TEMPERATURE-DEPENDENT VISCOELASTIC PROPERTIES OF UNIDIRECTIONAL COMPOSITE MATERIALS

Transcription

1 6 INERNAIONAL CONFERENCE ON COMPOSIE MAERIALS EMPERAURE-DEPENDEN VISCOELASIC PROPERIES OF UNIDIRECIONAL COMPOSIE MAERIALS Shuan Lu, Fengzh Wang Deparmen of Engneerng Mechancs, Sae Key Laboraory of Srucural Analyss for Indusral Equpmen, Dalan Unversy of echnology, Dalan, 64, Chna Keywords: compose maerals; emperaure-dependen vscoelascy; hermal expanson; consuve equaons; homogenzaon heory Absrac he undersandng of he vscoelasc behavor ha accouns for he heerogeneous compose maeral s mcrosrucure and he emperauredependency of properes sll requres furher sudy and developmen. A homogenzaon-based mehod s used o sudy he propery of me-emperauredependen vscoelasc behavor of undreconal fber renforced composes wh hermorheologcal smple marx, and a novel mul-scale mehod for analyzng he vscoelasc propery of undreconal compose maerals s developed. he characer of he vscoelasc relaxaon law subec o varaon n emperaure s nvesgaed, and he global effecve vscoelasc relaxaon modulus consderng he emperaure-dependency of properes was gven. Numercal examples for a plae of undreconal fber renforced compose maeral are presened. hrough he analyss of he hermal sran of he plae, s found ha he me effec of he relaxaon s small, so he coeffcen of hermal expanson may be gven by he nsananeous hermal deformaon. Inroducon Compose maerals have been wdely used n advanced ndusres for her nheren advanages, such as desgnable characer, hgh specfc srengh and specfc sffness. Researches have physcally shown ha compose maerals can exhb vscoelasc behavor, parcularly hose conanng polymers and vscoelascy marx [-]. he undersandng of he vscoelasc behavors for heerogeneous compose maerals ha accouns for her mcrosrucures sll requres furher sudy and developmen. herefore, s very mporan o sudy mcro-mechancal mehods for predcng numercally or analycally he meemperaure-dependen vscoelasc properes of compose maerals based on he heerogeneous mcrosrucural deals of compose maerals. Exsng mcromechancal mehods for predcng he properes of maerals nclude selfconssen heory [3], Eshelby-Mor-anaa s mehod [4,5], he cells model [6,7], homogenzaon mehod [8-] and ohers. hese mehods have been successfully used o predc he elasc properes of composes [3]. he mcromechancal mehods have been employed o predc he vscoelasc properes of compose maerals [4-9]. For he undreconal fber renforced polymerc marx composes, based on he mcromechancal heory and he elasc-vscoelasc correspondence prncple, L and Weng [4] suded he aenuaon laws wh me of he fve ndependen vscoelasc consans of compose maerals. Lang and Du [5] used he Eshelby equvalen ncluson mehod o sudy he creep consuve relaons of parculae renforced composes, and developed he varaon laws for he modul of maerals wh me, ncluson volume and load. Based on he homogenzaon mehods, Lu e al. [6] predced he vscoelasc properes of mul-layered maerals and undreconal fber renforced composes, and nvesgaed he effec of he ncluson s volume fracon on he relaxaon modulus. Chung e al. [7] proposed a mcro/macro homogenzaon approach for vscoelasc creep analyss wh dsspave correcors for hererogeneous woven-fabrc layered maerals. Sefera e al. [8] employed fne elemen mehod o predc he vscoelasc properes of an E- glass/vnyleser plan weave, woven rovng compose maeral a hree dfferen emperaures, and compared hem wh expermenal daa. hese

2 Shuan Lu, F.Z.Wang sudes gave he vscoelasc laws a consan emperaures, however he properes of compose maerals are emperaure-dependen. hough Sefera e al. [8] gave he vscoelasc properes of compose maeral for hree dfferen emperaures, he nfluence of he varyng emperaure hsory on he propery of compose maerals wasn consdered. he sudy on he vscoelasc laws under varyng emperaure saes has rarely been repored. In our prevous wor [9], he vscoelasc laws under a very parcularly varyng emperaure saes, n whch he emperaure has a ump a a specfc me (e.g, a he begnnng and hen eeps consan, were suded. he vscoelasc properes under general varyng emperaure sage should be nvesgaed. Zhang e al. [] sysemcally suded he vscoelasc properes of sngle phase maerals under me-dependen emperaure change, and expressed he vscoelasc consuve equaons under varyng emperaure saes n he same form as ha under consan emperaure saes. For compose maerals, alhough he consuve equaons under a me-ndependen emperaure sae can be expressed as he same form as ha of a sngle phase, s sll unnown f he consuve equaon under me-dependen varyng emperaure sae can be expressed as a smlar form. he man purpose of hs paper s o develop a novel mul-scale mehod for analyzng he vscoelasc properes of compose maerals, and o nvesgae he characerscs of he vscoelasc relaxaon law under me-dependen emperaure changes. Based on he homogenzaon heory, he mul-scale analyss mehods of he vscoelasc properes and he effecve hermal sress relaxaon laws are suded. Numercal examples are presened n he end of he paper. me-emperaure-dependen Consuve Equaon of Undreconal Compose Maerals In hs paper, we wll nvesgae he meemperaure-dependen consuve equaon of undreconal fber renforced compose maerals. he fbers are elasc maerals and he marx s a hermorheologcal smple maeral. he emperaure consdered here s me-dependen.. Vscoelasc Consuve Equaon Owng o he heerogeney of undreconal fber renforced composes, he consuve equaon s dfferen from poson o poson. Denoe he domans occuped by he marx and fber as m and f, respecvely, hen he oal doman of he composes s he sum of hs wo domans. = ( f In he fber phase doman, he maeral exhbs elasc propery, and he consuve equaon can be expressed as m σ ( x, = E ε ( x,, x ( l l f For he hermorheologcal smple maerals, hrough he horzonal funcon χ (, he curve of relaxaon a dfferen emperaures can be brough no a sngle one on he reduced-me scale ξ ( ξ = χ(, hereby formng he bass of he meemperaure superposon prncple. Expermenal daa of many polymers ndcaed he exsence of such a shf facor α, for ha log χ( = logα (3 Based on he Wllam-Landel-Ferry equaon (WLF, one has: logα C ( r = C r (4 Where, he parameer C and C are relaed o he free volume of he polymer and are dependen on he chosen reference emperaure r. For a emperaure change hsory ( relave o he reference emperaure r, he reduced-me scale ξ s gven by exp[ C ( ( C ( ] d (5 ξ = η η η hus, he consuve equaon of maerals n he marx doman can be expressed as: dεl ( τ d( τ σ ( = ( '[ ] l ξ ξ αl (6 where, τ exp[ C ( ( C ( ] d (7 ξ = η η η By he negral ranson, and, τ beng replaced by ξ, ξ respecvely, hen Eq.(5 s rewren as:

3 EMPERAURE-DEPENDEN VISCOELASIC PROPERIES OF UNIDIRECIONAL COMPOSIE MAERIALS ξ dεl ( ξ ' d( ξ ' σ ( = ( '[ ] ' l ξ ξ αl dξ (8 dξ' dξ' If we defne he relaxaon modulus of he fber maeral as s l u u v u v [( x y y y x l l l v α l ]d=, v( x, y V y Χ (5 ( f ( ξ = E (9 l hen, he consuve equaon of he fber can be expressed as a smlar form as Eq.(8. ξ f dεl ( ξ ' d( ξ ' σ ( = ( '[ ] ' l ξ ξ αl dξ ( dξ' dξ'. Effecve me-emperaure-dependen Vscoelasc Properes of Composes For a undreconal fber renforced compose maeral wh elasc fbers dsrbued perodcally n a hermorheologcally smple vscoelasc marx, he vscoelasc governng equaon of he maeral under he acon of body forces, racons and emperaure ncremens, can be expressed as v { σ ( } d f vd v ds, x = Γ ( v V= vx ( x, vγ = l { d } Subsung Eq.( no Eq.(, we hen have: ξ u ( x, ξ ( x, ξ (, l x ξ ξ αl dξ xl ξ ξ v d fv d v ds=, v V x Γ Laplace ransformaon of Eq. ( yelds ( ξ u (, x ξ (, (, l x s αl x s dξ xl ξ (3 v d fv d v ds=, v V x Γ Based on he dea of homogenzaon heory [8,,], dsplacemens can be expressed as a double-scale asympoc seres: u x = u x u x y u x y (4 (, (, ε (,, ε (,, Subsung Eq. (4 no Eq. (3, equang he erms wh he same power of ε yelds: u u v u u v s l[( ( xl yl x xl yl y v α l ]d f vd vds =, x (6 Γ v ( x,y V Where, Χ v( xy, V ={ v( x, y ( x, y Υ, vxy (, = vxy (,, vγ = } ang he lm ε and consderng he perodc characers of he funcons, Eqs. (5 and (6 become u u v s ( d, l lαl = xl yl y v(y V = v( y y Υ, v( y = v( y { } (7 u u v { s l[( αl ]d }d y xl yl x fv d v ds=, v( x, y V Γ Χ (8 Eq. (8 s lnear abou u, whose soluon can be expressed as: u = u u (9 u u where, u and u are respecvely he soluons of he followng wo equaons: θ u v s d, ( V l αl = yl y y v y ( u u u v s d, ( V l = yl xl y y v y ( Owng o he lneary of Eqs.( and (, Eq.(9 can be expressed as: l u = χ u x Ψ, =,,3 ( ( l d 3

4 Shuan Lu, F.Z.Wang Where, he generalzed funcons l χ ( y, s and Ψ ( y, s are respecvely he soluons of he followng wo equaons: mp ( v d, (, V mp χ l yl y = v x y (3 v s ( Ψ d, (, V l αl = v x y yq y (4 Inroducng Eq.( no Eq. (8 yelds u v s l β d f vd xl x v dγ=, vx ( V Γ where (5 l m χ l = [ ]d l mn yn (6 Ψ β = [ ]d lαl l y (7 Eq.(5 s smlar o Eq.(, and β has he same role as l α l n Eq. (. herefore, l represens he Laplace ransforman of he effecve vscoelasc relaxaon modulus of composes, and β s defned as he effecve me-dependen hermal relaxaon modulus (ERM. ERM can be expressed n erms of l χ ( y, s as β χm = [ l l mnl l ]d α α (8 y In fac, l χ ( y, s V, hus, Ψ m [ χ mnlαl mnl ] d =. yl yn Consderng hs fac, he followng equaons can be obaned: β χm = [ l l mnl l ]d α α (9 y Defne macroscopc sress σ ( by: = l l ( σ s u x s β (3 he nverse Laplace ransformaon yelds: l n n ξ ( u ( x, ξ σ ( = ( l ξ ξ ξ xl ξ ( ( x, ξ β ( ξ ξ dξ ξ x l (3 Eq. (3 represens he effecve vscoelasc consuve equaon of compose maerals, whch ncludes he hermal sress relaxaon. he sress and s Laplace ransformaon a any poson n he doman of compose maerals can be expressed as: σ x = σ x εσ x (3 (, (, (,y, σ = σ εσ (33 ( x, s ( x, s ( x,y, s where, ( ( u m um σ ( x, s = s mn s mnαmn (34 xn yn ( ( ( u m um σ ( x, s = s mn, =,, (35 yn yn Subsuon of Eq. ( no Eq. (34 yelds l χ m u σ ( s = s l mn yn xl Ψ m s mn αmn yn (36 ang he volumerc average of he equaon above over he base cell, we oban: σ (x, s = σ (x, s (37 whch means ha he macroscopc sress s he mean value of he frs approxmae sress over he base cell. In hs secon, he vscoelasc consuve equaon of compose maerals s presened. A homogenzaon-based mul-scale mehod s developed for predcng he effecve relaxaon modulus and ERM, and solvng vscoelasc problem of compose maeral srucures. he man schemes of hs mehod are summarzed as follows: Oban he generalzed dsplace funcon χ l (y by solvng Eq. (8; Calculae he effecve relaxaon modulus and ESRM by nversely ransformng Eqs. (6 and (8; 3 Solve Eq. (5 o ge he macroscopc dsplace n he ransformed space; 4

5 EMPERAURE-DEPENDEN VISCOELASIC PROPERIES OF UNIDIRECIONAL COMPOSIE MAERIALS 4 he sress whch shows he local heerogeneous nfluence s obaned from he nverse ransformaon of Eq. (36 5 he macroscopc sress and he consuve equaon are obaned by Eq. (3..3 hermal Expanson Propery For he convenonal maerals, f ( s he hsory of emperaure, he sran wll be gven by d ( τ ε ( = α (38 α represens he hermal expanson coeffcen and s a consan under consan emperaure. So he hermal sran complees nsananeously under he unformly ncreasng emperaure. hen he hermal sran of undreconal composes can be expressed n a smlar form under unformly ncreasng emperaure: d ( τ ε ( = α ( τ (39 If he effecve global sress equals o zero, he Laplace ransform of he consuve equaon (3 s gven by l l βl ε = (4 If ˆ α ( ξ ( s defned by ˆ αl ( s = l βl s (4 hen he nverse Laplace ransform s gven by ˆ α ( ξ( = Rev l βl s (4 Rev ( f ( s represens he nverse Laplace ransform for he funcon f (s. hen he hermal sran relaed wh emperaure s gven by ( ( ˆ d τ ε = α ( τ (43 ha equaon llusraes he hermal sran of undreconal composes can no complee nsananeously. If ˆ α ( s defned as he effecve hermal expanson coeffcen of undreconal composes, s dependen on emperaure, and s called as he effecve hermal expanson coeffcen a fnal emperaure. 3 hermal Deformaon Analyss of a Plae of Undreconal Fber Renforced Composes 3. Problem Descrpon As an example, he vscoelasc hermal deformaon of a undreconal fber renforced compose plae s consdered. he four edges of he plae are fxed as shown n Fg.. he undreconal fber renforced composes conan a nd of carbon fbers 3 embedded n a vscoelasc marx. he marx s a nd of resn called ED-6, and s volume deformaon s elasc and shear deformaon follows he hree-parameer sold model as shown n Fg.. Fg. hermal expanson of a plae Fg. hree-parameer sold model he relaonshp of he sress and he sran n he drecon of hcness s gven by ξ ε 33 ( ξ σ33 ( = 3333 ( ξ ξ ' ξ d ( ξ ' β33 ( ξ ξ ' ] dξ = dξ ' For he Laplace ransform, s wren as ( β hermal exp anson drecon (44 ε ( s = ( s ( s ( s (45 he ransform n he equaon s for he reduced-me scale ξ. Based on he resuls under he consan η 5

6 Shuan Lu, F.Z.Wang emperaure, he parameers n Eq.(45 are gven by followng equaons and able. ~ ~ β q ( s = s β q ( s = s q s p β q s p β (46 M a M ( =, ( = M b M ( =, ( = M c M 3 ( =, 3 ( = = = = ( ( ( a c ξ (5 e b e cξ c able.parameers of he relaxaon modulus under he reference emperaure q (Pa q(pa P(h β Laplace ransform of he sory of emperaure I s dffcul o ge he Laplace ransform of he hsory of emperaure. For he smplfcaon, ( s approxmaed as n ( = ( Δ ( (47 = Where, ( s he eavsde funcon. he approxmaed hsory of he emperaure s shown n Fg.3. hen, he varaon of he emperaure wh reduced-me scale s gven by ( ( ( ( (48 [ ξ ] = ξ ξ ξ = and he correspondng reduced-me scale becomes b c ( p p ( p A β β 3 = A β β ( p ( p A ( A p ( A A 3 = A β ( A( A p p A a = A c = p c = A = q p n Δ 3 β,, β p A q q Fg.3 Unformly ncreasng emperaure Δ n (53 [ ] = d = ξ χ ( τ τ χ( = [ ] d = = ξ χ ( τ τ χ( ( χ( =,3, n (49 Fg.4 Insananeously ncreasng emperaure he Laplace ransform of he emperaure and he sran can be expressed as ( ξ =, ( ξ ( e s = s s = = 3,,,n Where, c c ε ( ξ M M e M e ξ s (5 ξ ξ = ( Analyss of he resuls Frsly, he hermal sran of he plae nduced by a ump of emperaure a he begnnng are calculaed. Fg.5 shows he sran of he plae wh he ump of. hen, he hermal sran of he plae nduced by a unformly ncreasng emperaure a he begnnng sage of me wh a upper lm of emperaure of (as shown n Fg. 3 are calculaed. Fg. 6 and Fg. 7 show he sran hsores wh ncreasng raes of 6

7 EMPERAURE-DEPENDEN VISCOELASIC PROPERIES OF UNIDIRECIONAL COMPOSIE MAERIALS emperaure of /h, /h, and 4 /h. he varaon of he sran wh me a he las sep of emperaure change s shown n Fg.8. hrough he above curves, we found ha when he emperaure ncreased o unformly, he ncremen of he sran s more han ha under he nsananeously ncreasng o.he ncremen for he former s Δε =.4456e 4, whle s for he laer s Δε =.593e 6. ( (, α( ( α = (54 hus, he hermal sran s gven by ( τ ε (, = α ( τ d τ And he hermal sress s ( ( τ (55 ( ˆ ( τ σ (, = ( τ α( ( τ τ (56 ξ ( ξ = ( ξ ξ α ( ( ξ dξ ξ Fg.5 Sran curve under nsananeously ncreasng emperaure Fg.7 Sran curve under varous rae of emperaure Fg.6 Sran curve under unformly ncreasng emperaure Fg.7 shows he hermal sran ha under he varous rae of emperaure. he relaxaon s quc as he rae of he emperaure ncreasng, and he fnal values of he sran are he same as he curves end o consan. When he emperaure ncreased unformly sep by sep, he relaxaon rae s so hgh ha he ransen process s very quc. ha s, he me effec of he relaxaon s small, so he coeffcen of he hermal expanson may be gven by he ransen hermal deformaon. ha s Fg.8 Sran curve under he las sep of emperaure 4 Summary and Concluson In hs paper we have developed a novel mul-scale mehod for analyzng he vscoelasc propery of undreconal compose maerals, and have nvesgaed he characer of he vscoelasc relaxaon law under varyng emperaure. A homogenzaon-based mehod for predcng he effecve vscoelasc relaxaon modulus consderng he emperaure-dependency of properes was gven. 7

8 Shuan Lu, F.Z.Wang he numercal resuls show ha, when he emperaure ncreased unformly sep by sep, he relaxaon rae s so hgh ha he ransen process s very quc. ha s, he me effec of he relaxaon s small, so he coeffcen of he hermal expanson may be gven by he ransen hermal deformaon. 4 Acnowledgemens hs research s suppored n par by he NSFC hrough he ran No.s (573, 33 and 4, he Naonal Basc Research Program of Chna. he fnancal conrbuons are graefully acnowledged. References [] Wllams J.. On he predcon of resdual sresses n polymers. Plascs and Rubber Processng and Applcaons, Vol., pp , 98. [] Brnson L C. and Ln W S. Comparson of mcromechancs mehods for effecve properes of mulphase vscoelasc composes. Compose Srucures, Vol.4, pp , 998. [3] ll R. A self conssen mechancs of compose maerals. J. Mech. Phys. Solds, Vol.3, pp3-, 965. [4] Mor, anaa K. Average sress n marx and average elasc energy of maerals wh msfng nclusons. Aca Meallurgy, Vol., pp57-574, 973. [5] Benvense. A new approach o he applcaon of Mor-anaa s heory n compose maerals. Mech. Maer., Vol.6, pp47-57, 987. [6] Paley M. and Aboud J. Mcromechancal analyss of composes by he generalzed cells model. Mech. Maer. Vol.4, pp7-39, 99. [7] hosh S., Moorhy S. and Lee K. Mulple scale elasc-plasc analyss of heerogeneous maerals wh he vorono cell fne elemen model. Compuaonal Mehods n Mcromechancs, ASME, Vol., pp87-5, 995. [8] Bensoussan A., Lons J.L. and Papancolaou. Asympoc analyss for perodc srucures. Norh- olland, New or, 978. [9] ollser S J. and Kuch N. A comparson of homogenzaon and sandard mechancs analyses for perodc porous composes. Compuaon Mechancs, Vol., pp73-95, 99. [] Luassen D., Persson LE. and Wall P. Some engneerng and mahemacal aspecs on he homogenzaon mehod. Composes Engneerng, Vol.5, No.5, pp59-3, 995. [] Lu S.. and Cheng.D. omogenzaon-based mehod for predcng coeffcens of hermal expanson of compose maerals. Chnese Journal of Aca Maerae Composae Snca, Vol.4, pp76-8, 997. [] assan B. and non E. A revew of homogenzaon and opology opmzaon I: homogenzaon heory for meda wh perodc srucure. Compuers and srucures, Vol.69, pp77-77, 998. [3] Aboud J. Mechancs of Compose Maerals: A Unfed mcromechancal approach. Elsever, Amserdam, 99. [4] L J and Weng J. Effecve creep behavor and complex modul of fber and rbbon-renforced polymer-marx composes. Compose Scence and echnology, Vol.5, 65-69, 994. [5] Lang J. and Du S.. Sudy of mechancal properes of vscoelasc marx compose by mcromechancs. Chnese Journal of Aca Maerae Composae Snca, Vol.8, No., pp97-,. [6] Lu S.., Chen K.Z. and Feng X.A. Predcon of vscoelasc propery of layered maerals, Inernaonal Journal of Solds and Srucures. Vol.4, No.3, pp , 4. [7] Chung P. W., amma K. K. and Namburu R. R. A mcro/macro homogenzaon approach for vscoelasc creep analyss wh dsspave correcors for heerogeneous woven-fabrc layered meda. Composes Scence and echnology, Vol.6, pp33-53,. [8] Sefera O. E., Schumacherb S. C. and ansena A. C. Vscoelasc properes of a glass fabrc compose a elevaed emperaures: expermenal and numercal resuls. Composes Par B: Engneerng, Vol. 34, No.7, pp57-586, 3. [9] Shuan Lu and Nng Ma. he hermal sress and consuaon of vscoelasc composes (I: heoral Analyss. Aca Maerae Compose Snca, Vol., No., 5. [] Zhang.. eneral heory of Vscoelascy under Non-Consan emperaure Saes. he 999 Brsh Appled Mahemacs Colloquum, Bah, U.K. pp-5,