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The effects of matrix complex moduli on the dynamic properties of CFRP laminae
Composites Science and Technology 36 (1989) 77-94
The Effects of Matrix Complex Moduli on the Dynamic Properties of CFRP Laminae T. A. Willway & R. G. White Institute of Sound and Vibration Research, Universityof Southampton, Southampton, Hampshire SO9 5NH, UK (Received 23 July 1987; revised version received 7 February 1989; accepted 3 March 1989) A BS TRA C T Conventional composites in the form of carbon fibre reinforced plastic ( CFRP ) have high specific stiffness but low damping. This paper describes the effects of the complex moduli of resins on the moduli and damping of unidirectionally reinforced composite laminae. Both theoretical and experimental studies have been carried out in shear and in flexure. The properties of a number of resins and composite materials are reported upon and it is shown that considerable variation in energy dissipation is possible with epoxy resins.
1 INTRODUCTION Results of previous research suggest that high volume fraction CFRP composites have a stiffness to weight ratio many times that of aluminium alloys, but have low material damping. In high performance structures subjected to acoustic loads, fatigue life is a crucial parameter, Consequently, it is imperative that resonant responses are limited, i.e. there should be substantial damping. Gibson 1 states that assuming a perfect bond at the fibre-matrix interface, increased C F R P damping can only be achieved by increasing the energy dissipation as a result of shear in the visco-elastic matrix. He suggested that this should be effected by using short aligned fibre (SF) reinforcement. However, SF reinforcement has been shown to yield only slight increases in damping over continuous fibre reinforcement for a comparable sacrifice 77 Composites Science and Technology 0266-3538/89/$03"50 © 1989ElsevierSciencePublishers Ltd, England. Printed in Great Britain
78
T. A. Willway, R. G. White
modulus. 2'3 An alternative method of increasing matrix dissipation, not fully investigated previously, is the use of a more dissipative resin. In conventional materials, stiffness varying approximately inversely with damping appears to be inherent; consequently it is probable that a highly dissipative resin may have a low modulus. However, the stiffness of high performance (high volume fraction) C F R P composites derives predominantly from the fibres. It appears, therefore, that the use of a flexible but highly dissipative resin matrix may not have an appreciably adverse effect on composite stiffness and could be tolerated, if beneficial to composite damping. This paper describes an investigation of the effect of resin complex moduli on the moduli and damping of unidirectionally reinforced C F R P laminae. A subsequent paper will describe the second phase of the development of a stiff, light, highly-damped CFRP, i.e. the effect of flexible matrix laminae on C F R P laminate properties.
2 THEORY In predicting the moduli and damping (loss factors) of the laminae, the simple mechanics of materials approach was adopted. Although not ideal, 4 there is little evidence to suggest that more rigorous complicated theories yield greater accuracy.
2.1 The modulus of elasticity of the unidirectional CFRP laminae in longitudinal flexure (E~) E c was calculated for continuous fibre laminae from the well established law of mixtures equation: 4
Ec---
EfVf+Em(|
--
Vf)
(l)
where Ef = Young's modulus of fibres; E m = Young's modulus of matrix; Vf = volume fraction of fibres. This equation is modified to
I
Eo = E f v f 1 - -
tanh(fll/2)~
3 + E.,tl - V0
12)
to account for the effect of discontinuous fibres s where l = length of fibre;
ln(R/ro)
and
ro - x/ Vf
Effect of matrix complex moduli on CFRP laminae properties
79
shear modulus of the matrix; R = r a d i u s of cylindrical matrix; s r o = fibre radius.
Gm =
2.2 The loss factors of the unidirectional CFRP laminae in longitudinal flexure (qc) r/c was calculated for the continuous fibre laminae from eqn (3), derived using Hashin's visco-elastic correspondence principle 6
E. r/c = (1 - v,f)r/m-~-/~
(3)
where rim is the loss factor of the matrix under direct loads. However, the loss factors of the aligned discontinuous CFRP laminae were calculated from the real (E~) and imaginary (E~') parts of the complex modulus of the composite. 2 Considering the complex form of eqn (2) E*=ErV f 1
tanh (fl*l/2)] ~ . _ j + E * ( 1 - Vf)
i.e. E* = E~ + iE~'
(4)
and =
where * denotes complex quantities. 2.3 The moduli of elasticity of the unidirectional CFRP laminae in transverse flexure (ET) According to the mechanics of materials equation, which assumes the stresses in the fibres and matrix to be the same in transverse flexure and ET to be independent of fibre aspect ratio:
ET--
Em
( 1 - Vr)+ Vr(Em/Erv )
(5)
where ETF = transverse modulus of elasticity of the fibres. Equation (5) was used to predict ET for both continuous and discontinuous CFRP laminae. 2.4 The loss factors of the unidirectional CFRP laminae in transverse flexure (qT) In the absence of any existing reliable theories for predicting r/T, the
80
T. A. Willway, R. G. White
mechanics of materials approach was used to derive the following equation assuming all energy dissipation to occur in the resin 3 tiT =
rlmET(1- Vf) Em
(6)
Again this equation applies to both short aligned and continuous CFRP. 2.5 The transverse shear moduli of the unidirectional CFRP laminae (G~) According to the mechanics of materials approach 4 GmGf
Gc =
{7)
+ Vra m
(1 - - v f ) a f
where Gc = shear modulus of lamina; G r = shear modulus of fibres and is independent of fibre aspect ratio.
2.6 The transverse shear loss factor of the unidirectional CFRP laminae (~/s) Once more adopting the mechanics of materials approach and the assumption of all energy dissipation occurring in the resin, the following equation derived in [3] was used to predict t/~: tl -
q~=
18)
Vf),~,.Gc
Gm
Predictions according to eqns (1)-(8), using resin and fibre properties given in Table 1, are shown in Tables 2-4 and compared with experimentally measured values. These results and their comparison enabled conclusions to be drawn concerning both the effect of matrix properties on laminae properties and the accuracy of these simple predictions.
TABLE la Properties of Carbon Fibres Fibre type
Longitudinal modulus, GPa Transverse modulus, GPa Shear modulus, GPa Poisson's ratio, v UTS, GPa
HM-S
345 410 20 33.5 0-24 1.7-22
HT-S/XAS
215 245 14 23.4 0.24 3.12-3.13
TABLE lb Resin Beam Properties (at room temp. = 10 Hz) Resin
RB 1
Mix ratio (by wt)
Em GPa
qm
E J SG° GPa
Emqm GPa
Order of Emqm
Gm GPa b
EL5 EDI EHT3
100 25 75
1'66
0-0800
1-36
1-3280 x 10a
8
0.60
CY208 HY905 DY063
100 50 0"5
0"19
0-9500
0"16
1'8050 x 108
5
0"07
EL5 ED1 EHT3
100 50 75
0"78
0"3500
0"63
2"7300 × 108
3
0.28
MY750 HY905 DY063
100 100 0'5
3'45
0"0139
2.84
4"7950 x 107
12
1-25
CY208 HY956 EL5
100 17'5 25
0"98
0"3500
0'85
3"4300 x 108
2
0'35
MY750 CY208 HY956
25 75 ! 5'5
1"07
0-3500
0"80
3'7450 x 108
1
0-39
MY750 CY208 HY932
50 50 24
3"35
0"0204
2'88
6"8340 x 107
9
t'21
EL5 EDI EHT3
100 24 64
1"75
0-0774
1"44
1"3545 x 108
7
0'63
EL5 EDI EHT3
100 0 75
3-55
0"0398
2'94
1-4129 x 108
6
1'28
RB10
MY750 HY956
100 23"5
3'27
0"0148
2'86
4"8395 × 107
11
1'18
RB 11
MY750 HY932
100 32
3"08
0"0067
2"72
2"063× 107
13
1"11
RB 12
MY750 H Y905 DY063 DY040 Silica Flour
lO0 100 3 10 400
13"68
0"0176
7"37
2'4077 x l0 s
4
4'94
DX210 BF3400
100 3
2"98
0"0180
2"47
5"3640
10
1"08
RB 2
RB 3
RB 4
RB 5
RB 6
RB 7
RB 8
RB 9
RB 13
° SG = specific gravity. b Calculated assuming Vepoxy= 0.385.
× 10 7
RB5 + 6 0 % Vr aligned C C F R B 2 + 6 0 % Vf aligned C C F RBI 1 + 6 0 % Vr aligned C C F RB3 + 6 0 % Vf aligned C C F RB6 + 60% V r aligned C C F ~' RB13 + 6 0 % Vf aligned d i s c o n t i n u o u s fibres 2.25mm length R B 1 3 + 6 0 % Vt aligned C C F
CB 1 CB2 CB 3 CB 4 CB 5 CB6 c
123.10 115.10 121.48 112-66 104-61 78.65 128.90
0-0055 0.0100 0.0005 0-0096 0.004 8 0-001 5 0.001 2
127.36 208.78
GPa
193.60 182-68 194.00 176.79 186.63
GPa
204.61 208-1l
207-39 207.08 208.23 207.31 207.43
GPu × × × × ×
10 10 10 10 10
a ~ s 4 4 4-3 × 10 ~ 1 × 10 ~
6"6 35 4.0 5.3 7.2
Experimental Theoretical ". . . . . . . . . . . . . . . . . . . . . . . . . .
All fibres EHM-S. S G = specific gravity. h Fibre taken in tows from cloth. ' Specimen was made some time previously and may therefore have degraded slightly via m o i s t u r e a b s o r p t i o n , etc.
CB 7
Description
Composite beam no.
0.6224 1.0078
0-9335 0.8822 0931 7 0"8528 0.899 7
E.w/ E,h,.,,,
TABLE 2 Theoretical and Experimental Flexure M o d u l u s a n d Loss E'actor of Longitudinally Reinforced C F R P Beams
3.4884 12.0000
8-33 28.5714 12.5000 18.1132 6-66
q,.,.#/ tl,h,.,,,
B(
'~ .~
5,.
Effect of matrix complex moduli on CFRP laminae properties
83
TABLE 3 Theoretical and Experimental Modulus and Loss Reactor of Transversely Reinforced CFRP Beams Composite beam no.
TB 1 TB 2 TB 3 TB4
Description
R B 6 + 6 0 % Vf aligned CCF RB8 + 60% Vf aligned CCF RB5 + 60% Vr aligned CCF RB2 + 60% Vf aligned CCF
Experimental
Theoretical
Er GPa
qr
Er/SG° GPa
Er GPa
qr
1.89
0-3232
1"33
2.48
0.3244
2.42
0.066 0
1.65
3.87
0.068 5
--
--
--
2"28
0"325 7
--
--
--
0.47
0.949 2
a SG = specific gravity.
TABLE 4 Theoretical and Experimental Shear Modulus and Loss Factor of CFRP Rods Composite rod no.
CRN1 CRNIa CRN4 CRN5 CRN7 CRNI0 SCRN1
Description
RB6 + 60% aligned CCF RB6 + 60% Vf aligned CCF RB8 + 60% Vf aligned CCF RB3 + 60% Vf aligned CCF RB2 + 60% Vf aligned CCF RBI3 + 50% Vr aligned CCF RBI3 + 60% Vf aligned short (2.25 mm) CF
Experimental
Theoretical
Gc
q,
Gc
G Pa
(y = 200 tas)
G Pa
q,
0.52 1-22 1.45 0.95 0.24 4'12
0.658 9 0.3200 0.0740 0.33 0.4649 0"224
0.95 0.95 1.54 0.70 0.17 2"08
0.344 0 0.3440 0"075 3 0'345 6 0"947 2 0"0174
5'! 1
0"0087
2.57
0"0172
All fibres EHM-S.
3 EXPERIMENTS Prior to experiments being carried out on the unidirectional CFRP specimens, the flexural moduli and damping ofapproximately 40 resins were measured using the well-established, free-free beam method described later. Unfortunately, numerous problems were encountered when using this method as a result of the high flexibility of the resins) Results from these
T..4. Willway, R. G. White
84
I0.0
i.5
]
/ M(Idu/us
__ _ L ' ~ . : _ . - 7 -. . . .
8,0
'~ I 1
Resin Samo~
. /
7.0
~.~.~'-~.__"~_
.5"=:.:--~--~
~.o
1.4
RBI RB2 RB3 RB4
..
1 / I .o 4 o,~, ;.
Desw/~If~n
RB5
. . . . . . . .
i o.~
..............
4o.;,
7
~
'w"~t~t'..C..7,.~.=-
_
Loas Poctor
"-
"".',--,......,,..~ ~
.~
_-
..........
i 0..'~
2: .._-"==.-,,.,,~ 4~J
o.:~ 0.t
6,0 L l
2
3
4
5
Log Frequency (Hz)
Fig. la.
Variation of resin modulus and loss &ctor with frequency at 20<(L
tests did however, provide an insight into the relative stiffness and damping of the resins enabling a small number to be selected (see Table ] and Fig. 1) for further more sophisticated tests using a Polymer Laboratories dynamic mechanical thermal analyser (DMTA). This apparatus measures the power input to and the phase difference between the excitation and response of a
k
...........
x /
/ 50
60
70
80
90
q 2,0 4,.9
.....................................
io.o F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I00
, 110
120
130
140
150
\ \
/\ \/
t60
Temperofure f ~C)
Fig. lb.
Viscoelastic properties of RB4.
170
..,~oooo~, j o.,~ "~/ 7o-'
180
190
200
Effect of matrix complex moduli on CFRP laminae properties
85 1.5
10.0
1.4 1.3
1.2 1.1 1.0
9.0
0.9
0.8 0.7
~
0.6 0.5
8,0
0.4 0.3 0.2 0.1 7.0 -20
0 -10
0
10
20
30
40
50
60
70
80
90
1O0
110
120
Temperature ( "C )
Fig. lc. Viscoelastic properties of RB6.
non-resonant beam from which it calculates the dynamic moduli and loss factors of a small cantilevered specimen at various points in a specified temperature range. The effect on results of variation in frequency were predicted by computer. 7 Consideration of the results of these experiments enabled a number of highly dissipative resins to be identified for use in the CFRP composite experiments and theoretical investigations. The longitudinal flexural, transverse flexural and transverse shear moduli and loss factors of the various CFRP specimens described in Tables 2-4 were measured using the following experiments. Although no sophisticated N D T method was used to assess the quality of the specimens, careful manufacture, visual inspection and comparison between theoretical and measured elastic modul,i were considered to give an indication of specimen integrity. 3.1 Measurement of the flexural modulus and loss factor of a free-free beam The transverse and longitudinal flexural properties of the CFRP laminae and resin moduli and damping were measured using this technique, developed by Wright 9 and White s and shown in Fig. 2. The beam was suspended by cotton threads at the nodes of the first flexural mode of vibration. Any added mass would affect the natural frequency of the system and attachments could provide extraneous
86
T. A. Willway, R. G. White
Decade Oscillator
Frequency Meter
i l Power
Amplifier
2 Beam
Oscilloscope
I
~ Switch Ink-Jet Recorder
l
I
Vibrator Filter
Beam Optical Vibration Transducer
.... ] "
;F
OVT Driver and f i l t e r unit
Fig. 2. Block diagram of apparatus for free-free flexural tests. damping. A method of excitation was found which required no added mass. A plate attached to a small vibrator was placed in close proximity to the beam centre. Air coupling between the vibrating plate and the beam was sufficient to excite the beam at resonance. Similarly, a non-contacting transducer (optical vibration transducer) was used to avoid added mass for response measurement. The beam was excited at its first resonance tYequency via the decade oscillator-power amplifier-ammeter circuit shown in Fig. 2. The excitation frequency was measured using the frequency counter, accurate to 0-1 Hz, from which the dynamic flexural modulus was found from the natural frequency equation of a homogeneous Euler Beam:
48n2J~ L 3rn E = (22.3729)2bd 3
(9)
where E = dynamic flexural modulus of beam;j], = first natural frequency of
Effect of matrix complex moduli on CFRP laminae properties
87
beam; L = length of beam; m = mass of beam; b = width of beam; d = thickness of beam. Having noted the natural frequency, the excitation was switched off and the decay of the beam response signal, filtered to remove the low frequency rigid body motion and noise, recorded on the ink-jet recorder. The beam loss factor was then calculated from the decaying waveform as follows: ~/=
t(Q/QN) nN
00)
where Nis the number of cycles between the measured peaks of amplitude Q and QN. The suitability of the apparatus for measuring material properties was validated in Refs 8 and 9, by attempting to measure the modulus and damping of a duralumin (r/< 3 × 10-6) specimen. Extraneous losses were found to be negligible. Results for resin, longitudinal CFRP and transverse C F R P specimens are given in Tables 1, 2 and 3 respectively. 3.2 Measurement of the shear moduli and loss factors of the unidirectional CFRP laminae The apparatus, instrumention and procedures used to measure the shear moduli and loss factors of the CFRPs are detailed in Ref. 3. Once again, the extraneous losses in the apparatus were found to be negligible by means of an experiment with a duralumin specimen. 3 Experiments were carried out on solid cantilevered C F R P rods of 7 mm diameter undergoing steady state torsional vibration as shown in Fig. 3. The dynamic torsional load was applied by two coil-magnet systems, driving an inertia bar attached to the rod at its free end. The response was monitored by strain gauges on the specimen, being restricted to low amplitudes (maximum surface shear strain amplitudes < 300/~s) in order to avoid damage to the specimens. As the mass of the specimen was negligible compared with the mass of the driven system, the specimen and drive system could be considered as a single degree of freedom system with lumped parameters when driven at the first resonance frequency (taken as the frequency at which the exciting current was a minimum for the specified strain response). The shear moduli of the rods were then calculated from the natural frequency equation of an inertia loaded rod, represented as a single degree of freedom system as follows: 8~zf~2JL
Gc -
R4
(11)
where fr = measured resonance frequency; J = moment of inertia of drive
T. A. Wilhvay, R. G. White
88
C t,lj,H~
"/'~
~M~/IIIL/I
TOP CLAMP
SPECIMEN ( clamped-clamped or cantilevered )
STRAIN GAUGES
'"
INERTIA BAR
/ K, ~]_~
DYNAMIC FORCES SPECIMEN
STRAIN GAUGES
(clamped-clamped onlyl SHIM
1/llllTtflllf
7
BOTTOM CLAMP
Fig. 3a. Combined loading apparatus.
/
2o.o
/
:
DiJralumln end pieces
\. \
Working Section
~Oo-- 7.0
I.o.o I
10 i = 5 . o 1
t90.0 210.0
Fig. 3b. Cantilevered specimen.
system (measured as in Ref. 8); L = length of rod between clamps; R = radius of rod. Results are given in Table 4. The loss factor of the rod was calculated from measurements of the input current and output voltage, frequency and specimen surface shear strain amplitude at resonance. 3 These values enabled calculations of the energy dissipated and energy stored in this system at resonance to be made, from which the loss factor ~]s-
iVo (03j02 (derived in Ret: 3)
(12j
where Vo=oUtput voltage; / = i n p u t current; ~o=resonance frequency (rad s - ~); 0 = rms rotational displacement of inertia bar, was calculated and results compared with theoretical predictions in Table 4.
Effect of matrix complex moduli on CFRP laminae properties
89
4 DISCUSSION OF RESULTS The initial phase of the project was the selection of some highly dissipative matrix materials. Existing manufacturing equipment and experience restricted the study to epoxies although cursory examinations of new thermoplastics such as poly(ether ether ketone) (PEEK) revealed inferior damping properties. Experimental results for resins are summarised in Fig. 1 and Table l, illustrating the range of dynamic properties achievable by varying flexibiliser and hardener content. The energy dissipated in a material at a specific stress is proportional to the quotient of its loss factor and modulus. However, at a specific strain, energy dissipated is proportional to their product. In a unidirectional FRP lamina, the former situation occurs in longitudinal flexure, the latter in transverse flexure and shear. Consequently, with a view to subsequent flexural tests on CFRPs, Table 1 includes the product of E= and ~/m(Emr/m). Consideration of these results reveals that resins conventionally used in CFRP such as RB4, 7, 10, 11, 13 have a high modulus but low loss factor and Emt/m product. Composites incorporating such resins will therefore have a low flexural energy dissipation and loss factor. For a specified strain amplitude, optimum energy dissipation appears to occur in resins having intermediate values of modulus and damping, e.g. RB 3, 5, 6. Theoretically, therefore, CFRPs incorporating RB3, 5 and 6 would have increased longitudinal flexural energy dissipation and loss factors. However, it should be noted that resin properties are highly temperature and frequency dependent, as shown in Fig. 1. It is therefore essential that resin materials are selected after consideration of operational temperatures and frequencies. Having determined the properties of the resins, their effects on the stiffness and damping of unidirectional CFRP laminae were studied theoretically and experimentally. Theoretical predictions of moduli suggest that the modulus of the composite is not significantly affected by the modulus of the resin. For example, CB7 manufactured of high modulus RB13 has a predicted modulus of 208.11 G N m -2, whilst CB2 using RB2 (low stiffness) has a modulus of 207.08 GN m- 2, there being less than 0"5% difference in moduli. As expected, the theoretically predicted value of composite loss factor is proportional to the Emr/mproduct of the component resin. It is interesting to note that the order of increasing loss factor is not the inverse of the order of increasing modulus, thus giving evidence of the ability of FRPs to defy the 'law' of conventional materials that modulus is approximately inversely proportional to damping.
90
7-. A. Willway, R. G. White
CB6 consisted of RB13 and aligned discontinuous fibres. Whilst the predicted loss factor is greater than that of CB7 (RB13 with continuous fibres) it is not as great as CB4 and 6 which are made of continuous fibres in high dissipative resin matrices. Theoretically, the approach to increasing CFRP damping by using highly dissipative matrix materials therefore looks promising. Unlike the longitudinal flexural modulus, the transverse flexural modulus of a unidirectional lamina is theoretically dependent on both the matrix and fibre moduli. Furthermore, the transverse flexural loss factor is proportional to the quotient of the matrix loss factor and modulus (rl,~/Em) and the composite transverse flexural modulus. Therefore, in theory, matrix properties have a significant effect on both the transverse flexural moduli and loss factors of the C F R P laminae (see Table 3). The theoretical predictions of the shear moduli of the unidirectional CFRP specimens according to eqn (7) (Table 4) suggest the shear modulus of the C F R P to be highly dependent on that of the matrix material. Consequently, the predicted values of the shear modulus for a specimen manufactured from a low modulus resin is much lower than that ofa CFRP made from a conventional, stiff resin. Comparing for example, CRN la and SCRN 1, the ratios of the predicted composite moduli are 2"7:1. The ratio of the component resin moduli is 2"78:1. Whereas the theoretical predictions of the composite flexural loss l~actor are directly proportional to the matrix energy dissipated at a given strain (Emqm), eqn (8) includes a direct relationship between the composite shear loss factor and the matrix energy dissipated at a specific stress, i.e. (r/s ~ ~m/Gm), and the shear modulus of the composite. However, the interrelationship between the shear moduli of the composite and matrix results in ~/s being effectively proportional to the matrix loss factor, r/m, at practical volume fractions. This is reflected in the predicted values of q~ in Table 4. Comparison of theoretical results for CRN1 and CRN5 having identical matrix loss factors but differing matrix moduli shows the effect of the matrix modulus to be less than 0.5% (~/s= 0.3440-0.3456). Thus, in theory, the dynamic properties of a unidirectional CFRP in shear are derived from the corresponding matrix shear properties. As predicted theoretically, experimentally determined values of the longitudinal flexural moduli (Table 2) showed little variation with resin properties. They were, however, generally lower than predicted, possibly a result of the fibre modulus being lower than that used in the calculations, and/or slip at the fibre-matrix interface due to imperfect bonding resulting in friction losses. The especially low value of modulus for CB6 may have been attributed to water absorption and other ageing processes discussed
Effect of matrix complex moduli on CFRP laminae properties
91
later. The modulus measured in an identical manner at the time of manufacture 2 was similar to the predicted value. The measured values of loss factor do not, however, follow theoretically predicted trends. The maximum composite loss factor was obtained from the CFRP manufactured from resin RB2, i.e. that resin having the maximum loss factor. The general trend appeared to be loss factor increasing with increasing resin loss factor, or decreasing resin modulus (Table 2). The measured loss factor of CB2 was twenty times that of CB3, whilst the modulus was decreased by less than 6% (actual and specific, (i.e. E/(specific gravity)) values). Similar trends were shown by CB1, 4 and 5, slightly lower increases in loss factor being achieved for a lower sacrifice in stiffness. Comparing theoretical and experimental results, measured values of loss factor were always greater than predicted values, r/~.ffr/th,o~ being 28.5714 for CB2. There are three possible sources of this discrepancy, assuming the experiment and theories to be inherently accurate, as we shall now discuss. It is possible that, even in the thin CFRP beams used in the experiments (thickness d = 3 mm), shear deformation may have a significant effect on the measured natural frequencies and damping. Shear deformation in the matrix will dissipate a substantial amount of energy in addition to that dissipated in tension-compression due to flexure. An additional or alternative explanation would be an imperfect bond between the fibre and matrix. Slip and the resulting friction at the fibre-matrix interface ('interfacial slip') would result in increased energy dissipation and damping, but a decreased modulus, as observed in the experimental results. However, it is unlikely that interfacial slip alone is responsible for the difference in measured and predicted values, and the overestimates of the moduli and underestimates of damping may simply be a result of fibre data used being inaccurate, manufacturers supplying approximate ranges of properties only. It may be concluded that these theoretical and experimental results indicate that it is feasible to produce a unidirectional CFRP lamina having a high longitudinal flexural modulus and loss factor by using a highly dissipative resin. Experimentally measured values of the transverse flexural moduli of two CFRP laminae show qualitative agreement with the theoretical prediction of dependence on matrix modulus (Table 3). Measured values were, however, much lower than those predicted by the mechanics of materials approach, eqn (5), possibly as a result of an imperfect bond a t the fibre-matrix interface. Although insufficient specimens were tested to establish whether the transverse damping was governed by the matrix
92
T. A. Willway, R. G. White
modulus or loss factor, measured values were only slightly lower than those predicted using the simple energy equation (6). Unlike the measured values of moduli, the measured values of loss factor gave no indication of the occurrence of interfacial slip. It can therefore be concluded that the transverse flexural properties of unidirectional C F R P laminae are highly dependent on the corresponding matrix properties. Measured shear moduli follow trends similar to predicted values, i.e. the composite shear modulus increases with increasing resin shear modulus (Table 4). Measured values for SCRN1 and CRN10 did, however, appear high whilst that of CRN1 appeared low. A possible explanation of these results will be put forward later. Experimentally determined values of shear loss factor are given in Table 4. Note that investigations of the strain dependence of the CFRP shear loss factor, 3 not detailed here, revealed essentially C F R P linear shear damping. The experimentally determined values of material loss factor shown in Table 4 behave in the same manner as the experimental values of flexural loss factor, i.e. increasing with increasing resin loss factor or decreasing resin modulus as predicted theoretically. Consequently, the order of increasing modulus is approximately the inverse of the order of decreasing loss factor. However, the values of shear loss factor measured for CRN1, CRN7 and SCRN1 do not conform to the generally observed trends, SCRN1 and CRN7 having lower than expected loss factors whilst that of CRN1 and SCRN1 are high. Both CRN1 and C R N I a contain resin RB6 and 60% Vf aligned continuous fibres (EHM-S). Theory therefore predicts that the composite shear moduli and loss factors should be the same. This, however, was found not to be the case. The difference in composite properties was probably a result of different moulding techniques used in their manufacture? The discrepancies between the theoretical and experimental moduli and loss factors of SCRN1 and CRN10 may be attributed to the manufacture. Both specimens were made for a previous set of experiments, 2 at a time when moulding techniques were not so well established, especially when using short fibres. The volume fraction of each specimen may therefore not be as specified, possibly being higher than the 50% and 60% specified for CRN10 and SCRN1 respectively. Furthermore, any misalignment of fibres would further increase the shear modulus and decrease shear loss factor. These factors would explain the achievement of a shear modulus greater than that predicted for SCRN 1 and CRN10 and loss factor smaller than predicted for SCRN1. The high measured loss factor of CRN10 could possibly be explained by some ageing effect (see later) on the resin, significantly increasing its loss factor but not affecting its modulus, although a similar effect would be expected in the resin of SCRN 1.
Effect of matrix complex modufi on CFRP laminae properties
93
Measured loss factors were found to agree well with those predicted by the simple energy approach, e q n (8). It is obvious from both the predicted and measured shear dynamic properties of unidirectional CFRPs that the shear modulus and damping of the CFRPs are dependent on the corresponding properties of the resin matrix. Therefore, whilst the use of low modulus resin in CFRP composites in order to increase composite damping may have no significant effect on the stiffness of the composite in the fibre direction, the effects on the stiffness in other directions are substantial. Although this paper is primarily concerned with the effects of matrix moduli on CFRP dynamic properties, the modulus and, particularly, damping of any CFRP are sensitive to many other factors which are difficult to control. The quality of the laminae (e.g. void content, fibre misalignment) and the integrity of the bond between the fibre and matrix significantly affect composite properties. Furthermore, the properties of the resin matrix, in addition to being temperature and frequency dependent, may change with time as a result of ageing effects such as water absorption, chemical attack and UV degradation. Although these factors have been taken into consideration in the discussion of the results of this research, it may ultimately be necessary, not only to consider their influences on material properties, but to use them in achieving the structural properties required in a certain practical environment.
5 CONCLUSIONS From measurements of the energy dissipation of a number of resins it was concluded that there is a considerable variation in energy dissipation possible with epoxy resins. Conventional stiff, lightly damped resins were found to dissipate very little energy. Optimum energy dissipation occurred in the intermediate stiffness and damping resins, which were just within the glassy region of the glass transition peak at test frequency and temperature. Indeed, it was evident that the dynamic properties of visco-elastic resins are extremely frequency and temperature dependent. Flexural tests on unidirectional continuous CFRP specimens proved that, as theory predicts, the flexural moduli were almost independent of the matrix. The results of the theoretical and experimental studies show that it is feasible to produce a unidirectional CFRP lamina having a high longitudinal flexural modulus and loss factor by using a highly dissipating resin. The transverse flexural and shear properties of the unidirectional continuous CFRP specimens were found to be governed by the matrix
94
T. A. Willway, R. G. White
properties, the moduli and loss factors being proportional to the matrix moduli and loss factors respectively, verifying theoretical trends. Theory did, however, overestimate the modulus and underestimate the loss factor. Discrepancies between theory and experiment in both shear and flexure suggests other mechanisms of energy dissipation in the composite, such as slip at the fibre-matrix interface due to imperfect bonding resulting in frictional losses. It is evident that whilst the use of a low modulus, highly dissipative resin matrix in C F R P composites to increase laminae damping may have no significant effect on the stiffness of the composite in the fibre direction, the effects on the stiffness in other directions are substantial. However, if the laminae were laminated such (hat a fraction of the fibre content was orientated in each of the directions of the principal axes, the fibres would then contribute to composite stiffness in all directions under load. A study of the effect of flexible matrix C F R P laminae on C F R P laminate properties is the subject of a subsequent paper.
REFERENCES 1. Gibson, R. F., Development of damping composite materials. In Proceedings of the Advances in Aerospace Structures, Materials and Dynamics Symposium on Composites, Boston. ASME Aerospace Symposium Series AD, Vol. 067.
American Society of Mechanical Engineers, New York, 1983, pp. 89. 2. White, R. G. & Abdin, E. M. Y., Dynamic properties of aligned short carbon fibre reinforced plastic beams in flexure and torsion. Composites, 16(4) (1985) 293-306. 3. Willway, T. A., Stiff, light, highly damped CFRPs and the effect of complex loads on damping. PhD thesis, University of Southampton, UK, 1986. 4. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, 1975, 5. Cox, H. L., The elasticity and strength of paper and other fibrous materials. Brit. J. Appl. Phys., 3 (1952) 72-9. 6. Hashin, Z., Complex moduli of visco-elastic composites: II. Fibre reinforced materials. Int. J. Solid Struct., 6 (1970) 797-805. 7. Townend, D. J., A computerised technique for producing WLF shifted data from a single frequency temperature scan without graphical manipulation. Technical Memorandum AMTE(M) TM 83307, ARE (Holton Heath), 1983. 8. White, R. G., Some measurements of the dynamic properties of mixed carbon fibre reinforced plastic beams and plates. Aeronaut. J., 79 (775) (1975) 318 25. 9. Wright, G. C., Dynamic behaviour of fibre reinforced plastic beams and plates. PhD thesis, University of Southampton, UK, 1973.
The Effects of Matrix Complex Moduli on the Dynamic Properties of CFRP Laminae T. A. Willway & R. G. White Institute of Sound and Vibration Research, Universityof Southampton, Southampton, Hampshire SO9 5NH, UK (Received 23 July 1987; revised version received 7 February 1989; accepted 3 March 1989) A BS TRA C T Conventional composites in the form of carbon fibre reinforced plastic ( CFRP ) have high specific stiffness but low damping. This paper describes the effects of the complex moduli of resins on the moduli and damping of unidirectionally reinforced composite laminae. Both theoretical and experimental studies have been carried out in shear and in flexure. The properties of a number of resins and composite materials are reported upon and it is shown that considerable variation in energy dissipation is possible with epoxy resins.
1 INTRODUCTION Results of previous research suggest that high volume fraction CFRP composites have a stiffness to weight ratio many times that of aluminium alloys, but have low material damping. In high performance structures subjected to acoustic loads, fatigue life is a crucial parameter, Consequently, it is imperative that resonant responses are limited, i.e. there should be substantial damping. Gibson 1 states that assuming a perfect bond at the fibre-matrix interface, increased C F R P damping can only be achieved by increasing the energy dissipation as a result of shear in the visco-elastic matrix. He suggested that this should be effected by using short aligned fibre (SF) reinforcement. However, SF reinforcement has been shown to yield only slight increases in damping over continuous fibre reinforcement for a comparable sacrifice 77 Composites Science and Technology 0266-3538/89/$03"50 © 1989ElsevierSciencePublishers Ltd, England. Printed in Great Britain
78
T. A. Willway, R. G. White
modulus. 2'3 An alternative method of increasing matrix dissipation, not fully investigated previously, is the use of a more dissipative resin. In conventional materials, stiffness varying approximately inversely with damping appears to be inherent; consequently it is probable that a highly dissipative resin may have a low modulus. However, the stiffness of high performance (high volume fraction) C F R P composites derives predominantly from the fibres. It appears, therefore, that the use of a flexible but highly dissipative resin matrix may not have an appreciably adverse effect on composite stiffness and could be tolerated, if beneficial to composite damping. This paper describes an investigation of the effect of resin complex moduli on the moduli and damping of unidirectionally reinforced C F R P laminae. A subsequent paper will describe the second phase of the development of a stiff, light, highly-damped CFRP, i.e. the effect of flexible matrix laminae on C F R P laminate properties.
2 THEORY In predicting the moduli and damping (loss factors) of the laminae, the simple mechanics of materials approach was adopted. Although not ideal, 4 there is little evidence to suggest that more rigorous complicated theories yield greater accuracy.
2.1 The modulus of elasticity of the unidirectional CFRP laminae in longitudinal flexure (E~) E c was calculated for continuous fibre laminae from the well established law of mixtures equation: 4
Ec---
EfVf+Em(|
--
Vf)
(l)
where Ef = Young's modulus of fibres; E m = Young's modulus of matrix; Vf = volume fraction of fibres. This equation is modified to
I
Eo = E f v f 1 - -
tanh(fll/2)~
3 + E.,tl - V0
12)
to account for the effect of discontinuous fibres s where l = length of fibre;
ln(R/ro)
and
ro - x/ Vf
Effect of matrix complex moduli on CFRP laminae properties
79
shear modulus of the matrix; R = r a d i u s of cylindrical matrix; s r o = fibre radius.
Gm =
2.2 The loss factors of the unidirectional CFRP laminae in longitudinal flexure (qc) r/c was calculated for the continuous fibre laminae from eqn (3), derived using Hashin's visco-elastic correspondence principle 6
E. r/c = (1 - v,f)r/m-~-/~
(3)
where rim is the loss factor of the matrix under direct loads. However, the loss factors of the aligned discontinuous CFRP laminae were calculated from the real (E~) and imaginary (E~') parts of the complex modulus of the composite. 2 Considering the complex form of eqn (2) E*=ErV f 1
tanh (fl*l/2)] ~ . _ j + E * ( 1 - Vf)
i.e. E* = E~ + iE~'
(4)
and =
where * denotes complex quantities. 2.3 The moduli of elasticity of the unidirectional CFRP laminae in transverse flexure (ET) According to the mechanics of materials equation, which assumes the stresses in the fibres and matrix to be the same in transverse flexure and ET to be independent of fibre aspect ratio:
ET--
Em
( 1 - Vr)+ Vr(Em/Erv )
(5)
where ETF = transverse modulus of elasticity of the fibres. Equation (5) was used to predict ET for both continuous and discontinuous CFRP laminae. 2.4 The loss factors of the unidirectional CFRP laminae in transverse flexure (qT) In the absence of any existing reliable theories for predicting r/T, the
80
T. A. Willway, R. G. White
mechanics of materials approach was used to derive the following equation assuming all energy dissipation to occur in the resin 3 tiT =
rlmET(1- Vf) Em
(6)
Again this equation applies to both short aligned and continuous CFRP. 2.5 The transverse shear moduli of the unidirectional CFRP laminae (G~) According to the mechanics of materials approach 4 GmGf
Gc =
{7)
+ Vra m
(1 - - v f ) a f
where Gc = shear modulus of lamina; G r = shear modulus of fibres and is independent of fibre aspect ratio.
2.6 The transverse shear loss factor of the unidirectional CFRP laminae (~/s) Once more adopting the mechanics of materials approach and the assumption of all energy dissipation occurring in the resin, the following equation derived in [3] was used to predict t/~: tl -
q~=
18)
Vf),~,.Gc
Gm
Predictions according to eqns (1)-(8), using resin and fibre properties given in Table 1, are shown in Tables 2-4 and compared with experimentally measured values. These results and their comparison enabled conclusions to be drawn concerning both the effect of matrix properties on laminae properties and the accuracy of these simple predictions.
TABLE la Properties of Carbon Fibres Fibre type
Longitudinal modulus, GPa Transverse modulus, GPa Shear modulus, GPa Poisson's ratio, v UTS, GPa
HM-S
345 410 20 33.5 0-24 1.7-22
HT-S/XAS
215 245 14 23.4 0.24 3.12-3.13
TABLE lb Resin Beam Properties (at room temp. = 10 Hz) Resin
RB 1
Mix ratio (by wt)
Em GPa
qm
E J SG° GPa
Emqm GPa
Order of Emqm
Gm GPa b
EL5 EDI EHT3
100 25 75
1'66
0-0800
1-36
1-3280 x 10a
8
0.60
CY208 HY905 DY063
100 50 0"5
0"19
0-9500
0"16
1'8050 x 108
5
0"07
EL5 ED1 EHT3
100 50 75
0"78
0"3500
0"63
2"7300 × 108
3
0.28
MY750 HY905 DY063
100 100 0'5
3'45
0"0139
2.84
4"7950 x 107
12
1-25
CY208 HY956 EL5
100 17'5 25
0"98
0"3500
0'85
3"4300 x 108
2
0'35
MY750 CY208 HY956
25 75 ! 5'5
1"07
0-3500
0"80
3'7450 x 108
1
0-39
MY750 CY208 HY932
50 50 24
3"35
0"0204
2'88
6"8340 x 107
9
t'21
EL5 EDI EHT3
100 24 64
1"75
0-0774
1"44
1"3545 x 108
7
0'63
EL5 EDI EHT3
100 0 75
3-55
0"0398
2'94
1-4129 x 108
6
1'28
RB10
MY750 HY956
100 23"5
3'27
0"0148
2'86
4"8395 × 107
11
1'18
RB 11
MY750 HY932
100 32
3"08
0"0067
2"72
2"063× 107
13
1"11
RB 12
MY750 H Y905 DY063 DY040 Silica Flour
lO0 100 3 10 400
13"68
0"0176
7"37
2'4077 x l0 s
4
4'94
DX210 BF3400
100 3
2"98
0"0180
2"47
5"3640
10
1"08
RB 2
RB 3
RB 4
RB 5
RB 6
RB 7
RB 8
RB 9
RB 13
° SG = specific gravity. b Calculated assuming Vepoxy= 0.385.
× 10 7
RB5 + 6 0 % Vr aligned C C F R B 2 + 6 0 % Vf aligned C C F RBI 1 + 6 0 % Vr aligned C C F RB3 + 6 0 % Vf aligned C C F RB6 + 60% V r aligned C C F ~' RB13 + 6 0 % Vf aligned d i s c o n t i n u o u s fibres 2.25mm length R B 1 3 + 6 0 % Vt aligned C C F
CB 1 CB2 CB 3 CB 4 CB 5 CB6 c
123.10 115.10 121.48 112-66 104-61 78.65 128.90
0-0055 0.0100 0.0005 0-0096 0.004 8 0-001 5 0.001 2
127.36 208.78
GPa
193.60 182-68 194.00 176.79 186.63
GPa
204.61 208-1l
207-39 207.08 208.23 207.31 207.43
GPu × × × × ×
10 10 10 10 10
a ~ s 4 4 4-3 × 10 ~ 1 × 10 ~
6"6 35 4.0 5.3 7.2
Experimental Theoretical ". . . . . . . . . . . . . . . . . . . . . . . . . .
All fibres EHM-S. S G = specific gravity. h Fibre taken in tows from cloth. ' Specimen was made some time previously and may therefore have degraded slightly via m o i s t u r e a b s o r p t i o n , etc.
CB 7
Description
Composite beam no.
0.6224 1.0078
0-9335 0.8822 0931 7 0"8528 0.899 7
E.w/ E,h,.,,,
TABLE 2 Theoretical and Experimental Flexure M o d u l u s a n d Loss E'actor of Longitudinally Reinforced C F R P Beams
3.4884 12.0000
8-33 28.5714 12.5000 18.1132 6-66
q,.,.#/ tl,h,.,,,
B(
'~ .~
5,.
Effect of matrix complex moduli on CFRP laminae properties
83
TABLE 3 Theoretical and Experimental Modulus and Loss Reactor of Transversely Reinforced CFRP Beams Composite beam no.
TB 1 TB 2 TB 3 TB4
Description
R B 6 + 6 0 % Vf aligned CCF RB8 + 60% Vf aligned CCF RB5 + 60% Vr aligned CCF RB2 + 60% Vf aligned CCF
Experimental
Theoretical
Er GPa
qr
Er/SG° GPa
Er GPa
qr
1.89
0-3232
1"33
2.48
0.3244
2.42
0.066 0
1.65
3.87
0.068 5
--
--
--
2"28
0"325 7
--
--
--
0.47
0.949 2
a SG = specific gravity.
TABLE 4 Theoretical and Experimental Shear Modulus and Loss Factor of CFRP Rods Composite rod no.
CRN1 CRNIa CRN4 CRN5 CRN7 CRNI0 SCRN1
Description
RB6 + 60% aligned CCF RB6 + 60% Vf aligned CCF RB8 + 60% Vf aligned CCF RB3 + 60% Vf aligned CCF RB2 + 60% Vf aligned CCF RBI3 + 50% Vr aligned CCF RBI3 + 60% Vf aligned short (2.25 mm) CF
Experimental
Theoretical
Gc
q,
Gc
G Pa
(y = 200 tas)
G Pa
q,
0.52 1-22 1.45 0.95 0.24 4'12
0.658 9 0.3200 0.0740 0.33 0.4649 0"224
0.95 0.95 1.54 0.70 0.17 2"08
0.344 0 0.3440 0"075 3 0'345 6 0"947 2 0"0174
5'! 1
0"0087
2.57
0"0172
All fibres EHM-S.
3 EXPERIMENTS Prior to experiments being carried out on the unidirectional CFRP specimens, the flexural moduli and damping ofapproximately 40 resins were measured using the well-established, free-free beam method described later. Unfortunately, numerous problems were encountered when using this method as a result of the high flexibility of the resins) Results from these
T..4. Willway, R. G. White
84
I0.0
i.5
]
/ M(Idu/us
__ _ L ' ~ . : _ . - 7 -. . . .
8,0
'~ I 1
Resin Samo~
. /
7.0
~.~.~'-~.__"~_
.5"=:.:--~--~
~.o
1.4
RBI RB2 RB3 RB4
..
1 / I .o 4 o,~, ;.
Desw/~If~n
RB5
. . . . . . . .
i o.~
..............
4o.;,
7
~
'w"~t~t'..C..7,.~.=-
_
Loas Poctor
"-
"".',--,......,,..~ ~
.~
_-
..........
i 0..'~
2: .._-"==.-,,.,,~ 4~J
o.:~ 0.t
6,0 L l
2
3
4
5
Log Frequency (Hz)
Fig. la.
Variation of resin modulus and loss &ctor with frequency at 20<(L
tests did however, provide an insight into the relative stiffness and damping of the resins enabling a small number to be selected (see Table ] and Fig. 1) for further more sophisticated tests using a Polymer Laboratories dynamic mechanical thermal analyser (DMTA). This apparatus measures the power input to and the phase difference between the excitation and response of a
k
...........
x /
/ 50
60
70
80
90
q 2,0 4,.9
.....................................
io.o F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I00
, 110
120
130
140
150
\ \
/\ \/
t60
Temperofure f ~C)
Fig. lb.
Viscoelastic properties of RB4.
170
..,~oooo~, j o.,~ "~/ 7o-'
180
190
200
Effect of matrix complex moduli on CFRP laminae properties
85 1.5
10.0
1.4 1.3
1.2 1.1 1.0
9.0
0.9
0.8 0.7
~
0.6 0.5
8,0
0.4 0.3 0.2 0.1 7.0 -20
0 -10
0
10
20
30
40
50
60
70
80
90
1O0
110
120
Temperature ( "C )
Fig. lc. Viscoelastic properties of RB6.
non-resonant beam from which it calculates the dynamic moduli and loss factors of a small cantilevered specimen at various points in a specified temperature range. The effect on results of variation in frequency were predicted by computer. 7 Consideration of the results of these experiments enabled a number of highly dissipative resins to be identified for use in the CFRP composite experiments and theoretical investigations. The longitudinal flexural, transverse flexural and transverse shear moduli and loss factors of the various CFRP specimens described in Tables 2-4 were measured using the following experiments. Although no sophisticated N D T method was used to assess the quality of the specimens, careful manufacture, visual inspection and comparison between theoretical and measured elastic modul,i were considered to give an indication of specimen integrity. 3.1 Measurement of the flexural modulus and loss factor of a free-free beam The transverse and longitudinal flexural properties of the CFRP laminae and resin moduli and damping were measured using this technique, developed by Wright 9 and White s and shown in Fig. 2. The beam was suspended by cotton threads at the nodes of the first flexural mode of vibration. Any added mass would affect the natural frequency of the system and attachments could provide extraneous
86
T. A. Willway, R. G. White
Decade Oscillator
Frequency Meter
i l Power
Amplifier
2 Beam
Oscilloscope
I
~ Switch Ink-Jet Recorder
l
I
Vibrator Filter
Beam Optical Vibration Transducer
.... ] "
;F
OVT Driver and f i l t e r unit
Fig. 2. Block diagram of apparatus for free-free flexural tests. damping. A method of excitation was found which required no added mass. A plate attached to a small vibrator was placed in close proximity to the beam centre. Air coupling between the vibrating plate and the beam was sufficient to excite the beam at resonance. Similarly, a non-contacting transducer (optical vibration transducer) was used to avoid added mass for response measurement. The beam was excited at its first resonance tYequency via the decade oscillator-power amplifier-ammeter circuit shown in Fig. 2. The excitation frequency was measured using the frequency counter, accurate to 0-1 Hz, from which the dynamic flexural modulus was found from the natural frequency equation of a homogeneous Euler Beam:
48n2J~ L 3rn E = (22.3729)2bd 3
(9)
where E = dynamic flexural modulus of beam;j], = first natural frequency of
Effect of matrix complex moduli on CFRP laminae properties
87
beam; L = length of beam; m = mass of beam; b = width of beam; d = thickness of beam. Having noted the natural frequency, the excitation was switched off and the decay of the beam response signal, filtered to remove the low frequency rigid body motion and noise, recorded on the ink-jet recorder. The beam loss factor was then calculated from the decaying waveform as follows: ~/=
t(Q/QN) nN
00)
where Nis the number of cycles between the measured peaks of amplitude Q and QN. The suitability of the apparatus for measuring material properties was validated in Refs 8 and 9, by attempting to measure the modulus and damping of a duralumin (r/< 3 × 10-6) specimen. Extraneous losses were found to be negligible. Results for resin, longitudinal CFRP and transverse C F R P specimens are given in Tables 1, 2 and 3 respectively. 3.2 Measurement of the shear moduli and loss factors of the unidirectional CFRP laminae The apparatus, instrumention and procedures used to measure the shear moduli and loss factors of the CFRPs are detailed in Ref. 3. Once again, the extraneous losses in the apparatus were found to be negligible by means of an experiment with a duralumin specimen. 3 Experiments were carried out on solid cantilevered C F R P rods of 7 mm diameter undergoing steady state torsional vibration as shown in Fig. 3. The dynamic torsional load was applied by two coil-magnet systems, driving an inertia bar attached to the rod at its free end. The response was monitored by strain gauges on the specimen, being restricted to low amplitudes (maximum surface shear strain amplitudes < 300/~s) in order to avoid damage to the specimens. As the mass of the specimen was negligible compared with the mass of the driven system, the specimen and drive system could be considered as a single degree of freedom system with lumped parameters when driven at the first resonance frequency (taken as the frequency at which the exciting current was a minimum for the specified strain response). The shear moduli of the rods were then calculated from the natural frequency equation of an inertia loaded rod, represented as a single degree of freedom system as follows: 8~zf~2JL
Gc -
R4
(11)
where fr = measured resonance frequency; J = moment of inertia of drive
T. A. Wilhvay, R. G. White
88
C t,lj,H~
"/'~
~M~/IIIL/I
TOP CLAMP
SPECIMEN ( clamped-clamped or cantilevered )
STRAIN GAUGES
'"
INERTIA BAR
/ K, ~]_~
DYNAMIC FORCES SPECIMEN
STRAIN GAUGES
(clamped-clamped onlyl SHIM
1/llllTtflllf
7
BOTTOM CLAMP
Fig. 3a. Combined loading apparatus.
/
2o.o
/
:
DiJralumln end pieces
\. \
Working Section
~Oo-- 7.0
I.o.o I
10 i = 5 . o 1
t90.0 210.0
Fig. 3b. Cantilevered specimen.
system (measured as in Ref. 8); L = length of rod between clamps; R = radius of rod. Results are given in Table 4. The loss factor of the rod was calculated from measurements of the input current and output voltage, frequency and specimen surface shear strain amplitude at resonance. 3 These values enabled calculations of the energy dissipated and energy stored in this system at resonance to be made, from which the loss factor ~]s-
iVo (03j02 (derived in Ret: 3)
(12j
where Vo=oUtput voltage; / = i n p u t current; ~o=resonance frequency (rad s - ~); 0 = rms rotational displacement of inertia bar, was calculated and results compared with theoretical predictions in Table 4.
Effect of matrix complex moduli on CFRP laminae properties
89
4 DISCUSSION OF RESULTS The initial phase of the project was the selection of some highly dissipative matrix materials. Existing manufacturing equipment and experience restricted the study to epoxies although cursory examinations of new thermoplastics such as poly(ether ether ketone) (PEEK) revealed inferior damping properties. Experimental results for resins are summarised in Fig. 1 and Table l, illustrating the range of dynamic properties achievable by varying flexibiliser and hardener content. The energy dissipated in a material at a specific stress is proportional to the quotient of its loss factor and modulus. However, at a specific strain, energy dissipated is proportional to their product. In a unidirectional FRP lamina, the former situation occurs in longitudinal flexure, the latter in transverse flexure and shear. Consequently, with a view to subsequent flexural tests on CFRPs, Table 1 includes the product of E= and ~/m(Emr/m). Consideration of these results reveals that resins conventionally used in CFRP such as RB4, 7, 10, 11, 13 have a high modulus but low loss factor and Emt/m product. Composites incorporating such resins will therefore have a low flexural energy dissipation and loss factor. For a specified strain amplitude, optimum energy dissipation appears to occur in resins having intermediate values of modulus and damping, e.g. RB 3, 5, 6. Theoretically, therefore, CFRPs incorporating RB3, 5 and 6 would have increased longitudinal flexural energy dissipation and loss factors. However, it should be noted that resin properties are highly temperature and frequency dependent, as shown in Fig. 1. It is therefore essential that resin materials are selected after consideration of operational temperatures and frequencies. Having determined the properties of the resins, their effects on the stiffness and damping of unidirectional CFRP laminae were studied theoretically and experimentally. Theoretical predictions of moduli suggest that the modulus of the composite is not significantly affected by the modulus of the resin. For example, CB7 manufactured of high modulus RB13 has a predicted modulus of 208.11 G N m -2, whilst CB2 using RB2 (low stiffness) has a modulus of 207.08 GN m- 2, there being less than 0"5% difference in moduli. As expected, the theoretically predicted value of composite loss factor is proportional to the Emr/mproduct of the component resin. It is interesting to note that the order of increasing loss factor is not the inverse of the order of increasing modulus, thus giving evidence of the ability of FRPs to defy the 'law' of conventional materials that modulus is approximately inversely proportional to damping.
90
7-. A. Willway, R. G. White
CB6 consisted of RB13 and aligned discontinuous fibres. Whilst the predicted loss factor is greater than that of CB7 (RB13 with continuous fibres) it is not as great as CB4 and 6 which are made of continuous fibres in high dissipative resin matrices. Theoretically, the approach to increasing CFRP damping by using highly dissipative matrix materials therefore looks promising. Unlike the longitudinal flexural modulus, the transverse flexural modulus of a unidirectional lamina is theoretically dependent on both the matrix and fibre moduli. Furthermore, the transverse flexural loss factor is proportional to the quotient of the matrix loss factor and modulus (rl,~/Em) and the composite transverse flexural modulus. Therefore, in theory, matrix properties have a significant effect on both the transverse flexural moduli and loss factors of the C F R P laminae (see Table 3). The theoretical predictions of the shear moduli of the unidirectional CFRP specimens according to eqn (7) (Table 4) suggest the shear modulus of the C F R P to be highly dependent on that of the matrix material. Consequently, the predicted values of the shear modulus for a specimen manufactured from a low modulus resin is much lower than that ofa CFRP made from a conventional, stiff resin. Comparing for example, CRN la and SCRN 1, the ratios of the predicted composite moduli are 2"7:1. The ratio of the component resin moduli is 2"78:1. Whereas the theoretical predictions of the composite flexural loss l~actor are directly proportional to the matrix energy dissipated at a given strain (Emqm), eqn (8) includes a direct relationship between the composite shear loss factor and the matrix energy dissipated at a specific stress, i.e. (r/s ~ ~m/Gm), and the shear modulus of the composite. However, the interrelationship between the shear moduli of the composite and matrix results in ~/s being effectively proportional to the matrix loss factor, r/m, at practical volume fractions. This is reflected in the predicted values of q~ in Table 4. Comparison of theoretical results for CRN1 and CRN5 having identical matrix loss factors but differing matrix moduli shows the effect of the matrix modulus to be less than 0.5% (~/s= 0.3440-0.3456). Thus, in theory, the dynamic properties of a unidirectional CFRP in shear are derived from the corresponding matrix shear properties. As predicted theoretically, experimentally determined values of the longitudinal flexural moduli (Table 2) showed little variation with resin properties. They were, however, generally lower than predicted, possibly a result of the fibre modulus being lower than that used in the calculations, and/or slip at the fibre-matrix interface due to imperfect bonding resulting in friction losses. The especially low value of modulus for CB6 may have been attributed to water absorption and other ageing processes discussed
Effect of matrix complex moduli on CFRP laminae properties
91
later. The modulus measured in an identical manner at the time of manufacture 2 was similar to the predicted value. The measured values of loss factor do not, however, follow theoretically predicted trends. The maximum composite loss factor was obtained from the CFRP manufactured from resin RB2, i.e. that resin having the maximum loss factor. The general trend appeared to be loss factor increasing with increasing resin loss factor, or decreasing resin modulus (Table 2). The measured loss factor of CB2 was twenty times that of CB3, whilst the modulus was decreased by less than 6% (actual and specific, (i.e. E/(specific gravity)) values). Similar trends were shown by CB1, 4 and 5, slightly lower increases in loss factor being achieved for a lower sacrifice in stiffness. Comparing theoretical and experimental results, measured values of loss factor were always greater than predicted values, r/~.ffr/th,o~ being 28.5714 for CB2. There are three possible sources of this discrepancy, assuming the experiment and theories to be inherently accurate, as we shall now discuss. It is possible that, even in the thin CFRP beams used in the experiments (thickness d = 3 mm), shear deformation may have a significant effect on the measured natural frequencies and damping. Shear deformation in the matrix will dissipate a substantial amount of energy in addition to that dissipated in tension-compression due to flexure. An additional or alternative explanation would be an imperfect bond between the fibre and matrix. Slip and the resulting friction at the fibre-matrix interface ('interfacial slip') would result in increased energy dissipation and damping, but a decreased modulus, as observed in the experimental results. However, it is unlikely that interfacial slip alone is responsible for the difference in measured and predicted values, and the overestimates of the moduli and underestimates of damping may simply be a result of fibre data used being inaccurate, manufacturers supplying approximate ranges of properties only. It may be concluded that these theoretical and experimental results indicate that it is feasible to produce a unidirectional CFRP lamina having a high longitudinal flexural modulus and loss factor by using a highly dissipative resin. Experimentally measured values of the transverse flexural moduli of two CFRP laminae show qualitative agreement with the theoretical prediction of dependence on matrix modulus (Table 3). Measured values were, however, much lower than those predicted by the mechanics of materials approach, eqn (5), possibly as a result of an imperfect bond a t the fibre-matrix interface. Although insufficient specimens were tested to establish whether the transverse damping was governed by the matrix
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modulus or loss factor, measured values were only slightly lower than those predicted using the simple energy equation (6). Unlike the measured values of moduli, the measured values of loss factor gave no indication of the occurrence of interfacial slip. It can therefore be concluded that the transverse flexural properties of unidirectional C F R P laminae are highly dependent on the corresponding matrix properties. Measured shear moduli follow trends similar to predicted values, i.e. the composite shear modulus increases with increasing resin shear modulus (Table 4). Measured values for SCRN1 and CRN10 did, however, appear high whilst that of CRN1 appeared low. A possible explanation of these results will be put forward later. Experimentally determined values of shear loss factor are given in Table 4. Note that investigations of the strain dependence of the CFRP shear loss factor, 3 not detailed here, revealed essentially C F R P linear shear damping. The experimentally determined values of material loss factor shown in Table 4 behave in the same manner as the experimental values of flexural loss factor, i.e. increasing with increasing resin loss factor or decreasing resin modulus as predicted theoretically. Consequently, the order of increasing modulus is approximately the inverse of the order of decreasing loss factor. However, the values of shear loss factor measured for CRN1, CRN7 and SCRN1 do not conform to the generally observed trends, SCRN1 and CRN7 having lower than expected loss factors whilst that of CRN1 and SCRN1 are high. Both CRN1 and C R N I a contain resin RB6 and 60% Vf aligned continuous fibres (EHM-S). Theory therefore predicts that the composite shear moduli and loss factors should be the same. This, however, was found not to be the case. The difference in composite properties was probably a result of different moulding techniques used in their manufacture? The discrepancies between the theoretical and experimental moduli and loss factors of SCRN1 and CRN10 may be attributed to the manufacture. Both specimens were made for a previous set of experiments, 2 at a time when moulding techniques were not so well established, especially when using short fibres. The volume fraction of each specimen may therefore not be as specified, possibly being higher than the 50% and 60% specified for CRN10 and SCRN1 respectively. Furthermore, any misalignment of fibres would further increase the shear modulus and decrease shear loss factor. These factors would explain the achievement of a shear modulus greater than that predicted for SCRN 1 and CRN10 and loss factor smaller than predicted for SCRN1. The high measured loss factor of CRN10 could possibly be explained by some ageing effect (see later) on the resin, significantly increasing its loss factor but not affecting its modulus, although a similar effect would be expected in the resin of SCRN 1.
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Measured loss factors were found to agree well with those predicted by the simple energy approach, e q n (8). It is obvious from both the predicted and measured shear dynamic properties of unidirectional CFRPs that the shear modulus and damping of the CFRPs are dependent on the corresponding properties of the resin matrix. Therefore, whilst the use of low modulus resin in CFRP composites in order to increase composite damping may have no significant effect on the stiffness of the composite in the fibre direction, the effects on the stiffness in other directions are substantial. Although this paper is primarily concerned with the effects of matrix moduli on CFRP dynamic properties, the modulus and, particularly, damping of any CFRP are sensitive to many other factors which are difficult to control. The quality of the laminae (e.g. void content, fibre misalignment) and the integrity of the bond between the fibre and matrix significantly affect composite properties. Furthermore, the properties of the resin matrix, in addition to being temperature and frequency dependent, may change with time as a result of ageing effects such as water absorption, chemical attack and UV degradation. Although these factors have been taken into consideration in the discussion of the results of this research, it may ultimately be necessary, not only to consider their influences on material properties, but to use them in achieving the structural properties required in a certain practical environment.
5 CONCLUSIONS From measurements of the energy dissipation of a number of resins it was concluded that there is a considerable variation in energy dissipation possible with epoxy resins. Conventional stiff, lightly damped resins were found to dissipate very little energy. Optimum energy dissipation occurred in the intermediate stiffness and damping resins, which were just within the glassy region of the glass transition peak at test frequency and temperature. Indeed, it was evident that the dynamic properties of visco-elastic resins are extremely frequency and temperature dependent. Flexural tests on unidirectional continuous CFRP specimens proved that, as theory predicts, the flexural moduli were almost independent of the matrix. The results of the theoretical and experimental studies show that it is feasible to produce a unidirectional CFRP lamina having a high longitudinal flexural modulus and loss factor by using a highly dissipating resin. The transverse flexural and shear properties of the unidirectional continuous CFRP specimens were found to be governed by the matrix
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properties, the moduli and loss factors being proportional to the matrix moduli and loss factors respectively, verifying theoretical trends. Theory did, however, overestimate the modulus and underestimate the loss factor. Discrepancies between theory and experiment in both shear and flexure suggests other mechanisms of energy dissipation in the composite, such as slip at the fibre-matrix interface due to imperfect bonding resulting in frictional losses. It is evident that whilst the use of a low modulus, highly dissipative resin matrix in C F R P composites to increase laminae damping may have no significant effect on the stiffness of the composite in the fibre direction, the effects on the stiffness in other directions are substantial. However, if the laminae were laminated such (hat a fraction of the fibre content was orientated in each of the directions of the principal axes, the fibres would then contribute to composite stiffness in all directions under load. A study of the effect of flexible matrix C F R P laminae on C F R P laminate properties is the subject of a subsequent paper.
REFERENCES 1. Gibson, R. F., Development of damping composite materials. In Proceedings of the Advances in Aerospace Structures, Materials and Dynamics Symposium on Composites, Boston. ASME Aerospace Symposium Series AD, Vol. 067.
American Society of Mechanical Engineers, New York, 1983, pp. 89. 2. White, R. G. & Abdin, E. M. Y., Dynamic properties of aligned short carbon fibre reinforced plastic beams in flexure and torsion. Composites, 16(4) (1985) 293-306. 3. Willway, T. A., Stiff, light, highly damped CFRPs and the effect of complex loads on damping. PhD thesis, University of Southampton, UK, 1986. 4. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, 1975, 5. Cox, H. L., The elasticity and strength of paper and other fibrous materials. Brit. J. Appl. Phys., 3 (1952) 72-9. 6. Hashin, Z., Complex moduli of visco-elastic composites: II. Fibre reinforced materials. Int. J. Solid Struct., 6 (1970) 797-805. 7. Townend, D. J., A computerised technique for producing WLF shifted data from a single frequency temperature scan without graphical manipulation. Technical Memorandum AMTE(M) TM 83307, ARE (Holton Heath), 1983. 8. White, R. G., Some measurements of the dynamic properties of mixed carbon fibre reinforced plastic beams and plates. Aeronaut. J., 79 (775) (1975) 318 25. 9. Wright, G. C., Dynamic behaviour of fibre reinforced plastic beams and plates. PhD thesis, University of Southampton, UK, 1973.