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Inflation of elastomeric circular membranes using
network constitutive equations
Erwan Verron, Gilles Marckmann
To cite this version:
Erwan Verron, Gilles Marckmann. Inflation of elastomeric circular membranes using network
consti-tutive equations. International Journal of Non-Linear Mechanics, Elsevier, 2003, 38 (8), pp.1221-1235.
�10.1016/S0020-7462(02)00069-0�. �hal-01006966�
using network onstitutive equations
Verron E.
and Mar kmann G.
Laboratoire de Me anique et Materiaux, Division Stru tures
E ole Centrale de Nantes, BP92101, 44321 Nantes edex 3, Fran e
Tel: (33) 2.40.37.25.14 Fax: (33) 2.40.37.25.73
Abstra t
The present paper deals with the use of network-based hyperelasti onstitutive
equationsinthe ontext ofthinmembranesin ation.Thestudyfo usonthe
in a-tion of plane ir ularmembranesand the materialsareassumed to obey Gaussian
and non-Gaussian statisti al hains network models. The governing equations of
thein ation of axisymmetri thin rubber-like membranes arebrie y re alled. The
materialmodelsareimplementedina numeri al toolthat in orporatesan eÆ ient
B-spline interpolation method and a oupled Newton-Raphson/ar -length solving
algorithm.Twonumeri alexamplesarestudied:thehomogeneousin ationof
spher-i alballoonsand the in ation ofinitiallyplane ir ular membranes. In these ond
example, the in ation pro les and the distributions of extension ratios along the
membrane are extensively analyzed during the in ation pro ess. Both examples
highlight theneedof ana urate modeling ofthe strain-hardeningphenomenon in
elastomers.
Towhom orresponden e shouldbe addressed
In the last few years, lassi al phenomenologi al onstitutive equations for
rubber-likesolids, su h asMooney-Rivlin orOgden models, are progressively
repla ed by more physi al models based on statisti al onsiderations, in
var-ious engineering appli ations [1℄. The determination of the material
parame-tersofthe onstitutivemodelsisoftenperformedusing lassi alhomogeneous
strain experiments (uniaxial extension or pure shear tests for example). For
biaxial deformation, authors use frequently the bubble in ation te hnique,
that onsistsinin atinganinitiallyplane ir ularthin membrane(see [16,22℄
and more re ently [8,20℄). In this type of experiments, deformations are not
homogeneous and the analysis of experimental data needs eÆ ient numeri al
methodtosolvethein ationproblem.Theproblemof ir ularmembranes
in- ationis lassi allysolved for phenomenologi alrubber-likemodels. Here, we
present some solutions orresponding with new statisti al onstitutive
equa-tions.
The governing equations of the large strains in ation of axisymmetri
mem-branes were established by Green and Adkins[11℄. Then, some authorssolve
the orrespondingproblemby dire tlyintegratingthe system of ordinary
dif-ferentialequationsusingshootingmethods.Bothhyperelasti [17,33℄and
vis- oelasti [30,10℄ onstitutive equations are onsidered. Nevertheless, most of
the works are based on numeri al dis retized methods, su h as nite element
analysis [19,5,15,28℄.Most of the papers fo us onthe numeri al methodsand
onsider simple phenomenologi al onstitutive equations. Very re ently,
Has-sager et al. study the response of polymeri uid membranes under in ation
tomeri membranesusingnetworkbasedrubber-like onstitutiveequations.In
Se tion 2,the governing equations of the axisymmetri in ation problemare
re alledandthe onstitutiveequationsarepresented.Fourmodelsarestudied:
the lassi alneo-HookeanmodelbasedonGaussianstatisti sandthree hains
models that are based onnon-Gaussian statisti s.Se tion3 is devoted tothe
solving pro edure. The numeri als heme and our original B-spline
interpola-tion, detailed in [29℄, are brie y re alled.The numeri al analysis of spheri al
balloons and ir ular plane membranes are presented in Se tion 4. Finally,
on luding remarks are proposed inSe tion 5.
2 Problem formulation
2.1 Governing equations
We onsider an axially symmetri hyperelasti membrane of uniform
thi k-ness h 0
in its undeformed on guration. The geometry of this membrane
an be des ribed by the ylindri al oordinates of its mid-surfa e. In ea h
- onstant plane, both undeformed and deformed on gurations are redu ed
to one-dimensional urves, as shown in Figure 1. Consider a parti le of
un-deformed oordinates (r 0
;z 0
)of the undeformed membrane. Thispointis
dis-pla edtoanewposition(r;z)inthe deformed on guration. Thethi knessof
the deformedmembraneis denoted h.Every geometri aldata r 0
,z 0
,r, z and
h are fun tions of the ar -length oordinate in the undeformed on guration
s, that varies between 0 and l 0
,the lengthof the initialmembrane.As shown
m = v u u t r 2 ;s +z 2 ;s r 2 0;s +z 2 0;s ; = r r 0 ; n = h h 0 (1) inwhi h ;s
stands forthe di erentiationwith respe t tos. As the materialis
assumed in ompressible, thethi kness of the deformedbodyissimplyrelated
tothe thi kness inthe naturalstate by:
h= 1 m h 0 (2)
Underaquasi-stati pressure load, thePrin ipleofVirtual Work anbe
writ-ten in the following form:
Z V0 ÆWdV 0 Z S ÆuP ndS =0 8Æu (3) whereV 0
and Sare respe tivelythevolumeoftheundeformed membraneand
thesurfa e ofthe deformedmembrane.W isthe strainenergy,P nisthe
out-ward oriented blowing pressure and Æu is an admissiblevirtual displa ement
ve tor.Noting that the omponents of nare (z ;s = q r 2 ;s +z 2 ;s ; r ;s = q r 2 ;s +z 2 ;s ),
the previous equation (3)simpli es:
Z l 0 0 2r 0 h 0 (t m Æ m +t Æ ) ds Z l 0 0 2P r (Æuz ;s Ævr ;s )ds =0 (4) where t m and t
are the prin ipal rst Piola-Kir hho stressesrespe tively in
the meridian and ir umferential dire tions, and (Æu;Æv) stand for the
om-ponentsof the virtual displa ement ve tor.
2.2 Network onstitutive equations for elastomers
As mentioned above, the use of statisti al network models for rubber-like
they agree well with experiments invarious modes of deformation[32℄.
Inthestatisti alapproa h,polymersare onsideredasnetworksoflong exible
hains randomly oriented and joined together by hemi al ross-links. More
details onthis subje t an befound in[26,9℄.
2.2.1 Gaussianstatisti s model
Treloar rstproposedastatisti altreatmentofrubberelasti ity.He onsidered
thatthe on gurationof polymer hains an be des ribed by Gaussian
statis-ti s.Thisleadstothewell-knownneo-Hookean onstitutiveequation[25℄.The
orresponding strainenergy fun tionis:
W = 1 2 nkT(I 1 3) (5)
where n denotes the averagenumberof polymer hains per unit of volume, k
isthe Boltzmann onstant,T isthe absolutetemperatureand nallyI 1
isthe
rst invariant of the stret h tensor, said its tra e. This model only depends
ona unique parameter C R de ned as: C R =nkT (6)
2.2.2 Non-Gaussian statisti s models
Inordertoover omethelimitationsofthepreviousmodelthatisrestri tedto
smallstrains,authorsusenon-Gaussianstatisti stheorytodes ribemole ular
Gaussianstatisti stheorytomodelthelimitedextensionof hains[18℄. Their
approa his based onthe randomwalk statisti s of anideal phantom hain.
Consider a mole ular hain omposed by N monomer segments of length l.
Its average unstret hed length is p
Nl and its maximum stret hed length is
Nl. Moreover, the strain energyfun tion ofthe hainw an bewritteninthe
following form: w=NkT " p N +ln sinh # (7)
where is the extension of the hain and with = L 1 = p N . L 1 is
the inverse of the Langevin fun tion de ned by L(x) = oth(x) 1=x. The
rst Piola-Kir hho stress inthis single hain isobtained by derivation ofthe
strain energy with respe t tothe extension:
t = w =kT p NL 1 p N ! (8)
2.2.2.2 p- hains models In order to develop onstitutive equations, a
network of hains whi h strain energy fun tions are given by (7) should be
onsidered. Re ently, Wu and van der Giessen integrate the previous stress
(8) on a unit sphere de ned by its hains density [32℄. Their full network
model agrees well with experiments but ne essitates numeri al integration
pro edures on the sphere. This numeri al integration does not lead to an
eÆ ient implementationin numeri alsoftwares.
Otherauthorsusedthenon-Gaussianstatisti stheorytodevelopsimpler
mod-els whi h do not need numeri al integration. The most employed are the
3- hains and the 8- hains models. Moreover, we mention the approa hed full
by onsideringprivilegeddire tionsintheunitsphere. Denotingnthe density
ofpolymer hainsasintrodu edfortheneo-Hookeanmodel,itisassumedthat
n=p hainsperunitofvolumeareorientedinea hofthepprivilegeddire tions
inthe undeformed on guration. Then, the previous numeri alintegrationof
the strain energy on the unit sphere is advantageously repla ed by a sum of
p single hain strain energy fun tions weighted by the fa tor n=p. Parti ular
ases of p- hains models are presented inFigure 2.
Thesimplestp- hainsmodelisde nedby onsideringthethreeprin ipalstrain
axes as privileged dire tions. It was derived by James and Guth [13℄ and is
widely known asthe 3- hains model. Figure2(a) presents this model.
Prin i-palstresses(t 3- hains i
) i=1;3
are expressed as fun tionsof prin ipalstret h ratios
( i ) i=1;3 by: t 3- hains i = q i + 1 3 C R p NL 1 i p N ! (9)
where q is the hydrostati pressure introdu ed by the in ompressibility
as-sumption.
A more re ent model based on non-Gaussian statisti s is the 8- hains model
developed by Arruda and Boy e [2℄. Privileged dire tions are de ned by the
half diagonalsof a ube ontained inthe unit sphere. Figure 2(b) shows this
hains distribution of the model. Its major property is its symmetry with
respe t to the three prin ipalaxes. Therefore, the eight hains are stret hed
with the sameextension ratio, h
, that orresponds tothe stret hing of ea h
halfdiagonals of the ube:
h = q ( 2 1 + 2 2 + 2 3 )=3 (10)
t 8- hains i = q i + 1 3 C R p N i h L 1 h p N ! (11)
Finally, we an mention the approa hed full network model proposed by Wu
and vander Giessen [31℄. Authors use a linear ombination of the stresses of
the 3- hainsand 8- hainsmodelstoapproximatethe fullnetwork stressesby:
t fullnetwork i =(1 )t 3- hains i +t 8- hains i (12)
where isa parameter that isrelated tothe maximalprin ipal stret h ratio:
= 0:85 p N max( 1 ; 2 ; 3 ) (13)
in whi h the fa tor 0.85 is hosen to give the best orrelation with the full
integration onthe unit sphere.
3 Solution pro edure
In this part, the resolution pro edure used to integrate the problem (4)
as-so iated with one of the stress-strain relationships de ned above is brie y
presented. For more details,the reader an refer to[29℄.
3.1 Dis retized equationsand solution pro edure
Whatever the interpolation method adopted, the membrane is dis retized by
onsideringn+1nodes, (N i ) i=0;n . Ea hnode N i
isde ned by itsundeformed
ar -length oordinate s i
. Its displa ements in radial and axial dire tion are
respe tivelyu i
andv i
U 2i 1 =u i and U 2i =v i 8i=0;:::;n (14)
Using interpolation formulas (detailed in the next paragraph), the Prin iple
of Virtual Work (4)be omes:
ÆU T [F int (U) F ext (U;p)℄=0 8ÆU T (15) where F int (U) and F ext
(U;p) are the internal and external for es ve tors
re-spe tively, ÆU is the virtual nodal displa ement ve tor and T
denotes the
transposition. F int
is the dis retized ounterpart of the rst left-hand side
term in Eq. (4) and F ext
orresponds to the se ond term in this equation. It
is obvious that the system (15) is highly non-linear both geometri ally and
by the onstitutiveequation. Thus, aniterativemethodmust be employed to
solve this system and the tangent operator must be derived. This operator
is denoted K and represents the derivative of the out-of-balan e for e ve tor
(F int F ext ) with respe t toU: K= F int U F ext U (16)
The detailed ready-to-program formulas of F int
, F ext
and K are given in[23℄
forthe lassi altwonodes membrane niteelementandin[29℄forouroriginal
B-spline interpolation.
As shown by Beatty for stati in ation [4℄ and Verron et al. for dynami
in ation [27℄, the pressure-displa ement urves orresponding to rubber-like
membranesin ationexhibitlimitpoints.Consequentlya ontinuationmethod
has tobeadopted to ompute the equilibriumpath [24℄. Here, the ar -length
method is implemented. It onsists in ompleting the system of equilibrium
the present ase, the displa ement ontrol method is onsidered [3℄ and the
additionalequationis:
kUk 2
da 2
=0 (17)
where U is the displa ement in rement ve tor between two equilibrium
pointsand da is the pres ribed ar -length. This equationis addedtothe
pre-vioussystem (15) asa bordered equation and the assembled system is solved
by the lassi alNewton-Raphson algorithm[21℄.
3.2 B-splines interpolation method
In this study, the membrane is interpolated by two ubi splines de ned on
the set of nodes (N i
) i=0;n
, one for ea h oordinater 0 and z 0 : r 0 (s)= n+1 X i= 1 i B i (s=l 0 ) (18) z 0 (s)= n+1 X i= 1 i B i (s=l 0 ) (19) where ( i ) i= 1;n+1 and ( i ) i= 1;n+1
are the parameters of the two splinesand
fun tions B i
0 B i ()= 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : 0 i 2 ( i 2 ) 3 ( i+1 i 2 )( i i 2 )( i 1 i 2 ) i 2 < i 1 ( i 2) 2 (i ) ( i+1 i 2 )( i i 2 )( i i 1 ) + (i+1 )( i 1 )( i 2 ) ( i+1 i 1 )( i i 1 )( i+1 i 2 ) + ( i 1) 2 (i+2 ) ( i+1 i 2 )( i i 1 )( i+2 i 1 ) i 1 < i ( i+1 ) 2 ( i 2 ) (i+1 i 2
)(i+1 i 1)(i+1 i) + ( i+2 )( i+1 )( i 1 ) (i+2 i 1)(i+1 i 1)(i+1 i)
+ ( i+2 ) 2 ( i ) ( i+2 i 1 )( i+2 i )( i+1 i ) i < i+1 ( i+2 ) 3 ( i+2 i 1 )( i+2 i )( i+2 i+1 ) i+1 < i+2 0 i+2 (20) where i
is the redu ed ar -length oordinate of the node N i
and where the
following onventions are adopted:
i =s 0 =l 0 for i0 (21) i =s n =l 0 for in (22)
With these two onventions some denominators are equal to 0 and we adopt
the equality: 0=0=0.
In Eqs (18, 19), the n+3 parameters of ea h splines are de ned ina unique
mannerusingtheundeformed oordinatesofthen+1nodesandbytakinginto
a ountend onditions todetermine the twoextra parameters. Dependingon
1. if the extreme point does not lies in the symmetry axis, both r 0
and z 0
interpolation admit natural end onditions, i.e. the se ond derivative of
both fun tions are equalto zero atthis point,
2. ifthe extreme point liesinthe symmetry axis,the z 0
-spline admitsnatural
end onditionsbutther 0
-splineadmitsHermiteend onditions,i.e.the rst
derivative of the fun tion is pres ribed at the point and, in our ase, is set
tozero.
These boundary onditions provide two additional linear relations between
parameters for both splines, and interpolationformulas (18,19) redu e to:
r 0 (s)= n X i=0 i B r i (s=l 0 ) (23) z 0 (s)= n X i=0 i B z i (s=l 0 ) (24) where(B r i ) i=0;n and(B z i ) i=0;n
arethenewinterpolationbasis,de nedaslinear
ombinationsof the lassi al B-spline fun tionsB i
.
Similarly to isoparametri al nite elements, the displa ement eld (u,v) is
interpolatedby two splines developed on the same basis fun tions:
u(s)= n X i=0 i B r i (s=l 0 ) (25) v(s)= n X i=0 Æ i B z i (s=l 0 ) (26) inwhi h ( i ) i=0;n and (Æ i ) i=0;n
are the parameters of these twosplines.
Some al ulations are now ne essary to establish the dis retized problem
In this se tion, some numeri alexamples are presented.
Theaimofthispaperisthe omparisonofdi erentnetworkmodels.Thus,the
di erent onstitutive equations have to be tted for the same material using
the same type of experimental data. Here, experimental data orresponding
to the uniaxial extension of a naturalrubber as reported by James et al. are
onsidered[14℄.Moreover, weadoptthematerialparameterspreviously
deter-minedby Wuand vander Giessen[32℄. It istonote that network parameters
N and C R
must be determined for ea h model. But, as the small strain
be-haviouris governed primarilyby C R
, Wu and vander Giessen rst determine
it, and onsider an identi al value of C R
for allmodels, said C R
= 0:4 MPa.
Thenafterwards,the averagenumberof monomersper hain is ttedforea h
onstitutiveequation:N =80forthe3- hains model,N =25forthe 8- hains
model, and nallyN =50for the approa hed fullnetwork model.
4.1 Spheri al balloons
Before examiningthe ase of ir ularmembranes, we onsider the simple
ho-mogeneous in ation of spheri al balloons. This problem was widely studied
for lassi alhyperelasti models, both instati s[4℄and dynami s[27℄. Asthe
deformationremains biaxialinevery materialpoints of the sphere, the
prob-lemredu es toa simple analyti al governing equation that relies the blowing
pressure and the deformed radius. Here, our numeri al approa h is used to
solve the problem and the numeri alresults were su essfully ompared with
equal to1.0 and the uniform thi kness is 0.01. The semi-spheri al ap is
dis- retized using21nodes. It isassumed thatthe balloonremainsspheri al
dur-ing the in ation, so that no bifur ating phenomenon o urs. Consequently,
the extension ratios are:
m = ; = ; = 1 2 (27)
where istheadimensionaldeformedradius r=r 0
.The urvespressure versus
are presented in Figure 3 for the four onstitutive equations studied here.
Values of extension ratios and blowing pressures that orrespond to the two
limitpoints, ( I ;P I ) and ( II ;P II
), are presented in Table 1.As shown
else-where,the neo-Hookeanballoonexhibitsonlyone limitpointthat dividesthe
urve into a stable path and an unstable path. For the three other balloons,
the urvesadmittwolimitpointsandthreepaths,twostable(in reasingparts
ofthe urves) andone unstable(de reasingparts ofthe urves).A similar
be-haviour was highlighted previously for the Mooney-Rivlin model [27℄. Let us
examine more pre isely the rst part of the urves that orrespond to small
strains, i.e. for 2:0. First limit points o ur approximately for the same
values of the extension ratio, said =1:39. This value an be obtained
ana-lyti allyinthe ase ofthe neo-Hookean modeland isequalto 6 p
7.This result
veri esour previous statementthat the smallstrainsbehaviouronly depends
onthematerialparameterC R
,thatisidenti alforthe fourmodels.Forlarger
strains,in ation urves di er. In the ase of the neo-Hookean material, there
isnostrain-hardening(be auseofthe Gaussianassumption)andthe pressure
slowly de reases to zero as the membrane ontinues to in ate. For the three
other models, polymer hains rea h their limit of extensibility for di erent
The limitvalues of the adimensionalradius, i.e.values forwhi h the limiting
stret h of hains is rea hed, an be simply predi ted for both 3- hains and
8- hains models. In the ase of the 3- hains model, the limit of extensibility
is rea hed when one of the extension ratios i
tends to p
N in Eq. (9). Here,
re allingthe equibiaxial deformations (27), we have:
1 = m = ; 2 = = ; 3 = n =1= 2 (28)
Thus the limiting extensibility is obtained for = p
80 8:94. Note that
in our numeri al simulations, we are not able to over ome 8:0 due to
onvergen e problems. In the ase ofthe 8- hainsmodel,the stret hing ofthe
hains h
is de ned by Eq. (10). Using(28), it redu es to:
h = q (2 2 +1= 4 )=3 (29)
and the limit hain extensionis obtained by:
h
= p
N (30)
with N = 25. The solution of this equation is 6:12. This value an be
seenasthe asymptoti verti alline orrespondingtothepressure urveofthe
8- hains modelin Fig.3.Finally,the response of theapproa hed full network
model is onsidered. Asthe modelisalinear ombinationof the 3- hainsand
the 8- hains models, the model rea hes its limiting extensibility when one of
them rea hes its own limit.Thus, onsidering now that N =50, the limiting
extensibility is obtained for = p
50 7:07 for the 3- hains model. For the
8- hains model, Eqs (29) and (30) are solved with N = 50 and the limiting
extensibilityisapproximatelyequalto8:66.Thestresses(12)tendstoin nity
4.2 Cir ular plane membranes
Consider now an initiallyplane ir ularmembrane of radius 1.0and uniform
thi kness0.01.The exterioredge ofthemembraneis lamped.Themembrane
ismeshedusing21nodes.Inthisproblem,thedeformationisnothomogeneous
in the membrane. At the pole (the node inwhi h r =0:), the deformationis
equibiaxial: m ; = m ; n = 1 2 m (31)
and atthe lamped rim,the membrane is under pure shear onditions:
m ; =1 ; n = 1 m (32)
Moreover, it is obvious that the pole lies in the symmetry axis. With our
numeri al approa h, there is no diÆ ulty to satisfy the tangen y ondition,
said r 0
= 0, be ause this boundary ondition is dire tly taken into a ount
in the B r
i
interpolationfun tions (see Se tion 3.2). Using lassi al nite
ele-ments,this ondition an not be stri tly imposed, be ause there is no degree
of freedom that orresponds to r 0
(see [12℄ for example).
First, we study the urves that show the in ating pressure versus the axial
oordinate of the pole (denoted z pole
through the rest of the paper) for the
di erent onstitutive equations. These results are shown in Figure 4. The
orrespondingvaluesoflimitpoints(z poleI ;P I )and(z poleII ;P II )arepresented
inTable 2.Generalaspe ts of the urves are similartothose observed forthe
spheri al balloons with both in reasing and de reasing paths. In the present
Gaussian onstitutive equations is highly remarkable and the limiting hain
extension is rapidly rea hed as it an be shown by the nal verti al parts of
urves presented in Fig. 4.
Letusnowexaminethein ationpro lesshown inFigure5.Forthe four
mod-els,wepresentpro lesthat orrespondtoz pole
=1.0,2.0,3.0and4.0.Moreover,
the last pro les obtained before numeri al divergen e are also drawn for the
non-Gaussian onstitutive equations.It an beseen that forz pole
equalto1.0
and 2.0 the pro les orresponding to the four models are similar.
Neverthe-less, for higher values of the axial oordinateof the pole, said z pole
3:0, the
pro les orrespondingtothe three non-Gaussianmodelsare similar,but they
highly di er from the shape of the deformed neo-Hookean membranes. The
neo-Hookean bubble is more "spheri al" than the other ones. This example
showsthatthestrain-hardeningofthematerialhasaverysigni antin uen e
of the shapeof in ated membranes.
Thisremark is on rmedbythe thi kness distributionspresented inFigure6.
The thi kness is plottedasfun tionof theundeformed radiusr 0
. Asobserved
by Hassager et al., the neo-Hookean membrane exhibits a strong thinningat
the pole[12℄.Here, we showthatthe same observations anbe madefor
non-Gaussian statisti almodels. As predi ted by the examination of the pro les,
the threenon-Gaussian thi kness distributionsare almostsimilar atthe pole,
and areof thesame orderofmagnitude alongthe membranefor agiven value
of z pole
. The last pro le of ea h non-Gaussian membrane exhibits the same
thi kness distribution that an be seen as the ultimate state before the
o - urren eof hainsrupture. Moreover, thedi eren ebetween theneo-Hookean
pole
lamped rimare identi al and values at the pole tend tozero (see Fig. 6(a)).
Inthe aseofthethreeothermodels,the thi kness atthe lampedrimevolves
signi antlywithrespe ttoz pole
andthe thinningatthepoleremainssmaller
than inthe neo-Hookean ase.
In order to highlight the in uen e of the strain-hardening modeled by the
inverse Langevin fun tion in onstitutive equations (9), (11) and (12), the
distribution of the in-plane extension ratios m
and
along the membranes
arepresentedinFigure7.First,weverifythatthe ir umferentialstret hratio
variesbetween m
atthe poleand1.0atthe lampedrim.Fig. 7(a)reveals
that,fortheneo-Hookeanbubble,bothdistributionsare similar,aspreviously
demonstrated by Yang and Feng [33℄. Moreover, for z pole
= 5:0, extensions
ratios are equalto 25.0 at the pole. This result has nophysi al meaningand
provestheimportan eofagoodmodelingofstrain-hardeningofelastomersat
large strains. In the ase of the non-Gaussian models, distributions of
for
thethreemodelsare omparableforthesamedeformationlevel.Distributions
of m
arealsosimilaruntilthematerialrea hesitsstrain-hardeningbehaviour.
Inthis ase,the8- hainsmodelsdistribution(Fig.7( )forz pole
=4:02)di ers
fromthe3- hains andthe approa hedfullnetworkdistributions(Fig.7(b) for
z pole
=5:60 and Fig. 7(d) for z pole
=3:87). In the former ase, m
is greater
at the lamped extremity than at the pole; in the later, m
is approximately
uniform for both 3- hains and approa hed full network membranes.This last
observation an be explained by examining the maximum hain extensibility
along the membrane for the three non-Gaussian bubbles. Re all that for the
hmax =max( m ; ; n ) (33)
and in our ase, it is always equal to m
. For the 8- hains model, it is given
by: hmax = q ( 2 m + 2 + 2 n )=3 (34)
and nally,for the approa hed full network model, it is the greater values of
the 3- hains and 8- hains data:
hmax =max( m ; ; n ; q ( 2 m + 2 + 2 n )=3) (35)
and here,itis alwaysequalto m
.In thethree ases,as hmax
tends to p
N,
the material hardens and the asymptoti verti al line in the pressure versus
z pole
diagram is approa hed (Fig. 4). The distributions of hmax
along the
membranesareplottedversustheundeformedradiusinFigure8,forthe three
non-Gaussiannetworkmodels.Theyarehighlysimilarforthethreemodels:at
the beginningof the in ationthe distributionis uniform,then the membrane
extends more signi antly in the neighbourhood of the pole than near the
lampedrim,and nallyforhighervaluesofz pole
the hainextensibility tends
tobe omeuniformandapproa hesthe limitvalue p
N,whilefewevolutionof
strains takes pla e at the pole. This an be seen as the in ationof a ir ular
membrane with a rigid in lusion at the pole, onsidering that the radius of
this in lusionin reases duringthe in ation.
5 Con luding remarks
In this paper, we have presented some results on erning the in ation of
onsti-that was proved to be very eÆ ient for large strains in ation of
axisymmet-ri membranes. The non-Gaussian models are implemented and ompared
with the lassi alneo-Hookean onstitutive equation.The in uen e of
strain-hardening on the membrane response is highlighted. It is shown that both
in ation pro les and thi kness distribution are highly in uen ed by the
na-ture of the materialmodel.
These results are of interest in the ontext of the identi ation of material
parameters in biaxial onditions. In order to perform biaxial experiments on
elastomers or heat-softened plasti s, the bubble in ation te hnique is widely
used. In most of the ases, only experimental data obtained at the pole are
onsidered to determine parameters of onstitutive equations, be ause the
deformationisequibiaxialatthispoint.The presentworkshows thatthe
evo-lutionof the membranepro leduring in ationandsome data on erningthe
thi knessdistributionmaybeofgreatinterestinthe hoi eofthe onstitutive
equationthatmustbeadopted,andinthedeterminationofthe orresponding
material parameters. Thus, the numeri al analysis of the entire membrane is
highlysuitable inthe identi ationpro edure.
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Fig.1.Problem des ription.
Fig.2.Des riptionofthemostemployedp- hainsmodels:(a)3- hains,(b)8- hains.
Fig.3.Pressureversusadimensional deformedradiusforspheri al balloons.
Fig. 4.Pressureversusaxial oordinateof thepolefortheplane membranes.
Fig. 5. In ation pro les of the ir ular plane membranes: (a) neo-Hookean,
(b) 3- hains, ( ) 8- hains, (d) approa hed full network. Numbers on the urves
stand forthe orresponding in ating pressure.
Fig. 6. Thi kness distribution in the ir ular plane membranes: (a) neo-Hookean,
(b)3- hains,( )8- hains,(d)approa hedfullnetwork.Numbersonthe urvesstand
forthe orrespondingvalue ofz pole
.
Fig. 7. Extension ratios in the ir ular plane membranes (|) m
, ()
:
(a)neo-Hookean, (b) 3- hains, ( )8- hains,(d) approa hed fullnetwork.Numbers
on the urvesstandfor the orresponding valueof z pole
.
Fig.8.Maximum hainsextensibilityinthe ir ularplanemembranes(|)
distribu-tioninthemembrane,(--)limitvalue p
N:(a)3- hains,(b)8- hains,( )approa hed
fullnetwork.Numberson the urvesstand forthe orrespondingvalueof z pole
Model I P I II P II Neo-hookean 1.40 4.96e-03
3- hains 1.40 5.04e-03 5.71 1.97e-03
8- hains 1.39 5.13e-03 3.92 2.88e-03
Approa hedfullnetwork 1.39 5.08e-03 5.10 2.26e-03
Table 1
Limitpointsforspheri alballoons.Cir lesrepresenttheanalyti al solutions.
Model z poleI P I z poleII P II Neo-hookean 1.16 7.52e-03
3- hains 1.23 7.73e-03 3.16 6.09e-03
8- hains 1.17 8.01e-03 2.17 7.64e-03
Approa hed fullnetwork 1.26 7.84e-03 2.83 6.66e-03
Table 2
