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1 Zhang, W., Zhou, Z., Scarpa, F., & Zhao, S. (2016). A fatgue damage mesomodel for fber-renforced compostes wth stress rato effect. Materals and Desgn, 107, DOI: /j.matdes Peer revewed verson Lcense (f avalable): CC BY-NC-ND Lnk to publshed verson (f avalable): /j.matdes Lnk to publcaton record n Explore Brstol Research PDF-document Ths s the author accepted manuscrpt (AAM). The fnal publshed verson (verson of record) s avalable onlne va Elsever at Please refer to any applcable terms of use of the publsher. Unversty of Brstol - Explore Brstol Research General rghts Ths document s made avalable n accordance wth publsher polces. Please cte only the publshed verson usng the reference above. Full terms of use are avalable:

2 A fatgue damage meso-model for fber-renforced compostes wth stress rato effect Wenjao Zhang a,b,c*, Zhengong Zhou b, Fabrzo Scarpa c, Shuyuan Zhao b* a School of Engneerng, Northeast Agrcultural Unversty, Harbn, , Chna b Center for Composte Materals and Structures, Harbn Insttute of Technology, Harbn, , Chna c Advanced Compostes Center for Innovaton and Scence, Unversty of Brstol, Brstol BS8 1TR, UK Abstract Ths work presents a fatgue damage meso-model for fber-renforced plastc compostes, n whch the effect of stress ratos on the off-axs fatgue behavor s taken nto account. The non-dmensonal effectve stress concept s ntroduced n the contnuum damage mechancs method. Damage growths and fatgue falure are studed along axal, transverse and shear drectons at meso-scale level. The proposed model s valdated through numercal smulatons that descrbe the meso fatgue damage accumulaton and the fatgue lfe for off-axs undrectonal fber-renforced plastc composte lamnates of arbtrary fber orentaton under dfferent stress ratos. It s shown that the fatgue damage behavor and fatgue lfe for off-axs undrectonal glass/epoxy and carbon/epoxy composte lamnates are adequately descrbed by the proposed fatgue model over the range of dfferent stress ratos. Keywords: fber-renforced composte, fatgue damage, meso-model, contnuum damage mechancs, stress rato INTRODUCTION Fber-renforced compostes are wdely used n aerospace, marne, automotve and advanced engneerng applcatons n recent years, due to ther hgh-qualty mechancal propertes. However, these structures always suffer cyclc fatgue loadngs durng servce lfe, such as arcraft wngs, helcopter blades, wnd turbne blades and so on [1]. As a consequence, one mportant ssue durng the desgn of these composte structures s the fatgue damage assessment: the strength and durablty of the composte structural components must take nto account the typcal damage 1

3 phenomena occurrng under n-servce loadng. The fatgue behavor of fber-renforced compostes s qute dfferent from the one of metals [2], due to ther ansotropy and heterogenety characterstcs, and the mult-scale nature of the damage processes and non-lnear damage evoluton durng loadng [3]. Therefore, t s mportant to understand the mechansms assocated to fatgue damage and to predct the long-term fatgue strength and lfe for fber-renforced compostes under complex cyclc fatgue loadng. The fatgue damage falure process of fber-renforced compostes nvolves a number of dfferent falure mechansms and nteractve couplng effects. The dfferent types of damage nclude fber fracture, matrx crackng, matrx crazng, fber bucklng, fber matrx nterface falure, delamnaton among composte ples and the effect of shear-nduced dffuse damage on transverse cracks n fber-renforced compostes, whch has been already nvestgated through expermental [4] and theoretcal methods [5], respectvely. In addton, the fatgue performance of compostes s also affected by the consttuents of composte system, renforcement structure, lay-up sequence, resdual stress due to manufacturng process [6] and stress ratos [7-9] from external loadng condtons. In order to smulate the fatgue damage behavor and to predct fatgue lfe of fber-renforced compostes, n recent years several methodologes that mplement progressve falure analyss and approprate consttutve models wth damage accumulaton laws have been developed. In open lterature, fatgue progressve damage models have been extensvely establshed from macro to mcroscopc scales by means of theoretcal analyss methods, fnte element solutons and experments [10-17]. Montesano et al [18] have establshed a damage mechancs based model that takes nto account local multaxal stresses as well as varable ampltude cyclc loadng. The numercal results from that model showed the capablty of that approach to predct the evoluton of the damage and the degradaton of the materal propertes n a traxally braded carbon fber polymer matrx component. Krüger and Rolfes [19] have presented a new layer-based fatgue damage model (FDM) for lamnated multdrectonal lamnates n general states of plane stress. The stffness and strength degradaton were smulated usng a Fnte Element (FEM) 2

4 analyss, and the stress redstrbutons and sequence effects were also analyzed. Elopoulos and Phlppds [20] developed an ansotropc non-lnear consttutve model mplementng progressve damage concepts to predct the resdual strenghth/stffness and lfe of composte lamnates subjected to multaxal varable ampltude cyclc loadng. In-plane mechancal propertes of the materal were fully characterzed at the ply level whle statc or fatgue strength of any multdrectonal stackng sequence can be predcted. Paepegem and Degreck [21] establshed a phenomenologcal resdual stffness model to predct the stffness degradaton and possble permanent strans n fbre-renforced polymers under n-plane fatgue loadng. The stress-stran-damage relatonshps and the damage growth rate equatons were developed and explaned thoroughly. Montesano and Sngh [22] have developed a mult-scale damage model combnng synergstc damage mechancs wth an energy-based damage evoluton framework to predct the evoluton of sub-crtcal matrx cracks n dfferent ples under multaxal loadng, the ply crack densty evoluton and the lamnate stffness degradaton. Quaresmn et al [23] nvestgated the very early stages of the damage evoluton under a unaxal cyclc tensle loadng by testng [45/-45/0]s glass/epoxy specmens. In that work the frst event observed for the damage ntaton was multple mcro-cracks n the nterfber regon of the 45 ply, wth a specfc nclnaton wth respect to the fbers. However, all the studes cted above manly focus on the evaluaton of the fatgue damage behavor, lttle attempt has been made to nterpret the fatgue damage propagaton and the effect of complcated loadng mode such as stress rato on fatgue damage growth, as well as the fatgue damage mechansms for fber-renforced compostes at meso-scale. Contnuum damage mechancs (CDM) s a mathematcal and expermental descrpton of the damage accumulaton and growth due to changes of the materal mcrostructure. On the bass of CDM theory, Perre Ladevèze and hs group establshed meso-scale damage models to descrbe the strength deteroraton of compostes under statc loadng. It s assumed that the behavor of any stratfed structure can be descrbed through two famles of basc damageable consttuents: the elementary layer and the nterlamnar nterface, and damage s consdered unform 3

5 through the thckness of ndvdual layers of compostes [24-27]. In these models two damage mechansms are ntroduced. The frst s related to the dffuse ntralamnar damage assocated wth the fber/matrx debondng n the ply and wth small transverse cracks n the matrx. The second damage mechansm s assocated wth dffuse nterlamnar damage lnked to the formaton of mcro-vods n the matrx of the nterlamnar nterface, resultng n a reduced stffness of the nterlamnar nterface wth no vsble delamnaton (Fg. 1 [27]). Therefore, the dffuse damage at the elementary ply scale can be modelled by a stffness declne of the materal along the axal, transverse and shear drectons. It s essental to extend the meso-scale damage model assocated to statc loadng to complcated cyclc fatgue confguraton. Also, t s qute mportant to nvestgate the fatgue damage behavors and to develop new fatgue predcton methodologes for fber-renforced compostes at meso-scale levels. Fg. 1 Mechansms of degradaton on meso-scale: (a) transverse matrx mcrocrackng; (b) local delamnaton; (c) dffuse damage [27] In ths paper we am to establsh a new fatgue damage meso-model n whch the CDM theory s appled wth the use of damage varables at the meso-scale of elementary ples and stress ratos to account for the complex fatgue loadng hstory. The model s able to determne the fatgue damage growth at meso-scale and to predct the fatgue lfe of undrectonal composte lamnates wth arbtrary fber orentaton under dfferent stress ratos. In ths approach the progressve growth of dffuse damage s evaluated by establshng three groups of damage growth rate equatons (along the axal, transverse and shear drectons) accordng to contnuum damage mechancs. We also ntroduce a non-dmensonal effectve stress [8, 9] to buld a new fatgue dffuse damage meso-model that consders the effects of the fber orentaton and the stress ratos on the off-axs fatgue behavor of undrectonal 4

6 fber-renforced composte lamnates. We then evaluate the valdty of the proposed fatgue damage meso-model usng data from the prncpal damage varables occurrng n tenson-tenson cyclc loadng under hgh-low stress levels and dfferent stress ratos by GFRP and CFRP undrectonal composte lamnates under dfferent stress ratos wth constant ampltude and frequency condtons [7, 8]. The results from the model are therefore dscussed and show the vablty of the proposed approach to predct on and off-axs fatgue damage propagaton n compostes. FATIGUE DAMAGE MESO-MODEL The present fatgue damage meso-scale model for undrectonal ples s developed wthn the framework of the thermodynamcs n rreversble phenomena. Under the assumpton of plane stresses and small perturbatons, the stran energy of the ply can be wrtten n the followng form: W D E11(1 D11) E11 E11 E22(1 D22) E22 G12(1 D12) (1) Where 0 0 E 11, E22 and G 0 12 represents the ntals stffness of fber, transverse and shear drecton n plane, respectvely. s defned as the postve part and as the negatve parts. Consequently, when 22 0, mcro-cracks are closed and no notceable damage occurs. Three damage ndcators, whch are constant through the thckness, pertan to the followng mechansms: Fber breakage D 11 (along the axal drecton), matrx mcro-crackng D 22 (along the transverse drecton) and deteroraton of the fber-matrx bonds D 12 (along the shear drecton). From ths potental, thermodynamc forces assocated wth the tenson and shear nternal varables D (, j 1,2 and j) are defned: Y 2 WD D 0 2E 1 D 2 (2) The damage growth rates dd dn correspond to the damage knetcs and are expressed as a functon of the thermodynamc forcesy, whch are also connected to 5

7 the appled stress. Therefore, a typcal damage growth equaton for a contnuum fatgue damage varable D can be represented as: dd dn f D,, R, N, p (3) Where f defnes a fatgue damage functon, the parameters, R, N and p denote appled maxmum stress, stress rato, number of fatgue cycles and a hstory dependent parameter, respectvely. THE EFFECT OF STRESS RATIOS Under realstc servce condtons most structural components made from multdrectonal composte lamnates are subjected to complex fatgue loadng hstores characterzed by changes n the ampltude, mean stress, frequency and waveform of the cyclc loadng. As a sgnfcant nfluence factor the effect of the stress ratos on the off-axs fatgue behavor of undrectonal compostes should not be therefore gnored. Therefore, t s necessary to develop approprate fatgue damage models to study the fatgue damage accumulaton at meso-scale and to predct the fatgue lfe of off-axal undrectonal composte lamnates under dfferent stress rato condtons. In order to ncorporate the senstvty to dfferent loadng modes, the stress rato R, the alternatng stress a and mean stress m have the relatonshps as R, a 1 2(1 R ) max and m 1 2(1 R ) max. mn max A non-dmensonal scalar quantty [9] has been defned as: a (1 R) max 2 (1 R) B m B max (4) Where B s statc falure strength. The modfed fatgue strength rato s a useful measure for the off-axs fatgue behavor of undrectonal compostes and fatgue behavor of metals [28] at dfferent stress ratos. The non-dmensonal effectve stress [8, 9] for orthotropc materals based on the Tsa Hll statc quadratc nteracton falure crteron has been defned as: 6

8 2 2 2 * Gkl kl 2 (5) X Where X, Y and S represent the longtudnal, transverse and shear strength, respectvely, n the case of plane stress. Smlarly, the maxmum non-dmensonal effectve stress can be unquely decomposed asmax a m. The scalar quanttes X Y S a and m represent the normalzed alternatng stress and normalzed mean stress, respectvely. Consderng the statc falure condton max a m and the analogy wth Eq. 4, the modfed non-dmensonal effectve stress [9] s derved as follows: a (1 R) 1 2 (1 R) m max max When off-axs specmens are subjected to fatgue loadng along the axal drecton, the non-dmensonal effectve stress fatgue stress max can be expressed as: max (6) assocated wth the maxmum cos cos sn sn sn cos X X Y S max max (7) Therefore, the relatonshp between the maxmum stress max and the modfed non-dmensonal effectve stress s obtaned as follows: max 2 (8) ( )[(1 R) (1 R) ] Where cos cos sn sn sn cos ( ) s orentaton factor X X Y S In ths paper, the damage growth rate equaton s prescrbed followng the form of the Kachanov-Rabotnov equaton [29]: dd dn A 1 D n B (, j 1,2 and j) (9) Where A, n and B are materal parameters assocated to the axal, transverse and n-plane shear drectons, respectvely. 7

9 Replacng Eq. (2) and (8) nto Eq. (9), the meso fatgue damage growth equatons for three drectons n-plane are derved as: n max( )(1 ) B 1 D 2 max( )(1 R) dd A R dn (, j 1,2 and j) (10) Accordng to Eq. (10), the effects of stress ratos on fatgue damage growths for fber breakage, matrx crackng and fber/matrx debondng n-plane and on the fatgue behavors for off-axal undrectonal compostes under dfferent multaxal cyclc loadng modes are all taken nto account. APPLICATION TO OFF-AXIAL FATIGUE BEHAVIOR Materal parameter dentfcaton The materal parameters n these damage growth equatons were determned by fttng to the typcal S Nrelatonshps for three groups of fatgue experment data under R=0 n Ref [7, 8], shown n Fg. 2 and Fg. 3, respectvely. The sold lnes n Fg. 2 and Fg. 3 ndcate the master S N relatonshps dentfed for respectve groups of the basc fatgue experment data. The values of parameters nvolved by the master fatgue damage growth equatons are lsted n Table 1 and Table 2, for undrectonal glass/epoxy and carbon/epoxy lamnates, respectvely. Fg. 2. Fttng S-N relatonshps for GFRP composte lamna under R=0: (a) 0 o, (b) 45 o (c) 90 o 8

10 Table 1 Parameter determnaton for E-Glass Epoxy compostes parameter n A B 0 o (=j=1) o (=1, j=2) o (=j=2) Fg. 3. Fttng S N relatonshps for CFRP composte lamna under R=0 (a) 0 o (b) 45 o (c) 90 o Table 2 Parameter determnaton for Carbon/Epoxy compostes parameter n A B 0 o (=j=1) o (=1, j=2) o (=j=2) Falure Crteron In ths modellng, fatgue damage growths along axal, transverse and shear drectons on the ply scale (fber breakage, matrx mcro-crackng and fber/matrx debondng) are coexstng. Incorporatng three sorts of damage, the fatgue falure crteron of fber-renforced undrectonal composte lamnates s proposed as the determnant of coeffcent of damaged stffness matrx for the lamna s zero, whch s equvalent to damage varable D 1 (, j 1,2 and j). Off-axs fatgue smulaton In order to obtan the quantfcatonal fatgue behavor of off-axs undrectonal composte lamnates, the process of fatgue damage growth on meso-scale s 9

11 smulated n the followng procedure: (a) In vew of three damage modes n-plane, accordng to Eq. (10), two damage extremums for mnmum status and maxmum status are consdered n the frst calculaton, respectvely. It s assumed that the undrectonal compostes have no ntal ntal damage, namely D 0. Therefore, the ntal damage ncrement ether 0 D mn or could be determned by Eq. (10) and as a consequence, the 0 D max correspondng ntal cycle ncrement well. Then, the ntal cycle ncrement 0 N mn or 0 N mn or could be obtaned as 0 N max s substtuted nto the 0 N max other two damage evoluton equatons, respectvely, n order to get the rest damage ncrements D f ( N ) n other two dfferent drectons, where 0 max/mn, j 1,2 and j. So these three dffuse damage ncrements under the crcumstance of ether mnmum or maxmum extremums for frst cyclc number are all obtaned, respectvely. (b) Calculate the new damage felds for axal, transverse and shear drectons after the frst cyclc damage ncrement as follows: D' D D, j 1,2 and j (11) Where D refers to the new damage varables along axal, transverse and shear ' drectons, respectvely. Snce the structural changes on meso-scale are characterzed by a macroscopc stffness reducton, the value of damage varable D s located between zero (ntal materal state) and one (fnal falure). (c) Judge whether the new damage felds satsfy the falure crteron. If t s satsfed, the fatgue lfe of undrectonal composte lamnates s N N ; f not, the steps are repeated to step (a) untl the falure crteron s satsfed. It must be emphaszed that the value of damage ncrement D should be small enough to make sure a convergence result of N. The flow chart of ths procedure s shown n Fg

12 11, 22, 12 extremum( D ), N 0 0 D f ( N ); D D extremum( D ) D 11 N Y D 22 Y N D D D f ( N ) D f ( N ) D f ( N ) D D D f ( N ) D f ( N ) D f ( N ) D D D D D ' Falure crteron N Y N N Fg. 4 Flow chart of the present numercal smulaton 11

13 RESULTS AND DISCUSSION The parameters n fatgue damage growth equatons along axal, transverse and shear drectons are determned by the fatgue tests of 0 o, 90 o, 45 o GFRP and CFRP undrectonal composte lamnates, respectvely. Then the comparsons between the results of numercal predcton and experment for 19 o undrectonal glass/epoxy composte lamna under R 0.5 are reported n Table 3. Smlar comparsons for 10 o, 30 o undrectonal carbon/epoxy composte lamnas under R 0.5 are presented n Table 4 and Table 5 as well. In Table 3-Table 5, LogN refers to the mean exp 50, logarthmc fatgue lfe of the experment, Log N pred refers to the logarthmc fatgue lfe of predctng model, subscrpt refers to the stress level and error s the relatve error defned as error pred Log N Log N exp Log N50, exp 50, 2 (12) Table 3 Comparson between fatgue lfe predcton and experment data for 19ºundrectonal GFRP lamna under R=0.5 max LogN Log exp 50, error pred N e e e e-04 Table 4 Comparson for 10ºundrectonal CFRP lamna between fatgue lfe predcton and experment data under R=0.5 max LogN Log exp 50, error pred N e e e-04 12

14 Table 5 Comparson for 30ºundrectonal CFRP lamna between fatgue lfe predcton and experment data under R=0.5 max LogN Log exp 50, error pred N e e e-04 The predctng S N relatonshps of 19 o, 71 o GFRP undrectonal composte lamnates and 10 o, 15 o, 30 o CFRP undrectonal composte lamnates under dfferent stress ratos are gven n Fg. 5 and Fg.6, respectvely. Fg. 5 exhbts that the fatgue lfe and resdual strength of off-axs GFRP undrectonal composte lamnates decreases when the fber orentaton ncreases under the same stress rato, the smlar tendency s shown n Fg. 6 as well. The results ndcate that the predctons of fatgue lfe are n good agreement wth experment data of Ref [7, 8], the off-axs fatgue behavor of two types of undrectonal compostes and ts stress rato R-dependence have favorably been descrbed by the fatgue damage meso-model. Fg o and 71 o GFRP undrectonal lamnate predctng S N relatonshps (a) under R=0 (b) under R=0.5 Fg o, 15 o and 30 o CFRP lamna predctng S N relatonshps under (a) R=0.1 and (b) R=0.5 13

15 The off-axal fatgue behavors of 19 o undrectonal composte lamnate under stress ratos R=0 and R=0.5 are compared n Fg. 7 (a); furthermore, the predctng S-N curve group under dfferent stress ratos R=0, 0.3, 0.5, 0.8 s shown n Fg.7 (b) as well. For a gven maxmum stress n the tenson-tenson cyclc loadng, the off-axs fatgue lfe of the undrectonal composte lamnate ncreases wth the ncreasng stress rato R. Ths behavor s consstent wth the one observed n open lterature for contnuous and short fber renforced compostes predcted by dfferent methodologes [8, 11, 30, 31]. The nterpretaton for ths trend s that ncreasng the stress rato R, reduces the stress ampltude amp (for a constant maxmum stress, 1 R 2 amp max ) of the load regme, leads to a flatter S-N curve as well as a slower fatgue degradaton and damage accumulaton rates. At hgher stress ratos, the materal s beng subjected to lower stress ampltudes, and hence t wll have to endure lower stress/stran gradents n the fber, matrx and at the fber/matrx nterface. It would n turn lead to a reduced crack growth rates and less sgnfcant fatgue strength degradaton wth ncreasng number of cycles [32-34]. Fg. 7. Off-axs fatgue behavor for the E-Glass Epoxy 19 o undrectonal compostes: (a) comparson of the fatgue lfe between R=0 and R=0.5 (b) predcton S-N curves group at R=0, 0.3, 0.5, 0.8 The fatgue meso-damage growth for the 19 o GFRP undrectonal composte lamnate under hgh and low stress levels s shown n Fg.8. The fgure shows that the relatonshp between the transverse damage D 22 (a), shear damage D 12 (b), axal damage D 11 (c) and cyclc number N are almost the same durng the whole entre fatgue damage growth progress of the composte materals. One can observe a rapd 14

16 damage accumulaton durng the frst cycles, then a slow and steady damage growth followed by a sudden falure on the last stage. Fg. 8. Fatgue damage growths on meso-scale for GFRP 19 o undrectonal composte lamnate under R=0: (a) transverse damage D 22, (b) shear damage D 12, (c) axal damage D 11 The elastc modulus of the matrx s sgnfcantly smaller than the one of the fber and the nterface n meso-consttuents [24]. The transverse damage D 22 (.e., matrx mcrocrackng) s lower than 0.6 for the whole stress levels (Fg.8 (a)). Ths feature demonstrates that the transverse matrx mcrocracks always occur durng the early stages of the damage growth, and then ncrease and expand untl the cracks n each ply reach to equlbrum or saturaton, a stage whch s denomnated the characterstc damage state (CDS) [35]. The fatgue damage growth of the shear damage D 12 s however dfferent under hgh and low stress levels (Fg.8 (b)). At hgh stress levels ( max u 72% ) D 12 s wthn the range of , however, at low stress levels ( max u 47% ) D 12 now approaches the unty value, whch means that the shear damage D 12 (fber/matrx nterfacal debondng) plays a part at low cyclc loadng and provdes an effect to the fnal falure of the composte materals. Ths 15

17 phenomenon can be nterpreted by consderng that durng the second stage of the fatgue damage growth transverse mcrocracks extend through the thckness of the off-axs ples. Interfacal debondng mcrocracks perpendcular to transverse cracks then occur, whch are caused by the tensle stress along the crack axs ahead of the prmary transverse mcrocracks [35]. In Fg.8 (c) the axal damage D 11 almost reaches the value of 1 on both hgh and low stress levels, whch means that the fber breakage generates a sgnfcant effect on the damage growth progress and domnates the fnal falure of the 19 o GFRP undrectonal composte lamnate. Ths feature s nterpreted by consderng that scattered fber falures begn to ntate locally durng the early stage of the damage growth, followed by fber breakage growths wth cyclc loadng untl the neghborng zones of fber falures are joned together, and the specmen s weakened wth an eventual catastrophc falure [35]. It should be also emphaszed that these fatgue damage mechansms are nteractve and coupled together. No ndvdual damage mechansm sngle handedly domnates a dstnct phase of the total lfe of a component; these mechansms occur more or less smultaneously. The fatgue damage growths of D 22, D 12 and D 11 (n transverse, shear and axal drectons) for 19 o GFRP undrectonal composte lamnates under R=0.5 are dscussed as well and shown n Fg.9, respectvely, whch have the smlar tendency of damage growth under R=0. It s shown that axal damage D 11 and shear damage D 12 both ncrease wth the ncreasng of cyclc number N and contrbute to the fnal falure under low stress level the same as scenaro R=0. However, the transverse fatgue damage D 22 s almost nvarable n the whole damage growth stage. 16

18 Fg. 9. Fatgue damage growths on meso-scale for GFRP 19 o undrectonal composte lamnate under R=0.5: (a) transverse damage D 22, (b) shear damage D 12, (c) axal damage D 11 For most cases, the fatgue tests under dfferent stress ratos for the same stress levels are not usually carred out, therefore, the paper doesn t compare the fatgue damage D 22, D 12 or D 11 between R=0 and R=0.5 wth the same stress level. The proposed fatgue damage meso-model s also applcable to tenson-tenson (T-T) cyclc loadng, n partcular for the scenaros descrbng tenson-compresson (T-C) and compresson-compresson (C-C) cyclc loadng. The formulaton of Eq. (10) and the falure crteron should be however modfed to consder the compressve component of the fatgue damage [36]. CONCLUSIONS A new fatgue damage meso-model consderng the stress rato effect as well as the fber-orentaton effect on the off-axs fatgue behavor of undrectonal compostes was developed. The non-dmensonal effectve stress that accounts for the effect of stress rato was ntroduced and three groups of dffuse damage growth equatons on elementary ply scale were establshed based on contnuum damage mechancs theory. Three fatgue damage growths of axal, transverse and shear drectons under dfferent stress ratos were studed, respectvely. Fatgue lfe of off-axs undrectonal composte lamnates wth arbtrary fber orentaton under dfferent stress ratos were predcted by numercal smulaton and compared wth the expermental data as well. The results demonstrated that the proposed fatgue damage meso-model can adequately descrbe the off-axs fatgue damage behavors and predct off-axs fatgue lfe of glass/epoxy and carbon/epoxy undrectonal composte 17

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