Numerical evaluation of rheological experiment

NUMERICAL EVALUATION OF RHEOLOGICAL EXPERIMENT Pe Salaˇ c *I o Ma ouˇ sek Technical Uni e si y o Libe ec Facul y o Science, Humani ies and Educa ion S uden sk´ a 2, 461 17, Libe ec, Czech Republic pe [email p o ec ed] *Technical Uni e si y o Libe ec Facul y o Mechanical Enginee ing S uden sk´ a 2, 461 17, Libe ec, Czech Republic i [email p o ec ed] Abs ac A iscoelas ic simply suppo ed o a ionally symme ic body, ixed on a base, is conside ed. The body is loaded by a la plunge , which mo es in he di ec ion o he zaxis by a cons an eloci y . In his wo k he eac ion o ce is compu ed. This allows us o compa e nume ical esul s wi h da a om heological expe imen (see [6], [7]). The a ia ional o mula ion o he p oblem is de i ed and ans o med o cylind ical coo dina es. Some esul s o nume ical calcula ions a e p esen ed. Keywo ds: Viscoelas ici y; axisymme ic hype bolic p oblems; dimensional educ ion. In oduc ion Ma hema ical modeling o echnological p ocesses has been al eady ega ded as a powe ul ool o an op imiza ion o echnological p ocesses also in glass indus y. The undamen al issue o i ual modelling o silica glass o ming is an accu acy o nume ical ou pu s. C i ical ac o is no only de ini ion o bounda y condi ions, mainly he mal ones, bu also speci ica ion o ma e ial p ope ies. Numbe o me hods o an iden i ica ion o heological p ope ies exis s [1]. The disad an age o he mos o published models is independen desc ip ion o he bo h s ages - he s age wi h he dominan in luence o an elas ic componen o de o ma ion and ha one wi h a dominan iscous low [8]. One o he mos e ec i e me hods is iso he mal comp ession me hod which is based on he e alua ion o he o ce esponse on comp ession loading o cylind ical samples [4], [6], [7]. The ad an age o his me hod is i s ela i ely simplici y and possibili y o e alua e bo h elas ic and iscous p ope ies o glass mel simul aneously du ing one expe imen . Howe e he c i ical issue o his me hod is an accu acy o e alua ion o expe imen al ou pu s. Se e al me hods we e sugges ed. In his con ibu ion we in oduce he a ia ional o mula ion o he heological expe imen model, which includes iscoelas ic de o ma ions. This p oblem will be used as a s a e p oblem in o mula ions o a ious iden i ica ion p oblems o a ious model pa ame e s, ha a e planed as nex s ep o ou esea ch. Ne e heless ecen nume ical esul s a e oughly in con o mi y wi h esul s o expe imen s (see [6]). 129 1 Fo mula ion o he P oblem Fo ming o glass is qui e complica ed p ocess which con ains bo h elas ic and plas ic esponses o s ain om s ess. This is he main eason o using a iscoelas ic model o desc ibe ela ion- ship be ween s ess and s ain. y z x P2 h P3 P1 Ω Sou ce: Own Fig. 1.Scheme o glass sample We conside a iscoelas ic iso opic homogeneous cylind ical body Ωsymme ical acco d- ing o zaxis wi h bases o med by wo pa allel ci cles wi h adii R, and al i ude h. We conside he body which is ixed on bo h bases and which is ee on i s su ounding su ace. We deno e P1uppe , esp. P3bo om, base and P2su ounding su ace o he body. The body is de o med by la plunge mo ing by a cons an eloci y in he di ec ion o he zaxis placed on he uppe base. To ep esen changes o he shape o he body we de ine a de o ma ion enso by he o mula ε ij(x x x, ) = 1 2 ∂ ui(x x x, ) ∂ xj+ ∂ uj(x x x, ) ∂ xi.(1) The s ess is ep esen ed by symme ical s ess enso σ . We conside s ess s ain ela ion gi en by iscoelas ic gene alized Hook’s law in he o m σ ij(x x x, ) = δ ij Z −∞ λ ( − τ ) ∂ε kk(x x x, τ ) ∂τ d τ +2Z −∞ µ ( − τ ) ∂ε ij(x x x, τ ) ∂τ d τ ,(2) whe e λ ( )and µ ( )deno e elaxa ion unc ions desc ibing glass p ope ies in p essing, esp. in shea and δ ij is he K oneke symbol. The balance o a linea momen um o he dynamic p oblem has he o m ∂σ ij(x x x, ) ∂ xj+Fi(x x x, ) = ρ∂ 2ui(x x x, ) ∂ 2i=1,2,3,(3) whe e F F F(x x x, ) ep esen s he body o ces and ρ he densi y o glass. We use he ime disc e iza ion me hod: Le he ime in e al [0,T]be di ided in o he subin e als [ k−1, k], o k=1,2,...,p, hen (3) has a each ime le el he o m ∂σ k ij(x x x) ∂ xj+Fk i(x x x) = ρ zk i(x x x)−2zk−1 i(x x x)+zk−2 i(x x x) h2i=1,2,3,k=2,..., p,(4) whe e zk i(x x x) = u(x x x, k), σ k ij(x x x) = σ ij(x x x, k),Fk i(x x x) = Fi(x x x, k)and h=T p. Acco ding o he ac ha he body, suppo and load ha e o a ional symme y we ans o m 130 he p oblem o cylind ical coo dina es and apply dimensional educ ion o an angle o ge wo- dimensional p oblem. We a e going o sol e he p oblem in he egion Dk( α )dependen on ime he le el kbounded by he axis (pa Γ3), he axis z(pa Γ4), he s aigh line z=h− (pa Γ1) and he ee bounda y desc ibed by he unc ion α (z)o a iable z(pa Γ2). z Γ2 Γ3 Γ1 Γ4Dk( α ) Sou ce: Own Fig. 2.Domain a e he dimensional educ ion Acco ding o he ac ha he p oblem has o a ional symme y we assume ha a displace- men ec o componen in he di ec ion ϑ is ze o (zk ϑ (x x x) = 0), simila ly ∂ zk ∂ϑ =0, and ∂ zk z ∂ϑ =0. We deno e he physical componen s o he displacemen ec o by wo unc ions, e.g. zk (x x x) = uk, zk z(x x x) = wk, (zk ϑ (x x x) = 0). The ela ionship be ween he displacemen ec o and he s ain enso is in he o m ε k = ∂ uk ∂ ,(5) ε k zz = ∂ wk ∂ z,(6) ε k ϑϑ =uk ,(7) ε k z =1 2( ∂ uk ∂ z+ ∂ wk ∂ ),(8) ( ε k ϑ =0, ε k z ϑ =0).(9) The componen s o he s ess enso σ σ σ ha e he o m σ k =1 hZ −∞ λ ( − τ )(ek(x x x)−ek−1(x x x))+2 µ ( − τ )( ε k (x x x)− ε k−1 (x x x))d τ ,(10) σ k zz =1 hZ −∞ λ ( − τ )(ek(x x x)−ek−1(x x x))+2 µ ( − τ )( ε k zz(x x x)− ε k−1 zz (x x x))d τ ,(11) σ k ϑϑ =1 hZ −∞ λ ( − τ )(ek(x x x)−ek−1(x x x))+2 µ ( − τ )( ε k ϑϑ (x x x)− ε k−1 ϑϑ (x x x))d τ ,(12) 131 σ k z =2 hZ −∞ µ ( − τ )( ε k z(x x x)− ε k−1 z (x x x))d τ ,(13) ( σ k ϑ =0, σ k z ϑ =0),(14) whe e e= ε + ε zz + εϑϑ .(15) The bilinea o m ep esen ing mechanical wo k o inne o ces has he o m A(u u u, ϕ ϕ ϕ ) = −1 hZDk( α )Z −∞( λ ( − τ )+2 µ ( − τ )) ∂ uk(x x x) ∂ ∂ϕ 1(x x x) ∂ + + ∂ wk(x x x) ∂ z ∂ϕ 2(x x x) ∂ z +uk(x x x) ϕ 1(x x x)1 + + µ ( − τ ) ∂ wk(x x x) ∂ ∂ϕ 2(x x x) ∂ + ∂ wk(x x x) ∂ ∂ϕ 1(x x x) ∂ z + + ∂ uk(x x x) ∂ z ∂ϕ 2(x x x) ∂ + ∂ uk(x x x) ∂ z ∂ϕ 1(x x x) ∂ z d τ dx x x+ + ρ h2ZDk( α )uk(x x x) ϕ 1(x x x) +wk(x x x) ϕ 2(x x x) dx x x.(16) Linea unc ional ep esen ing mechanical wo k o ou wa d o ces has he o m hF F F( ), ϕ ϕ ϕ i=ZDk( α )[Fk 1(x x x) ϕ 1(x x x) +Fk 2(x x x) ϕ 2(x x x) ]− −1 hZ −∞( λ ( − τ )+2 µ ( − τ )) ∂ uk−1(x x x) ∂ ∂ϕ 1(x x x) ∂ + + ∂ wk−1(x x x) ∂ z ∂ϕ 2(x x x) ∂ z +uk−1(x x x) ϕ 1(x x x)1 + + µ ( − τ ) ∂ wk−1(x x x) ∂ ∂ϕ 2(x x x) ∂ + ∂ wk−1(x x x) ∂ ∂ϕ 1(x x x) ∂ z + + ∂ uk−1(x x x) ∂ z ∂ϕ 2(x x x) ∂ + ∂ uk−1(x x x) ∂ z ∂ϕ 1(x x x) ∂ z d τ − − ρ h2huk−2(x x x) ϕ 1(x x x) +wk−2(x x x) ϕ 2(x x x) − −2uk−1(x x x) ϕ 1(x x x) +wk−1(x x x) ϕ 2(x x x) idx x x.(17) Bounda y condi ions: The la plunge mo es in he di ec ion o he zaxis by he cons an eloci y ac ing on pa o bounda y Γ1, i.e. u=0 w= ( − k)on Γ1.(18) The pa o bounda y Γ2 ep esen s he so called ee bounda y which is de o med by he in lu- ence o he inne o ces and is no able o ca ch any o ce ( angen o no mal) σ =0 σ zz =0on Γ2.(19) 132 The pa o bounda y Γ3is ixed, i.e. u=0 w=0on Γ3.(20) The pa o bounda y Γ4is o med by he axis o symme y and has p ope ies o con ac wi h solid suppo (wi hou ic ion), i.e. u=0 ∂ w ∂ =0on Γ4.(21) We de ine he space W1,2, (Dk( α )) wi h he no m kuk1,2, = ZDk( α )" ∂ u ∂ 2 + ∂ u ∂ z2 +u2# dx x x!1 2 .(22) We de ine hespaceo unc ions wi h he ini e ene gyas heweigh ed Sobole spaceH(Dk( α )) H(Dk( α )) = {ˆ u u u≡(u,w)∈W1,2, (Dk( α ))×W1,2, (Dk( α ))}.(23) We deno e V1={u∈C∞(Dk( α )) |suppu∩Γ4=/0,u=0 on Γ1∪Γ3}.(24) Le V1be he closu e o he se V1in he space W1,2, (Dk( α )). Fu he we deno e V2={u∈C∞(Dk( α )) |u=0 on Γ1∪Γ3}.(25) Le V2be he closu e o he se V2in he space W1,2, (Dk( α )). We deno e by H H H={ˆ u u u≡(u,w)∈V1×V2}(26) he space o es unc ions (i.e. such unc ions wi h ini e ene gy which sa is y s able bounda y condi ions). We use he p inciple o i ual displacemen o ge a a ia ional o mula ion o he p oblem: Le ˆu0∈H(Dk( α )) be gi en, which speci ies he displacemen on he bounda y Γ1by i s aces. We a e looking o ˆu∈H(Dk( α )) such ha ˆ u u u−ˆ u u u0∈H H H,(27) A(ˆ u u u,ˆ ϕ ϕ ϕ ) = hF F F( ),ˆ ϕ ϕ ϕ i ∀ˆ ϕ ϕ ϕ ∈H H H,k=2,3,...,p.(28) Theo em. The p oblem (27) - (28) has he unique solu ion. P oo : The p oo based on he Lax-Milg am heo em is oo long and echnical o be published. 2 Nume ical Expe imen The nume ical model, desc ibing he cou se o expe imen al measu emen s o heological p op- e ies o mel glass, was c ea ed. The p inciple o expe imen is based on he e alua ion o he o ce ( iscoelas ic) esponse o iso he mal cylind ical mol en glass sample, which is comp essed a a cons an eloci y. Nu- me ical simula ion was ealized in he comme cial FEM (Fini e Elemen s Me hod) code MSC MARC. 133 The ini ial sample sizes we e 20,3 mm (diame e ) - 18,45 mm (heigh ), he eloci ies o comp ession we e aken om he ange 0,5 - 40 mm/s. The Maxwell model was used o desc ip ion o ma e ial beha io o FLOAT mel ed glass. Viscosi y o he shaped glass was de ined acco ding o he expe imen , i.e. η =107,52 [Pa.s]. The modulus o elas ici y was selec ed om he ange E1=2,5.1082,5.109[Pa], mol en glass was assumed o be incomp essible subs ance, i.e. The Poisson cons an ν =0,5. S icking condi ions we e p esumed be ween glass and me al punch con ac su aces. Sou ce: Own Fig. 3.Dis ibu ion o he s ess ields in he o m o equi alen Cauchy s ess o comp ession 2, 6 and 10 mm The cou se o dis ibu ion o he s ess ields in he o m o he equi alen Cauchy s ess o 3 di e en s ages (comp ession 2, 6 and 10 mm) a e p esen ed in Fig. 3 ( o eloci y =4 [mm/s]). The cou ses o he o ce esponse o di e en elas ic moduli a e shown in Fig. 4. F om he igu e i esul s ha he elas ic modulus in luences only he i s s age o he expe imen , second one is only con olled by iscous low. Conclusion In he con ibu ion he model o e alua ion o desc ip ion o iscoelas ic o ce esponse o comp ession loading was sugges ed. In eg a ion o he iscoelas ic model o he Maxwell ype 134 Sou ce: Own Fig. 4.Cou se o compu ed load o ces o he ma hema ical model allowed ai desc ip ion o he o ce esponse acco ding o cha ac e o he ealized expe imen s [6]. The shape cou se o de o med glass sample in he expe imen was isibly simila o he compu ed one. De elopmen o he measu ed load o ce showed he simila endency bu alues became mo e di e en om he compu ed ones du ing he expe i- men . Acknowledgmen s The pape was suppo ed by he ESF P ojec No.CZ.1.07/2.3.00/09.0155 “Cons i u ion and imp o emen o a eam o he demanding echnical compu a ions on pa allel compu e s a TU o Libe ec”. Li e a u e [1] DUFFRENE, L.; GY, R.; MASNIK, J. E.: Tempe a u e dependence o he high- equency iscoelas ic beha iou o a soda-lime-silica glass. J. Ame . Ce am. Soc. 81, 1998, no. 5, pp. 1278-1284. [2] GR ¨ AFE, W.: Time-dependen Mechanical P ope ies o Solids. T ans Tech Publica ions Inc, 2008, ISBN 978-0-87849-476-7, ISSN 1422-3597. [3] NEˇ CAS, J., HLAV ´ Aˇ CEK, I.: ´ U od do ma ema ick´ e eo ie p uˇ zn´ ych a p uˇ znˇ e plas ick´ ych ˇ eles. SNTL, P aha, 1983. [4] HESSEKEMPER, H.; BR ¨ UCKNER R.: Load-dependen low beha iou o silica e glass mel s. Glas echnische Be ich e, 1988, ol. 61, no. 11, pp. 312-322. [5] CHRISTENSEN, R. M.: Theo y o iscoelas ici y. Academic P ess, NewYo k, 2003, ISBN 0-486-42880-X. [6] MATOUˇ SEK, I.; V´ ITOV ´ A, M.: Rheological beha iou o silica glass mel s. Skl´ aˇ a ke amik, ol. 59, 2009, ˇ c. 10-12, pp. 217-223. ISSN 0037-637X. 135 [7] MATOUˇ SEK, I.; V´ ITOV ´ A, M.: Rheological Response o Glass Mel s. Ce amics – Si- lik´ a y, ol. 53, 2009, ˇ c. 1, pp. 59-62. ISSN 0862-5468. [8] YU-CHUNG,T.; CHINGHUAH.; JUNG-CHUNG H.: Glass ma e ialmodel o he o m- ing s age o he glass molding p ocess. J. Ma e . P oc. Technol., 2008, ol. 201, no. 1-3, pp. 751-754. RND . Pe Salaˇ c, CSc., Ing. I o Ma ouˇ sek, Ph.D. 136 NUMERICK´ E HODNOCEN´ I REOLOGICK´ EHO EXPERIMENTU U aˇ zujeme iskoelas ick´ e, p os ˇ e podepˇ en´ e, o aˇ cnˇ e syme ick´ e ˇ eleso pe nˇ e spojen´ e s podkla- dem. Tˇ eleso je za ˇ eˇ zo ´ ano plochou liso ac´ ıˇ celis ´ ı, k e ´ a se pohybuje e smˇ e u osy zkons an n´ ı ychlos ´ ı . V pˇ edloˇ zen´ em pˇ ´ ıspˇ e ku je poˇ c´ ı ´ ana silo ´ a odez a. To n´ am umoˇ zn´ ı po o n´ a a nu- me ick´ e ´ ysledky s e´ aln´ ymi da y z eologick´ ych expe imen ˚ u. Je od ozena a iaˇ cn´ ı o mulace ´ ulohy a ans o mo ´ ana do ´ alco ´ ych souˇ adnic. D´ ale jsou p ezen o ´ any nume ick´ e ´ ysledky. NUMERISCHE BEWERTUNG EINES RHEOLOGISCHEN EXPERIMENTS Wi be ach en einen iskoelas ischen, ein ach un e s ¨ u z en, d ehsymme ischen K¨ o pe , de es mi dem Un e g und e bunden is . De K¨ o pe wi d mi de Fl¨ ache eines P esskie e s beschwe , die sich in Rich ung de Achse aus de kons an en Geschwindigkei beweg . Im o liegenden Bei ag wi d das K a echo be echne . Dies e m¨ oglich uns einen Ve gleich de nume ischen E gebnisse mi den ealen Da en aus den heologischen Expe imen en. Da- aus wi d eine Va ian en o mulie ung de Au gabe abgelei e und in Walzenkoo dina en ans- o mie . Wei e we den nume ische E gebnisse p ¨ asen ie . NUMERYCZNA OCENA EKSPERYMENTU REOLOGICZNEGO W a ykule ozwa˙ zane jes wiskoelas yczne, p os o podpa y, o acyjnie syme yczne ciało na s ałe poł ֒ aczony z podło˙ zem. Na ciało oddziałuje powie zchnia szcz֒ ek p asuj ֒ acych, k ´ o a po usza si֒ e w kie unku osi zs ał ֒ a p ֒ edko´ sci ֒ a . W op acowaniu obliczana jes eakcja siłowa. Umo˙ zliwia o po ´ ownanie wynik´ ow nume ycznych z ealnymi danymi z ekspe ymen ´ ow eo- logicznych. Ok e´ slona jes zmienna o muła zadania, k ´ o a jes ans o mowana do wsp´ oł z֒ ed- nych cylind ycznych. Nas ֒ epnie zap ezen owano wyniki nume yczne. 137