G N. Morscher, J.D. Cawley /Journal of the European Ceramic Society 22(2002) 2777-2787 Table I Composite and constituent properties (16b) HN SYL Composite physical properties At this point it is illustrative to show the probability Tow ends per cm for fiber failure from Eqs. (16a)and (16b) and the hickness. mm number of predicted fiber failures in and around a single matrix crack for various applied rupture stress condi- No fibers per tow tions for the HN-BN-MI 4 composite system as a function of time and depth of embrittlement(Fig 5a and b, respectively). The variables used are listed in Table 1. most fiber failure would occur at short times and diminish with increasing time(Fig 5a); however, it takes a period of time to embrittle most fibers. For thi Room temperature cult reason, when predicting whether or not a fiber failure will occur for an embrittled fiber, it is absolutely neces- sary to take into account the probability that fibers had already failed prior to being embrittled. Fig. 5 depicts Fiber strength distribution properties e situation where rupture time reaches 21.5 hours(the 2800 2800 time to fully embrittle the HN-BN-MI composite at mm 815C in air). At a depth of 0. 2 mm from the composite urface, fibers were not embrittled for 0.9h. However, a Time dependent propertie greater fraction of fibers would be expected to fail prior 0215 0.125 to fiber embrittlement at 0. 2 mm depth, r, compared to the fraction of fibers that would be expected to fail after 56.5 embrittlement, emb. If fibers fail prior to embrittlemen hen the load shed from that fiber is shared globally, unbridged crack growth will not occur, and those fibers (15) that fail prior to embrittlement are removed from the OO(RT population of weak fibers that could fail. Therefore, the fraction of embrittled fibers that fail at a given The fraction of fibers that fail in a matrix crack as a region in a matrix crack can be estimated according to function of time can then be determined from Eqs. (10) the construct of Fig. 5 using Eq.(16b) for embrittle and(15)Eq(16a)]. However, for the purpose of deter- ment depth mining the fraction of fibers that fail as a function of the depth of oxidation embrittlement, it is advantageous convertino depth,x,from Eq.(1)asu(b.ab=pn、_的mCc2m1 ](17) (1)mn where max is the fraction of fibers that fail at the ●000015 000005 中 0246810121416182022000 0200.40 Time, hours (b Fig. 5. Predicted fraction of fiber failures for HN in a matrix crack of an HN/BN/MI composite at 815C (Table 1)as a function of (a)time and(b) embrittlement depth for an apl mposite stress of 150 MPa
0ðt;TÞ ¼
oðcompositeÞ
oðRTÞ Cf t 1=n ¼ Cfrupturet 1=n ð15Þ The fraction of fibers that fail in a matrix crack as a function of time can then be determined from Eqs. (10) and (15) [Eq. (16a)]. However, for the purpose of deter￾mining the fraction of fibers that fail as a function of the depth of oxidation embrittlement, it is advantageous to convert t into depth, x, from Eq. (11) as well [Eq. (16b)]. t;T ¼ K Cm frupture ð Þt m=n ð16aÞ t;T ¼ K Cm frupture x Cox
2m=n ð16bÞ At this point it is illustrative to show the probability for fiber failure from Eqs. (16a) and (16b) and the number of predicted fiber failures in and around a single matrix crack for various applied rupture stress condi￾tions for the HN-BN-MI 14 composite system as a function of time and depth of embrittlement (Fig. 5a and b, respectively). The variables used are listed in Table 1. Most fiber failure would occur at short times and diminish with increasing time (Fig. 5a); however, it takes a period of time to embrittle most fibers. For this reason, when predicting whether or not a fiber failure will occur for an embrittled fiber, it is absolutely neces￾sary to take into account the probability that fibers had already failed prior to being embrittled. Fig. 5 depicts the situation where rupture time reaches 21.5 hours (the time to fully embrittle the HN–BN–MI composite at 815
C in air). At a depth of 0.2 mm from the composite surface, fibers were not embrittled for 0.9 h. However, a greater fraction of fibers would be expected to fail prior to fiber embrittlement at 0.2 mm depth, t, compared to the fraction of fibers that would be expected to fail after embrittlement, emb. If fibers fail prior to embrittlement then the load shed from that fiber is shared globally, unbridged crack growth will not occur, and those fibers that fail prior to embrittlement are removed from the population of weak fibers that could fail. Therefore, the fraction of embrittled fibers that fail at a given region in a matrix crack can be estimated according to the construct of Fig. 5 using Eq. (16b) for embrittle￾ment depth: emb ¼ tmax t ¼ K Cm frupture C2m=n ox x2m=n max x2m=n  ð17Þ where tmax is the fraction of fibers that fail at the Table 1 Composite and constituent properties HN SYL Composite physical properties Tow ends per cm 6.7 7.1 No. Plies 8 8 Thickness, mm 2.1 2.1 Width, mm 12.5 10 No. fibers per tow 500 800 R, mm 6.5 5 F 0.17 0.17 Composite mechanical properties Ec, GPa 215 265 n 0.15 0.15 Room temperature sult 390 340 t, MPa 30 60 Ef, GPa 280 380 Em, GPa 202 242 Fiber strength distribution properties so, MPa 2800 2800 M 75 Lo, mm 25.4 25.4 Time dependent properties Cox 0.215 0.125 Cf 1761 2169 n 56.5 122 Fig. 5. Predicted fraction of fiber failures for HN in a matrix crack of an HN/BN/MI composite at 815
C (Table 1) as a function of (a) time and (b) embrittlement depth for an applied composite stress of 150 MPa. 2782 G.N. Morscher, J.D. Cawley / Journal of the European Ceramic Society 22 (2002) 2777–2787