Towards shock absorbing hyperelastic metamaterial design (II): a prospective multiscale buckling-lattice computational model

Compu a ional Mechanics
h ps://doi.o g/10.1007/s00466-024-02593-y
ORIGINAL PAPER
Towa ds shock abso bing hype elas ic me ama e ial design (II): A
p ospec i e mul iscale buckling-la ice compu a ional model
J. Can e1,2 ·A. Nuñez-Labielle1,2 ·A. E. Huespe4,5 ·J. Oli e 1,3
Recei ed: 24 Oc obe 2024 / Accep ed: 19 Decembe 2024
© The Au ho (s) 2025
Abs ac
As a con inua ion o a p e ious wo k o he au ho s, on Compu a ional Design o Shock-abso bing Me ama e ials (Pa I)
(NunezLabielle e al. in Compu Me hods Appl Mech Eng 393:114732, 2022), his wo k explo es he po en ial o compu a ional
mul iscale me hods, in combina ion wi h massi e buckling-la ice s uc u es a he me ama e ial co e (meso/mic o scale), o
ende a sui able amewo k o designing such a shock-abso bing me ama e ials ocusing on indus ial applica ions. In his
con ex , a p ospec i e compu a ional se ing is conside ed unde he hypo hesis ha , o a su icien ly complex mic ola ice
opology, some localized egions migh buckle wi hin he la ice-s uc u e co e and p opaga e h ough i , gi ing ise o
di e en loading-unloading pa hs, in such a way ha hys e e ic-like s uc u al beha iou s would ake place, hus a ising
dissipa i e beha iou s, e en i he base ma e ial a he buckling mic o-la ice beha es in a hype elas ic ( hus in insically non-
dissipa i e) manne . Using he s anda d Hill-Mandel homogeniza ion p inciple, and assuming ha he necessa y sepa a ion
o scales holds, he homogenized body, now li ing in a classical solid-mechanics se ing, displays a homogenized non-con ex
beha iou which, in ag eemen wi h he conclusions o Pa (I) o he wo k, exhibi s ex insic dissipa ion and, hus, could be
po en ially used (a educed compu a ional cos ) o shock abso bing me ama e ials analysis and design pu poses. A en a i e
indus ial applica ion, o a sneake ’s insole design, has been hen conside ed as a wo k’s a ge o e alua ion o he oom
o e ed by he explo ed se ing in he con ex o shock-abso bing me ama e ial design. Finally, ema ks on he scope and
limi a ions o he wo k, and i s signi icance o u he ad ances in he ield a e emphasized.
Keywo ds Shock abso bing me ama e ials ·Compu a ional me ama e ial design ·Mul iscale ma e ial modeling ·Buckling
mic o-la ice ma e ials
BJ. Oli e
oli[email p o ec ed]
J. Can e
[email p o ec ed]
A. Nuñez-Labielle
[email p o ec ed]
A. E. Huespe
[email p o ec ed]
1Cen e In e nacional de Mè odes Numè ics en Enginye ia
(CIMNE), C/ G an Capi á, S/N, Edi ici C1, 08034 Ba celona,
Spain
2Escola Supe io d’Enginye ies Indus ial Ae oespacial i
Audio isuals de Te assa (ESEIAAT), Uni e si a Poli ècnica
de Ca alunya - Ba celonaTech (UPC), Campus Te assa,
C/ Colom 11, 08222 Ba celona, Spain
3Escola Tècnica Supe io d’Enginye s de Camins, Canals i
Po s de Ba celona (ETSECCPB), Uni e si a Poli ècnica de
1 Mo i a ion
This pape co esponds o he second pa o a esea ch wo k
de o ed o explo ing he possibili ies and bene i s o com-
pu a ional mul iscale me hods o be used in he analysis and
design o shock-abso bing me ama e ials o indus ial appli-
ca ions. In Pa (I) o he wo k [1] a i s app oach o he
o e all p oblem was made based on a single-scale modeling
o a solid, endowed wi h a classic compu a ional la ge-s ain
hype elas ic model, wi h de ised pe u ba ions in he olu-
Ca alunya - Ba celonaTech (UPC), Campus No d, C/ Jo di
Gi ona 1, 08034 Ba celona, Spain
4CIMEC-UNL-CONICET, P edio Conice , Ru a Nac. 168 s/n
- Pa aje El Pozo, 3000 San a Fe, A gen ina
5P og ama de Engenha ía Mecânica, COPPE, Uni e sidade
Fede al do Rio de Janei o, Cidade Uni e si á ia, Rio de
Janei o, RJ 21941-972, B azil
123
Compu a ional Mechanics
me ic pa o he ee ene gy such us o o e ide he o iginal
con exi y o he model (loss o con exi y). The e, i was dis-
played ha his ac ion:
•B eaks he non-dissipa i e pa adigm o poly-con ex
hype elas ic models,
•Yields he o ma ion o mechanical shock wa es, p opa-
ga ing ac oss he solid, and
•Resul s in he p oduc ion o ex insic dissipa ion asso-
cia ed bo h o he p opaga ion speed o he wa es and
he in ensi y o he local Eshelby s ess enso jump
[2–4]. This ac po en ially endows homogenized con-
inuum hype elas ic models, wi h pa ially o e idden
con exi y in he ee ene gy, he abili y o be used in
shock-abso bing compu a ional models.
•In addi ion, hose models exhibi he null s ain eco e y
unde null s esses p ope y i.e.: he p oduced ex insic
dissipa ion, and he co esponding shock abso bing capa-
bili y, is no a e dependen (as, o ins ance, in classic
iscoelas ic dissipa o s), he dissipa ion cha ac e o any
shock abso bing de ice (shock abso be ) buil wi h ha
ype o ma e ial, is po en ially ins an aneous and indepen-
den o he a e o he p oduced s ains. In o he wo ds:
a e a i s shock, he abso be ge s eady o abso b a sub-
sequen shock immedia ely, wi h a ully in ac dissipa ion
po en ial. This is o c ucial ele ance in indus ial appli-
ca ions aiming a objec p o ec ion om as epea ed
impac s (i.e. package p o ec ion du ing anspo a ion)
as well as in o he ields (spo -wea ing ma e ial design
e c.) in ending o con ol he amoun o dissipa ion and
i s e olu ion.
Howe e , i is well known ha such hypo hesized elas-
ic dissipa i e ma e ials do no appea na u ally. Mos o
elas ic ma e ials1exhibi , a leas a ini ial s ages o de o -
ma ion, a pu e elas ic s ain-beha iou and hey beha e
as non-dissipa i e. Bu , wha i in he mode n se ing o
me ama e ials2one could alk o , and e en manu ac u e,
hype elas ic ma e ials exhibi ing some kind o dissipa ion?.
The pu pose o his second pa o he wo k is o explo e he
domain o exis ing compu a ional me hods and se ings o
compu a ional me ama e ial design, and p o ide some new
insigh s on he subjec o compu a ional design o shock-
abso bing me ama e ials.
1.1 The ole o scales
Me ama e ials, a i icially enginee ed ma e ials, ye manu-
ac u able, showcasing unusual (and ex eme) physical p op-
1His o ically ound in na u e o e en manu ac u ed along he las
decades o he indus ial e a.
2A i icially enginee ed ma e ials showcasing unusual (and ex eme)
physical p ope ies.
e ies, a e cu en ly a ocal poin in ma e ial science. This is
p ima ily because hese peculia p ope ies, hough displayed
a he obse able scale o he ma e ial ( he mac oscale), a e
achie ed h ough i s in e ac ion wi h a i icial s uc u es in
he lowe scales ( he meso/mic o scales) in a way ha de ies
con en ional physical in ui ion. In spi e ha hese s uc-
u es may some imes be e y in ica e, hey can nowadays
be manu ac u ed using ad anced echniques, like addi i e
manu ac u ing. The selec ion o he in ended uppe -scale
p ope ies, he obse ed p ope ies, and he co esponding
low-scale s uc u es as well as hei cause, a e he goals
o he concep o Ma e ials by Design [5]. Compu a ional
Mechanics me hods o e inc easing oom o such a design,
in combina ion wi h expe imen al, in-lab, me hods which
has coined he e m Compu a ional (Me a) ma e ial Design
(CMD) [6–8]. In he con ex o shock-abso bing me ama-
e ials design a commonly chosen low-scale ea u e, o be
esponsible o he shock-abso bing p ope ies o he me a-
ma e ial, is he s uc u al buckling [9–13].
1.1.1 Buckling as a sou ce o dissipa ion: he buckling
mic ola ice
Buckling (o s uc u al ins abili y in he mos gene al mean-
ing) is a well-known phenomenon in s uc u al mechanics
appea ing in slende s uc u es which, despi e being con-
s i u ed by a con ex hype -elas ic ma e ial, exhibi , beyond
ce ain load limi s, an uns able s uc u al beha io , no -
mally associa ed wi h sudden and la ge geome ical changes
in he s uc u al membe s: i.e. s uc u al ins abili y, snap-
h ough/snap-back esponses, which mani es as hys e e ic
ac ion- esponse his o ies. This sugges s ha i a massi e
buckling la ice3, hebuckling mic ola ice cons i u ing he
co e o a ce ain de ice, is obse ed as a solid mechanics
body subjec ed o ex e nal loading ac ions, i migh expe-
ience local buckling a ce ain egions o he mic ola ice
causing sudden ins abili ies and la ge geome ical changes
[14–18]. I he size o hese buckling egions is su icien ly
small conce ning he ypical size o he body, hese buck-
ling phenomena can p opaga e h oughou he mic ola ice
causing local buckling/unbuckling beha io s, which ans-
la es in o hys e e ic global mechanical esponses [19–21].
In o he wo ds: om he Solid Mechanics poin o iew, a
he obse able (mac oscale) some global mechanical dissi-
pa ion would occu .
In ac , his has been expe imen ally obse ed: in [22]
i is epo ed ha obse a ion o in-lab mechanical load-
ing/unloading expe imen s using lase mic oscopy (3D lase
li hog aphy), on expe imen al specimens endowed wi h a ce -
ain low-scale buckling la ice mo phology (i.e. a ce ain
3In he emaining o his wo k he e m massi e la ice should be unde -
s ood as a opologically dense la ice.
123
Compu a ional Mechanics
numbe o buckling cells wi h de e mined ini ial opol-
ogy and shapes), display hys e e ic ac ion- esponse ( o ce-
displacemen ) in loading/unloading his o ies. Howe e , upon
comple e emo al o he ac ions he esponse became null,
and he specimen mo phology emains unchanged conce n-
ing he o iginal one. In addi ion, epe i ion o he expe imen s
displays e y simila ac ion- esponse pa hs.
1.1.2 Hie a chical mul iscale modeling: linking he scales
Du ing he las decades, hie a chical mul iscale modeling
has gained inc easing c edi , in he Compu a ional Mechan-
ics communi y, as a powe ul nume ical ool o ob ain new
insigh s o modeling complex ma e ial beha io [23–25].
Hie a chical mul iscale modeling is an idealiza ion o a
highly complex body ( o be designed and manu ac u ed a
a low scale), which is eplaced by a much simple one: he
homogenized4ma e ial ep esen a ion a he high-scale ul-
illing:
•The Con inuum Mechanics conse a ion and ene gy bal-
ance laws
•A scale-b idging p inciple: ypically an ene ge ic equi -
alence p inciple linking he idealized (homogenized)
ma e ial ep esen a ion and he o iginal ma e ial
esul ing in a homogenized con inuum body model, e aining
he essen ial beha io o he o iginal ma e ial and p o iding
use ul insigh s o Compu a ional Ma e ial Design a a o d-
able compu a ional cos s.
The homogeniza ion p ocedu e can be, hen, summa ized as
ollows:
1. The highly complex, mechanical beha io o a de o mable
body a he obse able scale, is conside ed he esul o
he hie a chical in e ac ion o simple mechanical beha -
io s linked o a nes ed sequence o ep esen a i e ma e ial
scales, assumed sepa able in e ms o hei co esponding,
dec easing, ep esen a i e sizes.
2. The ma e ial beha io a e e y scale is go e ned by spe-
ci ic mechanical laws, in e ms o he alues o some,
con enien ly iden i ied, s a e and in e nal a iables, which
a e aimed a being ob ained ia he compu a ional model
equa ions.
3. A hie a chical-link p inciple is hen pos ula ed o e e y
wo consecu i e scales ( om op o bo om). Tha link
in ol es some ene gy-equi alence laws, exp essed as
4The concep o homogeniza ion e e s he e o he de e mina ion o
he s ess-s ain measu es e olu ion, a one poin o he uppe scale,
in e ms o hei ene ge ic equi alence wi h a po ion o he low scale
( he Rep esen a i e Volume Elemen , RVE) displaying a ce ain ac ion-
esponse acco ding o he so-called Hill-Mandel P inciple. De ails a e
gi en in sec ion 1.2.
a ia ional equa ions, o he e olu ion o he s a e a i-
ables a e e y ma e ial poin a he uppe scale and he
co esponding a e age densi y e olu ion, a he lowe
scale, in a con enien ly shaped and sized domain e med
he Rep esen a i e Volume Elemen (RVE). The adequacy
and physical signi icance o ha link, in e ms o he
ep esen a ion o he ma e ial beha io , is, ob iously, c u-
cial o he accu acy o he esul s, which is assumed o
imp o e asymp o ically wi h he so-called scale sepa a-
ion, unde s ood as he a io o each scale size and he
immedia e lowe -scale RVE size ( he la ge he be e ).
The solu ion o he esul ing a ia ional equa ions p o-
ides
•De e mina ion o he uppe -scale s a e a iables in
e ms o he lowe -scale ones ( he homogeniza ion
p ocedu e) and
•A nes ed a ia ional p oblem a he lowe scale ( he
RVE p oblem), o be sol ed, o e e y se o kinema ic
en ies downloaded om he uppe scale, in e ms
o some low-scale kinema ic a iables he luc ua ing
displacemen s,li inga heRVE.
Recu si e (bo om o op) esolu ion o he p oblem
h ough scales, allows s a ing he e olu ion p oblem a all
scales, now including all he physical in e ac ions among
ma e ial scales, which, once sol ed, p o ide he ( op o
bo om) s a e a iables e olu ion.
The e ec i eness o his idealized mul iscale ma e ial ep e-
sen a ion depends on he app op ia e alue o he so-called
scales-sepa a ion ( he la ge he be e ), which migh be
assessed by asymp o ic compa isons wi h simple Di ec
Nume ical Simula ions (DNS) a he lowe scale.
1.1.3 Hie a chical mul iscale modeling o a
buckling-mic ola ice shock abso be
In ligh o he abo e conside a ions, an app oach is explo ed
in his pape o he hie a chical mul iscale compu a ional
ma e ial design o a buckling mic ola ice o be used in 2D
ep esen a ions o ligh weigh shock-abso bing de ices. I is
conside ed ha :
1. A he buckling mic ola ice, he ma e ial poin s exhibi
alinea ma e ial beha io (elas ic s ains and con ex-
non-dissipa i e cons i u i e beha io ) in a geome ically
non-linea en i onmen (cha ac e ized by la ge displace-
men s and small s ains) which, o a su icien ly complex
mic ola ice mo phology and opology may a ise local
buckling- egions in he la ice. In hese condi ions he ol-
lowing assump ions a e made:
123
Compu a ional Mechanics
•Mechanical ins abili y (mic ola ice buckling) can
occu as dynamic e ec s, p oducing buckling phe-
nomena in egions o he la ice, which can buckle,
unbuckle, o exhibi a s able beha io o e ime.
In u n, hese buckling phenomena can p opaga e
h ough di e en egions o he la ice.
•This buckling p opaga ion may be i e e sible, i.e.:
o ce ain s uc u al loading/unloading pa hs he
buckling/unbuckling a ec ed mic ola ice egions a e
no he same, hus esul ing in a hys e e ic ac ion-
esponse his o y, i.e.: he global ac ion- esponse his-
o y is di e en o loading and unloading p ocesses.
2. Rega ding he o e all shock-abso bing specimen ( he
shock abso be ) i is assumed ha :
•The mic ola ice egions beha e as homogenized pa -
icles o a mac oscale con inuum, hus uled by he
s anda d conse a ion laws o solid mechanics and he
homogenized cons i u i e law, in e ms o he homog-
enized measu es (homogenized s esses and s ains).
•Acco ding o hese conse a ion laws he shock
abso be , endowed wi h he homogenized ma e ial,
exhibi s a global dissipa i e beha io , eme ging om
he hys e e ic ac ion- esponse (s uc u al esponse) o
he mic ola ice, and uled by a non-con ex homoge-
nized cons i u i e model. In o he wo ds: homogeniza-
ion ende s he s uc u al (con ex elas ic) beha iou
o he pa icles a he mic ola ice membe s in o a
non-con ex (solid mechanics-like) cons i u i e model
a he co esponding poin s in he homogenized
mac oscale con inuum. This ansla es in o an o e -
all dissipa ion in he (mac oscale) shock-abso bing
de ice p oblem.
1.2 P oposed wo-scale app oach o ligh weigh
buckling mic ola ice shock abso be s
In his wo k, he gene al mul iscale se ing desc ibed abo e
is speci ied as a wo-scale app oach as ollows:
1. Only wo scales a e conside ed in he shock-abso bing
me ama e ial i.e.:
•The (obse able) mac oscale: i cons i u es a
de o mable solid (shock abso be de ice) uled by
he classical la ge s ain kinema ics, in e ms o
(homogenized) s ains and s esses (he e e med he
homogenized measu es), and ul illing he s anda d
conse a ion laws in solid mechanics p oblems (i.e.
linea and angula momen um conse a ion). Thei
app oxima e solu ion, ia a ini e elemen disc e iza-
ion p oblem a he mac oscale ( he mac oscale p ob-
lem), p o ides a p edic ion o he displacemen ield,
and he homogenized s ains, and s esses in he global
sys em (impac o -abso be ). This solu ion also yields
he e alua ion o he accumula ed sys em dissipa ion,
in he impac o -abso be sys em. This dissipa ion is
an icipa ed o be a scala measu e, compu ed as he
di e ence be ween he ex e nal ene gy supplied o
he shock abso be and he sum o i s whole amoun
o in e nal ene gy ( ee ene gy plus kine ic ene gy),
and i should be ( o he modynamic easons) always
ime-inc easing and posi i e. I is also an icipa ed ha
in his wo k, de e mina ion, op imiza ion, and con ol
o he sys em dissipa ion a e conside ed he unda-
men al issues in he design o he shock-abso bing
me ama e ial.
•The lowe -scale: i is cons i u ed by he ma e ial pa -
icles a a massi e buckling mic ola ice in he shock
abso be , which is cha ac e ized by he mo phology
and opology o he a o emen ioned chosen egion
o he la ice, he RVE, he whole massi e buckling
la ice being conside ed amenable o be gene a ed
h ough sui able epe i ions o he RVE along he low-
scale dimensions.
2. The low-scale kinema ics (displacemen and s ains de i-
ni ion) is ob ained as a (linea ) Taylo ’s expansion, along
he RVE, o he mac oscale displacemen s and he co -
esponding homogenized s ain ields, supplemen ed by
an unknown luc ua ing displacemen s ield,li inga he
RVE and playing he ole o a co ec ion, a he low-scale,
o ha linea ized displacemen ield hus p o iding addi-
ional accu acy o he homogenized solu ion conce ning
he, heo e ically exac , DNS (Di ec Nume ical Simula-
ion) solu ion.
3. Imposi ion o he mechanical ene gy conse a ion
pa adigm (Hill-Mandel P inciple) ac oss he scales, which
is nume ically sol ed ia a a ia ional s a emen . The
esul ing solu ion p o ides [24]:
•The homogenized (RVE-a e aged) s ess and s ain
ields in he mac oscale
•The RVE p oblem: a a ia ional p oblem sol ing
he luc ua ing displacemen s, in a ini e elemen
disc e ized p oblem on he RVE domain. These
addi ional ( luc ua ing displacemen s) unknowns a e
assumed o co ec he mac oscopic p oblem solu-
ion in e ms o he homogenized solu ions, yielding
a highe accu acy, owa ds he (exac ) DNS solu-
ions, as he luc ua ions dec ease o ze o (i.e. as he
scale-sepa a ion inc eases). In a way, hey compen-
sa e o he e o o compu ing a p oblem solu ion, in
he educed-size RVE o he mic ola ice, ins ead o
sol ing he (non-a o dable in e ms o compu a ional
cos ) DNS solu ion.
123
Compu a ional Mechanics
4. The RVE p oblem is addi ionally subjec ed o some
es ic ions on he luc ua ing displacemen s: he consis-
ency condi ions. They en o ce he nulli ica ion, a he
RVE domain, o bo h he a e ages o he luc ua ing dis-
placemen s and hei spa ial g adien alues. In a way, hey
gua an ee he con e gence o he mul iscale solu ions in
e ms o he scale sepa a ion: he la ge he scale sepa-
a ion he mo e negligible he luc ua ing displacemen s
and s ains, and he highe he accu acy o he ob ained
homogenized quan i ies, conce ning he DNS solu ion.
5. In he hie a chical mul iscale app oach con ex , he ep-
esen a i e cha ac e o he chosen RVE is assumed5.In
his wo k, his is deemed achie ed by using, as RVE, a
sui able po ion o he ac ual buckling mic ola ice ul ill-
ing he ollowing condi ions:
•The ac ual mic ola ice opology and mo phology is
ob ainable by spa ial epe i ion o he RVE, endowed
wi h pe iodic bounda y condi ions, along all he
dimensions o he p oblem.
•A he same ime, and in combina ion wi h he ene gy
equi alence p inciple, he complexi y o he RVE is
enough o e icien ly cap u ing he physical phe-
nomena in e ening in he p opaga ion o buckling
ac oss he mic ola ice and, hus, o ep oduce he
co esponding hys e e ic beha io a he mac oscale
causing he dissipa ion. In addi ion, he RVE size
and complexi y keep in balance he accu acy o he
ob ained simula ions and hei compu a ional cos 6.
6. Due o he ligh weigh cha ac e o he conside ed
buckling mic ola ice, i is s a ed ha he homogenized
(mac oscale) ine ial o ces in he shock abso be a e
negligible in compa ison wi h he ones ha a ise in he
impac ing bodies. The e o e, he solid mechanics p ob-
lem in he shock-abso bing me ama e ial is conside ed
quasis a ic.
Such an app oach is adop ed in his wo k aiming a ende -
ing a sui able se ing o ace he compu a ional design o
shock-abso bing me ama e ials, balancing he accu acy o
he model wi h a mode a e compu a ional cos when com-
pa ed wi h he mo e ealis ic, bu una o dable high-cos ,
DNS models.
5This is o say: he RVE is su icien ly small (and opologically
de ailed), wi h espec o he o e all shock abso bing specimen, o yield
s uc u al esponses which a e s a is ically ep esen a i e o he con-
side ed mechanical de o ma ion p ocess.
6When compa ed wi h DNS simula ions.
2 Buckling mic ola ice modeling.
Dimensional educ ion o he RVE o a se
o buckling beams
Based on he mo i a ional a gumen s in Sec . 1,so a helow
scale has been conside ed a buckling mic ola ice, made o
slende componen s, which, consis en ly wi h he 2D cha -
ac e o he assumed Solid Mechanics p oblem, cons i u e
geome ical 2D objec s (see Fig. 1b). The e o e, in p inci-
ple, i should be consis en ly modeled as a 2D con inuum.
Howe e , o he p ospec i e pu poses o his wo k, he slen-
de cha ac e o he buckling componen s is hypo hesized o
be such ha hey could be eplaced, wi h li le e o , by 1D
buckling s uc u al elemen s (buckling-beams), which could
be modeled in he con ex o elas ic la ge-s ain beam heo y.
The co esponding Rep esen a i e Volume Elemen (RVE)
would hen become he se o s uc u al buckling beams in
Fig. 1c. This dimensional educ ion o he RVE model, con-
side ed so o h, will ansla e in o a e y ele an educ ion
in he compu a ional cos , bu s ill p o ides subs an ial accu-
acy o he esul s7.
In consequence, he plane beam model ske ched in Fig. 2,
aken om Felippa [26], will be used o cap u e he s uc u al
beha io o he RVE domain a he lowe scale. Fo easons
o comple eness, a b ie discussion abou his model will be
p esen ed in his sec ion.
The main ea u es o he model a e hen:
(1) he educed (1D), dimensional cha ac e , wi h espec o
he o iginal one (2D), which is assumed o yield enough
accu acy,
(2) la ge displacemen s and o a ions, bu small s ains, kine-
ma ics in a To al Lag angian o mula ion, and
(3) a con ex linea elas ic model (Sain -Venan ’s model) con-
side ed o he beam cons i u i e ma e ial.
2.1 Kinema ics
Le ’s conside a beam o leng h L, wi h c oss-sec ion A,
symme ic in he loading plane, ha ing a s aigh neu al axis
in he e e ence con igu a ion passing h ough he cen oid o
he c oss-sec ion and iden i ied by Y0(see Fig. 2). The uni
ec o ˆ
Nis pa allel o he beam neu al axis. Conside ing
a local coo dina e sys em (s,η) associa ed o he o hogonal
basis {ˆ
N,ˆ
T}, whe e ˆ
Tis he o hogonal uni ec o o ˆ
N,see
Fig. 2, he poin s o he beam neu al axis in he e e ence
con igu a ion, deno ed Y0, migh be w i en in e ms o he
local coo dina e (a c-leng h pa ame e ) s:Y0(s)=Y1+
7No ice ha modeling he la ge de lec ions o he buckling componen s
o he RVE, assumed endowed o e y small hickness, would imply
e y dense disc e iza ions o he co esponding (low-aspec - a io) 2D
ini e elemen s meshes o a oid nume ical ill-condi ioning.
123

Compu a ional Mechanics
Fig. 1 The wo scales p oblem. aThe mac oscale, consis ing o a
de o mable body go e ned by he classical la ge s ain kinema ics,
desc ibed in e ms o homogenized s ains and s esses, bRVE, com-
posed o slende s uc u al 2D s uc u al elemen s, c he simpli ied RVE
composed o 1D s uc u al buckling elemen s (beams)
Fig. 2 Felippa’s beam heo y.
(a) Kinema ic desc ip ion; (b)
Conside ed 1D linea ini e
elemen
s(Y2−Y1)/L=Y1+sˆ
N. Conside ing now a ma e ial
poin Yoccupying he ma e ial con igu a ion a :
Y=Y0+ηˆ
T(1)
and assuming a de o med beam displaying small axial
s e ches and shea dis o ions, he de o ma ion map o his
ma e ial poin migh be exp essed as:
y(Y, )=Y0+u0(Y0, )+ηK(Y0, );
k=−sin(θ) ˆ
N+cos(θ) ˆ
T;(2)
whe e he displacemen ec o o poin s on he beam neu-
al axis is deno ed u0, he uni ec o kis pa allel o he
beam c oss-sec ion in he cu en con igu a ion, while in he
e e ence con igu a ion is o hogonal o he neu al axis and
pa allel o ˆ
T. Thus, θis he c oss-sec ion o ien a ion angle in
he cu en con igu a ion wi h espec o i s di ec ion in he
e e ence con igu a ion, γ=θ−ψis he shea dis o ion
angle, and ψis he angle o he beam neu al axis di ec ion
in he ac ual con igu a ion and he di ec ion o he same axis
in he e e ence con igu a ion. Then, he displacemen ec o
a Yis de ined as:
u(Y, )=y−Y=u0(Y0, )+η(k−ˆ
T)(3)
and he de o ma ion g adien F=∇
Yyis e alua ed in con-
sequence, esul ing:
F=(ˆ
N+u0
,s+ηk,s)⊗ˆ
N+k⊗ˆ
T(4)
whe e no a ion (·),s e e s o he de i a i e o en i y (·)wi h
espec o s. Also, he displacemen g adien Jis compu ed
as J=F−1, whe e 1 ep esen s he second-o de iden i y
enso .
No e ha in his app oach, and om exp ession (2), he
kinema ic desc ip o s o he model p esen ed in [26]a eu0
and θ. Fu he mo e, based on he de ini ion o he dis o ion
angle γ, he cu a u e e m is, κ=θ,s, assuming small s ains
and small γ, and dis ega ding highe o de e ms. Felippa in
[26] de i es a consis en linea iza ion, which is he e adop ed.
Taking in o accoun all o hese ing edien s, he esul ing
G een-Lag ange de o ma ion enso E, a e linea izing he
con en ional exp ession E=1
2(FTF−1), can be w i en
123
Compu a ional Mechanics
as ollows:
E=e−ηκ γ/2
γ/20
(5)
whe e he axial, e, and shea , γ, s ain measu es a e gi en
by:
e=1+d(u·ˆ
N)
dS cos(θ) +d(u·ˆ
T)
dS sin(θ) −1
γ=−1+d(u·ˆ
N)
dS sin(θ) +d(u·ˆ
T)
dS cos(θ)
(6)
espec i ely. Addi ional de ails abou he linea iza ion used
o de i e he G een-Lag ange s ain enso (5) om he de o -
ma ion g adien (4) can be seen in [26]. The iple (e,γ,κ)
cons i u es he gene alized s ain measu es o he beam
model. Then, he gene alized s ain ec o will be deno ed
as
h=⎡
⎣
e
γ
κ⎤
⎦(7)
2.2 Cons i u i e model and s esses
By assuming a linea elas ic ma e ial, he i s Piola-
Ki chho s ess Pin he p oposed beam heo y is gi en by:
P=PnN(ˆ
n⊗ˆ
N)+PnT (ˆ
n⊗ˆ
T)
+P N(ˆ
⊗ˆ
N)+P T(ˆ
⊗ˆ
T)(8)
Equi alen ly he Second Piola-Ki chho s ess enso S
is:
S=SNN(ˆ
N⊗ˆ
N)+SNT(ˆ
N⊗sym ˆ
T)(9)
whe e he no a ion ⊗sym indica es he symme ic enso
(open) p oduc o he wo ec o s. Consis en ly wi h he beam
heo y de eloped in [26], he componen STT is equal o 0.
This non-null componen can be exp essed in e ms o he
gene alized s ains using he ollowing cons i u i e equa ion:
SNN
SNT =E(e−ηκ)
Gγ(10)
whe e he symbols Eand G ep esen he Young’s and
shea moduli o he beam ma e ial, espec i ely. Finally, he
s ess enso S(Y)can be in eg a ed h ough he beam c oss-
sec ion, p o iding he gene alized s esses:
N=A
SNNdA=EAe
V=A
SNTdA=GAγ
M=A
−ηSNNdA=EIκ
(11)
whe e Nis he axial o ce, Vis he shea o ce, Mis he
bending momen , Ais he a ea o he c oss sec ion and Iis
he second momen o ine ia.
2.3 Homogeniza ion o he buckling mic o-la ice.
Mul iscale o mula ion
As desc ibed in Sec . 1a mul iscale p oblem displaying
wo cha ac e is ic well-sepa a ed leng h scales is analyzed.
A he mac o o coa se scale, a con inuum model wi h an
exac non-linea kinema ic desc ip ion is conside ed. A he
mic o o ine scale, a beam ne wo k is modeled using he
app oach p esen ed in he p e ious sec ion. An idealized
ske ch o his p oblem is depic ed in Figu e 1. Fo simplic-
i y, i is also assumed ha , a he mic oscale, he sec ions
o each beam a e uni o m and ha he e a e no disc e e
o ces, disc e e momen s, o body o ces applied o he
mic os uc u e. The e o e, he in e nal gene alized o ces
emain uni o m along each mic os uc u al ame elemen .
Thus, he mechanical a iables a he mac oscale a e he posi-
ion o he e e ence poin , X, he spa ial posi ion o he poin ,
x(X), he displacemen ec o , uM=x−X, he g adien o
de o ma ion, FM, and he Piola s ess, PM.
The mic oscale mechanical s a e associa ed wi h he
mac oscale poin Xis de ined by he posi ion o he e e ence
poin Yand he spa ial posi ion o he poin y(Y). Addi ion-
ally, ollowing he beam heo y o Sec . 2, he displacemen
ec o o he beam neu al ibe s is u0
μ=y(Y0)−Y0, he
g adien o de o ma ion is Fμ, and he Piola s ess is Pμ.
2.3.1 Scale b idging equa ions
The displacemen ec o uμo any poin Ya he mic oscale,
in ag eemen wi h (2), can be w i en as ollows:
uμ=uM(X)+JM·(Y−YC)+˜
u0
μ(Y0)+ημ(˜
kμ−ˆ
T)
(12)
whe e, a e imposing he condi ion ha YCis he mass cen e
a he RVE, esul s:
μ
JM·(Y0−YC)dμ=0(13)
123
Compu a ional Mechanics
Unde he hypo hesis o pe iodic bounda y condi ions, he
ollowing ma hema ical exp essions a e ul illed (see Fig.
19):
u0
μY+=u0
μY−+JM·Y+−Y−,
˜
u0
μY+=˜
u0
μY−,
˜
θμY+=˜
θμY−
(14)
2.3.2 Hill-Mandel p inciple
The Mul iscale Vi ual Powe P inciple p oposed in [24]is
adop ed in his wo k. I co esponds o a a ia ional s a emen
o he classical Hill-Mandel p inciple [27,28], in which an
ene ge ic equi alency be ween he wo scales is pos ula ed.
I es ablishes ha he s ess powe a a gi en pa icle a he
mac oscale, X, equals he mean alue o he s ess powe a
he co esponding RVE domain, μ(X), i.e.:
PM(X):δFM(δuM)=1
|μ|μ
Pμ(Y):(δFM(δuM)
+∇Yδ˜
uμ)dμ(15)
o any admissible a ia ions, δFM, belonging o he ull
se o second o de enso s and o any i ual kinema ically
admissible luc ua ion,δ˜
uμ(see Appendix A o de ailed
in o ma ion on he no a ion and de ini ion o spaces). Equa-
ion (15) de ines a undamen al link be ween he wo scales.
Di ec consequences o i a e:
•The homogenized s ess exp ession,
PM=1
|μ|μ
Pμuμ(Y)dμ(16)
•The RVE p oblem a he mic oscale, i.e.: Find uμ ul ill-
ing:
μ
Pμ(uμ):∇
Yδ˜
uμdμ=0,
∀δ˜
uμ,(admissible luc ua ion)(17)
As a consequence o Eq. (17), he homogenized s ess (16)
can be exp essed in e ms o he ac ion ec o μ=Pμ·Nμ
e alua ed exclusi ely a he bounda y o he RVE, i.e.,
PM=1
μμ
μ⊗Ydμ(18)
whe e μ=∂ωs
μ∩∂μ, (see Fig. 1b).
2.3.3 F om con inuum o buckling mic o-la ice
Le ’s assume ha he a angemen o slende de o mable
solids can be ep esen ed by a s aigh beam ne wo k, μ
(Fig. 1c), modeled using he app oach p esen ed in Sec . 2.1.
By ollowing he ma hema ical manipula ions de ailed in
Appendix A, he di ec consequences o he Hill-Mandel
p inciple can be equi alen ly exp essed in e ms o he beam
e minology as ollows:
•The RVE p oblem a he mic oscale
Find (u0
μ,θ
μ) ul illing
μ
z(u0
μ,θ
μ)·δ˜
hμδ˜
u0
μ,δ˜
θμdμ=0,
∀δ˜
u0
μ∈˜
V0
μ,∀δ˜
θμ∈˜
μ(19)
whe e zμ=(Nμ,Vμ,Mμ)Tis he gene alized s ess
ec o , δ˜
hμ=(δ ˜eμ,δ˜γμ,δ˜κμ)Tis he a ia ion o he
gene alized luc ua ing s ain ec o ,
˜
V0
μ=δ˜
u0
μ:μ→R2|δ˜
u0+
μ=δ˜
u0−
μ,(20)
and
˜
μ=δ˜
θμ:μ→R|δ˜
θ+
μ=δ˜
θ−
μ.(21)
•Homogenized s ess exp ession compu ed in e ms o he
eac ion o ces a he bounda y o he RVE:
PM=1
μ
nb

i
Rμi⊗Y0i(22)
whe e
Rμi=μi
Pμ·Nμdμ(23)
ep esen s he eac ion o ce ec o a node i:1...nb.8
3 Fini e elemen app oach
Le
h
μ=
nbeams

α=1
α
μ(24)
deno e he ini e elemen disc e iza ion o he polygonal cu e
ha ep esen s he midline o all he beams comp ising he
8nb ep esen s he numbe o bounda y nodes in he RVE.
123
Compu a ional Mechanics
RVE. Based on [26], a C0con inuous in e pola ion wi h wo
nodes unde each elemen , α
μ, can be used o in e pola e
he mic o-displacemen s (o he neu al axis) and o a ions
as linea unc ions o he nodal pa ame e s
u0
μ(S, )
θμ(S, )α
=Nα
1(S)u0
μ( )
θμ( )1
+Nα
2(S)u0
μ( )
θμ( )2
(25)
whe e Nα
1and Nα
2a e he linea shape unc ions while
u0
μ( )=[
u0
μ,Y1u0
μ,Y2]Tand θμ( )9a e he in e pola-
ion pa ame e s associa ed o nodes 1 and 2, espec i ely.
Acco ding o he s ains de ini ion (6) and he in e pola ion
exp ession (25), he disc e e o m o he a ia ion o he gen-
e alized s ain ec o (7), can be exp essed as:
δ˜
hα
μ=B
B
Bα
μδ˜
dα
μ(26)
whe e B
B
Bα
μ, he s ain-displacemen ma ix, con ains he a i-
a ions o he s ains δ˜eμ,δ˜γμand δ˜κμwi h espec o gene al-
ized displacemen luc ua ions δ˜
dα
μ=δ˜
u1
μ,δ˜
θ1
μ,δ˜
u2
μ,δ˜
θ2
μ.
Fo he de ini ions o he componen s o B
B
Bα
μand a de ailed
explana ion o hei de i a ion, please e e o [26]. By apply-
ing he ini e elemen disc e iza ion (24), (25) and he elemen
Eq. (26), he a ia ional p oblem (19) can be e o mula ed in
i s disc e e o m as ollows:
•Disc e e a ia ional p oblem:
Find dμ,pe iodic, ul illing
δ˜
dμ
T μdμ=0∀δ˜
dμpe iodic (27)
whe e
μ(dμ)=
αh
α
B
B
BμαT(dα
μ)zμα(dα
μ)dα
μ.(28)
He e, zα
μ ep esen s he s ess- esul an ec o a elemen
le el.
3.1 Exac linea iza ion o he RVE p oblem
As a esul o he geome ically non-linea ea u es inhe en
o he beam heo y [26] used o cap u e he mechanical beha -
io o he RVE he p oblem in Eq. (27) becomes non-linea .
Consequen ly, an i e a i e app oach is c ucial o e ec i ely
sol ing he sys em. To his end, he New on-Raphson (NR)
me hod is p oposed, in ol ing he exac linea iza ion o he
p oblem a each i e a ion as ollows:
δ˜
dμ
TKTd(k)
μ,n+1d(k+1)
μ,n+1+ μd(k)
μ,n+1=0 (29)
9He e, is conside ed a pseudo- ime, ∈[0,T], go e ning he e olu-
ion o he de o ma ion p ocess.
whe e d(k+1)
μ,n+1 ep esen s he inc emen o he gene alized
displacemen ec o a load s ep n+1 and du ing he in e nal
New on-Raphson i e a ion k. The angen s i ness ma ix,
deno ed by KTis exp essed as:
KTd(k)
μ,n+1=Kma d(k)
μ,n+1+Kgeom d(k)
μ,n+1,(30)
whe e Kma is he ma e ial s i ness ma ix and Kgeom is he
geome ic s i ness ma ix. Fo de ailed compu a ion p oce-
du es, e e o [26]).
3.2 Imposi ion o pe iodici y condi ions
Conside ing JM
n+1=FM
n+1−1 he displacemen g adien
enso a load s ep n+1. Le us exp ess he luc ua ion dis-
placemen ec o in a mo e con enien o m:
˜
dμ=˜
din
μ˜
d−
μ˜
d+
μT
(31)
whe e ˜
din
μis he luc ua ions displacemen ec o o he
nodes in he in e io o he RVE, while ˜
d−
μand ˜
d+
μa e he luc-
ua ions o he nodes in −
μand +
μ, espec i ely (see Fig. 19).
In his con ex , he pe iodici y condi ions a e exp essed as
˜
d−
μ=˜
d+
μ,(32)
o , in componen o m,
˜
u0
μ
˜
θμi−
=˜
u0
μ
˜
θμi+
(33)
o i+:1...mand i−:1...m.10
This condi ion can be equi alen ly ew i en in e ms o
he gene alized mic o displacemen s:
u0
μ
θμi+
=u0
μ
θμi−
+JMT
n+1Y0+−Y0−
0(34)
o equi alen ly as
d+
μ=d−
μ+¯
dM(35)
whe e
¯
dMi−=JMT
n+1Y0+−Y0−
0(36)
10 mis he numbe o nodes in +
μ, o equi alen ly, in −
μ.
123
Compu a ional Mechanics
Fig. 11 S uc u al esponses
and e olu ion o he o al
ex insic dissipa ion ene gy o
he e ical loading and
unloading p ocess o a a iable
c oss-sec ion specimen, o each
o he RVEs complexi ies: aone
uni cell; b wo uni cells; c
h ee uni cells; d ou uni cells
Fig. 12 Compa ison o
s uc u al esponses and
accumula ed dissipa ion
e olu ion in he a iable
c oss-sec ion specimen o
di e en RVEs complexi ies
123

Compu a ional Mechanics
Fig. 13 Sneake -insole as a
shock abso bing me ama e ial: a
sneake insole, bsneake impac
on he g ound, ce ec i e
unne ’s mass, W, impac ing on
a speci ic clea ( aken om [34,
35])
5.1 Mic o-la ice-buckling as a sou ce o dissipa ion
in shock abso bing me ama e ials
The conside ed key aspec s o be analyzed in his p ospec i e
exe cise a e:
•The abili y o he shock abso bing me ama e ial o p o-
duce ex insic dissipa ion and i s ansla ion in o kine ic
ene gy a enua ion,
•The amoun and ime-e olu ion o he dissipa ion along
as ep
27. This dissipa ion will ansla e in o a enua ion
o he kine ic ene gy in oduced by he unne ’s e ec i e
weigh on he clea ( ep esen ed by he uppe impac ing
mass Win Fig. 13c), along a ypical unne ’s s ep,
•The quick shape eco e y p ope y o he clea a e he
impac (in o de o ge eady o abso b an impac e y
son a e ano he one).
•The amoun o ene gy dissipa ion and i s e olu ion o e
one unne ’s s ep a e conside ed he op imali y c i e ia28.
Figu e 14a ske ches he impac p ocess o he e ec i e
mass,W, o he unne impac ing on a single ep esen a-
i e clea , which is o be analyzed in e ms o wo possible
(cylind ical/ unk-conical) mac oscopic shapes. Figu e 14b,
ins ead, ocuses on he low-scale, displaying a gene ic RVE
consis ing o ou di e en cells/buckling-laye s29.
E e y cell o he RVE is assumed cons i u ed by (see
Fig. 14b):
27 Assumed no malized o one-second du a ion.
28 To be de e mined in he design p ocess h ough app op ia ed bio-
mechanical c i e ia.
29 E e y colo in he igu e ep esen s a di e en cell.
•Inclined buckling beams, wi h low s i ness, which a e
in cha ge o inducing e ical-buckling e ec s in he cell
and
•Ve ical and ho izon al s i ening ba s, wi h much la ge
s i ness, which a e in cha ge o p ecluding ho izon al
buckling in he cell.
The elas ic ma e ial p ope ies o beams and ba s a e he
same o all cells in he RVE, wi h he ollowing alues
•Inclined buckling beams. Young’s modulus: Eb=32.5
GPa.
•Ve ical and ho izon al s i ening beams. Young’s modu-
lus: Es=325 GPa.
The s uc u al p ope ies o he buckling ba s change om
laye o laye , since he hickness o he inclined beams is se
o inc ease by a 12% om one laye o he nex one ( om op
o bo om). This is done o induce a sequen ial buckling (also
om op o bo om) o he cells in he RVE. The esul ing
RVE-homogenized cons i u i e model is ob ained ollowing
he p ocedu e in Sec . 4.2.1, and i is shown in Fig. 14c.
Obse a ion o Fig. 14 yields he ollowing commen s:
•The homogenized cons i u i e model exhibi s a non-
con ex shape simila o he one in Fig. 5, hus displaying
se e al bumps, each o hem coinciding wi h he in ended
sequen ial buckling o he RVE laye s in Fig. 6.
•In u n, his ype o , non-con ex, cons i u i e esponse
ma ches he ones discussed in Pa (I) o his wo k [1],
conce ning single-scale models endowed wi h a i icially
imposed non-con ex ee ene gy, whe e di e en ia ion
123
Compu a ional Mechanics
Fig. 14 Shock-abso be
elemen s o design. acons an
c oss-sec ion I, s. a iable
c oss-sec ion II o a clea
impac ed by an e ec i e weigh
Wp oducing an ini ial kine ic
ene gy o 386 N·mm,bRVE
consis ing o ou di e en
buckling laye s (uni cells),
colo -coded o easy
iden i ica ion, cHomogenized
cons i u i e esponse o he
p oposed RVE
o he ee ene gy yields simila bumping cons i u i e
models, hus inducing p opaga ing mechanical shocks
and he co esponding ex insic dissipa ion.
•The e o e, in a mul iscale mic o-buckling la ice me a-
ma e ials con ex , his allows iden i ying he p opaga ing
buckling phenomena a he mic oscale as he sou ce o
he ex insic mechanical dissipa ion.
5.2 Room o design a he mac oscale (clea -shape
design)
Le us now explo e he oom o design p o ided by he
p esen p ospec i e app oach o he p oblem conside ed in
Fig. 13a in e ms o he clea shape.
Figu e 15a, displays he dynamic p oblem o he e ec i e
unne ’s mass, W, impac ing he clea o bo h con igu a-
ions. In Fig. 15b, c, d an oscilla o y e olu ion o he e ical
displacemen , δa he op o he clea , along wi h he kine ic
ene gy o he impac ing mass, and he accumula ed dissipa-
ion o e ime, a e, espec i ely, depic ed o he wo cases
unde analysis (case I, displayed wi h g een dashed lines,
and case II, wi h blue solid lines). The esul s indica e ha
he shape o he shock abso be signi ican ly in luences he
physical esponse. I s obse a ion indica es ha :
•The dissipa ion e ol es much as e in case II han in
case I. The amoun o accumula ed dissipa ion is simila
in bo h cases ( hough a li le highe o case II.
•A he end o he dissipa ion p ocess, some (small)
oscilla ions pe sis , hei magni ude being signi ican ly
smalle in case II han in case I. This could ha e
been expec ed, since he hys e e ic beha io in he
ac ion/ esponse cu es (see o ins ance Fig. 12a) ends
be o e he s a ic equilib ium poin , (F,δ) ≡(0,0), and
he model exhibi s a ( e y small) non-dissipa i e oscilla-
o y esponse a ound his poin .
•Finally, Fig. 15c con i ms ha he shape o he clea
signi ican ly in luences bo h he e olu ion o he accu-
mula ed dissipa ion and he o al dissipa ed ene gy. A
he same mass, a unk-conical shape o he clea p o-
duces as e and la ge dissipa ion han a cylind ical one.
In summa y: he explo ed amewo k p o ides a subs an-
ial oom o design in e ms o he clea ’s shape. The clea ’s
shape allows o con ol o bo h he e olu ion o he ex insic
dissipa ion and he o al inal amoun o dissipa ion.
5.3 Room o design a he mic oscale buckling
la ice (RVE mo phology design)
Figu e 16 displays di e en RVE-mo phologies o be con-
side ed, each one con aining a di e en numbe o buckling-
laye s/cells: 2, 4, and 8, espec i ely ( o enhance isualiza-
ion, e e y cell is shaded in a dis inc colo ). The hickness
o he buckling beams a each cell/buckling-laye is hen
adjus ed o ensu e ha he a e age densi y o he h ee RVEs
p ese es he same o al mass o all RVEs (same mass o
he o e all shock-abso be ), and ha , unde he ac ion o he
comp essi e load, he di e en cells o he RVE buckle in a
sequen ial ( op- o-bo om) manne (Fig. 17).
Figu e 18 p esen s he esul s o he e olu ion o he dis-
placemen s on he op ace o he shock abso be , he kine ic
ene gy, and he accumula ed dissipa ion o he h ee p o-
posed mic oscale con igu a ions. The esul s demons a e a
123
Compu a ional Mechanics
Fig. 15 Design a iables in he p ocess o designing a single clea acylind ical-shaped clea I, s., unk-conical shaped clea II; bdisplacemen
a enua ion cu e; caccumula ed dissipa ion; dhis o y o he kine ic ene gy
clea in luence o he mic oscale mo phology on he dissi-
pa ion p ope ies ea u ed by he shock abso be . Figu e 18c
con i ms ha he dissipa ion pe o mance, assessed h ough
bo h he o al accumula ed dissipa ion and he ime aken o
achie e his alue, s ongly a ec ed he RVE mo phology.
In hese e ms, case III appea s he mos e icien , while case
I appea s he leas e icien one. This conclusions a e sup-
po ed by he e olu ion o displacemen s (Fig. 18a) and he
co esponding changes in kine ic ene gy (18b). In case III,
kine ic ene gy eaches i s minimum apidly, and he ampli-
ude o subsequen oscilla ions is minimized. Finally, when
conside ing he numbe o le els in each RVE as a e e ence,
he p opo ion by which hese le els inc ease in each con-
igu a ion does no co espond o he inc ease in dissipa ed
ene gy. This emphasizes he high deg ee o non-linea i y and
complexi y o he phenomenon.
6 Final conside a ions and signi icance o
u u e de elopmen s
6.1 Conside a ions on he explo ed compu a ional
se ing o shock-abso bing me ama e ials
design
In p e ious sec ions, a en a i e compu a ional se ing o
modeling shock-abso bing me ama e ials has been explo ed.
I elies on wo main ing edien s:
1. Conside a ion o shock-abso bing de ices cons i u ed by
a massi e buckling-mic ola ice, in which buckling can
locally a ise and p opaga e (Sec . 1.1.1).
2. The use o hie a chical mul iscale modeling and he Hill-
Mandel p inciple o building an ene ge ically equi alen
homogenized solid o p o ide meaning ul esul s, a mod-
e a e compu a ional cos (Sec . 1.2).
123
Compu a ional Mechanics
Fig. 16 Sensi i i y o he
mic oscale mo phology. Th ee
di e en RVE mo phologies. o
a cons an c oss-sec ion clea . I:
2 cells RVE, II: 4 cells RVE, and
III: 8 cells RVE (e e y cell is
depic ed in a di e en colo ).
The hickness o he buckling
beams a each cell is adjus ed o
ensu e ha : (1) he a e age
densi y o he h ee RVEs
p ese es he same o al mass,
and (2) unde he ac ion o he
comp essi e e ical load, he
RVE cells buckle in a sequen ial
( om op o bo om) manne
Fig. 17 Sensi i i y o he
mic oscale mo phology: a
mac oscale impac p oblem, b
esul ing homogenized
cons i u i e models
co esponding o he h ee RVEs
in Fig. 16
In such a amewo k a shock abso bing me ama e ial,
cons i u ed by a massi e buckling mic ola ice, has been
de ised30 (Sec . 2), and i ually es ed in a po en ial indus-
ial applica ion: a sneake insole design. Then, he nume ical
simula ions display encou aging esul s o he pu poses o
compu a ional me ama e ial design, i.e.:
a) Signi ican mechanical (ex insic) dissipa ion akes place
in loading-unloading impac cycles (Sec . 4).
b) Dissipa ion is inc easingly sus ained unde quickly
epea ed impac s (Sec . 5)31.
30 Assuming ha su icien scale sepa a ion exis s o p o ide eliable
esul s.
31 A speci ic bene icial ea u e o some applica ions (e.g. packaging).
c) The e is clea oom o design a bo h scales, i.e.: he
esul ing mechanical dissipa ion32 s ongly depends on
he high-scale shape o he impac ed objec bu , also, o
he conside ed buckling la ice mo phology a he low
scale. This allows he inse ion o he explo ed amewo k
in o wide-pu pose me ama e ial design se ings.
6.2 Rema ks and signi icance o subsequen
de elopmen s on he opic
The au ho s a e awa e o he p ospec i e cha ac e o
his wo k on his speci ic b anch o me ama e ials design
(mul iscale-based compu a ional shock-abso bing models).
This is why, in he ollowing pa ag aphs, hey would like o
32 Conside ed as he goal unc ion o an assumed op imiza ion p oce-
du e
123
Compu a ional Mechanics
Fig. 18 E ec s o mic oscale mo phology on he shock-abso be dis-
sipa ion. aHomogenized cons i u i e esponses o he h ee analyzed
cases, bdynamic his o ies o he e ical (downwa ds) displacemen o
he op ace o he shock-abso be , caccumula ed ex insic dissipa ion
his o ies de olu ion cu es o he kine ic ene gy o he impac o
ansmi o he eade some addi ional ema ks a isen du ing
he esea ch wo k and w i ing o his pape i.e.:
•The s a emen , “ he ex insic dissipa ion a ises om
buckling p opaga ion a he mic oscale”, made in his
wo k (see Sec . 1.1.3), has gained much c edibili y a he
au ho s’ eyes along he w i ing. Howe e , hey ha e no
been able o p o ide g aphical e idence o i . F om one
side, consequences o ha p opaga ion ac oss he mic o-
la ice (p oduc ion o ex insic dissipa ion) ha e been
widely displayed along he p esen manusc ip by means
o he ep esen a i e examples. Howe e , g aphical ep-
esen a ion o such p opaga ion could no bee p esen ed
he e, since i would equi e a highly complex, and com-
pu a ional cos ly, calcula ion p ocess o he p opaga ing
buckling-bands a he mic oscale33. This is whe e Pa (I)
o his wo k [1] comes in o play h ough he ollowing
easoning:
1. Compa ison o he p esen ed examples in his Pa
(II) wi h he ones in Pa (I), bo h p oducing ex insic
33 May be a e being de-codi ied,in op opaga ing shock-wa e lines
a he mac oscale
dissipa ion, sugges s ha he ul ima e eason o ha
dissipa ion should be he same in bo h cases.
2. In Pa (I) (single-scale model) he dissipa ion was
iden i ied as a di ec consequence o he pe u ba ion
o he o iginal ( ully-con ex) hype elas ic cons i u-
i e ma e ial34. Then, he p opaga ion o he esul ing
shock-wa es was na u ally displayed and explained
in he con ex o he classical single-scale shock
wa es p opaga ion heo y [2,4,36].
3. In his Pa (II) (mul iscale model), i is clea ly
displayed he s ess homogeniza ion p ocedu e ans-
la es in o non ully-con ex homogenized cons i u i e
models as well (see, o ins ance, Fig. 14c).
4. The e o e, he mani es a ion o such lack o con exi y
should be he same, in bo h cases i.e.: shock-wa es
p opaga ion. In o he wo ds, he buckling p opaga-
ion appea ing in he ideally homogenized solid in his
Pa (II) and, he e o e, in he ackled buckling mic o-
la ice shock abso be de ice, could be explained,
h ough a sui able de-codi ica ion me hod o , on a
simila basis han in Pa (I), in he con ex o he clas-
sical single-scale shock wa es p opaga ion heo y,
34 Which o e ides i s o iginal ully-con ex hype elas ic cha ac e .
123

Compu a ional Mechanics
and no necessa ily by eso ing o hidden sou ces o
in insic local dissipa ion35.
5. In con as wi h he ex insic dissipa ion (associ-
a ed wi h he buckling p opaga ion mechanisms
accoun ed o he e), in insic36 dissipa ion sou ces
a e no conside ed ele an in he p esen ed idealized
scena io.
•In his con ex , issues such as he uniqueness and s abili y
o he mechanical sys em migh be mo e p ope ly exam-
ined in he p opaga ing shock wa es se ing ins ead o in
he, mo e s anda d, equilib ium/s abili y se ings.
•F om he Solid Mechanics poin o iew, he way in which
he con ex hype elas ic ma e ial ( hus non-dissipa i e in
e ms o classical s anda d ma e ials) endows he co e-
sponding shock-abso bing de ice wi h ene gy dissipa ion
capabili ies, can be a ibu ed o mul iscale coupling.
In he p oposed compu a ional se ing, his coupling is
aken in o accoun by he adop ed hie a chical mul iscale
amewo k37. The homogeniza ion p ocedu e esul s in
he homogenized s ess-s ain measu es in equa ions
which a e implici ly ela ed o each o he h ough he
esul ing homogenized cons i u i e model which u ns
ou o be non-con ex (see o ins ance Fig. 5).
•A p ope e alua ion o he ex insic dissipa ion can be
expec ed as a as a su icien scale sepa a ion exis s,
be ween he conside ed shock abso be and he buckling
la ice sizes. In simple wo ds: as a as he ypical mic o
buckling la ice size38 is su icien ly small conce ning he
de ice dimensions.
Appendix: A Compu a ional mul iscale
modeling: om con inuum o
la ice mic o-s uc u es
As in oduced in Sec . 1.2, a wo-scale app oach o
ligh weigh buckling-mic ola ice shock abso be s is p o-
posed. A he mac oscale, a con inuum model wi h an exac
non-linea kinema ic desc ip ion is employed. The mechani-
cal beha io o his model is cap u ed using adi ional 2D o
3D solid modeling echniques39. In con as , he mic oscale
35 Fo ins ance, high- equency mic o-oscilla ions in he buckling la -
ice, as some imes has been p oposed in he li e a u e on he subjec
(e.g., [37]).
36 Inelas ic-s ains-based dissipa ion o he mal-like dissipa ion i.e.:
amenable o be quan i ied in e ms o he buckling-la ice pa icles mass-
densi y.
37 And he es ic ion o scale-sepa a ion associa ed o i .
38 A measu e o he buckling-cell size
39 Fo he sake o simplici y in his wo k he p oblem will be es ic ed
o he 2D case
is cons i u ed by a massi e buckling mic ola ice composed
o slende de o mable solids a anged in a la ice s uc-
u e de ised o de elop a p opaga ing buckling beha io
wi hin he mic os uc u e. Fo compu a ional sa ing easons,
such mic o-s uc u es will be modeled using degene a ed
beam/ ame 1D kinema ics in he RVE (see Fig.1). The use
o he simpli ied 1D degene a ed kinema ics ansla es in o
la ge sa ings in he compu a ional cos o he RVE solu ion,
when compa ed wi h he ac ual 2D kinema ics, bu in oduces
an incong ui y when de eloping he classical mul iscale he-
o y40 which is he e sol ed by eso ing o he limi case
o an E sa z ma e ial con aining slende 1D solid elemen s
app oach in a 2D RVE (see Sec . Appendix: A.2). The model
de ails a e ou lined below, s a ing wi h a b ie desc ip ion o
he classical 2D mul iscale model used as a ounda ion.
Appendix: A.1 Classical compu a ional mul iscale
modelling
Appendix: A.1.1 P oblem se -up
Conside a mac oscopic solid body ep esen ing he shock
abso be ma e ial occupying a domain ⊂R2wi hasmoo h
bounda y ∂, and ma e ial pa icles labeled by X.Le
uM:→R2(A.1)
be he mac o displacemen ield,
JM(X)=uM(X)⊗∇
X≡∇
XuM(X)(A.2)
he co esponding displacemen g adien enso , and
FM(X)=1+JM(X)(A.3)
he de o ma ion g adien enso , whe e 1 ep esen s he
second-o de uni enso . He e, each ma e ial poin Xo he
mac oscale is associa ed wi h a ep esen a i e olume ele-
men (RVE) o he mic oscale. Mic oscale a iables will be
deno ed wi h he subsc ip μ, and mac oscale wi h he supe -
sc ip M. The RVE will be deno ed by μ⊂R2, wi h a
smoo h bounda y ∂μand i s ma e ial coo dina es by Y.
Fo eadabili y, he angle b acke •μwill deno e he RVE
olume a e age in eg al o he ield (•),
•μ≡1
μμ
(•)dμ.(A.4)
A kinema ic connec ion be ween bo h scales will be es ab-
lished by conside ing he i s -o de Taylo ’s expansion o
40 Whe e he spa ial dimensions o he mac o and mic o scales a e
assumed he same
123
Compu a ional Mechanics
he kinema ic a iables associa ed wi h poin Xin he
mac oscale. Thus,
uμ(Y)=uM(X)+JM(X)·(Y−YC)+˜
uμ(Y),
(A.5)
∇Yuμ(Y)=JM(X)+∇
Y˜
uμ(Y),(A.6)
and
Fμ(Y)=1+∇
Yuμ(Y)=FM(X)+∇
Y˜
uμ(Y),(A.7)
whe e YC ep esen s he coo dina es o he cen oid o he
RVE and ˜
uμdeno es he mic o-displacemen luc ua ion
ield.
To ensu e he consis ency condi ions, he ollowing kine-
ma ic homogeniza ion ela ions mus be sa is ied: 41
˜
uμ(Y)μ=0(A.8)
and
∇Y˜
uμ(Y)μ=0.(A.9)
O equi alen ly, (A.9) on he bounda y,
˜
uμ(Y)⊗Nμμ=0.(A.10)
By assuming con en ional pe iodic condi ions, he ol-
lowing exp essions should be sa is ied (see Fig. 19):
NμY+=−NμY−
˜
uμY+=˜
uμY−
uμ+=uμ−+FM−1·Y+−Y−
(A.11)
whe e he second e m in he las exp ession emains cons an
o each pai o bounda y poin s Y+and Y−.
Appendix: A.1.2 Hill-Mandel p inciple
The Mul iscale Vi ual Powe P inciple p oposed in [23]is
adop ed in his wo k. I co esponds o a a ia ional s a emen
o he classical Hill-Mandel p inciple in oduced in [27,28],
in which an ene ge ic equi alency be ween he wo scales is
es ablished. I es ablishes ha
PM(X):δFM=1
μμ
Pμ(Y):(δFM+∇
Yδ˜
uμ)dμ
(A.12)
41 Equa ions (A.8)and(A.9) a e o en e e ed o in he li e a u e as
minimal kinema ic es ic ions. He e, hey will be sa is ied by assuming
ha he RVE ul ills he pe iodical condi ions.
Fig. 19 Scheme o a squa e pe iodic RVE cell
∀δFMand δ˜
uμ∈˜
Vμ, whe e
˜
Vμ:= δ˜
uμ∈H1(μ)|δ˜
uμμ=0;∇Yδ˜
uμμ=0
(A.13)
deno es he i ual kinema ically admissible luc ua ion
space.InEq.(A.12), PM(X)and Pμ(Y) e e , espec i ely,
o he Fi s Piola-Ki chho s ess enso a he mac oscale
poin Xand a he co esponding RVE poin Y∈μ(X).
Equa ion (15) de ines a undamen al link be ween he wo
scales. Di ec consequences o i a e:
•The homogenized s ess exp ession,
PM=Pμuμ(Y)μ(A.14)
•The RVE p oblem a he mic oscale,
Find uμ ul illing
Pμ(uμ):∇
Yδ˜
uμμ=0,∀δ˜
uμ∈˜
Vμ.(A.15)
Finally, applying he gene al enso ela ion desc ibed in
[38] and he ob ained s ong o m in Eq. (17), yields an al e -
na i e exp ession o he homogenized s ess based solely on
he bounda y in o ma ion, i.e.:
PM=1
μ∂μ
μ⊗Yd∂μ(A.16)
whe e μ=Pμ·Nμis he Piola s ess ec o on he RVE
bounda y ∂μ.
Appendix: A.2 F om con inuum o la ice
mic o-s uc u es
The mic oscale consis s o a massi e se o slende
de o mable solids a anged in a la ice s uc u e, as illus a ed
in Fig. (20). As in oduced abo e, he use o simpli ied 1D
123
Compu a ional Mechanics
degene a ed kinema ics leads o signi ican sa ings in com-
pu a ional cos s bu in oduces an incong ui y when de elop-
ing he classical mul iscale heo y desc ibed in he p e ious
sec ion, in which he spa ial dimensions o he mac o and
mic o scales a e assumed o be he same. To add ess his
incong ui y, he E sa z ma e ial app oach (widely used in
he ield o opology op imiza ion, [39,40]) is empo a ily
applied.
Unde his app oach, i is assumed ha he RVE is com-
posed o wo ma e ial phases: a solid ma e ial phase, deno ed
as ωs
μ(see Fig. 20), which ep esen s he slende de o mable
solid o he mic oscale, and a subs i u e ma e ial phase,
deno ed as ω , ep esen ing a ic i ious low s i ness ma e ial
ha comple ely ills he ini ially emp y egions o he RVE.
Subsequen ly, he classical mul iscale model summa ized in
he p e ious sec ion can na u ally be applied, since he mac o
and mic o spa ial dimension scales a e he same. In his con-
ex , he kinema ic cons ain (A.9) can be eph ased as
μ
∇Y˜
uμdμ=ωs
μ
∇Y˜
uμdωs
μ+ω
∇Y˜
uμdω
μ=0,
(A.17)
o equi alen ly on he bounda y as
∂ωs
μ
˜
uμ⊗Nm
μd∂ωs
μ+∂ω
˜
uμ⊗N
μd∂ω
μ=0,
(A.18)
whe e Nm
μand N
μ e e o he ou bound uni no mals o he
solid ma e ial phase and he ic i ious phase, espec i ely.
Taking in o accoun ha Nm
μ=−N
μ, he las condi ion
(A.18) educes o
μ
˜
uμ(Y)⊗Nμdμ=0(A.19)
whe e μ=∂ωs
μ∩∂μ. Thus, he modi ied kinema ically
admissible luc ua ion space is gi en by
˜
Vω
μ=˜
uμ∈H1(μ)˜
uμωs
μ
=0;
μ
˜
uμ(Y)⊗Nμdμ=0 .(A.20)
Assume, wi hou loss o gene ali y, ha ψμand ψ
μdeno e
he s ain ene gy densi ies o he solid and subs i u e phases o
he RVE, espec i ely. Unde he e sa z assump ion, ψ
μcan
be exp essed in e ms o ψμas ψ
μ=ψμ, whe e  ep esen s
a e y small posi i e scala alue. Taking in o accoun ha
Pμcan be de ined h ough he co esponding s ain ene gy
unc ion, and making  end o ze o, he equilib ium condi-
ion (17) can be e o mula ed as:
Find uμ ul illing
1
μωs
μ
Pμ:∇
Yδ˜
uμdωs
μ=0∀δ˜
uμ∈˜
Vω
μ.(A.21)
Unlike p oblem (17), he equilib ium p oblem (A.21) needs
o be sol ed only wi hin he ma e ial domain ωs
μ.
Following a simila p ocedu e, he calcula ion o he
homogenized s ess enso PM educes o he ollowing
exp ession,
PM=1
μμ
μ⊗Yd∂μ(A.22)
Rema k 2 Fo con enience, p oblem (A.21) can be o mu-
la ed equi alen ly in e ms o he second Piola-Ki chho
s ess enso Sμand he G een-Lag ange de o ma ions δEμ
as ollows:
1
μωs
μ
Sμ:δEμδ˜
uμdωs
μ=0∀δ˜
uμ∈˜
Vω
μ.
(A.23)
Appendix: A.2.1 RVE ep esen a ion ia a mic ola ice o
slende buckling beams
Fo compu a ional sa ings easons, he slende de o mable
solids, ωs
μ, cons i u ing he RVE will be modeled using 1D
degene a e beam/ ame kinema ics. Assume ha ωs
μcan be
ep esen ed by a s aigh beam ne wo k, μ(Figu e 20b),
modeled wi h he app oach p esen ed in Sec . 2. Fo simplic-
i y, i is also assumed ha he sec ions o each beam, Aμ,
a e uni o m and ha he e a e nei he disc e e o ces no dis-
c e e momen s no body o ces applied on he mic os uc u e.
The e o e, he in e nal gene alized o ces emain uni o m
along each mic os uc u al ame elemen .
Thus, he mechanical a iables a he mac oscale a e he
posi ion o he e e ence poin , X, he spa ial posi ion o he
poin , x(X), he displacemen ec o , uM=x−X, he
g adien o de o ma ion, FM, and he Piola s ess, PM.
The mic oscale mechanical s a e, ela ed o he mac oscale
poin X, is de ined by he posi ion o he e e ence poin ,
Y, he spa ial posi ion o he poin , y(Y). Also, conside ing
he beam heo y o Sec . 2, he displacemen ec o o he
beam neu al ibe s is u0
μ=y(Y0)−Y0, he g adien o
de o ma ion, Fμ, and he Piola s ess, Pμ.
123
Compu a ional Mechanics
Fig. 20 Rep esen a i e RVE
cons i u ed equi alen ly by (1) a
se o slende de o mable solids
and (2) a se o slende buckling
beams
Scale b idging equa ions
The displacemen ec o uμo any poin Ya he mic oscale,
in ag eemen wi h 3, can be w i en as ollows:
uμ=uM(X)+JM·(Y0−YC)+˜
u0
μ(Y0)+ημ(˜
kμ−ˆ
T)
(A.24)
whe e he condi ion ha YCis he RVE mass cen e posi-
ion esul s in:
!JM·(Y0−YC)"=0(A.25)
Unde he hypo hesis o pe iodic bounda y condi ions, he
ollowing ma hema ical exp essions a e ul illed:
u0
μY+=u0
μY−+JM·Y+−Y−
˜
u0
μY+=˜
u0
μY−
˜
θμY+=˜
θμY−
(A.26)
RVE p oblem a he beam mic ola ice
Unde his beam ne wo k ep esen a ion, he a ia ional
p oblem s a ed in Eq. (A.23) can be ew i en as
ωs
μ
Sμ:δEμdωs
μ=μAμ
Sμ:δEμdA
μdμ(A.27)
An equi alen o mula ion o (A.27) using gene alized
s esses and s ains is p esen ed below ( e e o Appendix B
o de ails)
μAμ
Sμ:δEμdA
μdμ=μ
zμ·δhμdμ(A.28)
whe e zμ=(Nμ,Vμ,Mμ)Tis he gene alized s ess ec o
and δhμ=(δeμ,δγ
μ,δκ
μ)Tis he a ia ion o he gen-
e alized s ains ec o de ined in Eq. (7). F om (A.28) he
equi alen RVE p oblem can be s a ed as ollows:
Find (u0
μ,θ
μ) ul illing
μ
zμ(u0
μ,θ
μ)·δhμδ˜
u0
μ,δ˜
θμdμ=0,∀δ˜
u0
μ∈˜
V0
μ,
∀δ˜
θμ∈˜
μ(A.29)
whe e
˜
V0
μ=δ˜
u0
μ:μ→R2|δ˜
u0+
μ=δ˜
u0−
μ,(A.30)
and
˜
μ=δ˜
θμ:μ→R|δ˜
θ+
μ=δ˜
θ−
μ(A.31)
Homogeniza ion
Taking in o accoun he con ibu ion o he buckling beams
o he la ice, he homogenized s ess enso in Eq. (A.22)
can be exp essed as:
PM=1
μ
iμiPμ·Nμ⊗Yidμ(A.32)
whe e each in eg al ep esen s he con ibu ion o he i- h
beam ha in e sec s he bounda y o he RVE. Wi hou loss o
gene ali y, i is assumed ha Nμcoincides wi h he no mal
ec o o he c oss-sec ion, ˆ
Ni. Consequen ly, by (1), he
ma e ial poin s o he c oss-sec ion μican be exp essed as
123