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Theoretical tensile behaviour of brittle matrix many-fibre-type composites
THEORETICAL TENSILE BEHAVIOUR OF BRITTLE MATRIX MANY-FIBRE-TYPE COMPOSITES
J. SPURR1ER*
Department of Civil Engineering, University College of Swansea, Swansea SA2 8PP (Great Britain)
SUMMA R Y Equations are derived from first principles to predict the load extension curve of a brittle matrix composite reinforced by continuous high modulus fibres of many different O'pes aligned in the direction of the tensile load. The equations deal with composite extension from the initial elastic behaviour, through matrix multiple cracking, to ultimate failure of the fibres, and incorporate an expression for the suppression of matrix cracking by the constraining effect of the fibres. Some numerical results which couM be obtained from conventionally reinforced cement and gypsum plaster have been substituted into the equations, and possible modifications to deal with debonding and premature failure of the fibres are indicated. Some similar equations are then derived for discontinuous different fibre reinforcement aligned parallel to the applied load, and the ensuing complications are discussed and simplified where possible.
INTRODUCTION A considerable amount of theoretical work has been published recently on the behaviour of fibre-reinforced composite materials under load, particularly with the advent of finite element computing methods. Most of the models considered assume some form of uniform distribution of fibres to obviate the need for elaborate statistical formulae and complicated equations, and this work is no exception. Normally, however, the modelled composites contain only one type of fibre, hence the results are not readily applicable to practical systems where significant * Present address: Department of Materials, Cranfield Institute of Technology, Cranfield, Beds. (Great Britain). 103 Fibre Science and Technology (9) (1976)--O Applied Science Publishers Ltd, England, 1976 Printed in Great Britain
104
s. SPURRIER
fluctuations can occur as a result of accidental damage to fibres during the manufacturing process and imperfect bonding along the interfaces, as well as imperfect distribution. Models for reinforcement by many different types of fibre are better equipped to deal with such variations. In this work, equations for the load extension curve of such a composite are derived, assuming perfectly elastic behaviour of both the brittle matrix and the fibres. The simplest model is one of a uniform matrix containing aligned continuous fibres, but even then a number of assumptions and simplifications have to be made about the load transferred between matrix and fibres. This model is discussed in detail, then the same techniques are applied to the far more complicated case of aligned discontinuous fibres, where further simplifications are necessary.
A L I G N E D C O N T I N U O U S DIFFERENT FIBRES
Initial elastic behaviour Consider N parallel fibres embedded in a brittle matrix of Young's modulus E,,. Let each fibre have a modulus E: and an individual cross-sectional area A: = 7tr:z. Since the fibres are continuous, their length may be taken as the length, l, of the specimen (or of the gauge length being investigated). Assuming completely elastic behaviour, the initial modulus of the composite, when loaded in tension parallel to the fibres, is given by the rule of mixtures formula 1 , 2 .
E 1
+ A,
As
~u E:~r: 2l vs
Em(Vs-~u ~r:21) +
vs
(1)
where the volume of the specimen under test Vs = A s. l, and the volume of matrix V,, = (As - EN A:) • 1. The initial tensile behaviour of the specimen, when the whole composite obeys Hooke's Law, is thus given by the relationship: Lo.1 e°" =
l
A--7"E--71
(2)
where e o. 1 is the total (gauge-length) extension produced by the external application of a load L o. 1In the derivation of the formulae for the load extension curve beyond this initial behaviour, it is necessary to estimate what proportion of the total applied load is
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
105
being carried by each fibre, and by the matrix. One of the most convenient ways of doing this is to assume that the proportion of load carried by a particular phase, b, in relation to that carried by the total load-bearing phases, B, is:
EbAb
(3)
Hence the strain of this particular phase at an applied load L o. ~ is: Stress Lo.1 [ EbAb ] Strain=--~--b = ~ "[(~EA)J'~
1
= constant for load Lo.t. This method of dividing the load between the phases is thus equivalent to assuming that the cross-sectional extension is planar.
First crack Provided that both the matrix and fibres are uniform, the first crack across the matrix will occur at any point on the gauge length, when the matrix stress reaches its ultimate value a,.u. Let the load then be L~. Just before this crack forms, all the phases present are bearing load, hence:
This assumes, of course, that the reinforcing fibres all have a greater strain to failure than the brittle matrix: this is normally a valid assumption for practical systems where the matrix is being reinforced by relatively low amounts of fibre, and especially true for cementitious materials. Hence:
Lt EmAm ~mu =
- Am
"
EIAs
where A . = area of matrix in a cross-section. Thus:
trmE1As L 1 =
-
E.
-
t [
~rm
(4)
106
J. SPURRIER
Constraint effects Romualdi a and Romualdi and Batson 4 have pointed out two mechanisms by which the presence of fibres can affect the matrix cracking process. One of these deals with the uniform stripping of wires from the matrix following debonding. It is therefore more applicable to discontinuous fibres. Debonding of the continuous fibres is not considered here. The other mechanism, however, is concerned with the pinching effect exerted by fibres perpendicular to the plane of the crack, especially during the early stages of crack growth• This mechanism is particularly applicable to cementitious matrices, which exhibit controlled flaw and micro-crack growth just before the matrix fails, 5 and is analogous to the constraint effect of Cooper and Sillwood, 6 which predicts that the failure strain of the matrix will be:
when fibres are present. This expression is derived from a consideration of the energy-absorbing processes during crack propagation. Assuming that the surface energy per unit area of the matrix, 7,,, does not change, the strain to failure for a given volume fraction of fibres may be increased by using finer reinforcement, which implies that the fibres are more closely spaced. When more than one type of fibre is present, this expression must be modified to allow for variations in the value of r r, the limiting shear stress for each fibre at the fibre-matrix interface. For a total number of N fibres:
This approximation is derived by substituting for zs using the transfer length x z (see below), summing over N fibres, and replacing Xz by an expression for the mean transfer length 2z = (E N xs)/(N).
Transfer length From eqn. (3), the extra load carried by a particular fibre at the site of the crack when the matrix fails is given by: •
~muAm
~ EsAs N
By equating forces, this can be related to the distance along the fibre x s that is required for this extra load to be transmitted back to the fibre by limiting shear 7 :
107
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
a,,uAm
~ f A f = 27rrfTfXf EsAs N
This equation assumes that the force transferred per unit length is a constant. Several other expressions with more complicated forms have been derived, 8-11 usually by considering the load transfer for discontinuous fibres. Any such expressions may be substituted in the following formulae provided that the proportion of the load transferred at any distance from the crack can be calculated. The linear relationship for load transfer (Fig. 1) has been used here to simplify the
I I
LOAD ~ CARRIED| ACROSS| MATRIXJ~f CRACK -
-
,, ~ , ~
MATRIX SHARING LOAD FIBRE ~SHARING LOAD
-
"]- MATRIX ~LOAD
I I
~ FIBRE LOAD
I
.U
_
_
MATRIX CRACK
Fig. 1. A diagram to illustrate linear load transfer in the region of a matrix crack for a single continuous fibre loaded in tension. mathematical expressions. The proportion of the load recovered by the matrix at a distance x from a crack is then (x/x:) for 0 <_ x < x: where x: is the transfer length of the fibre. At load L1 when the matrix cracks, let the transfer length of each fibre be x:l.
x:l =
a.~ v.E:r:
(6)
2zyl ( ~ E:nr: 2)
The values of x:~ will be different for different fibre types. If the applied load is altered, the transfer distance for a particular fibre type will also change. The linear relationship adopted gives the simple expression:
L2 x:2 = -~l "x:l
(7)
The maximum value of x:l will be associated with the fibre-type which produces the maximum value for the expression
108
J. SPURRIER
Efrf "Cf /
and this distance will control the positions of any further matrix cracks.
Matrix multiple cracking--block formation At distances greater than xil~m~ I on either side of the first crack, the whole cross-section of the specimen is still loaded in the same manner as it was before the first crack formed, provided, of course, that the fibres bridging the matrix crack are sufficiently strong to carry all the load (i.e. EN as,nrs z > LI where Cry,, is the ultimate stress of the fibre). If the gauge length of the specimen is sufficiently long, multiple cracking of the matrix will occur, without any change of load if the specimen is ideally uniform, resulting in the matrix being divided into blocks by cracks crossing the specimen perpendicular to the fibres. Since the matrix is capable of cracking at load L 1 at any distance greater than Xy~lmax~from existing cracks, it follows that the resulting block size will be between xll (max)and 2xfl (max," The exact number of cracks which form in the gauge length will evidently affect the overall extension of the specimen. Since this number is open to statistical fluctuation, the best approximation is to assume that the cracks are regularly spaced, dividing the matrix into blocks of the most probable length, ~2 .f, where : .~xi~lm~ I (8) The second series of matrix cracks will form across the centre of these primary blocks, since it is here that most of the load previously carried by the matrix has been transferred back from the fibres by interfacial shear forces. 8'9 However, before this secondary cracking can take place, more load has to be applied externally to the specimen to raise the average stress of the matrix to a,,,, over the whole cross-section at these points. This will also change the values of the transfer lengths of the fibres, in accordance with eqn. (7). Let the load required for secondary cracking be L2. Matrix multiple cracking--load calculation In order to calculate how much load the matrix is carrying at any point, it is necessary to assume that load transfer takes place as quickly as possible, and to designate a certain cross-sectional area of matrix as being controlled by each of the fibres. In positions remote from matrix cracks, the load carried by each fibre for an externally applied load, L: Lf(s) = L • EyAs EIAs and the load carried by the whole matrix equals: E,,A,,
L . - -
EiAs
109
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
It is convenient to designate the area o f matrix associated with each fibre as:
.4.. E~a~
EsAs
N
such that the load carried by that part of the matrix associated with a particular fibre at load L 2, when transfer of load between t h e t w o is complete: Lm(s) 2 = L 2 • EmAm
(9)
EsAs
EIAs E EfAf N
PRIMARY
SITE OF SECONDARY RACK
CRACK
<~ LOAD L2
I
PRIMARY CRACK
II
I[
II
II
,
II
I!
>
- LOAD L2 _
t4~-X ---*
LB. ii MATRIX L ~ LOAD
\
Lm(s)2
N - T2 FIBRE
x~2 ~
z4
Xt2
MATRIX LOAD Lm(s)2
T2 FIBRE
x~2 >
zc/z
Fig. 2. A d i a g r a m to illustrate the load transferred to the matrix at a distance, x, from a primary matrix crack w h e n the specimen is carrying a load, L2, sufficient for secondary matrix cracking to occur, for both non-transferring a n d transferring fibre-types.
110
J. SPURRIER
However, the secondary cracks, which form at load L2, have predetermined sites at distances ½2 from the primary cracks. If x:2, the fibre transfer length at load L2, is less than this distance, the matrix associated with that fibre will be fully loaded at the site of the secondary crack. Fibres in this situation are classified as N-T (not transferring) at x = ½~?. If xf2 is greater than the distance from a crack to the centre of the primary block, these fibres are still transferring load to the matrix at this point and are grouped as T (transferring) fibres. (See Fig. 2.) This classification has to be made at all values of x, the distance from the nearest crack, hence the number of fibres in each of these groups depends on this distance. When x = 0, all fibres will be in category T. For a specimen with N fibres, let T z be transferring at a distance, x, from the primary cracks• Then the load transmitted back to their associated areas of matrix by these fibres is:
(~
E,,A,,
EfAf
x
/_, L 2 . - EIAs X~ EyA: x:2 T,2 N
Hence the total transmitted back to the matrix at a distance x from a crack
~")
E,.Am
E:A~
~
E,,Ar,.
E:A:
x
N-T,2 N
N
LzEraAm • e:A: + ~ E:A: "' EtA~~ EfAf (N2 T.2 T,2
(lo)
N
But at load L2, by definition, the maximum load transmitted back to the matrix must be just enough to crack it, and this occurs when x = ½~. Hence:
L2Em • El A~ ~, E:A: N
E:A: + ~ E:A: 7",2
= (a,..
[no constraints] or constraints]
\ E,,e,,p [with
(ll)
This equation, together with eqn. (7), form a simultaneous pair which may be solved for the load L2 required for secondary cracking. The solution, however, has to be found by iteration, since some fibres may pass from category N-T into category T as the load is increased from L~, as a result of the increase in transfer length with load. These fibres obey 1 > x:l/½.;c > L1/L2. A first estimate for L2, which will be lower than the true value, may be obtained by solving the equation with only Tz fibres transferring. Any fibres which obey the above inequality, when this estimate of L2 is used, are included with the transferring
TENSILE BEHA,VIOUR OF MANY-FIBRE-TYPE COMPOSITES
111
fibres, and a second estimate made. As soon as no additional fibres obey the revised inequality, the true value of L 2 has been found and all the transferring fibres have been correctly assigned to category T2.
Matrix multiple cracking--load extension curve The initial linear load extension curve before the matrix cracks is given by eqn. (2). Once cracking has occurred, the overall extension is no longer linear. The series of primary cracks, and each subsequent series of cracks, form without change of load in a uniform matrix, and hence produce horizontal plateaux on the load extension curve. An equation for the curve between these discontinuities is required.
vr- $~c N-T N-T
F't'M
LOAD L1. 2
F
N-T
= " N O N - TRANSFERRING"FIBRE
T _-"TRANSFERRING" F+M
II
=
TYPE.
FIBRE TYPE.
FIBRE AND ASSOCIATED MATRIX CONTRIBUTE MODULUS E(~)1"2
F = FIBRES ONLY
LOAD L1.2
T
CONTRIBUTE
TO
TO MODULUS.
Fig. 3. A diagram to illustrate the method of assessing the Young's modulus at a distance, x, from a matrix crack for an applied load, L I.2. The shaded portions represent the transfer lengths xj 1.2 of the various fibre types.
Consider a thin cross-sectional element of thickness 3x at a distance x from a crack (see Fig. 3). Let the external load be Lt. 2, such that L1 -< L~.2 < L2. Then the operating transfer length for each of the fibres is xii.2 where: L1.2 Xfl.
2 ~
L1
" Xfl
In general, some fibres in this element will have completed the transfer of load to their associated matrix (N-T) and the remainder (T) will be still transferring
112
J. SPURRIER
the excess load. Because of the assumed linear load transfer along the fibre, the matrix associated with these transferring fibres in the elemental cross-section will not be able to carry any additional load, if extra load is applied externally. The total load borne by the fibre-matrix pair is increased by such an application, and the transfer distance correspondingly lengthens, but these changes affect the matrix only at distances greater than x s. The effective Young's modulus of the element E(x)l.2 is thus given by: E(x),'2 = E
E:A:
(~) E,.A,.
--~- +
~
N
As
EIA:
(12)
~ EfAf
N - - TI. 2
N
and the variable load carried by the element at position x is:
(x~ E,,,A,, L(x)I.
2 =
L1. 2
1 -
.] E EsAs xsl,2 EIA:
'
TI.2 E1As
-
x
-
(13)
N
The extension of this element is thus: L(xh. •
.4sE(x) l. :
6x
The total extension of the specimen may then be obtained by summing for all the elements contained within half of one of the primary matrix blocks, and multiplying by 21/~ which is the number of half-blocks in the gauge length. The summation must be performed in two stages because, at the fixed load, L~.2, some of the fibres will change from category T to category N-T as the position of the element moves towards the centre of the block. This will affect the summations in both eqns. (12) and (13). If the fibre types are arranged in ascending order of transfer length xzl.z, the contents of the sums in these equations will change by a fibre type every time the position of the element, x, increases beyond one of the values xyl.2 (up to the centre of the block, when x = ½~?). Hence the total extension of the specimen at load L~.2(L~ <_ L~.2 < L2) is given by:
21 E el2
=
X (Axf1.2 )
((xfl'2)n-+tL(x)I2
J(X]l.2)n
"dx
(14)
msE(X)l.2
where the integral is taken over successive steps Axle. 2 as x increases from zero to ½~, and these steps are then summed for the final result. This equation gives the extension of the specimen for an externally applied load when the matrix is subdivided only by the primary cracks, such that the block size is ~.
113
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
LOAD
- COMPOSITE FAILURE
-T
PzPu~"
--
GENERAL RANGE
Pp.
~1
Po -
i _
_
Lend
R
Lo+I
L=
L1
I
---
_-
I 11 III II II
I I
li ',
GENERAL RANGE
I I I t
I J
I
L
f
I I EXTENSION DUE TO ~e'l~I III PRIMARY MATRIX I CRACKING.
i
r;
__E
i I
f ,i
_i___
I
--.~" LO'I
-- L-1
I
i I
i
el.2 RANGE
I I
GENERAL RANGE
e~.~÷ 1 Fig. 4.
RANGE
EXTENSION
"~1.2 RANGE GENERAL RANGE ~P'P
÷1
General load extension curve for a brittle matrix reinforced by aligned c o n t i n u o u s different fibres which remain perfectly elastic.
114
J. SPURRIER
The model indicates that there will be further cracking of the matrix until the external load is such that all the fibres are transferring excess load to the matrix along their entire length. When all the fibres are thus in category T, let the load be Lend. The following general formulae assume that the load at which the fibres start to fail, P1, is greater than Le. d. The generalised equations for the loads Lq+ 1 at which matrix multiple cracking occurs when constraint effects are present are:
Lq+ 1 EIA~ N EfA:
[_N r.+,
E:A:
E,:.,
E: :
(15)
rq+ I
where:
Lq+ 1 Lq+ 1 " E,,A,,, • E / A f x:q+l = L~ " x:l = 2rrryz:EtA ~ ~ EfAf N
(16)
and other terms are defined by eqns. (1), (4), (5), (6) and (8). These expressions are valid for q > 0 provided that Lq+l does not exceed Le, d, when all fibres go into category T. If constraint effects are ignored the right-hand side of eqn. (15) is replaced by a,, u. The general formulae describing the load extension curve between these loads (which are marked by horizontal regions where the extension increases at constant load due to matrix cracking on the general diagram, Fig. 4) becomes:
eq.q+ 1 --
1" 2 q .~
~(Xfqq +l)n4-1 L(x)q.q+ 1 ~ "..l(Xfq.q+1)n AsE(x)q'q+ dx (Axfq.q+ l) 1
(17)
valid f o r q _ > O,n_> 0 where"
L(x)q.q+ 1 = Lq.q+ 1
F•.Am E:A:
x ]
(18)
1 + Tq.q+I E1As " ~ EfAf Xfq'q+l N
and:
E:A: E(X)q'q+l = ~-'-~'-s
(x) EmAm. E:ay N- r~aq+l A~ ~ E f A f N
(19)
where the variable load Zq.q+1 is such that Lq < Lq.q+l,
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
1!5
Fibre extension Provided that the fibres are sufficiently strong to carry the entire load in the region of the cracks, multiple cracking of the matrix will occur in accordance with the above equations until all the fibres are transferring load back to the matrix even at the centre of the existing matrix blocks. This load is defined as Lend, and marks the point above which the load bearing capacity of the matrix is fixed. The load extension curve above this point is linear and depends on the moduli of the fibres.
_ _.
:::
LOAD
......
II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II ...... : : :
~
L~ AFTER
I
i
I
:i__ ................................... II ...........
i!
___
II . . . .
II
.
.
.
.
.
.
:::
oO.no
CASE (o) ,q
q,
I I I
I I I
LOAD
Lq.+ I CRACKING
..........................
<7:= i
I!....... ___TTT
!
,'~x~.q.+m.i.--)' ) LOAD
L~÷I AF'~R
II
CRACKING I. e n d
,
g
II
P
II
"-;
' :)Cj'q'+Itmin) '
CASE
(b)
Fig. 5. A diagram to show the two ways in which Lend is evolved: (a) by an increase in load; (b) by matrix cracking.
Whereas the initial size of the matrix blocks depended on the maximum transfer length X/l(ma.), the load Lend depends on the minimum value xIl(ml, ), since this fibre type will be the last to pass into category T. The two possible ways in which this change can take place are illustrated in Fig. 5. In the first diagram, which shows the last fibre type still not transferring, the transfer lengths (taken from each end of a matrix block) nearly meet after cracking at load L~ has occurred, and eventually meet (at Le.d) before the load across the matrix at the centre of the block is sufficient for further cracking to take place. In the second diagram, it is
116
J. SPURR1ER
the formation of a crack, and not an increase in the load, which causes the last fibre-type to become totally transferring. Lendis then taken as the load at which cracking occurred. In order to determine which of these possibilities occurs, and hence the value of Lend, use is made of the load at which the transfer length for the final fibre type would be the same as half the length of the matrix blocks: -'~
L1
(20)
Z(tryJq+l = 2q+ I " Xfl(min)
where q > 0. Let Q be the lowest value of q to make either of these inequalities true: L{try)q+l --- L q + l
L Q + 1 = Len d
~
Lq+l < L(trylq+l ----- Lq+2 ~ L ( t r y ) O + l = Le, d (21) The former inequality implies that cracking of the matrix, and not loading of the specimen, caused the final fibre-type to become transferring.
Fibrefailure Above Le,d, all additional load applied to the specimen is borne by the fibres only. Provided that the fibres are uniform, they will fail in one of the matrix cracks, where their load is greatest. Once one fibre has failed in a crack, that crack will then be the site of failure of all the other fibres. In these idealised equations, it is assumed that once a fibre breaks, it and the matrix associated with it become redundant. In order to distinguish the equations from those where the matrix plays an active part, the letters P, M and h are used for toad, Young's modulus of a thin section and extension, respectively. Let the fibres start to fail at load P~. Then the maximum load on each fibre is across a matrix crack and has a value
E:A: P1 ~ EfAf N
The stress in one of the fibre types is then just sufficient to fracture it, hence:
l(min)
N
where (trf/Ef)l~min)is the minimum value of (a:/E:) for all the fibre types. When this fibre type breaks, its share of the load has to be carried across the matrix cracks by the other fibres. It is possible, therefore, for some, or all, of the other fibres to break when this additional load comes on them. The general formula for the loads at which fibre groups fail is thus:
TENSILE BEHAV1OUR OF MANY-FIBRE-TYPE COMPOSITES
PP=(aE'~iS)p(mi,)
~w EsAf
forp>
1
117
(23)
where (aI/EI)p(mi,) is the minimum value of (ai/E I) for the Wp fibres remaining whole up to load Pp and N -- W1. To determine which fibres fall into category Wp+ 1, i.e. those which stay whole despite the extra load thrown onto them when one or more fibre types fail at load Pp, it is convenient to arrange the fibre types in ascending order of (ay/Ey). Consider the load carried by the second weakest of the fibres in group Wp (i.e. the weakest of the remaining Wp.p+ 1 fibres) when the weakest fibre in group Wp fails. If: PP >-- ~
p.p+l(min) W p p + 1
then this second weakest fibre will also fail, hence it may no longer be considered in the Wp.,+ 1 summation. Therefore if any of the fibres are to remain intact above load Pp, it will be possible to find a solution to the inequality: O'f Wp
(n -- 1)(rain)
where the fibres are arranged in ascending order of value (ariEs) and the nth in this order is the first to obey the inequality. This fibre, and those above it in the series, are the fibres which remain whole up to load Pp+ 1. The composite ultimately fails at load Pu, when no fibres are left intact for group 14I,+1. The load extension curve above Le.d is calculated on a similar principle to that of matrix multiple cracking. It is simplified by the fact that further subdivision of the matrix does not occur, the block size being that remaining after the cracking at load Lo+ t (-----Lend) is complete. Also, the matrix associated with a particular fibre retains a certain amount of load if the fibre is whole, instead of a certain fraction of the load. The effective modulus of a thin section of the specimen is thus constant, and equal to the modulus of the whole fibres only. The general formula is thus:
2°+ll.I(~/2°+l) P(x)p.V+ldx -x AsMp+ 1
hp.p+ 1 = - -
(25)
valid f o r p > 0 where:
P(x)v.v+ 1 = Pv.v+ 1
-
-
~r,..A,. Wp+l ~
EsA s x ] EIAI xft
N
and:
(26)
118
J. SPURRIER
Mp+ 1 = 2_,
EIAs
(27)
Wp+l As
where the variable load Pp.p+l is such that: Pp -< Pp.p+l < Pp+l and : Po = Le.d Because of the simple form of eqn. (27) compared with eqn. (19), eqn. (25) may be integrated to give: [
a,,,Aml
Pp'p+ 1l
_ hpp+ 1 = l ~-' "Wp+ 1
~ __.
~EIAs/xsl WpM. _ _
] .[
E+A+ N
(28)
Wp+ 1
A general form of the complete load extension curve is given in Fig. 4.
Modifications Throughout these equations, it has been assumed that matrix cracking has been completed and all the fibres have passed into the transferring category before any fibres fail. Such is normally the case for brittle matrices such as cement and plaster conventionally reinforced by steel wires and glass fibres. Even if this assumption is not valid, the equations, in a modified form, may still be applied if a broken fibre is considered as being completely redundant. All that is required then is a redefinition of the specimen every time a fibre breaks, since the volume of matrix remains the same although the number of fibres falls. The failure of a fibre will obviously cause an overall extension of the specimen without change in load, in the same way as the formation of a matrix crack, The load carried by the fibre will be transferred to the rest of the specimen, and some of this will be taken by the matrix if the load is below Le,~. This additional load may be sufficient to cause simultaneous matrix cracking. The formulation of the equations also assumes that debonding does not occur. The inclusion of debonding terms into the equations is theoretically fairly simple, but the resulting equations are long and tedious. I f a suitable expression is available to describe the variation in debonded length, dI, with the load carried by the fibre, and it is assumed that the debonded region of a fibre transmits no load back to the matrix, then the incorporation of debonding into the equations is equivalent to moving the origin for load transfer from the crack at x = 0 to x = d I. The inclusion of debonding implies that less load is being transferred to the matrix blocks, hence the fibres extend more. Friction over the debonded lengths can also be modelled by assigning the frictional load to the matrix over these regions. This
1 19
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
produces a reduction in the 'active' loads L(x) or P(x) over the debonded region of x, resulting in smaller extensions of the specimen than when friction is ignored. NUMERICAL SOLUTIONS
General data Numerical values 13-15 have been substituted into the equations to represent cement or gypsum plaster matrices reinforced by low volume fractions of conLOAD (NEWTONS)
-%
X
X
16,00(
WITH CONSTRAINTS /
12,000
/
/
/ /
EXCLUDING CONSTRAINTS /
/
8.000 /
//
,/
/
//
I-- _1-- - - J
4,000
t I
EXTEN$1ON(x lOAm) BEFORE AFTER 0,16 0.32 0'36 0-53 0.59 0 '76 6,41
8"03
15.73
(N)
LOAD (N) EXTENSION(x 104r~ AT CRACK BEFORE AFTER 0'6/. 1.26 1026 L1 4049 1111 1-42 2.0"7 L2 4387 119"7 L3 = L u 4717 2 "34 3"01 8482 7-46 P1 6462 5"92 16422
Pz = Pu 16421
15 15
EXCLUDING CONSTRAINTS .L iNCLUDING CONSTRAINTS
MATRIX ONLY FAILURE LOAD
I z, x 10-4
I 8 x 10-4
I 12 x 10-4
,
I 16x 10-/.
EXTENSION (METRES)
Fig. 6.
Load extension curves computed for two types of glass fibres in gypsum plaster, showing the effect of including the constraining influence of the fibres.
120
J. SPURRIER
tinuous glass fibres and steel wires, and the equations solved with the aid of a suitable computer program. 15 The figures used in the calculations are given below, and the resulting load extension curves are illustrated in Figs. 6-8. Both cement and gypsum plaster have similar moduli and tensile strengths, and so average values were used to represent both matrices and so simplify comparison between the inclusion of different types of fibres. Cylindrical specimens were considered, the dimensions being kept constant throughout.
LOAD (NEWTONS)
1,60(
1.200" -LQ+ 1 -L 2 -L1 _
800
f
f
/
_. _AT_RJ_X
0 NL¥
I
Fig. 7.
J
l
l
I
0.4 x 10-~' EXTENSION (METRES)
I
!
0.Sx 10.4-
Matrix multiple cracking for two types of glass fibres in gypsum plaster, ignoring constraint effects. (Enlarged section of Fig. 6.)
121
TENSILE BEI-IAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
Radius of specimen (A, = 7~Rs2) Length of specimen Ultimate tensile strength of matrix Young's modulus of matrix Fracture surface energy
R, = 10 -2 m l= 10-'m ainu = 2"75 x 1 0 6 N / m E . = 17 x 1 0 9 N / m 2 )'m = 6"25 J/m 2
2
Two types of glass fibres in gypsum plaster The figures used in this section represent a 5.4 per cent addition of aligned continuous glass fibres, a number of which have been damaged by handling, resulting in a lower ultimate tensile strength and significantly weaker bond strength with the matrix. 12 x 104 3 x 104 Number of fibres N I 76 x 1 0 9 76 × 1 0 9 Young's modulus E s (N/m 2) 6 x 10 - 6 6 x 10 -6 Radius r s (m) 5"52 x 1 0 6 1-5 x 106 Interfacial bond strength Ts (N/m 2) 1210 x 106 500 × 106 Ultimate tensile strength try (N/m 2) The computed results are shown in Fig. 6. Two curves are plotted to show the effect of constraints exerted on the matrix cracking by the fibres. The difference is obtained simply by replacing the constrained matrix failure stress Emempin eqn. (15) by the normal value ainu, and may be explained 4 in terms of forces parallel to the fibres opposing the opening of matrix cracks associated with crack propagation. The section of one of the curves dealing with matrix multiple cracking is enlarged in Fig. 7. It should be noted that, in this and all other matrix cracking regions, but not in other parts of the graphs, the lines joining the horizontal regions where cracks propagate are not linear, but have gradients which decrease slightly with increasing load (i.e. at a general load L~.q+ 1 where L, < Lq.q+ 1 < Lc,d). This is a result of assuming that the transfer lengths of all the fibres increase with load, whereas the matrix block size remains constant. With this arrangement o f data, LQ+, =- L,,d, hence the load extension relationships above LQ+ 1 are entirely linear except where the damaged fibres fail. The undamaged fibres are sufficiently strong and sufficiently numerous to take the excess load thrown onto them when this occurs. Steel fibres in cement Two sets of figures were used, both representing 4 per cent of continuous steel wire reinforcement. Constraint effects were not considered because of the coarse nature of the reinforcing wires. Number of fibres Nj, 64 Young's modulus Ej- (N/m 2) 2.0 x 101' Radius r s (m) 2.5 x 10-4 Interfacial bond strength TI (N/m 2) 11.04 x 106 Ultimate tensile strength trf (N/m 2) 2.1 × 1 0 9
122
J. SPURRIER
LOAD (NEWTONS) FOUR FiE RE TYPES LOAD (N.) EXTENSION(x 104rn) AT CRACKi BEFORE AFTER L1 1236 0-16 0.23 L2 1373 0"25 0-33 L3=L u 1511 0"37 0"48 P1 Pu 26389 10.3e
2500
pu : 26389
~-_~0~ ~"%
2000
~ . ~ " .....~i CURVE FOR FOUR FIBRE T~PES
1500
~
.
~
_
_
J"
/ dp/
_
...
_
_
~ ' ~ ~.,
_
~
~"
~"
~"
CURVE FOR ONE FIBRE TYPE
1000
500
_
= 26388 ~_. :1o`3, f~u = 10 '39
LoAQN~N~ ' BREExT~;sEION( x 10'~'m' AT CRACK L 1 =L u 1236 P1 =Pu 26389 I 0.2 x I0 -4
BEFORE AFTER 0.16 0'38 10 39
I 0.4 x I0 -4 EXTENSION (METRES)
I 0.6 x 10-4
ll-
Fig. 8. Matrix multiple cracking for 4 per cent of continuous steel wire reinforcement in cement. Four values for the interfacial bond strength were used in the upper curve, a single higher value for the lower curve. The second set o f figures split the fibres into four groups, with N s = 16 fibres in each group, with differences only in the interracial bond strengths, r I, which were ascribed four values: Zf = 1"5 × 106; 5'52 x 106; 7"75 × 106; 10'5 × 106 N / m 2 These curves, which are shown in Fig, 8, are basically similar except for the number o f loads at which the matrix cracks. The final size o f the matrix blocks, however, will be similar for the two curves since xI~(,~,) is approximately the same, and cracking continues until this fibre type is in category T. The general effect o f increasing the load carrying capacity o f the fibres with respect to the matrix--either by increasing the number o f fibres or increasing their Young's moduli--is to reproduce a similarly shaped curve at a higher load for the same extension. The extension when matrix cracking ceases will be slightly reduced because there will be more fibres to carry the applied load across the
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
123
matrix cracks, but the ultimate extension of the specimen will be slightly greater because the proportion of the total load removed from the fibres by the fully cracked matrix will be less.
ALIGNED DISCONTINUOUS DIFFERENT FIBRES
lnitial behaviour and transfer lengths Consider N discontinuous fibres in a specimen, all aligned parallel to the tensile axis. It is assumed that the distribution of each type of fibre is uniform, and that each fibre type is present in sufficient quantities for the chance 2 s of finding a fibre in a particular cross-section to be the same as the fraction of the number of fibres of that type to be found in any cross-section. 2s =/s/ts (29) where ls = length of discontinuous fibre; ls = specimen gauge length. A second useful parameter to define is:
Ks'=
2xs'
Is
(30)
which, for Ks -< 1, may be considered as the probability of being within that distance of a fibre end where the load is being transferred. Here, xsl is again a transfer length corresponding to an externally applied load, L~, but it has a different definition from the transfer length used with continuous fibres. In this section, it is the minimum distance from the end of the fibre, over which the fibre can gain all its share of the load from the uncracked matrix. Ks is thus a function of load, such that: Lq+ 1 Kfq + 1 =
L~ " XS~
(31)
since: Lq.+,
Lq " Xsq
Xfq + 1
(32)
For a general initial load Lo. , (insufficient to crack the matrix) applied to the specimen, a particular fibre and its associated matrix carry a load: 2sEsrtrs 2 Lo. t ~-,
/_._, 2:Eyrtr: z N
Let the load carried by the fibre be Ly(s) and the load carried by the associated matrix be Lm(sr If we assume that the area of matrix controlled by a particular fibre is:
124
J. SPURRIER
A.,2fEfrrr f z
•
:.sE:rs 2
N
then:
EmA,,
L/(s)O.l+
Lm(s)O.l =
_ Lo.1 2sE:rcr: 2
L:(~)o.1 1 + E )./Eyrir 2 N
E 2:E/rrr:2 N
if the ratio of matrix load to controlling fibre load is to be a constant when transfer of load between the two is complete. Hence: L o. 12sEsnr: 2 (33) Lf(s)°'l "= ~ , 2:Eynr:2 + E,.A., N
and:
Zra(s)O'l
= ~
L°'IEmA" • 2:Eyrir:2 2:E:nr: 2 + EmAm ~ 2:E/rrrs 2
N
(34)
N
(Note: these equations are written in such a way that the summation to express the behaviour of the composite is always over N fibres. Strictly speaking, at a chosen cross-section, a fibre which is known to be in that cross-section will carry a load: L o. iEyr~r:2 L ~/($)o.I
).:EfTtr:2 + EmAm N
and the composite behaviour is expressed by summing over the 2:N: fibres of each type in that cross-section.) If a particular fibre is known to be included in the chosen cross-section, its transfer length is related to the applied load by the equation:
Lo.1 "Eyrir:2 = 2~ryzyX:o.x E 2:E:nr: + EmA,,, N
i.e. X:o.l =
L°'lE:r¢
(35)
2z: ( ~ 2:E:A: + E,.A,.) It is also convenient to define a second transfer length x:~. 1 as the minimum distance along a fibre over which a load, equal to that share of an applied load, L 1, borne by both the fibre and its associated matrix, can be transmitted (see Fig. 9).
TENSILE
BEHAVIOUR
OF MANY-FIBRE-TYPE
125
COMPOSITES
FIBRE LOAD
LOAD L1
<
i
I
I~ztl" BEFORE
~
LOAD L1
CRACKING
FIBRE LOAD
I
I~
.MATRIX CRACK
s)1'I
L•.
/
.L~..(. $)1~ ¢
•"
LOAD <
=
"',
I
L1 ~-1r'1.1.1
AFTER
I
>LOAD L1
..-
I
CRACKING
Fig. 9. A diagram to illustrate the transfer lengths, x : 1 and x : ]. l on a discontinuous fibre which is long enough, and correctly positioned, to bridge the matrix crack when it forms.
First crack By virtue o f the definition o f 2/, the total load carried by a certain type o f fibre at a particular cross-section is equal to the average load carried by that type o f fibre, times the number o f fibres in the cross-section. The two possible situations which arise, depending on whether Z:o.~ is or is not greater than unity for an applied load, Lo.~, are shown in Fig. 10. When Z/O.l<- 1 mean load o f fibre = L:(s)o.l( 1 - ½Zro.1) m e a n load o f associated matrix = Lm(s)o.~ +
½)Cfo.ILf(s)O-I
126
J. SPURRIER FIBRE LOAD
LOAD
LOAD
LO.1
[ ''>
LO.1
:zl0.~' FOR
X.~.o. ~ < 1
FIBRE LOAD
[ LI"(s)0.1 .~
LOAD
LO. I
,
,
OAD >
LO.1
"r:ro. ~ FOR 7-.~0.~ > 1 Fig. 10.
A diagram to illustrate the significance of the value of Z on the load carried by a d i s c o n t i n u o u s fibre in an uncracked matrix.
When •fo.1 > 1, the m a x i m u m sharing load o f the fibre L:(s)O. ~ is not reached. Provided that there is no debonding from the ends o f the fibre, the m a x i m u m load, at the centre o f the discontinuous fibre _
Lf(s)O.1
Z.ro.1 Hence: mean load o f fibre _
Z f ( s ) o.1
2Z: o. 1 mean load o f associated matrix
TENSILE BEHAVIOUROF MANY-FIBRE-TYPECOMPOSITES
= Lmo)o.l + Lf(s)O.l
(
1
127
l)
2Z~0.1
The total matrix load over the whole cross-section:
Lmo.1 =
~ (Lmt~)o.~ + ½Zfo.lLf(~)o.1) ZfO.l-~1
+~(Lmfs)O.lq-Lf(s)o. 1 Lf(s) 1 O.
z/O.l> 1
2Zfo.l */
(Lm(s)o.1 "4- ½Z/o. IL/(s)o.1)
N
(36)
-- )~f0.1 1 ~> \'((~f°'l-1)2"Lf(s)°'l)--~f~l
The matrix first cracks when the load applied to the whole specimen reaches L~. Hence, putting:
Lml Am
~mu
give s:
~ ~fEfAf(1L1 = trmuAm 1 +
N fmAm + ~
N
½~fl) -~ ~ ZfI> 1
)':f/af(½•/l) -
?~
~fEfAf[()~f1 -
Xfl>1
l)2/2Zfl]
)'fffAf[(Zfl -
]
I)2/2Zfi]
where:
Zfl =
2Xfl
(37)
If
This equation, in conjunction with eqn. (35) (for an applied load L~) gives a complicated expression which may be solved for the load L~ at which matrix cracking first takes place.
Debonding Provided that the fibre bridges hold and the specimen is sufficiently long, when the matrix has cracked in one place at load L~ it will be sufficiently loaded to crack elsewhere. Until now, the matrix has been transferring load to the fibres. When a crack has formed, the bridging fibres must transmit all the extra load back to the matrix on either side until the total stress in the matrix again reaches its ultimate value.
128
J. SPURRIER
When a crack crosses the specimen, this analysis assumes that there are so many fibres of each type that it crosses all types of fibre at all possible points. Where the crack intersects a transferring fibre, or the transferring end of a fibre, it may be assumed that the fibre/end of fibre debonds, and takes no further load if frictional effects are ignored. The only fibres which actively bridge the crack are those with a length greater than 2xI1.1 which are intersected at distances greater than x;1.1 from either end. The main barrier to the formulation of simple equations for the tensile behaviour of a composite reinforced by aligned discontinuous fibres is the determination of a value for x i l . l , since it has to be found by an iterative process in conjunction with the number of bridging fibres. The equations are too long and too complicated to be worth recording, but it may be possible to obtain specific numerical solutions by computational methods. The cycle is to first take all the fibres which are capable of bridging when the transfer length is x i l , to share amongst them the excess load from the cracked matrix and from the debonded fibres and fibre ends, and then to calculate the first trial solution for the transfer length xsl. ~. Some fibres which were capable of bridging at xll are now likely to debond, and so this cycle has to be repeated until it is clear which, if any, fibre bridges hold. The assumption that Lits) 1 = Ls(s~l. 1 (see Fig. 9) would enable xil. ~ to be calculated directly from x I l , but this is equivalent to saying that no debonding occurs, or that its effect is insignificant. Especially in view of the fact that the lengths of fibres normally used in experimental composites are less than an order of magnitude greater than their transfer lengths, xy~, it does not appear to be a valid assumption. LOAD
CARRIED
BY INTERFACES
•
_*~rH FRICTION NO FRICTION
t DEBONDING
ll'
L LOAD APPLIED
OCCURS
Fig. 11. A schematic representation of debonding, followed by possible frictional load transfer along the fibre-matrix interface.
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
129
So far it has been assumed that the interfacial bond holds at all values of load provided that there is a sufficiently large distance for load transfer to be completed. If the concept of debonding as a result of the applied load exceeding the bond strength is introduced (see Fig. ll), with or without the subsequent possibility of frictional load transfer, the model rapidly becomes exceedingly complex. Nonplanar matrix cracking is then almost certain to occur.
Matrix multiple cracking--load calculation The technique of integration from the site of the crack to determine the load carried by each fibre type is not very suitable for application to discontinuous FIBRE LOAD
LOAD
EBONDED
M,IR,X CRACK
I
Fig. 12. Diagrams to show the fibre load distribution of a transferring fibre which has debonded at one end because of the formation of a matrix crack, and the range of such distributions which should be integrated to give the load distribution of that fibre type on each side of the matrix crack.
130
J. SPURRIER
fibres, because the maximum load carried by a fibre which is transferring over its entire length changes with the position of the crack, as shown diagrammatically in Fig. 12. There are also problems in describing the matrix load associated with a particular fibre, when the fibre is itself removing load from the matrix. The only simple solution appears to be one where the average load over the whole length of the fibres is calculated by double integration techniques (see Appendix) for those fibres which are not actively bridging. The load removed by these fibres in the vicinity of the crack is then represented by the constant value LIA v for a distance lI on each side of the crack. Two such square waveforms are shown in Fig. 13. The total load removed by the non-bridging fibres at some distance w from the crack is obtained by summing all such waveforms, and the difference between L 1 and this load represents the joint load carried by the bridging fibres and matrix. The total fraction of this load borne by the matrix, however, is still:
Ls(~) + L~(~)
~.
+
N
AVERAGE OF INTEGRATED FIBRE LOAD FOR DIFFERENT TYPE OF FIBRE
I "~ ,
FIBRE
~q"~-. MAT RI X LENGTH.~.-~ CRACK
I
Fig. 13. A diagram of the simplified load distribution employed for two fibre types, and the range of possible positions of the discontinuous fibre with respect to the matrix crack for their loads to be included in the calculations for the average load, LyAV,for that fibre type.
TENSILE BEHAV1OUR OF MANY-FIBRE-TYPE COMPOSITES
131
which is independent of the applied load. Clearly, this simple model implies that further matrix cracks cannot form closer to any existing matrix cracks, at load L1, than a distance equal to the longest fibre length, lltr,,x ). The most probable separation 12 will thus be ~lytm,~). If it is valid for this model to consider secondary matrix multiple cracking, then the secondary cracks will form in the centre of the blocks, and a rough guide to the load at which this cracking occurs is given by: Zl 2
L2
L 1 ~
L~/2
where L~/2 is the load removed by the non-bridging fibres in the position corresponding to the centre of the primary matrix blocks. Unless L 2 is very close to LI, however, it seems unlikely that secondary cracking will occur in a form which can be accurately predicted by a model which makes such drastic simplifications, especially if there is a possibility of longitudinal cracks being present as a result of debonding.
CONCLUSION
An acceptable working model has been formulated to describe the load extension curve o f a brittle matrix composite when the reinforcement consists of assorted continuous aligned fibres. It is theoretically possible for the sarhe techniques to be applied to discontinuous reinforcement, but severe complications arise as a result o f load transfer at the fibre ends, and these effects can neither be overlooked nor suitably simplified. Whereas the whole load extension'curve may be predicted for continuous reinforcement, with reasonable accuracy, the method is likely to yield only an approximation to the load at which the matrix first cracks when used with discontinuous fibres. Random orientations of fibres has not been considered. The incorporation of a suitable efficiency factor 16" 17 r/, into the equations for continuous reinforcement is not difficult but there is little practical interest in such formulae since it is desirable that continuous fibres are aligned in the direction of the tensile axis where this is known to avoid serious losses in strength and efficiency. The suggested model for aligned continuous different fibre reinforcement, together with the suggested modifications, should prove to be a fairly flexible system capable of giving reasonable predictions for a wide spectrum of possible composites, and it may be possible to use the formulae to design composites to certain stress-strain requirements. The likely effect of the inclusion of a certain proportion of damaged fibres is available directly from the formulae. It should also be possible to get some theoretical indication of the effects of fibre bunching in composites by splitting the composite into regions with various percentages of
J. SPURRIER
132
fibres in them, and treating each region in the manner in which a fibre and its associated matrix area were employed in formulating the original model.
APPENDIX" AVERAGE DISCONTINUOUS FIBRE LOADS
In calculating the average load by double integration for the non-bridging discontinuous fibres, five separate cases have to be considered: (a) l > 2 x : l , l I > 2x:1 + x:1.1 B (b) 1"> 2xf1.1 l < 2Xy I + x : l . 1 B (c) I < 2 x : 1 . 1 l < 2 x y l T (d) l < 2 x y l . 1 l > 4xyl T (e) l < 2xy1.1 2 x f l < l < 4Xyl T Those fibres labelled B are sufficiently long to be bridging fibres, but have to be included because the crack crosses some of them within the transfer distance of the fibre ends. Those labelled T are not capable of bridging the matrix crack.
FIBRE LOAD
~-g~
t
II ....
o
"rtl
I
i
j_
II
ii'--o',o MATRIX"~ CRACK
I)C1.1
Fig. 14. A diagram to show the method used for the double integration to calculate the average load for a fibre which, although long enough to bridge a matrix crack, has been crossed by the crack within a distance, x: 1.1, of an end, and has therefore debonded. (a) (See Figure 14): Average load for the proportion of these fibres which are not actively bridging:
"Uf-xyl.1)
2Ly(s)l . . . .
Therefore: average load for fibre type
Xf 1 If
+
.,xf I
2Lf(s)l
X f 1.1
133
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
2xfx ~f + x:"l] 2:N:" 2Xzl.tlf
= Lf(s)I(I
since the totalnumber of this type of fibrein a cross-section = 2/N; and the chance that the fibrewill partlydebond __ 2X$1"1
b
(b) Average load
2L:(~)I" Xf~l " ~f " dy
kd¢If-xf l'l)
f (':) (I~yx 2Ly(s)l z
+ J~2X:l~
dz
I
:'/2 2Ly¢s,xdz~f) ]
.X.f.I. If + ax$1 |
d?
Xfl.1
Therefore: average for fibre type Lf(s)___~l[Zxf13 -- (l: - x/l'')3 =
]2Vx71
Xfl'1
(l_f2 +
)] 2x:t" " 2:N: -
x:,
"
b
(c) Average load
=f(") f~22Lj,(,).._____..~l,z .dz d, J(J~lf) •fl ½~f If ½b Therefore: average load for fibre type = Lf(,)l.7 . 2 : N : Z:~
24
(d) Average load
z
=f,,,, J(*tS)
dz) dy
d~ + ( ,,22L:¢s)i~ 2Lsf,)1"--'~f Xf 1 ..,xf 1
V:
Therefore: average load for fibre type
= Ls(s),(~ - ~) "AfNf (e) Average load z =
L J(-t-lf)
dz
2Lf{s) 1 . .x/~ . . . l/ dy
~_{(If)J(2xf l,(ill 1
Z~,zdzf ,,2
d_~) ]
I X
--
½b
134
J. SPURRIER
Therefore: average load for fibre type
=
Lf(s)l [8xf13 _ ~153 ½1I L 121rxyl +(~-xI')]
"2'NJ"
The average load for the fibre types takes into consideration the n u m b e r o f fibres o f that type in a particular cross-section, a n d is thus a measure of the load per unit length removed from L1 by all fibres o f that type over a distance, lI, on b o t h sides o f the matrix crack. The resulting block diagram, similar to Fig. 13, will thus represent the load b o r n e by fibre types instead of by the individual fibres.
REFERENCES
1. N. F. ASTBURY,Advances in materials research in the N A T O nations, Pergamon Press, 1963, pp. 369-90. 2. G. S. HOUSTERand R. THOMAS,Fibre-reinforced materials, Applied Science Publishers Ltd, London, 1967. 3. J. P. ROMUALDI,Proc. Cement and Concrete Assn. Conference, 'Structure of Concrete', Imperial College, London, September, 1965. 4. J. P. ROMUALDIand G. B. BATSON,Jnl. Eng. Mech. Div., Am. Soc. Cir. Eng. EM3 (1963) p. 147. 5. M. F. KAPLAN,Jnl. Am. Concr. Inst., 58(5) (1961) p. 591. 6. G. A. COOPERand J. M. SILLWOOD,J. Mat. Sci., 7 (1972) p. 325. 7. A. KELLY,Strong solids, Clarendon Press, 1966. 8. H. L. Cox, Brit. J. AppL Phys., 3 (1952) p. 72. 9. N. F. Dow, GEC Missile and Space Div. Report R63SD61, pp. 1-42. 10. J. O. OUTWATER,Mod. Plast., 33 (1956) p. 156. 11. B. W. ROSEN,Fibre Comp. Mats., Am. Soc. Metals, (1965) p. 37. 12. J. SPURRIERand A. LUXMOORE,Fibre Sci. and Tech., 6 (1973) p. 281. 13. A. J. MAJUMDAR,Proc. Roy. Soc., 319A (1970) p. 69. 14. A. J. MAJOMDARand J. F. RYDER,Sci. Ceram., 5 (1970) p. 539. 15. I. W. NORRIS,Thesis for B.Sc. Degree, Dept. of Civil Eng., Univ. Coll. Swansea, CP/454/75, 1975. 16. H. KRENCrlEL,Fibre reinforcement, Akademisk Forlag, 1964. 17. V. LAWS,J. Appl. Phys., 4 (1971) p. 1737.
J. SPURR1ER*
Department of Civil Engineering, University College of Swansea, Swansea SA2 8PP (Great Britain)
SUMMA R Y Equations are derived from first principles to predict the load extension curve of a brittle matrix composite reinforced by continuous high modulus fibres of many different O'pes aligned in the direction of the tensile load. The equations deal with composite extension from the initial elastic behaviour, through matrix multiple cracking, to ultimate failure of the fibres, and incorporate an expression for the suppression of matrix cracking by the constraining effect of the fibres. Some numerical results which couM be obtained from conventionally reinforced cement and gypsum plaster have been substituted into the equations, and possible modifications to deal with debonding and premature failure of the fibres are indicated. Some similar equations are then derived for discontinuous different fibre reinforcement aligned parallel to the applied load, and the ensuing complications are discussed and simplified where possible.
INTRODUCTION A considerable amount of theoretical work has been published recently on the behaviour of fibre-reinforced composite materials under load, particularly with the advent of finite element computing methods. Most of the models considered assume some form of uniform distribution of fibres to obviate the need for elaborate statistical formulae and complicated equations, and this work is no exception. Normally, however, the modelled composites contain only one type of fibre, hence the results are not readily applicable to practical systems where significant * Present address: Department of Materials, Cranfield Institute of Technology, Cranfield, Beds. (Great Britain). 103 Fibre Science and Technology (9) (1976)--O Applied Science Publishers Ltd, England, 1976 Printed in Great Britain
104
s. SPURRIER
fluctuations can occur as a result of accidental damage to fibres during the manufacturing process and imperfect bonding along the interfaces, as well as imperfect distribution. Models for reinforcement by many different types of fibre are better equipped to deal with such variations. In this work, equations for the load extension curve of such a composite are derived, assuming perfectly elastic behaviour of both the brittle matrix and the fibres. The simplest model is one of a uniform matrix containing aligned continuous fibres, but even then a number of assumptions and simplifications have to be made about the load transferred between matrix and fibres. This model is discussed in detail, then the same techniques are applied to the far more complicated case of aligned discontinuous fibres, where further simplifications are necessary.
A L I G N E D C O N T I N U O U S DIFFERENT FIBRES
Initial elastic behaviour Consider N parallel fibres embedded in a brittle matrix of Young's modulus E,,. Let each fibre have a modulus E: and an individual cross-sectional area A: = 7tr:z. Since the fibres are continuous, their length may be taken as the length, l, of the specimen (or of the gauge length being investigated). Assuming completely elastic behaviour, the initial modulus of the composite, when loaded in tension parallel to the fibres, is given by the rule of mixtures formula 1 , 2 .
E 1
+ A,
As
~u E:~r: 2l vs
Em(Vs-~u ~r:21) +
vs
(1)
where the volume of the specimen under test Vs = A s. l, and the volume of matrix V,, = (As - EN A:) • 1. The initial tensile behaviour of the specimen, when the whole composite obeys Hooke's Law, is thus given by the relationship: Lo.1 e°" =
l
A--7"E--71
(2)
where e o. 1 is the total (gauge-length) extension produced by the external application of a load L o. 1In the derivation of the formulae for the load extension curve beyond this initial behaviour, it is necessary to estimate what proportion of the total applied load is
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
105
being carried by each fibre, and by the matrix. One of the most convenient ways of doing this is to assume that the proportion of load carried by a particular phase, b, in relation to that carried by the total load-bearing phases, B, is:
EbAb
(3)
Hence the strain of this particular phase at an applied load L o. ~ is: Stress Lo.1 [ EbAb ] Strain=--~--b = ~ "[(~EA)J'~
1
= constant for load Lo.t. This method of dividing the load between the phases is thus equivalent to assuming that the cross-sectional extension is planar.
First crack Provided that both the matrix and fibres are uniform, the first crack across the matrix will occur at any point on the gauge length, when the matrix stress reaches its ultimate value a,.u. Let the load then be L~. Just before this crack forms, all the phases present are bearing load, hence:
This assumes, of course, that the reinforcing fibres all have a greater strain to failure than the brittle matrix: this is normally a valid assumption for practical systems where the matrix is being reinforced by relatively low amounts of fibre, and especially true for cementitious materials. Hence:
Lt EmAm ~mu =
- Am
"
EIAs
where A . = area of matrix in a cross-section. Thus:
trmE1As L 1 =
-
E.
-
t [
~rm
(4)
106
J. SPURRIER
Constraint effects Romualdi a and Romualdi and Batson 4 have pointed out two mechanisms by which the presence of fibres can affect the matrix cracking process. One of these deals with the uniform stripping of wires from the matrix following debonding. It is therefore more applicable to discontinuous fibres. Debonding of the continuous fibres is not considered here. The other mechanism, however, is concerned with the pinching effect exerted by fibres perpendicular to the plane of the crack, especially during the early stages of crack growth• This mechanism is particularly applicable to cementitious matrices, which exhibit controlled flaw and micro-crack growth just before the matrix fails, 5 and is analogous to the constraint effect of Cooper and Sillwood, 6 which predicts that the failure strain of the matrix will be:
when fibres are present. This expression is derived from a consideration of the energy-absorbing processes during crack propagation. Assuming that the surface energy per unit area of the matrix, 7,,, does not change, the strain to failure for a given volume fraction of fibres may be increased by using finer reinforcement, which implies that the fibres are more closely spaced. When more than one type of fibre is present, this expression must be modified to allow for variations in the value of r r, the limiting shear stress for each fibre at the fibre-matrix interface. For a total number of N fibres:
This approximation is derived by substituting for zs using the transfer length x z (see below), summing over N fibres, and replacing Xz by an expression for the mean transfer length 2z = (E N xs)/(N).
Transfer length From eqn. (3), the extra load carried by a particular fibre at the site of the crack when the matrix fails is given by: •
~muAm
~ EsAs N
By equating forces, this can be related to the distance along the fibre x s that is required for this extra load to be transmitted back to the fibre by limiting shear 7 :
107
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
a,,uAm
~ f A f = 27rrfTfXf EsAs N
This equation assumes that the force transferred per unit length is a constant. Several other expressions with more complicated forms have been derived, 8-11 usually by considering the load transfer for discontinuous fibres. Any such expressions may be substituted in the following formulae provided that the proportion of the load transferred at any distance from the crack can be calculated. The linear relationship for load transfer (Fig. 1) has been used here to simplify the
I I
LOAD ~ CARRIED| ACROSS| MATRIXJ~f CRACK -
-
,, ~ , ~
MATRIX SHARING LOAD FIBRE ~SHARING LOAD
-
"]- MATRIX ~LOAD
I I
~ FIBRE LOAD
I
.U
_
_
MATRIX CRACK
Fig. 1. A diagram to illustrate linear load transfer in the region of a matrix crack for a single continuous fibre loaded in tension. mathematical expressions. The proportion of the load recovered by the matrix at a distance x from a crack is then (x/x:) for 0 <_ x < x: where x: is the transfer length of the fibre. At load L1 when the matrix cracks, let the transfer length of each fibre be x:l.
x:l =
a.~ v.E:r:
(6)
2zyl ( ~ E:nr: 2)
The values of x:~ will be different for different fibre types. If the applied load is altered, the transfer distance for a particular fibre type will also change. The linear relationship adopted gives the simple expression:
L2 x:2 = -~l "x:l
(7)
The maximum value of x:l will be associated with the fibre-type which produces the maximum value for the expression
108
J. SPURRIER
Efrf "Cf /
and this distance will control the positions of any further matrix cracks.
Matrix multiple cracking--block formation At distances greater than xil~m~ I on either side of the first crack, the whole cross-section of the specimen is still loaded in the same manner as it was before the first crack formed, provided, of course, that the fibres bridging the matrix crack are sufficiently strong to carry all the load (i.e. EN as,nrs z > LI where Cry,, is the ultimate stress of the fibre). If the gauge length of the specimen is sufficiently long, multiple cracking of the matrix will occur, without any change of load if the specimen is ideally uniform, resulting in the matrix being divided into blocks by cracks crossing the specimen perpendicular to the fibres. Since the matrix is capable of cracking at load L 1 at any distance greater than Xy~lmax~from existing cracks, it follows that the resulting block size will be between xll (max)and 2xfl (max," The exact number of cracks which form in the gauge length will evidently affect the overall extension of the specimen. Since this number is open to statistical fluctuation, the best approximation is to assume that the cracks are regularly spaced, dividing the matrix into blocks of the most probable length, ~2 .f, where : .~xi~lm~ I (8) The second series of matrix cracks will form across the centre of these primary blocks, since it is here that most of the load previously carried by the matrix has been transferred back from the fibres by interfacial shear forces. 8'9 However, before this secondary cracking can take place, more load has to be applied externally to the specimen to raise the average stress of the matrix to a,,,, over the whole cross-section at these points. This will also change the values of the transfer lengths of the fibres, in accordance with eqn. (7). Let the load required for secondary cracking be L2. Matrix multiple cracking--load calculation In order to calculate how much load the matrix is carrying at any point, it is necessary to assume that load transfer takes place as quickly as possible, and to designate a certain cross-sectional area of matrix as being controlled by each of the fibres. In positions remote from matrix cracks, the load carried by each fibre for an externally applied load, L: Lf(s) = L • EyAs EIAs and the load carried by the whole matrix equals: E,,A,,
L . - -
EiAs
109
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
It is convenient to designate the area o f matrix associated with each fibre as:
.4.. E~a~
EsAs
N
such that the load carried by that part of the matrix associated with a particular fibre at load L 2, when transfer of load between t h e t w o is complete: Lm(s) 2 = L 2 • EmAm
(9)
EsAs
EIAs E EfAf N
PRIMARY
SITE OF SECONDARY RACK
CRACK
<~ LOAD L2
I
PRIMARY CRACK
II
I[
II
II
,
II
I!
>
- LOAD L2 _
t4~-X ---*
LB. ii MATRIX L ~ LOAD
\
Lm(s)2
N - T2 FIBRE
x~2 ~
z4
Xt2
MATRIX LOAD Lm(s)2
T2 FIBRE
x~2 >
zc/z
Fig. 2. A d i a g r a m to illustrate the load transferred to the matrix at a distance, x, from a primary matrix crack w h e n the specimen is carrying a load, L2, sufficient for secondary matrix cracking to occur, for both non-transferring a n d transferring fibre-types.
110
J. SPURRIER
However, the secondary cracks, which form at load L2, have predetermined sites at distances ½2 from the primary cracks. If x:2, the fibre transfer length at load L2, is less than this distance, the matrix associated with that fibre will be fully loaded at the site of the secondary crack. Fibres in this situation are classified as N-T (not transferring) at x = ½~?. If xf2 is greater than the distance from a crack to the centre of the primary block, these fibres are still transferring load to the matrix at this point and are grouped as T (transferring) fibres. (See Fig. 2.) This classification has to be made at all values of x, the distance from the nearest crack, hence the number of fibres in each of these groups depends on this distance. When x = 0, all fibres will be in category T. For a specimen with N fibres, let T z be transferring at a distance, x, from the primary cracks• Then the load transmitted back to their associated areas of matrix by these fibres is:
(~
E,,A,,
EfAf
x
/_, L 2 . - EIAs X~ EyA: x:2 T,2 N
Hence the total transmitted back to the matrix at a distance x from a crack
~")
E,.Am
E:A~
~
E,,Ar,.
E:A:
x
N-T,2 N
N
LzEraAm • e:A: + ~ E:A: "' EtA~~ EfAf (N2 T.2 T,2
(lo)
N
But at load L2, by definition, the maximum load transmitted back to the matrix must be just enough to crack it, and this occurs when x = ½~. Hence:
L2Em • El A~ ~, E:A: N
E:A: + ~ E:A: 7",2
= (a,..
[no constraints] or constraints]
\ E,,e,,p [with
(ll)
This equation, together with eqn. (7), form a simultaneous pair which may be solved for the load L2 required for secondary cracking. The solution, however, has to be found by iteration, since some fibres may pass from category N-T into category T as the load is increased from L~, as a result of the increase in transfer length with load. These fibres obey 1 > x:l/½.;c > L1/L2. A first estimate for L2, which will be lower than the true value, may be obtained by solving the equation with only Tz fibres transferring. Any fibres which obey the above inequality, when this estimate of L2 is used, are included with the transferring
TENSILE BEHA,VIOUR OF MANY-FIBRE-TYPE COMPOSITES
111
fibres, and a second estimate made. As soon as no additional fibres obey the revised inequality, the true value of L 2 has been found and all the transferring fibres have been correctly assigned to category T2.
Matrix multiple cracking--load extension curve The initial linear load extension curve before the matrix cracks is given by eqn. (2). Once cracking has occurred, the overall extension is no longer linear. The series of primary cracks, and each subsequent series of cracks, form without change of load in a uniform matrix, and hence produce horizontal plateaux on the load extension curve. An equation for the curve between these discontinuities is required.
vr- $~c N-T N-T
F't'M
LOAD L1. 2
F
N-T
= " N O N - TRANSFERRING"FIBRE
T _-"TRANSFERRING" F+M
II
=
TYPE.
FIBRE TYPE.
FIBRE AND ASSOCIATED MATRIX CONTRIBUTE MODULUS E(~)1"2
F = FIBRES ONLY
LOAD L1.2
T
CONTRIBUTE
TO
TO MODULUS.
Fig. 3. A diagram to illustrate the method of assessing the Young's modulus at a distance, x, from a matrix crack for an applied load, L I.2. The shaded portions represent the transfer lengths xj 1.2 of the various fibre types.
Consider a thin cross-sectional element of thickness 3x at a distance x from a crack (see Fig. 3). Let the external load be Lt. 2, such that L1 -< L~.2 < L2. Then the operating transfer length for each of the fibres is xii.2 where: L1.2 Xfl.
2 ~
L1
" Xfl
In general, some fibres in this element will have completed the transfer of load to their associated matrix (N-T) and the remainder (T) will be still transferring
112
J. SPURRIER
the excess load. Because of the assumed linear load transfer along the fibre, the matrix associated with these transferring fibres in the elemental cross-section will not be able to carry any additional load, if extra load is applied externally. The total load borne by the fibre-matrix pair is increased by such an application, and the transfer distance correspondingly lengthens, but these changes affect the matrix only at distances greater than x s. The effective Young's modulus of the element E(x)l.2 is thus given by: E(x),'2 = E
E:A:
(~) E,.A,.
--~- +
~
N
As
EIA:
(12)
~ EfAf
N - - TI. 2
N
and the variable load carried by the element at position x is:
(x~ E,,,A,, L(x)I.
2 =
L1. 2
1 -
.] E EsAs xsl,2 EIA:
'
TI.2 E1As
-
x
-
(13)
N
The extension of this element is thus: L(xh. •
.4sE(x) l. :
6x
The total extension of the specimen may then be obtained by summing for all the elements contained within half of one of the primary matrix blocks, and multiplying by 21/~ which is the number of half-blocks in the gauge length. The summation must be performed in two stages because, at the fixed load, L~.2, some of the fibres will change from category T to category N-T as the position of the element moves towards the centre of the block. This will affect the summations in both eqns. (12) and (13). If the fibre types are arranged in ascending order of transfer length xzl.z, the contents of the sums in these equations will change by a fibre type every time the position of the element, x, increases beyond one of the values xyl.2 (up to the centre of the block, when x = ½~?). Hence the total extension of the specimen at load L~.2(L~ <_ L~.2 < L2) is given by:
21 E el2
=
X (Axf1.2 )
((xfl'2)n-+tL(x)I2
J(X]l.2)n
"dx
(14)
msE(X)l.2
where the integral is taken over successive steps Axle. 2 as x increases from zero to ½~, and these steps are then summed for the final result. This equation gives the extension of the specimen for an externally applied load when the matrix is subdivided only by the primary cracks, such that the block size is ~.
113
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
LOAD
- COMPOSITE FAILURE
-T
PzPu~"
--
GENERAL RANGE
Pp.
~1
Po -
i _
_
Lend
R
Lo+I
L=
L1
I
---
_-
I 11 III II II
I I
li ',
GENERAL RANGE
I I I t
I J
I
L
f
I I EXTENSION DUE TO ~e'l~I III PRIMARY MATRIX I CRACKING.
i
r;
__E
i I
f ,i
_i___
I
--.~" LO'I
-- L-1
I
i I
i
el.2 RANGE
I I
GENERAL RANGE
e~.~÷ 1 Fig. 4.
RANGE
EXTENSION
"~1.2 RANGE GENERAL RANGE ~P'P
÷1
General load extension curve for a brittle matrix reinforced by aligned c o n t i n u o u s different fibres which remain perfectly elastic.
114
J. SPURRIER
The model indicates that there will be further cracking of the matrix until the external load is such that all the fibres are transferring excess load to the matrix along their entire length. When all the fibres are thus in category T, let the load be Lend. The following general formulae assume that the load at which the fibres start to fail, P1, is greater than Le. d. The generalised equations for the loads Lq+ 1 at which matrix multiple cracking occurs when constraint effects are present are:
Lq+ 1 EIA~ N EfA:
[_N r.+,
E:A:
E,:.,
E: :
(15)
rq+ I
where:
Lq+ 1 Lq+ 1 " E,,A,,, • E / A f x:q+l = L~ " x:l = 2rrryz:EtA ~ ~ EfAf N
(16)
and other terms are defined by eqns. (1), (4), (5), (6) and (8). These expressions are valid for q > 0 provided that Lq+l does not exceed Le, d, when all fibres go into category T. If constraint effects are ignored the right-hand side of eqn. (15) is replaced by a,, u. The general formulae describing the load extension curve between these loads (which are marked by horizontal regions where the extension increases at constant load due to matrix cracking on the general diagram, Fig. 4) becomes:
eq.q+ 1 --
1" 2 q .~
~(Xfqq +l)n4-1 L(x)q.q+ 1 ~ "..l(Xfq.q+1)n AsE(x)q'q+ dx (Axfq.q+ l) 1
(17)
valid f o r q _ > O,n_> 0 where"
L(x)q.q+ 1 = Lq.q+ 1
F•.Am E:A:
x ]
(18)
1 + Tq.q+I E1As " ~ EfAf Xfq'q+l N
and:
E:A: E(X)q'q+l = ~-'-~'-s
(x) EmAm. E:ay N- r~aq+l A~ ~ E f A f N
(19)
where the variable load Zq.q+1 is such that Lq < Lq.q+l,
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
1!5
Fibre extension Provided that the fibres are sufficiently strong to carry the entire load in the region of the cracks, multiple cracking of the matrix will occur in accordance with the above equations until all the fibres are transferring load back to the matrix even at the centre of the existing matrix blocks. This load is defined as Lend, and marks the point above which the load bearing capacity of the matrix is fixed. The load extension curve above this point is linear and depends on the moduli of the fibres.
_ _.
:::
LOAD
......
II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II ...... : : :
~
L~ AFTER
I
i
I
:i__ ................................... II ...........
i!
___
II . . . .
II
.
.
.
.
.
.
:::
oO.no
CASE (o) ,q
q,
I I I
I I I
LOAD
Lq.+ I CRACKING
..........................
<7:= i
I!....... ___TTT
!
,'~x~.q.+m.i.--)' ) LOAD
L~÷I AF'~R
II
CRACKING I. e n d
,
g
II
P
II
"-;
' :)Cj'q'+Itmin) '
CASE
(b)
Fig. 5. A diagram to show the two ways in which Lend is evolved: (a) by an increase in load; (b) by matrix cracking.
Whereas the initial size of the matrix blocks depended on the maximum transfer length X/l(ma.), the load Lend depends on the minimum value xIl(ml, ), since this fibre type will be the last to pass into category T. The two possible ways in which this change can take place are illustrated in Fig. 5. In the first diagram, which shows the last fibre type still not transferring, the transfer lengths (taken from each end of a matrix block) nearly meet after cracking at load L~ has occurred, and eventually meet (at Le.d) before the load across the matrix at the centre of the block is sufficient for further cracking to take place. In the second diagram, it is
116
J. SPURR1ER
the formation of a crack, and not an increase in the load, which causes the last fibre-type to become totally transferring. Lendis then taken as the load at which cracking occurred. In order to determine which of these possibilities occurs, and hence the value of Lend, use is made of the load at which the transfer length for the final fibre type would be the same as half the length of the matrix blocks: -'~
L1
(20)
Z(tryJq+l = 2q+ I " Xfl(min)
where q > 0. Let Q be the lowest value of q to make either of these inequalities true: L{try)q+l --- L q + l
L Q + 1 = Len d
~
Lq+l < L(trylq+l ----- Lq+2 ~ L ( t r y ) O + l = Le, d (21) The former inequality implies that cracking of the matrix, and not loading of the specimen, caused the final fibre-type to become transferring.
Fibrefailure Above Le,d, all additional load applied to the specimen is borne by the fibres only. Provided that the fibres are uniform, they will fail in one of the matrix cracks, where their load is greatest. Once one fibre has failed in a crack, that crack will then be the site of failure of all the other fibres. In these idealised equations, it is assumed that once a fibre breaks, it and the matrix associated with it become redundant. In order to distinguish the equations from those where the matrix plays an active part, the letters P, M and h are used for toad, Young's modulus of a thin section and extension, respectively. Let the fibres start to fail at load P~. Then the maximum load on each fibre is across a matrix crack and has a value
E:A: P1 ~ EfAf N
The stress in one of the fibre types is then just sufficient to fracture it, hence:
l(min)
N
where (trf/Ef)l~min)is the minimum value of (a:/E:) for all the fibre types. When this fibre type breaks, its share of the load has to be carried across the matrix cracks by the other fibres. It is possible, therefore, for some, or all, of the other fibres to break when this additional load comes on them. The general formula for the loads at which fibre groups fail is thus:
TENSILE BEHAV1OUR OF MANY-FIBRE-TYPE COMPOSITES
PP=(aE'~iS)p(mi,)
~w EsAf
forp>
1
117
(23)
where (aI/EI)p(mi,) is the minimum value of (ai/E I) for the Wp fibres remaining whole up to load Pp and N -- W1. To determine which fibres fall into category Wp+ 1, i.e. those which stay whole despite the extra load thrown onto them when one or more fibre types fail at load Pp, it is convenient to arrange the fibre types in ascending order of (ay/Ey). Consider the load carried by the second weakest of the fibres in group Wp (i.e. the weakest of the remaining Wp.p+ 1 fibres) when the weakest fibre in group Wp fails. If: PP >-- ~
p.p+l(min) W p p + 1
then this second weakest fibre will also fail, hence it may no longer be considered in the Wp.,+ 1 summation. Therefore if any of the fibres are to remain intact above load Pp, it will be possible to find a solution to the inequality: O'f Wp
(n -- 1)(rain)
where the fibres are arranged in ascending order of value (ariEs) and the nth in this order is the first to obey the inequality. This fibre, and those above it in the series, are the fibres which remain whole up to load Pp+ 1. The composite ultimately fails at load Pu, when no fibres are left intact for group 14I,+1. The load extension curve above Le.d is calculated on a similar principle to that of matrix multiple cracking. It is simplified by the fact that further subdivision of the matrix does not occur, the block size being that remaining after the cracking at load Lo+ t (-----Lend) is complete. Also, the matrix associated with a particular fibre retains a certain amount of load if the fibre is whole, instead of a certain fraction of the load. The effective modulus of a thin section of the specimen is thus constant, and equal to the modulus of the whole fibres only. The general formula is thus:
2°+ll.I(~/2°+l) P(x)p.V+ldx -x AsMp+ 1
hp.p+ 1 = - -
(25)
valid f o r p > 0 where:
P(x)v.v+ 1 = Pv.v+ 1
-
-
~r,..A,. Wp+l ~
EsA s x ] EIAI xft
N
and:
(26)
118
J. SPURRIER
Mp+ 1 = 2_,
EIAs
(27)
Wp+l As
where the variable load Pp.p+l is such that: Pp -< Pp.p+l < Pp+l and : Po = Le.d Because of the simple form of eqn. (27) compared with eqn. (19), eqn. (25) may be integrated to give: [
a,,,Aml
Pp'p+ 1l
_ hpp+ 1 = l ~-' "Wp+ 1
~ __.
~EIAs/xsl WpM. _ _
] .[
E+A+ N
(28)
Wp+ 1
A general form of the complete load extension curve is given in Fig. 4.
Modifications Throughout these equations, it has been assumed that matrix cracking has been completed and all the fibres have passed into the transferring category before any fibres fail. Such is normally the case for brittle matrices such as cement and plaster conventionally reinforced by steel wires and glass fibres. Even if this assumption is not valid, the equations, in a modified form, may still be applied if a broken fibre is considered as being completely redundant. All that is required then is a redefinition of the specimen every time a fibre breaks, since the volume of matrix remains the same although the number of fibres falls. The failure of a fibre will obviously cause an overall extension of the specimen without change in load, in the same way as the formation of a matrix crack, The load carried by the fibre will be transferred to the rest of the specimen, and some of this will be taken by the matrix if the load is below Le,~. This additional load may be sufficient to cause simultaneous matrix cracking. The formulation of the equations also assumes that debonding does not occur. The inclusion of debonding terms into the equations is theoretically fairly simple, but the resulting equations are long and tedious. I f a suitable expression is available to describe the variation in debonded length, dI, with the load carried by the fibre, and it is assumed that the debonded region of a fibre transmits no load back to the matrix, then the incorporation of debonding into the equations is equivalent to moving the origin for load transfer from the crack at x = 0 to x = d I. The inclusion of debonding implies that less load is being transferred to the matrix blocks, hence the fibres extend more. Friction over the debonded lengths can also be modelled by assigning the frictional load to the matrix over these regions. This
1 19
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
produces a reduction in the 'active' loads L(x) or P(x) over the debonded region of x, resulting in smaller extensions of the specimen than when friction is ignored. NUMERICAL SOLUTIONS
General data Numerical values 13-15 have been substituted into the equations to represent cement or gypsum plaster matrices reinforced by low volume fractions of conLOAD (NEWTONS)
-%
X
X
16,00(
WITH CONSTRAINTS /
12,000
/
/
/ /
EXCLUDING CONSTRAINTS /
/
8.000 /
//
,/
/
//
I-- _1-- - - J
4,000
t I
EXTEN$1ON(x lOAm) BEFORE AFTER 0,16 0.32 0'36 0-53 0.59 0 '76 6,41
8"03
15.73
(N)
LOAD (N) EXTENSION(x 104r~ AT CRACK BEFORE AFTER 0'6/. 1.26 1026 L1 4049 1111 1-42 2.0"7 L2 4387 119"7 L3 = L u 4717 2 "34 3"01 8482 7-46 P1 6462 5"92 16422
Pz = Pu 16421
15 15
EXCLUDING CONSTRAINTS .L iNCLUDING CONSTRAINTS
MATRIX ONLY FAILURE LOAD
I z, x 10-4
I 8 x 10-4
I 12 x 10-4
,
I 16x 10-/.
EXTENSION (METRES)
Fig. 6.
Load extension curves computed for two types of glass fibres in gypsum plaster, showing the effect of including the constraining influence of the fibres.
120
J. SPURRIER
tinuous glass fibres and steel wires, and the equations solved with the aid of a suitable computer program. 15 The figures used in the calculations are given below, and the resulting load extension curves are illustrated in Figs. 6-8. Both cement and gypsum plaster have similar moduli and tensile strengths, and so average values were used to represent both matrices and so simplify comparison between the inclusion of different types of fibres. Cylindrical specimens were considered, the dimensions being kept constant throughout.
LOAD (NEWTONS)
1,60(
1.200" -LQ+ 1 -L 2 -L1 _
800
f
f
/
_. _AT_RJ_X
0 NL¥
I
Fig. 7.
J
l
l
I
0.4 x 10-~' EXTENSION (METRES)
I
!
0.Sx 10.4-
Matrix multiple cracking for two types of glass fibres in gypsum plaster, ignoring constraint effects. (Enlarged section of Fig. 6.)
121
TENSILE BEI-IAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
Radius of specimen (A, = 7~Rs2) Length of specimen Ultimate tensile strength of matrix Young's modulus of matrix Fracture surface energy
R, = 10 -2 m l= 10-'m ainu = 2"75 x 1 0 6 N / m E . = 17 x 1 0 9 N / m 2 )'m = 6"25 J/m 2
2
Two types of glass fibres in gypsum plaster The figures used in this section represent a 5.4 per cent addition of aligned continuous glass fibres, a number of which have been damaged by handling, resulting in a lower ultimate tensile strength and significantly weaker bond strength with the matrix. 12 x 104 3 x 104 Number of fibres N I 76 x 1 0 9 76 × 1 0 9 Young's modulus E s (N/m 2) 6 x 10 - 6 6 x 10 -6 Radius r s (m) 5"52 x 1 0 6 1-5 x 106 Interfacial bond strength Ts (N/m 2) 1210 x 106 500 × 106 Ultimate tensile strength try (N/m 2) The computed results are shown in Fig. 6. Two curves are plotted to show the effect of constraints exerted on the matrix cracking by the fibres. The difference is obtained simply by replacing the constrained matrix failure stress Emempin eqn. (15) by the normal value ainu, and may be explained 4 in terms of forces parallel to the fibres opposing the opening of matrix cracks associated with crack propagation. The section of one of the curves dealing with matrix multiple cracking is enlarged in Fig. 7. It should be noted that, in this and all other matrix cracking regions, but not in other parts of the graphs, the lines joining the horizontal regions where cracks propagate are not linear, but have gradients which decrease slightly with increasing load (i.e. at a general load L~.q+ 1 where L, < Lq.q+ 1 < Lc,d). This is a result of assuming that the transfer lengths of all the fibres increase with load, whereas the matrix block size remains constant. With this arrangement o f data, LQ+, =- L,,d, hence the load extension relationships above LQ+ 1 are entirely linear except where the damaged fibres fail. The undamaged fibres are sufficiently strong and sufficiently numerous to take the excess load thrown onto them when this occurs. Steel fibres in cement Two sets of figures were used, both representing 4 per cent of continuous steel wire reinforcement. Constraint effects were not considered because of the coarse nature of the reinforcing wires. Number of fibres Nj, 64 Young's modulus Ej- (N/m 2) 2.0 x 101' Radius r s (m) 2.5 x 10-4 Interfacial bond strength TI (N/m 2) 11.04 x 106 Ultimate tensile strength trf (N/m 2) 2.1 × 1 0 9
122
J. SPURRIER
LOAD (NEWTONS) FOUR FiE RE TYPES LOAD (N.) EXTENSION(x 104rn) AT CRACKi BEFORE AFTER L1 1236 0-16 0.23 L2 1373 0"25 0-33 L3=L u 1511 0"37 0"48 P1 Pu 26389 10.3e
2500
pu : 26389
~-_~0~ ~"%
2000
~ . ~ " .....~i CURVE FOR FOUR FIBRE T~PES
1500
~
.
~
_
_
J"
/ dp/
_
...
_
_
~ ' ~ ~.,
_
~
~"
~"
~"
CURVE FOR ONE FIBRE TYPE
1000
500
_
= 26388 ~_. :1o`3, f~u = 10 '39
LoAQN~N~ ' BREExT~;sEION( x 10'~'m' AT CRACK L 1 =L u 1236 P1 =Pu 26389 I 0.2 x I0 -4
BEFORE AFTER 0.16 0'38 10 39
I 0.4 x I0 -4 EXTENSION (METRES)
I 0.6 x 10-4
ll-
Fig. 8. Matrix multiple cracking for 4 per cent of continuous steel wire reinforcement in cement. Four values for the interfacial bond strength were used in the upper curve, a single higher value for the lower curve. The second set o f figures split the fibres into four groups, with N s = 16 fibres in each group, with differences only in the interracial bond strengths, r I, which were ascribed four values: Zf = 1"5 × 106; 5'52 x 106; 7"75 × 106; 10'5 × 106 N / m 2 These curves, which are shown in Fig, 8, are basically similar except for the number o f loads at which the matrix cracks. The final size o f the matrix blocks, however, will be similar for the two curves since xI~(,~,) is approximately the same, and cracking continues until this fibre type is in category T. The general effect o f increasing the load carrying capacity o f the fibres with respect to the matrix--either by increasing the number o f fibres or increasing their Young's moduli--is to reproduce a similarly shaped curve at a higher load for the same extension. The extension when matrix cracking ceases will be slightly reduced because there will be more fibres to carry the applied load across the
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
123
matrix cracks, but the ultimate extension of the specimen will be slightly greater because the proportion of the total load removed from the fibres by the fully cracked matrix will be less.
ALIGNED DISCONTINUOUS DIFFERENT FIBRES
lnitial behaviour and transfer lengths Consider N discontinuous fibres in a specimen, all aligned parallel to the tensile axis. It is assumed that the distribution of each type of fibre is uniform, and that each fibre type is present in sufficient quantities for the chance 2 s of finding a fibre in a particular cross-section to be the same as the fraction of the number of fibres of that type to be found in any cross-section. 2s =/s/ts (29) where ls = length of discontinuous fibre; ls = specimen gauge length. A second useful parameter to define is:
Ks'=
2xs'
Is
(30)
which, for Ks -< 1, may be considered as the probability of being within that distance of a fibre end where the load is being transferred. Here, xsl is again a transfer length corresponding to an externally applied load, L~, but it has a different definition from the transfer length used with continuous fibres. In this section, it is the minimum distance from the end of the fibre, over which the fibre can gain all its share of the load from the uncracked matrix. Ks is thus a function of load, such that: Lq+ 1 Kfq + 1 =
L~ " XS~
(31)
since: Lq.+,
Lq " Xsq
Xfq + 1
(32)
For a general initial load Lo. , (insufficient to crack the matrix) applied to the specimen, a particular fibre and its associated matrix carry a load: 2sEsrtrs 2 Lo. t ~-,
/_._, 2:Eyrtr: z N
Let the load carried by the fibre be Ly(s) and the load carried by the associated matrix be Lm(sr If we assume that the area of matrix controlled by a particular fibre is:
124
J. SPURRIER
A.,2fEfrrr f z
•
:.sE:rs 2
N
then:
EmA,,
L/(s)O.l+
Lm(s)O.l =
_ Lo.1 2sE:rcr: 2
L:(~)o.1 1 + E )./Eyrir 2 N
E 2:E/rrr:2 N
if the ratio of matrix load to controlling fibre load is to be a constant when transfer of load between the two is complete. Hence: L o. 12sEsnr: 2 (33) Lf(s)°'l "= ~ , 2:Eynr:2 + E,.A., N
and:
Zra(s)O'l
= ~
L°'IEmA" • 2:Eyrir:2 2:E:nr: 2 + EmAm ~ 2:E/rrrs 2
N
(34)
N
(Note: these equations are written in such a way that the summation to express the behaviour of the composite is always over N fibres. Strictly speaking, at a chosen cross-section, a fibre which is known to be in that cross-section will carry a load: L o. iEyr~r:2 L ~/($)o.I
).:EfTtr:2 + EmAm N
and the composite behaviour is expressed by summing over the 2:N: fibres of each type in that cross-section.) If a particular fibre is known to be included in the chosen cross-section, its transfer length is related to the applied load by the equation:
Lo.1 "Eyrir:2 = 2~ryzyX:o.x E 2:E:nr: + EmA,,, N
i.e. X:o.l =
L°'lE:r¢
(35)
2z: ( ~ 2:E:A: + E,.A,.) It is also convenient to define a second transfer length x:~. 1 as the minimum distance along a fibre over which a load, equal to that share of an applied load, L 1, borne by both the fibre and its associated matrix, can be transmitted (see Fig. 9).
TENSILE
BEHAVIOUR
OF MANY-FIBRE-TYPE
125
COMPOSITES
FIBRE LOAD
LOAD L1
<
i
I
I~ztl" BEFORE
~
LOAD L1
CRACKING
FIBRE LOAD
I
I~
.MATRIX CRACK
s)1'I
L•.
/
.L~..(. $)1~ ¢
•"
LOAD <
=
"',
I
L1 ~-1r'1.1.1
AFTER
I
>LOAD L1
..-
I
CRACKING
Fig. 9. A diagram to illustrate the transfer lengths, x : 1 and x : ]. l on a discontinuous fibre which is long enough, and correctly positioned, to bridge the matrix crack when it forms.
First crack By virtue o f the definition o f 2/, the total load carried by a certain type o f fibre at a particular cross-section is equal to the average load carried by that type o f fibre, times the number o f fibres in the cross-section. The two possible situations which arise, depending on whether Z:o.~ is or is not greater than unity for an applied load, Lo.~, are shown in Fig. 10. When Z/O.l<- 1 mean load o f fibre = L:(s)o.l( 1 - ½Zro.1) m e a n load o f associated matrix = Lm(s)o.~ +
½)Cfo.ILf(s)O-I
126
J. SPURRIER FIBRE LOAD
LOAD
LOAD
LO.1
[ ''>
LO.1
:zl0.~' FOR
X.~.o. ~ < 1
FIBRE LOAD
[ LI"(s)0.1 .~
LOAD
LO. I
,
,
OAD >
LO.1
"r:ro. ~ FOR 7-.~0.~ > 1 Fig. 10.
A diagram to illustrate the significance of the value of Z on the load carried by a d i s c o n t i n u o u s fibre in an uncracked matrix.
When •fo.1 > 1, the m a x i m u m sharing load o f the fibre L:(s)O. ~ is not reached. Provided that there is no debonding from the ends o f the fibre, the m a x i m u m load, at the centre o f the discontinuous fibre _
Lf(s)O.1
Z.ro.1 Hence: mean load o f fibre _
Z f ( s ) o.1
2Z: o. 1 mean load o f associated matrix
TENSILE BEHAVIOUROF MANY-FIBRE-TYPECOMPOSITES
= Lmo)o.l + Lf(s)O.l
(
1
127
l)
2Z~0.1
The total matrix load over the whole cross-section:
Lmo.1 =
~ (Lmt~)o.~ + ½Zfo.lLf(~)o.1) ZfO.l-~1
+~(Lmfs)O.lq-Lf(s)o. 1 Lf(s) 1 O.
z/O.l> 1
2Zfo.l */
(Lm(s)o.1 "4- ½Z/o. IL/(s)o.1)
N
(36)
-- )~f0.1 1 ~> \'((~f°'l-1)2"Lf(s)°'l)--~f~l
The matrix first cracks when the load applied to the whole specimen reaches L~. Hence, putting:
Lml Am
~mu
give s:
~ ~fEfAf(1L1 = trmuAm 1 +
N fmAm + ~
N
½~fl) -~ ~ ZfI> 1
)':f/af(½•/l) -
?~
~fEfAf[()~f1 -
Xfl>1
l)2/2Zfl]
)'fffAf[(Zfl -
]
I)2/2Zfi]
where:
Zfl =
2Xfl
(37)
If
This equation, in conjunction with eqn. (35) (for an applied load L~) gives a complicated expression which may be solved for the load L~ at which matrix cracking first takes place.
Debonding Provided that the fibre bridges hold and the specimen is sufficiently long, when the matrix has cracked in one place at load L~ it will be sufficiently loaded to crack elsewhere. Until now, the matrix has been transferring load to the fibres. When a crack has formed, the bridging fibres must transmit all the extra load back to the matrix on either side until the total stress in the matrix again reaches its ultimate value.
128
J. SPURRIER
When a crack crosses the specimen, this analysis assumes that there are so many fibres of each type that it crosses all types of fibre at all possible points. Where the crack intersects a transferring fibre, or the transferring end of a fibre, it may be assumed that the fibre/end of fibre debonds, and takes no further load if frictional effects are ignored. The only fibres which actively bridge the crack are those with a length greater than 2xI1.1 which are intersected at distances greater than x;1.1 from either end. The main barrier to the formulation of simple equations for the tensile behaviour of a composite reinforced by aligned discontinuous fibres is the determination of a value for x i l . l , since it has to be found by an iterative process in conjunction with the number of bridging fibres. The equations are too long and too complicated to be worth recording, but it may be possible to obtain specific numerical solutions by computational methods. The cycle is to first take all the fibres which are capable of bridging when the transfer length is x i l , to share amongst them the excess load from the cracked matrix and from the debonded fibres and fibre ends, and then to calculate the first trial solution for the transfer length xsl. ~. Some fibres which were capable of bridging at xll are now likely to debond, and so this cycle has to be repeated until it is clear which, if any, fibre bridges hold. The assumption that Lits) 1 = Ls(s~l. 1 (see Fig. 9) would enable xil. ~ to be calculated directly from x I l , but this is equivalent to saying that no debonding occurs, or that its effect is insignificant. Especially in view of the fact that the lengths of fibres normally used in experimental composites are less than an order of magnitude greater than their transfer lengths, xy~, it does not appear to be a valid assumption. LOAD
CARRIED
BY INTERFACES
•
_*~rH FRICTION NO FRICTION
t DEBONDING
ll'
L LOAD APPLIED
OCCURS
Fig. 11. A schematic representation of debonding, followed by possible frictional load transfer along the fibre-matrix interface.
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
129
So far it has been assumed that the interfacial bond holds at all values of load provided that there is a sufficiently large distance for load transfer to be completed. If the concept of debonding as a result of the applied load exceeding the bond strength is introduced (see Fig. ll), with or without the subsequent possibility of frictional load transfer, the model rapidly becomes exceedingly complex. Nonplanar matrix cracking is then almost certain to occur.
Matrix multiple cracking--load calculation The technique of integration from the site of the crack to determine the load carried by each fibre type is not very suitable for application to discontinuous FIBRE LOAD
LOAD
EBONDED
M,IR,X CRACK
I
Fig. 12. Diagrams to show the fibre load distribution of a transferring fibre which has debonded at one end because of the formation of a matrix crack, and the range of such distributions which should be integrated to give the load distribution of that fibre type on each side of the matrix crack.
130
J. SPURRIER
fibres, because the maximum load carried by a fibre which is transferring over its entire length changes with the position of the crack, as shown diagrammatically in Fig. 12. There are also problems in describing the matrix load associated with a particular fibre, when the fibre is itself removing load from the matrix. The only simple solution appears to be one where the average load over the whole length of the fibres is calculated by double integration techniques (see Appendix) for those fibres which are not actively bridging. The load removed by these fibres in the vicinity of the crack is then represented by the constant value LIA v for a distance lI on each side of the crack. Two such square waveforms are shown in Fig. 13. The total load removed by the non-bridging fibres at some distance w from the crack is obtained by summing all such waveforms, and the difference between L 1 and this load represents the joint load carried by the bridging fibres and matrix. The total fraction of this load borne by the matrix, however, is still:
Ls(~) + L~(~)
~.
+
N
AVERAGE OF INTEGRATED FIBRE LOAD FOR DIFFERENT TYPE OF FIBRE
I "~ ,
FIBRE
~q"~-. MAT RI X LENGTH.~.-~ CRACK
I
Fig. 13. A diagram of the simplified load distribution employed for two fibre types, and the range of possible positions of the discontinuous fibre with respect to the matrix crack for their loads to be included in the calculations for the average load, LyAV,for that fibre type.
TENSILE BEHAV1OUR OF MANY-FIBRE-TYPE COMPOSITES
131
which is independent of the applied load. Clearly, this simple model implies that further matrix cracks cannot form closer to any existing matrix cracks, at load L1, than a distance equal to the longest fibre length, lltr,,x ). The most probable separation 12 will thus be ~lytm,~). If it is valid for this model to consider secondary matrix multiple cracking, then the secondary cracks will form in the centre of the blocks, and a rough guide to the load at which this cracking occurs is given by: Zl 2
L2
L 1 ~
L~/2
where L~/2 is the load removed by the non-bridging fibres in the position corresponding to the centre of the primary matrix blocks. Unless L 2 is very close to LI, however, it seems unlikely that secondary cracking will occur in a form which can be accurately predicted by a model which makes such drastic simplifications, especially if there is a possibility of longitudinal cracks being present as a result of debonding.
CONCLUSION
An acceptable working model has been formulated to describe the load extension curve o f a brittle matrix composite when the reinforcement consists of assorted continuous aligned fibres. It is theoretically possible for the sarhe techniques to be applied to discontinuous reinforcement, but severe complications arise as a result o f load transfer at the fibre ends, and these effects can neither be overlooked nor suitably simplified. Whereas the whole load extension'curve may be predicted for continuous reinforcement, with reasonable accuracy, the method is likely to yield only an approximation to the load at which the matrix first cracks when used with discontinuous fibres. Random orientations of fibres has not been considered. The incorporation of a suitable efficiency factor 16" 17 r/, into the equations for continuous reinforcement is not difficult but there is little practical interest in such formulae since it is desirable that continuous fibres are aligned in the direction of the tensile axis where this is known to avoid serious losses in strength and efficiency. The suggested model for aligned continuous different fibre reinforcement, together with the suggested modifications, should prove to be a fairly flexible system capable of giving reasonable predictions for a wide spectrum of possible composites, and it may be possible to use the formulae to design composites to certain stress-strain requirements. The likely effect of the inclusion of a certain proportion of damaged fibres is available directly from the formulae. It should also be possible to get some theoretical indication of the effects of fibre bunching in composites by splitting the composite into regions with various percentages of
J. SPURRIER
132
fibres in them, and treating each region in the manner in which a fibre and its associated matrix area were employed in formulating the original model.
APPENDIX" AVERAGE DISCONTINUOUS FIBRE LOADS
In calculating the average load by double integration for the non-bridging discontinuous fibres, five separate cases have to be considered: (a) l > 2 x : l , l I > 2x:1 + x:1.1 B (b) 1"> 2xf1.1 l < 2Xy I + x : l . 1 B (c) I < 2 x : 1 . 1 l < 2 x y l T (d) l < 2 x y l . 1 l > 4xyl T (e) l < 2xy1.1 2 x f l < l < 4Xyl T Those fibres labelled B are sufficiently long to be bridging fibres, but have to be included because the crack crosses some of them within the transfer distance of the fibre ends. Those labelled T are not capable of bridging the matrix crack.
FIBRE LOAD
~-g~
t
II ....
o
"rtl
I
i
j_
II
ii'--o',o MATRIX"~ CRACK
I)C1.1
Fig. 14. A diagram to show the method used for the double integration to calculate the average load for a fibre which, although long enough to bridge a matrix crack, has been crossed by the crack within a distance, x: 1.1, of an end, and has therefore debonded. (a) (See Figure 14): Average load for the proportion of these fibres which are not actively bridging:
"Uf-xyl.1)
2Ly(s)l . . . .
Therefore: average load for fibre type
Xf 1 If
+
.,xf I
2Lf(s)l
X f 1.1
133
TENSILE BEHAVIOUR OF MANY-FIBRE-TYPE COMPOSITES
2xfx ~f + x:"l] 2:N:" 2Xzl.tlf
= Lf(s)I(I
since the totalnumber of this type of fibrein a cross-section = 2/N; and the chance that the fibrewill partlydebond __ 2X$1"1
b
(b) Average load
2L:(~)I" Xf~l " ~f " dy
kd¢If-xf l'l)
f (':) (I~yx 2Ly(s)l z
+ J~2X:l~
dz
I
:'/2 2Ly¢s,xdz~f) ]
.X.f.I. If + ax$1 |
d?
Xfl.1
Therefore: average for fibre type Lf(s)___~l[Zxf13 -- (l: - x/l'')3 =
]2Vx71
Xfl'1
(l_f2 +
)] 2x:t" " 2:N: -
x:,
"
b
(c) Average load
=f(") f~22Lj,(,).._____..~l,z .dz d, J(J~lf) •fl ½~f If ½b Therefore: average load for fibre type = Lf(,)l.7 . 2 : N : Z:~
24
(d) Average load
z
=f,,,, J(*tS)
dz) dy
d~ + ( ,,22L:¢s)i~ 2Lsf,)1"--'~f Xf 1 ..,xf 1
V:
Therefore: average load for fibre type
= Ls(s),(~ - ~) "AfNf (e) Average load z =
L J(-t-lf)
dz
2Lf{s) 1 . .x/~ . . . l/ dy
~_{(If)J(2xf l,(ill 1
Z~,zdzf ,,2
d_~) ]
I X
--
½b
134
J. SPURRIER
Therefore: average load for fibre type
=
Lf(s)l [8xf13 _ ~153 ½1I L 121rxyl +(~-xI')]
"2'NJ"
The average load for the fibre types takes into consideration the n u m b e r o f fibres o f that type in a particular cross-section, a n d is thus a measure of the load per unit length removed from L1 by all fibres o f that type over a distance, lI, on b o t h sides o f the matrix crack. The resulting block diagram, similar to Fig. 13, will thus represent the load b o r n e by fibre types instead of by the individual fibres.
REFERENCES
1. N. F. ASTBURY,Advances in materials research in the N A T O nations, Pergamon Press, 1963, pp. 369-90. 2. G. S. HOUSTERand R. THOMAS,Fibre-reinforced materials, Applied Science Publishers Ltd, London, 1967. 3. J. P. ROMUALDI,Proc. Cement and Concrete Assn. Conference, 'Structure of Concrete', Imperial College, London, September, 1965. 4. J. P. ROMUALDIand G. B. BATSON,Jnl. Eng. Mech. Div., Am. Soc. Cir. Eng. EM3 (1963) p. 147. 5. M. F. KAPLAN,Jnl. Am. Concr. Inst., 58(5) (1961) p. 591. 6. G. A. COOPERand J. M. SILLWOOD,J. Mat. Sci., 7 (1972) p. 325. 7. A. KELLY,Strong solids, Clarendon Press, 1966. 8. H. L. Cox, Brit. J. AppL Phys., 3 (1952) p. 72. 9. N. F. Dow, GEC Missile and Space Div. Report R63SD61, pp. 1-42. 10. J. O. OUTWATER,Mod. Plast., 33 (1956) p. 156. 11. B. W. ROSEN,Fibre Comp. Mats., Am. Soc. Metals, (1965) p. 37. 12. J. SPURRIERand A. LUXMOORE,Fibre Sci. and Tech., 6 (1973) p. 281. 13. A. J. MAJUMDAR,Proc. Roy. Soc., 319A (1970) p. 69. 14. A. J. MAJOMDARand J. F. RYDER,Sci. Ceram., 5 (1970) p. 539. 15. I. W. NORRIS,Thesis for B.Sc. Degree, Dept. of Civil Eng., Univ. Coll. Swansea, CP/454/75, 1975. 16. H. KRENCrlEL,Fibre reinforcement, Akademisk Forlag, 1964. 17. V. LAWS,J. Appl. Phys., 4 (1971) p. 1737.