Shear Stress-Slip Model for Steel-CFRP Single-Lap Joints under Thermal Loading

Transcription

1 Shear Sress-Slp Model for Seel-CFRP Sngle-Lap Jons under Thermal Loadng *Ank Agarwal 1), Eha Hamed 2) and Sephen J Foser 3) 1), 2), 3) Cenre for Infrasrucure Engneerng and Safey, School of Cvl and Envronmenal Engneerng, The Unversy of New Souh Wales Sydney, UNSW Sydney, Ausrala * ank.agarwal@unsw.edu.au ABSTRACT The shear sress-slp model (SSM) has een nensvely used n he pas for dfferen knd of adhesve jons and wh varous sress-slp relaons. However n hose sudes, he fleural sffness of he adherends was negleced. In hs paper, oh he adherends (seel and CFRP) n a sngle-lap jon are modelled as lnear elasc Bernoull-Euler eams wh aal and fleural sffness, whle he adhesve layer and s nerfaces are modelled as an arfcal nerface ha s governed y a lnear relaon eween he slp and he nerfacal shear sress. Two ypes of hermal loadng, namely freeze-haw cyclng and hermal cyclng, are nvesgaed, whose effecs are nroduced n erms of reduced elasc modulus of he adhesve. Engneerng shear sran ased falure creron s hen used o predc he falure load. The predced srenghs of he jons are compared wh epermenal resuls and are found o e n good agreemen. Paramerc sudes were hen conduced o nvesgae he nfluence of adhesve hckness. 1. INTRODUCTION The ehavor of adhesve jons have een nensvely nvesgaed n recen years usng oh fne elemen modellng and analycal modellng approaches (Chew e al. 211; Yu e al. 211; Yu e al. 212; da Slva e al. 29; da Slva e al. 29; Mancus and Ascone 213). However, here s no heorecal model n he leraure capale of predcng he falure load ncludng he effecs of freeze-haw (F-T) cycles and hermal cycles. The applcaons of hese cycles have sgnfcan nfluence on he ond srengh of seel-cfrp sngle-lap jon (Agarwal e al. 214 and 215). 1) Research Assocae; 2) Senor Lecurer; 3) Professor and Head of School

2 The modellng of sngle-lap jons requres he use of a sress analyss phase and a falure analyss phase ha can e ased on lm sress, lm sran, or fracure mechancs approaches. In hs paper, shear sress-slp model (SSM) s used wh he srengh ased falure creron o predc he ond srengh of seel-cfrp sngle lap jons, ncludng he effecs of freeze-haw cycles and hermal cycles, whose effecs are nroduced n erms of reduced elasc modulus of he adhesve. 2. SHEAR STRESS-SLIP MODEL (SSM) The shear sress-slp model or SSM has een nensvely used for dfferen knds of adhesve jons and wh varous sress-slp relaons (Teng e al., 26; Ferracu e al., 26; Wu e al., 22; Czadersk e al. 21). However, n all prevous sudes, he fleural sffness of he adherends was negleced. In a sngle lap-jon, here s a consderale amoun of endng n he sysem whch needs o e consdered durng modellng. Thus n he SSM adoped here, oh he CFRP and he seel plaes are modelled as lnear elasc Bernoull-Euler eams wh aal and fleural sffnesses, whle he adhesve layer s modelled as an arfcal nerface. The nerface s governed y a lnear or non-lnear relaon eween he slp and he nerfacal shear sress. Slp s defned as he relave longudnal dsplacemen eween he op layer of he seel and he oom layer of he CFRP. The reason for hs defnon s ecause hese are he wo quanes ha are ypcally measured durng ess. The sgn convenons of he model for lap jons wh edge loadng only are shown n Fg. 1. The man advanage of he SSM s he ease of mplemenaon of lnear/non-lnear shear sress-slp relaons and he aly o acheve closed form soluons. where 2.1 Knemac relaons The knemac relaons for he adherends are gven y: u (, z ) u ( ) z w ( ), (, z ) u ( ) z w ( ) (1) o, o,, u s he horzonal dsplacemen of he CFRP (=) and he seel (=) plaes; u o and w are he horzonal and vercal dsplacemens a he cenrodal (reference) as of he CFRP (=) and he seel (=) plaes (Fg. 1a), respecvely; ε s he longudnal sran, z s measured downwards from he reference as; s he longudnal coordnae, and ( ), denoes a dervave wh respec o. 2.2 Equlrum equaons The equlrum equaons for regons 1 and 3 (Fg. 1a) are he classcal eam equaons ha ake hs form: N ( ) ; Q ( ) ; M ( ) Q ( ) ; (2),,,

3 a) uo Seel z,w uo CFRP Q M N z,w Regon 1 Regon 2 Regon 3 L ) M M +M,d N N +N, d Q Q +Q,d ) d M M +M,d N d N +N, d Q Q +Q,d Fg. 1: Shear sress-slp model: a) Sgn convenon; and ) nernal resulans. For Regon 2, he equlrum equaons accoun for he nerfacal shear sresses as follows:,, N ( ) ( ) ; N ( ) ( ) ; (3),, Q ( ) ; Q ( ) ; (4) where N, d d Q ( ) M, ( ) ( ) ; Q ( ) M, ( ) ( ) ; (5) 2 2 M, and Q are he aal force, endng momen, and shear force, respecvely; s he wdh of he lap jon; τ s he nerfacal shear sress; d and d are he hckness of he op adherend (CFRP) and he oom adherend (seel), respecvely. For smplcy, s assumed ha he shear sress-slp relaon can e appromaed as lnear efore deondng and ha he shear sress mmedaely drops

4 o zero when he magnude of he slp eceeds a characersc value ( f ). Thus, he shear sress akes hs form: ( ) G ( N) d d ( ) G ( N) u u w w a a o, o,,, ca c a 2 2 (6) where G a s he shear modulus of he adhesve, whch s a funcon of he numer of F-T cycles or hermal cycles (N); c a s he hckness of he adhesve. 2.3 Consuve relaons and oundary condons The consuve relaons for he adherends (seel and CFRP) are: o,, N ( ) ( EA) u ( ); M ( ) ( EI ) w ( ) (7) where (EA) and (EI) are he aal and fleural rgdes, respecvely. In he specfc geomery and loadng of he esed specmens (Fg. 1a), he followng oundary condons are used, whch nclude he effec of he grps: N ( L) N; Q ( L) Q ; M ( L) M (8) u ( ) ; w ( ) ; w ( ) ; (9) o, where N, Q, and M are eernally appled aal load, shear force, and endng momen, respecvely (Fg. 1); and L s he oal lengh of he lap shear specmen. 2.4 Falure creron Srengh ased falure creron s used o predc he falure n he lap-shear specmen. For hs, / c defnes a crcal engneerng shear sran of he adhesve, f a whch s used as he falure creron wh he SSM. Usng he equlrum equaons along wh he consuve relaons, he governng dfferenal equaons of he lap-jon can e derved n erms of he unknown deformaons, u o, u o, w, w, whch are analycally solved along wh he oundary and he connuy condons eween he regons. The resuls are analysed n he followng secon.

5 3. NUMERICAL STUDY The geomery and he maeral properes of he seel-cfrp lap-shear jon used for he numercal smulaon are aken from Agarwal e al. (214, 215) and are shown n Fg. 2. The hcknesses of he CFRP (d ) and he seel plaes (d ) are 1.4 mm and 3 mm, respecvely. The wdh of he jon () s 25 mm, and he onded lengh s 25 mm. The elasc modulus of he CFRP (E ) and he seel (E ) are 165 GPa and 27 GPa, respecvely. Seel E = 27 GPa d = 3 mm Regon 2 85 mm 85 mm Regon 1 25 mm Regon 3 = 25 mm Adhesve E a ( N ) c a =.5 mm CFRP E = 165 GPa d = 1.4 mm Fg. 2: Geomercal and maeral properes of he lap-shear specmen used for he numercal sudy. 3.1 Deermnaon of falure load of seel-cfrp jon afer F-T cyclng The elasc modulus of adhesve, whch s a funcon of he numer of F-T cycles (N), s aken from Agarwal e al. (214) and s shown n Eq. (1). Ea (.55 N ) e (1) The shear sress-slp model (SSM) developed efore s solved analycally n a commercally avalale sofware, Maple. Usng he falure load of 7,95 N, aken from Agarwal e al. (214) for conrol specmens whou any cycles, he magnude of he mamum slp oaned from he model s.2 mm and yelds a crcal engneerng shear sran of.4 for he adhesve. Usng he reduced values of he elasc modulus, assumng ha Posson s rao s unchanged due o F-T cycles (for oanng he shear modulus) and adopng a crcal engneerng shear falure sran of.4, he soluon of Eq. (2) o Eq. (6) gves he load versus eenson predcons as shown n Fg. 3. Noe ha he model curves are ploed unl he engneerng shear sran n he adhesve reaches.4. I can e seen ha he resuls from he model are n good agreemen wh hose measured n he ess y Agarwal e al. (214). The mamum dfference eween he heorecal and he epermenal resuls n erms of he falure load s less han 15%, whch can e due o many facors u manly due o he presence of vercal normal sresses n he adhesve layer, whch are no consdered n he SSM.

6 Load (N) Load (N) Load (N) Load (N) Conrol Specmens ( F-T cycle) LS-FT-1 LS-FT-3 LS-FT-4 SSM F-T cycles LS-FT2-4 LS-FT2-5 LS-FT2-6 SSM Eensomeer Readng (mm) LS-FT3-1 LS-FT3-3 LS-FT3-5 SSM 3 F-T cycles Eensomeer Readng (mm) Fg. 3: Load versus eensomeer measuremens: shear sress-slp model compared wh es daa for conrol specmens; afer 2 F-T cycles, 3 F-T cycles and 4 F-T cycles. 3.2 Deermnaon of falure load of seel-frp jon afer hermal cyclng only The falure load of seel-cfrp jons afer eposure o hermal cyclng only s also predced from SSM and compared wh he es resuls of Agarwal e al. (215). Lke n he case of F-T cyclng, he effec of hermal cyclng s nroduced n erms of reduced elasc modulus of he adhesve. For he falure load of 4,61 N (as measured n Agarwal e al. 215), n case of conrol specmens of seel-cfrp jons, and wh elasc modulus of adhesve n he conrol case as 4,62 MPa, he magnude of he mamum slp oaned from he model s.11 mm and yelds a crcal engneerng shear sran of.22. Afer he applcaon of 18 hermal cycles eween 1ºC and 5ºC wh wo and half hours each for ho and cold cycle (for more deals on he hermal cycles, see Agarwal e al. 215), he elasc modulus of he adhesve reduced o 3,46 MPa, as measured from he epermens. Adopng a crcal engneerng shear falure sran of.22 and usng he reduced value of elasc modulus of he adhesve, he predced falure load s oaned as 4, N, whch s close o he falure load of 3,925 N measured from he epermens y Agarwal e al. (215). 3.3 Paramerc sudy: Effec of adhesve hckness on shear sresses n he adhesve layer from SSM Egh dfferen adhesve hcknesses varyng from.25 mm o 2. mm wh a sep sze of.25 mm were analysed usng SSM o oan mamum shear sress a he lef and he rgh edge of he adhesve layer n he case of conrol specmens (no F-T Eensomeer Readng (mm) LS-FT4-1 LS-FT4-2 LS-FT4-3 SSM 4 F-T cycles Eensomeer Readng (mm)

7 cyclng and no hermal cyclng). The analyss was performed for a ensle load of 7,95 N, whch was he falure load of he conrol specmens n Agarwal e al. (214). The mamum shear sress s normalzed wh respec o he shear sress oaned for.5 mm adhesve hckness, and s shown n Fg. 4. I can e seen ha he shear sress decreases eponenally wh he ncrease n adhesve hckness. Normalzed Ma shear sress Rgh end Lef end Adhesve Thckness (mm) Fg. 4: Normalzed mamum shear sress n he adhesve layer versus adhesve hckness oaned from SSM a a load level of 7,95 N. 4. CONCLUSIONS The seel-cfrp sngle-lap jon was modelled usng a smple shear sress-slp relaon, whch models he seel and CFRP plaes as eam memers, whle he adhesve layer s replaced wh an arfcal nerface ha accouns for s shear deformaly. The effecs of F-T cycles and hermal cycles were nroduced n erms of reduced elasc modul of he adhesve. The reduced modul were hen used o heorecally esmae he falure load afer F-T cycles and hermal cycles. A falure creron, ased on ulmae engneerng shear sran of he adhesve layer, was adoped and calraed from he srengh of jons aken from Agarwal e al (214, 215). The predced resuls for he falure load due o F-T cycles and hermal cycles were found o e n good agreemen wh he correspondng srenghs from he epermens (Agarwal e al., 214, 215).

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