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A ailable online a www.sciencedi ec .com
P ocedia S uc u al In eg i y 23 (2019) 553–558
2452-3216
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2019 The Au ho s. Published by Else ie B.V.
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Pee - e iew unde esponsibili y o he scien i ic commi ee o he ICMSMF o ganize s
10.1016/j.p os .2020.01.144
10.1016/j.p os .2020.01.144 2452-3216
© 2019 The Au ho s. Published by Else ie B.V.
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h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
)
Pee - e iew unde esponsibili y o he scien i ic commi ee o he ICMSMF o ganize s
A ailable online a www.sciencedi ec .com
ScienceDi ec
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2019 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o he scien i ic commi ee o he IC MSMF o ganize s.
9 h In e na ional Con e ence on Ma e ials S uc u e and Mic omechanics o F ac u e
In luence o he cell geome y on he ensile s eng h o open-cell
ce amic oams
Oldřich Še ečeka*, Roman Papšíka, Zdeněk Maje a, Michal Ko oula
a B no Uni e si y o Technology, Facul y o Mechanical Enginee ing, Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Technická
2896/2, 616 69 B no, Czech Republic
Abs ac
Nowadays used open cell oam ce amic ma e ials a e mos ly o i egula s uc u e which means ha he shape o pa icula oam
cells does no exhibi any egula pa e n. On one hand, such oam s uc u es lead o only e y sligh aniso opic o e en iso opic
beha io upon he mechanical loading, bu on he o he hand hey do no ha e an op imal esis ance o ailu e upon gi en loading
condi ions and le el o po osi y. The s eng h o he ce amic oam s uc u e can be hus u he imp o ed by design o cells ha ing
a ious egula shapes. Such oams can inally exhibi an o ho opic beha io om bo h he elas ic and s eng h poin o iew. To
unde s and how di e en ypes o cells in luence he oam cha ac e is ics in a ious di ec ions, oam s uc u es wi h a ious cell
shapes we e hus s udied and in es iga ed in e ms o hei ensile s eng h wi hin his con ibu ion. The s uc u es we e modelled
by means o beam elemen based FE models and by u iliza ion o he s ess c i e ion de ining ailu e o pa icula s u s. To ally
six di e en cell ypes we e analyzed unde conside a ion o he same po osi y o he inal oam s uc u e and amoun o he s eng h
aniso opy was quan i ied. Rela ion be ween o ien a ion o s u s wi h espec o a loading di ec ion and he oam s eng h was
discussed in mo e de ails. Recommenda ions o an employmen o pa icula cell ypes o speci ic loading condi ions we e gi en.
© 2019 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o he scien i ic commi ee o he IC MSMF o ganize s.
Keywo ds: ce amic oam; Kel in cell; Vo onoi essela ion; ensile s eng h; FE model
1. In oduc ion
Open cell oam ce amic ma e ials a e nowadays widely used in a ious ligh weigh , high empe a u e o il e ing
applica ions bu s ill no e y o en in mechanically loaded applica ions due hei ela i ely low esis ance o ailu e.
* Co esponding au ho . Tel.: +420 5 4114 2857
E-mail add ess: se ece[email p o ec ed].cz
A ailable online a www.sciencedi ec .com
ScienceDi ec
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2019 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o he scien i ic commi ee o he IC MSMF o ganize s.
9 h In e na ional Con e ence on Ma e ials S uc u e and Mic omechanics o F ac u e
In luence o he cell geome y on he ensile s eng h o open-cell
ce amic oams
Oldřich Še ečeka*, Roman Papšíka, Zdeněk Maje a, Michal Ko oula
a B no Uni e si y o Technology, Facul y o Mechanical Enginee ing, Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Technická
2896/2, 616 69 B no, Czech Republic
Abs ac
Nowadays used open cell oam ce amic ma e ials a e mos ly o i egula s uc u e which means ha he shape o pa icula oam
cells does no exhibi any egula pa e n. On one hand, such oam s uc u es lead o only e y sligh aniso opic o e en iso opic
beha io upon he mechanical loading, bu on he o he hand hey do no ha e an op imal esis ance o ailu e upon gi en loading
condi ions and le el o po osi y. The s eng h o he ce amic oam s uc u e can be hus u he imp o ed by design o cells ha ing
a ious egula shapes. Such oams can inally exhibi an o ho opic beha io om bo h he elas ic and s eng h poin o iew. To
unde s and how di e en ypes o cells in luence he oam cha ac e is ics in a ious di ec ions, oam s uc u es wi h a ious cell
shapes we e hus s udied and in es iga ed in e ms o hei ensile s eng h wi hin his con ibu ion. The s uc u es we e modelled
by means o beam elemen based FE models and by u iliza ion o he s ess c i e ion de ining ailu e o pa icula s u s. To ally
six di e en cell ypes we e analyzed unde conside a ion o he same po osi y o he inal oam s uc u e and amoun o he s eng h
aniso opy was quan i ied. Rela ion be ween o ien a ion o s u s wi h espec o a loading di ec ion and he oam s eng h was
discussed in mo e de ails. Recommenda ions o an employmen o pa icula cell ypes o speci ic loading condi ions we e gi en.
© 2019 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o he scien i ic commi ee o he IC MSMF o ganize s.
Keywo ds: ce amic oam; Kel in cell; Vo onoi essela ion; ensile s eng h; FE model
1. In oduc ion
Open cell oam ce amic ma e ials a e nowadays widely used in a ious ligh weigh , high empe a u e o il e ing
applica ions bu s ill no e y o en in mechanically loaded applica ions due hei ela i ely low esis ance o ailu e.
* Co esponding au ho . Tel.: +420 5 4114 2857
E-mail add ess: se ecek@ me. u b .cz
554 Oldřich Še eček e al. / P ocedia S uc u al In eg i y 23 (2019) 553–558
2 Oldřich Še eček / S uc u al In eg i y P ocedia 00 (2019) 000–000
Nomencla u e
a dimensions o he FE model
DC cell size (diame e o he sphe e insc ibed in he cell)
DS s u diame e
Ebulk Young´s modulus o he bulk ce amic
F applied o ce upon he oam ailu e
ui displacemen in he di ec ion o he coo dina e sys em axes (i=x,y,z)
x, y, z Ca esian coo dina e sys em axes

i o a ional deg ee o eedom on he nodes o he beam elemen s (i=x,y,z)

bulk Poisson´s a io o he bulk ce amic

c ensile s eng h o he bulk ce amic ma e ial

appa en ailu e s ess o he oam s uc u e
Mos o hese s uc u es a e composed o cells ha ing i egula shapes coming mos ly om he p ocessing echnology
(e.g. he eplica me hod). Thanks o a signi ican de elopmen in he ield o addi i e manu ac u ing i is howe e
possible o p epa e ce amic oam s uc u es also by 3D p in ing echnology whe e a bi a y shape o pa icula cells
can be p epa ed. Such echnology enables hus o design oams wi h signi ican ly be e mechanical p ope ies
(including ensile s eng h) han he cu en ly used i egula oam s uc u es. An example o possible designs o open
cell oam s uc u es is shown in Fig. 1. These designs will be also u he in es iga ed wi hin his wo k.
Unde s anding and p edic ion o ac u e o such oams unde a ious loading condi ions is hus essen ial o enable
hei employmen in mechanically loaded applica ions. In he i s app oxima ion, he mechanical s eng h o hese
s uc u es can be oughly es ima ed using ela ions p esen ed e.g. by Gibson and Ashby (1999), Fleck and Qiu (2007),
Romijn and Fleck (2007), Quin ana Alonso and Fleck (2007) which we e mainly de i ed o simple 2D o 3D oam
s uc u es wi h egula cell pa e n. Ano he po en ial and mo e p ecise way how o es ima e he s eng h o he open
cell oam s uc u es is o use hei 3D Fini e Elemen (FE) models based on he beam elemen s ( ep esen ing pa icula
s u s) which can ake in o accoun he eal size o he specimen, eal a e age cell size o s u hickness, e en ually
also shape o he s u c oss-sec ion. Such an app oach can be ound e.g. in Še eček e al. (2019), Jin and Wang (2012),
Še eček e al. (2017) o Še eček e al. (2018). I s main ad an age lays in s ill ela i ely low compu a ional cos s when
also bigge oam olumes a e assessed (since each s u is ep esen ed usually jus by one o ew beam elemen s). The
simples condi ion o he de ini ion o he s u ailu e is he s ess condi ion de ining ailu e o he co esponding
Fig. 1. Analysed shapes o cells wi hin he open-cell oam s uc u es: (a) cubic, (b) hexagonal p ism, (c) iangula p ism, (d)
hombododecahed al, (e) e akaidecahed al (Kel in cell), ( ) i egula .
a
b
c
e
d
Oldřich Še eček e al. / P ocedia S uc u al In eg i y 23 (2019) 553–558 555
Oldřich Še eček / S uc u al In eg i y P ocedia 00 (2019) 000–000 3
s u when he maximum ensile s ess on i exceeds he ensile s eng h

c o he ma e ial. I was shown ha e en his
simple me hod p o ides ela i ely good app oxima ions o he oam ensile s eng h p edic ions in compa ison wi h
expe imen al obse a ions Še eček e al. (2019).
Main aim o his pape is o u ilize he abo e men ioned modelling me hod o analyze he in luence o geome y o
cells shown in Fig. 1 on he mechanical s eng h o open cell oam s uc u es composed o hese cells. The oam
s eng h will be s udied in a ious di ec ions in o de o cha ac e ize he le el o aniso opic esponse o pa icula
cells. Ob ained esul s will help o ind a design o he mos sui able oam s uc u e o a gi en mechanically loaded
componen ha ing a desi ed eliabili y.
2. Nume ical simula ions
2.1. FE model
Foam s uc u es wi h a ious shapes o cells shown in Fig. 1 we e c ea ed using he Vo onoi essella ion echnique
in ma hema ical so wa e MATLAB and hen expo ed in o he FE sys em ANSYS. He e, pa icula s u s we e
meshed using beam elemen s. An app oach o modelling o oam s uc u es based on beam elemen s (al eady
p esen ed in p e ious wo k o au ho s - Še eček e al. (2019)) was employed. Namely, each s u was modelled using
h ee beam elemen s. A he ends o each s u igid beam elemen s MPC184 o leng h 0.3Ds (s u diame e ) we e
used in o de o co ec ly cap u e he s i ness o he connec ion o mul iple s u s in one node. The cen al pa o he
s u was modelled using s anda d quad a ic beam elemen s BEAM183 wi h 6 deg ees o eedom in each node. Mo e
de ails abou his modelling app oach can be ound in he abo e men ioned e e ence. To in es iga e he ensile
s eng h, a cubic model o size 10x10x10mm (Fig. 2a) wi h inne oam s uc u e shown in Fig. 2(b)-(g) was employed.
Dimensions o each cell wi hin his model we e de ined by a sphe e o a uni diame e Dc=1mm insc ibed in his cell.
Diame e o s u s Ds was se so as o ob ain o al po osi y o he oam s uc u e o 85% in all cases depic ed in Fig. 1.
The po osi y o 85% co esponds o he po osi y o a ailable (i egula cell) ce amic oam specimens p epa ed by he
eplica me hod om VUCOPOR®A ma e ial. The whole model was always clamped on one side and on he opposi e
side displacemen load in z (e en ually x and y) di ec ion was applied – see de ined bounda y condi ions in Fig. 2(a).
Fig. 2. (a) Bounda y condi ions applied on he bounda y o he beam elemen based FE model o he oam s uc u e; Beam elemen based
FE model o (b) cubic, (c) hexagonal p ism, (d) iangula p ism, (e) hombododecahed al, ( ) Kel in, (g) i egula oam s uc u e.
u
z
a= 10 mm
u
x
= u
y
= u
z
=0
x=y =0
u
x
= u
y
=0; u
z
>0
x=y =0
a
a
x
z
y
a
a
b
c
d
e
g
DC
DS
556 Oldřich Še eček e al. / P ocedia S uc u al In eg i y 23 (2019) 553–558
4 Oldřich Še eček / S uc u al In eg i y P ocedia 00 (2019) 000–000
The Young’s modulus o he ce amic ma e ial was conside ed o be Ebulk=90 000 MPa, Poisson´s a io bulk=0.25
and ensile s eng h o he bulk ma e ial

c=60MPa (which all co esponds o a “VUCOPOR®A” - Al2O3 based
ce amic oam) – see Še eček e al. (2019). The c ea ed models we e o med o 2k-14k s u s (depending on he ype
o he s uc u e shown in Fig. 2(b)-(g).
2.2. Simula ion o he ensile es – de e mina ion o he ensile s eng h
In o de o de e mine a ensile s eng h o , in o he wo ds, he c i ical applied o ce leading o b eakage o he
specimen, a s ess c i e ion, deciding abou ailu e o pa icula s u s, was employed. The model was subjec ed o
s epwise displacemen load (in z, x and y di ec ion) and in each loading sub-s ep he s ess condi ions in all s u s we e
moni o ed. In case when he ensile s ess on he s u su ace exceeded i s c i ical alue (in ou case co esponding o
he s eng h o he bulk ce amic

c=60MPa) he co esponding elemen o he s u was emo ed and a new FE solu ion
wi h he same applied load was pe o med. I he e we e, wi hin he nex simula ion s ep, no s u s wi h s esses highe
han

c, he applied displacemen was inc eased and he whole p ocedu e epea ed. Such a simula ion p ocess was
i e a ed un il he whole oam s uc u e was di ided in o 2 pieces o a leas un il ha momen when he eac ion o ce
a op nodes o he model s a ed o d op (which indica ed ha he ensile s eng h o he oam s uc u e was a ained)
– o mo e in o ma ion see Še eček e al. (2019). An example o he inal ac u e su ace ob ained on he beam
elemen based model o he hombododecahed al and i egula oam s uc u e is shown in Fig. 3(a)-(b). The amoun
o b oken s u s was a ound 490 in case o he i egula oam s uc u e and a ound 540 in case o he Kel in cell
s uc u e. These numbe s co espond also app oxima ely o he amoun o simula ion s eps which had o be pe o med
o ecei e inal ac u e su ace and whole loading cu e shown in Fig. 3(c). The g aph in Fig. 3(c) shows ypical
loading cu es ob ained om he ensile es simula ion (again o he case o hombododecahed al and i egula oam
mesh). Namely, i is he o al eac ion o ce in op nodes calcula ed o a gi en displacemen load. One can also obse e
in his g aph ha he i egula oam s uc u e is almos 3 imes s i e han he hombododecahed al upon conside a ion
o he same le el o po osi y o bo h meshes (85%). This can be explained by a p esence o la ge amoun o sho
s u s which makes he nodal poin s o he oam s uc u e s i e and also by p esence o s u s o ien ed in a ious
di ec ion (in case o he hombododecahed al mesh all s u s a e inclined by 45° wi h espec o he loading di ec ion).
Fig. 3. Examples o he inal oam ac u e a e he whole i e a i e simula ion p ocess o he case o (a) hombododecahed al oam
s uc u e and (b) i egula oam s uc u e; (c) example o he F(u) loading cu es o hombododecahed al and i egula oam s uc u e
ob ained om he FE simula ion.
a
b
Rhombododecahed al cells
I egula cells
c
F
=24.7N
Oldřich Še eček e al. / P ocedia S uc u al In eg i y 23 (2019) 553–558 557
Oldřich Še eček / S uc u al In eg i y P ocedia 00 (2019) 000–000 5
3. Resul s and discussion
The g aph in Fig. 4 shows esul s o he no malized ensile s eng hs calcula ed o all in es iga ed oam s uc u es
shown in Fig. 2 by means o he uniaxial ensile es simula ion desc ibed in he p e ious sec ion (and pe o med o
an uniaxial loading o he oam s uc u e in x, y and z axis). Fo easie compa ison o s eng hs o pa icula cell shapes,
all ensile s eng hs we e di ided by he minimal ensile s eng h de e mined on he hombododecahed al mesh which
was, o he geome y shown in Fig. 1(d) o Fig. 3(a), de e mined o be F = F ,min=24.7N (see Fig. 3(c)). In e ms o
s ess ( ela ed o he model c oss-sec ional a ea) has he ensile s eng h o hombododecahed al mesh alue o

=

,min=F ,min/S=0.247 MPa ( o specimen c oss-sec ional a ea S=a

a=10

10=100mm2 – see Fig. 2(a)).
By quali a i e compa ison o all in es iga ed s uc u es (ha ing always he same po osi y 85%) i was ound ha
he highes esis ance o uniaxial mechanical load is ob ained in case o oams ha ing mos o s u s o ien ed in he
loading di ec ion so ha p ima ily pu e ensile mode is induced on hem (which is a case o cubic and hexagonal cells
in he z-di ec ion). The cubic mesh exhibi ed he same mechanical s eng h in all 3 p incipal axes while he hexagonal
mesh had signi ican ly lowe ac u e esis ance in x and y di ec ion (due o p esence o inclined s u s in x and y
di ec ion). On he inclined s u s o he oam s uc u e bending loading is induced which leads o signi ican ly highe
su ace s esses ( esponsible o a s u ailu e upon lowe applied load) in compa ison wi h s u s loaded by pu e
ension upon he same ex e nal load. The only excep ion is he iangula p ism cell which exhibi s highe s eng h
e en when i is loaded in he (y) di ec ion whe e inclined cells a e p esen – as shown in Fig. 4. The eason o such
beha iou is he ac ha he inclined cells lays only in he plane xy and o m a uss s uc u e on whose s u s jus
pu e ensile o comp essi e loading occu s (in o he wo ds, no bending componen on s u s due hei special o ma ion
in he xy plane is p esen ).
One has o no e ye ha all he in es iga ed oam s uc u es we e loaded and analysed always in he di ec ion o
main p incipal axes x, y and z (see Fig. 2). Ne e heless, some o he s uc u es (such as he cubic cell mesh) can
exhibi signi ican ly lowe s eng h when hey a e loaded in o he di ec ion han he p incipal one (since he s u s will
become subjec ed o bending and no only o a pu e ension). The cubic s uc u e is hus sui able only o cases whe e
he loading is o ien ed always in he di ec ion o p incipal axes. In ha case he s eng h o he cubic s uc u e is
app oxima ely 12 imes highe han o he hombododecahed al cell mesh (upon he same oam po osi y) which is a
signi ican di e ence. I jus a uniaxial loading is expec ed wi hin he designed componen , he mos ac u e esis an
oam upon he gi en po osi y is he hexagonal p ism oam (in he di ec ion z - pe pendicula o a hexagon plane). On
Fig. 4. Compa ison o he ela i e ensile s eng h o all in es iga ed oam s uc u es in di ec ions o he Ca esian coo dina e sys em axes.
The alues o ensile s eng hs a e no malized by he minimal ensile s eng h de e mined on he hombododecahed al cell oam s uc u e.
1
1.7
1.2
1.1
Cubic cell
Hexagonal
p ism cell
T iangula
p ism cell
Rhombic
dodecahed al
cell
Kel in cell
I egula cell
Rela i e ensile s eng h

/

,min [-]
18.7
12
7
8.5
5.5
5.5
y
x
y
x
Load di . x
Load di . y
Load di . z
Tensile s eng h

[MPa]
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
20
18
16
14
12
10
8
6
4
2
0
(

,min
=0.247 MPa)

558 Oldřich Še eček e al. / P ocedia S uc u al In eg i y 23 (2019) 553–558
6 Oldřich Še eček / S uc u al In eg i y P ocedia 00 (2019) 000–000
he o he hand, he iangula p ism cell mesh is he bes op ion o s uc u es whe e loading di ec ion may a y du ing
ope a ion. The s eng h o he hombododecahed al, Kel in and i egula cell mesh is he lowes (due o absence o
s u s o ien ed in he di ec ion o loading and due o a p esence o inclined s u s), bu hei s eng h can be conside ed
o be mo e o less iso opic (no di ec ionally dependen ).
4. Conclusions
Based upon he 3D FE beam elemen based model he ensile s eng h o open cell oam s uc u es was in es iga ed
o a ious shapes o cells. The ensile s eng h was e alua ed in di e en di ec ions since some o he cell shapes
exhibi aniso opic beha io . The size o all s udied shapes o cells was designed so ha he insc ibed sphe e in each
cell had a uni diame e and he c oss-sec ional a ea o s u s was se so ha he inal oam s uc u e had always he
same po osi y. Failu e o s u s upon he mechanical es simula ion was modelled by employmen o he s ess
c i e ion. Namely, he s u was conside ed o ail when he maximal ensile p incipal s ess on i exceeded he ensile
s eng h o he ma e ial. Using he i e a i e simula ion p ocedu e, he comple e o ce displacemen cu e o he ensile
es was ob ained. The maximal peak o his cu e de e mines he s eng h o he oam s uc u e. By quali a i e
compa ison o all in es iga ed s uc u es i was ound ha he highes esis ance o uniaxial mechanical load is
ob ained in case o oams ha ing mos o he s u s o ien ed in he di ec ion o he loading so ha p ima ily he ensile
loading is induced on hem. The oam s uc u es wi h inclined s u s a e less esis an o ac u e since bending
componen s and hus highe ensile s esses on hei su ace a e induced. The only excep ion was he iangula p ism
cell which exhibi ela i ely high s eng h also in a di ec ion whe e inclined s u s a e p esen . Fo he pu ely
unidi ec ional loading he oam wi h highes ac u e esis ance is he hexagonal p ism oam, ne e heless i s s eng h
in o he wo di ec ions is signi ican ly lowe and is hus no sui able o mul iaxial loading. The bes op ion o he
o ho opic i-axial loading is he cubic oam mesh whe e he s eng h in all h ee p incipal di ec ions is he same and
app oxima ely wel e imes highe in compa ison wi h he weakes hombododecahed al mesh. Mo e o less iso opic,
bu signi ican ly lowe s eng h has he Kel in, i egula and hombododecahed al mesh.
Acknowledgemen s
Financial suppo o he Czech Science Founda ion unde he p ojec no. 17-08447Y is g a e ully acknowledged.
Au ho s g a e ully acknowledge also inancial suppo p o ided by he ESIF, EU Ope a ional P og amme
Resea ch, De elopmen and Educa ion wi hin he esea ch p ojec "A chi ec u ed ma e ials designed o addi i e
manu ac u ing", Reg. No.: CZ.02.1.01/0.0/0.0/16_025/0007304.
Compu a ional esou ces we e p o ided by he CESNET LM2015042 and he CERIT Scien i ic Cloud
LM2015085, p o ided unde he p og amme "P ojec s o La ge Resea ch, De elopmen , and Inno a ions
In as uc u es".
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