2780 G N Morscher, J.D. Cawley / Journal of the European Ceramic Society 22(2002)2777-2787 With time, fibers exposed by a matrix crack will fail. Eq (5a) then simply bec Fiber failure is assumed to follow a weibull distribution that can be used to determine the probability for fiber (m+1) failure φ(6)= (8) m+1 P(σ,L)=1-e-中 The fraction of fibers that fail in a matrix crack can be o is the fraction of failed fibers according to determined by summing Eqs.( 8)and(5b)and simplify ing with Eq(4) where m is the Weibull modulus, oo is the reference y, m+/x stress and Lo is the reference length that corresponds to the average fiber strength determined in single fiber tensile tests. L is the effective gage length in the matrix oo(t, T crack. It proved useful to adopt the formulation of urtin's s characteristic stress, Oe, and characteristic where Ks (10) gage length, 8c, where (oc, Sc)=l and Sc is twice the m+1 The operative mechanism to be modeled to R composite rupture is the failure of the strongly fiber that triggers the growth of an unbridged crack fiber slip length, by definition, R is the fiber diameter, through the embrittled fibers in a matrix crack. The time and t is the interfacial shear stress [Eq. (4) dependence for the depth of embrittlement into the The fiber stress around the matrix crack varies composite determines the number of embrittled fibers because of load transfer due to friction(Fig. 2). The available in a matrix crack. The region of fiber embrit- fibers are subject to the maximum fiber stress in the tlement often appears as a"picture frame"4 of strongly crack opening. To determine the total fraction of fiber bonded fibers around the rim of the cross-section of a failures, o can be integrated over the stress transfer specimen fracture surface for a composite with through length, Eo and added to the fraction of fiber failures in thickness matrix cracks. The time dependence for fiber the crack opening width, o to-fiber fusion was determined empirically from the depth of this"picture frame"by examination of rup a(z)"dz (5a) tured-specimen fracture surfaces for two different MI composite systems, one reinforced with HN fibers and =()(∞ Corp, Midland, MI) fibers \4is/mmic (Dow Corning that a semi-empirical parabolic time-dependence was an adequate description: where z is the stress transfer length and u is the crack opening width which can be approximated by: 21 (6) where Cox is an empirical coefficient that best fits the 4Tf2E measured oxidation depth data for a given composite system Since the fiber strength degradation of pu Eq (Sa)was solved by othersfor the case where z is fibers in a through-thickness cracked composite is con- qual to twice the fiber slip length, 8, assuming the far sistent with the measured degradation in fiber strengths field stress on the fibers to be zero. This is an appro- of as-produced fibers, the descriptive expression for time priate assumption because there is a negligible con- dependent fiber strength degradation developed by Yun tribution to from the low far field fiber stress(1/5 and DiCarlo 6 for the latter was used. Their data for ofc).8 Can then be approximated assuming a constant t rupture strength of three Sic type fibers are plotted in from the relationship(see Fig. 2) Fig 4 as a Larson-Miller plot. The conditions for our
f ¼
f ð1Þ With time, fibers exposed by a matrix crack will fail. Fiber failure is assumed to follow a Weibull distribution that can be used to determine the probability for fiber failure: Pð
; LÞ ¼ 1e ð2Þ is the fraction of failed fibers according to: ¼ L Lo
f
o
m ð3Þ where m is the Weibull modulus, so is the reference stress and Lo is the reference length that corresponds to the average fiber strength determined in single fiber tensile tests. L is the effective gage length in the matrix crack. It proved useful to adopt the formulation of Curtin’s 18 characteristic stress, sc, and characteristic gage length, dc, where (sc,dc)=1 and dc is twice the
c ¼
m o Lo R
1 mþ1 ; c ¼ R
c  ð4Þ fiber slip length, by definition, R is the fiber diameter, and t is the interfacial shear stress [Eq. (4)]. The fiber stress around the matrix crack varies because of load transfer due to friction (Fig. 2). The fibers are subject to the maximum fiber stress in the crack opening. To determine the total fraction of fiber failures, can be integrated over the stress transfer length, and added to the fraction of fiber failures in the crack opening width, u: ¼ ðZ dz ¼ ðZ
ðzÞ
o
m dz Lo ð5aÞ u ¼ u Lo

f
o
m ð5bÞ where z is the stress transfer length and u is the crack opening width which can be approximated by:21 u ¼
2R 4f 2Ef 1 þ Ef f Emð Þ 1f
ð6Þ Eq. (5a) was solved by others 22 for the case where z is equal to twice the fiber slip length, d, assuming the far field stress on the fibers to be zero. This is an appro￾priate assumption because there is a negligible con￾tribution to from the low far field fiber stress (1/5 sfc).  Can then be approximated assuming a constant t from the relationship (see Fig. 2):  ¼ R
f  ð7Þ Eq. (5a) then simply becomes: ðÞ ¼
f
c
ð Þ mþ1 m þ 1 ð8Þ The fraction of fibers that fail in a matrix crack can be determined by summing Eqs. (8) and (5b) and simplify￾ing with Eq. (4): t;T ¼
f
c
ð Þ mþ1 m þ 1 þ u Lo
f
o
m ¼ 1 Lo
f
oðt;TÞ
m  m þ 1 þ u
¼ K
m oðt;TÞ ð9Þ where K ¼
m f Lo
 m þ 1 þ u
ð10Þ The operative mechanism to be modeled to predict composite rupture is the failure of the strongly bonded fiber that triggers the growth of an unbridged crack through the embrittled fibers in a matrix crack. The time dependence for the depth of embrittlement into the composite determines the number of embrittled fibers available in a matrix crack. The region of fiber embrit￾tlement often appears as a ‘‘picture frame’’ 4 of strongly bonded fibers around the rim of the cross-section of a specimen fracture surface for a composite with through￾thickness matrix cracks. The time dependence for fiber￾to-fiber fusion was determined empirically from the depth of this ‘‘picture frame’’ by examination of rup￾tured-specimen fracture surfaces for two different MI composite systems, one reinforced with HN fibers and the other reinforced with Sylramic (Dow Corning Corp., Midland, MI) fibers 14,15 (Fig. 3). It was found that a semi-empirical parabolic time-dependence was an adequate description: x ¼ Coxt 1=2 ð11Þ where Cox is an empirical coefficient that best fits the measured oxidation depth data for a given composite system. Since the fiber strength degradation of pulled out fibers in a through-thickness cracked composite is con￾sistent with the measured degradation in fiber strengths of as-produced fibers, the descriptive expression for time dependent fiber strength degradation developed by Yun and DiCarlo 16 for the latter was used. Their data for rupture strength of three SiC type fibers are plotted in Fig. 4 as a Larson–Miller plot. The conditions for our 2780 G.N. Morscher, J.D. Cawley / Journal of the European Ceramic Society 22 (2002) 2777–2787