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Fracture toughness of plain concrete from compression splitting tests
The .;n~ematlonal Journal of Cement Composites and Lightweight Concrete, Volume 8, Number4
November 7986
Fracture toughness of plain concrete from compression splitting tests B. L. Karihaloo*
Synopsis The initiation (K~) and asymptotic (KR) values of fracture toughness of plain concretes are determined from compression splitting tests. These tests are simple to perform and do not require specimens of massive proportions even for the determination of KR. The fracture toughness so determined does not vary with the shape and the size of the specimens, provided that the width of the compressing platen and the depth of the pre-crack are within well defined limits ensuring failure of specimens by axial splitting. The tests reported here demonstrate conclusively the existence of R-curve behaviour in plain concrete. Keywords Fracture tests, concretes, compression splitting tests, fracture toughness, cracking (fracturing), fracture strength, water-cement ratio, compressive strength, crack propagation, strength of materials, failure.
INTRODUCTION The resistance of plain concretes to crack propagation (i.e. fracture toughness) is commonly determined from tension, compact tension and three-point bend tests [1-6]. The fracture toughness is usually expressed through critical stress intensity factor (K~c), critical energy release rate (G~c) or work of fracture (GF). It has been consistently noticed that the fracture toughness so determined varies not only with the mix variables but also with the test specimen dimensions and geometry [7-8], thereby casting doubt on the accuracy of the toughness value. There is as yet no agreement as to the test specimen geometry or test technique best suited for determining the intrinsic resistance of plain concretes to fracture, although discussions currently underway may prove successful [9J. However, it is now generally accepted that the concepts of linear elastic fracture mechanics are applicable to cementitious materials provided due allowance is made for the substantial slow, stable crack growth (process zone) preceding fracture [10-11] and for the true state of stress existing at a propagating crack front [12]. The former is achieved by augmenting the pre-crack length appearing in the expressions for fracture toughness by the size of the process zone, while the latter is accomplished by an appropriate modification of these expressions. The fracture toughness so determined is designated Ko or Gc [12]. Several techniques have been proposed for determining the size of the process zone, but until now they have only had partial success. It is also clear that cementitious * School of Civiland Mining Engineering,The Universityof Sydney. Sydney2006, Australia © Longman Group UK Ltd 1986 0262-5075/86/08405251/$02.00
materials do exhibit crack growth resistance behaviour [13, 14], i.e. their fracture toughness achieves an asymptotic value designated KR, after substantial stable crack growth. However, the experimental determination of KR from existing test specimen geometries requires specimens of gargantuan proportions and the measurement of additional quantities such as the crack mouth opening displacement, which can present significant problems. In this paper an alternative test procedure is proposed for determining the fracture toughness of plain concretes. It permits determination of both the initiation (K~c)and the asymptotic values (KR) of fracture toughness from a single test in compression. In this test procedure a sharp starting crack is introduced into the test specimen of concrete. Propagation of this flaw is induced by loading a narrow steel platen onto the upper surface of the specimen. The flaw propagates more or less in line with itself, eventually splitting the specimen into halves. Experience with this experimental procedure has shown [16-18] that the fracture toughness so determined is independent of specimen geometry, provided that the platen is rather narrow and the starting crack is sufficiently deep to ensure that failure in compression occurs by axial splitting and not by crushing under the platen or by buckling. Other main advantages of the compression splitting test are the relative ease of specimen preparation and testing and its similarity to the commonly used quality control test for concrete. Moreover, the specimens do not have to be very large even when the KR value is being determined.
SPECIMEN PREPARATION AND TEST PROCEDURE The specimen geometry is shown in Figure l a. This geometry was chosen because of ease of preparation and testing. It should be stressed, however, that there is
251
Fracture toughness of plain concrete from compression splitting tests
no reason w h y other geometries and/or sizes may not be adopted. In fact, several different geometries and sizes have been tried in the past [17-18] with no noticeable effect on the fracture toughness. In order to ensure failure by axial splitting it is necessary that c/d/> 3/2 and w/d ~< V2 (see below). The mix proportions, w a t e r cement ratios and the type of coarse aggregate used in the 12 mixes are listed in Table 1. In each experimental factorial group of tests, five specimens and two cylinders generally formed the sample. The test specimens w e r e cast in steel moulds with steel bases. The geometrical properties of the test specimens of various sizes are given in Table 2. The pre-crack (notch) was introduced using a 5 mm thick mild steel strip milled to a very sharp edge. The edge was remilled as often as necessary to retain its sharpness. The concrete was poured and vibrated with the steel w e d g e in position. To reduce the occurrence of prema ture cracking during stripping the wedge was gently
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pulled out from the test specimen after the initial setting time (approximately 7 hours). The specimens w e r e stripped after 24 hours and placed inside a fog chamber for curing. Two standard cylinders (diameter 150mm) w e r e also cast from each mix and cured together with the test specimens. These were used to determine the compressive strength (~Tc)and elastic modulus (E) of the mix. The m a x i m u m size of coarse aggregate in each of the mixes was 20ram. The crushed coarse aggregate used was dacite. The specimens were removed from the fog chamber on the 28th day of curing and air dried at room temperature for 2 hours. Any accumulated debris was removed from the front of the notch in each of the specimens, and the notched sides were painted with t w o coats of w h i t e paint. The specimens were left at room temperature for a further 1 hour to allow the paint to dry. Propagation of the sharp notch front was induced by
Table 1 Mix proportion, kg
Batch no.
Dacite 20 mm
1 2 3 4 5 6
37.5 37.5 37.5 375
7
375
8 9 10 11 12
37.5
Table2
375 37.5
River gravel 20 mm
River gravel 10 mm
Beach sand
River sand
Cement
Water
Slump, mm
-
37.5 37.5 37.5 37.5 37.5 37.5
18.75 18.75 20.75 -23.25 23.25
18.75 18.75 20.75 37.5 2075 20.75 230 23.0 23.75 23.75 20.75 23.0
26.55 26.55 22.55 26.55 20.0 20.0 18.2 18.2 16.6 16.6 16.6 18.2
9.75 10.65 10.65 !065 10.0 10.0 10.0 10.0 100 10.0 10.0 100
80 75 75 85 75 75 85 80 85 80 85 80
--
37.5 375 37.5 375 -
37.5
23.0
37.5 37.5 37.5 37.5 37.5
23.0 20.75 20.75 23.75 23.0
Geometrical and selected mix properties of test specimens
Batch
Pre crack length, c mm
Width, d mm
Depth, b mm
Platen width, w mm
Watercement
Coarse aggregate type
1 2 3 4 5 6 7 8 9 10 1i 12
145 145 145 145 145 145 145 145 145 145 175 175
95 95 95 95 95 95 95 95 95 95 95 95
100 100 100 100 100 100 100 100 100 100 100 100
38 38 38 38 38 38 38 38 50 50 38 38
0.37 0.40 0.47 040 0.50 0.50 0.55 0.55 0.60 0.60 060 0.55
CRG* CRG CRG RRG** CRG RRG RRG CRG CRG RRG CRG CRG
CRG Crushed River Gravel (Dacite) * RRG Rounded Rwer Gravel
252
Fracture toughness of ptain concrete from compression splitting tests
F
F
1
"
II:,
C
v
"i
///
F
Figure 1 Possible geometries of test specimens. Specimens of type (a) were used in the present study
1 ,,
IIII II It
/4
/ (a)
F
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)
l/
/ (b)
(,:::)
loading a flat steel platen onto the upper surface of the specimen, taking care to make contact over the entire area of compression. The painted surfaces of the specimen were illuminated by powerful spotlights in order to monitor the progress of the advancing crack front. The compressive force was applied symmetrically (with respect to the notch) and gradually at a nominal rate of 1 kN/sec. The force at which the pre-existing notch front started to propagate was noted. This force is used to calculate the initiation value of fracture toughness K~c. Thereafter, the force was increased in steps of 5-10 kN and at the end of each increment the position of the crack front was marked on both painted surfaces. If it was not possible to locate clearly the crack front on either of the surfaces, the load was further incremented until the front became visible again on at least one of the surfaces. The procedure of incrementing the load and marking the position of the crack front was continued until no further load increment was possible. At this moment, the central crack grew rapidly, eventually splitting the specimen into halves. It should be mentioned that until this moment of instability, the crack advanced stably almost along the centre of the specimen. A typical crack growth pattern is visible on the photograph of Figure 2a, while Figure 2b highlights the location of the crack front at various load levels. Notice
(d)
the increase in the crack extension between successive load levels. The position of the crack front at each load level was established after the completion of the test from the recordings made during the test. Because the crack propagation path was usually tortuous, it was divided into linear segments for measurement purposes. Table 3 shows the mean crack extension and deviation as obtained from recordings over the two faces of each of the five specimens tested from a batch. At some load levels it was not possible to record crack extension in all specimens; in such cases no deviation has been reported. For each mix (batch) the entries in Table 3 begin at no crack extension, i.e. at the load level corresponding to the onset of pre-crack growth, and end at the load level corresponding to unstable crack growth. The elastic modulus (E) and compressive strength ((r~) of each mix were determined from separate cylinder tests. The cylinders were 150mm in diameter and 3 0 0 m m tall. It should be noted that (re is not required in the calculation of K~cor KR from the compression splitting tests; only E appears in the expression for fracture toughness (see below). The compressive strength of mix was determined with a view to relating it to KR. Two electrical strain gauges to measure the compressive strain were glued on diametrically opposite points at mid height of the cylinder. The load was
253
Fracture tough,hess of p/a~n concrete from compression sphtbng tests
increased in steps of 50 kN and the strain readings were automatically recorded. E was calculated from the first few recorded readings. The variation of ~T~and E of the several mixes used in this study is shown in Figure 3 as a function of the w a t e r - c e m e n t ratio and coarse aggregate type.
C O M P R E S S I O N SPLITTING M O D E L
The initial flaw (pre-crack) effectively divides the test specimen into two identical struts which bend when compressed to open up the central gap. If the compressing platen is fairly narrow and the central gap sufficiently deep, then the specimen fails invariably by axial splitting and not by crushing under the platen or by buckling, In fact, it has been shown [17-18] that failure by axial splitting is assured provided w/d ~< 1/2 and c/d ~ 3/2. Moreover, it is known [18] that K~cis insentitive to loading
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rate provided c/d ~> 3/2. However, it is important not to choose a compressing platen that is so narrow as to lead to local crushing of the material. Should such a situation arise then either the platen width or the pre-crack length or both should be increased within the above limits. For example, from tests on the first specimen from each of the batches 9 and 10 (Table 3) it quickly became apparent that the local compressive stress under the narrow platen (w = 38 mm) exceeded the compressive strength of these weak mixes, leading to failure by crushing under the platen rather than by axial splitting. To remedy this situation a wider platen (width 50ram) was used. On the other hand, for weak mixes such as batches 11 and 12 the same problem was resolved not by increasing the platen width but by increasing the depth of central cut from 1 4 5 m m t o 175mm. The propagation of the pre-crack under compression is due to tensile stresses that arise because of the flexing
Table3 Compressive force and crack extension
Batch No./Crack Extension, mm Force F kN
1
2
3
4
5
5O
6
7
8
0
0
9
0
0
7O
1821
~+33 ,~u 22 +12 31_16
33*4
10
13_+0
51_+5
31 -+12 8
+7 31 -13
49
45,10
+14 63_20
48*8
80
0
0
0
3128
85 17+t5' -12
15
17_+~
5~+15 b--11
60_+4
73,4
97
85_+0
+9" 19_12
~+10 Jb_l 4
68_+12
96_+~
90 +11 -17162.3
109_+2
95 100 105 110
__+12 L~-24
34+12 _13
+15
59-12
78*7
_+14 +32 12Y_9 127_37
51 +16 -18
+4 62-10
93-+13
166-+7
125 57_+17
~+15 bZ--14
91+4 --6
+18208.1 221 -28
135 67
-+16 10Z_14
131
150
89
+56 151_62
164+18
155 160 170 180
106 132_+1 123+22 +60 138__48 211.18
190
161.19
254
197 205
+36 -31
141_+7 ~+16
1/"_25 21129
0
0 1-+b_513
+7
42_12 23+2 0
67*8
55+~
13
75+19_ 75_10
17
+6
57
119_+2 152217
7028165+3121_+11 .+19
159
84_29 _+26 1061 0~ 33
215
__+17
1/u_27
225
1301 81 _+12 155
140
12
27_+~
89
t20
11
+1211~+14 30-20 ~-2598 41.16
115
130
10
0
6O
90
9
153-+12
195
Fracture toughness of p/,~u~concrete from compress/on sp//tt/ng tests
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(a)
(b}
Figure2 a) Typical crack growth pattern; b) location of crack front at load levels up to the instant of unstable crack growth
60
1
,xl
I
/+5
I
×
X
n 3-Jr-
50-
~
C aJ
i./)
~0--
aJ
i./1 Q.I
2
\ \
, c R G
-
+ ~~ ~
30--
X
~0
×
.~ 35
¢-
>-
30-
E 0 LJ
20 0.2
1 0.3
I
I
0.4 0.5 Wafer/Cement
\ 0.6
25 0.2
I 0.3
I
I
0./. 0.5 Water/Cement
I 0.6
Figure3 Variation of compressive strength and modulus of elasticity with water-cement ratio for mixes prepared with two different types of coarse aggregate
255
Fracture toughness of p/a/n concrete from compression sphtt/ng tests
of the specimen's arms (two struts). Elastic finiteelement and boundary-element calculations [17-181 have confirmed the existence of such stresses for w/d <~ V2 and c/d = 3/2. According to linear elastic fracture mechanics, crack propagation occurs when the energy released from the struts balances the energy required to create new crack surfaces. Using this energy balance principle it has been shown [17] that the Griffith surface energy per unit area R is given by F2e2 kc R = 3-~C-(2-kc tan-~-) sec 2 2
(1)
where F is the compressive force corresponding to central gap length c, k 2 = F/(2EI) and E is Young's modulus of the material. For rectangular specimens e = (d-w)/4, I -- bdS/96 and C = EIb. For circular specimens of diameter d, the load eccentricity e = [(2d/3~r) - w/4], i = d ~ (9Tr2 - 64)/(1152Tr) and C = EId. The load eccentricity, e and second moment of area of each strut about the axis of bending, t are easily calculated for other cross-sectional shapes. It should be mentioned that the (small) contribution from shear deformations has not been included in Equation (1). It is also worth remembering that the above relation between R, w/d, c/d and F applies only to compression splitting mode of failure. If plane stress conditions are assumed to prevail in the test specimen at the crack front, the elastic fracture surface energy, R, may be expressed in terms of the stress intensity factor K~, whereupon kc K~.,'D = B ( 2 - k c tan-~-) sec 2 2
(2)
For rectangular specimens D = (F/bd)2d and B = 13 (I - w,"d)2}.'16 , and for circular specimens D = (4F/~d2)2d and B =
8 (9-~ (3~
wd )2}/{64 ( 9 -
6Tr42) I .
The replacement of R with opening mode stress intensity factor K~ implies that the pre-crack growth results from tensile stresses developed at the crack front by the bending of the struts. As mentioned previously, the existence of such stresses has been confirmed by elastic finite-element and boundary-element calculations [17-181 for the range of w/d and c/d used in the present study.
RESULTS AND DISCUSSION Equation (2) was used to calculate the values of K, from the onset of pre-crack growth right up to the moment of fast, unstable crack growth. To calculate the initiation value of K~, namely K~,:, the compressive force at the onset of pre-crack growth and the initial value of precrack length were used. However, the values of Kr during the slow, stable crack growth were calculated using the actual load level and the corresponding augmented crack length equal to the sum of pre-crack length and mean
256
Kanhaloo
crack extension. This is necessarily an approximation in that the crack front does not advance exactly along the middle of the specimen. However, it should be noted that Equation (2) remains valid even at the full crack extension because the buckling load of the struts is still well in excess of the load at the onset of instability. The results of these calculations are shown in Figure 4 for typical mixes used in this study. For each mix the R-curve behaviour is quite evident, allowing the estimation of the asymptotic value, KR. It is worth mentioning that specimens from identical mixes (mixes 8 and 12, and 9 and 11) gave fairly close values of K~c and KR, although the geometrical test conditions were different. Thus, specimens from mix 8 contained a 145mm deep pre-crack, whereas specimens from the identical mix 12 contained a 175mm deep pre-crack. In both cases, however, the platen was 38mm wide. On the other hand, specimens from mix 9 contained a 145mm deep pre-crack but were loaded by a platen 50mm wide, whereas specimens from the identical mix 11 contained a 175 mm deep pre-crack but were loaded by a narrower platen (width 38mm). Based on the values of K~cand the estimated values Of KR obtained from the tests, it is possible to judge their variation with one of the important mix variables, namely the water to cement ratio. This variation is shown in Figure 5. Two interesting results emerge from this figure. First, the initiation value of K~ for mixes using rounded river aggregate is practically independent of w a t e r - c e m e n t ratio. This is not surprising because the initiation value of K~ depends on local stress gradients in the vicinity of coarse aggregate particles. With rounded river aggregate the gradients are generally not steep, with the result that K~c of mixes using this type of aggregate is almost equal to that of the cement/sand matrix. On the other hand, because of steep local gradients at crushed aggregate sites it is easier for crack growth to initiate in weaker mixes at these sites, resulting in a decrease of K~,: with increasing w a t e r cement ratio. Secondly, the asymptotic value of K~, namely KR decreases rapidly with increasing water-cement ratio; the rate of decrease is practically independent of the type of coarse aggregate used in the mix, although fora given w a t e r - c e m e n t ratio the KR value of mixes using CRG is higher than that of mixes using RRG. The latter is a result of the relative ease with which cracks can propagate in mixes using RRG than in mixes using CRG. The decrease Of KR with increasing water-cement ratio is a consequence of the weakness of the mix which reduces its resistance to crack propagation KR decreases almost linearly with increasing water--cement ratio, i.e with decreasin 9 compressive strength (Figure 6). Depending on the type of coarse aggregate, K~ may be estimated from the following expressions, once the compressive strength (~: or the water-cement ratio 11 of the mix is known: For mixes using crushed river gravel K~ = 4.10 - 4 5 ~q, 0 3 ~< ~/ ~< 0.7 K~ = ~r + 205),'33, 125 ~<,n ~<60
(3)
Fracture toughness of plato concrete from compression spkttmg tests
2.5 KR
I
I
I ,
2.0--
/
I
Kanhaloo
2.5
I
I
I
I
I
I
- KR
o ~
2,0o/o
Batch 1 Water/Cement:O 37 Crushed Graver
E
2 Z
15--
_
b
E ro o-
Batch 2 Water/Cement=O 40 Crushed Gravel
~__o._~/~-~-~ /
15 /
-7"
1.0 /
1 0 ~//Kk
. Kk
0.5
I
I
50
0
I
]
I
100 150 200 Crack Extension (mm)
250
,l
300
50
2.5 2.0
K~
I
I
I
I
,]
I
100 150 200 Crack Extension (mm)
I
I
I
250
300
I
I o ____-.~I -
KR
20 ~-"
15
2 "r
:~
/
1.0
b E to
Batch 3 Water/Cement:O 47 Crushed 5ravel
E
1.0:
i
0.5
I
I
- KR
I
I
100 150 200 [rack Extension {mm)
5O
1.75
Batch 4 Water/Cement=O 4 Rounded Gravel
~--~-~ /~_~_..,--o--~
Kic
0
15
:E
I
I
I
o2~"
I
250
300
•
I
05
l
50
175
I
-
Kk
I
I
100 150 200 Crack Extension (mm)
I
I
I
I
I
250
300
I
KR
15
1.5
f
/
E
2 1.0
'~°
/
E to Q_
Batch 5 Water/Cement=O 5 Crushed Gravel
/ Batch 6 Water/Cement=0 50 Rounded Gravel
~,~ 1.0
/ ~Ic
I
050
Figure4
50
I
l
i
100 150 200 Crack Extension (ram)
I
250
300
05
I
50
I
I
100 I50 200 Crack Extension (mm}
I
250
300
I n c r e a s e in Ki w i t h c r a c k e x t e n s i o n f o r v a r i o u s r n i x e s
257
Fracture toughness of pta~n concrete from compress,'on sphtbng tests
For mixes using rounded river gravel: KR = 4 4 5 -- 4.75 q, 0.3 ~< q ~< 0.7 KR = (m + 7)/26, 19 ~< ~o ~< 57
(4)
In Equations (3) and (4), KR is in MPa m 12 ( M N m 32) and {rc is in MPa (MN/m2).
2S
2.0
D
E (3_ :IE
115 -
X
O5 0.3
04
x
0.5 Water/Cement
06
Figure 5 Variation of Kic and KR with water-cement ratio for mixes prepared with two different types of coarse aggregate
60
CONCLUSIONS 1. The compression splitting test for determining the fracture toughness of plain concretes is simple to perform and does not require specimens of massive proportions even when the asymptotic value of toughness is to be determined. The results are independent of the shape and size of the specimens, provided that the width of the compression platen and the depth of pre-crack are within certain limits in order to ensure failure by axial splitting. The rough manual procedure for locating the position of the advanced crack front used in this study can obviously be improved, without jeopardising the simplicity of the actual test method. Likewise, it is possible to improve the expression for KI using the two-strut model, by including non-linear material properties. 2. The results of the present investigation prove conclusively the existence of R-curve behaviour in concrete. In fact, the slow crack growth responsible for this behaviour is a reservoir of latent strength in the material; the KR value for a given water-cement ratio is roughly three times the initiation value. 3. The compression splitting test is currently being applied to fibre reinforced concretes. The results of this investigation will be reported in a future communication.
\
1.0-
RRG--x
Karihaloo
,
,
I
I
REFERENCES 1. Nallathambi, PI, Karihaloo, BILI and Heaton, B. S i 'Effect of specimen and crack sizes, water-cement ratio and coarse aggregate texture upon fracture toughness of concrete', Magazine of Concrete Research, Vol. 36, No. 129, December 1984, pp. 227-36. 2. Nallathambi, P., Karihaloo, B. L. and Heaton, B. S. 'Various size effects in fracture of concrete', Cement and Concrete Research, Vol 15, No. 1, February 1985, pp. 117-26. 3. Legendre, D. and Mazars, J. 'Damage and fracture mechanics for concrete (a combined approach)', Advances in Fracture Research: Proceedings of the
j..-o
Figure 6 Variation of KR with compressive strength of mixes prepared with two different types of coarse aggregate
SO
40 I0
3O
RRG/XERGX
20 1.0
1.5
2.0 KR (HPa.m l/z)
258
2.5
Fracture toughness of plain concrete from compression spl/tting tests
4. 5.
6.
7.
8. 9.
10.
6th International Conference on Fracture (ICF6), Pergamon Press. Editors: S. R. Valluri et al., Vol. 4, 1984, pp. 2841-8. Petersson, P. E. 'Fracture energy of concrete: method of determination', Cement and Concrete Research, Vol. 10, No. 1, February 1980, pp. 78-89. Rossi, P., Acker, P. and Francois, D. 'Measurements of fracture toughness Kjc of concrete', Advances in Fracture Research: Proceedings of the 6th International Conference on Fracture (ICF6), Pergamon Press, Editors: S. R. Valluri et al., Vol. 4, 1984, pp. 2833-40. Barker, D. B., Hawkins, N. M., Jeang, F. L, Cho, K. Z. and Kobayashi, A. S. 'Concrete fracture in a CWL-DCB specimen', Journal of Engineering Mechanics, Proceedings, American Society of Civil Engineers, Vol. 111, No. EM5, May 1985, pp. 623-38. Hillsdorf, H. K. and Brameshuber, W. 'Size effects in the experimental determination of fracture mechanics parameters', NATO Advanced Research Workshop: Application of fracture mechanics to cementitious materials, 4-7 September 1984. Editor: S. P. Shah, Northwestern University, Evanston, U.S.A. pp. 213-54. Bazant, Z. P., Kim, J. K. and Pfeiffer, P. 'Determination of nonlinear fracture parameters from size effect tests', /bid, pp. 143-70. Hillerborg, A. 'Results of three comparative test series for determining the fracture energy GF of concrete', Materiaux et Constructions, Vol. 18, No. 107, September-October 1985, pp. 407-13. Wecharatana, M. and Shah, S. P. 'Predictions of
11.
12.
13.
14.
15.
16. 17. 18.
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nonlinear fracture process zone in concrete', Journal of Engineering Mechanics, Proceedings, American Society of Civil Engineers, Vol. 109, No. EM5, May 1983. pp. 1231-46. Cho, K. A., Kobayashi, A. S., Hawkins, N. M., Barker, D. B. and Jeang, F. L 'Fracture process zone of concrete cracks', Journal of Engineering Mechanics, Proceedings, American Society of Civil Engineers, Vol. 109, No. EM8, August 1984. pp. 1174-84. Nallathambi, P. and Karihaloo, B. L 'Determination of specimen-size independent fracture toughness of plain concrete', Magazine of Concrete Research Vol. 38, No. 135, June 1986. pp.67-76. Wecharatana, M. and Shah, S. P. 'Slow crack growth in cement composites', Journal of Structural Division, Proceedings, American Society of Civil Engineers, Vol. 108, No. ST6, June 1982. pp. 1400-13. Wecharatana, M. and Shah, S. P. 'A model for predicting fracture resistance of fiber reinforced concrete', Cement and Concrete Research, Vol. 13, No. 6, December 1983. pp. 819-29. Kendall, K. 'Complexities of compression failure', Proceedings of Royal Society London, Vol. A361. 1978, pp. 245-63. Karihaloo, B. L. 'A note on complexities of compression failure', ibid. Vol. A368. 1979 pp. 483-93. Karihaloo, B. L. 'Compressive failure of brittle materials', ibid. Vol. A396. 1984. pp. 297-314. Guest, B. W. and Karihaloo, B L. 'Fracture of brittle materials', Journal of Materials Science Letters, Vol. 4, No. 10, October 1985. pp. 1285-9.
259
November 7986
Fracture toughness of plain concrete from compression splitting tests B. L. Karihaloo*
Synopsis The initiation (K~) and asymptotic (KR) values of fracture toughness of plain concretes are determined from compression splitting tests. These tests are simple to perform and do not require specimens of massive proportions even for the determination of KR. The fracture toughness so determined does not vary with the shape and the size of the specimens, provided that the width of the compressing platen and the depth of the pre-crack are within well defined limits ensuring failure of specimens by axial splitting. The tests reported here demonstrate conclusively the existence of R-curve behaviour in plain concrete. Keywords Fracture tests, concretes, compression splitting tests, fracture toughness, cracking (fracturing), fracture strength, water-cement ratio, compressive strength, crack propagation, strength of materials, failure.
INTRODUCTION The resistance of plain concretes to crack propagation (i.e. fracture toughness) is commonly determined from tension, compact tension and three-point bend tests [1-6]. The fracture toughness is usually expressed through critical stress intensity factor (K~c), critical energy release rate (G~c) or work of fracture (GF). It has been consistently noticed that the fracture toughness so determined varies not only with the mix variables but also with the test specimen dimensions and geometry [7-8], thereby casting doubt on the accuracy of the toughness value. There is as yet no agreement as to the test specimen geometry or test technique best suited for determining the intrinsic resistance of plain concretes to fracture, although discussions currently underway may prove successful [9J. However, it is now generally accepted that the concepts of linear elastic fracture mechanics are applicable to cementitious materials provided due allowance is made for the substantial slow, stable crack growth (process zone) preceding fracture [10-11] and for the true state of stress existing at a propagating crack front [12]. The former is achieved by augmenting the pre-crack length appearing in the expressions for fracture toughness by the size of the process zone, while the latter is accomplished by an appropriate modification of these expressions. The fracture toughness so determined is designated Ko or Gc [12]. Several techniques have been proposed for determining the size of the process zone, but until now they have only had partial success. It is also clear that cementitious * School of Civiland Mining Engineering,The Universityof Sydney. Sydney2006, Australia © Longman Group UK Ltd 1986 0262-5075/86/08405251/$02.00
materials do exhibit crack growth resistance behaviour [13, 14], i.e. their fracture toughness achieves an asymptotic value designated KR, after substantial stable crack growth. However, the experimental determination of KR from existing test specimen geometries requires specimens of gargantuan proportions and the measurement of additional quantities such as the crack mouth opening displacement, which can present significant problems. In this paper an alternative test procedure is proposed for determining the fracture toughness of plain concretes. It permits determination of both the initiation (K~c)and the asymptotic values (KR) of fracture toughness from a single test in compression. In this test procedure a sharp starting crack is introduced into the test specimen of concrete. Propagation of this flaw is induced by loading a narrow steel platen onto the upper surface of the specimen. The flaw propagates more or less in line with itself, eventually splitting the specimen into halves. Experience with this experimental procedure has shown [16-18] that the fracture toughness so determined is independent of specimen geometry, provided that the platen is rather narrow and the starting crack is sufficiently deep to ensure that failure in compression occurs by axial splitting and not by crushing under the platen or by buckling. Other main advantages of the compression splitting test are the relative ease of specimen preparation and testing and its similarity to the commonly used quality control test for concrete. Moreover, the specimens do not have to be very large even when the KR value is being determined.
SPECIMEN PREPARATION AND TEST PROCEDURE The specimen geometry is shown in Figure l a. This geometry was chosen because of ease of preparation and testing. It should be stressed, however, that there is
251
Fracture toughness of plain concrete from compression splitting tests
no reason w h y other geometries and/or sizes may not be adopted. In fact, several different geometries and sizes have been tried in the past [17-18] with no noticeable effect on the fracture toughness. In order to ensure failure by axial splitting it is necessary that c/d/> 3/2 and w/d ~< V2 (see below). The mix proportions, w a t e r cement ratios and the type of coarse aggregate used in the 12 mixes are listed in Table 1. In each experimental factorial group of tests, five specimens and two cylinders generally formed the sample. The test specimens w e r e cast in steel moulds with steel bases. The geometrical properties of the test specimens of various sizes are given in Table 2. The pre-crack (notch) was introduced using a 5 mm thick mild steel strip milled to a very sharp edge. The edge was remilled as often as necessary to retain its sharpness. The concrete was poured and vibrated with the steel w e d g e in position. To reduce the occurrence of prema ture cracking during stripping the wedge was gently
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pulled out from the test specimen after the initial setting time (approximately 7 hours). The specimens w e r e stripped after 24 hours and placed inside a fog chamber for curing. Two standard cylinders (diameter 150mm) w e r e also cast from each mix and cured together with the test specimens. These were used to determine the compressive strength (~Tc)and elastic modulus (E) of the mix. The m a x i m u m size of coarse aggregate in each of the mixes was 20ram. The crushed coarse aggregate used was dacite. The specimens were removed from the fog chamber on the 28th day of curing and air dried at room temperature for 2 hours. Any accumulated debris was removed from the front of the notch in each of the specimens, and the notched sides were painted with t w o coats of w h i t e paint. The specimens were left at room temperature for a further 1 hour to allow the paint to dry. Propagation of the sharp notch front was induced by
Table 1 Mix proportion, kg
Batch no.
Dacite 20 mm
1 2 3 4 5 6
37.5 37.5 37.5 375
7
375
8 9 10 11 12
37.5
Table2
375 37.5
River gravel 20 mm
River gravel 10 mm
Beach sand
River sand
Cement
Water
Slump, mm
-
37.5 37.5 37.5 37.5 37.5 37.5
18.75 18.75 20.75 -23.25 23.25
18.75 18.75 20.75 37.5 2075 20.75 230 23.0 23.75 23.75 20.75 23.0
26.55 26.55 22.55 26.55 20.0 20.0 18.2 18.2 16.6 16.6 16.6 18.2
9.75 10.65 10.65 !065 10.0 10.0 10.0 10.0 100 10.0 10.0 100
80 75 75 85 75 75 85 80 85 80 85 80
--
37.5 375 37.5 375 -
37.5
23.0
37.5 37.5 37.5 37.5 37.5
23.0 20.75 20.75 23.75 23.0
Geometrical and selected mix properties of test specimens
Batch
Pre crack length, c mm
Width, d mm
Depth, b mm
Platen width, w mm
Watercement
Coarse aggregate type
1 2 3 4 5 6 7 8 9 10 1i 12
145 145 145 145 145 145 145 145 145 145 175 175
95 95 95 95 95 95 95 95 95 95 95 95
100 100 100 100 100 100 100 100 100 100 100 100
38 38 38 38 38 38 38 38 50 50 38 38
0.37 0.40 0.47 040 0.50 0.50 0.55 0.55 0.60 0.60 060 0.55
CRG* CRG CRG RRG** CRG RRG RRG CRG CRG RRG CRG CRG
CRG Crushed River Gravel (Dacite) * RRG Rounded Rwer Gravel
252
Fracture toughness of ptain concrete from compression splitting tests
F
F
1
"
II:,
C
v
"i
///
F
Figure 1 Possible geometries of test specimens. Specimens of type (a) were used in the present study
1 ,,
IIII II It
/4
/ (a)
F
Karihaloo
)
l/
/ (b)
(,:::)
loading a flat steel platen onto the upper surface of the specimen, taking care to make contact over the entire area of compression. The painted surfaces of the specimen were illuminated by powerful spotlights in order to monitor the progress of the advancing crack front. The compressive force was applied symmetrically (with respect to the notch) and gradually at a nominal rate of 1 kN/sec. The force at which the pre-existing notch front started to propagate was noted. This force is used to calculate the initiation value of fracture toughness K~c. Thereafter, the force was increased in steps of 5-10 kN and at the end of each increment the position of the crack front was marked on both painted surfaces. If it was not possible to locate clearly the crack front on either of the surfaces, the load was further incremented until the front became visible again on at least one of the surfaces. The procedure of incrementing the load and marking the position of the crack front was continued until no further load increment was possible. At this moment, the central crack grew rapidly, eventually splitting the specimen into halves. It should be mentioned that until this moment of instability, the crack advanced stably almost along the centre of the specimen. A typical crack growth pattern is visible on the photograph of Figure 2a, while Figure 2b highlights the location of the crack front at various load levels. Notice
(d)
the increase in the crack extension between successive load levels. The position of the crack front at each load level was established after the completion of the test from the recordings made during the test. Because the crack propagation path was usually tortuous, it was divided into linear segments for measurement purposes. Table 3 shows the mean crack extension and deviation as obtained from recordings over the two faces of each of the five specimens tested from a batch. At some load levels it was not possible to record crack extension in all specimens; in such cases no deviation has been reported. For each mix (batch) the entries in Table 3 begin at no crack extension, i.e. at the load level corresponding to the onset of pre-crack growth, and end at the load level corresponding to unstable crack growth. The elastic modulus (E) and compressive strength ((r~) of each mix were determined from separate cylinder tests. The cylinders were 150mm in diameter and 3 0 0 m m tall. It should be noted that (re is not required in the calculation of K~cor KR from the compression splitting tests; only E appears in the expression for fracture toughness (see below). The compressive strength of mix was determined with a view to relating it to KR. Two electrical strain gauges to measure the compressive strain were glued on diametrically opposite points at mid height of the cylinder. The load was
253
Fracture tough,hess of p/a~n concrete from compression sphtbng tests
increased in steps of 50 kN and the strain readings were automatically recorded. E was calculated from the first few recorded readings. The variation of ~T~and E of the several mixes used in this study is shown in Figure 3 as a function of the w a t e r - c e m e n t ratio and coarse aggregate type.
C O M P R E S S I O N SPLITTING M O D E L
The initial flaw (pre-crack) effectively divides the test specimen into two identical struts which bend when compressed to open up the central gap. If the compressing platen is fairly narrow and the central gap sufficiently deep, then the specimen fails invariably by axial splitting and not by crushing under the platen or by buckling, In fact, it has been shown [17-18] that failure by axial splitting is assured provided w/d ~< 1/2 and c/d ~ 3/2. Moreover, it is known [18] that K~cis insentitive to loading
Karihaloo
rate provided c/d ~> 3/2. However, it is important not to choose a compressing platen that is so narrow as to lead to local crushing of the material. Should such a situation arise then either the platen width or the pre-crack length or both should be increased within the above limits. For example, from tests on the first specimen from each of the batches 9 and 10 (Table 3) it quickly became apparent that the local compressive stress under the narrow platen (w = 38 mm) exceeded the compressive strength of these weak mixes, leading to failure by crushing under the platen rather than by axial splitting. To remedy this situation a wider platen (width 50ram) was used. On the other hand, for weak mixes such as batches 11 and 12 the same problem was resolved not by increasing the platen width but by increasing the depth of central cut from 1 4 5 m m t o 175mm. The propagation of the pre-crack under compression is due to tensile stresses that arise because of the flexing
Table3 Compressive force and crack extension
Batch No./Crack Extension, mm Force F kN
1
2
3
4
5
5O
6
7
8
0
0
9
0
0
7O
1821
~+33 ,~u 22 +12 31_16
33*4
10
13_+0
51_+5
31 -+12 8
+7 31 -13
49
45,10
+14 63_20
48*8
80
0
0
0
3128
85 17+t5' -12
15
17_+~
5~+15 b--11
60_+4
73,4
97
85_+0
+9" 19_12
~+10 Jb_l 4
68_+12
96_+~
90 +11 -17162.3
109_+2
95 100 105 110
__+12 L~-24
34+12 _13
+15
59-12
78*7
_+14 +32 12Y_9 127_37
51 +16 -18
+4 62-10
93-+13
166-+7
125 57_+17
~+15 bZ--14
91+4 --6
+18208.1 221 -28
135 67
-+16 10Z_14
131
150
89
+56 151_62
164+18
155 160 170 180
106 132_+1 123+22 +60 138__48 211.18
190
161.19
254
197 205
+36 -31
141_+7 ~+16
1/"_25 21129
0
0 1-+b_513
+7
42_12 23+2 0
67*8
55+~
13
75+19_ 75_10
17
+6
57
119_+2 152217
7028165+3121_+11 .+19
159
84_29 _+26 1061 0~ 33
215
__+17
1/u_27
225
1301 81 _+12 155
140
12
27_+~
89
t20
11
+1211~+14 30-20 ~-2598 41.16
115
130
10
0
6O
90
9
153-+12
195
Fracture toughness of p/,~u~concrete from compress/on sp//tt/ng tests
Kanha/oo
(a)
(b}
Figure2 a) Typical crack growth pattern; b) location of crack front at load levels up to the instant of unstable crack growth
60
1
,xl
I
/+5
I
×
X
n 3-Jr-
50-
~
C aJ
i./)
~0--
aJ
i./1 Q.I
2
\ \
, c R G
-
+ ~~ ~
30--
X
~0
×
.~ 35
¢-
>-
30-
E 0 LJ
20 0.2
1 0.3
I
I
0.4 0.5 Wafer/Cement
\ 0.6
25 0.2
I 0.3
I
I
0./. 0.5 Water/Cement
I 0.6
Figure3 Variation of compressive strength and modulus of elasticity with water-cement ratio for mixes prepared with two different types of coarse aggregate
255
Fracture toughness of p/a/n concrete from compression sphtt/ng tests
of the specimen's arms (two struts). Elastic finiteelement and boundary-element calculations [17-181 have confirmed the existence of such stresses for w/d <~ V2 and c/d = 3/2. According to linear elastic fracture mechanics, crack propagation occurs when the energy released from the struts balances the energy required to create new crack surfaces. Using this energy balance principle it has been shown [17] that the Griffith surface energy per unit area R is given by F2e2 kc R = 3-~C-(2-kc tan-~-) sec 2 2
(1)
where F is the compressive force corresponding to central gap length c, k 2 = F/(2EI) and E is Young's modulus of the material. For rectangular specimens e = (d-w)/4, I -- bdS/96 and C = EIb. For circular specimens of diameter d, the load eccentricity e = [(2d/3~r) - w/4], i = d ~ (9Tr2 - 64)/(1152Tr) and C = EId. The load eccentricity, e and second moment of area of each strut about the axis of bending, t are easily calculated for other cross-sectional shapes. It should be mentioned that the (small) contribution from shear deformations has not been included in Equation (1). It is also worth remembering that the above relation between R, w/d, c/d and F applies only to compression splitting mode of failure. If plane stress conditions are assumed to prevail in the test specimen at the crack front, the elastic fracture surface energy, R, may be expressed in terms of the stress intensity factor K~, whereupon kc K~.,'D = B ( 2 - k c tan-~-) sec 2 2
(2)
For rectangular specimens D = (F/bd)2d and B = 13 (I - w,"d)2}.'16 , and for circular specimens D = (4F/~d2)2d and B =
8 (9-~ (3~
wd )2}/{64 ( 9 -
6Tr42) I .
The replacement of R with opening mode stress intensity factor K~ implies that the pre-crack growth results from tensile stresses developed at the crack front by the bending of the struts. As mentioned previously, the existence of such stresses has been confirmed by elastic finite-element and boundary-element calculations [17-181 for the range of w/d and c/d used in the present study.
RESULTS AND DISCUSSION Equation (2) was used to calculate the values of K, from the onset of pre-crack growth right up to the moment of fast, unstable crack growth. To calculate the initiation value of K~, namely K~,:, the compressive force at the onset of pre-crack growth and the initial value of precrack length were used. However, the values of Kr during the slow, stable crack growth were calculated using the actual load level and the corresponding augmented crack length equal to the sum of pre-crack length and mean
256
Kanhaloo
crack extension. This is necessarily an approximation in that the crack front does not advance exactly along the middle of the specimen. However, it should be noted that Equation (2) remains valid even at the full crack extension because the buckling load of the struts is still well in excess of the load at the onset of instability. The results of these calculations are shown in Figure 4 for typical mixes used in this study. For each mix the R-curve behaviour is quite evident, allowing the estimation of the asymptotic value, KR. It is worth mentioning that specimens from identical mixes (mixes 8 and 12, and 9 and 11) gave fairly close values of K~c and KR, although the geometrical test conditions were different. Thus, specimens from mix 8 contained a 145mm deep pre-crack, whereas specimens from the identical mix 12 contained a 175mm deep pre-crack. In both cases, however, the platen was 38mm wide. On the other hand, specimens from mix 9 contained a 145mm deep pre-crack but were loaded by a platen 50mm wide, whereas specimens from the identical mix 11 contained a 175 mm deep pre-crack but were loaded by a narrower platen (width 38mm). Based on the values of K~cand the estimated values Of KR obtained from the tests, it is possible to judge their variation with one of the important mix variables, namely the water to cement ratio. This variation is shown in Figure 5. Two interesting results emerge from this figure. First, the initiation value of K~ for mixes using rounded river aggregate is practically independent of w a t e r - c e m e n t ratio. This is not surprising because the initiation value of K~ depends on local stress gradients in the vicinity of coarse aggregate particles. With rounded river aggregate the gradients are generally not steep, with the result that K~c of mixes using this type of aggregate is almost equal to that of the cement/sand matrix. On the other hand, because of steep local gradients at crushed aggregate sites it is easier for crack growth to initiate in weaker mixes at these sites, resulting in a decrease of K~,: with increasing w a t e r cement ratio. Secondly, the asymptotic value of K~, namely KR decreases rapidly with increasing water-cement ratio; the rate of decrease is practically independent of the type of coarse aggregate used in the mix, although fora given w a t e r - c e m e n t ratio the KR value of mixes using CRG is higher than that of mixes using RRG. The latter is a result of the relative ease with which cracks can propagate in mixes using RRG than in mixes using CRG. The decrease Of KR with increasing water-cement ratio is a consequence of the weakness of the mix which reduces its resistance to crack propagation KR decreases almost linearly with increasing water--cement ratio, i.e with decreasin 9 compressive strength (Figure 6). Depending on the type of coarse aggregate, K~ may be estimated from the following expressions, once the compressive strength (~: or the water-cement ratio 11 of the mix is known: For mixes using crushed river gravel K~ = 4.10 - 4 5 ~q, 0 3 ~< ~/ ~< 0.7 K~ = ~r + 205),'33, 125 ~<,n ~<60
(3)
Fracture toughness of plato concrete from compression spkttmg tests
2.5 KR
I
I
I ,
2.0--
/
I
Kanhaloo
2.5
I
I
I
I
I
I
- KR
o ~
2,0o/o
Batch 1 Water/Cement:O 37 Crushed Graver
E
2 Z
15--
_
b
E ro o-
Batch 2 Water/Cement=O 40 Crushed Gravel
~__o._~/~-~-~ /
15 /
-7"
1.0 /
1 0 ~//Kk
. Kk
0.5
I
I
50
0
I
]
I
100 150 200 Crack Extension (mm)
250
,l
300
50
2.5 2.0
K~
I
I
I
I
,]
I
100 150 200 Crack Extension (mm)
I
I
I
250
300
I
I o ____-.~I -
KR
20 ~-"
15
2 "r
:~
/
1.0
b E to
Batch 3 Water/Cement:O 47 Crushed 5ravel
E
1.0:
i
0.5
I
I
- KR
I
I
100 150 200 [rack Extension {mm)
5O
1.75
Batch 4 Water/Cement=O 4 Rounded Gravel
~--~-~ /~_~_..,--o--~
Kic
0
15
:E
I
I
I
o2~"
I
250
300
•
I
05
l
50
175
I
-
Kk
I
I
100 150 200 Crack Extension (mm)
I
I
I
I
I
250
300
I
KR
15
1.5
f
/
E
2 1.0
'~°
/
E to Q_
Batch 5 Water/Cement=O 5 Crushed Gravel
/ Batch 6 Water/Cement=0 50 Rounded Gravel
~,~ 1.0
/ ~Ic
I
050
Figure4
50
I
l
i
100 150 200 Crack Extension (ram)
I
250
300
05
I
50
I
I
100 I50 200 Crack Extension (mm}
I
250
300
I n c r e a s e in Ki w i t h c r a c k e x t e n s i o n f o r v a r i o u s r n i x e s
257
Fracture toughness of pta~n concrete from compress,'on sphtbng tests
For mixes using rounded river gravel: KR = 4 4 5 -- 4.75 q, 0.3 ~< q ~< 0.7 KR = (m + 7)/26, 19 ~< ~o ~< 57
(4)
In Equations (3) and (4), KR is in MPa m 12 ( M N m 32) and {rc is in MPa (MN/m2).
2S
2.0
D
E (3_ :IE
115 -
X
O5 0.3
04
x
0.5 Water/Cement
06
Figure 5 Variation of Kic and KR with water-cement ratio for mixes prepared with two different types of coarse aggregate
60
CONCLUSIONS 1. The compression splitting test for determining the fracture toughness of plain concretes is simple to perform and does not require specimens of massive proportions even when the asymptotic value of toughness is to be determined. The results are independent of the shape and size of the specimens, provided that the width of the compression platen and the depth of pre-crack are within certain limits in order to ensure failure by axial splitting. The rough manual procedure for locating the position of the advanced crack front used in this study can obviously be improved, without jeopardising the simplicity of the actual test method. Likewise, it is possible to improve the expression for KI using the two-strut model, by including non-linear material properties. 2. The results of the present investigation prove conclusively the existence of R-curve behaviour in concrete. In fact, the slow crack growth responsible for this behaviour is a reservoir of latent strength in the material; the KR value for a given water-cement ratio is roughly three times the initiation value. 3. The compression splitting test is currently being applied to fibre reinforced concretes. The results of this investigation will be reported in a future communication.
\
1.0-
RRG--x
Karihaloo
,
,
I
I
REFERENCES 1. Nallathambi, PI, Karihaloo, BILI and Heaton, B. S i 'Effect of specimen and crack sizes, water-cement ratio and coarse aggregate texture upon fracture toughness of concrete', Magazine of Concrete Research, Vol. 36, No. 129, December 1984, pp. 227-36. 2. Nallathambi, P., Karihaloo, B. L. and Heaton, B. S. 'Various size effects in fracture of concrete', Cement and Concrete Research, Vol 15, No. 1, February 1985, pp. 117-26. 3. Legendre, D. and Mazars, J. 'Damage and fracture mechanics for concrete (a combined approach)', Advances in Fracture Research: Proceedings of the
j..-o
Figure 6 Variation of KR with compressive strength of mixes prepared with two different types of coarse aggregate
SO
40 I0
3O
RRG/XERGX
20 1.0
1.5
2.0 KR (HPa.m l/z)
258
2.5
Fracture toughness of plain concrete from compression spl/tting tests
4. 5.
6.
7.
8. 9.
10.
6th International Conference on Fracture (ICF6), Pergamon Press. Editors: S. R. Valluri et al., Vol. 4, 1984, pp. 2841-8. Petersson, P. E. 'Fracture energy of concrete: method of determination', Cement and Concrete Research, Vol. 10, No. 1, February 1980, pp. 78-89. Rossi, P., Acker, P. and Francois, D. 'Measurements of fracture toughness Kjc of concrete', Advances in Fracture Research: Proceedings of the 6th International Conference on Fracture (ICF6), Pergamon Press, Editors: S. R. Valluri et al., Vol. 4, 1984, pp. 2833-40. Barker, D. B., Hawkins, N. M., Jeang, F. L, Cho, K. Z. and Kobayashi, A. S. 'Concrete fracture in a CWL-DCB specimen', Journal of Engineering Mechanics, Proceedings, American Society of Civil Engineers, Vol. 111, No. EM5, May 1985, pp. 623-38. Hillsdorf, H. K. and Brameshuber, W. 'Size effects in the experimental determination of fracture mechanics parameters', NATO Advanced Research Workshop: Application of fracture mechanics to cementitious materials, 4-7 September 1984. Editor: S. P. Shah, Northwestern University, Evanston, U.S.A. pp. 213-54. Bazant, Z. P., Kim, J. K. and Pfeiffer, P. 'Determination of nonlinear fracture parameters from size effect tests', /bid, pp. 143-70. Hillerborg, A. 'Results of three comparative test series for determining the fracture energy GF of concrete', Materiaux et Constructions, Vol. 18, No. 107, September-October 1985, pp. 407-13. Wecharatana, M. and Shah, S. P. 'Predictions of
11.
12.
13.
14.
15.
16. 17. 18.
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nonlinear fracture process zone in concrete', Journal of Engineering Mechanics, Proceedings, American Society of Civil Engineers, Vol. 109, No. EM5, May 1983. pp. 1231-46. Cho, K. A., Kobayashi, A. S., Hawkins, N. M., Barker, D. B. and Jeang, F. L 'Fracture process zone of concrete cracks', Journal of Engineering Mechanics, Proceedings, American Society of Civil Engineers, Vol. 109, No. EM8, August 1984. pp. 1174-84. Nallathambi, P. and Karihaloo, B. L 'Determination of specimen-size independent fracture toughness of plain concrete', Magazine of Concrete Research Vol. 38, No. 135, June 1986. pp.67-76. Wecharatana, M. and Shah, S. P. 'Slow crack growth in cement composites', Journal of Structural Division, Proceedings, American Society of Civil Engineers, Vol. 108, No. ST6, June 1982. pp. 1400-13. Wecharatana, M. and Shah, S. P. 'A model for predicting fracture resistance of fiber reinforced concrete', Cement and Concrete Research, Vol. 13, No. 6, December 1983. pp. 819-29. Kendall, K. 'Complexities of compression failure', Proceedings of Royal Society London, Vol. A361. 1978, pp. 245-63. Karihaloo, B. L. 'A note on complexities of compression failure', ibid. Vol. A368. 1979 pp. 483-93. Karihaloo, B. L. 'Compressive failure of brittle materials', ibid. Vol. A396. 1984. pp. 297-314. Guest, B. W. and Karihaloo, B L. 'Fracture of brittle materials', Journal of Materials Science Letters, Vol. 4, No. 10, October 1985. pp. 1285-9.
259