Tensile and Compressive Damage Coupling for Fully-reversed Bending Fatigue of Fibre-reinforced Composites

Transcription

1 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibre-reinfored Composies W. VAN PAEPEGEM * and J. DEGRIECK Ghen Universiy, Dep. of Mehanial Consruion and Produion, Sin-Pieersnieuwsraa 41, B-9000 Gen, Belgium ABSTRACT Due o heir high speifi siffness and srengh, fibre-reinfored omposie maerials are winning hrough in a wide range of appliaions in auomoive, naval and aerospae indusy. Their design for faigue is a ompliaed problem and a large researh effor is being spen on i oday. However here is sill a need for exensive experimenal esing or large safey faors o be adoped, beause numerial simulaions of he faigue damage behaviour of fibre-reinfored omposies are ofen found o be unreliable. This is due o he limied appliabiliy of he heoreial models developed so far, ompared o he omplex muli-axial faigue loadings ha omposie omponens ofen have o susain in in-servie loading ondiions. In his paper a new phenomenologial faigue model is presened. I is basially a residual siffness model, bu hrough an appropriae hoie of he sress measure, he residual srengh and hus final failure an be predied as well. Two oupled growh rae equaions for ensile and ompressive damage desribe he damage growh under ension-ompression loading ondiions and provide a muh more general approah han he use of he sress raio R. The model has been applied o fully-reversed bending of plain woven glass/epoxy speimens. Sress redisribuions and he hree sages of siffness degradaion (sharp iniial deline gradual deerioraion final failure ould be simulaed saisfaorily. Keywords faigue; omposies; residual siffness; ensionompression. INTRODUCTION Fibre-reinfored omposie maerials are finding more and more appliaions in auomoive, naval and aerospae indusry. They have quie a good raing as regards o life ime in faigue, bu he same does no apply o he number of yles o iniial damage. Alhough he faigue behaviour of fibre-reinfored omposies has been sudied for many years, i is so diverse and omplex ha presen knowledge is far from omplee. Moreover he design experiene wih meals anno be simply ransposed o he design of omposies, beause omposies are inhomogeneous and anisoropi in naure. There are a number of imporan differenes beween he faigue behaviour of meals and of fibrereinfored omposies. In meals, he sage of gradual and invisible deerioraion spans nearly * Corresponding auhor (Fax: +3-( , Wim.VanPaepegem@rug.a.be.

2 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, he whole faigue life ime. No signifian reduion of siffness is observed in meals during he faigue proess. The final sage of he proess sars wih he formaion of small raks, whih are he only form of marosopially observable damage. Gradual growh and oalesene of hese raks quikly produe a large rak and final failure of he sruural omponen. Fibre-reinfored polymers are made of reinforing fibres embedded in a polymer marix. This makes hem heerogeneous and anisoropi. The firs sage of deerioraion by faigue is observable by he formaion of damage zones, whih onain a muliude of mirosopi raks and oher forms of damage, suh as debonding and pull-ou of fibres from he marix. I is imporan o observe ha damage sars very early, afer only a few or a few hundred loading yles. This early damage is followed by a seond sage of very gradual deerioraion of he maerial, haraerized by a gradual reduion of he siffness. More serious ypes of damage appear in he hird sage, suh as fibre breakage and unsable delaminaion growh, leading o an aeleraed deline and final failure. These major differenes in damage behaviour mus be refleed in he modelling approah. Indeed, as he siffness of a meal remains quasi unaffeed during faigue life, he linear relaion beween sress and srain remains valid. Alhough more and more ompliaed faigue laws for meals are developed due o heir omplex maerial design and he severe in-servie faigue loadings, he faigue proess an sill be simulaed in mos ommon ases by one finie elemen alulaion. Then in he pos-proessing sage, he faigue life of he individual nodes of he finie elemen mesh is assessed aking ino aoun he (muli-axial sress sae in ha pariular node. Ofen he riial plane onep is used for his purpose [1]. On he onrary, he gradual deerioraion of a fibre-reinfored omposie an lead o oninuous sress redisribuions during faigue life, and a reduion of sress onenraions in a sruural omponen. To simulae hese sress redisribuions auraely, he simulaion should follow he pah of suessive damage saes and evaluae he hanging siffness disribuion a regular imes during faigue life. Faigue models for fibre-reinfored omposies an be generally lassified ino hree aegories []: (i faigue life models, whih do no ake ino aoun he aual degradaion mehanisms bu use S-N urves or Goodman-ype diagrams and inrodue some sor of faigue failure rierion; (ii phenomenologial models for residual siffness/srengh; and (iii progressive damage models whih use one or more damage variables relaed o measurable manifesaions of damage (ransverse marix raks, delaminaion size. As repored earlier by he auhors [3][4], sress redisribuion an indeed be an imporan phenomenon wih experimenal faigue esing, and hene he residual siffness approah was hosen as he basis of he presen model. Suh residual siffness models propose an evoluion law whih desribes he (gradual deerioraion of he siffness of he omposie speimen in erms of marosopially observable properies. As a onsequene hey are able o simulae sress redisribuion. Moreover siffness an be measured nondesruively and he saer is small. On he oher hand, residual siffness models have no disin rierion for faigue failure as ompared o residual srengh models. Faigue failure is mos ofen assumed o our when he modulus has degraded o a riial level whih is no unique and has been defined by many invesigaors. Already in he early 70s, Salkind [5] suggesed o draw a family of S-N urves, being onours of a speified perenage of siffness loss, o presen faigue daa. Hahn and Kim [6] and O Brien and Reifsnider [7] proposed ha faigue failure ours when he faigue sean modulus degrades o he sean modulus a he momen of failure in a sai es. Aording o Hwang and Han [8], faigue failure ours when he faigue resulan srain reahes he ulimae sai srain. To alleviae his problem, he sai Tsai-Wu failure rierion will be used in a differen way in he presen model. I will provide a measure for he applied faigue sress level, bu wih a

3 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, orrelaion o he remaining srengh. This oupled model of residual siffness and srengh an simulae he hree sages of siffness degradaion: sharp iniial deline, gradual deerioraion and final failure. Furher he damage growh rae equaion will be spli up in wo differenial equaions, desribing he developmen of ensile and ompressive damage, respeively. Through he oupling of hese wo equaions, he aggravaing effe of fully-reversed ension-ompression faigue loadings an be simulaed. MATERIALS AND EXPERIMENTAL SETUP The maerial used in he faigue experimens, was a glass fabri/epoxy omposie. The fabri was a Roviglass R40 plain woven glass fabri (Synoglas and he epoxy was Araldie LY 556 (Ciba-Geigy. The plain woven glass fabri was saked in eigh layers, denoed as [#0º] 8, where 0 means ha he warp direion of eah of he eigh layers has been aligned wih he loading direion and where he symbol # refers o he fabri reinforemen ype. All omposie speimens were manufaured using he resin-ransfer-moulding ehnique. Afer uring hey had a hikness of.7 mm. The samples were u o dimensions of 145 mm long by 30 mm wide on a waer-ooled diamond saw. The fibre volume fraion V f was The in-plane elasi properies of he [#0 ] 8 omposie laminaes were deermined using he dynami modal analyis mehod desribed by Sol e al. [9][10], while he sai ensile srengh X T and ompressive srengh X C were deermined wih an Insron hydrauli esing mahine (ASTM D All relevan maerial properies are lised in Table 1. Table 1 Measured in-plane properies of he [#0 ] 8 omposie laminaes E [GPa] 4.57 E [GPa] 3.94 ν 1 [-] G 1 [GPa] 4.83 X T [MPa] X C [MPa] The experimenal resuls were obained from displaemen-onrolled anilever bending faigue experimens. One side of he speimen was lamped, while a sinusoidal displaemen was imposed a he oher side of he speimen. Figure 1 shows a shemai drawing. Fixed lamp L = 54.0 mm Moving lamp Composie speimen =.7 mm u u( Figure 1 Shemai drawing of he bending faigue seup.

4 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, The esing frequeny for all faigue experimens was. Hz. The ampliude u max of he imposed displaemen is a onrollable parameer and he displaemen raio R d (analogous o he sress raio R is defined as R d = u min /u max. For single-sided bending R d = 0.0, while for fully-reversed bending R d = Of ourse, due o he varying bending momen along he speimen lengh and he varying sresses and srains along he speimen lengh and hrough is hikness, he maximum sress ampliude an be differen in eah maerial poin. FATIGUE DAMAGE MODEL FOR SINGLE-SIDED BENDING Reenly he auhors have proposed a phenomenologial faigue damage model for uni-axial loading ondiions [][1]. This model was developed for single-sided bending and did no ake ino aoun he effe of negaive displaemen raios R d (or equivalen: negaive sress raios R. This model will firs be explained and hen be exended for negaive sress raios (ension-ompression faigue loading. Definiion of he faigue failure index Σ(σ, D As menioned earlier, he modified inerpreaion of he sai Tsai-Wu failure rierion mus provide a sress value, whih is a measure for he aually applied sress ampliude in he onsidered maerial poin, bu now in relaion wih he omposie s aual srengh. To ha purpose, he definiion of he failure index, as proposed by Liu and Tsai [13], is realled here. The failure index is he inverse value of he srengh faor Φ, he laer being he reserve o failure in he sai Tsai-Wu failure rierion. The srengh (or safey faor Φ is alulaed from he Tsai-Wu equaion as follows: σ Φ + σ Φ 1 = 0 X X XT X (1 T C C where X T and X C are he sai ensile and ompressive srengh and he srengh raio Φ defines he reserve faor o failure. If Φ = 1, failure ours, while if for example Φ =, he faor of safey is. The failure index Σ(σ is hen defined as 1/Φ. To use his rierion for faigue modelling, he effeive sress σ ~ is used. This sress measure originaes from he oninuum damage mehanis heory and is defined as: σ σ ~ = = E0ε 1 D ( where, for uni-axial loading ondiions, he damage variable D represens he degradaion of he longiudinal siffness E 0 (D = 1 E/E 0, σ is he applied nominal sress and ε is he nominal srain. When D is now onsidered as a measure for faigue damage, he applied nominal sress σ is replaed by he effeive sress σ ~ in he Equaion (1 for alulaion of he srengh faor Φ: σ 1 σ 1 1 Φ + 1 D X X 1 D XT X T C C Φ 1 = 0 (3

5 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, Thus, he failure index Σ beomes a funion Σ(σ, D of boh he applied nominal sress ampliude σ and he faigue damage D. I ould be argued ha he use of he onsan sai srengh insead of he faigue srengh in he definiion of he faigue failure index Σ(σ, D, ould lead o non-onservaive prediions. However, here is an indire relaion wih residual srengh. In he one-dimensional ase, i an be easily alulaed ha: σ 1 σ ~ 1 D E0 ε Σ ( σ,d = = = = (X = XT,XC (4 Φ X X X Thus, in he one-dimensional ase, he proposed definiion of he faigue failure index Σ(σ, D is equivalen wih he raio of he applied nominal sress σ o he residual srengh X (1-D (X = X T, X C, beause he effeive sress σ ~ is defined as σ/(1-d (Equaion (. The only differene is he inerpreaion of he residual srengh onep: here, i is assumed ha he apparen loss of srengh is due o he fa ha he aual load-bearing area is dereased due o faigue damage, and ha he effeive sress σ ~ in ha ross-seion is raised. If he effeive sress σ ~ hen equals he sai srengh X T or X C, he safey faor Φ and he faigue failure index Σ(σ, D boh equal 1.0 and final failure ours for ha pariular maerial poin. The advanage of his approah is ha he inroduion of he srengh properies ino he faigue model does no require any new laws for residual srengh o be esablished. The effeive sress σ ~ is used o aoun for presen faigue damage. Furher, as he effeive sress σ ~ equals E 0 ε (Equaion (, he faigue failure index Σ(σ, D is also a measure for he applied faigue srain. Consiuive equaions From Equaion (, he damage D is defined as a measure for he siffness degradaion, lying beween zero (undamaged maerial and one (omplee failure. The remaining equaion o omplee he faigue damage model, is he damage growh rae equaion dd/dn, where N is he yle number. This faigue model has been reenly proposed by he auhors and is derivaion an be found in deail in referene [1]. The model disinguishes beween a growh rae for ensile sresses (σ 0 and one for ompressive sresses (σ < 0 [][1]: dd dn = D 1 Σ exp Σ 3 D 1 Σ exp Σ iniiaion + D Σ 3 [ 1+ exp( ( Σ ] D Σ 1 + exp propagaion 5 4 if σ 0 ( Σ if σ < 0 (5 Boh growh rae equaions onsis of wo erms, separaely aouning for damage iniiaion and damage propagaion. If damage is very small, he seond erm is negligible (D 0 and only he firs iniiaion erm is aing, while for larger values of D, he exponenial funion in he firs erm fores his erm o diminish, while he seond erm is inreasingly dominaing. The damage iniiaion rae is differen in ension and ompression, beause i was observed from he experimenal faigue ess ha he ompressive damage iniiaion rae is muh smaller under he following resriive ondiions:

6 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, he displaemen raio R d = 0.0, whih means ha he bending experimens are singlesided. As a onsequene, eah maerial poin is subjeed o sresses whih do no hange sign during one yle. Beause u min = 0.0, he sress raio R is zero for all he maerial poins involved, - here are no delaminaions. Due o he hosen saking sequene, no delaminaions have been observed during faigue loading. I was a deliberae hoie o exlude delaminaions from he sope of modelling for he momen, beause i would be nearly impossible o disinguish beween he onribuion of eah damage ype o he bending faigue degradaion of he sudied maerial, espeially in fully-reversed bending. All five onsans i (i = 1,, 5 in he faigue damage model have a disinive meaning: - 1 regulaes he growh rae of he damage iniiaion regime (and hus he sharp iniial deline of he modulus degradaion urve, - is suffiienly large, so ha he firs erm is disappearing when damage inreases. Then, he Charaerisi Damage Sae of marix raking [14][15][16] has been reahed, - 3 represens he growh rae in he seond sage of modulus degradaion, where addiional damage mehanisms (fibre/marix inerfae failure, fibre pull-ou lead o a gradual deline of he siffness, - 4 and 5 express he explosive damage growh one ha he faigue failure index Σ(σ, D approahes is failure value 1.0. Indeed, if he faigue failure index rosses he hreshold 4, he power of he exponenial funion hanges from negaive o posiive, ausing a fas aeleraing damage growh. In his sage, fibre fraure iniiaes and quikly loalizes o ause final failure. This faigue damage model has been implemened in he ommerial finie elemen ode SAMCEF TM. The inegraion of he damage growh rae equaion for eah Gauss-poin of he finie elemen mesh has been done wih he yle jump approah whih has also been reenly proposed by he auhors [17]. Briefly he yle jump approah means ha he ompuaion is done for a erain se of loading yles a deliberaely hosen inervals, and ha he effe on he siffness degradaion of hese loading yles is exrapolaed over he orresponding inervals in an appropriae manner. To his purpose, eah Gauss-poin has been assigned beside he damage variable D a seond sae variable NJUMP1, whih is he number of yles over whih exrapolaion of he damage D is possible wihou losing reliabiliy and auray for ha pariular Gauss-poin. This loal yle jump NJUMP1 is alulaed by imposing a maximum allowed inrease in damage D for eah pariular Gauss-poin when he exrapolaion would proeed for NJUMP1 yles. When he inrease D is limied o for example 0.01, his is equivalen o a pieewise inegraion of he damage evoluion law for ha Gauss-poin by dividing he ordinae axis of he damage-yle hisory ino 100 segmens. Afer looping over all Gauss-poins, a umulaive relaive frequeny disribuion of he NJUMP1 values is alulaed and he overall yle jump NJUMP (whih will be applied o he whole finie elemen mesh is deermined as a perenile of his frequeny disribuion. By dereasing he upper hreshold for D for eah Gauss-poin, he damage evoluion law dd/dn will be inegraed more auraely, bu he global NJUMP a perenile of he umulaive frequeny disribuion of all NJUMP1 values will be smaller and he alulaion will proeed more slowly. As suh, he differenial equaion dd/dn is inegraed for eah Gauss-poin of he finie elemen mesh, bu depending on he loading ondiions, he faigue failure index Σ(σ, D an be differen for eah Gauss-poin, and hene he damage growh rae dd/dn. Afer eah yle jump, he simulaion of he faigue loading yle is resared wih alered damage and siffness disribuions in he omposie sruure. As a onsequene, sress redisribuions and relaxaions of sress onenraions due o damage developmen an be aouned for by simulaing he suessive damage saes.

7 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, The parameers i (i=1,,5 in Equaion (5 were deermined for he bending faigue es Pr05_ whih shows a lear disinion beween he hree sages in (bending siffness degradaion: sharp iniial deline, gradual (almos linear reduion, and final aeleraed deline. Sine he faigue damage model is no a all a urve-fiing model, he values of he onsans i (i=1,,5 were of ourse reained when simulaing oher loading ondiions. Figure shows he experimenal and simulaed fore-yle hisory for he Pr05_ experimen. The imposed displaemen varied beween zero (sress raio R = 0 for all Gauss-poins and u max = 30.4 mm. The fore was experimenally measured by a srain gauge bridge and represens he fore neessary o impose he bending displaemen wih onsan ampliude u max. Due o he (bending siffness degradaion, his fore dereases during faigue life Experimenal and simulaed fore-yle hisory for [#0 ] 8 speimen, single-sided bending R d = 0.0, u max = 30.4 mm Fore [N] Pr05_, experimenal fore-yle hisory Pr05_, finie elemen simulaion No. of yles [-] Figure Experimenal and simulaed fore-yle hisory for single-sided bending (u max = 30.4 mm. The model parameers were deermined wih a non-linear opimizaion proedure. The deerminaion of he parameers i (i=1,,5 an be spli up in wo pars in order o redue he opimizaion ime. Indeed, sine he damage iniiaion funion should be able o aoun for he iniial siffness degradaion, he firs sharp deline of he fore-yle hisory an be used o deermine he parameers 1 and. Then he parameers 3, 4 and 5 an be deermined for he full fore-yle hisory. The non-linear opimizaion proedure ieraively alls he finie elemen ode, and opimizes he error beween he experimenally measured fore-yle hisory and he predied one for a predeermined number of loading yles. The final values of all onsans in he model are lised in Table. Table Model onsans. 1 [1/yle] 0.00 [-] [1/yle] [-] [-] 93.0

8 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, I is imporan o noe ha he appliaion of he faigue damage model (Eq. (5 is no limied o he simulaion of bending experimens. As he faigue model preends o be a rue maerial model, i.e. inrinsi o he maerial used, i predis he siffness degradaion in eah maerial poin for he applied uni-axial nominal sress σ whih an hange during faigue life and whih migh be differen in eah maerial poin. However bending faigue experimens were preferred beause eah maerial poin hrough he hikness susains a differen sress ampliude σ (due o he bending momen, so ha he faigue damage model is esed for a wide range of ensile and ompressive sress ampliudes. Moreover, due o he sress redisribuion, he sress ampliude in he maerial poins hanges during faigue life, and a sor of variable-ampliude loading is simulaed. These advanages suppor he hoie of he bending experimens as a powerful validaion experimen for he model developed. The same argumens apply for he exended damage model for fully-reversed bending whih will be oulined below. EXTENSION OF THE MODEL FOR FULLY-REVERSED BENDING For fully-reversed bending, he displaemen raio R d = -1.0 and as a onsequene, he applied sress ampliude hanges sign during one faigue loading yle. I has been ofen repored in lieraure ha his sign reversal of he applied sress has a devasaing effe on faigue life [18][19][0][1][]. Indeed, in Figure 3 he experimenal fore-yle hisory of he singlesided bending es Pr05_ (see also Figure is ompared wih he experimenal fore-yle hisory of he fully-reversed bending es Pr08_6. For he single-sided bending es (u max = 30.4 mm, he faigue life was abou 700,000 yles. When fully-reversed bending was applied wih even a lower displaemen ampliude (u max = 7.0 mm, he faigue life was drasially redued wih a faor 10 o abou 70,000 yles. Derimenal effe of fully-reversed bending on faigue life Pr05_ single-sided u max = 30.4 mm Pr08_6 fully-reversed u max = 7.0 mm Fore [N] No. of yles [-] Figure 3 Comparison of experimenal daa from single-sided and fully-reversed bending ess. When he faigue damage model for single-sided bending (Eq. (5 is applied o his fullyreversed bending faigue experimen Pr08_6, he faigue life is onsiderably overesimaed,

9 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, beause he ineraing effes of ensile and ompressive damage have no been inluded in he growh rae equaions so far. Figure 4 illusraes he experimenal and simulaed resuls. Finie elemen simulaion of fully-reversed bending es wih single-sided faigue damage model Pr08_6, experimenal resuls fully-reversed bending Pr08_6, simulaion wih single-sided faigue model 60 Fore [N] No. of yles [-] Figure 4 Simulaion of he fully-reversed bending faigue experimen wih he faigue damage model for single-sided bending. The overesimaion an be easily explained by he fa ha in he single-sided bending model, he ompressive par of he loading yle auses negligible damage, due o he hird power of he damage iniiaion funion for ompressive sresses (Eq. (5. So only he ensile par of he loading yle auses damage, and as u max (= 7.0 mm for he fully-reversed faigue es is smaller han u max (= 30.4 mm for he experimen in Figure, he faigue life mus be longer han he faigue life of 700,000 yles whih was predied and experimenally observed for he faigue es in Figure. Therefore he faigue damage model has been exended o inlude he ineraing effe of ensile and ompressive damage. Firs he general modelling framework for reversed faigue loadings will be explained, based on he oninuum damage mehanis heory. Seondly, he modified growh rae equaions will be invesigaed in deail. General modelling framework The ommon faigue modelling approah makes use of he sress raio R = σ min /σ max o haraerize he naure of he faigue es (ension-ension, ension-ompression, ompressionompression,. In he auhors opinion, his approah is no generally suied, for several reasons: - he sress raio is only valid as a haraerizing parameer of he faigue es when uni-axial ension, ompression or ension-ompression ess are performed. In more ompliaed uniaxial loading ondiions (e.g. bending, he sress raio an be differen for eah maerial poin and due o sress redisribuion, he sress raio for one pariular maerial poin an even hange during faigue life. For muli-axial loading ondiions, he use of one single sress raio does lose is sense ompleely, beause he definiion of a minimum and maximum sress level makes no sense,

10 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, he sress raio does no ake ino aoun he underlying damage mehanisms. A faigue damage model should be based on a sound modelling of he aual damage kineis in order o orrely model he derimenal ineraion beween ensile and ompressive damage. Therefore i is proposed here ha he ension-ompression faigue loading yle for eah maerial poin should be reaed as wo separae loading yles, one in he ensile regime and one in he ompressive regime, whereby he muual ineraion beween hese wo loading yles will be deermined by he naure of he damage ha is already presen. In Figure 5 he priniple is illusraed for a pariular maerial poin ha is subjeed o a ension-ompression loading yle. Of ourse, for oher maerial poins in he sruure, he sequene migh be ompression-ension insead of ension-ompression. u u max σ (+ = T 4 d(d dn u min R d = u max = -1 0 d(d dn T = 1 f u min σ (- = 3T 4 Figure 5 Inerpreaion of he ension-ompression faigue loading yle. For eah maerial poin, he uni-axial damage variable D (furher designaed as D is disriminaed ino (damage aused by ensile sresses and (damage aused by d ompressive sresses. The sum of boh onribuions d and d sill equals he oal damage D. In Figure 5 he firs half of he loading yle is in he ensile regime and he maximum ensile sress is σ (+ ( = T/4 where T is he period of he full ension-ompression loading yle; he orresponding growh rae is d d / dn. In he seond loading yle he maximum ( ompressive sress is σ (- ( = 3T/4 and he orresponding growh rae is d(d / dn. The exa expression for hese growh rae equaions will be disussed furher. Nex, a new faigue loading yle mus be evaluaed wih alered damage sae and siffness disribuion. The yle jump approah [17] sill remains valid. From he maximum allowed inrease for D (= d + d, he loal yle jump is esimaed for eah Gauss-poin of he finie elemen mesh. The global yle jump is hen deermined for he whole finie elemen mesh as a erain perenile of he umulaive relaive frequeny disribuion of all loal yle jump values. Figure 6 illusraes he yle jump priniple applied o ension-ompression loading. d

11 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, u dd yle dd yle dd dn jump 1 dn jump dn N=1 N=3 N=6 simulaed yle exrapolaed yles d d d d d d ime N = 1 N = 3 N = 6 Figure 6 The yle jump approah for ension-ompression loading. The las problem in he exposed modelling approah for ension-ompression loading is he rak losure problem. This phenomenon was already enounered by Lemaire when he defined he effeive sress onep [3]. He observed ha he effeive sress σ ~ should be differen in ension and ompression ess on onree speimens for wo imporan reasons [4]: - sai rupure was differen in ension and ompression, - he elasiiy modulus was differen in ension and ompression. These effes are assoiaed wih rak losure. Due o he losure of miro-raks and miroaviies in ompression, he abiliy of he ross-seion o arry loads depends upon he sign of he applied sress. Lemaire suggesed o define he effeive sress in he one-dimensional ase as follows [4]: ~ σ σ = 1 D ~ σ σ = 1 hd if if σ 0 σ < 0 (6 where h is a losure oeffiien whih haraerizes he losure of he miro-raks and miro-avaies. Lemaire onluded from experimenal resuls ha h = 0. is an aepable value for several maerials. Laer on Lemaire ranslaed he onep o he hree-dimensional ase for quasi-sai loading of meals [5]. I is imporan o noe ha he damage variable D in he definiion by Lemaire is he damage aused by ensile sresses in onree. For onree, i was assumed ha he damage aused by ompressive sresses was negligible. Here he effeive sress mus be ransformed ino: ~ σ σ = 1 (d + d ~ σ σ = 1 (d + h d if σ 0 if σ < 0 (7 The value of 0. for he rak losure oeffiien has been reained for all subsequen alulaions. I is imporan o noe ha hrough his definiion, he faigue failure index Σ beomes a funion of σ, and d. d

12 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, Now ha he global framework for he modelling approah of he ension-ompression faigue loading ondiions has been explained, he modified growh rae equaions will be disussed. Growh rae equaions for ension-ompression faigue loading From Figure 3 and Figure 4, i is lear ha he addiional damage mehanisms due o fullyreversed bending, are muh more damage-driven han sress-driven, even hough no delaminaions are presen. The imposed displaemen u max is so small ha he indued sresses and srains are very small ompared o he sai srenghs in ension and ompression. The damage growh rae equaions of he originally proposed model are governed by he faigue failure index Σ(σ, D, bu when applied o he fully-reversed bending experimen, he degradaion is very small and seriously underesimaes he real damage growh. Moreover here is numerous evidene in open lieraure ha ension-ompression faigue is more derimenal han ension-ension faigue, even wihou delaminaions. Two major reasons an be disinguished: inreased debond growh and rak losure phenomena. Very reenly Gamsed and Sjögren [] have published a sound heory o explain he inreased debond growh wih ension-ompression faigue. They observed ha, when a debond is subjeed o ompressive loading, an opening zone appears a he ips of he inerfaial rak. Sine debond propagaion is more susepible o mode I loading, he debond growh rae will be larger in ompression. Indeed, for global ension, rak propagaion would be in pure shear mode beause he rak ip is losed. They furher showed ha debonding is he iniiaing mehanism o ransverse raking. One he ransverse raks have been iniiaed, rak losure phenomena play an imporan role as well. Wevers e al. [6][7] proved ha during he growh of 90 raks in [0 /90 ] s arbon epoxy omposies, maerial debris is formed beween he rak faes. When he rak is losed, his exess of maerial auses ompressive fores in he 90 plies for perpendiular raks, or sliding fores for inlined marix raks. Thus, due o rak losure, addiional damage is inrodued. The faigue damage model whih was proposed for single-sided bending, will now be exended o inlude he damaging effe of ension-ompression faigue loading. Thereo he differenial growh rae equaions for ensile damage d and ompressive damage d will be oupled. Damage iniiaion funion As repored by Gamsed and Sjögren [], here is an aeleraed iniiaion of ransverse raks under ension-ompression loading. They observed ha he oal lengh of ransverse raks grew o be more han wie as large for ension-ompression ompared o ensionension. This is onfirmed by he presen bending faigue experimens. When he derease of he bending fore during he firs,00 yles is alulaed for several bending faigue experimens, i appears ha he fore differene F is onsiderably larger for fully-reversed bending han for single-sided bending. In Table 3, some values are presened, and alhough here is some saer on he F value, i learly shows ha he damage iniiaion phase is more damaging for fully-reversed bending. Table 3 Degradaion of bending fore during firs,00 loading yles.

13 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, Single-sided bending Fully-reversed bending Speimen u max F (N=,00 Speimen u max F (N=,00 Pr08_ 39.0 mm 7.33 N Pr10_ mm 5.68 N Pr05_3 9.5 mm 5.1 N Pr05_1 7.7 mm 3.30 N Pr08_6 7.0 mm 6.1 N Pr09_5 1.8 mm 3.9 N Pr08_ mm 4.53 N Hene he model onsans 1 and in Eq. (5 should be affeed by he ransiion o ensionompression loading. However, as a sound modelling requires a onsan value of he parameers 1 and for boh single-sided and fully-reversed loading, he onsans 1 and are no modified hemselves, bu are muliplied wih a damage-dependen funion. The firs funion should inrease he parameer 1, beause he iniiaion rae for reversed loading is higher, while he seond funion should derease he parameer, beause he sauraion sae is reahed a a higher marix rak densiy. Neverheless, he inroduion of hese funions alone did no suffie o simulae he aeleraed iniiaion phase saisfaorily. From a deailed inspeion of he model equaions, i was proved ha he hird power of he damage iniiaion funion for ompressive sresses (Eq. (5 should derease o simulae he aeleraed damage iniiaion phenomenon adequaely. Indeed, when he hird power is mainained for fully-reversed bending, he ompressive damage remains very small, and he resuling bending fore degradaion remains limied, beause he dereasing load-bearing apaiy a he ensile side is ompensaed by higher sresses a he ompressive side. However, when he hird power is lowered, he damage iniiaion phase an be simulaed very well. The new equaions for he damage iniiaion phase wih ension-ompression faigue loading are: d(d dn d(d dn iniiaion iniiaion = = 1 1 ( 1+ ( d ( 1+ ( d Σ exp Σ exp Σ Σ d ( 1+ ( d d ( 1+ ( d 1+ exp( 6 d if if σ 0 σ < 0 (8 Only one new onsan 6 has been inrodued, whih regulaes he derease of he hird power and he aeleraed iniiaion of ompressive damage. I is worhwhile o noe ha if here is only one ype of damage (ompressive or ensile in eah maerial poin, he equaions redue o he ones for single-sided bending (Eq. (5. Figure 7 shows he fore-yle hisory of he Pr08_4 speimen for he firs 150,000 loading yles under fully-reversed bending, ogeher wih wo finie elemen simulaions. The firs simulaion used he oupled equaions for damage iniiaion in Eq. (8, while he seond simulaion used he unoupled erms for damage iniiaion under single-sided bending in Eq. (5. The displaemen raio R d was 1.0 and u max was 19.5 mm. The onsan 6 was opimized o be 13.0, while he oher onsans 1 and were reained (see Table.

14 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, Damage iniiaion funion for fully-reversed bending Pr08_4, experimenal resuls Pr08_4, iniiaion funion fully-reversed Pr08_4, iniiaion funion single-sided 54 Fore [N] No. of yles [-] Figure 7 Damage iniaion for fully-reversed bending wih unoupled and oupled growh rae equaions. Damage propagaion funion I appears from Figure 3 ha he slope of he seond sage of he fore-yle hisory is inreased for fully-reversed bending. This has also been observed wih oher faigue experimens: he downhill slope of he fore-yle hisory for he fully-reversed bending experimens is larger han for he single-sided bending experimens. Therefore i seems eviden ha also he seond erm in he growh rae equaions whih aouns for damage propagaion, should be inreased. For single-sided bending, damage propagaion was aeleraed fas as soon as he hreshold 4 was rossed (iniiaion of fibre fraure leading o final failure. From various simulaions, i appeared ha for fully-reversed bending, he damage propagaion is also aeleraed by he presene of he oher ype of damage ( or, bu only if a erain hreshold for he faigue failure index Σ(σ, Eq. (5 is replaed by: d(d dn d(d dn propagaion propagaion = = d 3 d 3 d, Σ Σ d d d has been rossed. Hene he damage propagaion erm in d exp exp 5 d exp exp 5 ( 8 d ( ( Σ 7 ( 8 d ( ( Σ 7 [ 1+ exp( ( Σ ] exp ( Σ if σ 0 if σ < 0 Alhough his new damage aeleraion faor may seem ompliaed, i simply expresses ha due o he muual ineraion of boh damage ypes, he damage propagaion erm is muliplied wih he faor [1+ index Σ(σ, d, d d exp( 8 d ] or [1+ d exp( 8 d (9 ], one he faigue failure has rossed he hreshold 7. The presene of his hreshold 7 is modelled

15 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, by he sigmoidal funion 1/[1+exp(- 5 (Σ- 7 ]. This funion is widely used in neural neworking algorihms, where i behaves as an aivaion funion in eah node of he neural nework opology. Here i as in he same way: one ha he faigue failure index Σ(σ, d, d has rossed he hreshold 7, damage propagaion will be aeleraed due o he presene of he oher damage ype ( d or d. Figure 8 shows he behaviour of he sigmoidal funion, he onsans 5 and 7 being 93.0 (see Table and 0.5, respeively. Sigmoidal funion 1+ exp 1 [ ( Σ ] 5 7 Sigmoidal funion [-] Faigue failure index Σ [-] Figure 8 Sigmoidal funion used o model he hreshold exisene. Alhough i seems from Eq. (9 ha, one he hreshold 7 has been rossed, he damage aeleraion faor depends no longer on he aual value of he faigue failure index Σ, his is no he ase. If he faigue failure index Σ inreases, he inreased growh raes (d / dn d and d(d / dn will onribue o a higher value of d and d, and hene o a larger value of he oupled damage aeleraion faors. Again he sysem of equaions (Eq. 9 redues o he original model for single-sided bending, when only one ype of damage is presen in eah maerial poin. Furher i is imporan o noe ha even if he faigue failure index Σ is below he damage aeleraion hreshold 7, here is sill a oupling beween he wo expressions d(d / dn and d ( d / dn for damage propagaion, beause he faigue failure index is funion of he effeive sress σ ~ and his effeive sress σ ~ aouns for boh ensile and ompressive damage (Eq. (7. Final layou of he faigue damage model The final sysem of damage growh rae equaions an be wrien as:

16 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, d(d dn d(d dn D = d = = + d ( 1+ ( d 3 3 d ( 1+ ( d d Σ Σ Σ exp d exp exp 5 Σ exp Σ d exp exp 5 ( 1+ ( d ( 8 d ( ( Σ Σ d ( 1+ ( d ( 8 d ( ( Σ d 7 7 [ 1+ exp( ( Σ ] 1+ exp( d exp ( Σ if if σ 0 σ < 0 (10 As ompared o he faigue damage model for single-sided bending, hree addiional onsans 6, 7 and 8 have been inrodued. Eah of hese onsans has again a very lose relaion wih he underlying damage mehanisms. The onsan 6 expresses he experimenal observaion ha damage iniiaion in he ompressive regime is aeleraed by he presene of ensile damage, as repored by Gamsed and Sjögren []. The onsans 7 and 8 are inrodued in he damage aeleraion faor, whih simulaes ha one he faigue failure index Σ has rossed he hreshold 7, he damage propagaion is aeleraed beause of he presene of he oher damage ype ( d or d. Of ourse, he onsans i (i=1,,5 keep heir values lised in Table. FINITE ELEMENT SIMULATIONS The hree onsans 6, 7 and 8 have been deermined for he fully-reversed bending faigue es Pr08_4. The imposed displaemen ampliude u max was 19.5 mm and he displaemen raio R d was equal o 1.0 (fully-reversed bending. In Figure 9, he experimenal and simulaed resuls are shown. The onsans 6, 7 and 8 were deermined o be respeively: 13.0 [-], 0.5 [-] and 9.7 [-]. The onsans i (i = 1,,8 were of ourse reained for all subsequen simulaions.

17 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, Fully-reversed bending of [#0 ] 8 speimen, u max = 19.5 mm 60 Pr08_4, experimen Pr08_4, FEM resuls Fore [N] No. of yles [-] Figure 9 Experimenal and simulaed resuls for a [#0 ] 8 speimen wih u max = 19.5 mm and R d = I is worhwhile o noe ha here is indeed a hreshold for damage propagaion, beause he faigue life now exeeds 1,00,000 yles for u max = 19.5 mm, while i was redued o abou 70,000 yles for he faigue es Pr08_6 wih u max = 7.0 mm (see Figure 3. Figure 10 shows he experimenal and simulaed fore-yle hisory for his Pr08_6 speimen. The faigue damage model simulaes he observed experimenal behaviour saisfaorily, he more so as he maerial onsans i (i = 6,7,8 were opimized for a faigue life of more han 1,00,000 yles. Fully-reversed bending of [#0 ] 8 speimen, u max = 7.0 mm Pr08_6, experimen Pr08_6, FEM resuls 60 Fore [N] No. of yles [-] Figure 10 Experimenal and simulaed resuls for a [#0 ] 8 speimen wih u max = 7.0 mm and R d = I was saed earlier ha he sress raio R (= σ min /σ max is no a suiable parameer o desribe he faigue loading ondiions in he ension-ompression regime. This an be learly demonsraed by he simulaed resuls of his las faigue es. The Gauss-poin 160 was

18 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, hosen as he poin of ineres as i is lying a he speimen surfae in he lamped rossseion. Figure shemaially shows is posiion in he finie elemen mesh. Clamped ross-seion GP mm Imposed displaemen mm Figure Posiion of he Gauss-poin of ineres in he lamped ross-seion. In Figure 1 he ime-hisory of he nominal sress σ for his pariular Gauss-poin is shown a several sages of faigue life. The faigue es frequeny was menioned o be. Hz, so one faigue yle orresponds o 0.45 seonds. A he firs loading yle, he Gauss-poin is loaded wih a ensile sress during he firs half of he period and wih an equal ompressive sress during he seond half of he period. The sress raio R is indeed 1.0. However when damage is iniiaing, he sress magniudes diverge for ension and ompression, beause he damage growh rae expressions are differen and he ompression-loaded speimen side is behaving siffer due o he rak losure phenomenon (see Equaion (7. So he sress raio R is oninuously hanging during faigue life, as is he general ase in real sruures. A yle 33,918, he sress raio R for his pariular Gauss-poin has beome 1.7 (= /79., while he sress raio R was 1.0 for he firs loading yle. Normal sress σ [MPa] Sress-ime hisory during a single loading yle in he Gauss-poin 160 yle 1 yle 714 yle 18,918 yle 8,918 yle 33, Time [s]

19 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, Figure 1 Sress-ime hisory during one single faigue loading yle for a pariular Gauss-poin. Finally, Figure 13 shows for he same Gauss-poin 160 he evoluion of he oal damage D, he ensile damage d and he ompressive damage d Evoluion of oal damage D, ensile damage d and ompressive damage d oal damage ensile damage ompressive damage D, d, d [-] No. of yles [-] Figure 13 Evoluion of oal, ensile and ompressive damage in a pariular Gauss-poin. I is worhwhile o noe ha alhough he Gauss-poin 160 is predied o fail a yle 38,918 (and hus also he Gauss-poin a he reverse surfae of he speimen, he bending fore is mainained for 17,000 yles more a a relaively high level. This is of ourse due o he sress redisribuion in he ross-seions near he lamped end of he omposie speimen, as illusraed by Figure 14. Sress redisribuion a he lamped ross-seion yle 1 yle 18,918 yle 43,918 yle 5,16 yle 55,933 heigh y [mm] sress [MPa] Figure 14 Sress redisribuion a he lamped ross-seion during faigue life.

20 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, The las simulaion is for he speimen Pr09_5. This speimen was subjeed o a slighly larger displaemen ampliude u max = 1.8 mm han he Pr08_4 experimen (see Figure 9. In Figure 15, he experimenal and simulaed resuls are shown. Fully-reversed bending of [#0 ] 8 speimen, u max = 1.8 mm 60 Pr09_5, experimen Pr09_5, FEM resuls 50 Fore [N] No. of yles [-] Figure 15 Experimenal and simulaed resuls for a [#0 ] 8 speimen wih u max = 1.8 mm and R d = I appears ha he degradaion is overpredied by he faigue damage model. This is no surprising, beause i ould already be seen from Table 3 ha he Pr09_5 speimen shows a smaller iniial deline of he bending fore han he oher speimens. Afer he iniial deline, he experimenal fore-yle hisory remains almos horizonal, while here was a signifian slope in he fore-yle hisory of he Pr08_4 speimen (see Figure 9 alhough he laer was subjeed o a smaller displaemen ampliude u max. This proves ha he value of he hreshold for damage propagaion is subje o some experimenal saer, bu his is an inrinsi propery of faigue experimens. CONCLUSIONS A faigue damage model for simulaion of ension-ompression faigue loadings has been proposed. The innovaive aspes of he model are: (i hrough he modified use of he sai Tsai-Wu failure rierion, here is a orrelaion beween residual siffness and srengh. Neverheless he omplee framework of he model is onsisen wih he oninuum damage mehanis heory; (ii he sress raio R as a haraerizaion of ension-ompression faigue ess has been replaed by a oupled sysem of growh rae equaions for ensile and ompressive damage. This approah is far more general in naure han he sress raio R, as i an keep rak of sress redisribuions during faigue life (whih implies varying sress raios; (iii he model has been implemened in a ommerial finie elemen ode and hene allows for simulaion of he faigue behaviour of more omplex omposie omponens. The model has been validaed by finie elemen simulaions of displaemen-onrolled bending faigue experimens on plain woven glass/epoxy speimens. The yle-hisory of he bending fore, damage variables and sress omponens ould be simulaed saisfaorily.

21 Van Paepegem, W. and Degriek, J. (00. Tensile and Compressive Damage Coupling for Fully-reversed Bending Faigue of Fibrereinfored Composies. Faigue and Fraure of Engineering Maerials & Sruures, 5(6, ACKNOWLEDGEMENTS The auhor W. Van Paepegem graefully aknowledges his finane hrough a gran of he Fund for Sienifi Researh Flanders (F.W.O., and he advie and ehnial suppor of he SAMTECH ompany. The auhors also express heir graiude o Synoglas for heir suppor and ehnial ollaboraion. REFERENCES [1] Carme, A., Weber, B. and Rober, J.L. (000. Faigue Life Assessmen of Componens and Sruures Under Muliaxial Servie Loading. In : Bahe, M.R. e al. (eds.. Faigue 000 : Faigue & Durabiliy assessmen of maerials, omponens and sruures. Proeedings. Cambridge, 10-1 April 000, Chameleon Press Ld., pp [] Degriek, J. and Van Paepegem, W. (001. Faigue Damage Modelling of Fibre-Reinfored Composie Maerials: Review. Applied Mehanis Reviews, 54(4, [3] Van Paepegem, W. and Degriek, J. (001. Experimenal seup for and numerial modelling of bending faigue experimens on plain woven glass/epoxy omposies. Composie Sruures, 51(1, 1-8. [4] Van Paepegem, W. and Degriek, J. (000. Numerial modelling of faigue degradaion of fibrereinfored omposie maerials. In: Topping, B.H.V. (ed.. Proeedings of he Fifh Inernaional Conferene on Compuaional Sruures Tehnology. Volume F: Compuaional Tehniques for Maerials, Composies and Composie Sruures, Leuven, 6-8 Sepember 000, Civil-Comp Press, pp [5] Salkind, M.J. (197. Faigue of omposies. In: Coren, H.T. (ed.. Composie Maerials Tesing and Design (Seond Conferene. ASTM STP 497. Balimore, Amerian Soiey for Tesing and Maerials, pp [6] Hahn, H.T. and Kim, R.Y. (1976. Faigue behaviour of omposie laminaes. Journal of Composie Maerials, 10, [7] O'Brien, T.K. and Reifsnider, K.L. (1981. Faigue damage evaluaion hrough siffness measuremens in boron-epoxy laminaes. Journal of Composie Maerials, 15, [8] Hwang, W. and Han, K.S. (1986. Cumulaive damage models and muli-sress faigue life prediion. Journal of Composie Maerials, 0, [9] Sol, H. and de Wilde, W.P. (1988. Idenifiaion of elasi properies of omposie maerials using resonan frequenies. In : Brebbia, C.A., de Wilde, W.P. and Blain, W.R. (eds.. Proeedings of he Inernaional Conferene "Compuer Aided Design in Composie Maerial Tehnology", Souhampon, 1988, Springer-Verlag, pp [10] Sol, H. (1990. Idenifiaion of he omplex moduli of omposie maerials by a mixed numerial/experimenal mehod. In : de Wilde, W.P. and Blain, W.R. (eds.. Proeedings of he seond Inernaional Conferene on Compuer Aided Design in Composie Maerial Tehnology, Brussels, 5-7 April 1990, Springer-Verlag, pp [] Van Paepegem, W. and Degriek, J. (001. New oupled oupled approah of residual siffness and srengh for faigue of fibre-reinfored omposies. Eigh Inernaional Conferene on Composies Engineering (ICCE/8. Proeedings. Tenerife, 5- Augus 001. [1] Van Paepegem, W. and Degriek, J. (001. A New Coupled Approah of Residual Siffness and Srengh for Faigue of Fibre-reinfored Composies. Aeped for publiaion in Inernaional Journal of Faigue.