Materials Science and Engineering A

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1 Materials Science and Engineering A 528 (211) Contents lists availale at ScienceDirect Materials Science and Engineering A journal homepage: Study o strength and its reliaility o SiC ier undle y experimental and Monte-Carlo simulation approach Yun Lu a,, Liang Hao, Mitsuji Hirohashi a a Graduate School & Faculty o Engineering, Chia University, 1-33, Yayoi-cho, Inage-ku, Chia , Japan Graduate school, Chia University, 1-33, Yayoi-cho, Inage-ku, Chia , Japan article ino astract Article history: Received 21 Septemer 21 Received in revised orm 11 January 211 Accepted 4 Feruary 211 Availale online 15 Feruary 211 Keywords: SiC Fier Fier undle Tensile test Monte-Carlo simulation Strength Reliaility Tensile tests and Monte-Carlo simulation on the strength and its reliaility o the SiC CVD (ormed y chemical vapor deposition) ier undle were perormed. The experimental and simulation results were compared and analyzed y ier undle strength theories. The experimental results showed that the strength o the SiC CVD ier undle was decreased as the numer o iers in the undle was increased; while in the process, Weiull shape parameter o the undle strength was increased. The experimental results were in good agreement with the Monte-Carlo simulation. In addition, Monte-Carlo simulation was used to clariy the detailed relationship etween the strength o the SiC CVD undle and the numer o iers in the undle and the simulation results were compared with that o Coleman theory. The comparison revealed that the strength o the SiC CVD undle converged to the value o Coleman theory as the numer o iers in the undle increased; at the same time, the rate o convergence was increased as Weiull shape parameter was increased. Furthermore, the relationship etween the strength o the SiC CVD ier undle and Weiull shape parameter o the SiC CVD ier strength was also examined. It was ound that the strength o the SiC CVD ier undle increased as Weiull shape parameter was increased although the numer o iers in the undle was countale. Finally, the reaking-down process and the numer o roken-iers in the undle depended on the numer o iers the undle and Weiull shape parameter o the SiC CVD ier strength. 211 Elsevier B.V. All rights reserved. 1. Introduction In recent years, ier-reinorced composites are increasingly applied in the industries o aerospace and aviation or their excellent mechanical properties including high speciic stiness, high speciic strength and so on. However, the properties o iers adopted in ier-reinorced composites are distinct rom that o the matrix and the strength o iers shows divergence. Thereore, it is very important to evaluate strength and its reliaility o iers and ier undles when mechanical properties o ier-reinorced composites are discussed. In evaluation o ier strength distriution, Weiull distriution [1] is one o the most requently used theories. Ater that, Daniel [2] developed his classic theory on ier undle strength which greatly improved the simple Weiull theory. Daniels theory proved that the undle strength was asymptotically Corresponding author at: Department o Mechanical and Engineering, Graduate School & Faculty o Engineering, Chia University, 1-33, Yayoi-cho, Inage-ku, Chia , Japan. Tel.: ; ax: addresses: luyun@aculty.chia-u.jp (Y. Lu), haoliang25acd@gmail.com (L. Hao), hirohashi@aculty.chia-u.jp (M. Hirohashi). distriuted with a normal distriution. Thereater, Coleman [3] estalished his theory on the asis o Weiull theory and Daniels theory and represented the relationship etween the strength o ier undle and strength o the individual iers as indicated in Eq. (1). = ( 1 ) 1/m 1 m e (1 + 1/m ) where and are undle strength and strength o ier in the undle, m is Weiull shape parameter o ier strength, e is the ase o the natural logarithms, and is gamma unction. From Eq. (1),it can e known that the ratio o strength o ier undle and strength o ier in the undle only depends on Weiull shape parameter m. Since then, Coleman s theory has een oten used to discuss ier undle strength as the ase o mechanical models o ierreinorced composite [4 8]. In the last ew years, many researchers have made great progresses in the study o strength and its reliaility o ier-reinorced composites, or example, Hwang [9] studied size eect on ier strength o composite; Porwal [1] applied computer simulation method to analyze the statistical strength o ier undles; the attention o Vorechovsky [11] is ocus on small ailure proailities and the related proailistic distriutions o the strength o composites and presented an extension o Weiull (1) /$ see ront matter 211 Elsevier B.V. All rights reserved. doi:1.116/j.msea

2 Y. Lu et al. / Materials Science and Engineering A 528 (211) Tale 1 Parameters o the single SiC CVD ier or the present Monte-Carlo simulation. Sample numer E (GPa) (GPa) m (GPa) E : elastic modulus o the constituent ier; : single ier strength. m : Weiull shape parameter o the constituent ier strength. : Weiull scale parameter o the constituent ier strength. Fig. 1. Tensile specimens and loading modes o the individual SiC CVD ier and undle: (a) individual SiC CVD ier and () SiC CVD ier undle. theory y introducing into a statistical length scale; Dassios [12] investigated the strength o Al 2 O 3 ier undle at room and elevated temperatures. However, Daniels theory and Coleman s theory were ased on the assumption that the numer o iers in the undle is numerous. While, till now, any report on the exact numer o iers in the undle has not appeared. In the study, a series o tensile tests and Monte-Carlo simulation were carried out or SiC ier undle. Bundle strength and its reliaility were examined and discussed. Besides, the inluence o the numer o iers in the undle on the strength and its reliaility o ier undle were also investigated. Finally, the results o experimental and Monte-Carlo simulation were compared with Coleman s theory. 2. Experimental and Monte-Carlo simulation approach 2.1. Tensile test o SiC CVD ier undle SiC CVD ier, produced y Textron Specialty Materials (type: SCS-9, diameter: 7 m), was used as the ier in the studied SiC undle. A series o tensile tests were carried out at room temperature in a screw-driven constant crosshead tensile testing machine (Shimadzu, AG-5ES) with a constant tensile speed o.5 mm/min and a gauge length o 3 mm or the iers in the undles which included 13, 5, 1 and 2 iers, respectively. Fig. 1 shows tensile specimens and their loading modes o the SiC CVD ier and the undle. From Fig. 1(a), it can e seen that Al tas with a length o 3 mm ware adhered to oth o the tensile specimens ends in the case o individual ier; as or the undle, iers were arranged in parallel along the tensile direction and ormed a undle o iers. To avoid any damage to the individual ier or the ier undle eore tensile tests, supporting paperoards were used to one side o the tensile specimens and were urnt just eore tensile tests egan. All the studied iers were selected at random rom numerous numers o SiC CVD ilaments which were cut into ones with a length o 7 mm. Besides, the numer o the tensile specimens or each group o ier undle was over Monte-Carlo simulation approach A series o Monte-Carlo simulation was carried out or tensile tests o the SiC CVD ier undles. In the simulation, it was assumed that the ier strength complied with two-parameter Weiull distriution shown as Eq. (2). Thereore, the strength o the individual ier can e otained in computer program rom the inverse unc- tion o Weiull distriution shown as Eq. (3). { ) F( ) = 1 exp ( } m (2) ( 1 )} 1/m = {ln (3) 1 Z where is strength o ier, m and are Weiull shape parameter and scale parameter o ier strength, respectively. Z is uniorm random numer in (, 1). It was assumed that the diameter and gauge length o the ier were constant. Also, it was presumed that the given load applied to each o the ier in the undle uniormly and the load applying to the reaking ier would redistriute amongst the surviving iers. The parameters otained rom the tensile tests o individual SiC CVD ier were used in the present simulation as listed in Tale 1. In the simulation, Weiull scale parameter, was a constant and sustituted y the experimental value, GPa. Based on the experimental values, Weiull shape parameter, m was given values o 1, 3, 6, 9.95 and 15, respectively. Fig. 2 shows the low chart o the present Monte-Carlo simulation. Where,, ε, E, P and A are mean strength, strain, Young s modulus, load and sectional area, respectively. Also, suscript, s, Start Input Weiull scale and shape parameter o ier strength, m and ier numer N Assign strength o the i th ier i (1, 2,, i,, N ) in a undle y Weiull random numers Assign strain o the i th ier sj (1, 2,, i,, N ) produced y slack in a undle y multiplying uniormly random numer Z y product o constant r and ultimate strain o ier in the undle P Make the order statistic o the ier reaking strain including slack strain in a undle. The total strain o ier in a undle is given y sk = i (ier strength) /E (ier modulus) + sj (strain y slack) (1, 2,, k,, N ) Give a displacement increment to the undle ollowing the order statistic and otain load increment Strain o undle ier i = si(strain o the i th ier y slack) Load on the undle when the j th ier reaking P j E A j i si N P j i r i=n Yes si u (1, 2,, i,, N )(i+1,,j,, N) No i=i+1 Bundle maximum load P max = max { P j (load on the j th ier)} Bundle strength = P max (maximum load)/a (undle sectional area) s Ye j=5 End No u j=j+1 Fig. 2. Flow chart o the present Monte-Carlo simulation.

3 482 Y. Lu et al. / Materials Science and Engineering A 528 (211) σ a σ 4 σ a ε σ slacks ε Bun d le stress, GPa σ u σ u Strength decrease. 1. Strain, % ε u ε ε u ε Fig. 4. Stress strain curves o the SiC CVD ier undles in tensile tests. Fig. 3. Relationships etween the racture process o the SiC CVD ier undle and the iers in the undles without or with slacks: (a) without the slacks and () with the slacks. Denotation ε,,, and u represent strain, stress, undle, ier and ultimate, respectively., u and max mean ier, slack, undle, racture and maximum, respectively. For example, i expresses the strength o the ith ier in a ier undle, and is otained rom Eq. (3). The procedure o the simulation is shown as ollows. Firstly, strength o each ier is given ater inputting the parameters o the SiC CVD ier. Secondly, to simulate tensile test o the ier undles, which have slack or individual ier and the slack is added to the racture strain or individual ier comparing with experimental results. The slacks are given y multiplying r ε u to uniorm random numer Z in (, 1), r is the constant. Where, ε u is the ultimate racture strain o the SiC CVD ier and equals to the ultimate strength divided y elastic modulus o the ier, u /E. rε u represents the level o the slacks, and was sustituted y.55 in the case o slacks or y zero in the case o no slacks according to presimulation. Thereater, the order statistic o ier reaking strains including the slacks o the iers in the undle is given. Then the displacement increment is given to the ier undle at the step y ollowing the order statistic o all the ier-reaking including the slacks. At the same time, the load increment is given. Ater a ier reaks, the load acting on the reaking ier will redistriute to the surviving iers which have a smaller slack strain than the undle strain at that step. Stress and strain o the ier undle can e calculated y the accumulation o displacement increment and load increment at each step, respectively. The strength o ier undle deined as the maximum load divided y sectional area o the ier undle. Monte-Carlo simulations were carried out or 5 times repeatedly. Relationship etween the racture process o the SiC CVD ier undle and the iers in the undle with or without slacks is shown as Fig. 3. From the igures, it can e seen that slacks had strongly eected on the racture process, the ultimate strength o the ier undle with slacks was lower than that without slacks. The result was in agreement with the study o Phoenix [6]. The tensile tests in the study were more close to the situation o Fig. 3(). That proved that slacks exactly existed in the practical tensile tests. In the ollowing discussion, slacks were also considered to make the simulation and the experiment agreed with each other. 3. Results and discussion Fig. 4 shows stress strain curves o the SiC CVD ier undles in tensile tests. Though the undle strain was calculated y using displacement o the crosshead during tensile tests, it can e seen clearly that the slope o curves ecame moderate as the numer o iers in the undles increased. It hints that the slacks o iers in the undles went up as the numer o iers in the undles increased. Also, eore the maximum stress o the undles appeared, there were arupt drops on the curves due to the racture o iers in the undles. Relationship etween the undle strength and the numer o iers in the undles in experiment and simulation is shown in Fig. 5. From this igure, it can e seen that the undle strength decreased with the increase o the numer o iers in the undles, ut the descent rate ecame lower when the numer o iers in the undles was over 1. Although the undle strength otained rom the simulation was larger than the experimental ones in the case o no slacks, they were well consistent with each other y introducing the slacks. The result that undle strength decreased as the numer o iers in the undle increased was also consistent with the correction descried y Phoenix [13]. Also, the relationship etween Weiull shape parameter o undle strength and the numer o iers in the undles is shown in Fig. 6. Weiull shape parameter o undle strength increased with the increase o the numer o iers in the undles or oth o experiments and Monte-Carlo simulation. Ater introducing slacks, the simulation results were more closed to the experimental ones. Bundle strength, GPa Experimental Simulation r=, m =9.95 r=.55, m =9.95 The numer o iers in the undle, N Fig. 5. Relationship etween the undle strength and the numer o iers in the undles, r represents level o slacks and m is Weiull shape parameter o the SiC CVD ier strength. 3

4 Y. Lu et al. / Materials Science and Engineering A 528 (211) Weiull shape parameter o undle strength m Experimental Simulation r=, m =9.95 r=.55, m =9.95 The numer o iers in the undles, N Fig. 6. Relationship etween Weiull shape parameter o undle strength and the numer o iers in the undles, r represents level o slacks and m is Weiull shape parameter o the SiC CVD ier strength. From the results aove, it can e concluded that slacks existed in the ier undles o the experiments. It also proved that the present Monte-Carlo approach was applicale to simulate the tensile tests o the SiC CVD ier undles. However, the simulation was only carried out or no slacks elow in order to compare with the results o Coleman s theory ecause there was no the concept o slacks in Coleman s theory. To compare with Coleman s theory, the relationship etween the strength o the undle and the numer o iers in the undles was examined y the simulation y changing Weiull shape parameter o the ier strength. The results are showed in Fig. 7. Bundle strength was denoted as the ratio o undle strength rom the simulation to that rom Coleman s theory. It can e noted that undle strength decreased and converged to Coleman s theory as the numer o iers in the undles increased. Besides, undle strength converged to Coleman s theory with a higher rate when Weiull shape parameter was larger. Fig. 8 shows the relationship etween the undle strength and Weiull shape parameter o ier strength. Here, undle strength is denoted as the ratio o undle strength ( ) and ier strength ( ). From the igure, it can e seen that undle strength went up as Weiull shape parameter increased even though the numer o 3 σ /σ The ier numer Coleman theory Weiull shape parameter o the ier strength m Fig. 8. Relationship etween the undle strength and Weiull shape parameter o the SiC CVD ier strength, and are undle strength and ier strength. iers in the undles was countale; esides, the curve o undle strength was more closed to that o Coleman s theory with larger numer o iers in the undles. According to the assumption and method o Daniels [2] and Coleman [3], coeicient o variation (C.V.) o undle strength is given as Eq. (4). [ ( 1 ) 1/2 1/2 C.V. = exp 1] N m (4) where N is the numer o iers in the undle. From this equation, it can e understood that the divergence o undle strength is only related to Weiull shape parameter o the ier strength and the numer o iers in the undle. From the equation, it can e concluded that, on one hand, C.V. will decrease as the numer o iers in the undle increases when m is constant; on the other hand, when the numer o iers in the undle is constant, C.V. will also decrease with the increase o m. Thereore, it can e concluded rom Eq. (4) and Fig. 6 that C.V. o the ier undle was decreased as the numer o iers in the undle was increased. Coeicient o variation o undle strength rom Monte-Carlo simulation and the calculated results rom Eq. (4) is shown in Fig σ / σ c m =1 m =3 m =6 m =9.95 m =15 Coleman theory Coeicient o variation C.V., % m =9.95 Simulation Coleman theory The numer o iers in the undles, N The numer o iers in the undles, N Fig. 7. Relationship o undle strength and the numer o iers in the undles, m is Weiull shape parameter o the SiC CVD ier strength, and c undle strength rom simulation and rom Coleman s theory. Fig. 9. Relationship etween C.V rom the simulation and Coleman theory and the numer o iers in the undles, m is Weiull shape parameter o the SiC CVD ier strength.

5 484 Y. Lu et al. / Materials Science and Engineering A 528 (211) Average ier Stress, GPa m =9.95 The ier numer The order numer o roken-iers Fig. 1. Relationship etween the average ier stress and the order numer o roken-iers, m is Weiull shape parameter o the SiC CVD ier strength. Coeicient o variation o undle strength went down with the increase o the numer o iers in the undles and the results rom the simulation and the calculation are relatively similar with each other. To discuss the relationship etween the racture process, the numer o roken-iers in the undles against the numer o iers in the undles, racture stress o ier was recorded in the order numer o roken-ier in each simulation. Then average ier stress and Weiull shape parameter were calculated y Weiull distriution. Relationships etween the average ier stress, Weiull shape parameter o roken-ier stress against the order numer o roken-iers are shown in Figs. 1 and 11, respectively. From Fig. 1, it can e seen that the average ier stress increased as the order numer o roken-iers, that means racture stress o the undle would e increased as the numer o roken-iers increased. From Fig. 11, Weiull shape parameter o roken-iers stress was ound to increase as the order numer o roken-iers increased. It hints that the divergence o ier undle strength was decreased. Based on the results aove, it was ovious that undle strength depended on the numer o iers in the undles more Weiull shape parameter o Broken-ier stress m =9.95 The ier numer The order numer o roken-iers Fig. 11. Relationship etween Weiull shape parameter o roken-ier stress and the order numer o roken-iers, m is Weiull shape parameter o the SiC CVD ier strength. 6 Numer o roken-ier till maximum load, N 2 1 m =1 m =3 m =6 m =9.95 m = The numer o iers in the undles, N Fig. 12. Relationship etween the numer o roken-iers till maximum load and the numer o iers in the undle, m is Weiull shape parameter o the SiC CVD ier strength. increasingly with the increasing o the numer o roken-iers in the undles. In racture process, strength o ier undle ecame higher and its divergence ecame lower. Fig. 12 shows the relationship etween the numer o iers in the undles and the numer o roken-iers till maximum stress appeared. It can e ound that the numer o roken-iers in the undle went up as the numer o iers in the undles was increased or with the decrease o Weiull shape parameter o the SiC CVD ier strength. From the results aove, it can e concluded that the reaking-down process o the SiC CVD ier undle and the numer o roken-iers in the undles eore maximum stress appeared depended on the numer o iers in the undles and Weiull shape parameter, m o the SiC CVD ier strength. 4. Conclusions Based on the experiment and Monte-Carlo simulation aove, some important results can e concluded as ollows. (1) In the process o the numer o iers in the undles was increased, the strength o the SiC CVD ier undle decreased and Weiull shape parameter o undle strength increased. (2) The strength o the SiC CVD ier undle converged to Coleman theory as the numer o iers in the undles was increased and the rate o convergence was higher when Weiull shape parameter o the SiC CVD ier strength had a larger value. (3) The strength o the SiC CVD ier undle was increased with the increase o Weiull shape parameter o the SiC CVD ier strength as Coleman theory s description even when the numer o iers in the undles was countale. (4) The racture process and the numer o roken-iers in the undle eore maximum stress appeared depended on the numer o iers in the undles and Weiull shape parameter, m o the SiC CVD ier strength. Reerences [1] W. Weiull, J. Appl. Mech. 18 (1951) [2] H.E. Daniels, Proc. R. Soc. Lond. A 183 (1945) [3] B.D. Coleman, J. Mech. Phys. Solids 7 (1958) 6 7. [4] B.W. Rosen, Am. Inst. Aeronaut. Astronaut. J. 2 (1964) [5] C. Zween, Am. Inst. Aeronaut. Astronaut. J. 6 (1968)

6 Y. Lu et al. / Materials Science and Engineering A 528 (211) [6] S.L. Phoenix, Fier Sci. Technol. 7 (1974) [7] D.G. Harlow, S.L. Phoenix, J. Compos. Mater. 12 (1978) [8] H. Fukuda, K. Kawata, Trans. Jpn. Soc. Compos. Mater. 2 (1976) [9] T.K. Hwang, C.S. Hong, C.G. Kim, Compos. Struct. 59 (23) [1] P.K. Porwal, I.J. Beyerlein, S.L. Phoenix, J. Mech. Mater. Struct. 2 (27) [11] M. Vorechovsky, Compos. Struct. 92 (21) [12] K.G. Dassios, M. Steen, C. Filiou, Mater. Sci. Eng. A 349 (23) [13] S.L. Phoenix, Compos. Sci. Technol. 48 (1993) 65 8.