Delamination propagation in rotating carbon-epoxy composite shaft

Engineering Fracrure Methanics Vol. 49, No. I, pp. 121-132, 1994 Copyright ‘c 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0013-7944(94)EOO73-P 0013-7944,‘94 $7.00 + 0.00

Pergamon

DELAMINATION PROPAGATION IN ROTATING CARBON-EPOXY COMPOSITE SHAFT G. CHANDRAMOULI Defence

Research

and Development

Laboratory,

Hyderabad.

India

K. GUPTA* Department

of Mechanical

Engineering,

Indian

Institute

of Technology,

New Delhi, India

R. K. PANDEY Centre

for Materials

Science and Technology,

Indian

Institute

of Technology,

New Delhi,

India

Abstract-A carbon
INTRODUCTION FIBRE REINFORCED composite materials are being used increasingly in the aerospace, automobile, chemical and marine industries due to their superior properties like high strength to weight ratio, low thermal conductivity, high corrosion resistance, etc. Advanced composite materials like graphite-epoxy exhibit better fatigue (tension-tension) resistance than steel, aluminium or glass reinforced plastics. Delamination, which is also known as interlaminar cracking, is the separation of two adjacent layers in the thickness direction and is considered to be one of the major flaws in composite components. Delamination can occur either during manufacturing or during usage of the component. The presence of such flaws in a component make it fail much earlier than its designed life as achieved by the conventional strength of materials approach. Study of flaws like delamination and through laminar cracks in composite components helps one to arrive at a good design and prediction of the life of a component. Fracture analyses of composites are being carried out in laboratory scale modelled laminates in each of the three (I, IT or III) modes of failure. However, mixed mode fracture is the general phenomenon in actual problems as is also the case in the present study and cannot be simulated on laminates. Laksimi et al. [l] studied experimentally and analytically a Double Cantilever Beam (DCB) specimen of carbon+poxy laminate for appearance of microcracking and onset of delaminations in Mode I. Expressions for mode I fatigue delamination resistance are obtained [2] from experiments on DCB laminates. Lu and Liu [3] found that delamination caused by impact has a peanut like shape and is generated as a result of mismatch of bending stiffness between adjacent layers. Kim and Hang [4] developed analytical expressions for strain energy release rates using the Finite Element Method for various fibre angle ply laminates. Studies have also been carried out [5] on effects of delamination length, fibre orientation and ply thickness on interlaminar fracture. It has been shown analytically [6] that the total strain energy release rate for an interfacial crack between two anisotropic solids may be defined though it is not well defined for individual modes. Using the “Strain Energy Density factor approach” by Sih [7], Pate1 and Pandey [8] studied mixed mode crack growth and gave a procedure for estimation of fatigue life. Tensile fracture in a circumferentially cracked filament wound GRP tube has been reported in a recent work [9], *To whom

correspondence

should

be addressed 121

G. CHANDRAMOLJLI

172 350

_+_-------A-____ 1

%P9-

200

I

PI c/l.

DELAMINATION

1-

/

2

__

______‘--f-

8108.6

A__Iz-_-_

___.!+_____J \ FIBRE DIRECTION

/-Pi

1010 pDELAMINATION 1. 10x10 2. 10x20 3. 10x15 4. 10x25

Fig. 1. Delaminations

in composite

SIZES

mm mm mm mm

shaft

whereas fatigue crack propagation behaviour in carbon-epoxy composites, etc. has been presented by Kobayashi er al. [lo], Hang and Han [I 1] and Kim [12]. Dentsoras and Dimaroganas [13] have investigated fatigue crack propagation of a metallic rotor with multiple disks mounted and excited at one of its resonances. As the crack propagates, the stiffness of the system comes down increasing the flexibility, thus causing a gradual shift from resonance and reducing the dynamic response. Some recent studies of applications of nonmetallic composite shafts are to helicopter tail drive rotors [14], drive shafts for automobiles [15], aircraft generators [ 161 and several other rotor systems [I 71. The various research trends in composite shafting are towards optimization studies [18], rotor dynamic analysis [ 191 and supercritical operations [14], balancing [20] and modal testing [21, 221. In view of the above applications, the study on aspects of fatigue and fracture in rotating composite shafts is considered to be of primary importance and has been the motivation for the present work. This paper first presents some background studies on delamination in a simple composite specimen. Fabrication of a tubular composite shaft and assembly of an experimental rotor setup is described. Details of tests on propagation of delamination in a rotating shaft along with theoretical stress analysis and calculations based on a macromechanics approach are given. Material constants c and n are evaluated from the test data. Direction of delamination propagation is studied and experimental results are compared with theoretical predictions. FABRICATION

OF COMPOSITE

SHAFT AND TEST SETUP

The carbonepoxy composite shaft is fabricated by the “Filament Winding Technique” using a 1250 mm long tapered aluminium mandrel. The mandrel is rotated about one axis as continuous fibers are passed through a bath of resin and wet wound helically on the mandrel. When half the thickness is achieved, a thin film (0.14 mm) of Teflon is introduced deliberately to create the delaminations at selected locations as shown in Fig. 1. Wet winding is continued until the required thickness is achieved. Coordinates of the delamination zones were noted with respect to the mandrel

HORIZONTAL

PROBE

OlGlTAL VECTOR FILTER

VIBRATION MONITOR

VERTfCAL PROBE u

MASS SPECTRUM ANALYSER

+ -9

COMP. SHAFT 1. /SURFACE

PLATE

Fig. 2. Schematic

’

COMPUTER

RIG10 - FRAME

I diagram

of experimental

setup and instrumentation.

Delamination

in carbon-epoxy

Fig. 3. Experimental

composite

test rig.

shaft

123

124

G. CHANDKAMOULI

--m---e

<‘I d

,RADIOGRAPHED I------l I I -me -m---BI_ --r

AREA

__

p-----------c---, I

I

I

I-

A7 ZERO

I

CYCLES

I L-----,,-------,.A

I-

-----w.‘------‘y

I

I

I

I

I I

I

I I-

AFTER

-I

-------I------

AFTER L---------em

-

c--------

CYCLES

-A

1

I

I

I

I

--------a

r -I

104~10~

w-c--

I

L

6.8 x,O'CYCLES

-I

---_

1

-----c--c--

Y , $ii,, na 4

AFTER16-8 X 10' CYCLES

D /T'

I I

I -.. l- _----_---____

I -I

AFTER

20.0 ~10' CYCLES

V-VOID O-DELAMINATION

Fig. 4. Delamination

growth

at various

cycles.

Delamination

in carbon+poxy composite shaft

125

during wet winding. After the final machining of the shaft is completed, the coordinates of the delamination zones are transferred back and marked on the shaft. Hercules IM6-G carbon fibers were used with epoxy resin LY556 and hardener HT972. A fiber orientation of f45” was selected. The composite shaft had eight layers. A PULTREX filament winding machine with CADFIL software was used for fabrication of the shaft. After wet winding, the mandrel was placed in the oven for 4 h for permanent setting. The composite shaft was turned to its final dimensions. The length of the finished tubular shaft was 1008 mm with an average wall thickness of 4.8 mm. Through the lower diameter end of the mandrel, the shaft was extracted. The hollow composite shaft was press fitted on the knurled surfaces of the two metallic flanges one at each end. A cylindrical mass was mounted on the composite shaft with the help of a tapered split sleeve, SKF H 3124. An internal matching taper was provided on the mass to facilitate its clamping on the shaft through the sleeve. As the nut of the sleeve is rotated, the sleeve end with higher diameter is pulled into the mass tightening itself onto the shaft. Two self-aligning rolling element bearings, SKF 1209 K, are mounted on the two flanges. The rotor is then placed in the bearing housing which in turn is clamped onto the T-slots of the surface plate. The rotor is driven by a variable speed d.c. motor through a belt drive. Two eddy current probes are mounted through a rigid frame, one vertically and another horizontally to measure shaft displacements at the lumped mass as shown in Fig. 2. One eddy current probe is used at one of the end flanges to act as a keyphasor for reference purposes and for picking up the actual speed of the shaft. Figure 3 shows the experimental test rig.

TESTS

AND EXPERIMENTS

Various tests were carried out before starting the experiment on crack propagation. Rotor dynamic measurements comprising steady state unbalance were made during continuous operation of the shaft. Material

testing

The tests were carried out as per ASTM standards on the samples cut during the machining stage. Composite shaft density of 1.35 x lo3 kg/m3 and resin content of 44.45% by weight were obtained. Equivalent volume fraction of resin was 55%. Radiography

of the composite

shqft

Radiography of the delaminated zones was carried out by using tangential shots. The radiography level used for carbon-epoxy material was 65 kV, 4 mA, at a distance of 0.8 m. The radiography test was repeated at regular intervals after subjecting the composite shaft to a selected number of rotational cycles. Figure 4 shows delamination lengths at various cyclic loadings. Estimation

of the natural frequency

of the composite

shaft

The natural frequency of the composite shaft without the lumped mass is estimated as 135 Hz by conducting the rap test under non-rotating conditions with the help of an accelerometer. The stiffness k of the shaft assuming simply supported conditions is calculated from the following relation,

(1) where m is uniformly distributed shaft mass and equal to 2.06 kg. Thus, k works out to be 0.74 MN/m. Selecting first the critical speed to be around 2800 r.p.m., the lumped mass, M, required to be added at the midspan of the shaft is estimated from the following relation, k A4 + 0.5m

= 46.7,

which works out to A4 = 7.5 kg. Accordingly, a mass was fabricated and mounted at the midspan of the shaft with the help of a tapered split sleeve. The total mass of the lumped mass and sleeve

G. CHANDRAMOULI

126

CI ul.

UP

.

m

i 11

I

II

1

I

11

11

AMP SCALE = 70 um/div ROTATION: CCW

AMP SCALE

= 70 um/div

TIME SCALE = 25 ms/div RPM (START)

Fig. 5. Unfiltered

uncompensated

= 1280

RPM (END) = 1280

whirl orbit.

with nut was more than 7.5 kg. The natural frequency of the rotor with lumped mass and sleeve was measured by the rap test using an accelerometer. It was found to be 42.1 Hz which corresponds to 2526 r.p.m. Experiment ,for crack propagation

The composite shaft was run at various speeds and deflections were measured in the horizontal and vertical planes with the help of two eddy current probes. From the observed deformations, a speed of 1280 r.p.m. (21.3 Hz) was selected for continuous rotation. Vibrations were monitored with the help of a digital vector filter (DVF). After each 5 x 10’ cycles, the composite shaft was taken out from the rig and delaminations were radiographed and corresponding crack lengths measured. At 1280 r.p.m., orbital analysis of the shaft was performed by using a microprocessor (HP 9836) based data acquisition and analysis system for rotating machinery dynamics. The unfiltered and uncompensated orbit generated along with the time base plot for one revolution is shown in Fig. 5. The corresponding filtered but uncompensated components in horizontal and vertical directions are given in Table 1. The run out component was fairly large and was obtained by measurements at a slow roll speed of 237 r.p.m. The synchronous compensated whirl orbit (Fig. 6) is obtained by subtracting the run out component from the filtered uncompensated components at 1280 r.p.m. The elliptical orbit gives rise to a fluctuating stress. From the vibration signals (Table 1) obtained through both the horizontal and the vertical probes, the 1 x component (filtered at the same frequency as the rotational speed) was found to be predominant and other components like 2 x , etc. were negligibly small. Thus the whirl orbit was predominantly synchronous. The nature of the fluctuating load at any instant is different at different locations along the shaft length and the circumference. In the present experiment it was ensured that the delamination under study was always in the same orientation (Fig. 7) as that of unbalance and the delamination zone was always under tensile loading. The magnitude of tensile stress thus varied between the two limits o,, (tensile) to u,, (tensile) which correspond to deflections represented by semi-minor and semi-major axes of the whirl orbital ellipse. The stress variation can be represented by a mean stress (T and a fluctuating component gv. It may be noted that a non-rotating shaft/beam subjected to dynamic transverse loading in one plane only will have Table Speed (r.p.m.)

1. Steady

state unbalance

1 x component pm (PP)

Phase (deg.)

237 231 1280

308 303 499

-40 - 127 - 3I

1280

430

- 121

response

measurements Remarks

Run out horizontal Run out vertical Horizontal dynamic signal including run out Vertical dynamic signal including run out

Delamination in carbon-epoxy

composite shaft

127

TENSILE

\

=rnin

TENSILE

0’

t

Fig. 6. Delamination orientation in one whirl orbit.

alternating stress with a zero mean value. The composite shaft was loaded for a total of 23 x 10’ cycles, during which time radiographs were taken at regular intervals to monitor the delamination growth.

STRESS ANALYSIS The gross mechanical follows,

properties

E, = E2 = E,, = E, = E,,, = V, = V, = vr = v, =

OF THE COMPOSITE

(from the Rule of Mixtures)

Young’s modulus of the composite in principal Young’s modulus of the composite in principal fiber longitudinal Young’s modulus (275 GPa) fiber transverse Young’s modulus (32 GPa) Youngs modulus for matrix (3.5 GPa) fiber volume fraction (0.42*) matrix volume fraction (0.55*) fiber Poisson’s ratio (0.2) matrix Poisson’s ratio (0.35) V ,* = 0.28 V2, = 0.334 Gr = fiber shear modulus = 127 GPa G, = matrix shear modulus = 1.35 GPa Glz = shear modulus of the composite = 3.23 GPa.

SHAFT of the composite

axis 1 direction axis 2 direction

used are as

(117 GPa) (7.0 GPa)

The stress analysis was performed using the Classical Laminate Theory. The composite laminate under consideration consists of eight homogeneous orthotropic layers of alternate +45” fiber angles. Stress-strain relations for orthotropic materials in a lamina are given by Jones [23],

where Q, are reduced stiffness coefficients for two-dimensional plane stress condition. Here, 0, and g2 are normal stresses in principal directions 1, 2 of the fibers and rIZ is the shear stress, L, , c2 and *Void fraction is estimated to be 0.026.

G. CHANDRAMOULI

THIN

LAYER

P/ (I/.

OF EPOXY

CARBON

Fig. 7. Matrix

cracking

FIBRE

model

rIz are the corresponding strains. Substituting the values for E, , E2, v,*, v2, and Glz, the reduced stiffness coefficients are calculated from the following expressions,

e,,= 1 - E1 ; \1,*I’?,

a*=*,,

E,

m

YE = , _ “,2v2, 1 Yhh = bl? which work out to, Q,, = 127 MN/m

Qz2 = 7.6 MN/m

Q,* = 2.1 MN/m

Qss = 3.2 MN/m.

Bauchau [ 181 has given an expression shaft as,

for the longitudinal

E= where I,, Z2, R, and R, are invariants ( f 45”). These are given by,

elastic modulus,

E, for a graphiteepoxy

4z,(z,+R,s,)-s:R; I, +I,-s,R,

+s,R,

(5)

’

and s, and s2 are stiffnesses,

dependent

on fiber orientations

Z, = (Q,, + Qz + 2e,d/4 1, = tQ,, + Qz - ~Q,z + 4Qd8 R, = @,I - Qzz)P & = (Q,, + Qx - 29,~ - 4QdP s, = i ; cos 28, i=l

(6)

sz= i f’cos4e,, j=, t where, t, = thickness

of the ith layer

t = total thickness

of the shaft = 5 mm

Bi = fiber angle of the ith layer N = number

of layers in the thickness

of shaft = 8.

Delamination

in carbon-epoxy

composite

129

shaft

Substituting these values, E is calculated to be 28 GPa. The second area moment of the cross-section of the shaft, Z, is 2.13 x lo6 mm4. Thus the flexural stiffness EZ is 6.0 x 10” N mm*. The composite shaft supported on two bearings is assumed to be a simply supported beam. From the dynamic deflections recorded in the experiment, the dynamic load is calculated to be 285 N and the maximum bending moment which occurs at the midspan is calculated to be 71,250 N mm. Accordingly, the maximum and minimum bending stresses are 1.80 N/mm* and 1.14 N/mm’, respectively.

FRACTURE

ANALYSIS

OF THE

COMPOSITE

SHAFT

Even though substantial tests have been carried out on composite materials, not much progress has been made towards the development of predictive failure procedures. Some fundamental problems exist in incorporating non-homogeneity and anisotropy of the material into continuum mechanics analysis. Additional uncertainties arise from voids and the defects that occur during manufacturing. General failure modes in composites are matrix cracking, fiber breakage, fiber matrix debonding and delamination. As the loads are increased gradually, the complex phenomena of combination of more than one mode of failure may take place. As we are interested only in one failure mode, i.e. delamination in the present work, the loading on the composite shaft is chosen at constant speed. Considering the analytical model (Fig. 7) suggested by Sih [7] for the crack propagation in a layer of isotropic matrix material sandwiched between two anisotropic solids, the stress intensity factors for mode I and mode II can be written as

K,= C#J (1)~ &

sin* /?

K,=$(l)g&sinflcos/?,

(7)

where u is half delamination crack length and p is the angle between the direction of loading and the crack plane. 4 (1) and $ (1) depend on the elastic constants and geometric parameters of the composite. For graphite-epoxy, $J (1) = 0.22 and $ (1) = 0.07. The delamination at zone 3 is similar to the analytical model in Fig. 7, where /? = 45”; a = 7.5 mm and is subjected to mixed mode (mode I and mode II) loading. Fracture is assumed to occur when a certain combination of stress intensity factors K,and K,,reaches a critical value. Assuming that the crack is propagating in a self-similar manner, the fracture condition would be [3]

where

K,=&cK,,

K,,=fiK,.

(8)

However, experiments conducted so far reveal that in mixed mode, the crack growth takes place at a certain angle with respect to the original crack. To deal with such a situation, the strain energy density factor approach [7] or maximum principal stress criterion [24] may be applied. The maximum principal stress criterion postulates that the crack growth will occur in a direction perpendicular to the maximum principal stress. If a crack is loaded in the combined mode I and mode II, the stresses at the crack tip can be expressed [25] as

e

cosT 1 r=QZ

cost

e

3

K,cos*?- jK,,sin 8

1

[K,sin 8 + K,, (3 cos 8 - l)].

(9)

The stress oB will be the principal stress if t = 0. This is the case for 0 = 8, where 8, is found by equating the shear stress to zero. Thus,

G. CHANDRAMOULI

130

el ul.

in the present By substituting values of K,, K,, and 0,. the direction of propagation evaluated to be -30”. The strain energy density factor, S, [7] for mixed mode fracture is given by, S = (a,, k ; + 2a,$,

kz + az2k ; + q3k :),

where the terms are defined in ref. [7]. In the direction to 0 = 0,, . Thus

as %i= O;

case is

(11)

of crack initiation,

e>O 802

S is minimum

with respect

(12)

at B =BO.

Rapid crack growth occurs when the minimum strain energy density factor rcachesazritical y&e. Substituting c = 1.8 MN/m*, a = 7.5 mm and the values of K, and K,, , the minimum strai~~nergy density function is evaluated through a computer program by varying vaIues of 8 from -9(3+-d +90” in increments of 5”. The direction of propagation of delamination using the above approach in the present case is found to be at -60”. In the present investigation it was observed from the radiographs taken at various angles that the propagation of delamination continued in the same pIane, and was caused due to cracking of the thin layer of resin between the laminas. It was also observed that the general direction of propagation is parallel to the fiber orientation. Thus the actual direction of delamination propagation is neither in agreement with the SED theory nor with the Maximum Principal stress theory. In fact, the actual direction of crack propagation lies at 45”, i.e. in between the directions predicted by the maximum principal stress and SED theories. Estimation

of material

The Paris equation

constunts [25] for fatigue crack propagation da dN

=

under mixed mode can be expressed

C (A&)“,

as (13)

where da/dN is the rate of crack propagation and C and n are material constants to be evaluated for the composite. The K,, is the equivalent stress intensity factor and AK,, is the range of K,, and can be expressed [25] as, AK,, = AK, cos3 % - 3AK,, cos2 % sin $.

(14)

Substituting the values of 0,, K, and K,, , the K,, is calculated for the maximum and the minimum tensile stress for different delamination lengths, as reported in Table 2. The half delamination crack length with respect to the number of fatigue cycles is plotted in Fig. 8 which can be used to obtain the da/dN values. The material constants C and n are determined from the graph in Fig. 9. The C and n values can be evaluated from the straight line drawn in Fig. 9 on the basis of least square curve fitting [26]. The material constants are obtained as, C = 3.59 x 10pl’ and 7 = 5.912 where da/dN

and AK,, are expressed

Table 2. Variation For stress Half delam. length (mm) 7.5 9.5 12.0 15.0

AK,

of stress intensity

1.80 MPa

AK,,

in m x 10-S/cycles

factors

9.67 10.89 12.24 13.69

with delamination

For stress 1.14 MPa

AK,,

AK,

34.32 38.65 43.44 48.55

19.25 21.67 24.34 27.22

(kNm-I-‘) 30.40 34.24 38.48 43.00

and kNm-‘.5,

AK,,

length AK,,

and stress da/dN

AK,,

(kNm-‘-5) 6.13 6.90 7.74 8.66

respectively.

(kNm-I.‘) 21.74 24.47 27.48 30.73

12.58 14.18 15.96 17.82

(m x IO-‘/cycles) 1.02 2.64 4.92 8.19

x x x x

10-4 1O-4 IO-’ 1O-4

Delamination

0

in carbon+poxy

5

10 NO. OF CYCLES

131

composite shaft

15 I xl’? 1

20

25

Fig. 8. Delamination length vs fatigue cycles

The above material constants can be used for estimating the number of fatigue cycles that a component can withstand once we know the crack length and stress levels. For example, if the limiting delamination crack length is assumed to be half that of the inner diameter (53 mm) of the composite shaft, the number of cycles that the delaminated shaft can withstand, at the same speed of rotation and stress levels, is estimated to be about 32 x lo5 cycles. If the shaft speed is increased from 1280 r.p.m. (used in the experiment) to say about 2450 r.p.m. (critical speed), the stresses in the shaft will increase considerably and the rate of delamination propagation is also expected to increase. Also the interlaminar matrix cracking which is taking place at 1280 r.p.m. may get converted into through lamina cracking (fiber breakage) and lead to ultimate failure of the shaft. Mechanism of cracking It is observed from the radiographs taken at various intervals (through 23 x lo5 cycles) that the delamination located below the central mass started propagating in the same plane by fracturing the thin film of resin between the adjacent layers. The rate of propagation is observed to be low for the speeds selected. No surface cracks appeared even after 23 x lo5 cycles of loading at the frequency of 1280 c.p.m. Just below the central mass, a void started growing at about 4.8 x lo5

1

10 b KIo kNm

-1.5

Fig. 9. Log-log plot; da/dN vs AK,,.

100

I32

G. CHANDRAMOULI

PI ~1.

cycles in addition to the created delamination. The void became fully fledged delamination after 16.8 x 10’ cycles. This may be due to the stress concentration present at that location. Out of the four delaminations created in the shaft, only the delamination below the mass started propagating and others did not propagate during the course of observation. CONCLUSIONS Based on the investigation, the following conclusions may be drawn. (i) A test setup for propagation of delaminations in a rotating composite shaft has been developed. The shaft is rotated in its subcritical range and shaft whirl amplitudes in horizontal and vertical directions are measured. It has been shown that the bending loads due to unbalance excitation will be quite different in comparison to bending loads in a non-rotating shaft/beam. (ii) From the values of material constants (C and n) generated for carbon-epoxy composite under fatigue bending loads due to unbalance excitation, the rate of delamination propagation can be estimated, if the stresses in the shaft are known. The life of the shaft can also be estimated for a known delamination length. (iii) A comparison between the theoretical and experimental studies shows that the direction of propagation of delamination in a rotating shaft does not change drastically and is in between the directions predicted by the “maximum principal stress” and the SED theories. REFERENCES []I A. Laksimi, Mode I interlaminar fracture of symmetrical cross-ply composites. Compos. Sci. Techno/. 41, 147 164 (199 I). PI R. H. Martin and T. K. O’Brien, Characterizing mode I fatigue delamination of comnosite materials. Proceedines of the American Society for Composites, 4th Technical Conference, Blacksburg, VA, pp. 257-263 (1989). front, Proceedings of the American Society for ]31 X. Lu and D. Liu. Strain energy release rate at delamination Composites, 4th Technical Conference, Blacksburg, VA. pp. 2777286 (1989). growth in angle-ply laminated composites. J. Compos. Mater. 20, 423 438 (41 K. S. Kim and C. S. Hang. Delamination (1986). problems in composite materials. J. Compo.~. Mu&. 17,2 IO 223 (I 983). [51 S. S. Wang, Fracture mechanics for delamination and C. T. Sun, Strain energy release rates of an interfacial crack between two anisotropic solids Kl M. G. Manoharan under uniform axial strain. Compos. Sci. Tech&. 39, 9991 I6 (1990). [71 G. C. Sih and E. P. Chen, Cracks in composite materials, in Mechunics o/‘Fructuw. Vol. 6. Martinus Nijholf. Dordrecht (1981). PI A. B. Pate] and R. K. Pandey, Fatigue crack growth under mixed mode loading. Fu/igue Engng Muter. SIrwmre.\ 4( I). 65-77 (1981). cracked filament wound GRP tubes under [91 Xue-Ning Huang, M. Kumosa and D. Hull, Fracture of circumferentially uniform tensile loading. Proceedings of 7th International Conference on Composites. Guangzhou, pp. 578 583 (I 989). N. Ohtni and S. Oghihara, Matrix effects on fatigue damage in composites. Proceedings of 7th [lOI A. Kobayashi, International Conference on Composites, Guangzhou, pp. 547-552 (1989). fracture and fatigue crack propagation of composite materials. Proceedings of 11 II W. Hang and K. S. Han, lnterlaminar the 7th International Conference on Composites, Guangzhou, pp. 6066613 (1989). composite laminates. Proceedings of 7th International [I21 Y. R. Kim, Effect of mean stress on fatigue behavior-of Conference on Composites, Guangzhou, pp. 621-626 (1989). Fatigue crack propagation in resonating structures. Oigng Frwrwe MN/~. [I31 A. J. Dentsoras and A. D. Dimaroganas, 34(3). 72ll728 (1989). R. F. Kraus and M. S. Darlow, Demonstration of a super critical composite helicopter power [I41 P. L. Hetherington, transmission shaft. J. Am. Helicopter Sot. 23-28 (1990). (1989). [I51 W. Hoffman. Fibre composite in the drive line. Plasrics Rubber Inr. 14(5). 4649 Development of a filament wound composite for an aircraft generator. 39th [IhI R. S. Raghava and R. S. Hammond, Annual Conference of the Reinforced Plastics/Composites Institute, New York, pp. l-24 (1984). in power transmission using filament wound composites. 34th International SAMPE 1171 B. E. Spencer. Advances Symposium and Exhibition, Vol. _. 34, Tomorrow’s Marerids: To&v. pp. 1109-l I I7 (1989). [I 81 A. 0. Bauchau, Optimal design of high speed rotating graphiteepoxy shafts. J. Compos. Marer. 17, I70 I8 I ( 1983). [19] L. M. dos Reis Henrique et al., Thin walled laminated composite cylindrical tubes-Part 111. Critical speed analysis. ASTM J. Compos. Technol. Res. 9(2), 5842 [20] E. S. Zorzi and J. C. Giordano, Composite

[21] [22] [23] [24] [25] [26]

(1987).

shaft rotor dynamic evaluation. ASME paper No. 85-DET-114 (1985). J. B. Andruilli, Measured damping and modulus ofcomposite cylinders. Proceedings of Damping ‘89, BCC-l-26 (1989). S. P. Singh and K. Gupta, Modal testing of tubular composite shafts. Proceedings of 34th International Modal Analysis Conference, Florida, U.S.A. (February 1993). M. R. Jones, Mechanics of Composire Muterids, McGraw-Hill/Kogakusha, Tokyo (1975). F. Erdogan and G. C. Sih. On the crack extension in plates under plane loading and transverse shear. J. Basic Engng 85, 519-527 (1963). B. David, Elementary Engineering Fracrure Mechanics. Martinus Nijhoff. Dordrecht (I 986). H. D. Young, Statistical treatment of experimental data, p. 121. McGraw-Hill, New York (1962). (Receioed

4 May

1993)