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The mechanics of matrix cracking in brittle-matrix fiber composites
ACIU metalf. Vol. 33, No. Il. pp. 2013-2021. Printed in Greai Britain
OKl1-6160/85 s3.00 +o.ao Pergmon Press Ltd
i985
THE MECHANICS OF MATRlX CRACKING BRITTLE-MATRIX FIBER COMPOSITES D. B. MARSHALL
IN
and B. N. COX
Rockwell International Science Center, 1049 Camino DOS Rios. Tbousanci Oaks, CA 9I360, U.S.A. and A. G. EVANS Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720, U.S.A. (Receiued 31 January 1985; in revised firm
9 April 1985)
Abstract-Matrix fracture- in brittle-matrix fiber composites is analyzed for composites that exhibit multiple matrix cracking prior to fiber failure and have purefy frictional bonding bctwecn the fibers and matrix. The stress for matrix cracking is evaluated using a stress intensity approach, in which the influcncc of the fibers that bridge the matrix crack is represented by closure tractions at the crack surfaces. Long and short cracks arc di~ing~shed. Long cracks approach a steady-state ~nfi~~tion, for which the stress intensity anaIysis and a previous energy balance analysis arc shown to predict idcntica1 dcpendenoe of matrix cracking stress on material properties. A numerical solution and an approximate analytical solution are obtained for smaller cracks and used to estimate the range of crack sizes over which the steady-state solution applies. R&urn&-Nous avons budit: la rupture de la matricc dans da composites fibrcux g mat+ fragile prtsentant une fissuration multiple de la matrice avant la rupture des tibrcs et pour lesquels la liaison mtre les fibres et la matricc se. faisait par frottement pur. Nous avons CvaluCla contraintc de fissuration de la matrice en utilisant une approchc par l’intcnsitk de la contrainte dans laquelle on rcprtsentait l’influcncc des fibres qui traverscnt la fissure de la mat& par des tractions de fermcturc sur la surf= de la fissure. Nous distinguons lcs &suns longucs et lcs courtcs. Lcs fissures longucs sent prochcs d’une con!iguration stationnairc pour laquelle i’analyse par l’intensiti de la contrainte et une anaIysc ant&urc de l’tquilibn d%ncrgie pcrmcttent toutcs deux de p&voir unc variation identiquc de la contra& de &u&on de la mat&c en fonction dcs propri&,t&sdu mat&iau. Nous avons obtenu unc sohttion num&rique et une solution analytique approch&c pour lcs pctitcs fissures; nous les avons utilisixs pour cstimcr k domaine de taitles de fissures dans lcquel on pcut appliquer la solution stationnain. faservcntiirktcn Werkstoffcn mit spriider Matrix wird dcr Bruch in der Matrix fur den Fall untersucht. daB mehrfacher Matrixbruch auf&t, bcvor die Fascr bricht, und daB zwischen Fascr und Matrix Bindung durch rcine Reibung bcsteht. Die Spannung f”ur Matrixbruch wird in der Niiherung dcr Spannungsintensitiit bcrcchnet. In dieser Nahcrung wird der EinfluB dcr Fascm, die den RiB iiberbriickcn, mit SchlicIlungsspannungen an den RiDobcrfllchen angcniihert. Lange und kurzc Rissc werden unterschieden. Lange Rissc gchen in eine stationPre Konfiguration Bbcr. Fiir dicsen Zustand sagcn Spannungsintensitiits-Analyst und tine friihere Analyse der Energicbilanz identischc Zusammenhlnge zwischen der Spannung fir Matrixbruch und den Materialeigenschaften voraus. Fiir schmalerc Risse wcrden eine numcrischc und tine angeniihertc analytische &sung erhalten. Mit dicscn Liisungcn wird dcr Bereich der RiBr%en abgcschiitzt, in dem die L&sung der station&n Konfiguration gilt. Zusammenfasang--An
1. INTRODUCTION
The reinforcement of brittle materials with high strength brittle fibers can yield composites that undergo large tensile strains prior to failure [Fig. l(a)]. In such materials prefailure damage initiates with the formation of multiple, regularly spaced cracks in the matrix [l] [Fig. l(b)]. This damage mode requires the strain-to-failure of the matrix to be less than that of the fibers, and the fibers to have sufficient strength to remain intact after a crack passes completely through the matrix. Materiais that have been observed to behave in this manner include cement and plaster reinforced by glass, steel or asbestos fibers
[Z--6] and glasses and glass ceramics reinforced by carbon [7-91 and Sic [IO,ll] fibers. The ultimate load-bearing capacity of the composite may substantially exceed the load for matrix cracking [Fig. l(a)]. Nevertheless, the first matrix crack is of prime concern because matrix fracture signifies the onset of permanent damage, the loss of protection provided by the matrix against corrosion and oxidation of the fibers, and the likelihood of an enhanced susceptibility to degradation due to cyclic loading. Despite the importance of the first matrix crack, a complete fracture mechanics analysis does not exist. Some aspects of the problem have been
2013
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MARSHALL et al.: MATRIX CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
c Fig. 2. Sdicmatic rqnwentation of a steady-state matrix crack in a fiber composite.
It is convenient to distinguish between large and small cracks as a basis for further analysis. Large cracks experience a crack opening, U, which asymptotically approaches (but cannot exceed) the equilibrium separation, ~0,of the completely failed matrix [i.e. two half plants connected by fibers (Fig. 211.This limiting separation is approached beyond a characteristic distance, co, from the crack tip. Within that region, the net force in the intact fibers that bridge the crack must exactly balance the applied force. In this case, the crack tip w concentration is induced exclusively over the length co, and the stress needed to extend the crack must be independent of the total crack length. Crack growth in this region is referred 09 Fig. 1. (a) Load&k&on emve for tension test and (a) to as steady-stare growth. Conversely, for short cracks (i.e. c < co), the entire crack contributes to the multiple matrix crackiag in ~Gg~~~~rna~ composite (after Ref. [ill) Width of field I.Sarm. stress concentration, so that the stress required to propagate a crack is sensitive to the crack length (as it is in monolithic materials). addressed by Aveston et al. [I]. Specifically, assuming Crack growth can be evaluated using either energy that the matrix possesses a characteristic strength balance or stress intesity considerations. The general independent df st~ngth~ntrolling defects, they formalism for the stress intensity approach is rationalized the regularity of the matrix cracking. presented in the following section. Then the steadyThey also used an energy balance analysis to intro- state crack is examined, using both stress intensity duce a dependence of strength on microstructure and and energy balance approaches in order to establish account for certain increases in matrix strength ob- the equivalence of the two independent methods of served in systems containing small diameter fibers. analysis. The growth characteristics of short cracks However, this analysis has two limitations. First, the are then evaluated using the stress intensity method. cracking strengths of glasses and ceramics are not Finally, several impliitions of the analysis for matrix generally deterministic, but rather are sensitive to cracking in various mmposite systems are discussed. distxibutions of flaws. Second, the energy balance analysis was based on a comparison of energies 2. STRESS INTENSITY FORMULATION before and after complete cracking of the matrix, OF MATRIX CRACKING whereas a rigorous thermodynamic treatment of crack growth requires e~mi~tion of energy changes gyration of the surfaces of a matrix crack which that accompany incremental changes in crack length. is bridged by uniaxiafly aligned reinforcing fibers Thm deficiencies are addressed in the present study. (Fig. 2) requires some sliding of the matrix ovef the
MARSHALL ef a/.:
MATRIX CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
2015
crack surfaces. Therefore, with the crack surfaces being subjected to net pressure [Q, --p(x)], a composite stress intensity factor can be defined as [12, 131
-/WI KL = 2(c/7r)“2 I O’]uwJD fat
dX
(2a)
for a straight crack embedded in an infinite medium, or
Jo
Jl-Xl
for a penny crack, where X =x/c. The stress intensity KL characterizes the composite stress and strain fields in the region immediately ahead of the matrix crack. In this region, net relative displacements between the fibers and the matrix are not permitted. Consequently, the matrix and fiber strains must be compatible, whereupon the stresses exhibit the composite relationship GM/& = a/E,
(3a)
where #‘is the matrix stress, Qis the composite stress and E, is the composite modulus Fig. 3. (a), (b) Hypothetical operations wad to evaluate the closure e&et of the fibers in the stress intensity analysis.
iibcrs. In gcnerai, this process would require debonding foliowed by sliding against frictional forces. However, it appears that in successfully reinforced ceramic-matrix composites, them is no chemical bond between the fibers and matrix [II]. The present analysis is concerned with such tmbonded composites, in which the sliding of the matrix over the fibers is resisted only by frictional forces. The restraining effect of the fibers causes a reduction in both the crack-surface displacements and the crack-tip stresses. Relations between these quantities and the remotely applied stress, Q, , can be evaluated by imagining the crack in Fig. 2 to be formed in two steps. First, all of the bonds across the prospective crack plane (in the fibers as well as the matrix) are cut and the stress Q, is applied [Fig. 3(a)], causing the crack to open. In the second step tractions, T, arc applied to the end of each fiber. The magnitude of T is chosen such that the fiber ends displace relative to the mattix and allow the fibers to be rejoined [Fig. 3(b)]. In a continuum approximation (c>>tlber spacing), this procedure is equivalent to applying a distribution of closing pressure p(x) to the crack surfaces P(X) = WW, (1) where x represents the position on the crack surface [Fig. 3(b)] and V,is the volume fraction of fibers. The closure induced by the pressure p(x) opposes the opening due to the applied stress 0,. The influence of the applied stress on the crack tip stress intensity can be exactly evaluated by regarding the stres= as a uniform opening pressure, Q_,, acting along the
E,=E,,,V,+
E,V,
with E, and E, tefming to the Young’s moduhts of the matrix and fibers, mspectively, and V,(m 1 - I$)
the volume fraction of matrix. The matrix and composite stress intensities scale with the stresms, such that KL = K”EJEm
(3b)
where KM is the stress intensity factor in the matrix. The condition for quilibrium crack growth (in the absence of environmental effects) is given by setting KM equal to the critical stress intensity factor, K,M, for the matrix. Therefore, the criterion for crack growth can be expressed in terms of KL KL = K: = KfEJEm.
(41
Thus, equations (2) and (4) relate the matrix cracking condition to the applied stress 0,. Evaluation of KL in equation (2) requites a separate calculation of the pressure distribution p(x). Analysis of tiber pullout from the matrix (Appendix 1) reveals that the closure pressure is related to the crack opening, a, at a given location by
P=
IzIr+K,(l + rt)/Rl’”
(5)
where ?r = Ef V//E,,, V,,,, R is the fiber radius, and T is the sliding frictional stress at the interface. However, the crack opening at a given position is determined by the entire distribution of surface tractions [la J
U(X) = y-;y c
a! S2- X2
2016
MARSHALL
et (il.: MATRIX CRACKING
IN BRITTLE-MATRIX
FIBER COMPOSITES
strain energy in the fibers increases by dU,, and the
potential energy of the loading system decreases by di/,. These energy changes can be calculated from
the results presented by Aveston et al. (I] dU,=[o;R/GrE,Vj(l
+q)‘]dc
(7a)
dU,,,= [c&Rq/3rE/Vj(l -I-qy]dc
(7b)
dU,= [a%R(3~ + l)~~~,V~~l+ #I dc(7c) dU, = [&,R/2rE,V;(l
+ q)‘] de.
(7d)
The net energy change of the system resulting from the incremental crack extension is dU=ur,V,dc~dU,+dU,-dU,-dU, Fig. 4. Steady-state era& configuration used for energy balaaa analysis.
for a strai@ crack, or
(8)
where 2I’,,,= Cl: = KY (1 - v’)/& is the fracture surface energy of the matrix. Setting u, = u. at Griffith equilibrium (dU/dc = 0) and combining equations (7) and (8) yields a crack extension stress + q)2/E,,,R]“3 (9)
a, = S[(l - v')K~rE,V'jV,(l
where 6 = 6’9
Jo
Js'-t2
(6b) . .
for a penny crack, where s and t are normalized position coordinates and v is the Poisson’s ratio of the wmpo$itc. Therefore, analysis of matrix cracking by the stress intensity approach requires solution of equations (5) and (6) to obtain the crack surface tractions, followed .by evaluation of the integral in equation (2) and combination with equation (4). 3 BTEADYrnATE CRACKS
The intent of this section is to demonstrate that the ener8y balance and stress intensity approaches predict equivalent dependence of the stress for steady-state crack growth, oo, on microstructural parameters. The energy balance solution derives from the earlier analysis by Aveston et al. (11, but is expressed in terms of incremental crack extension. The stress intensity approach has not been considered previously.
3.2. Stress intensity analysis
A simple dimensional analysis is used in this set+ tion to demonstrate that the crack extension stress obtained from stress intensity considerations exhibits the same form as equation (9). Spe&c numerical quantities for the steady-state crack are obtained in the following section. The stress system for a steady-state crack is depicted in. Fig. 5. As noted previously, the closure pressure exactly balances the applied loading in the region x c c - co. Co~uen~y, the net crack surface tractions, 0, --p, are zero. However, over the area c - co< x < c net opening tractions exist. In this region, p(x) must vary smoothly between zero at the crack tip, x = c, and u, at x = c - co, i.e. fJ=%J@)
PGl
(10)
where p = r/co = (c -x)/co and f@) is a function that varies between zero at p = 0 and unity at p = 1. Thus, for steady-state cracks [c X, and (g_, - p) = .O
t
The energy changes occurring in the specimen and loading system during an incremental crack extension define the crack growth behavior. These energy changes can be calculated by employing the hypothetical operation depicted in Fig. 4 to extend the crack by dc. A strip of material of width dc ahead of the crack (arv AA’B’B) is removed, a cut is made in the matrix along CC’, the matrix is allowed to relax causing the ends of the strip to displace, and the strip is attached at the mouth of the crack. When the cut is made in the matrix, the matrix must slide back over the fibers while the fibers also extend. When this occurs work, dU,<,is done against frictional forces, the strain energy in the matrix decreases by dU,,, the
00
-
PW
-
0 00
x
t-
I
-
ptxt
I
Fig. 5. Net surface tractions acting on steady-state crack.
MARSHALL
et al.:
MATRIX
CRACKING
for r > co], equations (2a) and (2b) both reduce to KL = (Z/n)“2 I
-P(r)]r-“Zdr
7%
where u is a dimensionless
a~vu,c;~
(II)
eonstam
cu=(2,71) li2 1 ’ [I -f(p)]p
- “* dp
The length e. is evaluated by considering the crack opening displa~ments within the range r d c,. For the steady-state crack, equations (6a) and (6b) both reduce to
u(P)=
1
x
where d, = c(l - t)/co and 1 = c( 1 - s)/c,,. An alternative expression for u@) is obtained from equations (5) and (10) @co) = ~~[~~~12~/4~~~~,(1 Thus, equating
equations
+ tl).
x
I
v')KrrE~yl/V,(1+~1)2/E,Rj'"
(14)
fW
where a = 8( 1 - v2)zVfE,( 1 + q)/Eclw2.
t17b)
For x x c, equation (16) is an exact solution for the present problem (since displacements near the crack tip must be a unique function of KL). However, it is clearly not appropriate at small x for large cracks because, as already noted, the crack opening must asymptotically approach u,,, whereas the opening expressed by equation (16) is unbounded at large c. The timiting displacement is given by setting p = 5, in equation (5) +q)
dA
(15)
4 SHORT CRACKS
Evaluation of KL for short cracks requires expIicit determination of the crack opening displacements from equations (5) and (6). However, rigorous analytic solutions for u(x) cannot be obtained. Consequently, an approximate analytical solution is used initially to gain insight into the mechanics of crack growth. Then, a numerical solution is evaluated. solution
The approximate solution invokes an assumption that, at small crack sixes, the crack profile does not differ greatly from that of a crack subject to uniform
(19)
Therefore, the displacement expressed by equation (16) is used for cracks smaller than c,, (i.e. II < ug). For larger cracks, an approximate limiting solution is obtained from a crack prolle with uniform opening (u = %) near the center (X KC - cg) and a near-tip opening (at x > c - co) that is the same as that for a crack of length co. With this approximation, the stress intensity factor for c d c, is given by substituting equation (17a) into equations @a) or (2b) KL = Ru,c’~
Comparison of equations (9) and (15) indicates that the energy balance and stress intensity analyses predict identical relations between the matrix-cracking stress and m.ierostructuraI parameters.
(18)
and the corresponding transition crack length co is given by equation (16) with u = u, and x = 0 c, = a;“,/a2Ku.
where
4. I. Approximate udytid
- xz/cyy
Mo=5’,R/4?VjE/(l
1* The matrix cracking stress, cr,, is now obtained by evaluating KL from equations (11) and (14) and setting 5, = 5@at KL = K,dE,/E, a0 = &‘[(I -
p(x) = [aKW~(l
constant
’ 11-ftd; 11d4 l&-z
(16)
(13)
fl)(l - v2)
where w is another dimensionless
2017
pressure (with the magnitude of the pressure adjusted to give a stress intensity factor equal to KL). Then, an analytical solution for u(x) is obtained from equation (6a) or (6b) in terms of KL
(12) and (13) yields
c, = n5,E,RfrV$,(l+
COMPOSITES
The actual pressure distribution is then obtained by combining equations (5) and (16) to give
. >
O 5 pm
FIBER
u(x) = 2(1 - v2)KLc”‘(1 --x~/c~)‘~/&A”~.
0
2(1 -vZ)5,co nE c
fN BRITTLE-MATRIX
- (~c+c)‘~(K~)‘~c”‘I
c G co (20a) where a and I arc dimensionless crack geometry constants. For straight cracks R = ~‘12and
I=
‘(1 -X2)-“‘dXr1.20 i0 whereas for penny cracks, Q = 2/7r”2 and
(2Ob)
f=
‘(I-X2)-‘“XdX=2/3. (2@) I0 The mechanics of crack growth can be investigated by setting KL = Kk in equation (20a) and solving for 5, to obtain an equilibrium-stress/crack&e function u, = Kf/fk’” + [4aKf12/nS12]“2c”4
(c
This function can be conveniently malized form 5/a, = (1/3)(c/cJ-‘”
(21)
expressed in nor-
f (2~3)~c/c~)“’
(22)
2018
MARSHALL
et al.:
MATRIX
CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
where c,,,= (nK~/a12)2/3
(234
and
COMWSITIE:
APPROXIMATE
ANALYTICAL
SOLUTION
ti =&“[(I -v2)Kf%E,YjV,(l
+q)2,‘E,,,R]“3
(23b)
where 6” = 6P/Rn 1/z. Equation (22) provides a relation between normalized stress and crack length parameters, S = o/a,,, and C = c/c,,,, without explicit reference to material and micros~ctur~ properties (these properties enter only in their influence on the normalizing factors c, and u,,,). Thus, the mechanics of crack growth may he examined independently of the specific composite system. For c > c, the region of crack where u = or, does not contribute to the stress intensification at the crack tip. Therefore, for a straight crack, which maintains geometrical similarity as it extends, the stress intensity factor for c > co is given by equation (20) with c = co.t The corresponding equilibrium stress, co, which is independent of crack length, is given by equations (19), (21) and (23b)
NORMAWED
CRACK LENGTH. C/Cm &a) i
I
PENNY CRACK
and the transition crack length is given by [equations (19), (23a), and (24a)] c, = [Z/n”2n(l
-
21/z’“R)]4’3cm.
(24b)
For straight cracks, Qrr, = 1.02 and c, Jc,,,= 1.88, whereas for penny cracks, or, = o,,, and c,, = cm. The complete equilibrium-stress/crack-size functions for straight cracks and penny cracks are plotted in Fig. 6(a) and (b). It is noted that the stress required to propagate a matrix crack is almost independent of crack length for cracks larger than about c,,,/3. This defines the range of crack sizes over which steadystate conditions apply. The crack response in this region contrasts with the behavior of cracks in unreinforced brittle materials, for which the strength decreases with c-‘~. For cracks shorter than -c,,,/3, the equiIib~um stress increases as the crack length decreases, with a crack-length dependence resembling that of a monolithic brittle materia1.
2
1 NOMALUED
CRACK LENOTM. CfCm (b)
Fig: 6. Equilibrium-stnsslcracic-size functions for matrix cracks in composites and in a monoiithjc matrix: (a) straight embedded crack in infinite medium; (b) penny crack in infinite medium. putting p = (T, in equation (5) u, =[2(1 -vZ)/a.E,7f1/*]0~.
(25)
Thus, with U(X) = u(X)/u,, S = ~,/a, and C = c/c,, equations (6a) and (6b) can be re-expressed as
4.2. Numerical solution It is convenient to normalize the stresses, crack lengths and displacements in equations (6a) and (6b) by the parameters c,,, and a, defined in the previous section, and a limiting displacement I(, defined by
(264 and
tThe corresponding region of a penny crack does not maintain geometrical similarity. In this case, the region that determines the stress intensity factor can be viewed as an annular crack around a cylindrical hole. At c z c,, this con~guration approaches a penny crack, whereas at large crack lengths, c >>c,, it approaches an edge crack. Therefore. c0 and o0 vary between these two limits as the crack grows. The relative values of u0 and c0 for straight cracks and penny cracks are 0.97 and 0.87, respectively [Equations (19). (23b) and (24)].
f2W where y = 2rr’/2Q/3/2. For penny cracks, y = 3 and for straight cracks, y = 3.02.
MARSHALL
er. al.:
MATRIX
IN BRITTLE-MATRIX
CRACKING
Equations (2,h) and (26b) were solved by iteration to self-consistency in the function U(X), beginning with the approximate crack profile obtained in the previous section. The integrals were evaluated after t =s sin0 and making the transformations s = X cash 4. This procedure is more straightforward than previously published solutions of related crack opening problems [17j. After obtaining a selfconsistent crack profile, the stress intensity factor was computed from equation (2a) or (2b). Then, from sets of solutions for various values of C at each value of S, the function S(C) at quiWrium (i.e. KL = Kt). was evaluated. The results are compared with the approximate analytical solutions in Fig. 6(a) and (b). The two solutjuns show similar trends; at small crack lengths, they converge, but at large crack lengths the numerical steady-state stress is lower than the approximate analytical value (as expected from consideration of the procedure used to obtain the analytical solution) by -20%.
5. IMPLICATIONS AND DI&ZUSSION One of the important results of the present analysis is the definition of the approximate crack dimensions over which the transition from short-crack behavior to steady-state response occurs. The matrixcracking stress approaches the steady-state value for crack lengths &,/3 (Fig. 6). The crack length c, can be evaluated from quation (23a), in conjunction with equation (17b) c, = (7r/41’n)[K:E,
v:(l+
q)R/?Y;E/ x (1 - v’)P.
(27)
Values of c, calculated for two composite systems in which the parameters of equation (26) are known with reasonable accuracy are shown in Table 1. For both composites, cJ3 is several fiber spacings. Since the sizes of inherent flaws in brittle materials are usually about the same as microstructural dimensions, these results imply that the stress for matrix cracking in these composites is not substantially reduced by the further introduction of larger flaws during fabrication or service (e.g. mechanical contact
FIBER
SiC/@aw ccramiti propmties Kf
Ef
JL L? * Matrix cracking stress Caladated Properties %I 00 ‘Data
from Refs [IO] and [II].
bData from Refs [I and [S].
2019
damage), or by the extension of pre-existing flaws in thermal shock or environmentally assisted slow crack growth. It is also noted that, with the continuum approximation adopted in Section 2, the solutions in Fig. 6(a) and (b) hold for crack lengths larger than several fiber spacings. The crack-size independence of the matrixcracking stress at large crack lengths justifies, in part, the earlier analysis of Aveston et al. [l] for it is now possible to define a matrix-cracking stress as an intrinsic property of the composite, as required for their explanation of the regularity of the multiple matrix cracking. Moreover, for steady-state crack growth, their analysis based on a comparison of the energy of the untracked system and the energy after complete cracking is valid, and their result is quivalent to equation [9]. However, their analysis differs in one important respect from the present work. They prescribed the matrix-cracking strain E,,,as a characteristic of the matrix rather than the composite, and proposed that equation (9) applies only at small fiber diameters for which a, exceeds the stress corresponding to E,. Such a restriction is not required in the present analysis which takes into account the cracklength dependence of the strength of the tinforced matrix. In fact, the reinforcing e&t of the fibers can be inferred directly from quation (21). The 6rst term on the right side represents the product of the strength of the unreinforced matrix (Kf/Qc’“) and the modulus ratio, EC/E,,,. This term is plotted in Fig. 6(a) and (b). The difference between this curve and the soh&ions for the matrixaacking stress of the composite represents the reinforcing effect of the fibers for a composite with equal fiber and matrix moduli (E/= E,,,). For typical compositea, the fiber modulus is the larger, and the matrixcracking stress of the composite is always higher than the strength of the unreinforced matrix for a given crack length. However, for composites with E/-C E,,,, the relative strengths are dependent upon the crack length, The energy balance and stress intensity analyses provide equivalent relations between the steady-state matrix cracking stress and microstructural parameters, as indicated in Section 3.2. 4 quantitative comparison of the two results can be obtained by estimating the dimensionless constant, a’, in equation
Table 1. Fiber/matrix
Meawed
COMPOSITES
2 h4Pa.m’”
cnrb0n/glassb
200 GPa as GPa 0.5 8pm 2 MPa 270-300 MPa
0.75 MPa.m’” 380 GPa 70 GPa 0.4 4pm 10 MPa 340-430 MPa
325 pm 265 tiPa
44um 420 ‘MPa
2020
MARSHALL et al.: MATRIX CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
(15) from the analysis of Section 4.2. Thus, from equation (23b) and Fig. 6(a) or (b), we obtain 6’ = 1.83, which yields a stress essentially identical to that obtained from the energy balance analysis (6 = 6”‘). Predicted values of the stress for matrix cracking [from equation (911 in the SiC/glass-ceramic and carbon/glass composite systems are shown in Table 1. In both cases, agreement with experimentally measured values is evident. Hence, the predicted relation between the critical stress for matrix cracking and microstructural parameters appears to provide a basis for design of optimum microstmctums. Specifically, the criticfd stress increases with the toughness of the matrix, the modulus and volume fraction of fibe* the frictional stress at the fiber/matrix interface, and decmasing fiber diameter. However, it should also be appreciated that, as the interface frictional resistance increases, the net tensile stress on the fibers ahead of the crack also tends to increase. Thisstress eAmcement must eventually result in fiber failures associated with the advancing matrix crack and a consequent change in the failure mode. An optimum frictional resistance is thus anticipated, coincident with the maximum matrix cracking stress that prohibits fiber failure. This aspect of the problem remains to be resolved.
bc small and that the fibers remain intact after a crack passes completely through the matrix (i.e. co < a, V,, where trb is the bundle strength of the fibers in the presence of interfacial frictional forces). The most significant microstructural restriction concerns the volume fraction of fibers aligned in the principal stress axis, because an increase in V, beneficially inlluences all of the parameters that determine opti-
mum steady-state properties (i.e. both co and trbV, increase, while c, decreases). Thus, optimum properties are most likely in uniaxially reinforced composites. In multiaxially reinforced systems (such as 3-D or random whisker composites), the smaller volume fraction of fibers aligned in any direction limits the value of do that can be achieved without causing fiber failure and increases c,, making a crack-length-dependent matrix fracture stress more likely. The existence of optimum vahtes of matrix toughness, K,” , and interfacial shear resistance, r, can be inferred from the analysis. Increasing f beneficially increases u. and decreases c,,,, but the maximum increase in u. is’ limited by the fiber-failure stress. Increasing Kr increases u. but also has the detrimental effect of increasing c,. Thus, the maximum acceptable KF could be dictated either by the fiber-failure stress or by the requirement that c, be less than a pre-existing flaw size. The preceding restrictions account for the brittle response observed in a number of fiber or whisker6. CONCLUDING REMARKS reinforced brittle systems, and place important The principal implications of the present analysis bounds on the design of optimum microstructures. concern the predicted transition to a crack-length- Furthermore, implicit in the ability to design materiindependent matrix-cracking stress, uo, for cracks als and to interpret results within the context of the longer than a characteristic length -c,,,/3, and the present analysis is the availability of methods for associated trends in matrix fracture. The matrix- measuring T, K,Mand eb. Developments pertinent cracking stress (i.e. the first deviation from linearity to such measurement have been reported in recent in the stress-strain curve) should be both damage studies [1I]. tolerant and independent of specimen size, provided for this work was supplied by the characteristic length is smaller than preexisting Acknowledgement-Funding the Rockwell International Independent Research and flaws in the matrix. In this sense, ceramic matrix Development Program and the U.S. O&e of Naval ‘compositescan be more like metaIs than ceramics in Research, Contract No. NOQOW79-C-0159. their tensile mechanical behavior. The analysis indicates that the attainment of REFERENCE3 steady-state cracking at high stress levels is likely to be restricted to a narrow range of microstructures. 1. J. Aveston, G. A. Cooper and A. Kelly, in The Properties of Fiber Composites, Gmf. Pruc. pp. 15-26. These restrictions arise from the requirements that c,
Fig.
7. Analysis
of fiber pullout
mechanics.
MARSHALL
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16.
17.
et al.:
MATRIX CRACKING
National Physical Laboratory, IPC Science and Technology Press (1971). M. A. Ali and F. J. Grimer, J. Mater. Sci. 4,389 (1969). A. J. Majumdar, Proc. R. Sot. A319, 69 (1970). R. C. deVekey and A. J. Najumdar, Msg. Concrete Res. 20, 229 ( 1968). H. G. Allen, J. Camp. Mater. 5, 194 (1971). J. P. Romualdi, U.S. Patent 3429094 (1969). R. A. J. Sambell, A. Briggs. D. C. Phillips and D. H. Bowen, J. Ma&r. Sci. 7, 676 (1972). D. C. Phillips, J. Marer. Sci. 7, 1175 (1972). J. J. Brennan and R. M. Prewo, f. Mater. Sci. 17, 2371 ( 1982). K. M. Prewo and J. J. Brennan. J. Mater. Sci. 17. 1201 (1982). D. B. Marshall and A. G. Evans, J. Am. Cerwn. Sot. 68, 225 (1985). B. R. Lawn and T. R. Wilshaw, Fracture of Brirrle Solia!s. Cambridge Univ. Press (1975). G. C. Sib, Handbook of Stress Intensity Factors, Lehigh University, Bethlehem (1973). S. M. Wiederhom, in Fracture Mechanics of Ceramics (edited by R. C. Bradt, 0. P. H. Hasselman and F. F. Lange), Vol. 2, pp. 613-646. Plenum, New York (1974). B. J. Pletka and S. M. We&horn. in Fracture Mech&es of Cerumics (edited by R. 6. Bradt, 0. P. H. Has&man and F. F. Lange), Vol. 4, pp. 745-759. Plenum, New York (I978). I. N. Sneddon and M. Lowengrub, Crack ProbIems in the Classical Theory of Elasticity. Wiley, New York (1969). C. Atkinson, Znt. J. Fract. Nech. 6, 193 (1970). APPENDIX
1
Mechanics of fiber pullout The application of tractions 2’ to the end of the fiber in Fig. 7 causes sliding between the matrix and fiber over a distance f, and allows the fiber to pull out of the matrix a distance u. For a purely frictional matrix-t&r bond, the
IN BRIDLE-MATRIX
FIBER CGMPDSITES
202 1
sliding distance is determined by the length over which the interface shear stresses exceed the frictional stress T. A relation between T and u is required for the stress intensity analysis. The mechanics of fiber pullout can be conveniently analyzed by applying tractions T, and T,, equal and opposite to the stresses in the matrix and fibers, along the plane AA’ at the end of the slipped region, and removing the section A’C’C A [Fig. 7(b)]. If we neglect the effect of shear stresses above AA’ in Fig. 7(a) (i.e. assume that the strains in the matrix and fiber are equal above AA’), these tractions are related by
TmIEm = TfIEI. Equations relating the stresses and displacements in Fig. 7(b) are obtained by considering the equilibrium of the matrix and fiber separately and also by calculating the extensions S and d + u of the matrix and fiber T, A, = 2nRlr
(A2)
TA,,= 2n Rls + T/A,
(A3)
6/l = nRk/A,E,,,
644)
(6 + u)/l = T//E/+
G?k/A,E,
W)
where A,= nA” is the fiber cross-sectional area and A, is the area of matrix per fiber. The requisite relation between T and u is then obtained from equations (AI) to (AS). First, equations (Al) to (A3) combine to give T=21r(l
+rf)/E
(A6)
whem 4 = Efvf/E,,, F,. Then equations (Al), (A2), (A4) and (As) combine to give I2 = uRE,/r(l + q). Finally, from equations (A6) and (A7) we obtain T = Z[uE,r(l + q)/R]“2
(A@
and, with equation (1) p = 2[urf’;E,(I +q)/R]“2.
649)
OKl1-6160/85 s3.00 +o.ao Pergmon Press Ltd
i985
THE MECHANICS OF MATRlX CRACKING BRITTLE-MATRIX FIBER COMPOSITES D. B. MARSHALL
IN
and B. N. COX
Rockwell International Science Center, 1049 Camino DOS Rios. Tbousanci Oaks, CA 9I360, U.S.A. and A. G. EVANS Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720, U.S.A. (Receiued 31 January 1985; in revised firm
9 April 1985)
Abstract-Matrix fracture- in brittle-matrix fiber composites is analyzed for composites that exhibit multiple matrix cracking prior to fiber failure and have purefy frictional bonding bctwecn the fibers and matrix. The stress for matrix cracking is evaluated using a stress intensity approach, in which the influcncc of the fibers that bridge the matrix crack is represented by closure tractions at the crack surfaces. Long and short cracks arc di~ing~shed. Long cracks approach a steady-state ~nfi~~tion, for which the stress intensity anaIysis and a previous energy balance analysis arc shown to predict idcntica1 dcpendenoe of matrix cracking stress on material properties. A numerical solution and an approximate analytical solution are obtained for smaller cracks and used to estimate the range of crack sizes over which the steady-state solution applies. R&urn&-Nous avons budit: la rupture de la matricc dans da composites fibrcux g mat+ fragile prtsentant une fissuration multiple de la matrice avant la rupture des tibrcs et pour lesquels la liaison mtre les fibres et la matricc se. faisait par frottement pur. Nous avons CvaluCla contraintc de fissuration de la matrice en utilisant une approchc par l’intcnsitk de la contrainte dans laquelle on rcprtsentait l’influcncc des fibres qui traverscnt la fissure de la mat& par des tractions de fermcturc sur la surf= de la fissure. Nous distinguons lcs &suns longucs et lcs courtcs. Lcs fissures longucs sent prochcs d’une con!iguration stationnairc pour laquelle i’analyse par l’intensiti de la contrainte et une anaIysc ant&urc de l’tquilibn d%ncrgie pcrmcttent toutcs deux de p&voir unc variation identiquc de la contra& de &u&on de la mat&c en fonction dcs propri&,t&sdu mat&iau. Nous avons obtenu unc sohttion num&rique et une solution analytique approch&c pour lcs pctitcs fissures; nous les avons utilisixs pour cstimcr k domaine de taitles de fissures dans lcquel on pcut appliquer la solution stationnain. faservcntiirktcn Werkstoffcn mit spriider Matrix wird dcr Bruch in der Matrix fur den Fall untersucht. daB mehrfacher Matrixbruch auf&t, bcvor die Fascr bricht, und daB zwischen Fascr und Matrix Bindung durch rcine Reibung bcsteht. Die Spannung f”ur Matrixbruch wird in der Niiherung dcr Spannungsintensitiit bcrcchnet. In dieser Nahcrung wird der EinfluB dcr Fascm, die den RiB iiberbriickcn, mit SchlicIlungsspannungen an den RiDobcrfllchen angcniihert. Lange und kurzc Rissc werden unterschieden. Lange Rissc gchen in eine stationPre Konfiguration Bbcr. Fiir dicsen Zustand sagcn Spannungsintensitiits-Analyst und tine friihere Analyse der Energicbilanz identischc Zusammenhlnge zwischen der Spannung fir Matrixbruch und den Materialeigenschaften voraus. Fiir schmalerc Risse wcrden eine numcrischc und tine angeniihertc analytische &sung erhalten. Mit dicscn Liisungcn wird dcr Bereich der RiBr%en abgcschiitzt, in dem die L&sung der station&n Konfiguration gilt. Zusammenfasang--An
1. INTRODUCTION
The reinforcement of brittle materials with high strength brittle fibers can yield composites that undergo large tensile strains prior to failure [Fig. l(a)]. In such materials prefailure damage initiates with the formation of multiple, regularly spaced cracks in the matrix [l] [Fig. l(b)]. This damage mode requires the strain-to-failure of the matrix to be less than that of the fibers, and the fibers to have sufficient strength to remain intact after a crack passes completely through the matrix. Materiais that have been observed to behave in this manner include cement and plaster reinforced by glass, steel or asbestos fibers
[Z--6] and glasses and glass ceramics reinforced by carbon [7-91 and Sic [IO,ll] fibers. The ultimate load-bearing capacity of the composite may substantially exceed the load for matrix cracking [Fig. l(a)]. Nevertheless, the first matrix crack is of prime concern because matrix fracture signifies the onset of permanent damage, the loss of protection provided by the matrix against corrosion and oxidation of the fibers, and the likelihood of an enhanced susceptibility to degradation due to cyclic loading. Despite the importance of the first matrix crack, a complete fracture mechanics analysis does not exist. Some aspects of the problem have been
2013
2014
MARSHALL et al.: MATRIX CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
c Fig. 2. Sdicmatic rqnwentation of a steady-state matrix crack in a fiber composite.
It is convenient to distinguish between large and small cracks as a basis for further analysis. Large cracks experience a crack opening, U, which asymptotically approaches (but cannot exceed) the equilibrium separation, ~0,of the completely failed matrix [i.e. two half plants connected by fibers (Fig. 211.This limiting separation is approached beyond a characteristic distance, co, from the crack tip. Within that region, the net force in the intact fibers that bridge the crack must exactly balance the applied force. In this case, the crack tip w concentration is induced exclusively over the length co, and the stress needed to extend the crack must be independent of the total crack length. Crack growth in this region is referred 09 Fig. 1. (a) Load&k&on emve for tension test and (a) to as steady-stare growth. Conversely, for short cracks (i.e. c < co), the entire crack contributes to the multiple matrix crackiag in ~Gg~~~~rna~ composite (after Ref. [ill) Width of field I.Sarm. stress concentration, so that the stress required to propagate a crack is sensitive to the crack length (as it is in monolithic materials). addressed by Aveston et al. [I]. Specifically, assuming Crack growth can be evaluated using either energy that the matrix possesses a characteristic strength balance or stress intesity considerations. The general independent df st~ngth~ntrolling defects, they formalism for the stress intensity approach is rationalized the regularity of the matrix cracking. presented in the following section. Then the steadyThey also used an energy balance analysis to intro- state crack is examined, using both stress intensity duce a dependence of strength on microstructure and and energy balance approaches in order to establish account for certain increases in matrix strength ob- the equivalence of the two independent methods of served in systems containing small diameter fibers. analysis. The growth characteristics of short cracks However, this analysis has two limitations. First, the are then evaluated using the stress intensity method. cracking strengths of glasses and ceramics are not Finally, several impliitions of the analysis for matrix generally deterministic, but rather are sensitive to cracking in various mmposite systems are discussed. distxibutions of flaws. Second, the energy balance analysis was based on a comparison of energies 2. STRESS INTENSITY FORMULATION before and after complete cracking of the matrix, OF MATRIX CRACKING whereas a rigorous thermodynamic treatment of crack growth requires e~mi~tion of energy changes gyration of the surfaces of a matrix crack which that accompany incremental changes in crack length. is bridged by uniaxiafly aligned reinforcing fibers Thm deficiencies are addressed in the present study. (Fig. 2) requires some sliding of the matrix ovef the
MARSHALL ef a/.:
MATRIX CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
2015
crack surfaces. Therefore, with the crack surfaces being subjected to net pressure [Q, --p(x)], a composite stress intensity factor can be defined as [12, 131
-/WI KL = 2(c/7r)“2 I O’]uwJD fat
dX
(2a)
for a straight crack embedded in an infinite medium, or
Jo
Jl-Xl
for a penny crack, where X =x/c. The stress intensity KL characterizes the composite stress and strain fields in the region immediately ahead of the matrix crack. In this region, net relative displacements between the fibers and the matrix are not permitted. Consequently, the matrix and fiber strains must be compatible, whereupon the stresses exhibit the composite relationship GM/& = a/E,
(3a)
where #‘is the matrix stress, Qis the composite stress and E, is the composite modulus Fig. 3. (a), (b) Hypothetical operations wad to evaluate the closure e&et of the fibers in the stress intensity analysis.
iibcrs. In gcnerai, this process would require debonding foliowed by sliding against frictional forces. However, it appears that in successfully reinforced ceramic-matrix composites, them is no chemical bond between the fibers and matrix [II]. The present analysis is concerned with such tmbonded composites, in which the sliding of the matrix over the fibers is resisted only by frictional forces. The restraining effect of the fibers causes a reduction in both the crack-surface displacements and the crack-tip stresses. Relations between these quantities and the remotely applied stress, Q, , can be evaluated by imagining the crack in Fig. 2 to be formed in two steps. First, all of the bonds across the prospective crack plane (in the fibers as well as the matrix) are cut and the stress Q, is applied [Fig. 3(a)], causing the crack to open. In the second step tractions, T, arc applied to the end of each fiber. The magnitude of T is chosen such that the fiber ends displace relative to the mattix and allow the fibers to be rejoined [Fig. 3(b)]. In a continuum approximation (c>>tlber spacing), this procedure is equivalent to applying a distribution of closing pressure p(x) to the crack surfaces P(X) = WW, (1) where x represents the position on the crack surface [Fig. 3(b)] and V,is the volume fraction of fibers. The closure induced by the pressure p(x) opposes the opening due to the applied stress 0,. The influence of the applied stress on the crack tip stress intensity can be exactly evaluated by regarding the stres= as a uniform opening pressure, Q_,, acting along the
E,=E,,,V,+
E,V,
with E, and E, tefming to the Young’s moduhts of the matrix and fibers, mspectively, and V,(m 1 - I$)
the volume fraction of matrix. The matrix and composite stress intensities scale with the stresms, such that KL = K”EJEm
(3b)
where KM is the stress intensity factor in the matrix. The condition for quilibrium crack growth (in the absence of environmental effects) is given by setting KM equal to the critical stress intensity factor, K,M, for the matrix. Therefore, the criterion for crack growth can be expressed in terms of KL KL = K: = KfEJEm.
(41
Thus, equations (2) and (4) relate the matrix cracking condition to the applied stress 0,. Evaluation of KL in equation (2) requites a separate calculation of the pressure distribution p(x). Analysis of tiber pullout from the matrix (Appendix 1) reveals that the closure pressure is related to the crack opening, a, at a given location by
P=
IzIr+K,(l + rt)/Rl’”
(5)
where ?r = Ef V//E,,, V,,,, R is the fiber radius, and T is the sliding frictional stress at the interface. However, the crack opening at a given position is determined by the entire distribution of surface tractions [la J
U(X) = y-;y c
a! S2- X2
2016
MARSHALL
et (il.: MATRIX CRACKING
IN BRITTLE-MATRIX
FIBER COMPOSITES
strain energy in the fibers increases by dU,, and the
potential energy of the loading system decreases by di/,. These energy changes can be calculated from
the results presented by Aveston et al. (I] dU,=[o;R/GrE,Vj(l
+q)‘]dc
(7a)
dU,,,= [c&Rq/3rE/Vj(l -I-qy]dc
(7b)
dU,= [a%R(3~ + l)~~~,V~~l+ #I dc(7c) dU, = [&,R/2rE,V;(l
+ q)‘] de.
(7d)
The net energy change of the system resulting from the incremental crack extension is dU=ur,V,dc~dU,+dU,-dU,-dU, Fig. 4. Steady-state era& configuration used for energy balaaa analysis.
for a strai@ crack, or
(8)
where 2I’,,,= Cl: = KY (1 - v’)/& is the fracture surface energy of the matrix. Setting u, = u. at Griffith equilibrium (dU/dc = 0) and combining equations (7) and (8) yields a crack extension stress + q)2/E,,,R]“3 (9)
a, = S[(l - v')K~rE,V'jV,(l
where 6 = 6’9
Jo
Js'-t2
(6b) . .
for a penny crack, where s and t are normalized position coordinates and v is the Poisson’s ratio of the wmpo$itc. Therefore, analysis of matrix cracking by the stress intensity approach requires solution of equations (5) and (6) to obtain the crack surface tractions, followed .by evaluation of the integral in equation (2) and combination with equation (4). 3 BTEADYrnATE CRACKS
The intent of this section is to demonstrate that the ener8y balance and stress intensity approaches predict equivalent dependence of the stress for steady-state crack growth, oo, on microstructural parameters. The energy balance solution derives from the earlier analysis by Aveston et al. (11, but is expressed in terms of incremental crack extension. The stress intensity approach has not been considered previously.
3.2. Stress intensity analysis
A simple dimensional analysis is used in this set+ tion to demonstrate that the crack extension stress obtained from stress intensity considerations exhibits the same form as equation (9). Spe&c numerical quantities for the steady-state crack are obtained in the following section. The stress system for a steady-state crack is depicted in. Fig. 5. As noted previously, the closure pressure exactly balances the applied loading in the region x c c - co. Co~uen~y, the net crack surface tractions, 0, --p, are zero. However, over the area c - co< x < c net opening tractions exist. In this region, p(x) must vary smoothly between zero at the crack tip, x = c, and u, at x = c - co, i.e. fJ=%J@)
PGl
(10)
where p = r/co = (c -x)/co and f@) is a function that varies between zero at p = 0 and unity at p = 1. Thus, for steady-state cracks [c X, and (g_, - p) = .O
t
The energy changes occurring in the specimen and loading system during an incremental crack extension define the crack growth behavior. These energy changes can be calculated by employing the hypothetical operation depicted in Fig. 4 to extend the crack by dc. A strip of material of width dc ahead of the crack (arv AA’B’B) is removed, a cut is made in the matrix along CC’, the matrix is allowed to relax causing the ends of the strip to displace, and the strip is attached at the mouth of the crack. When the cut is made in the matrix, the matrix must slide back over the fibers while the fibers also extend. When this occurs work, dU,<,is done against frictional forces, the strain energy in the matrix decreases by dU,,, the
00
-
PW
-
0 00
x
t-
I
-
ptxt
I
Fig. 5. Net surface tractions acting on steady-state crack.
MARSHALL
et al.:
MATRIX
CRACKING
for r > co], equations (2a) and (2b) both reduce to KL = (Z/n)“2 I
-P(r)]r-“Zdr
7%
where u is a dimensionless
a~vu,c;~
(II)
eonstam
cu=(2,71) li2 1 ’ [I -f(p)]p
- “* dp
The length e. is evaluated by considering the crack opening displa~ments within the range r d c,. For the steady-state crack, equations (6a) and (6b) both reduce to
u(P)=
1
x
where d, = c(l - t)/co and 1 = c( 1 - s)/c,,. An alternative expression for u@) is obtained from equations (5) and (10) @co) = ~~[~~~12~/4~~~~,(1 Thus, equating
equations
+ tl).
x
I
v')KrrE~yl/V,(1+~1)2/E,Rj'"
(14)
fW
where a = 8( 1 - v2)zVfE,( 1 + q)/Eclw2.
t17b)
For x x c, equation (16) is an exact solution for the present problem (since displacements near the crack tip must be a unique function of KL). However, it is clearly not appropriate at small x for large cracks because, as already noted, the crack opening must asymptotically approach u,,, whereas the opening expressed by equation (16) is unbounded at large c. The timiting displacement is given by setting p = 5, in equation (5) +q)
dA
(15)
4 SHORT CRACKS
Evaluation of KL for short cracks requires expIicit determination of the crack opening displacements from equations (5) and (6). However, rigorous analytic solutions for u(x) cannot be obtained. Consequently, an approximate analytical solution is used initially to gain insight into the mechanics of crack growth. Then, a numerical solution is evaluated. solution
The approximate solution invokes an assumption that, at small crack sixes, the crack profile does not differ greatly from that of a crack subject to uniform
(19)
Therefore, the displacement expressed by equation (16) is used for cracks smaller than c,, (i.e. II < ug). For larger cracks, an approximate limiting solution is obtained from a crack prolle with uniform opening (u = %) near the center (X KC - cg) and a near-tip opening (at x > c - co) that is the same as that for a crack of length co. With this approximation, the stress intensity factor for c d c, is given by substituting equation (17a) into equations @a) or (2b) KL = Ru,c’~
Comparison of equations (9) and (15) indicates that the energy balance and stress intensity analyses predict identical relations between the matrix-cracking stress and m.ierostructuraI parameters.
(18)
and the corresponding transition crack length co is given by equation (16) with u = u, and x = 0 c, = a;“,/a2Ku.
where
4. I. Approximate udytid
- xz/cyy
Mo=5’,R/4?VjE/(l
1* The matrix cracking stress, cr,, is now obtained by evaluating KL from equations (11) and (14) and setting 5, = 5@at KL = K,dE,/E, a0 = &‘[(I -
p(x) = [aKW~(l
constant
’ 11-ftd; 11d4 l&-z
(16)
(13)
fl)(l - v2)
where w is another dimensionless
2017
pressure (with the magnitude of the pressure adjusted to give a stress intensity factor equal to KL). Then, an analytical solution for u(x) is obtained from equation (6a) or (6b) in terms of KL
(12) and (13) yields
c, = n5,E,RfrV$,(l+
COMPOSITES
The actual pressure distribution is then obtained by combining equations (5) and (16) to give
. >
O 5 pm
FIBER
u(x) = 2(1 - v2)KLc”‘(1 --x~/c~)‘~/&A”~.
0
2(1 -vZ)5,co nE c
fN BRITTLE-MATRIX
- (~c+c)‘~(K~)‘~c”‘I
c G co (20a) where a and I arc dimensionless crack geometry constants. For straight cracks R = ~‘12and
I=
‘(1 -X2)-“‘dXr1.20 i0 whereas for penny cracks, Q = 2/7r”2 and
(2Ob)
f=
‘(I-X2)-‘“XdX=2/3. (2@) I0 The mechanics of crack growth can be investigated by setting KL = Kk in equation (20a) and solving for 5, to obtain an equilibrium-stress/crack&e function u, = Kf/fk’” + [4aKf12/nS12]“2c”4
(c
This function can be conveniently malized form 5/a, = (1/3)(c/cJ-‘”
(21)
expressed in nor-
f (2~3)~c/c~)“’
(22)
2018
MARSHALL
et al.:
MATRIX
CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
where c,,,= (nK~/a12)2/3
(234
and
COMWSITIE:
APPROXIMATE
ANALYTICAL
SOLUTION
ti =&“[(I -v2)Kf%E,YjV,(l
+q)2,‘E,,,R]“3
(23b)
where 6” = 6P/Rn 1/z. Equation (22) provides a relation between normalized stress and crack length parameters, S = o/a,,, and C = c/c,,,, without explicit reference to material and micros~ctur~ properties (these properties enter only in their influence on the normalizing factors c, and u,,,). Thus, the mechanics of crack growth may he examined independently of the specific composite system. For c > c, the region of crack where u = or, does not contribute to the stress intensification at the crack tip. Therefore, for a straight crack, which maintains geometrical similarity as it extends, the stress intensity factor for c > co is given by equation (20) with c = co.t The corresponding equilibrium stress, co, which is independent of crack length, is given by equations (19), (21) and (23b)
NORMAWED
CRACK LENGTH. C/Cm &a) i
I
PENNY CRACK
and the transition crack length is given by [equations (19), (23a), and (24a)] c, = [Z/n”2n(l
-
21/z’“R)]4’3cm.
(24b)
For straight cracks, Qrr, = 1.02 and c, Jc,,,= 1.88, whereas for penny cracks, or, = o,,, and c,, = cm. The complete equilibrium-stress/crack-size functions for straight cracks and penny cracks are plotted in Fig. 6(a) and (b). It is noted that the stress required to propagate a matrix crack is almost independent of crack length for cracks larger than about c,,,/3. This defines the range of crack sizes over which steadystate conditions apply. The crack response in this region contrasts with the behavior of cracks in unreinforced brittle materials, for which the strength decreases with c-‘~. For cracks shorter than -c,,,/3, the equiIib~um stress increases as the crack length decreases, with a crack-length dependence resembling that of a monolithic brittle materia1.
2
1 NOMALUED
CRACK LENOTM. CfCm (b)
Fig: 6. Equilibrium-stnsslcracic-size functions for matrix cracks in composites and in a monoiithjc matrix: (a) straight embedded crack in infinite medium; (b) penny crack in infinite medium. putting p = (T, in equation (5) u, =[2(1 -vZ)/a.E,7f1/*]0~.
(25)
Thus, with U(X) = u(X)/u,, S = ~,/a, and C = c/c,, equations (6a) and (6b) can be re-expressed as
4.2. Numerical solution It is convenient to normalize the stresses, crack lengths and displacements in equations (6a) and (6b) by the parameters c,,, and a, defined in the previous section, and a limiting displacement I(, defined by
(264 and
tThe corresponding region of a penny crack does not maintain geometrical similarity. In this case, the region that determines the stress intensity factor can be viewed as an annular crack around a cylindrical hole. At c z c,, this con~guration approaches a penny crack, whereas at large crack lengths, c >>c,, it approaches an edge crack. Therefore. c0 and o0 vary between these two limits as the crack grows. The relative values of u0 and c0 for straight cracks and penny cracks are 0.97 and 0.87, respectively [Equations (19). (23b) and (24)].
f2W where y = 2rr’/2Q/3/2. For penny cracks, y = 3 and for straight cracks, y = 3.02.
MARSHALL
er. al.:
MATRIX
IN BRITTLE-MATRIX
CRACKING
Equations (2,h) and (26b) were solved by iteration to self-consistency in the function U(X), beginning with the approximate crack profile obtained in the previous section. The integrals were evaluated after t =s sin0 and making the transformations s = X cash 4. This procedure is more straightforward than previously published solutions of related crack opening problems [17j. After obtaining a selfconsistent crack profile, the stress intensity factor was computed from equation (2a) or (2b). Then, from sets of solutions for various values of C at each value of S, the function S(C) at quiWrium (i.e. KL = Kt). was evaluated. The results are compared with the approximate analytical solutions in Fig. 6(a) and (b). The two solutjuns show similar trends; at small crack lengths, they converge, but at large crack lengths the numerical steady-state stress is lower than the approximate analytical value (as expected from consideration of the procedure used to obtain the analytical solution) by -20%.
5. IMPLICATIONS AND DI&ZUSSION One of the important results of the present analysis is the definition of the approximate crack dimensions over which the transition from short-crack behavior to steady-state response occurs. The matrixcracking stress approaches the steady-state value for crack lengths &,/3 (Fig. 6). The crack length c, can be evaluated from quation (23a), in conjunction with equation (17b) c, = (7r/41’n)[K:E,
v:(l+
q)R/?Y;E/ x (1 - v’)P.
(27)
Values of c, calculated for two composite systems in which the parameters of equation (26) are known with reasonable accuracy are shown in Table 1. For both composites, cJ3 is several fiber spacings. Since the sizes of inherent flaws in brittle materials are usually about the same as microstructural dimensions, these results imply that the stress for matrix cracking in these composites is not substantially reduced by the further introduction of larger flaws during fabrication or service (e.g. mechanical contact
FIBER
SiC/@aw ccramiti propmties Kf
Ef
JL L? * Matrix cracking stress Caladated Properties %I 00 ‘Data
from Refs [IO] and [II].
bData from Refs [I and [S].
2019
damage), or by the extension of pre-existing flaws in thermal shock or environmentally assisted slow crack growth. It is also noted that, with the continuum approximation adopted in Section 2, the solutions in Fig. 6(a) and (b) hold for crack lengths larger than several fiber spacings. The crack-size independence of the matrixcracking stress at large crack lengths justifies, in part, the earlier analysis of Aveston et al. [l] for it is now possible to define a matrix-cracking stress as an intrinsic property of the composite, as required for their explanation of the regularity of the multiple matrix cracking. Moreover, for steady-state crack growth, their analysis based on a comparison of the energy of the untracked system and the energy after complete cracking is valid, and their result is quivalent to equation [9]. However, their analysis differs in one important respect from the present work. They prescribed the matrix-cracking strain E,,,as a characteristic of the matrix rather than the composite, and proposed that equation (9) applies only at small fiber diameters for which a, exceeds the stress corresponding to E,. Such a restriction is not required in the present analysis which takes into account the cracklength dependence of the strength of the tinforced matrix. In fact, the reinforcing e&t of the fibers can be inferred directly from quation (21). The 6rst term on the right side represents the product of the strength of the unreinforced matrix (Kf/Qc’“) and the modulus ratio, EC/E,,,. This term is plotted in Fig. 6(a) and (b). The difference between this curve and the soh&ions for the matrixaacking stress of the composite represents the reinforcing effect of the fibers for a composite with equal fiber and matrix moduli (E/= E,,,). For typical compositea, the fiber modulus is the larger, and the matrixcracking stress of the composite is always higher than the strength of the unreinforced matrix for a given crack length. However, for composites with E/-C E,,,, the relative strengths are dependent upon the crack length, The energy balance and stress intensity analyses provide equivalent relations between the steady-state matrix cracking stress and microstructural parameters, as indicated in Section 3.2. 4 quantitative comparison of the two results can be obtained by estimating the dimensionless constant, a’, in equation
Table 1. Fiber/matrix
Meawed
COMPOSITES
2 h4Pa.m’”
cnrb0n/glassb
200 GPa as GPa 0.5 8pm 2 MPa 270-300 MPa
0.75 MPa.m’” 380 GPa 70 GPa 0.4 4pm 10 MPa 340-430 MPa
325 pm 265 tiPa
44um 420 ‘MPa
2020
MARSHALL et al.: MATRIX CRACKING IN BRITTLE-MATRIX FIBER COMPOSITES
(15) from the analysis of Section 4.2. Thus, from equation (23b) and Fig. 6(a) or (b), we obtain 6’ = 1.83, which yields a stress essentially identical to that obtained from the energy balance analysis (6 = 6”‘). Predicted values of the stress for matrix cracking [from equation (911 in the SiC/glass-ceramic and carbon/glass composite systems are shown in Table 1. In both cases, agreement with experimentally measured values is evident. Hence, the predicted relation between the critical stress for matrix cracking and microstructural parameters appears to provide a basis for design of optimum microstmctums. Specifically, the criticfd stress increases with the toughness of the matrix, the modulus and volume fraction of fibe* the frictional stress at the fiber/matrix interface, and decmasing fiber diameter. However, it should also be appreciated that, as the interface frictional resistance increases, the net tensile stress on the fibers ahead of the crack also tends to increase. Thisstress eAmcement must eventually result in fiber failures associated with the advancing matrix crack and a consequent change in the failure mode. An optimum frictional resistance is thus anticipated, coincident with the maximum matrix cracking stress that prohibits fiber failure. This aspect of the problem remains to be resolved.
bc small and that the fibers remain intact after a crack passes completely through the matrix (i.e. co < a, V,, where trb is the bundle strength of the fibers in the presence of interfacial frictional forces). The most significant microstructural restriction concerns the volume fraction of fibers aligned in the principal stress axis, because an increase in V, beneficially inlluences all of the parameters that determine opti-
mum steady-state properties (i.e. both co and trbV, increase, while c, decreases). Thus, optimum properties are most likely in uniaxially reinforced composites. In multiaxially reinforced systems (such as 3-D or random whisker composites), the smaller volume fraction of fibers aligned in any direction limits the value of do that can be achieved without causing fiber failure and increases c,, making a crack-length-dependent matrix fracture stress more likely. The existence of optimum vahtes of matrix toughness, K,” , and interfacial shear resistance, r, can be inferred from the analysis. Increasing f beneficially increases u. and decreases c,,,, but the maximum increase in u. is’ limited by the fiber-failure stress. Increasing Kr increases u. but also has the detrimental effect of increasing c,. Thus, the maximum acceptable KF could be dictated either by the fiber-failure stress or by the requirement that c, be less than a pre-existing flaw size. The preceding restrictions account for the brittle response observed in a number of fiber or whisker6. CONCLUDING REMARKS reinforced brittle systems, and place important The principal implications of the present analysis bounds on the design of optimum microstructures. concern the predicted transition to a crack-length- Furthermore, implicit in the ability to design materiindependent matrix-cracking stress, uo, for cracks als and to interpret results within the context of the longer than a characteristic length -c,,,/3, and the present analysis is the availability of methods for associated trends in matrix fracture. The matrix- measuring T, K,Mand eb. Developments pertinent cracking stress (i.e. the first deviation from linearity to such measurement have been reported in recent in the stress-strain curve) should be both damage studies [1I]. tolerant and independent of specimen size, provided for this work was supplied by the characteristic length is smaller than preexisting Acknowledgement-Funding the Rockwell International Independent Research and flaws in the matrix. In this sense, ceramic matrix Development Program and the U.S. O&e of Naval ‘compositescan be more like metaIs than ceramics in Research, Contract No. NOQOW79-C-0159. their tensile mechanical behavior. The analysis indicates that the attainment of REFERENCE3 steady-state cracking at high stress levels is likely to be restricted to a narrow range of microstructures. 1. J. Aveston, G. A. Cooper and A. Kelly, in The Properties of Fiber Composites, Gmf. Pruc. pp. 15-26. These restrictions arise from the requirements that c,
Fig.
7. Analysis
of fiber pullout
mechanics.
MARSHALL
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16.
17.
et al.:
MATRIX CRACKING
National Physical Laboratory, IPC Science and Technology Press (1971). M. A. Ali and F. J. Grimer, J. Mater. Sci. 4,389 (1969). A. J. Majumdar, Proc. R. Sot. A319, 69 (1970). R. C. deVekey and A. J. Najumdar, Msg. Concrete Res. 20, 229 ( 1968). H. G. Allen, J. Camp. Mater. 5, 194 (1971). J. P. Romualdi, U.S. Patent 3429094 (1969). R. A. J. Sambell, A. Briggs. D. C. Phillips and D. H. Bowen, J. Ma&r. Sci. 7, 676 (1972). D. C. Phillips, J. Marer. Sci. 7, 1175 (1972). J. J. Brennan and R. M. Prewo, f. Mater. Sci. 17, 2371 ( 1982). K. M. Prewo and J. J. Brennan. J. Mater. Sci. 17. 1201 (1982). D. B. Marshall and A. G. Evans, J. Am. Cerwn. Sot. 68, 225 (1985). B. R. Lawn and T. R. Wilshaw, Fracture of Brirrle Solia!s. Cambridge Univ. Press (1975). G. C. Sib, Handbook of Stress Intensity Factors, Lehigh University, Bethlehem (1973). S. M. Wiederhom, in Fracture Mechanics of Ceramics (edited by R. C. Bradt, 0. P. H. Hasselman and F. F. Lange), Vol. 2, pp. 613-646. Plenum, New York (1974). B. J. Pletka and S. M. We&horn. in Fracture Mech&es of Cerumics (edited by R. 6. Bradt, 0. P. H. Has&man and F. F. Lange), Vol. 4, pp. 745-759. Plenum, New York (I978). I. N. Sneddon and M. Lowengrub, Crack ProbIems in the Classical Theory of Elasticity. Wiley, New York (1969). C. Atkinson, Znt. J. Fract. Nech. 6, 193 (1970). APPENDIX
1
Mechanics of fiber pullout The application of tractions 2’ to the end of the fiber in Fig. 7 causes sliding between the matrix and fiber over a distance f, and allows the fiber to pull out of the matrix a distance u. For a purely frictional matrix-t&r bond, the
IN BRIDLE-MATRIX
FIBER CGMPDSITES
202 1
sliding distance is determined by the length over which the interface shear stresses exceed the frictional stress T. A relation between T and u is required for the stress intensity analysis. The mechanics of fiber pullout can be conveniently analyzed by applying tractions T, and T,, equal and opposite to the stresses in the matrix and fibers, along the plane AA’ at the end of the slipped region, and removing the section A’C’C A [Fig. 7(b)]. If we neglect the effect of shear stresses above AA’ in Fig. 7(a) (i.e. assume that the strains in the matrix and fiber are equal above AA’), these tractions are related by
TmIEm = TfIEI. Equations relating the stresses and displacements in Fig. 7(b) are obtained by considering the equilibrium of the matrix and fiber separately and also by calculating the extensions S and d + u of the matrix and fiber T, A, = 2nRlr
(A2)
TA,,= 2n Rls + T/A,
(A3)
6/l = nRk/A,E,,,
644)
(6 + u)/l = T//E/+
G?k/A,E,
W)
where A,= nA” is the fiber cross-sectional area and A, is the area of matrix per fiber. The requisite relation between T and u is then obtained from equations (AI) to (AS). First, equations (Al) to (A3) combine to give T=21r(l
+rf)/E
(A6)
whem 4 = Efvf/E,,, F,. Then equations (Al), (A2), (A4) and (As) combine to give I2 = uRE,/r(l + q). Finally, from equations (A6) and (A7) we obtain T = Z[uE,r(l + q)/R]“2
(A@
and, with equation (1) p = 2[urf’;E,(I +q)/R]“2.
649)