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Tensile stability of cylindrical membranes
0020 7462'X353oO+.CM Prrp.mm rrr\*LIJ
TENSILE
STABILITY
OF CYLINDRICAL A. M.
Civil Engineering
Department,
MEMBRANES
RATNER
Stevens Institute of Technology,
Hohohen.
NJ 07030.
U.S.A.
Abstract-The tensile stability of rotationally symmetric thin membranes composed of isotropic, incompressible and elastic materials is considered by investigating under what conditions the initial equilibrium conliguration can hirurcatc IO anorhcr rotationally symmetric equilibrium mode. The general cqurlihrium equation\ of a rotationally symmetric membrane are first derived in cylindrical coordinates. The initially cylindrical membrane is studied. The classic solution. which assumes homogeneous deformations. is shown IO be a special case ofthe general equations. Perturbation theory is employed lo find the bifurcation points Born the homogeneous mode. This study shows that, (or the chosen boundary conditions. no rotationally symmetric equilibrium mode exists near the cylindrical mode except the cylindrical mode itself. This corresponds IO all experimental data that the author is aware of. The initially cylindrical membrane either remains cylindrical or goes into a non-rotationally symmetric mode.
INTRODUCTION
theory has been widely used as a first approximation in the analysis of membranes (such as tents and balloons) and shells, and in various analogs to solve other problems, for example the torsion problem in classical elasticity. In many (if not most) problems non-linearity of deformations are important. The classic reference in non-linear membrane theory is Green and Adkins [l]. Currently there are many important problems involving membranes. They include fluid containers, weather balloons, inflatable structures, oil slick containment apparatus, and many problems in biomedical engineering. In all of these the non-linearities of deformation and material response are very important. While much work has been done on non-linear membranes, most investigators have been concerned with finding equilibrium configurations. The present study investigates the stability of the equilibrium configurations. Rotationally symmetric thin membranes composed of isotropic, incompressible, elastic material under uniform internal pressure and axial forces are considered. (Elastic is used here to denote completely recoverable deformation, the constitutive equation is not necessarily linear.) Although this reduces the generality of the equations derived, the methods employed can be used in the analysis of much more complex problems without extensive modification. The theorem of stationary potential energy is used to derive the general equilibrium equations for a rotationally symmetric membrane in cylindrical coordinates, using the deformation quantities as the variables with a general constitutive relation. One specific problem, the initially cylindrical membrane is then studied. The classic solution to this problem which assumes no change in mode (shape) is rederived from the general equations. The stability of the solution is then investigated, noting that the physically observed configurations point toward a bifurcation type instability. Instability is used here to denote a change in mode. Perturbation theory is used to determine the bifurcation points from the classic, “zero order” solutions. Particular attention is paid to the parameter perturbed. Care is also taken not to linearize the equations prematurely. Both of these points are explained by Soifer [Z] and Kerr and Soifer [3]. Classical
membrane
EQUILIBRIUM
EQUATIONS. ROTATIONALLY SYMMETRIC IN CYLINDRICAL COORDINATES
MEMBRANE
Since the materials of greatest interest in large deformation membrane problems are elastomers (i.e. rubber-like materrals) it is assumed that the material is isotropic, elastic, 133
A. M. RATNEW
I.34
and incompressible. These assumptions are justified by experimental results. (See Alexander c41, C51). With the above assumptions Rivlin ([6] equation 6.3) showed that the most general constitutive relation can be derived from a strain energy density function, W, as, c!W
oi=2 [
1 dW
j&f----j.f21, 81,
(1)
+p
I
where gi is the principal true stress, p is an arbitrary hydrostatic pressure (which does not affect the deformations due to incompressibility), i.i is the principal extension ratio in the i direction, and I, and 1, are the first and second invariants of the extension tensor respectively. They are defined as,
As was shown by Alexander [43, [S], the indeterminancy eliminated by use of the stress deviator, Cij, defined as,
aij
=
aij
--6ija~l,
introduced
by p can be
1 3
which make the constitutive relation (1) (3) with the indices cyclic in the form 1, 2, 3, 1, 2, 3, etc. In the usual cylindrical coordinate system X = rcose
Y = rsintl
(4)
z = z. If s is the arc length along the undeformed membrane and S the arc length along the deformed membrane, in a rotationally symmetric system a( ) = 0 and ( do
-
a( ) )’ = ds.
It is convenient to use s as the independent variable. The position of any point on the membrane is specified by the coordinates r(s) and z(s). In the undeformed state r(s) = r,(s) and z,(s). See Fig. 1. Then
Therefore: (d$12= (dr2 + dz2)
2: =
(r’)2
+ (+!T’)~
Tensile stability
of cylindrical
ROTATIONALLY
SYMMETRIC
135
membranes
LL MEMBRANE
I
DEFORMED MEMBRANE
UNDEFORMED
ho ro
Fig. I.
-
r
.
x2
=
‘heridional
=
-
0 r.
(6) j,22-
0
2
L
r,
From the condition of incompressibility
r’
From the above it is obvious that the strain energy density, W, is a function of z’, r, and only. Therefore the elastic strain energy, E, is the volume integral of W, which becomes,
+ s b/2
E=
21rr, W(z', r, r’)h,ds
(8)
+w2
where h, is the initial thickness of the membrane and 1, is the total initial length. The work term, P, is, P
=
PA
v
+
W,,2
-
W
+ 1,/z
+
Fzol- 1,/2
-
Fzl - l,/2
(9)
where F is the axial load and p is the uniform internal pressure. AV is the volume change of the membrane, i.e. the volume enclosed by the deformed membrane minus the volume enclosed by the undeformed membrane. s=
AV=
V-
V, =
I
+1,/2
I= -la/2
+ s
[nr’ dz - m-,’ dz,].
(10)
Since dz = z’ds, the total work becomes, I,12
P=
- I,12
nplr2z’-riz~lds+Fz/~~,2-Fzl_~~,2-Fz”l~~,2+Fiol_,o,i
(11)
136
M. RATNEH
A.
The total potential cncrgy, ft. is dclincd as the difference between the elastic strain energy and the work, yielding [2nh,r,, W - np(r2z’ - t-f&)] ds
rl=E-p= I,,/2
(12)
By the principle of stationary
potential energy, for equilibrium configurations al
= 0.
(13)
Taking the first variation of (12), +L/2 6rl=
[2nh,r,( - 1,312
WZ. 6~’ + W,. br’ + W, 6r)
(14) - np(r2 62’ + 2rz’ dr)]
ds - F bz
+F6z
II,/2
where (
(?( 1
=0
I-
I,‘2
etc.
),. = F,
Integrating by parts,
si + I”12
6rl
=
2n [ - (h,r, W,,)‘] 6: + 271[- (hot-, W,.y + hero W,] 6r
- I,>/2
-
2np [ - rr' Sz + rz’ 6r] 1 ds + (2rch,r, W,. - npr2 - F) 6z
(1% I,/2
+ L/2 +
= 0
(27&r, W,.)& - I”/2
from which of the following differential equations and boundary conditions are obtained by the fundamental lemma of variational calculus: Differential equations: (h,r, W,.)’ = prr’
(164
hero W, - (h,r, W,.)’ = prz’.
(16b)
Boundary conditions:
I(
2nr,WZ.-pr2-5
+L/2 7T)
6z
=O
(174
I -L/2
(17b) These are the general equilibrium equations and boundary conditions for an axially symmetric membrane of an isotropic, incompressible, elastic material with a uniform internal pressure. The strain invariants are defined by equation (2). Making use of equations (5) and (6),
Tensile stability of cylindrical
137
membranes
(19) From equations (18) and (19), with some slight manipulation
w,. -=
we obtain:
w_.
(20)
L
r’
2‘
as was also noted by Isaacson [7] and Alexander [4] and is valid for any form of W, using this equation (16a) can be integrated directly giving,
where A, is a constant of integration. Using the last two equations, equation (16b) can be expressed as, horoW,-[J$$+~]=prz.
(22)
The boundary conditions (equation 17) become, (23a)
GW To apply these equations we must assume a form for the strain energy density function, which effectively specifies the membrane skin material. Assuming the membrane skin is an elastomer, an accurate strain energy density function is that presented by Alexander [S] equation (53) which in dimensionless form is, W=f
c,~ek(‘~-3)*dl, +P,In c”2-:‘+7]
+F,I11-3)l
and therefore,
aw -c,e p-
-
al,
=
dW dl,
P =
k(l,-3)’
2
T
1 (I2
c2 -
3) + i’ + c3I .
(25)
The problem of a rotationally symmetric membrane of isotropic incompressible elastic material, loaded with a uniform internal pressure and an axial load, is now completely formulated in cylindrical coordinates. The rotationally symmetric deformed shape of this membrane is not necessarily the same as the undeformed shape. INITIALLY
CYLINDRICAL
MEMBRANE
In this section deformation of an initially cylindrical membrane due to a uniform internal pressure and an axial load is considered, subject to the same material assumptions as in
13x
A. M.
RATNEK
the previous section. Since a perturbation analysis is used the “zero order” solution is needed, i.e. the solution for the case where the membrane stays cylindrical. This is the solution whose stability is being tested. Therefore the first task is to solve the problem of the deformation of a cyiindrica~ (not only in~tial[y, but always cyljndrical) membrane in large deformations. The equations derived earlier can be specialized to solve this problem. First, due to the initial configuration being a cylinder, it is seen that
r-6= 0 rb’ = 0.
(26)
Now assuming the membrane is always a cylinder,
r’ = 0 f‘
= 0
(27)
and I, = 2’ =
c0n.g
= f *
Using equations (28) in equations (21) and (22) along with the strain derivative definitions yields for the governing equations:
~(~~+~=2[~-~~~~~]~+2[(~)it-~~~]~
(2%)
and
f29b) where
(304
Tensile stability of cylindrical and
the
boundary
conditions
membranes
139
(23) bccomc,
(31)
where F is the axial force. The second boundary condition is identically satisfied along the whole cylinder. Applying the first in (29a) and using (29b) to eliminate p yields,
+ 2g[;(;>‘;
- (;)a
+ ;(:)‘;I.
(32)
Therefore, in dimensionless form, (29b) and (32) become,
P* =f(~-&){C,exp[k(i*2+r*z+r*21* c2
+ P2 [
pzf42
+
r*-2
+
c,
+
p-2
C,
exp [k(I*2 + r*’ + r*-2/*-3
+3+y
I)
(33)
and
,*-&A
- 3)2]
where r* =
F*
.L. r,
(354
F
= -
6pnroho P
pro
* _
-5g-(
(354
These relations can be checked against the results of a simple force equilibrium approach to the problem, which works for this case due to its simple geometry [8]. Using the standard approximation of membrane theory that 0, is negligible, we see (Fig. 2) that,
Lr#J =
Pr
-
h
(36)
A. M.
140
Fig. 2. Cylindrical
membrane
,tATN,X
loaded by a uniform internal pressure and an axial force.
a_=P’+ -
211
F
rn’
(37)
Using equation (3) and the definition of 5 as ii,, =
where fiij is the Kronecker
rJij
-
ldijq,, 3
delta, and equation (28) and the fact that
5, =
+oe +
a, = $25
a,)
- a,)
a, = $20, - a:) and the definitions of the strain invariants equations (2), equations (33) and (34) are obtained as before. Figure 3 shows p* vs r* for various values of F *. Note that the relationship is highly non-linear. Now to investigate the stability of the cylindrical equilibrium configuration as given by equations (33) and (34) perturbation theory will be used.
Tensile stability of cylindrical
membranes
141
P’
0 F*= 0.0, 04
c
i? F’=O
F+=0216,
434,
0
F’z0644
(From
Ref
61
/
I .o
2.0
30
Fig. 3. Pressure vs radius characteristic
4.0
50
60
70
for thin walled latex tube at various values of axial load
The basis of perturbation theory is that any parameter, y may be written as 4 + 4, where 4 is a known quantity or function. This is called perturbing (1,and q is called the perturbation quantity. This yields a new equation for q’ that may be easier to solve than the original equation. An order of magnitude analysis on q’may also yield valuable results. (See Roorda and Chilver 193.) In this problem linearized perturbation theory is used. In this q is written as 4 + 4’ and q’ is assumed to be very small, and 4’& 4. The resulting equation is then linearized in q’. It is very important not to linearize 4. Soifer [2] showed that linearizing 4 gives highly inaccurate results. This investigation is concerned with where non cylindrical axisymmetric solutions exist near the cylindrical solution, and specifically where such solutions bifurcate from the cylindrical, or zero order solution. Linearized pertubation theory is well suited to finding these bifurcation points. As was shown by Soifer 123 the choice of perturbation parameters is important. In general the “best results” are obtained by perturbing the parameters which specify the configuration most simply. In this problem these are the cylindrical coordinates (r, z). The initially cylindrical membrane is perturbed, i.e. r=i+r’
z=f+i.
(38)
This implies r’=T+f ).n
=
r””
y
=
f’
+
+
r’”
2
2” = 2” + z”‘,
(39)
from which, using the cylindrical membrane zero order solution z” =
A,
i:
-_=)
r. r = *,I
‘B
r”
=
2”
=
0
(40)
142
A. M. RATNEW
yields z’ = 2; + $’
=
z
Z’
r = roi8 +
r’
r’ = J r” = J’.
(41)
These expressions are now plugged into equations (18)-(25), and quantities of second order or higher in F, J, r”, F’, and I are neglected. Terms containing powers ofthe perturbed quantities are written in a binomial series. Thus,
(l+i 1-‘=;-n(l_$+j,o.)
r -n=(ro+fl-n=~-n
(42)
where h.o. stands for higher order terms in the tilde quantities which are ignored, and n is any positive or negative interger. Similarly z
t-n
=
(z”‘+
fry”
=
it-.
(, +;)-” =i.“(l -$+h.,).
(43)
Equation (21) becomes
(44) which implies
(45) which is the zero order differential equation and p
_Pf_f *’ - h,r,'
Equation (22) becomes
h,r,(Gr+ @A-
p(r’+f12J
2(? + Z)
+-
A,7
.? + F
(46)
1= ’
p(; + F)(f’ + 3).
(47)
Going through the necessary algebra (p, - 3) = i’2 + (J’+(-r+3 I.
r; =2i.[*-(-$J]i.-2(-&j I, - 3 = (P, - 3) + I;
(48)
(49) (50)
Tensile
stability
of cylindrical
143
membranes
(51)
aw
SW
(>
dl,= dl,
+ /Iv, (r”, - 3)ek”1- 3’*1;
(45)q* _(.!.)-4~l-2]
(53)
~=(~)+g$+;(~+~-~) 81 -! = 2z”’ I --& 2 ( (t&
4r2J % -g > + A+21 3393
(52)
(54
(55) ) I+,, (
3r,Z rz
(56) > (57)
(58)
(59)
(60)
(61)
(62) 4z”‘H 61 -+_+-_ rt jr4
2rz.T
(63)
f2 (64)
(65)
From equation (22)
w,= (gy$) etc. The system is made dimensionless solution) by defining,
+($($)
(consistent with the final form of the zero order
s*
2 10
NLM
Vol. 18. No. 2-D
(66)
144
A. M. RATN~;R
(67) Letting
and
p
1 dr’ = --
rds r’*”
1 d2?
= --
ads’
(70) Then equation (2 I) becomes
(B, - 2p*&)r"* -t B2t*’ =
0
(71)
and equation (22) becomes * 4
0 z
2 ap*z*”
$
I@*’
+ $.iip*
z where p* and F* are defined by equation (35) and,
$3F*)F’=
0
(72)
Tensile stability of cylindrical
+ (- c22&i.,
B,
=
c,
(f2 2 3) + y + ”
&WI
2 3P
1 + [
&
145
-.
>
c21.2ro
- [(l,
2k(I,
+
membranes
3 (i.: - i;“) 2 3) + ;1] ( ?r >
4. 3)(i,
- j.i 3i.;2)2
‘B -_
(75)
1 (76)
The boundary conditions (equation (23)) become at s* = fi.z, P = 0 f*’
-- 0 .
(77)
Solving equation (71) for Z*’ and plugging into equation (72) yields c?Y’,J*” + aY,J*’
+ \ycr’* = 0
(78)
where
Yc = 2i,p*
- B, + 2(2&p*
- 8,).
(79)
2
This is a second order, homogeneous, constant coefficient differential equation with homogeneous boundary conditions (equation (77)). The solution form is r’* = 1$,&1~*+ #j2eP2s+
(gOa)
r’* = ~$~e~‘* + d2s*eps*
POW
if PI # P2 or
if /I, = p2 = j?. For a nontrivial solution to exist, the Wronskian of the system, A@*), must equal zero.
A(s*) = ~,+ZeP1s*ePzs*(/.12 - /II) = 0
(82)
which implies /3r = p2. Therefore A(s*) = ~,42ePs*[(s*/I + 1) - s*/I] = 0
(83)
146
A. M. RATNIX
which implies either C#I,= 0 or $z = 0. From the boundary conditions (equation 77) either of these requires the other to also equal zero, and therefore r’* = 0. which is the trivial solution. Therefore no nontrivial solution exists. To restate the result obtained above, an initially cylindrical thin membrane of an isotropic. incompressible, elastic material, inflated by a uniform internal pressure and an axial load, with the ends free to expand, has no rotationally symmetric equilibrium configuration near the cylindrical mode except the cylindrical mode itself. DISCUSSION
OF
RESULTS
The results obtained indicate that if a bifurcation to a non-homogeneous state of stress occurs in a cylindrical membrane under uniform internal pressure and axial load the membrane must go into a non-rotationally symmetric mode. This is what is observed in practice. Tests of cylindrical mcmbrancs by Alexander [4], [8] have shown the passing into a non-rotationally symmetric mode upon tensile instability. Yang and Feng [IO] have found a rotationally symmetric noncylindrical mode for an initially cylindrical membrane by numerical methods, but the boundary conditions they assumed were very different, r = r. at the ends, rather than r’ = 0 at the ends as was assumed in this study. Physically their assumption is equivalent to holding the ends clamped (with fixed clamps at the initial position). Therefore the membrane is forced to leave the cylindrical mode if there is to be any expansion. This is not a bifurcation from the cylindrical mode, but a different problem with a different zero order mode. Since the observed tensile instabilities are not rotationally symmetric, the results of this paper cannot predict when they will occur, but the method employed can be used to solve the problem of the initially cylindrical membrane without the assumption of rotational symmetry with some modification, and is a logical extension of this work. The solution to this problem will be of considerable practical importance both in aerospace problems (balloons) and biomedical problems, particularly in the problem of aneurysms, or local instability of blood vessel walls. The method illustrated is applicable to a much wider range of problems in stability. without major modification. REFERENCES 1. A. E. Greene and J. E. Adkins. Laryc Elusr~c Dc~&vwrwm and Non-Linear Continuum Mcchamcs. Clarendon Press, Oxrord ( 1960). 2. M. P. Soiler. Sludies in the stability theory ol’ elastic solids. PhD Dissertation. New York University (1961) 3. M. T. Soifer and A. D. Kerr, The linearization of the prebuckling state and its eNect on the determined instability loads. J. uppl. Mech. 36. 775-783 (1969). 4. H. Alexander, Deformations and stress analysis of balloons. PhD Dissertation. New York University (1967). 5. H. Alexander, A constituative relation for rubber-like materials. Inr. J. Enyng Sci. 6. 549-563 (1968). 6. R. S. Rivlin, Large elastic deformations of isotropic materials. Part IV. Further developments in the general theory. Phil. Truns. Roy. Sot. Lmd., Series A. 241, 379 ( 1948). 7. E. Isaacson, The shape of a balloon. Comntuns pure appl. Mech. 18. 163 (1965). 8. H. Alexander, The tensile instability or an intlated cylindrical membrane as affected by axial load. Inr. J. nwch. Sci. 13, 87-99 (1971). 9. J. Roorda and A. H. Chilver, Frame buckling. an illustration of the perturbation technique. Inf. J. Non-Linrur Mech. 5, 235-246 (1970). 10. W. H. Yang and W. W. Feng. On axisymmetric deformations of non-linear membranes. J. appl. Mech. 37. 1002-1011 (1970).
Tensile stahllity of cylindrical
membranes
On considsre la stabiliti en tension de membranes minces symgtriques par rotation et colnposees de mat6riaux isotropes, incompressibles et ilastiques en examinant sous quelles conditions la configuration d’&quilibre initiale peut devier vcrs un autrc rnodc d’equilibre cgalement sym&triqut par rotation. On e’tablit d’abord les 6quations d’equilibre generaies d’une telie membrane en coordonnies cylindriques. On e’tudie d’abord la membrane initialement cylindrique. On montre que la solution classique, qui suppose des d6formations homog;nes, est un cas particulier des Equations g6ne’rales. On uti I ise la theorie des perturbations pour trouver les points de deviation par rapport au mode homogsne. Cette 6tude montre que, pour les conditions aux limites choisies, il n’existe pas de mode d’iquilibre symitrique par rotation au voisinage du mode cylindrique mis a part le mode cylindrique lui-mcme. Ceci correspond a toutes les don&es expgrimentales dont I’auteur a connaissance. La membrane initialement cylindrique soit reste cylindrique soit prend un mode non symitrique par rotation. Zusammenfassung: Die Stabilitaet unter Zug rotationssymmetrischer duenner Mcmbranen. deren Material isotropisch, inkompressibel und elastisch ist, wi rd untersucht. Dabei wird beruecksichtigt unter welchen Bedingungen die urspruengliche Gleichgewichtsanordnung in eine andere rotationssymmetrische Gleichgewichtsform abzweigen kann. Zuerst werden die allgemeinen Gleichgewichtsgleichungen einer rotationssymmetrischen Nembran in zylindrischen Koordinaten hergeleitet. Die urspruenqliche zylindrische Membran wi rd untersucht. Es wird gezeigt, dass die klassische Loesung, fuer die homogen Verformungen angenommen werden, ein Spezialfall der allgemeinen Gleichungen ist. Die Perturbationstheorie wird verwendet. urn die Abzweigungspunkte von der homogenen Form zu finden. Diese Untersuchung zeigt, dass fuer die gcwaehlten Randbedingungen keine rotationssymmetrische Gleichgewichtsform in der Nachbarschaft der zylindrischen Form existiert ausser der zylindischen Form selbst. Dies Tatsache stimmt mit allen dem Verfasser bekannten experimentellen Ergebnissen ueberein. Die urspruenglich zylindrische Membran bleibt entweder zylindrisch oder geht in eine nichtrotationssymmetrische Form ueber.
147
TENSILE
STABILITY
OF CYLINDRICAL A. M.
Civil Engineering
Department,
MEMBRANES
RATNER
Stevens Institute of Technology,
Hohohen.
NJ 07030.
U.S.A.
Abstract-The tensile stability of rotationally symmetric thin membranes composed of isotropic, incompressible and elastic materials is considered by investigating under what conditions the initial equilibrium conliguration can hirurcatc IO anorhcr rotationally symmetric equilibrium mode. The general cqurlihrium equation\ of a rotationally symmetric membrane are first derived in cylindrical coordinates. The initially cylindrical membrane is studied. The classic solution. which assumes homogeneous deformations. is shown IO be a special case ofthe general equations. Perturbation theory is employed lo find the bifurcation points Born the homogeneous mode. This study shows that, (or the chosen boundary conditions. no rotationally symmetric equilibrium mode exists near the cylindrical mode except the cylindrical mode itself. This corresponds IO all experimental data that the author is aware of. The initially cylindrical membrane either remains cylindrical or goes into a non-rotationally symmetric mode.
INTRODUCTION
theory has been widely used as a first approximation in the analysis of membranes (such as tents and balloons) and shells, and in various analogs to solve other problems, for example the torsion problem in classical elasticity. In many (if not most) problems non-linearity of deformations are important. The classic reference in non-linear membrane theory is Green and Adkins [l]. Currently there are many important problems involving membranes. They include fluid containers, weather balloons, inflatable structures, oil slick containment apparatus, and many problems in biomedical engineering. In all of these the non-linearities of deformation and material response are very important. While much work has been done on non-linear membranes, most investigators have been concerned with finding equilibrium configurations. The present study investigates the stability of the equilibrium configurations. Rotationally symmetric thin membranes composed of isotropic, incompressible, elastic material under uniform internal pressure and axial forces are considered. (Elastic is used here to denote completely recoverable deformation, the constitutive equation is not necessarily linear.) Although this reduces the generality of the equations derived, the methods employed can be used in the analysis of much more complex problems without extensive modification. The theorem of stationary potential energy is used to derive the general equilibrium equations for a rotationally symmetric membrane in cylindrical coordinates, using the deformation quantities as the variables with a general constitutive relation. One specific problem, the initially cylindrical membrane is then studied. The classic solution to this problem which assumes no change in mode (shape) is rederived from the general equations. The stability of the solution is then investigated, noting that the physically observed configurations point toward a bifurcation type instability. Instability is used here to denote a change in mode. Perturbation theory is used to determine the bifurcation points from the classic, “zero order” solutions. Particular attention is paid to the parameter perturbed. Care is also taken not to linearize the equations prematurely. Both of these points are explained by Soifer [Z] and Kerr and Soifer [3]. Classical
membrane
EQUILIBRIUM
EQUATIONS. ROTATIONALLY SYMMETRIC IN CYLINDRICAL COORDINATES
MEMBRANE
Since the materials of greatest interest in large deformation membrane problems are elastomers (i.e. rubber-like materrals) it is assumed that the material is isotropic, elastic, 133
A. M. RATNEW
I.34
and incompressible. These assumptions are justified by experimental results. (See Alexander c41, C51). With the above assumptions Rivlin ([6] equation 6.3) showed that the most general constitutive relation can be derived from a strain energy density function, W, as, c!W
oi=2 [
1 dW
j&f----j.f21, 81,
(1)
+p
I
where gi is the principal true stress, p is an arbitrary hydrostatic pressure (which does not affect the deformations due to incompressibility), i.i is the principal extension ratio in the i direction, and I, and 1, are the first and second invariants of the extension tensor respectively. They are defined as,
As was shown by Alexander [43, [S], the indeterminancy eliminated by use of the stress deviator, Cij, defined as,
aij
=
aij
--6ija~l,
introduced
by p can be
1 3
which make the constitutive relation (1) (3) with the indices cyclic in the form 1, 2, 3, 1, 2, 3, etc. In the usual cylindrical coordinate system X = rcose
Y = rsintl
(4)
z = z. If s is the arc length along the undeformed membrane and S the arc length along the deformed membrane, in a rotationally symmetric system a( ) = 0 and ( do
-
a( ) )’ = ds.
It is convenient to use s as the independent variable. The position of any point on the membrane is specified by the coordinates r(s) and z(s). In the undeformed state r(s) = r,(s) and z,(s). See Fig. 1. Then
Therefore: (d$12= (dr2 + dz2)
2: =
(r’)2
+ (+!T’)~
Tensile stability
of cylindrical
ROTATIONALLY
SYMMETRIC
135
membranes
LL MEMBRANE
I
DEFORMED MEMBRANE
UNDEFORMED
ho ro
Fig. I.
-
r
.
x2
=
‘heridional
=
-
0 r.
(6) j,22-
0
2
L
r,
From the condition of incompressibility
r’
From the above it is obvious that the strain energy density, W, is a function of z’, r, and only. Therefore the elastic strain energy, E, is the volume integral of W, which becomes,
+ s b/2
E=
21rr, W(z', r, r’)h,ds
(8)
+w2
where h, is the initial thickness of the membrane and 1, is the total initial length. The work term, P, is, P
=
PA
v
+
W,,2
-
W
+ 1,/z
+
Fzol- 1,/2
-
Fzl - l,/2
(9)
where F is the axial load and p is the uniform internal pressure. AV is the volume change of the membrane, i.e. the volume enclosed by the deformed membrane minus the volume enclosed by the undeformed membrane. s=
AV=
V-
V, =
I
+1,/2
I= -la/2
+ s
[nr’ dz - m-,’ dz,].
(10)
Since dz = z’ds, the total work becomes, I,12
P=
- I,12
nplr2z’-riz~lds+Fz/~~,2-Fzl_~~,2-Fz”l~~,2+Fiol_,o,i
(11)
136
M. RATNEH
A.
The total potential cncrgy, ft. is dclincd as the difference between the elastic strain energy and the work, yielding [2nh,r,, W - np(r2z’ - t-f&)] ds
rl=E-p= I,,/2
(12)
By the principle of stationary
potential energy, for equilibrium configurations al
= 0.
(13)
Taking the first variation of (12), +L/2 6rl=
[2nh,r,( - 1,312
WZ. 6~’ + W,. br’ + W, 6r)
(14) - np(r2 62’ + 2rz’ dr)]
ds - F bz
+F6z
II,/2
where (
(?( 1
=0
I-
I,‘2
etc.
),. = F,
Integrating by parts,
si + I”12
6rl
=
2n [ - (h,r, W,,)‘] 6: + 271[- (hot-, W,.y + hero W,] 6r
- I,>/2
-
2np [ - rr' Sz + rz’ 6r] 1 ds + (2rch,r, W,. - npr2 - F) 6z
(1% I,/2
+ L/2 +
= 0
(27&r, W,.)& - I”/2
from which of the following differential equations and boundary conditions are obtained by the fundamental lemma of variational calculus: Differential equations: (h,r, W,.)’ = prr’
(164
hero W, - (h,r, W,.)’ = prz’.
(16b)
Boundary conditions:
I(
2nr,WZ.-pr2-5
+L/2 7T)
6z
=O
(174
I -L/2
(17b) These are the general equilibrium equations and boundary conditions for an axially symmetric membrane of an isotropic, incompressible, elastic material with a uniform internal pressure. The strain invariants are defined by equation (2). Making use of equations (5) and (6),
Tensile stability of cylindrical
137
membranes
(19) From equations (18) and (19), with some slight manipulation
w,. -=
we obtain:
w_.
(20)
L
r’
2‘
as was also noted by Isaacson [7] and Alexander [4] and is valid for any form of W, using this equation (16a) can be integrated directly giving,
where A, is a constant of integration. Using the last two equations, equation (16b) can be expressed as, horoW,-[J$$+~]=prz.
(22)
The boundary conditions (equation 17) become, (23a)
GW To apply these equations we must assume a form for the strain energy density function, which effectively specifies the membrane skin material. Assuming the membrane skin is an elastomer, an accurate strain energy density function is that presented by Alexander [S] equation (53) which in dimensionless form is, W=f
c,~ek(‘~-3)*dl, +P,In c”2-:‘+7]
+F,I11-3)l
and therefore,
aw -c,e p-
-
al,
=
dW dl,
P =
k(l,-3)’
2
T
1 (I2
c2 -
3) + i’ + c3I .
(25)
The problem of a rotationally symmetric membrane of isotropic incompressible elastic material, loaded with a uniform internal pressure and an axial load, is now completely formulated in cylindrical coordinates. The rotationally symmetric deformed shape of this membrane is not necessarily the same as the undeformed shape. INITIALLY
CYLINDRICAL
MEMBRANE
In this section deformation of an initially cylindrical membrane due to a uniform internal pressure and an axial load is considered, subject to the same material assumptions as in
13x
A. M.
RATNEK
the previous section. Since a perturbation analysis is used the “zero order” solution is needed, i.e. the solution for the case where the membrane stays cylindrical. This is the solution whose stability is being tested. Therefore the first task is to solve the problem of the deformation of a cyiindrica~ (not only in~tial[y, but always cyljndrical) membrane in large deformations. The equations derived earlier can be specialized to solve this problem. First, due to the initial configuration being a cylinder, it is seen that
r-6= 0 rb’ = 0.
(26)
Now assuming the membrane is always a cylinder,
r’ = 0 f‘
= 0
(27)
and I, = 2’ =
c0n.g
= f *
Using equations (28) in equations (21) and (22) along with the strain derivative definitions yields for the governing equations:
~(~~+~=2[~-~~~~~]~+2[(~)it-~~~]~
(2%)
and
f29b) where
(304
Tensile stability of cylindrical and
the
boundary
conditions
membranes
139
(23) bccomc,
(31)
where F is the axial force. The second boundary condition is identically satisfied along the whole cylinder. Applying the first in (29a) and using (29b) to eliminate p yields,
+ 2g[;(;>‘;
- (;)a
+ ;(:)‘;I.
(32)
Therefore, in dimensionless form, (29b) and (32) become,
P* =f(~-&){C,exp[k(i*2+r*z+r*21* c2
+ P2 [
pzf42
+
r*-2
+
c,
+
p-2
C,
exp [k(I*2 + r*’ + r*-2/*-3
+3+y
I)
(33)
and
,*-&A
- 3)2]
where r* =
F*
.L. r,
(354
F
= -
6pnroho P
pro
* _
-5g-(
(354
These relations can be checked against the results of a simple force equilibrium approach to the problem, which works for this case due to its simple geometry [8]. Using the standard approximation of membrane theory that 0, is negligible, we see (Fig. 2) that,
Lr#J =
Pr
-
h
(36)
A. M.
140
Fig. 2. Cylindrical
membrane
,tATN,X
loaded by a uniform internal pressure and an axial force.
a_=P’+ -
211
F
rn’
(37)
Using equation (3) and the definition of 5 as ii,, =
where fiij is the Kronecker
rJij
-
ldijq,, 3
delta, and equation (28) and the fact that
5, =
+oe +
a, = $25
a,)
- a,)
a, = $20, - a:) and the definitions of the strain invariants equations (2), equations (33) and (34) are obtained as before. Figure 3 shows p* vs r* for various values of F *. Note that the relationship is highly non-linear. Now to investigate the stability of the cylindrical equilibrium configuration as given by equations (33) and (34) perturbation theory will be used.
Tensile stability of cylindrical
membranes
141
P’
0 F*= 0.0, 04
c
i? F’=O
F+=0216,
434,
0
F’z0644
(From
Ref
61
/
I .o
2.0
30
Fig. 3. Pressure vs radius characteristic
4.0
50
60
70
for thin walled latex tube at various values of axial load
The basis of perturbation theory is that any parameter, y may be written as 4 + 4, where 4 is a known quantity or function. This is called perturbing (1,and q is called the perturbation quantity. This yields a new equation for q’ that may be easier to solve than the original equation. An order of magnitude analysis on q’may also yield valuable results. (See Roorda and Chilver 193.) In this problem linearized perturbation theory is used. In this q is written as 4 + 4’ and q’ is assumed to be very small, and 4’& 4. The resulting equation is then linearized in q’. It is very important not to linearize 4. Soifer [2] showed that linearizing 4 gives highly inaccurate results. This investigation is concerned with where non cylindrical axisymmetric solutions exist near the cylindrical solution, and specifically where such solutions bifurcate from the cylindrical, or zero order solution. Linearized pertubation theory is well suited to finding these bifurcation points. As was shown by Soifer 123 the choice of perturbation parameters is important. In general the “best results” are obtained by perturbing the parameters which specify the configuration most simply. In this problem these are the cylindrical coordinates (r, z). The initially cylindrical membrane is perturbed, i.e. r=i+r’
z=f+i.
(38)
This implies r’=T+f ).n
=
r””
y
=
f’
+
+
r’”
2
2” = 2” + z”‘,
(39)
from which, using the cylindrical membrane zero order solution z” =
A,
i:
-_=)
r. r = *,I
‘B
r”
=
2”
=
0
(40)
142
A. M. RATNEW
yields z’ = 2; + $’
=
z
Z’
r = roi8 +
r’
r’ = J r” = J’.
(41)
These expressions are now plugged into equations (18)-(25), and quantities of second order or higher in F, J, r”, F’, and I are neglected. Terms containing powers ofthe perturbed quantities are written in a binomial series. Thus,
(l+i 1-‘=;-n(l_$+j,o.)
r -n=(ro+fl-n=~-n
(42)
where h.o. stands for higher order terms in the tilde quantities which are ignored, and n is any positive or negative interger. Similarly z
t-n
=
(z”‘+
fry”
=
it-.
(, +;)-” =i.“(l -$+h.,).
(43)
Equation (21) becomes
(44) which implies
(45) which is the zero order differential equation and p
_Pf_f *’ - h,r,'
Equation (22) becomes
h,r,(Gr+ @A-
p(r’+f12J
2(? + Z)
+-
A,7
.? + F
(46)
1= ’
p(; + F)(f’ + 3).
(47)
Going through the necessary algebra (p, - 3) = i’2 + (J’+(-r+3 I.
r; =2i.[*-(-$J]i.-2(-&j I, - 3 = (P, - 3) + I;
(48)
(49) (50)
Tensile
stability
of cylindrical
143
membranes
(51)
aw
SW
(>
dl,= dl,
+ /Iv, (r”, - 3)ek”1- 3’*1;
(45)q* _(.!.)-4~l-2]
(53)
~=(~)+g$+;(~+~-~) 81 -! = 2z”’ I --& 2 ( (t&
4r2J % -g > + A+21 3393
(52)
(54
(55) ) I+,, (
3r,Z rz
(56) > (57)
(58)
(59)
(60)
(61)
(62) 4z”‘H 61 -+_+-_ rt jr4
2rz.T
(63)
f2 (64)
(65)
From equation (22)
w,= (gy$) etc. The system is made dimensionless solution) by defining,
+($($)
(consistent with the final form of the zero order
s*
2 10
NLM
Vol. 18. No. 2-D
(66)
144
A. M. RATN~;R
(67) Letting
and
p
1 dr’ = --
rds r’*”
1 d2?
= --
ads’
(70) Then equation (2 I) becomes
(B, - 2p*&)r"* -t B2t*’ =
0
(71)
and equation (22) becomes * 4
0 z
2 ap*z*”
$
I@*’
+ $.iip*
z where p* and F* are defined by equation (35) and,
$3F*)F’=
0
(72)
Tensile stability of cylindrical
+ (- c22&i.,
B,
=
c,
(f2 2 3) + y + ”
&WI
2 3P
1 + [
&
145
-.
>
c21.2ro
- [(l,
2k(I,
+
membranes
3 (i.: - i;“) 2 3) + ;1] ( ?r >
4. 3)(i,
- j.i 3i.;2)2
‘B -_
(75)
1 (76)
The boundary conditions (equation (23)) become at s* = fi.z, P = 0 f*’
-- 0 .
(77)
Solving equation (71) for Z*’ and plugging into equation (72) yields c?Y’,J*” + aY,J*’
+ \ycr’* = 0
(78)
where
Yc = 2i,p*
- B, + 2(2&p*
- 8,).
(79)
2
This is a second order, homogeneous, constant coefficient differential equation with homogeneous boundary conditions (equation (77)). The solution form is r’* = 1$,&1~*+ #j2eP2s+
(gOa)
r’* = ~$~e~‘* + d2s*eps*
POW
if PI # P2 or
if /I, = p2 = j?. For a nontrivial solution to exist, the Wronskian of the system, A@*), must equal zero.
A(s*) = ~,+ZeP1s*ePzs*(/.12 - /II) = 0
(82)
which implies /3r = p2. Therefore A(s*) = ~,42ePs*[(s*/I + 1) - s*/I] = 0
(83)
146
A. M. RATNIX
which implies either C#I,= 0 or $z = 0. From the boundary conditions (equation 77) either of these requires the other to also equal zero, and therefore r’* = 0. which is the trivial solution. Therefore no nontrivial solution exists. To restate the result obtained above, an initially cylindrical thin membrane of an isotropic. incompressible, elastic material, inflated by a uniform internal pressure and an axial load, with the ends free to expand, has no rotationally symmetric equilibrium configuration near the cylindrical mode except the cylindrical mode itself. DISCUSSION
OF
RESULTS
The results obtained indicate that if a bifurcation to a non-homogeneous state of stress occurs in a cylindrical membrane under uniform internal pressure and axial load the membrane must go into a non-rotationally symmetric mode. This is what is observed in practice. Tests of cylindrical mcmbrancs by Alexander [4], [8] have shown the passing into a non-rotationally symmetric mode upon tensile instability. Yang and Feng [IO] have found a rotationally symmetric noncylindrical mode for an initially cylindrical membrane by numerical methods, but the boundary conditions they assumed were very different, r = r. at the ends, rather than r’ = 0 at the ends as was assumed in this study. Physically their assumption is equivalent to holding the ends clamped (with fixed clamps at the initial position). Therefore the membrane is forced to leave the cylindrical mode if there is to be any expansion. This is not a bifurcation from the cylindrical mode, but a different problem with a different zero order mode. Since the observed tensile instabilities are not rotationally symmetric, the results of this paper cannot predict when they will occur, but the method employed can be used to solve the problem of the initially cylindrical membrane without the assumption of rotational symmetry with some modification, and is a logical extension of this work. The solution to this problem will be of considerable practical importance both in aerospace problems (balloons) and biomedical problems, particularly in the problem of aneurysms, or local instability of blood vessel walls. The method illustrated is applicable to a much wider range of problems in stability. without major modification. REFERENCES 1. A. E. Greene and J. E. Adkins. Laryc Elusr~c Dc~&vwrwm and Non-Linear Continuum Mcchamcs. Clarendon Press, Oxrord ( 1960). 2. M. P. Soiler. Sludies in the stability theory ol’ elastic solids. PhD Dissertation. New York University (1961) 3. M. T. Soifer and A. D. Kerr, The linearization of the prebuckling state and its eNect on the determined instability loads. J. uppl. Mech. 36. 775-783 (1969). 4. H. Alexander, Deformations and stress analysis of balloons. PhD Dissertation. New York University (1967). 5. H. Alexander, A constituative relation for rubber-like materials. Inr. J. Enyng Sci. 6. 549-563 (1968). 6. R. S. Rivlin, Large elastic deformations of isotropic materials. Part IV. Further developments in the general theory. Phil. Truns. Roy. Sot. Lmd., Series A. 241, 379 ( 1948). 7. E. Isaacson, The shape of a balloon. Comntuns pure appl. Mech. 18. 163 (1965). 8. H. Alexander, The tensile instability or an intlated cylindrical membrane as affected by axial load. Inr. J. nwch. Sci. 13, 87-99 (1971). 9. J. Roorda and A. H. Chilver, Frame buckling. an illustration of the perturbation technique. Inf. J. Non-Linrur Mech. 5, 235-246 (1970). 10. W. H. Yang and W. W. Feng. On axisymmetric deformations of non-linear membranes. J. appl. Mech. 37. 1002-1011 (1970).
Tensile stahllity of cylindrical
membranes
On considsre la stabiliti en tension de membranes minces symgtriques par rotation et colnposees de mat6riaux isotropes, incompressibles et ilastiques en examinant sous quelles conditions la configuration d’&quilibre initiale peut devier vcrs un autrc rnodc d’equilibre cgalement sym&triqut par rotation. On e’tablit d’abord les 6quations d’equilibre generaies d’une telie membrane en coordonnies cylindriques. On e’tudie d’abord la membrane initialement cylindrique. On montre que la solution classique, qui suppose des d6formations homog;nes, est un cas particulier des Equations g6ne’rales. On uti I ise la theorie des perturbations pour trouver les points de deviation par rapport au mode homogsne. Cette 6tude montre que, pour les conditions aux limites choisies, il n’existe pas de mode d’iquilibre symitrique par rotation au voisinage du mode cylindrique mis a part le mode cylindrique lui-mcme. Ceci correspond a toutes les don&es expgrimentales dont I’auteur a connaissance. La membrane initialement cylindrique soit reste cylindrique soit prend un mode non symitrique par rotation. Zusammenfassung: Die Stabilitaet unter Zug rotationssymmetrischer duenner Mcmbranen. deren Material isotropisch, inkompressibel und elastisch ist, wi rd untersucht. Dabei wird beruecksichtigt unter welchen Bedingungen die urspruengliche Gleichgewichtsanordnung in eine andere rotationssymmetrische Gleichgewichtsform abzweigen kann. Zuerst werden die allgemeinen Gleichgewichtsgleichungen einer rotationssymmetrischen Nembran in zylindrischen Koordinaten hergeleitet. Die urspruenqliche zylindrische Membran wi rd untersucht. Es wird gezeigt, dass die klassische Loesung, fuer die homogen Verformungen angenommen werden, ein Spezialfall der allgemeinen Gleichungen ist. Die Perturbationstheorie wird verwendet. urn die Abzweigungspunkte von der homogenen Form zu finden. Diese Untersuchung zeigt, dass fuer die gcwaehlten Randbedingungen keine rotationssymmetrische Gleichgewichtsform in der Nachbarschaft der zylindrischen Form existiert ausser der zylindischen Form selbst. Dies Tatsache stimmt mit allen dem Verfasser bekannten experimentellen Ergebnissen ueberein. Die urspruenglich zylindrische Membran bleibt entweder zylindrisch oder geht in eine nichtrotationssymmetrische Form ueber.
147