Con en s lis s a ailable a ScienceDi ec
Eu opean Jou nal o Mechanics / A Solids
jou nal homepage: www.else ie .com/loca e/ejmsol
Full leng h a icle
A modi ica ion o Holzap el–Ogden hype elas ic model o myoca dium
be e desc ibing i s passi e mechanical beha io
Jiří Va e ka ∗, Jiří Bu ša
Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Facul y o Mechanical Enginee ing, B no Uni e si y o Technology, B no, Czech Republic
ARTICLE INFO
Keywo ds:
Ca diac mechanics
Myoca dium
Hype elas ici y
Cons i u i e model
O ho opy
ABSTRACT
The passi e mechanical beha io o he myoca dium is usually ma hema ically desc ibed wi hin he amewo k
o hype elas ici y. One o he mos popula models o his kind is ha p oposed by Holzap el and Ogden
in 2009. I is an o ho opic model o mula ed in e ms o a easonably selec ed se o scala in a ian s
ep esen ing di e en componen s o he myoca dium. Se e al modi ica ions o he model ha e eme ged
o e he yea s. In his pape , we p esen ano he one which is cha ac e ized by an inno a i e app oach o
he modeling o myoca dial ‘‘shee s’’, i.e. lamella collagenous s uc u es ha endow he myoca dium wi h
o ho opic mechanical p ope ies. We desc ibe hei con ibu ion by means o a less common scala in a ian
which exp esses he change o a ea o an o ien ed plana elemen ( ep esen ing he plane o a shee ). To
compa e ou o mula ion wi h he o iginal model, we ma ched bo h o hem o he biaxial ension and
simple shea expe imen al da a om he li e a u e using a nonlinea leas -squa es op imiza ion algo i hm. The
objec i e unc ion o each model included bo h biaxial and simple shea da a in o de o ob ain a single se
o pa ame e s o bo h de o ma ion modes. The esul s show ha ou modi ied model can accu a ely desc ibe
bo h ypes o es s. The o al esidual is lowe ed by app oxima ely 80% by ou modi ica ion and 𝑅2inc eases
om 0.877 o 0.978 which demons a es he signi icance o ou modi ica ion on he quali y o he i .
1. In oduc ion
Accu a e ma hema ical desc ip ion o he mechanical beha io o
passi e ( elaxed) myoca dium is essen ial in o de o s udy mechani-
cal phenomena in he hea by compu a ional modeling. Cons i u i e
equa ions p o iding such desc ip ion a e pa icula ly impo an o he
de elopmen o compu e models based on he ini e elemen me hod.
Compu a ional s udies published o e he las ew decades p o ed ha
such models can subs an ially inc ease ou knowledge and unde s and-
ing o he issue p ope ies and physiological p ocesses in a heal hy
hea (e.g., Xi e al.,2019;Ke ckho s e al.,2003;McE oy e al.,2018),
help us o explain mechanisms unde lying hea diseases and in es i-
ga e hei consequences (e.g., Ko ache a e al.,2021;Mojumde e al.,
2023;Cos abal e al.,2019), as well as allow us o p edic he ou comes
o medical ea men s and in e en ions (e.g., Xu e al.,2020;Li e al.,
2020a;Dabi i e al.,2019). The as majo i y o hese wo ks ea
myoca dium as a hype elas ic ma e ial despi e he epo ed iscoelas ic
ea u es ( a e-dependence, hys e esis) exhibi ed by myoca dium du ing
mechanical es s (see, e.g., Somme e al.,2015;Dokos e al.,2002).
Howe e , a ecen compu a ional s udy by Tikenoğulla ı e al. (2022)
has jus i ied his simpli ica ion by showing ha iscous elaxa ion o
∗Co espondence o: Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, B no Uni e si y o Technology, Technická 2896/2, 616 69, B no, Czech
Republic.
E-mail add ess: [email p o ec ed] (J. Va e ka).
he myoca dium has a negligible e ec on he o e all beha io o he
whole hea du ing physiologically ele an ime scales o he ca diac
cycle. Thus he app oxima ion by a hype elas ic model is accep able
o mos p ac ical applica ions, p o ided he chosen model can accu-
a ely cap u e he o ho opic and highly nonlinea (elas ic) esponse
o myoca dium e ealed by expe imen s (Somme e al.,2015;Dokos
e al.,2002). Conside ing u he ha nonlinea iscoelas ic models o
he myoca dium a e highly complex and include addi ional ma e ial
pa ame e s ha a e di icul o es ima e (see, e.g., Gül ekin e al.,2016;
Zhang e al.,2023), i can be expec ed ha hype elas ic models will
emain dominan in he ield o compu a ional ca diac biomechanics
wi hin he nea u u e; hus hei u he de elopmen is s ill impo an .
Ob iously, a hype elas ic model is mo e likely o p o ide an accu-
a e mechanical desc ip ion o myoca dium i he s ain–ene gy unc-
ion (SEF) de ining he model e lec s he in e nal mic os uc u e o he
issue. This can be e icien ly achie ed by o mula ing he unc ion as
a sum o sepa a e e ms, each o which depends only on a single scala
in a ian wi h a clea physical in e p e a ion. Each e m o he model
hen ep esen s ce ain cons i uen (s) o he issue and he cha ac e o
h ps://doi.o g/10.1016/j.eu omechsol.2025.105586
Recei ed 18 Augus 2024; Recei ed in e ised o m 30 Decembe 2024; Accep ed 19 Janua y 2025
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
A ailable online 27 Janua y 2025
0997-7538/© 2025 The Au ho s. Published by Else ie Masson SAS. This is an open access a icle unde he CC BY license
( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
J. Va e ka and J. Bu ša
i s dependence on he co esponding in a ian (e.g., quad a ic o expo-
nen ial) is de e mined on he basis o expe imen al s ess–s ain da a.
Such a a ional, s uc u ally and expe imen ally mo i a ed app oach
was adop ed also by Holzap el and Ogden (2009) who p oposed an
o ho opic hype elas ic model o myoca dium (he eina e abb e i-
a ed as he “HO” model) which has since hen become one o he mos
popula models in he ield. Howe e , se e al modi ied e sions o he
model ha e eme ged o e he yea s and some o hem a e p e e ed
nowadays by many esea che s o e he o iginal o mula ion. One o
he i s (and mos no able) modi ica ions was p oposed by Gök epe
e al. (2011) who in oduced he adi ional isocho ic– olume ic de-
coupling (see Holzap el,2000) in o he (o iginally incomp essible) HO
model. This was e ec ed by eplacing all in a ian s in he SEF wi h
hei isocho ic e sions and ex ending he unc ion wi h a olume ic
e m. The esul ing nea ly-incomp essible o mula ion was hen used
by many o he au ho s (e.g., E iksson e al.,2013a;Pali e al.,2018);
besides, i was also implemen ed in comme cial ini e elemen so wa e
Abaqus®. In some o he wo ks he isocho ic in a ian s we e in oduced
only o he iso opic pa o he SEF while he aniso opic pa was
le unchanged (e.g., Ge bi e al.,2019;McE oy e al.,2018); his
was mo i a ed by he wo ks o Nolan e al. (2014) and Ve go i e al.
(2013) who ha e shown ha isocho ic aniso opic in a ian s p oduce
unphysical esponses unde ce ain de o ma ion modes. An impo an
imp o emen was made by E iksson e al. (2013b) and Melnik e al.
(2018) who in oduced ibe dispe sion in o aniso opic in a ian s.
O he signi ican modi ica ions include a subs i u ion o he expo-
nen ial iso opic e m by a polynomial one (McE oy e al.,2018),
simpli ica ion o he model by omi ing wo o i s h ee aniso opic
e ms (Chapelle and Le Gall,2023) o , in con as , i s ex ension by wo
mo e aniso opic e ms (wi h ou addi ional ma e ial pa ame e s) (Li
e al.,2020b). The mos gene al HO model wi h 6 aniso opic e ms
(ins ead o 3) was conside ed by Guan e al. (2019) who s udied i s
abili y o desc ibe h ee di e en expe imen al da a se s om li e a u e.
By analyzing he con ibu ion o each e m o he o e all goodness
o i , hey p oposed h ee educed e sions o he model (wi h 3–4
aniso opic e ms) su icien o an accu a e desc ip ion o he h ee
da a se s. The same gene al model was in es iga ed also by Ma ono á
e al. (2024).
The abo e examples show ha many di e en e sions o he HO
model can al eady be ound in he li e a u e, each o hem designed
o i he speci ic needs o i s au ho s and, he e o e, mo e sui able
o pa icula applica ions han he o he s. In his pape , we p esen
a new modi ica ion which is dis inc i e om any o he o mula ion
p esen ed in he li e a u e by he manne in which i deals wi h he
ac ha he en icula muscle ibe s a e agg ega ed in o he so-called
“shee s” (e.g., S ephenson e al.,2018) ha p o ide he myoca dium
wi h o ho opic a he han ans e sely iso opic mechanical p ope -
ies. Unlike Holzap el and Ogden (2009), we desc ibe he mechanical
con ibu ion o shee s by an exponen ial s ain–ene gy e m p oposed
by Balzani e al. (2006) which is o mula ed in e ms o a (no e y
usual) scala in a ian 𝐾1which is a measu e o change in ela ion o a
de o ma ion o he a ea o a p e e ed plana elemen . We conside his
in a ian o be a sui able measu e o he de o ma ion o shee s, simila
in i s na u e o he analogous in a ian s ha exp ess s e ch o ibe s o
change o olume and ha a e egula ly used in cons i u i e equa ions.
Since we in oduce no o he change o he model, ou modi ica ion
e ains he ma hema ical s uc u e o he HO model, i s exponen ial
o m, as well as he numbe o ma e ial pa ame e s. Howe e , we will
show ha i has conside ably imp o ed i ing capabili ies compa ed o
he o iginal o mula ion. We demons a e his by i ing he model o
da a om i e di e en biaxial ension es s and six di e en simple
shea es s (all aken om Somme e al.,2015). The compa ison
wi h he o iginal HO model e eals ha ou modi ica ion enables a
desc ip ion o all he da a wi h a single se o ma e ial pa ame e s,
which he o iginal o mula ion canno sa is ac o ily achie e.
The pape is o ganized as ollows. Sec ion 2b ie ly summa izes
he mos impo an ana omical ea u es o he en icula myoca dium
and desc ibes how Holzap el and Ogden (2009) inco po a ed hem in o
hei success ul model. Sec ion 3p esen s ou app oach o he cons i-
u i e modeling o he myoca dium, cha ac e ized by he employmen
o in a ian 𝐾1whose physical meaning we explain in mo e de ail. We
in oduce a modi ied SEF o he myoca dium and de i e he co e-
sponding ela ion o he Cauchy s ess. Following Holzap el and Ogden
(2009), we assume ully incomp essible beha io . In Sec ion 4we i s
desc ibe he biaxial and simple shea da a published by Somme e al.
(2015), which a e sui able o he e alua ion o ma e ial pa ame e s o
o ho opic models, and hen we discuss he con o e sy su ounding
he co ec in e p e a ion o he c oss- ibe esponse ob ained om
biaxial es s. In Sec ion 5we i ou model and he HO model o he
da a om Somme e al. (2015) in o de o demons a e he bene i s o
ou app oach. Finally, in Sec ion 6, we discuss some p oblema ic poin s,
make u he sugges ions and o mula e conclusions.
2. Ana omy o en icula myoca dium and he model by
Holzap el & Ogden
Myoca dium is composed mos ly o ca diomyocy es, i.e. elonga ed
con ac ile cells embedded in an ex acellula ma ix o connec i e
issue (Sme up e al.,2009). Ca diomyocy es a e linked end- o-end
in o longi udinal chains, o ibe s, which a e in e connec ed by side-
b anches (Sme up e al.,2009). The ibe s join and spli along hei
pa hs and ha e nei he a disce nible beginning no ending (Lunkenheime
and Niede e ,2012;S ephenson e al.,2018); ne e heless, in a ypical
small olume o he ne wo k, i is possible o dis inguish he p e-
dominan ibe di ec ion, which can be ep esen ed by a uni ec o
𝐟(see Fig. 1). Adjacen ibe s also ha e a highe deg ee o o ganiza ion
since hey a e la e ally agg ega ed ( h ough endomysial connec i e
issue) in o la ge blocks wi h lamella appea ance (Lunkenheime and
Niede e ,2012;LeG ice e al.,1995;S ephenson e al.,2018). These
highe -o de s uc u es, which a e equen ly called “shee s”, p o-
ide he myoca dium wi h o ho opic mechanical p ope ies (Somme
e al.,2015;Dokos e al.,2002) a he han ans e sely iso opic.
The s uc u al in eg i y o laye s is p o ided mainly by endomysial
collagen ibe s which ac as la e al connec ions be ween muscle ibe s
and hus enable ansmission o o ces in axial as well as in ans e se
di ec ions (Lunkenheime and Niede e ,2012;Webe ,1989). Adja-
cen shee s a e sepa a ed by elonga ed pe imysial spaces (also called
pe imysial cle s o clea age planes) illed wi h gela inous lub ica ing
luid which is supposed o acili a e shea ing o shee s ela i e o each
o he (Lunkenheime and Niede e ,2012;Webe ,1989). Howe e , in
o de o p e en an excessi e slippage o laye s o e en a up u e o
he issue, he clea age planes a e spanned by a spa se ne wo k o
pe imysial collagen ibe s which connec adjacen laye s and hence
s eng hen he s uc u e (Webe e al.,1994;Webe ,1989, see also
mic og aphs in Lunkenheime and Niede e ,2012 and LeG ice e al.,
1995). The o ien a ion o a shee in he myoca dium can be ma he-
ma ically desc ibed by choosing one o he wo uni ec o s pa allel
o he plane o he shee and pe pendicula o he 𝐟di ec ion. The
chosen ec o is hen e e ed o as he shee ec o , 𝐬. The di ec ions
o ma e ial o ho opy a e comple ed wi h he shee -no mal ec o , 𝐧,
which is he ec o o which he iple (𝐟,𝐬,𝐧) o ms a igh -handed
o hono mal basis.
Respec ing he s uc u al o ganiza ion o he myoca dium desc ibed
abo e, Holzap el and Ogden (2009) desc ibed i s mechanical beha io
in e ms o in a ian s
𝐼1∶=
𝐼1(𝐂) ∶= 𝐂,(1)
𝐼4 ∶=
𝐼4 (𝐂) ∶= 𝐟⋅(𝐂𝐟),(2)
𝐼4s∶=
𝐼4s(𝐂) ∶= 𝐬⋅(𝐂𝐬),(3)
𝐼8∶=
𝐼8(𝐂) ∶= 𝐟⋅(𝐂𝐬),(4)
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
2
J. Va e ka and J. Bu ša
Fig. 1. Schema ic ep esen a ion o he lamella s uc u e o en icula myoca dium
wi h co esponding s uc u al ec o s 𝐟( ibe ), 𝐬(shee ), and 𝐧(no mal o shee ). C e-
a ed on he basis o scanning elec on mic oscopic images published by Lunkenheime
and Niede e (2012) and LeG ice e al. (1995).
which depend on he de o ma ion g adien 𝐅 h ough he igh Cauchy–
G een enso 𝐂∶= 𝐅⊤𝐅. (No e ha in (1)−(4), and in some o he
equa ions below, we use he ci cum lex “ˆ” o dis inguish a unc ion
om i s alue.) As he au ho s explained, 𝐼1was employed o desc ibe
he de o ma ion o all non-collagenous and non-muscula cons i uen s
(like elas in and luids) which can easonably be conside ed iso opic.
The mo e p onounced s i ness exhibi ed by he myoca dium in he
𝐟di ec ion was assumed o be due o he muscle ibe s as well as o
he collagen ne wo k (bo h endomysial and pe imysial). The combined
e ec o hese wo cons i uen s was desc ibed by 𝐼4 , which exp esses
he squa e o s e ch in he 𝐟di ec ion since
𝐼4 =𝐟⋅(𝐂𝐟) =𝐟⋅(𝐅⊤𝐅𝐟) = (𝐅𝐟)⋅(𝐅𝐟) =‖𝐅𝐟‖2.(5)
Addi ional s i ness in he 𝐬di ec ion, associa ed wi h he collagen
ibe s connec ing he muscle ibe s, was modeled h ough 𝐼4swhose
physical meaning is, o cou se, analogous o 𝐼4 . Finally, 𝐼8was in-
cluded in o de o s i en he esponse o he simple shea modes ( s)
and (s ), and he eby o dis inguish hem om he modes ( n) and (sn),
see Fig. 2. The decision o Holzap el and Ogden (2009) o inc ease
s i ness in modes ( s) and (s ) was mo i a ed by he expe imen al da a
o Dokos e al. (2002) who es ed cubic samples om pig myoca dium
in all he six possible simple shea modes; hey epo ed conside ably
s i e beha io o he myoca dium in he ( s) mode compa ed o he
( n) mode, and in he (s ) mode compa ed o he (sn) mode. Al hough
he esul s om he simple shea es s o human myoca dium, published
la e by Somme e al. (2015), did no con i m such signi ican di e -
ences, no comple e da a se compa able o ha o Dokos e al. (2002)
was a ailable a he ime o publica ion o he HO model, he e o e he
inclusion o he 𝐼8in a ian seemed necessa y. The sensi i i y o 𝐼8 o
he shea modes ( s) and (s ) becomes e iden i we ew i e (4) as
𝐼8=𝐟⋅(𝐂𝐬) =𝐟⋅(𝐅⊤𝐅𝐬) = (𝐅𝐟)⋅(𝐅𝐬) = cos (𝜃)‖𝐅𝐟‖‖𝐅𝐬‖,(6)
whe e 𝜃is he angle be ween 𝐅𝐟 and 𝐅𝐬. Thus 𝐼8= 0in all simple shea
modes excep o ( s) and (s ).
In o de o ep oduce exponen ial ends shown by expe imen al
da a, Holzap el and Ogden (2009) embedded all ou in a ian s in o
exponen ial unc ions based on ha p oposed by Demi ay (1972). The
esul ing SEF 𝜓HO o he a gumen
HO ∶=
HO(𝐂) ∶= (𝐼1, 𝐼4 , 𝐼4s, 𝐼8)(7)
is gi en by
𝜓HO(HO) ∶= 𝑎1
2𝑏1
(exp(𝑏1(𝐼1− 3)) − 1)
+𝑎
2𝑏
(exp(𝑏 (𝐼4 − 1)2) − 1)
+𝑎s
2𝑏s
(exp(𝑏s(𝐼4s− 1)2) − 1)
+𝑎 s
2𝑏 s
(exp(𝑏 s𝐼2
8) − 1),
(8)
whe e 𝑎1,𝑎 ,𝑎sand 𝑎 s a e s ess-like ma e ial pa ame e s, while 𝑏1,
𝑏 ,𝑏sand 𝑏 s a e dimensionless. To ensu e he exis ence o minimize s
in bounda y alue p oblems and o ob ain physically meaning ul e-
sponses, he 𝑎-pa ame e s mus be non-nega i e and he 𝑏-pa ame e s
mus be posi i e (see he discussion in Sec ion 6 o Holzap el and
Ogden,2009). Also, each o he middle wo e ms in (8) should be
se o ze o when i s co esponding in a ian (𝐼4 o 𝐼4s) is less han
1 which ensu es con exi y o bo h e ms in he ull ange o de o -
ma ions (Balzani e al.,2006). This deac i a ion is also physically
easonable because muscle and collagen ibe s canno be expec ed o
suppo signi ican comp ession.
Iden i ica ion o he pa ame e s o he HO model on he basis o
expe imen al da a has been pe o med by se e al au ho s. Howe e ,
mos o hem ei he used only simple shea da a (e.g., Gök epe e al.,
2011;Wang e al.,2013;McE oy e al.,2018), o hey used bo h simple
shea da a and biaxial da a bu hey p o ided di e en pa ame e s o
each es ype (e.g., Holzap el and Ogden,2009;Gül ekin e al.,2016).
This can be a ibu ed o he ac ha a usion o biaxial and simple
shea da a in one i ing p ocess does no gi e sa is ac o y esul s wi h
he HO model, as we will show in Sec ion 5. This de iciency o he
model was ecognized ea lie by Gül ekin e al. (2016) (al hough in
he con ex o iscoelas ici y) and i is also appa en om he i ing
esul s o Guan e al. (2019) and Ma ono á e al. (2024). Ou app oach
p esen ed in his pape aims o o e come his de iciency o he HO
model wi hou inc easing he numbe o pa ame e s.
3. The p oposed model
I is easonable o expec ha a ypical shee (composed o en-
domysial issue and muscle ibe s) esis s o ces ac ing in any di ec ion
wi hin i s plane. This assump ion is consis en wi h Holzap el and
Ogden (2009) who assumed ha collagen ibe s inc ease he s i ness
in bo h 𝐟and 𝐬di ec ions. The mechanical p ope ies o endomysial
connec i e issue migh , o cou se, be aniso opic (because o he po en-
ially nonuni o m di ec ional dis ibu ion o collagen ibe s) bu he e
seems o be no ele an in o ma ion on his issue; hus we ind i app o-
p ia e o idealize a ypical shee as an iso opic plane ein o ced by a
amily o pa allel muscle ibe s. This plana s uc u e can hen be imag-
ined as being embedded in a (p esumably) iso opic h ee-dimensional
ma ix ep esen ing he es o he ex acellula componen s (including
he pe imysial issue).
This modeling app oach lea es us wi h h ee idealized s uc u al
componen s o he myoca dium whose mechanical beha io should
now be desc ibed in e ms o sui able in a ian s inse ed in o sepa a e,
addi i ely con ibu ing s ain–ene gy e ms. We see no good eason o
change any hing on he desc ip ion o ex acellula ma ix and muscle
ibe s p oposed by Holzap el and Ogden (2009); hus we e ain he
i s wo e ms in Eq. (8) bu we associa e he 𝐼4 e m solely wi h
muscle ibe s and no wi h collagen ibe s. Howe e , we ake a di e en
app oach o he desc ip ion o de o ma ion and mechanical esponse
o collagenous endomysium, which is based on a no e y common
in a ian which Sch öde and Ne (2003) and Balzani e al. (2006)
deno e as 𝐾1. Using he de ini ion o he co ac o o any in e ible
enso 𝐋,
co (𝐋) ∶= de (𝐋)𝐋−⊤,(9)
he in a ian is de ined by
𝐾1∶=
𝐾1(𝐂) ∶= 𝐧⋅(co (𝐂)𝐧).(10)
Vec o 𝐧in (10) could be an a bi a y ec o de e mining a p e e ed
ma e ial di ec ion, bu he e we will ega d i as ep esen ing he shee -
no mal di ec ion illus a ed in Fig. 1. This choice will gi e 𝐾1a physical
meaning sui able o ou pu pose, which can be disclosed using he
ollowing esul (c . Eq. (5)):
𝐾1=𝐧⋅(co (𝐂)𝐧) =𝐧⋅(co (𝐅)⊤co (𝐅)𝐧) = (co (𝐅)𝐧)⋅(co (𝐅)𝐧)
=‖co (𝐅)𝐧‖2.(11)
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
3
J. Va e ka and J. Bu ša
Fig. 2. Six possible modes o simple shea de o ma ion o a cubic myoca dial specimen ela i e o he p incipal ma e ial di ec ions 𝐟,𝐬and 𝐧. We speci y a simple shea mode
by a pa en he ical symbol in which he i s le e e e s o he no mal o he ace o he cubic specimen ha is shi ed by he simple shea and he second deno es he di ec ion
o shi . Dashed lines ep esen cube edges in he e e ence con igu a ion. The igu e was d awn on he basis o simila illus a ions used elsewhe e; e.g., in Holzap el and Ogden
(2009).
Fig. 3. Illus a ion o he ac ion o 𝐅and co (𝐅). De o ma ion o a neighbo hood o a
ma e ial poin 𝐗on o a neighbo hood o he co esponding poin 𝐱 ans o ms ma e ial
ec o s 𝐟,𝐬and 𝐧 espec i ely o 𝐅𝐟,𝐅𝐬 and 𝐅𝐧. Howe e , co (𝐅)maps 𝐧( ega ded
as a ea ec o ) o co (𝐅)𝐧which is pe pendicula o he pa allelog am wi h a ea
𝑃
(ini ially 𝑃) and i s magni ude equals
𝑃.
Since 𝐧=𝐟×𝐬, we can ega d 𝐧as he a ea ec o o he pa allelog am
de ined by 𝐟and 𝐬. Then, using he iden i y
co (𝐅)(𝐟×𝐬) = (𝐅𝐟) × (𝐅𝐬)(12)
(c ., e.g., Gu in e al.,2010, p. 23), we can see ha co (𝐅)𝐧is he
a ea ec o o he pa allelog am de ined by 𝐅𝐟 and 𝐅𝐬 (i.e., he ec o
o hogonal o he pa allelog am, di ec ed acco ding o he igh -hand
sc ew ule, and wi h magni ude equal o he a ea o he pa allelog am;
c . Fig. 3). Thus, by (11),𝐾1is he squa e o he a ea o he pa allelo-
g am, o , since ‖𝐧‖= 1, he squa e o he ela i e change o he a ea
o a plana elemen ini ially pe pendicula o 𝐧.
We use 𝐾1in a unc ion ha ing he same exponen ial o m as he
e ms wi h 𝐼4 and 𝐼4sin he HO model (8). The usage o his e m ( he
hi d one in Eq. (14)) in SEFs o so biological issues was sugges ed
by Balzani e al. (2006) bu we a e no awa e o any p e ious a emp
o employ i o he desc ip ion o he myoca dium. Since he e m can
cap u e he collagen-induced s i ening in he 𝐬di ec ion, he e is no
u he need o he 𝐼4sin a ian and hus we omi he co esponding
e m o he HO model. Howe e , in o de o make he model adap able
o quali a i ely di e en simple shea da a a ailable in he li e a u e
(c . Sec ion 2), we main ain he las e m o he HO model con aining
he 𝐼8in a ian . The esul ing modi ied model is hen de ined in e ms
o he amily o ou in a ian s
∶=
(𝐂) ∶= (𝐼1, 𝐼4 , 𝐾1, 𝐼8)(13)
by a SEF
𝜓() ∶= 𝑎1
2𝑏1
(exp(𝑏1(𝐼1− 3)) − 1)
+𝑎
2𝑏
(exp(𝑏 (𝐼4 − 1)2) − 1)
+𝑎nn
2𝑏nn
(exp(𝑏nn(𝐾1− 1)2) − 1)
+𝑎 s
2𝑏 s
(exp(𝑏 s𝐼2
8) − 1).
(14)
Ma e ial pa ame e s 𝑎1,𝑎 ,𝑎nn and 𝑎 s a e s ess-like, while 𝑏1,𝑏 ,
𝑏nn and 𝑏 s a e uni less. In o de o sa is y he equi emen s ela ed
o con exi y, he 𝑎-pa ame e s mus be non-nega i e, he 𝑏-pa ame e s
mus be posi i e, and each o he middle wo e ms in (14) should be
se o ze o when i s co esponding in a ian (𝐼4 o 𝐾1) is less han
1 (Balzani e al.,2006).
Fo cla i y, we will deno e by
𝛹 he SEF 𝜓 exp essed as a unc ion
o 𝐂ins ead o , i.e.:
𝛹∶= 𝜓◦
.(15)
Then, wi h he assump ion o incomp essibili y, he second Piola–
Ki chho s ess 𝐒co esponding o 𝜓 can be exp essed as
𝐒∶=
𝐒(𝐂) ∶= 2∇𝐂
𝛹−𝑝𝐽𝐂−1 (16)
and he Cauchy s ess 𝝈is gi en by
𝝈∶= 𝐽−1𝐅𝐒𝐅⊤= 2𝐽−1𝐅(∇𝐂
𝛹)𝐅⊤−𝑝𝟏.(17)
The p esence o he olume change 𝐽∶= de (𝐅)in (16) and (17)
may seem unnecessa y since 𝐽= 1 o incomp essible ma e ials, bu
main aining 𝐽is impo an o he co ec e alua ion o elas ici y
enso s (c . Sec ion 6.5.1 o Bone and Wood,2008). Fo his eason,
we keep 𝐽(o equi alen ly de (𝐂)) in se e al equa ions in his pape .
The scala 𝑝in (16) and (17) is an inde e mina e p essu e (meaning
ha i canno be de e mined om a cons i u i e equa ion). No e ha
in his pa icula case, i does no coincide wi h he p essu e pa o 𝝈,
i.e., 𝑝≠−1
3 𝝈(c . Sec ion 6.5.1 o Bone and Wood,2008).
In he ollowing, we will use he abb e ia ions
∶= (𝐂−1𝐧)⊗(𝐂−1𝐧),(18)
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
4
J. Va e ka and J. Bu ša
∶= (co (𝐅)𝐧)⊗(co (𝐅)𝐧).(19)
Using he chain ule, we can exp ess
∇𝐂
𝛹= ∇𝜓∇𝐂
=𝜓,𝐼1()∇𝐂
𝐼1+𝜓,𝐼4 ()∇𝐂
𝐼4 +𝜓,𝐾1()∇𝐂
𝐾1+𝜓,𝐼8()∇𝐂
𝐼8.(20)
G adien s in he las line o (20) sa is y he iden i ies
∇𝐂
𝐼1=𝟏,(21)
∇𝐂
𝐼4 =𝐟⊗𝐟,(22)
∇𝐂
𝐾1=𝐾1𝐂−1 −de (𝐂),(23)
∇𝐂
𝐼8= (1∕2)(𝐟⊗𝐬+𝐬⊗𝐟),(24)
and pa ial de i a i es a e gi en by
𝜓,𝐼1() = (𝑎1∕2) exp(𝑏1(𝐼1− 3)),(25)
𝜓,𝐼4 () =𝑎 (𝐼4 − 1) exp(𝑏 (𝐼4 − 1)2),(26)
𝜓,𝐾1() =𝑎nn(𝐾1− 1) exp(𝑏nn(𝐾1− 1)2),(27)
𝜓,𝐼8() =𝑎 s𝐼8exp(𝑏 s𝐼2
8).(28)
Eqs. (21)–(24) can now be subs i u ed in o (20) and he esul ing
exp ession can be used in (17) o ob ain he Cauchy s ess ela ion in
an ex ended o m
𝝈= 2𝜓,𝐼1𝐁+ 2𝜓,𝐼4
𝐟⊗
𝐟+ 2𝜓,𝐾1(𝐾1𝟏−) +𝜓,𝐼8(
𝐟⊗
𝐬+
𝐬⊗
𝐟) −𝑝𝟏,(29)
whe e
𝐟∶= 𝐅𝐟 and
𝐬∶= 𝐅𝐬. No e ha we omi ed he a gumen om
he pa ial de i a i es in (29) in o de o a oid clu e .
The gene al exp ession (29) can be used, e.g., o de i e explici
s ess–s ain ela ions o de o ma ion modes o which specimens a e
commonly subjec ed du ing mechanical es s. Such ela ions can hen
be ma ched wi h co esponding expe imen al da a in o de o e alua e
he ma e ial pa ame e s o he model, which is wha we do in Sec ion 5.
4. Expe imen al da a o i ing, in e p e a ion o he c oss- ibe
esul s o biaxial es s
In o de o compa e ou modi ica ion 𝜓, Eq. (14), wi h he o iginal
SEF 𝜓HO, Eq. (8), we will i hei co esponding s ess ela ions o
he da a ex ac ed om Figs. 9 and 13 in Somme e al. (2015) which
con ain a e aged esul s o i e di e en biaxial es s (Fig. 9) and six
di e en simple shea es s (Fig. 13) o human myoca dium. The biaxial
es s we e pe o med on hin squa ed specimens wi h dimensions o
25 × 25 mm and a hickness o app oxima ely 2.3mm. They we e
cu pa allel o he en icula wall so ha one loading di ec ion o
each specimen was aligned wi h he p edominan 𝐟di ec ion and he
o he one was he e o e app oxima ely pe pendicula o he ibe s. The
o me di ec ion was called he mean- ibe di ec ion (MFD) and he
la e he c oss- ibe di ec ion (CFD). Fig. 9 o Somme e al. (2015)
con ains biaxial esponses o i e di e en s ain a ios be ween he
MFD and he CFD, namely: 0.5∶1,0.75∶1,1∶1 (equibiaxial es ), 1∶0.75
and 1∶0.5. The maximum applied s ain was 10% in all cases. The
simple shea es s cap u ed in Fig. 13 o Somme e al. (2015) we e
pe o med on small cubic specimens excised om egions adjacen
o he biaxially es ed specimens in o de o ensu e uni o m ma e ial
p ope ies. The edges o each specimen we e app oxima ely 4mm long
and hey we e o ien ed along he ma e ial di ec ions 𝐟,𝐬and 𝐧. The
specimens we e es ed in all six possible simple shea modes associa ed
wi h hese di ec ions (c . Fig. 2). The maximum applied amoun o
shea was 0.5 in all modes.
Be o e using he abo e-desc ibed expe imen al da a o ob ain he
ma e ial pa ame e s o he in es iga ed models, we mus i s es ab-
lish how o ea he esponses in he CFD ob ained om biaxial
es s. Speci ically, we mus decide whe he i is mo e app op ia e o
conside hese esul s as ep esen ing he mechanical beha io o he
myoca dium in he 𝐬di ec ion o in he 𝐧di ec ion. As we discuss be-
low, his is no a s aigh o wa d ask because he o ien a ion o shee s
inside he wall is inhomogeneous and highly a iable and he e o e i
is di icul o make a conclusi e assessmen o he ela ion be ween he
CFD and he a e age 𝐬o 𝐧di ec ion in a ypical biaxial specimen.
As a consequence, some au ho s in he pas associa ed he CFD wi h
he 𝐬di ec ion (e.g., Holzap el and Ogden,2009;McE oy e al.,2018)
while o he s wi h he 𝐧di ec ion (e.g., Gül ekin e al.,2016;Guan e al.,
2019). We will now p o ide a sho summa y o ele an in o ma ion
om he li e a u e abou he shee o ien a ion ha has in luenced ou
decision on his issue.
The o ien a ion o shee s in he le en icle is he e ogeneous, as is
e iden , e.g., om he esul s o Agge e al. (2017) who analyzed in de-
ail he myoca dial a chi ec u e in en icles by means o he di usion
enso magne ic esonance imaging echnique. Simila indings we e
epo ed also by Dokos e al. (2002) who s udied ansmu al segmen s
o he le en icula wall be o e hey cu hem in o cubic simple-shea
specimens. They w o e: “In all hea s es ed, midwall muscle laye s
we e inclined a ∼45◦ o he adial di ec ion. Al hough i was possible o
iden i y a p edominan midwall o ien a ion o myoca dial laminae on
he ansmu al cu su aces, he e we e commonly egions whe e laye s
we e o ien ed a ∼90◦ o his p incipal di ec ion.” Besides con i ming
he he e ogeneous o ien a ion o shee s, his desc ip ion asse s ha
hey a e ei he app oxima ely pa allel o he local epica dial angen ial
plane, o inclined om i by app oxima ely 45◦o less, bu hey a e
unlikely o ex end in a adial ashion. Con a y o his, Somme e al.
(2015) in hei Fig. 3ma ked he shee di ec ions in myoca dial samples
by a ows di ec ed almos adially, bu un o una ely, hey did no
o mula e any gene al summa y compa able o ha o Dokos e al.
(2002). Also, since hey es ed specimens om he ou e , middle and
inne po ion o he wall, i is likely ha he p edominan o ien a ion
conside ably a ied be ween specimens (because o he he e ogenei y).
Ano he impo an inding abou he o ien a ion o shee s is ha he
angle be ween 𝐬and he local wall angen plane signi ican ly changes
du ing he ca diac cycle. In pa icula , Fe ei a e al. (2014) conduc ed
in- i o di usion enso magne ic esonance imaging o heal hy human
hea s and obse ed he mean angle o 24.0◦a end-dias ole and o 56.4◦
a end-sys ole. A simila s udy was pe o med by Nielles-Vallespin e al.
(2017) who epo ed he mean alues o 18◦and 65◦a end-dias ole and
end-sys ole, espec i ely. Finally, he abo e-men ioned s udy by Agge
e al. (2017) used he same echnique o measu e he angle be ween 𝐧
( a he han 𝐬) and he angen plane in excised pig hea s ixed a he
end-dias olic s a e. Thei angle his og ams show ha he angle alues
we e mos equen ly be ween 60◦and 90◦(c . hei Fig. 3) which is
consis en wi h he low end-dias olic alues (ci ed abo e) o he angle
be ween 𝐬and he angen ial plane.
All hese esul s sugges ha he answe o he ques ion o whe he
a biaxial es specimen is mo e likely o be spanned by local 𝐟and 𝐬
ec o s o by 𝐟and 𝐧 ec o s depends on he ac ual con igu a ion o
he myoca dial sample a he momen when he specimen is being cu
om i . In his ega d, i should be ecalled ha expe imen e s always
do wha hey can o keep hei issue samples in a non-con ac ed s a e
(see, e.g., he desc ip ion o he p epa a ion p ocess in Somme e al.,
2015). Consequen ly, he samples a e always in a elaxed s a e. I we
add o his he ac ha he myoca dium is ully con ac ed a he end
o sys ole, i ollows ha he issue samples o expe imen s should
be much close o he end-dias olic s a e o he myoca dium, which is
cha ac e ized by he angen ial o ien a ion o shee s a he han adial.
The same esul can be ob ained by compa ing he le en icle ee
wall hicknesses o he 28 hea s used by Somme e al. (2015) (c . hei
Table 1) wi h no mal a e age hickness alues a end-dias ole and end-
sys ole. Since he e is a signi ican co ela ion be ween wall hickness
changes and shee angle changes du ing sys ole (Nielles-Vallespin e al.,
2017), i is possible o in e he p edominan o ien a ion o shee s in
he samples om hei hicknesses. Acco ding o Peshock e al. (1989)
and Dawson e al. (2011), he mean end-dias olic ee wall hickness
in a no mal heal hy hea is app oxima ely 10 mm and he pe cen ual
sys olic wall hickening eaches 50 − 60%. These e e ence alues can be
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
5
J. Va e ka and J. Bu ša
compa ed wi h he mean wall hickness o he hea s used by Somme
e al. (2015), which is 12.5mm. I we add he ac ha he majo i y o
he pa ien s whose hea s we e es ed had some eco d o hea - ela ed
disease(s) (including in pa icula 5 hea s wi h diagnosed hype ophy
and wall hickness as high as 19 mm), i can be concluded ha he wall
samples we e in a s a e which was much close o he end-dias olic
con igu a ion o he en icle han o he end-sys olic one.
The abo e ac s and conside a ions lead us o belie e ha he s ess
esponses ob ained om biaxial es s ep esen he beha io o he 𝐟 𝐬
plane o myoca dium a he han he 𝐟 𝐧plane, o a leas ha he
esponses in he CFD a e indeed signi ican ly in luenced by he s i ness
o collagenous shee s. The e o e, we would ind i inapp op ia e o
neglec he mechanical con ibu ion o shee s du ing biaxial ex ension
by assuming ha hey a e pe pendicula o he CFD (and hus also
pe pendicula o he hea wall which, as he abo e- epo ed angle
measu emen s clea ly show, i ually ne e occu s in a heal hy hea ).
Fo hese easons, we decided o ma ch he biaxial esul s in he CFD
wi h he myoca dial 𝐬di ec ion and in he nex sec ion we will i bo h
𝜓 and 𝜓HO using his assump ion.
5. Fi ing o expe imen al da a
We u ilized bo h biaxial and simple shea da a in a single op i-
miza ion p oblem aimed a minimizing he objec i e unc ion de ined
as he sum o weigh ed squa es o di e ences be ween he measu ed
and model-p edic ed s ess alues. The s ess esponses o ou SEF 𝜓
we e calcula ed om he gene al s ess ela ion (29). The biaxial s ess
componen s co esponding o he applied s e ches 𝜆 and 𝜆sa e gi en
by
𝜎 = 2((𝜆2
−𝜆−2
𝜆−2
s)𝜓,𝐼1+𝜆2
𝜓,𝐼4 +𝜆2
𝜆2
s𝜓,𝐾1),(30)
𝜎s= 2((𝜆2
s−𝜆−2
𝜆−2
s)𝜓,𝐼1+𝜆2
𝜆2
s𝜓,𝐾1),(31)
while he shea esponses co esponding o he amoun o shea 𝛾a e
gi en by
( s) ∶𝜏= 2𝛾(𝜓,𝐼1+𝜓,𝐼4 ) +𝜓,𝐼8,(32)
( n) ∶𝜏= 2𝛾(𝜓,𝐼1+𝜓,𝐼4 +𝜓,𝐾1),(33)
(s ) ∶𝜏= 2𝛾 𝜓,𝐼1+𝜓,𝐼8,(34)
(sn) ∶𝜏= 2𝛾(𝜓,𝐼1+𝜓,𝐾1),(35)
(n ) ∶𝜏= 2𝛾 𝜓,𝐼1,(36)
(ns) ∶𝜏= 2𝛾 𝜓,𝐼1.(37)
Analogous exp essions o he HO model can be ound in he o iginal
pape (Holzap el and Ogden,2009).
F om each o he six a ailable simple shea esponses, we ex ac ed
𝑛p∶= 21 da a poin s, e enly spaced in he es ed ange [0,0.5] o he
amoun o shea . Thus, in gene al, o he 𝑗 h simple shea mode, whe e
𝑗∈ {1,…,6}, we ob ained a amily ((𝛾(𝑗)
𝑖, 𝜏(𝑗)
𝑖) ∣𝑖∈ {1,…, 𝑛p}) o
𝑛ppai s (𝛾(𝑗)
𝑖, 𝜏(𝑗)
𝑖), each consis ing o he amoun -o -shea alue, 𝛾(𝑗)
𝑖,
and he co esponding shea s ess alue, 𝜏(𝑗)
𝑖. (To a oid con usion, we
no e ha we use 𝑗∈ {1,…,6} me ely as a dis inguishing label, he
enume a ion o he shea modes may be a bi a y. No ice also ha
we use a supe posed ba o dis inguish expe imen al s esses om he
p edic ed ones.) Simila ly, o each o he i e a ailable biaxial es
eco ds, we ex ac ed 𝑛pda a poin s om he esponse in he MFD
and he same numbe o poin s om he esponse in he CFD (which
we iden i y wi h he 𝐬di ec ion). In his manne , o he 𝑘 h biaxial
es , wi h 𝑘∈ {1,…,5}, we ob ained a amily (((𝜀 )(𝑘)
𝑖,(𝜎 )(𝑘)
𝑖) ∣𝑖∈
{1,…, 𝑛p}) o 𝑛ppai s ((𝜀 )(𝑘)
𝑖,(𝜎 )(𝑘)
𝑖), each consis ing o he alue o
enginee ing s ain in he 𝐟di ec ion, (𝜀 )(𝑘)
𝑖, and he co esponding
alue o Cauchy s ess, (𝜎 )(𝑘)
𝑖( he o de o es s is again imma e ial).
Analogous da a se s o he o m (((𝜀s)(𝑘)
𝑖,(𝜎s)(𝑘)
𝑖) ∣𝑖∈ {1,…, 𝑛p}), o
𝑘∈ {1,…,5}, we e ob ained also o he 𝐬di ec ion.
The objec i e unc ion 𝛺 o be minimized wi h espec o he ec o
𝐩∶= (𝑎1, 𝑏1, 𝑎 , 𝑏 , 𝑎nn, 𝑏nn, 𝑎 s, 𝑏 s)o ma e ial pa ame e s in he SEF 𝜓
(Eq. (14)) was o mula ed as
𝛺(𝐩) ∶=
6
∑
𝑗=1 ‖√10
6(
𝝉(𝑗)−𝝉(𝑗)(𝐩))‖2+
5
∑
𝑘=1
(‖
𝝈(𝑘)
−𝝈(𝑘)
(𝐩)‖2+‖
𝝈(𝑘)
s−𝝈(𝑘)
s(𝐩)‖2),
(38)
whe e
𝝉(𝑗),
𝝈(𝑘)
and
𝝈(𝑘)
sa e ec o s con aining 𝑛pexpe imen al s ess
alues (e.g.,
𝝉(𝑗)= (𝜏(𝑗)
𝑖∣𝑖∈ {1,…, 𝑛p})), while 𝝉(𝑗)(𝐩),𝝈(𝑘)
(𝐩)and
𝝈(𝑘)
s(𝐩)con ain he co esponding model alues, calcula ed om (29).
We decided o mul iply he shea s ess e o s in (38) by he ac o o
√10
6(and hence he squa es o e o s by he ac o o 10
6) o elimina e
he ini ial bias a ising om he ac ha 𝛺(𝐩)is in luenced by only
6 simple shea da a se s bu , in e ec , by 10 biaxial da a se s (5 o
each loading di ec ion). The minimiza ion o 𝛺was ealized using he
Le enbe g–Ma qua d algo i hm.
The same minimiza ion p ocess was applied also o he HO model.
The s ess ela ion o he model, analogous o ou Eq. (29), can
be ound in he o iginal pape by Holzap el and Ogden (2009). The
objec i e unc ion 𝛺HO was de ined he same way as 𝛺in (38) and
i was minimized wi h espec o he amily 𝐩HO ∶= (𝑎1, 𝑏1, 𝑎 , 𝑏 ,
𝑎s, 𝑏s, 𝑎 s, 𝑏 s)o ma e ial pa ame e s occu ing in he SEF 𝜓HO (Eq. (8)).
The Le enbe g–Ma qua d algo i hm always con e ged owa d he
same minimizing solu ion, e en om s a ing poin s chosen a om
he minimum. Howe e , his solu ion con ained a nega i e alue o he
pa ame e 𝑏 s, as can be seen om he i s ow o Table 1. Nega i e
pa ame e s should gene ally no be accep ed (c . he end o Sec ion 2)
and he e o e we decided o modi y he de ining exp ession o he
objec i e unc ion 𝛺HO by adding o i a penal y e m
8
∑
𝑚=1
𝛾𝑚min{0,(𝐩HO)𝑚}2(39)
which, by means o he penal y coe icien s 𝛾𝑚>0, penalizes any
nega i e componen (𝐩HO)𝑚o he ec o o ma e ial pa ame e s. A e
his modi ica ion, he solu ion was epea ed and we ob ained a second
minimizing ec o ( he second ow o Table 1) in which, howe e ,
𝑎 s and 𝑏 s we e bo h ze o which means ha he 𝐼8- e m in 𝜓HO was
inac i a ed by he cons ain s imposed by (39). The s ess esponses
co esponding o bo h calcula ed pa ame e ec o s a e plo ed in
Fig. 4agains he expe imen al da a. Fo each cu e in he igu e,
we calcula ed he coe icien o de e mina ion 𝑅2; he ob ained alues
a e lis ed in Table 2 oge he wi h he mean 𝑅2 o each minimizing
solu ion. In addi ion, he las column o Table 1shows he minima
𝛺HO(𝐩HO)a ained by he HO model in bo h analyses.
Wi h ou modi ied e sion 𝜓 he algo i hm did no gene a e any
nega i e pa ame e s, as can be seen om he calcula ed alues in
he hi d ow o Table 1. Howe e , i is also desi able o p o ide a
compa ison be ween he cons ained HO model in Table 1and ou
modi ied SEF wi h in a ian 𝐾1. Fo his eason, we decided o pe o m
ano he solu ion in which we excluded he e m wi h in a ian 𝐼8 om
𝜓, hus c ea ing a six-pa ame e model compa able o he cons ained
HO model in Table 1. Minimiza ion o he objec i e unc ion (38) wi h
espec o he educed pa ame e ec o 𝐩∶= (𝑎1, 𝑏1, 𝑎 , 𝑏 , 𝑎nn, 𝑏nn) hen
p oduced ma e ial pa ame e s which a e lis ed in he ou h ow o
Table 1. The s ess esponses ob ained by bo h i s o 𝜓 a e shown in
Fig. 5. The 𝑅2 alues o indi idual esponses a e gi en in Table 2and
he weigh ed e o s 𝛺(𝐩)a e in he las column o Table 1.
I can be seen om Fig. 5 ha ou modi ied model 𝜓 desc ibes
all he expe imen al se s e y well. This is also e lec ed in he alue
o he minimum 𝛺(𝐩)in Table 1which is abou 80% lowe han he
co esponding minimum 𝛺HO(𝐩HO)a ained by he o iginal model 𝜓HO.
The 𝑅2 alues in Table 2show ha he HO model has a good o e lap
wi h he simple shea da a. I is also possible o achie e an accu a e
desc ip ion o biaxial da a by assigning su icien ly high weigh s o
biaxial esiduals in 𝛺HO, bu hen he shea esponses o 𝜓HO become
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
6
J. Va e ka and J. Bu ša
Table 1
Es ima ed pa ame e s o he s ain–ene gy unc ion 𝜓HO p oposed by Holzap el and Ogden (2009) and o he modi ied unc ion 𝜓 p oposed in his pape . The a ained minima o
objec i e unc ions a e deno ed as WSSR (weigh ed sum o squa ed esiduals). Row 1: Uncons ained op imiza ion wi h 𝜓HO led o 𝑏 s <0. Row 2: Cons ained analysis p oduced
𝑎 s =𝑏 s = 0which, in e ec , made 𝜓HO independen o 𝐼8. Row 3: Ou model p oduced a much lowe WSSR han 𝜓HO. Row 4: The 𝐼8-dependen e m was emo ed om 𝜓 o
make i compa able wi h he cons ained 𝜓HO ( ow 2); he WSSR is again lowe o 𝜓.
𝑎1𝑏1𝑎 𝑏 𝑎s𝑏s𝑎nn 𝑏nn 𝑎 s 𝑏 s WSSR
(kPa) (kPa) (kPa) (kPa) (kPa) (kPa2)
𝜓HO 1.287 4.888 2.498 30.266 1.067 28.749 – – 0.615 −14.617 49.094
𝜓HO cons ained 1.332 4.760 2.462 30.489 1.031 29.272 – – 0.000 0.000 49.214
𝜓 0.907 7.347 1.298 34.348 – – 0.335 11.111 0.705 0.899 7.520
𝜓 wi hou 𝐼81.074 6.846 1.291 34.962 – – 0.276 11.896 – – 10.124
Table 2
𝑅2 alues o all cu es displayed in Figs. 4and 5. The i s i e columns, which cha ac e ize biaxial es s, con ain wo alues in each cell, whe e he uppe ep esen s he MFD
and he lowe he CFD.
0.5:1 0.75:1 1:1 1:0.75 1:0.5 ( s) ( n) (s ) (sn) (n ) (ns) Mean
𝜓HO 0.845 0.881 0.872 0.960 0.869 0.988 0.972 0.983 0.977 0.967 0.954 𝟎.𝟖𝟕𝟕
0.583 0.910 0.794 0.774 0.710
𝜓HO cons ained 0.843 0.881 0.872 0.959 0.869 0.985 0.972 0.987 0.975 0.968 0.954 𝟎.𝟖𝟕𝟔
0.582 0.909 0.794 0.775 0.715
𝜓 0.924 0.972 0.993 0.990 0.981 0.994 0.984 0.985 0.985 0.993 0.997 𝟎.𝟗𝟕𝟖
0.944 0.988 0.983 0.986 0.952
𝜓 wi hou 𝐼80.922 0.972 0.993 0.989 0.980 0.973 0.985 0.981 0.991 0.976 0.978 𝟎.𝟗𝟕𝟑
0.940 0.985 0.982 0.984 0.944
a wo se han hose shown in Fig. 4and he ma e ial pa ame e s so
ob ained canno be used o modeling o shea beha io . The capabili y
o he HO model o desc ibe bo h biaxial and simple shea da a wi h
one se o pa ame e s was es ed ea lie by Gül ekin e al. (2016), Guan
e al. (2019) and Ma ono á e al. (2024) bu he ob ained i s we e also
no en i ely sa is ac o y. Howe e , his inaccu acy o he HO model
should no be exagge a ed because i could be a consequence o he
di e ences in he p ope ies o he biaxial and shea specimens. These
di e ences may be signi ican despi e he ac ha Somme e al. (2015)
educed hem by cu ing he biaxial and shea specimens om adjacen
egions o he hea wall.
Guan e al. (2019) and Ma ono á e al. (2024) also in es iga ed
se e al modi ied e sions o he HO model wi h imp o ed i ing capa-
bili ies bu none o hem eached he le el o accu acy compa able o
ou model. In pa icula , Ma ono á e al. (2024) i ed he same da a
as we did using modi ied models wi h 14, 7 and 5 pa ame e s, and
he esul ing i s we e cha ac e ized by he mean 𝑅2o 0.876, 0.894
and 0.872, espec i ely. Ou model ou pe o ms hese models because
i p oduces 𝑅2= 0.978 wi h 8 pa ame e s and 𝑅2= 0.973 wi h only 6
pa ame e s.
6. Discussion and conclusions
The modi ied model p oposed in his pape is, like he o iginal HO
model, inspi ed by he ue myoca dial a chi ec u e as desc ibed in
se e al his ological s udies, some o which we e men ioned in Sec-
ion 2. We wan ed o a oid making any excessi e changes in he
o iginal o mula ion, hus we p ese ed i s cha ac e is ic addi i ely
decoupled s uc u e composed o se e al exponen ial e ms, each o
which ep esen s a po ion o he cons i uen s o he myoca dium.
We also main ained he exponen ial e m wi h coupling in a ian 𝐼8,
e en hough i s omission o eplacemen by a simple e m (wi h only
one pa ame e ) migh be app op ia e, a leas o he human da a
om Somme e al. (2015) ha we used o i he model pa ame e s.
The dis inc i e cha ac e is ic o ou app oach is ha we u ilized he
aniso opic in a ian 𝐾1(whose po en ial applicabili y in biomechanics
was sugges ed ea lie by o he au ho s) o desc ibe he mechanics o
plana myoca dial shee s which he o me au ho s desc ibed pa ly
h ough in a ian 𝐼4 , and addi ionally by in a ian 𝐼4s. Al hough ou
modi ica ion consis s me ely in subs i u ing 𝐼4s by 𝐾1in he SEF 𝜓HO,
he esul s we p esen in Sec ion 5p o e ha he e ec o such a small
change is subs an ial. I is impo an o no e ha his imp o emen was
achie ed wi hou inc easing he numbe o ma e ial pa ame e s o he
model.
Un o una ely, he e is some con o e sy ega ding he in e p e a-
ion o he CFD esul s in e ms o he p e e ed ma e ial di ec ions.
In some o he p e ious pape s he CFD was iden i ied wi h he 𝐬
di ec ion (e.g., Holzap el and Ogden,2009;McE oy e al.,2018), while
in some o he s wi h he 𝐧di ec ion (e.g., Gül ekin e al.,2016;Guan
e al.,2019). F om ou expe ience he e is usually no in-dep h discus-
sion on his issue in he esea ch pape s. Usually, he au ho s seem o
a bi a ily choose one o he wo op ions wi hou e en men ioning he
unce ain y su ounding his opic, which is e iden om he incon-
clusi e in o ma ion and con adic o y s a emen s ha can be ound in
he li e a u e. O cou se, unde such condi ions, bo h iewpoin s can be
de ended by ci ing a ew suppo i e li e a u e sou ces. We sea ched he
li e a u e and collec ed a bulk o ele an in o ma ion on his ma e in
Sec ion 4. Based on his in o ma ion, we ound i mo e app op ia e o
assign he CFD esul s o he 𝐬di ec ion. This choice has he addi ional
ad an age ha i enables o ully exploi he i ing po en ial o bo h
in es iga ed SEFs, 𝜓HO and 𝜓. I , ins ead, we associa ed he CFD wi h
he shee -no mal di ec ion 𝐧, hen he e ms wi h 𝐼4s and 𝐾1in 𝜓HO
and 𝜓, espec i ely, would be inac i a ed in all conside ed biaxial es s
because simul aneous ex ension in 𝐟and 𝐧di ec ions leads o 𝐼4s <1
and 𝐾1<1. The capabili y o hese e ms o imp o e he quali y o
i s would hus be conside ably limi ed. One can ind he i s o he
HO model ob ained unde such limi ing condi ions, e.g., in he wo k
o Guan e al. (2019). In o de o inc ease he quali y o i s, he au ho s
p oposed a modi ica ion consis ing o a eplacemen o he in a ian
𝐼4s by 𝐼4n. This change, howe e , makes he biaxial esponse o he
esul ing model in he 𝐟 𝐧plane iden ical o he biaxial esponse o
he o iginal HO model (used by us) in he 𝐟 𝐬plane. The only s ain–
ene gy e m capable o p oducing any di e ence be ween he models is
he e o e he 𝐼8- e m whose impac , howe e , is gene ally e y mino ,
as can be seen om ou Figs. 4and 5. This small di e ence be ween
he wo models is pe haps bes demons a ed by he ac ha he mean
𝑅2o 0.877 ha we eached wi h he o iginal HO model (see Table 2)
is almos he same as he alue o 0.867 ha Ma ono á e al. (2024)
ob ained when hey i ed he abo e-men ioned model o Guan e al.
(2019) o he same da a ha we used. Thus, bo h models a e simila ,
excep ha he modi ied e sion wi h 𝐼4n exhibi s s i e beha io
along he 𝐧di ec ion han along he 𝐬di ec ion, which is un ealis ic
conside ing he s uc u al o ganiza ion o myoca dium shown in Fig. 1.
I , on he o he hand, we admi ha he con ibu ion o shee s o he
s i ness in he CFD is signi ican , we can ob ain he same esul s using
he o iginal HO model which also co ec ly assumes ha he s i ness
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
7
J. Va e ka and J. Bu ša
Fig. 4. Expe imen al da a o Somme e al. (2015) i ed by he HO model (Eq. (8)). An uncons ained p oblem was sol ed i s , ollowed by a cons ained p oblem allowing
only non-nega i e pa ame e alues. The op i e panels con ain expe imen al esponses o biaxial ex ension in he mean- ibe di ec ion (MFD) and he c oss- ibe di ec ion (CFD),
and he co esponding model esponses in he ibe di ec ion 𝐟and he shee di ec ion 𝐬. The i le o each g aph speci ies he applied MFD o CFD s ain a io. The bo om six
panels con ain expe imen al and model esponses o he six simple shea modes speci ied in he i les.
in ension is highes along 𝐟, medium along 𝐬and lowes along 𝐧. This
beha io o he myoca dium is also consis en wi h ou model which,
howe e , achie es be e i ing esul s han he HO model wi h he
same numbe o ma e ial pa ame e s. We also wan o no e ha anyone
who eels ha he CFD should a he be iden i ied wi h he 𝐧di ec ion
(and also does no insis on ha ing 𝐬di ec ion s i e han 𝐧) can ollow
he app oach o Guan e al. (2019) and simply de ine 𝐾1in 𝜓 using 𝐬
ins ead o 𝐧, he eby ein o cing (again) he CFD. The impo an hing
is ha one ob ains much be e biaxial esponses wi h he combina ion
o 𝐼4and 𝐾1 han wi h he combina ion o wo in a ian s 𝐼4(used
by Holzap el and Ogden,2009;Guan e al.,2019). This unde lines
he impo ance o ou wo k because, o he bes o ou knowledge,
he in a ian 𝐾1has no been used so a in any cons i u i e equa ion
o myoca dium, which is su p ising conside ing i s g ea po en ial
demons a ed by ou esul s. On he o he hand, i mus be admi ed
ha he e is cu en ly no expe imen al e idence ha he combina ion
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
8
J. Va e ka and J. Bu ša
Fig. 5. Expe imen al da a o Somme e al. (2015) i ed by he modi ied model 𝜓 p oposed in his pape and by i s educed e sion ob ained by excluding in a ian 𝐼8 om
(14). The op i e panels con ain expe imen al esponses o biaxial ex ension in he mean- ibe di ec ion (MFD) and he c oss- ibe di ec ion (CFD), and he co esponding model
esponses in he ibe di ec ion 𝐟and he shee di ec ion 𝐬. The i le o each g aph speci ies he applied MFD o CFD s ain a io. The bo om six panels con ain expe imen al and
model esponses o he six simple shea modes speci ied in he i les.
o 𝐼4 and 𝐾1 e lec s he ue s uc u e o en icula myoca dium
be e han he adi ional app oach.
Ano he aspec ha dese es u he discussion is ou choice o
shea da a wi h a maximum applied shea o 0.5. Rega ding he biaxial
es s, Somme e al. (2015) p o ide comple e esul s only o he max-
imum applied s e ch o 1.1, bu hey show ou di e en collec ions
o simple shea esponses, ob ained o he subsequen ly applied shea
le els o 0.2, 0.3, 0.4 and 0.5. The eason o his mul is ep loading was
he p onounced s ain-induced so ening (Mullins e ec ) exhibi ed by
he samples in bo h biaxial and simple shea es s. This means ha each
inc ease in load le el was associa ed wi h an i e e sible dec ease in
s i ness. In o he wo ds, di e en mechanical esponses we e ob ained
o di e en shea le els, which ob iously complica es i ing by a
hype elas ic model because i is no clea which shea le el should
be combined wi h a gi en se o biaxial da a. I would be easonable
o use ha shea le el in which he cons i uen s mos esponsible
Eu opean Jou nal o Mechanics / A Solids 111 (2025) 105586
9
