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A re-examination of the analysis of toughening in brittle-matrix composites
OOOl-6160/89 $3.00+ 0.00 Copyright 0 1989Pergamon Press plc
Acra metall. Vol. 37, No. 9, pp. 2297-2304, 1989 Printed in Great Britain. All rights reserved
A RE-EXAMINATION OF THE ANALYSIS OF TOUGHENING IN BRITTLE-MATRIX COMPOSITES M. D. THOULESS
IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Received
31 October
1988; in revisedform
13 March 1989)
Abstract-The effect of interfacial friction on the use of the J-integral for calculating the toughness of a brittle-fibre reinforced composite has been examined. A simple micro-mechanical model that includes friction as the only dissipative mechanism yields a lower bound on the toughness. When the same constitutive equations are used with a J-integral formulation, an upper bound is obtained which incorporates other energy losses associated with the mechanics of fibre failure. Finally, a method is provided by which the J-integral can be correctly combined with statistical models of fibre failure to incorporate the effects of fibre pull-out on the toughness of a composite. R&m&-Nous reaxaminons I’usage des l’inttgrale J pour calculer la composante de contact de la rtsistance dans les composites renforcts par des fibres fragiles. Le glissement de friction au niveau de l’interface fibre/matrice est responsable du durchissement, de sorte que l’intkgrale J dipend du trajet dans la zone proche de la fissure oi la fibre est arrachie. Nous analysons dans cet article l’importance de cet effet et l’erreur que I’on commet si l’on n’en tient pas compte. Zusnmmenfassung-Die Nutzung des J-Integrales bei der Berechnung der Zghigkeitskomponente, die von der oberbtickung in spriiden faserverstiirkten Verbundwerkstoffen herriihrt, wurde nochmals untersucht. Die Ursache der Ziihigkeit ist die reibende Gleitung an der Grenzfllche zwischen Faser und Matrix, so dal3 das J-Integral in dem Gebiet in der Nlhe des Risses, in dem die Faser herausgezogen wird, nicht wegunabhangig ist. In dieser Arbeit wird untersucht, wie wichtig dieser Effekt ist und welcher Fehler bei der Nichtbeachtung auftritt.
1.
INTRODUCTION
One toughening mechanism that has received a great deal of attention recently is that of crack bridging [l-6]. A number of possible microstructural entities may produce ligaments that bridge the surfaces of a crack. For example, in polycrystalline materials there may be grains that do not initially rupture as the crack passes around them [7-91. There are other systems, such as WC/Co, in which the ligaments consist of a ductile second phase dispersed in a brittle matrix [l&13]. In a third category, which includes brittle-fibre reinforced composites, the second phase is weakly bonded to the matrix [14-211. Energy is then absorbed not only by deformation of the ligaments, but also in the debonding or/and sliding processes that may occur at the interface between the ligaments and matrix. An elegant formulation of the procedure used to determine the toughness of an elastic body toughened by crack-surface tractions can be made using the J-integral [22] J=
Qdx
2
-T.*ds ax,
where r is a path that starts from one crack surface and ends on the other, s is the arc length, Q is the strain-energy density, T is the traction vector and u is the displacement vector on the contour. The coordinate system and the sense in which the contour A.M. ,,,%A
is traversed is shown in Fig. l(a). In an elastic material, the integration is independent of the path taken and gives the rate of change of the potential energy of the system as the crack advances (i.e. the crack-driving force). Under equilibrium conditions, when the load applied to the system is just sufficient to cause crack propagation, the crack-driving force, J,, is equal to the crack-resistance force, i.e. J,=&,+@
(2)
where ~8~ is the intrinsic toughness for the system (the energy required to create unit area of new crack surface) and Bb is an additional bridging term which can be obtained by performing the integration of equation (1) along the crack surfaces [l] Bb = 2 the area fraction of ligaments on the crack surface,fr{u} relates the average stress on the crack surface to the semi crack-opening displacement, u, and tl is the value of u at the end of the traction zone [l, 4,5] [Fig. l(b)]. In the approach adopted in this paper, the value of Re depends upon whether or not ligament failure accompanies crack propagation. If ligament failure occurs, then the intrinsic toughness of the composite system is given by the rule of mixtures wherefis
2291
(4a)
2298
THOULESS:
ANALYSIS OF TOUGHENING
IN BRITTLE-MATRIX
(a)
COMPOSITES
2. CALCULATION OF TOUGHNESS
In this section, a procedure is described to determine, from micro-mechanical considerations, the toughness of a system containing inelastic regions resulting from such processes as frictional-bridging, micro-cracking, phase transformations or plasticity. In principle, if the constitutive behaviour is known, it should be possible to calculate the stress distribution throughout the body under the conditions at which crack advance is about to occur, perform the J-integral along a contour in the elastic region, and deduce 9s from equation (1). An equivalent approach, which utilises the path-independence of the J-integral, is to imagine a “black box” enclosing the crack tip and region of hysteresis [Fig. 2(a)]. Provided that any examination is limited to the exterior of the box, it is impossible to distinguish the “real” system from any other in which the stresses and displacements are matched on the boundaries of the box. In particular, the J-integral will be identical for all such systems. If it is possible to replace the “real” material inside the box by elastic material and a single crack with surface tractions [Fig. 2(b)], equation (3a) could then be used to determine the toughness. This procedure, as used to evaluate the toughness of a brittlematrix composite with a particular form of assumed constitutive behaviour, is further elaborated below.
fb)
Fig. 1. (a) The contour of integration used for computing the J-integral. (b) The contour used for calculating the bridging contribution to toughening in a completely elastic material, where L( is the crack-opening displacement and fr{u) is the average stress on the crack surfaces.
(a) ,i_M__t_L_t-, -I
where 9: and 9;” are the intrinsic toughnesses of the matrix and fibres. If ligament failure does not accompany crack propagation
p
then
we = (I -f)wl.
b
m-aterial
t4b)
The term 9&,in equation (3) represents the energy absorbed from the system by the ligaments and is therefore unavailable for crack propagation. The particular utility of this equation is that it allows the toughness of a composite to be predicted from a knowledge of the micro-mechanical behaviour of the system. However, a problem arises in its use for composites because of the hysteretic region surrounding the crack which has a different constitutive law from the bulk of the material. It is the energy dissipated by frictional sliding within this region that results in the enhanced toughness of the composite. &, cannot be obtained by simply taking the J-integral along the crack surfaces, because path-inde~ndency no longer applies and the equality of equation (2) at equilibrium is not satisfied. It is the purpose of the paper to consider this problem and to illustrate one method by which the analysis using the J-integral can be modified. This may be of particular importance when trying to incorporate the effects of statistical variations in fibre strengths [23,24].
(bl
Fig. 2. (a) The properties of a system containing a crack with an associated non-elastic region will be indistinguishable from (b) a perfectly elastic system, provided that the stresses and displacements are exactly matched along a given contour in the elastic region and that any examination is limited to the exterior of this contour. The contours illustrated in this figure by dashed lines can be identified with the “black box” referred to in the text.
THOULESS:
ANALYSIS OF TOUGHENING
The micro-mechanical model chosen to examine the toughness of a composite is that described by Marshall et al. [17] [Fig. 3(a)]. The interface between the fibre and matrix is assumed to be unbonded with a constant shear resistance oft. A consequence of this assumption is that as the fibre is pulled a distance u out of the matrix, the stress in the fibre decreases linearly from a maximum value of T in the plane of the matrix surface while the matrix stress increases in a linear fashion from zero. Within a region known as the slip length, there is a difference in the strains in the fibre and matrix; this results in relative slip between the two components. The extent of the slip length I(T), in which this linear variation of stress occurs, can be determined by calculating the position at which the strains in the fibre and matrix are equal. Beyond this point the strains in both components are matched so the stresses are in proportion to the moduli. Reference (171 provides the equations that govern fibre pull-out; two of particular importance are I(T)=
w
TR + tl)
(54
IN BRITTLE-MATRIX
COMPOSITES
where S is the fibre strength which is assumed to be
a constant. Frictional dissipation in the sliding zone provides one argument for treating this equation with suspicion. An alternative form of expressing this same objection is to realise that the position of the crack surface has been assumed to be given by the location of the cracked portion of the matrix, whereas one might expect some average position which includes both the fibres and matrix to better reflect an effective crack surface. Such a modi~~tion would result in a smaller crack-opening displacement, and would consequently imply that equation (6) is an over-estimate of the toughness. Each element illustrated in Fig. 3(a) can be separated from the composite by a cut along the slip-zone boundary, and tractions can be applied to the cut so that the stresses and displacements in the bulk of the composite remain unchanged. The elements can then be replaced by equivalent ones of the same length lf Tj and with the same average stressfron the crack plane [Fig. 3(b)]. If the stresses in the new elements are distribute in such a way that the strains in the fibre and matrix are equal everywhere, i.e.
Tr Tm Em
(74
-=I-
7’(u) = 2[rE,( 1 -I- q)/R]“’ u “*
4 Pb)
where R is the fibre radius, n =f&/[(l -f)E,,,], and E, and E,,, are Young’s moduli of the fibre and matrix respectively. The toughness of the composite has been obtained by identifying equation (5b) as the function relating the tractions on the crack surface to the crack-opening displa~ment required in equation (3) [4, 171.Such an approach would suggest that the contribution of bridging to steady-state toughness is
2299
and .P’r+ (1 -f)T,
=fT
(7b)
where T, and T,,, are the stresses in the fibre and matrix, then there will be no losses inside the system. A relationship can then be obtained betweenfT and the crack-opening displacement u * of the new system, so that equation (3) can be used to determine gs. The crack-owning displacement u*{ T) is related to u {T) by
u*fTj
= u{T} + 6(T)
- P{Tj
@a)
(6) where &{7’) and 6*{ T) are defined in Fig. 3. 6{T) is given by [17]
rRT2
6{T}=
4rE,(l + ?)2
@b)
and d*{T} is determined by calculating the strain in the composite element of Fig. 3(b) a*{ T} zflT E
(8~)
where EC, the average modulus of the composite, equals fE, + (1 -S)&, . Hence, using equation (5), it can be shown that T{u*f
I++
LCrackplane
Fig. 3. (a) The micro-mechanical model used for anaiysing fibre pull-out [17]. (b) An equivalent element for a system in which there are no losses, and for which the stresses and dispia~ments at the boundary of the slip zone match those of the original system.
= 2(1 + v)(~E,,‘R)“~ u*‘/~.
(9)
Therefore, the contribution of bridging to the toughness of a composite system when fibre failure occurs is
(10) so that, as predicted, equation (6) is an over-estimate of the toughness. This is identical to the results of
2300
THOULESS:
ANALYSIS OF TOUGHENING
McCartney [20] and Budiansky and Amazigo [21] which were obtained by slightly different approaches. The error in equation (6) is directly att~butable to the over-estimation, by a factor (1 f Q), of the effective crack-opening displacement. The error disappears as q -+O, i.e. when the compliance of the fibres is much greater than that of the matrix. The opposite limit, in which the relative errors are very large, corresponds to the trivial limit of no slipping and zero toughening so that the absolute discrepancy is again zero.
3. ENERGY PARTITION
Having used the J-integral to calculate the increase in toughness associated with the application of tractions, it is also of interest to repeat the calculations by considering the partition of the energy terms involved in crack propagation. Steady-state conditions, in which the extent of bridging is independent of the crack length, will be considered first. The energy required to advance a crack by an incremental amount 6A is identically equal to that involved in cycling an element of the composite (of crosssectional area 0) through the stresses and displacements it would experience upon passing though the bridging zone. It will be recognised that the first term contributing to the toughness must be i@&A [equation (4a)], since in steady state both the matrix and fibres are broken. In the equivalent system of Fig. 3(b), there are no residual stresses once the tractions are removed (when the fibres are assumed to be broken). Therefore, other than the creation of new surface area, the work done by the tractions is the only mechanism of energy dissipation. The total work done by the tractions acting on an area 6A of the composite is
IN BRITTLE-MATRIX
COMPOSITES
assumptions involved in the constitutive laws of both bounds can be understood by reference to Fig. 4. The rising portion of this curve is a plot of equation (Sb), and the fibres break at a stress T = S, when u = a’. Were it possible for the crack to close during fibre failure, the stress could fall to zero along the solid line of Fig. 4 [25]. This path represents a rather unphysical failure process, but it is one for which the hysteresis is completely attributable to frictional losses. In practice, the tractions will fall to zero along the path represented by the dotted line, and this is the form of the constitutive law implicitly assumed in the J-integral calculations. The difference in energy dissipation between these two paths represents other forms of hysteresis which cannot be accounted for by the simple model used in this analysis.? The work done against friction during the initial part of the cycle, while the fibres are being pulled out of the matrix, is given by
fRS3
J/*= 62E,(l + q)’
6A
’
When the fibre breaks, there is some reverse slip as the fibre attempts to slip back into the matrix. The region of reverse slip only extends half way up the original slip length, and consequently there is residual tension in the tibre and residual compression in the matrix 1251.If the unloading process is represented by the solid line of Fig. 4, there are three terms that constitute additional sinks of energy: the work done against friction upon reverse sliding
fRs3
,/, &,# ’ - 24rE,( 1 + q)2
(13b)
W,=cIA2
‘fT{U*]du*=&YA (11) s0 where Q’ is the value of the semi crack-opening displacement for which T = S. The J-integral ap preach therefore predicts a toughness of
Equation (12) is essentially an upper bound in which it is assumed that none of the work done by the tractions applied to the crack surface can be returned to the system. A lower bound on the toughness can be obtained using the micro-mechanical model of Fig. 3(a) and ass~ing that, apart from fictional losses, all other processes are reversible. The relevant equations are summarised in the Appendix, and the
Tit should be noted that it is conceivable that some of this energy may in fact be returned to the system in the form of elastic waves. The toughness could then be slightly less than predicted by equation (12).
Crack-opening displacement,
u
Fig. 4. A stress/displacement curve illustrating the fibre stress in the plane of the crack as a function of the separation of the matrix surfaces. The dotted line illustrates the form of the curve that would give rise to an upper bound for the toughness. The solid line yields a lower bound in which the crack is permitted to close as fibre failure occurs.
ANALYSIS
THOULESS:
OF TOUGHENING
fRs’rl
=
m
24zE,(l
SA
+ u)j
(13c)
Uf=
SA. lt>3
24zE,(l+
A lower bound on the toughness of the composite under steady-state conditions is obtained by adding up the four terms of equation (13)
%==:(I -fPc+fse
t
fRS3 +4rEf(* +r)*’
[see also equation (I l)]. Under these conditions, there is also some strain energy associated with this element {S c
} f
=I(fSJ2 2T(21(W4) c
-f&S?
g
SA_
2tE, (1 + n)2
UW
Since this last term involving the elastic deformations in the composite is completely recoverable, it does not contribute to the toughening. The toughness of the composite in the absence of any fibre failure is therefore
SA
fRsi SA + q)’
hE,(l
.
(17c)
The only term in equation (17) that can unambiguously be considered to be dissipative is the one involving friction. It is not obvious to what extent the other terms contribute to the toughness, but it should be noted that
The energy required to propagate the crack therefore consists of the frictional losses plus the difference between the stored strain energy and the strain energy of the equivalent, perfectly elastic system. There is now no ambiguity caused by uncertainty about the details of fibre failure. 4. FlBRE FAILURE
IN
MATRIX
Fibres exhibiting a unique strength will always fail in the crack plane where the stress is a maximum. Although this was an underlying assumption of the previous two sections, in practice, the strength of a brittle fibre is not constant and the fibre may fail inside the matrix [23]. The broken portion of the fibres may subsequently be pulled out of the matrix so that the tractions on the crack face will not drop precipitously to zero. The work done in pulling the broken fibres out of the matrix may contribute substant~alIy to the toughness of the system. Again, the J-integral can be used to compute this contribution, provided that a relationship can be found between the tractions and an effective crack-opening displacement. In calculations that follow, it will be assumed that S{T) is always much smaller than the slip length at fibre failure, Is, and that the region of reverse slip at the end of the embedded portion of the broken fibre, I{T}, is always contained in the original slip zone [Fig. 5(a)]. Furthermore, it will also be assumed that
The energy terms associated with the “real” element of Fig. 3(a) are calculated in the Appendix. When the peak fibre stress is S,, the energy dissipated as work done against friction is $#,(S,) =
(17b)
(14)
It should be noted that equations (12) and (14) are almost identical. The small difference between them is a measure of the irreversibilities, other than friction, that are associated with fibre failure and may contribute to the toughness. Equation (14) includes only friction as a mechanism of energy dissipation; it is therefore a lower bound on the toughness, whereas equation (12) is an upper bound within the constraints of the particular model used. The partition of the energy terms is fairly obvious once stready state has been reached; it is somewhat more ambiguous when fibre failure does not occur. As an example, consider the case where the semi crack-opening displacement at the mouth of the bridging zone is ec, and the maximum stress in the fibres is S,, which is less than S. The work done by the tractions in loading the equivalent element of Fig. 3(b) up to an average stress of $S, is
fJ
2301
and the strain energy in the matrix is fR.2 r Um{S.) =-6zE, (1 +r,~)~
and the residual energy stored in the fibres fRS'
COMPOSITES
f,,,(S ) _fRS: 3r?*+ 3rl+ 1 sA f ? -(1 + r~)~ 6zEt
the residual strain energy stored in the matrix u
IN BRITTLE-MATRIX
(1W
the strain energy in the fibres within the slip length is
Matrix
Stress
(bl
Fig. 5. (a) The geometry for analysing the pull-out of a broken fibre. (b) The stress distribution in the matrix during this process.
2302
ANALYSIS OF TOUGHENING
THOULESS:
the strain in the fibre can be neglected. The traction in the fibre is Iimited by the shear resistance of the remaining interface between the fibre and matrix, so that
T=2(h-u)
(1%
where h is the length of the broken fibre [Fig. 5(a)]. The relative displa~ment between the matrix surface and the original slip boundary is 6{7-)=&,1T)
UE,z,
(20)
where cf,{Tf is the mean stress in the matrix. By considering the stress distribution of Fig. 5(b), it can be shown that
IN BRITTLE-MATRIX
COMPOSITES
equivalent crack in calculations of the composite toughness. When fibre failure accompanies crack advance, it may occur either in the crack plane at a well-defined stress or in the matrix at a stress determined by the statistics of the problem. In the former case, the J-integral provides an upper bound on the toughness, as compared to the energy-balance approach which provides a lower bound. This difference can be rationalised by appreciating that the micromechanical model used to describe the fibre pull-out includes no description of how energy might be dissipated during the fibre-failure process, except by friction. Fibre pull-out provides an additional contribution to the toughening when fibres fail in the matrix. In principle, an effective crack position can be determined for this process which, when combined with statistical effects, will allow the toughness of the composite J-integral.
system
to be computed
using
the
Acknowledgements-ThE author acknowledges useful discussions with Drs P. A. Mataga, A.G. Evans, R. F. Cook and D. R. Clarke.
+ T,(l, - h - f{T}) where [equation (7)]
--- f
REFERENCES
T
CW
Tm-tl-f)(l+‘l) and
l{T}=~~T,
Wb)
27Cl+ tt>
Therefore
46 V’) II
t,zhT =11+11)+
I,T (1 +q)
RR1 470
t22j
+%I. +rlY
Combining this with equations (8), the relationship between the tractions and the effective crack-opening displacement can be found as
u*{T}=h
-g
+ htt*-4(1 -rl)-
RR1 + 2rl)tl 47t1
+11>
1 T
x------. W1 +a)
(23)
A complete calculation of the toughness must incorporate this equation with equation (9) for the unbroken fibres, and the statistics linking h, 1, and the probability of fibre failure. Existing analyses do this only approximately by using equation (19) rather than equation (23) [23,24]. 5. CONCLUSIONS
The analysis conducted in this paper has considered a method of calculating the toughness of a brittle-matrix composite in which hysteresis effects beyond the plane of the crack are significant. It is possible to define an effective position for the surface of a crack in a composite material and to use this
I. B. Budiansky, Micromechanics II, pp. 2532, in Proc. Tenth U.S. National Congr. Appl. Mech., Austin, Texas (1986). 2. L. R. F. Rose, Int. J. Fract. 33, 145 (1987). 3. L. R. F. Rose, J. Mech. Phys. Solidr 35, 383 (1987). 4. A. G. Evans and R. M. McMeeking, Acta MetaN. 34, 2435 (1986). 5. B. Budiansky, J. C. Amazigo and A. G. Evans, J. Mech. Phys. Sol& 36, 167 (1988). 6. R. F. Cook, J. Mater. Res. 2, 345 (1987). 7, R. G. Hoagland, A. R. Rosenfeld and G. T. Hahn, Metalf. Trans. A 3, 123 (1972). 8. P. L. Swanson, C. J. Fairbanks, 8. R. Lawn, Y-W Mai and B. J. Hockey, f. Am. Ceram. Sot. 70, 279 (1987). 9. Y.-W. Mai and B. R. Lawn, J. Am. Ceram. Sot. 70,289 (1987). 10. S. Kunz-Douglass, P. W. R. Beaumont and M. F. Ashby, J. Mater. Sci. 15, 1109 (1980). 11. L. S. Sigl and H. E. Exner, Metall. Tram A 18, 1299 (1987). 12. L. S. Sigl, A. G. Evans, P. A. Mataga, R. M. McMeeking and B. J. Dalgleish, Acta metail. 36, 945 (1988). 13. L. S. Sigl and H. F. Fishmeister, Acta metall. 36, 887 (1988). 14. J. Aveston, G. A. Cooper and A. Kelly, in The Properties of Fiber Comaosites. Conf. Proc.. National Phvsical Laboratory, p~.~l5-26: IPC Science and Tech&logy Press (t971). 1.5. J. Aveston and A. Kelly, J. Mater. Sei. 8, 352 (1973). 16. J. Bowlinn and G. W. Groves. J. Mater. Sci. 14. 443 (1979). 17. D. B. Marshall, B. N. Cox and A. G. Evans, Actu Metall. 33, 2013 (1985). 18. B. Budiansky, J.‘ W. ‘Hutchinson and A. G. Evans, J. Mech. Phys. Solids 34, 167 (1986). 19. D. B. Marshall and A. G. Evans, Proc. F$th Int. Co& on Composite Materials (Edited by W. C. Harrigan and J. Strife), pp. 557-568. Am. Inst. Min. Engrs, New York (1985).
20. L. N. McCartney, Froc. R. SW. A4Q9, 329 (1987). 21. B. Budiansky and J. C. Amazigo, J. Mech. Phys. Solids 37, 93 (1989). 22. J. R. Rice, J. Appf. Me&. 35, 379 (1968).
THOULESS:
ANALYSIS OF TOUGHENING
23. M. D. Thouless and A. G. Evans, Acta metall. 36, 517 (1988). 24. B. N. Cox, D. B. Marshall and M. D. Thouless, Acta metall. 37, 1933 (1989). 25. D. B. Marshall and W. C. Oliver, J. Am. Gram. Sot. 70, 542 (1987).
IN BRIDLE-MATRIX
COMPOSITES
2303
The work done against friction is &I $,{&}
=(hcRr)2
"
~:~I#
(A54
5 where IL{<} is the relative displacement between the tibre and matrix at a point z = 5. Therefore
APPENDIX Fibre Pull-Out
The equations used to describe the micro-mechanics of fibre putl-out are summa~sed in this Appendix. There are two distinct regimes of behaviour: fibre pull-out and reverse slipping after the fibres fail. The former regime was modelled in Ref. 1171and the latter can be modelled using minor modifications to the original analysis, When the peak stress in the fibre is T the slip length extends a distance /{T) on both sides of the crack plane [Fig. 3(a)][l7]. TR
I{T} =p 20
+ vl)
where T, R, 7, and n have been defined in the text. Inside the slip length, the strains in the matrix and fibre are given by 1171
A term involving the strain energy in the portion of the fibre that extends outside the matrix might also be included; however, this can be ignored if E,>>S,. Fibre Failure
If it is assumed that the fibres have a unique strength of S, they will fail and attempt to slip back into the matrix when T = S. A region of reverse slip extends a distance I, {T} from the crack surface into the former slip region [Fig. Al(a) [25]. The boundary between the slip and reverse-slip regions is marked by an abrupt change in the direction of the interfacial shear stresses. If, after failure, T decreases smoothly from S to zero along the solid line of Fig. 4 [25], then the stresses in the fibre and matrix in the reverse-slip region can be found by a force-balance calculation
where .z is the distancesmeasured from the slip boundary [Fig. 3(a)]. When T = S,, the strain energy in a single fibre within the slip region on both sides of the crack plane is
s vz I
U,{S,}
= (nR*)2
,
2E,t:{S,,
z} dz
where .Y is the distance from the reverse-slip boundary [Fig. Al(b)]. The corresponding strains are therefore
0
fRS; =_
3q= + 3rj + 1 6A
(A3) (I + q)’ where 6A = nR’/f, and is the cross-sectional area of composite containing a single fibre. The strain energy of the matrix in the corresponding volume is 65E,
The extent over which reverse slip takes place can be shown to be
(A81
(A4)
where 1, is the slip length when the fibre tractions equal S [equation (Al)] (A9) In particular, the reverse-slip region only extends over a distance I,/2 when the tractions have been reduced to zero. Residual stresses therefore exist in the composite as shown in Fig. At(c). From these equations it is possible to compute the energy changes undergone by an element of material as the fibres are loaded up to S and back to zero again. The work done against friction includes that done on the Loading cycle [equation AS(b)] and an additional reverse-slip term upon unloading
‘r’ T (a)
(b)
(cl
Fig. Al. (a) The distribution of shear stresses when unloading of a fibre begins. The slip length at the peak applied load is I,, and the extent of reverse slip is I,{T}. (b) The stress distribution in the fibre and matrix during the reverse-slip process. (c) The residual stress distribution in the fibre and matrix after fibre failure, when T = 0.
hZ $, = (2nRz) 2
d5 s”
‘(~I(O>~.}--6,{0,x})dx s0
(AlO) When T = 0, the residual stresses in the fibre and matrix inside the original slip length are symmetrical about the
2304
THOULESS:
ANALYSIS OF TOUGHENING
slip/reverse-slip boundary [Fig. Al(c)]. Consequently, the total residual strain energy in the fibre on both sides of the crack can be determined as cJ,=(nP)4
b/l 1 -.f+:{o,x)dx s cl 2
IN BRIDLE-MATRIX
COM~SITES
The strain energy in the matrix is u _ &(l m
-f)
f
4
‘& 1
- E,c;{O, f0 2
2nRZEf ‘d2 =c;{O,x}dx rl s0
x) dx
Acra metall. Vol. 37, No. 9, pp. 2297-2304, 1989 Printed in Great Britain. All rights reserved
A RE-EXAMINATION OF THE ANALYSIS OF TOUGHENING IN BRITTLE-MATRIX COMPOSITES M. D. THOULESS
IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Received
31 October
1988; in revisedform
13 March 1989)
Abstract-The effect of interfacial friction on the use of the J-integral for calculating the toughness of a brittle-fibre reinforced composite has been examined. A simple micro-mechanical model that includes friction as the only dissipative mechanism yields a lower bound on the toughness. When the same constitutive equations are used with a J-integral formulation, an upper bound is obtained which incorporates other energy losses associated with the mechanics of fibre failure. Finally, a method is provided by which the J-integral can be correctly combined with statistical models of fibre failure to incorporate the effects of fibre pull-out on the toughness of a composite. R&m&-Nous reaxaminons I’usage des l’inttgrale J pour calculer la composante de contact de la rtsistance dans les composites renforcts par des fibres fragiles. Le glissement de friction au niveau de l’interface fibre/matrice est responsable du durchissement, de sorte que l’intkgrale J dipend du trajet dans la zone proche de la fissure oi la fibre est arrachie. Nous analysons dans cet article l’importance de cet effet et l’erreur que I’on commet si l’on n’en tient pas compte. Zusnmmenfassung-Die Nutzung des J-Integrales bei der Berechnung der Zghigkeitskomponente, die von der oberbtickung in spriiden faserverstiirkten Verbundwerkstoffen herriihrt, wurde nochmals untersucht. Die Ursache der Ziihigkeit ist die reibende Gleitung an der Grenzfllche zwischen Faser und Matrix, so dal3 das J-Integral in dem Gebiet in der Nlhe des Risses, in dem die Faser herausgezogen wird, nicht wegunabhangig ist. In dieser Arbeit wird untersucht, wie wichtig dieser Effekt ist und welcher Fehler bei der Nichtbeachtung auftritt.
1.
INTRODUCTION
One toughening mechanism that has received a great deal of attention recently is that of crack bridging [l-6]. A number of possible microstructural entities may produce ligaments that bridge the surfaces of a crack. For example, in polycrystalline materials there may be grains that do not initially rupture as the crack passes around them [7-91. There are other systems, such as WC/Co, in which the ligaments consist of a ductile second phase dispersed in a brittle matrix [l&13]. In a third category, which includes brittle-fibre reinforced composites, the second phase is weakly bonded to the matrix [14-211. Energy is then absorbed not only by deformation of the ligaments, but also in the debonding or/and sliding processes that may occur at the interface between the ligaments and matrix. An elegant formulation of the procedure used to determine the toughness of an elastic body toughened by crack-surface tractions can be made using the J-integral [22] J=
Qdx
2
-T.*ds ax,
where r is a path that starts from one crack surface and ends on the other, s is the arc length, Q is the strain-energy density, T is the traction vector and u is the displacement vector on the contour. The coordinate system and the sense in which the contour A.M. ,,,%A
is traversed is shown in Fig. l(a). In an elastic material, the integration is independent of the path taken and gives the rate of change of the potential energy of the system as the crack advances (i.e. the crack-driving force). Under equilibrium conditions, when the load applied to the system is just sufficient to cause crack propagation, the crack-driving force, J,, is equal to the crack-resistance force, i.e. J,=&,+@
(2)
where ~8~ is the intrinsic toughness for the system (the energy required to create unit area of new crack surface) and Bb is an additional bridging term which can be obtained by performing the integration of equation (1) along the crack surfaces [l] Bb = 2 the area fraction of ligaments on the crack surface,fr{u} relates the average stress on the crack surface to the semi crack-opening displacement, u, and tl is the value of u at the end of the traction zone [l, 4,5] [Fig. l(b)]. In the approach adopted in this paper, the value of Re depends upon whether or not ligament failure accompanies crack propagation. If ligament failure occurs, then the intrinsic toughness of the composite system is given by the rule of mixtures wherefis
2291
(4a)
2298
THOULESS:
ANALYSIS OF TOUGHENING
IN BRITTLE-MATRIX
(a)
COMPOSITES
2. CALCULATION OF TOUGHNESS
In this section, a procedure is described to determine, from micro-mechanical considerations, the toughness of a system containing inelastic regions resulting from such processes as frictional-bridging, micro-cracking, phase transformations or plasticity. In principle, if the constitutive behaviour is known, it should be possible to calculate the stress distribution throughout the body under the conditions at which crack advance is about to occur, perform the J-integral along a contour in the elastic region, and deduce 9s from equation (1). An equivalent approach, which utilises the path-independence of the J-integral, is to imagine a “black box” enclosing the crack tip and region of hysteresis [Fig. 2(a)]. Provided that any examination is limited to the exterior of the box, it is impossible to distinguish the “real” system from any other in which the stresses and displacements are matched on the boundaries of the box. In particular, the J-integral will be identical for all such systems. If it is possible to replace the “real” material inside the box by elastic material and a single crack with surface tractions [Fig. 2(b)], equation (3a) could then be used to determine the toughness. This procedure, as used to evaluate the toughness of a brittlematrix composite with a particular form of assumed constitutive behaviour, is further elaborated below.
fb)
Fig. 1. (a) The contour of integration used for computing the J-integral. (b) The contour used for calculating the bridging contribution to toughening in a completely elastic material, where L( is the crack-opening displacement and fr{u) is the average stress on the crack surfaces.
(a) ,i_M__t_L_t-, -I
where 9: and 9;” are the intrinsic toughnesses of the matrix and fibres. If ligament failure does not accompany crack propagation
p
then
we = (I -f)wl.
b
m-aterial
t4b)
The term 9&,in equation (3) represents the energy absorbed from the system by the ligaments and is therefore unavailable for crack propagation. The particular utility of this equation is that it allows the toughness of a composite to be predicted from a knowledge of the micro-mechanical behaviour of the system. However, a problem arises in its use for composites because of the hysteretic region surrounding the crack which has a different constitutive law from the bulk of the material. It is the energy dissipated by frictional sliding within this region that results in the enhanced toughness of the composite. &, cannot be obtained by simply taking the J-integral along the crack surfaces, because path-inde~ndency no longer applies and the equality of equation (2) at equilibrium is not satisfied. It is the purpose of the paper to consider this problem and to illustrate one method by which the analysis using the J-integral can be modified. This may be of particular importance when trying to incorporate the effects of statistical variations in fibre strengths [23,24].
(bl
Fig. 2. (a) The properties of a system containing a crack with an associated non-elastic region will be indistinguishable from (b) a perfectly elastic system, provided that the stresses and displacements are exactly matched along a given contour in the elastic region and that any examination is limited to the exterior of this contour. The contours illustrated in this figure by dashed lines can be identified with the “black box” referred to in the text.
THOULESS:
ANALYSIS OF TOUGHENING
The micro-mechanical model chosen to examine the toughness of a composite is that described by Marshall et al. [17] [Fig. 3(a)]. The interface between the fibre and matrix is assumed to be unbonded with a constant shear resistance oft. A consequence of this assumption is that as the fibre is pulled a distance u out of the matrix, the stress in the fibre decreases linearly from a maximum value of T in the plane of the matrix surface while the matrix stress increases in a linear fashion from zero. Within a region known as the slip length, there is a difference in the strains in the fibre and matrix; this results in relative slip between the two components. The extent of the slip length I(T), in which this linear variation of stress occurs, can be determined by calculating the position at which the strains in the fibre and matrix are equal. Beyond this point the strains in both components are matched so the stresses are in proportion to the moduli. Reference (171 provides the equations that govern fibre pull-out; two of particular importance are I(T)=
w
TR + tl)
(54
IN BRITTLE-MATRIX
COMPOSITES
where S is the fibre strength which is assumed to be
a constant. Frictional dissipation in the sliding zone provides one argument for treating this equation with suspicion. An alternative form of expressing this same objection is to realise that the position of the crack surface has been assumed to be given by the location of the cracked portion of the matrix, whereas one might expect some average position which includes both the fibres and matrix to better reflect an effective crack surface. Such a modi~~tion would result in a smaller crack-opening displacement, and would consequently imply that equation (6) is an over-estimate of the toughness. Each element illustrated in Fig. 3(a) can be separated from the composite by a cut along the slip-zone boundary, and tractions can be applied to the cut so that the stresses and displacements in the bulk of the composite remain unchanged. The elements can then be replaced by equivalent ones of the same length lf Tj and with the same average stressfron the crack plane [Fig. 3(b)]. If the stresses in the new elements are distribute in such a way that the strains in the fibre and matrix are equal everywhere, i.e.
Tr Tm Em
(74
-=I-
7’(u) = 2[rE,( 1 -I- q)/R]“’ u “*
4 Pb)
where R is the fibre radius, n =f&/[(l -f)E,,,], and E, and E,,, are Young’s moduli of the fibre and matrix respectively. The toughness of the composite has been obtained by identifying equation (5b) as the function relating the tractions on the crack surface to the crack-opening displa~ment required in equation (3) [4, 171.Such an approach would suggest that the contribution of bridging to steady-state toughness is
2299
and .P’r+ (1 -f)T,
=fT
(7b)
where T, and T,,, are the stresses in the fibre and matrix, then there will be no losses inside the system. A relationship can then be obtained betweenfT and the crack-opening displacement u * of the new system, so that equation (3) can be used to determine gs. The crack-owning displacement u*{ T) is related to u {T) by
u*fTj
= u{T} + 6(T)
- P{Tj
@a)
(6) where &{7’) and 6*{ T) are defined in Fig. 3. 6{T) is given by [17]
rRT2
6{T}=
4rE,(l + ?)2
@b)
and d*{T} is determined by calculating the strain in the composite element of Fig. 3(b) a*{ T} zflT E
(8~)
where EC, the average modulus of the composite, equals fE, + (1 -S)&, . Hence, using equation (5), it can be shown that T{u*f
I++
LCrackplane
Fig. 3. (a) The micro-mechanical model used for anaiysing fibre pull-out [17]. (b) An equivalent element for a system in which there are no losses, and for which the stresses and dispia~ments at the boundary of the slip zone match those of the original system.
= 2(1 + v)(~E,,‘R)“~ u*‘/~.
(9)
Therefore, the contribution of bridging to the toughness of a composite system when fibre failure occurs is
(10) so that, as predicted, equation (6) is an over-estimate of the toughness. This is identical to the results of
2300
THOULESS:
ANALYSIS OF TOUGHENING
McCartney [20] and Budiansky and Amazigo [21] which were obtained by slightly different approaches. The error in equation (6) is directly att~butable to the over-estimation, by a factor (1 f Q), of the effective crack-opening displacement. The error disappears as q -+O, i.e. when the compliance of the fibres is much greater than that of the matrix. The opposite limit, in which the relative errors are very large, corresponds to the trivial limit of no slipping and zero toughening so that the absolute discrepancy is again zero.
3. ENERGY PARTITION
Having used the J-integral to calculate the increase in toughness associated with the application of tractions, it is also of interest to repeat the calculations by considering the partition of the energy terms involved in crack propagation. Steady-state conditions, in which the extent of bridging is independent of the crack length, will be considered first. The energy required to advance a crack by an incremental amount 6A is identically equal to that involved in cycling an element of the composite (of crosssectional area 0) through the stresses and displacements it would experience upon passing though the bridging zone. It will be recognised that the first term contributing to the toughness must be i@&A [equation (4a)], since in steady state both the matrix and fibres are broken. In the equivalent system of Fig. 3(b), there are no residual stresses once the tractions are removed (when the fibres are assumed to be broken). Therefore, other than the creation of new surface area, the work done by the tractions is the only mechanism of energy dissipation. The total work done by the tractions acting on an area 6A of the composite is
IN BRITTLE-MATRIX
COMPOSITES
assumptions involved in the constitutive laws of both bounds can be understood by reference to Fig. 4. The rising portion of this curve is a plot of equation (Sb), and the fibres break at a stress T = S, when u = a’. Were it possible for the crack to close during fibre failure, the stress could fall to zero along the solid line of Fig. 4 [25]. This path represents a rather unphysical failure process, but it is one for which the hysteresis is completely attributable to frictional losses. In practice, the tractions will fall to zero along the path represented by the dotted line, and this is the form of the constitutive law implicitly assumed in the J-integral calculations. The difference in energy dissipation between these two paths represents other forms of hysteresis which cannot be accounted for by the simple model used in this analysis.? The work done against friction during the initial part of the cycle, while the fibres are being pulled out of the matrix, is given by
fRS3
J/*= 62E,(l + q)’
6A
’
When the fibre breaks, there is some reverse slip as the fibre attempts to slip back into the matrix. The region of reverse slip only extends half way up the original slip length, and consequently there is residual tension in the tibre and residual compression in the matrix 1251.If the unloading process is represented by the solid line of Fig. 4, there are three terms that constitute additional sinks of energy: the work done against friction upon reverse sliding
fRs3
,/, &,# ’ - 24rE,( 1 + q)2
(13b)
W,=cIA2
‘fT{U*]du*=&YA (11) s0 where Q’ is the value of the semi crack-opening displacement for which T = S. The J-integral ap preach therefore predicts a toughness of
Equation (12) is essentially an upper bound in which it is assumed that none of the work done by the tractions applied to the crack surface can be returned to the system. A lower bound on the toughness can be obtained using the micro-mechanical model of Fig. 3(a) and ass~ing that, apart from fictional losses, all other processes are reversible. The relevant equations are summarised in the Appendix, and the
Tit should be noted that it is conceivable that some of this energy may in fact be returned to the system in the form of elastic waves. The toughness could then be slightly less than predicted by equation (12).
Crack-opening displacement,
u
Fig. 4. A stress/displacement curve illustrating the fibre stress in the plane of the crack as a function of the separation of the matrix surfaces. The dotted line illustrates the form of the curve that would give rise to an upper bound for the toughness. The solid line yields a lower bound in which the crack is permitted to close as fibre failure occurs.
ANALYSIS
THOULESS:
OF TOUGHENING
fRs’rl
=
m
24zE,(l
SA
+ u)j
(13c)
Uf=
SA. lt>3
24zE,(l+
A lower bound on the toughness of the composite under steady-state conditions is obtained by adding up the four terms of equation (13)
%==:(I -fPc+fse
t
fRS3 +4rEf(* +r)*’
[see also equation (I l)]. Under these conditions, there is also some strain energy associated with this element {S c
} f
=I(fSJ2 2T(21(W4) c
-f&S?
g
SA_
2tE, (1 + n)2
UW
Since this last term involving the elastic deformations in the composite is completely recoverable, it does not contribute to the toughening. The toughness of the composite in the absence of any fibre failure is therefore
SA
fRsi SA + q)’
hE,(l
.
(17c)
The only term in equation (17) that can unambiguously be considered to be dissipative is the one involving friction. It is not obvious to what extent the other terms contribute to the toughness, but it should be noted that
The energy required to propagate the crack therefore consists of the frictional losses plus the difference between the stored strain energy and the strain energy of the equivalent, perfectly elastic system. There is now no ambiguity caused by uncertainty about the details of fibre failure. 4. FlBRE FAILURE
IN
MATRIX
Fibres exhibiting a unique strength will always fail in the crack plane where the stress is a maximum. Although this was an underlying assumption of the previous two sections, in practice, the strength of a brittle fibre is not constant and the fibre may fail inside the matrix [23]. The broken portion of the fibres may subsequently be pulled out of the matrix so that the tractions on the crack face will not drop precipitously to zero. The work done in pulling the broken fibres out of the matrix may contribute substant~alIy to the toughness of the system. Again, the J-integral can be used to compute this contribution, provided that a relationship can be found between the tractions and an effective crack-opening displacement. In calculations that follow, it will be assumed that S{T) is always much smaller than the slip length at fibre failure, Is, and that the region of reverse slip at the end of the embedded portion of the broken fibre, I{T}, is always contained in the original slip zone [Fig. 5(a)]. Furthermore, it will also be assumed that
The energy terms associated with the “real” element of Fig. 3(a) are calculated in the Appendix. When the peak fibre stress is S,, the energy dissipated as work done against friction is $#,(S,) =
(17b)
(14)
It should be noted that equations (12) and (14) are almost identical. The small difference between them is a measure of the irreversibilities, other than friction, that are associated with fibre failure and may contribute to the toughness. Equation (14) includes only friction as a mechanism of energy dissipation; it is therefore a lower bound on the toughness, whereas equation (12) is an upper bound within the constraints of the particular model used. The partition of the energy terms is fairly obvious once stready state has been reached; it is somewhat more ambiguous when fibre failure does not occur. As an example, consider the case where the semi crack-opening displacement at the mouth of the bridging zone is ec, and the maximum stress in the fibres is S,, which is less than S. The work done by the tractions in loading the equivalent element of Fig. 3(b) up to an average stress of $S, is
fJ
2301
and the strain energy in the matrix is fR.2 r Um{S.) =-6zE, (1 +r,~)~
and the residual energy stored in the fibres fRS'
COMPOSITES
f,,,(S ) _fRS: 3r?*+ 3rl+ 1 sA f ? -(1 + r~)~ 6zEt
the residual strain energy stored in the matrix u
IN BRITTLE-MATRIX
(1W
the strain energy in the fibres within the slip length is
Matrix
Stress
(bl
Fig. 5. (a) The geometry for analysing the pull-out of a broken fibre. (b) The stress distribution in the matrix during this process.
2302
ANALYSIS OF TOUGHENING
THOULESS:
the strain in the fibre can be neglected. The traction in the fibre is Iimited by the shear resistance of the remaining interface between the fibre and matrix, so that
T=2(h-u)
(1%
where h is the length of the broken fibre [Fig. 5(a)]. The relative displa~ment between the matrix surface and the original slip boundary is 6{7-)=&,1T)
UE,z,
(20)
where cf,{Tf is the mean stress in the matrix. By considering the stress distribution of Fig. 5(b), it can be shown that
IN BRITTLE-MATRIX
COMPOSITES
equivalent crack in calculations of the composite toughness. When fibre failure accompanies crack advance, it may occur either in the crack plane at a well-defined stress or in the matrix at a stress determined by the statistics of the problem. In the former case, the J-integral provides an upper bound on the toughness, as compared to the energy-balance approach which provides a lower bound. This difference can be rationalised by appreciating that the micromechanical model used to describe the fibre pull-out includes no description of how energy might be dissipated during the fibre-failure process, except by friction. Fibre pull-out provides an additional contribution to the toughening when fibres fail in the matrix. In principle, an effective crack position can be determined for this process which, when combined with statistical effects, will allow the toughness of the composite J-integral.
system
to be computed
using
the
Acknowledgements-ThE author acknowledges useful discussions with Drs P. A. Mataga, A.G. Evans, R. F. Cook and D. R. Clarke.
+ T,(l, - h - f{T}) where [equation (7)]
--- f
REFERENCES
T
CW
Tm-tl-f)(l+‘l) and
l{T}=~~T,
Wb)
27Cl+ tt>
Therefore
46 V’) II
t,zhT =11+11)+
I,T (1 +q)
RR1 470
t22j
+%I. +rlY
Combining this with equations (8), the relationship between the tractions and the effective crack-opening displacement can be found as
u*{T}=h
-g
+ htt*-4(1 -rl)-
RR1 + 2rl)tl 47t1
+11>
1 T
x------. W1 +a)
(23)
A complete calculation of the toughness must incorporate this equation with equation (9) for the unbroken fibres, and the statistics linking h, 1, and the probability of fibre failure. Existing analyses do this only approximately by using equation (19) rather than equation (23) [23,24]. 5. CONCLUSIONS
The analysis conducted in this paper has considered a method of calculating the toughness of a brittle-matrix composite in which hysteresis effects beyond the plane of the crack are significant. It is possible to define an effective position for the surface of a crack in a composite material and to use this
I. B. Budiansky, Micromechanics II, pp. 2532, in Proc. Tenth U.S. National Congr. Appl. Mech., Austin, Texas (1986). 2. L. R. F. Rose, Int. J. Fract. 33, 145 (1987). 3. L. R. F. Rose, J. Mech. Phys. Solidr 35, 383 (1987). 4. A. G. Evans and R. M. McMeeking, Acta MetaN. 34, 2435 (1986). 5. B. Budiansky, J. C. Amazigo and A. G. Evans, J. Mech. Phys. Sol& 36, 167 (1988). 6. R. F. Cook, J. Mater. Res. 2, 345 (1987). 7, R. G. Hoagland, A. R. Rosenfeld and G. T. Hahn, Metalf. Trans. A 3, 123 (1972). 8. P. L. Swanson, C. J. Fairbanks, 8. R. Lawn, Y-W Mai and B. J. Hockey, f. Am. Ceram. Sot. 70, 279 (1987). 9. Y.-W. Mai and B. R. Lawn, J. Am. Ceram. Sot. 70,289 (1987). 10. S. Kunz-Douglass, P. W. R. Beaumont and M. F. Ashby, J. Mater. Sci. 15, 1109 (1980). 11. L. S. Sigl and H. E. Exner, Metall. Tram A 18, 1299 (1987). 12. L. S. Sigl, A. G. Evans, P. A. Mataga, R. M. McMeeking and B. J. Dalgleish, Acta metail. 36, 945 (1988). 13. L. S. Sigl and H. F. Fishmeister, Acta metall. 36, 887 (1988). 14. J. Aveston, G. A. Cooper and A. Kelly, in The Properties of Fiber Comaosites. Conf. Proc.. National Phvsical Laboratory, p~.~l5-26: IPC Science and Tech&logy Press (t971). 1.5. J. Aveston and A. Kelly, J. Mater. Sei. 8, 352 (1973). 16. J. Bowlinn and G. W. Groves. J. Mater. Sci. 14. 443 (1979). 17. D. B. Marshall, B. N. Cox and A. G. Evans, Actu Metall. 33, 2013 (1985). 18. B. Budiansky, J.‘ W. ‘Hutchinson and A. G. Evans, J. Mech. Phys. Solids 34, 167 (1986). 19. D. B. Marshall and A. G. Evans, Proc. F$th Int. Co& on Composite Materials (Edited by W. C. Harrigan and J. Strife), pp. 557-568. Am. Inst. Min. Engrs, New York (1985).
20. L. N. McCartney, Froc. R. SW. A4Q9, 329 (1987). 21. B. Budiansky and J. C. Amazigo, J. Mech. Phys. Solids 37, 93 (1989). 22. J. R. Rice, J. Appf. Me&. 35, 379 (1968).
THOULESS:
ANALYSIS OF TOUGHENING
23. M. D. Thouless and A. G. Evans, Acta metall. 36, 517 (1988). 24. B. N. Cox, D. B. Marshall and M. D. Thouless, Acta metall. 37, 1933 (1989). 25. D. B. Marshall and W. C. Oliver, J. Am. Gram. Sot. 70, 542 (1987).
IN BRIDLE-MATRIX
COMPOSITES
2303
The work done against friction is &I $,{&}
=(hcRr)2
"
~:~I#
(A54
5 where IL{<} is the relative displacement between the tibre and matrix at a point z = 5. Therefore
APPENDIX Fibre Pull-Out
The equations used to describe the micro-mechanics of fibre putl-out are summa~sed in this Appendix. There are two distinct regimes of behaviour: fibre pull-out and reverse slipping after the fibres fail. The former regime was modelled in Ref. 1171and the latter can be modelled using minor modifications to the original analysis, When the peak stress in the fibre is T the slip length extends a distance /{T) on both sides of the crack plane [Fig. 3(a)][l7]. TR
I{T} =p 20
+ vl)
where T, R, 7, and n have been defined in the text. Inside the slip length, the strains in the matrix and fibre are given by 1171
A term involving the strain energy in the portion of the fibre that extends outside the matrix might also be included; however, this can be ignored if E,>>S,. Fibre Failure
If it is assumed that the fibres have a unique strength of S, they will fail and attempt to slip back into the matrix when T = S. A region of reverse slip extends a distance I, {T} from the crack surface into the former slip region [Fig. Al(a) [25]. The boundary between the slip and reverse-slip regions is marked by an abrupt change in the direction of the interfacial shear stresses. If, after failure, T decreases smoothly from S to zero along the solid line of Fig. 4 [25], then the stresses in the fibre and matrix in the reverse-slip region can be found by a force-balance calculation
where .z is the distancesmeasured from the slip boundary [Fig. 3(a)]. When T = S,, the strain energy in a single fibre within the slip region on both sides of the crack plane is
s vz I
U,{S,}
= (nR*)2
,
2E,t:{S,,
z} dz
where .Y is the distance from the reverse-slip boundary [Fig. Al(b)]. The corresponding strains are therefore
0
fRS; =_
3q= + 3rj + 1 6A
(A3) (I + q)’ where 6A = nR’/f, and is the cross-sectional area of composite containing a single fibre. The strain energy of the matrix in the corresponding volume is 65E,
The extent over which reverse slip takes place can be shown to be
(A81
(A4)
where 1, is the slip length when the fibre tractions equal S [equation (Al)] (A9) In particular, the reverse-slip region only extends over a distance I,/2 when the tractions have been reduced to zero. Residual stresses therefore exist in the composite as shown in Fig. At(c). From these equations it is possible to compute the energy changes undergone by an element of material as the fibres are loaded up to S and back to zero again. The work done against friction includes that done on the Loading cycle [equation AS(b)] and an additional reverse-slip term upon unloading
‘r’ T (a)
(b)
(cl
Fig. Al. (a) The distribution of shear stresses when unloading of a fibre begins. The slip length at the peak applied load is I,, and the extent of reverse slip is I,{T}. (b) The stress distribution in the fibre and matrix during the reverse-slip process. (c) The residual stress distribution in the fibre and matrix after fibre failure, when T = 0.
hZ $, = (2nRz) 2
d5 s”
‘(~I(O>~.}--6,{0,x})dx s0
(AlO) When T = 0, the residual stresses in the fibre and matrix inside the original slip length are symmetrical about the
2304
THOULESS:
ANALYSIS OF TOUGHENING
slip/reverse-slip boundary [Fig. Al(c)]. Consequently, the total residual strain energy in the fibre on both sides of the crack can be determined as cJ,=(nP)4
b/l 1 -.f+:{o,x)dx s cl 2
IN BRIDLE-MATRIX
COM~SITES
The strain energy in the matrix is u _ &(l m
-f)
f
4
‘& 1
- E,c;{O, f0 2
2nRZEf ‘d2 =c;{O,x}dx rl s0
x) dx