K Goda/Composites Science and Technology 59(1999)1871-1879 concerned with decreasing a scatter in composite strength. Such a situation accompanied with the more ErA cumulative fiber breaks is generated by small-scale debondings,and therefore contribute to improving the Ea, d+1-2u; -li-1)=0 strength and reliability of composites. Changes in the strength and reliability can also be explained using a stochastic model, in which the failure EyA/ (uN-lN-1)=0 probability of a fibrous composite can be estimated As described in the previous chapter, the small inter- facial shear strength results in a poor load-bearing where ui(i=2, 3. N-1): displacement of fiber capacity. a decrease in the capacity corresponds to an for x axis, E: elastic modulus of the fiber,A:cross- increase in the ineffective length [1-4], i.e. the stress sectional area of the fiber, Gm: shear modulus of the region unrecovered from its broken point. It is reported matrix, h: thickness of the monolayer, d: distance that a decrease in ineffective length of broken fibers between the fibers. The finite difference approximation brings an increase in scatter in composite strength, as for the differential terms of Eq (2)enables the problem well as a decrease in strength, because the number of to solve it as linear equations. By giving appropriate conceptual bundle elements consisting of ineffective boundary conditions to some of the finite difference length is reduced [17]. Therefore, large-scale debonding terms, one can simulate the composite fracture process which leads to an increase in ineffective length, reduces In this study a fiber element corresponds to the part the strength and reliability of composites. Incidentally, between two axial finite difference nodes, and a matrix an increase in the local stress concentrations around element does to the part between two transverse nodes broken fibers also reduces the strength and reliability The simulation procedure of this model follows the Pitt and Phoenix [18] developed a stochastic model previously proposed technique [19]. In order to compare under a tapered load sharing rule in a planar composite, with the present finite element simulation, an interfacial in which the four nearest fibers around broken fibers condition without frictional stress was assumed for the support the lost loads. They compared the stochastic finite difference shear-lag simulation. Fig. 8 shows the behavior with the conventional model with a local load results of five hundreds of the shear-lag simulation. The sharing rule, in which only the nearest two fibers support average strength shows almost a constant at the rela the lost loads. Accordingly in this rule stress concentra- tively low interfacial shear strengths and begins to ion acts in the fibers more locally and intensively. The increase monotonically around ti= 30MPa with further results showed that the local load sharing rule reduced increase in the interfacial shear strength. Also, the coef- composite strength and reliability, i.e. decreased the ficient of variation begins to decrease around the same strength and increased the scatter in strength. It is pre- value, but never increases. In the shear-lag simulati dicted from the fact that matrix cracking following broken fibers, which leads to a further increase in stress 300 2 concentration, also reduces the strength and reliability Average strength of composites. On the other hand, small-scale debonding produces a short ineffective length and a relaxed stress 290 11 concentration such as a tapered load sharing rule, and consequently leads to a high strength and reliability 280 00-0OO 105 4. 2. Comparison with shear -lag model a shear-lag model is often used in estimating stress 70 distributions around broken fibers in a composite and composite strength properties, as described earlier. In this section the effect of interfacial shear strength on the 260 8 tensile strength of a boron/epoxy composite monolayer is determined with aid of a monte- Carlo simulation -A- Coefficient of variation based on a shear-lag model, and compared with the 250 finite element simulation results. This model assumes 01020304050 that only the fibers in the composite sustain tensile Interfacial shear strength MPa load, and the matrix deforms only in shear and trans- Fig. 8. Effect of interfacial shear strength on the strength and relia- mits tensile load to the fibers. The force equilibrium bility of boron/epoxy composites by shear-lag model simulation quations for a composite monolayer with N fibers are Reprinted with on from Trans JSME 1997: 63A: 445-452 given as simultaneous differential equations as follows: o 1999 The Japan Society of Mechanical Engineers [161concerned with decreasing a scatter in composite strength. Such a situation accompanied with the more cumulative ®ber breaks is generated by small-scale debondings, and therefore contribute to improving the strength and reliability of composites. Changes in the strength and reliability can also be explained using a stochastic model, in which the failure probability of a ®brous composite can be estimated. As described in the previous chapter, the small inter￾facial shear strength results in a poor load-bearing capacity. A decrease in the capacity corresponds to an increase in the ine€ective length [1±4], i.e. the stress region unrecovered from its broken point. It is reported that a decrease in ine€ective length of broken ®bers brings an increase in scatter in composite strength, as well as a decrease in strength, because the number of conceptual bundle elements consisting of ine€ective length is reduced [17]. Therefore, large-scale debonding which leads to an increase in ine€ective length, reduces the strength and reliability of composites. Incidentally, an increase in the local stress concentrations around broken ®bers also reduces the strength and reliability. Pitt and Phoenix [18] developed a stochastic model under a tapered load sharing rule in a planar composite, in which the four nearest ®bers around broken ®bers support the lost loads. They compared the stochastic behavior with the conventional model with a local load sharing rule, in which only the nearest two ®bers support the lost loads. Accordingly in this rule stress concentra￾tion acts in the ®bers more locally and intensively. The results showed that the local load sharing rule reduced composite strength and reliability, i.e. decreased the strength and increased the scatter in strength. It is pre￾dicted from the fact that matrix cracking following broken ®bers, which leads to a further increase in stress concentration, also reduces the strength and reliability of composites. On the other hand, small-scale debonding produces a short ine€ective length and a relaxed stress concentration such as a tapered load sharing rule, and consequently leads to a high strength and reliability. 4.2. Comparison with shear-lag model A shear-lag model is often used in estimating stress distributions around broken ®bers in a composite and composite strength properties, as described earlier. In this section the e€ect of interfacial shear strength on the tensile strength of a boron/epoxy composite monolayer is determined with aid of a Monte-Carlo simulation based on a shear-lag model, and compared with the ®nite element simulation results. This model assumes that only the ®bers in the composite sustain tensile load, and the matrix deforms only in shear and trans￾mits tensile load to the ®bers. The force equilibrium equations for a composite monolayer with N ®bers are given as simultaneous di€erential equations as follows: EfAf d2u1 dx2 ‡ Gmh d …u2 ÿ u1† ˆ 0 EfAf d2ui dx2 ‡ Gmh d …ui‡1 ÿ 2ui ÿ uiÿ1† ˆ 0 EfAf d2uN dx2 ÿ Gmh d …uN ÿ uNÿ1† ˆ 0 …2† where ui…i ˆ 2; 3; ...... ; N ÿ 1†: displacement of ®ber for x axis, Ef: elastic modulus of the ®ber, Af: cross￾sectional area of the ®ber, Gm: shear modulus of the matrix, h: thickness of the monolayer, d: distance between the ®bers. The ®nite di€erence approximation for the di€erential terms of Eq. (2) enables the problem to solve it as linear equations. By giving appropriate boundary conditions to some of the ®nite di€erence terms, one can simulate the composite fracture process. In this study a ®ber element corresponds to the part between two axial ®nite di€erence nodes, and a matrix element does to the part between two transverse nodes. The simulation procedure of this model follows the previously proposed technique [19]. In order to compare with the present ®nite element simulation, an interfacial condition without frictional stress was assumed for the ®nite di€erence shear-lag simulation. Fig. 8 shows the results of ®ve hundreds of the shear-lag simulation. The average strength shows almost a constant at the rela￾tively low interfacial shear strengths and begins to increase monotonically around I ˆ 30MPa with further increase in the interfacial shear strength. Also, the coef- ®cient of variation begins to decrease around the same value, but never increases. In the shear-lag simulation Fig. 8. E€ect of interfacial shear strength on the strength and relia￾bility of boron/epoxy composites by shear-lag model simulation. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. K. Goda / Composites Science and Technology 59 (1999) 1871±1879 1877