

Ci a ion: Kochano , V.; Píš ˇek, V.;
Kond a ie , A.; Yu esko, T.; Kuˇce a, P.
In luence o Geome ic Pa ame e s o
Conical Ac ylic Po holes on Thei
S ess–S ain Beha iou . Polyme s
2022,14, 1041. h ps://doi.o g/
10.3390/polym14051041
Academic Edi o : Beom Soo Kim
Recei ed: 15 Feb ua y 2022
Accep ed: 3 Ma ch 2022
Published: 5 Ma ch 2022
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polyme s
A icle
In luence o Geome ic Pa ame e s o Conical Ac ylic Po holes
on Thei S ess–S ain Beha iou
Vladimi Kochano 1, Václa Píš ˇek 2,* , And ii Kond a ie 3, Te yana Yu esko 1and Pa el Kuˇce a 2
1Depa men o Design and P oduc ion o S uc u es om Composi e Ma e ials, Admi al Maka o Na ional
Uni e si y o Shipbuilding, He oi Uk ainy A e. 9, 54025 Mykolayi , Uk aine; [email p o ec ed] (V.K.);
[email p o ec ed] (T.Y.)
2Ins i u e o Au omo i e Enginee ing, B no Uni e si y o Technology, Technická2896/2,
616 69 B no, Czech Republic; [email p o ec ed].cz
3
Depa men o Building Technology and Cons uc ion Ma e ials, O.M. Beke o Na ional Uni e si y o U ban
Economy in Kha ki , Ma shal Bazhano S . 17, 61002 Kha ki , Uk aine; [email p o ec ed]
*Co espondence: [email p o ec ed].cz; Tel.: +420-541-142-271
Abs ac :
T anslucen elemen s in he o m o unca ed cones, which a e made o o ganic glass, a e
widely used in he s uc u es o po holes, subme sible essels, space ehicles, p essu e chambe s,
eleboxes and o he ypes o echnical equipmen . The decisi e ac o in designing po holes is o
ensu e he s eng h o hei anslucen elemen s. In o de o educe he weigh o po holes and,
acco dingly, o inc ease he payload, i is necessa y o op imise he geome ic pa ame e s o he
anslucen elemen s, which include he ape ing angle and he a io o hickness o adius o he
smalle base. The pape deals wi h de elopmen o he applied (enginee ing) me hod o de e mining
he s ess–s ain beha iou o he conical anslucen elemen s o po holes made o o ganic glass
unde he ac ion o a uni o m hyd os a ic p essu e. Fini e-elemen modelling o he anslucen
elemen o he conical po hole is pe o med, wi h he calcula ion o i s s ess–s ain beha iou .
Ex e nal hyd os a ic p essu e o 10 MPa, absence o loads om he inside and con inuous sliding o
he anslucen elemen wi h ic ion along he conical suppo ing su ace o he po hole me al body
a e he bounda y condi ions o he compu a ional model. Full-scale es s o anslucen elemen s o
po holes made o o ganic glass we e pe o med unde he ac ion o uni o m hyd os a ic p essu e.
Analysis o he in luence o geome ic cha ac e is ics o he po holes on s ess–s ain beha iou
showed ha he inc ease in he ape ing angle a he cons an ela i e hickness o he anslucen
elemen educed i s axial displacemen in all cases. Equi alen s esses acqui e minimum alues
when he ape ing angle is in he ange om 75
◦
o 105
◦
(when he ela i e hickness inc eases, he
op imal ape ing angle becomes smalle ). I is shown ha he de eloped me hod o de e mina ion
o he s ess–s ain beha iou o he conical anslucen elemen s o po holes made o o ganic glass
e lec s he eal pic u e o de o ma ion and ag ees wi h he esul s o ull-scale es s. Resul s o
he wo k allow us o choose he a ional pa ame e s o he anslucen elemen o inc easing he
eliabili y o po holes h ough he c ea ion o an e ec i e dis ibu ion o s esses and s ains in he
anslucen elemen , and imp o ing i s op ical cha ac e is ics due o a ela i ely small de lec ion
in ope a ion.
Keywo ds: o ganic glass; polyme hyl me hac yla e plas ic; anslucen elemen ; hyd os a ic load
1. In oduc ion
The e a e a ious designs o po holes adop ed in cu en echnology, which can be
classi ied acco ding o he shape and ma e ial o he anslucen elemen [
1
,
2
]. Such di e si y
is a ibu ed o he di e ences in ope a ing condi ions o he po holes: wo king p essu e;
empe a u e and agg essi eness o he en i onmen ; load pa e n; image- egis a ion
me hod; and so on [
3
,
4
]. The anslucen elemen o he po hole as a pa o an op ical
sys em has a di ec e ec on image quali y, and i s s eng h p ede e mines he eliabili y
Polyme s 2022,14, 1041. h ps://doi.o g/10.3390/polym14051041 h ps://www.mdpi.com/jou nal/polyme s
Polyme s 2022,14, 1041 2 o 15
o he echnical equipmen as a whole [
5
–
7
]. The anslucen elemen is subjec o s ic
equi emen s on he c i e ia o s eng h, igh ness, de o ma ion o op ical su aces unde
load and op ical and he mophysical p ope ies o he ma e ial [8–10].
Conside ing he indica o s o s eng h, ailu e pa e n, manu ac u ing and p ocessing
echnology, i is mo e app op ia e o use he polyme hyl me hac yla e plas ic (PMMA o
o ganic glass o Ac ylic) as a ma e ial o anslucen elemen s [
11
]. The cha ac e is ics o
s eng h o anslucen elemen s made o PMMA ( he alue o p essu e a ailu e) depend
p ima ily on he a io o hickness h o diame e Do he po hole; and o conical and
sphe ical elemen s, on he ape ing angle αo he suppo ing su ace.
The mos undamen al ecommenda ions o he design and choice o he s uc u al
ype and geome ic pa ame e s o he ac ylic po holes o di e en s uc u al ypes a e
gi en in [
12
,
13
]. Pape [
14
] summa ises he in o ma ion abou s uc u al ypes o ac ylic
po holes bu does no gi e any algo i hms o selec ion o hei op imal shape depending
on ope a ing condi ions. Conside able a en ion in [
15
] is paid o he design o he po hole
body and ways o ensu e i s igh ness; howe e , no op imal enginee ing solu ions a e
p oposed. The au ho s o [
16
] ound ha a he ape ing angle o
α
= 90
◦
and ela i e
hickness o h/D= 0.41 no ensile s ains we e obse ed in he po hole, bu hey did no
assess he e ec o design pa ame e s on he po hole s ess s a e.
Calcula ion o he s ess–s ain beha iou o a anslucen elemen using classical
me hods o elas ici y heo y is e y ime-consuming in e ms o implemen a ion, since in
he p ocess o inding a solu ion, he mixed bounda y condi ions on h ee su aces ( wo
end su aces and a suppo ing conical su ace) should be ul illed [
17
]. Fu he mo e, i
is necessa y o conside he in luence o design pa ame e s, as well as he coe icien o
ic ion on he suppo [18].
Rega ding use o he bending heo y o hin pla es in he calcula ions o s ess–s ain
beha iou o he po holes, he e a e always signi ican e o s in he de e mina ion o
displacemen s wi h sligh ly ewe e o s han s ess de e mina ion. This is because ypical
dimensions o he po hole anslucen elemen ( hickness hand minimum diame e 2R
0
)
a e compa able, and he heo y o hin pla es can be applied a h/(R0≤1/5) [19,20].
Cu en ly, he s ess–s ain beha iou o anslucen elemen s is o en de e mined
using nume ical me hods [
21
]. The ini e-elemen me hod is he mos accu a e and uni e -
sal algo i hm o calcula ing he s ess–s ain beha iou o he class o s uc u es unde
s udy [
22
,
23
]. The me hods o calcula ion o conical po holes based on he ini e-elemen
me hod a e conside ed in [
24
–
26
]. I is shown ha he anslucen elemen unde he ac ion
o hyd os a ic p essu e slides wi h he ic ion o he conical suppo ing su ace o he
po hole body. A he same ime, he elas ic-plas ic s a e o he po hole made o ac ylic
glass is conside ed in [
24
], bu he c eeping a cyclic loads is no analysed. The au ho s
o [
25
] pe o med calcula ions o he po holes wi h h/D= 1,
α
= 90
◦
and coe icien o
ic ion on he suppo ing su ace = 0
−
0.2 unde he ac ion o p essu e P= 70 MPa,
which ag eed well wi h he esul s o expe imen s [
12
]. Howe e , he e is no da a on he
applicabili y o his calcula ion me hod o he po holes wi h h/D≤1.
Pape [
27
] shows ha he mos loaded elemen s o conical po holes a e he edges
o he low-p essu e su ace. Fo con i ma ion o he esul s, an expe imen al s udy was
conduc ed. A ew po holes we e manu ac u ed and es ed unde a hyd os a ic p essu e
o 60 MPa. Pape [
28
] deals wi h he de elopmen o a “design by analysis” me hod and
analysis o he s ess s a e o he po holes conside ing he c i e ia o ailu e o PMMA o
de e mine he p essu e and a ea o ailu e. The de ailed ini e-elemen model o he conical
po holes made o PMMA is de eloped in [
29
]. Based on his model, he s ess–s ain
beha iou o he po hole is analysed, and he causes o i s ailu e a e in es iga ed. The
esul s show ha local ailu e ends o occu a he co ne o he po hole cone whe e he
equi alen s esses acco ding o he Hube – on Mises s ain ene gy heo y a e maximal.
The expe imen al s udy o [
30
] shows he abili y o PMMA o “adap ” o he shape o
he po hole body. I was ound ha due o he plas ic beha iou o PMMA, he highes
mechanical s esses a dange ous poin s o he po hole a e educed by 64–71%. I con i med
Polyme s 2022,14, 1041 3 o 15
he easibili y o use o PMMA in he op ical sys ems’ echnology, bu he me hod o
de e mine i s eliabili y ac o is no speci ied in his pape .
The expe imen s and nume ical calcula ions o he s ess–s ain beha iou o PMMA
po holes a di e en le els o hyd os a ic p essu e o de elop a me hod o op imal design-
ing ega ding ope a ing condi ions a e desc ibed in [
31
]. Expe imen al dependences o
displacemen s o po holes made o PMMA a long- e m and cyclic impac s o hyd os a ic
p essu e a e ob ained.
Nume ical me hods a e known o hei e sa ili y, bu hey a e a he cumbe some
and di icul o apply in p ac ice [
32
,
33
]. A he same ime, explici solu ions o he
conical anslucen elemen s o he po holes made o PMMA unde he ac ion o uni o m
hyd os a ic p essu e a e limi ed by he necessi y o ul il he mixed bounda y condi ions and
ake in o conside a ion he in luence o s uc u al pa ame e s o a anslucen elemen and
coe icien o ic ion wi h he po hole body. These di icul ies, oge he wi h he limi ed
choice o undamen al unc ions om he ull ange o solu ions o he axisymme ic
p oblem o he elas ici y heo y, do no allow he bounda y condi ions o he anslucen
elemen o be ully me , while he mos s ess poin s a e always on he su ace o he
de o mable body. Howe e , based on hese me hods i is possible o ob ain ela i ely
simple bu ai ly accu a e applied solu ions o he p oblems [31,34].
Based on he analysis, he cu en ask is o de elop an applied (enginee ing) me hod
o he de e mina ion o he s ess–s ain beha iou o conical anslucen elemen s o he
po holes made o PMMA unde he ac ion o uni o m p essu e.
2. Ma e ials and Me hods
The conical anslucen elemen s o po holes made o PMMA a e s udied unde he
ac ion o uni o m p essu e. Classical me hods o elas ici y heo y (axisymme ic p oblem)
and he bending heo y o hick pla es a e used o de elop he applied (enginee ing) me hod
o he de e mina ion o he s ess–s ain beha iou o he conical anslucen elemen s. We
accep ed he ollowing hypo heses: In de o ma ion unde uni o m p essu e he anslucen
elemen bends and slides in he po hole body; he elemen mo es on he conical suppo ing
su ace wi h ic ion and wi hou sepa a ion om he su ace; he suppo eac ion on he
conical suppo ing su ace decomposes a each poin in o no mal and angen componen s.
To ind an analy ical solu ion, we used Lo e’s unc ion om polynomials o deg ees 3, 4 and
6, which we e ob ained using Legend e polynomials. Ful ilmen o he kinema ic and o ce
bounda y condi ions is achie ed by minimising he s anda d de ia ions o hese condi ions
along he gene a ing conical suppo ing su ace. Nume ical s udies we e conduc ed in he
so wa e package o he ini e-elemen analysis ANSYS Mechanical 2019 R1. In iew o
he axial symme y o he anslucen -elemen shape and ex e nal load, a na ow sec o
was used as a compu a ional model. The egula g id size was aken equal o 5 mm. Fo
he conical suppo ing su ace, a hal -sized g id was used, which allowed us o mo e
accu a ely model he anslucen elemen ’s in e ac ion wi h he me al ( i anium) body o
he po hole. Fo he ini e elemen o he po hole body, size o g id in he cen al zone
was aken equal o 10 mm; on he suppo ing su ace, whe e a sha p change in s ess could
be expec ed, 5 mm g id was used. The cons uc ed ini e-elemen model con ained mo e
han 10,000 Te a10 elemen s ( e ahed on wi h 10 nodes). Size o he g id was chosen
acco ding o dimensions o he eal po hole. S udy o he con e gence o he nume ical
solu ion showed ha wi h his numbe o ini e elemen s in he models he no mal and
shea s esses a ied sligh ly (by 5% a mos ). Analysis o he quali y o he cons uc ed
ini e-elemen models did no e eal any c i ical e o s. The p oblem o de e mina ion o
he po hole s ess–s ain beha iou was sol ed in he linea se ing. Full-scale es s we e
pe o med using he specialised expe imen al complex o hyd os a ic loads o anslucen
elemen s o he po holes. Tes s o po holes unde he ac ion o he hyd os a ic p essu e
we e pe o med in he high-p essu e chambe (chambe olume was 0.06 m
3
, maximum
p essu e was 150 MPa). Axial displacemen s we e measu ed by he mechanical dial gauge
ICh-10 (Mic o ech, Kyi , Uk aine) and elec ic-con ac manome e EKM-2U (UAM, Kyi ,
Polyme s 2022,14, 1041 4 o 15
Uk aine) was used o p essu e measu emen . P essu e in he chambe was c ea ed by he
pumping s a ion UNGR–2500R (UAM, Kyi , Uk aine). Specimens o anslucen elemen s
we e made o SO-120 PMMA (Admi al Maka o Na ional Uni e si y o Shipbuilding,
Mykolai , Uk aine) pla es o 50 mm hickness by u ning wi h subsequen polishing o he
op ical su aces.
3. Theo e ical Backg ound
Ma e ial o he anslucen elemen in he wo king posi ion is unde s a ic p essu e P
om he end su ace o high p essu e, as well as con ac (no mal and angen ial)-dis ibu ed
loads om he conical suppo ing su ace (Figu e 1).
Polyme s 2022, 14, x FOR PEER REVIEW 4 o 16
ac ion o he hyd os a ic p essu e we e pe o med in he high-p essu e chambe (chambe
olume was 0.06 m
3
, maximum p essu e was 150 MPa). Axial displacemen s we e meas-
u ed by he mechanical dial gauge ICh-10 (Mic o ech, Kyi , Uk aine) and elec ic-con ac
manome e EKM-2U (UAM, Kyi , Uk aine) was used o p essu e measu emen . P essu e
in he chambe was c ea ed by he pumping s a ion UNGR–2500R (UAM, Kyi , Uk aine).
Specimens o anslucen elemen s we e made o SO-120 PMMA (Admi al Maka o Na-
ional Uni e si y o Shipbuilding, Mykolai , Uk aine) pla es o 50 mm hickness by u n-
ing wi h subsequen polishing o he op ical su aces.
3. Theo e ical Backg ound
Ma e ial o he anslucen elemen in he wo king posi ion is unde s a ic p essu e P
om he end su ace o high p essu e, as well as con ac (no mal and angen ial)-dis ib-
u ed loads om he conical suppo ing su ace (Figu e 1).
Figu e 1. Po hole compu a ional model: P—hyd os a ic p essu e; —sliding wi h ic ion; 1— ans-
lucen elemen ; 2—po hole body.
The e a e no ex e nal loads on he end su ace o low p essu e. The e o e, bo h he
geome ic shape o he anslucen elemen and ex e nal su ace loads a e symme ic
a ound he cen al axis o he po hole, so he s ess–s ain beha iou o he anslucen
elemen is de e mined by sol ing he axisymme ic p oblem o he elas ici y heo y.
In case o he axisymme ic p oblem, nonze o componen s o he s ess–s ain beha -
iou in he cylind ical coo dina es can be exp essed in e ms o a biha monic Lo e’s unc-
ion [35–37] F( ,z):
𝑢=−1
2𝐺𝜕𝐹
𝜕𝑟𝜕𝑧;𝑤= 1
2𝐺󰇧2󰇛1−𝜇󰇜∇−𝜕
𝜕𝑧󰇨𝐹+𝑏
𝜎=𝜕
𝜕𝑧󰇧𝜇∇−𝜕
𝜕𝑟󰇨𝐹; 𝜎=𝜕
𝜕𝑧𝜇∇−1
𝑟𝜕
𝜕𝑟𝐹
𝜎=𝜕
𝜕𝑧󰇛2−𝜇󰇜∇−𝜕
𝜕𝑧𝐹; 𝜏 =𝜕
𝜕𝑟󰇛1−𝜇󰇜∇−𝜕
𝜕𝑧
(1)
whe e ∇=



+

+



is Laplace ope a o in he cylind ical coo dina es; u,w a e a-
dial and axial displacemen s, espec i ely; σ
, σ
θ
, σ
z
, τ
z
a e adial, angen ial, axial and
shea s esses; b
0
is ee cons an co esponding o he axial displacemen o he body.
Biha monic na u e o Lo e’s unc ion ensu es he exac ul ilmen o he equilib ium
equa ions


 +

 +
󰇛𝜎−𝜎󰇜=0;


 +

 +
𝜏 =0;
(2)
and equa ions o s ain compa ibili y

𝜎−𝜇󰇛𝜎−𝜎󰇜+
󰇛𝜎−𝜎󰇜=0; (3)
Figu e 1.
Po hole compu a ional model: P—hyd os a ic p essu e; —sliding wi h ic ion; 1—
anslucen elemen ; 2—po hole body.
The e a e no ex e nal loads on he end su ace o low p essu e. The e o e, bo h he
geome ic shape o he anslucen elemen and ex e nal su ace loads a e symme ic a ound
he cen al axis o he po hole, so he s ess–s ain beha iou o he anslucen elemen is
de e mined by sol ing he axisymme ic p oblem o he elas ici y heo y.
In case o he axisymme ic p oblem, nonze o componen s o he s ess–s ain be-
ha iou in he cylind ical coo dina es can be exp essed in e ms o a biha monic Lo e’s
unc ion [35–37]F( ,z):
u=−1
2G
∂2F
∂ ∂z;w=1
2G2(1−µ)∇2−∂2
∂z2F+b0
σ =∂
∂zµ∇2−∂2
∂ 2F;σθ=∂
∂zµ∇2−1
∂
∂ F
σz=∂
∂z(2−µ)∇2−∂2
∂z2F;τ z =∂
∂ (1−µ)∇2−∂2
∂z2
(1)
whe e
∇2=∂2
∂ 2+1
∂
∂ +∂2
∂z2
is Laplace ope a o in he cylind ical coo dina es; u,wa e
adial and axial displacemen s, espec i ely;
σ
,
σθ
,
σz
,
τ z
a e adial, angen ial, axial and
shea s esses; b0is ee cons an co esponding o he axial displacemen o he body.
Biha monic na u e o Lo e’s unc ion ensu es he exac ul ilmen o he equilib ium
equa ions
∂σ
∂ +∂τ z
∂z+1
(σ −σθ)=0;
∂τ z
∂ +∂σz
∂z+1
τ z =0;
(2)
and equa ions o s ain compa ibili y
∂
∂ σθ−µ(σ −σz)+1+µ
(σθ−σ )=0;
∂2
∂z2(σθ−µ(σ −σθ)) −2(1+µ)∂τ z
∂z+∂
∂ (σz−µ(σ +σθ)) =0.
(3)
Polyme s 2022,14, 1041 5 o 15
The e o e, solu ion o he p oblem o he axisymme ic s ess–s ain beha iou s a e o
he body o o a ion is educed o de e mina ion o Lo e’s s ess unc ion. Un o una ely, a
p esen he e is no analy ical ep esen a ion o Lo e’s unc ion sa is ying any p ede e mined
bounda y condi ions [35,36]. The explici solu ion o biha monic equa ion
∇2∇2F=0 (4)
is possible in he cylind ical coo dina es in in ini e se ies on he Bessel unc ions [
35
,
36
].
Howe e , his solu ion is incon enien o p ac ical use because o slow con e gence o
he se ies (o he o de 1
/√n
,n–se ial numbe o e ms o he se ies), pa icula ly unde
di icul bounda y condi ions and o noncylind ical bodies [
38
,
39
]. In addi ion, hey a e
inconsis en a he o igin o coo dina es a = 0.
The solu ion o Equa ion (4) in he sphe ical coo dina es in se ies using Legend e
polynomials wi h he u he ans o ma ion o he unc ion Fin o cylind ical coo dina es
is ela i ely simple; howe e , i is always necessa y o limi he numbe o e ms o se ies
in he expansion o he unc ion Fand sa is ac o y solu ions a e ob ained o he simple
bounda y condi ions only.
Ob aining dependences o he anslucen -elemen s ess–s ain beha iou , which can
be applied in p ac ice, in ol es a ce ain comp omise be ween accu acy and simplici y o
he solu ion. When cons uc ing an analy ical solu ion, i was necessa y o conside he
equi emen s below:
•
class o unc ions desc ibing he s ess–s ain beha iou o a anslucen elemen
should sa is y he biha monic Equa ion (4);
•
solu ion should ake in o accoun he in e ac ion o he anslucen elemen wi h eal
suppo , i.e., ensu e compliance wi h he bounda y condi ions he eon;
•
dependences o he componen s o he anslucen -elemen s ess–s ain beha iou
should be ela i ely simple.
S ess–s ain beha iou o he anslucen elemen depends on i s geome ic cha ac-
e is ics, which include he ape ing angle
α
and he a io o he smalle base diame e o
hickness Rmin/c(Figu e 2).
Polyme s 2022, 14, x FOR PEER REVIEW 5 o 16
𝑟



𝜎−𝜇󰇛𝜎−𝜎󰇜−2󰇛1+𝜇󰇜

 +
𝜎−𝜇󰇛𝜎+𝜎󰇜=0.
The e o e, solu ion o he p oblem o he axisymme ic s ess–s ain beha iou s a e
o he body o o a ion is educed o de e mina ion o Lo e’s s ess unc ion. Un o u-
na ely, a p esen he e is no analy ical ep esen a ion o Lo e’s unc ion sa is ying any
p ede e mined bounda y condi ions [35,36]. The explici solu ion o biha monic equa ion
∇∇𝐹=0 (4)
is possible in he cylind ical coo dina es in in ini e se ies on he Bessel unc ions [35,36].
Howe e , his solu ion is incon enien o p ac ical use because o slow con e gence o
he se ies (o he o de 1√𝑛
⁄, n–se ial numbe o e ms o he se ies), pa icula ly unde
di icul bounda y condi ions and o noncylind ical bodies [38,39]. In addi ion, hey a e
inconsis en a he o igin o coo dina es a = 0.
The solu ion o Equa ion (4) in he sphe ical coo dina es in se ies using Legend e
polynomials wi h he u he ans o ma ion o he unc ion F in o cylind ical coo dina es
is ela i ely simple; howe e , i is always necessa y o limi he numbe o e ms o se ies
in he expansion o he unc ion F and sa is ac o y solu ions a e ob ained o he simple
bounda y condi ions only.
Ob aining dependences o he anslucen -elemen s ess–s ain beha iou , which
can be applied in p ac ice, in ol es a ce ain comp omise be ween accu acy and simplici y
o he solu ion. When cons uc ing an analy ical solu ion, i was necessa y o conside he
equi emen s below:
• class o unc ions desc ibing he s ess–s ain beha iou o a anslucen elemen
should sa is y he biha monic Equa ion (4);
• solu ion should ake in o accoun he in e ac ion o he anslucen elemen wi h eal
suppo , i.e., ensu e compliance wi h he bounda y condi ions he eon;
• dependences o he componen s o he anslucen -elemen s ess–s ain beha iou
should be ela i ely simple.
S ess–s ain beha iou o he anslucen elemen depends on i s geome ic cha ac-
e is ics, which include he ape ing angle α and he a io o he smalle base diame e o
hickness R
min
⁄ c (Figu e 2).
Figu e 2. T anslucen -elemen loading diag am: h—hal - hickness o he anslucen elemen ; R
0
—
a e age adius.
When we choose he o igin o coo dina es in he cen e o he median su ace o he
anslucen elemen , bounda y condi ions a ends a e as ollows:
𝜎󰇡𝑟;−
󰇢=−𝑝; 𝜎󰇡𝑟;
󰇢=𝜏󰇡𝑟;
󰇢=0. (5)
Figu e 2.
T anslucen -elemen loading diag am: h—hal - hickness o he anslucen elemen ; R
0
—
a e age adius.
When we choose he o igin o coo dina es in he cen e o he median su ace o he
anslucen elemen , bounda y condi ions a ends a e as ollows:
σz ;−c
2=−p;σz ;c
2=τ z ;c
2=0. (5)

Polyme s 2022,14, 1041 6 o 15
In o de o ully de e mine he s ess–s ain beha iou o he anslucen elemen ,
we add he condi ions on he conical suppo ing su ace ad (Figu e 2) o he bounda y
condi ions (5)
Pτ= Pn, (6)
whe e is coe icien o ic ion.
The ollowing hypo heses a e in oduced:
•
in de o ma ion unde he uni o m p essu e P he anslucen elemen bends and slides
in he po hole body along he axis OZ;
•
he anslucen elemen mo es on he conical suppo ing su ace ad (i.e., a
=
R0− g α
2) wi h ic ion and wi hou sepa a ion om he su ace;
•
suppo eac ion P
on he conical suppo ing su ace ad decomposes in each poin
in o no mal Pnand angen ial Pτcomponen s.
Using he la e condi ion, i is possible o de e mine he ela ionship be ween he
adial P
and axial P
z
eac ions o he suppo ing su ace. Necessa y ope a ions o his
cons uc ion a e shown in Figu e 3.
Polyme s 2022, 14, x FOR PEER REVIEW 6 o 16
In o de o ully de e mine he s ess–s ain beha iou o he anslucen elemen , we
add he condi ions on he conical suppo ing su ace ad (Figu e 2) o he bounda y condi-
ions (5) 𝑃=
𝑓
𝑃, (6)
whe e is coe icien o ic ion.
The ollowing hypo heses a e in oduced:
• in de o ma ion unde he uni o m p essu e P he anslucen elemen bends and
slides in he po hole body along he axis OZ;
• he anslucen elemen mo es on he conical suppo ing su ace ad (i.e., a 𝑟=𝑅−
𝑡𝑔
) wi h ic ion and wi hou sepa a ion om he su ace;
• suppo eac ion P
on he conical suppo ing su ace ad decomposes in each poin
in o no mal P
n
and angen ial P
τ
componen s.
Using he la e condi ion, i is possible o de e mine he ela ionship be ween he
adial P
and axial P
z
eac ions o he suppo ing su ace. Necessa y ope a ions o his
cons uc ion a e shown in Figu e 3.
Figu e 3. Componen s o he ec o o suppo eac ions.
The e o e,
𝑃 =𝑃
𝑠𝑖𝑛
+𝑃𝑐𝑜𝑠
=𝑃
󰇡𝑠𝑖𝑛
+
𝑓
𝑐𝑜𝑠
󰇢;
𝑃 =𝑃
𝑐𝑜𝑠
−𝑃𝑠𝑖𝑛
=𝑃
󰇡𝑐𝑜𝑠
−
𝑓
𝑠𝑖𝑛
󰇢;
(7)
ha is,




=






. (8)
The anslucen -elemen suppo ing condi ions may be di e en , bu hey a e ul i-
ma ely educed o he o ce bounda y condi ions on he conical suppo ing su ace (i.e.,
he equilib ium equa ion o an elemen a y e ahed on) and a e w i en as:
𝜎𝑐𝑜𝑠
+𝜏𝑠𝑖𝑛
+𝑝 =0;
𝜎𝑠𝑖𝑛
+𝜏𝑐𝑜𝑠
+𝑝 =0; (9)
hence, we ob ain he ollowing ela ionship:




=













. (10)
A e Equa ions (8) and (10), he bounda y condi ion on he conical su ace o he
anslucen elemen is
Figu e 3. Componen s o he ec o o suppo eac ions.
The e o e,
P z =Pnsin α
2+Pτcosα
2=Pnsin α
2+ cos α
2;
P =Pncos α
2−Pτsin α
2=Pncosα
2− sin α
2;
(7)
ha is,
P z
P
= gα
2+
1− g α
2
. (8)
The anslucen -elemen suppo ing condi ions may be di e en , bu hey a e ul i-
ma ely educed o he o ce bounda y condi ions on he conical suppo ing su ace (i.e., he
equilib ium equa ion o an elemen a y e ahed on) and a e w i en as:
σ cos α
2+τ zsinα
2+p =0;
σzsin α
2+τ zcos α
2+p z =0;
(9)
hence, we ob ain he ollowing ela ionship:
p z
P
=σz gα
2+τ z
σ +τ z gα
2
. (10)
Polyme s 2022,14, 1041 7 o 15
A e Equa ions (8) and (10), he bounda y condi ion on he conical su ace o he
anslucen elemen is
σ +τ zβ−1− β
β+ (τ z +σzβ)=0, (11)
whe e β= g α
2.
Equa ion (11) is a condi ion o ic ion o he anslucen elemen on he conical
suppo ing su ace. The condi ion o con inuous sliding o he anslucen elemen is
w i en as
u+βw=0. (12)
Condi ions (5), (11) and (12) ully de e mine he su ace loads and displacemen s o
he anslucen elemen .
Pape s [
35
,
36
,
38
] show ha o exac co espondence o he bounda y condi ions (5)
on he end su aces o hick pla es, he deg ee o polynomials en e ing he unc ion in F
should no be highe han 6. To ind an analy ical solu ion, i is con enien o o m Lo e’s
unc ion o polynomials o deg ees 3, 4 and 6, ob ained wi h he use o he co esponding
Legend e polynomials [35,36]:
F=F3+F4+F6;
F3=a32z3−3 2z+b3 2z+z3;
F4=a48z4−24 2z2+3 4+b42z4+ 2z2− 4;
F6=1
3a616z6−120z4 2+90z2 4−5 6+b68z6−16z4 2−21z2 4+3 6;
(13)
whe e
a3
,
a4
,
a6
,
b3
,
b4and b6
a e a bi a y cons an s de e mined du ing ul ilmen o
bounda y condi ions.
Subs i u ion o (13) in (1) gi es he exp essions o s esses as shown below:
σz=−12a3+2(7−5µ)b3−192a4z+8(8−7µ)b4z+320a6−2z3+3 2z+
b664(7−11µ)z3−96(18 −11µ) 2z;
τ z =96a4 −4(8−7µ)b4z+240a64 z2− 3+
b6−96(7−11µ) z2+24(18 −11µ) 3.
(14)
Ful ilmen o bounda y condi ions o ype (5) a he ends a z=
±
callows us o
de e mine he a bi a y cons an s a3,a4,a6and b6:
a3=4(7−5µ)b3+p
24 ;a4=32(8−7µ)hb4−3p
768c;
a6=18 −11µ
40·704h3p;b6=p
4·704h3
(15)
A e subs i u ion o (15) in (13) we ob ain Lo e’s unc ion as
F=20(2−µ)b3+p
12 z3−20(1−µ)b3+p
8z 2+224(2−µ)hb4−3p
96hz4+224(1−µ)hb4−3p
32hz2 2−
224µhb4+3p
96h 4+p
256h38(3−µ)
15 z6−4(2−µ)z4 2+3(1−µ)z2 4+µ
6 6(16)
The co esponding unc ions (1) o he componen s o s ess–s ain beha iou o he
anslucen elemen a e w i en as ollows:
Polyme s 2022,14, 1041 8 o 15
2Gu =p
4 +5(1−µ)b3 +28(1−µ)b4−3p
8hz +p
32h34(2−µ)z3 −3(1−µ)z 3;
2Gw =b0−10b3µz−p
2z+3p
8hz2−14b42µz2+(1−µ) 2−3p
16h 2+p
128h3−8(1+µ)z4+24µz2 2+3(1−µ) 4;
σz=−p
2+pz
4h3−z2
h2;
τ z =3p
8hz2
h2−1;
σ =p
4+5b3(1+µ)+28(1+µ)b4z−3p
8hz+pz
32h34(2+µ)z2−3(3+µ) 2;
σθ=p
4+5b3(1+µ)+28(1+µ)b4z−3p
8hz+pz
32h34(2+µ)z2−3(1+3µ) 2.
(17)
Exp essions (17) show ha he coe icien b
3
de e mines he uni o m adial comp es-
sion, while b
4
is a pu e axisymme ic bending o he ound pla e. Coe icien b
0
co esponds
o he mo emen o he anslucen elemen as a solid objec in he po hole body along he
cen al axis z.
I is no possible o accu a ely implemen he o ce (11) and kinema ic (12) bounda y
condi ions. The e o e, we de ine a bi a y cons an s b
0
,b
3
and b
4
by minimisa ion o he
s anda d de ia ions o hese condi ions along he gene a ing conical suppo ing su ace.
The gene a ix equa ion in he accep ed coo dina e sys em (see Figu e 1) is gi en by:
=R0−zβ. (18)
Conside ing he mixed ( o ce and kinema ic) bounda y condi ions (11) and (12), we
can con enien ly p oceed o dimensionless exp essions o displacemen s and s esses (17).
To achie e his we in oduce new dimensionless a iables ζand ρ:
ζ=z
h;ρ=
R0
; (19)
and adop he no a ion
R0
h=γ. (20)
Conside ing (19) and (20), he gene a ix equa ion (18) is w i en as
ρ=1−βζ
γ. (21)
Subs i u ing (18) in o (17) and p oceeding o dimensionless coo dina es acco ding o
(19)
−
(21), we ob ain he exp essions o dimensionless displacemen s u,wand s esses
σz
,
σ ,σθand τ z on he conical suppo ing su ace a µ=1/3as below:
u=2Gu
ph =1
4(γ−βζ)+10
3(γ−βζ)b3+56(γ−βζ)ζb4−3
8(γ−βζ)ζ+5(γ−βζ)ζ3
24 −ζ(γ−βζ)3
16 ;
w=2Gw
ph =b0−10
3b3ζ−1
2ζ−28
3b4ζ2+3
8ζ2−28
3(γ−βζ)2b4−3
16(γ−βζ)2−1
12ζ4+1
16(γ−βζ)2ζ2+1
64(γ−βζ)4;
σz=σz
p=−1
2+3
4ζ−1
4ζ2;
τ z =τ z
p=3
8(γ−βζ)ζ2−1;
σ =σ
p=1
4+20
3b3+112
3b4−3
8ζ+7
24ζ3−5
16(γ−βζ)2ζ;
σθ=σ
p=1
4+20
3b3+112
3b4ζ−3
8ζ+7
24ζ3−3
16(γ−βζ)2ζ;
(22)
whe e b0=b0
pc ;b3=b3
p;b4=hb4
p.
A bi a y cons an
b3
is de e mined om he condi ion o s a ic equilib ium o he
anslucen elemen along he axis oz. The main ec o o hyd os a ic-p essu e o ces, P,
ac ing on he anslucen elemen , is balanced by he axial componen s P
z
o he conical
Polyme s 2022,14, 1041 9 o 15
suppo ing-su ace eac ions, which in u n a e co ela ed by ela ionship (8) wi h he eac-
ions o adial comp ession P
. Conside ing he equilib ium o one hal o he anslucen
elemen and es ablishing he dependence o he no mal s esses
σ
(p), being a e age on
diame ic c oss sec ion, and conside ing (22), we ob ain
b3=−3
80 (γ+β)2
λ!1− β
β+ +1. (23)
The emaining a bi a y cons an s a
0
and b
4
a e de e mined om he condi ion o
con inuous sliding o he anslucen elemen on he conical suppo ing su ace (12), and
a e w i en as:
Z1
−1(u+βw)2∂ζ →min. (24)
We de e mine he coe icien s b
0
,b
3
and b
4
om he condi ion o sa is ac ion o he
bounda y condi ions o sliding wi h ic ion on he conical suppo ing su ace a he
minimum dispe sion D. Wi h he use o he leas -squa es me hod in he in eg al o m, he
equa ion will be as ollows:
D=Z1
−1 (u−βw)2+σ +βτ z −1− β
+β(τ z +βσz)2!∂ζ →min. (25)
Thus, he unknown coe icien s a e de e mined om he sys em o equa ions:
∂D
∂b0
=0; ∂D
∂b3
=0; ∂D
∂b4
=0. (26)
4. Nume ical Implemen a ion
In iew o he axial symme y o he anslucen -elemen shape and ex e nal load, a
na ow sec o was used as a compu a ional model (Figu e 4a). I allowed us o educe
he numbe o ini e elemen s in he compu a ional model o he anslucen elemen ,
i.e., o educe he ime and o inc ease he accu acy o calcula ions wi h he same size o
he ini e-elemen g id as in he model as a whole. Since he e o o he ini e-elemen
me hod in de e mina ion o componen s o he s ess–s ain beha iou o s uc u es is o
o de 1
/n2
[
40
,
41
] (n–o de o di ision in o elemen s on one coo dina e), wi h a ull-size
anslucen -elemen hickness o 48 mm, he g id size was aken equal o 5 mm. Fo he
conical suppo ing su ace, a hal -sized g id was used; i allowed us o mo e accu a ely
model he anslucen elemen ’s in e ac ion wi h he me al ( i anium) body o he po hole.
Fo he ini e elemen o he po hole body, he size o g id in he cen al zone was aken
equal o 10 mm; on he suppo ing su ace, whe e a sha p change in s ess could be expec ed,
a 5 mm g id was used. The size o he g id was chosen acco ding o dimensions o he
eal po hole. The s udy o he con e gence o he nume ical solu ion showed ha wi h
his numbe o ini e elemen s in he models, he no mal and shea s esses a ied sligh ly
(by 5% a mos ). Analysis o he quali y o he cons uc ed ini e-elemen models did no
e eal any c i ical e o s. The p oblem o de e mina ion o he s ess–s ain beha iou o
he po hole was sol ed in he linea se ing.
As a esul , we ob ained he ollowing b eakdown o he po hole s uc u e in o ini e
elemen s (Figu e 4b).
Bounda y condi ions (5), (11) and (12) emained he same, bu since a model o he
na ow sec o o he anslucen elemen was used o he calcula ions by he ini e-elemen
me hod, bounda y condi ions o he F ic ionless Suppo ype we e se on ee-symme y
planes. This condi ion allowed us o ix he compu a ional model on he adial sec ions
o med by sepa a ion o a sec o om he ull-sized anslucen elemen wi hou ic ion
(i.e., a = 0) and wi h no no mal displacemen s o hese su aces. The F ic ionless Suppo
condi ion is used on he conical suppo ing su ace as well, a he coe icien s o ic ion