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Approximate upper and lower bounds for the strength of unidirectional composites
Composites Science and Technology 59 (1999) 89±95
Approximate upper and lower bounds for the strength of unidirectional composites Atsushi Wada, Hiroshi Fukuda* Department of Materials Science and Technology, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Received 30 April 1997; received in revised form 29 January 1998; accepted 26 February 1998
Abstract This paper addresses the problem of determining the probabilistic strength of unidirectional composites. We ®rst calculated the stress-concentration factors (SCFs) around broken ®bers in a three-dimensional, hexagonal-array model. Two approaches were tried to calculate the SCFs. One is a shear-lag analysis originated by Hedgepeth and Van Dyke and the other is close to a local loadsharing rule (LLS). These two sets of SCFs provide the upper and lower bounds of the strength. With these SCFs, we next carried out a Monte Carlo simulation to evaluate the strength of unidirectional composites in which a chain-of-bundles probability model and a Weibull distribution were used. Parametric studies were ®rst conducted and a comparison with experiments and other existing theories was next done. Our `approximate' upper bound was lower than Rosen's prediction and lower bound, higher than Zweben's value. The value from the simulation was also close to the experimental results. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Unidirectional composites; Stress concentration factor; Shear-lag analysis; Monte Carlo simulation; Rosen model; Zweben model; Weibull distribution; Parametric study; Comparison with experiment
1. Introduction Composite materials are heterogeneous materials composed of reinforcements and matrices and it therefore, becomes necessary to deal with these materials from a micromechanical view point. Since the original report of Rosen [1], much work have been done on the micromechanical and statistical evaluation of the strength of unidirectional composites. The characteristics of individual ®bers, matrix, and ®ber/matrix interface are the most important factors for evaluating FRPs from a micromechanical view point. Because some synergistic eects at the ®ber/matrix interface occur in ®ber-reinforced composites, the strength cannot be predicted by simply averaging the strength of its constituents. Prediction of the strength of advanced composite materials requires the factor of stress concentrations around broken ®bers and the eect of the ®ber/matrix interface the latter of which can be derived from the tensile test of a mono®lament embedded in a matrix.
* Corresponding author. Tel.: +81-471-241501-4308; fax: +81471-23-9362; e-mail: [email protected].
Among the various existing models for the strength of unidirectional composites, the Rosen model [1] and the Zweben model [2] would be fundamental and most important. When ®bers are broken in a unidirectional composite, stress concentrations occur around the broken ®bers. The Rosen model does not take the stress-concentration factors (SCFs) into consideration. Harlow and Phoenix [3] called this idea the equal load-sharing (ELS) rule. The Rosen model predicts higher strength than that of the actual composite. Zweben [2], on the other hand, introduced the SCFs calculated by Hedgepeth [4] on the basis of a shear-lag analysis. However, Hedgepeth's original work was for a unidirectional sheet rather than three-dimensional composites. Two-dimensional calculation ends up with larger SCFs and therefore, the Zweben model leads to lower composite strength. The Rosen and Zweben models are eectively upper and lower bounds of the strength although this classi®cation is not made in a precise manner. The objective of the present paper is to predict the strength more accurately than either the Rosen or the Zweben model. To this end, we need more accurate SCFs. After calculating the SCFs in a three-dimensional,
0266-3538/99/$Ðsee front matter # 1999 Elsevier Science Ltd.. All rights reserved PII: S0266 -3 538(98)00052 -9
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A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
hexagonal-array model, we try a Monte Carlo simulation for the strength of composites.
H. This does not coincide with the LLS rule in a precise manner and therefore, we call it here a quasi-LLS rule.
2. Stress-concentration factors
2.2. Shear-lag analysis
If we estimate larger SCFs than actual values, the consequent strength becomes small and vice versa. In the present study, we took into consideration two kinds of SCFs. One is close to the local load-sharing (LLS) rule [3]. As will be described later, the SCFs derived from this idea are rather large and hence, the strength will be smaller. Another idea relating to SCFs is based on a shear-lag analysis; this leads smaller SCFs and larger strength. Therefore, the actual strength will be between the upper and lower bounds.
Hedgepeth and Van Dyke [6] already derived the SCFs in a hexagonal-array model, Fig. 2, although their numerical results are for limited cases. Suemasu [7] also derived similar equations, although the expression is dierent from Ref. [6]. What we need to add to Refs. [6] or [7] is to calculate the SCFs at various combination of broken ®bers. Fig. 3 is an example of the SCFs by means of the shear-lag analysis. This ®gure shows a cross section of a unidirectional composite. Each circle corresponds to each mono®lament and the three hatched ®bers are assumed to be broken in this example. Each numeral is the SCF of each ®ber. Although the calculation was done in in®nite region, only the SCFs of ®bers nearest to the broken ®bers are shown in this ®gure. As is seen, the SCF diers each other even though these ®bers are all next to the broken ®bers. Thus we need to know the SCFs of all combination of broken ®bers to deal with the forthcoming simulation. As will be described later, simulations were conducted for 7, 19, . . ., 91-®ber models. Fibers are assumed to be broken within this region only. Since all ®bers surrounding this region are assumed to be continuous, the SCFs calculated here may lead somewhat smaller
2.1. Quasi-LLS rule Harlow and Phoenix [3] introduced an idea of LLS rule which assumes that the load initially sustained by a broken ®ber will be shared with the nearest neighboring, surviving ®bers. For example, if one ®ber is broken in a unidirectional sheet, the SCFs of adjacent ®bers should be 1.5. In the case of a hexagonal-array, the SCF will be 1.166 (1 + 1/6) because there are six surrounding ®bers. Lienkamp et al. [5], for example, adopt this idea for a seven ®ber microcomposite. Although we are aware that this is not the case as was shown in the works of Hedgepeth [4,6], we adopt this idea to calculate an approximate lower bound of the strength. As will be described later, we assume the failure surface is ¯at in the simulation. If we allow other failure patterns such as ®ber pull out or ®ber-matrix interfacial debonding, the strength may decrease. That is, even if we use the LLS rule, there is no guarantee that the result is absolutely the lower bound. This is the reason why we use the word `approximate' in the title. It is rather time-consuming task to calculate each time the SCFs at each failure pattern. Then we adopted an approximate method as shown in Fig. 1. That is, when the ®ber #1 is broken, each 1/6 of load is allocated to the surrounding six ®bers; this is exactly the same as the LLS rule. But when the ®ber #2 is next broken, we assume that the load, 1.166, is equally shared with the surrounding and still surviving ®ve ®bers, A, B, F, G,
Fig. 1. Concept of SCF (quasi-LLS rule).
Fig. 2. Shear-lag model of hexagonal array.
A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
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3.2. Simulation program
Fig. 3. SCFs by means of shear-lag analysis (example).
SCFs than the actual case and therefore, the estimated strength may be larger than the actual strength. 3. Monte Carlo simulation 3.1. Weibull distribution The strength of many high-performance ®bers can be well represented by a Weibull distribution, which, in one functional form, may be written as F x 1 ÿ exp ÿ x= m ;
1
where F(x) is the cumulative distribution function of the strength, x is the strength, and m and are, respectively, the shape and scale parameters. Large m corresponds to a small scatter and large indicates large strength. Eq. (1) is known as a two-parameter Weibull distribution. If the shape and scale parameters are known, and if we specify a probability of failure F(x), then we can solve Eq. (1) for the strength associated with that probability by x ÿ ln 1 ÿ F x1=m :
In the simulation, it is assumed that the in¯uence of broken ®ber is limited to a critical length, lc of Fig. 4, and that the stress concentration occurs to only neighbouring ®bers. The basic idea of the failure process is the same as Rosen [1] or Zweben [2] model. That is, when all ®bers within a critical length are broken, the specimen is considered to break. The formulation of the computer program is close to Oh's work [8] which may be described as follows. First, a model of MF ®bers with NL links is settled as shown in Fig. 5. The total number of links (NL) is MF NL. Then each Weibull random number is allocated to each link. This represents the strength of the link, denoted by STR (i, j), where i 1; . . . ; MF indicates each ®ber and j=1,. . ., NL shows the link number. The stress concentration factors of links, denoted by SCF (i, j), were originally settled to unity. In the actual simulation, MF=7,19,37,61,91 were chosen. A link (i, j) which has the smallest STR (i, j)/SCF (i, j) is searched. This link is considered to break at the normalized stress of STR (i, j)/SCF (i, j). If the link breaks, the stress redistribution will take place, and hence, the SCF (i, j) should be changed. Again the link of the smallest STR (i, j)/SCF (i, j) is looked for among the surviving ®bers; this link should break next. Fig. 5 also shows the ¯ow chart of this simulation. This process should be continued until all links in the same critical length are broken. However, in the present simulation, the following procedure was adopted to reduce the computational time. Fig. 6 is an example of the load versus the number of broken links in a 91-®ber model where the quasi-LLS rule is applied. In this case, the maximum load appears when the 4th link breaks.
2
For a Monte Carlo analysis, in order to simulate ®ber strengths that follow the Weibull distribution, uniform random numbers between 0 and 1 are ®rst generated. These random numbers are substituted into Eq. (2) producing a composite whose ®ber strengths follow the Weibull distribution.
Fig. 4. Critical length, lc
Fig. 5. Monte Carlo simulation of three-dimensional model.
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A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
After that, the load decreases and therefore, it is not necessary to continue the computation. Then the present computer program was revised so that it stops when the load reduces to 70% of maximum load. The simulation was carried out on a WorkStation. 4. Results and discussion One hundred iterations were conducted for each case and in most cases, the NL of each ®ber was ®xed to 50. We ®xed the scale parameter of link at 7.55 GPa, which was derived from our experimental data as will be described later. Throughout this chapter, the notation Fig. 7(a) shows results of quasi-LLS rule simulation and Fig. 7(b), results of simulation based on shear-lag analysis. 4.1. Failure process
4.2. Parametric study Figs. 8(a) and (b) show the dependence of the bundle shape parameter for strength upon the ®ber shape parameter for the strength at various number of ®bers where the word ``bundle'' means a unidirectional composite and ``®ber'' means mono®lament. Both ®gures show that the bundle shape parameter for the strength increases when the ®ber shape parameter increases. That is, the variation of the bundle strength decreases when the variation of the ®ber strength decreases. According to Fig. 8(a), the bundle shape parameter increases with the increase of the number of ®bers. In Fig. 8(b), on the other hand, this tendency is not remarkable. But the bundle shape parameter is not less than 40 in Fig. 8(b) which means very low variation. Figs. 9(a) and (b) show the dependence of the bundle scale parameter for strength upon the ®ber shape
Fig. 7(a) shows an example of the successive failure of individual ®bers in the same case as Fig. 6 where numerals are the order of breakage. In this case, the maximum load takes place when 4 ®bers break as was pointed out before. It is clearly shown that successive failure takes place after 4th link breaks. Fig. 7(b) is the case where the SCFs by means of the shear-lag analysis are adopted. The numeral 00 means that this link is not broken yet until the termination of the simulation. Before 12th ®ber breaks, we can ®nd a group of broken ®bers 1, 4, 5, 7, 8 at the top-right corner of Fig. 7(b) and another group 6, 9, 10 in the middle. This may correspond to a cluster [5] although details are not clear.
Fig. 6. Failure sequence of ®ber bundle.
Fig. 7. Location and order of broken ®bers: (a) quasi-LLS rule; (b) shear-lag rule.
A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
Fig. 8. Bundle shape parameter versus ®ber shape parameter: (a) quasi-LLS rule, (b) shear-lag rule.
parameter for strength at various number of ®bers. In these ®gures, the bundle strength increases when the ®ber shape parameter for strength increases. That is, the bundle strength increases when the variation of the ®ber strength is small. One point to be noted is that the strength of Fig. 9(b) is higher than that of Fig. 9(a) which is the re¯ection of dierence of SCFs. Figs. 10(a) and (b) are the rearrangement of Figs. 9(a) and (b) and it clearly shows that the bundle for strength decreases when the number of ®bers increases. This ®gure should be compared with the work of Lienkamp et al. [5] where only seven ®ber models were discussed. Our
93
Fig. 9. Scale parameter of bundle versus ®ber shape parameter: (a) quasi-LLS rule, (b) shear-lag rule.
present simulation is a step forward to their work from a view point of the number of ®bers. By careful observation, we ®nd that the relation between the scale parameter of bundle and logarism of number of ®bers is not linear but it tends to saturate. Although the number of ®bers cannot be increased so much because of computational time, 91-®ber model might be applicable to estimate the strength of unidirectional composites. So far we ®xed the NL to 50. Now we examine the eect of specimen length. Figs. 11(a) and (b) show the NL versus the bundle scale parameter where the shape parameter of ®ber
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A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
Fig. 10. Scale parameter of bundle versus number of ®bers: (a) quasiLLS rule, (b) shear-lag rule.
was 5. The scale parameter decreases monotonically with increasing the NL, which is a so-called size eect. 4.3. Comparison of experiment, traditional model and present model So far we have done a parametric study. In this section, we compare the present results with experiment, and results of traditional models.
Fig. 11. Eect of size, increasing bundle length: (a) quasi-LLS rule, (b) shear-lag rule.
In order to deal with a micromechanical study, we need to know the Weibull parameters as well as the critical length. We already measured the scale and shape parameters for the strength of mono®laments (T300) of 25 mm length [9] and they were =3.17 GPa and m=5.1. Although the critical length was also measured in Ref. [9], we again tried the fragmentation test. Fig. 12 demonstrates the midway of the test and ®nally we decide the critical length to be 0.3 mm. We also carried out tensile tests of 3000 ®lament-carbon/epoxy rod
A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
95
not done the simulation of large number bundle, say, 3000 ®laments. This is another reason why we use the word approximate. But 91-®ber model is already an advancement to Lienkamp and Schwartz [5] who conducted the simulation of 7-®ber composite. Through the present simulation, the gap has got narrow, and the value of the present estimation is close to the experimental value. 5. Conclusions
Fig. 12. Fragmentation test.
of 200 mm length [10], and the average strength of four specimens was 2.88 GPa. Using =3.17 GPa and m=5.1 for the strength of mono®lament of 25 mm length, we ®rst estimated Weibull parameters of a link of 0.3 mm length. Assuming the weakest link model which is an essence of the Weibull distribution, n 1 nÿ1=m
The SCFs at various combination of broken ®bers in a hexagonal-array were calculated applying a quasi-LLS rule and a shear-lag analysis. Using these SCFs, a parametric study on the strength of unidirectional composites was conducted by means of Monte Carlo simulation. Eects of mono®lament scale and shape parameter, number of ®bers and NL on the bundle strength were made clear. The present estimation of the bundle strength was close to the experimental value. Also the gap of strength between Rosen model and Zweben model got narrow by the present simulation.
3
is obtained where 1 is the scale parameter of a link, an is that of a chain of n links, and m is the shape parameter. Substituting an=3.17 GPa, m=5.1 and n=83 (=25/ 0.3) into Eq. (3), 1=7.55 GPa is obtained. Next, substituting =7.55 GPa, m=5.1 and NL=667 (200 mm/0.3 mm) in our simulation in the 91®ber model, we got Fig. 13. The present upper-bound simulation predicts the strength lower than the Rosen's value and the lower-bound simulation, higher than Zweben's prediction. In the above simulation, we used the 91-®ber model. If we use larger model, both upper and lower bound will decrease a little and the experimental value will fall just in between the two simulations. However, we do not know how much value will decrease because we have
Fig. 13. Comparison of experiment, traditional model and present model.
Acknowledgements This work is supported by the Special Coordination Funds of the Science and Technology Agency of the Japanese Government.
References [1] Rosen BW. Tensile failure of ®brous composites. AIAA J 1964;2:1985±91. [2] Zweben C. Tensile failure of ®ber composites. AIAA J 1968;6:2325±31. [3] Harlow DG, Phoenix SL. The chain-of-bundles probability model for the strength of ®brous materials I: Analysis and conjectures. J Composite Mater 1978;12:195±214. [4] Hedgepeth JM. Stress concentrations in ®lamentary structures. NASA Technical Note, TN D-882, 1961. [5] Lienkamp M, Schwartz P. A Monte Carlo simulation of the failure of a seven ®ber microcomposite. Composite Sci Tech 1993;46:139±46. [6] Hedgepeth JM, Van Dyke P. Local stress concentrations in imperfect ®lamentary composite materials. J Composite Mater 1967;1:294±309. [7] Suemasu H. An analytical study of probabilistic aspects of unidirectional ®ber reinforced composites under tensile loads. Trans Japan Soc Composite Mater 1982;8:29±36. [8] Oh KP. A Monte Carlo study of the strength of unidirectional ®ber reinforced composites. J Composite Mater 1979;13:311±27. [9] Fukuda H, Miyazawa T. Micromechanical approach to the tensile strength of unidirectional composites, I. Adv Composite Mater 1991;1:39±53. [10] Fukuda H, Miyazawa T. Micromechanical approach to the tensile strength of unidirectional composites, II. Adv Composite Mater 1995;4:287±95.
Approximate upper and lower bounds for the strength of unidirectional composites Atsushi Wada, Hiroshi Fukuda* Department of Materials Science and Technology, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Received 30 April 1997; received in revised form 29 January 1998; accepted 26 February 1998
Abstract This paper addresses the problem of determining the probabilistic strength of unidirectional composites. We ®rst calculated the stress-concentration factors (SCFs) around broken ®bers in a three-dimensional, hexagonal-array model. Two approaches were tried to calculate the SCFs. One is a shear-lag analysis originated by Hedgepeth and Van Dyke and the other is close to a local loadsharing rule (LLS). These two sets of SCFs provide the upper and lower bounds of the strength. With these SCFs, we next carried out a Monte Carlo simulation to evaluate the strength of unidirectional composites in which a chain-of-bundles probability model and a Weibull distribution were used. Parametric studies were ®rst conducted and a comparison with experiments and other existing theories was next done. Our `approximate' upper bound was lower than Rosen's prediction and lower bound, higher than Zweben's value. The value from the simulation was also close to the experimental results. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Unidirectional composites; Stress concentration factor; Shear-lag analysis; Monte Carlo simulation; Rosen model; Zweben model; Weibull distribution; Parametric study; Comparison with experiment
1. Introduction Composite materials are heterogeneous materials composed of reinforcements and matrices and it therefore, becomes necessary to deal with these materials from a micromechanical view point. Since the original report of Rosen [1], much work have been done on the micromechanical and statistical evaluation of the strength of unidirectional composites. The characteristics of individual ®bers, matrix, and ®ber/matrix interface are the most important factors for evaluating FRPs from a micromechanical view point. Because some synergistic eects at the ®ber/matrix interface occur in ®ber-reinforced composites, the strength cannot be predicted by simply averaging the strength of its constituents. Prediction of the strength of advanced composite materials requires the factor of stress concentrations around broken ®bers and the eect of the ®ber/matrix interface the latter of which can be derived from the tensile test of a mono®lament embedded in a matrix.
* Corresponding author. Tel.: +81-471-241501-4308; fax: +81471-23-9362; e-mail: [email protected].
Among the various existing models for the strength of unidirectional composites, the Rosen model [1] and the Zweben model [2] would be fundamental and most important. When ®bers are broken in a unidirectional composite, stress concentrations occur around the broken ®bers. The Rosen model does not take the stress-concentration factors (SCFs) into consideration. Harlow and Phoenix [3] called this idea the equal load-sharing (ELS) rule. The Rosen model predicts higher strength than that of the actual composite. Zweben [2], on the other hand, introduced the SCFs calculated by Hedgepeth [4] on the basis of a shear-lag analysis. However, Hedgepeth's original work was for a unidirectional sheet rather than three-dimensional composites. Two-dimensional calculation ends up with larger SCFs and therefore, the Zweben model leads to lower composite strength. The Rosen and Zweben models are eectively upper and lower bounds of the strength although this classi®cation is not made in a precise manner. The objective of the present paper is to predict the strength more accurately than either the Rosen or the Zweben model. To this end, we need more accurate SCFs. After calculating the SCFs in a three-dimensional,
0266-3538/99/$Ðsee front matter # 1999 Elsevier Science Ltd.. All rights reserved PII: S0266 -3 538(98)00052 -9
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A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
hexagonal-array model, we try a Monte Carlo simulation for the strength of composites.
H. This does not coincide with the LLS rule in a precise manner and therefore, we call it here a quasi-LLS rule.
2. Stress-concentration factors
2.2. Shear-lag analysis
If we estimate larger SCFs than actual values, the consequent strength becomes small and vice versa. In the present study, we took into consideration two kinds of SCFs. One is close to the local load-sharing (LLS) rule [3]. As will be described later, the SCFs derived from this idea are rather large and hence, the strength will be smaller. Another idea relating to SCFs is based on a shear-lag analysis; this leads smaller SCFs and larger strength. Therefore, the actual strength will be between the upper and lower bounds.
Hedgepeth and Van Dyke [6] already derived the SCFs in a hexagonal-array model, Fig. 2, although their numerical results are for limited cases. Suemasu [7] also derived similar equations, although the expression is dierent from Ref. [6]. What we need to add to Refs. [6] or [7] is to calculate the SCFs at various combination of broken ®bers. Fig. 3 is an example of the SCFs by means of the shear-lag analysis. This ®gure shows a cross section of a unidirectional composite. Each circle corresponds to each mono®lament and the three hatched ®bers are assumed to be broken in this example. Each numeral is the SCF of each ®ber. Although the calculation was done in in®nite region, only the SCFs of ®bers nearest to the broken ®bers are shown in this ®gure. As is seen, the SCF diers each other even though these ®bers are all next to the broken ®bers. Thus we need to know the SCFs of all combination of broken ®bers to deal with the forthcoming simulation. As will be described later, simulations were conducted for 7, 19, . . ., 91-®ber models. Fibers are assumed to be broken within this region only. Since all ®bers surrounding this region are assumed to be continuous, the SCFs calculated here may lead somewhat smaller
2.1. Quasi-LLS rule Harlow and Phoenix [3] introduced an idea of LLS rule which assumes that the load initially sustained by a broken ®ber will be shared with the nearest neighboring, surviving ®bers. For example, if one ®ber is broken in a unidirectional sheet, the SCFs of adjacent ®bers should be 1.5. In the case of a hexagonal-array, the SCF will be 1.166 (1 + 1/6) because there are six surrounding ®bers. Lienkamp et al. [5], for example, adopt this idea for a seven ®ber microcomposite. Although we are aware that this is not the case as was shown in the works of Hedgepeth [4,6], we adopt this idea to calculate an approximate lower bound of the strength. As will be described later, we assume the failure surface is ¯at in the simulation. If we allow other failure patterns such as ®ber pull out or ®ber-matrix interfacial debonding, the strength may decrease. That is, even if we use the LLS rule, there is no guarantee that the result is absolutely the lower bound. This is the reason why we use the word `approximate' in the title. It is rather time-consuming task to calculate each time the SCFs at each failure pattern. Then we adopted an approximate method as shown in Fig. 1. That is, when the ®ber #1 is broken, each 1/6 of load is allocated to the surrounding six ®bers; this is exactly the same as the LLS rule. But when the ®ber #2 is next broken, we assume that the load, 1.166, is equally shared with the surrounding and still surviving ®ve ®bers, A, B, F, G,
Fig. 1. Concept of SCF (quasi-LLS rule).
Fig. 2. Shear-lag model of hexagonal array.
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3.2. Simulation program
Fig. 3. SCFs by means of shear-lag analysis (example).
SCFs than the actual case and therefore, the estimated strength may be larger than the actual strength. 3. Monte Carlo simulation 3.1. Weibull distribution The strength of many high-performance ®bers can be well represented by a Weibull distribution, which, in one functional form, may be written as F x 1 ÿ exp ÿ x= m ;
1
where F(x) is the cumulative distribution function of the strength, x is the strength, and m and are, respectively, the shape and scale parameters. Large m corresponds to a small scatter and large indicates large strength. Eq. (1) is known as a two-parameter Weibull distribution. If the shape and scale parameters are known, and if we specify a probability of failure F(x), then we can solve Eq. (1) for the strength associated with that probability by x ÿ ln 1 ÿ F x1=m :
In the simulation, it is assumed that the in¯uence of broken ®ber is limited to a critical length, lc of Fig. 4, and that the stress concentration occurs to only neighbouring ®bers. The basic idea of the failure process is the same as Rosen [1] or Zweben [2] model. That is, when all ®bers within a critical length are broken, the specimen is considered to break. The formulation of the computer program is close to Oh's work [8] which may be described as follows. First, a model of MF ®bers with NL links is settled as shown in Fig. 5. The total number of links (NL) is MF NL. Then each Weibull random number is allocated to each link. This represents the strength of the link, denoted by STR (i, j), where i 1; . . . ; MF indicates each ®ber and j=1,. . ., NL shows the link number. The stress concentration factors of links, denoted by SCF (i, j), were originally settled to unity. In the actual simulation, MF=7,19,37,61,91 were chosen. A link (i, j) which has the smallest STR (i, j)/SCF (i, j) is searched. This link is considered to break at the normalized stress of STR (i, j)/SCF (i, j). If the link breaks, the stress redistribution will take place, and hence, the SCF (i, j) should be changed. Again the link of the smallest STR (i, j)/SCF (i, j) is looked for among the surviving ®bers; this link should break next. Fig. 5 also shows the ¯ow chart of this simulation. This process should be continued until all links in the same critical length are broken. However, in the present simulation, the following procedure was adopted to reduce the computational time. Fig. 6 is an example of the load versus the number of broken links in a 91-®ber model where the quasi-LLS rule is applied. In this case, the maximum load appears when the 4th link breaks.
2
For a Monte Carlo analysis, in order to simulate ®ber strengths that follow the Weibull distribution, uniform random numbers between 0 and 1 are ®rst generated. These random numbers are substituted into Eq. (2) producing a composite whose ®ber strengths follow the Weibull distribution.
Fig. 4. Critical length, lc
Fig. 5. Monte Carlo simulation of three-dimensional model.
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A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
After that, the load decreases and therefore, it is not necessary to continue the computation. Then the present computer program was revised so that it stops when the load reduces to 70% of maximum load. The simulation was carried out on a WorkStation. 4. Results and discussion One hundred iterations were conducted for each case and in most cases, the NL of each ®ber was ®xed to 50. We ®xed the scale parameter of link at 7.55 GPa, which was derived from our experimental data as will be described later. Throughout this chapter, the notation Fig. 7(a) shows results of quasi-LLS rule simulation and Fig. 7(b), results of simulation based on shear-lag analysis. 4.1. Failure process
4.2. Parametric study Figs. 8(a) and (b) show the dependence of the bundle shape parameter for strength upon the ®ber shape parameter for the strength at various number of ®bers where the word ``bundle'' means a unidirectional composite and ``®ber'' means mono®lament. Both ®gures show that the bundle shape parameter for the strength increases when the ®ber shape parameter increases. That is, the variation of the bundle strength decreases when the variation of the ®ber strength decreases. According to Fig. 8(a), the bundle shape parameter increases with the increase of the number of ®bers. In Fig. 8(b), on the other hand, this tendency is not remarkable. But the bundle shape parameter is not less than 40 in Fig. 8(b) which means very low variation. Figs. 9(a) and (b) show the dependence of the bundle scale parameter for strength upon the ®ber shape
Fig. 7(a) shows an example of the successive failure of individual ®bers in the same case as Fig. 6 where numerals are the order of breakage. In this case, the maximum load takes place when 4 ®bers break as was pointed out before. It is clearly shown that successive failure takes place after 4th link breaks. Fig. 7(b) is the case where the SCFs by means of the shear-lag analysis are adopted. The numeral 00 means that this link is not broken yet until the termination of the simulation. Before 12th ®ber breaks, we can ®nd a group of broken ®bers 1, 4, 5, 7, 8 at the top-right corner of Fig. 7(b) and another group 6, 9, 10 in the middle. This may correspond to a cluster [5] although details are not clear.
Fig. 6. Failure sequence of ®ber bundle.
Fig. 7. Location and order of broken ®bers: (a) quasi-LLS rule; (b) shear-lag rule.
A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
Fig. 8. Bundle shape parameter versus ®ber shape parameter: (a) quasi-LLS rule, (b) shear-lag rule.
parameter for strength at various number of ®bers. In these ®gures, the bundle strength increases when the ®ber shape parameter for strength increases. That is, the bundle strength increases when the variation of the ®ber strength is small. One point to be noted is that the strength of Fig. 9(b) is higher than that of Fig. 9(a) which is the re¯ection of dierence of SCFs. Figs. 10(a) and (b) are the rearrangement of Figs. 9(a) and (b) and it clearly shows that the bundle for strength decreases when the number of ®bers increases. This ®gure should be compared with the work of Lienkamp et al. [5] where only seven ®ber models were discussed. Our
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Fig. 9. Scale parameter of bundle versus ®ber shape parameter: (a) quasi-LLS rule, (b) shear-lag rule.
present simulation is a step forward to their work from a view point of the number of ®bers. By careful observation, we ®nd that the relation between the scale parameter of bundle and logarism of number of ®bers is not linear but it tends to saturate. Although the number of ®bers cannot be increased so much because of computational time, 91-®ber model might be applicable to estimate the strength of unidirectional composites. So far we ®xed the NL to 50. Now we examine the eect of specimen length. Figs. 11(a) and (b) show the NL versus the bundle scale parameter where the shape parameter of ®ber
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A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
Fig. 10. Scale parameter of bundle versus number of ®bers: (a) quasiLLS rule, (b) shear-lag rule.
was 5. The scale parameter decreases monotonically with increasing the NL, which is a so-called size eect. 4.3. Comparison of experiment, traditional model and present model So far we have done a parametric study. In this section, we compare the present results with experiment, and results of traditional models.
Fig. 11. Eect of size, increasing bundle length: (a) quasi-LLS rule, (b) shear-lag rule.
In order to deal with a micromechanical study, we need to know the Weibull parameters as well as the critical length. We already measured the scale and shape parameters for the strength of mono®laments (T300) of 25 mm length [9] and they were =3.17 GPa and m=5.1. Although the critical length was also measured in Ref. [9], we again tried the fragmentation test. Fig. 12 demonstrates the midway of the test and ®nally we decide the critical length to be 0.3 mm. We also carried out tensile tests of 3000 ®lament-carbon/epoxy rod
A. Wada, H. Fukuda/Composites Science and Technology 59 (1999) 89±95
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not done the simulation of large number bundle, say, 3000 ®laments. This is another reason why we use the word approximate. But 91-®ber model is already an advancement to Lienkamp and Schwartz [5] who conducted the simulation of 7-®ber composite. Through the present simulation, the gap has got narrow, and the value of the present estimation is close to the experimental value. 5. Conclusions
Fig. 12. Fragmentation test.
of 200 mm length [10], and the average strength of four specimens was 2.88 GPa. Using =3.17 GPa and m=5.1 for the strength of mono®lament of 25 mm length, we ®rst estimated Weibull parameters of a link of 0.3 mm length. Assuming the weakest link model which is an essence of the Weibull distribution, n 1 nÿ1=m
The SCFs at various combination of broken ®bers in a hexagonal-array were calculated applying a quasi-LLS rule and a shear-lag analysis. Using these SCFs, a parametric study on the strength of unidirectional composites was conducted by means of Monte Carlo simulation. Eects of mono®lament scale and shape parameter, number of ®bers and NL on the bundle strength were made clear. The present estimation of the bundle strength was close to the experimental value. Also the gap of strength between Rosen model and Zweben model got narrow by the present simulation.
3
is obtained where 1 is the scale parameter of a link, an is that of a chain of n links, and m is the shape parameter. Substituting an=3.17 GPa, m=5.1 and n=83 (=25/ 0.3) into Eq. (3), 1=7.55 GPa is obtained. Next, substituting =7.55 GPa, m=5.1 and NL=667 (200 mm/0.3 mm) in our simulation in the 91®ber model, we got Fig. 13. The present upper-bound simulation predicts the strength lower than the Rosen's value and the lower-bound simulation, higher than Zweben's prediction. In the above simulation, we used the 91-®ber model. If we use larger model, both upper and lower bound will decrease a little and the experimental value will fall just in between the two simulations. However, we do not know how much value will decrease because we have
Fig. 13. Comparison of experiment, traditional model and present model.
Acknowledgements This work is supported by the Special Coordination Funds of the Science and Technology Agency of the Japanese Government.
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