THREE-DIMENSIONAL CONSTITUTIVE EQUATION OF SHAPE MEMORY POLYMERS AND THEIR COMPOSITES

Transcription

1 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS THREE-DIMENSIONAL CONSTITUTIVE EQUATION OF SHAPE MEMORY POLYMERS AND THEIR COMPOSITES H. Pak 1, W.-R. Yu 1*, P. Hason, Z. Guo 3 1 Depatment of Mateals Scence and Enneen, Seoul Natonal Unesty, Seoul, Republc of Koea, School of Enneen, Unesty of Glasow, UK, 3 Depatment of Enneen Mechancs, Chonqn Unesty, Chna * Coespondn autho (woonyu@snu.ac.k) Keywods: shape memoy polyme, consttute equatons, non-mechancal stan 1 Intoducton Shape memoy polymes (SMPs) ae smat mateals that can ecoe the pemanent shape fom tempoaly fxed one when a stmulus such as heat s en. Vaous attempts hae been made to deelop aous smat deces and applcatons usn SMPs. Howee, the consttute modeln of SMPs s slowly poessed. Most of the consttute models deeloped so fa assume small defomatons, een thouh SMPs featue the ablty to ecoe the pemanent shape fom moe than a few hundeds pecentae of defomaton. In addton, many models ae lmted to one-dmensonal defomaton case. In ths eseach, thee dmensonal consttute model fo shape memoy polyme was deeloped by usn the non-mechancal stan. Bascally ths model s a type of two phase descpton wth scohypeelastcty. The man featue of the model s a theoetcal descpton of the non-mechancal stan that nokes shape memoy effect n the thee dmensonal defomaton. We assumed that the noechancal stan of SMP s eneated adually, the eneaton ate of whch depends on tempeatue; the hhe tempeatue, the lae ate ncease. Fnally non-mechancal stan was descbed usn the pncpal axes and pncpal stetches of the total defomaton. Fo aldaton puposes, the shape memoy tests wee smulated and compaed wth expemental esults of themomechancal cyclc test of coss-lnked poly(cyclooctene) (PCO). Consttute equatons.1 Model descpton based on two phase F. 1 shows a phenomenolocal model of SMPs whch we employed to deelop the consttute equaton. To teat the lae defomaton of SMPs, we assumed that the total defomaton adent could be decomposed multplcately [1]: e FSMP F F (1) e FFF whee the subscpts,, and epesent the ubbey and lassy phases, and non-mechancal stan, espectely. Note that the supescpts e and epesent hypeelastc spn and Vot elements, espectely, n ou phenomenolocal model as shown n F. 1. If and ae Helmholtz fee eney of ubbey and lassy phases, espectely, the total fee eney ( ) can be expessed usn a ule of mxtue as follows. () whee and ae the olume factons of ubbey and lassy phases, espectely. Note that the sum of these two quanttes should be always one. At the tempeatue hhe than the tanston tempeatue ( T ), e.., T T 15 o C, the olume facton of the lassy phase s zeo,.e., only ubbey phase exsts. At low tempeatue (e.., T T 15 o C ), the opposte stuaton s assumed. The Clausus-Duhem nequalty based on second Pola Kchhoff stess ( S ) s expessed as follows q T 1 T S : C 0 (3) T whee T s the tempeatue, s the mass densty of the body, and q s the heat flux ecto.

2 If the ule of mxtue fo the Cauchy stess s expessed as σσ σ, the second Pola- Kchhoff stess can be en by T S JFσF (4) T T JFσF JFσ F S S Substtutn eqs. () and (4) nto (3) es th 0 (5) whee s ae the mechancal dsspatons defned as follows, and assumed equal o lae than zeo: 1 : : S C C C C (6) 1 S : C : C C C : C C (7) T th T q (8) T T To exst s, the follown elatons hae to be satsfed: 1 S C C (9) 1 S C C (10) C C (11) C C (13). Glassy phase Fo the lassy phase consstn of hypeelastc spn (fst ode Mooney-Rln), Vot element, and non-mechancal stan element, the fee eney can be decomposed as follows e CC,, C, C C (14) wth whee I, II, III I, II, III IP e e e e I, II, J C10, I 3 k k k k k k k k k k C01, II J k I, 3 1 k II, and (15) k J ae ncompessble naants. Supescpt k s e o. Accodn to [3] and [4], the pseudo potental of dsspaton can be expessed by 1 kc : C (16) whee k s the scosty of the Vot element. Then, eqs. (11) and (1) ae ewtten as : /3 e 1 e e e J C10, I C10, e /3 e 1 e 1 J C01, C C C F S F I T 1 I C II C 3 1 e e e T J J 1C e e e e 10, 01, (17) C C (1)

3 THREE-DIMENSIONAL CONSTITUTIVE EQUATION OF SHAPE MEMORY POLYMERS AND THEIR COMPOSITES /3 1 10, 01, /3 1 1 e e e J C01, C C C k C J C C C I C I e e e e e o smply : e e e e e e e I C10, II C01, J J 1C I /3 4/3 10, 01, 01, C J C I C J C Rubbey phase 1 I C10, II C01, J J 1 1 C (18) S f C, F, F (19) S C k C f C, F, F (0) As shown n F. 1, ubbey phase does not hae non-mechancal stan element. Theefoe, lettn F I and eplacn the subscpts wth n eqs. (17) and (18) es S /3 e e e e J C10, I C10, e e J C01, C C C / I C II C 3 1 e e e T J J 1C e e e e 10, 01, /3 1 10, 01, /3 1 1 e e J C01, C CC k C J C C C I C I e e e e o smply : e e e e e e e I C10, II C01, J J 1C I /3 4/3 10, 01, 01, C J C I C J C I C 1 I C10, II C01, J J 1 1 (1) () S f C, F (3) S k C f C, F (4) C.3 Non-mechancal stan The non-mechancal stan s a defomaton eneated n SMP whch s elated the tempoay shape of unloaded sample afte defomaton. Bascally the non-mechancal stan can be eneated only when the total defomaton occus and ts amount and decton ae detemned by the total defomaton. If the sample s unloaded at T T l afte the non-mechancal stan s eneated, t does not ecoe ts onal shape but nstead the ecoey s stopped at cetan shape whch coesponds the non-mechancal stan and s called as tempoay shape. Unloadn means the stess elease so that othe elements n the sample affected by the mechancal stess ecoe the onal shapes. Only emann defomaton s the noechancal stan whch s not dectly affected by stess. Othe elements (n ubbey phase) cannot etun to the onal shape by the ntefeence of the non-mechancal stan. In addton, fo the fxed total defomaton F, f the non-mechancal stan becomes lae, then the esultn stess on the sample becomes elaxed. Consde the defomaton adent of the noechancal stan ( F RU ) and assume no d body otaton. Then, U s calculated fom C. If the bass ectos of the ht Cauchy Geen tenso C s M, the spectal decomposton of C can be expessed by 3 C M M 1 (5) whee ae the pncpal stetches whch can be calclated by the Laane stan alon the pncpal 1 decton defned as E 1. When the mateal s extended up to E, noechancal stan E appoaches to ae adually. 3

4 But f E s lae than ae, E does not ncease anymoe. Ths can be expessed as de E ae (6) dt fo ae E whee a s the poducton ato of non-mechancal stan to en extenson whle s the poducton and extncton ate of non-mechancal stan whch s the functon of stan and olume facton: E, E (7) 1 E whee exp wth E E : E. Eo In any case, E cannot exceed E because the non-mechancal stan s one of component of total defomaton, as mentoned aboe. If extended mateal stats to contact and E become smalle than E appoachn to E : de dt E E fo E E (8) Wth E 1, the fnal fomula of the defomaton adent of non-mechancal stan s en as follows : F RU U (9) 3 1/3 M M J 1 whee J, e 1 s the olume chane of the 1/3 non-mechancal stan. The facto 1 J s ntoduced fo ncompessblty of all elements e e ( det F = det F = det FF = det FFF 1). 3 Expemental detals 3.1 Mateal pepaatons Syntheszn PCO sample s based on [5]. PCO was dssoled n Tetahydofuan (THF) to fom a PCO soluton at 80. Afte 4h, dcumyl peoxde (DCP) was added nto PCO-THF soluton. The ato of DCP to PCO was 1wt%. At oom tempeatue fo 4h, the soluton was placed n a fume hood to apoze THF. Afte that, t was put nto acuum oen fo 1h at 80. Then, t was cut nto specmen wth a dmenson of cm wth the aue lenth of 3cm. A peheatn was appled to the specmens fo 5mn at 80 afte whch a heatn fo 30mn at 170 was futhe teated fo the fnal samples. 3. Themomechancal cyclc unaxal test fo shape memoy effect Shape memoy test conssts of fou steps: (a) Extenson: the sample was extended up to 00% at a ate of 33.6mm/mn at T h. (b) Relaxaton and cooln: Unde keepn stan, the sample was cooled down to Tl 30 at a ate of 1 / mn. Afte the tempeatue was eached to T l, the elaxaton state was contnued fo about 300s. (c) Unloadn and zeo stess: The stess was eleased and the state was mantaned dun 300s. (d) Heatn: Unde zeo stess, the tempeatue was nceased up to T h at a ate of 0 / mn and the chane of stan was obseed. Dun step (a) and (b), the extenson and stesss wee nput and output whle these wee toled dun step (c) and (d). 3.3 Themomechancal cyclc punchn test wth punchn plana sample PCO-DCP sheet wth a dmenson of cm was pepaed. The fou sdes of the sheet wee fxed. The cente of the sheet was pessed up to cetan heht by d sphee wth the adus of 0.65cm. Thee pessn dsplacement wee set: 30mm, 37.5mm, 45mm. In ths expemental, the dsplacements of the punch was measued.

5 THREE-DIMENSIONAL CONSTITUTIVE EQUATION OF SHAPE MEMORY POLYMERS AND THEIR COMPOSITES 4 Smulatons Two knds of smulatons wee caed out to aldate the cuent consttute equatons by compan the esults wth expements n secton Unaxal tensle and punchn smulaton A ectanula sheet was modeled as shown n F.. Punchn smulaton of a SMP sheet was caed out (see F. 7 fo detaled meshes). A ball n adus of 0.65cm was used to pess the flm upwad at cente pont. All steps of smulaton wee the same as those n secton Results and dscussons 5.1 Unaxal tensle and punchn smulatons F. 3 and F. 4 ae the esults of themomechancal cyclc test. The defomaton fues ae n F. 5. The modeln esults ft well expemental esults. By ths calculaton, dstbuton of the noechancal stan can be obtaned. F. 6 shows the dstbuton of E 1 alon the cental lne whch 15 mm,0,0 and connects the ponts 15 mm,0,0. t 11s means the tme pont when stan eached to maxmum stan at T Th. t 511s s the tme just befoe unloadn whch means last mnute of eneatn non-mechancal stan. Thouh the mechancal behaou of the sample s not affected by non-mechancal stan at t 11s accodn to eq. (4), t was aleady eneated. And the dstbuton of non-mechancal stan between the two tme steps s not much dffeent. Thus t means that the eneaton ate of non-mechancal stan at T T s ey fast, that s, the non-mechancal stan s eneated almost mmedately when the defomaton occus. The noechancal stan was eneated n close to cente of the sample than n constaned eon. It s consstent wth moe extenson occun n cental eon athe than n ede aea. If the mateal defoms unfomly, the ede aea would be elmnated so that the fxty of SMP wll ncease. h befoe t 511s, afte whch esults ae shown. The fxtes detemned n smulaton ae always hhe than expemental one. The smulaton shows the ecoey of the onal shape whle unecoeed defomaton emaned n the expements. 6 Conclusons A thee-dmensonal consttute model wth two phase was deeloped. Themodynamcally deed equatons and nequalty fomed the consttute equatons of ubbey and lassy phases. The noechancal stan was added nto lassy phase and was detemned fom pncpal axes and stetches of the total defomaton. The consttute models was used to smulate unaxal and punchn tests, demonstatn that the deeloped consttute equaton s sutable to pedct the defomaton behaou of SMPs. Refeences [1] E.H. Lee, Elastc-Plastc Defomaton at Fnte Stans. Jounal of Appled Mechancs, Vol. 36, pp 1-6, [] H. Pak, W.-R. Yu, Themo-mechancal consttute modeln of shape memoy polymes and ts applcaton to desnn smat compostes. Intenatonal Symposum on Plastcty, 013. [3] A. Boukamel, C. Gabel, S. Méo, Modélsaton en andes défomatons scoélastques des élastomèes. Actes du 3eme Colloque Natonal en Calcul des Stuctues, pp , [4] S. Meo, A. Boukamel, O. Debodes, Analyss of a themoscoelastc model n lae stan. Computes and Stuctues, Vol. 80, pp , 00. [5] C. Lu, S.B. Chun, P.T. Mathe, L. Zhen, E.H. Haley, E.B. Couhln, Chemcally Coss- Lnked Polycyclooctene: Synthess, Chaactezaton, and Shape Memoy Behao. Macomolecules, Vol. 35, pp , 00. F. 8 shows the z -dsplacement of the cental pont accodn to tme. The defomed shapes ae shown n F. 9. Because the z -dsplacement s the nput 5

6 F. 1. Phenomenolocal model of SMP. F.. Geomety and mesh confuaton of the ectanula sample. F. 3. Compason of unaxal tensle smulaton of SMP wth expement F. 4. Compason of unaxal tensle smulaton of SMP wth expement: Stess-stan cue

7 THREE-DIMENSIONAL CONSTITUTIVE EQUATION OF SHAPE MEMORY POLYMERS AND THEIR COMPOSITES 15mm, 0, 0 to 15mm, 0, 0 n mateal coodnate). t 0 t 11 s F. 7. Geomety and mesh confuaton of the SMP sheet. t 511s t 811s t 1351s F. 5. Thee dmensonal shape chanes of SMP sample dun unaxal tensle test. Colou means the mantude of non-mechancal stan,.e. E. F. 8. Expemental and modeln esults of themomechancal and punchn test: Tme eoluton of z-decton dsplacement of cente pont. F. 6. Dstbuton of x-component of noechancal stan alon the cental lne (fom 7

8 t 0 t 11s t 511s t 811s t 1351s F. 9. Thee dmensonal shape chanes of SMP sheet dun punchn test when the maxmum z- dsplacement s 45mm. Colou means the mantude of non-mechancal stan,.e. E.