1876 K Goda/Composites Science and Technology 59(1999)1871-1879 300 12 0.7 a290○ 11 09 0.6/-A2-break ○3- break or more 0.5 280 △105 04 270 0.3 260 8 -O- Average strength 0.2 △ Coefficient of variation 250 0.1 01020304050 Interfacial shear strength MPa 0.0 01020304050 Fig. 6. Effect of interfacial shear strength on the strength and reliability of boron/epoxy composites. Reprinted with permission Interfacial shear strength MPa from Trans JSME 1997: 63A: 445-452. c 1999 The Japan Society of Fig. 7. Ratio of the number of j-break occurrence to the total num ber [/=297MPa and more. In these levels many of fiber mode, shows the minimum value at t/=175MPa, and breaks are followed by matrix cracking, not interfacial increases up to around 0.5 with an increase in interfacial debonding, because ti is beyond the shear strength of the shear strength. On the other hand, the ratio of 3-break matrix. Consequently the effect of interfacial debonding or more cumulative failure mode behaves to be con- trary to the case of l-break. Namely, this ratio gives the peak at tI=17.5MPa, decreases to around 0. 15, and then shows almost a constant. In the case of 2-break 4. Discussion intermediate failure mode, the ratio shows approxi- ightly higher level 4.1. Concerns of interfacial shear debonding with respect at t/=11.7MPa to 23.3 MPa. Such a correspondence to to strength and reliability Fig. 6 tells us that composites with large-scale debond ing and matrix cracking yield a smaller i-break than a It was shown that the small and large interfacial shear composite with small-scale debonding. If failure of the strengths cause large-scale debonding and matrix weakest fiber in a composite immediately leads to com- cracking, respectively, and they both reduced the com- posite fracture without accumulation of fiber breaks, a posite's average strength On the other hand, the inter- variability in composite strength would reflect that in mediate values of interfacial shear strengths resulted fiber strength. Because the scatter of the weakest fiber in increasing the average strength. Why do the inter- strength is the same as that of the whole fiber strength. mediate values raising the average strength also produce Thus the scatter is large, corresponding to an upper the low coefficients of variation? The author considers bound. A less cumulative failure mode will approach to that the strength and reliability of fibrous composites is this degree of scatter. Therefore both small and large closely related with a damage quantity accumulated up interfacial shear strengths induce significant scatter in to the maximum stress, the number of broken fiber ele- composite strength On the other hand, more cumula ments. For example, in Fig. 3(a(c) these numbers are tive fiber breaks further increases the stress level, so M Fig. 7 shows the ratio of the number of i-break width toward the upper side. The shrinkage might be 2 and 1, respectively(referred to as 'i-break) that the survival fiber distribution would shrink in occurrence to the total number of simulations versus the interfacial shear strength. In the present simulation most of composites indicated 1-, 2-or 3-break, but four a composite consists of N fibers and the fiber st or more breaks was not a major figure (in any lows a 2-parameter Weibull distribution, the cumulative distribution function for the weakest fiber strength. i.e. for the first order statistic approximately I to 3% The transition of the ratio is given as follows: A()=1-( -Fo)]=1-exp(-N(a is comparable to the pattern shown in Fig. 6. That is This form is also a 2-parameter Weibull distribution, and this shape to say, the ratio of l-break, non-cumulative failure parameter, m, agrees with that of its population F(a)I ˆ 29:7MPa and more. In these levels many of ®ber breaks are followed by matrix cracking, not interfacial debonding, because I is beyond the shear strength of the matrix. Consequently the e€ect of interfacial debonding disappears. 4. Discussion 4.1. Concerns of interfacial shear debonding with respect to strength and reliability It was shown that the small and large interfacial shear strengths cause large-scale debonding and matrix cracking, respectively, and they both reduced the com￾posite's average strength. On the other hand, the inter￾mediate values of interfacial shear strengths resulted in increasing the average strength. Why do the inter￾mediate values raising the average strength also produce the low coecients of variation? The author considers that the strength and reliability of ®brous composites is closely related with a damage quantity accumulated up to the maximum stress, the number of broken ®ber ele￾ments. For example, in Fig. 3(a)±(c) these numbers are 1, 2 and 1, respectively (referred to as `i-break'). Fig. 7 shows the ratio of the number of i-break occurrence to the total number of simulations versus the interfacial shear strength. In the present simulation most of composites indicated 1-, 2- or 3-break, but four or more breaks was not a major ®gure (in any case approximately 1 to 3%.). The transition of the ratio is comparable to the pattern shown in Fig. 6. That is to say, the ratio of 1-break, non-cumulative failure mode, shows the minimum value at I ˆ 17:5MPa, and increases up to around 0.5 with an increase in interfacial shear strength. On the other hand, the ratio of 3-break or more, cumulative failure mode, behaves to be con￾trary to the case of 1-break. Namely, this ratio gives the peak at I ˆ 17:5MPa, decreases to around 0.15, and then shows almost a constant. In the case of 2-break, intermediate failure mode, the ratio shows approxi￾mately a constant, though it keeps a slightly higher level at I ˆ 11:7MPa to 23.3 MPa. Such a correspondence to Fig. 6 tells us that composites with large-scale debond￾ing and matrix cracking yield a smaller i-break than a composite with small-scale debonding. If failure of the weakest ®ber in a composite immediately leads to com￾posite fracture without accumulation of ®ber breaks, a variability in composite strength would re¯ect that in ®ber strength. Because the scatter of the weakest ®ber strength is the same as that of the whole ®ber strength.1 Thus the scatter is large, corresponding to an upper bound. A less cumulative failure mode will approach to this degree of scatter. Therefore both small and large interfacial shear strengths induce signi®cant scatter in composite strength. On the other hand, more cumula￾tive ®ber breaks further increases the stress level, so that the survival ®ber distribution would shrink in width toward the upper side. The shrinkage might be Fig. 6. E€ect of interfacial shear strength on the strength and reliability of boron/epoxy composites. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. Fig. 7. Ratio of the number of i-break occurrence to the total number of simulations vs. interfacial shear strength. 1 When a composite consists of N ®bers and the ®ber strength fol￾lows a 2-parameter Weibull distribution, the cumulative distribution function for the weakest ®ber strength, i.e. for the ®rst order statistic, is given as follows: F…1† …† ˆ 1 ÿ f g 1 ÿ F…† Nˆ 1 ÿ exp ÿN  0 n o  m . This form is also a 2-parameter Weibull distribution, and this shape parameter, m, agrees with that of its population F…†. 1876 K. Goda / Composites Science and Technology 59 (1999) 1871±1879