The critical constrained crack length in fibre-reinforced cementitious matrices

T H E C R I T I C A L C O N S T R A I N E D C R A C K L E N G T H IN FIBRE-REINFORCED C E M E N T I T I O U S M A T R I C E S

J. SPURR1ER* and A. R. LUXMOORE

Department of Civil Engineering, University College of Swansea, Singleton Park, Swansea SA2 8PP (Wales)

SUMMAR Y Cementitious matrices do not behave as ideal brittle materials, since they exhibit a period of controlled crack growth prior to failure. This paper brings together some of the theories on the effects of fibres in cementitious materials, including the ability of -fibres to reinforce the composite by constraining this crack growth, and shows that the theories are compatible provided certain numerical modifications are made. The combined equations can be used to give a guide to the-fibre size and concentration needed for a required improvement in the tensile strength of the matrix.

1NTRODUCTION Virtually all the important theories for the brittle fracture of materials under conditions of static and quasi-static loading are based on the Griffith criterion. In order to calculate the condition for unstable propagation of a crack, G r i ~ t h considered a wide plate of unit thickness under uniaxial tension tr, containing a perpendicular crack of length 2a. (See Fig. l.) By calculating the elastic strain energy, W, in the plate, he showed that the rate of strain energy released when the crack extends by an amount 2da is given by: (dW)

°r2~a

2Ta = E= where E,, is the Young's modulus of the material. The rate at which work has to be supplied to create new crack surfaces against a surface tension term, V, is: * Present address: Department of Materials, Cranfield Institute of Technology, Cranfield, Beds. MK43 0AL, Great Britain. 225 Fibre Science and Technology (9) 0976)--© Applied Science Publishers Ltd, England, 1976 Printed in Great Britain

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J. SPURRIER, A. R. LUXMOORE

(d2-~a) = 2 7 = constant Hence, for instability: _>

At the critical crack length 2a¢: ac 2~a c -

E~ -

=

27

and the crack becomes self-propagating (Fig. 2). O"

Failure stress

Crack width 2a Fig. 1.

T h e Griffith relationship between failure stress a n d crack width for a wide plate under uniaxial tension a containing a transversal crack.

This basic concept can be extended to non-ideal brittle solids by assuming that y consists of both a surface tension and a plastic work component, where the latter is normally dominant. A direct application of the Griffith criterion for cracking, as extended to cementitious materials by Kaplan, 2 reveals that the tensile strengths of such matrices are limited by surface cracks and internal flaws. However, it does not predict the period of controlled crack growth which can occur in cementitious materials, but indicates that failure should be catastrophic. Glucklich 3 has explained this period in terms of the energy required by microcracks growing ahead of the moving crack tip, this extra energy increasing with crack size.

CONSTRAINED CRACK LENGTH IN CEMENTITIOUS MATRICES

227

If the controlled crack growth can be impeded, so that it takes place at higher stresses, then the ultimate strength of the material can be raised--in a manner analogous to dislocation locking in metals. One way of achieving this objective is by fibre reinforcement, and two main theories have been put forward to explain the effect of fibres in cementitious materials. This paper attempts to correlate these theories by relating them to experimental data, and the calculated size of flaws or cracks inherent in reinforced cementitious materials. The resulting equations are then used to indicate the necessary fibre concentration for improvements in the strength of the matrix.

Energy per unit length Crack C r a c k \o~;/" stable propagates ~b'/

Sergy requiredfor. Jl~ crackpropagationI_du ac Fig. 2.

Crack half-length a

Crack stability diagram.

THEORIES OF FIBRE REINFORCEMENT

Romualdi (see Romualdi and Batson 4) first suggested the use of steel fibres to increase the tensile strength of concrete. He based his theory on the fact that the material around a propagating crack tip is displaced perpendicular to the plane of the crack, and suggested that a perpendicular stiffener bonded to the matrix in this vicinity would increase the resistance to crack propagation. In his analysis he used an idealised arrangement in which a circular crack is contained centrally within a square array of very stiff fibres, the spacing of these fibres being equal to the diameter of the crack (Fig. 3). The compressive stresses caused by the forces along the four adjacent wires are computed over the area of the crack, and a comparison of the resulting total stress intensity factor K with Kc indicates if the chosen crack

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J. SPURRIER, A. R. LUXMOORE

size is stable at the selected applied stress. Thus, for a given wire spacing, it is possible to calculate the critical applied stress causing instability of the largest flaw which could be contained wholly within the square 'box' of four adjacent fibres. Flaws spreading beyond these limits would overlap, hence the localising effect of the fibres is then assumed to have failed. Bowing of the crack front between fibres is not considered.

2~a Cross-section Fig. 3.

Section AB

R o m u a l d i ' s flow-localisation. A cross-section a n d section AB t h r o u g h a circular crack localised by a square array of fibres.

Romualdi identified this point of instability as the point of'first crack spreading' associated with local, non-catastrophic, failure of the matrix. In subsequent experimental work 5 he attempted to correlate this point with deviations from linearity in the load displacement curves of concrete specimens, but there are serious doubts cast on the validity of his results. 6'7 In his calculations for the onset of instability, Romualdi used an expression derived by Sneddon 8 for the lowering of the potential energy, W, of a specimen by a disc-shaped crack of radius a. Since the crack surface area A = ~a 2 and: 8(1 W=

--

v2)a2a 3 3E~

then the crack extension force for a newly created area of crack is given by: dW

d W da

dA

da

dA

4(1

--

v2)oEa

~zEm

where v is Poisson's ratio (which is approximately 0.3 for cementitious materials). It can be seen that this is fundamentally similar to the expression for the Griffith crack. A second paper by Romualdi 5 deals with a different aspect of fibre reinforcement of cementitious matrices. Here he points out that the fibres continue to play a role when they are no longer able to Iocalise the flaws and crack growth has initiated,

CONSTRAINED CRACK LENGTH IN CEMENTITIOUS MATRICES

229

since then they cause an increase in the critical energy required to open a new unit area of crack surface Go. By considering the work done when discontinuous wires strip uniformly and cleanly from the matrix in association with an advancing perpendicular crack front, of acute angle 0, (see Fig. 4), he demonstrates that:

rrrlV:O 7.6d

Gc -

where z is the limiting bond strength, and 1, d and V/are respectively the length, diameter and volume fraction of the wire reinforcement. This latter mechanism of reinforcement is analogous to the 'constraint effect', i.e. the suppression of matrix cracking under conditions of small fibre size and high concentration, postulated by Aveston et al. 9 They consider the energy which must be dissipated when a reinforced matrix fails and the load is borne by the fibres. For unstable cracking, the energy balance gives the inequality: 127V,.z < Ef2Vfe30t2(l h- ~)r

~Iength Fibre

t

/-Crack _ w

O posir t~ion- !-~.

~ U Matrix/fibre

Fig. 4.

Crackingplane

~

position

slippage

Romualdi's growth inhibition caused by wire slippage associated with crack advances.

where: ct = EmVmfE:V:, r = radius of reinforcing fibres, e = matrix strain when the crack propagated and the subscripts m and f on the Young's moduli E and volume fractions V denote matrix and fibre respectively. For constant V:, the matrix cracking can be suppressed by choosing a fibre radius (and hence fibre spacing) given by: 127rEfV$ 2 r < EmZe, 3Vm(EmV," + E:V:) where e,. is the unreinforced matrix cracking strain. This expression can also be used to predict the critical constrained matrix cracking strain emp in the reinforced matrix:

(

12yzEfVf 2

.~

e"P = \E,.2V,.r(EmV,. + EfVf)]

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J. S P U R R I E R , A. R. L U X M O O R E

The equation above cannot be compared directly with those derived from Romualdi's theories because of basic differences in the analyses, but both predict an increase in strength with decreasing fibre radius. Romuaidi's equation can be used to predict the strength of tensile specimens by treating the composite as a homogeneous body containing a crack with a diameter equal to the fibre spacing. Hence, using Griffith's equation:

r~lViO

2ga2ac Gc

--

_

_

E,,

--

15"2r

where a is the strength of the reinforced matrix.

A P P L I C A T I O N TO E X P E R I M E N T A L RESULTS

Application of these principles and equations to the reinforcement of cements and gypsum plaster by glass fibres yields some interesting results. Much experimental work in this field has been performed by Majumdar and Ryder at the Building Research Establishment, l° and their experimental values have been used in the calculations. The glass reinforcement used was in the form of rovings, containing 204 filaments (each of radius approximately 6 x 10-6 m) in a strand which had a radius of approximately I0-4 m, and the strands were chopped to be incorporated into a two-dimensionally random mat. Some experiments have also been performed on reinforced gypsum plaster using similar glass strands in a square array, as well as basic research to test the compatibility of these results with those of Majumdar and Ryder. For glass fibre-reinforced gypsum plaster and cement, the following data were used throughout the calculations: Gypsum tru E,, e,, 7 r v

= = = = = =

2"75 17 x 1.62 6"25 5"52 0'3

x 106 N / m 2 109 N / m E x 10 - 4 J/mE x 106N/m E

Glass fibre try Ej, rl r204

- 1210 = 76 x = 6 x = 1 x

x 106N/m 2 109N/m 2 10 - 6 m 10 - 4 m

Cement tru E,. e,. 7 z v

= 2"75 x 106 N / m 2 - 17 x 109 N / m 2 = 1 ' 6 2 x 10 - 4 = 15 J / m E ] 1 = 10.34 x l 0 6 N / m 2 --0"3

Using these figures in the Griffith equation the critical crack length 2a¢ ~ 18 x I0-3 m for unreinforced gypsum plaster and : 2ac ~ 43 x 10-3 m for unreinforced Portland cement For Romualdi's equations where Sneddon's expression is used, these lengths are increased by a factor of 2.7. In the case of gypsum plaster especially, these calculated figures are clearly much greater than the size of flaw which would be anticipated in such materials. The equations used to obtain these figures, however, predict an instantaneous

CONSTRAINED CRACK LENGTH IN CEMENTITIOUS MATRICES

231

brittle failure at the matrix ultimate stress ac, and take no account of the period of slow micro-crack growth which precedes fracture in cementitious materials. 3 The micro-cracks themselves originate from flaws in the material. Ultimate failure takes place only when a sufficient number of these micro-cracks interact with each other or become incorporated into the main crack to produce an effective critical crack length which is unstable. If it is possible to isolate the individual flaws (postulated to be disc-shaped) by incorporating fibres along the lines suggested by Romualdi and Batson, 4 it should be possible to increase significantly the matrix strength of gypsum plaster by this technique. Postulating an expected flaw size of 3 x 10- 3 m, and using the Griffith equation in reverse indicates a fourfold increase in the stress required to cause crack propagation. The glass reinforced gypsum specimens tested experimentally i o contained either 0.047 or 0.089 volume fraction of glass in a two-dimensionally random mat of filaments of length 43 x 10- 3 m (greater than the critical stress transfer length of 22.2 x 10- 3 m). Under test, these materials began to suffer matrix failure at stresses of approximately 8 M N / m 2 and 10 M N / m 2 respectively. Without the constraint effect, the predicted failure stresses would have been (Em V m + q E I V I ) e m or approximately 2.8 M N / m 2 in both cases,12 once an efficiency factor 13 r/is incorporated to allow for the effect of orientation of the fibres. However, substitution into the inequality for the constraint effect derived by Aveston et al.9 indicates that a constraint effect is predictable. Rearrangement of this equation enables the predicted failure strain of the matrix to be calculated assuming that there is a constraint. (Because of the manufacturing method, it is not certain to what extent the 204 filaments of glass separate once mixed in the material. This difficulty can be partially circumvented i 2 by using the definition of the transfer length 14: It -- ~£ " r

which is a parameter that can be measured by pull-out tests for fibre strands embedded in the matrix, thus giving an effective strand radius under conditions approximating to those found in the fabricated composite. This also eliminates the uncertainty in the value of z, which is discussed later.) The predicted failure stresses then become 8-9 M N / m 2 and 13-9 M N / m 2 for the two gypsum composites. Similarly calculated results for the cement composite containing glass fibres of length 34 x 10 -3 m (transfer length 11.8 × 10-3 m) predict failure stresses of about 2.8 M N / m 2 without constraint, 11.6 M N / m 2 allowing for constraints, and experimentally the matrix begins to fail at about 10.5 M N / m 2.

Volume fraction o f glass in composite Experimental matrix failure stress, M N / m 2 Predicted values, without constraint, M N / m 2 Predicted constrained values of stress, M N / m 2

Gypsum plaster 0-047 0.089 8"O 10-0 2"7 2"8 8-9 13.9

Cement 0.0325 10-5 2"8 I 1"6

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J. S P U R R I E R , A. R. L U X M O O R E

Clearly the formulae including the predicted constraint effect more accurately reflect the experimental behaviour, although in all cases these equations are rather optimistic on the expected failure stress of the reinforced matrix. The final phases of matrix failure, however, usually occur close to the predicted values of stress. This indicates that the reinforcing fibres are unable to isolate the matrix flaws at higher stresses, probably due to imperfect bonding between the components. The restraining effect of fibres during crack growth as calculated by Romualdi 5 gives:

z~zlVsO G~

7'6 x 2r

As typical values, however, Romualdi quotes 0 = 0.1 radians in concrete, which seems excessively large. A direct comparison with the Griffith criterion 2rcac2ac/E,,, > Gc = 4~, and substitution in this'inequality of the experimentally determined values for glass fibre-reinforced gypsum plaster, given above, gives unrealistic figures (825 m!) for the critical half-crack size ac if this value of 0 is "used. For wires of diameter 1.5 × 10 -4 m (0.006 in) in concrete Romualdi's value of 0 indicates that Gc = 6073.0 J/m 2 (34.7 in-lbf/in2). In an earlier paper, 4 Romualdi's experiments indicated a value of Gc = 3.5 J/m z (0.02 in-lbf/in 2) which implies 0 ~ 6 × 10-5 radians. If this new value is used for gypsum, more acceptable figures are obtained for the size of flaw which could be propagated by an applied stress or. (For example, an applied stress of 7.8 × l 0 6 N/m z should propagate a crack with ac = 3.0 × 10 - 3 m.) In order to compare Romualdi's expression for composite failure with the Griffith prediction for brittle fracture, it is necessary to regard the fibre-reinforced composite as a continuum with a Young's modulus E. Suppose that the reinforced matrix fails at a stress or, and the unreinforced matrix at a stress at(where a > c%). Then:

2rca2a

rTrlViOq

E

15'2r

where the efficiency factor r/ = ½ has been introduced to take account of the two-dimensional randomness of the fibres in the experimental composites. ~3 Since the flaw localisation effect of the fibres has been overcome when the matrix finally fails, an approximate comparison of a and ac may be obtained by substituting a ~ a C and, for low Vf, E ~ E,,,. Hence: 2 ~ 0 "2

Em

i.e.

2yE,, ac2~z~

rTrlVsOq 15-2r

CONSTRAINED CRACK LENGTH IN CEMENTITIOUS MATRICES

233

The ratios (a/trc) are given below for reinforced gypsum plaster. Vs = 0.047 17-5 4-3 3-0

r = f i l a m e n t r a d i u s rl r = s t r a n d r a d i u s r204 Experimental ratios

Vs = 0"089 24-1 5.9 3"5

In these calculations it has not been possible to circumvent the uncertainty in the value of z by substitution, since this would also eliminate r from the equation. If allowance is made for incomplete fibre-matrix bonding by reducing the value used for the bond strength z (from the limiting bond strength 5-52 x l 0 6 N / m 2 used in the calculation to a suggested effective bond strength around 1.5 x 1 0 6 N / m 2) the experimental ratios would then lie between the results obtained using the filament radius and strand radius, and would indicate poor separation of the strands. Optical microscopy has revealed that strand separation is incomplete in the experimental materials. A further indication that the theoretical bond strength should be reduced may be obtained by substituting the experimental value of the transfer length It = 22.2 × 10 -3 m in Kelly's defining equation. 14 It --

Gf

. r

Then for r = r I (filaments) z ~ 0"3

x

10 6

N/m 2

r --- r2o 4 (strands) z ~ 5"5 X 106 N / m 2 Since it is probable that the strands separate at least partially into filaments during the manufacturing process, a reduction in the limiting bond strength from 5.52 × 1 0 6 N / m 2 to an intermediate value is indicated, to take account of the experimental bonding conditions existing in the tested composite.

EFFECTIVE REINFORCEMENT OF CEMENTITIOUS MATRICES

Combining these equations gives a guide to the size of reinforcement and the volume fraction needed to reinforce the matrix to a certain required stress level a, when certain assumptions are made about the flaw size and fibre distribution. If the strength of the unreinforced matrix is assumed to be limited by the presence of internal flaws, 2 the Griffith equation then enables the maximum permitted flaw size to be calculated as: 4~E,.

2ac = nau2 In Romualdi's theory, 4 the localising effect of the reinforcement is assumed to have

234

J. S P U R R I E R ,

A . R. L U X M O O R E

failed once the flaw expands beyond the confines of four adjacent fibres in a square array. If the fibres are to be used efficiently against flaw expansion, their closest spacing must be set equal to the flaw diameter. Thus, for a square array of continuous fibres, the volume fraction of reinforcement: /l-r 2

V:

a-r 2

-- (2ac)2 -- S2

while for random or discontinuous fibres: (0 < Vf < 1) (0 < r < a<)

nr 2 nr 2 Vf = q(2a<) 2 = qs---5 i.e.

(1)

V: = v,#r 2

This assumes that the presence of the reinforcement does not alter the maximum inherent flaw size o f the matrix, nor the value of ?. The condition for constraint 9 can be rewritten as: r <_ 12yzEfVfZEcZ E,.2o'3(1- V:)

where: E< = (E,.V,. + E : V : )

Since economically only low volume fractions of reinforcement are at present worth considering, Ec ~ E.,, hence an approximate expression for the critical fibre radius is : 12?rE: ( V: 2 1

r-

a~

kl - Vii

i.e.

r=X.( tJ

v:~ 1

-

vsi

(2)

If the reinforced matrix is to crack at the failure stress, a,, of the unreinforced matrix, the critical radius, r0, is given by the simultaneous solution of eqn. (1) and eqn. (2) with a = a u. This yields a unique solution for both r and V: because of the stipulation that the fibre spacing (or effective fibre spacing in the case of random or discontinuous fibres) is equal to the expected maximum flaw diameter (s = 2at). When the spacing of the fibres is fixed in this way, and it is desired to increase the failure stress of the matrix in the composite by making use of the constraint effect, the radius of the reinforcement must be increased. At first glance, this is a contravention of the constraint inequality, but it must he remembered that under these conditions V: is not a constant, but increases in proportion to the square of the radius.

235

CONSTRAINED CRACK LENGTH IN CEMENTITIOUS MATRICES

The minimum volume fraction of reinforcement needed to obtain a desired matrix failure stress, tr, and also the minimum consistent fibre radius, when the fibre separation is fixed at 2ac, may be found from a graphical plot of the two simultaneous equations in r and Vy (Fig. 5). The lack of a solution indicates that too optimistic a value of tr has been chosen, or insufficiently rigid fibres used. Generally, however, the uncertainty in the value of z will make these results unreliable. Eliminating z by using the definition of transfer length/t yields a quadratic equation in VI with only one positive solution which approximates to the minimum volume fraction required for the reinforcement. If the centre-to-centre fibre spacing is allowed to reduce such that s < 2ac, it is necessary to assume that the reinforcement affects the size of at least the largest flaws and micro-cracks in the matrix, so that they are still constrained within the V o l u m e f r a c t i o n Vf

1.0

0,5

Minimum Vf- ~

- - - ~ '

Minimum r

I

r=12~ml~E f ( Vf= 0"3 kl -V.

ac

Fibreradius

r.

Fig. 5. Graphicalsolution for approximate minimum fibre radius and minimum volume fraction of reinforcement required for a chosen matrix cracking stress a, when the fibre spacing s is held constant. bounds of adjacent fibres although these are closer together than the maximum flaw diameter in the unreinforced matrix. Provided that this condition is satisfied, and that Vs is kept low, then eqn. (1) (with variables in volume fraction Vs, fibre radius r and inter-fibre spacing s) and eqn. (2) (with variables in VI, r and the reinforced mb.trix failure stress tr) may be used to obtain approximate solutions for any two of these four variables when the other two are ascribed chosen, or experi-

236

J. SPURRIER, A. R. LUXMOORE

mentally enforced, values. The normal use of the equations would be to obtain a value for the volume fraction of fibres of a given size to reinforce the matrix to a chosen failure stress.

CONCLUSION

Modifications in the quoted angle 0 at the tip of a brittle crack and in the bond shear strength r make Romualdi's theories on flaw localisation and fibre influence mutually compatible with the constraint effect postulated by Aveston e t al. 9 and also with the Griffith criterion for brittle crack propagation. The combined equations may be used to obtain an indication of the fibre size and concentration required for a certain improvement in the 'first crack strength' of the reinforced brittle matrix.

REFERENCES

I. A. A. GRIFFITH, Phil. Trans. R. Soc., A221 (1920) p. 163. 2. M. F. KAPLAN,Jnl Amer. Contr. Inst., 58 (5) (November, 1961) p. 591. 3. J. GLUCKLICH, Jet Propulsion Lab, California Inst. of Tech., Pasadena, California; Tech. Report 32-1438 (August, 1970). 4. J. P. ROMUALDIand G. B. BATSON, A S C E J n l o f E n g . Mech. Div., EM3 (June, 1963) p. 147. 5. J. P. ROMUALDI, Fibres Concretes, USA and UK, Symposium, University of Birmingham (September, 1972). 6. S. P. SHAH and B. V. RANGAN,Jnl Amer. Contr. Inst., 68 (February, 1971) p. 126. 7. B. B. BROMS, S. P. SHAH and P. W. ABELES, A S C E J n l o f E n g . Mech. Div., EM1 (February, 1964) p. 167. 8. I. R. SNEDDON, Proc. Roy. Soc. London, 187A (1946) p. 229. 9. J. AVESTON, G. A. COOPERand A. KELLY, Nat. Phys. Lab. Conf., London, November, 1971. 10. A. J. MAJUMDARand J. F. RYDER, Sci. Ceram., 5 (1970) p. 539. I 1. G. A. COOPER and J. FIGG, Trans. Brit. Ceram. Soc., 71 (1972) p. 1. 12. J. Sr'URRIER and A. R. LUXMOORE,Fibre Sci. and Tech., 6 (1973) p. 281. 13. V. LAws, J. Phys. D: Appl. Phys., 4 (1971) p. 1737. 14. A. KELLY, Strong Solids, Clarendon Press, 1966.