Mechanics o Ma e ials 95 (2016) 28–48
Con en s lis s a ailable a ScienceDi ec
Mechanics o Ma e ials
jou nal homepage: www.else ie .com/loca e/mechma
No el app oach o FE solu ion o c ack p oblems in he
Laplacian-based g adien elas ici y
Pe Skalka, Pe Na á il, Michal Ko oul∗
B no Uni e si y o Technology, Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Technicka 2, 61669 B no, Czech Republic
a icle in o
A icle his o y:
Recei ed 17 June 2015
Re ised 13 Decembe 2015
A ailable online 30 Decembe 2015
Keywo ds:
Ce amic ma e ial
Foam ma e ial
G adien elas ici y
S ess concen a ions
Fini e elemen s
Nume ical algo i hms
abs ac
S ess/s ain concen a ion a ound a c ack ip in ce amic oam-like s uc u es, whe e he
cha ac e is ic size o he oam cell is compa able wi h a ypical leng h o e which field
quan i ies change significan ly, is analysed. I is con enien o eplace he oam s uc-
u e by an e ec i e con inuum which e ains all necessa y cha ac e is ic ea u es o he
oam. To his end a homogenized c acked open cell ce amic oam is analysed using he
s ain/s ess model based upon an implici dependence o he non-local s ess and s ain
on he local s ess and s ain in he o m o an inhomogeneous Helmhol z equa ion which
is sol ed oge he wi h he equilib ium equa ions. To p ese e he ad an age o he un-
coupled sys em o equa ions, which means ha one se o equa ions is sol ed p io he
o he one and ha wo se s o shape unc ions can be chosen independen ly, a ecu en
sequence o well posed bounda y alue p oblems is cons uc ed which can be sol ed us-
ing a classical 8-nodes isopa ame ic elemen o 2-D p oblems. The key idea behind he
sugges ed p ocedu e consis s in eplacing he scale pa ame e lby a pa ame e inc emen
lchosen a bi a ily small. The gi en small dis u bance causes a edis ibu ion o he local
s ess/displacemen field. A e he new local s ess/displacemen field is known, ano he
dis u bance is in oduced and so on. The p ocedu e is s opped when he solu ion, exhibi -
ing bounda y laye beha iou , leads o he cohesi e-like zone wi h leng h eaching a alue
uniquely ela ed o he scale pa ame e l.
© 2015 Else ie L d. All igh s ese ed.
1. In oduc ion
Ce amic oam-like s uc u es a e inc easingly used in a
numbe o indus ial b anches. Manu ac u ed cellula ma-
e ials ha e been de eloped o a ange o applica ions like
ca aly ic subs a es, high empe a u e fil e s o mel ed al-
loys, issue enginee ing as bone eplacemen ma e ial, in-
sula ion ma e ials, ligh -weigh ein o cemen e c., see e.g.
(Gibson and Ashby, 1997; Baumann and He be g-Lied ke,
1994; Boccaccini e al., 2005; Coch an, 1998). These ap-
plica ions equi e high pe meabili y, high su ace a ea and
good insula ion cha ac e is ics bu also a good esponse
∗Co esponding au ho . Tel.: +420 54114 2889.
E-mail add ess: ko[email p o ec ed].cz (M. Ko oul).
o di e en ypes o mechanical loading co esponding
o gi en applica ions. Also hei na u al coun e pa s (e.g.
bone, sponge and wood) ha e a cellula s uc u e ha op i-
mises pe o mance in a pa icula se ing. The use ul p op-
e ies o cellula solids depend on he ma e ial om which
hey a e made, hei ela i e densi y, and hei in e nal ge-
ome ical s uc u e.
Modelling o oam mechanical beha iou was a -
emp ed by many esea che s al hough mos ly limi ed o
he p edic ion o jus he ini ial elas ic cons an s (e.g.
elas ic modulus, Poisson’s a io and shea modulus). The
mos ep esen a i e models assume some cellula mi-
c os uc u e (usually egula ), ea he ligamen s as beam
columns, and use elemen a y s eng h o ma e ials o e al-
ua e he de o ma ion o ep esen a i e mic o sec ions. P e-
dic ion o elas ic cons an s o Kel in cell oams ha e been
h p://dx.doi.o g/10.1016/j.mechma .2015.12.007
0167-6636/© 2015 Else ie L d. All igh s ese ed.
P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48 29
Nomencla u e
A,B enso influence unc ions
(displacemen and ac ion)
Abinfluence unc ion co e-
sponding o cells loca ed in
he bulk
Aeinfluence unc ion co e-
sponding o cells adjacen
o edges
bkj Eshelby s ess enso
bT,b
x,b
ybody o ces field
Bus ain-displacemen ma ix
Cmic o-scale s i ness enso
C1,C
2nume ical cons an s in Eq.
(11) pe inen o he e ec-
i e elas ic p ope ies
Ce e ec i e elas ici y s i ness
enso
{C} symme ic ma ix o elas ic
moduli
Cijkl elas ic s i ness enso com-
ponen s
dspec cha ac e is ic specimen di-
mension
d he dis ance o poin o in-
flec ion om he c ack ip
dn he dis ance o poin o in-
flec ion om he c ack ip
compu ed in n h s ep o he
i e a ion p ocedu e
Dmic o-scale compliance
enso
De e ec i e elas ici y compli-
ance enso
{D} symme ic ma ix o elas ic
compliances
Ee ,E1e ,E2e ,E3e young modulus o he e -
ec i e con inuum
egunknown nodal nonlocal
s ains
e sj Le i-Ci i a enso
Esyoung modulus o elas ici y
o s u s
Ge ,G1e ,G2e ,G3e shea modulus o he e ec-
i e con inuum
G’ modulus measu ing he de-
g ee o cubic aniso opy
Djsu ace g adien ope a o
componen s
Hcha ac e is ic cell size
Kuu S i ness ma ix in he o -
mula ion o fini e elemen
disc e iza ion
Kεε1,Kεε2,Kεuma ices occu ing in he
o mula ion o fini e ele-
men disc e iza ion
Lsleng h o beams (s u s o
cell)
lleng h scale pa ame e o
he g adien elas ici y
Lde i a i e ope a o
nuni ec o no mal o he
infini esimal a ea
nj,n
x,n
yuni no mal ec o compo-
nen s
NC numbe o cells pe uni
leng h o specimen edge
Numa ix o shape unc ions
app oxima ing displace-
men s
Nεma ix o shape unc ions
app oxima ing nonlocal
s ains
ˆ
Piauxilia y o ce ac ions
cunknown nodal displace-
men s o classical elas ici y
ˆ
Riauxilia y double o ce ac-
ions
Sc oss sec ion o beams
(s u s o cell)
ue s ess ec o
ˆ
ip esc ibed ac ions on
he Neumann pa o he
bounda y σ
uc
iclassical elas ici y displace-
men componen s
ug
ig adien -en iched displace-
men componen s
ûip esc ibed essen ial bound-
a y condi ions on u
ˆ
gip esc ibed essen ial bound-
a y condi ions on Du
ux, uy, uz, o x, o y, o z deg ees o eedom in FEM
V olume o he specimen
Vb olume ac ion o cells no
adjacen o he specimen
edges
Ve olume ac ion o cells
adjacen o he specimen
edges
Ws ain ene gy densi y
x, xjCa esian coo dina es
lscale pa ame e inc emen
uDi ichle pa o hebound-
a y
σNeumann pa o he
bounda y
α(x) nonlocal weigh unc ion
δjk K onecke del a
δui i ual displacemen com-
ponen s
δεijg i ual nonlocal s ain en-
so componen s
δεij i ual local s ain compo-
nen s
30 P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48
λlamé cons an
ɛp esc ibed ole ance
εmmic o-scale s ain field
εkl,εklclocal s ain enso compo-
nen s
εklgnonlocal s ain enso com-
ponen s
{εg} nonlocal s ain in ma ix
no a ion
νe ,ν1e ,ν2e ,ν3e Poisson’s a io o he e ec-
i e con inuum
νsPoisson’s a io o s u s
∇nabla ope a o
σmmic o-scale s ess field
σij,σijcCauchy s ess enso
{σg} nonlocal s ess in ma ix
no a ion
pe o med amongs o he s by Demen ’e and Ta akano
(1970), Wa en and K aynik (1997), Zhu e al., (1997)and
P adel, (1998).
Since a di ec nume ical app oxima ion o enginee ing
s uc u es composed o cellula o he e ogeneous ma e ials
is compu a ionally p ohibi i e, homogeniza ion p ocedu es
a e o g ea impo ance. Analyses based on idealized pe i-
odic oam models in gene al yield good app oxima ions o
he mean e ec i e p ope ies. Howe e , idealized pe iodic
oam models do no allow ob aining he sca e and he
unce ain y o he e ec i e ma e ial esponse. Al hough
he sca e dec eases wi h inc easing specimen sizes due
o sel -a e aging e ec s, nume ical s udies by Kanaun and
Tkachenko (2006), Van De Bu g e al. (1997)andZhu e al.
(2000) indica e ha specimen sizes in he ange o se -
e al hund eds o cells migh be equi ed in o de o ob-
ain s able esul s. Mechanical p ope ies o cellula solids
show a s ong size-dependence in leng h scales when he
cell size is o he o de o he specimen size. The indi-
idual esponse o cells o a loading is s ongly influenced
by hei loca ion in he ma e ial: de o ma ion o cells lo-
ca ed in he bulk su ounded by o he cells, o loca ed
nea he edges whe e kinema ic bounda y condi ions a e
applied, a e mo e cons ained compa ed o cells adjacen
o s ess- ee bounda ies. As a consequence, he size e ec
occu s because o he inc eased a ea ac ion o bounda y
laye s in small specimens. Howe e , i he dimensions o
a oam specimen a e la ge enough o include many cells,
he di e ences in pe -cell esponse o a mac oscopic load-
ing a e a e aged ou , leading o size-independen e ec i e
p ope ies.
Nume ical homogeniza ion echniques o mic o-mac o
scale ansi ion ha e been de eloped o ins ance in (Fish
and Belsky, 1995; Gila e al., 2009; Hain and W igge s,
2008; Ka i e al., 2007; Loehne and W igge s, 2008;
Miehe and Koch, 2002; Oden and Zohdi, 1997; Zohdi and
W igge s, 2005). Commonly used is wo-scale nume ical
homogeniza ion app oach based upon FEM, see (Feyel and
Chaboche, 2000), whe e a mic os uc u e is embedded in o
a mac oscopic fini e elemen amewo k ia p ojec ion and
homogeniza ion schemes using he concep o a ep esen-
a i e olume elemen (RVE) in oduced by Nema -Nasse
and Ho i (1999). The applica ion o he wo-scale nume -
ical homogeniza ion me hod o classical con inua can be
ound in (Miehe and Koch, 2002), (Nema -Nasse and Ho i,
1999; Miehe e al., 1999; Miehe e al., 2002a; Miehe e al.,
2002b; Sch öde , 1996; Sch öde , 2000a; Sch öde , 2000b),
o g adien -enhanced homogeniza ion in (Kouzne so a,
2002; Kouzne so a e al., 2002) o o Cosse a and mi-
c omo phic con inua in (Feyel, 2003; Hi schbe ge e al.,
2008; Jänicke e al., 2009), among o he . The e a e homog-
eniza ion schemes based on highe o de bounda y condi-
ions o a single ep esen a i e olume elemen (RVE), e.g.
(Gologanu e al., 1995).
Chen e al. (1998), de eloped a con inuum model o
cellula ma e ials and ound ou ha he con inuum de-
sc ip ion o hese ma e ials obeys a g adien elas ici y he-
o y. In he la e s udy, he in insic ma e ial leng h was
na u ally iden ified wi h he cell size.
S ess/s ain concen a ions due o no ches and holes in
oams depends on he hole adius, ela i e o he cell size,
(An oniou e al., 2004). Di ec nume ical modelling o c ack
in a oam specimen shows ha he c ack aces close mo e
smoo hly and exhibi a cusp-like closu e. The simila be-
ha iou is obse ed in homogenized solids when g adien
elas ici y heo ies a e applied o desc ibe he s ess/s ain
field wi h emo ed singula i ies a ound sha p c ack ips.
Fo he o e iew o g adien elas ici y heo ies see e.g.
(Askes and Ai an is, 2011). Wi hin he amewo k o he
g adien elas ici y heo ies, highe o de displacemen g a-
dien s a e inco po a ed in o he o mula ion o he mac o-
scopic s ain ene gy densi y. I equi es in oducing addi-
ional pa ame e s which ela e o mic os uc u al leng h
p ope ies. The majo challenge, howe e , is he de e mi-
na ion o hese addi ional pa ame e s. Also compu a ional
difficul ies a ise since he go e ning di e en ial equa ions
and bounda y condi ions a e complica ed in compa ison o
he classical elas ici y. No e ha analy ical solu ion is e-
s ic ed o he simples cases. The highe o de g adien s
equi e ha ei he fini e elemen o mula ions inco po a e
C1displacemen field wi h deg ees o eedom including
nodal displacemen s and displacemen g adien s o mixed
C0-con inui y fini e elemen o mula ions a e necessa y,
whe e e e y elemen includes high numbe o nodal de-
g ees o eedom. E.g. (Amana idou and A a as, 2002)p o-
posed mixed C0-con inui y fini e elemen wi h 70 DOF in
2-D p oblems. The lack o obus C1-con inuous elemen s
and/o C0-con inui y fini e elemen s including high num-
be o nodal deg ees show, ha cu en ly no efficien fini e
elemen me hods a e a ailable o s ain g adien heo y
o mula ions.
Ai an is (1992), Al an and Ai an is (1992)andRu and
Ai an is (1993) sugges ed a simple a ian o g adien elas-
ici y which seems o be e y powe ul. Specifically, he
s esses a e ela ed o he s ains and o he Laplacian o
he s ains, wi h only one in e nal leng h scale which is
easie o be de e mined. To a oid difficul ies connec ed
wi h FEM solu ion o ou h-o de pa ial di e en ial equa-
ions in e ms o displacemen s, Ru and Ai an is, (1993)
sugges ed he solu ion s a egy whe eby he ou h-o de
pa ial di e en ial equa ions a e spli in o wo se s o
second-o de pa ial di e en ial equa ions which can be
P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48 31
sol ed in a decoupled manne . This pa icula s a egy en-
ables he use o s anda d fini e elemen disc e isa ion wi h
con inui y o he displacemen s only, (Ai an is, 1992). The
au ho s (Ru and Ai an is, 1993) claim ha hei app oach
implies ha he s ess field o classical elas ici y coin-
cides wi h ha o g adien elas ici y, howe e , he co e-
sponding displacemen fields a e di e en in gene al. An-
o he o mula ion o g adien elas ici y which also p o ides
wo se s o second-o de and fi s o de pa ial di e en ial
equa ions was sugges ed in by Askes and Gu ié ez (2006)
who sugges ed an implici dependence o he non-local
s ess and s ain on he local s ess and s ain in he o m
o an inhomogenous Helmhol z equa ion. The Helmhol z
equa ion is sol ed oge he wi h he equilib ium equa ion.
The same model was ob ained by E ingen, (1983) om
his heo y o nonlocal elas ici y whe e he in eg al a e e-
placed by g adien s. The ansi ion om in eg al- ype non-
locali y o g adien - ype nonlocali y depends on he choice
o he weigh unc ion. I as he weigh unc ion he mod-
ified Bessel’s unc ion o he second kind o he Gaussian
unc ion is chosen, he o me inhomogeneous Helmhol z
equa ion is ob ained. Con a y o he s a egy o Ru and
Ai an is (1993), he esul ing sys em o pa ial di e en ial
equa ions is coupled. Howe e , he a o emen ioned s a e-
gies may also lead o compu a ional e o s connec ed wi h
a poo sa is ac ion o na u al bounda y condi ions o he
o iginal ou h-o de pa ial di e en ial equa ions and/o
oscilla ions in displacemen s a ee bounda ies like c ack
aces closely behind he c ack ip p obably associa ed wi h
b eaching he Babuska-B ezzi in -sup condi ion (Babuška,
1973; Ba he, 2001). In his pape a special i e a ion p o-
cedu e is p oposed which sa isfies na u al bounda y con-
di ions and allows using a classical 8-nodes isopa ame ic
elemen o 2-D p oblems.
In he fi s pa o he pape a fini e-elemen based
mic omechanics p ocedu e is used o calcula e he elas-
ic p ope ies o open oam ma e ials. An idealized pe i-
odic e akaidecahed al oam is conside ed. A c i ical size
o he oam specimen, which leads o size-independen e -
ec i e elas ic p ope ies, is sough . In he second pa a
c ack is in oduced in he oam specimen. C ack is c ea ed
by emo ing ce ain numbe o cells pe aining o a c ack
leng h. The c acked oam specimen is loaded and he c ack
opening is eco ded. Then he oam is modelled in e ms
o he implici g adien elas ici y heo y wi h one in e nal
leng h scale. P e iously de e mined e ec i e elas ic p op-
e ies a e employed. The c ack p ofiles compu ed using he
disc e e model o oam mic os uc u e a e compa ed wi h
hose compu ed using he g adien elas ici y heo y.
2. Nume ical homogeniza ion o pe iodic
e akaidecahed al oam
An idealized pe iodic e akaidecahed al oam consis s
o equisided e akaidecahed on cells, see Fig. 1.Thes u s
o he equisided e akaidecahed on model a e consid-
e ed as iso opic. The beam elemen s a e used o model
he s u s. Timoshenko beam elemen s wi h h ee nodes
and quad a ic in e pola ion unc ions a e used o allow
o bending and ans e se shea de o ma ions. Th ee ele-
men s a e used o disc e iza ion o a single s u . The me-
Fig. 1. Scheme o he equisided e akaidecahed on cell.
chanical pa ame e s o s u s a e assumed o co espond
o Young modulus Es=23.42 GPa and Poisson’s a io νs=
0.33.
E ec i e con inuum model is ob ained by homogeniza-
ion o he oam. The e ec i e elas ici y enso Ce ela es
he olume a e age o he mic o-scale s esses and he ol-
ume a e age o he mic o-scale s ains
σm(x)=Ce :εm(x),(1)
whe e
·=1
VV
(·)dV (2)
s ands o he olume a e age o he mic o-scale quan i-
ies. S anda d no a ion is used h oughou . Bold ace sym-
bols deno e enso s o he o de indica ed by he con ex .
The usual Eins ein summa ion con en ion is used. The ol-
lowing p oduc s a e used in he ex : (ab)ij =aibj,a
jAji =
(a.A)i,A
ijBjk =(A.B)ik,A
ijBij =A:B,CijklAkl =(C:A)ij,εijσij =
ε:σe c. The a e ages a e aken o e he whole specimen
since he aim is o analyse he size e ec and also o es i-
ma e he c i ical size o he specimen abo e which he e -
ec i e elas ic p ope ies a e p ac ically size-independen .
A linkage be ween mic o- and mac o-fields can be ound
nume ically using FE analysis. Fo mally, his link can be
es ablished by in oducing unknown enso influence unc-
ions A(x), B(x)
εm(x)=A(x):εm,σm(x)=B(x):σm.(3)
Themic o-scalefieldsεm(x), σm(x) co espond o he
field mac o-scale a iables εm,σmdependen only
on he imposed bounda y condi ions. The enso influ-
ence unc ion A(x) co esponds o displacemen bounda y
32 P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48
Fig. 2. Two se s o bounda y condi ions applied o he oam specimen. (a) Foam specimen unde uniaxial comp ession. (b) Foam specimen unde shea .
condi ions
um(x)=um(x)=εm·x,εm=cons ,x∈(4)
and he enso influence unc ion B(x) co esponds o ac-
ion bounda y condi ions
σm(x)·n(x)= (x)=σm(x)·n(x),
σm=cons ,x∈.(5)
I he igh -hand sides o Eqs. (3) a e sub-
s i u ed in o he a e aged cons i u i e equa ions
σm(x)=C(x):εm(x),εm(x)=D(x):σm(x),D(x)
is he mic o-scale compliance, one ob ains
σm(x)=C(x):A(x):εm,
εm(x)=D(x):B(x):σm,(6)
whe e om he e ec i e s i ness and he compliance
ollow
Ce =C(x):A(x),De =D(x):B(x).(7)
The homogeniza ion p ocedu e has o ulfil he Hill–
Mandel condi ion (Hill, 1963; 1967) alid asymp o ically o
o de H/dspec (dspec is he cha ac e is ic specimen dimen-
sion, His he cha ac e is ic cell size)
σm(x):εm(x)=σm:εm+O(H/dspec),(8)
which enables one o spli a olume a e age o he en-
e gy in o a p oduc o olume a e ages o s ess and s ain
fields. The iola ion o he Hill–Mandel condi ion leads
o a dec ease o meaning ulness o he homogeniza ion
p ocedu e.
The influence unc ions p o ide a ool o simple ea-
soning conce ning he size-dependence o elas ic p ope -
ies. As al eady men ioned in he In oduc ion, he indi id-
ual esponse o cells o a loading is s ongly influenced by
hei loca ion in he ma e ial, namely de o ma ion o cells
loca ed nea he edges whe e kinema ic bounda y condi-
ions a e applied, is mo e cons ained compa ed o cells in
he bulk. Appa en ly, he influence unc ions ha e o ake
significan ly di e en alue nea he specimen edges com-
pa ed o he specimen bulk. Conside Eq. (7)1,deno eby
Ab he influence unc ion co esponding o cells loca ed
in he bulk, by Ae he influence unc ion co esponding o
cells adjacen o edges and ob ain
Ce =1
VV=Vb⊕Ve
C(x):A(x)dV =1
VVb
C(x):Ab(x)dV
+1
VVe
C(x):Ae(x)dV,(9)
whe e Vbdeno es he olume ac ion o cells which a e
no adjacen o he specimen edges and Vedeno es he
olume ac ion o cells adjacen o he specimen edges.
Wi h e e ence o Fig. 2 one can see ha Vb∼NC2H3and
Ve∼NC.H3,whe eNC s ands o he numbe o cells pe
leng h o specimen edge. Making use o Eq. (9) one can
show ha Ce depends on NC as ollows
Ce =C1+C2
NC ,(10)
whe e C1and C2a e defined as ollows
C1=1
VbVb
C(x):Ab(x)dV,C2=1
VeVe
C(x):Ae(x)dV.
The model o pe iodic e akaidecahed al oam speci-
men was subjec ed o wo kinds o kinema ic bounda y
condi ions, see Fig. 2, co esponding o uniaxial p essu e
loading and o shea loading a he uppe edge o he spec-
imen. The lowe edge o specimen was fixed. The le -hand
edge and he igh -hand edge we e s ess- ee howe e
wi h p esc ibed coupling o displacemen o nodes in he

P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48 33
Fig. 3. Shea modulus no malized by he bulk alue plo ed agains he
numbe o cells.
di ec ion o x-axis. In he case o uniaxial p essu e loading
only hal o he specimen is conside ed and he symme-
y bounda y condi ion is p esc ibed along he y-axis, see
Fig. 2a. The same bounda y condi ions we e p esc ibed on
he con inuum model o he specimen. The goal is o each
a coincidence in he de o ma ion esponse o he oam
s uc u e and he e ec i e con inuum by a sui able choice
o he elas ic cons an s o he con inuum. When he coin-
cidence o he de o ma ion fields is eached, he deduced
elas ic cha ac e is ics (Ee ,νe ,G
e )a e akenas hecha -
ac e is ics o he homogenized oam specimen. I is an in-
e se p oblem which is ela i ely cos ly. Fo ha eason
homogenized specimen is analysed unde plane s ain con-
di ions while he oam s uc u e is modelled by a single
laye o 3D cells subjec ed o plane s ain condi ions, oo,
see Fig. 2.
The Young modulus Ee and Poisson’s a io νe o he
e ec i e con inuum a e sough using a s ep by s ep p o-
cedu e which s ops when he change o he Young modu-
lus Ee be ween las wo s eps is smalle han 10 Pa and
he change o Poisson’s a io νe is smalle han 10−4.The
shea modulus Ge o he e ec i e con inuum is sough
in he same manne unde shea loading condi ions, see
Fig. 2b and he p ocedu e s ops when he change o Ge
is smalle han 10 Pa. In o de o analyze he size e ec ,
he homogeniza ion was pe o med o specimens wi hin a
wide ange o sizes s a ing wi h he specimen o 1×1×1
cell up o he specimen consis ing o 140×140×1 cells.
Two di e en si ua ions we e conside ed: he size o he
specimen 1, see he inle in Fig. 3, was inc eased in sel -
simila manne , while only he heigh o he specimen 2
was inc eased. Fig. 3 shows he e ec o he specimen size
on he shea modulus. I is seen, ha he e ec o less
cons ained cells adjacen o s ess- ee bounda ies in he
specimen 1 is s onge han he e ec o mo e cons ained
cells nea he edges whe e kinema ic bounda y condi ions
a e applied leading o he lowe shea s i ness o small
specimens. The size e ec anishes o he numbe o cells
pe edge ∼
=80. In he case o specimen 2 he mo e con-
s ained cells nea he edges whe e kinema ic bounda y
condi ions a e applied cause a highe alue o he shea
s i ness o small specimens. The size e ec anishes o
he numbe o cells pe edge ∼
=40.
The Young modulus and Poisson’s a io depend on he
specimen size in a simila way, see Fig. 4, which shows he
e ec o specimen size i he specimen dimensions a e in-
c eased sel -simila ly.
Acco ding o Eq. (10) one can exp ess he e ec i e elas-
ic p ope ies Ee ,νe ,G
e o pe iodic e akaidecahed al
oam specimen in he ollowing o m
Ee ,νe ,Ge =C1+C2
NC .(11)
Nume ical alues o he cons an s C1,C2 o a pa icula
ma e ial pa ame e a e gi en in he Table 1.
The cons an s C1and C2we e ob ained di ec ly om
Eq. (11) whose le -hand side con ains he e ec i e elas ic
p ope ies compu ed nume ically o specimens wi hin a
wide ange o sizes as desc ibed abo e. The e ec i e elas ic
p ope ies Ee ,νe ,Ge we e ound using wo simple in-
dependen loading s a es. I is impo an o e i y he cal-
cula ed alueso Ee ,νe ,Ge unde mo e complex load-
ing. The loading specified in Fig. 5, which combines ension
and shea , was conside ed. The displacemen uyo he e -
e ence poin , see Fig. 5, was moni o ed bo h o he oam
specimen ha ing he size 80×80×1 cells (3D model o a
single plane o cells) and o he e ec i e con inuum spec-
imen (2D model) o he same size. A di e ence o he dis-
placemen uycalcula ed in he oam specimen and in he
e ec i e con inuum specimen ela i e o he displacemen
o he oam specimen cha ac e izes he ela i e e o o
homogeniza ion. The displacemen fields uycalcula ed in
bo h oam specimen and he e ec i e con inuum specimen
espec i ely is shown in Fig. 6.
Fo he specimen consis ing o 80×80×1 cells he e ec-
i e elas ic p ope ies ake sa u a ed alues, c . Fig. 4,and
he ela i e e o o homogeniza ion defined as
e o [%]=u oam
y−usolid
y
u oam
y·100 (12)
is app oxima ely 0.39%. The e y small ela i e e o
leads o he conclusion ha he in es iga ed pe iodic
e akaidecahed al oam can be modelled as an e ec i e
cubic elas ic con inuum wi h elas ic p ope ies Ee
1=Ee
2=
Ee
3,νe
1=νe
2=νe
3and Ge
1=Ge
2=Ge
3( he subsc ip s
1,2,3 e e o he p incipal axis o symme y) aking alues
p esen ed in Fig. 4.
Fo gi en bounda y condi ions, he co esponding max-
imum mic o-scale ensile s esses in he s u s o he oam
can be calcula ed and compa ed wi h he ensile s eng h
o s u s hus allowing o p edic s u s b eaking and a
c ack o ma ion in he oam.
34 P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48
Fig. 4. Elas ic p ope ies o he e ec i e con inuum plo ed agains he numbe o cells. (a) Ee – Young modulus o elas ici y. (b) νe – Poisson’s a io.
(c) Ge – Shea modulus.
Table 1
Nume ical alues o he cons an s C1and C2in Eq. (11).
C1C2
Ee [MPa] 694.25 −61.45
Ge [MPa] 280.76 −125.41
νe [-] 0.3659 0.0355
3. C acked pe iodic e akaidecahed al oam
Suppose ha he oam specimen al eady con ains an
exis ing c ack loca ed a he plane o oam symme y x-
z,seeFig. 7. Nea he c ack ip a h ee dimensional s ess
s a e de elops ( ela i e o he scale o oam cells). Assum-
ing he plane s ain condi ions one can educe he nume -
ical simula ion o a single laye o cells by p esc ibing a
all nodes on he laye aces he condi ion uz=0, see Fig. 7.
Mo eo e , due o symme y only one qua e o he speci-
men is sufficien o be analysed.
Among o he s, he nume ical simula ion o he c acked
oam was ocused on he c ack opening and ensile s esses
in he s u s o he oam. As expec ed, he peak al-
Fig. 5. Tes loading o he specimen.
ues o mic o-scale s esses in he s u s a e ound o
cells in on o he c ack ip, see Fig. 8.I shouldbe
howe e no ed ha he magni ude o mic o-scale s esses
significan ly depends on he complexi y o FE model o
P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48 35
Fig. 6. Displacemen fields uywi hin he es ing specimen. (a) Foam s uc u e. (b) E ec i e con inuum.
Fig. 7. Scheme o he oam s uc u e wi h p ima y c ack. (a) Global iew. (b) De ail iew.
indi idual s u s. In he gi en case, as al eady men ioned,
he beam elemen s we e used o model he s u s. I is he
simples model which only ensu es he displacemen and
o a ion con inui y, no he con inui y o s esses. Hence
he Kel in cell hen beha es as a beam s uc u e. Ap-
pa en ly, he beam model is insufficien o eliably p e-
dic s u s ailu e. To efine he calcula ion o s esses,
he s u s and he geome y o s u join s a e needed
o be modelled by 3D elemen s. Howe e his efinemen
makes nume ical simula ion e y cos ly and e en p o-
hibi s i . On his accoun i is con enien o eplace he
oam s uc u e by an e ec i e con inuum which e ains all
necessa y cha ac e is ic ea u es o he oam. The classi-
cal e ec i e cubic con inuum model in oduced abo e is
sufficien o simula e a esponse o he examined oam
wi h ela i ely small g adien s o he esul ing s ess-s ain
field.
Ne e heless, i ails o eliably desc ibe he s ess-
s ain field wi h high g adien s such hose nea he c ack
ip, whe e he cha ac e is ic size o he oam cell is com-
pa able wi h a ypical leng h o e which field quan i ies
change significan ly. The simple a ian o iso opic g a-
dien elas ici y based on he ollowing s ain-ene gy den-
si y unc ion (Geo giadis and G en zelou, 2006; Maugin,
1995)
W=1
2λεiiεjj +Gεijεij +l21
2λεii, εjj, +l2Gεij, εij, (13)
whe e lis he scale pa ame e ha ing dimension o
[leng h] and (λ,G) a e he s anda d Lamé cons an s, can
36 P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48
Fig. 8. Equi alen mic o - scale s ess in s u s ahead o he c ack ip.
emedy his p oblem while s ill allowing making use o he
e ec i e elas ic p ope ies calcula ed in he p eceding pa .
Only one new ma e ial cons an – he in e nal leng h scale
lis needed. We ha e a emp ed o employ he Ru–Ai an is
(Ru and Ai an is, 1993) me hod o solu ion o he g adien
elas ici y based upon an ope a o spli by which he ou h
o de pa ial di e en ial equa ions can be sol ed as an un-
coupled sequence o wo se s o second-o de equa ions.
Howe e , his me hod does no p edic cusp-like closu e o
c ack aces. I is due o an imp ope dis ibu ion o non-
local s esses because he supplemen ed na u al bound-
a y condi ions di e om hose pe aining o he o iginal
ou h o de p oblem. Hence, we es ed he E ingen he-
o y (E ingen, 1983) which possesses e y simila o ma as
he Ai an is heo y, i.e. wo se s o second-o de di e en-
ial equa ions, one o which a e he balance o momen um
equa ions exp essed in e ms o he di e gence o he non-
local s esses. Howe e , in con as o he Ai an is heo y,
hese wo se s o second-o de di e en ial equa ions a e
coupled and mus hus be sol ed simul aneously. I is a
mixed o mula ion which o en b ings p oblems wi h e-
spec o obus ness o mixed fini e elemen me hod. To
o e come his p oblem, a special i e a ion p ocedu e wi h
espec o he scale pa ame e lwas de ised. I will be
shown ha a e y good ag eemen be ween di ec mod-
elling o oam and he e ec i e con inuum based upon he
E ingen model can be ound.
4. C acked oam modelling using g adien heo y o
elas ici y
I is well-known ha g adien elas ici y heo ies p o-
ide an efficien ool o desc ip ion o he s ess-s ain
fields nea concen a o s which, when ea ed in e ms o
he classical elas ici y, possess p ope ies o singula be-
ha iou (c acks, sha p no ches) o discon inuous beha iou
(bima e ial in e aces). Equa ions o g adien elas ici y he-
o y a e equipped wi h highe o de spa ial de i a i es o
ele an a iables. The e ms wi h highe -o de de i a i es
also con ain addi ional cons i u i e pa ame e s ha ha e
he dimension o leng h and in a ce ain sense cha ac e -
ize he mic os uc u e o he ma e ial unde in es iga ion.
The highe o de spa ial de i a i es help o smoo h ou he
singula beha iou o s ess-s ain fields.
As al eady men ioned in he In oduc ion, a ious g a-
dien elas ici y models exis in li e a u e. One o he sim-
ples bu s ill efficien models was p oposed by Ru and
Ai an is (1993). They ex ended he s ain ene gy densi y
wi h an addi ional e m and a i ed o he ollowing con-
s i u i e equa ion o he Cauchy s ess σij
σij =Cijkl(εkl −l2εkl,mm),(14)
whe e Cijkl is a elas ic s i ness enso and in he case o
cubic symme y i may be w i en as ollows
Cijkl =λδijδkl +G(δikδjl +δilδjk)+G
2(δikδjl +δilδjk)
×[1 −|eijm|(δm+3,4+δm+3,5+δm+3,6)],(15)
whe e G’ measu es he deg ee o cubic aniso opy ( o
b e i y he supe sc ip ‘e ’ is al eady omi ed), eijm is he
Le i-Ci i a an isymme ic enso and δij and δm+3,jis K o-
necke del a, lis he abo e men ioned leng h scale pa-
ame e , and εkl is he local s ain enso which e ains i s
usual meaning
εkl =1
2(uk,l+ul,k),(16)
whe e uia e displacemen componen s. Obse e ha he
e ec i e s i ness enso o he homogenized oam calcu-
la ed in he p eceding sec ion can be used as he elas ic
s i ness enso in Eq. (14). Using he equilib ium equa ions
(conside ing ze o body o ces)
σij,j=0,(17)
and kinema ic Eq. (16) he ollowing ou h o de sys em
in displacemen s is ob ained
1
2Cijkl(uk,jl +ul,jk −l2(uk,jl +ul,jk)mm)=0.(18)
(Ru and Ai an is, 1993) sugges ed ew i ing he sys em
o he ollowing o m
1
2Cijkl((uk−l2uk,mm)jl +(ul−l2ul,mm)jk)=0,(19)
which allows applying C0-con inuous shape unc ions o
FE calcula ions. I he e ms in he inne pa en heses in
Eq. (19) a e iden ified as he componen s o ano he dis-
placemen field uc
i hen Eq. (19) can be spli in o wo se s
o second o de di e en ial equa ions as ollows
1
2Cijkluc
k,jl +uc
l,jk=0,(20)
P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48 43
Fig. 14. G adien -elas ic s ess componen s (a) σg
xx,(b)σg
yy and (c) σg
xy o (A) l=0.0025 mm, (F) l=0.05 mm and (CE) he classical elas ici y
solu ion.
his so o beha iou was sough by analysing nume i-
cal da a. I was ound ou ha i ela es o a poo sa -
is ac ion o he highe o de na u al bounda y condi ion
(l)2nmnjCijklεg
kl,m=0 (see Eq. (44)) closely behind he
c ack ip o l>0.01 mm.
We ha e seen ha dec easing scale pa ame e inc e-
men p o ides a be e ag eemen wi h he disc e e model.
Ne e heless, he compu a ional cos s a e educed nea ly
16 imes i he scale pa ame e inc emen is inc eased
om l=0.0025 mm o l=0.01 mm while s ill sus ain-
ing a sa is ac o y ag eemen wi h he disc e e model cal-
cula ions o he c ack p ofile. Wi hin his con ex i is im-
po an o know how he maximum o he nonlocal s ess
componen s ahead o he c ack depends on he scale pa-
ame e inc emen . Fig. 15 illus a es he a ia ion o he
maximum o nonlocal s ess componen s wi h l. Disc e e
FE da a deno ed by black bulle s we e in e pola ed using
quad a ic polynomial in l, see he legend in Fig. 15.I is
seen ha o l→0σg
xx →1.17 MPa and σg
yy →1.69 MPa.
No e ha he limi l→0 does no mean ha he p o-
posed g adien elas ici y model con e ges o he classical
elas ici y model! I only implies he numbe o i e a ions
would go o infini y. Obse e ha he maximum o he
s ess componen σg
yy calcula ed o l=0.01 mm di e s
only by 4% om he maximum alue 1.69 MPa. This find-
ing is impo an e.g. o a eliable ac u e c i e ion based
upon he maximum o ensile s ess ahead o he c ack
ip.
Fig. 16 shows con ou s o he nonlocal s ess com-
ponen s σg
xx and σg
yy. I can be seen ha he fini e

44 P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48
Fig. 15. The a ia ion o he maximum o he nonlocal s ess componen s σg
xx and σg
yy ahead o he c ack wi h l.
Fig. 16. Con ou plo s o he nonlocal s ess componen s a he c ack ip egion (a) σg
xx and (b) σg
yy.
maximum o bo h s ess componen s is a ained a he
c ack ip.
I is impo an o compa e he esul s ob ained by
means o he p oposed app oach and by means o he
o iginal Ru-Ai an is app oach. The COD p ofiles com-
pu ed ia he Ru-Ai an is app oach acco ding o Eqs.
(20),(22) and ia he p oposed app oach a e compa ed
in Fig. 17.
I is seen ha he COD compu ed acco ding o Eqs. (20),
(22) is somewha educed in compa ison o he classical
elas ici y solu ion, howe e s ill asymp o ically scales as
1/2.Fig.(18) compa es g adien en iched nonlocal s ess
componen s σg
xx,σg
yy and σg
xy compu ed by means o he
Ru-Ai an is app oach and by means o he p oposed ech-
nique. The classical elas ici y solu ion is also included
o compa ison. As expec ed, he dis ibu ion o nonlocal
s esses compu ed by means o he o iginal Ru-Ai an is ap-
p oach di e s significan ly om he dis ibu ion o non-
local s esses compu ed ia he p oposed app oach. Con-
side ing he al eady men ioned ela ion be ween in eg al-
ype nonlocali y and g adien - ype nonlocali y o mula ed
by E ingen, (1983),
σg
ij (x)=
α(s;l)σc
ij (x−s)d,(60)
wi h an app op ia ely chosen weigh unc ion α,whose
suppo closely ela es o he scale pa ame e l,one
can easily p o e ha he nonlocal s esses compu ed by
means o he o iginal Ru-Ai an is app oach co espond
o he weigh ed a e age o classical s ess dis ibu ion
P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48 45
Fig. 17. Compa ison o COD p ofiles close behind he c ack ip.
which asymp o ically scales as −1/2. Obse e ha he
COD p ofiles and he dis ibu ion o nonlocal s esses
compu ed by means o he o iginal Ru-Ai an is app oach
p esen ed in Figs. 17 and 18 espec i ely, co espond o
ou p e ious findings which show ha o he scale pa-
ame e inc emen l>0.01 mm no cusp-like closu e
zone occu s and nonlocal s esses di e om ze o o e
a ela i ely long dis ance behind he c ack ip due o
a poo sa is ac ion o he highe o de na u al bound-
a y condi ion l2nmnjCijklεg
kl,m=0 closely behind he c ack
ip.
6. Conclusions
The main pu pose o his pape is o o e a no el
app oach o solu ion o c ack p oblems in he Laplacian-
based g adien elas ici y wi h educing con inui y equi e-
men s omC
1 o C0andsubsequen ly oapply hisap-
p oach o modelling o homogenized c acked open cell ce-
amic oams. The absence o C1con inui y equi emen s
allows using con en ional FEM packages o sol ing g a-
dien elas ici y p oblems. As he s a ing poin he Ru-
Ai an is me hod o solu ion o he g adien elas ici y, based
upon an ope a o spli by which he ou h o de pa -
ial di e en ial equa ions can be sol ed as an uncoupled
sequence o wo se s o second-o de equa ions, was a -
emp ed. Un o una ely, he o iginal Ru-Ai an is app oach
does no p edic a cusp-like closu e o he c ack which
asymp o ically scales as 3/2and he calcula ed g adi-
en en iched non local s ains/s esses consequen ly co e-
spond o he weigh ed a e age o classical s ain/s ess dis-
ibu ion which asymp o ically scales as −1/2.Hence, he
E ingen heo y (E ingen, 1983) was es ed, which possesses
e y simila o ma as he Ai an is heo y, i.e. wo se s o
second-o de di e en ial equa ions, one o which a e he
balance o momen um equa ions exp essed in e ms o he
di e gence o he non-local s esses. Howe e , in con as
o he Ai an is heo y, hese wo se s o second-o de di -
e en ial equa ions a e coupled and mus hus be sol ed si-
mul aneously. I is a mixed o mula ion which o en b ings
p oblems wi h espec o obus ness o mixed fini e ele-
men me hod. To o e come his p oblem, a special i e a-
ion p ocedu e wi h espec o he scale pa ame e lwas
de ised. The i e a ion p ocedu e is ca ied ou wi h espec
o he scale pa ame e lwhich is eplaced by a sequence o
small scale pa ame e inc emen s l. Physically, he g adi-
en e m (l)2Cijklεg
kl,mm is conside ed as a small dis u -
bance because he pa ame e inc emen lis chosen a -
bi a ily small. As a esul , a sequence o bounda y alue
p oblems is ob ained. Thei solu ion p o ides a sequence o
quasi-equilib a ed s a es { c(i),eg(i)}. Fo sufficien ly small
l he p ocedu e is capable o ep oduce he ze o ac ions
on he c ack aces, con a y o he classical Ru-Ai an is o -
mula ion, and o sa is y he highe o de na u al bounda y
condi ion (l)2nmnjCijklεg
kl,m=0 closely behind he c ack
ip.
The co espondence be ween he g adien elas ici y
heo y and he ela ed disc e e model o oam is used o
es ima e a sui able land he numbe o i e a ions which
makes he c ack opening displacemen compu ed ia he
g adien elas ici y model e y close o he c ack open-
ing displacemen compu ed ia he disc e e model. The
disc e e model oge he wi h a sui able homogeniza ion
scheme also p o ide he e ec i e elas ic p ope ies Ee ,
νe ,G
e used in FE calcula ions and, as a side-e ec , i p o-
ides he size-dependence o he e ec i e elas ic p ope -
ies.
The unique ela ionship be ween he dis ance do he
poin o inflexion, whe e he cusp-like closu e zone an-
si s o he a field 1/2, and a cha ac e is ic leng h o oam
s uc u e p o ides an efficien ool o modelling o oam
s uc u es wi h di e en size o cells wi hou pe o ming
FE calcula ions on he disc e e model. As a consequence,
he compa ison o he g adien elas ici y solu ion wi h he
disc e e model solu ion is pe o med once o all and i is
no necessa y o epea FE compu a ions on he disc e e
model o oam case by case i he a chi ec u e o he cell is
he same and he o e all elas ic p ope ies do no change
wi h he cell size.
A s udy o he influence o he scale pa ame e in-
c emen l e eals ha he necessa y numbe o i e a-
ions dec eases apidly wi h he inc easing scale pa am-
e e inc emen land app oxima ely scales as (l/l)2.
The compu a ional cos s can be educed significan ly i in
he gi en case, he alue o he scale pa ame e inc e-
men l=0.01 mm is selec ed which s ill p o ides he
cusp-like closu e zone behind he c ack ip in a sa is ac-
o y ag eemen wi h he disc e e model. Nume ical analy-
sis e ealed ha he cusp-like closu e zone does no de-
elop o l>0.01 mm, albei he c ack is globally closed
in compa ison wi h he classical elas ici y solu ion. F om
he poin o compu a ional cos s he sugges ed p ocedu e
is pa icula ly con enien in he case o 3D p oblems in
compa ison o he FE solu ion o he o iginal ou h-o de
model.
46 P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48
Fig. 18. Compa ison o g adien en iched s ess componen s (a) σg
xx,(b)σg
yy and (c) σg
xy.
Acknowledgmen
The au ho s g a e ully acknowledge a financial suppo
o Czech Science Founda ion unde he p ojec No. 14-
11234S
Appendix
I is con enien o o mula e FEM models in compac
ma ix no a ion. S ain field and s ess field a e p esen ed
in he o m o h ee dimensional ec o {ε}={εxx,εyy,
2εxy}T,{σ}={σxx,σyy,σxy}T.{C}, o {D} espec i ely, is a
symme ic 3 ×3 ma ix o elas ic moduli, o a symme -
ic ma ix o compliances o a ma e ial, espec i ely. (Com-
pound pa en heses dis inguish he 3 dimensional ec o
om he co esponding enso o 2nd o de and he sym-
me ic 3 ×3 ma ix om he co esponding enso o 4 h
o de .) The usual de i a i e ope a o s ∇and La e in o-
duced
∇T=∂
∂x,∂
∂y,LT=∂
∂x0∂
∂y
0∂
∂y
∂
∂x(A1)
P. Skalka e al. / Mechanics o Ma e ials 95 (2016) 28–48 47
and ∇2=∇T∇. Ma ix o di ec ion cosines o ou e no -
mal o he  eads
{n}T=nx0ny
0nynx.(A2)
Displacemen field and he body o ces field a e p e-
sen ed in he usual no a ion as uT={ux,uy},
bT={bx,by}. The equilib ium equa ion (32) in ma ix
no a ion ead:
LT·{C}·L·u=0.(A3)
The weak o m o Eq. (32) gi es a e employing in e-
g a ion by pa s

(L·δu)T·{C}·L·ud=σ
δuT·
d=0,(A4)
The ma ix– ec o e sion o Eq. (30)1 eads
{C}·({εg}−l2∇2{εg})={C}·L·u(A5)
and i s weak o m, c . Eq. (39),is

.{δεg}T·{C}·{εg}+l2∂{δεg}T
∂x·{C}·∂{εg}
∂x
+∂{δεg}T
∂y·{C}·∂{εg}
∂yd−
(δεg)T·{C}·L·ud
−
l2δ∂ug
∂nT
·{n}T·{C}·∂{εg}
∂nd
−
(δug)T(Dpnp)l2·{n}T·{C}·∂{εg}
∂nd
+
(δug)T·DT·l2{C}·∂{εg}
∂nd=0,(A6)
whe e ∂
∂n=nj∂
∂xjis he no mal de i a i e on and Dis
he ma ix o he su ace g adien ope a o on 
DT=∂
∂x−nx∂
∂n0∂
∂y−ny∂
∂n
0∂
∂y−ny∂
∂n
∂
∂x−nx∂
∂n.(A7)
The FE equa ions ollow s aigh o wa dly om
Eqs. (A4) and (A6). App oxima e displacemen s u
a e ela ed o he unknown nodal displacemen s
={ 1x, 1y, 2x, 2y,...}Tby u=Nu· ,whe e Nuis he
ma ix o shape unc ions
Nu=Nu10Nu20....
0Nu10Nu2.....(A8)
The i ual displacemen s δua e ela ed o he dis-
c e ized nodal alues δ in simila way by δu=Nu·δ .In-
oducing he s ain-displacemen ma ix Bu=L·Nu, he fi-
ni e elemen disc e iza ion o Eq. (A4) hen eads
δ T·
BT
u·{C}·Bu· d=δ T·σ
NT
u·
d=0,(A9)
o
Kuu · =0,(A10)
whe e he s i ness ma ix Kuu is defined.
Eq. (A6) is disc e ized in a simila ashion. App oxi-
ma e nonlocal s ains {εg} a e ela ed o he unknown
nodal nonlocal s ains eg={eg
1x,eg
1y,eg
1xy,eg
2x,eg
2y,eg
2xy, ...}T
by {εg}=Nε·eg, whe e he ma ix o he shape unc ions
Nεis
Nε=Nε100Nε200...
0Nε100Nε20...
00Nε100Nε2...
.(A11)
The same shape unc ions a e used o disc e ize he
i ual s ains {δεg}=Nε·δeg. Conside ing homogeneous
na u al bounda y condi ions, he esul ing FE equa ions
ollow as

.NεT·{C}·Nε+l2∂NεT
∂x·{C}·∂Nε
∂x+∂NεT
∂y·{C}·∂Nε
∂y
×egd=
NεT·{C}·Bu·ud,(A12)
o
Kεε1+l2Kεε2·eg=Kεu· ,(A13)
whe e he ma ices Kεε1,Kεε2, and Kεua e defined. Two se s
o FE Eqs. (A10) and (A13) a e coupled and o m a mixed
o mula ion. Hence, he shape unc ions Nuand Nεcanno
be chosen independen ly because he es ic ions gi en by
he Babuška-B ezzi condi ion (Babuška, 1973; Ba he, 2001;
B ezzi, 1974) apply.
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