- Email: [email protected]
The elastic properties of carbon fibres and their composites
The elastic properties of carbon fibres and their composites G. D. DEAN and P. TURNER
Using a technique based upon the measurement of elastic wave velocity, values have been recorded for all the elastic stiffness components of a number of type 1, type 2 and Rolls Royce carbon fibre reinforced epoxy samples covering a range of fibre fraction. Sources of inaccuracy in the results have been considered. Analysis of the results using a theory interpreting composite properties in terms of phase properties, concentration and geometry has enabled a set of stiffness components to be derived for type 1 and type 2 carbon fibres.
The determination of the properties of a unidirectional fibre composite from the properties of its constituents may be considered as the initial phase in the structural design cycle. The analysis includes a calculation of a set of elastic moduli for the composite from a knowledge of the elastic moduli of the fibre and matrix phases, and a number of theories have been proposed 1 for predicting these moduli. The application of these theories, however, requires a knowledge of the fibre and matrix properties. Properties for the matrix may be determined bymeasurements on bulk material although the applicability of such values assumes the morphology of the matrix in the composite to be the same as that of the bulk material, an assumption that is apparently not valid in the vicinity of the fibre surface. 2 Fibre properties are less readily determined. Measurements applied to single fibres have been virtually restricted to the determination of the axial Young's and axial shear moduli. This data is sufficient for fibres assumed to be isotropic but not for carbon fibres where consideration of the microstructure implies the existence of high anisotropy. If carbon fibres are assumed to be transversely isotropic, five independent components of elastic modulus need to be measured before the elastic properties of the fibres may be defined. Thus, in addition to the axial Young's and shear moduli,/:a and Ga, the transverse Young's and shear moduli, E t and G t , and the axial Poisson ratio, Va, must be determined. Direct measurement of these latter components on individual fibres would be extremely difficult to perform owing to their small diameter. A set of compliance components for a single fibre, from which the above moduli may be deduced, has been obtained by Reynolds 3 using a model based upon an assumption of uniform stress in the crystallites of a fibre under load.
National Physical Laboratory,Teddington, MiddlesexTW11 OLW, England
174
The rotated compliance components for single crystal graphite are combined using crystallite orientation functions, obtained from an x-ray analysis on individual fibres, to give fibre properties. The significance of the derived fibre compliances is dependent upon the validity of the assumptions of the model which gives accurately only a lower bound to fibre properties and does not allow for porosity in the fibres. In addition, there is some uncertainty in the values that should be used for the compliance components of the graphite crystallite, especially in S44 which is sensitive to the concentration of dislocations in the crystallites. Ruland 4 has established an empirical relationship between fibre modulus and crystallite orientation which considers porosity but this has only been successful for the prediction of Young's modulus. In this report, the dependence of the elastic stiffness components of type 1 and type 2 carbon fibre composites upon fibre fraction has been studied. The measured data, together with a published theory predicting this dependence, are used to obtain a set of values for the elastic moduli of single type 1 and type 2 fibres. THE/VIA TERIA LS STUDIED
Most of the samples studied were supplied by Fothergill and Harvey Ltd and Morganite Ltd and contained uniaxially aligned Modmor type 1 and type 2 carbon fibres in Ciba LY558 epoxy resin. Sample sheets were supplied which nominally covered a range of fibre fraction between 0.3 and 0.7. Specimens were prepared from those areas of each sheet which appeared most uniform on a c-scan record. Specimen dimensions were typically 10 mm x 40 mm x 2.5 ram. The c-scan records indicated a slight variation in transmission properties over some sheets, and subsequent examination in an optical microscope of polished sections revealed the presence of voids which, however, were not quantified. In .the low fibre fraction material, some non,
COMPOSITES. JULY 1973
14'0
I
I
I
I
50
24-0,
12"0
J
22'0 c22
/
I0'0
/ 400
®
200
C33
20-0
180
% z8'O kO
~160-~ 300
-
nl u
c./
u
6"0
J,s
- -
z
L9
,~ 14.0
g
~ 120
20O
~o.o,
--
,,j
-,oo
80--
4"0
6,0
C 66
~
C44
- I00
C66
~ - 50
/
4.0
2-0
i
2' 0 I ~ 7 ~ - , - -
I 0"2
I
L
0"4 0"6 Fibre fraction
t
I 0"8
1.0
0
0.2
1 0.4 06 Fibre fraction
1
I.O
0"8
Fig.1 Variation of stiffness components of type 1 carbon fibre reinforced epoxy with fibre fraction
Fig,2 Variation of stiffness components of type 2 carbon fibre reinforced epoxy with fibre fraction
uniformity in fibre concentration with position was observed. Some specimens, supplied by Rolls Royce Ltd and containing Rolls Royce fibres in LY558 resin, were also studied but these covered a limited range of fibre fraction.
density. Similarly, c44 and c66 may be deduced from the velocity of shear waves polarised normal to the fibre axis and travelling along and transverse to the fibre axis respectively. The component c13 is obtained from velocity measurements at 45 ° to the fibre axis. Wave velocities are determined from measurements of the time difference between a zero crossover position of a short ultrasonic pulse (~0.5/lsec) which has travelled through water with and without the specimen in the beam. A more detailed description of the experimental arrangement and procedure and the interpretation of results is given in references 5 and 6.
EXPERIMEN TA TION
The elastic properties of each specimen were obtained using an ultrasonic pulse transmission technique. Detailed descriptions of the application of this technique to the determination of the elastic moduli of composite materials have been given. 5,6 The technique is based upon the measuremerit of the velocities of high frequency elastic waves propagating along known directions in the composite. For an arbitrary direction, the velocity is dependent upon the character of the wave, the material density and one or more elastic stiffness components. Assuming the composite symmetry to be hexagonal, five independent stiff ness components are then necessary to define the elastic anisotropy of the material. Taking the 3-axis parallel to the fibres, the 1 - 2 plane is then isotropic and the components e l l = c22, c33, c44 = c55, c13 = c23 and c66 = (ell - c12)/2 are the only non-zero terms in the elastic stiffness matrix. For a specimen with the assumed symmetry, the expressions for the velocity of elastic waves are greatly simplified, especially for propagation along symmetry axes. Thus, values for ¢33 and c 11 may be obtained from the velocity of longitudinal waves propagating along and transverse to the fibre axis respectively, by expressions of the form c = p V 2 where p is the material
C O M P O S I T E S . J U L Y 1973
RESUL -IS A N D ACCURACY
Measured elastic stiffness components for the specimens studied are recorded in Tables 1,2 and 3, and illustrated in Figs I, 2 and 3. The 2-axis is taken as the direction in the specimen along which the moulding pressure was applied. Fibre volume fractions were calculated from acid digestion 7 and density measurements. The following single fibre density values were used: Type 1 p = 1 9 8 0 k g / m 3 Type2 p = 1 780kg/m 3 RR p = 1 750 kg/m 3 The calculation of void content from these measurements is not accurate owing to the sensitivity of the calculation to the values chosen for fibre densities. The components of the elastic compliance matrix, sij, may be obtained by inverting the stiffness matrix. The moduli considered
175
Table 1. Elastic stiffness components for Wpe 1 carbon fibre reinforced epoxy resin.
Density kg/m 3
Fibre fraction, %
1514 1579 1592 1608 1684 1700 1731 1751 1771
37 46 48 50 60 63 67 69 72
c22
c33
c44
c66
Cl 2
c23
u13
P31
P12
9.6 9.9 10.3
9.8 10.15 10.25 9.7 10.6 10.2 10.5 10.90 10.8
154 197 218 212 261 264 277 284 299
3.4 4.05 4.35 4.2 5.0 5.5 5.6 6.2 6.7
2.1 2.2 2.2
5.5 5.6 5.9
0.5 0.29 0.23
0.02 0.01 0.01
0.55 0.56 0.57
2.3
5.9
2.4
6.1
7.6 4.6 3.8 9.0 5.6 9.9 9.5 5.4 7.2
0.34 0.61 0.58 0.32 0.43
0.01 0.02 0.02 0.01 0.01
0.56 0.54 0.54 0.56 0.55
c23
P13
v31
P12
5.8 6.7 5.5 4.8 5.2 9.6 4.9 6.1 7.3 8.3
0.34 0.35 0.29 0.24 0.25 0.48 0.23 0.28 0.33 0.37
0.03 0.03 0.02 0.02 0.02 0.03 0.02 0.02 0.02 0.03
0.53 0.51 0.51 0.50 0.51 0.46 0.50 0.49 0.49 0.47
11.05
Table 2. Elastic stiffness components for type 2 carbon fibre reinforced epoxy resin.
Fibre fraction, %
1432 1438 1494 1503 1559 1564 1577 1582 1611 1615 1634
35 36 47 48 58 59 62 63 68 69 72
Cll 11.5 11.5 12.7
14.4 14.6 15.4
c22
c33
c44
c66
c12
11.2 10.9 12.2 12.4 13.3 13.7 13.6 14.1 14.7 14.7 15.2
116 117 125 146 142 150 152 170 167 171
3.8 4.25 4.6 4.8 6.5 5.9 6.2 6.9 7.25 7.4 7.7
2.7 2.6 3.0
5.8 5.7 6.5
3.3
6.7
3.5 3.5
6.6 7.1
3.9
7.4
IO'O
earlier are related to stiffness and compliance components 1 by the fol,owing relationships.
CpqG N / m
I
2
I
I
~
1
®
®
8"O-
® ®
1
C33(Cll +C12)--2C~3
s33
6'0-
(la)
Ea-
•
Cll + Cl2
o
4.O ~ 1 Et-
2
Cl I
10.5
Density kg/m 3
CpqG N / m
(Cl1 -
c12)(CllC33
+c12c33 -2C~3 )
-
Sll
o
®
Type I
t
(lb)
2.0-
CllC33 - c23 ~E
Va
s13._
= P13-
S33
c13
(lc)
Cll + c12
--.. O Z IO'O
I
I
~
I
I
1
I
I
I
u
8"0 1 Ga -
-
c44
( 1 d)
S44 1 Gt -
S66
1 -
2(Sll - S12 )
c66 = -c12 ) 2 (cll
(le)
where the notation V a b implies the direction b as the direction o'f applied load. In the specimen, along no direction is the velocity of elastic waves related solely to the value of the component c13. The expression for wave velocities at an angle to the
176
o
4'0 -
l -
6"0
Type 2
2"0 O
O
I
I
I
I
02
0'4
0"6
O'8
Fibre fraction
Fig.3 Variation of c]3 GN/m2 with fibre fraction for type 1 and type 2 fibre specimens. Smooth lines give ~'13 = 0.35
COMPOSITES. JULY 1973
I'O
Table 3. Elastic stiffness components for Rolls Royce carbon fibre reinforced epoxy resin.
Density kg/m 3
Fibre fraction, %
1250 1549 1557 1557 1558 1563 1565 1577 1586
0 60 62 62 62 63 63.5 66 67.5
Cll
c22
c33
8.65
8.65 14.5 14.5 14.6 14.2 14.9 14.2 15.1 15.8
123 125 124 123 127 128 133 135
15.3
l
I
I
I
I
I
1
2
c44
c66
c12
c23
v13
v31
1)12
1.95 6.2
1.95 3.6
4.75 6.9
4.75 8.75
0.35 0.41
0.35 0.04
0.35 0.45
6.55
3.6
7.4
6.4
0.29
0.03
0.50
6.7
3.6
7.7
7.4
0.33
0.03
0.50
7.0 7.2
4.0 4.1
7.3 7.5
8.3 10.4
0.37 0.45
0.03 0.04
0.46 0.45
fibre axis contains c13 together with other stiffness components. Although at 45 ° to the fibre axis the expression is most sensitive to the magnitude of c 13, this sensitivity is low so that any reasonable accuracy in c13 can only be achieved if measurements o f velocity and the other stiffness components are made very accurately. Accordingly, the values used for the stiffness components in the calculation o f c l 3 for each specimen were obtained from the 'best fit' lines to all data o f Tables 1 and 2 shown in Figs 1 and 2. The scatter in values is shown in Fig.3. The low accuracy in the measurement o f c l 3 gives rise to the recorded scatter in the axial Poisson ratio,/213 , see (lc). It may be observed in Tables 1 and 2 that PI3 shows no marked dependence upon fibre fraction and has a value around 0.35 which is also independent of fibre type and equal to the matrix value. The smooth curves in Fig.3 pass through values for ('13 giving a value for/)13 of 0.35. Values for c22 I
CpqG N / m
were calculated which would yield these curves and are plotted in Fig.4. The derived c22 data shows only a small (less than experimental) scatter from expected values, illustrating that the c13 values lying on the smooth curves are reasonable. This analysis therefore confirms the insensitivity of the calculation for Cl3 to the magnitude of c 13. Whilst considering accuracy, it should be mentioned that one might expect experimental data to depart slightly from a smooth distribution with fibre fraction owing to the probable sensitivity of elastic properties to specimen structure which could differ from specimen to specimen even if the fibre fraction were constant. The other Poisson ratios are defined by:
V12 -
I
s12
c12c33
c~3
Sll
c11c33
c~3
s13
c13(c11 - c12 )
P31 Sll
12"O
c11c33
('~3
These components also show little dependence upon fibre fraction but differ slightly for the types of composite studied.
IO'O 8"0
from the smooth fit to c13 data
The stiffness components derived from measurements using the pulse technique relate to that part of the specimen through which the beam has travelled (approximately) 100 mm 2 cross section) so that, for inhomogeneous material, values for stiffness will depend upon the position of the beam in the specimen. The values listed in Tables 1, 2 and 3 are averages for a number of positions, but where significant scatter was recorded, the limits are indicated in Figs 1 and 2. The highest scatter was observed for the component c22. Although a variation in fibre packing arrangement within a specimen is likely to contribute to this scatter and to the difference between c I 1 and c22, the presence of defects, voids for example, is probably the major cause. Defects act as scattering centres which give rise to frequency dependent attenuation o f the pulse. Analysis of the spectrum of a pulse having travelled through a poor specimen (as indicated by a c-scan record) shows attenuation of the higher frequency components resulting in a change of pulse shape and a corresponding shift in zero cross-over positions. This shift gives rise to an error in recorded delay time and a reduction in the deduced value for elastic stiffness. The error will depend upon detect size, distribution and concentration and upon the
COMPOSITES. JULY 1973
177
Type I
6"0 4'0
(.9 a - o u
1
I
I
I
l
I
[
I
I
20"0 16'0 12'0 8"0
Type 2 4"0
0
0
0.2
0.4
0.6
0.8
Fibre fraction
Fig.4
Comparisonof experimentalc22 data with valuescalculated
I.O
bO
by eye. The solid lines are based upon theoretical expressions, derived by Hashin and Rosen, 8 which predict the dependence of composite properties upon the properties and concentration of the constituents• The theory is based upon a model to represent the composite which makes certain assumptions concerning the transverse phase geometry. The model considers the composite made up of parallel cylindrical units which are composed of a fibre surrounded by a coaxial column of matrix. In order to fill all space, the units are allowed to vary in diameter but are assumed to have a constant fibre fraction equal to that of the composite. In this way, the model approximates the random packing arrangement observed in real materials. Using this model, simple, closed-form expressions are derived for the moduli E a, v a, G a and k t which are exact for the model, k t is the transverse plane strain bulk modulus for the composite and is related to elastic stiffness components by:
c4c . c4 m c 4c + c4: d-6
0',~
,(+ 0.2
O
0"2
0.4
I 0.6
I 0-8
1 k t = ~(Cll +c12)=Cll
Fibre fraction
Fig.5 Variation of (c~4 --cr~44)/(cC, +cn444) and (k~-/~t)[ (k~ + cn444)with fibre fraction for type 1 fibre composites
direction of wave propagation. A calculation based upon ~ertain assumptions shows that, for the poor areas of the supplied sample sheets, an error (reduction) in transverse stiffness would be typically 5%. Since the specimens studied were cut from higher quality areas, the error in recorded values for c22 arising from defects is expected to be less than 5%. For propagation along the fibre axis, the velocity and hence delay time are greater than for propagation transverse to this direction, so that any error in c33 from defects is less important. However, elongation of the transmitted pulse was observed in some specimens, especially those of low fibre fraction, and this may be interpreted in terms of a variation in fibre concentration present in the specimen within the cross-section of the beam. In order to obtain a more homogeneous test specimen, its dimejasions transverse to the fibre axis were reduced to approximately 2.5 mm x 2.5 mm. The dependence of mode of propagation upon specimen dimensions had been previously studied by performing a series of measurements on a high quality specimen, moulded at the Royal Aircraft Establishment, as the specimen thickness was progressively reduced• Using a pulse frequency centred at 5 MHz, no significant variation in the magnitude of the components c 11, c22, c33 a n d C44 was observed down to a specimen thickness of 1.2 mm. Any comment concerning the change of mode arising along the fibre direction as the thickness is reduced would be inconclusive since the velocities of body waves and waves in the vicinity of the free surface differ by less than 2% in most of the specimens studied owing to their high anisotropy in stiffness.
CALCULA TION OF FIBRE PROPERTIES Figs 1 and 2 illustrate the variation of measured values for c22, c33, c44 and c66 with fraction of type 1 and type 2 fibres. The broken curve through data for c66 is fitted
178
c66
(2)
I.O An expression for the lower bound to the transverse shear modulus was also derived. Using the Hashin-Rosen expressions for E a, G a and k t, the following equations may be deduced:
c ~ 4 - c4//~ _ ( c f 4 -- c~4 ~
~kcf44+(-'~41 pf
cC44 + c ~ 4
t
"'t
_
V,( + c 41 where subscripts c, f, and m refer to the composite, the fibre and the matrix respectively, and v f is the fibre volume fraction. From these equations, it is apparent that plots of
4,, 4 ,
C~4 + C~4
kL L? k~+c~4
with fibre fraction should be linear. For the type 1 and type 2 fibre composite specimens, these plots are illustrated in Figs 1 2, 5 and 6 using the data for ,~thematrix, recorded in Table 3. The single fibre properties ~33' c~4 and k ft may then be deducted by extrapolating the data to a fibre fraction of unity. Since bounds for c66 have been derived by the theory and experimental data increases only slightly with fibre fraction,.~a value for the fibre transverse shear modulus, cJ 6, may be obtained with reasonable accuracy by extrapolating the data by eye as shown by the broken lines in Figs 1 and 2. The values for cf66 may then be used to obtain values for the fibre transverse stiffness, ell t = k ft + cf66 , and since k ft ~" cf66 the accuracy in c f 11 "~'ill not be critical to the accuracy in ~f'66The scatter in measured values for c~3 is too high to allow extrapolation. However, it was notedearlier that the composite axial Poisson ratio, v13, is essentially independent of fibre fraction and fibre type at a value of 0.35 + 0.05 which is the matrix value• Bearing in mind that the Hashin-
COMPOSITES . JU LY 1973
I-0
I
I
the deduced values for fibre transverse stiffness are reasonable, lower bounds to composite transverse stiffness were calculated from the fibre values using the Reuss average given by the expression:
I
O~
1 =~f+l-vf
c{2
c,:+ c,:
0-6
Composite values obtained from this expression were equal to or below the measured values. In fact, the Reuss values for the transverse stiffness of type 1 fibre composites coincide very closely with values derived using the HashinRosen expression and, for type 2 composites, Reuss values were slightly lower. The accuracy in the component c~3 is/elatively much lower and so the values quoted for cJ13 are as valid as the assumption that the axial Poisson ratiq of type 1 and type 2 composites is independent of fibre fraction and equal to the matrix value.
0,4
0"2
b¢~ O
I 0.2
I
I
0.4 06 Fibre fraction
0.8
I.O
Fig.6 Variation of (c~14- ~ ) / ( c C 4 + cn444) and (k~- kr~)/ {k~+ cn444)with fibre fraction for type 2 fibre composites
Rosen theory predicts a linear dependence of axial Poisson io with fibre fraction, we have assumed a fibre value of = 0.35 for type 1 and type 2 fibres. It is then possible om (1 c) to deduce values for cfl3. These five independent stiffness components for type 1 and type 2 fibres are listed in Table 4, together with compliance components deduced from them. Also shown in Table 4 is limited data on the Rolls Royce fibres obtained from Fig.7. DISCUSSION The reliability of the values given in Table 4 will depend upon the accuracy of the measurements made on the composite specimens and upon the validity of the theoretical expressions used to derive data for the fibre. The scatter in experimental data for c~3 and c,~4 is small and the theoretical curves fit the data closely. The scatter in values for c~2 is higher which introduces some uncertainty in the de.duced fibre values. An estimate of the uncertainty in c~2 may be obtained from Figs 3 and 4. To check whether
The manufacturers quoted values for fibre Young's modulus, obtained from tensile tests on individual fibres, vary from batch to batch but fall, typically, between 400 and 410 GN/m 2 for type 1 fibres and between 255 and 270 GN/m 2 for type 2 fibres. Since these values refer to large strain levels compared with the data derived from ultrasonic measurements, 400 GN/m 2 and 233 GN/m 2 respectively, it may be concluded that some non-linearity exists in the elastic behaviour of type 2 fibres whereas there is very little in the behaviour of type 1 fibres. This may be reasonable when consideration is taken of the high crystallite orientation in the latter. There is little published data on direct measurements of fibre axial shear modulus and a value of 24.8 GN/m 2 quoted for type 1 Harwell fibres 3 appears somewhat high. This value was used by Reynolds to assist in the assignment of compliance components for the graphite t:rystallite of type 1 fibres from which a set of fibre stiffness components was derived. 3 In this set, the components c 11 and c66 are also considerably higher than values quoted in Table 4. CONCL USION Data has been obtained illustrating the dependence of the elastic stiffness components of type 1 and type 2 carbon fibre composite materials upon fibre fraction. By choosing particular properties for the fibres, the expressions derived
Table 4. Elastic stiffness and compliance components for type 1 and type 2 carbon fibres.
Type 1 fibres
Type 2 fibres
CpqGN/m 2 Spqm2/GN
Cll = c22
c33
c44 = c55
c66
c13 = c23
c12
Sll = $22
$33
$44 = $55
$66
s13 = s23
$12
12.1
410
13.7
2.8
6.5
6.5
0.116
0.0025
0.073
0.357
-0.00086
-0.062
20.4
240
24
5.5
10.5
9.4
0.064
0.0043
0.042
0.182
-0.0015
-0.030
200
26
Rolls Royce fibres
COM POSITES . JU LY 1973
P13
~31
v12
0.35
0.01
0.53
0.35
0.02
0.45
179
1.0
200
C m c 4 4 - c44 c c44+c4~
0'5
"- I00 Z L9 u u
o Tensile
to calculate the elastic properties of a transversely isotropic, unidirectional composite of any fibre fraction once the properties of the matrix are known. Such information could be used for assessing the degree of orientation in material containing a distribution in fibre orientation. It is probable that the difference between the elastic properties of type l, type 2 and Rolls Royce fibres can only partly be explained in terms of crystallite orientation, h addition, the different heat treatments experienced by each fibre type will affect the fibre porosity and the properties of the crystallites, both being factors which also contribute to fibre properties. We can therefore expect fibre properties to vary slightly from batch to batch, although the extent of the variation ['or any particular modulus cannot, as yet, be assessed.
o Shear REFERENCES
0
0"2
0'4
1
1
0
0"6
0'8
I'O
Fibre f r a c t i o n Fig.7 Variation of longitudinal tensile and shear moduli with volume fraction of Rolls Royce fibres
by Hashin and Rosen have been found to closely fit experimental data. The fibres are highly anisotropic in elastic properties, the degree of anisotropy depending upon the heat treatment of the fibre. The derived elastic stiffness'values may be used with the Hashin-Rosen expressions
180
1 Chamis, C. C., Sendeckyj, G. P. J Camp Mat 2 (1968) 332 2 Schrager, M., Carey, J. Po(vmer l:'ngincering and Science ! 0 (1970) 369 3 Reynolds, W. N. Proceedings o f 3rd Con terence on Industrial Carbon and Graphite (April 1970) 427 4 Ruland, W. Applied Polymer Symposia, No 9 (1969) 293 5 Markham, M. F. Composites I (1970) 145 6 Dean, G. D., Lockett, F. J. Analysis of test methods for high modulus fibres and composites (April 1972). To be published in an ASTM Special Technical Publication 7 Crossland, B., Fennell, T. R. RAE Technical Memo MAT 105 (January 1971) 8 Hashin, Z., Rosen, B. W. Journal o f Applied Mechanics 31 (1964) 1
COMPOSITES . JULY
1973
Using a technique based upon the measurement of elastic wave velocity, values have been recorded for all the elastic stiffness components of a number of type 1, type 2 and Rolls Royce carbon fibre reinforced epoxy samples covering a range of fibre fraction. Sources of inaccuracy in the results have been considered. Analysis of the results using a theory interpreting composite properties in terms of phase properties, concentration and geometry has enabled a set of stiffness components to be derived for type 1 and type 2 carbon fibres.
The determination of the properties of a unidirectional fibre composite from the properties of its constituents may be considered as the initial phase in the structural design cycle. The analysis includes a calculation of a set of elastic moduli for the composite from a knowledge of the elastic moduli of the fibre and matrix phases, and a number of theories have been proposed 1 for predicting these moduli. The application of these theories, however, requires a knowledge of the fibre and matrix properties. Properties for the matrix may be determined bymeasurements on bulk material although the applicability of such values assumes the morphology of the matrix in the composite to be the same as that of the bulk material, an assumption that is apparently not valid in the vicinity of the fibre surface. 2 Fibre properties are less readily determined. Measurements applied to single fibres have been virtually restricted to the determination of the axial Young's and axial shear moduli. This data is sufficient for fibres assumed to be isotropic but not for carbon fibres where consideration of the microstructure implies the existence of high anisotropy. If carbon fibres are assumed to be transversely isotropic, five independent components of elastic modulus need to be measured before the elastic properties of the fibres may be defined. Thus, in addition to the axial Young's and shear moduli,/:a and Ga, the transverse Young's and shear moduli, E t and G t , and the axial Poisson ratio, Va, must be determined. Direct measurement of these latter components on individual fibres would be extremely difficult to perform owing to their small diameter. A set of compliance components for a single fibre, from which the above moduli may be deduced, has been obtained by Reynolds 3 using a model based upon an assumption of uniform stress in the crystallites of a fibre under load.
National Physical Laboratory,Teddington, MiddlesexTW11 OLW, England
174
The rotated compliance components for single crystal graphite are combined using crystallite orientation functions, obtained from an x-ray analysis on individual fibres, to give fibre properties. The significance of the derived fibre compliances is dependent upon the validity of the assumptions of the model which gives accurately only a lower bound to fibre properties and does not allow for porosity in the fibres. In addition, there is some uncertainty in the values that should be used for the compliance components of the graphite crystallite, especially in S44 which is sensitive to the concentration of dislocations in the crystallites. Ruland 4 has established an empirical relationship between fibre modulus and crystallite orientation which considers porosity but this has only been successful for the prediction of Young's modulus. In this report, the dependence of the elastic stiffness components of type 1 and type 2 carbon fibre composites upon fibre fraction has been studied. The measured data, together with a published theory predicting this dependence, are used to obtain a set of values for the elastic moduli of single type 1 and type 2 fibres. THE/VIA TERIA LS STUDIED
Most of the samples studied were supplied by Fothergill and Harvey Ltd and Morganite Ltd and contained uniaxially aligned Modmor type 1 and type 2 carbon fibres in Ciba LY558 epoxy resin. Sample sheets were supplied which nominally covered a range of fibre fraction between 0.3 and 0.7. Specimens were prepared from those areas of each sheet which appeared most uniform on a c-scan record. Specimen dimensions were typically 10 mm x 40 mm x 2.5 ram. The c-scan records indicated a slight variation in transmission properties over some sheets, and subsequent examination in an optical microscope of polished sections revealed the presence of voids which, however, were not quantified. In .the low fibre fraction material, some non,
COMPOSITES. JULY 1973
14'0
I
I
I
I
50
24-0,
12"0
J
22'0 c22
/
I0'0
/ 400
®
200
C33
20-0
180
% z8'O kO
~160-~ 300
-
nl u
c./
u
6"0
J,s
- -
z
L9
,~ 14.0
g
~ 120
20O
~o.o,
--
,,j
-,oo
80--
4"0
6,0
C 66
~
C44
- I00
C66
~ - 50
/
4.0
2-0
i
2' 0 I ~ 7 ~ - , - -
I 0"2
I
L
0"4 0"6 Fibre fraction
t
I 0"8
1.0
0
0.2
1 0.4 06 Fibre fraction
1
I.O
0"8
Fig.1 Variation of stiffness components of type 1 carbon fibre reinforced epoxy with fibre fraction
Fig,2 Variation of stiffness components of type 2 carbon fibre reinforced epoxy with fibre fraction
uniformity in fibre concentration with position was observed. Some specimens, supplied by Rolls Royce Ltd and containing Rolls Royce fibres in LY558 resin, were also studied but these covered a limited range of fibre fraction.
density. Similarly, c44 and c66 may be deduced from the velocity of shear waves polarised normal to the fibre axis and travelling along and transverse to the fibre axis respectively. The component c13 is obtained from velocity measurements at 45 ° to the fibre axis. Wave velocities are determined from measurements of the time difference between a zero crossover position of a short ultrasonic pulse (~0.5/lsec) which has travelled through water with and without the specimen in the beam. A more detailed description of the experimental arrangement and procedure and the interpretation of results is given in references 5 and 6.
EXPERIMEN TA TION
The elastic properties of each specimen were obtained using an ultrasonic pulse transmission technique. Detailed descriptions of the application of this technique to the determination of the elastic moduli of composite materials have been given. 5,6 The technique is based upon the measuremerit of the velocities of high frequency elastic waves propagating along known directions in the composite. For an arbitrary direction, the velocity is dependent upon the character of the wave, the material density and one or more elastic stiffness components. Assuming the composite symmetry to be hexagonal, five independent stiff ness components are then necessary to define the elastic anisotropy of the material. Taking the 3-axis parallel to the fibres, the 1 - 2 plane is then isotropic and the components e l l = c22, c33, c44 = c55, c13 = c23 and c66 = (ell - c12)/2 are the only non-zero terms in the elastic stiffness matrix. For a specimen with the assumed symmetry, the expressions for the velocity of elastic waves are greatly simplified, especially for propagation along symmetry axes. Thus, values for ¢33 and c 11 may be obtained from the velocity of longitudinal waves propagating along and transverse to the fibre axis respectively, by expressions of the form c = p V 2 where p is the material
C O M P O S I T E S . J U L Y 1973
RESUL -IS A N D ACCURACY
Measured elastic stiffness components for the specimens studied are recorded in Tables 1,2 and 3, and illustrated in Figs I, 2 and 3. The 2-axis is taken as the direction in the specimen along which the moulding pressure was applied. Fibre volume fractions were calculated from acid digestion 7 and density measurements. The following single fibre density values were used: Type 1 p = 1 9 8 0 k g / m 3 Type2 p = 1 780kg/m 3 RR p = 1 750 kg/m 3 The calculation of void content from these measurements is not accurate owing to the sensitivity of the calculation to the values chosen for fibre densities. The components of the elastic compliance matrix, sij, may be obtained by inverting the stiffness matrix. The moduli considered
175
Table 1. Elastic stiffness components for Wpe 1 carbon fibre reinforced epoxy resin.
Density kg/m 3
Fibre fraction, %
1514 1579 1592 1608 1684 1700 1731 1751 1771
37 46 48 50 60 63 67 69 72
c22
c33
c44
c66
Cl 2
c23
u13
P31
P12
9.6 9.9 10.3
9.8 10.15 10.25 9.7 10.6 10.2 10.5 10.90 10.8
154 197 218 212 261 264 277 284 299
3.4 4.05 4.35 4.2 5.0 5.5 5.6 6.2 6.7
2.1 2.2 2.2
5.5 5.6 5.9
0.5 0.29 0.23
0.02 0.01 0.01
0.55 0.56 0.57
2.3
5.9
2.4
6.1
7.6 4.6 3.8 9.0 5.6 9.9 9.5 5.4 7.2
0.34 0.61 0.58 0.32 0.43
0.01 0.02 0.02 0.01 0.01
0.56 0.54 0.54 0.56 0.55
c23
P13
v31
P12
5.8 6.7 5.5 4.8 5.2 9.6 4.9 6.1 7.3 8.3
0.34 0.35 0.29 0.24 0.25 0.48 0.23 0.28 0.33 0.37
0.03 0.03 0.02 0.02 0.02 0.03 0.02 0.02 0.02 0.03
0.53 0.51 0.51 0.50 0.51 0.46 0.50 0.49 0.49 0.47
11.05
Table 2. Elastic stiffness components for type 2 carbon fibre reinforced epoxy resin.
Fibre fraction, %
1432 1438 1494 1503 1559 1564 1577 1582 1611 1615 1634
35 36 47 48 58 59 62 63 68 69 72
Cll 11.5 11.5 12.7
14.4 14.6 15.4
c22
c33
c44
c66
c12
11.2 10.9 12.2 12.4 13.3 13.7 13.6 14.1 14.7 14.7 15.2
116 117 125 146 142 150 152 170 167 171
3.8 4.25 4.6 4.8 6.5 5.9 6.2 6.9 7.25 7.4 7.7
2.7 2.6 3.0
5.8 5.7 6.5
3.3
6.7
3.5 3.5
6.6 7.1
3.9
7.4
IO'O
earlier are related to stiffness and compliance components 1 by the fol,owing relationships.
CpqG N / m
I
2
I
I
~
1
®
®
8"O-
® ®
1
C33(Cll +C12)--2C~3
s33
6'0-
(la)
Ea-
•
Cll + Cl2
o
4.O ~ 1 Et-
2
Cl I
10.5
Density kg/m 3
CpqG N / m
(Cl1 -
c12)(CllC33
+c12c33 -2C~3 )
-
Sll
o
®
Type I
t
(lb)
2.0-
CllC33 - c23 ~E
Va
s13._
= P13-
S33
c13
(lc)
Cll + c12
--.. O Z IO'O
I
I
~
I
I
1
I
I
I
u
8"0 1 Ga -
-
c44
( 1 d)
S44 1 Gt -
S66
1 -
2(Sll - S12 )
c66 = -c12 ) 2 (cll
(le)
where the notation V a b implies the direction b as the direction o'f applied load. In the specimen, along no direction is the velocity of elastic waves related solely to the value of the component c13. The expression for wave velocities at an angle to the
176
o
4'0 -
l -
6"0
Type 2
2"0 O
O
I
I
I
I
02
0'4
0"6
O'8
Fibre fraction
Fig.3 Variation of c]3 GN/m2 with fibre fraction for type 1 and type 2 fibre specimens. Smooth lines give ~'13 = 0.35
COMPOSITES. JULY 1973
I'O
Table 3. Elastic stiffness components for Rolls Royce carbon fibre reinforced epoxy resin.
Density kg/m 3
Fibre fraction, %
1250 1549 1557 1557 1558 1563 1565 1577 1586
0 60 62 62 62 63 63.5 66 67.5
Cll
c22
c33
8.65
8.65 14.5 14.5 14.6 14.2 14.9 14.2 15.1 15.8
123 125 124 123 127 128 133 135
15.3
l
I
I
I
I
I
1
2
c44
c66
c12
c23
v13
v31
1)12
1.95 6.2
1.95 3.6
4.75 6.9
4.75 8.75
0.35 0.41
0.35 0.04
0.35 0.45
6.55
3.6
7.4
6.4
0.29
0.03
0.50
6.7
3.6
7.7
7.4
0.33
0.03
0.50
7.0 7.2
4.0 4.1
7.3 7.5
8.3 10.4
0.37 0.45
0.03 0.04
0.46 0.45
fibre axis contains c13 together with other stiffness components. Although at 45 ° to the fibre axis the expression is most sensitive to the magnitude of c 13, this sensitivity is low so that any reasonable accuracy in c13 can only be achieved if measurements o f velocity and the other stiffness components are made very accurately. Accordingly, the values used for the stiffness components in the calculation o f c l 3 for each specimen were obtained from the 'best fit' lines to all data o f Tables 1 and 2 shown in Figs 1 and 2. The scatter in values is shown in Fig.3. The low accuracy in the measurement o f c l 3 gives rise to the recorded scatter in the axial Poisson ratio,/213 , see (lc). It may be observed in Tables 1 and 2 that PI3 shows no marked dependence upon fibre fraction and has a value around 0.35 which is also independent of fibre type and equal to the matrix value. The smooth curves in Fig.3 pass through values for ('13 giving a value for/)13 of 0.35. Values for c22 I
CpqG N / m
were calculated which would yield these curves and are plotted in Fig.4. The derived c22 data shows only a small (less than experimental) scatter from expected values, illustrating that the c13 values lying on the smooth curves are reasonable. This analysis therefore confirms the insensitivity of the calculation for Cl3 to the magnitude of c 13. Whilst considering accuracy, it should be mentioned that one might expect experimental data to depart slightly from a smooth distribution with fibre fraction owing to the probable sensitivity of elastic properties to specimen structure which could differ from specimen to specimen even if the fibre fraction were constant. The other Poisson ratios are defined by:
V12 -
I
s12
c12c33
c~3
Sll
c11c33
c~3
s13
c13(c11 - c12 )
P31 Sll
12"O
c11c33
('~3
These components also show little dependence upon fibre fraction but differ slightly for the types of composite studied.
IO'O 8"0
from the smooth fit to c13 data
The stiffness components derived from measurements using the pulse technique relate to that part of the specimen through which the beam has travelled (approximately) 100 mm 2 cross section) so that, for inhomogeneous material, values for stiffness will depend upon the position of the beam in the specimen. The values listed in Tables 1, 2 and 3 are averages for a number of positions, but where significant scatter was recorded, the limits are indicated in Figs 1 and 2. The highest scatter was observed for the component c22. Although a variation in fibre packing arrangement within a specimen is likely to contribute to this scatter and to the difference between c I 1 and c22, the presence of defects, voids for example, is probably the major cause. Defects act as scattering centres which give rise to frequency dependent attenuation o f the pulse. Analysis of the spectrum of a pulse having travelled through a poor specimen (as indicated by a c-scan record) shows attenuation of the higher frequency components resulting in a change of pulse shape and a corresponding shift in zero cross-over positions. This shift gives rise to an error in recorded delay time and a reduction in the deduced value for elastic stiffness. The error will depend upon detect size, distribution and concentration and upon the
COMPOSITES. JULY 1973
177
Type I
6"0 4'0
(.9 a - o u
1
I
I
I
l
I
[
I
I
20"0 16'0 12'0 8"0
Type 2 4"0
0
0
0.2
0.4
0.6
0.8
Fibre fraction
Fig.4
Comparisonof experimentalc22 data with valuescalculated
I.O
bO
by eye. The solid lines are based upon theoretical expressions, derived by Hashin and Rosen, 8 which predict the dependence of composite properties upon the properties and concentration of the constituents• The theory is based upon a model to represent the composite which makes certain assumptions concerning the transverse phase geometry. The model considers the composite made up of parallel cylindrical units which are composed of a fibre surrounded by a coaxial column of matrix. In order to fill all space, the units are allowed to vary in diameter but are assumed to have a constant fibre fraction equal to that of the composite. In this way, the model approximates the random packing arrangement observed in real materials. Using this model, simple, closed-form expressions are derived for the moduli E a, v a, G a and k t which are exact for the model, k t is the transverse plane strain bulk modulus for the composite and is related to elastic stiffness components by:
c4c . c4 m c 4c + c4: d-6
0',~
,(+ 0.2
O
0"2
0.4
I 0.6
I 0-8
1 k t = ~(Cll +c12)=Cll
Fibre fraction
Fig.5 Variation of (c~4 --cr~44)/(cC, +cn444) and (k~-/~t)[ (k~ + cn444)with fibre fraction for type 1 fibre composites
direction of wave propagation. A calculation based upon ~ertain assumptions shows that, for the poor areas of the supplied sample sheets, an error (reduction) in transverse stiffness would be typically 5%. Since the specimens studied were cut from higher quality areas, the error in recorded values for c22 arising from defects is expected to be less than 5%. For propagation along the fibre axis, the velocity and hence delay time are greater than for propagation transverse to this direction, so that any error in c33 from defects is less important. However, elongation of the transmitted pulse was observed in some specimens, especially those of low fibre fraction, and this may be interpreted in terms of a variation in fibre concentration present in the specimen within the cross-section of the beam. In order to obtain a more homogeneous test specimen, its dimejasions transverse to the fibre axis were reduced to approximately 2.5 mm x 2.5 mm. The dependence of mode of propagation upon specimen dimensions had been previously studied by performing a series of measurements on a high quality specimen, moulded at the Royal Aircraft Establishment, as the specimen thickness was progressively reduced• Using a pulse frequency centred at 5 MHz, no significant variation in the magnitude of the components c 11, c22, c33 a n d C44 was observed down to a specimen thickness of 1.2 mm. Any comment concerning the change of mode arising along the fibre direction as the thickness is reduced would be inconclusive since the velocities of body waves and waves in the vicinity of the free surface differ by less than 2% in most of the specimens studied owing to their high anisotropy in stiffness.
CALCULA TION OF FIBRE PROPERTIES Figs 1 and 2 illustrate the variation of measured values for c22, c33, c44 and c66 with fraction of type 1 and type 2 fibres. The broken curve through data for c66 is fitted
178
c66
(2)
I.O An expression for the lower bound to the transverse shear modulus was also derived. Using the Hashin-Rosen expressions for E a, G a and k t, the following equations may be deduced:
c ~ 4 - c4//~ _ ( c f 4 -- c~4 ~
~kcf44+(-'~41 pf
cC44 + c ~ 4
t
"'t
_
V,( + c 41 where subscripts c, f, and m refer to the composite, the fibre and the matrix respectively, and v f is the fibre volume fraction. From these equations, it is apparent that plots of
4,, 4 ,
C~4 + C~4
kL L? k~+c~4
with fibre fraction should be linear. For the type 1 and type 2 fibre composite specimens, these plots are illustrated in Figs 1 2, 5 and 6 using the data for ,~thematrix, recorded in Table 3. The single fibre properties ~33' c~4 and k ft may then be deducted by extrapolating the data to a fibre fraction of unity. Since bounds for c66 have been derived by the theory and experimental data increases only slightly with fibre fraction,.~a value for the fibre transverse shear modulus, cJ 6, may be obtained with reasonable accuracy by extrapolating the data by eye as shown by the broken lines in Figs 1 and 2. The values for cf66 may then be used to obtain values for the fibre transverse stiffness, ell t = k ft + cf66 , and since k ft ~" cf66 the accuracy in c f 11 "~'ill not be critical to the accuracy in ~f'66The scatter in measured values for c~3 is too high to allow extrapolation. However, it was notedearlier that the composite axial Poisson ratio, v13, is essentially independent of fibre fraction and fibre type at a value of 0.35 + 0.05 which is the matrix value• Bearing in mind that the Hashin-
COMPOSITES . JU LY 1973
I-0
I
I
the deduced values for fibre transverse stiffness are reasonable, lower bounds to composite transverse stiffness were calculated from the fibre values using the Reuss average given by the expression:
I
O~
1 =~f+l-vf
c{2
c,:+ c,:
0-6
Composite values obtained from this expression were equal to or below the measured values. In fact, the Reuss values for the transverse stiffness of type 1 fibre composites coincide very closely with values derived using the HashinRosen expression and, for type 2 composites, Reuss values were slightly lower. The accuracy in the component c~3 is/elatively much lower and so the values quoted for cJ13 are as valid as the assumption that the axial Poisson ratiq of type 1 and type 2 composites is independent of fibre fraction and equal to the matrix value.
0,4
0"2
b¢~ O
I 0.2
I
I
0.4 06 Fibre fraction
0.8
I.O
Fig.6 Variation of (c~14- ~ ) / ( c C 4 + cn444) and (k~- kr~)/ {k~+ cn444)with fibre fraction for type 2 fibre composites
Rosen theory predicts a linear dependence of axial Poisson io with fibre fraction, we have assumed a fibre value of = 0.35 for type 1 and type 2 fibres. It is then possible om (1 c) to deduce values for cfl3. These five independent stiffness components for type 1 and type 2 fibres are listed in Table 4, together with compliance components deduced from them. Also shown in Table 4 is limited data on the Rolls Royce fibres obtained from Fig.7. DISCUSSION The reliability of the values given in Table 4 will depend upon the accuracy of the measurements made on the composite specimens and upon the validity of the theoretical expressions used to derive data for the fibre. The scatter in experimental data for c~3 and c,~4 is small and the theoretical curves fit the data closely. The scatter in values for c~2 is higher which introduces some uncertainty in the de.duced fibre values. An estimate of the uncertainty in c~2 may be obtained from Figs 3 and 4. To check whether
The manufacturers quoted values for fibre Young's modulus, obtained from tensile tests on individual fibres, vary from batch to batch but fall, typically, between 400 and 410 GN/m 2 for type 1 fibres and between 255 and 270 GN/m 2 for type 2 fibres. Since these values refer to large strain levels compared with the data derived from ultrasonic measurements, 400 GN/m 2 and 233 GN/m 2 respectively, it may be concluded that some non-linearity exists in the elastic behaviour of type 2 fibres whereas there is very little in the behaviour of type 1 fibres. This may be reasonable when consideration is taken of the high crystallite orientation in the latter. There is little published data on direct measurements of fibre axial shear modulus and a value of 24.8 GN/m 2 quoted for type 1 Harwell fibres 3 appears somewhat high. This value was used by Reynolds to assist in the assignment of compliance components for the graphite t:rystallite of type 1 fibres from which a set of fibre stiffness components was derived. 3 In this set, the components c 11 and c66 are also considerably higher than values quoted in Table 4. CONCL USION Data has been obtained illustrating the dependence of the elastic stiffness components of type 1 and type 2 carbon fibre composite materials upon fibre fraction. By choosing particular properties for the fibres, the expressions derived
Table 4. Elastic stiffness and compliance components for type 1 and type 2 carbon fibres.
Type 1 fibres
Type 2 fibres
CpqGN/m 2 Spqm2/GN
Cll = c22
c33
c44 = c55
c66
c13 = c23
c12
Sll = $22
$33
$44 = $55
$66
s13 = s23
$12
12.1
410
13.7
2.8
6.5
6.5
0.116
0.0025
0.073
0.357
-0.00086
-0.062
20.4
240
24
5.5
10.5
9.4
0.064
0.0043
0.042
0.182
-0.0015
-0.030
200
26
Rolls Royce fibres
COM POSITES . JU LY 1973
P13
~31
v12
0.35
0.01
0.53
0.35
0.02
0.45
179
1.0
200
C m c 4 4 - c44 c c44+c4~
0'5
"- I00 Z L9 u u
o Tensile
to calculate the elastic properties of a transversely isotropic, unidirectional composite of any fibre fraction once the properties of the matrix are known. Such information could be used for assessing the degree of orientation in material containing a distribution in fibre orientation. It is probable that the difference between the elastic properties of type l, type 2 and Rolls Royce fibres can only partly be explained in terms of crystallite orientation, h addition, the different heat treatments experienced by each fibre type will affect the fibre porosity and the properties of the crystallites, both being factors which also contribute to fibre properties. We can therefore expect fibre properties to vary slightly from batch to batch, although the extent of the variation ['or any particular modulus cannot, as yet, be assessed.
o Shear REFERENCES
0
0"2
0'4
1
1
0
0"6
0'8
I'O
Fibre f r a c t i o n Fig.7 Variation of longitudinal tensile and shear moduli with volume fraction of Rolls Royce fibres
by Hashin and Rosen have been found to closely fit experimental data. The fibres are highly anisotropic in elastic properties, the degree of anisotropy depending upon the heat treatment of the fibre. The derived elastic stiffness'values may be used with the Hashin-Rosen expressions
180
1 Chamis, C. C., Sendeckyj, G. P. J Camp Mat 2 (1968) 332 2 Schrager, M., Carey, J. Po(vmer l:'ngincering and Science ! 0 (1970) 369 3 Reynolds, W. N. Proceedings o f 3rd Con terence on Industrial Carbon and Graphite (April 1970) 427 4 Ruland, W. Applied Polymer Symposia, No 9 (1969) 293 5 Markham, M. F. Composites I (1970) 145 6 Dean, G. D., Lockett, F. J. Analysis of test methods for high modulus fibres and composites (April 1972). To be published in an ASTM Special Technical Publication 7 Crossland, B., Fennell, T. R. RAE Technical Memo MAT 105 (January 1971) 8 Hashin, Z., Rosen, B. W. Journal o f Applied Mechanics 31 (1964) 1
COMPOSITES . JULY
1973