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Micromechanical aspects of the fibre-cement bond
Micromechanical aspects of the fibre-cement bond v. LAWS The nature and properties of the interfacial region determine stress transfer between components of a composite and the development of the stress/strain curve, strength and the toughness of the composite. In this paper, the effect of the bonding on fibre pull-out, multiple cracking and strength, is considered. The measurement of the strength of the bonds is discussed. Key words: composite materials; reinforced cement; fibre/matrix interface; bond strength; toughness; test procedure In recent years considerable progress has been made in understanding the mechanism of reinforcement of cements by fibres and the factors that contribute to the properties of the composite. Of these factors, the interface between the fibre and cement is of prime importance: the shape of the tensile stress/strain curve, the strength of the composite and the ability to absorb energy depend critically on the mechanism of stress transfer between the reinforcement and the matrix. Theories of fibre strengthening commonly assume linear transfer of stress between fibre and matrix, that is frictional stress transfer. Practical bond strength measurements and the determination of the critical fibre length, are usually based on the uniaxial pull-out test and also assume linear stress transfer. But stress transfer might be non-linear, if for example there is a strong bond between fibre and matrix and elastic continuity at the interface is maintained. The assumption of linear stress transfer could then cause the calculated bond strength to differ greatly from the true bond strength, and depend on embedment length. In predicting composite strength properties from bond strength data calculated simply from pull-out tests, there are two further problems. Firstly even at relatively short embedment lengths, the fibres often break rather than pull-out; and secondly the fibres in commercial composites are usually distributed in all directions in a plane rather than in one direction only. To avoid these problems Allen I related the tensile strength of a two dimensional randomly arranged short fibre composite, to an average fibre stress obtained from pull-out tests carried out over a range of embedment lengths. In his analysis no assumptions were needed about the size or distribution of the bond stresses involved, but it was assumed that the effectiveness of fibre in supporting load decreased as their angle of orientation increased. The efficiency of inclined fibres in supporting load has been the subject of much debate; Naaman and Shah 2 have measured the peak load of steel fibres oriented at an angle to the applied stress and have shown that it is of the same order as for parallel fibres. They showed also that the peak load per fibre decreased as the number of fibres pulling out from the same area increased, although it did not when the fibres were aligned in the pull-out direction. While such studies more nearly simulate the conditions in a composite
they present problems in interpretation, in particular in separating the effect of the 'hinge' from that of the bond. Other workers have used pull-out of single fibres, for instance de Vekey & Majumdar 3 and Walton & Majumdar. 4 Greszczuks and Lawrence6 derived theoretical expressions for the distribution of load and shear stress at the fibre/ matrix interface during pull-out of a fibre from an elastic matrix. Lawrence also considered the effect of a frictional stress opposing pull-out after failure of the interfacial bond. He showed how both interfacial and frictional bonds can be calculated from pull-out data over a range of embedment lengths. Bartos 7 later used a treatment very similar to that of Lawrence and has shown how the bond strengths can be calculated from pull-out traces at a single pull-out length. In this paper, Lawrence's theory is extended to calculate the load/displacement curve during pull-out, the crack spacing and strength of an aligned short fibre composite. The effect of the bonds, interfacial and frictional, on fibre pull-out, crack spacing and strength is outlined; and the calculation of the strength of the bonds is discussed. An attempt is made to interpret previous work, in particular the pull-out curves of de Vekey and Walton mentioned earlier, in terms of the individual bonds operating.
FIBRE PULL-OUT The analyses of Oreszczuks and Lawrence6, mentioned above, show that the shear stress developed when a load is applied to the fibre is a maximum at the point where the fibre enters the matrix. When a load is applied such that the shear stress at this point reaches the shear strength of the interface, the fibre debonds completely and pulls out with no further increase in load. Bond strength calculations which assume a constant shear stress along the embedded fibre length, are therefore likely to be low even if the embedded fibre length is short. Lawrence also considered what would happen if there were a frictional stress r i opposing pull-out after the interfacial bond (strength rs) has failed. He showed that, provided the embedded fibre length is greater than a critical value Xmax debonding would stop, and a further increase in load would be needed to continue debonding and lead to complete pull-
0010-4361/82/020145-07$03.00 © 1982 Butterworth & Co (Publishers) Ltd COMPOSITES. APRIL 1982
145
out; and suggested how the bond strengths, 1"s and Ti could be estimated from the maximum load/embedded fibre length curve.
ex X Xo
In considering the interpretation of pull-out curves it is useful to extend Lawrence's analysis to calculate the shape of the load/extension curve. Before debonding begins (ie when the maximum shear stress developed, l'max < l's, the shear strength of the interface), the fibre load distribution Px is given by, Px = Pf sinh sinh ~,6flt
where x is the distance from the embedded end of the fibre, ~ the embedded fibre length, and Pf is the load applied (see Fig. 1). fl depends on the elastic contents of the fibre and matrix, the fibre cross-sectional area and the fibre volume fraction.
(2)
and the load/extension curve is linear in this region.
Load/extension curves during pull-out are shown in Fig. 2. Point A is the departure from linearity, and the point where debonding begins (ie where Pf (prJfl) tanh I/4/70. Once debonding has commenced, either it is catastrophic and the fibre pulls out without further increase in load (curves (i)) or it is progressive arid requires a further increase in load to overcome the frictional forces and achieve complete debonding and pull-out (curves ii)). Points marked B are the maximum loads achieved for the relevant conditions (pfmax). After the maximum load is reached, the load decreases until debonding is complete at points C (load Pf = F~pri/). The extension also decreases, in some cases after the maximum load has been reached.
Suppose now, that the fibre has debonded from the free end a distance (V2/- Xo) into the matrix under an applied load Pf (Fig. 1). The load distribution in the fibre is, Px = Pf - P~'i ( 1/2/-- X)
xot ], =
It follows from Equation (1) that the extension, A, of the fibre of length 1A/is,
EfA f sinh V49/)
(4)
-x
For any debonded length Xo, the corresponding load Pf can be calculated from Equation (3) and the extension follows from Equation (4).
(1)
A = Pf (cosh i/~t - 1)/~
(to)
[ p'rs (1 -- sech fl0Co) +Pr
Xo ~
ex' = [gf - P~'i ( ~12/- Xo)] sinh fl0c/sinh~0c0 0 • x ~ x 0
Pull-out tests are normally carried out at a constant rate of cross-head travel, so that the predicted decrease in extension would not occur; instead, the load would drop suddenly to
where ri is the frictional stress opposing slipping and p is the fibre perimeter. A t x =Xo
•~S,,,,>¢B pT"max = pT"s = fl[Pf - pT"i (~¢z/- Xo)] COth~o¢0
#.,
(3)
so that
20 /
Pf = (pTs/fl)tanh/~Xo + pr i (Vd - Xo)
/
//
The average fibre load is then: ,/
I=
/
/
A
/
/
Xo
/
/
-'---:
o
/
A P,
5
/
• ~---~__
)
C/ /
(')
Fig. 1
146
Geometry of fibre debonding and pull-out
0 Fig. 2
/
"PTI
2NMrnrn-I
Volues used:
/7)
///
/ //JPrl=0
X
/ f
x/(ii) / / 7 B C/
/
1
--.
pri=4Nmrn ~
/
/lii)
15
I/2l I I
/
//C
o;,
E, 76GNr~2 Af .027mmz /3 1.3mm-I pv$ IONmm-I I/ 113mrn solid line 2 15mm brokenline
o'oa
(ram) Calculated load/extension curves during pull-out Extension
0:03
C O M P O S I T E S . A P R I L 1982
a value corresponding to that extension, on the 'postdebond' part of the pull-out curve. The following facts emerge: (1)
(2)
(3)
The load to start debonding (point A) approaches prs//3 as tanh Y~l ~ 1. It is insensitive to fibre length for realistic values of/3. The average bond strength ~(= e~nax/b~l), is an underestimate of the interfacial bond strength ~'s, and an overestimate of the frictional bond r i. This is illustrated in Fig. 3. As the fibre length is reduced, ~-approaches rs, but even for very short pull-out lengths, the error can be large. At long fibre lengths, r approaches r i.
Here, a composite in which the fibres are continuous and aligned is considered, and the minimum fibre length, ½l, needed to transfer sufficient load to break the fibre is calculated. This occurs when the load Pmu is the maximum load the fibre of length ½1 can support. The minimum crack spacing is then either,
1
/Pmufl/
Prs
½l, = ~- tanh -] \ Prs ] , ifPmu ~< ~ -
tanh ~Xmax
(5) or
½12 = ~
For fibre lengths ~>Xmax the drop in load (B to C) after the maximum load is reached, to the completion of debonding is,
1
pT"s
Pmu 1> ~
P% /3 tanh~xmax)'if
(Pmu +PriXmax
(6) tanh/3xma x
Prs
pfmax _ ],~pril = -~-- tanh/3 Xmax - Pri Xmax
where Xmax
_ 1 cosh_l/~-s /3
~] Ti
and is equal to the intercept on the load/pull-out length curve. It is independent of fibre length. In practice, the drop in load at constant extension, will differ from this if the post debond load is not constant. THE MULTIPLE CRACKING REGION In the above analysis, it was assumed that the embedded end of the fibre carries no load, an assumption that does not apply when multiple cracking of the matrix is considered. In this case, the fibre is effectively stressed at both 'ends'; that is, at the 'free' end at a matrix crack, and at a distance into the matrix such that the transfer of load from the fibre to the matrix is sufficient to break the matrix. At this latter point, the fibre and matrix strains are the same and equal to the matrix failure strain. However, it can be shown that the different end conditions for fibre pull-out and multiple cracking lead only to slight differences in the analyses, and Lawrence's theory can be applied, as a good approximation, to calculate the crack spacing.
½l~ describes the case where there is no debonding, for instance where the matrix is weak and/or the interfacial bond is strong and/or the fibre volume fraction is high. If Pmu/3/pr s is small, ½11 -+Pmu/Prs. This crack spacing, which will be denoted x(rs), corresponds to the minimum crack spacing calculated using the model of Aveston et al, 8 ie assuming linear stress transfer, for a bond strength of r s. Provided there is some debonding, the crack spacing ½12 (Equation (6)) applies, and it follows that ([Pmu/Pri] - ½12) is independent of the matrix strength and depends on the fibre volume fraction only through the constant/3. Pmu/PT"iis the minimum crack spacing for linear stress transfer and a bond strength 7"i,and will be denoted x(r-t). Fig. 4 shows the minimum crack spacing defined by Equa-
/c / / /
/ /
~6
/
/
A
U //
6
/// b4
ri
5O
4
Values used: Ef 76 GNrr{2 Af .027rnrn 2 Gm 8 GNm-2 e'mu 6 Nmm-2
E E
2
p'% p'r i I
2
Olo 0
I0
I0 Nmrn"1 2 N rnnrn"t
20
50
40
5O
I/2 BL
Fig. 3 Ratio of average bond strength ~to interfaeial bond strength r s and frictional bond strength ri, as a function of embedded fibre length factor, for various ratios rs/f I
COMPOSITES. A P R I L 1982
I 50
I I00 Volume ratio
(Vr./Vf)
Fig. 4 Minimum crack spacing as a function of volume ratio V m / V F. Curve A assumes elastic stress transfer and r~ ~ r i (crack spacing ½ 11,2); curves B and C assume linear stress transfer with bond strengths r$ and ri respectively (spacings x ( r s) and x ( r i) respective ly)
147
tion (5) or (6) above as a function of volume ratio Vm/Vf, for a glass fibre/cement composite, together with x(rs) and x(rO for comparison (curves A, B and C respectively).
40
The difference between the 'actual' crack spacing shown in curve A, ½1t or ½12, and that calculated on the frictional bond ri (curve C) is near constant over a wide range. If the minimum crack spacing is used to calculate an apparent bond strength rAre assuming linear stress transfer, ie PrAee =Pmu/gala,2, the apparent bond strength will overestimate the frictional bond ri, and underestimate the interracial bond rs; and Fig. 3 applies.
30 vE
g
o) (u 2, 2O
COMPOSI TE STRENG TH
Lawrence's analysis can also be applied to the calculation of the strength of short fibre composites. The critical fibre length, lc, is the minimum length of fibre such that the fibre will break rather than pull-out from the matrix and, if the shear stress r s developed at the interface is uniform, is given by:
E} t~
,ff
pT, ( N tam't)
[0
2o
½l c = Ofu A f / p r s
(7) 5O
The reinforcement efficiency factors for an aligned composite containing fibres of length l are, 9 0 le
n = 1 - ~- (2-ri/rs)
l > 2l'o
l 7? - 2lc( 2 _ ri/rs)
l<2l;
[0
15
p-ci(Nmm-l) Fig. 5 Critical fibre length as a function o f p r I for.various values o f p r $ (Data as f o r Fig. 4)
or
able embedments lengths, and calculated the average bond strength, 7 from the maximum load achieved. Pinchin 1° on the other hand used long embedment lengths, and also calculated the average bond strength r. At short embedment lengths, r approaches rs; at long embedment lengths r approaches r i. Neither method allows both r s and r i to be calculated.
where l'c = ½l c (2 - Ti/Ts). If elastic continuity prevails, Equation (7) does not apply, and,
1
½!c = ~ tanh-1
ofuA f
)
½l c ~
or
~lc
_
1 pT-i
Prs tanh (3Xmax) (°fuAf +PriXmax -- ---if1Alc i> Xmax
The critical fibre length is a function of both interfacial bond strength rs, and the frictional bond 7"i,and Equation (7) applies only if r s = r i. Fig. 5 shows the critical fibre length le, plotted as a function of r s and r i. The frictional bond is seen to be the more effective in reducing lc and hence in increasing composite strength. EXPERIMENTAL METHODS OF DETERMINING BOND S TRENG TH The most common method o f determining bond strength is the pull-out test, either on single 'fibres' or on arrays o f
fibres. De Vekey & Majumdar a reported pull-out results for single fibres from cement. They used the smallest practic-
148
Aveston et al, ~1 Baggott & Gandhi a2 and others have calculated average bond strengths from measurements of crack spacing. Again r s and r i cannot be found. Lawrence 6 described how both r s and zi can be calculated from pull-out data over a range of embedded fibre lengths. Bartos 7 has used a similar analysis to show how the bond strengths can be determined from pull-out curves at a single fibre length. Both methods require that (3 is known. This is problematical as/3 depends on the shear modulus of the matrix in the interracial region, which is usually not known. Steel fibres
De Vekey & Majumdar 3 reported average bond strengths for steel/cement ranging from 5.5 Nmm -2 for bright high tensile steel wires after 28 days water storage, to 11 Nmm -1 for air storage. The embedment lengths used were 1 and 2 mm. The pull-out curves, while showing considerable variability, generally showed the expected shape, although in some cases the first deviation from linearity was marked and in other cases it could not be detected. After the maximum load was reached, there was a sudden drop in load and then a more gradual drop in many cases, to what was apparently a constant pull-out load, as the fibre which protruded from the matrix, pulled through. Unfortunately the recorder was turned off early in this region, and details of this region were lost.
COMPOS ITES. AP R I L 1982
There was evidence of 'stick-slip' behaviour in some cases, notably for bright high tensile steel, and usually in the 'constant' pull-out region. In a few cases however, it occurred before the maximum load was reached, and in those cases, failure was catastrophic. The curves for rusty high tensile steel, on the other hand, did not show stickslip behaviour. The pull-out curves for rusty high tensile steel after 28 days storage in air showed either a discontinuity (peak) or a change of slope before the maximum load was reached. Assuming this point marks the beginning of debonding, it can be used, together with the maximum load, and the 'constant' frictional load during pull-out, to calculate the three unknowns, rs, r i and/3. The results obtained are/3 = 1.65 mm -1 ; r s = 22 N mm -a and r i -- 8 N mm -2 (Table 1). The interfacial bond Ts is the total bond, both the chemical or cohesive bond, and the frictional bond. It follows that the cohesive bond r e (= r s - r'O has strength " q 4 N mm -2. The curves for pull-out of bright high tensile steel wire did not show an obvious debond point, and without a measurement of this point there is insufficient information to allow the calculation of/3 as well as Ts and r i as before. It was assumed therefore that the value ~ calculated for rusty wire in cement applied to the bright wire also, and to both conditions of storage, air and water. The calculated values of rs and r i are shown in Table 1. The drop of the maximum load is very much reduced in the case of the water stored specimens compared with that for air storage, and the calculated bond strengths are lower. This is consistent with Pinchin's conclusion l° that the cohesive bond as well as the frictional bond depends on the normal pressure at the interface, arising in the case of the air stored specimens, from drying shrinkage. The values deduced for the cohesive bond strength Te (Table 1) compare with the values of shear strength of the cement/cement interface deduced by Pinchin l° from friction experiments. Pinchin 1° measured load as a function of cross-head movement for wires pulled through cylinders of cement paste or mortar. At the long embedment length used (42.5 mm), the maximum load would be expected to approach that due to the frictional bond alone (Fig. 3). The load/displacement curves should therefore rise to a maximum and thereafter the load should remain constant as the wires pull through. Such curves were observed only in a few cases, notably smooth wires pulled through wet-cured cement mortar cylinders. With increasing roughness, the peak load increased and there was a sudden large drop and an approach to a much lower, approximately constant load. Pinchin also showed that, if a confining pressure is applied immediately after debonding, the load drops smoothly to
Table 1.
Calculated values of ~, rs, r i and r c T
Ts
Ti
Tc
8
14
(N turn -2) Rusty HT wire in air
10.4
22.0
Bright HT wire in air
11.0
15.5
7.7
7.7
5.5
7.4
4.9
2.5
Bright HT wire in water
COMPOSITES, APRIL 1982
approach a constant value. It appears then that the sudden large drop he observed for roughened wires during pull-out is the result partly of the method of test, and partly of the frictional stress decreasing with increasing wire movement. It follows from the analysis outlined in an earlier section that the frictional stress/displacement part of the curve should be extrapolated back to approach the initial (rising) part of the curve, in order to estimate the load and hence r i at the beginning of slip. While with the limited information available, it is not possible to do this with any confidence, extrapolation of the post-slip regions suggests initial values of 7i of 1.2 to 2.7 N mm -2 respectively. No estimate can be made of the total bond r s. The average values of bond strength calculated from the maximum loads, range from 1.2 N mm -2 for highly polished to 3.9 N mm -2 for the most highly roughened wires in water cured cement paste at 16 weeks. These bond strengths are much lower than those calculated from de Vekey & Majumdar's data. While there are differences in materials, it seems more likely that the discrepancy arises from differences in sample preparation. In particular, Pinchin used a vibration method which led to increased porosity and decreased hardness of the cement near the wires.
Polypropylene fibres Walton & Majumdar 4 reported the results of pull-out tests of single polypropylene fibres from a cement matrix. The fibres extended through the matrix so that both ends protruded from the matrix. They found two types of pullout curves: in one the load rose smoothly to the maximum; in the other, there was a 'yield region' which in many cases was extensive and thereafter a reduced slope up to the maximum load. In some cases there was a drop in load in the yield region. Pull-out of the fibre, defined as obvious movement of the free (unloaded) end of the fibre, was observed early in this region, although there was some indication that total debonding and movement of the fibre through the matrix might have begun even earlier. The free end of the fibre was observed entering the matrix, often before the maximum load was reached. Walton & Majumdar 4 calculated average bond strengths based on the yield point and on the maximum load, and pointed out that because a large number of fibres broke, the calculated bond strengths are not true bond strengths but may be regarded as lower bounds only. The analysis of the results to calculate details of the interfacial bond r s and the frictional bond r i is open to doubt. Since in many cases the fibres broke, the test is not a true pull-out test. Where the load/extension curve rose smoothly to a maximum, all the fibres broke; and the curve resembled that of the fibre alone, except that the breaking load appeared to be approximately 30% lower. The fibres appeared to be anchored in the cement and there was no sign of slip relative to the matrix. One possible interpretation of the pull-out curve is that the yield region marks the maximum load achieved before complete debonding, and thereafter the frictional force opposing fibre movement increased for some as yet undefined reason.
149
Assuming then that the 'yield point' is the 'maximum load' of Lawrence's model, from the 10 and 20 mm data, 7"i = 0.07 N mm ~ provided that %l > Xmax. Since/3 is not known, r s cannot be calculated; but we know that the average bond strength r'(0.6 N mm -2 at 10 mm, 0.34 N mm -2 at 20 mm) underestimates r s and overestimates I"i.
region to be seen. It appears however, that there is an interfacial bond that is greater than the frictional bond, at least at the beginning of major slip. Its value will be higher than the corresponding average bond strength. The frictional bond at the beginning of major slip will be lower than ~. The rising load which was observed in some cases after major slip, could be the result of the taper produced in the fibre by the hand-drawing process.
Or if ½l ~
Bartos 7 reported the results of pull-out tests of glass fibre strands from a cement matrix. He used a relatively long embedment length (8.6 mm) with a water/cement ratio of 0.4, (dry-cured for 7 days). The curves showed a maximum load indicating the completion of debonding, and then a drop and a fairly uniform pull-out against the frictional forces. Assuming a value of matrix shear modulus in the calculation of/3, he was then able to calculate the 'ultimate shear flow' Prs and the 'frictional flow' pr i (6 and 1.8 N mm -1 respectively). The average shear flow p 7 (maximum load divided by embedded length) was 2.1 N mm -1 , not far different from the frictional flow Pri. The calculation of bond strength requires that the perimeter p can be measured. For glass fibre strands it is impossible to get an accurate estimate of the perimeter in contact with the matrix, but on average, it is unlikely to be less than 2 to 3 mm for the E-glass strand that Bartos used. The bond strengths are obviously low. De Vekey & Majumdar 3 used single glass filaments and the problem of measuring the perimeter did not arise. They used a very short embedment length (1 mm) and calculated the average bond strength r. The values were high - 6 to 9 N mm -2 for E-glass in cement (0.3 water/cement ratio) after 28 days storage. Further analysis of the pull-out traces they obtained is difficult. There is some ivdication on some traces of a discontinuity which might mark the beginning of debonding, but others do not show it or show a peak near the maximum load. After the maximum load is reached, the load drops. The load which presumably then reflects a frictional bond ri, sometimes rises again, but unfortunately the recorder was switched off too soon to allow details of this
150
In principle it is possible to calculate rs, ri and/3 from pullout tests, but the results will apply to the conditions obtained in the pull-out test, and not necessarily to those in the composite. In particular the shear modulus of the matrix in the vicinity of the fibre, might be very different. There is no information available on the effect of embedment length in a single fibre pull-out test, on the properties of the matrix in the interfacial zone; nor on the effect of other fibres in an actual composite. In an effort to reduce these (and other) uncertainties BRE has developed a multiple fibre pull-out test, in which the production of the test specimens more nearly resembles that of an aligned fibre composite. Inevitably the averaging effect of pulling out an array of 'fibres' leads to problems in interpreting the pull-out traces. The possibility of using crack spacings rather than pull-out tests is attractive for this reason, but crack spacings alone are not sufficient to allow calculation of both r s and ri, even if/3 were 'known'. Another problem arises in that actual pull-out tests suggest that r i is not necessarily constant and might decrease (eg Pinchin's studies of steel wires), or increase (Walton & Majumdar on polypropylene fibres) with increasing fibre movement. It is interesting to speculate on the effect of such changes: a decrease in r i after debonding has started would reduce the stress opposing further debonding, and lead to catastrophic failure and pull-out at an earlier stage. An increase in r i would lead firstly to an increase in the load required to continue the debonding. As the displacement increases and r i continues to increase debonding would stop and further increase in load would lead to fibre failure rather than pull-out. At the displacements obtaining during debonding, it might well be that the effect of a non-constant frictional bond strength is small. But once the whole fibre is slipping it is obviously important since it must modify the strength, the multiple cracking and the total energy to break. This and previous studies suggest that the frictional bond is more important in determining composite properties than the total interfacial bond, which is the sum of the (initial) frictional bond and the cohesive bond. In view of this and of the difficulty in determining the interfacial bond strength, it might be better to concentrate studies on the frictional components, 'static', 'dynamic' and 'ploughing'; and on modifying the interfacial zone to improve these properties. A CKNOWL EDGEMENTS
I am grateful to Dr R.C. de Vekey and Mr P. Walton for
COMPOSITES. APRIL 1982
providing data for analysis. The work described has been carried out as part of a research programme being conducted by the Building Research Establishment of the Department of the Environment and this paper is published by courtesy of the Director, Building Research Establishment, and by permission of the Controller HMSO, holder of Crown copyright.
6 7 8 9 10
© Copyright Controller HMSO, London 1981.
11
REFERENCES
12
1 2 3 4 5
Allen,H.G.JPhysD:AppIPhys 5 (1972) pp 331-343 Naaman, A.E. and Shah, S.P. 'Fibre reinforced cement and concrete' RILEM Symposium 19 75 (The Construction Press) pp 171-178 de Vekey, R.C. and Majumdar, AJ. Mag ConcreteRes 20 (1968) pp 229-234 Walton,P.L. and Majumdar, AJ. Composites 6 (1975) pp 209-216 Greszczuk, L.B. Interfaces in Composites, ASTM STP 452 (1969) pp 42-58
COMPOSITES. A P R I L 1982
Lawrence,P. JMater Sc/7 (1972) pp 1-6 Bartos,P.JMaterSci 15 (1980) pp 3122-3128 Aveston, J., Cooper, G.A. and Kelly, A. 'The Properties of Fibre Composites' Conf Proc, National Physical Laboratory, 4 November 1971, (IPC, Guildford, UK, 1971) pp 15-26 Laws,V.YPhysD:ApplPhys4(1971) pp1737-1746 Pinehin,DJ. "The cement steel interface: friction and adhesion' Ph D Dissertation (University of Cambridge, 1977) Aveston, J., Mercer, R.A. and Sillwood, J.M. 'Composites - Standards Testing and Design' ConfProc, NPL 8/9 April 1974 (IPC, Guildford, UK, 1974) pp 93-102 Baggott, R. and Gandhi, D. JMater Sci 16 (1981) pp 65-74
A UTHOR
The author is a researcher at the UK Department of the Environment's Building Research Establishment. Inquiries should be addressed to: Miss V. Laws, Building Research Establishment, Building Research Station, Garston, Watford WD2 7JR, England.
151
they present problems in interpretation, in particular in separating the effect of the 'hinge' from that of the bond. Other workers have used pull-out of single fibres, for instance de Vekey & Majumdar 3 and Walton & Majumdar. 4 Greszczuks and Lawrence6 derived theoretical expressions for the distribution of load and shear stress at the fibre/ matrix interface during pull-out of a fibre from an elastic matrix. Lawrence also considered the effect of a frictional stress opposing pull-out after failure of the interfacial bond. He showed how both interfacial and frictional bonds can be calculated from pull-out data over a range of embedment lengths. Bartos 7 later used a treatment very similar to that of Lawrence and has shown how the bond strengths can be calculated from pull-out traces at a single pull-out length. In this paper, Lawrence's theory is extended to calculate the load/displacement curve during pull-out, the crack spacing and strength of an aligned short fibre composite. The effect of the bonds, interfacial and frictional, on fibre pull-out, crack spacing and strength is outlined; and the calculation of the strength of the bonds is discussed. An attempt is made to interpret previous work, in particular the pull-out curves of de Vekey and Walton mentioned earlier, in terms of the individual bonds operating.
FIBRE PULL-OUT The analyses of Oreszczuks and Lawrence6, mentioned above, show that the shear stress developed when a load is applied to the fibre is a maximum at the point where the fibre enters the matrix. When a load is applied such that the shear stress at this point reaches the shear strength of the interface, the fibre debonds completely and pulls out with no further increase in load. Bond strength calculations which assume a constant shear stress along the embedded fibre length, are therefore likely to be low even if the embedded fibre length is short. Lawrence also considered what would happen if there were a frictional stress r i opposing pull-out after the interfacial bond (strength rs) has failed. He showed that, provided the embedded fibre length is greater than a critical value Xmax debonding would stop, and a further increase in load would be needed to continue debonding and lead to complete pull-
0010-4361/82/020145-07$03.00 © 1982 Butterworth & Co (Publishers) Ltd COMPOSITES. APRIL 1982
145
out; and suggested how the bond strengths, 1"s and Ti could be estimated from the maximum load/embedded fibre length curve.
ex X Xo
In considering the interpretation of pull-out curves it is useful to extend Lawrence's analysis to calculate the shape of the load/extension curve. Before debonding begins (ie when the maximum shear stress developed, l'max < l's, the shear strength of the interface), the fibre load distribution Px is given by, Px = Pf sinh sinh ~,6flt
where x is the distance from the embedded end of the fibre, ~ the embedded fibre length, and Pf is the load applied (see Fig. 1). fl depends on the elastic contents of the fibre and matrix, the fibre cross-sectional area and the fibre volume fraction.
(2)
and the load/extension curve is linear in this region.
Load/extension curves during pull-out are shown in Fig. 2. Point A is the departure from linearity, and the point where debonding begins (ie where Pf (prJfl) tanh I/4/70. Once debonding has commenced, either it is catastrophic and the fibre pulls out without further increase in load (curves (i)) or it is progressive arid requires a further increase in load to overcome the frictional forces and achieve complete debonding and pull-out (curves ii)). Points marked B are the maximum loads achieved for the relevant conditions (pfmax). After the maximum load is reached, the load decreases until debonding is complete at points C (load Pf = F~pri/). The extension also decreases, in some cases after the maximum load has been reached.
Suppose now, that the fibre has debonded from the free end a distance (V2/- Xo) into the matrix under an applied load Pf (Fig. 1). The load distribution in the fibre is, Px = Pf - P~'i ( 1/2/-- X)
xot ], =
It follows from Equation (1) that the extension, A, of the fibre of length 1A/is,
EfA f sinh V49/)
(4)
-x
For any debonded length Xo, the corresponding load Pf can be calculated from Equation (3) and the extension follows from Equation (4).
(1)
A = Pf (cosh i/~t - 1)/~
(to)
[ p'rs (1 -- sech fl0Co) +Pr
Xo ~
ex' = [gf - P~'i ( ~12/- Xo)] sinh fl0c/sinh~0c0 0 • x ~ x 0
Pull-out tests are normally carried out at a constant rate of cross-head travel, so that the predicted decrease in extension would not occur; instead, the load would drop suddenly to
where ri is the frictional stress opposing slipping and p is the fibre perimeter. A t x =Xo
•~S,,,,>¢B pT"max = pT"s = fl[Pf - pT"i (~¢z/- Xo)] COth~o¢0
#.,
(3)
so that
20 /
Pf = (pTs/fl)tanh/~Xo + pr i (Vd - Xo)
/
//
The average fibre load is then: ,/
I=
/
/
A
/
/
Xo
/
/
-'---:
o
/
A P,
5
/
• ~---~__
)
C/ /
(')
Fig. 1
146
Geometry of fibre debonding and pull-out
0 Fig. 2
/
"PTI
2NMrnrn-I
Volues used:
/7)
///
/ //JPrl=0
X
/ f
x/(ii) / / 7 B C/
/
1
--.
pri=4Nmrn ~
/
/lii)
15
I/2l I I
/
//C
o;,
E, 76GNr~2 Af .027mmz /3 1.3mm-I pv$ IONmm-I I/ 113mrn solid line 2 15mm brokenline
o'oa
(ram) Calculated load/extension curves during pull-out Extension
0:03
C O M P O S I T E S . A P R I L 1982
a value corresponding to that extension, on the 'postdebond' part of the pull-out curve. The following facts emerge: (1)
(2)
(3)
The load to start debonding (point A) approaches prs//3 as tanh Y~l ~ 1. It is insensitive to fibre length for realistic values of/3. The average bond strength ~(= e~nax/b~l), is an underestimate of the interfacial bond strength ~'s, and an overestimate of the frictional bond r i. This is illustrated in Fig. 3. As the fibre length is reduced, ~-approaches rs, but even for very short pull-out lengths, the error can be large. At long fibre lengths, r approaches r i.
Here, a composite in which the fibres are continuous and aligned is considered, and the minimum fibre length, ½l, needed to transfer sufficient load to break the fibre is calculated. This occurs when the load Pmu is the maximum load the fibre of length ½1 can support. The minimum crack spacing is then either,
1
/Pmufl/
Prs
½l, = ~- tanh -] \ Prs ] , ifPmu ~< ~ -
tanh ~Xmax
(5) or
½12 = ~
For fibre lengths ~>Xmax the drop in load (B to C) after the maximum load is reached, to the completion of debonding is,
1
pT"s
Pmu 1> ~
P% /3 tanh~xmax)'if
(Pmu +PriXmax
(6) tanh/3xma x
Prs
pfmax _ ],~pril = -~-- tanh/3 Xmax - Pri Xmax
where Xmax
_ 1 cosh_l/~-s /3
~] Ti
and is equal to the intercept on the load/pull-out length curve. It is independent of fibre length. In practice, the drop in load at constant extension, will differ from this if the post debond load is not constant. THE MULTIPLE CRACKING REGION In the above analysis, it was assumed that the embedded end of the fibre carries no load, an assumption that does not apply when multiple cracking of the matrix is considered. In this case, the fibre is effectively stressed at both 'ends'; that is, at the 'free' end at a matrix crack, and at a distance into the matrix such that the transfer of load from the fibre to the matrix is sufficient to break the matrix. At this latter point, the fibre and matrix strains are the same and equal to the matrix failure strain. However, it can be shown that the different end conditions for fibre pull-out and multiple cracking lead only to slight differences in the analyses, and Lawrence's theory can be applied, as a good approximation, to calculate the crack spacing.
½l~ describes the case where there is no debonding, for instance where the matrix is weak and/or the interfacial bond is strong and/or the fibre volume fraction is high. If Pmu/3/pr s is small, ½11 -+Pmu/Prs. This crack spacing, which will be denoted x(rs), corresponds to the minimum crack spacing calculated using the model of Aveston et al, 8 ie assuming linear stress transfer, for a bond strength of r s. Provided there is some debonding, the crack spacing ½12 (Equation (6)) applies, and it follows that ([Pmu/Pri] - ½12) is independent of the matrix strength and depends on the fibre volume fraction only through the constant/3. Pmu/PT"iis the minimum crack spacing for linear stress transfer and a bond strength 7"i,and will be denoted x(r-t). Fig. 4 shows the minimum crack spacing defined by Equa-
/c / / /
/ /
~6
/
/
A
U //
6
/// b4
ri
5O
4
Values used: Ef 76 GNrr{2 Af .027rnrn 2 Gm 8 GNm-2 e'mu 6 Nmm-2
E E
2
p'% p'r i I
2
Olo 0
I0
I0 Nmrn"1 2 N rnnrn"t
20
50
40
5O
I/2 BL
Fig. 3 Ratio of average bond strength ~to interfaeial bond strength r s and frictional bond strength ri, as a function of embedded fibre length factor, for various ratios rs/f I
COMPOSITES. A P R I L 1982
I 50
I I00 Volume ratio
(Vr./Vf)
Fig. 4 Minimum crack spacing as a function of volume ratio V m / V F. Curve A assumes elastic stress transfer and r~ ~ r i (crack spacing ½ 11,2); curves B and C assume linear stress transfer with bond strengths r$ and ri respectively (spacings x ( r s) and x ( r i) respective ly)
147
tion (5) or (6) above as a function of volume ratio Vm/Vf, for a glass fibre/cement composite, together with x(rs) and x(rO for comparison (curves A, B and C respectively).
40
The difference between the 'actual' crack spacing shown in curve A, ½1t or ½12, and that calculated on the frictional bond ri (curve C) is near constant over a wide range. If the minimum crack spacing is used to calculate an apparent bond strength rAre assuming linear stress transfer, ie PrAee =Pmu/gala,2, the apparent bond strength will overestimate the frictional bond ri, and underestimate the interracial bond rs; and Fig. 3 applies.
30 vE
g
o) (u 2, 2O
COMPOSI TE STRENG TH
Lawrence's analysis can also be applied to the calculation of the strength of short fibre composites. The critical fibre length, lc, is the minimum length of fibre such that the fibre will break rather than pull-out from the matrix and, if the shear stress r s developed at the interface is uniform, is given by:
E} t~
,ff
pT, ( N tam't)
[0
2o
½l c = Ofu A f / p r s
(7) 5O
The reinforcement efficiency factors for an aligned composite containing fibres of length l are, 9 0 le
n = 1 - ~- (2-ri/rs)
l > 2l'o
l 7? - 2lc( 2 _ ri/rs)
l<2l;
[0
15
p-ci(Nmm-l) Fig. 5 Critical fibre length as a function o f p r I for.various values o f p r $ (Data as f o r Fig. 4)
or
able embedments lengths, and calculated the average bond strength, 7 from the maximum load achieved. Pinchin 1° on the other hand used long embedment lengths, and also calculated the average bond strength r. At short embedment lengths, r approaches rs; at long embedment lengths r approaches r i. Neither method allows both r s and r i to be calculated.
where l'c = ½l c (2 - Ti/Ts). If elastic continuity prevails, Equation (7) does not apply, and,
1
½!c = ~ tanh-1
ofuA f
)
½l c ~
or
~lc
_
1 pT-i
Prs tanh (3Xmax) (°fuAf +PriXmax -- ---if1Alc i> Xmax
The critical fibre length is a function of both interfacial bond strength rs, and the frictional bond 7"i,and Equation (7) applies only if r s = r i. Fig. 5 shows the critical fibre length le, plotted as a function of r s and r i. The frictional bond is seen to be the more effective in reducing lc and hence in increasing composite strength. EXPERIMENTAL METHODS OF DETERMINING BOND S TRENG TH The most common method o f determining bond strength is the pull-out test, either on single 'fibres' or on arrays o f
fibres. De Vekey & Majumdar a reported pull-out results for single fibres from cement. They used the smallest practic-
148
Aveston et al, ~1 Baggott & Gandhi a2 and others have calculated average bond strengths from measurements of crack spacing. Again r s and r i cannot be found. Lawrence 6 described how both r s and zi can be calculated from pull-out data over a range of embedded fibre lengths. Bartos 7 has used a similar analysis to show how the bond strengths can be determined from pull-out curves at a single fibre length. Both methods require that (3 is known. This is problematical as/3 depends on the shear modulus of the matrix in the interracial region, which is usually not known. Steel fibres
De Vekey & Majumdar 3 reported average bond strengths for steel/cement ranging from 5.5 Nmm -2 for bright high tensile steel wires after 28 days water storage, to 11 Nmm -1 for air storage. The embedment lengths used were 1 and 2 mm. The pull-out curves, while showing considerable variability, generally showed the expected shape, although in some cases the first deviation from linearity was marked and in other cases it could not be detected. After the maximum load was reached, there was a sudden drop in load and then a more gradual drop in many cases, to what was apparently a constant pull-out load, as the fibre which protruded from the matrix, pulled through. Unfortunately the recorder was turned off early in this region, and details of this region were lost.
COMPOS ITES. AP R I L 1982
There was evidence of 'stick-slip' behaviour in some cases, notably for bright high tensile steel, and usually in the 'constant' pull-out region. In a few cases however, it occurred before the maximum load was reached, and in those cases, failure was catastrophic. The curves for rusty high tensile steel, on the other hand, did not show stickslip behaviour. The pull-out curves for rusty high tensile steel after 28 days storage in air showed either a discontinuity (peak) or a change of slope before the maximum load was reached. Assuming this point marks the beginning of debonding, it can be used, together with the maximum load, and the 'constant' frictional load during pull-out, to calculate the three unknowns, rs, r i and/3. The results obtained are/3 = 1.65 mm -1 ; r s = 22 N mm -a and r i -- 8 N mm -2 (Table 1). The interfacial bond Ts is the total bond, both the chemical or cohesive bond, and the frictional bond. It follows that the cohesive bond r e (= r s - r'O has strength " q 4 N mm -2. The curves for pull-out of bright high tensile steel wire did not show an obvious debond point, and without a measurement of this point there is insufficient information to allow the calculation of/3 as well as Ts and r i as before. It was assumed therefore that the value ~ calculated for rusty wire in cement applied to the bright wire also, and to both conditions of storage, air and water. The calculated values of rs and r i are shown in Table 1. The drop of the maximum load is very much reduced in the case of the water stored specimens compared with that for air storage, and the calculated bond strengths are lower. This is consistent with Pinchin's conclusion l° that the cohesive bond as well as the frictional bond depends on the normal pressure at the interface, arising in the case of the air stored specimens, from drying shrinkage. The values deduced for the cohesive bond strength Te (Table 1) compare with the values of shear strength of the cement/cement interface deduced by Pinchin l° from friction experiments. Pinchin 1° measured load as a function of cross-head movement for wires pulled through cylinders of cement paste or mortar. At the long embedment length used (42.5 mm), the maximum load would be expected to approach that due to the frictional bond alone (Fig. 3). The load/displacement curves should therefore rise to a maximum and thereafter the load should remain constant as the wires pull through. Such curves were observed only in a few cases, notably smooth wires pulled through wet-cured cement mortar cylinders. With increasing roughness, the peak load increased and there was a sudden large drop and an approach to a much lower, approximately constant load. Pinchin also showed that, if a confining pressure is applied immediately after debonding, the load drops smoothly to
Table 1.
Calculated values of ~, rs, r i and r c T
Ts
Ti
Tc
8
14
(N turn -2) Rusty HT wire in air
10.4
22.0
Bright HT wire in air
11.0
15.5
7.7
7.7
5.5
7.4
4.9
2.5
Bright HT wire in water
COMPOSITES, APRIL 1982
approach a constant value. It appears then that the sudden large drop he observed for roughened wires during pull-out is the result partly of the method of test, and partly of the frictional stress decreasing with increasing wire movement. It follows from the analysis outlined in an earlier section that the frictional stress/displacement part of the curve should be extrapolated back to approach the initial (rising) part of the curve, in order to estimate the load and hence r i at the beginning of slip. While with the limited information available, it is not possible to do this with any confidence, extrapolation of the post-slip regions suggests initial values of 7i of 1.2 to 2.7 N mm -2 respectively. No estimate can be made of the total bond r s. The average values of bond strength calculated from the maximum loads, range from 1.2 N mm -2 for highly polished to 3.9 N mm -2 for the most highly roughened wires in water cured cement paste at 16 weeks. These bond strengths are much lower than those calculated from de Vekey & Majumdar's data. While there are differences in materials, it seems more likely that the discrepancy arises from differences in sample preparation. In particular, Pinchin used a vibration method which led to increased porosity and decreased hardness of the cement near the wires.
Polypropylene fibres Walton & Majumdar 4 reported the results of pull-out tests of single polypropylene fibres from a cement matrix. The fibres extended through the matrix so that both ends protruded from the matrix. They found two types of pullout curves: in one the load rose smoothly to the maximum; in the other, there was a 'yield region' which in many cases was extensive and thereafter a reduced slope up to the maximum load. In some cases there was a drop in load in the yield region. Pull-out of the fibre, defined as obvious movement of the free (unloaded) end of the fibre, was observed early in this region, although there was some indication that total debonding and movement of the fibre through the matrix might have begun even earlier. The free end of the fibre was observed entering the matrix, often before the maximum load was reached. Walton & Majumdar 4 calculated average bond strengths based on the yield point and on the maximum load, and pointed out that because a large number of fibres broke, the calculated bond strengths are not true bond strengths but may be regarded as lower bounds only. The analysis of the results to calculate details of the interfacial bond r s and the frictional bond r i is open to doubt. Since in many cases the fibres broke, the test is not a true pull-out test. Where the load/extension curve rose smoothly to a maximum, all the fibres broke; and the curve resembled that of the fibre alone, except that the breaking load appeared to be approximately 30% lower. The fibres appeared to be anchored in the cement and there was no sign of slip relative to the matrix. One possible interpretation of the pull-out curve is that the yield region marks the maximum load achieved before complete debonding, and thereafter the frictional force opposing fibre movement increased for some as yet undefined reason.
149
Assuming then that the 'yield point' is the 'maximum load' of Lawrence's model, from the 10 and 20 mm data, 7"i = 0.07 N mm ~ provided that %l > Xmax. Since/3 is not known, r s cannot be calculated; but we know that the average bond strength r'(0.6 N mm -2 at 10 mm, 0.34 N mm -2 at 20 mm) underestimates r s and overestimates I"i.
region to be seen. It appears however, that there is an interfacial bond that is greater than the frictional bond, at least at the beginning of major slip. Its value will be higher than the corresponding average bond strength. The frictional bond at the beginning of major slip will be lower than ~. The rising load which was observed in some cases after major slip, could be the result of the taper produced in the fibre by the hand-drawing process.
Or if ½l ~
Bartos 7 reported the results of pull-out tests of glass fibre strands from a cement matrix. He used a relatively long embedment length (8.6 mm) with a water/cement ratio of 0.4, (dry-cured for 7 days). The curves showed a maximum load indicating the completion of debonding, and then a drop and a fairly uniform pull-out against the frictional forces. Assuming a value of matrix shear modulus in the calculation of/3, he was then able to calculate the 'ultimate shear flow' Prs and the 'frictional flow' pr i (6 and 1.8 N mm -1 respectively). The average shear flow p 7 (maximum load divided by embedded length) was 2.1 N mm -1 , not far different from the frictional flow Pri. The calculation of bond strength requires that the perimeter p can be measured. For glass fibre strands it is impossible to get an accurate estimate of the perimeter in contact with the matrix, but on average, it is unlikely to be less than 2 to 3 mm for the E-glass strand that Bartos used. The bond strengths are obviously low. De Vekey & Majumdar 3 used single glass filaments and the problem of measuring the perimeter did not arise. They used a very short embedment length (1 mm) and calculated the average bond strength r. The values were high - 6 to 9 N mm -2 for E-glass in cement (0.3 water/cement ratio) after 28 days storage. Further analysis of the pull-out traces they obtained is difficult. There is some ivdication on some traces of a discontinuity which might mark the beginning of debonding, but others do not show it or show a peak near the maximum load. After the maximum load is reached, the load drops. The load which presumably then reflects a frictional bond ri, sometimes rises again, but unfortunately the recorder was switched off too soon to allow details of this
150
In principle it is possible to calculate rs, ri and/3 from pullout tests, but the results will apply to the conditions obtained in the pull-out test, and not necessarily to those in the composite. In particular the shear modulus of the matrix in the vicinity of the fibre, might be very different. There is no information available on the effect of embedment length in a single fibre pull-out test, on the properties of the matrix in the interfacial zone; nor on the effect of other fibres in an actual composite. In an effort to reduce these (and other) uncertainties BRE has developed a multiple fibre pull-out test, in which the production of the test specimens more nearly resembles that of an aligned fibre composite. Inevitably the averaging effect of pulling out an array of 'fibres' leads to problems in interpreting the pull-out traces. The possibility of using crack spacings rather than pull-out tests is attractive for this reason, but crack spacings alone are not sufficient to allow calculation of both r s and ri, even if/3 were 'known'. Another problem arises in that actual pull-out tests suggest that r i is not necessarily constant and might decrease (eg Pinchin's studies of steel wires), or increase (Walton & Majumdar on polypropylene fibres) with increasing fibre movement. It is interesting to speculate on the effect of such changes: a decrease in r i after debonding has started would reduce the stress opposing further debonding, and lead to catastrophic failure and pull-out at an earlier stage. An increase in r i would lead firstly to an increase in the load required to continue the debonding. As the displacement increases and r i continues to increase debonding would stop and further increase in load would lead to fibre failure rather than pull-out. At the displacements obtaining during debonding, it might well be that the effect of a non-constant frictional bond strength is small. But once the whole fibre is slipping it is obviously important since it must modify the strength, the multiple cracking and the total energy to break. This and previous studies suggest that the frictional bond is more important in determining composite properties than the total interfacial bond, which is the sum of the (initial) frictional bond and the cohesive bond. In view of this and of the difficulty in determining the interfacial bond strength, it might be better to concentrate studies on the frictional components, 'static', 'dynamic' and 'ploughing'; and on modifying the interfacial zone to improve these properties. A CKNOWL EDGEMENTS
I am grateful to Dr R.C. de Vekey and Mr P. Walton for
COMPOSITES. APRIL 1982
providing data for analysis. The work described has been carried out as part of a research programme being conducted by the Building Research Establishment of the Department of the Environment and this paper is published by courtesy of the Director, Building Research Establishment, and by permission of the Controller HMSO, holder of Crown copyright.
6 7 8 9 10
© Copyright Controller HMSO, London 1981.
11
REFERENCES
12
1 2 3 4 5
Allen,H.G.JPhysD:AppIPhys 5 (1972) pp 331-343 Naaman, A.E. and Shah, S.P. 'Fibre reinforced cement and concrete' RILEM Symposium 19 75 (The Construction Press) pp 171-178 de Vekey, R.C. and Majumdar, AJ. Mag ConcreteRes 20 (1968) pp 229-234 Walton,P.L. and Majumdar, AJ. Composites 6 (1975) pp 209-216 Greszczuk, L.B. Interfaces in Composites, ASTM STP 452 (1969) pp 42-58
COMPOSITES. A P R I L 1982
Lawrence,P. JMater Sc/7 (1972) pp 1-6 Bartos,P.JMaterSci 15 (1980) pp 3122-3128 Aveston, J., Cooper, G.A. and Kelly, A. 'The Properties of Fibre Composites' Conf Proc, National Physical Laboratory, 4 November 1971, (IPC, Guildford, UK, 1971) pp 15-26 Laws,V.YPhysD:ApplPhys4(1971) pp1737-1746 Pinehin,DJ. "The cement steel interface: friction and adhesion' Ph D Dissertation (University of Cambridge, 1977) Aveston, J., Mercer, R.A. and Sillwood, J.M. 'Composites - Standards Testing and Design' ConfProc, NPL 8/9 April 1974 (IPC, Guildford, UK, 1974) pp 93-102 Baggott, R. and Gandhi, D. JMater Sci 16 (1981) pp 65-74
A UTHOR
The author is a researcher at the UK Department of the Environment's Building Research Establishment. Inquiries should be addressed to: Miss V. Laws, Building Research Establishment, Building Research Station, Garston, Watford WD2 7JR, England.
151