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