A n a ly s is o f + 6 ': A c c e le r a t io n In d u c e d S t r e s s e s
in t h e H u m a n V e n tr ic le M y o c a r d iu m
by
James Ernest Moor-.
M .E n g ., C arleton U n iv e rs ity , 1985 B .E ng.. C a rle ton U n iv e rs ity , 1985
A D isse rta tion S u b m itte d in P a rtia l F ulfillm ent, o f the R equirem ents fo r tlr * Degree o f
D O C T O R O F P H IL O S O P H Y
in th e D e o a rtm e n t o f M echanical E ng in e ering
We accept th is d isse rtatio n as co n fo rm in g to the* re quired standard
D r. J . f l H addow , S upervisor (M echanical E ng in e ering )
D r. J .W t Provan, M e m b e r (M echanical E n g in e ering )
D r. ( l. W V ickers. M e m b e r (M echanical E n g in e e rin g )
D r. R. Miner. O utsid e M e m b e r (M a th e m a tic s )
D r. R A Y . Ogden, E x te rn a l E x a m in e r (M a th e m a tic s ) U n iv e rs ity o f Clasgow
© James Ernest M oore, 1991 U n iv e rs ity o f V ic to ria
A l l riyhts n s t r r t d . D/.sst rt at ion may not Ix n p r o d i u i d in t hole o r in part, by photocopyiny o r ntht r an aim iri ihout t in p tr mi sa io n c f t in author.
I l a m e ” • L - ’ v ‘ , * • . Iv ^ _ A l j - . h r j r f', / r i f e r r i a / i o n o r a r r a n g e d b y b r o a d , g e n e r a l s u b j e t . c a t e g o r i e s . P l e a s e s e l e c t t h e o n e s u b j e c t w h i c h m o s t n e a r l y d e v r i b e s t h e ' o n t ' - e t o f y o u r d i s s e r t a t i o n E n t e r t h e c o r r e s p o n d i n g f o u r d i g i t c o d e i n t h e s p a c e s p r o v i d e d I r ■- v v ' n - , t SUBJECT TERM * /
SUBJECT CODE
UMI
Subject Categories
THE HUMANITIES AND SOCIAL SCIENCES
C O M M U N IC A T IO N S A N D THE ARTS All t l:.|. y ' n i f ‘i n ’ 1 I MU' •• I Ml. . A t V. l t d ' Mt .1 i t ’ . M l ]■ r . j t n o l r . m M r l " , '! r,M M ED U C A TIO N A t Itr III II *> ■ it- A.lull .u,-i < •• A l J I 11 .,|t. Mill A,t M -. s I l*ii' ci n ' * t .tl It t n II lit . ' ‘ 1 it I Ii n il III. r ir l n . l v . 1111> j l ■■ 11 I It-i nt i it, it , j I r i ‘ 11 ■ ■ I • I i t <r JI ' .uuf, Hr- It I it Ml. Hm.MI .1 ' I . I n* ji a ft ti tl M u t l . M . i - . t U n i,. u | i u n | . . J /7 v ■ , . / / - ' / ! / , I 1.1 i M . 1 t'--/. I,'" !- ’. ) / f r . ’l l ' l t l i q j*.,.;!!)!..!!', V r i yt i' i. ir y l - p / f.t f t T ftin iiiis i 7 * 0 .... Ill It < Ml I ..I.tt.M-.tr,.'.. M . , K . .v mI M . 7 - r r . A f r 1; ■ Hi An n 'i a mi t n l i b 111 (f hi l i t '/ ;■ ld l 'I 'l (f M-ni I 1 111 •. Att 11 ■ f ; 1 , jrt A V . i J l r f 1 J‘,t. -t 11 0:7 S 0 7 U u iy n 7V14 (■>440 U 'jJ V 0 / 1 0 O '/H H 7 ’ 4 / LA NG UA G E , LITERATURE A N D LIN G U ISTIC S (V . .'v ( tJHV <•7 VO 0 7 * 0 ( M M o ; v 4 n w ; (.7 7 .’ 0.4 1 8 O .YO o li.'i4) 0 4 . 7 0 7 7 o .V '4 i ) -i 1 104 !
( )4 ! S l) 7 4, P H IL O S O P H Y , R E L IG IO N A N D T H EO LO G Y ,j,o n ( 1 f - n n n l BitJr ; 1 \ r u t f hat-Ty V P t „ l r , - . ' I , : , ■ o l o i i y Tl,, S O C IA L SCIENCES A t T l t t r t C MP mM Jm ■ A n t f i r c . p ' / u ' j y A , , l , . „ , i , „ C_t Ju ui mI f ’ t i VMC Ml B-jmpcs'. ( jf-nf.T mI A ' . . . H i . j , i t i n M [ V j n k ' t u j Mm, k i M f M P f i d M i . l i k . 4 t n ' ! < . C l P M t i l M P ‘ j ti 1 J i t ■ f t - J POt TM. S ' jf.n.TMl A - j n - ' u i t u m l ( 1 i i r i t p f p » fv, • . ' i n ' F ifn 1,p ... ! 11 s t r, | v' l u U / TllCMP. r.fjP.lf I O l l t r J1 Hi Mm M i n i I M' .t f 1 i f >t p t n i n 0 4 1 4 ( ; ' i ] H 7 7 1 04 IV ' v7'0 0 : 7 7 0 4 8 7 O 'C'ft U 4 1 0 0 /7 U - J O O J 4 8 O Ji.j 1 0 .0 7 i 7 0 v '. 7 10 Tim U J Wr MpJm'v mI VmJE.-r, ; K'mO A f r I f ] J, Amm A '.'-.tifiV j ' '■ . illiM Jn ?'■ I If ■ d it: J> ■ ! I. ] t l : I A r n r - I !■ Ml I M . J J n - I ' M' , r--M' . U , : l t f J mt: l bltf 7
( , , -rJ.»j , Ijr’f ) t i - J I mv k / ' l . i t .m i' 0.7:,. A ir • . •• -.'M-nti-in fn-|! W .-l I. r - " j i , M l y I. n-M )'H Mf d' r t t h n ; . - t n J Pm.. IM! Mi. i 111J 1 . 1M M ! i . r v . l 0 i n : , ! r Of.,.. lie-" InJt.'Mnn! .” Mi L/jKc -y . - l ' l*,r Jtv. f ' u M u - :r : . I !>c-c m l W > . ■Mo,;irji bt-t" Mil.' ;; 1■ 1 * 1 f M' srt . - Jf : I M t. - tt . I.-]-' r-.[ i ■ frtfi.... "Mil M,|J pf jjion I1 r'fli.THE SCIENCES A N D EN G IN E ER IN G
B IO L i G IC A l SCIENCES > , I. v Ami , '.n . 1' V Ar nit n il ' , 4( 1. s) 11. U i - t : l i a r f w , ( M , n - M ■ - »,) H ,,!" fl. ,.,t t U K, :: It Jl* \V 1 K h |: \ N . , . i 1 1 - t , - . . . . 'm. A- - .M . -, X lA-.hM-.M,. . H, d. -v . M . .-!■ • '• t' M, A'. : N .V '.' <"1-M !' H i','. -. . W’M ’ EARTH SCIENCES .1 'P 1" . ' )*. 1 v M s - I n A , |V M m > 1 il. \ \ > U M.t n -tl III » f 'mIp- >■" ■ A '*!' f MIC, i n .!■ >■ 1 ‘ ' S ! ' ‘I- *- J . >M r :i Jiv | * 'Ml 1, - M i M f ' P v ■ HMH 041 i 0 (4.4 0 4 . 7 1 M) H i t V H * - D41 ' 0 4 * .,-s 0 4 ’ ' H EALTH A N D E N V IR O N M E N T A L SCIENCES A -.J -.Aimv . f l C. M, f f l . u O i i M l t i . n H.' Mmi|M.)i',' hrn- I 11 , ^ !JV M r-J . m mJ !nl' W - M i H i ' . i i r 1- r^,- . n.; , ‘ V ’ V Mi' . 1 hi'iTlttl M'hf t \lt ’\ ’\\) v I >\ jr i " . \ f ' * \ r m m. v k‘ * " . . t ■ T i , f i . i p v t’.M'. y Mt\H ^ Km.*v' . ‘.jv Kt-v • ’440 I'A V < ivV J y f ' f ' - ' 1 - < 4 7 l 7^ t' ’ 4h .K 'H J1
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0 4 4 ’ Or>r.y 0 4 4 4 0 7 i c ^ 0 0 4 1 ^ .it .ft', t- f'jt: : M, Tu«;,.M|V H . M Pi. f . ■. . f V . M i i ‘ P H Y S IC A L SCIENCES P u r e S c ie n c e s 0 7 . I P , ' . t i y A n d i v t i i ,ii b 1 ->< fii'Pi-‘.t, *■ I n , * mmmm N u r l r . l , L. )( ij.Hlk f'lli K Ml( |( EM tl- -ill T’f i y siv'ui 0 t i y n n ' t K.uj:;Ttiori ' itflOTlHltlC s OtP-VlLS CwfH'KlI Ac m; ;s*iC s Asti..,nomv cinti AvtfolJ'ysi<:> A tiPost” ip rk' Sc ii'th t> A\tc),Pk[ IfOntnic s cTH J C liYtr i.Mtv F !t‘iT'rfltctr> Ptlftn.lns vj. nf H ujfi tni'ujy f b i - d ; t n j P l u 5. r ’ i i i M cJlk CUM, Nuc ! t \r OptkS KouMt'on Solui Statr 7dt.ct;vs A p p l ie d S c ie n c e s AppiirJ M t * . . J i i m c s s O'tUH'tt'f Ss‘ t'lVt* 04M S 1 ,'M v 0 4 8 7 0 4 8 -'' - ,488 | 0 ’ 4H 0 4 0 0 0 4 v ! 0 4 V 4 0 4 ^ 5 O ' 7 * 4 0404 oofifr 17 0 '* OcOvS 0 7 5 8 ' . 7 0 4 0 UPr.O i 7 10 0 ' 7 ; 0 " . 7 O M 1 0 4 f t 4 044c* A , A. 81 1. Mf'Pii. . 11 t I y ji f" 't'( t' , 11 H , - a 7 T 7 H .O in O i,- l n , I n 1 tr •• li M . - t t i t : . - M - i t * 111. <: ■ S c ; Mt i h u n p . a l . M . v t d l l u ' d v M i n u u t N u d i - . ir kcKki.ici-ftc: P f ’ tf o l f ' i - r f t S c U i ' t a r y ; i n O S v h i P M , I vJ'O es up01 -*cj> ' .‘ f t n r . ' l t i v VT S R * ' . , f Fisr.hv-s Tn.-fneJ. ■ !r*t,|n Tc-i:t1 •,1 \ P S Y C H O L O G Y L.?t*ru'rcil b o h c T v k i ’ Cll Oimisijl P ‘'v i'lop rn *'n f7 E^|H‘ri,TH‘,PTcl! Inuustr cj! Forsoiuikty PHvsioli'qidM' Fsvcf-.obioiouv L yvchMfPOtric McM1
®
S u p e rv is o r: Dr. J.B. Haddow ii
A b str a c t
Il is well know n th a t physiological problem s occur when p ilo ts are su b je c te d to m o d e ra te -to -h ig h + ( r z accelerations. T h e goal o f th is research is to
develop a q u a n tita tiv e m odel th a t provides a d d itio n a l in s ig h t in to th e adverse effects o f - f ( ! - accelerations 011 a p ilo t ’s cardiovascular system . T h e m e th o d o f in v e s tig a tio n is in co n tra st to previous studies, w h ich have m a in ly re lie d on e x p e rim e n ta l techniques.
T h is w ork focuses on the developm ent o f a th re e -d im e n sio n a l fin ite clem ent m odel to analyse + (7 . induced stresses in th e h um an le ft and rig h t v e n tric le s . T h e c o m p u ta tio n a l m odel is based on n o n -lin e a r c o n tin u u m th e o ry , where th e effects o f fin ite d e fo rm a tio n , irre g u la r shape o f th e h ea rt, and (n e a rly ) in c o m pressible b e h a vio u r o f m y o c a rd iu m tissue are take n in to account. T h e fin ite elem ent fo rm u la t ion is developed using th e G a le rk in weight ed re sid u a l m e th o d w ith a ]>( nalty tre a tm e n t o f th e in c o m p re s s ib ility c o n d itio n . In th is stud y, an e x p o n e n tia l ty p e stra in energy fu n c tio n is used to m odel th e ca rd iac tissue.
T h is tech n iq ue provides a new perspective fo r th e m echanical s tu d y o f +(7~ acceleration 011 the hum an h ea rt. Results presented d e m o n stra te th e a b ility o f tin* fin ite elem ent m odel to p ro v id e q u a n tita tiv e d a ta on th e effects o f g ra v ita tio n a l lo a d in g 011 th e cardiovascular system . T h e analysis p re d ic ts gross d is to rtio n and stress d a ta fo r th e hum an h eart u nd e r sustained exposure to in e rtia loading up to + 5 (7,.
E x a m in e rs :
D r. J .B . H addow . S upervisor al E ngineering)
D r. J .W . P rovan, M e m b e r (M ech a nica l E ngineering)
D r. G .W . V icke rs, M e m b e r (M ech a nica l Engine/ rin g )
D r. H. Illn e r, O u ts id e M e m lx r (M a th e n ia t ics)
D r. R .W . O gden, E x te rn a l E xa m in e r (M a th e m a tic s ) U n iv e rs ity o f Glasgow
A c k n o w le d g e m e n ts
I w ould like to express m y sincere g ra titu d e to m y supe rviso r, Professor J .B . H addow . for his valuable guidance and assistance in th is endeavour.
I also wish to acknowledge' M r. vV. Fraser fo r suggesting th e research to p ic and M r. M in li L y for his assistance and te ch n ica l su p p o rt.
This research was p a rtia lly funded by D r. B. T a b a rro k in c o n ju n c tio n w ith th e Defence and C iv il In s titu te for E n v iro n m e n ta l M e d ic in e ( I ) C IE M ) u n d e r ( Ira n i No. W771 l-D -7077/01.
C o n te n ts
A b s tra c t ii A c k n o w le d g e m e n ts iv T a b le o f C o n te n ts v L is t o f F ig u re s v ii 1 In tr o d u c tio n I 2 T h e o r e tj • 1 F o rm u la tio n and D e v e lo p m e n t 8 2.1 I n t r o d u c t i o n ... S 2.2 K in e m a lic s o f D e fo rm a tio n ... 10 2.2 S tra in M e a s u re s ... I I 2.1 B alance Laws 2.1.1 C onservation o f M a s s ... IL 2.1.2 C onservation o f M om ent u n i ... Hi I <1 2.5 V a ria tio n a l (W eak) Form o f Linear M o m e n tu m Balance . . . 2.G C o n sisten t L in e a riz a tio n o f th e Balance o f M o m e n lu m ...212.7 A C o n s titu tiv e R e la tio n s h ip for C a rd ia c Tissue... ... 21
2.8 Stress and S tra in R a t.e s ... 28
2.0 In c re m e n ta l C o n s litu t ive R e la tion for C ardiac 1 is s u e ...21
2.10 S u m m a ry o f the In c re m e n ta l Equal i o n s ... 25
3 N o n - L in e a r F in ite E le m e n t F o rm u la tio n 37 2.1 I n t r o d u c t i o n ... 27
2.2 Space T im e D is c r e t iz a t io n ... 28
2.2 C o v e rn in g F in ite Elem ent E quations ... 20 2.1 A n In c re m e n ta l S olution P r o c e d u r e ... T l
4 H u m a n H e a rt: M o d e llin g Aspects 4P
1.1 Stages ol D e v e lo p m e n t... 16
1.1.1 M H I i m a g i n g ... 47
1.1.2 I'M Disced izat ion of t lie Vent ■ride M y o c a rd iu m . . . . 47
1.2 A ( '( >nsi it uf iv<’ lic la l ion ... 51
1.2.1 M y o c a r d iu m ... 51
1.2.2 P e r ic a r d iu m ... 52
5 A n aly sis S o ftw are and V e rific a tio n 54 5 1 Soft w a r - 'M o d u le 's ... 51
5.2 C o m p u ta tio n a l V e rific a tio n and T e s t in g ... 57
5.2.1 L inear E x a m p l e s ... 59
5.2.2 N o n -lin e a r (reomet r i r E x a m p le s ... 59
5.2 5 N o n -lin e a r ( le o m e tric and M a te u a l E x a m p l e s ...62
6 H u m a n H e a r t: A n alysis and Discussion 71 7 Conclusions and F u tu re W o r k 78 R eferences 80 A L in e a riz a tio n T h e o ry 90 I I E le m e n t M a tric e s 91 11.1 Int ro d in t i o n ... 91
B.2 A 5 1) Iso p a ra m e tlie Solid E le m e n t ... 92
11.2.1 I n t r o d u c t io n ... 92
11.2.2 (ie o m e try / K in e m a tic D e s c r ip t io n ...92
B .2.4 F in ite E lem ent M a t r i c e s ... 95
B .2 .1 E xte rn a l Force V ecto r . . . . 97
11.4 A .4 I) C o n tin u u m Based Shell E lem ent... 99
11.4.1 I n t r o d u c t io n ... 99
1121.2 (Ie o m e try / K in e m a tic D e s c r i p t i o n ... 100
11.525 F in ite r o t a t io n s ... 106
11.4.1 Lam ina Stress and S tra in ...110
11.4.5 F in ite E lem ent M a t r i c e s ...111
L ist o f F ig u res
1.1 A section a l view o f the heart w a l l ...
2.1 R e fe re n tia l and sp atia l co nfig u ra tion s o f a body
1.1 M R I o f th e hum an h e a r t . ... •1.2 Im a g in g p l a i n ' s ... 1.3 F in ite elem ent d is c re tiz a tio n ol the heart m odel
m l F F G ra ph ics P ro c e s s o r...
5.2 A n eng in e ering P atch test ... 59 5.3 B e n d in g o f a c a n tile v e r heam ...
5.1 Post B u cklin g o f a c irc u la r arch s t r u c t u r e ... 5.5 F q u i-b ia x ia l response o f m y o c a r d iu m ... 5.0 F F mesh o f a th ic k w a lk'd c y lin d e r ... 5.7 A th ic k c y lin d e r under in te rn a l p re s s u re ... 5.5 F F mesh o f a th in sphere ... 5.9 In fla tio n response o f a th in s p h e r e ...
0.1 S olid m odel o f tin* hum an h e a r t ... 0.2 D e fo rm a tio n profiles o f the heart ... 0.3 P re d ic te d stress v a ria tio n along the FV endocardium 0.1 P re d ic te d stress d is trib u tio n for the heart at f5 <!
B . l A 3-1) c o n tin u u m based solid e le m e n t... B .2 A 3 I) c o n tin u u m based shell e le m e n t... B .3 F in ite ro ta tio n s in 3-1) s p a c e ...
C h a p te r 1
I n tr o d u c tio n
T h e ongoing developm ent o f high p erform ance a irc ra ft capable o f p r n id i n g
s iilis ta u ti« l p o s itiv e accelerations ( + ( / - ) has created th e need fo r a b e tte r
u n d e rs ta n d in g o f th e p hysiological responses th a t can affect, a p ilo t's ju d g
m e n t. T h e te rm (positive* (1) c o m m o n ly used to denote in e rtia forces
th a t act along the v e rtic a l axis u f an u p rig h t body. E ffe c tiv e ly , these* feirces
cause the* heart and either beiely p arts to elisplace elowmvarels, inelucing an
e levnled stress state*.
Kxpe’rim e n la l inve*stigatiem has he*e*n the* p iin c ip a l nie*thoel use*el to s tu d y
the* physieilogical c!iange-s asse>ciate*el w ith a p ilo t's e*xpe>sure> te) mode'rate* to
higl f t / , aece'h'rations. The*se* expe'rime’id s a.*e designed in part to s im u la te
in High! acce*le*rat ie>n cemditieuis th a t can occur d u rin g aerial cennbat m a-
neie'uvre's o r e*me*rgency situatienis. Frenn e*xpe*riine uts eonduete'el in a .safe
envirem m ent using a e*e*nt rifu g e , th e ave*rage b la c k o u t le*ve*l eihserveel fo r an
unpro'.t cUd ineli vielnal is betwe'on T o to 1.0 (7- . W h e n litte'el w ith an a n ti-G
suit and using s tra in in g manenuivre's (m uscle tea sin g w ith e*emtredle*el b re a th
in g ) a pile>t ra n often a tta in f ! ) ( / . few a slm rt duratiem [n, 10, l l j . In fu tu re ,
1. Jntnuiurijon ‘2
a m i inzioased perform ance ol newer lig h te r a in i'a h , com bat m a n o e u v rin g in
excess i)l’ + 9 ( j - m ay he a tta in a b le w ith o u t (1 .O l (loss o f > a ■nsciousuees i
fT, ■)!). 971.
P h ysiolo g ica l responses 1o i ' r . a c v lc t . i t ion range from tin* less sever''
te m p o ra ry loss o f p e rip h e ra l vision to unconsciousness and. in very severe
cases, possible damage to heart tissue. For safely reasons, there is p a rtic u la r
in terest in th e effects o f f d b in e rtia forces on the cardiovas; u lar s y t<mi (see
[ti, 17, ho]). C onsequently, e i.v a te d s* r< ss le v Is in I lie vascular due to high
+ ( > - lo a d in g have a ttra c te d the a tte n tio n o f researchers (eg. j"> 1. n o . S i j ) .
In such research, high - K / , loading has Pi en associated it h a lm o rm a lit tes
o f th e e le c tro c a rd io g ra m in man [ o l], as well as suh endocardial haem orrhage
and p a th o lo g ic a l changes o f the m yo ca rd ial tissue in anim als l"> [. T hough
most researchers have assumed that iscuem ia is the cause ol tissue dam age
in a n im a ls sul ject to high +C b lo a din g co n d itio n s js7|. detailed o nth olog ica l
e x a m in a tio n o f sw ine has in d icate d te a rin g o f the heart libres m l her tln m
dam age consistent, w ith a h ypo xic o r ischem ic in s u lt. It r probable that
th e observed dam age is due to th e high stresses and st rains re s u ltin g b o m
a c o m b in a tio n o f i) high +C b loading a c tin g d ire lly on the heart libres,
ii) elevated hydros1 a i ic pressures in I lie vasculature, and i i i j stresses from
n o rm a l c o n tra c tio n o f 1 lie heart.
N a tu ra lly , t hese le p o rts o f endocardial haem o rrh a ging and in yo lib rilla t
d e g ra d a tio n in swine undergoing high sustained } (!,. accelerations raise .-pies
lio n s re g a rd in g the p o s s ib ility o f cardiac tissue dam age in hum ans subjected
to s im ila r + Ch forces. N om m vasive cardiological technbjues used d u rin g
e x p e rim e n ts on hum ans seem too insensil i ve to p ro vid e su llicie n i P a t a t o de
I hit Ullllll l lull
;i < 1 for a ( o m p u ta t ional m odel to predict stress levels in the hum an heart
under high \ ( i accelerations. In 1 his study. such a m odel capable ol p ro v id
ing a d d itio n a l insight in to th e effects of -f-.'V. induced stresses on th e h ijin a n
heart is developed. I lie c o m p u ta tio n a l m odei hased on th e fin ite element
m e tho d w here the effects o f fin ite d e fo rm a tio n for an in co m p re ssib le elastic
m a te ria l are accounted for.
Ih e stu d y details the developm ent o f a fin ite clem ent m odel to riel e rm in e
the s tre s s /s tra in state o f the hum an left v e n tric le (IA ) and rig h t v e n tric le
(R Y ) m yo c a rd iu m d u rin g sustained - K b acceleration. W hereas in previous
studies, fin ite element models o f the heart have considered o n ly tin* passive
d ia sto le and act ive systole c y c lic responses o f the left vent rich* in a n o rm a l
( f 1 K . ) e nviron m e n t (eg. [S. 22. '2-1. 25. 2N. 50. .‘11. In, IN, (i l, (in, 07. !)X. 101)]).
T h e fle x ib ility o f the fin ite elem ent m e tho d to deal w ith c o m p lic a te d shapes
and account fo r the effects o f m a te ria l and g e o m e trica l n o n -liir* a ritie s makes
it an ioeal teehni<|Ue to s tu d y the s tre s s /s tra in b eh a v io u r o f th e h eart. In
a d d itio n , w ith the advent o f c o m p u te r-a id e d to m o g ra p h y am i d a ta im a g in g ,
an accurate g e o m e tric re co n stru ctio n o f the heart, is possible [1. 15. 70].
I lie heart w all is com post'd of continuous in te r tw in in g m y tx —vdial fibre's
fo llo w in g a helical path th ro u g h the w a ll thickness [MS], T h e a rrangem ent o f
the m yo c a rd ia l fibres, which form s p ira lin g bands o f m uscle re s u ltin g in an
in tric a te w ebbed s tru c tu re across tin ' v e n tric le w a ll, is d e p ic t(><1 in F ig u re 1.1.
l'hest* com plex lib ie bundles fo rm the left and rig h t v e ntricles. In essence, the*
heart w all behaves as a non lin e a r anF iro p ic com posite m a te ria l [till]. In the
lite ra tu re some data on the p ro pe rtie s o f heart tissue is available' [50, 0!), 72].
In a d d itio n , there exists a n u m b e r o f p ro m isin g c o n s titu tiv e re la tio n s h ip s ,
1. I n t r o d u c t i o n ■i E P IC A R D IU M - i M Y O C A R D IU M E N D O C A R D IU M Aorta S u p erio r v e n a cavity Pulm onary artery Lett atrium Right atrium R ight - ventricle Left ventricle P E R IC A R D IA L C A V IT Y P E R IC A R D IU M Inferior x v e n a cavity
F ig u re 1.1: A S chem atic d ra w in g d e p ic tin g an excised th ro u g h thickness sec tio n o f th e h um an v e n tric le : (a) Show ing the d is t r ib i: '' : r j l . ■ e n d o c a rd iu m , m y o c a rd iu m , and e p ic a rd iu m o f th e v e n tric le w all w ith s u p p o rtin g p c ric a r d ia l sac. (b ) T h e arrangem ent o f the m yo ca rd ia l fib re 1 "e s, where the angle o f o rie n ta tio n , n . gives the sm ooth tra n s itio n o f tin* fib re d ire c tio n am i
is m easured counterclockw ise from v in the local axis system j.
6786 5
I. hi t r o d i n t inn 5
ofi. fi'J jj. These ndat ionships arc based cm a pxt udo-strain energy fu n c tio n ,
w h ir}; represents a Ix.si Jit for e-vpeudment a lly collected m at ('ria l d a ta . It
is. therefore, bot.h feasible' and desirable to in co rpo ra te 1 such a c o n s titu tiv e 1
re la tio n in to the1 finite1 cleanout, m odel. Furl her, to ensure a re a lis tic moded
for th e analysis o f \ (!~ induced stresses in the hum an ''(M ilric h ' m y o c a rd iu m ,
co nside ra tion o f th e s tru c tu ra l in te ra c tio n o f the s u rro u n d in g a n a to m y is
desirable. Since u nd e r se've're + (7- lo a d in g co n d itio n s th e v e n tric le s undergo
s ig n ilic nt d e fo rm a tio n s , it is expected th a t load tra n s fe r occurs betwe'eui the
heart and s u p p o rtin g p e ric a rd ia l m em brane [95].
In ('hapten 2, the theore'tical fo u n d a tio n on w h ich a c o m p u ta tio n a l moded
for t lie dot or m i nnt ion o f t he stress and stra in state in t lie1 ventricle 1 m y o c a rd i
um d u rin g sustained + (7 ; accederation is pre'sented. The1 m odel u tiliz e s the
lin i.e element teediniepie (C h a p te r 3) w h e n 1 t i l . 1 effects o f fin ite d e fo rm a tio n ,
non lin e a r m a te ria l b e h a vio u r, and the1 irre g u la r shape1 o f tlie1 heuirf are- ac
count e'd for. Km phasis is placenl on the1 ine’o rp o ra tio u o f a rendistie- e'onstitu-
live- n d a tio n in to the1 fin ite element fo rm u la tio n . A fu rth e 'r e h'scrip tion eT the
g e o m e tric iv c o n s tru c tio n and m o d e llin g considerations o f the1 h um an he'art is
g iven in ( 'liapteT 1.
In th is w o rk , a spevial purpose1 emmputeT p ro gram is elevedeipe'el in haunt
for the1 nem line'ar analysis e>f the1 hum an ventricle's su b je cte d te> + C k in e 'rtia
force's. This approaedi lends its e lf to an e'asier process o f moded redinennent as
im prove'd analysis te'ediiiiejne's are1 dewedeipe-el a m i nenv m a te ria l/p h y s io lo g ic a l
data be'cemie's available1. A eliscussiem eif the1 non-line'ar finite1 eleuiK'nt s o ft
ware1 package- devedope'd for th e analy.-.is o f sustaine'el + (7 . ineluevel stross in
the1 hum an 1A’ and PA’ m yeicardium is given in C h a p te r 5. T h e fin ite 1 ede-
1. I n t r o u u c t i o n (»
p ro g ra m m in g language' [19]. Also, the convergence- e lia ra etu rist ies and p ie
d ie t ive c a p a b ilit ies o f t lie m odel are1 discussed, and co m p nt at io iia l ve rilic a t i .»n
o f tin ' software' is asse'sse'el n u m t-ric a lly using a se’rie's o f f is t problem s. I'hest-
prohle'ins c o n ta in both line'ar and n o n -lin e a r gt'omet r"n■ (tin ite elisplae e'uie'iits
/ ro ta tio n s ) anel mate-rial (ine-eJinpre'ssihle') e-flects. ( 'eunparison eif num eri
c a lly gen'-rate-d re'sults w ith clost-el fo rm so lu tio n s a n d /o r e*x|>e'iinu'iilal data
o f o th e r re'searehe'rs is ineluele'el w lie'ii available'. In aeIdit it>i 1, an inlt-r.ie I ive
th n 'e '-d im e n s io n a l ( id ) ) g ra ph ica l m o d -'lle r w ritte n in C1 using I ’ l l H IS graph
ie lib ra ry routine's [<S9] is also disi usse-d in C h a p te r a. T h e graph-cal m o d eller
w ith co lo u r c a p a b ilitie s is eeuisiele'red im p e ra tiv e lor I lie v is u a liz a tio n o f I lie
'l l ) heart m odel d u rin g c o n s tru c tio n anel. nmst im pendanlly. for the- post pro
cessing o f the vast cjuant.it,y o f stre-ss ami displaceine'iit data hi a p ic to ria l
h <rm at.
In t he* fnt in i', it shouhi also be- possible- to compare' n u m e ric a lly generated
ge oi -trie- re-sponse-s w it h those- me-asuri-d e x pe rim en t a l'y using ult ra sound
im a g in g te'chni pies in a centrifuge- [11], Howe-ve-r, whe-n e-xpermients < a n im t
be ju s tiln d due- to safety consiele-rations, the- c o m p u ta tio n a l nioeh-l ta n pro
v id e valuable ejuant it at ive (gross d is to rtio n anel pre-ilieti-el st ress) data <ui t lit*
e'ffe'cts o f r(r'~ in d u ct'd stresses in hum ans (se-e C h a p te r ti).
T h o u g h th e task o f p e rfo rm in g a re a lis tic s im u la tio n o f th e cardiac re
sjjonse- u nd e r suslained f ( ! . acceleration is an e n o rtm m sly d ib it nil tnie.
jrrogress over the- past t wo decaele-s in both c o m jm te r and m edical technology
has reached the stage wlie-re a s u flic ie n tly accurate- c o m p u la tie m a l m odel of
th e heart can ntnv be- developeel. d he c o m jm ta tio u a l m odel eh-ve mped t an
be used to fa c ilita te th e ■••ollectiem ami analysis of e x p e rim e n ta l data. pe>s
I. h i t n x i i K ' t i o n 7
u n fit I ain.-dde e x p e rim e n ta lly . W h ile m a n y questions re m a in unanswered, th e
progress i n . i h ’i!e d w ill hopefuhy p ro v id e the s tim u lu s fo r fu tu re n u m e ric a l
8
C h a p te r 2
T h e o r e tic a l F o rm u la tio n an d
D e v e lo p m e n t
2.1
In tr o d u c tio n
T h e n o n -lin e a r g e o m e tric and m a te ria l b e h a vio ur o f the hum an heart, de
fo rm in g u n d e r sustained + (7 , acceleration requires careful s tu d y from both
a th e o re tic a l and c o m p u ta tio n a l s ta n d p o in t. In th is ch ap ter, th e o re tic a l as
poets o f th e p ro b le m are presented w hich subsequently p ro vide the founda
tio n for th e fin ite ('le n ie n t c o m p u ta tio n a l m odel. F u rth e r, a statem ent, o f the
lin e a riz e d w a l k o r variat ional fo rm o f th e g overning I xilaiia o f irionn nluni
equations is presented. In a d d itio n , the rate form o f these e quations are
g ive n for com parison purposes w ith th e ir lin e a r c o u n te rp a rt. F u rth e rm o re ,
special a tte n tio n is given to th e in c o rp o ra tio n o f a re a lis tic incom pressible
h y p e re la s tic c o n s titu tiv e re la tio n in to th e cardiac m a te ria l m odel.
In o rd e r to develop an effe ctive so lu tion procedure lo solve th is com
p le x p ro b le m , m any s ta te -o f-th e -a rt c o m p u ta tio n a l techniques are em ployed.
F ro m a th e o re tic a l s ta n d p o in t, it is essential th a t the fo llo w in g issues are
2. 7'hfoff'Uccil l 'o r m u l a t i o i i mi d D c v d o p n u ' i i t 9
( i) . Il is necessary th a t any system atic lin e a riz a tio n o f th e weak o r v a ri
a tio n a l form o f th e bale rr o f mom en turn equations agrees w ith the?
corresponding rate h u m ,
( ii) . The* use o f an objectirt measure* o f stre*ss-rate? and d e fo rm a tio n to en
sure- that ce rta in physical q u a n titie s are in d ep e nd e nt o f th e choice o f
observer [.‘I 1, ( il].
( iii) . A va lid me-thod fo r stre-ss inte-gration o f ra te expiations (ie. in c re m e n -
ta l u p d a tin g o f stre-sse*s) th a t m a in ta in s i ncrement al o b je c tiv ity in the
presene-e* o f fin ite stretches and ro ta tio n s [41, 68].
( i v ). In c o rp o ra tio n o f a re a lis tic in cre n u -n ta l h y p e re la stic c o n s titu tiv e re la
tio n to m odel th e incom pn-ssihh* Ix-h aviou r o f passive* m y o c a rd iu m and
pe*rie*arelium.
(v ). I’he* a b ility to handle* non-conservative o r d(*form a tion d ep e nd e nt lo a d
ing. In p a rtic u la r, th e blooel-volum e in th e le ft and rig h t v e n tric le
e-hamhe-rs o f th e he*art exerts a hydrost a tic surface pressure on th e ve n
2. T h eo re ti ca l Formu la ti on and Development
2 .2
K in e m a tic s o f D e fo r m a tio n
10
C o n side r a sim p le d e fo rm a b le body, B . o ccu p yin g a region B in F.uclidean
space 8 , w here B C 8 w ith a body particle given by \ t B . T h e b ody is
i n it ia lly unstressed in *he n a tu ra l c o u lig u ra tio n at tim e / t), and is denoted
b y B°. Let B r represent th e re fe re n tia l c o u lig u ra tio n w ith X d e n o tin g th e
p o s itio n v e c to r o f th e p a rtic le A". A fte r ih e body undergoes a d e fo rm u 'io n , it
occupies th e current c o u lig u ra tio n denoted by B 1, w here the particle' A now
has a s p a tia l p o s itio n ve cto r x . O ne may note th a t th e re feren tia l co id ig u ra
tio n m ay o r m ay n o t correspond to the n a tu ra l c o n fig u ra tio n o f th e b o d , at
t = 0 (see F ig u re 2 .1),
M a th e m a tic a lly , th e d e fo rm a tio n process can be represented by th e one
to one and onto (in v e rtib le ) m a p p in g x : B ' — > o ', such th a t
x = X ( X , / ) . (2 .1)
A s s u m in g t he same o rig in fo r X and x , the mot ion o f a p a rtic le can be w ritte n
as
x =- X ( X , 0 = I X + u ( X , /) , (2.2)
w here u is th e v e c to r displacem ent o f the p a rtic le an<l th e id e n tity tensor,
I , is used to denote a tw o -p o in t s h i f h r re la tin g vectors in tw o re c ta n g u la r
c o o rd in a te system s [18], A lte rn a tiv e ly , in com ponent form
Xi - \ , ( X , /) = (2.:{)
w here / , a € { 1 , 2 , 3 } and <*>,„ is the K ronecker d e lta . T h e low er case Lat.iii
and C reek ch a ra cte r indices are used for the C artesian c o o rd in a te system
Theoret ical F o rmu la ti on and Development 11
N,
5 ? ( N )
B
F ig u re *2.1: A sketch d e p ic tin g the nat ural B ‘\ re fe re n tia l B r , and s p a tia l B 1 c o n fig u ra tio n s o f a b ody undergoing a d e fo rm a tio n . A lso , e le m e n ta l surfaces
d V and </r w ith surface tra c tio n s T ‘ ‘v ’ - T r N and t (,,) = a 1 n a rc shown in
th e re fe re n tia l and s p a tia l co n fig u ra tion s, respectively. T h e o rig in O is used to define th e g lobal fram e o f reference.
2. Theoretical F o rm n l a t u m and Development 12
T h e def or mat ion g r a d u a l o f the body. b~, is a second ord er (tw o p o in t)
tensor g ive n by
F = v > c-: x * /•<.. "
where V x denotes th e gradient o p e ra to r w ith respect tc. the reference co ulig
u ra tio n and e,, jE iV an* base vectors in the sp atia l and re le re n tia l co nligura
lio n s , re sp e ctive ly. In a d d itio n , the displan mt nt u r a d i t n l . denoted by 1), is
given b y
D = v x - F - I,
w here
I
is the u n it tensor.In th e analysis, th e suit cardiac tissue is m odelled as an iu co m p re sible
solid caoable o f fin ite d e fo rm a tio n . For an isoehorie d e fo rm a tio n process
th e t o t a l rare o f change (or m a te ria l d e riv a tiv e ) o f d en sity is subject to the
c o n s tra in t
I ) p { x , t )
- i n - - “ • ' - ' i)
N o tin g th a t F is n o n -sin g u la r, a vo lu m e elem ent hi the deform ed sp a tia l
c o n fig u ra tio n , du, is given b y
du = dot F d l l \ (2.7)
w here d i l T is th e elem ent in the reference c o n fig u ra tio n . A lso, from mass
co nservation
f)r = def F (>, I'l.H)
where p is the d e n s ity in th e sp atia l c o n fig u ra tio n , and
2. 'I ln-nrciit hI Formu la ti on m i d I)('V<dopui<'ui 13
(T h e superscript /■ is used to denote th e re fe re n tia l c o n fig u ra tio n .) F o r the
is o rh o ric b e h a vio ur o f th e soft b io log ica l tissue,
del F = 1. (2.10)
In a d d itio n , i t is convenient to define th e ve lo city, v , am v e lo c ity g ra d ie n t,
L , o f a p a rtic le as
v(x,/)
= ^ ( x , / )= *(X./),
andL
= V r >v.
(2.11)w here V , dem .tes th e gradient o p e ra to r w ith respect to th e c u rre n t c o n fig u
ra tio n . F u rth e r, ih e velocity gra die n t can be decomposed in to a s y m m e tric a l
and a n ti-s y m m e t ric p a rt, resp ective ly as
L = d + co. (2.12)
In part ic u la r, th e s y m m e tric p a ri corresponds t o the s p a tia l rate o f d e f o r m a
t ion tensor
d = l- ( L + L r ), (2.13)
and the a n ti s y m m e tric p a rt :s th e sp a tia l spin tensor
« = ^ ( L - L r ), (2.14)
where supe rscript r (hmotes transpose. A lso , a useful re la tio n fo r the tim e -
ra te o f the d e fo rm a tio n g ra d ie n t. F , is given by
F = L F . (2.15)
In o rd e r to develop a fin ite elem ent m e th o d o lo g y fo r th e p ro b le m con
2. 1'heorrtic.i! Fo -imitation and Drv cl opnu' ti t l-l
c u rr- n it) approach is adopted. T h e case whore all variables are referred to
th e unstressed n a tu ra l e o n lig u ra tio n at t 0. say F , is term ed the l o l a l
Lagrangian m e th o d . In th is w o rk, the reference eonligurat ion is chosen to
move' w ith the change o f geom etry o f the body. F \ and corresponds to 1 lit*
m ost re c e n tly know n eonligurat ion. This approach has been term ed the I p
d a t f d Lagrangian m e tho d . B o th o f these approaches can be shown to be
m a th e m a tic a lly e q u iv a le n t; however, e ith e r m ay possess some c o m p u ta tio n a l
advantage's over th e o th e r depending on the problem under conside ra tion .
2 .3
S tr a in M e a su r e s
T h e d e fo rm a tio n gradient , lx i:ig in v e rtib le , possesses a polar d e co m p o sitio n ,
w h ic h is u n iq u e and defined by
F - V R - R I J , C h ib ;
w here R is a p ro p e r o rtho g on a l tensor, and V ( I J ) an* the s y m m e tric p os itiv e
d e fin ite !< ft {r ig ht ) stretch measure's. A lte rn a tiv e ly , in com ponent form
F = F i„e ; (•) E „ = I ■,/<„,«,■ E.. I U . .. e , E „.
It is also useful to represent some com m on d e fo rm a tio n t ensues in te rm s o f
th e d e fo rm a tio n g ra d ie n t, that is, th e l i f t (r ight ) (' auchg ( i n i n d e fo rm a tio n
measures A (C ) defined by
A - V 2 - F F 7 , and 0 - U 2 F 7 F . (2 ,i7 )
A s s o c ia t'd w ith these tensors are tlie p n n c ip a l in va ria n ts . /,, for i < { I.2 C 5 }.
2. I h r o i f l i t n l I ' o n n u l n t i o n a n d D e v e /o p m e n f 1 5
/ , l \ ( l r A ) 2 - t r A 2] = l7)\ ( t r C ) 2 - i r C 2}. (2.1N)
/ 5 det A del C . ( Incoi tiprc s s i b i l i t y : / :j — 1)
w li'T * 1 l r denotes th e trace.
O ik* can express (L a gran g ian ) ( I r a n ' s strain. E , in te rm s o f th e rig h t
C auchy (Jreen d e fo n n a tio n measure as
E = ^ ( C - I ) . (2.19)
A lte rn a tiv e ly , the (H u le ria n ) Al nnni si ' s strain tensor. e. expressed in te rm s
o f th e inverse* left C auchy (Jreen d e fo rm a tio n m easure is defined by
c - ^ (1 - A - 1 ). (2.20)
F rom equations (2.19) and (2.20) th e re la tio n s h ip betw een th e L ag ra n gia n
and K u le ria n s tia in tensors i.; g i\(*n , respectively, by
E - F 7 e. F . and e = F ^ ' E F ' 1. (2.21)
2 .4
B a la n c e L aw s
2 .4 .1
C o n s e r v a tio n o f M a ss
From the* consei ra tio n o f mass flow
/ x t ' , ( x - ' )' ' " = "• ( 2 - 2 2 )
A p p ly in g th e R eynold's tra n s p o rt theorem and th e divergence th e o re m to
('([n a tio n (2.22) yie ld s
2. riu'i>v('tical Font ill ation and Dt'vclopun'iit 10
and since th e d om ain f i l is a rb itra ry , the integ ra n d must he /e ro e v e ry w h c ro.
T h is in te g ra n d is often referred to as tin ' c o n tin u ity e q u a tio n , and can he
re w ritte n as
o r in term s o f tin ' m a te ria l d e riv a tiv e o f the density. /•>/>( x , / ) / / ’/. as
Feu- an in co m pressible m a te ria l, as is the case w ith soft c a rd iac tissue, equa
tio n (2.0) im p lie s th a t the divergence o f the' ve io c iH lield is zero. ir.
w h ic h e ffe c tiv e ly becomes th e c o n tin u ity c o n d itio n f o r an i n c o m p r e s s i b l e m . i
te ria l.
2 .4 .2
C o n s e r v a tio n o f M o m e n tu m
S p a tia l F o r m
T h e balance e q u a tion for th e lin e a r m o m e n tu m of a bodv is equaled to J la
v e d o r sum o f the e xte rn a l forces a c t i n g o n t h a t body
w h e n ' b ( x . / ) is th e body force per unit mass, and t,,n )( x . / ) is the surface
tra c tio n on part o f the b oundary, denoted by ( f f i / . such tha t
l i v v \ / - v 0,
2. I l i r a i i l i t , i l I n r m u h t l i m i a m / I ) <' Yi ' h>i >ni cnt 1 7
w l n ' i c c r i s t l i e ( ' i n i ' h y ( 1 M U ') s1 r r s s t I 'l i s u r , I h r p o s i t i o n . x . r a n h r p r e s c r i b e d
• >: i l l i r p a r i . i f I h r b o u n d a r y . d e n o t e d b y i ) B [ . as
X X ft;*' X M ()B'U. ( 2 .2 !) j
w ith the fo llo w in g in it iid ro n d itio n s
u( X. O) ti ( X ) . and v ( X . O) v " ( X) . (‘JdlO)
1' III t I l l ' l l I |i )1 r.
i > B [ n i ) L V r i ) B \ am i ()
M> in ' iD ilu rin u i^ ry n o ld 's tra n s p o rl theorem and a p p ly in g th i' divergence
■Dirni to t l i r linear m om ent uni e q u a tio n . e q u a tio n (2.27) becomes
P\
/)- - - iib - d i\ <t
/<■ L D i ' <hi - 0. (2.21)
w h rM ' div <t h { \ \ - V ’i 'fT- N o tin g th a t t h r inttyiyral must vanish for <m a11>i11a i\ region o f B ' iiin l. replacing P v / P t hy v. 1 h r F.ultviau (qutilion
r/' mot ion ra n hr w rit ten
HfT,,
d iv cr t /»b /iv. + />/», = / n \. (2.22)
(Jj-j
l o r tin - special r.is r o f ,i ho-q at rest or u nd e rg oin g u n ild r in tra n s la tio n a l
m o tio n , rip ia tio n 12.22) is r r f r r r r d to as t l i r n j u i l i b r i i i n i K/ iui li on w ith the
rig h t hand s id r b rin g / r r o .
c u r t lie r u n re. f r o m t h r l a w o f c o n s e r v a t i o n o f a n g u l a r m o m e n t u m , t h e < ’n u r h \ s tre s s t r n s o r is s v n i m ' * t r i c i n the* a h s r n c r o f h o d v i n o m r n t s . a 1 — <r.
2. ' I ' h r o r r t i c , t l F n n n u h i t i o i i a n d D o v r l ( ml 18
M a t e r i a l F o rm
I * sin u, e q u a tio n (2.7) and not ini* the k in e m a tic re la tio n s h ip between a eurrent
and reference e iem eutal surface's
m/1 . / ( F l ) N d V r . r.1 F il
th e m a te ria l form o f the lin e a r m o m e n tu m e qu a tion can he w ritte n as
() [ / m x )v i x . /)</(>'
[ i>r( X ) h ( X j ) t n r \ [
T ' ( X . / i < / r . p.’a d
d i .n>' J i t ’ .1 ‘i f ,
w here T denotes th e (non s y m m e tric ) n o m in a l stress tensor (and its lra n s
pose is the.//V s/ 1’iola I \ / r r h l i o j j si less tensor) as
T ./ F 1 <r
A p p ly in g the divergence tlie o rem to equation (L’ .d !) ainl pro< -ceding in a
s im ila r fashion to th e d e riv a tio n o f the sp a tia l lin e a r m o m e n tu m e qu a tion s
one o b ta in s the Ltujvdiujiiiii i i /iintion o f motion
i ) l
D iv T - f p ' b / / V , ) f i l l , / / I . r.’ .dti)
D A , ,
w here V ( X . /) — \ ( X , t ) and D iv (w ith uppercase' |J ) denotes t he d i\ e ig e rn e
o p e ra to r w ith respect to X and it is assumed that I) does not depend on the
d e fo rm a tio n . I lie b o u n d a ry c o n d itio n s in th e reference Iru in e are given by
X X for X * i W ‘n.
T ' N '[ •' lo r X < i)B'r
A lte r n a tiv e ly , the m a te ria l lo rm oi the bahmce e q u a tio n - can be o b ta in e d
using tlie a p p ro p ria t e tra nsfo rm a t ions from t he spat ini to m a te i ini re le ie iu e
Ira n u .
2. I l i c o r c tif ul I ' o n n i i h i t i o n n.nl D r v c l o p n i r u t 19
'I he fn con (I (s y m m e tric ) I ’ lolo K i rc h ho j f slress tensor S is given by
S T ( F 1 )r o r T - S F 7 . (2.37)
and is co n ju g a te to ( Ireen's s tra in ra le tensor ( i(*.. tlie stress pow er per u n it
vo lu m e in the reference eonligurat ion is given hy / c { S E } ) . 'This stress m e a
sure does not have a direct physical in te rp re ta tio n , h u t. its im p o rta n c e w ill
become evident when fo rm u la tin g a fin ite clem ent m e th o d olog y. F u rth e r,
one may note that the second Piola K ir c lih o lf stress is related to th e C a uchy
st ress tensor by
a - ./ 1 F S F 7 or S = ./ F 1 <r ( F ~ ‘ )r . (2.3S)
2 .5
V a r ia tio n a l (W ea k ) F orm o f L in ea r
M o m e n tu m B a la n c e
C onsider th e sfront/ fo rm o f the b o u n da ry-valu e p ro b le m o f m o m e n tu m b a l
ance (w here <T1 (r)
d iv a 4 p b = p x in B 1,
<r 1 n = t " ' 1 on
x = x on <)B‘U.
From calculus o f v a ria tio n s , an equivalent wt ak or v a ria tio n a l fo rm o f th e
b o u n d a ry value’ p ro b le m o f m o m e n tu m balance can be c o n s tru c te d fro m
f (d iv a + pb - px)-t'-x<lu - 0. (2.39)
Jiv
whore C y is an a rb itra ry k in e m a tic a lly adm issible displacem ent fie ld . A p p ly
in g the divergence theorem to e q u a tio n (2.39), th e weak fo rm o f the v a ria
tio n a l (’qua! ion is given by
2. T h c o r r t i c n l F or mu la ti on mu l /)e \ol opmr nt 20 w h (> n ‘ Q { x , b x ) >s d e l i n e d as / / ; - { < r ( y <vy)}</w-+ / / M v h) - i ' \ i hi f V"' - b\ , h J i r Jiv J u<’r fo r a ll U -- | x | x t ( ' ( B 1), x x on H B 1., and V ~ j V y | ^ v o ( ' ( B 1). (s y 0 o n i ) F ln j - . T h e sot o f t r i a l s o l n t i o n s , d e n o t e d h y M . a rc k i n e m a t i c a l l y a d m i s s i b l e I'iiiic l i o n s o f y f X , / ) t h a l s a t i s f y t h e e s s e n tia l b o u n d a r y c o n d i t i o n s . In a d d i t i o n , t h e set. o f v a r i a t i o n s , d e n o t e d b y V . a re th e h o m o g e n e o u s c o u n t e r p a r t o f th e t r i a l f u n c t i o n s . F o r t h e s p e c ia l ease w h e r e th e V is ta k e n as a v i r t u a l t l t s / i l a r t ti n u l lie l d . ((‘n i = b \ ) o v e r D B 1 a n d u s in g t h e p r o p e r t y t h a t cr is s y m m e t r i c , one o b t a i n s t h e v i r t u a l s t r a i n e n e r g y b a la n c e l b r a h y p e r e l a s t i c s o li d as [ t r { ( r b e ] ( h i — f n ( b - v ) - b u i h i 1 / t."" J i r > nr,
b X . s t r a in energ y X , \ X . b od y and in e r t ia l forces y . \ Y . s n rfa r e t i a i t i o ic o w l u ' r e be. is t h e v i r t u a l r l r a / n t e n s o r 1 a n d is d e fin e d as th e s y m m e t r i c p a rt o f t h e v i r t u a l d is p l a c e m e n t g r a d i e n t 1 be. ( V, ^ u ) I (V, • b.k l'dj F o r c o m p le t e n e s s o f t h e w e a k f o r m o f th e b o u n d a r y v a lu e p r o b l e m an a p p r o p r i a t e c o n s t i t u t i v e r e l a t i o n b e tw e e n (T a n d be. needs t o be p r o v i d e d . 1 his is d is c u s s e d i n t h e f o l l o w i n g s e c tio n .
1 I he tensor be should not be confused with the linear u d im li simal si rain tensor, bt is t lie variational si rain tensor conjugate to the Can liy stress
2. l l ii -orrt ic al For mu la ti on and Development 21
2 .6
C o n s iste n t L in e a r iz a tio n o f t h e B a la n c e
o f M o m e n tu m
In o rd e r to solve th e strong form o f th e governing b o u n d a ry -v a lu e p ro b le m ,
a lin e a r a p p ro x im a tio n to an equivalent weak fo rm is sought. T h e lin e a r
a p p ro x im a tio n w hich provides a basis fo r the fin ite element te c h n iq u e is d is
cussed in some d e ta il in th is section. This approach is equivalent to the su
p e rim p o s in g o f in cre m e n ta l displacem ents on a fin ite ly defo rm e d b o d y [2(i].
'The term consist< til is used to in d ic a te that tlie* system at ic lin e a riz a tio n o f
th e weak fo rm o f th e v a ria tio n a l equations is consistent w ith the co rre s p o n d
in g rate form [10]. Consistent lin e ariza t ion allows fo r th e o p t i m a l in c re m e n ta l
lin e a r a p p ro x im a tio n to th e set o f non lin e a r equations w it h in a s m a ll n e ig h
b ou rh o od o f some know n reference state. T he in c re m e n ta l (lin e a riz e d ) equa
tio n s are o n ly a first order a p p ro x im a tio n , whereas the co rre sp o n d in g ra te
e quations are exact in form . Before com m encing lin e a riz a tio n o f th o g o ve rn
in g equations, the fo llo w in g te rm in o lo g y is in tro d u c e d to assist in d e fin in g
th e d e fo rm a tio n process.
( 'onsider t lie o rig in a l undeform ed b ody o ccup yin g B° at t im e / = 0. u n d e r
g oin g a d e fo rm a tio n and now o ccup yin g the reference c o n fig u ra tio n denoted
by B 1. N e x t, consider a superim posed d e fo rm a tio n from th e B r c o n fig u ra
tio n such th a t the b ody occupies the s p a tia l c o n fig u ra tio n , B {. R e fe rrin g to
F ig u re 2.1, the b o d y m o tio n can be expressed as
x - ,v ( X .O = I X ( X , „ / ) + u ( X , / ) , (2.-13)
w here u is a superim posed m o tio n fro m the re fe re n tia l c o n fig u ra tio n .
2. Theoret ical Tonn ul at i on and Development 22
m o re convenient to w ork w ith th e m a te ria l rathe.- tlia u the s p a tia l form
o f these equations. For the reference eonligurat ion tin* in te g ra l d om ain is
independent, o f body m o tio n , w hich somewhat s im p lilie s the lin e a riz a tio n
process. F u rth e r, it is assumed that b and Tv " are independent o f the
k in e m a tic s . T h e n , equation (2.10) can he expressed in m a te ria l form .is
G(x.' f i x ) = F-’ . i n
f t r { T b ¥ } d n r + [ />r ( V b ) - ^ x < I D r I T ' 1 y d l 1 0.
.18' .18' j;>8’r
w h e re AF = \JX Isy . In order to find a solution to equation (2.11) it is
d esirab le to o b ta in a local a p p ro x im a tio n to t/ about some reference p o in t, A .
(H e n c e fo rth , A’ is replaced by X .) One such local a p p ro x im a tio n is o bta in ed
u sin g a consistt id approach to lin e a riz a tio n o f the balance o f m o m e n tu m
e q u a tio n [10], and is based on T a y lo r's Form ula (see A p p e n d ix A ).
A local lin e a r a p p ro x im a tio n can be o bta in ed by a p p ly in g a lin e a r oper
a to r L (as delined in ('([nation (A d i) ) to a le n sor held, say f , as follow s
/ , [f . a ]j. = f ( . r ) + ( v f ( - i ' ) ) - a f(.r) I <!,[f(./-)]. (2.1b)
w here ( [ = a - V , defines th e scalar d iffe re n tia l o p e ra to r for any a rb itra ry
v e c to r a in E uclidean v e cto r space.
A p p ly in g th e lin e a r o p e ra to r /. to the weak form o f the m o m e n tu m bal
ance e quations O ( x . ^ X ) — 0. about some fixed point X • yields
L[Q. u]jj- -- t / ( X ) f ('-A A / ( X ) j u (2. Ki) - C?(X) f C ,[fv (X )]
2. I ln-oret ienl 1'brmnlnt ion and Development 23
( I lf iK efort h. 1 he superscript ( ) is o m itte d .) Now, co nsidering th e gradient
o f t / ( X ) in t lie d ire c tio n u . as defined in equ a tion ( A . 2), one m ay w rite
C iv f ^ f X )] / ( h [ / r { T * F } ] d n ' + / >r ( ix [ ( V - b )]-fix<KV
J P ’ J p r
- (2.47)
where b \ is an a rb itra ry k in e m a tic a lly adm issible displacem ent fie ld . A s
su m in g that b and T ' V) are independent o f th e d e fo rm a tio n , th e n e q u a tio n
(2.17) reduces to
U [ t 7 ( X ) ] - [ t r { D x [ T ] b F } < i n r + f Pr u - b x < i n r . (2.48)
J P ’ J p r
where (>v[V] — i i fro m equation ( A . 2). Hy s u b s titu tin g th e corre sp on d ing
expressions fo r f7 (X ) and (;x [t/( X ) ] in to (2.-Hi), a local lin e a r a p p ro x im a tio n
to th e weak fo rm o f tin* m o m e n tu m balance e q u a tio n about some1 fixe d p o in t
X is o b ta in e d as
/ / / • { ( ; x [T]<s>F}</Sr + [ P r i i - b x d W = (2.49)
J p ’ J p ’
I i r { Tb F\ < n r - f Pr( v - b ) - b x < i n r + [
T v,- < ^ / r \
J P ’ J P ’ J : i P ’r
A lte rn a tiv e ly , the s p a tia l form o f (2.-19) can be w ritte n using th e tra n s fo rm a
tio n equations (2.7), (2.S), (2.22), and (2.5S) as follow s
I . / * / / • { ( !X[ T ] <^F}(/u + l ^ p u - b x < l u = (2.50)
f ./ 1 l r { T b F } ( h i --- f p(v - b)-bx<hi + f V n)-bx<h\
,lp< J p ‘ JoP 'j
r it im a t e ly , t in ' te rm / r { ( ! x [T ] (''F} defined fo r the re fe re n tia l c o n fig u ra tio n
w ill require an equivalent sp atia l fo rm for c o m p u ta tio n a l im p le m e n ta tio n ,
w hich is o b ta in e d by s u b s titu tio n o f the c o n s titu tiv e re la tio n . In th is s tu d y,
a hype re la stie c o n s titu tiv e re la tio n capable o f m o d e llin g th e (n e a rly ) in c o m
2. T h eo re ti ca l F or mu la ti on and Development 2-1
2 .7 A C o n s titu tiv e R e la tio n s h ip for
C a rd ia c T issu e
In th is s tu d y, the- ca rd iac tissue is assumed to behave as an in co m pressible
h y p e re la s tic so lid 2. Kve-n tho u gh cardiac (issue e x h ib its a n is o tro p ic and vis
coelastic b eh a v io u r, successful p re d ic tio n s o f m a te ria l response have' be-e-n re1
p o rte d by re’searchers whe-n iele-alizing the- careliac tissue' as hennoge-ne-eius .uiel
iseitropic 1 [Id , 2d], 'F lic assnmptieins e>f m ate'rial he>me>ge-ne-ity and ise>tle>|>\
are' adopte-d in th is the-sis. In additiem , the* tissue- is eemsidei e-d le> unde-ny>
o n ly isothe'rm al defen'inatiem pmce-sse-s anel te> 1 >e* e-he-mie a lly ine-rl.
For a hvpere'lastic m ate'rial the-re- e’xists a p o te n tia l e-ne-rgy fu n c tio n , eh-
n o te d by I F , w hich is takeui as the* stem’d e-ne-rgy pe>r unit ie-fereu<<- vo lu m e
fo r is o th e rm a l de-format iem. Ile-ne-e-ldrth, IF e-an he- e-xpie-sseel as
IF - l f ( F ) . (2..r»|)
In th is se-clieui, it is im plie-d th a t the- re-le-re-nev slate eone-spouds to the o
rig in a l unelefeirmed eem liguralieni where- stress and s tra in vanish. F u rth e r,
invariane-e- iinde-r rigiel bexly nieition (ie>. ej|»je-ctivity [ t il] ) ree|iiiie-s the s tra in
e ne rg y fune-tiexi te> eibey
IF - I F ( Q F ) , (2.52)
fo r a ll prope-r eirthogonal t<*iise»rs Q . In p a rtie u la r, i f Q is ehose-n to he R /
and using (2.1(i) eme- o b ta in s
l-F (F ) I F ( U ) . f2 .d d )
" A m a t e r ia l is de-scri b e d as //.?//« n l a s t i c i f a si i n d e n e rg y fun< lio n < x is ls l o r isof lir r iu a l o r a d ia b a tic d e f o r m a t io n .
•JA c o n s tit u t iv e re -Ia tio n £ ( C ) is di-fine-il its i s o h n j m i f a n d o n ly if tlx - pr<>p» r t y o f £ ( Q C Q 7 ) = C (C ) h o ld s , w h o re Q is a p ro p e r o rl h o g o n a l te n s o r
2. 7 h c o i f t i r n l I'oi niuhit ion and Dcvclopim' ii t 25
Since there exists a one-to-one correspondence between th e tensors U and C
fie . O U ^ ). equ a tion (2.55) can be re w ritte n as
W =
l i ’ (C ). (2.54)M o re co n ve n ie ntly, for an is o tro p ic m a te ria l (2.54) can be expressed as
W
= W { I x J 2, h ) , C2.55)w here ( / i . I-a) are the p rin c ip a l in va ria n ts o f C . as defined p re vio u sly.
f o r a (7 m » d a s i i c o r h ype re la stic m a te ria l th e (n o m in a l) stress d e fo r
m a tio n re la tio n is given by
T . i t ' M .
7:„ _ f l .
( ) l 0 l u,
and from e quations (2.16) and (2.54) th is becomes, fo r an is o tro p ic m a te ria l
t
.
I ’ sing (2.17) and (2 J i7 ) th e above1 can be expressed in te rm s o f th e second
P io la K irch h olF stress and rig h t ( ’auehy (freen d e fo rm a tio n tensors as
<riv(C)
S = (2.58)
A n a lte rn a tiv e expression in term s o f th e in v a ria n ts o f C can be o b ta in e d by
2. Theoret ical F or mu la ti on a m i Development 20
I f tlu> reference c o n fig u ra tio n coincides w ith the n a tu ra l eon lig u ra t ion (ie.
C = I ) , fo r w hich th e stress vanishes, then
lc=I
m v
m r
m yi o f
---<)lx <)12 DI., c i 0.
In th e special case o f soft b io log ica l m a te ria ls, such as cardiac tissue,
th e in c o m p re s s ib ility c o n stra in t given in equation ( 2 .IS) is im posed. T h is
c o n s tra in t in h ib its any stress measure from being solely d e te rm in e d fro m the
d e fo rm a tio n . In fa c t, th e d e fo rm a tio n determ ines the C auchy stress o n ly to
some a d d itiv e h y d ro s ta tic value. Hy in tro d u c in g an a rb itra ry Lagrangian
m u ltip lie r —~p and re w ritin g IT in the form
IT II (Cl) - \ l> (dot. C (2.(i I )
an a lte rn a tiv e fo rm o f ('([nation (2.m l) results. T h erefo re , for an incom press
ib le m a te ria l, ('([nation (2.5S) becomes
(2. (12)
o r in te rm s o f tin ' in v a ria n ts o f C ,
i ) M ’ ( M V , M Y '
ur1c +
U +'
I /'< (2 .(id)Due to th e lack o f a re lia b le const it ut i ve re la tio n s h ip to m odel t he anisol rop
ic p o m -v is c o e la s tic m at(*rial beh a vio ur o f th e hum an hea rt, a s im p le r ap
proa eh is taken w here the ca rd iac tissue is cotisid* red to be a non lin e ar
hom ogeneous is o tro p ic elastic m a te ria l. In th is w ork, an e x p o n e n tia l stra in
energy fu n c tio n proposed for soft biological m a te ria ls (21J and subsequently
used to m o d e l cardiac tissue (1 1. !)?)] o f the form
2. I I i c o r d i m l bon nnh it ion and Development 27
is u s ffl, where a find h arc* m a te ria l constants d erived fro m e x p e rim e n ta l
dat a. T h e const ant s a and b represent, the best j i t param ete rs for passive*
ca rd iac tissue data o bta in ed e x p e rim e n ta lly . In c o rp o ra tin g e q u a tio n (2.64)
in to (2.fid) yields
S = ‘l a b ( hd i - ' A) I _ p C ~ l . (2.65)
h or e q u a t i o n (2.60) at I = I) to hold true* when the* s tra in energy fu n c tio n is
given hy <*quation (2.61), one* obtains
p |f=» = Pd = 2ab. (2.66)
T he* stress d efo rm a tio n re la tio n given in e q u a tio n (2.65) gives r i e to a
U nite elem ent fo rm u la tio n , th a t is, b o th disp lace m e n t and pressure
(;/,/>) arc* lake'll as independent unknow ns [27]. H ow ever, th e e lim in a tio n
o f p from she system o f equations can be accom plished by re la x in g th e in-
c o m p re s s ib ility c o n s tra in t, / ;J — 1, using a p e n a lty fu n c tio n . In the* p e n a lty
fo rm u la tio n o f the incom pressible p ro b le m the* c o n s tra in t given b y (2 .26 ) is
relaxed using
V -r'V + ^ = 0, (2.67)
w here A is a p e n a lty p aram ete r. A ssum in g the* pressure to be a fu n c t io n o f th e
d ila ta tio n , the* so lu tio n o f (2.67) can be expressed in te rm s o f th e in v a ria n t
/;i OK
/» = /> ..- h ) . (2.68)
KH'eetively, th e pro ble m becomes one o f m o d e llin g a n e a rly in c o m p re s s ib le
2. Theoret ical F o r mu l at i on and Development 28
to in iin ity . This approach allows the n um ber o f system equations to be
d ra s tic a lly reduced w h ile m a in ta in in g (near) in c o m p re s s ib ility . In a d d itio n ,
due to th e fin ite d ig ita l accuracy o f c o m p u ta tio n , the' m a g n itu d e o f \ is
re s tric te d so th a t the problem is not ill c o n d itio n e d [.'iV].
2 .8
S tr e ss a n d S tra in R a te s
R e la tio n sh ip s in term s (if stress and stra in raies should be independent o f an
observer tra n s fo rm given by ( x . /) - > ( x * , / * ) w it h
x * = c ( /) + Q ( / ) x , /* I (/. (2.(i!))
w h e n ’ c is a tra n s la tio n and Q is a proper o rtho g on a l tensor.
T h e te rm o b j t d i r t is used to describe a tensor w hich is in d ep e nd e nt o f
an observer tra n s fo rm as defined in Ref. jf ilj . One such fram e in va ria n t
ra te used for th e c o n s titu tiv e law in the reference c o n fig u ra tio n is th e rate
o f tht* second P iola K irc h h o ff stress and (Ireen s stra in rate tensors. For
e q u a tio n (2.58) the ra te form is s im p ly expressed as
()2W
•
•
i)2\Y
S = : C - S‘d = “ r,T7 V, , < r j . 7 ( h
c C ( ) ( ,vr
wh(*re th e sym b o l : is used to denote the in n e r product o f a fourt h and s e c o n d
o rd e r tensor. A lte rn a tiv e ly , the ra te form o f S given in term s o f ( b e e n ' s s tra in
ten so r E fo r an incom pressible m a te ria l governed by equal ion (2 .(i2 j b e c o m e s
H ow ever, for th e fin ite element im p le m e n ta tio n one m ay consider a m ore
useful fo rm o f (2.71), in term s o f (Ire e n ’s stra in ra te , that is
2. I h c o r d u ni Ft n n u h i t i n i i mi d D r v c l o p u u m t 29
w here (’ is a fo u rth order m a te ria l response tensor in the reference c o n fig u
ra tio n . A lso, (Jreen's stra in rate is related to the ra te o f d e fo rm a tio n tensor
by
E = = F r r iF . (2.73)
A n a lte rn a tiv e fo rm o f (2.71) can be w ritte n in co m p on e nt fo rm as
A V , (2.74) ,s'
-w h e r e th e te rm inside the square brackets is e qu iva le nt t o t ’ in e q u a tio n (2.72). F u rth e r, the in c o m p re s s ib ility co nstra in t (2.2(i) m ust also h old fro m mass
conservat ion.
For the fin ite elem ent fo rm u la tio n th e sp a tia l fo rm o f th e ra te equa
tio n (2.72) is re quired. A d o p tin g th e T ruesdell stress ra te w h ic h is a s u ita b le
o b je ctive 1 stress rate1 [91], th e corresponding ra te -ty p e c o n s titu tiv e e q u a tio n
is given by
o <1 / )>
(T - £ \ d, (2.75)
w here £ is th e fo u rth order instantaneous m a te'ria l response tensor, and d is
th e (s p a tia l) d e fo rm a tio n ra te tensor defined in e q u a tio n (2.13). In a d d itio n ,
<r can be relat ed t o t he i at e o f change' o f t lie' see'onel Pierla KirchherfF stre*ss
tensor by
< T r " = , r l F S F r . (2.7(5)
N ow , ta k in g the' tim e' ele'rivative erf eepiation (2.38) anel s u b s titu tin g in to
c'epiation (2.7(5) for S yiedels
