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A physically based micromechanical theory of macroscopic stress-corrosion cracking in aligned continuous glass-fibre-reinforced polymer laminates
(1998) 1659±1665
# 1998 Elsevier Science Ltd. All rights reserved
Composites Science and Technology
58
Printed in Great Britain 0266-3538/98 $Ðsee front matter
P I I : S 0 2 6 6 - 3 5 3 8 ( 9 7 ) 0 0 2 3 6 - 4
A PHYSICALLY BASED MICROMECHANICAL THEORY OF MACROSCOPIC STRESS-CORROSION CRACKING IN ALIGNED CONTINUOUS GLASS-FIBRE-REINFORCED POLYMER LAMINATES Hideki Sekine * & Peter W. R. Beaumont a
b
a
Department of Aeronautics and Space Engineering, Tohoku University, Sendai 980-8579, Japan
b
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
(Received 10 December 1996; revised 19 November 1997; accepted 11 December 1997)
environment. For example, their susceptibility to stresscorrosion cracking in acidic environments has received wide attention.1±15 Hogg and Hull,1 Noble et al.6 and Price and Hull7,10 studied the actual propagation mechanism of a stress-corrosion crack in individual glass ®bres by means of scanning electron microscopy. They observed the fracture surface of each glass ®bre to be characterized by a smooth mirror region surrounded by `mist' and `hackle' regions. They concluded10 from this observation that a stress-corrosion crack in a glass ®bre initiates and propagates to produce the mirror surface, followed by propagation across the remainder of the ®bre forming the mist and hackle regions, and ®nally extending into the surrounding matrix. Recently, one of the authors (HS) and his coworkers16 carried out a direct numerical simulation of the propagation of a stress-corrosion crack in a glass ®bre. For macroscopic propagation of stress-corrosion cracks in GRP, there are several reported experimental studies of a variety of GRPs in the presence of dilute hydrochloric,4,7,10,14,15 and sulphuric acids.4,5,9 As a result, it was found that the relationship between macroscopic crack propagation rate, da*/dt, and apparent stress-intensity factor for opening mode, K , at the macroscopic crack tip over a few orders of magnitude of macroscopic crack propagation rate can be expressed by a power law:
Abstract This paper is concerned with a micromechanical theory of macroscopic cracking
in
crack
propagation
aligned
due
continuous
to
stress-corrosion
glass-®bre-reinforced
polymer laminates, in particular, the relationship between the macroscopic crack propagation rate and the apparent stress-intensity factor at the macroscopic crack tip. A physically based micromechanical model of crack propagation
in
a
glass
®bre
as
a
result
of
stress-corrosion
cracking is proposed. It is based on the premise that a stress-corrosion crack initiates at a pre-existing surface ¯aw in a glass ®bre, grows and ®nally leads to breakage of the ®bre. We derive theoretically an equation for the macroscopic crack propagation rate as a function of the apparent stress-intensity factor. The theoretical prediction is in good agreement with experimental measurement. It is emphasized that the size of the inherent surface ¯aw aects signi®cantly the macroscopic crack propagation rate. For glass ®bres free of pre-existing ¯aws, the relationship can be represented by a simple power law with the value of power of 2. Finally, the macroscopic
#
crack propagation rate in aligned short-glass-®bre-reinforced
polymers
is
brie¯y
discussed.
Science Ltd. All rights reserved
1998
Elsevier
3 I
: A. polymer-matrix composites (PMCs), A. glass ®bres, B. corrosion, crack, C. micromechanical theory Keywords
d d
a
3
t
AK
3m I
1
1 INTRODUCTION
where A and m are constants which depend on the material and environment. In this paper, we establish a micromechanical theory of the macroscopic crack propagation rate in cross-ply and unidirectional GRP laminates in corrosive environments. Using a physically based micromechanical model of crack propagation due to stress-corrosion cracking of glass ®bre, we deduce theoretically the relationship
Glass-®bre-reinforced polymer (GRP) laminates are widely used in corrosive environments because of their superiority over metals or alloys in resisting corrosive attack. Nevertheless, it is known that they can be weakened by the combined in¯uence of stress and corrosive *To whom correspondence should be addressed. 1659
1660
H. Sekine, P. W. R. Beaumont
between the macroscopic crack propagation rate and the apparent stress-intensity factor for opening mode at the macroscopic crack tip in the laminate. Comparison is made between the theoretical results and experimental data. The relationship is expressed as a power law with a power of 2 for the limiting case of no pre-existing surface ¯aw.
2 MICROMECHANICAL THEORY OF MACROSCOPIC CRACK PROPAGATION RATE
Following the work by Wiederhorn and Bolz,17 the crack propagation rate da/dt due to stress-corrosion cracking in bulk glass is given by:
d d
a t
exp
E ÿ
ÿ
K
2
I
I
ÿ
E ÿ
Scanning electron micrograph of fracture surfaces of glass ®bres.
RT
where E is the activation energy, K is the stressintensity factor for opening mode, R is the gas constant, T is absolute temperature, and and are empirical constants. On the surface of a commercial glass ®bre, there exist a large number of surface ¯aws of variable size.18,19 Our model is based on the premise that a stress-corrosion crack initiates at a pre-existing surface ¯aw, grows perpendicular to the ®bre direction with time and ®nally breaks the glass ®bre (Fig. 1). The fracture surfaces of many glass ®bres in an epoxy matrix were observed by scanning electron microscopy. Figure 2 shows a micrograph of a typical fracture surface of composite. The fracture surface of each glass ®bre is characterized by a mirror region surrounded by a hackle region. In view of the results16 of numerical simulation of the propagation of a stress-corrosion crack in a glass ®bre, the shape of the front of the stress- corrosion crack is assumed to be at every moment a circular arc of radius r, where r is equal to the ®bre radius. The average crackpropagation rate due to stress-corrosion cracking in a glass ®bre can then be written from eqn (2) as: 1 dY exp 2r dt
Fig. 2.
K 1
RT
should be interpreted as the average value of K , along the front of the stress-corrosion crack. Since the stressintensity factor for opening mode is constant more or less over a large part of the central portion of the circular crack front,16 we will represent K by the stressintensity factor at the maximum depth of the stresscorrosion crack:20 I
I
KI
p F 2r
4
where is the tensile stress acting on the ®bre and is written as:
1 ÿ cos f1112 ÿ 3 1 40 1 ÿ cos
p
F
13 1 87 1 ÿ cos 2 ÿ 14 1 37 1 ÿ cos 3 g
3 I
3
I
Model of crack propagation in glass ®bres.
5
For aligned continuous glass ®bres, which are assumed to be distributed in doubly periodic array and perpendicular to the macroscopic stress-corrosion crack, the relationship between the tensile stress and the apparent stress-intensity factor for opening mode K at the macroscopic crack tip in the laminates is given by15
where Y is the area of the stress-corrosion crack in the glass ®bre, is half the angle which is made by two ®bre radii on the edges of the stress-corrosion crack and t is time (Fig. 3). In eqn (3), the stress-intensity factor K ,
Fig. 1.
F
Fig. 3.
Shape of stress-corrosion crack in glass ®bres.
1661
A physically based theory of macroscopic stress-corrosion cracking in GRP
K
6
3
I
where
8 > > > > > > > < > > > > > > > :
r 1 VVf
2V1=2
r 1=2
23=2 V
1 VVf
for square array of fibre for face-centred square array of fibre for face-centred hexagonal array of fibre
f
3=2 r
f
3=2 r
r 23=2 V1=2 31=4 3=2 r
1
f
VVf
7
in sequence. This mechanism of crack propagation was ®rst identi®ed by Hogg and Hull.1 Aveston and Sillwood5 pointed out that a feature of a macroscopic stress-corrosion crack is the planar nature of the crack surfaces of the ®bre and matrix with little ®bre pull-out at low macroscopic crack propagation rates, but with longer ®bre pull-out lengths with increasing propagation rate. The time required to the brittle fracture of a single glass ®bre and surrounding matrix is much shorter than the time t due to stress-corrosion cracking. Therefore, the macroscopic crack propagation rate da*/dt in the laminate is approximately given by F
Here V is the volume fraction of glass ®bres and V is the fraction of the plies in which the glass ®bres are perpendicular to the macroscopic stress-corrosion crack in the laminate. Meanwhile, geometrical consideration of the area of the stress- corrosion crack as shown in Fig. 3 gives
d d
f
d d
Y t
4r2 sin2
d d
8
a
t
8 q r > > > > > < q
d
t
2r sin2
k
exp
ÿ
p F 2r RT
K
d
3 I
9
D
tF
for square array of fibre for face-centred square array of fibre for face-centred hexagonal array of fibre
Vf
Substituting eqn (8) into eqn (3) and using eqns (4) and (6), we obtain
13
where D is the distance between the neighbouring row of ®bres in doubly periodic array (see Fig. 4), and is written as:
t
3
D
2V
r
> > q > > > : 321=2 r f
Vf
14
Substituting eqn (11) into eqn (13), we obtain
where
exp
k
ÿ
E
RT
10
By performing the integration of eqn (9), the time required to propagate a crack slowly in a single glass ®bre by stress-corrosion cracking, t , is given by F
tF ÿ
2r
k
F
0
sin2 exp
ÿ
p F 2r
3
K
I
RT
d
11
F
IC
F
!
1
F
F
1
KIC p
K
3 I
2r
12
when F is the inverse function of F given by eqn (5). When a macroscopic stress-corrosion crack propagates in aligned continuous GRP, glass ®bres are broken ÿ
3
t
2r
sin2 exp
F
0
kD ÿ
p
F 2r
3
K
I
RT
d
15
At this point we introduce the following quantities:
kD
where 0 is half the angle made by two ®bre radii on the edges of the pre-existing surface ¯aw and is that of the stress-corrosion crack at the onset of brittle fracture of the glass ®bre. The brittle fracture of the glass ®bre takes place when the stress intensity factor at the front of the stress-corrosion crack in the ®bre attains the fracture toughness of glass K . By combining eqns (4) and (6), is given by ÿ
d d
a
2r
;
p 2r
RT
16
so that eqn (15) can be rewritten as
d d
a
3
t
F
0
sin2 expfÿK
3
I
d
17
F g
Consider the integral of the denominator. By taking values of =0.1100.216 m5/2/mol17, V =0.400.57, V=0.5 for the cross-ply laminate and 1.0 for the unidirectional laminate, R=8.31 J/(mol K) and T=298 K at room temperature, the value of is estimated as f
102 619 MPa m1=2
ÿ
1
18
Generally speaking, the range of apparent stress-intensity factor for opening mode K at the macroscopic 3 I
1662
H. Sekine, P. W. R. Beaumont
Performing the integration, we obtain
1 I ÿ4 1158K
exp
3
1 1158K
3
I
2
I
2 2
1 582K
3
ÿ 1
I
F
20
2 20
where
20 sin 20 ; 2
F
sin 2
F
21
Since the angle 0 is small and much smaller than , it follows that F
exp
1 1 5820 K
ÿ
3 I
exp
1 582
ÿ 1
3 F KI
22
Therefore, eqn (20) reduces to: I
4 1158K
3 I
1 1158 K
3 I
0
2
exp
ÿ
0 1790 K
3 I
23
Table 1 shows the approximate values of I calculated from eqn (23) together with the exact values of I for a unidirectional laminate with V =0.5. The apparent stress-intensity factor, K , for the laminate is ®xed as 5 MPa m1/2. For the fracture toughness of glass, K , we use 0.73 MPa m1/2 which corresponds to a fracture energy c=3.7 J m 2 for SiO2 glass.21 Thus, the angle calculated using eqn (12) is about 4 (6.98210 2 rad). It is recognized from Table 1 that the approximate values of I agree to the exact values of I with satisfactory 2 . rad). accuracy for 0 < 2 1 5 (4 36210 Substituting eqn (23) into the denominator of eqn (17), we obtain the macroscopic crack propagation rate, da /dt, in the form: f
3 I
IC
ÿ
F
ÿ
ÿ
3
d d
a
3
t
Glass ®bres distributed in doubly periodic array: (a) Square array; (b) face-centred square array (c); face-centred hexagonal array. Fig. 4.
crack tip is experimentally set between 2 and 15 MPa m1/2. Therefore, the integrand of the denominator of eqn (17) tends to zero, except for very small values of . Thus, the denominator, which is denoted by I, is written approximately as
F
I
0
11252 K
2
3 I
1 2 11580 K
sin exp
ÿ
1 1 12K
3 I
19
exp 0 790
1
3
K I
24
I
F
ÿ
ÿ
3
This equation expresses the relationship between the macroscopic crack propagation rate and the apparent stress-intensity factor for opening mode at the macroscopic crack tip in the aligned continuous GRP laminate. Since is inversely proportional to the square root of ®bre radius, r, and D is proportional to the ®bre radius r, and are independent of r. Therefore, the macroscopic crack propagation rate is independent of ®bre radius. It is worthwhile noting that the macroscopic crack propagation rate is independent of the fracture toughness of glass ®bre since is not present in eqn (24). Figure 5 shows the relationship eqn (24) in a logarithmic plot for 0=0 , 0.2 (3.49210 3 rad), 0.3 (5.24210 3 rad) and 0.4 (6.98210 3 rad). The values of and are set as =5210 15 m s 1 and =230 (MPa m1/2) 1. As can be seen from the ®gure, the
1 ÿ cos d
p
ÿ
ÿ
ÿ
ÿ
1663
A physically based theory of macroscopic stress-corrosion cracking in GRP
Table 1. Values of I
=150 (MPa m1/2) Approximate 0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5
1.1048210 2.3381210 4.1456210 6.7450210 1.0428210 8.6259210 6.4355210 4.5283210
ÿ
6
ÿ
7
ÿ
8
ÿ
9
ÿ
9
ÿ
12
ÿ
14
ÿ
16
=350 (MPa m1/2)
1
ÿ
Exact
Error% Approximate
1.0958210 2.3091210 4.0783210 6.6146210 1.0205210 8.4477210 6.3813210 4.6163210
ÿ
6
ÿ
7
ÿ
8
ÿ
9
ÿ
9
ÿ
12
ÿ
14
ÿ
16
0.8 1.3 1.7 2.0 2.2 2.1 0.8 ÿ1.9
ÿ
8
ÿ
10
ÿ
12
ÿ
14
ÿ
16
ÿ
21
ÿ
26
ÿ
31
2.4445210 3.5844210 4.1771210 4.3944210 4.3619210 3.7187210 2.8367210 2.0341210
macroscopic crack propagation rate, da*/dt, increases with the apparent stress-intensity factor, K . Moreover, the larger the pre-existing surface ¯aw size, the higher the macroscopic crack propagation rate. If the apparent stress-intensity factor is experimentally set over a small range, log(da /dt) is observed to be essentially linear with logK . This means that the relationship between da /dt and K is represented by a simple power law. Assuming that a power law is correct, the ®gure reveals that the value of power is smallest at 0=0 . 3 I
3
3 I
3
3 I
3 DISCUSSION 3.1 Veri®cation by experiment
The laminate used in tests is a cross-ply GRP laminate which was manufactured by Nippon Steel Chemical
=550 (MPa m1/2)
1
ÿ
Exact
Error% Approximate
2.4089210 3.4937210 4.0315210 4.2069210 4.1515210 3.5360210 2.7669210 2.1073210
ÿ
8
ÿ
10
ÿ
12
ÿ
14
ÿ
16
ÿ
21
ÿ
26
ÿ
31
1.5 2.6 3.6 4.5 5.1 5.2 2.5 ÿ3.5
9.2532210 8.8675210 6.6281210 4.4492210 2.8116210 2.4416210 1.8928210 1.3780210
ÿ
10
ÿ
13
ÿ
16
ÿ
19
ÿ
22
ÿ
30
ÿ
38
ÿ
46
1
ÿ
Exact
Error%
9.0577210 8.5297210 6.2761210 4.1583210 2.6030210 2.2548210 1.8167210 114514210
ÿ
10
ÿ
13
ÿ
16
ÿ
19
ÿ
22
ÿ
30
ÿ
38
ÿ
46
2.2 4.0 5.6 7.0 8.0 8.3 4.2 ÿ5.1
Corporation. It contains 52% by volume of continuous E-glass ®bres in a diglycidyl ether of bisphenol-A epoxyresin matrix, and consists of 44 plies of pre-preg with ply thickness of around 0.14 mm in the stacking sequence (0/90)11S. Compact tension specimens with side grooves were machined from the laminate. The apparent fracture toughness, K , of the laminate was measured by means of 5% oset load procedure, and found to be 18.2 MPa m1/2 in air at room temperature, which is a conservative value. Stress-corrosion crack propagation tests were carried out in 0.5N HCl at 298K under constant loading.14,15 The macroscopic crack length was measured by means of an indirect method, i.e. the crack mouth displacement of the specimen was measured at intervals of time by a linear variable displacement transducer. The macroscopic crack length was calculated from the crack mouth displacement by use of a ®nite element method. Data for the macroscopic crack propagation rate were obtained by averaging the derivative of the macroscopic crack length with respect to time over the interval of 4± 24 h. Figure 6 shows a logarithmic plot of the experimental data of macroscopic crack propagation rate against the apparent stress-intensity factor at the macroscopic crack tip. The ®gure reveals the tendency of log(da /dt) to increase approximately linearly with log K , over a factor of 10 in da /dt. 3
Q
3
3 I
3
Fig.
5.
Macroscopic crack propagation rate vs apparent stress-intensity factor.
Fig. 6.
Experimental results of macroscopic crack propagation rate vs apparent stress-intensity factor.
1664
H. Sekine, P. W. R. Beaumont
Table 2. Values of power in simple power law
Material Unidirectional E-glass/polyester Unidirectional E-glass/polyester Cross-ply short E-glass/epoxy Unidirectional E-glass/polyester Cross-ply E-glass/epoxy
Environment
Value of power
Reference number
1N H2SO4 0.6N HCl 1N H2SO4 0.6N HCl 0.5N HCl
3.1 3.57 4.22 6.34 2.56 2.73 3.99 5.3 3.6
5 7 9 10 15
By setting the values of , and 0 , the macroscopic crack propagation rate da /dt can be calculated for the apparent stress intensity factor K at the macroscopic crack tip using eqn (24). Following Ref. 14 where a best ®t method was used, we take , and 0 as =5.01210 3 m s 1, =228 (MPa m1/2) 1 and 0=0.215 (3.75210 3 rad). The prediction of macroscopic crack propagation rate is shown by the solid line in Fig. 6, and is in good agreement with experimental measurement. This gives con®dence in our physical interpretation of macroscopic crack propagation in aligned continuous GRP laminates.
glass ®bres of 10 m diameter is less than 0.01 m. For this example, the angle 0 can be roughly estimated at less than 2.5 (4.36210 2 rad). When 0 is large and comparable with 2.5 , 0K becomes much larger than one. Then, eqn (24) reduces to
3.2 Macroscopic crack propagation rate for the limiting
3.3 Macroscopic crack propagation rate in aligned short
y0 The size of pre-existing surface ¯aw aects signi®cantly the macroscopic crack propagation rate. From eqn (24), it can be seen that the macroscopic crack propagation rate increases monotonically with the angle 0 of the pre-existing surface ¯aw of glass ®bre. First, consider the case where 0=0 , i.e., the glass ®bres are free of pre-existing surface ¯aws. In this case, eqn (24) reduces simply to
GRP
3
3 I
ÿ
ÿ
d d
a
ÿ
values of
d d
a
3
t
0 1 6252 K
2
3
I
25
By use of eqns (7) and (14), substitution of eqn (16) into eqn (25) yields the formula which is identical for three kinds of ®bre arrangement, as follows:
d d
a
3
t
1 1 25 2 vk V
2 V 2 R2 T2
3 I
3
t
0 1 79
0
3
K
I
exp 0 790
1
3
K I
27
In this case, the macroscopic crack propagation rate da /dt is faster than that estimated using a simple power law. 3
The study of macroscopic crack propagation in aligned short glass-®bre-reinforced epoxy was made by Hsu et 9 al. They observed that the surface of the macroscopic stress-corrosion crack is planar with little ®bre pull-out. It can be easily demonstrated that for the condition of little or no ®bre pull-out, the relationship between the macroscopic crack propagation rate and the stress intensity factor at the macroscopic crack tip, given by eqn (24), is also valid for uniformly distributed and aligned short GRP. However, if the condition is such that a large number of ®bres pull-out without breakage, eqn (24) has to be modi®ed by reducing provisionally the value of the volume fraction of glass ®bres.
4 CONCLUSIONS AND IMPLICATIONS
2
3
K
I
26
f
Equation (26) gives the minimum macroscopic crack propagation rate for an aligned continuous GRP laminate with respect to 0, and demonstrates that the variation of da /dt with K obeys the simple power law with the value of power of 2. The value of power is independent of the laminate and acidic environment. Table 2 shows the values of power given in previous studies. We see that all values of power are larger than 2. As a consequence, the experimental results of the previous studies are consistent with the result that the smallest value of power is 2. Meanwhile, let us consider the case of large values of 0. Bartenev18 pointed out that the depth of pre-existing surface ¯aws generated during the drawing of commercial 3
3 I
ÿ
ÿ
A physically based micromechanical theory of macroscopic crack propagation rate due to stress-corrosion cracking in aligned continuous GRP laminates has been established. Good agreement between experiment and theory give con®dence in our model. When the relationship is expressed for the limiting case where the glass ®bres are free of pre-existing surface ¯aws, the relationship is represented by the simple power law with the value of power of 2. The theory can be applied to GRP with uniformly distributed and aligned short ®bres. The eects of ®bre/matrix interface debonding and matrix toughness on the macroscopic crack propagation rate could be included in the theory.22 When ®bre/ matrix interface debonding occurs, the macroscopic crack propagation rate is even higher with increasing
A physically based theory of macroscopic stress-corrosion cracking in GRP
apparent stress intensity factor. Concerning the eect of matrix toughness, the macroscopic crack propagation rate is even slower with decreasing the apparent stressintensity factor because the matrix bridging eect is more pronounced at low values of K . In both cases, the further large values of power might be found by experiment. 3 I
ACKNOWLEDGEMENTS
One of the authors (H.S.) acknowledges the support of the Japan Society for the Promotion of Science in the form of a JSPS Fellowship for Research at Centres of Excellence Abroad and the Ministry of Education, Science and Culture of Japan in the form of an MESCJ Fellowship for Research Abroad during the research stay at the University of Cambridge.
REFERENCES
1. Hogg, P. J and Hull, D., Micromechanisms of crack growth in composite materials under corrosive environments. Met. Sci., 1980, 14, 441±449. 2. Carswell, W. S. and Roberts, R. C., Environmental fatigue stress failure mechanism for glass ®bre mat reinforced polyester. Composites, 1980, 11, 95±99. 3. Hogg, P. J., Hull, D. and Spencer, B., Stress and strain corrosion of glass-reinforced plastics. Composites, 1981, 12, 166±172. 4. Friedrich, K., Stress corrosion crack propagation in glass ®bre reinforced/thermoplastic PET. J. Mater. Sci., 1981, 16, 3292±3302. 5. Aveston, J. and Sillwood, J. M., Long-term strength of glass-reinforced plastics in dilute sulphuric acid. J. Mater. Sci., 1982, 17, 3491±3498. 6. Noble, B., Harris, S. J. and Owen, M. J., Stress corrosion cracking of GRP pultruded rods in acid environments. J. Mater. Sci., 1983, 18, 1244±1254. 7. Price, J. N. and Hull, D., Propagation of stress corrosion cracks in aligned glass ®bre composite materials. J. Mater. Sci., 1983, 18, 2798±2810. 8. Lhymn, C. and Schultz, J. M., Chemically-assisted fracture of thermoplastic PET reinforced with short E-glass ®bre. J. Mater. Sci., 1983, 18, 2923±2938.
1665
9. Hsu, P.-L., Yau, S.-S. and Chou, T.-W., Stress-corrosion cracking and its propagation in aligned short-®bre composites. J. Mater. Sci., 1986, 21, 3703±3709. 10. Price, J. N. and Hull, D., Eect of matrix toughness on crack propagation during stress corrosion of glass reinforced composites. Compos. Sci. Technol., 1987, 28, 193± 210. 11. Kumosa, M., Hull, D. and Price, J. N., Acoustic emission from stress corrosion cracks in aligned GRP. J. Mater. Sci., 1987, 22, 331±336. 12. Miyanaga, T. and Sekine, H., Micromechanics study of crack propagation rate in glass ®ber reinforced plastics under acid environments. J. Soc. Mater. Sci., Jpn, 1990, 39, 737±741. 13. Sekine, H. and Miyanaga, T., A micromechanics study of crack propagation rate by stress corrosion in glass ®ber reinforced plastics in consideration of pre-existing microcracks on the surface of ®bers. J. Soc. Mater. Sci., Jpn, 1990, 39, 1545±1548. 14. Sekine, H. and Miyanaga, T., A micromechanics study of the eects of concentration and temperature on the propagation rate of stress corrosion crack in cross ply glass/ epoxy laminates under acid environments. J. Soc. Mater. Sci., Jpn, 1992, 41, 1132±1135. 15. Sekine, H., Micromechanics study of the propagation rate of a stress corrosion crack in cross ply glass/epoxy laminates under acid environments. Proc. 8th Int. Conf. Comp. Mater., 1991, pp. 27.K.1±10. 16. Sekine, H., Hu, N. and Fukunaga, H., Direct numerical simulation of the extension of stress-corrosion cracks in glass ®bers embedded in laminates in acid environments. Compos. Sci. Technol., 1995, 53, 317±323. 17. Wiederhorn, S. M. and Bolz, L. H., Stress corrosion and static fatigue of glass. J. Am. Ceram. Soc., 1970, 53, 543± 548. 18. Bartenev, G. M., Constitution and strength of glass ®bers. Int. J. Fract. Mech., 1969, 5, 179±186. 19. Piggott, M. R. and Yokom, J. C., The weakening of silica ®bres by heat treatment. Glass Technol., 1968, 9, 172±175. 20. Kiuchi, A., Aoki, M., Kobayashi, M. and Ikeda, K., Evaluation of brittle fracture strength of surface notched round bar. J. Iron Steel. Inst. Jpn, 1982, 68, 1830±1838. 21. Mecholsky, J. J., Rice, R. W. and Freiman, S. W., Prediction of fracture energy and ¯aw size in glasses from measurements of mirror size. J. Am. Ceram. Soc., 1974, 57, 440±443. 22. Sekine, H. and Beaumont, P. W. R., A physically-based micromechanical theory of macroscopic stress corrosion cracking in glass ®bre-polymer laminates. Cambridge University Engineering Department Rep. CUED/CMATS/TR237, 1997.
# 1998 Elsevier Science Ltd. All rights reserved
Composites Science and Technology
58
Printed in Great Britain 0266-3538/98 $Ðsee front matter
P I I : S 0 2 6 6 - 3 5 3 8 ( 9 7 ) 0 0 2 3 6 - 4
A PHYSICALLY BASED MICROMECHANICAL THEORY OF MACROSCOPIC STRESS-CORROSION CRACKING IN ALIGNED CONTINUOUS GLASS-FIBRE-REINFORCED POLYMER LAMINATES Hideki Sekine * & Peter W. R. Beaumont a
b
a
Department of Aeronautics and Space Engineering, Tohoku University, Sendai 980-8579, Japan
b
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
(Received 10 December 1996; revised 19 November 1997; accepted 11 December 1997)
environment. For example, their susceptibility to stresscorrosion cracking in acidic environments has received wide attention.1±15 Hogg and Hull,1 Noble et al.6 and Price and Hull7,10 studied the actual propagation mechanism of a stress-corrosion crack in individual glass ®bres by means of scanning electron microscopy. They observed the fracture surface of each glass ®bre to be characterized by a smooth mirror region surrounded by `mist' and `hackle' regions. They concluded10 from this observation that a stress-corrosion crack in a glass ®bre initiates and propagates to produce the mirror surface, followed by propagation across the remainder of the ®bre forming the mist and hackle regions, and ®nally extending into the surrounding matrix. Recently, one of the authors (HS) and his coworkers16 carried out a direct numerical simulation of the propagation of a stress-corrosion crack in a glass ®bre. For macroscopic propagation of stress-corrosion cracks in GRP, there are several reported experimental studies of a variety of GRPs in the presence of dilute hydrochloric,4,7,10,14,15 and sulphuric acids.4,5,9 As a result, it was found that the relationship between macroscopic crack propagation rate, da*/dt, and apparent stress-intensity factor for opening mode, K , at the macroscopic crack tip over a few orders of magnitude of macroscopic crack propagation rate can be expressed by a power law:
Abstract This paper is concerned with a micromechanical theory of macroscopic cracking
in
crack
propagation
aligned
due
continuous
to
stress-corrosion
glass-®bre-reinforced
polymer laminates, in particular, the relationship between the macroscopic crack propagation rate and the apparent stress-intensity factor at the macroscopic crack tip. A physically based micromechanical model of crack propagation
in
a
glass
®bre
as
a
result
of
stress-corrosion
cracking is proposed. It is based on the premise that a stress-corrosion crack initiates at a pre-existing surface ¯aw in a glass ®bre, grows and ®nally leads to breakage of the ®bre. We derive theoretically an equation for the macroscopic crack propagation rate as a function of the apparent stress-intensity factor. The theoretical prediction is in good agreement with experimental measurement. It is emphasized that the size of the inherent surface ¯aw aects signi®cantly the macroscopic crack propagation rate. For glass ®bres free of pre-existing ¯aws, the relationship can be represented by a simple power law with the value of power of 2. Finally, the macroscopic
#
crack propagation rate in aligned short-glass-®bre-reinforced
polymers
is
brie¯y
discussed.
Science Ltd. All rights reserved
1998
Elsevier
3 I
: A. polymer-matrix composites (PMCs), A. glass ®bres, B. corrosion, crack, C. micromechanical theory Keywords
d d
a
3
t
AK
3m I
1
1 INTRODUCTION
where A and m are constants which depend on the material and environment. In this paper, we establish a micromechanical theory of the macroscopic crack propagation rate in cross-ply and unidirectional GRP laminates in corrosive environments. Using a physically based micromechanical model of crack propagation due to stress-corrosion cracking of glass ®bre, we deduce theoretically the relationship
Glass-®bre-reinforced polymer (GRP) laminates are widely used in corrosive environments because of their superiority over metals or alloys in resisting corrosive attack. Nevertheless, it is known that they can be weakened by the combined in¯uence of stress and corrosive *To whom correspondence should be addressed. 1659
1660
H. Sekine, P. W. R. Beaumont
between the macroscopic crack propagation rate and the apparent stress-intensity factor for opening mode at the macroscopic crack tip in the laminate. Comparison is made between the theoretical results and experimental data. The relationship is expressed as a power law with a power of 2 for the limiting case of no pre-existing surface ¯aw.
2 MICROMECHANICAL THEORY OF MACROSCOPIC CRACK PROPAGATION RATE
Following the work by Wiederhorn and Bolz,17 the crack propagation rate da/dt due to stress-corrosion cracking in bulk glass is given by:
d d
a t
exp
E ÿ
ÿ
K
2
I
I
ÿ
E ÿ
Scanning electron micrograph of fracture surfaces of glass ®bres.
RT
where E is the activation energy, K is the stressintensity factor for opening mode, R is the gas constant, T is absolute temperature, and and are empirical constants. On the surface of a commercial glass ®bre, there exist a large number of surface ¯aws of variable size.18,19 Our model is based on the premise that a stress-corrosion crack initiates at a pre-existing surface ¯aw, grows perpendicular to the ®bre direction with time and ®nally breaks the glass ®bre (Fig. 1). The fracture surfaces of many glass ®bres in an epoxy matrix were observed by scanning electron microscopy. Figure 2 shows a micrograph of a typical fracture surface of composite. The fracture surface of each glass ®bre is characterized by a mirror region surrounded by a hackle region. In view of the results16 of numerical simulation of the propagation of a stress-corrosion crack in a glass ®bre, the shape of the front of the stress- corrosion crack is assumed to be at every moment a circular arc of radius r, where r is equal to the ®bre radius. The average crackpropagation rate due to stress-corrosion cracking in a glass ®bre can then be written from eqn (2) as: 1 dY exp 2r dt
Fig. 2.
K 1
RT
should be interpreted as the average value of K , along the front of the stress-corrosion crack. Since the stressintensity factor for opening mode is constant more or less over a large part of the central portion of the circular crack front,16 we will represent K by the stressintensity factor at the maximum depth of the stresscorrosion crack:20 I
I
KI
p F 2r
4
where is the tensile stress acting on the ®bre and is written as:
1 ÿ cos f1112 ÿ 3 1 40 1 ÿ cos
p
F
13 1 87 1 ÿ cos 2 ÿ 14 1 37 1 ÿ cos 3 g
3 I
3
I
Model of crack propagation in glass ®bres.
5
For aligned continuous glass ®bres, which are assumed to be distributed in doubly periodic array and perpendicular to the macroscopic stress-corrosion crack, the relationship between the tensile stress and the apparent stress-intensity factor for opening mode K at the macroscopic crack tip in the laminates is given by15
where Y is the area of the stress-corrosion crack in the glass ®bre, is half the angle which is made by two ®bre radii on the edges of the stress-corrosion crack and t is time (Fig. 3). In eqn (3), the stress-intensity factor K ,
Fig. 1.
F
Fig. 3.
Shape of stress-corrosion crack in glass ®bres.
1661
A physically based theory of macroscopic stress-corrosion cracking in GRP
K
6
3
I
where
8 > > > > > > > < > > > > > > > :
r 1 VVf
2V1=2
r 1=2
23=2 V
1 VVf
for square array of fibre for face-centred square array of fibre for face-centred hexagonal array of fibre
f
3=2 r
f
3=2 r
r 23=2 V1=2 31=4 3=2 r
1
f
VVf
7
in sequence. This mechanism of crack propagation was ®rst identi®ed by Hogg and Hull.1 Aveston and Sillwood5 pointed out that a feature of a macroscopic stress-corrosion crack is the planar nature of the crack surfaces of the ®bre and matrix with little ®bre pull-out at low macroscopic crack propagation rates, but with longer ®bre pull-out lengths with increasing propagation rate. The time required to the brittle fracture of a single glass ®bre and surrounding matrix is much shorter than the time t due to stress-corrosion cracking. Therefore, the macroscopic crack propagation rate da*/dt in the laminate is approximately given by F
Here V is the volume fraction of glass ®bres and V is the fraction of the plies in which the glass ®bres are perpendicular to the macroscopic stress-corrosion crack in the laminate. Meanwhile, geometrical consideration of the area of the stress- corrosion crack as shown in Fig. 3 gives
d d
f
d d
Y t
4r2 sin2
d d
8
a
t
8 q r > > > > > < q
d
t
2r sin2
k
exp
ÿ
p F 2r RT
K
d
3 I
9
D
tF
for square array of fibre for face-centred square array of fibre for face-centred hexagonal array of fibre
Vf
Substituting eqn (8) into eqn (3) and using eqns (4) and (6), we obtain
13
where D is the distance between the neighbouring row of ®bres in doubly periodic array (see Fig. 4), and is written as:
t
3
D
2V
r
> > q > > > : 321=2 r f
Vf
14
Substituting eqn (11) into eqn (13), we obtain
where
exp
k
ÿ
E
RT
10
By performing the integration of eqn (9), the time required to propagate a crack slowly in a single glass ®bre by stress-corrosion cracking, t , is given by F
tF ÿ
2r
k
F
0
sin2 exp
ÿ
p F 2r
3
K
I
RT
d
11
F
IC
F
!
1
F
F
1
KIC p
K
3 I
2r
12
when F is the inverse function of F given by eqn (5). When a macroscopic stress-corrosion crack propagates in aligned continuous GRP, glass ®bres are broken ÿ
3
t
2r
sin2 exp
F
0
kD ÿ
p
F 2r
3
K
I
RT
d
15
At this point we introduce the following quantities:
kD
where 0 is half the angle made by two ®bre radii on the edges of the pre-existing surface ¯aw and is that of the stress-corrosion crack at the onset of brittle fracture of the glass ®bre. The brittle fracture of the glass ®bre takes place when the stress intensity factor at the front of the stress-corrosion crack in the ®bre attains the fracture toughness of glass K . By combining eqns (4) and (6), is given by ÿ
d d
a
2r
;
p 2r
RT
16
so that eqn (15) can be rewritten as
d d
a
3
t
F
0
sin2 expfÿK
3
I
d
17
F g
Consider the integral of the denominator. By taking values of =0.1100.216 m5/2/mol17, V =0.400.57, V=0.5 for the cross-ply laminate and 1.0 for the unidirectional laminate, R=8.31 J/(mol K) and T=298 K at room temperature, the value of is estimated as f
102 619 MPa m1=2
ÿ
1
18
Generally speaking, the range of apparent stress-intensity factor for opening mode K at the macroscopic 3 I
1662
H. Sekine, P. W. R. Beaumont
Performing the integration, we obtain
1 I ÿ4 1158K
exp
3
1 1158K
3
I
2
I
2 2
1 582K
3
ÿ 1
I
F
20
2 20
where
20 sin 20 ; 2
F
sin 2
F
21
Since the angle 0 is small and much smaller than , it follows that F
exp
1 1 5820 K
ÿ
3 I
exp
1 582
ÿ 1
3 F KI
22
Therefore, eqn (20) reduces to: I
4 1158K
3 I
1 1158 K
3 I
0
2
exp
ÿ
0 1790 K
3 I
23
Table 1 shows the approximate values of I calculated from eqn (23) together with the exact values of I for a unidirectional laminate with V =0.5. The apparent stress-intensity factor, K , for the laminate is ®xed as 5 MPa m1/2. For the fracture toughness of glass, K , we use 0.73 MPa m1/2 which corresponds to a fracture energy c=3.7 J m 2 for SiO2 glass.21 Thus, the angle calculated using eqn (12) is about 4 (6.98210 2 rad). It is recognized from Table 1 that the approximate values of I agree to the exact values of I with satisfactory 2 . rad). accuracy for 0 < 2 1 5 (4 36210 Substituting eqn (23) into the denominator of eqn (17), we obtain the macroscopic crack propagation rate, da /dt, in the form: f
3 I
IC
ÿ
F
ÿ
ÿ
3
d d
a
3
t
Glass ®bres distributed in doubly periodic array: (a) Square array; (b) face-centred square array (c); face-centred hexagonal array. Fig. 4.
crack tip is experimentally set between 2 and 15 MPa m1/2. Therefore, the integrand of the denominator of eqn (17) tends to zero, except for very small values of . Thus, the denominator, which is denoted by I, is written approximately as
F
I
0
11252 K
2
3 I
1 2 11580 K
sin exp
ÿ
1 1 12K
3 I
19
exp 0 790
1
3
K I
24
I
F
ÿ
ÿ
3
This equation expresses the relationship between the macroscopic crack propagation rate and the apparent stress-intensity factor for opening mode at the macroscopic crack tip in the aligned continuous GRP laminate. Since is inversely proportional to the square root of ®bre radius, r, and D is proportional to the ®bre radius r, and are independent of r. Therefore, the macroscopic crack propagation rate is independent of ®bre radius. It is worthwhile noting that the macroscopic crack propagation rate is independent of the fracture toughness of glass ®bre since is not present in eqn (24). Figure 5 shows the relationship eqn (24) in a logarithmic plot for 0=0 , 0.2 (3.49210 3 rad), 0.3 (5.24210 3 rad) and 0.4 (6.98210 3 rad). The values of and are set as =5210 15 m s 1 and =230 (MPa m1/2) 1. As can be seen from the ®gure, the
1 ÿ cos d
p
ÿ
ÿ
ÿ
ÿ
1663
A physically based theory of macroscopic stress-corrosion cracking in GRP
Table 1. Values of I
=150 (MPa m1/2) Approximate 0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5
1.1048210 2.3381210 4.1456210 6.7450210 1.0428210 8.6259210 6.4355210 4.5283210
ÿ
6
ÿ
7
ÿ
8
ÿ
9
ÿ
9
ÿ
12
ÿ
14
ÿ
16
=350 (MPa m1/2)
1
ÿ
Exact
Error% Approximate
1.0958210 2.3091210 4.0783210 6.6146210 1.0205210 8.4477210 6.3813210 4.6163210
ÿ
6
ÿ
7
ÿ
8
ÿ
9
ÿ
9
ÿ
12
ÿ
14
ÿ
16
0.8 1.3 1.7 2.0 2.2 2.1 0.8 ÿ1.9
ÿ
8
ÿ
10
ÿ
12
ÿ
14
ÿ
16
ÿ
21
ÿ
26
ÿ
31
2.4445210 3.5844210 4.1771210 4.3944210 4.3619210 3.7187210 2.8367210 2.0341210
macroscopic crack propagation rate, da*/dt, increases with the apparent stress-intensity factor, K . Moreover, the larger the pre-existing surface ¯aw size, the higher the macroscopic crack propagation rate. If the apparent stress-intensity factor is experimentally set over a small range, log(da /dt) is observed to be essentially linear with logK . This means that the relationship between da /dt and K is represented by a simple power law. Assuming that a power law is correct, the ®gure reveals that the value of power is smallest at 0=0 . 3 I
3
3 I
3
3 I
3 DISCUSSION 3.1 Veri®cation by experiment
The laminate used in tests is a cross-ply GRP laminate which was manufactured by Nippon Steel Chemical
=550 (MPa m1/2)
1
ÿ
Exact
Error% Approximate
2.4089210 3.4937210 4.0315210 4.2069210 4.1515210 3.5360210 2.7669210 2.1073210
ÿ
8
ÿ
10
ÿ
12
ÿ
14
ÿ
16
ÿ
21
ÿ
26
ÿ
31
1.5 2.6 3.6 4.5 5.1 5.2 2.5 ÿ3.5
9.2532210 8.8675210 6.6281210 4.4492210 2.8116210 2.4416210 1.8928210 1.3780210
ÿ
10
ÿ
13
ÿ
16
ÿ
19
ÿ
22
ÿ
30
ÿ
38
ÿ
46
1
ÿ
Exact
Error%
9.0577210 8.5297210 6.2761210 4.1583210 2.6030210 2.2548210 1.8167210 114514210
ÿ
10
ÿ
13
ÿ
16
ÿ
19
ÿ
22
ÿ
30
ÿ
38
ÿ
46
2.2 4.0 5.6 7.0 8.0 8.3 4.2 ÿ5.1
Corporation. It contains 52% by volume of continuous E-glass ®bres in a diglycidyl ether of bisphenol-A epoxyresin matrix, and consists of 44 plies of pre-preg with ply thickness of around 0.14 mm in the stacking sequence (0/90)11S. Compact tension specimens with side grooves were machined from the laminate. The apparent fracture toughness, K , of the laminate was measured by means of 5% oset load procedure, and found to be 18.2 MPa m1/2 in air at room temperature, which is a conservative value. Stress-corrosion crack propagation tests were carried out in 0.5N HCl at 298K under constant loading.14,15 The macroscopic crack length was measured by means of an indirect method, i.e. the crack mouth displacement of the specimen was measured at intervals of time by a linear variable displacement transducer. The macroscopic crack length was calculated from the crack mouth displacement by use of a ®nite element method. Data for the macroscopic crack propagation rate were obtained by averaging the derivative of the macroscopic crack length with respect to time over the interval of 4± 24 h. Figure 6 shows a logarithmic plot of the experimental data of macroscopic crack propagation rate against the apparent stress-intensity factor at the macroscopic crack tip. The ®gure reveals the tendency of log(da /dt) to increase approximately linearly with log K , over a factor of 10 in da /dt. 3
Q
3
3 I
3
Fig.
5.
Macroscopic crack propagation rate vs apparent stress-intensity factor.
Fig. 6.
Experimental results of macroscopic crack propagation rate vs apparent stress-intensity factor.
1664
H. Sekine, P. W. R. Beaumont
Table 2. Values of power in simple power law
Material Unidirectional E-glass/polyester Unidirectional E-glass/polyester Cross-ply short E-glass/epoxy Unidirectional E-glass/polyester Cross-ply E-glass/epoxy
Environment
Value of power
Reference number
1N H2SO4 0.6N HCl 1N H2SO4 0.6N HCl 0.5N HCl
3.1 3.57 4.22 6.34 2.56 2.73 3.99 5.3 3.6
5 7 9 10 15
By setting the values of , and 0 , the macroscopic crack propagation rate da /dt can be calculated for the apparent stress intensity factor K at the macroscopic crack tip using eqn (24). Following Ref. 14 where a best ®t method was used, we take , and 0 as =5.01210 3 m s 1, =228 (MPa m1/2) 1 and 0=0.215 (3.75210 3 rad). The prediction of macroscopic crack propagation rate is shown by the solid line in Fig. 6, and is in good agreement with experimental measurement. This gives con®dence in our physical interpretation of macroscopic crack propagation in aligned continuous GRP laminates.
glass ®bres of 10 m diameter is less than 0.01 m. For this example, the angle 0 can be roughly estimated at less than 2.5 (4.36210 2 rad). When 0 is large and comparable with 2.5 , 0K becomes much larger than one. Then, eqn (24) reduces to
3.2 Macroscopic crack propagation rate for the limiting
3.3 Macroscopic crack propagation rate in aligned short
y0 The size of pre-existing surface ¯aw aects signi®cantly the macroscopic crack propagation rate. From eqn (24), it can be seen that the macroscopic crack propagation rate increases monotonically with the angle 0 of the pre-existing surface ¯aw of glass ®bre. First, consider the case where 0=0 , i.e., the glass ®bres are free of pre-existing surface ¯aws. In this case, eqn (24) reduces simply to
GRP
3
3 I
ÿ
ÿ
d d
a
ÿ
values of
d d
a
3
t
0 1 6252 K
2
3
I
25
By use of eqns (7) and (14), substitution of eqn (16) into eqn (25) yields the formula which is identical for three kinds of ®bre arrangement, as follows:
d d
a
3
t
1 1 25 2 vk V
2 V 2 R2 T2
3 I
3
t
0 1 79
0
3
K
I
exp 0 790
1
3
K I
27
In this case, the macroscopic crack propagation rate da /dt is faster than that estimated using a simple power law. 3
The study of macroscopic crack propagation in aligned short glass-®bre-reinforced epoxy was made by Hsu et 9 al. They observed that the surface of the macroscopic stress-corrosion crack is planar with little ®bre pull-out. It can be easily demonstrated that for the condition of little or no ®bre pull-out, the relationship between the macroscopic crack propagation rate and the stress intensity factor at the macroscopic crack tip, given by eqn (24), is also valid for uniformly distributed and aligned short GRP. However, if the condition is such that a large number of ®bres pull-out without breakage, eqn (24) has to be modi®ed by reducing provisionally the value of the volume fraction of glass ®bres.
4 CONCLUSIONS AND IMPLICATIONS
2
3
K
I
26
f
Equation (26) gives the minimum macroscopic crack propagation rate for an aligned continuous GRP laminate with respect to 0, and demonstrates that the variation of da /dt with K obeys the simple power law with the value of power of 2. The value of power is independent of the laminate and acidic environment. Table 2 shows the values of power given in previous studies. We see that all values of power are larger than 2. As a consequence, the experimental results of the previous studies are consistent with the result that the smallest value of power is 2. Meanwhile, let us consider the case of large values of 0. Bartenev18 pointed out that the depth of pre-existing surface ¯aws generated during the drawing of commercial 3
3 I
ÿ
ÿ
A physically based micromechanical theory of macroscopic crack propagation rate due to stress-corrosion cracking in aligned continuous GRP laminates has been established. Good agreement between experiment and theory give con®dence in our model. When the relationship is expressed for the limiting case where the glass ®bres are free of pre-existing surface ¯aws, the relationship is represented by the simple power law with the value of power of 2. The theory can be applied to GRP with uniformly distributed and aligned short ®bres. The eects of ®bre/matrix interface debonding and matrix toughness on the macroscopic crack propagation rate could be included in the theory.22 When ®bre/ matrix interface debonding occurs, the macroscopic crack propagation rate is even higher with increasing
A physically based theory of macroscopic stress-corrosion cracking in GRP
apparent stress intensity factor. Concerning the eect of matrix toughness, the macroscopic crack propagation rate is even slower with decreasing the apparent stressintensity factor because the matrix bridging eect is more pronounced at low values of K . In both cases, the further large values of power might be found by experiment. 3 I
ACKNOWLEDGEMENTS
One of the authors (H.S.) acknowledges the support of the Japan Society for the Promotion of Science in the form of a JSPS Fellowship for Research at Centres of Excellence Abroad and the Ministry of Education, Science and Culture of Japan in the form of an MESCJ Fellowship for Research Abroad during the research stay at the University of Cambridge.
REFERENCES
1. Hogg, P. J and Hull, D., Micromechanisms of crack growth in composite materials under corrosive environments. Met. Sci., 1980, 14, 441±449. 2. Carswell, W. S. and Roberts, R. C., Environmental fatigue stress failure mechanism for glass ®bre mat reinforced polyester. Composites, 1980, 11, 95±99. 3. Hogg, P. J., Hull, D. and Spencer, B., Stress and strain corrosion of glass-reinforced plastics. Composites, 1981, 12, 166±172. 4. Friedrich, K., Stress corrosion crack propagation in glass ®bre reinforced/thermoplastic PET. J. Mater. Sci., 1981, 16, 3292±3302. 5. Aveston, J. and Sillwood, J. M., Long-term strength of glass-reinforced plastics in dilute sulphuric acid. J. Mater. Sci., 1982, 17, 3491±3498. 6. Noble, B., Harris, S. J. and Owen, M. J., Stress corrosion cracking of GRP pultruded rods in acid environments. J. Mater. Sci., 1983, 18, 1244±1254. 7. Price, J. N. and Hull, D., Propagation of stress corrosion cracks in aligned glass ®bre composite materials. J. Mater. Sci., 1983, 18, 2798±2810. 8. Lhymn, C. and Schultz, J. M., Chemically-assisted fracture of thermoplastic PET reinforced with short E-glass ®bre. J. Mater. Sci., 1983, 18, 2923±2938.
1665
9. Hsu, P.-L., Yau, S.-S. and Chou, T.-W., Stress-corrosion cracking and its propagation in aligned short-®bre composites. J. Mater. Sci., 1986, 21, 3703±3709. 10. Price, J. N. and Hull, D., Eect of matrix toughness on crack propagation during stress corrosion of glass reinforced composites. Compos. Sci. Technol., 1987, 28, 193± 210. 11. Kumosa, M., Hull, D. and Price, J. N., Acoustic emission from stress corrosion cracks in aligned GRP. J. Mater. Sci., 1987, 22, 331±336. 12. Miyanaga, T. and Sekine, H., Micromechanics study of crack propagation rate in glass ®ber reinforced plastics under acid environments. J. Soc. Mater. Sci., Jpn, 1990, 39, 737±741. 13. Sekine, H. and Miyanaga, T., A micromechanics study of crack propagation rate by stress corrosion in glass ®ber reinforced plastics in consideration of pre-existing microcracks on the surface of ®bers. J. Soc. Mater. Sci., Jpn, 1990, 39, 1545±1548. 14. Sekine, H. and Miyanaga, T., A micromechanics study of the eects of concentration and temperature on the propagation rate of stress corrosion crack in cross ply glass/ epoxy laminates under acid environments. J. Soc. Mater. Sci., Jpn, 1992, 41, 1132±1135. 15. Sekine, H., Micromechanics study of the propagation rate of a stress corrosion crack in cross ply glass/epoxy laminates under acid environments. Proc. 8th Int. Conf. Comp. Mater., 1991, pp. 27.K.1±10. 16. Sekine, H., Hu, N. and Fukunaga, H., Direct numerical simulation of the extension of stress-corrosion cracks in glass ®bers embedded in laminates in acid environments. Compos. Sci. Technol., 1995, 53, 317±323. 17. Wiederhorn, S. M. and Bolz, L. H., Stress corrosion and static fatigue of glass. J. Am. Ceram. Soc., 1970, 53, 543± 548. 18. Bartenev, G. M., Constitution and strength of glass ®bers. Int. J. Fract. Mech., 1969, 5, 179±186. 19. Piggott, M. R. and Yokom, J. C., The weakening of silica ®bres by heat treatment. Glass Technol., 1968, 9, 172±175. 20. Kiuchi, A., Aoki, M., Kobayashi, M. and Ikeda, K., Evaluation of brittle fracture strength of surface notched round bar. J. Iron Steel. Inst. Jpn, 1982, 68, 1830±1838. 21. Mecholsky, J. J., Rice, R. W. and Freiman, S. W., Prediction of fracture energy and ¯aw size in glasses from measurements of mirror size. J. Am. Ceram. Soc., 1974, 57, 440±443. 22. Sekine, H. and Beaumont, P. W. R., A physically-based micromechanical theory of macroscopic stress corrosion cracking in glass ®bre-polymer laminates. Cambridge University Engineering Department Rep. CUED/CMATS/TR237, 1997.