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Effects of pull-out on the mechanical properties of ceramic-matrix composites
Acta metall. Vol. 36, No. 3, pp. 517-522, 1988 Printed in Great Britain. All rights reserved
Copyright
0
OOOI-6160/88 %3.00 + 0.00 1988 Pergamon Journals Ltd
EFFECTS OF PULL-OUT ON THE MECHANICAL PROPERTIES OF CERAMIC-MATRIX COMPOSITES M. Materials
and A. G.
D. THOULESS
EVANS
Department, College of Engineering, University Santa Barbara, CA 93106, U.S.A. (Received
of California,
6 July 1987)
Ahstract-The influence of fiber pull-out on the mechanical properties of fiber reinforced ceramics has been analyzed using an approach based on weakest-link statistics. The essential physics contributed by the statistics is the establishment of a relationship between the fiber failure site, which governs the pull-out length, and the properties of the fibers, the matrix, and the interface. This analysis involves the development of a stress/displacement law for fibers in the bridging zone of a matrix crack, thereby permitting a discussion of the crack growth resistance and its dependence on relevant microstructural variables. R&sum&Nous avons analyst l’influence du tirage des fibres sur les proprietes mecaniques des dramiques renforcees par des fibres, a l’aide d’une approche basie sur la statistique des liaisons les plus faibles. Le point de physique essentiel apporte par cette statistique est l’ttablissement dune relation entre l’endroit od la fibre va rompre (ce qui controle la longueur du tirage) et les propribtes des fibres, de la matrice et de l’interface. Cette analyse suppose le dtveloppement d’une loi contrainte/deplacement pour les fibres dans la zone qui enjambe une fissure de la matrice; ceci permet ainsi une discussion de la resistance a la croissance des fissures et de la faGon dont elle depend des variables microstructurales correspondantes. Zusammenfamung-Der EinfluB des HerausreiDens der Fasem aus der Matrix auf die mechanischen Eigenschaften einer faserverstarketen Keramik wird mit einer Naherung, die auf der Statistik iiber die schwbhste Verbindung beruht, analysiert. Der wesentliche von der Statistik gelieferte physikalische Aspekt besteht in der Aufstellung einer Beziehung zwischen dem Ort des Faserbruches, der die Lange des herausgezogenen Faserstiickes bestimmt, und der Eigenschaften der Fasem, der Matrix und der Faser/Matrix-Grenzfllche. Die Analyse erfordert die Aufstellung eines Spannungs-Verschiebungsgesetzes fiir die Fasem in der Uberbriickungszone des Matrixrisses. Hierdurch wird die Diskussion des Widerstandes gegen Rigwachstum und dessen Abhlngigkeit von den wichtigen Variablen der Mikrostruktur ermiiglicht
1. INTRODUCTION
Ceramic materials toughened by the incorporation of fibers and whiskers have a toughness, AC,, which is governed by the net traction 5 exerted on the matrix crack by fibers in the crack wake [l-4] (Fig. 1): AG, = 2f
(1)
wherefis the volume fraction of fibers, u is the crack opening and u. is the opening either at the crack mouth or at the edge of the bridging zone (Fig. l), whichever is smaller. The tractions arise from both intact fibers circumvented by the matrix crack [l, 41 and from fibers that fail away from the crack plane (Fig. 1). The broken fibers resist crack opening by frictional sliding which occurs as the broken ends pull out of the matrix [5]. A comprehensive analysis of the mechanical behavior of ceramic composites must involve coupled consideration of both contributions to the crack growth resistance. However, existing analyses [l, 41 have the restriction that the fibers are assumed to exhibit a unique strength S. A consequence of this assumption is that the fibers must fail A.M.36,>!J
517
at the location of highest stress, which is between the surfaces of the matrix crack. The pull-out contribution to toughness has therefore not been formally analyzed. Statistical considerations of fiber failure provide the basis for incorporating pull-out effects and thus govern some of the essential fracture physics. The purpose of the present article is to establish the principal statistical concepts involved in the fracture of ceramic composites. Specifically, equation (1) will be used with the requisite statistics in order to predict trends in crack growth resistance with the relevant properties of the matrix, fibers, and interfaces. Transitions to steady state cracking [6, 71 will also be examined.
2. THE STRESS/DISPLACEMENT 2.1.
LAW
Assumptions
A number of simplifications and assumptions are used in the statistical analysis of the stress/displacement law such that analytic tractability is afforded while the essential physics is retained. These simpli-
518
THOULESS and EVANS:
. .
MECHANICAL
I Fiber Failure
Crack Front
~ LOCA of Mean Fiber Failure Site
thatrix
Fiber
Fig. 1. A schematic indicating crack bridging by intact and fractured fibers. Also shown is the locus of the mean fiber failure site and a typical variation in crack surface traction 5 with crack opening, u, that governs the fracture resistance, AG, fications are presented first in order to clarify the limitations of the resulting solutions. The fibers are assumed to have a sufficiently low
debond resistance so that they slip with respect to the matrix in the crack wake at a constant shear resistance, r [S]. The resulting slip length, 2, is presumed to be large enough to allow all of the interfacial shear to be accommodated along the zone of sliding; elastic shear stresses are therefore ignored [7& The axial stress in each intact fiber varies, within the slip length,? in accordance with [6,7] u = T(l -z/l),
I= RTj2z = ,iT
PROPERTIES
OF COMPOSITES
crack plane. Neglect of stresses normal to the fiber axis in the crack wake is implicit in the use of equation (2), in accordance with the usual shear lag assumption [6,7,9]. The preceding set of assumptions become most restrictive [q when the slip length, I, and pull-out length, h, become small. Such characteristics obtain when either the fiber strengths are low or the shear resistance is large; this will be evident from the analytical solutions developed below. A further assumption of the analysis is the use of a fiber strength distribution that satisfies weakest-link statistics. The explicit form is given by the twoparameter Weibull distribution, in which the probability that the flaws in a surface area of fibers,$ A,, have a strength
2.2, Some basic statistical results By adopting weakest-link statistics and following the analysis of Oh and Finnie [l I] (Appendix I), it is possible to derive a probability density function, @(T, z), for fiber failure as a function of the peak stress, T, and the distance from the crack plane, z: @(T, z) = exp -2 ‘2nR[(a(T, i s0
z)~S*~dz~~~
(2xRlAo)WT,
~)/~ol’“/~T.
x
I
(4)
The factor 2 in the exponential arises because the fiber on both sides of the crack must be considered. Using equation (2) in injunction with equation (4), it is possible to eliminate cr and i such that
(2)
where T is the stress in the fiber at the crack plane, R is the fiber radius, and z is the distance from the x [2nRm(T IStrictly speaking, the slip length is I = RT/%(l + <) where i: = E&/&(1 -f) such that S is the volume fraction of fibers, E is Young’s modulus and the subscripts m and f refer to the matrix and fiber, respectively. Also, within the slip length, a = T[l -z/&l + t)], but beyond, D = T
- z/~)~-‘/A~P].
(5)
This expression for @(T, z) is the basic formula that governs several statistical parameters involved in determining the toughness of fiber composites, notably the mean pull-out length, (h), the mean strength, (S), and the net stress, ii. The average failure position for all fibers that have failed at a stress less than T = S is given by I
s 2
SI
z@(T, z) dz dT
Om of 2 @(T, z) dz dT s0 s0
THOULESS and EVANS:
=
MECHANICAL
PROPERTIES
OF COMPOSITES
519
s s
exp{ -(T/Z)m+1}4nR
II
x (T -z/A)“-
dz dT
(6)
where c = [A,s;2:;*
+ I)]“(*+“.
Remote Failure
f f(h)dh
Recalling yields
that I = AT, integration
of equation
(6)
h-w
(7) where /3 = (T/C)“+‘, c1= (S/Z)‘“+‘, and the integral is the incomplete gamma function, r[(m + 2)/ (m + l), a]. The average failure location when ail the fibers have failed is therefore
?i=& 2 mr[(m
+2)/(m
+ l)]
(8)
0
where r is the complete gamma function. The corresponding solution for the mean strength of all fibers is s=2
I w T@(T, z) dz dT ss0 0
111. (9)
law
Determination of the stress/displacement law that governs toughness requires consideration of the fraction of fibers that fail at each location behind the crack tip and the associated failure site. For present purposes, since it has been assumed that fiber debonding occurs readily, the fraction of failed fibers near the matrix crack front is essentially zero. Then, as the matrix crack extends and opens, the “weaker” fibres begin to fail in the immediate crack wake, near the crack plane. Thereafter, upon additional crack extension, further fibers fail in the wake, at locations further from the crack plane, as depicted in Fig. 2. Fiber failure behavior is examined most conveniently in terms of the crack opening displacement, u, which varies monotonically with the distance from the crack front [l, 41. Then, for intact fibers, the peak stress on the fiber is [6, 7] T = 2[&(1
+ [)/R]“*u”~
(W
while, for failed fibers, the stress at the crack plane
is [2, 51 or = (2r/R)(h bP = 0
- u)
(h < u).
Near Tip Failures
Fig. 2. A schematic indicating the frequency distribution of fiber failure sites,f(h)dh, and trends in h with distance from the crack tip.
The average stress i? on the fibers at an opening u is thus a@)=(1
= CT [(m + 2)/(m + 2.3. Stress/displacement
lh
(h > u)
-q)T+q(e,)
(11)
where q is the fraction of fibers that have failed at opening, u, and (a,) is the average value of the pull-out stress [equation (lob)] based on the average failure length, (h), associated with all prior failures. Implicit in the application of equation (11) is the existence of sufficient fibers along each section to allow use of average values based on large sample statistics. With this premise, 0 can be deduced from equation (11) by noting that the fraction of failed fibers q is just the cumulative fiber failure probability at T = S. s
q-
I
0
s I
2
@(T,z)dzdT=
1 -exp(-cr).
(12)
0
Then, a nondimensional equations (9)-(12) is
result
obtained
from
(C/Z) = (u/v)‘12exp[ -(u/v)(~+‘)~~] 1 +(1 +T)(m
+ 1)
{I -exp[-(u/v)(“+‘)12]}
x r[(m + 2)/(m + l), (u/v)(~+‘)‘*] {
(lob) (13)
TI-IOULESS and EVANS: MECHANICAL PROPERTIES OF COMPOSITES
520 where
v = C2Rl[4Err(l + 01, r[(m +2)&l
+ l), (U/v)(m+‘)‘21 =
O1 B l/(m+i)e-fld/I s0
and
and is thus proportional to [Rm-5/tm-2]1’(m+1)for given (U./V). It is notable that an inversion in the trend with z occurs at m = 2, and with R at m = 5. The corresponding pull-out contribution presented in its entirety is rather unwieldy. However, the trends can be examined by recognizing that the result has the form AC, w
Using choices for the nondimensional parameters, Z/E, and {, appropriate for typical ceramic composites [S] (5 x 10m3 and 2, respectively), trends in the nondimensional stress (ii/C) with the nondimensional displacement (u/v) can be plotted (Fig. 3).
3. TRENDS IN MECHANICAL BEHAVIOR 3.1. Crack growth resistance Selection of fiber properties (m and So) within the typical range obtained experimentally [ 12, 131enables prediction of specific stress/displacement characteristics for a crack surface bridged by fibers (Fig. 4). It is apparent from such plots that the toughness quantity, r 2f a du J [equation (l)], is very sensitive to the properties of both the fiber and the interface. Furthermore, inspection of equation (13) reveals that this quantity always increases as So increases, thereby establishing that high fiber strengths are invariably desirable. However, most important, the dependence on z and R is ambivalent. The essential details are highlighted by considering separately the bridging and pull-out contributions to the toughness integral. The bridging component is given by “‘(1 -q)Tdu
AG,=2f
I = 4L
y[3,(m + 1),
(,*/v)‘m+r)‘2]/(m+ 1) (14)
intact Fiber Bridging
Pull-Out
which, with equation (7) indicates a toughness proportional to [Rm-3/~m-‘]‘~(mf’). The foregoing reveals the trends in toughness with r, R, and m. The toughness increases with increasing R when m > 5, and decreases when m < 3. Conversely, it increases with increasing r when m is very small (< l), and decreases when m > 2. These limits arise because of the competing importance of the contribution to toughness from the intact bridging fibers and the failed fibers that experience pull-out. Knowledge of the magnitude of the statistical shape parameter, m, for the fibers within the composite is therefore a prerequisite to optimizing the shear properties of the interface and the fiber radius. While the above trends are expected to be reflected in the fracture behavior of ceramic composites, it should be emphasized that the actual fracture resistance is also influenced by the crack mouth opening, u., which is often much smaller than the mean pull-out length, h. This is tantamount to expecting resistance curves that rise for very large changes in crack length. The relevant failure resistance characteristics can then only be ascertained by the numerical evaluation of u.. The requisite numerical procedure has been previously developed by Marshall and Cox [4] and will be used subsequently to predict these resistance curves. 3.2. Steady state cracking Other aspects of the fracture process can be qualitatively addressed by examination of Fig. 4. For conditions under which a matrix crack is bridged by intact fibers and loaded by a remote stress urn, steady state cracking can occur with 0 = a,/J This is the behavior observed in some composite systems when a series of parallel cracks develop in the matrix [5,6,9]. It can only occur if a,/f is less than 6, the peak stress capable of being supported by the fibers (Figs 2 and 4). The contribution of fiber pull-out is minimal when the average fiber stress is less than 8. Consequently, prior steady state cracking solutions are deemed to be valid even when fiber failure is statistical. These solutions predict that matrix cracking occurs at a stress e,, = WzK;f(l
2 Displacement (u/u)
3
Fig. 3. Trends in nondimensional stress with nondimensional opening displacement for X/E, FZ5 x lo-’ and 5 = 2.
(15)
-f)(l
+ 5)/R]“3
(16)
where & is the matrix toughness. Steady state matrix cracking cannot occur when co/f> 8. Hence, by adopting the reasonably accurate assumption that, at the maximum, only bridging contributes substantially to the stress, the first term
THOULESS and EVANS:
MECHANICAL
PROPERTIES
OF COMPOSITES
000
m=5
10 S,,=140MPa Ill
521
So=22MPa
=
7=25MPa ,R=Spm)
(h=20Ctm)
500
0 Crack
Opening,
I_
I
0
1 Crack
W/pm
(a)
3
2 Opening,
2u//..Lm
@I m=2
0
m=l So=16Pa
So=O.l MPa
0
~~~~~~ (F;=250pm) 0
Crack
Opening,
0
I 1
Crack
2u/pm
Opening,
I
I
2
3
2u/pm
(d)
(c)
Fig. 4. Stress/displacement characteristics for a crack surface bridged by fibers. The purpose of these graphs is to illustrate the dependence of toughness on r and R at particular values of m; and, in particular, the inversion on the dependence between high and low values of m. Consequently, S, has been varied to keep the axes about the same (A, = 1 m*). As stated in the text, it will be recognized that increasing S, invariably increases toughness. The value of the mean pull-out length is given as !i [equation (S)] for each curve.
in equation
(13) gives
B xC(m
+ I)-“(“+‘)exp[-I/(m+1)] exp[- l/(m + j~j.
(17)
Therefore, from equation (16), steady state matrix cracking can only occur when p(Zm-2j~m-5)1:(m+I)
<
(A,Sg2n)3’“+” W
-f)U
Steady state behavior is thus encouraged by small values of the quantity [K,$S~“‘@’ +I)I(2rn-2/~rn-5]l/(rnfl), and once again the importance of m is emphasized. Notice., however, that equation (18) is a necessary but not sufficient criterion for steady state cracking. More complex considerations apply when the fibers are initially broken in the crack wake. 4. CONCLUDING REMARKS
+
x exp[-3/(nz
t)
+ I)].
(18)
The statistical analysis of the location of fiber failure in the presence of a matrix crack provides new
THOULESS and EVANS: MECHANICAL PROPERTIES OF COMPOSITES
522
physical insights and associated statistical models of the mechanical behavior of ceramic-matrix composites. In particular, the coupled effect on crack openings of intact fibers and of fibers that have failed, but are subject to pull-out, has been addressed for the first time, resulting in the derivation of a stress/ displacement law for matrix cracks. This law has been used to obtain preliminary estimates of trends in the crack growth resistance and in the steady state cracking behavior of ceramic composites. In particular, trends with such variables as fiber strength, interfacial shear resistance, fiber diameter, and fiber strength variability have been elucidated. The principal limitation of the present analysis has been the neglect of fiber debonding effects. The results are thus strictly applicable to composites with weakly bonded fibers that exhibit a low interfacial shear resistance. The character of the problem is expected to change when interfacial debonding is limited and fibers fail at the end of the debond zone, rather than in accordance with weakest link statistics. Analysis of this debonding regime is the subject of further
investigation.
F. F. Lange), Vol. 7, pp. 33-51. Plenum, New York (1986). 13. G. Simon and A. R. Bunsell, J. Mater. Sci. 19, 3649 (1984).
APPENDIX Consider a fiber divided into 2N elements each of length, 6.~. The probability that an element will fail when the stress < u can be expressed as
&$(a)=ZxR$
1. D. B. Marshall and A. G. Evans, Fracture Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H.
Ha&man and F. F. Lange), Vol. 7, pp. l-15. Plenum Press, New York (1986). 2. A. G. Evans and R. M. McMeeking, Acta metall. 34, 2435 (1986).
3. B. Budiansky, Proc. Tenth U.S. Natn. Congr. of Applied Mech., pp. 25-32. Austin, Texas (1986). 4. D. B. Marshall and B. N. Cox, Acta metall. To be published. 5. D. C. Phillips, J. Mater. Sci. 7, 1175 (1972). 6. D. B. Marshall, B. N. Cox and A. G. Evans, Acta
p,(T, 2) = 1 - &J(T, 2). (A2) Consequently, the probability that all the elements in the fiber survive, being the product of the survival probabilities, is P,(T, 0 = i] 11-S&T, 211 (A3) n--N where, z = n&z, and I = Ndz. Furthermore, the probability that the element at .zwill fail when the peak stress is between T and T + 6T, but not when the stress is less than T, is [1 1] aQ(T,
+(T,z)]-+-
z) aT
J. Am. Ceram. Sot. 59, 304 (1976).
11. H. L. Oh and I. Finnie, Int. J. Fract. Mechs. 6, 287 (1970). 12. R. L. Stewart, K. Chyung, M. P. Taylor and R. F. Cooper, Fracture Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H. Hasselman and
(A4)
f/ [I - &~(T,z)l FaT. W’, z)aTaz= “-;I”_ s4(T,
Consequently, upon adopting [equation (311that
the Weibull assumption
62 o(T,z) &$(T, z) = 2nR A s 0[ 0
and noting that fi “Z-N
(A5)
z)l
[1-&#~(T,z)]%exp L
c--n
(fw
1 m
68, 225 (1985).
9. J. Aveston, G. A. Cooper and A. Kelly, In The Properties of Fibre Composit&, Conf. Proc., pp. 15-26. Natn. Phvsical Lab. IPC Sci. Technol. Press (1971). 10. J. R. Matthews, F. A. McClintock and W.‘J. Shack,
.
Denoting the probability density function of fiber failure by @(T, z), the probability that fracture occurs at a location z, when the peak stress is T, is equal to @(T, z)6T6z. This probability is governed by the probability that all elements survive up to a peak stress T, but that failure occurs at z when the stress reaches T. It is given by the product of equations (A3) and (A4)
metall. 33, 2013 (1985).
I. B. Budiansky, J. W. Hutchinson and A. G. Evans, J. Mech. Phys. Solids 34, 167 (1986). 8. D. B. Marshall and A. G. Evans, J. Am. Ceram. Sot.
t-41)
where g(S)dS/A, represents the number of flaws per unit surface of fiber having a “strength” between S and S + dS. The local stress, u, is a function of both the distance from the center of the fiber, I, and the peak stress, T. The probability of survival of an element located at z is simply
Pr(T,r)=[l REFERENCES
‘g(S)dS 0s 0
&$(T,z)
1 -I
(.47)
for small values of 64(T, z), one obtains (in the limit, N+co)
Copyright
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OOOI-6160/88 %3.00 + 0.00 1988 Pergamon Journals Ltd
EFFECTS OF PULL-OUT ON THE MECHANICAL PROPERTIES OF CERAMIC-MATRIX COMPOSITES M. Materials
and A. G.
D. THOULESS
EVANS
Department, College of Engineering, University Santa Barbara, CA 93106, U.S.A. (Received
of California,
6 July 1987)
Ahstract-The influence of fiber pull-out on the mechanical properties of fiber reinforced ceramics has been analyzed using an approach based on weakest-link statistics. The essential physics contributed by the statistics is the establishment of a relationship between the fiber failure site, which governs the pull-out length, and the properties of the fibers, the matrix, and the interface. This analysis involves the development of a stress/displacement law for fibers in the bridging zone of a matrix crack, thereby permitting a discussion of the crack growth resistance and its dependence on relevant microstructural variables. R&sum&Nous avons analyst l’influence du tirage des fibres sur les proprietes mecaniques des dramiques renforcees par des fibres, a l’aide d’une approche basie sur la statistique des liaisons les plus faibles. Le point de physique essentiel apporte par cette statistique est l’ttablissement dune relation entre l’endroit od la fibre va rompre (ce qui controle la longueur du tirage) et les propribtes des fibres, de la matrice et de l’interface. Cette analyse suppose le dtveloppement d’une loi contrainte/deplacement pour les fibres dans la zone qui enjambe une fissure de la matrice; ceci permet ainsi une discussion de la resistance a la croissance des fissures et de la faGon dont elle depend des variables microstructurales correspondantes. Zusammenfamung-Der EinfluB des HerausreiDens der Fasem aus der Matrix auf die mechanischen Eigenschaften einer faserverstarketen Keramik wird mit einer Naherung, die auf der Statistik iiber die schwbhste Verbindung beruht, analysiert. Der wesentliche von der Statistik gelieferte physikalische Aspekt besteht in der Aufstellung einer Beziehung zwischen dem Ort des Faserbruches, der die Lange des herausgezogenen Faserstiickes bestimmt, und der Eigenschaften der Fasem, der Matrix und der Faser/Matrix-Grenzfllche. Die Analyse erfordert die Aufstellung eines Spannungs-Verschiebungsgesetzes fiir die Fasem in der Uberbriickungszone des Matrixrisses. Hierdurch wird die Diskussion des Widerstandes gegen Rigwachstum und dessen Abhlngigkeit von den wichtigen Variablen der Mikrostruktur ermiiglicht
1. INTRODUCTION
Ceramic materials toughened by the incorporation of fibers and whiskers have a toughness, AC,, which is governed by the net traction 5 exerted on the matrix crack by fibers in the crack wake [l-4] (Fig. 1): AG, = 2f
(1)
wherefis the volume fraction of fibers, u is the crack opening and u. is the opening either at the crack mouth or at the edge of the bridging zone (Fig. l), whichever is smaller. The tractions arise from both intact fibers circumvented by the matrix crack [l, 41 and from fibers that fail away from the crack plane (Fig. 1). The broken fibers resist crack opening by frictional sliding which occurs as the broken ends pull out of the matrix [5]. A comprehensive analysis of the mechanical behavior of ceramic composites must involve coupled consideration of both contributions to the crack growth resistance. However, existing analyses [l, 41 have the restriction that the fibers are assumed to exhibit a unique strength S. A consequence of this assumption is that the fibers must fail A.M.36,>!J
517
at the location of highest stress, which is between the surfaces of the matrix crack. The pull-out contribution to toughness has therefore not been formally analyzed. Statistical considerations of fiber failure provide the basis for incorporating pull-out effects and thus govern some of the essential fracture physics. The purpose of the present article is to establish the principal statistical concepts involved in the fracture of ceramic composites. Specifically, equation (1) will be used with the requisite statistics in order to predict trends in crack growth resistance with the relevant properties of the matrix, fibers, and interfaces. Transitions to steady state cracking [6, 71 will also be examined.
2. THE STRESS/DISPLACEMENT 2.1.
LAW
Assumptions
A number of simplifications and assumptions are used in the statistical analysis of the stress/displacement law such that analytic tractability is afforded while the essential physics is retained. These simpli-
518
THOULESS and EVANS:
. .
MECHANICAL
I Fiber Failure
Crack Front
~ LOCA of Mean Fiber Failure Site
thatrix
Fiber
Fig. 1. A schematic indicating crack bridging by intact and fractured fibers. Also shown is the locus of the mean fiber failure site and a typical variation in crack surface traction 5 with crack opening, u, that governs the fracture resistance, AG, fications are presented first in order to clarify the limitations of the resulting solutions. The fibers are assumed to have a sufficiently low
debond resistance so that they slip with respect to the matrix in the crack wake at a constant shear resistance, r [S]. The resulting slip length, 2, is presumed to be large enough to allow all of the interfacial shear to be accommodated along the zone of sliding; elastic shear stresses are therefore ignored [7& The axial stress in each intact fiber varies, within the slip length,? in accordance with [6,7] u = T(l -z/l),
I= RTj2z = ,iT
PROPERTIES
OF COMPOSITES
crack plane. Neglect of stresses normal to the fiber axis in the crack wake is implicit in the use of equation (2), in accordance with the usual shear lag assumption [6,7,9]. The preceding set of assumptions become most restrictive [q when the slip length, I, and pull-out length, h, become small. Such characteristics obtain when either the fiber strengths are low or the shear resistance is large; this will be evident from the analytical solutions developed below. A further assumption of the analysis is the use of a fiber strength distribution that satisfies weakest-link statistics. The explicit form is given by the twoparameter Weibull distribution, in which the probability that the flaws in a surface area of fibers,$ A,, have a strength
2.2, Some basic statistical results By adopting weakest-link statistics and following the analysis of Oh and Finnie [l I] (Appendix I), it is possible to derive a probability density function, @(T, z), for fiber failure as a function of the peak stress, T, and the distance from the crack plane, z: @(T, z) = exp -2 ‘2nR[(a(T, i s0
z)~S*~dz~~~
(2xRlAo)WT,
~)/~ol’“/~T.
x
I
(4)
The factor 2 in the exponential arises because the fiber on both sides of the crack must be considered. Using equation (2) in injunction with equation (4), it is possible to eliminate cr and i such that
(2)
where T is the stress in the fiber at the crack plane, R is the fiber radius, and z is the distance from the x [2nRm(T IStrictly speaking, the slip length is I = RT/%(l + <) where i: = E&/&(1 -f) such that S is the volume fraction of fibers, E is Young’s modulus and the subscripts m and f refer to the matrix and fiber, respectively. Also, within the slip length, a = T[l -z/&l + t)], but beyond, D = T
- z/~)~-‘/A~P].
(5)
This expression for @(T, z) is the basic formula that governs several statistical parameters involved in determining the toughness of fiber composites, notably the mean pull-out length, (h), the mean strength, (S), and the net stress, ii. The average failure position for all fibers that have failed at a stress less than T = S is given by I
s 2
SI
z@(T, z) dz dT
Om of 2 @(T, z) dz dT s0 s0
THOULESS and EVANS:
=
MECHANICAL
PROPERTIES
OF COMPOSITES
519
s s
exp{ -(T/Z)m+1}4nR
II
x (T -z/A)“-
dz dT
(6)
where c = [A,s;2:;*
+ I)]“(*+“.
Remote Failure
f f(h)dh
Recalling yields
that I = AT, integration
of equation
(6)
h-w
(7) where /3 = (T/C)“+‘, c1= (S/Z)‘“+‘, and the integral is the incomplete gamma function, r[(m + 2)/ (m + l), a]. The average failure location when ail the fibers have failed is therefore
?i=& 2 mr[(m
+2)/(m
+ l)]
(8)
0
where r is the complete gamma function. The corresponding solution for the mean strength of all fibers is s=2
I w T@(T, z) dz dT ss0 0
111. (9)
law
Determination of the stress/displacement law that governs toughness requires consideration of the fraction of fibers that fail at each location behind the crack tip and the associated failure site. For present purposes, since it has been assumed that fiber debonding occurs readily, the fraction of failed fibers near the matrix crack front is essentially zero. Then, as the matrix crack extends and opens, the “weaker” fibres begin to fail in the immediate crack wake, near the crack plane. Thereafter, upon additional crack extension, further fibers fail in the wake, at locations further from the crack plane, as depicted in Fig. 2. Fiber failure behavior is examined most conveniently in terms of the crack opening displacement, u, which varies monotonically with the distance from the crack front [l, 41. Then, for intact fibers, the peak stress on the fiber is [6, 7] T = 2[&(1
+ [)/R]“*u”~
(W
while, for failed fibers, the stress at the crack plane
is [2, 51 or = (2r/R)(h bP = 0
- u)
(h < u).
Near Tip Failures
Fig. 2. A schematic indicating the frequency distribution of fiber failure sites,f(h)dh, and trends in h with distance from the crack tip.
The average stress i? on the fibers at an opening u is thus a@)=(1
= CT [(m + 2)/(m + 2.3. Stress/displacement
lh
(h > u)
-q)T+q(e,)
(11)
where q is the fraction of fibers that have failed at opening, u, and (a,) is the average value of the pull-out stress [equation (lob)] based on the average failure length, (h), associated with all prior failures. Implicit in the application of equation (11) is the existence of sufficient fibers along each section to allow use of average values based on large sample statistics. With this premise, 0 can be deduced from equation (11) by noting that the fraction of failed fibers q is just the cumulative fiber failure probability at T = S. s
q-
I
0
s I
2
@(T,z)dzdT=
1 -exp(-cr).
(12)
0
Then, a nondimensional equations (9)-(12) is
result
obtained
from
(C/Z) = (u/v)‘12exp[ -(u/v)(~+‘)~~] 1 +(1 +T)(m
+ 1)
{I -exp[-(u/v)(“+‘)12]}
x r[(m + 2)/(m + l), (u/v)(~+‘)‘*] {
(lob) (13)
TI-IOULESS and EVANS: MECHANICAL PROPERTIES OF COMPOSITES
520 where
v = C2Rl[4Err(l + 01, r[(m +2)&l
+ l), (U/v)(m+‘)‘21 =
O1 B l/(m+i)e-fld/I s0
and
and is thus proportional to [Rm-5/tm-2]1’(m+1)for given (U./V). It is notable that an inversion in the trend with z occurs at m = 2, and with R at m = 5. The corresponding pull-out contribution presented in its entirety is rather unwieldy. However, the trends can be examined by recognizing that the result has the form AC, w
Using choices for the nondimensional parameters, Z/E, and {, appropriate for typical ceramic composites [S] (5 x 10m3 and 2, respectively), trends in the nondimensional stress (ii/C) with the nondimensional displacement (u/v) can be plotted (Fig. 3).
3. TRENDS IN MECHANICAL BEHAVIOR 3.1. Crack growth resistance Selection of fiber properties (m and So) within the typical range obtained experimentally [ 12, 131enables prediction of specific stress/displacement characteristics for a crack surface bridged by fibers (Fig. 4). It is apparent from such plots that the toughness quantity, r 2f a du J [equation (l)], is very sensitive to the properties of both the fiber and the interface. Furthermore, inspection of equation (13) reveals that this quantity always increases as So increases, thereby establishing that high fiber strengths are invariably desirable. However, most important, the dependence on z and R is ambivalent. The essential details are highlighted by considering separately the bridging and pull-out contributions to the toughness integral. The bridging component is given by “‘(1 -q)Tdu
AG,=2f
I = 4L
y[3,(m + 1),
(,*/v)‘m+r)‘2]/(m+ 1) (14)
intact Fiber Bridging
Pull-Out
which, with equation (7) indicates a toughness proportional to [Rm-3/~m-‘]‘~(mf’). The foregoing reveals the trends in toughness with r, R, and m. The toughness increases with increasing R when m > 5, and decreases when m < 3. Conversely, it increases with increasing r when m is very small (< l), and decreases when m > 2. These limits arise because of the competing importance of the contribution to toughness from the intact bridging fibers and the failed fibers that experience pull-out. Knowledge of the magnitude of the statistical shape parameter, m, for the fibers within the composite is therefore a prerequisite to optimizing the shear properties of the interface and the fiber radius. While the above trends are expected to be reflected in the fracture behavior of ceramic composites, it should be emphasized that the actual fracture resistance is also influenced by the crack mouth opening, u., which is often much smaller than the mean pull-out length, h. This is tantamount to expecting resistance curves that rise for very large changes in crack length. The relevant failure resistance characteristics can then only be ascertained by the numerical evaluation of u.. The requisite numerical procedure has been previously developed by Marshall and Cox [4] and will be used subsequently to predict these resistance curves. 3.2. Steady state cracking Other aspects of the fracture process can be qualitatively addressed by examination of Fig. 4. For conditions under which a matrix crack is bridged by intact fibers and loaded by a remote stress urn, steady state cracking can occur with 0 = a,/J This is the behavior observed in some composite systems when a series of parallel cracks develop in the matrix [5,6,9]. It can only occur if a,/f is less than 6, the peak stress capable of being supported by the fibers (Figs 2 and 4). The contribution of fiber pull-out is minimal when the average fiber stress is less than 8. Consequently, prior steady state cracking solutions are deemed to be valid even when fiber failure is statistical. These solutions predict that matrix cracking occurs at a stress e,, = WzK;f(l
2 Displacement (u/u)
3
Fig. 3. Trends in nondimensional stress with nondimensional opening displacement for X/E, FZ5 x lo-’ and 5 = 2.
(15)
-f)(l
+ 5)/R]“3
(16)
where & is the matrix toughness. Steady state matrix cracking cannot occur when co/f> 8. Hence, by adopting the reasonably accurate assumption that, at the maximum, only bridging contributes substantially to the stress, the first term
THOULESS and EVANS:
MECHANICAL
PROPERTIES
OF COMPOSITES
000
m=5
10 S,,=140MPa Ill
521
So=22MPa
=
7=25MPa ,R=Spm)
(h=20Ctm)
500
0 Crack
Opening,
I_
I
0
1 Crack
W/pm
(a)
3
2 Opening,
2u//..Lm
@I m=2
0
m=l So=16Pa
So=O.l MPa
0
~~~~~~ (F;=250pm) 0
Crack
Opening,
0
I 1
Crack
2u/pm
Opening,
I
I
2
3
2u/pm
(d)
(c)
Fig. 4. Stress/displacement characteristics for a crack surface bridged by fibers. The purpose of these graphs is to illustrate the dependence of toughness on r and R at particular values of m; and, in particular, the inversion on the dependence between high and low values of m. Consequently, S, has been varied to keep the axes about the same (A, = 1 m*). As stated in the text, it will be recognized that increasing S, invariably increases toughness. The value of the mean pull-out length is given as !i [equation (S)] for each curve.
in equation
(13) gives
B xC(m
+ I)-“(“+‘)exp[-I/(m+1)] exp[- l/(m + j~j.
(17)
Therefore, from equation (16), steady state matrix cracking can only occur when p(Zm-2j~m-5)1:(m+I)
<
(A,Sg2n)3’“+” W
-f)U
Steady state behavior is thus encouraged by small values of the quantity [K,$S~“‘@’ +I)I(2rn-2/~rn-5]l/(rnfl), and once again the importance of m is emphasized. Notice., however, that equation (18) is a necessary but not sufficient criterion for steady state cracking. More complex considerations apply when the fibers are initially broken in the crack wake. 4. CONCLUDING REMARKS
+
x exp[-3/(nz
t)
+ I)].
(18)
The statistical analysis of the location of fiber failure in the presence of a matrix crack provides new
THOULESS and EVANS: MECHANICAL PROPERTIES OF COMPOSITES
522
physical insights and associated statistical models of the mechanical behavior of ceramic-matrix composites. In particular, the coupled effect on crack openings of intact fibers and of fibers that have failed, but are subject to pull-out, has been addressed for the first time, resulting in the derivation of a stress/ displacement law for matrix cracks. This law has been used to obtain preliminary estimates of trends in the crack growth resistance and in the steady state cracking behavior of ceramic composites. In particular, trends with such variables as fiber strength, interfacial shear resistance, fiber diameter, and fiber strength variability have been elucidated. The principal limitation of the present analysis has been the neglect of fiber debonding effects. The results are thus strictly applicable to composites with weakly bonded fibers that exhibit a low interfacial shear resistance. The character of the problem is expected to change when interfacial debonding is limited and fibers fail at the end of the debond zone, rather than in accordance with weakest link statistics. Analysis of this debonding regime is the subject of further
investigation.
F. F. Lange), Vol. 7, pp. 33-51. Plenum, New York (1986). 13. G. Simon and A. R. Bunsell, J. Mater. Sci. 19, 3649 (1984).
APPENDIX Consider a fiber divided into 2N elements each of length, 6.~. The probability that an element will fail when the stress < u can be expressed as
&$(a)=ZxR$
1. D. B. Marshall and A. G. Evans, Fracture Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H.
Ha&man and F. F. Lange), Vol. 7, pp. l-15. Plenum Press, New York (1986). 2. A. G. Evans and R. M. McMeeking, Acta metall. 34, 2435 (1986).
3. B. Budiansky, Proc. Tenth U.S. Natn. Congr. of Applied Mech., pp. 25-32. Austin, Texas (1986). 4. D. B. Marshall and B. N. Cox, Acta metall. To be published. 5. D. C. Phillips, J. Mater. Sci. 7, 1175 (1972). 6. D. B. Marshall, B. N. Cox and A. G. Evans, Acta
p,(T, 2) = 1 - &J(T, 2). (A2) Consequently, the probability that all the elements in the fiber survive, being the product of the survival probabilities, is P,(T, 0 = i] 11-S&T, 211 (A3) n--N where, z = n&z, and I = Ndz. Furthermore, the probability that the element at .zwill fail when the peak stress is between T and T + 6T, but not when the stress is less than T, is [1 1] aQ(T,
+(T,z)]-+-
z) aT
J. Am. Ceram. Sot. 59, 304 (1976).
11. H. L. Oh and I. Finnie, Int. J. Fract. Mechs. 6, 287 (1970). 12. R. L. Stewart, K. Chyung, M. P. Taylor and R. F. Cooper, Fracture Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H. Hasselman and
(A4)
f/ [I - &~(T,z)l FaT. W’, z)aTaz= “-;I”_ s4(T,
Consequently, upon adopting [equation (311that
the Weibull assumption
62 o(T,z) &$(T, z) = 2nR A s 0[ 0
and noting that fi “Z-N
(A5)
z)l
[1-&#~(T,z)]%exp L
c--n
(fw
1 m
68, 225 (1985).
9. J. Aveston, G. A. Cooper and A. Kelly, In The Properties of Fibre Composit&, Conf. Proc., pp. 15-26. Natn. Phvsical Lab. IPC Sci. Technol. Press (1971). 10. J. R. Matthews, F. A. McClintock and W.‘J. Shack,
.
Denoting the probability density function of fiber failure by @(T, z), the probability that fracture occurs at a location z, when the peak stress is T, is equal to @(T, z)6T6z. This probability is governed by the probability that all elements survive up to a peak stress T, but that failure occurs at z when the stress reaches T. It is given by the product of equations (A3) and (A4)
metall. 33, 2013 (1985).
I. B. Budiansky, J. W. Hutchinson and A. G. Evans, J. Mech. Phys. Solids 34, 167 (1986). 8. D. B. Marshall and A. G. Evans, J. Am. Ceram. Sot.
t-41)
where g(S)dS/A, represents the number of flaws per unit surface of fiber having a “strength” between S and S + dS. The local stress, u, is a function of both the distance from the center of the fiber, I, and the peak stress, T. The probability of survival of an element located at z is simply
Pr(T,r)=[l REFERENCES
‘g(S)dS 0s 0
&$(T,z)
1 -I
(.47)
for small values of 64(T, z), one obtains (in the limit, N+co)