co!llpuuu!on ot the rtsulWit aspl't~cal states. In the present Pll'U⢠we present computational te$Ulta for certain .sphericAl col\flguruions of lnidall:y s~hcnclll ...
L. Wheeler, T.E. Tezduyar and L. Graux, "Asymmetric Equilibria of Naturally Spheroidal Elastic Membranes", Proceedings of the Twelfth Canadian Congress of Applied Mechanics, Ottawa, Canada (1989).
Corrected/Updated References 4. T.E. Tezduyar, L.T. Wheeler and L. Graux, "Finite Deformation of a Circular Elastic Membrane Containing a Concentric Rigid Inclusion", International Journal of Nonlinear Mechanics, 22 (1987) 6172.
ASYMf...mnUC taQUU.WRIA OF NATliRAU.Y SPHEROIDAL EU.ST1C MEMBRANES
Lewis WI!Wet. ~pc1men1 ot Mechanical £n&in«rinlo Univmily of Houswn. HoiiSUlft, TX, 7nDH79Z
Tayfuft E. Tezduyar, ~~fA MQI:1181llcs and Aerospace Englnurin~o Un.ivenity of Minllescu. MiMeapolis, MN SS4SS Lue OI'IUI. SKF Bwina Industria. Towt, France
1. IN'l'RODUCltON.
3, LIN'EARlZATJON.
The poNibiUty that &1\ elastic membrane, sphetl.calln its reference confiaumtion. may possen nonspherical ~qllillbriudl confi&URtiOII$ when 111b)ectcd to u~it'orm in1emal prc:ssutt wu flrst rlllscd by Peodos'cv [1). This counterlntuitiverssibillty wq arrived ac mathemalically by Feodos'ev an Iacer confirmed, also mathematically, by H1usht0n {2]. Exp~ntal ovldence was put forth by Alexander [3].
Hcte we present che linc:ari:ted equations found by Feodos'cv. Con$ider 1 spherical suue and choose as iodependenl vuinblc the lllglc 'oerwctn .. die normal Cllld the pilUle z .. 0. Lee ... and w denote the projections of the displacement from this St:lte onto the t4n~entinl (8) and normal directions. Then. we h11vc: [1]
While Fc:odos'cv'a resulu detc:nnlne condldons under which bit'lucation fot 1 spherical &tate Is possible. his wotk does notlnolude co!llpuuu!on ot the rtsulWit aspl't~cal states. In the present Pll'U• we present computational te$Ulta for certain .sphericAl col\flguruions of lnidall:y s~hcnclll membnnel as well as n:swu concemlna tho loss of l}'lMle&:ry abolll the midplane of spheroidal mcmbnl.l\0$. Feodos'.v [lJ considered a class of materials which lncludes tho$e of the Mooney-Rivlin ~ but oonclud~ mat asph~al cquUibriJ are not possible for Mooney-·Rlvtln materials. In the prc.scllt paper, we demons ~rate th.c (ot a ranae of elastic mod11U evidently noc considered by Feodos'~. sucb bU\acltJon to asphmcal modo1 it possible. 'I'M resuiCI prcunted ln chis paper wen: obtained oomputadonally
by mean• or a predlc10t'-znu1tlcomc;cot method combmed wtcl\ ahootina schcmt~. Pot a more dewltd discussiOn, sec (4),
a
a
(a- 2)(2w + v' + vcote) •
b(v'.., vcoce) • w'COI8- w", where I d 1... " II. 4(1..) T()..) dr().- l (l.)), T(A.) .. t().,l.).
t,
R. • R(t), Z • Z(t), ~ t S tz. Let Th T2 denote th~! stress resuhants ln the meridional and clrcumt'erential dir«tiQ~ let ). 1 and~ stand tor the corresponding su-etchu and te., "2 tbo COTTespnnding curvatun~s. The IOVernins e(ll&a.tions are
T1tCt + T2"2 • p, T11Cz • . . ; + 21t1' d8 dr sine l(,a(di'/dt')~$e, 1(2• II
f,
A.
a
A
t'
l, • cO'se di'[(di'") +('dt) J 2 ' ~ * R'
where p s~Jnds tor the applied PfC$$utC, ho che initiallhick.ne$$, r the n~dlal coordinate ln the defonned configumtion,IUid 0 the angle made by the membraM normal and the axis of symmetry. Here we are in~sr.od ln tesponsc of che Btdcrman type (S« (I))
·
Thc,y)=- llloC (x2-h-)(lof'kY~'+'t1U1'Ylu~. ~U .. (x 2+y,.+.~ "Y Xy XY The quiU\title1 C, y, T , and y ~~re the elastic tnodLdi. For "f "'Y • we 1 2 2 1 hAve the Mooney-RMln rtsponse ~w. For the nntutally spheroidnl merobnlnct prescntl:y under considenltion, wo rake R • Acoaht(l-t)/21. Z • Bsin(n{l-t)/2), -1 ~ t s 1. The boundA.ry conditions are dr r • -2
1. 7
H7Y7rr.~~~~~~~~~~
Sr-------------------------~
0
1.4
1.7
Fi$ure 3. Baker·Erick;en domo.in £or y • ..().1.
Bidermon
4.1
1.2
(.02;.03;.00 1)
q _ .58 .J . .300 ,........________....;......._..........
),
~----~-~
Fi,surc 1. a~) vt ~ for various v11lucs of y,
aspherlcal moele
, .290
~.-~;::....
o.oo
__________________....J 1 .:;7 ~., o4
Millie 4. Stretch~~ in an initially sphcrioidal membrane.
Bidermcn
(.02;.03:.00 1) ~e:~,58
0.00
spher"ICal mode 2.32$
_, .65
'------~
o.oo
.o. 75
, .50
Fii'UM 2. Promo, :r. vs t, for initially sphcriclll Mooncy-IUvUn membrane wld'l y • •0, 1.
