短繊維補強複合材料の単調及び繰返し載荷下ひび割れ架橋則

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(1)Journal ofAppliedMechanicsVol.11,pp.891-902 (August2008). Effect of fiber fatigue rupture on monotonic in discontinuous. JSCE. and cyclic crack bridging laws. fiber reinforced composites. 短繊維補強複合材料の単調及び繰返し載荷下ひび割れ架橋則における繊維疲労破断の影響. Takashi Matsumoto* 松本. *. 高志. Member,Ph.D.,Assoc.Prof, Divisionof BuiltEnvironment,HokkaidoUniversity(Kita-13,Nishi-8,Kita-ku,Sapporo060-8628). This paper presentsthe micromechanicalmodelingof fiberbridgingconstitutivelaws of a discontinuousfiberreinforcedcomposite(DFRC)undermonotonicand cyclicloadingwith the effects of fiber fatigue rupture. Based on the considerationsof fatigue damage on randomlydistributedfibersunderthe fatigueloadingof constantcrackopeningamplitude,the monotonicand cyclicfiberbridgingconstitutivelaws are derived. It is shownthat, as fiber fatigueruptureproceeds,maximumbridgingstressas well as crack openingdisplacementat whichbridgingstressvanishesare decreased. Also,the criticalnormalizednumberof cycles, n*crit, at which crack bridgingdegradationtakes place, is analyticallyderived,and its role is explainedfor the designof a DFRCagainstfatigue. Key Words:Bridginglaw, discontinuous fiber reinforcedcomposite,cyclicloadingfiber fatiguerupture. 1. Introduction. Recently, short fiber reinforced cementitious composites are increasingly. applied into civil engineering. structures.. These. composites are placed in structural members in such a way that their high strength, ductility, and fracture toughness in tension are utilized in order to increase the safety, ductility, and durability of structures.. It is known that those beneficial composite properties. are thanks to bridging fibers that transfer stresses across a crack, and it is important to understand a fiber bridging constitutive law, which is the relation between bridging stress and crack opening displacement,. based on micromechanical. parameters. such as. fiber length, fiber diameter, fiber modulus, interfacial fictional bond strength, and so on. Some of the structural applications show that the composites are used in structural members where severe load repetitions are present, and that the composites are expected to yield long-term durabilities with their high fatigue strength, high crack resistance, and crack width. control ability.. In order to evaluate. the. long-term durabilities under load repetitions, it is necessary to. ―891―. understanda fiberbridgingconstitutivelaw undercyclicloading and also understandthe long-termdegradationof fiberbridging itself. The fiber bridgingconstitutivelawunder cyclicloading has been derived already,and it has shown the validity in comparisonto experimentalmeasurements1).The next step is the considerationof the long-termdegradationin fiberbridging itself This is because fiberbridgingitselfcan be consideredto degradein the longrun,namelythe transferredstressesdecay as load repetitionscontinue. For degradation,there can be two possiblemechanisms. One is interfacialbond degradation,and the otheris fiberfatiguerupture. Under cyclicloading,there is ample evidenceto show that interfacialfictionalbond strengthchangesundera largenumber of cyclic loads, e.g. in aligned continuous fiber reinforced ceramic matrix composites such as SiC/CAS2)3)4)5)6) and SiC/SiC7)8)and in steel fiber reinforced concrete9). The interfacialbond degradationhas been implementedinto the bridgingconstitutivelaw in a fatiguecrack growthanalysis,and an experimentalcrack growth measurementwas successfully reproduced10).Furthermore,the stress-life(S-N) relation of.

(2) various this. kinds. of. fiber. reinforced. concrete. were. reproduced. with. analysis11)12). On. the. loading,. other. there. Aligned to. study. the. with. shows. bridging. The. presence. the. comparison. by. this. under. fiber. of final. stress. cyclic. and. composite. of. is. namely,. in turn. leads. crack. fiber growth.. This. failure.. can. surfaces. fracture. to the degradation. to the final rupture. were Crack. significantly. the crack. rupture. fatigue. alloy. loading13).. accelerated. fatigue. of fiber. a titanium. fatigue. fracture,. the fiber. of crack. rupture. in. tensile. fiber. the onset. that. fatigue. fibers. under of. by. fiber. SiC. fracture. behavior. influenced coincides. for. evidences.. continuous. observed growth. hand,. are fewer. be. of. readily. proven. and. fatigue. static. Fig.. Specimens14)15)16). Randomly cement. distributed. matrix. Close-up. exhibited. pictures are clearly. The. specimen. rupture. as the. specimen. final. failure. of. because. the. although. the. can. place. take. than. the. friction. fiber. more. so that. Therefore, it is. rupture. on. bridging. to. crack. account,. and opening. by. brief. fiber. review. bridging. individual. fibers,. displacement. (P. P - ƒÐrelation. during. will. stress single. fiber. - ƒÐ) relation. be. done. is a sum. in. where. rupture. Ef =. frictional. is less. strength. pull-out. load. is reviewed17)18). - crack. (see. - crack. length,. debonding. stage. is given. by. fiber. bond. modulus,. df = fiber. diameter,. strength,•@. and during. sliding. r = ,l=. stage. interfacial. embedment. by. the. effect. crack. The. the. rupture. fatigue during. of the. damage. on. fatigue. expressions (•@. of. fiber. fiber. have been. pull-out. loading,. we. in. - crack. is given. debonding. opening. as follows1).. stage. under. the. have:. (3). pull-out loading, the fibers undergo unstretching and contracting according to. of into. stress. bridging. amplitude. that. amplitude. and for fibers that have been in sliding stage under the preceding. fibers.. taken. bridging. fibers. load. ) relation. in. loading is. fiber. ) and. displacement rupture. fatigue. pull-out (•@. will. model. the. loading,. amplitude. fatigue laws. evaluation. amplitude. cyclic. preceding. fatigue. fiber. bridging. composites.. For. loading.. fiber. of. Under displacement. composite. static of. suffer. in the static. this. evidences. displacement. opening. Fig.. and. matrix. that. dtu-ing. cyclic. on. analytical. this. of the pull-out. (1). laws.. fatigue. fiber. be noted. evaluate. displacement. with. a. Since. the. length. than. a micromechanical. opening the. specimens. these. reinforced. addresses. fiber. constant. The. loading,. fiber. in a cement. is allowed. and. constructing. Progressive. relation. rupture. necessary. paper. amplitude. it should. observed. constitutive. This. crack. fiber. fiber. fibers. in the fatigue. monotonic. by. curves)17). (2). having. discontinuous done. no. fiber. corresponding. damage,. so that. diameter,. for complete. with. 1).. the. reveals. rupture,. carried. opening. during. fatigue fatigue. the. fiber. loads. no. by. fatigue. section1)17)18).. fatigue. ruptured. during. (i.e.. on. alcohol. However,. rupture,. loading. based. (insert. curves. consideration).. rupture. is designed. ruptured. with. ruptured. to. fiber. length.. the. comparison. loading-unloading. equations. in a. fatigue. out. hand,. were. fibers. is designed. static. polyvinyl. specimens15)16).. be. are gradually. length. other. all the fibers This. and. pulled. fiber. tests14).. protrusion. gradually. polyethylene. of fibers. were. the. were. specimen.. during. Similarly, from. almost. length. critical. interface. that. static. of fiber. On. fibers. in fatigue. of. fibers. up.. fibers. the. fibers. surfaces. that. that. the. of. rupture. in terms. opened. and. susceptibility. crack. showed. crack. loading. the. polyethylene. fatigue. different. showed. fatigue. severe. of. specimens static. discontinuous. 1 Single. (4). -. stress. (•@. Furthermore,. ) after. are obtained.. unstretching. when. these. and. contracting,. fibers. slide the•@. back. into relation. the. matrix is given. by 2. Review Monotonic. of and. Single. Fiber. and. Cyclic. Loading. the derivation. of af-. CompositeBehavior. under. (5) where ƒÐmax. Before. Sand. My-. AS. relations. with. •\ 892•\. = crack. opening. displacement. at which. unloading.

(3) starts. and Pmax=ƒÎƒÑdf(l-ƒÑƒÂmax) Based. on the. relations), and. the. cyclic. found. above. fiber. loading. were. elsewhere1).. introduced crack. for. Under. later. the corresponding together. where. Vf = fiber. corresponds all. Fig.. the. fibers. have. on the. are. denoted. law. is defined. are measured. The. fiber. this. point,. with. value, ƒÐ0, *, and. fiber. at. it decays. length,. cyclic. loading, starts. the coordinates fiber. bridging. stress. amplitude, ƒ¢ƒÐ. in Fig. laws. a. constitutive. Namely,. as shown. we from. of the point. bridging. displacement. constitutive. and ƒÐ*. Lf / 2, at which. unloading. the origin.. opening. bridging. -. Fig. 2 Definitions of parameters. Lf=. The and. crack. 8. For. reloading.. this point. increases. of ƒÐo with l=. relation,. from. normalized. and. value. and ƒÐmax.. and. stress. Here,•@. monotonic. with. be are. composite.. a peak. displacement,. by ƒÐmax. amplitude, ƒ¢ƒÐ,. stress. debonding.. and. bridging. a cracked. it reaches. fraction. completed. can. of parameters the. of. monotonic. of the laws. shows. pull-out. volume. unloading. 2. bridging. opening. fiber. to the maximum. consider point. with. details. behavior. until. crack. to zero. under. the definitions. use.. displacement. (P-ƒÂand •¢P-•¢ƒÂ. laws. The. only. loading,. opening. behavior. derived.. displacement. monotonic. crack. fiber. constitutive. Here,. the. opening. single. bridging. 2.. Therefore,. will. the. be expressed. in Fig. 3 Assumed. terms. On 3. Fatigue. It. Damage. is. on. assumed. Randomly. that. fiber. Distributed. basis. fiber. (Fig.. 3) and. exert. fiber. bridging. fatigue. the. other. rupture. takes. place. on. (S-N). relation. fiber. composite. crack. opening. or cyclic cis. according that. to the. surviving. stress. loading.. stress-life. fibers. across. determined. a crack. fiber. Here, ƒ¢ƒÐsf is stress. ultimate. strength,. coefficient.. The. effect. fatigue. of fiber. the derivation Under fibers. under. in this either. is. manner. with. monotonic. location the. fatigue. conditions. and. viewpoint. of. orientation. life. each. fiber. number. of. cyclically. conditions:. with. b. with. and. is a the. details. or. composite,. of. single. fiber. load. loaded. fiber, load. also. plane,. can. two. load. due. to gradually. surviving. fibers. example,. this. profile a. takes. the. the. case. is cyclically. bridged. constant. fatigue. under is. crack cyclic. place. load. same when. loaded with. among. ruptured or. fibers,. fibers. has. larger. load. crack. the. lost. to be taken. crack. constant. non-uniform. when. overall. a bridged under. happens. load. with. by For. loading. conditions on fibers. since. when. a. of constant profile. (with. or. crack.. without. is certainly. surviving. as. In this. redistribution. amplitude).. load. fibers. study,. or. load. fibers. are. more. severe. to. sustain. have. gradually. lost. due. to. is because. fundamental. we. will. •\ 893•\. focus. insights. and. on. the. opening. the problem. consider. and. rupture.. For. uniform. cyclic. the. the. can. a basis. case. of. no. displacement. be simplified, even. for. but. the. is. as w) and monotonic. assumed,. are included. it. case. of. loading. those. and, in the. with. short. place. before domain. and. two:. sliding•@. to the. two. divided. can domains. be. loading location. fibers. in Fig.. of. length, l. fibers. respectively.. fiber by. which. of remaining and. bridging integrating Upon. is. w, note. in debonding•@. obtained. up. fibers. (large. domain. 5.. is opened. remaining. pull-out. embedment. and loading. any of fiber. the crack of. The into. fatigue. is. domain. that•@ is divided. fiber. shown. due. from. without. under. hereafter•@. takes. reduced•@. fibers. orientation,•@. orientation•@. the. starts. or. z, and. ),. denoted. of remaining. with. location,. randomness all the fibers. group. loading. fiber. (•@. the. amplitude.. will. crack. redistribution.. monotonic. grows. we. (constant. This. First,. fibers. uniform. amplitude profile. the condition of a uniform. damage. amplitude. rupture.. load. When. among. fibers. redistribution. under. amplitude. to•@. Load. redistribution. loaded. redistribution,. happens,. the. be. redistribution. on. initial. From. there. two. fatigue. monotonic. and,. on these. applied.. of. depend. but. crack. is dependent. without. fatigue. The. load. magnitude,. cycles. each. loading.. at a designated. of each. the. fiber. and. load. fiber,. and. laws. below,. pull-out/push-in. amplitude. no. cyclically. these. provides. of the resulting load. and. failure,. constitutive. a given. cyclic. and magnitude. the fiber. loading. of. to. the. hand,. displacement. redistribution),. in Appendix.. loading. on the overall. under. of a fiber. of a. in a single. to. is summarized. can be found. not only. in turn,. of. rupture. subjected. amplitude. nf is cycles. derivation. cyclic. amplitude. when. relation. a. increasing. crack. (S-N). Fibers. Comparing deterministic. load. stress-life. of ƒ¿=ƒÂmax/ƒÂ*and ƒÀ=ƒ¢ƒÂ/ƒÂmax.. stress (1). unloading. and. under (2). in. of this.

(4) loaded into. fiber. three. debonding unloaded. composite,. with. respect. (•@ according. (•@ by. been. 5).. is not The. reduced, fibers. ) in the preceding. sliding-in. in sliding. domain. to (3), and ) undergo. followed have. the. to w (Fig.. (5). is now. the. fibers. but. that. monotonic that. unstretching The. subdomain. divided. into two. have and for. divided. have. loading been. in are. bridging. integrating. fibers. (4),. by fiber stress. (6). fatigue. under. that. but under. fatigue. fatigue. relation. (5). proceeds,. the. resulting. fibers. ruptured. are. S-N. be. obtained. by. divided. domains. is gradually. reduced. in degradation. not. under. bridging. In this. study,. monotonic. is reduced. discounted. relation. of fiber. loading.. so the domain. fibers The. domain. or cyclic. are ruptured. loading,. can. three. on ƒÐmax. the. monotonic. of a fiber.. loading. in. it is dependent. rupture,. either. cyclic. and. loading. is assumed. that. under. (4),. and thus. As fatigue. that. by. stress (3),. respectively,. in sliding. contracting the. Fiber. been. loading,. in such. according. of a fiber. it. a way. to the. is assumed. S-N. to be. (7). (see Fig. 3). This simple assumption does not account for fatiguelimit,belowwhich fatiguefailurenever occurs,so every fiberfailsin fatigueat a finitenumberof cycles. It alsodoesnot accountfor mean stresseffect,which changesthe coefficient,b, of the S-N relationunderdifferentmean stress of a load cycle. Accordingto the assumedS-N relationof a fiber,the condition for fibersto havesurvivedfor n cyclesof fatigueloadingis given by (8). Fig.4 Fibercentroidaldistance,; and orientation,q5 The. stress. amplitude. through. dividing. sectional. area:. in. the. fiber. a. single. fiber,ƒ¢ƒÐsf,. pull-out/push-in. can. load. be. by the. obtained fiber. cross. (9). where ƒ¢P cycles. = fiber and. is. embedment. Fig. 5 Domain of the fiber centroidal distance, z, and orientation,. length. snubbing. effect. between. a loaded. assumed. that. by fiber. of. a fiber. which. load. (Fig.. fiber. From. n cycles. (8). the and. factor. mechanical. the matrix. is approximated and. (9),. the. applied. depending. 5). The. describes. inclined. amplitude. or ƒ¢P3. the ƒ¢P-ƒ¢ƒÐrelation. Matsumoto1). after. pull-out/push-in. either ƒ¢P1, ƒ¢P2,. survival. for on. efƒ³ refers. n the. to a. interactions material19).. It is. in the same condition. way of a. becomes. (10). First, then. fibers from. within (10). we. w = 0 •` z0 are. loaded. according. to AP. = ƒ¢P1,. have. (11) Fig.6 Reduceddomainof the fibercentroidaldistance,z, and orientation,4, dueto fiberfatiguerupture where•@. •\ 894•\. normalized.

(5) crack. opening. form,. we. displacement. amplitude. applied. for. Alternatively,. 4. Monotonic. n cycles,•@. The. monotonic. fiber. fatigue. rupture. surviving. fibers. is shown. of. (12). When prepeak. and•@. Therefore,. the. domain. of surviving. fibers. Law. with. Fiber. Fatigue. Rupture. in a normalized. have. where•@. Constitutive. fiber. a fiber. bridging is. given. in Fig.. composite. constitutive below,. 7 (see. under. for the. prepeak•@. and. monotonic. and. Appendix. is fatigue-loaded. region•@. loaded. law. loading,. with the. the effect domain. of. for details). for. n cycles. afterwards. the monotonic. in the. it is further. constitutive. law. .. is reduced,. and. only. for. n cycles. is given. by,. for•@. the ,. fibers. with•@. under. the. and•@ normalized. amplitude. of•@. are also. can. Hence. by ƒ³1. survive. , and (Fig.. , when. 6).. according. . limited. constant (Fig.. loaded. crack. the. Second,. fibers. Finally,. belong. of. fibers. displacement. within•@. , since. domain. the. survive. opening. to•@. only 6).. they. can. it is assumed. surviving. fibers. with•@. fibers. that. is. again •@. and•@. are. loaded. to•@. according .. Here,. to•@. from. (10),. we have. (13). (15) or, in a normalized form,. and,. for•@. (14) Therefore, (14),. and. the only. Based constant. survive also. crack. found. bridging. surviving. that. (Fig.. on•@. on the. , we. fibers. is limited. by. (13). satisfy •@ 6).. or. , and. In the case. of •@. where. , the limit •@. .. domain. (17). of the. opening. and•@ fiber. of. the fibers. can is dependent. domain. (16). surviving. displacement can. constitutive. construct laws.. fibers. after. amplitude the. Details. n cycles. loading. monotonicand of the. of with. and. cyclic •@. derivation. can. be. in Appendix.. (18). And. the monotonic is given. Fig.. 7 Domain. of the fiber. , for monotonic. bridging. centroidal constitutive. distance, law. z, and after. bridging by,. for•@. constitutive. law. for the postpeak•@ ,. orientation,. fiberƒ³ fatigue. rupture. (19). •\ 895•\.

(6) 5. Cyclic. (20). The fiber. When. a fiber. composite. is fatigue-loaded. in the. Constitutive. cyclic. fatigue. (1•…ƒ¿f). with. the. crack. opening. displacement. than. the value. further. for all-fibers-sliding-in. loaded. constitutive and. law. under. by (20). is given. below. law. in. Rupture. with. a similar. the. effect. manner. of. to the. The. Appendix. domain. of surviving. fibers. is shown. in. for details).. (ƒÀf<ƒÀo) and afterwards. monotonic. (1•…ƒ¿f<ƒ¿). Fatigue. constitutive. is given. subsection.. 8 (see When. it is. rupture. bridging. Fiber. amplitude Fig.. less. fiber. with. postpeak previous. region. Law. loading, by(19). the. a fiber. composite. is fatigue-loaded. for. n cycles. in the. monotonic prepeak. region. loading,. the cyclic. (ƒ¿f•…1). and. afterwards. it is loaded. under. cyclic. for ƒ¿<(1-z1(ƒÀf))/ƒÂ*. for(1-z1(ƒÀf))/ƒÂ*•…ƒ¿.ƒÀo. is given. constitutive. law. is given. by, for ƒÀ•…ƒÀf,. by. (21). When region. a fiber. (1•…ƒ¿f). greater. than. afterwards monotonic. composite. with. the. the. value. it is. further. is. crack for. constitutive. fatigue-loaded opening. in the. displacement. postpeak amplitude. all-fibers-sliding-in (ƒÀo•…ƒÀf). loaded. under. monotonic. law (1•…ƒ¿f<ƒ¿). is given. and loading,. the. by. (24) and,. for ƒÀf<ƒÀ,. (22) It should. be noted. that. for all ƒÓi's. (23). where. i = 1•`5. (Appendix. for definitions).. (25) When postpeak loading. a fiber region. 8 Domain. , for cyclic. of the fiber bridging. centroidal. constitutive. distance, law. after. z, and fiber. When. orientation,. fatigueƒÓ rupture. •\ 896•\. (1<ƒ¿f). (ƒÀf•…ƒÀo),the by (25). Fig.. composite. a fiber. is fatigue-loaded and. cyclic. afterwards. constitutive. for. n cycles. it is loaded law. is given. under. region. (1<ƒ¿f). (ƒÀo<ƒÀf),. the. for ƒÀ •…ƒÀf and. composite. loading. cyclic. by (24). for ƒÀf<ƒÀ.. postpeak. in the. is fatigue-loaded and. cyclic. afterwards constitutive. for n cycles. it is loaded law. is given. in the. under. cyclic. by,. for ƒÀ•….

(7) (26) and,. for ƒÀf<ƒÀ,. Fig. 9 Monotonic. bridging. stress. degradation (ƒ¿f=0.1, ƒÀf=0.5). (27). It should. be noted. that. for all ƒÓ'is. (28). where. i =1•`5. 6. Bridging. (Appendix. Stress. Bridging during loading. with. Fatigue. loading. the. the. amplitude,. and. the. of. displacement. of. the. is the. fatigue. by. crack. loading. fiber. opening crack. means. fatigue. laws. derived. opening. a cracked. cyclic above.. which. is. in a load displacement. number due. rupture. and. displacement. constant to. Rupture. monotonic. degradation. applied. Fatigue. parameters: ƒ¿f,. is the normalized. bridging. amplitude. three. normalized. n*, which crack. by. under. constitutive. defined. maximum. which. rupture,. use. to Fiber. induced. is evaluated. can be. normalized. due. degradation. loading. cycle, ƒÀf,. study. Degradation. stress. fatigue. for definitions).. of cycles. to. fiber. crack fiber. In. fatigue opening. composite,. Fig.. •\ 897•\. 10 Monotonic. bridging. stress. degradation. (ƒ¿f=0.1, ƒÀf=1.0).

(8) Fig.. 11 Monotonic. bridging. stress. degradation. (ƒ¿f=. 100.0, ƒÀf=. Fig.. 12 Cyclic. bridging. stress. degradation. (ƒ¿f= 0.1, ƒÀf=. 0.5). Fig.. 13 Cyclic. bridging. stress. degradation. (ƒ¿f= 0.1, ƒÀf=. 1.0). 1.0). where ƒ¿f the. and ƒÀf. composite. case. are. fixed. system. parameter. of a polyethylene. Fig.. 9 through. Fig.. is applied and. the. the normalized. changes. with. the normalized. monotonic. various. conditions. referred are no. surviving. and ƒÀf=. 1.0 (full. 0.1 smaller. (n*crit =. happens. n*. critical. value. (ƒÀf= 0.5) One. bridging. that. at a larger. the. original. shorter Fig. stress. amplitude, ,. crack. opening fatigue. 100.0, ƒÀf. length. as. degrades The cyclic stresses. can. and. monotonic. value. of n*crit. are exerted. stress. to. crack. the. cases.. Also,. 100.0,. 1.0, n*crit is. stress. vanishes only. decreased. reaches of. n*crit,. fiber. Fig.. 11.. at a smaller fibers. as. fatigue. at which. 9 through. since. under. with. the normalized. fiber. with. the Note. a than relatively. Fig.. various. cyclic. conditions The. cases.. The. bridging. 14 Cyclic. domain.. the normalized. under. n* = 0.1•`10.0).. bridging. stress. (ƒ¿f=. trends bridging. the. by the same. same. for. monotonic surviving. both. of and. degradation. (ƒ¿f=100.0, ƒÀf=1.0). the. amplitude. fibers. loading. loading is. is small. can. be. is applied applied. in. (ƒÀf<ƒÀo),n*crit. readily. obtained. in the prepeak the. (ƒ¿f<1). postpeak(ƒ¿f>1) is given. (Appendix).. by,. or when and. its. in a normalized. form,. stress. monotonic cyclic. fatigue. fatigue. 0.1,. are. Indeed, n*crit. When. loading,. or n*crit is increased. is the since. the. displacement. in Fig.. loading. either ƒ¿f, ƒÀf,. case,. bridging. For ƒ¿f=. is. n*. amplitude, ƒÀ,. is applied. when. loading. When ofƒ¿f= n* is. for ƒ¿f=. when. how. displacement. in. 10.0.. survive.. 14 show. 1.0,. 1.8. value. increases.. changes. = 0.5,. crack. reaches. due. rupture. opening. as seen. stress,. loading. the is. and,. (half. 1.0,. of cycles, n*crit),. consequence crack. Fig.. n*. in the two. fatigue. n* the bridging. 12 through. same. fiber. appears. vanishes,. embedment. of. difference. when ƒ¿f. loading 0.5,. (ƒÀf=1.0).. which. bridging. loss. 0.4 (ƒÀf= 1.0),. of. is the decreased stress. opening. 0.5. the critical. applied. above. Another. bridging. after. and. 0.33. strength. rupture. The. crack fatigue. number. total. of n* decreases. and. exemplified. the. bridging. n* surpasses. closure),. amplitude. consequence. when. when. crack. = 5.0.. (ƒÀf= 0.5). place. the. fiber. 100.0, ƒÀf=. and ƒÀf=. noimalized fibers. and. displacement. n*crit is 0.57 0.36. 1.27),. above. opening the. to critical. 0.1,. for. composite.. after. = 0.1. takes. and. there. (ƒ¿f=. When ƒ¿f. (hereafter. cementitious. loading,. Also,. = 0.0028. how. degradation. loading.. is set to ƒÂ*. 11 show. n* = 0.1•`10.0).. closure),. fatigue. reinforced. a; under under. the. fiber. stress,ƒÐf=ƒÐf/ƒÐo, displacement,. throughout. (29). and. bridging or. in the reduced. ― 898―.

(9) (30) When its amplitude. fatigue. loading. is applied. is large(ƒÀo<ƒÀf), n*crit,. in the postpeak is given. by,. (ƒ¿f>1). and. in a normalized. form,. (31). (32) As shown above, the degradationof fiberbridgingstresses starts when n*crit. exceedsn*crit.The two equationsabove are the conditionfor a single fiber inclinedat 90 degreesto the crack plane to be fatigue-ruptured and are for a fiberin debondingand slidingrespectively. Thisn*crit tellsthe point at whichfibersstart to be ruptured not only under constant crack opening displacementamplitudecondition,but also underconstantstress amplitudecondition,sincetwo conditionsare the same untilload redistributiontakes place upon the rupture of the first fiber. However, when n*>n*crit under constant stress amplitude condition,bridgingstresscurves are differentfrom those under constantcrack openingdisplacementamplitudecondition. Also, it shouldbe notedthat,undervariablestressamplitudecondition, n*crit doesnot holdvalid.. measurementsto showthe validityof the derivedfiberbridging constitutivelaw, to apply the fiberbridgingconstitutivelaw to progressivecrack occurrenceand growthin a structuralanalysis, and alsoto extendthe constitutivelawbeyondthe assumptionsof frictionalbond controlledinterfaceand no fiber rupture under staticloading. Appendix Derivation of the Monotonic and Cyclic Fiber Bridging Constitutive Laws with Effect of Fiber Fatigue Rupture The monotonicand cyclic fiber bridgingconstitutivelaws witheffectof fiberfatiguerupturecan be derivedin a similarway to the cyclic fiber bridging constitutive law derived by Matsumoto1).The differenceis the integrationdomainreduced due to progressivefiberfatiguerupture(Fig.7 for monotonicand Fig. 8 for cyclicconstitutivelaw). definedinFig. 7 and Fig. 8 determine the integrationdomain and have expressionsas follows: (A1). (A2). (A3). 7. ConcludingRemarks This paper presenteda theoreticalformulationof the cyclic constitutivelaw for a discontinuousfiberreinforcedcomposite withthe effectsof fiberfatiguerupture.The formulationis based on the micromechanicsof fiber bridgingunder cyclicloading, enabling the effects of microstructuralparameters to be evaluated. The singlefiber behavior( relation)was reviewed. Also, the notationsof fiberbridgingconstitutivelaw under cyclic loading( ) were introducedbased on the previousstudies. Effectsof fiberfatigueruptureare includedin the constitutive laws under monotonicand cyclic loading,accountingfor the fatiguerupture of randomly distributedfibers under constant crackopeningdisplacementamplitude. As fiberfatiguerupture proceeds,maximum bridging stress as well as crack opening displacementat which bridging stress vanishesare decreased. The criticalnormalizednumber of cycles,n*criat, twhich crack bridgingdegradationtakesplace,is analyticallyderived. For the futuretasks, it is necessaryto conductexperimental. •\ 899•\. (A4) (A5). (A6) (A7) and. (A8) It should. be noted. that,. for all ƒÓi's in the equations. below,.

(10) (A9). wherei=1〜5.. Monotonic. Constitutive. Law. (A12) The. monotonic. be obtained the. by. constitutive integrating. integration. domain. surviving. fibers. debonding. and. monotonic. loading,. pulled-out. Only. the bridging. stress.. law. the of. again. w. and. divided. sliding.. As. fibers. after. fiber. contribution. the. will. fatigue. rupture. of surviving shown. in. two. groups:. into bridged. crack. undergo. can. fibers. Fig.. 7.. The. fibers. opens. debonding,. whichreduces (19), to and,for. in. up. in. (A13). under. sliding,. and which. fibers. in debonding. The. relation. and. between. sliding the. fiber. contnbute pull-out. to. reduces When. and. the. crack. opening. displacement, ƒÂ,. of. a single. fiber. in. for. a single. fiber. in. less debonding. is given. by. (1). and. the. relation. (1•…ƒ¿f) than. it is sliding. is given. a fiber. with. the value. further. loaded. fatigue-loaded opening. in. the. displacement. postpeak. under. amplitude and. monotonic. loading,. afterwards. the. monotonic. by (2).. a fiber. composite. is fatigue-loaded. region under. is. crack. for all-(ƒÀf<ƒÀo). for. n cycles. law. (1•…ƒ¿f<ƒ¿). and monotonic. loading,. for the prepeak. afterwards. the monotonic. it is further. constitutive. can be obtained. can be. obtained. by (A12). for. in the and. prepeak. the. loaded. constitutive When. composite. load, region. P,. to (20).. When. a fiber. by (A13). for. composite. is. fatigue-loaded. in. the. postpeak. law. by,. region. (1<ƒ¿f). greater. than. with. the. crack. opening. displacement. amplitude. for ƒ¿ afterwards monotonic. (A10) whichreducesto (15),and,for. all-fibers-sliding-in. loaded. under. (ƒÀo•…ƒÀf). and. loading,. the. monotonic. law (1•…ƒ¿f<ƒ¿). can. be obtained. by. whichreduces to(22). Constitutive. The. Law. constitutive a similar. fibers. unstretching made. cyclic in. surviving. by. are. and. law. way. z1 (ƒÀ) in Fig.. 8.. stress,. none. pull-out. in cyclic. since. after. to the. divided. contracting. bridging. When. •\ 900•\. for. constitutive. obtained. whichreducesto (16). And the monotonicbridgingconstitutive law for the postpeak( ) canbe obtainedby,. value further. (A14). Cyclic. (A11). the it is. fiber. into. and All. fatigue. monotonic two. surviving them. and. fibers will. can. above.. groups:. in sliding-in,. of. rupture law. the. fibers. in. division. is. contribute. be. be The. to the. discounted. (no. n cycles. in the. loading).. a fiber. prepeak. region. loading,. the cyclic. composite. (ƒ¿f<1). and. constitutive. is fatigue-loaded afterwards law. for. it is loaded. can be obtained. under. cyclic. by, for ƒÀ•…ƒÀf,.

(11) which. reduces. to(26)and,for βf<β,. (A18) (A15). whichreduces to(27). References whichreducesto(24),and,for βf<β,. (A16). which. reduces When. postpeak. (A15). a fiber. (ƒÀf<ƒÀo),. loading. composite (1<ƒ¿f). is fatigue-loaded and. the cyclic. for ƒÀ•…ƒÀf and by (A16). When postpeak. to (25).. region. loading. 1) Matsumoto,T., CrackBridgingLaw in DiscontinuousFiber ReinforcedCompositesunder Cyclic Loading,Journal of AppliedMechanics,10,923-933,2007. 2) Evans,A.G. (1997)."Designand Life PredictionIssuesfor High-Temperature Engineering Ceramics and Their Composites." ActaMaterialia,45(1),23-40. 3) Evans,A.G., Zok, F.W., and McMeeking,R.M. (1995). "Fatigueof CeramicMatrixComposites." ActaMetallurgica etMaterialia,43(3),859-875. 4) Zok, F.W., McNulty,J.,Du, Z.Z., and Evans,A. G.(1994). "Effects of InterfacialWear on Fatigue Failure in Fiber. a fiber region (ƒÀo<ƒÀf),. composite (1<ƒ¿f). afterwards constitutive. law. and. can. in the. under. cyclic. be obtained. by. for ƒÀf<ƒÀ. is fatigue-loaded. the cyclic. for n cycles. it is loaded. afterwards constitutive. for n cycles. it is loaded law. can. in the. under. cyclic. be obtained. by,. for ƒÀ•…ƒÀf,. (A17). ― 901―. ReinforcedCMCs."InternationalConferenceon Composites Engineering-ICCE/1, 597-598. 5) Cho,C., Holmes,J.W., and Barber,J.R. (1991)."Estimation of InterfacialShear in CeramicCompositesfrom Frictional HeatingMeasurements."Journal of the AmericanCeramic Society,74(11),2802-2808. 6) Holmes, J.W., and Cho, C. (1992). "Experimental Observationsof Frictional Heating in Fiber-Reinforced Ceramics."Journal of theAmericanCeramicSociety,75(4), 929-938. 7) Dambrine, B. (1994). "Low Cycle Fatigue of SiC/SiC Compositesat High Temperature- Roleof the Oxidationof the Interface on the Cyclic Behaviour and Modelling." International Conference on Composites Engineering-ICCE/1, 679-680. 8) Rouby, D., and Reynaud, P. (1993). "Fatigue Behavior Related to InterfaceModificationduring Load Cycling in Ceramic-Matrix FibreComposites."CompositesScienceand Technology, 48, 109-118. 9) Zhang,J., Stang,H., and Li, V.C., ExperimentalStudyon Crack Bridging in FRC under Uniaxial Fatigue Tension, Journal of Materials in Civil Engineering, 12(1), 66-73, 2000. 10)Li, V.C. and Matsumoto,T., FatigueCrackGrowthAnalysis of FiberReinforcedConcretewithEffectof InterfacialBond Degradation,Journal of Cementand ConcreteComposites,.

(12) 20(5),339-351,1998. 11)Matsumoto,T. and Li, V.C., FatigueLife Analysisof Fiber Reinforced Concrete with a Fracture Mechanics Based Model,Journal of Cementand ConcreteComposites,21(4), 249-261,1999. 12)Matsumoto,T. and Li, V.C., Fracture MechanicsBased Fatigue Life Model of DiscontinuousFiber Reinforced Composites,Proceedingsof the Japan ConcreteInstitute, 20(3), 163-168,1998. (in Japanese) 13)Walls, D.P., Bao, G., and Zok, F.W., Mode I Fatigue Cracking in a Fiber ReinforcedMetal Matrix Composite, ActaMetallurgicaet Materialia,41(7),2061-2071,1993. 14)Matsumoto,T., Fracture MechanicsApproachto Fatigue Life of Discontinuous Fiber Reinforced Composites, Doctoral Thesis,The Universityof Michigan,Ann Arbor. 1998. 15)Suthiwarapirak,P., Matsumoto,T., and Kanda,T., Flexural Fatigue Failure Characteristics of an Engineered CementitiousComposite and Polymer Cement Mortars, Journal of Materials,ConcreteStructuresand Pavements, No.718/V-57,121-134,2002.. ― 902―. 16)Suthiwarapirak,P., Matsumoto,T., and Kanda,T., Multiple Crackingand Fiber BridgingCharacteristicsof Engineered CementitiousCompositesunderFatigueFlexure,Journal of Materialsin CivilEngineering,16(5),433-443,2004. 17)Matsumoto,T. and Li, V.C., UniaxialCyclic Behaviorof DiscontinuousFiber ReinforcedComposites.Proc. ASCE 4th Materials EngineeringConference,WashingtonD.C., USA,426-435,1996. 18)Li, V.C., PostcrackScalingRelationsfor Fiber Reinforced CementitiousComposites,Journal of Materials in Civil Engineering,4(1),41-57,1992. 19) Li,V. C., Wang,Y., and Backer,S., Effectof IncliningAngle, Bundling,and SurfaceTreatmenton SyntheticFiberPull-Out from a Cement Matrix, Journal of Composites,21(2), 132-140.1990. (Received:April 14,2008).

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