Stress Redistribution in Skin/Flange Assemblies

Transcription

1 Mechanics of Advanced Maeials and Sucues, 9: , 2002 Copyigh C 2002 Taylo & Fancis /02 $ DOI: / Sess Redisibuion in Skin/Flange Assemblies KAREL MATOUÏS GEORGE J. DVORAK Depamen of Mechanical, Aeospace & Nuclea Engineeing, Rensselae Polyechnic Insiue, Toy, New Yok, USA ABSTRACT A pesessing pocedue is poposed fo educion of sess concenaions a he leading edges of adhesive bondlines in composie skin/ ange assemblies. A nieelemenanalysisof a speci c geomeyof such an assemblyis pesened,whichaccouns fo nonlinea viscoelasic defomaion of he adhesive. Design diagams based on elasic analysis ae consuced fo evaluaion of pesess foces ha educe o compleely eliminae adhesive sess concenaions caused by eihe ension sesses o bending applied o he skin. Many composie sucues conain skin/ ange assemblies ha povide aachmen and suppo fo singes and ohe pas of he sucue. Adhesive joins which ae ypically used in such applicaions ae exposed o lage sess concenaions a he leading edges of he bondline and he adjacen apeed end of he laminaed ange. Join failues ofen oiginae a hese locaions, as shown in numeous expeimens by Kevin O Bien s goup a NASA-Langley [1, 2]. The pesen wok shows ha he sess concenaionsconibuingo such join failues can be educed o eliminaed alogeheby ceain pesess foces applied o he skin/ ange assembly pio o and hen emoved afe adhesive cue. Pesess applicaion can be accomplished by simple xues, as descibed in he s secion. Since a sessed adhesive can undego nonlinea viscoelasicdefomaion ove ime, we hen discuss he elevan consiuive elaions fo he adhesive, and deails of he nie-elemen analysis [3]. As expeced,he nonlinea viscoelasic defomaion is shown o educe he adhesive sess concenaions. Nex, a speci c skin/ ange assembly is seleced and subjeced o pesess followed by adhesive cue, pesess elease, and subsequen mechanical loading by ension and bending applied o he skin. Design diagams based on elasic analysis ae consuced fo evaluaion of pesess foces ha educe o compleely cancel adhesive sess componens induced a he leading edges of he bondline by seleced skin loads. PRESTRESSED SKIN/FLANGE ASSEMBLY Figue 1 shows a secion of uni widh b of a ypical skin/ ange geomey consideed heein. Exending ogehe wih he skin in he x 2 diecion, he ange seves o aach a The auhos appeciae nancial suppo of his wok by he Ship Sucues and Sysems S&T Division of he Of ce of Naval Reseach. D. Yapa D. S. Rajapakse seved as pogam monio. The auhos would also like o acknowledge he CRC Pess LLC, fo pemission o epoduce confeence maeial. Addess coespondenceo Geoge J. Dvoak, 5012 JEC Rensselae Polyechnic Insiue, Toy, NY , USA. dvoak@pi.edu 257

2 258 K. MaouÏs and G. J. Dvoak Figue 1. Geomey of he skin/ ange assembly. fame o a siffene o he skin, say, of a bonded fuselage panel. Complex in-plane and ouof-plane loadingof he panel is usually pesen in sevice; howeve, he load componensha conibuemos o local failue of he adhesive bond a ange ends ae in-planeension sess F 1 D F 1=bh S, pependiculao he leading edge of he ange, and ansvese bendingsess M 1 D 6M 1=b(h S ) 2, applied o he skin ouside he ange pope. These loads geneae shea and peel sess concenaions a he ends (poins Q) of he adhesive laye, making hem pefeed sies of adhesive failue o delaminaion a ply ends in a composie ange. Ou objecive is o popose and analyze a conolled pesessing sequence applied duing adhesive bonding of he ange, which inoduces a favoable esidual sess disibuion a boh ends of he bondline, and hus should enhance he load beaing capaciy of he join. Figue 2 oulines he poposed pocedue. Immediaely afe applicaion of he liquid adhesive a he bondline, he skin is loaded by a line foce P applied a poin S, while he ange is suppoed by a unifom compessive sess p a he apeed ends. In pacice, his may be accomplished by suppoing he ange ends by sealed liquid- lled pads. A small bending de ecion in he posiive diecion of he x 3 axis is impaed o he assembly, and is mainained while he adhesive cues. Afe cue, all loads ae emoved. A ceain pa of he Figue 2. Pesessing pocedue.

3 Sess Redisibuion in Skin/Flange Assemblies 259 bending de ecion is ecoveed, bu anohe pa is eained by he esidual shea and peel sesses in he adhesive. I is pobably obvious ha he esidual peel sess is compessive and he esidual shea sess has an opposie sign o ha geneaed by he ensile foce F 1 and/o momen M 1. The analysis ha follows is concened wih evaluaion of he adhesive sesses applied by he foce F 1 and momen M 1, and of he adhesive esidual sesses caused by elease of he pesess foce P. Resuls of he elasic soluion ae combined in design diagams which allow selecion of he pesess foce such ha he supeimposed applied and esidual sesses in he adhesive each a seleced minimum. Moeove, he effec of nonlinealy viscoelasic defomaion of he adhesive on he local sess disibuion is illusaed by seveal examples. ANALYSIS The elaively complex geomey pecludes eliable analyical evaluaion of he sesses and defomaions in he join skin/ ange assembly. Theefoe, a wo-dimensional nieelemen soluion has been used in he sess and de ecion evaluaions using he Fem2D code, by MaouÏs [4]. A mesh consising of hee node elemens based on linea shape funcions was geneaed using he T 3D geneao developed by Rypl [5]. A oal of 8,737 nodes and 16,908 iangles wee used in disceizaion of he domain. Six ows of elemens wee used o disceize he adhesive laye. Inceased e nemen o 100 ows of elemens was used a he leading edges, o appoximae he possibly singula sesses. Sess oscillaions associaed wih he singulaiies wee deeced in vey small volumes of he adhesive and adheends. These feaues depend on he ne-scale deails of he local geomey ha canno be eliably epoduced in an acual sucue. Theefoe, hickness aveages of he individual componens wee adoped in evaluaion of he local sesses. Boh adheends wee egaded as homogeneous, ohoopic elasic solids, wih elasic consans equal o hose of a woven composie laminae. A moe e ned analysis of a layeed laminae would be needed in a speci c applicaion of he pesess echnique. In developmen of he design diagams ha involved supeposiion of he esidual and applied sess elds, he adhesive was assumed o be isoopic and elasic. Moeove, a nonlinealy viscoelasic adhesive was used in examples ha illusae pesess and sess elaxaion. The viscoelasic model adoped fo he isoopic adhesive elies on he consiuive equaion suggesed by Schapey [6, 7], Z D g0 q D 0 C g1 q J (w w 0 s ) d d s g q s 2 s d s 2 R C (1) whee he supescips ; ; q efe o he value a cuen sess, ime 2 R C and empeaue q, especively. The symbol D 0 denoes he elasic compliance enso, J D D w c D is he ceep compliance enso, D w denoes a ansien ceep compliance funcion, and c m m m D D (1 C m ) (1 C m ) 0 5 sym: 2(1 C m ) (2)

4 w 260 K. MaouÏs and G. J. Dvoak Poisson s aio m can be geneally ime-dependen. Hence he pesen fomulaion is applicable o any hemoheologically simple isoopic viscoelasic adhesive. The g0 q ; g1 q ; g2 q denoe nonlinea kenel funcions of sess and empeaue q in he ime. The funcion w is de ned by D Z 0 ds a q s (3) whee a q s is a ime-shif faco depending on he sess and empeaue q, especively. The ime-dependen sain in (1) hus depends on one funcion of ime and fou funcions of sess and empeaue. Speci c evaluaion of hese funcions appeas in he Appendix, fo he FM-73 adhesive chaaceized by Peez and Weisman [8]. As suggesed by Heniksen [9] and Reddy and Roy [10, ], he ansien ceep funcion can be expessed by he Pony seies, D w c D NX D 1 e k w D1 k D 1 s in which s ae consan eadaion imes and N denoes numbe of Pony coef ciens 2 h1; Ni. The values of D, which descibe he sain incemen duing he ime peiod s, can be deemined by ing of he measued ceep cuves. The s can be seleced subjec o ceain well-known esicions. Numeical sudies con med ha he eadaion imes can be chosen as s D 10 2 s 2 fo D 2; : : : ; N, whee s 2 and N have o cove he ime ange of inees, and s 1 s 2, e.g., s 1 ¼ 10 5 s 2. The eadaion specum ploed as 1=D vesus s ( D 1; 2; : : : ; N ) fully chaaceize he maeial ceep popeies [12, 13]. Equaion (1) can be expessed in a sess opeao fom, (4) D F( ) (5) whee F( ) D J C E (6) and he insananeous compliance enso J eads J D D I D D I D g X 0 q E 1 C g1 q g2 q D 1 0 (7) E is Young s modulus and 0 is he elaxaion coef cien in he ceep compliance seies, whee 0 D 1 e k 1w (8) k 1w 1w D Z 1 ds (9) a q s The heediay sain componen E is de ned by X E D Dg q q ( 1 ) 1 D g e k 1w q 1 (10)

5 \ \ Sess Redisibuion in Skin/Flange Assemblies 261 and he h componen, q 1, of he heediay inegal seies a he end of he pevious load sep is q 1 D Z 1 e k (w 1 w 0 s ) d ds g q s 2 s ds () The heediay inegal a he end of he cuen sep can be deived fom he ecuence fomula, q D e k 1w q 1 C g q 2 g q ( 1 ) (12) whee 0 is de ned by Eq. (8). Once he sain veco (10) is known, he consiuiveelaion can be wien as D L ( E ) L D (J ) 1 (13) whee L is he insananeous siffness enso. In he nie-elemenimplemenaionof he aboveheoy,hepoenialenegy funcional is based on (13), 5 D 1 2 Z ( E ) T L (e Z E ) d\ \ Z u T Xd\ u T p d0 (14) 0 Afe execuing he paial deivaive of 5 wih espec o he veco u, and using sandad isopaameic nie-elemen mapping, one obains a sysem of algebaic equaions, K d D R C R E (15) whee K is he symmeic, posiive-de nie global siffness maix afe assembly and imposing he essenial bounday condiions. The d denoes he global veco of fee nodal paamees, and R is he veco of nodalfoces due o exenallyappliedmechanicalloading and hemal effecs. The load veco R E associaed wih he viscoelasicsain E is given by R E D Z B T L E d\ (16) whee B T denoes sain displacemen ansfomaion opeao. Noe ha in he nonlinealy viscoelasic case, he K conains embedded nonlinea maeial kenel funcions g0 q ; g1 q ; g2 q in he maix L. A modi ed Newon-Raphson ieaion echniquewas used o solve (15); he incemenal displacemen 1d obained a he end of he h ieaion is used o updae he local displacemen fo he nh ime sep, d n D dn 1 C 1dn (17) The ieaion coninues unil convegence is achieved a each ime sep. The pesess iniial condiion d 0 D d pesess was used in he s ime sep.

6 262 K. MaouÏs and G. J. Dvoak Figue 3. The skin/ ange con guaion seleced fo sess analysis. EXAMPLES 1. Adhesive sesses afe pesessing As an illusaion of he advanages of he pesessing concep, we analyzed he con- guaion shown in Figue 3, subjeced o he loading sequence of Figue 2. Boh skin and ange wee egaded as homogeneous, ohoopic elasic solids, and he adhesive as a homogeneous isoopic solid wih eihe elasic o nonlinealy viscoelasic esponse. Maeial selecion was moivaed by availabiliy of maeial popey daa fo he adhesive and adheends, Table 1. The viscoelasic popeies of FM-73 adhesive ae lised in he Appendix. No claim is made ha he paicula adhesive is suiable; in ou elaed wok [14], we found ha he Dexe-Hysol; 9339 adhesive povided a song bond of he adheend maeials. Howeve, no ime-dependen popey daa wee found fo his paicula adhesive. Shown s ae sess disibuions in he elasic adhesive, which should be good appoximaions of sho-ime esponse. The speci c applied load magniudes wee seleced wih egad o he esimaed allowable sesses in he adheends. Figue 4 descibes he adhesive sess componens befoe and afe pesess elease, Figue 2, in he absence of any exenal loads. In he s pa of he pesessing sequence shown in he op half of Figue 2, he adhesive is liquid and hus capable of ansmiing only compessive nomal sess beween skin and ange. This 33 componen, shown in Figue 4, has local maxima a ange ends and a he cene secion, and sess is suppoed by he adhesive a he end of adhesive cue. Addiional sesses ha may be caused by he cuing pocess wee no consideed in he absence of elevan maeial daa. Pesess elease is epesened by he loads in he boom dawing of Figue 2, which ae now applied o he bonded sucue. Signi can sess changes ae now obseved in he cued adhesive laye, Figue 4. In addiion o a diffeen peel sess 33, he adhesive now suppos a shea sess 31 and a longiudinal nomal sess. The peel sess emains compessive, and he shea sess sign is opposie o ha caused by he subsequen ension applied o he skin. The has vey small effec on join failue. Table 1 Elasic moduli of he adheends and adhesive [GPa] Maeial E 1 E 3 G 31 m Woven E-glass/vinyl ese FM-73 adhesive

7 Sess Redisibuion in Skin/Flange Assemblies 263 Figue 4. Adhesive sesses befoe and afe pesess elease, x D 0.36 NM=m. 2. Adhesive sesses due o pesess and skin ension Figue 5 shows he adhesive sesses caused by skin ension sess D 90 MPa, applied in he absence of pesess. As expeced in his adhesive join con guaion, hee ae lage ensile peel sess and shea sess concenaionsa he leading edge of he ange. Numeical values ae lised in he second column of Table 2. Supeposiion of he adhesive sesses due o pesess and skin ension is pesened in Figue 6; he maxima ae lised in he hid elasic column of Table 2. These esuls F 1 Figue 5. Sesses induced in he adhesive by ension sess ¾ F 1 D 90 MPa.

8 264 K. MaouÏs and G. J. Dvoak Table 2 Adhesive sess maxima caused by diffeen ensile loading hisoies [MPa] Tension wihou pesess Tension afe pesess Sess Elasic Elasic Nonlinea 24-h Nonlinea 48-h clealy demonsae he vey lage educion of he adhesive peel sess in he pesessed join, in he elasic case by a faco of 5:82. The shea sess maximum is also educed, and so ae boh componens by he nonlinea viscoelasic defomaion afe 24 o 48 h, Table 2. The vey small peel sess value a he leading edge of he bondline also implies educed likelihood of ply delaminaion in he apeed end of he ange, and also in he skin. The sesses obainedin he elasic analysiscan be scaled and used o consuc a design diagam fo selecion of he pesess foce needed o couneac he effec of he applied skin ension sess, Figue 7. The maxima of he adhesive sess componens ae ploed as linea F funcions of he ansvese ension sess 1 in he op half of he Figue 7. The boom half conains plos of he local sess maxima as funcions of he pesess foce P applied pio F and duing adhesive cue. Fo example, fo 1 D 30 MPa, we daw he hoizonal line A 1! A 2! A 3 o nd he especive sess componenmaxima in he skin/ ange sucue wihou pesess. If zeo peel sess is desied unde he pescibed ansvese ension in he pesessed sucue, we daw he veical line A 3 # B 3, o inecep wih he 33 line in he lowe half of he gue. Poceeding now hoizonallyalong B 3! B 2! B 1, we nd Figue 6. Sesses in he adhesive afe supeposiion of pesess and skin ension conibuions fom Figues 4 and 5.

9 Sess Redisibuion in Skin/Flange Assemblies 265 Figue 7. Pesess design diagam fo he skin/ ange assembly subjeced o skin ension. he equied pesess foce P D 0:15 MN/m. The maximum shea sess is obained by subacing he values of 31 a poin C 2L fom ha a C 2P. In he pesen case, 31 (C 2L ) 31 (C 2P ) D 4:4 MPa. Nex, o illusae he effec of nonlinea viscoelasic defomaion in he adhesive, we seleced he loading hisoy shown in Figue 8. The join of Figue 3 was pesessed by he same foce P D 0:36 MN/m, and his foce was emoved afe adhesive cue. A 24-h F elaxaion peiod was allowed befoe applicaion of he skin ension sess 1 D 90 MPa, which was hen kep a a consan level fo he nex 24 h. The sess changes occuing in he adhesive duing hese wo 24 h inevals wee monioed, as shown fo he 33 componen in Figue 9. The sess elaxaion aes end o zeo befoe he end of each ineval. Figues display he sess disibuions found in he adhesive laye afe hee speci ed ime inevals. These epesen changes in he iniial elasic disibuions of Figue 6; he sess maxima ae also lised fo easy compaison in Table 2. As expeced, all sesses ae educed. The lages diffeence, fom 33 D 4:05 MPa o 33 D 1:61 MPa is found in he peel sess, bu his is a addiionaleducion fom he elasic value of 23:59 MPa in he absence of pesess. Figue 8. The pesess and skin ension loading hisoy applied in nonlinea viscoelasic analysis of he adhesive.

10 266 K. MaouÏs and G. J. Dvoak Figue 9. Time disibuionof sess ¾ 33 a diffeensages of he loadinghisoy of Figue 8. Figue 13 pesens he veical de ecion of he sucue along he lengh of he ange. F If zeo de ecionsae imposed a he fa ends of he skin, he ension sess of 1 D 90 MPa causes he midsecion o ise by 1:3 mm. The pesess foce causes a downwad de ecion of 6:0 mm, which is educed o abou 4:4 mm by pesess elease and fuhe o 3:0 mm afe he 48-h loading sequence shown in Figue 8. These illusaive values sugges ha he esidual de ecion is of he same ode of magniude as ha imposed by he ansvese ension sess. Howeve, he diffeence in sign appeas o be impove sucual inegiy, as i pomoes compession in he bondline beween he skin and he fame o singe ha may be aached o he ange. Figue 10. Adhesive peel sess ¾ 33 a diffeen sages of he loading hisoy of Figue 8.

11 Sess Redisibuion in Skin/Flange Assemblies 267 Figue. Adhesive shea sess ¾ 31 a diffeen sages of he loading hisoy of Figue Adhesive sesses due o pesess and skin bending In addiion o he ension sess, he skin may also be subjeced o bending momens, as shown in Figue 1. The momen sign may change o oscillae, bu of inees hee is he momen diecion ha exposes he boom laye of he skin and he ange o ensile sess 1 M. The opposiemomen diecion would acually eun he sucue o a con guaion simila o ha unde pesess, wih compessive peel sess and favoable shea sess disibuion in he adhesive. Since we have aleady esablishedhe bene cial effec of viscoelasicdefomaion of he adhesive, we pesen only he elasic sess disibuionsand a design diagam. Figue 12. Adhesive nomal sess ¾ a diffeen sages of he loading hisoy of Figue 8.

12 268 K. MaouÏs and G. J. Dvoak Figue 13. De ecion of he skin unde skin ension and afe he loading sequence of Figue 8. The saing poin hee is again he sess disibuion in he adhesive afe pesess elease, Figue 4. The effec of he bending momen on adhesive sesses was evaluaed in he bonded skin/ ange assembly, unde M D 1: MNm/m, which caused M he ensile sess 1 D 40 MPa in he fee skin. Defomaion of he boom skin laye, indicaed by he ensile sess ahe han he momen value iself, should have a decisive effec on he adhesive sesses. These sesses ae shown in Figue 14. The maximum peel Figue 14. Sesses induced in he adhesive by bending sess ¾ M 1 D 40 MPa applied o he skin.

13 Sess Redisibuion in Skin/Flange Assemblies 269 Figue 15. Sesses in he adhesive afe supeposiion of pesess and bending momen conibuions fom Figues 4 and 14. sess value is simila o ha caused by he skin ension, bu he shea and nomal sesses exhibi lowe maxima. Supeposiion of he sesses in Figue 14 wih hose caused by he oiginal pesess in Figue 4 is pesened in Figue 15. Sess maxima ae compaed in Table 3. A design diagam indicaing he desied pesess fo a seleced maximum value of he bending-induced ensile sess is consuced in Figue 16. The design sequence sas M again a poin A 1, which coesponds o he seleced bending sess 1. The design pah is hen analogous o ha followed in he design diagam fo ension load, Figue 7. Since boh design diagams in Figues 7 and 16 ae based on elasic sess analysis, hey can be used joinly in nding he pesess needed o couneac he effec of supeimposed ension and bending. Such pesess foce could be found by simply adding he foces found o educe he expeced maxima of he wo loading componens. If he assembly wee loaded by skin compession and ension-inducing bending, o vice vesa, he design diagams could be easily adoped fo loading by evese bending and/o skin compession. In such an applicaion, he op sess banches in hese gues could be exended as saigh lines ino he negaive ension o bending sess egions, and he boom sess banches ino he negaive pesess egions, implying pesessing in a diecion opposie o ha shown Table 3 Adhesive sess maxima caused by diffeen bending loading hisoies [MPa] Bending wihou pesess Bending afe pesess Sess Elasic Elasic

14 270 K. MaouÏs and G. J. Dvoak Figue 16. Pesess design diagam fo he skin/ ange assembly subjeced o he bending. in Figue 2. The pesess magniude needed fo he combined loading can hen be found by supeimposing he values found fo each of he wo loading componens. Moeove, since he op pas of he diagams show he sess maxima caused by he especive loading componens, hey could also be used o deemine, fo example, how much skin compession is needed o couneac he effec of bending, o vice vesa, wihou he bene of pesess. CONCLUSION The esuls sugges a elaively simple mehod of adhesive sess educion in a skin/ ange assembly loaded eihe by skin ension and/o bending,acing ansvese o he longiudinal axis of he ange. While ceain special xues would be equied fo pesessing, he expeced enhancemen of load-beaing capaciy and/o enduance may well be woh he exa cos. The pesen numeical sudy should be also con med by expeimenal sain measuemen echniques fo adhesive bonds, such as Moié inefeomey. Possible viscoelasic defomaion of he adhesive ends o educe ove ime he sess maxima a he leading edges of he ange bondline. Theefoe, design diagams based on elasic sess analysis should suf ce, and lead o consevaive designs in mos applicaions. This would also obviae he need fo a deailed evaluaionof he maeial paamees equied in he nonlinea adhesive analysis. Since he adhesive sess disibuions depend on boh adhesive and adheend elasic moduli and deails of he join geomey, consucion of he design diagams should be based on a nie-elemen evaluaion of he sess disibuions. Scaling of soluions obained fo a single load magniude is pemissible in he elasic case. In an acual composie sucue, boh ange and skin ae made of a laminae consising of seveal bous layes. Layup deails may in uence he adhesive sesses a he leading edge of he bondline, and also he inelamina sesses a he apeed fee edges of he laminaed ange. Indeed, failue of he join ofen oiginaes in he apeed ange end, and exends along ply inefaces befoe eaching he adhesive laye [1, 2]. Since he laminaes wee homogenized in ou analysis, he esuls do no e ec ha level of deail. Howeve, inasmuch as he goal was o minimize he sess concenaionsa he bondline leading edge by supeposiion of he pesess and applied loading sess disibuions, he diffeences

15 Sess Redisibuion in Skin/Flange Assemblies 271 beween he layeed and homogenized soluionsshould no have a lage effec on he loading combinaions found o geneae he minimized sess disibuions. REFERENCES [1] J. Li, T. K. O Bien, and C. Q. Rousseau, J. Am. Helicope Soc., vol. 42, no. 4, pp , [2] M. K. Cvikovich, T. K. O Bien, and P. J. Mingue, Faigue Debonding Chaaceizaion in Composie Skin/Singe Con guaions, in R. B. Bucinell (ed.), Composie Maeials: Faigue and Facues (ASTM STP 1330), vol. 7, pp , [3] T. J. R. Hughes, The Finie Elemen Mehod: Linea Saic and Dynamic Finie Elemen Analysis, Penice-Hall, Englewood Cliffe, NJ, [4] K. MaouÏs, Analysis and Opimizaion of Composie Maeials and Sucues, Ph.D. hesis, CTU Repos, vol. 4, no. 3/2000, Pague, Czech Republic, [5] D. Rypl, Sequenial and Paallel Geneaion of Unsucued 3D Meshes, Ph.D. hesis, CTU Repos, vol. 2, no. 3/1998, Pague, Czech Republic, [6] R. A. Schapey, A Mehod of Viscoelasic Sess Analysis Using Elasic Soluions, J. Fanklin Ins., vol. 279, pp , [7] R. A. Schapey, On he Chaaceizaionof Non-linea Viscoelasic Maeials, Polym. Eng. Sci., vol. 9, no. 4, pp , [8] D. Peez and Y. Weisman, Nonlinea Viscoelasic Chaaceizaion of FM-73 Adhesive, J. Rheology, vol. 26, no. 3, pp , [9] M. Heniksen, Nonlinea Viscoelasic Sess Analysis A Finie Elemen Appoach, Compu. Suc., vol. 18, no. 1, pp , [10] S. Roy and J. N. Reddy, Finie-Elemen Models of Viscoelasiciy and Diffusion in Adhesively Bonded Joins, In. J. Num. Meh. Eng., vol. 26, pp , [] J. N. Reddy and S. Roy, Finie-Elemen Analysis of Adhesive Joins, in L. H. Lee (ed.), Adhesive Bonding, pp , Plenum Pess, New Yok, [12] Z. P. BaÏzan and S. Pasannan, Solidi caion Theoy fo Concee Ceep. II: Vei caion and Applicaion, J. Eng. Mech., vol. 5, no. 8, pp , [13] Z. P. BaÏzan and Y. Xi, Coninuous Readaion Specum fo Solidi caion Theoy of Concee Ceep, J. Eng. Mech., vol. 121, no. 2, pp , [14] G. J. Dvoak, J. Zhang, and O. Canyu, Adhesive Tongueand GooveJoins fo Thick Composie Laminaes, Composies Sci. Technol., vol. 61, pp , APPENDIX Nonlinea viscoelasic chaaceizaion of he FM-73 adhesive Hee we descibe viscoelasic maeial popeies of he FM-73 adhesive obained by Peez and Weisman [8]. The viscoelasic behavio of his adhesive can be compleely chaaceized by he compliance funcion D w c shown in Figue 17, epesening he sain e of adhesive a ime. Poisson s aio is assumed consan fo his adhesive. The nonlinea effecs fo he consan empeaue q D 303 K ae expessed by means of sess-dependen maeial kenel funcions, g q 0 D 1 C 0:2 (18) g q 1 D 1 C 1:435 g2 q D 1 C 0:75 ul ul 2:4 (19) ul 2:0 (20)

16 272 K. MaouÏs and G. J. Dvoak Figue 17. The ansien ceep compliance D Ã c. whee he ulimae sess ul D 50 MPa and equivalen Mises sess D shif faco a q is given by q 3 2 s i js i j. The a q D e 1:5 ul (21)