Loughborough Univrsity Institutionl Rpository Post-lol bukling-drivn dlmintion in bilyr omposit bms This itm ws submittd to Loughborough Univrsity's Institutionl Rpository by th/n uthor. Cittion: WA, S....t l., 6. Post-lol bukling-drivn dlmintion in bilyr omposit bms. Composit Struturs,, pp.58-66. Additionl Informtion: This ppr ws ptd for publition in th journl Composit Struturs nd th dfinitiv publishd vrsion is vilbl t http://dx.doi.org/.6/j.ompstrut.5.8. Mtdt Rord: https://dsp.lboro..uk/4/887 Vrsion: Aptd for publition Publishr: Elsvir Ltd. Rights: This work is md vilbl ording to th onditions of th Crtiv Commons Attribution-onCommril-oDrivtivs 4. Intrntionl (CC BY-C-D 4.) lin. Full dtils of this lin r vilbl t: https://rtivommons.org/linss/by-n-nd/4./ Pls it th publishd vrsion.
Post-lol bukling-drivn dlmintion in bilyr omposit bms S. Wng, C. M. Hrvy*, B. Wng, A. Wtson Dprtmnt of Aronutil nd Automotiv Enginring, Loughborough Univrsity, Loughborough, Listrshir LE TU, UK Abstrt Anlytil thoris r dvlopd for post-lol bukling-drivn dlmintion in bilyr omposit bms. Th totl nrgy rls rt (ERR) is obtind mor urtly by inluding n xil strin nrgy ontribution from th intt prt of th bm nd by dvloping mor urt xprssion for th post-bukling mod shp thn tht in th work by Chi t l. (98) nd Huthinson nd Suo (99). Th totl ERR is prtitiond by using prtition thoris bsd on th Eulr bm, Timoshnko bm nd D-lstiity thoris. Indpndnt xprimntl tsts by Kutlu nd Chng (995) show tht, in gnrl, th nlytil prtitions bsd on th Eulr bm thory prdits th propgtion bhviour vry wll nd muh bttr thn th prtitions bsd on th Timoshnko bm nd D-lstiity thoris. Kywords: Composit mtrils, Dlmintion propgtion, Mixd-mod prtition, Post-lol bukling * Corrsponding Author Emil ddrsss:.m.hrvy@lboro..uk (C. M. Hrvy), s.wng@lboro..uk (S. Wng)
omnltur b A A, A, A E rk lngth width of bm mplitud of bukld mod shp Efftiv ross-stionl rs of uppr, lowr nd intt bms E, Young s modulus of uppr nd lowr bms, I, II totl, mod I nd mod II ERRs h, h, h thiknsss of uppr, lowr nd intt bms L totl lngth of bm M bnding momnt in uppr bm M B dlmintion tip bnding momnt in uppr bm,, xil fors in uppr, lowr nd intt bms B, B, B dlmintion tip xil fors on uppr, lowr nd intt bms u V nd-shortning displmnt dfltion of bukld uppr bm α ritil bukling strin orrtion ftor β, β pur-mod-ii mods γ thiknss rtio, γ h h ritil lol-bukling strin nd-shortning omprssiv strin, u L,, omprssiv xil strins in th uppr, lowr nd intt bms η Young s modulus rtio, η E E θ, θ pur-mod-i mods. Introdution Intrf dlmintion in lyrd mtrils is oftn drivn by bukling nd post-bukling. Som xmpls inlud th dlmintion of lmintd omposit bms, plts nd shlls undr in-pln omprssion, nd th surf splling of thrml nd nvironmntl brrir otings. This topi hs ttrtd th ttntion of mny rsrhrs for dds. Rf. [] givs rnt rviw. Although post-bukling-drivn dlmintion gnrlly ours s mixd-mod frtur with ll thr opning, shring nd tring tions (i.. mod I, II nd III), post-bukling-drivn ondimnsionl (D) dlmintion hs rivd mor ttntion bus it is simplr, still pturs th ssntil mhnis, nd lso srvs s stpping ston towrds th study of gnrl mixdmod dlmintion. Th trm D dlmintion mns tht dlmintion propgts in on dirtion with mod I opning nd mod II shring tion only. Som xmpls of D
dlmintion inlud through-width dlmintion in bms, nd blistrs in lmintd omposit plts nd shlls. Th fous of th prsnt work is D post-lol bukling-drivn dlmintion. A dtild dfinition of this will b givn in th nxt stion. Ky tsks in studying D post-lol buklingdrivn dlmintion inlud: () dtrmining th ritil bukling strin nd th post-bukling dformtion, () lulting th post-lol bukling totl nrgy rls rt (ERR), () prtitioning th totl ERR into its individul mod I nd II ERR omponnts, I nd whih govrn th propgtion of mixd-mod dlmintion, nd (4) prditing th dlmintion propgtion bhviour. Anlytil, numril nd xprimntl pprohs r ll ommonly usd for this kind of study. Som rprsnttiv nlytil studis, numril studis nd xprimntl studis r givn in Rfs. [,], [4-] nd [8,9] rsptivly. Rf. [] is rgrdd s pionring nd instrumntl study. It givs full nlytil dvlopmnts for lulting th totl ERR for ss of thin-film, thik-olumn nd gnrl post-lol bukling-drivn dlmintion in lmintd bm-lik plts by using Eulr bm thory. o prtition of th totl ERR into its individul mod I nd II ERR omponnts, I nd nlytil lultions for both th totl ERR nd its omponnts, II, II, is ttmptd in Rf. []. Rf. [] givs I nd II, for th s of thin-film post-lol bukling-drivn dlmintion. Th prtition is bsd on D lstiity thory []. Th numril studis in Rfs. [4-7] r dvlopd by using lyr wis plt/shll thory. Th studis in Rfs. [8-] r bsd on D lstiity nd th study in Rf. [] lso uss th D finit lmnt mthod. Th virtul rk losur thniqu is usd to lult th ERRs in Rfs. [4,5,8-] nd th ohsiv zon modl is usd in Rfs. [6,7]. Th prsnt work ims to dvlop n improvd nlytil mthod to omplt th four ky tsks sttd bov, bsd on th work in Rfs. [,] nd [-]. Th strutur of th ppr is s follows: th nlytil dvlopmnt is givn in Stion, nd in Stion, th numril vrifition nd xprimntl vlidtion r rportd. Finlly, onlusions r givn in Stion 4.. Anlytil dvlopmnt [,,-] Fig. shows post-lolly bukld bilyr omposit bm. Th Young s moduli of th uppr nd lowr lyrs r E nd E rsptivly, nd th orrsponding thiknsss r h nd h with h >> h. Th bm hs totl lngth L nd width b with ntrl through-width intrfil dlmintion of lngth. Th dlmintion tips r lblld B. Th bm is lmpd t both nds nd is undr uniform nd-shortning omprssion. Th lol bukling, s
shown, divids th bm into thr prts, nmly, th lolly-bukld prt lblld, th substrt prt lblld nd th intt prts lblld. Th following dvlopmnt ssums tht th whol pross of bukling, post-bukling nd dlmintion propgtion is lolisd in th uppr lyr, tht is, th bnding tion in both prts nd is ngligibl... Dformtion, intrnl fors nd bnding momnts Th uniform nd-shortning omprssion is rprsntd by strin, dfind s 4 with u bing th nd-shortning displmnt nd L bing th totl lngth of th bm. Th omprssiv xil strins of th nutrl surfs of h th thr prts of th bm r rprsntd by i (with i,, ). Similrly, i nd ( ) i x i u / M rprsnt th xil fors nd bnding momnts rsptivly in h prt, whr x i is th xil xis on h nutrl surf. Th dirtions of th xs of th thr prts togthr r shown in Fig. whr only thir dirtions r inditd. Th xil fors whr th fftiv ross-stionl rs bh i n b xprssd s E () i A i A i r givn by A ( +ηγ ) A ηγ bη i A () bη nd η E E nd γ h h, whih r th modulus nd thiknss rtios rsptivly. Bfor th lol bukling of prt, nd M ( ) i, i E A i undr onstnt uniform xil omprssiv strin nd thr is no bnding. i x i, tht is, ll thr prts r Aftr th lol bukling of prt, prt is undr both xil omprssion nd bnding tion whil prts nd r still ssumd to b undr xil omprssion only without bnding tion. Th xil strin is ssumd to rmin onstnt t th ritil lol-bukling strin throughout [, ], tht is, () Th xil strin n b xprssd by using th xil quilibrium ondition, +, giving + (4) ηγ from whih it is obvious tht >. Also th xil strin should b smllr thn th ndshortning strin ftr lol bukling, tht is, <. From ths two obsrvtions, it is rsonbl to ssum tht th following is good pproximtion: L
Thn Eq. (4) givs In ordr to dtrmin th ritil lol bukling strin (5) + ηγ (6) + ηγ nd bnding momnt M ( ) urtly, it is ssntil to find n urt post-lolly bukld mod shp. Hr, it is ssumd to b A απx ( x ) os os( απ ) x V (7) α whr α is th orrtion ftor for th qulity of th lmpd nd ondition t th rk tip. In Rfs. [,], th vlu of α is tkn s. Th ritil lol-bukling strin n b dtrmind by onsidring th fr-body digrm of symmtril hlf of th bukld uppr lyr shown in Fig.. Horizontl quilibrium ombind with Eqs. () nd () givs B E A nd bnding momnt quilibrium givs M, whih togthr giv M M MB E A V M B V. Clssil bm thory nd Eq. (7) giv ( απ α) E I A( απ ) os( απ ) 4E I V. Thrfor th ritil lol-bukling strin α is obtind s ( π ) h (8) Th vlu of th orrtion ftor α for th problm undr onsidrtion n b dtrmind ithr from numril simultions or from xprimntl tsts. Mor dtils bout th vlu of α will b givn in Stion whih dls with th xprimntl vlidtion. Th mplitud A is now dtrmind by using th following ssumption, whr ( ) t th instnt of lol bukling, ( ) rprsnts hlf-lngth of prt rprsnts th hlf-lngth of prt during post-lol bukling, nd ds rprsnts th diffrntil r lngth of prt s bukld mod shp: ( ) ( ) ds dv + dx dx ot tht this ssumption implis tht th urvd hlf-lngth of th bukld prt rmins onstnt t ( ) during post-bukling. In ordr to dtrmin th mplitud A urtly, (9) 5
prtiulrly in th dp post-bukling rgion, third-ordr sris xpnsion bsd on ( dv dx ) is usd to xpnd th intgrnd on th right-hnd-sid of Eq. (9), whih rsults in th following: Lt 4 6 ( ) dv dv dv + dx () dx 4 dx 8 dx ( ), whih rprsnts th dditionl nd-shortning strin byond th ritil bukling, nd pproximt th uppr limit on th intgrtion s ( ). 4 6 dv dv dv + dx () dx 4 dx 8 dx Substituting Eq. (7) into Eq. () nd vluting th intgrtion givs 6 4 C A C A + C A () 6 + 4 whr Aπ A () ( απ ) sin C (4) απ ( απ ) sin( 4απ ) sin C 4 + (5) 4 απ 4 4απ ( απ ) sin( 4απ ) sin( 6απ ) 5 5 sin C 6 + (6) 4 8 απ 4 4απ 8 6απ Sin α is typilly los to, th hrmoni trms n b ngltd s furthr pproximtion. Th polynomil in Eq. () n thn b solvd, whih givs th mplitud A s whr ( + 5) 5α A (7) 5απ 675 4 + 7 + 5 4 (8) Th bnding momnt t th dlmintion tip B is thn obtind by using Eqs. (7), (8), nd (7), whr M B Ebh (9) ( + 5) 5 α os( απ ) () 5 α 6
7.. Strin nrgy nd totl nrgy rls rt By using th intrnl bnding momnt in prt nd th intrnl xil fors in prts, nd, but nglting th intrnl bnding momnts in prts nd, th strin nrgy U in on hlf of th symmtril post-bukld bm is + + + + + + / L A E A E dx I E M A E L A E A E dx dx V d I E A E U () Ths ssumptions r onsistnt with Stion.. Th totl ERR is thn lultd s + + B B B B A A A I M be () It is worth noting tht ERR rprsnts th strin nrgy dnsity diffrn or prssur ross th dlmintd nd intt prts. Sin uniform xil omprssion rsults in no strin nrgy dnsity diffrn, it dos not produ ny ERR. Thrfor, n fftiv xil for B is dfind s ( ) B bh E A E () Th totl ERR in Eq. () thn boms + + 6 B B B B h M h E b A A I M be λ (4) whr ( ) ηγ ηγ λ +. Substituting B M from Eq. (9) nd B from Eq. () into Eq. (4) givs ( ) E h λ + 4 (5) ot tht whn λ, α nd α Eq. (9) boms th sm s tht in Rfs. [,]... Prtitions of nrgy rls rt... Eulr bm prtition From th uthors prvious work [4-7,], th Eulr bm prtition of th totl ERR in Eq. (4) n b writtn s β β B B B B IE IE M M (6)
whr ( θ, β ) nd ( θ β ) IIE IIE B B M B M B (7) θ θ, r th two sts of orthogonl pur mods. Th θ nd β pur mods orrspond to zro rltiv shring displmnt nd zro rltiv opning displmnt rsptivly just hd of th rk tip [4-7,]. Using th bm mhnis in Stion. in onjuntion with ths onditions, nd thn th orthogonlity ondition [4-7,] through th ERR in Eq. (4) to obtin th orthogonl θ nd β pur mods, givs th following: 6 θ (8) (, β ), h λh (,β ) (, ) θ (9) ot tht th zro vlu of θ rsults from th pproximt ntur of th totl ERR in Eq. (4) nd is du to nglting th bnding tion in prts nd of th bilyr bm. This dos not prvnt from th mod II ERR whn th mod I ERR IIE from bing obtind s it is rdily obtind s IE IE is known. Th offiint IE in Eq. (6) is lultd by using Eqs. (4) nd (6) togthr, nd noting tht IE whn M B nd B θ, giving ow th ERR prtitions, momnt 6 λh + θ θ Eb h θ b IE nd IIE 6 E b h (), r known in trms of th dlmintion tip bnding M B in Eq. (9) nd th fftiv xil for B in Eq. (). For th sk of onvnin, thy r lso givn blow in trms of th ritil bukling strin nd th dditionl nd-shortning strin. ot tht whn omprssiv, nd so IE Eh ( λ ) IIE E h λ ( ) + > M or ( α ) ( λ ) B β B () () α >, th rk tip norml strss boms IE is tkn to b zro with IIE.... Timoshnko bm prtition From th uthors prvious work [4-7,], th Timoshnko bm prtition of th totl ERR in Eq. (4) n b writtn s 8
B B IT IT M () β whr B B IIT IIT M (4) θ IT 6 λh θ 6 E b h + θ b ( + λ ) E b h (5) 6 λh b 6 IIT b E b h + θ ) ( + /( λ ) E b h In trms of th ritil bukling strin nd th dditionl nd-shortning strin, thy bom (6) Agin not tht whn IT IIT ( + λ ) ( λ ) Eh (7) B βmb ( + λ ) ( ) λ Eh + (8) > or ( α ) ( λ ) α >, th rk tip norml strss boms omprssiv, nd so IT is tkn to b zro with IIT.... D lstiity prtition In gnrl, if thr is mtril mismth ross th intrf nd Young s modulus rtio η E E is not qul to, thn th D-lstiity-bsd prtition of ERR is rk xtnsion siz-dpndnt ERR du to th omplx strss intnsity ftor []. It hs bn on most hllnging frtur mhnis problms to obtin nlytil solutions for th ERR prtition nd th strss intnsity ftors. Rntly Hrvy t l. [,] hv solvd this problm by using novl nd powrful mthodology. It is xptd, howvr, tht th fft of mtril mismth ross th dlmintion is not signifint in this study s th lol dformtion in th uppr lyr domints th frtur. Thrfor th D-lstiity-bsd prtition thory in Rfs. [,9] for homognous bms with no mtril mismth ross th intrf is usd instd. Th totl ERR in Eq. (4) n b prtitiond s M B I D I D B β D M B II D II D B θd (9) (4) 9
whr I D II D.697 4.45 θ (4) ( ) D, βd, h λh 6 λh θ D θ D Eb h + b D 6 λh b D b D Eb h + θ D 4.45.697 ( +.697 λ) ( 4.45 +.697λ) E b h ( λ + 4.45 ) λ ( 4.45 +.697λ) E b h In trms of th ritil bukling strin nd th dditionl nd-shortning strin, thy bom Agin not tht whn ( 4.45 +.697λ) ( 4.45 λ ) I D 6 6 (4) (4) +.697 λ E h (44) 6 ( λ + 4.45 ) ( 4.45 +.697λ) (.697 ) λ (45) 6 II D E h + B β DMB > or (.45α ) ( λ ) α >, th rk tip norml strss 4 boms omprssiv, nd so th is tkn to b zro nd I D II D..4. Crk propgtion nd stbility In gnrl th propgtion ritrion n b xprssd in th form whr I nd (,,, ) f (46) I II I II II r th rsptiv ritil mod I nd II ERRs. Th form of Eq. (46) is not uniqu but is rk intrf-dpndnt nd is dtrmind from xprimntl tsting for givn intrf. At th instnt whn Eq. (46) is mt, two snrios ould our. On is unstbl rk propgtion in whih th rk ontinus to dvn without inrsing nd-shortning. Th othr is th stbl rk propgtion in whih th rk stops propgting unlss furthr ndshortning is pplid. Mthmtilly, ths two snrios n b xprssd s f unstbl stbl Altrntivly, th stbility of rk propgtion n b hkd by finding th vlu of f t th ritil nd-shortning strin for propgtion t th initil dlmintion lngth nd thn t slightly inrsd dlmintion lngth. An inrsing vlu of f indits unstbl propgtion. (47)
. umril vrifition nd xprimntl vlidtion This stion ims to xmin th pbility of th nlytil dvlopmnt in Stion for prditing th propgtion bhviour of post-lol buking-drivn dlmintion by mking omprisons with indpndnt numril [4,5] nd xprimntl dt [8,9]. Th quntitis of intrst r th ritil propgtion nd-shortning strin, th ERR prtitions during propgtion nd th propgtion stbility. Two omposit bms [8,9] r studid, whih both ontin singl through-width dlmintion, nd whih r subjtd to uniform nd-shortning displmnt t th lmpd nds, s shown in Fig.. Th omposit bms r md from T/976 grphit/poxy plis nd hv totl lngth L qul to 5.8 mm, nd width b qul to 5.8 mm. Tbl givs mor dtils of th two ss. Th doubl slshs // dnot th lotion of th dlmintd intrf. All plis hv qul thiknss. Th ply longitudinl modulus E is 9. P. Th ritil ERR for mod I I is 87.6 /m nd for mod II II is qul to 5. /m. Exprimntl studis in Rfs. [8,9] suggst tht th mtril hs linr filur ritrion, tht is, Eq. (46) tks th form I II f ( I, II, I, II ) + (48) whih will b usd in th following studis. For ths two ss, n mpiril formul for th ritil bukling strin orrtion ftor α in Eq. (8) is obtind by using finit lmnt mthod simultions nd is givn by I II h h.78.6654 +.9755 (49).. Comprison of totl ERR in Eq. (5) with indpndnt numril rsults [5] Aurt lultion of totl ERR is ruil pr-rquisit stp towrds th urt prdition of propgtion bhviour. Th following xris ims to xmin th ury of th totl ERR givn by Eq. (5) nd th solutions in Rfs. [,] by ompring thm ginst indpndnt numril rsults in Rf. [5]. Tbls nd rord th omprisons for Cs nd Cs rsptivly. In gnrl, good grmnt is obsrvd btwn th prsnt solutions nd th numril rsults in Rf. [5] for both ss. Th solutions from Rfs. [,] hv rsonbl grmnt for Cs nd vry poor grmnt for Cs.
.. Comprison of dlmintion propgtion bhviour with indpndnt xprimntl rsults [9] It is wll known tht frtur toughnss dpnds on frtur mod prtition. Th vlidity of prtiulr mixd-mod prtition thory n only b vlidtd ginst xprimntl tsts. Thorough nd omprhnsiv xprimntl tst dt from svrl indpndnt rsrh groups [4-9] shows [7,] tht Wng nd Hrvy s Eulr bm prtition thory givs th most urt prdition of mixd-mod frtur toughnss. Th xris in this stion ims to stblish whthr this prtition thory lso govrns th propgtion of mixd-mod dlmintion drivn by post-lol bukling. Cs is onsidrd first. Tbl 4 nd Fig. rord th dlmintion propgtion bhviour prditd by th thr prtition thoris dsribd in Stion. Th symbol f in Tbl 4 rprsnts th propgtion ritrion in Eq. (48) with f < inditing no propgtion nd f inditing stbl propgtion. ot tht th bold vlus of th nd-shortning strins in Tbl 4 r thos tht r disussd hr. Both th Eulr nd Timoshnko bm prtition thoris prdit n initil mixd-mod dlmintion followd by pur-mod-ii dlmintion, with dlmintion propgtion bginning in th pur-mod-ii rgion t n nd-shortning strin of nd rhing th lmpd nds t n nd-shortning strin of..76 Although th propgtion is stbl, it tks only.7.9 of xtr nd-shortning strin (or.85 mm of nd-shortning displmnt) to xtnd th dlmintion by.7 mm. This might suggst n obsrvtion of unstbl propgtion in xprimntl tsts. Th D lstiity prtition thory prdits mixd-mod dlmintion whih bgins to propgt t n nd-shortning strin of nd rhs th lmpd nds t n nd-shortning strin of.5.9. It tks n xtr nd-shortning strin of.9 (or. mm of ndshortning displmnt) to xtnd th dlmintion by th sm.7 mm, whih is muh lrgr thn th.7 of xtr nd-shortning strin prditd by th Eulr nd Timoshnko prtition thoris. This might suggst n obsrvtion of stbl propgtion in xprimntl tsts. Th propgtion bhviour is lso shown grphilly in Fig. s dlmintion lngth vs. ndshortning strin. Th two bm prtition thoris prdit muh stpr growth rt thn th D lstiity prtition thory dos. It is sn tht th prditions from th two bm prtition thoris r onsidrbly diffrnt from tht of th D lstiity prtition thory.
Exprimntl tst dt in Rf. [9] r usd nxt to ssss th ury of h prtition thory. Th tsts rord th history of th omprssion for pr unit width F ginst th uppr surf mid-spn xil strin s. Th omprssion for pr unit width is lultd nlytilly s ( + ) b E η ( + ) F ηγ (5) nd th uppr surf mid-spn xil strin is lultd nlytilly s s h d V dx x h A π Fig. 4 omprs th thr prtition thoris with th tst rsults [9]. Th following points r notd: () th nlytil ritil lol-bukling omprssion for is muh smllr thn th xprimntl on. On possibl rson for this is th stiking of th spimn s sub-lmints through th Tflon film insrtd to rt th initil dlmintion during mnufturing, thus inrsing th bukling lod [9]. ot tht both th nlytil nd xprimntl rsults disply bifurtion-typ lol bukling, whih pprs s th first shrp ornr in th figur. () By ross-ompring with th rsults in Tbl 4, th two bm prtition thoris prdit pur-mod-ii propgtion, bginning t th sond shrp ornr nd nding t th third on, whih orrsponds to th omplt dlmintion. During th dlmintion propgtion pross, th omprssion for dos not hng vry muh, whih quts to n lmost-unstbl propgtion. On th othr hnd, th D lstiity prtition thory prdits mixd-mod propgtion, strting smoothly nd nding t bout th sm point prditd by th two bm prdition thoris. During th dlmintion propgtion, th omprssion for dos hng signifintly, whih quts to stbl propgtion. () Th xprimntl rsults [9] do show n lmost-unstbl propgtion nd both th initil- nd nd-propgtion omprssion fors gr vry wll with th prditions of th two bm prtition thoris. (4) Th signifint disrpny btwn th nlytil nd xprimntl ritil lol-bukling omprssion fors rsults in signifint diffrn btwn th prditd nd xprimntl loding urvs. This nds to b invstigtd in ordr to xmin th prtition thoris mor thoroughly. In th following, n pproximt xprssion for th ritil lol-bukling nd-shortning strin is drivd whr th subsript indits tht it is bsd on xprimntl rsults. Similr to in Eq. (8), is writtn s ( π ) (5) h (5)
whr th orrtion ftor α nds to b dtrmind bsd on xprimntl rsults. It is prhps th s tht, in gnrl, th rtio α α vris with th rtio h ; howvr, α α is ssumd hr to b onstnt t its vlu t th initil-bukling dlmintion lngth du to lk of xprimntl rsults for othr rk lngths. Th ury of this ssumption will b xmind shortly. It is now only rquird to dtrmin th vlu of α t th point of initil bukling. From Fig. 4, two pproximt ritil lol-bukling nd-shortning strins r found from th uppr-surf mid-spn xil strin nd th omprssion for t th bifurtion point of th xprimntl rsults: () sin bfor th lol bukling of prt, t this lotion s.748. () Bfor th lol bukling of prt, F ( +ηγ ) or F [ E η ( + ηγ )].9 E η lso, giving t this lotion. By vrging ths two vlus, n pproximt ritil lol-bukling nd-shortning strin is obtind s.85. Thrfor th vlu of α t th ritil lol-bukling point is dtrmind from Eq. (5) to b α. 6 nd th rtio α α. 7. Th ritil lol-bukling strin t ny dlmintion lngth is thn lultd from Eq. (5) s.7. Fig. 5 omprs th tst rsults [9] with th thr prtition thoris, whih now us th ritil lol-bukling nd-shortning strin bsd on xprimntl rsults. Th two bm prtition thoris prdit th propgtion bhviour vry wll nd muh bttr thn th D lstiity prtition thory dos. Th dlmintion propgtion is indd th pur-mod-ii propgtion prditd by th two bm prtition thoris. It is now lr tht th D lstiity prtition thory dos not provid th right prtition for prditing th propgtion bhviour of bukling-drivn dlmintion for Cs. Th qustion of whih bm prtition thory provids th right prtitions whn th propgtion is not pur mod II, howvr, still nds to b nswrd. Cs is onsidrd nxt to nswr this qustion. Cs is now onsidrd in th sm mnnr. Tbl 5 nd Fig. 6 rord th dlmintion propgtion bhviour prditd by th thr prtition thoris. ot tht th bold vlus of th nd-shortning strins in Tbl 5 r thos tht r disussd hr. All thr prtition thoris prdit n initil mixd-mod dlmintion ftr th lol bukling of th uppr lyr t.7, followd by unstbl mixd-mod dlmintion propgtion nd thn stbl propgtion. Th Eulr bm prtition thory prdits mod-i-domintd unstbl propgtion ourring t n nd-shortning strin of.46 4, during whih th dlmintion xtnds to totl lngth of 8.99 mm. Thn th dlmintion propgts stbly s mod-ii-domintd to
totl lngth of 9.67 mm orrsponding to nd-shortning strin of.69 ftr whih th dlmintion propgts stbly s pur-mod-ii to th lmpd nds t n nd-shortning strin of.97. Th Timoshnko bm prtition thory prdits mod-ii-domintd unstbl propgtion ourring t n nd-shortning strin of.9, during whih th dlmintion xtnds to totl lngth of 46.68 mm. Thn th dlmintion propgts s purmod-ii to th lmpd nds t n nd-shortning strin of.97. Th D lstiity prtition thory prdits firly mixd-mod unstbl propgtion ourring t n ndshortning strin of.56, during whih th dlmintion xtnds to totl rk lngth of 7.45 mm. Thn th dlmintion propgts s mod-ii-domintd to th lmpd nds t n nd-shortning strin of.96. In sns, th D lstiity prtition thory is n vrg of th two bm prtition thoris. Th propgtion bhviour is lso shown grphilly in Fig. 6 s dlmintion lngth vs. th nd-shortning strin. It is sn tht th prditions from th thr prtition thoris r onsidrbly diffrnt from h othr. In ontrst with th prdition for Cs, for Cs th Timoshnko bm prtition thory givs vry diffrnt prditions to thos from th Eulr bm prtition thory. Similr to th study for Cs, xprimntl tst dt in Rf. [9] r usd to ssss th ury of h prtition thory. Fig. 7 shows th historis of th omprssion for pr unit width F ginst th uppr surf mid-spn xil strin s s msurd in tsting nd s prditd by th thr prtition thoris. In gnrl, it is sn tht th prditions from th Eulr bm prtition thory gr quit wll with th tst rsults, tht th prditions from th Timoshnko bm prtition thory r poor, nd tht th prditions from th D-lstiity prtition thory r somwhr in th middl. As ws sn for Cs, th ritil lol-bukling omprssion for prditd nlytilly my not gr vry wll with th xprimntlly obsrvd vlu. In ordr to xmin th prtition thoris mor thoroughly, it is nssry to orrt for ny disrpny btwn th nlytil nd xprimntl ritil lol-bukling omprssion fors. Fig. 7, howvr, shows tht n imprftion-typ initil bukling is obsrvd in xprimnts (whrs bifurtion-typ initil bukling is prditd by th nlytil thoris). To ount for this, th intrstion point of th linr rgions of th pr-bukling nd post-bukling rsponss in th xprimntl dt in Fig. 7 (dt mrkrs to 6, nd 5 to 7 rsptivly) is usd to pproximt th xprimntl vlus of th uppr-surf mid-spn xil strin s nd th omprssion for F t th point of 5
bifurtion-typ lol bukling, whih r found to b.84 nd F 67. As bfor for Cs, ths vlus giv two pproximt ritil lol-bukling nd-shortning strins. Whn vrgd,.867 is obtind with. 89 s α nd α α. 949. Th ritil lol-bukling strin t ny dlmintion lngth is thn lultd from Eq. (5) s.949. Fig. 8 shows th omprisons btwn th thr prtition thoris nd th tst rsults [9]. In gnrl, it is sn tht th prditions from th Eulr bm prtition thory gr wll with th tst rsults, tht th prditions from th Timoshnko bm prtition thory r poor, nd tht th prditions from th D-lstiity prtition thory r, gin, somwhr in th middl. 4 Conlusions Bsd on th Eulr bm, Timoshnko bm nd D-lstiity mixd-mod frtur prtition thoris [,-7], nlytil thoris hv bn dvlopd for prditing th propgtion bhviour of post-lol bukling-drivn dlmintion in bilyr omposit bms. Th onlusions r s follows: () urt lultion of th totl ERR is ssntil in ordr to obtin urt prditions. This work hs prsntd mor urt nlytil formul for totl ERR thn tht in Rfs. [,] by dvloping mor urt xprssion for th post-bukling mod shp nd lso by inluding th xil strin nrgy ontribution from th intt prt of bm. Vry good grmnt is obsrvd btwn th prsnt nlytil rsults nd th numril rsults [5]. () Th ury of ritil lol-bukling strin is lso ky ftor in mking urt prditions. Empiril vlus, obtind ithr numrilly or xprimntlly for prtiulr ss, giv mor urt prditions. () Th mthod usd to prtition th totl ERR into I nd II is nothr ky ftor for mking urt prditions. This work prsnts thr prtition thoris, nmly, th Eulr bm, Timoshnko bm nd D lstiity prtition thoris. Indpndnt xprimntl tsts by Kutlu nd Chng [9] show tht, in gnrl, th nlytil thory bsd on th Eulr bm prtition thory prdits th propgtion bhviour vry wll nd muh bttr thn th thoris bsd on th Timoshnko bm nd D lstiity prtition thoris, whn using th ritil lol-bukling strin drivd with th id of xprimntl rsults. (4) Bukling-drivn dlmintion is mjor form of filur in nginring struturs md of omposit mtrils. On importnt xmpl is th thrml bukling-drivn rking of thrml brrir otings usd in ro-ngins. Th prsnt Eulr bm nlytil thory provids vlubl tool for th nginring dsign of suh mtril struturs. Th 6
prsnt work is bing xtndd to bukling-drivn dlmintion in gnrlly lmintd omposit bms nd will b rportd in th nr futur. Rfrns [] Xu J, Zho Q, Qio P. A ritil rviw on bukling nd postbukling nlysis of omposit struturs. Frontirs in Arosp Enginring ;:57 68. [] Chi H, Bbok CD, Knuss W. On dimnsionl modlling of filur in lmintd plts by dlmintion bukling. Intrntionl Journl of Solid nd Struturs 98;7:69 8. [] Huthinson JW, Suo Z. Mixd mod rking in lyrd mtrils. Advns in Applid Mhnis 99;9:6 9. [4] Zhng Y, Wng S. Bukling, post-bukling nd dlmintion propgtion in dbondd omposit lmints: Prt Thortil dvlopmnt. Composit Struturs 9;88:. [5] Wng S, Zhng Y. Bukling, post-bukling nd dlmintion propgtion in dbondd omposit lmints: Prt umril pplitions. Composit Struturs 9;88: 46. [6] Hossini-Toudshky H, Hossini S, Mohmmdi B. Dlmintion bukling growth in lmintd omposits using lyrwis-intrf lmnt. Composit Struturs ;9:846 856. [7] Btr RC, Xio J. Anlysis of post-bukling nd dlmintion in lmintd omposit St. Vnnt Kirhhoff bms using CZM nd lyr-wis TSDT. Composit Struturs ;5:69 84. [8] Kutlu Z, Chng FK. Composit pnls ontining multipl through-th-width dlmintions nd subjtd to omprssion. Prt I: nlysis. Composit Struturs 995;:7 96. [9] Kutlu Z, Chng FK. Composit pnls ontining multipl through-th-width dlmintions nd subjtd to omprssion: Prt II: xprimnts nd vrifition. Composit Struturs 995;:97 4. [] Liu PF, Hou SJ, Chu JK, Hu XY, Zhou CL, Liu YL, Zhng JY, Zho A, Yn L. Finit lmnt nlysis of postbukling nd dlmintion of omposit lmints using virtul rk losur thniqu. Composit Struturs ;9:549 56. 7
[] Wng S, Hrvy CM. Mixd mod prtition in on dimnsionl frtur. Journl of Ky Enginring Mtrils ;46-6:66 6. [] Hrvy CM. Mixd-mod prtition thoris for on-dimnsionl frtur. PhD Thsis. Mrh, Loughborough Univrsity, UK. [] Wng S, un L. On frtur mod prtition thoris. Computtionl Mtril Sins ;5:4 45. [4] Wng S, Hrvy CM. A thory of on-dimnsionl frtur. Composit Struturs ;94:758 767. [5] Wng S, Hrvy CM. Mixd mod prtition thoris for on dimnsionl frtur. Enginring Frtur Mhnis ;79:9 5. [6] Hrvy CM, Wng S. Mixd-mod prtition thoris for on-dimnsionl dlmintion in lmintd omposit bms. Enginring Frtur Mhnis ;96:77 759. [7] Hrvy CM, Wng S. Exprimntl ssssmnt of mixd-mod prtition thoris. Composit Struturs ;94:57 67. [8] Wng S, Hrvy CM, un L. Prtition of mixd mods in lyrd isotropi doubl ntilvr bms with non-rigid ohsiv intrfs. Enginring Frtur Mhnis ;: 5. [9] Hrvy CM, Wood JD, Wng S, Wtson A. A novl mthod for th prtition of mixdmod frturs in D lsti lmintd unidirtionl omposit bms. Composit Struturs 4;6:589 594. [] Hrvy CM, Epltt MR, Wng S. Exprimntl ssssmnt of mixd-mod prtition thoris for gnrlly lmintd omposit bms. Composit Struturs 5;4: 8. [] Willims ML. Th strsss round fult or rk in dissimilr mdi. Bulltin of th Sismologil Soity of Amri 959;49:99 4. [] Hrvy CM, Wood JD, Wng S. Brittl intrfil rking btwn two dissimilr lsti lyrs: Prt Anlytil dvlopmnt. Composit Struturs 5 (in prss). DOI:.6/j.ompstrut.5.6.8. [] Hrvy CM, Wood JD, Wng S. Brittl intrfil rking btwn two dissimilr lsti lyrs: Prt umril vrifition. Composit Struturs 5 (in prss). DOI:.6/j.ompstrut.5.6.79. [4] Chrlmbids M, Kinloh AJ, Wng Y, Willims J. On th nlysis of mixd-mod filur. Intrntionl Journl of Frtur 99;54:69 9. 8
[5] Hshmi S, Kinloh AJ, Willims. Mixd-mod frtur in fibr-polymr omposit lmints. In: O Brin TK, ditor. Composit mtrils: ftigu nd frtur (third volum), ASTM STP. Phildlphi, PA: Amrin Soity for Tsting nd Mtrils, 99. pp. 4 68. [6] Dvidson BD, Frillo PL, Hudson RC, Sundrrmn V. Aury ssssmnt of th singulr-fild-bsd mod-mix domposition produr for th prdition of dlmintion. In: Hoopr SJ, ditor. Composit mtrils: tsting nd dsign (thirtnth volum), ASTM STP 4. Amrin Soity for Tsting nd Mtrils, 997, pp. 9 8. [7] Dvidson BD, hribin SJ, Yu LJ. Evlution of nrgy rls rt-bsd pprohs for prditing dlmintion growth in lmintd omposits. Intrntionl Journl of Frtur ;5:4 65. [8] Dvidson BD, Bilszwski RD, Sinth SS. A non-lssil, nrgy rls rt bsd pproh for prditing dlmintion growth in grphit rinford lmintd polymri omposits. Composits Sin nd Thnology 6;66:479 496. [9] Conroy M, Sørnsn BF, Ivnkovi A. Combind numril nd xprimntl invstigtion of mod-mixity in bm lik gomtris. In: Prodings of th 7th Annul Mting of th Adhsion Soity, Fburry 4, Sn Digo, Cliforni, USA. 9
Fig. : A post-lolly bukld bilyr omposit bm du to dlmintion undr omprssion.
Fig. : Fr-body digrm of symmtril hlf of th bukld uppr lyr.
Fig. : Dlmintion lngth vs. nd-shortning strin for Cs.
Fig. 4: Comprssion for pr unit width F vs. uppr-surf mid-spn strin s for Cs using th nlytil bukling strin.
Fig. 5: Comprssion for pr unit width F vs. uppr-surf mid-spn strin s for Cs using th xprimntl bukling strin. 4
Fig. 6: Dlmintion lngth vs. nd-shortning strin for Cs. 5
Fig. 7: Comprssion for pr unit width F vs. uppr-surf mid-spn strin s for Cs using th nlytil bukling strin. 6
Fig. 8: Comprssion for pr unit width F vs. uppr-surf mid-spn strin s for Cs using th xprimntl bukling strin. 7
Tbl : Configurtions of two omposit bms ontining ntrl through-width dlmintion. Cs Ly-up (mm) h (mm) [ / // ] 4 4 [ / // ] h (mm) 8..58.59 9.5.58.54 4 4 8
Tbl : Totl ERR rsults for Cs. ( - ) (/mm) Rf. [5] Eq. (5) Rfs. [,]..44.45.99..66.68.646.4.9.874.9.6.8.4.5.8.499.45.559..796.75.9..5.88..4.44.45.7 9
Tbl : Totl ERR rsults for Cs. ( - ) (/mm) Rf. [5] Eq. (5) Rfs. [,]..4.65...64.658..4.949.958..5.5.6.54.6.76.57.887
Tbl 4: Dlmintion propgtion bhviour of Cs. ( - ) Eulr Timoshnko D Elstiity (mm) f II (%) (mm) f II (%) (mm) f II (%).6 8. < 8.6 8. < 8. 8. < 4.6. 8. < 66.5 8. < 96. 8. < 69.4.4 8. < 88.4 8. < 99.5 8. < 8..8 8. <. 8. <. 8. < 86.6. 8. <. 8. <. 8. < 9.8.5 8. <. 8. <. 8. 9..6 8. <. 8. <. 9.96 95..7 8. <. 8. <. 4.5 96.9.76 8.. 8.. 44.4 97.8.8 4.. 4.. 45.7 98.4.9 48.75. 48.75. 5. 99.6.9 49.59. 49.59. 5.8 99.7.9 5.8. 5.8. - - -
Tbl 5: Dlmintion propgtion bhviour of Cs. ( - ) Eulr Timoshnko D Elstiity (mm) f II (%) (mm) f II (%) (mm) f II (%). 9.5 < 5. 9.5 < 8. 9.5 < 47..4 9.5 < 7. 9.5 < 84. 9.5 < 48.5.8 9.5 < 9. 9.5 < 84.8 9.5 < 49.6.4 9.5 <. 9.5 < 85.6 9.5 < 5.7.46 9.5. 9.5 < 86.4 9.5 < 5.8.46 8.99 9.7 9.5 < 86.4 9.5 < 5.8.5 9. 94. 9.5 < 86.9 9.5 < 5.6.54 9.6 95.4 9.5 < 87.6 9.5 < 5.5.56 9. 95.9 9.5 < 87.8 9.5 5.9.56 9. 95.9 9.5 < 87.8 7.45 9.4.6 9.44 97. 9.5 < 88.4 8.4 94.4.64 9.55 98.5 9.5 < 88.9 9. 95..68 9.64 99.6 9.5 < 89.4 4.9 96..7.56. 9.5 < 89.8 4. 96.7.76 5.55. 9.5 < 9. 4.45 97.4.8 8.7. 9.5 < 9.7 4.69 98..84 4.98. 9.5 < 9. 45.8 98.6.88 4.8. 9.5 < 9.5 46.67 99..9 46.68. 9.5 9.8 48.4 99.5.9 46.68. 46.68. 48.4 99.5.94 48.5. 48.5. 49.67 99.7.96 5.. 48.5. 5.8 99.8.97 5.8. 5.8. - - -