- Email: [email protected]
Slow crack growth and fracture instability of cement composites
The International Journal of Cement Composites and Lightweight Concrete, Volume 4, Number 1
February 1982
Slow crack growth and fracture instability of cement composites
SYNOPSIS A crack instability anlysis is presented which allows the maximum load stress intensity factor (Ko) to be predicted from a predetermined correlation between Kc and a geometric parameter, @. Experimental results on asbestos cements are available in the literature and they show that there is a good correlation between K~ and reasonably independent of size effects. However, specimen geometry seems to have some influence on the K~-@relationship. This is suggested to be caused by a geometric effect on the KR-curve.
Y. W. Mai* and B. Cotterell*
INTRODUCTION It has been shown by several investigators [1-5] that in order to characterise the influence of slow crack growth on the fracture behaviour of fibre cement composites, the crack growth resitance (KR)curve may be used. Large KR values on the curve are always associated with large separation of the crack faces. However, for many practical applications, such excessive deformations are not usually permitted and thus the beneficial effects of fibre reinforcement may not be fully utilized. Indeed, a suitable design criterion could be one that is based on either the maximum load bearing capacity of the structure or some deformation such as deflection. This means that only a small amount of crack growth can be allowed before the maximum load is reached. For a given crack geometry and loading configuration of the structure the KR-curve may be used to determine the instability Kc value. Thus the maximum load (Pm~×)or the amount of slow crack growth (~'amax)can be calculated depending on which one of these two quantities is known. In this communication, a crack instability analysis is presented which relates the maximum load Kc value to a geometric parameter ~. This is followed by a discussion on the experimental results for asbestos cements taken from a few previous publications [1-3].
KEYWORDS Composite materials, fracture strength, asbestos cement, fracture properties, fibre reinforced cement, cracking (fracturing), crack propagation, slow cracking, failure, fracture tests, scale effect, strength of materials.
PREDICTION OF INSTABILITY Kc FROM KR-CURVE The conditions for crack stability to be achieved have been discussed in great detail in literature [6,7]. Briefly, a crack will propagate slowly and steadily when the energy released at the crack tip equals that required for crack growth. However, when the energy supply exceeds the demand unstable cracking will follow. Mathematically, the crack instability conditions can be written as [6,7]: KA = KR = Ko
"Department of Mechanical Engineering, University of Sydney, Sydney, NSW 2006, Australia. @ Construction Press 1982 0142-095X/82/04130033502.06
and
aKA aa
(1) _>--
dKR da
(2)
The R.H.S. term of expression (2) represents the variation of the crack growth resistance as the crack extends. To a first approximation the material crack growth resistance curve is given by:
33
Slow crack growth
ia
and fracture instability
= (a - aO) = p KG
equations (2) to (41, the obtained from:
(3)
where a0 and a are the initiatand effective crack lengths, p and m are constants. The L.H.S. term gives the variation of the applied stress intensity factor (KA) as the crack length increass and this can be evaluated at either constant load or constant displacement depending on the constraint of the testing machine. For engineering design purposes, load-controlled conditions are of more critical significance and it can be shown that:
(4)
where P is the applied load and F(a) is a function of crack length [8]. F’(a) denotes the first derivative of F(a). Using
0.8
CRACKING +
CT(Foote
IN LONGITUDINAL
DIRECTION
et al)
DCB (Foote
et al)
W=380mm
,
/
0.6
CRACKING
,/
IN
TRANSVERSE
I
DIRECTION CT (Foote
et al)
W=200mm @ CT (Lenain 8. Bunsell) W-120mm h 3-PT NOTHED
NORMALISED
34
INITIAL
CRACK
can be
Figure 1 Correlations between normafised initial crack length (a/WjO and crack length at maximum load talW)max.
W=200mm I
Kc value
where CD= F(a)/F’(a) is a geometric parameter. If the KRcurve is both geometry and size independent the maximum load Kc value will increase with Cp. Table 1 gives the non-dimensionalised (@/WI values for three specimen geometries: compact tension, three-point notch bend and single edge notch. Clearly, @ increases directly with W (except for low values of a/W in two cases) which is an indication of a size effect on Kc. Equation (5) suggests that for a material with agiven KRcurve characteristic such as that given by equation (3) a direct correlation should exist between the maximum load Kc and @. Thus, if Q>is known at crack instability Kc can be established and used for design calculations. However, since @ is a function of a, this will involve
1
I
instability
(Si
Ka = P F(a)
= P F’(a)
Mai and Cottereii
of cement composites
DEPTH, (a/W
lo
/
Slow crack growth and fracture instability of cement composites
measurements of slow crack growth (z~a) up to the maximum load point. To avoid tedious crack length measurements, it is useful to establish a relationship between the initial (ao)and the maximum load (area×)crack lengths.
Mai and Cotterell
appropriate for load-controlled tests. This approach is reasonable since the maximum loads obtained in both displacement and load-controlled machines are equal. Figure 1 shows that there is an approximate straight line relationship between (a/W)o and (a/W)ma,. All the experimental data [1-3] fall within the _+10% scatter band but there is a bias tendency for the data to lie below the best fit line for (a/W)o >/ 0.5. It also appears that specimen size, geometry and crack propagation direction do not affect this linear relationship. However, it is desirable to await more experimental data from other specimen geometries to validate this conclusion. The instability Kc values obtained for the asbestos cements studied in reference [3] contained 5.34% of asbestos fibres and these are plotted against @as shown in Figure 2. Results obtained for a similar cement composite ~ [1] with 5% asbestos fibres from single edge notch and three-point notch bend specimens are also shown in this figure. The correlation between Kc and is reasonable with a correlation coefficient of 0.85 but half of the experimental data are outside the _+10% scatter band of the mean line. It is not possible to comment conclusively on size and geometry effects from this figure since the asbestos cements studied in these two previous investigations [1,3] are not exactly
COMPARISON OF INSTABILITY ANALYSIS WITH EXPERIMENTAL RESULTS To find a correlation between the instability Kc and • it is necessary to have as many experimental data as possible on (Pmax, amax)for several specimen geometries with varying starter crack lengths and sizes. A search into the published literature reveals that only a few previous investigations contain such detailed results. One is on steel fibre reinforced mortar by Velazco et al. [4] using four-point notch bend beams of 76 mm depth (W). Although the normalised initial notch depths (a/W)o were varied between 0.13 to 0.83 the normalised crack depths at maximum load (a/W)ma* always exceeded 0.60 which made the calculation of Kc based on the four-point bend equation dubious [1]. More complete information can be obtained from previous work on asbestos-cements by Lenain and Bunsell [3] and by Mai and co-workers [1,2] using compact tension and three-point notch bend specimens. Although these latter investigations were conducted in displacement-controlled testing machines the experimental data of (P,a) at maximum load can be used here for the correlation analysis which is
~Theproductionmethodsfor thesetwo asbestoscements[1,3] are not the same.
Figure 2 A natural logarithmic plot of maximum load Kc against geometric parameter ~b for asbestos cements with approximately 5% fibres.
t10% 1/,
/
/ / ~2
E)
~ L E AST SQUARE LINE
/ 10 // -/
0.8
// /
~ 5% ASBESTOS CEMENT
/// A
06 .-% Z J
E) CT (Lenoin & Bunsell) • SEN (Mei et at) 3 PT NOTCH BEND SIZE A (smell) (Moi et el) ® 3 PT NOTCH BEND SIZE B (Iorge) (Mai et al)
/
E} EL
/
/
/ OZ, F
/
J/ /
02 I
0
l__
I
1
I
J
L
[
2
3
z.
5
6
7
LN ((~)(mm)
35
Slow crack growth and fracture instability of cement composites
Mai and Cottereli
identical and the available data are sparse. More definite conclusions can be drawn from Figure 3 in which Kc is plotted against @ for three-point notch bend Hardiflex cement sheets containing 8% asbestos and 7% cellulose fibres [1,2]. The beam depth (W) varied in these tests from 25 mm to 200 mm. There is now a very good correlation between Kc and ~, the correlation coefficient being 0,91. Only three of the 17 data points lie outside the ---10% scatter band of the least square line. It is obvious from this figure that the instability Kc values are larger for bigger size specimens which have correspondingly larger q~values. To gain some insight into geometric effects, a few additional experimental data have been obtained for the compact tension geometry at a/W -0.30 [2] for the same Hardiflex sheets. They are plotted in Figure 3 and fall outside the scatter band of the three-point notch bend data. If these additional data are included in the linear regression analysis the correlation coefficient drops to 0.73 which is rather poor. This indicates a possibly geometry-dependent K~-curve for the material, resulting in different m and /3 values in equation (5). It is thus necessary to obtain different correlational relationships for different specimen geometries if equation (5) is to be used for engineering design.
Table 1 Variation of geometric parameter ,b with (a/W)
CONCLUDING REMARKS
correlational relationship between Kc and ~. Although this relationship is apparently size-independent there are preliminary results which indicate that it is possibly
The crack instability analysis presented here enables the maximum load Kc value to be predicted from a
~/W a/W
Compact tension *
Single edge notch+
3-point notch bends
0.1 0.2 0,3 0.4 0,5 0.6 0.7
17.43 2.75 0.53 0.40 0.34 0.25 0.19
0,163 0,237 0.264 0,256 0.234 0.2t6 0 209
0.219 0.353 0.389 0.364 0.3t 1 0.265 0,236
*F(a) = 29.6 (a/W)'
;
-F(a) = 199
(a/W) '~
~F(a) -
1 93 (a/W) '~
3 . 0 7 ( a / W P ~2 -
The span/beam
depth
1.0 / / /
/
/
/
/OC~~ / ~) 9 /
LEAST SQUARE
LINE
-
Z
....I
0.2-
V/ 0
-0.1
36
/ 2/&//
3 LN {~J
# (ram)
1017 (a/W}'
38.48
( a / W F ~"
5
( a / W P '" -
2 5 . 1 1 ( a / W ) 7,'~
l a t W ) 9'2
Figure 3 A natural logarithmic plot of maximum load Kcagainst geometric parameter ~Pfor Hardiflex. Crack propagation is in transverse direction. Three-point notched bend data; •,W = 25 mm; ~,W = 50 mm; ~],
W -~ 75 mm; 4~, W = 100mm;~-1%
;E
~o.~
-
raUo = 4
+_10%
o//
]453
4- 2 5 8 0
1.2
o. 0 . 6 -
( a / W ) ~'~'
- 53 85 (a/WP"
I~CT
L~ <~
( a / W P ~;
- 638.9
0 . 4 1 l a / W P ~ -,- ; 8 . 7 Ca/WP '=
1.4.
0.8
1 8 5 . 5 { a l W P '~ - 6 5 5 . 7
6
W = 150ram; Q,W = 200 ram. Compact tension data: O:
Slow crack growth and fracture instability of cement composites
dependent on specimen geometry implying that the KR-curve has a geometry effect. A basic design procedure given in this paper involves determining the crack length at maximum load (am~×)from a given initial crack length (ao) using Figure 1 and calculating ¢ at am~× from Table 1. Figure 3 can then be used to evaluate the effective Kc at maximum load which in turn allows the maximum load (Pm~×)the structure can withstand to be determined. It must be finally emphasized that the results shown here are not yet conclusive and further work must be conducted to vindicate (1) the apparently geometry- and size-independent relationship between (a/W)o and (a/W)ma× as well as to determine (2) the correlations between Kc and ¢ for different specimen geometries in order to confirm if a geometry effect exists.
ACKNOWLEDGEMENT The work reported in this paper is supported by a research grant on cement-based composites from James Hardie & Coy. Pty. Ltd., Australia. REFERENCES 1. Mai, Y. W., Foote, R. M. L. and Cotterell, B., 'Size effects and scaling laws of fracture in asbestos cement', International Journal of Cement Composites, Vol. 2, No. 1, February 1980, pp. 23-34. 2. Foote, R. M. L., Cotterell, B. and Mai, Y. W., 'Crack
Mai and Cotterell
growth in cement composites', in 'Advances in Cement-Matrix Composites', Editor: D. M. Roy, Materials Research Society, 1980, pp. 135-44. 3. Lenain, J. C. and Bunsell, A. R, 'The resistance to crack growth of asbestos cement', Journal of Materials Science, Vol. 14, No. 2, February 1979, pp. 321-32. 4. Velazco, G., Visalvanich, K., Shah, S. P. and Naaman, A. E., 'Fracture behaviour and anlysis of fibre reinforced concrete beams', Progress Report for National Science Foundation Grant ENG 77-23534, University of Illinois at Chicago, USA, March 1979, p. 115. 5. Swamy, R. N., 'Influence of slow crack growth on the fracture resistance of fibre cement composites', International Journal of Cement Composites, Vol. 2, No. 1, February 1980, pp. 45-53. 6. Mai, Y. W. and Atkins, A. G., 'Crack stability in fracture toughness testing', Journal of Strain Analysis, Vol. 15, No. 2, April 1980, pp. 63-74. 7. Gurney, C. and Mai, Y. W., 'Stability of cracking', Engineering Fracture Mechanics, Vol. 4, No. 3, June 1972, pp. 853-64. 8. Srawley, J. E. and Brown, W. F., 'Plain strain crack toughness testing', Special Publication No. 410, American Society for Testing and Materials, Philadelphia, 1966, p. 65.
37
February 1982
Slow crack growth and fracture instability of cement composites
SYNOPSIS A crack instability anlysis is presented which allows the maximum load stress intensity factor (Ko) to be predicted from a predetermined correlation between Kc and a geometric parameter, @. Experimental results on asbestos cements are available in the literature and they show that there is a good correlation between K~ and reasonably independent of size effects. However, specimen geometry seems to have some influence on the K~-@relationship. This is suggested to be caused by a geometric effect on the KR-curve.
Y. W. Mai* and B. Cotterell*
INTRODUCTION It has been shown by several investigators [1-5] that in order to characterise the influence of slow crack growth on the fracture behaviour of fibre cement composites, the crack growth resitance (KR)curve may be used. Large KR values on the curve are always associated with large separation of the crack faces. However, for many practical applications, such excessive deformations are not usually permitted and thus the beneficial effects of fibre reinforcement may not be fully utilized. Indeed, a suitable design criterion could be one that is based on either the maximum load bearing capacity of the structure or some deformation such as deflection. This means that only a small amount of crack growth can be allowed before the maximum load is reached. For a given crack geometry and loading configuration of the structure the KR-curve may be used to determine the instability Kc value. Thus the maximum load (Pm~×)or the amount of slow crack growth (~'amax)can be calculated depending on which one of these two quantities is known. In this communication, a crack instability analysis is presented which relates the maximum load Kc value to a geometric parameter ~. This is followed by a discussion on the experimental results for asbestos cements taken from a few previous publications [1-3].
KEYWORDS Composite materials, fracture strength, asbestos cement, fracture properties, fibre reinforced cement, cracking (fracturing), crack propagation, slow cracking, failure, fracture tests, scale effect, strength of materials.
PREDICTION OF INSTABILITY Kc FROM KR-CURVE The conditions for crack stability to be achieved have been discussed in great detail in literature [6,7]. Briefly, a crack will propagate slowly and steadily when the energy released at the crack tip equals that required for crack growth. However, when the energy supply exceeds the demand unstable cracking will follow. Mathematically, the crack instability conditions can be written as [6,7]: KA = KR = Ko
"Department of Mechanical Engineering, University of Sydney, Sydney, NSW 2006, Australia. @ Construction Press 1982 0142-095X/82/04130033502.06
and
aKA aa
(1) _>--
dKR da
(2)
The R.H.S. term of expression (2) represents the variation of the crack growth resistance as the crack extends. To a first approximation the material crack growth resistance curve is given by:
33
Slow crack growth
ia
and fracture instability
= (a - aO) = p KG
equations (2) to (41, the obtained from:
(3)
where a0 and a are the initiatand effective crack lengths, p and m are constants. The L.H.S. term gives the variation of the applied stress intensity factor (KA) as the crack length increass and this can be evaluated at either constant load or constant displacement depending on the constraint of the testing machine. For engineering design purposes, load-controlled conditions are of more critical significance and it can be shown that:
(4)
where P is the applied load and F(a) is a function of crack length [8]. F’(a) denotes the first derivative of F(a). Using
0.8
CRACKING +
CT(Foote
IN LONGITUDINAL
DIRECTION
et al)
DCB (Foote
et al)
W=380mm
,
/
0.6
CRACKING
,/
IN
TRANSVERSE
I
DIRECTION CT (Foote
et al)
W=200mm @ CT (Lenain 8. Bunsell) W-120mm h 3-PT NOTHED
NORMALISED
34
INITIAL
CRACK
can be
Figure 1 Correlations between normafised initial crack length (a/WjO and crack length at maximum load talW)max.
W=200mm I
Kc value
where CD= F(a)/F’(a) is a geometric parameter. If the KRcurve is both geometry and size independent the maximum load Kc value will increase with Cp. Table 1 gives the non-dimensionalised (@/WI values for three specimen geometries: compact tension, three-point notch bend and single edge notch. Clearly, @ increases directly with W (except for low values of a/W in two cases) which is an indication of a size effect on Kc. Equation (5) suggests that for a material with agiven KRcurve characteristic such as that given by equation (3) a direct correlation should exist between the maximum load Kc and @. Thus, if Q>is known at crack instability Kc can be established and used for design calculations. However, since @ is a function of a, this will involve
1
I
instability
(Si
Ka = P F(a)
= P F’(a)
Mai and Cottereii
of cement composites
DEPTH, (a/W
lo
/
Slow crack growth and fracture instability of cement composites
measurements of slow crack growth (z~a) up to the maximum load point. To avoid tedious crack length measurements, it is useful to establish a relationship between the initial (ao)and the maximum load (area×)crack lengths.
Mai and Cotterell
appropriate for load-controlled tests. This approach is reasonable since the maximum loads obtained in both displacement and load-controlled machines are equal. Figure 1 shows that there is an approximate straight line relationship between (a/W)o and (a/W)ma,. All the experimental data [1-3] fall within the _+10% scatter band but there is a bias tendency for the data to lie below the best fit line for (a/W)o >/ 0.5. It also appears that specimen size, geometry and crack propagation direction do not affect this linear relationship. However, it is desirable to await more experimental data from other specimen geometries to validate this conclusion. The instability Kc values obtained for the asbestos cements studied in reference [3] contained 5.34% of asbestos fibres and these are plotted against @as shown in Figure 2. Results obtained for a similar cement composite ~ [1] with 5% asbestos fibres from single edge notch and three-point notch bend specimens are also shown in this figure. The correlation between Kc and is reasonable with a correlation coefficient of 0.85 but half of the experimental data are outside the _+10% scatter band of the mean line. It is not possible to comment conclusively on size and geometry effects from this figure since the asbestos cements studied in these two previous investigations [1,3] are not exactly
COMPARISON OF INSTABILITY ANALYSIS WITH EXPERIMENTAL RESULTS To find a correlation between the instability Kc and • it is necessary to have as many experimental data as possible on (Pmax, amax)for several specimen geometries with varying starter crack lengths and sizes. A search into the published literature reveals that only a few previous investigations contain such detailed results. One is on steel fibre reinforced mortar by Velazco et al. [4] using four-point notch bend beams of 76 mm depth (W). Although the normalised initial notch depths (a/W)o were varied between 0.13 to 0.83 the normalised crack depths at maximum load (a/W)ma* always exceeded 0.60 which made the calculation of Kc based on the four-point bend equation dubious [1]. More complete information can be obtained from previous work on asbestos-cements by Lenain and Bunsell [3] and by Mai and co-workers [1,2] using compact tension and three-point notch bend specimens. Although these latter investigations were conducted in displacement-controlled testing machines the experimental data of (P,a) at maximum load can be used here for the correlation analysis which is
~Theproductionmethodsfor thesetwo asbestoscements[1,3] are not the same.
Figure 2 A natural logarithmic plot of maximum load Kc against geometric parameter ~b for asbestos cements with approximately 5% fibres.
t10% 1/,
/
/ / ~2
E)
~ L E AST SQUARE LINE
/ 10 // -/
0.8
// /
~ 5% ASBESTOS CEMENT
/// A
06 .-% Z J
E) CT (Lenoin & Bunsell) • SEN (Mei et at) 3 PT NOTCH BEND SIZE A (smell) (Moi et el) ® 3 PT NOTCH BEND SIZE B (Iorge) (Mai et al)
/
E} EL
/
/
/ OZ, F
/
J/ /
02 I
0
l__
I
1
I
J
L
[
2
3
z.
5
6
7
LN ((~)(mm)
35
Slow crack growth and fracture instability of cement composites
Mai and Cottereli
identical and the available data are sparse. More definite conclusions can be drawn from Figure 3 in which Kc is plotted against @ for three-point notch bend Hardiflex cement sheets containing 8% asbestos and 7% cellulose fibres [1,2]. The beam depth (W) varied in these tests from 25 mm to 200 mm. There is now a very good correlation between Kc and ~, the correlation coefficient being 0,91. Only three of the 17 data points lie outside the ---10% scatter band of the least square line. It is obvious from this figure that the instability Kc values are larger for bigger size specimens which have correspondingly larger q~values. To gain some insight into geometric effects, a few additional experimental data have been obtained for the compact tension geometry at a/W -0.30 [2] for the same Hardiflex sheets. They are plotted in Figure 3 and fall outside the scatter band of the three-point notch bend data. If these additional data are included in the linear regression analysis the correlation coefficient drops to 0.73 which is rather poor. This indicates a possibly geometry-dependent K~-curve for the material, resulting in different m and /3 values in equation (5). It is thus necessary to obtain different correlational relationships for different specimen geometries if equation (5) is to be used for engineering design.
Table 1 Variation of geometric parameter ,b with (a/W)
CONCLUDING REMARKS
correlational relationship between Kc and ~. Although this relationship is apparently size-independent there are preliminary results which indicate that it is possibly
The crack instability analysis presented here enables the maximum load Kc value to be predicted from a
~/W a/W
Compact tension *
Single edge notch+
3-point notch bends
0.1 0.2 0,3 0.4 0,5 0.6 0.7
17.43 2.75 0.53 0.40 0.34 0.25 0.19
0,163 0,237 0.264 0,256 0.234 0.2t6 0 209
0.219 0.353 0.389 0.364 0.3t 1 0.265 0,236
*F(a) = 29.6 (a/W)'
;
-F(a) = 199
(a/W) '~
~F(a) -
1 93 (a/W) '~
3 . 0 7 ( a / W P ~2 -
The span/beam
depth
1.0 / / /
/
/
/
/OC~~ / ~) 9 /
LEAST SQUARE
LINE
-
Z
....I
0.2-
V/ 0
-0.1
36
/ 2/&//
3 LN {~J
# (ram)
1017 (a/W}'
38.48
( a / W F ~"
5
( a / W P '" -
2 5 . 1 1 ( a / W ) 7,'~
l a t W ) 9'2
Figure 3 A natural logarithmic plot of maximum load Kcagainst geometric parameter ~Pfor Hardiflex. Crack propagation is in transverse direction. Three-point notched bend data; •,W = 25 mm; ~,W = 50 mm; ~],
W -~ 75 mm; 4~, W = 100mm;~-1%
;E
~o.~
-
raUo = 4
+_10%
o//
]453
4- 2 5 8 0
1.2
o. 0 . 6 -
( a / W ) ~'~'
- 53 85 (a/WP"
I~CT
L~ <~
( a / W P ~;
- 638.9
0 . 4 1 l a / W P ~ -,- ; 8 . 7 Ca/WP '=
1.4.
0.8
1 8 5 . 5 { a l W P '~ - 6 5 5 . 7
6
W = 150ram; Q,W = 200 ram. Compact tension data: O:
Slow crack growth and fracture instability of cement composites
dependent on specimen geometry implying that the KR-curve has a geometry effect. A basic design procedure given in this paper involves determining the crack length at maximum load (am~×)from a given initial crack length (ao) using Figure 1 and calculating ¢ at am~× from Table 1. Figure 3 can then be used to evaluate the effective Kc at maximum load which in turn allows the maximum load (Pm~×)the structure can withstand to be determined. It must be finally emphasized that the results shown here are not yet conclusive and further work must be conducted to vindicate (1) the apparently geometry- and size-independent relationship between (a/W)o and (a/W)ma× as well as to determine (2) the correlations between Kc and ¢ for different specimen geometries in order to confirm if a geometry effect exists.
ACKNOWLEDGEMENT The work reported in this paper is supported by a research grant on cement-based composites from James Hardie & Coy. Pty. Ltd., Australia. REFERENCES 1. Mai, Y. W., Foote, R. M. L. and Cotterell, B., 'Size effects and scaling laws of fracture in asbestos cement', International Journal of Cement Composites, Vol. 2, No. 1, February 1980, pp. 23-34. 2. Foote, R. M. L., Cotterell, B. and Mai, Y. W., 'Crack
Mai and Cotterell
growth in cement composites', in 'Advances in Cement-Matrix Composites', Editor: D. M. Roy, Materials Research Society, 1980, pp. 135-44. 3. Lenain, J. C. and Bunsell, A. R, 'The resistance to crack growth of asbestos cement', Journal of Materials Science, Vol. 14, No. 2, February 1979, pp. 321-32. 4. Velazco, G., Visalvanich, K., Shah, S. P. and Naaman, A. E., 'Fracture behaviour and anlysis of fibre reinforced concrete beams', Progress Report for National Science Foundation Grant ENG 77-23534, University of Illinois at Chicago, USA, March 1979, p. 115. 5. Swamy, R. N., 'Influence of slow crack growth on the fracture resistance of fibre cement composites', International Journal of Cement Composites, Vol. 2, No. 1, February 1980, pp. 45-53. 6. Mai, Y. W. and Atkins, A. G., 'Crack stability in fracture toughness testing', Journal of Strain Analysis, Vol. 15, No. 2, April 1980, pp. 63-74. 7. Gurney, C. and Mai, Y. W., 'Stability of cracking', Engineering Fracture Mechanics, Vol. 4, No. 3, June 1972, pp. 853-64. 8. Srawley, J. E. and Brown, W. F., 'Plain strain crack toughness testing', Special Publication No. 410, American Society for Testing and Materials, Philadelphia, 1966, p. 65.
37