the efficiency of fibrous reinforcement of brittle matrices

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The efficiency of fibrous reinforcement of brittle matrices

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1971 J. Phys. D: Appl. Phys. 4 1737

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J. Phys. D: Appl. Phys., 1971, Vol. 4. Printed in Great Britain

The efficiency of fibrous reinforcement

of brittle matricesV. LAWS

Department of the Environment, Building Research Station, Garston , Watford, Herts

M S . received 10th May 1971

Abstract. Efficiency fac tors a re proposed describing the effectiveness of utilization offibre stiffness and strength when a random distribution of extensible fibres is incor-porated in a brittle matrix.

A length efficiency factor for strength calculation is derived, which includes theeffect of sliding friction during fibre pull-ou t; this reduces to th e two previously appliedefficiency factors in the limits.

The interaction of fibre length and orientation is considered, and general efficiencyfactors describing the effect of both length and orienta tion on the strength of randomlyreinforced short fibre brittle matrices are derived.

1. Introduction

In t he field of com posite m aterials, especially those used in building, increasing attentionis being given to the possibility of improv ing the properties of a brittle m atrix by the incor-poration of fibrous reinforcement, notably metals, polymers and glass. Fibre reinforced

gypsum plaster and cement differ from most other fibre reinforced materials in that thematrix failure strain is very much lower than that of the fibre; the matrix is highly porousand the bond between fibre and cement is discontinuous; practical volume fibre fractionsare low; a nd the o rientation of the fibres in composite now being produced is approximatelyrando m in either two or three dimensions (Ryder 1969). Theories of the elastic responseand strength of composites based on the assumption of perfect bonding between fibre andmatrix, and alignm ent of the fibres with the stress axis, do not apply. The problem the narises of determining the efficiency factors of length and orientation that should be appliedto describe the effectiveness of the reinforcement when the fibre length is finite, the bondimperfect and the orientation random .

2. Notation

The subscripts c, f and m refer to composite, fibre and matrix; o and I refer to orientationand length.

The basic symbols are as follows: E is the elastic modulus; G the shear modulus;v Poisson’s ratio; nz (=.&/Em) he modular ratio; U and u u the stress and ultimate, orbreaking, (tensile) stress; L the load; E and EU the strain and ultimate, or breaking, strain;U the volume fraction; I the fibre length ; a the fibre cross-sectional area ; p the fibre perim eter;7 the interfacial bond strength; 9 the efficiency factor. Other sym bols are defined inthe text.

3. The theoretical model

Krenchel (1964) has proposed a theory of reinforcement in the elastic range that isbased on the assumption that the fibres in the composite support a load only along theirlongitudinal axes. He assumed that matrix and fibre extend together. If there is no

1737



1738 V . Laws

lateral contraction of the composite, the modulus of elasticity of the composite in thedirection of the applied stress is given by

Ec VoEfuf +E m v m (1)

where yodescribes the effect of fibre orientation an d has a value between zero (for orienta-

tion perpendicular to the stress axis) and unity (for alignment with the stress axis).The stress supported by the composite at the failure strain of the matrix is then

u c= VoEfOfEm' + Om'Um. (2)

After the matrix failure strain has been reached, it is assumed that the fibres alone supportthe stress, so tha t th e ultimate failure stress of the composite is simply

ucu = 70 f"Vf. ( 3 )

Equation (3) implies tha t the fibres break when the extension in the direction of applicationof the stress is equal to the brea king extension of the fibre, irrespective o f the fibre orientation ,and applies only if this condition holds.

These relations (1 ) to ( 3 ) are modified for short fibre composites by the inclusion of alength efficiency facto r 71 in the term describing the fibre orientation.

In this paper, the efficiency factors that should be applied to fibre reinforced brittlebuilding materials are considered. Com bined efficiency facto rs describing the effect oforientation and length on the composite strength are derived and lead to conditions forreinforcement.

4. The length efficiency factor

Previously two different efficiency factors have been used to describe the effect of thefibre length on the efficiency of reinforcemen t. Krenchel (1964) has used a factor77 1= 1- c/Z, where lC/2 s the minimum length required to allowl the fibre to reach its

breaking stress without slipping, and is given by

Bortz and Blum (1968) have used a fac tor of 1 -lc/21 in their determination of the strengthof metal fibre reinforced ceramics.

A consideration of the simple model-that is, an elastic response to stress of bo th com-ponents up t o the failure strain of the matrix, followed by a region such that the stress atfailure is determined by the fibres alone-leads to a n efficiency facto r tha t varies withextension. The two regions, elastic defo rmation an d final failure, will be consideredseparately.

4.1. Elastic region

Fo r sm all extensions of the composite, bo th matrix an d reinforcement can be consideredto deform elastically. Provided the shear strength a t the interface is sufficient, stress willbe transferred fro m matrix to fibre by shear. Cox (1952), Dow (1963) an d others haveshown that the tensile stress in the fibre builds up from zero at the ends to a maximumat the midpoint. Th e average stress in a fibre of length 1 subjected to a strain E $ is

where /3= (Gm/Ef)'/' F(v f ) ,F(v f )being a function of volume fibre fraction.

The length efficiency factor is given by the bracketed term in (5) and applies for smallextensions where the interfacial bond remains intact. A t matrix strains such tha t themaximum shear stress developed at the interface becomes comparable to the interfacial or



The eficiency of jibrous reinforcemen t of brittle matrices 1739

matrix shear strength, the two materials will debond progressively fro m the fibre ends, a ndstress transfer can then occur only if there is an effective frictional ‘bond’ at the interface.This could arise for instance if there is a difference in Poisson’s ratio of fibre and matrix.Let the frictional bond be described by a (static) frictional bond strength r s . The stressin the fibre will then build up linearly from zero at the ends to EfE, over a distance1,/2= EfExa/prs and the average stress in the fibre is

(5= 1- s/21) Efe,.

The length efficiency factor to be applied in the elastic region if this mechanism occurs is~2 = 1- /21 and is a function of the extension cX .

A t matrix failure

It follows from (6) that a factor 1-lc/21 could apply only if the fibre and matrix failurestrains were equal, a condition not met in the case of glass reinforced gypsum plaster orcement, since E f U C r : 100EmU.

Using the data given by Majumdar and Ryder (1970), the length efficiency factors thatapply at composite strains up to the matrix failure strain can be calculated according to(5) or (6) fo r typical glass reinforced gypsum plaster and cement composites. Calculationshows th at , over this region, th e length efficiency facto r is unlikely to be less th an 0.98.

4 . 2 . Composite strength

After the matrix has failed, the situation is very different.Consider fibres of length I aligned along the stress axis and arranged so that the fibre

ends are uniformly d istributed alo ng the length of the composite. A crack perpendicularto the stress axis is assumed t o release the fibres over a cr ack w idth (separation of the faces

of the crack) small compared with the fibre length. Th e fibres are intersected by th e crackso th at the sho rter lengths of fibre embedded in the matrix a re uniformly distributed between0 and 112. At a crack extension e x , those fibres will slip that have a shorter embeddedlength less than Ix/2, where Ix/2= EfExa/prS, nd the probability that a fibre will slip isI d .

The average stress suppor ted by all the fibres at an extension e x is

where IC is defined by (4) above.

value ( 6 z ) ma x at an extension ( E g ) m a x , where

From (7) it follows that the average stress increases with strain to reach a maximum

and

If this maximum (t;s)max is reached at an extension (Ez)max which is less than the fibrebreaking strain E f U , the breaking stress of the ma t is defined by this maximum (equation (8)).In this case the comp osite fails by fibre pull-out.

If, however, the fibre breaking strain is reached before this maximu m stress is developed,the fibres th at have no t slipped will break, an d this maximum is no t achieved. Th e strengthof the composite is then given by (7), where e x = qu.



1740 V,Laws

Th us there are tw o possibilities :

(i) If 1<21,,

($1.If, after the (static) interfacial bond T~ fails, there is a sliding or ploughing frictional

force between fibre and m atrix described by a bond strength T d , the average stress supportedby the fibres that have slipped at a crack extension eZ is

where i s the average length of the fibre ends that have slipped; that is, i= 12/4.

The total average stress supported by the fibres at a n extension ex s then

where Zc‘=$lc(2- T ~ / T ~ )nd the efficiency factors (9) and (10) are modified accordingly:

The length efficiency factor for fibres satisfying the condition 12 2Ic’ varies between

1 -lc/21 and 1-lc/l depending on the ratio T d / T s . The bond strength measurements ofde Vekey and M ajum dar (1968) suggest that 5-4/78 for cem ents could be as large as # leadingto an efficiency facto r of 1 -2lC/3l.

Bortz (1969) measured the maximum bond strength by pull-out tests and reported that,for uncoated stainless steel fibres in a glass-ceramic matrix, the pull-out stress dropped tozero after the bon d was broken. Putting T d = 0 in (1 1) leads t o a leng th efficiency factor of1- c/Z and overcomes the large discrepancy between his theoretical and experimentalstrength results for uniaxially aligned composites. Using this facto r, it is unnece ssary toinclude the matrix strength contribution in the calculation, an assumption that it is difficultto accept at the high strains involved.

5. The orientation factor

Cox (1952) has derived expressions for the elastic moduli of mats of fibres, assunungtha t the fibres are stiff an d that they sup por t a load only in tension. The two cases ofinterest here are the plan ar m at a nd the solid mat, in which the fibres are arranged randomlyin two and three dimensions respectively. Cox has shown tha t the ratio of the specificYoung’s modulus of the mat to that of the fibre is Q for the planar random mat and +for the solid rand om m at ; the Poisson’s ratios of the mats a re Q and 4 espectively. Krenchel(1964) has likewise considered the deformation of the fibrous component, and has added amatrix contribution, assuming that both matrix and fibrous mat suffer equal extensionsan d th at Hooke’s law applies. H e showed that the Young’s modulus of the composite is



The eficiency of fibrous reinforcement of brittle matrices 1741

Th e efficiency fac tor d efined by (1) and derived from (12) varies with volume fibre fractionand with the modular ratio m (=Ef/E,). The departure from Q is, however, insignificantin most practical cases. If the Poisson’s ratio of the matrix is zero and the fibrous ma t isconstrained so that i t is subjected t o deformation only in the direction of the applied stress,the ‘Young’s modulus’ of the composite is simply

Ec=$Ef~f+m J “

an d the efficiency factor is obviously 9 .At m atrix failure, it is assumed th at a crack is formed perpendicular to the stress direction

and that the composite strength is determined by the strength of the released fibrous mat.The fibres will fail in sequence as they reach their breaking strain, beginning with thosealigned with the stress axis. It can be shown th at the maxim um stress th at the fibrousmat can support is reached when the fibres aligned with the stress axis reach their breakingstrain. Th e efficiency factors tha t apply a t com posite failure are therefore also + and 3for the unconstrained and the constrained cases respectively.

Krenchel derived an efficiency factor of Q for a composite containing fibres randomlydistributed in three dimensions, and constrained to prevent lateral contraction, but did not

consider the unconstraine d case . This follows simply froin Cox’s analysis, when a matrixcontribution is added in a manne r similar to th at used by Krenchel in the two-dimensionalcase, and is outlined below.

Consider a composite consisting of an (unconstrained) fibrous mat of volume fraction u f .

Let a stress uz be applied to the composite in the x direction. The strains eC, c y and eZ

of the matrix are assumed to be equal to those of the fibrous mat. Assuming that thematrix is iso tro pic an d the strains are small, the stresses in the matrix can be calculatedfrom th e generalized H ooke’s Law. Th e stresses induced by the strains e x , eZ / and eZ are

and similarly for a m y and u m z .

total stress in the x direction is ux and in the y and z directions is zero, so thatWriting k ’ = v m / ( l + vm ) (1 -2vm) and using the elastic constants derived by Cox, the

uz= k’Em{ ex ( 1- m) + vm ( ~y + e t ) }++Eftif(2+ Q U ++EZ)

O = k ’ E m { ~ y ( l - v m ) + v m ( e z + ~ z ) ) + + E f v f ( + ~ z +y + + e z ) .

Putting ey= eZ and solving for Ec (= uZ / ep ) nd v c (= - y/cZ), we have

2( 1 5 k ’ vm +mvf)’

15(15k‘+ 4”)’(1- m) ++mvf-

and

1 5 k ’ vm +mvfvc = *

15k‘+4mvf

The efficiency factor of orientation over the elastic range of the composite, derived from(13), is a function of volume fibre fraction and of modular ratio m . The departure fromQ is, however, insignificant in most practical cases. Since the composite ‘breaks’ when theextension of the mat is equal to the fibre breaking strain E f U , the efficiency factor of Q alsoapplies in the calculation of the strength of the composite. The factor is obviously + if themat is constrained to prevent lateral contraction.

6. The interaction of orientation and length efficiency factors

arranged randomly in two or three dimensions.We consider now the efficiency factor for strength of composites containing short fibres



1742 V.Laws

6 . 1 . Fibres random in tw o dimensions

extension of the fibre along its axis is related t o the extensions e x andConsider a fibre in a (two-dimensional) composite a t an angle 8 to the stress axis. The

by

Ef= ex cos2 8+ sin2 8.

It follows that, since the length efficiency factors all depend on the extension q, he ytherefore depend on the orientation 8. We consider here the combined orientation andlength efficiency factor for the strength of a two-dimensional rand om fibre composite whenlateral contraction is prevented (eY= 0) and the fibre length is finite.

After the matrix h as failed by the opening of a crack perpendicular t o the applied stress,it is assumed th at the extension of the fibrous mat d etermines the strength of the composite.

Fo r an extension of E$ of the fibrous mat held across the crack, the load supported by afibre at an angle 8 is

Lo=Ef €fa=Efa ex cos2 8.

The fibre will pull out from the matrix if its shorter embedded length is less than Z0/2, where

1eP=LeipT8.The average stress supp orted by all the fibres at 8 is

If there are n fibres per unit length in any direction uniformly separated, the number offibres at 8 crossing unit length of crack face is n cos 8. The stress contribution of thefibres a t 8 to the total stress in the x direction is therefore nGo cos2 8. The average stressof all fibres over all angles is then

On integration, this becomes

where IC = 2 ufua/prgThe behaviour then follows that of the aligned s ho rt fibre composite outlined in 54 above.

The composite fails when ex reaches its maximum, at an extension ( E & , % ~ < q u . Thereare two possibilities :

(i) Ifl<%,

and

(ii) If 1>$I , ,

Comparison of (14) and (15) with (9) and (10) respectively (length efficiency factors foraligned fibre composites) shows that the combined factor differs from that obtained fromthe two separate factors ( q= q0ql) . The difference is 20 % when I < 51,/3; when 125IC/3the difference depends on th e ratio Ic/l.

The theory can be modified to take into account a sliding or ploughing frictional bondstress 7 6 after the fibre starts to slip. Reasoning analogous to that outlined in $4 aboveleads to the two possibilities:



The eficiency of fibrous reinforcement of brittle matrices 1743

2D random

ill,

Figure 1. The total efficiency factor 17 as a function of the ratio of fibre length to critical fibrelength, when 7 d= T~ (broken curves) and 76=0 full curves).

The total efficiency factors as a function of the ratio of fibre length to critical fibre lengthdefined by (4) are plotted i n figure 1 for Td= 0 (full curves) and Td= ~ (brok en curves). Th edifference between th e tw o estimates is large at low values of l/lc,decreasing as l/lc ncreases.

6 . 2 . Fibres random in three dimensions

An analysis similar to that above, for a composite containing short fibres randomlydistributed in three dimensions, leads to the following expressions for the efficiency factorfor the breaking stress :

(i) If Z<a&',

These to tal efficiency factors are also shown in figure 1.

7. Conditions for reinforcement

Th e conditions fo r reinforcement of the elastic modulus and the strength of the compositefollow simply from equations (1) to (3) above, modified to take into account both fibre



1744 V.Laws

length and fibre orientation. These can be written

Ec = E m +Vf(7’EfE m )

a c u = 7UfUZ‘f

(16)

(1 7)(sC= o m ” + U f ( 7 ’ E f E m U - UmU)

where 7’ and 7 are the combined efficiency factors a t the elastic limit an d at failure of thefibrous mat released after the matrix has cracked.

Obviously the elastic modulus Ec is greater than E m , and the stress supported by thecomposite at the matrix failure strain, oc, s greater than amu provided that ~ ‘ E E >m , acondition that is independent of volume fibre fraction cf. Since the composite breaks at astress equal to or greater than ac, this is the only condition for reinforcement. However,if ( q ’ E f - E m ) is small, the im pro ve m en t given by (16) an d (17) is small. Au sef ul improve-ment in breaking strength will result only if, on matrix failure, the fibres can support astress suitably greater than a m u . The conditions for an improvement of x(with x > 1)

then is

a c U = ~ ~ f U ~ f 2 ~ ~ m U .

This condition is obviously dependent on v f ; f we put x= 1, it also leads to a minimumvolume fibre fraction necessary to maintain a breaking stress equal to that of the matrix,provided the condition q‘Ef>E m does not apply. This situation could arise, for example,for a three-dimensionally random fibre composite of low modular ratio Ef /Em.

The da ta qu oted by Ma jum da r and Ryde r (1970) have been used t o calculate the tensilestrength o f glass reinforced gypsum plaster as a func tion of volum e fibre fraction (figure 2).

I I I I

0.05 0.1 0.15

Vo lume fibre fract ion

The composite tensile strength as a function of volume fibre fraction when Ic/l = 2 .The broken lines refer to the case when T d= T~ the full lines when T d = 0.

Figure 2.

The ratio of fibre length to critical length l/lc s given as 1.95. Since no d ata is available onthe values to be assigned to the ratio T d / T s , two limits have been chosen, namely Td= 0 (fulllines) an d T d = T s (bro ken lines). The difference between the two estimates is large. Apartfrom the need to have measurements of both T d and T~ in order to test the theory, other

difficulties arise when glass reinforced gypsum plaster and cement composites are con-sidered. The c ritical length almost certainly varies with volume fibre fraction, sincedifficulties in compaction at higher volume fibre fractions lead to increasing porosity



The eflciency of Jibrous reinforcement of brittle matrices 1745

(Majumdar et al . 1968), and with fibre length, since increased separation of the fibres in thebundle might be expected as the fibre length is reduced. Experimental results (Ryd er1969) strongly suggest that both factors are operating.

8. Conclusions

Although the mechanism of fibrous reinforcement of brittle building materials is notyet fully understood, some estimate can be made of likely efficiency factors of length andorientation th at should be applied. In the 'elastic' region, defined as the region before thematrix failure strain is reached, the lengthefficiency factor fo r a n aligned sho rt fibre comp ositeis likely to be very nearly unity for practical composites, provided there is a frictional'bond' at the interface. The orientation factor in this region for a composite containingfibres randomly arranged in a plane is 5 (unconstrained) and + (constrained); for a three-dimensionally random arrangement of fibres the corresponding factors are & and +respectively. These values are app roxima te only, since the tota l efficiency fac tor a t thesmall strains involved depend s on fibre length a nd , if the co mpo site is free to move laterally,on the m odular ratio a nd volume fibre fraction.

If the composite strength is determined by the maximum stress supported by the fibrousmat held across a continuous crack perpendicular to the stress direction, efficiency factorsof strength can be derived. These are summarized in table 1 for the case where the fibrous

Table 1. Efficiency factors for strength restrained fibrous mat

Orienta tion Efficiency facto rContinuous Short fibres

fibres

Aligned 1 tlll'c/ (1< 21c ')1- c / l ( la 1c')

Random, % & l l l f c ( I< IC '>two-dimensional +(l- l ' C / l ) (13$1.')

Random, J i&#'C (1< 'r"c ')three-dimensional &(1- l ' C / l ) (1k +Qlc ')

where IC ' l c ( 2- d /Ts )

m at is constrained to prevent lateral movem ent. It is seen tha t the combined efficiencyfactor for strength is not simply the product of orientation and length efficiency factors ashas previously been assumed.

The formulae derived allow for the frictional force required to pull out a fibre from thematrix after it has begun to slide. I n the case of the aligned sho rt fibre com posite, theefficiency fac tors redu ce to 1/41, fo r 1<21, and 1- c/l for I > 21, when the sliding frictionalbond strength Td is zero ; and 1/21, for 1<1, and 1- ,/21 for 121, when the, frictional bo nd

strength T d is equal to the bond strength T~ operating before the fibre begins to move.The theory thus reduces to the previously applied factors in the limits. Th e differencebetween the two estimates depends on the ratio Ic/l an d ca n be large-for typical glassreinforced gypsum plaster composites with random planar fibre orientation and a fibrelength approximately twice the critical length, the difference is about 35 %.

Apparently no attention has hitherto been given to the role of the sliding frictional forcedu ring fibre pull-out on the strength efficiency factor of fibrous composites, a nd n o m easure-ments of the sliding frictional bond strength have been reported. This would app ea r tobe an important point, warranting further attention.

Acknowledgments

The w ork described in this paper is carried o ut as part of the research programme of theBuilding Research Station of the Department of the Environment and is published bypermission of the Director.



1746 V.Laws

References

BORTZ, . A., 1969, Structural Ceramic Composite Systems (Chicago: IIT Research Institute),

BORTZ,S. A., and BLUM, . L., 1968, Special Ceramics, 1967, ed. P. Popper (London: British

Cox, H. L., 1952, Br. J. appl. Phys., 3, 72-9.

DOW,N. F., 1963, General Electric C o. Report R63 SD 61, AD 414 673, pp. 1-42.KRENCHEL,., 1964, Fibre Reinforcement (Copenhagen: Akademisk Forlag).MAJUMDAR,. J., and RYDER,. F., 1970, Sci. Ceram., 5, 539-64.MAJUMDAR,. J., RYDER, . F., and RAYMENT,. L., 1968,J. mater. Sci., 3, 561-2.RYDER,. F., 1969,Properties of Glass Reinforced G ypsum as a M aterial, Building Research S tation

DE VEKEY, . C., and MAJUMDAR,. J., 1968, Mag. Concr. Res., 20, 229-34.

AD 692 149, pp. 1-62.

Ceramic Research Association), pp. 29-36.

Seminar, Notes No. 21 1/69.

the efficiency of fibrous reinforcement of brittle matrices

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