2Qxy 1 4Qss Þ 8
(7.22)
1 U2 5 ðQxx 2 Qyy Þ 2
(7.23)
1 U3 5 ðQxx 1 Qyy 2 2Qxy 2 4Qss Þ 8
(7.24)
1 U4 5 ðQxx 1 Qyy 1 6Qxy 2 4Qss Þ 8
(7.25)
1 U5 5 ðQxx 1 Qyy 2 2Qxy 1 4Qss Þ (7.26) 8 and Qxx, Qxy, Qyy and Qss can be expressed as (Fu and Lauke, 1998c,d) Qxx 5
Ec 1 2 νxνy
Qxy 5 ν y Qxx Qyy 5
Ecy 1 2 νxνy
Qss 5 G s
(7.27) (7.28) (7.29) (7.30)
where Ec, Ecy, Gs, ν x and ν y are longitudinal tensile modulus, transverse tensile modulus, shear modulus, longitudinal Poisson’s ratio and transverse Poisson’s ratio of a unidirectional short fibre composite with fibres of length l, respectively. They can be expressed as functions of the fibre modulus, Ef, matrix modulus, Em, shear modulus, Gm, of the matrix, Poisson’s ratio, ν f, of the fibre, Poisson’s ratio, ν m, of the matrix and fibre volume fraction, v (Fu and Lauke, 1998c,d). When an FLD exists in the unidirectional SFRP composite, Dij in Eqs (7.127.17) should be integrated to obtain the overall plate flexural stiffness matrix, Aij, so that ð lmax Aij 5 Dij f ðlÞdl i; j 5 1; 2; 6 (7.31) lmin
where f(l) has been defined in Chapter 3 Major factors affecting the performance of short fibre–reinforced polymers (see Eq (3.9)) as the fibre length
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probability density function, in which a and b are size and shape parameters, respectively. Consequently, the effective flexural modulus of SFRP composites can be evaluated using Eq. (7.11) by replacing Dij with Aij.
7.3 Short fibre-reinforced polymer composite with a uniform fibre orientation distribution When both FLD and FOD are in an SFRP composite and fibres are distributed uniformly through the whole thickness of the composite, Dij must be integrated to obtain the overall plate flexural stiffness matrix: ð θmax ð lmax Aij 5 Dij f ðlÞgðθÞdl dθ i; j 5 1; 2; 6 (7.32) θmin
lmin
We can use Eq. (7.11) to evaluate the flexural modulus of the SFRP composite by replacing Dij with Aij. Notice that when θ 5 θmin 5 θmax, the uniform case reduces to one limiting case, namely the unidirectional fibre alignment, and when g(θ) 5 2/π, the uniform FOD reduces to another limiting case, namely the 2D random fibre alignment. A comparison of the theoretical results with the existing experimental results for a random, in-plane, SGF-reinforced polypropylene composite (Thomason and Vlug, 1996) is given in Fig. 7.3 for various fibre lengths and various fibre weight fractions (Fu et al., 1999), where Ef 5 75 GPa, ν f 5 0.25, Gf 5 Ef/(2(1 1 ν f)) 5 30 GPa, Em 5 1.6 GPa, ν m 5 0.35, Gm 5Em/(2(1 1 ν m))50.59 GPa and rf 5 6.5 µm. For the transformation between the volume fraction and the weight fraction, the following densities of the fibres and matrix are used: the density of glass fibres 5 2.620 g/cm3, the density of polypropylene resin 5 0.905 g/cm3. Fig. 7.3 shows that the theoretical results predicted by the theory are consistent with the existing experimental results. These results show that the composite flexural modulus increases with the increase of fibre weight fraction (or volume fraction) (Fig. 7.3A). The composite flexural modulus also increases with increasing mean fibre length when it is small but becomes insensitive to the fibre length (or aspect ratio) when the aspect ratio is large enough (e.g. $ B100) (Fig. 7.3B). Fig. 7.4 shows the flexural modulus of composites based on acrylonitrile butadiene styrene (ABS) reinforced with SGF and glass beads (GBs) as a function of filler content (Hashemi, 2008). The figure shows that the flexural modulus of the two composite systems increases with increasing filler volume fraction. Moreover, SGF-reinforced composites possess a higher
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11 (A)
+ 0.1 0.8
9
+
3
4.5
6 12
l = 12 mm
7
5
3 mm
+
Composite flexural modulus (GPa)
Fibre length (mm)
3
0.8 mm
+ 0.1 mm
1 0
10
20
30
40
50
Fibre content (% w/w)
(B)
Composite flexural modulus (GPa)
Fibre content (% w/w) 7 10
20
30 40 v = 0.4
5
0.3 0.2
3
0.1
1 0
3
6
9
12
Fibre length (mm)
Figure 7.3 Comparison between the predicted results and the existing experimental results for a random, in-plane, short glass fibre-reinforced polypropylene composite. (A) composite flexural modulus, Eflex, versus fibre weight fraction, (B) Eflex versus fibre length. Adapted from Thomason, J.L., Vlug, M.A., 1996. Influence of fibre length and concentration on the properties of glass fibre-reinforced polypropylene: 1. Tensile and flexural modulus. Composites A 27 (6), 477484 and Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiberreinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
flexural modulus than GB-filled composites because SGF has an aspect ratio much larger than unity for GBs. Further, the glass fibre aspect ratio is far less than 100 (Hashemi, 2008), which is why the composite flexural modulus increases dramatically with increasing fibre volume fraction.
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Composite flexural modulus (GPa)
10 GF 8
GB
6
4
2
0 0
6
12
18
Glass volume fraction (%)
Figure 7.4 Composite flexural modulus based on acrylonitrile butadiene styrene (ABS) reinforced with short glass fibres and glass beads. Adapted from Hashemi, S., 2008. Tensile and flexural properties of injection-moulded short glass fibre and glass bead ABS composites in the presence of weldlines. J. Mater. Sci. 43 (2), 721731.
7.4 Short fibre-reinforced polymer composite with a continuous fibre orientation distribution Assume fibre orientation in an SFRP composite changes continuously from the centre to the surface. The fibre orientation angle is assumed to be the minimum at the surface (z 5 h/2) and the maximum at the centre (z 5 0) of the specimen. The relationships of Dij with Vi can be obtained and are the same as those (Eqs 7.127.17) for the unidirectional composite but Vi (i 5 14) is different because Qij varies with θ. Here we evaluate Vi first and get ðθ h=2 2 z gðθÞdθ (7.33) 5 h=2 θmin So h h z5 2 2 2 and dz 5 2
ðθ θmin
gðθÞdθ
h gðθÞdθ 2
(7.34)
(7.35)
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Therefore we obtain ðθ 2 ð h3 θmax V½1;2;3;4 5 cosð2θÞ; cosð4θÞ; sinð2θÞ; sinð4θÞ 12 gðθÞdθ gðθÞdθ 4 θmin θmin (7.36) Dij can be solved by inserting Eq. (7.36) into Eqs (7.127.17), and Aij can be obtained using Eq. (7.32). Finally, the flexural modulus of SFRP composites can be determined from Eq. (7.11) by replacing Dij with Aij. The following data of the parameters are used in the following except stated specially: Ef 5 82.7 GPa, Gf 5 27.6 GPa, ν f 5 0.22, Em 5 2.18 GPa, Gm 5 1.03 GPa, ν m 5 0.35, df 5 10 µm and v 5 0.30. The effect of mean fibre orientation angle, θmean, on the composite flexural modulus is shown in Fig. 7.5 for the cases of uniform and continuous FODs (Fu et al., 1999), where p 5 0.5 and various q. In the continuous case, the fibre orientation angle changes continuously from the surface (minimum) to the core (maximum). Fig. 7.5 shows that the composite flexural modulus decreases with increasing mean fibre orientation angle for the two cases. This indicates that the unidirectional SFRP composite containing short fibres having an angle of 0 degree with the reference direction ‘1’ 27
Composite flexural modulus (GPa)
24 Continuous FOD
21 18 15 12 9
Uniform FOD
6
lmean = 3.198 mm
3
lmean = 0.466 mm
0 0
9
18
27
36
45
Mean fibre orientation angle (degree)
Figure 7.5 Effect of mean fibre orientation angle on the flexural modulus of SFRP composites for the cases of uniform and continuous fibre alignments for lmean 5 3.198 mm (a 5 0.15, b 5 1.5) and lmean 5 0.466 mm (a 5 5, b 5 2.5). Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiberreinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
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corresponds to the maximum composite flexural modulus. The larger the deviation in the mean fibre orientation angle with the reference direction, the lower is the composite flexural modulus. Moreover, the flexural modulus for continuous FOD is larger than that of uniform FOD. This is because in the continuous case, the fibre orientation angle is assumed to change continuously from the sample surfaces (minimum: 0 degree) to the core (maximum: 90 degrees) and thus the short fibres having a small fibre orientation angle near the sample surfaces for the continuous case would make a bigger contribution to the composite flexural strength than that by the short fibres with a mean fibre orientation angle near the sample surfaces for the uniform case. This leads to a higher composite flexural modulus for the continuous case than for the uniform case. The continuous case is well known for natural bamboos that evolve for many years to withstand flexural forces from wind. The effect of mode fibre orientation angle on the flexural modulus of SFRP composites is presented in Fig. 7.6 for the cases of uniform and continuous fibre alignments (Fu et al., 1999), where lmean 5 3.198 mm (a 5 0.15, b 5 1.5), θmean 5 12.95 degrees (both p and q vary). Fig. 7.6 shows that the composite flexural modulus decreases when the mode fibre orientation angle increases for the two cases. And the influence of mode fibre orientation angle is marked for the continuous case which is small
Composite flexural modulus (GPa)
24 Continuous FOD 22
20
18
Uniform FOD
16
14 6
8
10
12
14
Mode fibre orientation angle (degree)
Figure 7.6 Effect of mode fibre orientation angle on the flexural modulus of SFRP composites for the cases of uniform and continuous fibre alignments. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
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for the uniform case. This is because the mean fibre orientation angle near the surface layers does not change for the uniform case but increases with the mode fibre orientation angle for the continuous case. For example within the one-quarter surface layers, θmean 5 0.8133 degrees for θmod 5 6.59 degrees (p 5 6 and q 5 8) while θmean 5 2.761 degrees for θmod 5 12.866 degrees (p 5 18 and q 5 336.3). The fibres near the surface layers play a more important part in determining the composite flexural modulus. Similar to the Young’s modulus of SFRP composites in the paper physics approach (PPA), the modified rule-of-mixtures is also employed for estimating the flexural modulus of SGF-reinforced polyoxymethylene composites by considering the fibre length and orientation factors (Hashemi et al., 1997). Thus we have Eflex 5 ηl ηθ Ef v 1 Em vm
(7.37)
where ηl and ηθ respectively are the fibre length and orientation factors for the composite flexural modulus. Their expressions are given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh ðηl=2Þ 2Gm ηl 5 1 2 where η 5 (7.38) 2 ηl=2 Ef rf lnðR=rf Þ where Gm is the shear modulus of the matrix, Ef is the Young’s modulus of the fibre and R is the mean separation of the fibres normal to their length. The fibre orientation factor, ηθ , has to be determined from experiments. Note that Eq. (7.38) is only an approximate expression for estimating the flexural modulus of SFRP composites since, strictly speaking, both ηl and ηθ are in general dependent on each other and cannot be separately evaluated.
7.5 Short fibre-reinforced polymer composite with a layered structure Fig. 7.7 shows where in the SFRP composite plate there are two skin layers and one core layer with the thickness hc. In the skin and core layers, the FODs are different. In the skin layers, there is one FOD and fibres are distributed uniformly while in the core layer, there is another FOD and fibres are also distributed uniformly. To derive the expression for the flexural modulus of the SFRP with a layered structure, we consider first the case of the fibres in the core layer
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Figure 7.7 SEM micrograph of the layered structure of an injection moulded 45 wt. % SGF-reinforced polyethyleneterephthalate (PET) composite. Adapted from Friedrich, K., 1998. Mesoscopic aspects of polymer composites: processing, structure and properties. J. Mater. Sci. 33(23), 55355556.
having one orientation angle, θ1, and those fibres in the skin layers having another orientation angle, θ2. Then we can get Vi 5 Vi1 1 Vi2
i5124
(7.39)
and 1 V½1;2;3;4 5 Ic ½cos 2θ1 ; cos 4θ1 ; sin 2θ1 ; sin 4θ1 2 5 ðI 2 Ic Þ ½cos 2θ2 ; cos 4θ2 ; sin 2θ2 ; sin 4θ2 V½1;2;3;4
(7.40) (7.41)
where Ic 5
h3c 12
(7.42)
Thus we have Dij 5 Dij1 1 Dij2
i; j 5 1; 2; 6
(7.43)
where Dij1 as a function of Ic and θ1, and Dij2 as a function of (I 2 Ic ) and θ2 are the same as Eqs (7.127.17). The integration of Dij should be carried out respectively for the core layer and the surface layers. Therefore we obtain ð θ1 max ð lmax ð θ2 max ð lmax 1 Aij 5 Dij f ðlÞgðθ1 Þdl dθ1 1 Dij2 f ðlÞgðθ2 Þdl dθ2 (7.44) θ1
min
lmin
θ2
min
lmin
Finally the flexural modulus of SFRP composites can be evaluated using Eq. (7.11) by replacing Dij with Aij.
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The effect of mean fibre orientation angle on the composite flexural modulus for the case of layered structure is displayed in Fig. 7.8 (Fu et al., 1999), where lmean 5 3.198 mm (a 5 0.15, b 5 1.5), p 5 0.6 and various q except at θmean 5 0 degree (p 5 0.5 and q 5 N) and at θmean 5 45 degrees (p 5 0.5 and q 5 0.5) when θmean varies and (A) θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the skin layers and (B) θmean 5 0 degree (p 5 0.5 and q 5 N) in the core layer. Other parameters are the same as in Fig. 7.5; this is also true for other figures except when stated otherwise. Fig. 7.8 reveals again that the composite flexural modulus decreases with increasing mean fibre orientation angle. Moreover, when the FOD changes only in the skin layers (see curve (B)), the mean fibre orientation angle has a larger influence on the composite flexural modulus than when the FOD changes only in the core layer (see curve (A)). This result indicates that fibres in the skin layers play a more important role in determining the composite flexural modulus than the fibres in the core layer. This can be used to explain the above observation in Fig. 7.5 in which the composite flexural modulus for the continuous case is higher than that for the uniform case. The reason for this is that for the continuous case there are more fibres of small fibre orientation angle than those for the uniform
Composite flexural modulus (GPa)
30
24
18
(a) (b)
12
6
0 0
9
18
27
36
45
Mean fibre orientation angle (degree)
Figure 7.8 Effect of mean fibre orientation angle on the flexural modulus of SFRP composites for the case of layered structure: (A) θmean varies only in the core layer; and (B) θmean varies only in the skin layers. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
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203
Composite flexural modulus (GPa)
20
18 (b) 16
14 (a) 12
10 6
8
10
12
Mode fibre orientation angle (degree)
Figure 7.9 Effect of mode fibre orientation angle on the flexural modulus of SFRP for the case of layered structure: (A) θmod varies only in the core layer; and (B) θmod varies only in the skin layers. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
case near their skin layers, and the fibres near the skin layers are more important for determining the composite flexural modulus. Fig. 7.9 shows the effect of mode fibre orientation angle on the flexural modulus of SFRP composites for the case of a layered structure (Fu et al., 1999), where a, b, p and q are the same as in Fig. 7.6 and (A) θmean 5 12.95 degrees (p and q vary) in the core layer and θmean 5 25.81 degrees (p 5 0.6 and q 5 2.0) in the skin layers, (B) θmean 5 25.81 degrees (p 5 0.6 and q 5 2.0) in the core layer and θmean 5 12.95 degrees (p and q vary) in the skin layers. It can be seen from Fig. 7.9 that the mode fibre orientation angle has a small influence on the composite flexural modulus. This is clear because we can assume the fibres in the core and skin layers are distributed uniformly as for the uniform case. The flexural modulus of SFRP composites versus the core layer thickness for the case of layered structure is shown in Fig. 7.10 (Fu et al., 1999), where lmean 5 3.198 mm (a 5 0.15, b 5 1.5), and (A) θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the core layer and θmean 5 0° (p 5 0.5 and q 5 N) in the skin layers and (B) θmean 5 45 degrees (p 5 0.5 and q 5 0.5) in the core layer and θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the skin layers where a smaller mean fibre orientation angle in the skin layers than that in the core layer is assumed since this is consistent with experimental
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Composite flexural modulus (GPa)
30
(a)
25
20 (b) 15
10
5 0
0.2
0.4
0.6
0.8
1
Normalised core thickness, (hc/h)
Figure 7.10 The flexural modulus of SFRP composites versus the core layer thickness for the case of layered structure: (A) θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the core layer and θmean 5 0 degree (p 5 0.5 and q 5 N) in the skin layers and (B) θmean 5 45 degrees (p 5 0.5 and q 5 0.5) in the core layer and θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the skin layers. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
observations (Xia et al., 1995; Friedrich, 1985,1998). Fig. 7.10 reveals that the composite flexural modulus decreases slowly with the increase of the core thickness when the core thickness is small (e.g. ,0.4 h), but it decreases markedly with increasing the normalised core thickness when the core is large (e.g. .0.4 h). Moreover, the composite flexural modulus of (A) is higher than that of (B) because the mean fibre orientation angles of (A) are smaller than those of (B). The effect of mean fibre length and hence mean fibre aspect ratio on the composite flexural modulus is shown in Fig. 7.11 (Fu et al., 1999), where θmean 5 12.95 degrees (p 5 0.6, q 5 8), and a 5 0.15 and various b for lmean $ 1.0 mm (or mean aspect ratio $ 100) and a 5 5 and different b for lmean , 1.0 mm (or mean aspect ratio , 100), and moreover, θmean 5 25.81 degrees (p 5 0.6 and q 5 2.0) in the skin layers for the case of layered structure. Fig. 7.11 shows that the flexural modulus of SFRP composites increases dramatically with increasing mean fibre length or mean aspect ratio when lmean # 1 mm (or mean aspect ratio # 100) but is insensitive to mean fibre length (or mean fibre aspect ratio) when lmean . 1 mm (or mean aspect ratio . 100). In addition, the composite flexural modulus for the case of continuous FOD is the highest since it
Flexural modulus of short fibre-reinforced polymers
205
Composite flexural modulus (GPa)
26 (a) 22
18
(c)
14
(c)
10
6 0
1
2 3 4 Mean fibre length (mm)
5
6
Figure 7.11 Effect of mean fibre length (or mean fibre aspect ratio) on the flexural modulus of SFRP for the cases of (A) continuous FOD, (B) uniform FOD and (C) layered structure. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
has more fibres of small orientation angle at the skin layers than other two cases while the flexural modulus for the layered structure case is the lowest as its mean fibre orientation angle (525.81 degrees) in the skin layers is assumed to be less than that (512.95 degrees) of the uniform case. The effect of mode fibre length and hence mode fibre aspect ratio on the composite flexural modulus is shown in Fig. 7.12 for the two cases of a large and a small mean fibre length (or mean fibre aspect ratio) (Fu et al., 1999), where the parameters are the same as in Fig. 7.11 except a and b. The composite flexural modulus increases slightly with the increase of mode fibre length or mode fibre aspect ratio when mean fibre length (or mean fibre aspect ratio) is large (see Fig. 7.12A). However, when lmean (or lmean/d) is small, the effect of mode fibre length (or mode fibre aspect ratio) on the composite flexural modulus is noticeable (see Fig. 7.12B) and the composite flexural modulus increases with increasing mode fibre length (or mode fibre aspect ratio). Xia et al. (1995) proposed a model for predicting the flexural modulus of SFRP composites in which the composite is treated as a sandwich beam. First, the elastic moduli for the skin and core layers are obtained. Then, the composite flexural modulus is obtained from composite beam theory. Depending on the fibre orientation of the skin layer, the elastic modulus of the skin layer is obtained by
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Composite flexural modulus (GPa)
(A) 30 lmean = 3.198 mm 25 Continuous FOD 20 Uniform FOD 15
Layered structure
10 0
1 2 Mode fibre length (mm)
3
Composite flexural modulus (GPa)
(B) 18 lmean = 0.2 mm 16 Continuous FOD 14 Uniform FOD
12
Layered structure
10 8 6 0
0.0 3
0.0 6 0.0 9 0.12 0.15 Mode fibre length (mm)
0.18
0.21
Figure 7.12 Effect of mode fibre length (or mode fibre aspect ratio) on the flexural modulus of SFRP for (A) a large mean fibre length with a value of 3.198 mm (or a large mean fibre aspect ratio with a value of 319.8); and (B) a small mean fibre length with a value of 0.2 mm (or a small mean fibre aspect ratio with a value of 20). Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned shortfiber-reinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
Ecs
5
ð lmax
E11 ðlÞf ðlÞdl
(7.45)
lmin
Similarly, based on the fibre orientation in the core layer, the elastic modulus, Ecc , of the core layer can be obtained using the modified
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Flexural modulus of short fibre-reinforced polymers
HalpinTsai equation and Eq. (7.45). Finally, the flexural modulus of the SFRP composites is obtained according to composite beam theory as Ec 5
Ecs Is 1 Ecc Ic I
(7.46)
where Is, Ic and I are, respectively, the moments of inertia of the cross section of the skin layer, the core layer and the injection moulded composite part, and are represented by 3 h hs ðhc 1hs Þ2 Is 5 b s 1 (7.47) 6 2 3 h Ic 5 b s 12
(7.48)
3 h I 5b 12
(7.49)
where hs and hc are, respectively, the thickness of the skin and core layers, and b and h are the width and thickness of the injection moulded composite part. The experimental results of the flexural modulus in the 90 degrees (flow direction) and 0 degree (transverse to resin flow direction) for the long fibre reinforced polyamide (PAL) and short fibre reinforced polyamide (PAS) materials as well as the calculated results using Eq. (7.46) are listed in Table 7.1 (Xia et al., 1995). PAL shows about the same value in the flexural modulus as those of PLS. And the theoretical results coincide with the experimental data. This is explained by the fibre orientation in the skin and Table 7.1 Comparison of the experimental and theoretical results for the flexural modulus of SFRP. Material Type
PAS PAL
90-Degree Direction (Main Flow Direction, MFD)
0-Degree Direction (Transverse to MFD)
Experimental
Theory
Experimental
Theory
7.533 7.221
7.703 7.713
4.402 4.107
5.041 4.940
Source: Adapted from Xia, M., Hamada, H., Maekawa, Z., 1995. Flexural stiffness of injection molded glass fiber reinforced thermoplastics. Int. Polym. Process 10 (1), 7481.
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Flexural modulus Eflex (GPa)
20 15 10 5 0 –5 0
10
20 30 40 Glass fibre volume fraction v (%)
50
60
Figure 7.13 Composite flexural modulus versus fibre content. Adapted from Thomason, J.L., 2005. The influence of fibre length and concentration on the properties of glass fibre-reinforced polypropylene. 6. The properties of injection moulded long fibre PP at high fibre content. Compos. Part A: Appl. Sci. Manu. 36 (7), 9951003.
core layers and the fibre orientation is regarded as the important factor influencing the flexural modulus of the moulded composites. Moreover, since the fibres in the surface layers are well oriented along the flow direction, the anisotropy of the flexural modulus occurs in both the moulded PAL and PAS composites. The transverse composite modulus is different from and lower than the longitudinal composite flexural modulus. Fig. 7.13 gives the results for the flexural modulus of injection moulded discontinuous long glass fibre reinforced polypropylene composites (Thomason, 2005). Note that the composite flexural modulus increases almost linearly with increasing glass fibre content. In general, the composite flexural modulus increases with increasing fibre content and increasing fibre length but decreasing fibre orientation angle. The mean fibre length decreases while the fibre orientation angle parallel to the flow direction increases with increasing fibre content (Thomason, 2005). Therefore, the effects of both fibre length and orientation are levelled out as fibre content increases. Consequently, the composite flexural modulus increases linearly with increase of fibre content.
7.6 Hybrid short fibre-reinforced polymer composite The flexural modulus of a hybrid SFRP composite can generally be predicted by the rule of hybrid mixtures (RoHM). For example the flexural
Flexural modulus of short fibre-reinforced polymers
209
modulus of the SCFSGFPES hybrid composite can be estimated using the RoHM: EH 5 EG VG 1 EC VC where EH is the flexural modulus of the hybrid composite, and EG and EC are the flexural moduli of SGFPES and SCFPES composites, respectively. VG and VC are the relative volume fractions of SCFs and SGFs in the hybrid filler, respectively, and VG 1 VC 5 1. Fig. 7.14 exhibits the result of the flexural modulus of the SCFSGFPES hybrid composites measured (Li et al., 2018). The result of the experiment is consistent with the value predicted by the RoHM, and no obvious positive or negative synergistic effect is present. This observation is similar to that of the Young’s modulus reported in Chapter 5, Tensile strength of short fibre-reinforced polymer composites. Therefore, the SGFSCF hybrid does not show any hybrid effect on the moduli of the hybrid SFRP composites.
Figure 7.14 Flexural modulus of SCFSGFPES hybrid composites with different SCF volume fraction in fillers. Error bars indicate standard differentiations, connected data lines indicate data trend while dashed straight line is obtained from the RoHM. Adapted from Li, Y.Q., Du, S.S., Liu, L.Y., Li, F., Liu, D.B., Zhao, Z.K., et al., 2018. Synergistic effects of short glass fiber/short carbon fiber hybrids on the mechanical properties of polyethersulfone composites, Polym. Compos. 39, doi:10.1002/pc.
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References Baily, R.S., Davies, M., Moore, D.R., 1989. Processing property characteristics for long glass fiber reinforced polyamide. Composites 20 (5), 453460. Chin, W.K., Liu, H.T., Lee, Y.D., 1988. Effects of fiber length and orientation distribution on the elastic modulus of short fiber reinforced thermoplastic. Polym. Compos. 9 (1), 2735. Fakirov, S., Fakirova, C., 1985. Direct determination of the orientation of short glass fibers in an injection-molded poly(ethylene terephthalate) system. Polym. Compos. 6 (1), 4146. Friedrich, K., 1985. Microstructural efficiency and fracture toughness of short fiber/thermoplastic matrix composites. Compos. Sci. Technol. 22 (1), 4374. Friedrich, K., 1998. Mesoscopic aspects of polymer composites: processing, structure and properties. J. Mater. Sci. 33 (23), 55355556. Fu, S.Y., Lauke, B., 1997. Analysis of mechanical properties of injection molded short glass fibre/calcite/ABS composites. J. Mater. Sci. Technol. 13 (5), 389396. Fu, S.Y., Lauke, B., 1998a. Characterization of tensile behaviour of hybrid short glass fiber/calcite particle/ABS composites. Compos. Part A: Appl. Sci. Manu. 29 (56), 575583. Fu, S.Y., Lauke, B., 1998b. Fracture resistance of unfilled and calcite-filled ABS composites reinforced by short glass fibers (SGF) under impact load. Compos. Part A: Appl. Sci. Manu. 29 (56), 631641. Fu, S.Y., Lauke, B., 1998c. The elastic modulus of misaligned short fibre reinforced polymers. Compos. Sci. Technol. 58 (34), 389400. Fu, S.Y., Lauke, B., 1998d. An analytical characterization of the anisotropy of the elastic modulus of misaligned short fibre reinforced polymers. Compos. Sci. Technol. 58 (12), 19611972. Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiberreinforced polymers. Compos. Sci. Technol. 59 (10), 15331542. Gupta, V.B., Mittal, R.K., Sharma, P.K., 1989. Some studies on glass fibre-reinforced polypropylene. Part I: reduction in fibre length during processing. Polym. Compos. 10 (1), 815. Harmia, T., Friedrich, K., 1995. Fracture toughness and failure mechanisms in unreinforced and long-glass-fiber-reinforced PA6/PP blends. Compos. Sci. Technol. 53 (4), 423430. Hashemi, S., 2008. Tensile and flexural properties of injection-moulded short glass fibre and glass bead ABS composites in the presence of weldlines. J. Mater. Sci. 43 (2), 721731. Hashemi, S., Elmes, P., Sandford, S., 1997. Hybrid effects on mechanical properties of polyoxymethylene. Polym. Eng. Sci. 37 (1), 4558. Hine, P.J., Davidson, N., Duckett, R.A., Ward, I.M., 1995. Measuring the fibre orientation and modelling the elastic properties of injection-molded long-fibre-reinforced nylon. Compos. Sci. Technol. 53 (2), 125131. Karger-Kocsis, J., 1993. Instrumented impact fracture and related failure behavior in shortand long-glass-fiber-reinforced polypropylene. Compos. Sci. Technol. 48 (14), 273283. Li, Y.Q., Du, S.S., Liu, L.Y., Li, F., Liu, D.B., Zhao, Z.K., et al., 2018. Synergistic effects of short glass fiber/short carbon fiber hybrids on the mechanical properties of polyethersulfone composites. Polym. Compos. 39. Available from: https://doi.org/ 10.1002/pc. McGrath, J.J., Wille, J.M., 1995. Determination of 3D fibre orientation distribution in thermoplastic injection molding. Compos. Sci. Technol. 53 (2), 133143.
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O’Connell, P.A., Duckett, R.A., 1991. Measurement of fiber orientation in short-fiberreinforced thermoplastics. Compos. Sci. Technol. 42 (4), 329347. Ramsteiner, F., Theysohn, R., 1979. Tensile and impact strengths of unidirectional, short fiber-reinforced thermoplastics. Composites 10 (2), 111119. Spahr, D.E., Friedrich, K., Schultz, J.M., Bailey, R.S., 1990. Microstructure and fracture behavior of short and long fiber-reinforced polypropylene. J. Mater. Sci. 25 (10), 44274439. Takahashi, K., Choi, N.S., 1991. Influence of fibre weight fraction on the failure mechanisms of poly(ethylene terephthalate) reinforced by short-glass fibres. J. Mater. Sci. 26 (17), 46484656. Thomason, J.L., 2005. The influence of fibre length and concentration on the properties of glass fibre-reinforced polypropylene. 6. The properties of injection moulded long fibre PP at high fibre content. Compos. Part A: Appl. Sci. Manu. 36 (7), 9951003. Thomason, J.L., Vlug, M.A., 1996. Influence of fibre length and concentration on the properties of glass fibre-reinforced polypropylene: 1. Tensile and flexural modulus. Composites A 27 (6), 477484. Tsai, S.W., 1980. Introduction to Composite Materials. Technomic Publishing Inc, Pennsylvania, LA. Ulrych, F., Sova, M., Vokrouhlecky, J., Turcic, B., 1993. Empirical relations of the mechanical properties of polyamide 6 reinforced with short glass fibers. Polym. Compos. 14 (3), 229237. Xia, M., Hamada, H., Maekawa, Z., 1995. Flexural stiffness of injection molded glass fiber reinforced thermoplastics. Int. Polym. Process 10 (1), 7481.
Flexural modulus of short fibre-reinforced polymers 7.1 Introduction Short fibre-reinforced polymer (SFRP) composites have many applications as engineering materials. And short fibre-reinforced thermoplastic composites are frequently processed by extrusion compounding and injection moulding techniques (Fu and Lauke, 1997,1998a,b; Ulrych et al., 1993; Takahashi and Choi, 1991; Gupta et al., 1989; Xia et al., 1995; Chin et al., 1988). Injection moulding-grade thermoplastic polymers are usually highviscosity materials and this results in considerable shear-induced fibre breakage during compounding and moulding. This finally results in a fibre length distribution (FLD). During the injection moulding process, progressive and continuous changes in fibre orientation throughout the moulded components take place. As a result, final SFRP composite parts display a fibre orientation distribution (FOD) (Xia et al., 1995; Chin et al., 1988; Hine et al., 1995; Fakirov and Fakirova, 1985; McGrath and Wille, 1995). The mechanical properties of SFRP composites depend not only on the properties and volume fractions of constituent materials but also on the FLD and FOD in the final injection moulded SFRP composite parts. In general the fibre content (volume fraction) and the FLD are regarded explicitly or implicitly as uniform throughout the whole thickness of the product of SFRP composites (Ulrych et al., 1993; Gupta et al., 1989; Xia et al., 1995; Chin et al., 1988: 13; Baily et al., 1989). Unlike the fibre content and the FLD, the FOD may not only be non-uniform but also vary through the thickness of the product of SFRP composites. The FOD can be classified into three major cases. The first case is the uniform fibre alignment in which fibres are distributed uniformly through the thickness. And one limiting case of it is the unidirectional fibre alignment. In injection moulded dumbbell-shaped specimens most fibres are aligned along the mould fill direction (Fu and Lauke, 1998b; Takahashi and Choi, 1991; Ramsteiner and Theysohn, 1979), thus the fibre alignment in these
Science and Engineering of Short Fibre-Reinforced Polymer Composites DOI: https://doi.org/10.1016/B978-0-08-102623-6.00007-X
© 2019 Elsevier Ltd. All rights reserved.
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specimens can be regarded approximately as the unidirectional case. Another limiting case is the random fibre alignment (O’Connell and Duckett, 1991) in which the fibres are distributed uniformly and randomly throughout the specimen. The second case is the continuous fibre alignment in which the fibre orientation changes continuously from the centre to the surface of the specimen. For example the fibre orientation angle with respect to the mould fill direction was observed to increase from the surface to the centre of the plaque specimens of short glass fibre (SGF)reinforced poly(ethylene terephthalate) composites (Fakirov and Fakirova, 1985); in injection moulded plaques of short carbon fibre-reinforced PEEK composites, the fibre orientation averages change continuously across the thickness of the plaque (O’Connell and Duckett, 1991). These can be roughly regarded as examples of the continuous case. In the third case, namely the skincore layered structure in fibre orientation, was also often observed in injection moulded plaque or disc samples (Xia et al., 1995; Karger-Kocsis, 1993; Spahr et al., 1990; Harmia and Friedrich, 1995). In the core layer the fibres have a distinct FOD and are distributed uniformly while in the skin layers the fibres have a different FOD and are also distributed uniformly. The flexural stiffness of an SFRP composite plate with a given fibre orientation and FLD is discussed in this chapter. The flexural modulus of a unidirectional SFRP composite will be evaluated first as this will serve as the basis for the latter cases. Second, we will estimate the flexural stiffness of the SFRP composite with a uniform FOD. Third, we will study the SFRP composite with a continuous FOD. Lastly, we will consider the SFRP composite with a layered structure in fibre orientation. The planar FOD is considered for estimating the flexural modulus of SFRP composites. When there is a three-dimensional FOD, the out-of-plane fibres can be projected onto one plane so that all fibres in SFRP composites lie in the same plane and thus we can perform an approximate estimation of the flexural modulus in the same way.
7.2 Flexural modulus of unidirectional short fibrereinforced polymer composite Here the unidirectional case is considered first as shown in Fig. 7.1, where the short fibres are aligned with an angle θ with the reference direction ‘1’. The flexural modulus of such an SFRP composite plate containing
Flexural modulus of short fibre-reinforced polymers
191
1
Fibre
θ
3
2
Figure 7.1 Schematic drawing of a unidirectional SFRP composite. P
L
Figure 7.2 Schematic drawing of three-point bend testing for an SFRP composite beam with a span length of L under the applied load, P, at the mid-span.
aligned short fibres is analysed first. The constitutive relations for bending of a plate of thickness h as shown in Fig. 7.2 are of the form 2 3 2 32 3 D11 D12 D16 k1 M1 4 M2 5 5 4 D12 D22 D26 54 k2 5 (7.1) M6 D16 D26 D66 k6 where M1, M2 and M6 are the resultant bending and twisting moments per unit width. k1, k2 and k6 are the bending and twisting curvatures of the plate. Dij are the plate bending stiffnesses and are of the form ð h=2 Dij 5 Qij z2 dz (7.2) 2h=2
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Science and Engineering of Short Fibre-Reinforced Polymer Composites
where z is the axis along the thickness direction and varies from 2 h/2 to h/2. Qij (i, j 5 1, 2, 6) are the tensile stiffness constants in the off-axis system (Tsai, 1980). Inverting the momentcurvature relation (Eq. 7.1) one can obtain the following relation in terms of flexural compliance: 2 3 2 32 3 d11 d12 d16 M1 k1 4 k2 5 5 4 d12 d22 d26 54 M2 5 (7.3) k6 d16 d26 d66 M6 where dij are flexural compliances which can be obtained from Dij. d11 is written as follows: 2 2 2 d11 5 ðD26 2 D22 D66 Þ=ðD16 D22 2 2D12 D16 D26 1 D11 D26 2 1 D12 D66 2 D11 D22 D66 Þ
(7.4)
Under simple pure bending of moment M, for example in a threepoint bend test (Fig. 7.2), when a load is applied at the mid-span and the longitudinal direction of the beam is assumed in the reference direction ‘1’ (see Fig. 7.1), for a beam with width b we have M1 5
M b
(7.5)
M2 5 M6 5 0
(7.6)
the resulting momentcurvature relation is k1 5 d11 M1 5
d11 M b
(7.7)
From elementary theory, the rigidity of beam is Eflex I 5
M b 5 k1 d11
(7.8)
where Eflex is the flexural modulus of the SFRP composite plate. I is the moment of inertia of the cross section of plate about the centroidal axis: I5
bh3 12
(7.9)
So Eflex 5
12 h3 d
11
5
1 I d11
(7.10)
Flexural modulus of short fibre-reinforced polymers
193
where I 5 I/b. Finally, we obtain the expression for the flexural modulus of the SFRP composite plate by Eflex 5
2 2 2 D16 D22 2 2D12 D16 D26 1 D11 D26 1 D12 D66 2 D11 D22 D66 (7.11) 2 I ðD26 2 D22 D66 Þ
where Dij can be expressed as D11 5 U1 I 1 U2 V1 1 U3 V2
(7.12)
D22 5 U1 I 2 U2 V1 1 U3 V2
(7.13)
D12 5 U4 I 2 U3 V2
(7.14)
D66 5 U5 I 2 U3 V2
(7.15)
1 D16 5 U2 V3 1 U3 V4 2
(7.16)
1 D26 5 U2 V3 2 U3 V4 2
(7.17)
where Vi can be expressed as ð h=2 V1 5 cos ð2θÞ z2 dz 5 I cos ð2θÞ 2h=2
V2 5
V3 5
V4 5
ð h=2 2h=2
ð h=2 2h=2
ð h=2 2h=2
(7.18)
cos ð4θÞ z2 dz 5 I cos ð4θÞ
(7.19)
sin ð2θÞ z2 dz 5 I sin ð2θÞ
(7.20)
sin ð4θÞ z2 dz 5 I sin ð4θÞ
(7.21)
and Ui are functions of the tensile stiffness constants (Qxx, Qyy, Qxy and Qss) in the on-axis system and can be expressed as (Tsai, 1980)
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Science and Engineering of Short Fibre-Reinforced Polymer Composites
1 U1 5 ð3Qxx 1 3Qyy 1 2Qxy 1 4Qss Þ 8
(7.22)
1 U2 5 ðQxx 2 Qyy Þ 2
(7.23)
1 U3 5 ðQxx 1 Qyy 2 2Qxy 2 4Qss Þ 8
(7.24)
1 U4 5 ðQxx 1 Qyy 1 6Qxy 2 4Qss Þ 8
(7.25)
1 U5 5 ðQxx 1 Qyy 2 2Qxy 1 4Qss Þ (7.26) 8 and Qxx, Qxy, Qyy and Qss can be expressed as (Fu and Lauke, 1998c,d) Qxx 5
Ec 1 2 νxνy
Qxy 5 ν y Qxx Qyy 5
Ecy 1 2 νxνy
Qss 5 G s
(7.27) (7.28) (7.29) (7.30)
where Ec, Ecy, Gs, ν x and ν y are longitudinal tensile modulus, transverse tensile modulus, shear modulus, longitudinal Poisson’s ratio and transverse Poisson’s ratio of a unidirectional short fibre composite with fibres of length l, respectively. They can be expressed as functions of the fibre modulus, Ef, matrix modulus, Em, shear modulus, Gm, of the matrix, Poisson’s ratio, ν f, of the fibre, Poisson’s ratio, ν m, of the matrix and fibre volume fraction, v (Fu and Lauke, 1998c,d). When an FLD exists in the unidirectional SFRP composite, Dij in Eqs (7.127.17) should be integrated to obtain the overall plate flexural stiffness matrix, Aij, so that ð lmax Aij 5 Dij f ðlÞdl i; j 5 1; 2; 6 (7.31) lmin
where f(l) has been defined in Chapter 3 Major factors affecting the performance of short fibre–reinforced polymers (see Eq (3.9)) as the fibre length
Flexural modulus of short fibre-reinforced polymers
195
probability density function, in which a and b are size and shape parameters, respectively. Consequently, the effective flexural modulus of SFRP composites can be evaluated using Eq. (7.11) by replacing Dij with Aij.
7.3 Short fibre-reinforced polymer composite with a uniform fibre orientation distribution When both FLD and FOD are in an SFRP composite and fibres are distributed uniformly through the whole thickness of the composite, Dij must be integrated to obtain the overall plate flexural stiffness matrix: ð θmax ð lmax Aij 5 Dij f ðlÞgðθÞdl dθ i; j 5 1; 2; 6 (7.32) θmin
lmin
We can use Eq. (7.11) to evaluate the flexural modulus of the SFRP composite by replacing Dij with Aij. Notice that when θ 5 θmin 5 θmax, the uniform case reduces to one limiting case, namely the unidirectional fibre alignment, and when g(θ) 5 2/π, the uniform FOD reduces to another limiting case, namely the 2D random fibre alignment. A comparison of the theoretical results with the existing experimental results for a random, in-plane, SGF-reinforced polypropylene composite (Thomason and Vlug, 1996) is given in Fig. 7.3 for various fibre lengths and various fibre weight fractions (Fu et al., 1999), where Ef 5 75 GPa, ν f 5 0.25, Gf 5 Ef/(2(1 1 ν f)) 5 30 GPa, Em 5 1.6 GPa, ν m 5 0.35, Gm 5Em/(2(1 1 ν m))50.59 GPa and rf 5 6.5 µm. For the transformation between the volume fraction and the weight fraction, the following densities of the fibres and matrix are used: the density of glass fibres 5 2.620 g/cm3, the density of polypropylene resin 5 0.905 g/cm3. Fig. 7.3 shows that the theoretical results predicted by the theory are consistent with the existing experimental results. These results show that the composite flexural modulus increases with the increase of fibre weight fraction (or volume fraction) (Fig. 7.3A). The composite flexural modulus also increases with increasing mean fibre length when it is small but becomes insensitive to the fibre length (or aspect ratio) when the aspect ratio is large enough (e.g. $ B100) (Fig. 7.3B). Fig. 7.4 shows the flexural modulus of composites based on acrylonitrile butadiene styrene (ABS) reinforced with SGF and glass beads (GBs) as a function of filler content (Hashemi, 2008). The figure shows that the flexural modulus of the two composite systems increases with increasing filler volume fraction. Moreover, SGF-reinforced composites possess a higher
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Science and Engineering of Short Fibre-Reinforced Polymer Composites
11 (A)
+ 0.1 0.8
9
+
3
4.5
6 12
l = 12 mm
7
5
3 mm
+
Composite flexural modulus (GPa)
Fibre length (mm)
3
0.8 mm
+ 0.1 mm
1 0
10
20
30
40
50
Fibre content (% w/w)
(B)
Composite flexural modulus (GPa)
Fibre content (% w/w) 7 10
20
30 40 v = 0.4
5
0.3 0.2
3
0.1
1 0
3
6
9
12
Fibre length (mm)
Figure 7.3 Comparison between the predicted results and the existing experimental results for a random, in-plane, short glass fibre-reinforced polypropylene composite. (A) composite flexural modulus, Eflex, versus fibre weight fraction, (B) Eflex versus fibre length. Adapted from Thomason, J.L., Vlug, M.A., 1996. Influence of fibre length and concentration on the properties of glass fibre-reinforced polypropylene: 1. Tensile and flexural modulus. Composites A 27 (6), 477484 and Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiberreinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
flexural modulus than GB-filled composites because SGF has an aspect ratio much larger than unity for GBs. Further, the glass fibre aspect ratio is far less than 100 (Hashemi, 2008), which is why the composite flexural modulus increases dramatically with increasing fibre volume fraction.
Flexural modulus of short fibre-reinforced polymers
197
Composite flexural modulus (GPa)
10 GF 8
GB
6
4
2
0 0
6
12
18
Glass volume fraction (%)
Figure 7.4 Composite flexural modulus based on acrylonitrile butadiene styrene (ABS) reinforced with short glass fibres and glass beads. Adapted from Hashemi, S., 2008. Tensile and flexural properties of injection-moulded short glass fibre and glass bead ABS composites in the presence of weldlines. J. Mater. Sci. 43 (2), 721731.
7.4 Short fibre-reinforced polymer composite with a continuous fibre orientation distribution Assume fibre orientation in an SFRP composite changes continuously from the centre to the surface. The fibre orientation angle is assumed to be the minimum at the surface (z 5 h/2) and the maximum at the centre (z 5 0) of the specimen. The relationships of Dij with Vi can be obtained and are the same as those (Eqs 7.127.17) for the unidirectional composite but Vi (i 5 14) is different because Qij varies with θ. Here we evaluate Vi first and get ðθ h=2 2 z gðθÞdθ (7.33) 5 h=2 θmin So h h z5 2 2 2 and dz 5 2
ðθ θmin
gðθÞdθ
h gðθÞdθ 2
(7.34)
(7.35)
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Therefore we obtain ðθ 2 ð h3 θmax V½1;2;3;4 5 cosð2θÞ; cosð4θÞ; sinð2θÞ; sinð4θÞ 12 gðθÞdθ gðθÞdθ 4 θmin θmin (7.36) Dij can be solved by inserting Eq. (7.36) into Eqs (7.127.17), and Aij can be obtained using Eq. (7.32). Finally, the flexural modulus of SFRP composites can be determined from Eq. (7.11) by replacing Dij with Aij. The following data of the parameters are used in the following except stated specially: Ef 5 82.7 GPa, Gf 5 27.6 GPa, ν f 5 0.22, Em 5 2.18 GPa, Gm 5 1.03 GPa, ν m 5 0.35, df 5 10 µm and v 5 0.30. The effect of mean fibre orientation angle, θmean, on the composite flexural modulus is shown in Fig. 7.5 for the cases of uniform and continuous FODs (Fu et al., 1999), where p 5 0.5 and various q. In the continuous case, the fibre orientation angle changes continuously from the surface (minimum) to the core (maximum). Fig. 7.5 shows that the composite flexural modulus decreases with increasing mean fibre orientation angle for the two cases. This indicates that the unidirectional SFRP composite containing short fibres having an angle of 0 degree with the reference direction ‘1’ 27
Composite flexural modulus (GPa)
24 Continuous FOD
21 18 15 12 9
Uniform FOD
6
lmean = 3.198 mm
3
lmean = 0.466 mm
0 0
9
18
27
36
45
Mean fibre orientation angle (degree)
Figure 7.5 Effect of mean fibre orientation angle on the flexural modulus of SFRP composites for the cases of uniform and continuous fibre alignments for lmean 5 3.198 mm (a 5 0.15, b 5 1.5) and lmean 5 0.466 mm (a 5 5, b 5 2.5). Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiberreinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
Flexural modulus of short fibre-reinforced polymers
199
corresponds to the maximum composite flexural modulus. The larger the deviation in the mean fibre orientation angle with the reference direction, the lower is the composite flexural modulus. Moreover, the flexural modulus for continuous FOD is larger than that of uniform FOD. This is because in the continuous case, the fibre orientation angle is assumed to change continuously from the sample surfaces (minimum: 0 degree) to the core (maximum: 90 degrees) and thus the short fibres having a small fibre orientation angle near the sample surfaces for the continuous case would make a bigger contribution to the composite flexural strength than that by the short fibres with a mean fibre orientation angle near the sample surfaces for the uniform case. This leads to a higher composite flexural modulus for the continuous case than for the uniform case. The continuous case is well known for natural bamboos that evolve for many years to withstand flexural forces from wind. The effect of mode fibre orientation angle on the flexural modulus of SFRP composites is presented in Fig. 7.6 for the cases of uniform and continuous fibre alignments (Fu et al., 1999), where lmean 5 3.198 mm (a 5 0.15, b 5 1.5), θmean 5 12.95 degrees (both p and q vary). Fig. 7.6 shows that the composite flexural modulus decreases when the mode fibre orientation angle increases for the two cases. And the influence of mode fibre orientation angle is marked for the continuous case which is small
Composite flexural modulus (GPa)
24 Continuous FOD 22
20
18
Uniform FOD
16
14 6
8
10
12
14
Mode fibre orientation angle (degree)
Figure 7.6 Effect of mode fibre orientation angle on the flexural modulus of SFRP composites for the cases of uniform and continuous fibre alignments. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
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for the uniform case. This is because the mean fibre orientation angle near the surface layers does not change for the uniform case but increases with the mode fibre orientation angle for the continuous case. For example within the one-quarter surface layers, θmean 5 0.8133 degrees for θmod 5 6.59 degrees (p 5 6 and q 5 8) while θmean 5 2.761 degrees for θmod 5 12.866 degrees (p 5 18 and q 5 336.3). The fibres near the surface layers play a more important part in determining the composite flexural modulus. Similar to the Young’s modulus of SFRP composites in the paper physics approach (PPA), the modified rule-of-mixtures is also employed for estimating the flexural modulus of SGF-reinforced polyoxymethylene composites by considering the fibre length and orientation factors (Hashemi et al., 1997). Thus we have Eflex 5 ηl ηθ Ef v 1 Em vm
(7.37)
where ηl and ηθ respectively are the fibre length and orientation factors for the composite flexural modulus. Their expressions are given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh ðηl=2Þ 2Gm ηl 5 1 2 where η 5 (7.38) 2 ηl=2 Ef rf lnðR=rf Þ where Gm is the shear modulus of the matrix, Ef is the Young’s modulus of the fibre and R is the mean separation of the fibres normal to their length. The fibre orientation factor, ηθ , has to be determined from experiments. Note that Eq. (7.38) is only an approximate expression for estimating the flexural modulus of SFRP composites since, strictly speaking, both ηl and ηθ are in general dependent on each other and cannot be separately evaluated.
7.5 Short fibre-reinforced polymer composite with a layered structure Fig. 7.7 shows where in the SFRP composite plate there are two skin layers and one core layer with the thickness hc. In the skin and core layers, the FODs are different. In the skin layers, there is one FOD and fibres are distributed uniformly while in the core layer, there is another FOD and fibres are also distributed uniformly. To derive the expression for the flexural modulus of the SFRP with a layered structure, we consider first the case of the fibres in the core layer
Flexural modulus of short fibre-reinforced polymers
201
Figure 7.7 SEM micrograph of the layered structure of an injection moulded 45 wt. % SGF-reinforced polyethyleneterephthalate (PET) composite. Adapted from Friedrich, K., 1998. Mesoscopic aspects of polymer composites: processing, structure and properties. J. Mater. Sci. 33(23), 55355556.
having one orientation angle, θ1, and those fibres in the skin layers having another orientation angle, θ2. Then we can get Vi 5 Vi1 1 Vi2
i5124
(7.39)
and 1 V½1;2;3;4 5 Ic ½cos 2θ1 ; cos 4θ1 ; sin 2θ1 ; sin 4θ1 2 5 ðI 2 Ic Þ ½cos 2θ2 ; cos 4θ2 ; sin 2θ2 ; sin 4θ2 V½1;2;3;4
(7.40) (7.41)
where Ic 5
h3c 12
(7.42)
Thus we have Dij 5 Dij1 1 Dij2
i; j 5 1; 2; 6
(7.43)
where Dij1 as a function of Ic and θ1, and Dij2 as a function of (I 2 Ic ) and θ2 are the same as Eqs (7.127.17). The integration of Dij should be carried out respectively for the core layer and the surface layers. Therefore we obtain ð θ1 max ð lmax ð θ2 max ð lmax 1 Aij 5 Dij f ðlÞgðθ1 Þdl dθ1 1 Dij2 f ðlÞgðθ2 Þdl dθ2 (7.44) θ1
min
lmin
θ2
min
lmin
Finally the flexural modulus of SFRP composites can be evaluated using Eq. (7.11) by replacing Dij with Aij.
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The effect of mean fibre orientation angle on the composite flexural modulus for the case of layered structure is displayed in Fig. 7.8 (Fu et al., 1999), where lmean 5 3.198 mm (a 5 0.15, b 5 1.5), p 5 0.6 and various q except at θmean 5 0 degree (p 5 0.5 and q 5 N) and at θmean 5 45 degrees (p 5 0.5 and q 5 0.5) when θmean varies and (A) θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the skin layers and (B) θmean 5 0 degree (p 5 0.5 and q 5 N) in the core layer. Other parameters are the same as in Fig. 7.5; this is also true for other figures except when stated otherwise. Fig. 7.8 reveals again that the composite flexural modulus decreases with increasing mean fibre orientation angle. Moreover, when the FOD changes only in the skin layers (see curve (B)), the mean fibre orientation angle has a larger influence on the composite flexural modulus than when the FOD changes only in the core layer (see curve (A)). This result indicates that fibres in the skin layers play a more important role in determining the composite flexural modulus than the fibres in the core layer. This can be used to explain the above observation in Fig. 7.5 in which the composite flexural modulus for the continuous case is higher than that for the uniform case. The reason for this is that for the continuous case there are more fibres of small fibre orientation angle than those for the uniform
Composite flexural modulus (GPa)
30
24
18
(a) (b)
12
6
0 0
9
18
27
36
45
Mean fibre orientation angle (degree)
Figure 7.8 Effect of mean fibre orientation angle on the flexural modulus of SFRP composites for the case of layered structure: (A) θmean varies only in the core layer; and (B) θmean varies only in the skin layers. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
Flexural modulus of short fibre-reinforced polymers
203
Composite flexural modulus (GPa)
20
18 (b) 16
14 (a) 12
10 6
8
10
12
Mode fibre orientation angle (degree)
Figure 7.9 Effect of mode fibre orientation angle on the flexural modulus of SFRP for the case of layered structure: (A) θmod varies only in the core layer; and (B) θmod varies only in the skin layers. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
case near their skin layers, and the fibres near the skin layers are more important for determining the composite flexural modulus. Fig. 7.9 shows the effect of mode fibre orientation angle on the flexural modulus of SFRP composites for the case of a layered structure (Fu et al., 1999), where a, b, p and q are the same as in Fig. 7.6 and (A) θmean 5 12.95 degrees (p and q vary) in the core layer and θmean 5 25.81 degrees (p 5 0.6 and q 5 2.0) in the skin layers, (B) θmean 5 25.81 degrees (p 5 0.6 and q 5 2.0) in the core layer and θmean 5 12.95 degrees (p and q vary) in the skin layers. It can be seen from Fig. 7.9 that the mode fibre orientation angle has a small influence on the composite flexural modulus. This is clear because we can assume the fibres in the core and skin layers are distributed uniformly as for the uniform case. The flexural modulus of SFRP composites versus the core layer thickness for the case of layered structure is shown in Fig. 7.10 (Fu et al., 1999), where lmean 5 3.198 mm (a 5 0.15, b 5 1.5), and (A) θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the core layer and θmean 5 0° (p 5 0.5 and q 5 N) in the skin layers and (B) θmean 5 45 degrees (p 5 0.5 and q 5 0.5) in the core layer and θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the skin layers where a smaller mean fibre orientation angle in the skin layers than that in the core layer is assumed since this is consistent with experimental
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Science and Engineering of Short Fibre-Reinforced Polymer Composites
Composite flexural modulus (GPa)
30
(a)
25
20 (b) 15
10
5 0
0.2
0.4
0.6
0.8
1
Normalised core thickness, (hc/h)
Figure 7.10 The flexural modulus of SFRP composites versus the core layer thickness for the case of layered structure: (A) θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the core layer and θmean 5 0 degree (p 5 0.5 and q 5 N) in the skin layers and (B) θmean 5 45 degrees (p 5 0.5 and q 5 0.5) in the core layer and θmean 5 12.95 degrees (p 5 0.6 and q 5 8) in the skin layers. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
observations (Xia et al., 1995; Friedrich, 1985,1998). Fig. 7.10 reveals that the composite flexural modulus decreases slowly with the increase of the core thickness when the core thickness is small (e.g. ,0.4 h), but it decreases markedly with increasing the normalised core thickness when the core is large (e.g. .0.4 h). Moreover, the composite flexural modulus of (A) is higher than that of (B) because the mean fibre orientation angles of (A) are smaller than those of (B). The effect of mean fibre length and hence mean fibre aspect ratio on the composite flexural modulus is shown in Fig. 7.11 (Fu et al., 1999), where θmean 5 12.95 degrees (p 5 0.6, q 5 8), and a 5 0.15 and various b for lmean $ 1.0 mm (or mean aspect ratio $ 100) and a 5 5 and different b for lmean , 1.0 mm (or mean aspect ratio , 100), and moreover, θmean 5 25.81 degrees (p 5 0.6 and q 5 2.0) in the skin layers for the case of layered structure. Fig. 7.11 shows that the flexural modulus of SFRP composites increases dramatically with increasing mean fibre length or mean aspect ratio when lmean # 1 mm (or mean aspect ratio # 100) but is insensitive to mean fibre length (or mean fibre aspect ratio) when lmean . 1 mm (or mean aspect ratio . 100). In addition, the composite flexural modulus for the case of continuous FOD is the highest since it
Flexural modulus of short fibre-reinforced polymers
205
Composite flexural modulus (GPa)
26 (a) 22
18
(c)
14
(c)
10
6 0
1
2 3 4 Mean fibre length (mm)
5
6
Figure 7.11 Effect of mean fibre length (or mean fibre aspect ratio) on the flexural modulus of SFRP for the cases of (A) continuous FOD, (B) uniform FOD and (C) layered structure. Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned short-fiber-reinforced polymers. Compos. Sci. Technol. 59(10), 15331542.
has more fibres of small orientation angle at the skin layers than other two cases while the flexural modulus for the layered structure case is the lowest as its mean fibre orientation angle (525.81 degrees) in the skin layers is assumed to be less than that (512.95 degrees) of the uniform case. The effect of mode fibre length and hence mode fibre aspect ratio on the composite flexural modulus is shown in Fig. 7.12 for the two cases of a large and a small mean fibre length (or mean fibre aspect ratio) (Fu et al., 1999), where the parameters are the same as in Fig. 7.11 except a and b. The composite flexural modulus increases slightly with the increase of mode fibre length or mode fibre aspect ratio when mean fibre length (or mean fibre aspect ratio) is large (see Fig. 7.12A). However, when lmean (or lmean/d) is small, the effect of mode fibre length (or mode fibre aspect ratio) on the composite flexural modulus is noticeable (see Fig. 7.12B) and the composite flexural modulus increases with increasing mode fibre length (or mode fibre aspect ratio). Xia et al. (1995) proposed a model for predicting the flexural modulus of SFRP composites in which the composite is treated as a sandwich beam. First, the elastic moduli for the skin and core layers are obtained. Then, the composite flexural modulus is obtained from composite beam theory. Depending on the fibre orientation of the skin layer, the elastic modulus of the skin layer is obtained by
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Science and Engineering of Short Fibre-Reinforced Polymer Composites
Composite flexural modulus (GPa)
(A) 30 lmean = 3.198 mm 25 Continuous FOD 20 Uniform FOD 15
Layered structure
10 0
1 2 Mode fibre length (mm)
3
Composite flexural modulus (GPa)
(B) 18 lmean = 0.2 mm 16 Continuous FOD 14 Uniform FOD
12
Layered structure
10 8 6 0
0.0 3
0.0 6 0.0 9 0.12 0.15 Mode fibre length (mm)
0.18
0.21
Figure 7.12 Effect of mode fibre length (or mode fibre aspect ratio) on the flexural modulus of SFRP for (A) a large mean fibre length with a value of 3.198 mm (or a large mean fibre aspect ratio with a value of 319.8); and (B) a small mean fibre length with a value of 0.2 mm (or a small mean fibre aspect ratio with a value of 20). Adapted from Fu, S.Y., Hu, X., Yue, C.Y., 1999. The flexural modulus of misaligned shortfiber-reinforced polymers. Compos. Sci. Technol. 59 (10), 15331542.
Ecs
5
ð lmax
E11 ðlÞf ðlÞdl
(7.45)
lmin
Similarly, based on the fibre orientation in the core layer, the elastic modulus, Ecc , of the core layer can be obtained using the modified
207
Flexural modulus of short fibre-reinforced polymers
HalpinTsai equation and Eq. (7.45). Finally, the flexural modulus of the SFRP composites is obtained according to composite beam theory as Ec 5
Ecs Is 1 Ecc Ic I
(7.46)
where Is, Ic and I are, respectively, the moments of inertia of the cross section of the skin layer, the core layer and the injection moulded composite part, and are represented by 3 h hs ðhc 1hs Þ2 Is 5 b s 1 (7.47) 6 2 3 h Ic 5 b s 12
(7.48)
3 h I 5b 12
(7.49)
where hs and hc are, respectively, the thickness of the skin and core layers, and b and h are the width and thickness of the injection moulded composite part. The experimental results of the flexural modulus in the 90 degrees (flow direction) and 0 degree (transverse to resin flow direction) for the long fibre reinforced polyamide (PAL) and short fibre reinforced polyamide (PAS) materials as well as the calculated results using Eq. (7.46) are listed in Table 7.1 (Xia et al., 1995). PAL shows about the same value in the flexural modulus as those of PLS. And the theoretical results coincide with the experimental data. This is explained by the fibre orientation in the skin and Table 7.1 Comparison of the experimental and theoretical results for the flexural modulus of SFRP. Material Type
PAS PAL
90-Degree Direction (Main Flow Direction, MFD)
0-Degree Direction (Transverse to MFD)
Experimental
Theory
Experimental
Theory
7.533 7.221
7.703 7.713
4.402 4.107
5.041 4.940
Source: Adapted from Xia, M., Hamada, H., Maekawa, Z., 1995. Flexural stiffness of injection molded glass fiber reinforced thermoplastics. Int. Polym. Process 10 (1), 7481.
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Science and Engineering of Short Fibre-Reinforced Polymer Composites
Flexural modulus Eflex (GPa)
20 15 10 5 0 –5 0
10
20 30 40 Glass fibre volume fraction v (%)
50
60
Figure 7.13 Composite flexural modulus versus fibre content. Adapted from Thomason, J.L., 2005. The influence of fibre length and concentration on the properties of glass fibre-reinforced polypropylene. 6. The properties of injection moulded long fibre PP at high fibre content. Compos. Part A: Appl. Sci. Manu. 36 (7), 9951003.
core layers and the fibre orientation is regarded as the important factor influencing the flexural modulus of the moulded composites. Moreover, since the fibres in the surface layers are well oriented along the flow direction, the anisotropy of the flexural modulus occurs in both the moulded PAL and PAS composites. The transverse composite modulus is different from and lower than the longitudinal composite flexural modulus. Fig. 7.13 gives the results for the flexural modulus of injection moulded discontinuous long glass fibre reinforced polypropylene composites (Thomason, 2005). Note that the composite flexural modulus increases almost linearly with increasing glass fibre content. In general, the composite flexural modulus increases with increasing fibre content and increasing fibre length but decreasing fibre orientation angle. The mean fibre length decreases while the fibre orientation angle parallel to the flow direction increases with increasing fibre content (Thomason, 2005). Therefore, the effects of both fibre length and orientation are levelled out as fibre content increases. Consequently, the composite flexural modulus increases linearly with increase of fibre content.
7.6 Hybrid short fibre-reinforced polymer composite The flexural modulus of a hybrid SFRP composite can generally be predicted by the rule of hybrid mixtures (RoHM). For example the flexural
Flexural modulus of short fibre-reinforced polymers
209
modulus of the SCFSGFPES hybrid composite can be estimated using the RoHM: EH 5 EG VG 1 EC VC where EH is the flexural modulus of the hybrid composite, and EG and EC are the flexural moduli of SGFPES and SCFPES composites, respectively. VG and VC are the relative volume fractions of SCFs and SGFs in the hybrid filler, respectively, and VG 1 VC 5 1. Fig. 7.14 exhibits the result of the flexural modulus of the SCFSGFPES hybrid composites measured (Li et al., 2018). The result of the experiment is consistent with the value predicted by the RoHM, and no obvious positive or negative synergistic effect is present. This observation is similar to that of the Young’s modulus reported in Chapter 5, Tensile strength of short fibre-reinforced polymer composites. Therefore, the SGFSCF hybrid does not show any hybrid effect on the moduli of the hybrid SFRP composites.
Figure 7.14 Flexural modulus of SCFSGFPES hybrid composites with different SCF volume fraction in fillers. Error bars indicate standard differentiations, connected data lines indicate data trend while dashed straight line is obtained from the RoHM. Adapted from Li, Y.Q., Du, S.S., Liu, L.Y., Li, F., Liu, D.B., Zhao, Z.K., et al., 2018. Synergistic effects of short glass fiber/short carbon fiber hybrids on the mechanical properties of polyethersulfone composites, Polym. Compos. 39, doi:10.1002/pc.
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