Novel approach to FE solution of crack problems in the Laplacian-based gradient elasticity

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. Re e ences Ai an is, E.C., 1992. On he ole o g adien s in he localiza ion o de o - ma ion and ac u e. In . J. Eng. Sci. 30, 1279–1299. 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