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Phil. Trans. R. Soc. A (2012) 370, 1896–1911
doi:10.1098/rsta.2011.0297
Effects of imperfections on localized bulging in
inflated membrane tubes
BY Y. B. FU1, *
AND
Y. X. XIE2,3
1 Department
of Mathematics, Keele University, Keele, Staffordshire
ST5 5BG, UK
2 Department of Mechanics, and 3 Tianjin Key Laboratory of Modern
Engineering Mechanics, Tianjin University, Tianjin 300072,
People’s Republic of China
The problem of localized bulging in inflated membrane tubes shares the same features
with a variety of other localization problems such as formation of kink bands in fibrereinforced composites and layered structures. This type of localization is known to be
very sensitive to imperfections, but the precise nature of such sensitivity has not so far
been quantified. In this paper, we study effects of localized wall thinning/thickening on
the onset of localized bulging in inflated membrane tubes as a prototypical example. It
is shown that localized wall thinning may reduce the critical pressure or circumferential
stretch by an amount of the order of the square root of maximum wall thickness reduction.
As a typical example, a 10 per cent maximum wall thinning may reduce the critical
circumferential stretch by 19 per cent. This square root law complements the wellknown Koiter’s two-thirds power law for subcritical periodic bifurcations. The relevance
of our results to mathematical modelling of aneurysm formation in human arteries is also
discussed.
Keywords: imperfection sensitivity; membrane tubes; localization; bifurcation; aneurysm
1. Introduction
Most studies on elastic buckling have focused on buckling into sinusoidal patterns.
Buckling of an Euler strut and a cylindrical thin shell typifies two possible
scenarios in this case: one is supercritical and the other subcritical. Subcritical
bifurcations are known to be sensitive to imperfections. For elastic structures,
Koiter [1] (see also Hutchinson & Koiter [2]) demonstrated that geometrical or
material imperfections in the form of a buckling mode may reduce the critical
load by an amount of the order of the two-thirds power of the imperfection
amplitude. This is now known as the two-thirds power law for imperfectionsensitive structures and it provided an explanation on why the critical loads
recorded in actual experiments on compressed cylindrical shells were much lower
than the theoretical values predicted by the linear buckling analysis.
*Author for correspondence ([email protected]).
One contribution of 15 to a Theme Issue ‘Geometry and mechanics of layered structures and
materials’.
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Elastic localization
1897
Sinusoidal buckling patterns may localize, either owing to the existence of
a small or large parameter in the linearized problem or owing to nonlinear
modulation. For instance, when a thick-walled cylindrical shell is everted, a linear
buckling analysis would predict sinusoidal buckling (wrinkling) localized near
the inner surface because of the largeness of the critical mode number in the
circumferential direction [3] (also A. Juel 2009, personal communication, with
photos showing that localization may also take place along the circumferential
direction due to infinitely many nearby modes interacting resonantly). The other
localization mechanism, localization owing to nonlinear modulation, has been
systematically studied in recent decades by Hunt and co-workers (see [4,5] and
references therein).
The present paper is concerned with another type of localization that does
not involve any sinusoidal oscillations at all. Examples of this type include
necking of a polymeric strip under stretching [6,7], Lüders band formation in
steel strips [8,9], localized bulging in inflated membrane tubes [10–12], kink band
formation in fibre-reinforced composites [13–19] and layered structures [20,21],
and stress-induced phase transformations [22–24]. For each of these problems,
the actual load deformation diagram typically contains four distinctive sections:
a section of uniform deformation that is terminated when the load reaches a
maximum, a snap-back section, growth of the localized deformation as the load
decreases, and propagation of the localized deformation at a constant value of
the load. The nature of the latter constant load value is now well understood:
it can be determined by the Maxwell equal area rule. It is also known that the
stage corresponding to the snap-back section is usually abrupt and that the load
maximum is very sensitive to geometrical and material imperfections. However,
much less is known about the nature of the load maximum. For instance, it is
not yet completely understood how the load maximum can be determined when
there are no imperfections and how the imperfection sensitivity can be quantified
when imperfections are present. For the necking problem, it was shown by Mielke
[25] that the load maximum corresponds to a bifurcation at zero mode number
(see also Fu [26] and Dai et al. [27]). For the tube inflation problem, recent
studies by Fu et al. [28] and Pearce & Fu [29] show that the load maximum
also corresponds to a bifurcation at zero mode number, but it may or may not
equal the load maximum in uniform inflation depending on how the tube is loaded
axially. The occurrence of the latter load maximum is usually referred to as limitpoint instability [30–32]. For kink band formation in fibre-reinforced composites,
the load maximum is believed to correspond to a loss of ellipticity [33–35].
In this paper, we aim to quantify imperfection sensitivity for the class of
problems listed above. This will be achieved through an analysis of the model
problem of inflation of a membrane tube. This is a simple enough problem for
which almost the entire inflation process can be described analytically, and there
also exists an extensive literature on its numerical and experimental aspects (see
Kyriakides & Chang [12], Pamplona et al. [36] and Shi & Moita [37] and references
therein).
This paper can be viewed as the fourth of a series of studies devoted to
an improved understanding of the tube inflation process. In the first of this
series [28], the initial bulging/necking was recognized as a bifurcation problem and
the corresponding bifurcation condition was derived using two different methods.
The two subsequent papers—Pearce & Fu [29] and Fu & Xie [38]—then addressed
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Y. B. Fu and Y. X. Xie
the question of characterization of the fully nonlinear bulging solutions and their
stability properties under pressure or volume control. The present paper is a first
study on the effects of material and geometrical imperfections. We shall focus on
the case when the membrane wall suffers from a localized thinning or thickening.
This is particularly pertinent to the continuum-mechanical modelling of aneurysm
formation in arterial walls, where it is believed that localized wall thinning owing
to pathological changes is a precursor of aneurysm formation, and aneurysm
repairs invariably result in localized material or geometrical imperfections.
The rest of this paper is organized as follows. In the following section, we
write down the governing equations and summarize previously known results for
perfect membrane tubes. In §3, we carry out a weakly nonlinear analysis for the
case when the membrane tube suffers localized wall thinning or thickening. We
derive the amplitude equation for the radius variation along the axis and use its
solutions to quantify the effects of imperfections. The paper is concluded with
some additional remarks and a discussion on the relevance of our results to the
mathematical modelling of aneurysm formation in arteries.
2. Governing equations and known results for a perfect membrane tube
We consider the problem of inflation of an infinite cylindrical membrane tube
that is incompressible, isotropic and hyperelastic. The tube is assumed to have
variable thickness H and uniform mid-plane radius R before inflation. To simplify
analysis, we assume that the remote axial stretch is maintained at unity all the
time, but our results can easily be modified for other cases such as that of closed
ends or when the remote axial stretch is maintained at a non-unit constant value
(the latter case would pertain to human arteries). We also assume that when
the tube is inflated by an internal pressure, the inflated configuration maintains
axial symmetry. Thus, in general, the axisymmetric deformed configuration may
be described by
r = r(Z ) and z = z(Z ),
(2.1)
where Z and z are the axial coordinates of a representative material particle
before and after inflation, respectively, and r is the mid-plane radius after inflation
(figure 1).
Since the deformation is axially symmetric, the principal directions of stretch
coincide with the lines of latitude, the meridian and the normal to the deformed
surface. Thus, the principal stretches are given by
r
deformed wall thickness
,
(2.2)
l1 = , l2 = r 2 + z 2 and l3 =
R
H
where the indices (1, 2, 3) are used for the latitudinal, meridional and normal
directions, respectively, and the primes indicate differentiation with respect to Z .
In the following analysis, we use R as the unit of length, which is equivalent to
setting R = 1.
The principal Cauchy stresses s1 , s2 , s3 in the deformed configuration for an
incompressible material are given by
si = li Ŵ i − p,
Phil. Trans. R. Soc. A (2012)
i = 1, 2, 3 (no summation),
(2.3)
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Elastic localization
s2
ds
φ r'(Z) dZ
z'(Z ) dZ
internal pressure P
s2
Figure 1. Axisymmetric deformation of a thin-walled rubber tube.
where Ŵ = Ŵ (l1 , l2 , l3 ) is the strain energy function, Ŵ i = vŴ /vli and p is
the pressure associated with the constraint of incompressibility [39]. Using the
incompressibility constraint l1 l2 l3 = 1 and the membrane assumption s3 = 0,
we find
si = li Wi , i = 1, 2 (no summation),
(2.4)
where W (l1 , l2 ) = Ŵ (l1 , l2 , l1−1 l2−1 ) and W1 = vW /vl1 , etc. In our numerical
illustrations, we shall assume that the membrane material is described by the
Ogden strain energy function
W=
3
mr
r=1
ar
(l1ar + l2ar + l3ar − 3),
(2.5)
where a1 = 1.3, a2 = 5.0, a3 = −2.0, m1 = 1.491, m2 = 0.003 and m3 = −0.023 are
material constants given by Ogden [40], and the m’s have been scaled by the
ground state shear modulus.
By considering equilibrium of an infinitesimal volume element in the r- and
z-directions, respectively, we obtain
HW2 r + Prz − HW1 = 0
(2.6)
l2
and
HW2 z l2
− Prr = 0.
(2.7)
H r W1 = 0
l2
(2.8)
HW2 (r z − z r ) − Prl22 = 0,
l2
(2.9)
These two equations are equivalent to
(HW2 ) −
and
Hz W1 +
which represent equilibrium of an infinitesimal volume element in the meridional
and normal directions, respectively. With the help of these two sets of equations
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Y. B. Fu and Y. X. Xie
and the additional relations r = l2 sin f, z = l2 cos f (figure 1), we may obtain
⎫
l1 = l2 sin f,
⎪
⎪
⎪
⎪
⎪
W
W
H
−
l
W
1
2
12
2⎬
l2 =
sin f −
(2.10)
HW22 ⎪
W22
⎪
⎪
⎪
W1
Pl1 l2
⎪
⎭
,
and
f =
cos f −
HW2
W2
where W12 = v2 W /vl1 vl2 , etc. This system of equations will be integrated later
numerically to obtain fully nonlinear solutions.
The equilibrium equation (2.7) can be integrated straight away to yield
HW2 z 1 2
− Pr = constant ≡ C2 ,
(2.11)
l2
2
which reflects the fact that the resultant in the Z -direction at any cross section
must be a constant. Mathematically, this corresponds to invariance of the
governing equations with respect to translations in z. When H is a constant, the
equations are also invariant with respect to translations in Z , leading to another
integral
(2.12)
W − l2 W2 = constant ≡ C1 .
This integral was first obtained by Pipkin [41] by integrating (2.8) directly.
We now summarize what is known for a perfect membrane tube, that is, when
H is a constant. On taking the limit Z → ∞ in (2.6), (2.11) and (2.12), we obtain
(∞)
W1
1 2
(∞)
(∞)
, C2 = W2 − Pr∞
and C1 = W (∞) − W2 ,
r∞
2
where the superscript (∞) signifies evaluation at
P=
l1 = r∞
and
l2 = 1,
(2.13)
(2.14)
r∞ being the radius at infinity.
Equations (2.11) and (2.12) with (2.13) always admit the trivial solution (2.14)
as a solution, but as r∞ is increased from unity (with the corresponding pressure
calculated according to (2.13)1 ), other non-trivial solutions may bifurcate from
this trivial solution. For a bifurcated solution symmetric about Z = 0, we have
r (0) = 0 so that
(2.15)
l1 (0) = r0 and l2 (0) = z0 ,
where r0 = r(0) and z0 = z (0). On evaluating (2.11) and (2.12) at Z = 0, we obtain
two algebraic equations that can be solved numerically to find r0 and z0 in terms
of the single control parameter r∞ . Figure 2 shows the dependence of r0 on r∞
for the Ogden material model.
Once r0 and z0 are known for each r∞ , we may integrate the first-order
system of differential equations (2.10) subjected to the initial conditions (2.15)
and f(0) = 0. It is found that only the solid segment in figure 2 corresponds to
localized solutions. Figure 2 can be interpreted as follows. Uniform inflation would
start from the origin, where l1 = l2 = 1, and then follow the horizonal axis where
r0 ≡ r∞ . As inflation reaches the bifurcation point B, where r∞ = rcr , a bulge
will initiate at Z = 0 and its subsequent growth follows the curve BT. Growth
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Elastic localization
r0 − r•
8
6
T
4
2
B
1.0
1.2
1.4
1.6
1.8
r•
2.0
2.2
2.4
Figure 2. Amplitude diagram for a perfect membrane tube based on the Ogden material model.
Localized bulging would initiate when r∞ = 1.6955.
in the radius r0 stops at the turning point T, where dr0 /dr∞ → ∞. It can be
shown [29] that the latter condition implies that the right-hand side (r.h.s.) of
(2.10)3 vanishes and thus the corresponding point (l1 , l2 , f) = (r0 , z0 , 0) becomes
another fixed point of the dynamical system (2.10) (in addition to the fixed point
(r∞ , 1, 0)). In dynamical system theory terms, each localized solution before
point T is reached corresponds to a homoclinic orbit, whereas at T, the localized
solution has flattened out at Z = 0 and represents two heteroclinic solutions
(kink solutions) stitched together at Z = 0. If the tube is inflated further, the
localization will propagate in both directions, while the radius at Z = 0 remains
unchanged.
It was shown by Fu et al. [28] that if r = r∞ + y(Z ) and |y| 1, then y must
satisfy the differential equation
(y )2 = u(r∞ )y 2 + g(r∞ )y 3 + O(y 4 ),
(2.16)
where explicit expressions for u(r∞ ) and g(r∞ ) in terms of the strain energy
function can be found in the studies of Fu et al. [28] and Fu & Il’ichev [42],
respectively. It can be seen from (2.16) that there is a bifurcation when u(r∞ ) = 0,
the first root of which then defines the rcr appearing in the previous paragraph.
Writing r∞ = rcr + er1 , y = ey1 + O(e2 ), where e is a small positive constant and
r1 is an O(1) constant, then to leading order (2.16) gives
3
d 2 y1
= ucr
r1 y1 + gcr y12 ,
2
dx
2
x=
√
eZ ,
(2.17)
which has a localized solution given by
r1
ucr
2 1
y1 = −
sech
ucr r1 x ,
gcr
2
(2.18)
= du(rcr )/drcr , gcr = g(rcr ). This solution is valid provided ucr
(r∞ −
where ucr
rcr ) > 0. For all materials that we have considered so far, ucr corresponding to
the first bifurcation point is always negative and so bifurcation into the above
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Y. B. Fu and Y. X. Xie
(a) 2.0
(b)
1.9
1.0
1.8
r(Z)
r0 − r•
1.5
1.7
1.6
0.5
1.5
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7
r•
2
4
6
Z
8
10
Figure 3. Results for the case h0 = −1, e = 0.2. (a) Comparison of amplitude diagrams for a perfect
membrane tube (dashed line) and for an imperfect membrane tube (solid line, weakly nonlinear
result given by (3.6); solid dots, fully nonlinear result obtained by integrating (2.10)). (b) Profiles
of r(Z ) for an imperfect membrane tube, corresponding to r∞ = 1.4955 on the upper and lower
branches of the amplitude diagram, respectively (solid lines, fully nonlinear result; dashed lines,
weakly nonlinear result).
localized solution must necessarily be subcritical. For the Ogden material, we
= −3.2329, gcr = −1.3369 and so the localized solution (2.18) represents
have ucr
a bulge (since y1 > 0).
We observe that for subcritical bifurcations into sinusoidal patterns, a weakly
nonlinear near-critical analysis that takes into account amplitude modulation
would lead to an equation similar to (2.17), but with the quadratic term replaced
by a cubic term (see [43] or [44]). As a result, the sech2 on the r.h.s. of (2.18)
would be replaced by the less-localized sech.
Since any localized bulging/necking solutions must tend to the uniform state
r = r∞ in the limit Z → ±∞, for a tube with variable thickness H (Z ) whose
variation is localized (2.16) is valid sufficiently away
from the site of localized
wall thinning/thickening. Thus, we have r ∼ − u(r∞ )(r − r∞ ) as Z → ±∞.
The (symmetric) localized bulging/necking solutions can then be determined
by integrating the system (2.10) from Z = 0 towards ∞ subject to the initial
conditions
l1 (0) = r0 , l2 (0) = z0 and f(0) = 0,
where r0 is to be guessed in our shooting procedure and the constant z0 is related
to r0 by the integral (2.11). Thus, for each specified r∞ and a guess for r0 , we solve
(2.11) to find the corresponding z0 . We iterate on r0 so that the decay condition
r (L) + u(r∞ )(r(L) − r∞ ) = 0
(2.19)
is satisfied for a sufficiently big positive number L. As localized solutions
correspond to homoclinic orbits in the phase plane that usually separate
unbounded solutions from periodic solutions, a reasonable starting guess for r0 is
required to ensure the success of the above shooting method. We start from a case
for which the imperfection is of small amplitude and r∞ is close to the bifurcation
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Elastic localization
(a)
(b) 2.0
1.9
r 0–r•
1.0
r(Z)
1.8
0.5
1.7
1.6
1.5
1.2
1.4
1.6
1.8
1.4
2.0
0
r•
2
4
6
8
Z
10
12
14
Figure 4. Results for the case h0 = 1, e = 0.2. (a) Comparison of amplitude diagrams for a perfect
membrane tube (dashed line) and for an imperfect membrane tube (solid lines, weakly nonlinear
result given by (3.6); solid dots, fully nonlinear result obtained by integrating (2.10)). (b) Three
solutions of r(Z ) for an imperfect membrane tube when r∞ = 1.55 (solid lines, fully nonlinear
result; dashed lines, weakly nonlinear result).
value. The weakly nonlinear result to be presented in the following section then
provides a very good initial guess for r0 . Once a solution is found, we increase
the amplitude of imperfection and decrease r∞ in small steps, using the solution
from the previous step as the initial guess for the current step. A selection of such
fully nonlinear numerical solutions are presented as dots in figures 3 and 4, and
will be discussed together with weakly nonlinear solutions in §3.
3. Effects of localized wall thinning
The weakly nonlinear solution (2.18) indicates that when r∞ − rcr is of order
√ e,
the localized bulging solution has an O(e) amplitude and varies over an O(1/ e)
length scale. We now assume that the variable thickness H takes the form
H = H0 (1 + e2 h(x)),
h(±∞) → 0,
(3.1)
where H0 is the constant wall thickness at infinity and the function h(x) is to be
prescribed. The order of deviation of the wall thickness from the constant value
H0 is dictated by the requirement that h(x) must appear in our final amplitude
equation.
When H is not constant, the integral (2.12) no longer holds and so the simple
procedure of Fu et al. [28] for deriving (2.16) cannot be applied. Instead, we shall
use the equilibrium equation (2.6) and integral (2.11) to derive the amplitude
equation.
We thus look for an asymptotic solution of the form
z = 1 + ev1 (x) + e2 v2 (x) + · · · ,
(3.2)
where y1 (x), y2 (x), v1 (x), v2 (x), . . . are to be determined at successive orders of
approximations. As we are interested only in localized solutions, the asymptotic
expansion for the pressure is simply the Taylor series of (2.13)1 .
r = rcr + er1 + ey1 (x) + e2 y2 (x) + · · ·
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Y. B. Fu and Y. X. Xie
On substituting (3.2) into (2.6) and (2.11), and equating the coefficients of e,
we find a relation between y1 (x) and v1 (x) and reproduce the bifurcation condition
u(rcr ) = 0, whereas on equating the coefficients of e2 , we find a relation between
y2 (x) and v2 (x) and the amplitude equation
d 2 y1
3
= u (rcr )r1 y1 + gcr y12 + zh(x),
2
2
dx
(3.3)
where the only new coefficient z is given by
−1
z = W2−1 W22
(W1 W22 + W1 W2 − W2 W12 )|l1 =rcr ,
l2 =1 ,
(3.4)
and takes the value 2.0328 for the Ogden material. The amplitude equation (2.17)
for a perfect membrane tube is recovered by setting h(x) = 0. For each specified
h(x), the amplitude equation (3.3) is to be integrated subject to the decay
condition y1 (±∞) → 0.
There are obviously three special cases, corresponding to h(x) being a constant
multiple of y12 , y1 , y1 , respectively, for which closed-form localized solutions can
be obtained as (3.3) can then be reduced to the form (2.17) by substitution of
coefficients or rescaling of x. We first assume that
3
h(x) = d1 y12 ,
2
where d1 is a constant. By making the substitution gcr → gcr + zd1 in (2.18), we
obtain the solution
er1
ucr
2 1
sech
ucr r1 x .
(3.5)
r − r∞ = ey1 = −
gcr + zd1
2
Denoting h(0) by h0 , we have
r1 )2
3
3d1 (ucr
h0 = d1 y12 (0) =
,
2
2(gcr + zd1 )2
which can be solved to express d1 , and hence h(x) and y1 (x), in terms of h0 . We
then obtain
(r∞ − rcr )
ucr
8gcr h0 ze2
r0 − r∞ = ey1 (0) = −
1± 1−
.
(3.6)
2 (r − r )2
2gcr
3ucr
∞
cr
When h0 < 0, which corresponds to localized wall thinning, the graph of (3.6)
in the (r∞ , r0 − r∞ ) plane is a parabola opening to the left (figure 3a), similar
to the classical amplitude diagrams for imperfection-sensitive structures. As we
move away from the bifurcation value rcr , we have (rcr − r∞ )/e → ∞. The upper
and lower branches then have the asymptotic behaviour
r0 − r∞ ∼ −
ucr
(r∞ − rcr )
,
gcr
−
2h0 ze2
,
(r − r )
3ucr
∞
cr
(3.7)
respectively. We note that the upper asymptotic limit is the amplitude of the
localized solution for a perfect membrane tube (see equation (2.18)), whereas
the lower asymptotic limit is of the same order as the original wall thinning. In
figure 3a, we have shown the weakly nonlinear solution (3.6) together with a
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Elastic localization
1905
portion of figure 2 near the bifurcation point. On the same figure, we have also
used dots to indicate a set of fully nonlinear solutions obtained by integrating the
system (2.10); these fully nonlinear solutions will be discussed in detail later.
We see that localized wall thinning has the effect of reducing the bifurcation
value of r∞ , and hence the bifurcation pressure. The reduced bifurcation value of
r∞ corresponds to the turning point of the parabola in figure 3a, and is given by
2 e gcr zh0
.
(3.8)
r∞ = rcr + 2
3 ucr
Since gcr < 0 and z > 0 for the Ogden material, the above expression is real only
if h0 < 0, that is, only if the
wall suffers localized thinning. In this case, the r.h.s.
reduces to rcr (1 − 0.4935 −e2 h0 ) so that the bifurcation value of r∞ is reduced by
an amount proportional to the square root of the amplitude of localized thinning.
As an illustrative example, when e2 h0 = −0.1, which corresponds to a 10 per cent
maximum wall thinning, the reduction would be 16 per cent, which would bring
the critical stretch from 1.6873 down to 1.4240.
When h0 > 0, which corresponds to localized wall thickening, the two branches
of the weakly nonlinear solution (3.6) correspond to the two solid lines in
figure 4a where again the dashed line signifies the result for the perfect tube
and the solid dots correspond to fully nonlinear solutions obtained by integrating
(2.10). We note that the solid lines beyond the critical value r∞ = rcr are
meaningless as the corresponding weakly nonlinear solution (3.5) is no longer real.
We had originally expected that in a small neighbourhood of r∞ = rcr , the
fully nonlinear solution would be well approximated by the weakly nonlinear
solution (3.6). But figure 4a shows that the upper branch of the fully nonlinear
solution comes down and then turns left, instead of proceeding along the
weakly nonlinear solution further. As a result, for values of r∞ less than 1.567
approximately, the fully nonlinear solution has three branches. The top and
bottom branches correspond to localized bulging and necking, respectively, and
are well approximated by the weakly nonlinear theory, whereas the middle
solution has no counterpart in the weakly nonlinear theory. Further examination
of the profiles corresponding to the middle branch reveals that each of them has
a double hump structure, with the two humps symmetrically located on the two
sides of Z = 0. The valley at Z = 0 has zero depth at the turning point (where
the top and the middle branches join) and it gets deeper and deeper as we move
left on the middle branch (figure 5). As r∞ decreases, we are confident that the
top branch tends to that for the perfect tube and the bottom branch tends to
the trivial state, but so far we have not yet been able to determine the fate of
the middle branch. The reason is that the solution becomes increasingly sensitive
to the guess of r(0) and, as a result, it gets more and more difficult to decide
whether the solution for r(Z ) is a true localized solution, a periodic solution or
an unbounded solution. For instance, for the last point on the middle branch,
a variation in r(0) of the order of 10−7 can change the solution from one type
to another.
The second special case that we consider corresponds to h(x) = d2 y1 (x), where
d2 is a constant. In this case, the effect of imperfection can be scaled out by the
simple substitution
r1 → ucr
r1 + d2 z.
ucr
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1906
Y. B. Fu and Y. X. Xie
r (Z)
2.5
2.0
1.56
1.5
1.45
1.3
–15
–10
–5
5
10
15
Z
Figure 5. Profiles of r(Z ) corresponding to the middle branch for r∞ = 1.56, 1.45, 1.3.
Thus, the localized solution is given by
ucr
r1 + d 2 z
2 1
y1 = −
sech
ucr r1 + d2 zx .
gcr
2
(3.9)
Denoting h(0) again by h0 , we have
h0 = d2 y1 (0) = −
r1 + d2 z)
d2 (ucr
,
gcr
which can be solved to yield
r1
ucr
u r1
+ d2 z = cr
2
1±
4gcr h0 z
1 − 2 2 .
ucr r1
(3.10)
It then follows that
(r∞ − rcr )
ucr
4gcr h0 ze2
1 ± 1 − 2
.
r0 − r∞ = ey1 (0) = −
2gcr
ucr (r∞ − rcr )2
(3.11)
This expression has the same structure as (3.6). The two solutions are real only
if the expression under the radical is non-negative, that is
2e gcr h0 z
r∞ ≤ rcr +
,
(3.12)
ucr
where the r.h.s. defines the new critical principal stretch which may be compared
with the r.h.s. of (3.8). For the Ogden material, the new critical stretch takes
the form
(3.13)
rcr (1 − 0.604421 −e2 h0 ),
Phil. Trans. R. Soc. A (2012)
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Elastic localization
1907
where −e2 h0 represents maximum wall thinning. This type of wall thinning is
slightly more effective in reducing the critical principal stretch. For instance, a
10 per cent maximum wall thinning would now reduce the critical stretch by
19 per cent.
Finally, we consider the case when h(x) is proportional to y1 and write h(x) =
d3 y1 , where d3 is a constant. The last term in (3.3) can be eliminated by rescaling
x and we have
r
ucr
r1
ucr
1
2 1
sech
x .
(3.14)
y1 = −
gcr
2 1 − d3 z
From h0 = d3 y1 (0) we obtain
d3 =
2gcr h0
,
2 r 2
2zgcr h0 + ucr
1
u2 r 2 + 2zgcr h0
ucr
r1
.
= cr 1 1 − d3 z
ucr r1
(3.15)
When h0 < 0, the above solution is valid for all r1 < 0 and so this kind of
localized thinning has no effect on the onset of localized bulging. When h0 > 0,
the expression under the radical in (3.14) is positive only if (assuming r1 < 0)
2z(−gcr )h0
,
r1 < −
|
|ucr
that is,
e 2(−gcr )h0 z
.
r∞ ≤ rcr +
ucr
(3.16)
Since this is also the condition under which the expression for the imperfection
is real, this case is of no practical interest.
We note that the wall thickness variation function h(x) associated with the
exact solutions (3.6) and (3.11) is dependent on r∞ through the appearance of r1
in the argument of the sech function. Thus, for instance, the weakly
√ and fully
nonlinear results in figure 3a correspond to h(x) = −sech2 (0.8990 rcr − r∞ Z ),
which is continually changing as r∞ is varied. To show that this weak dependence
on r∞ does not qualitatively change the main characteristics of the amplitude
2
diagrams, we now consider the case when h(x) is given by h(x) = −e−x , which
does not vary with respect to r∞ . In this case the amplitude equation (3.3) can
be solved numerically subject to the initial conditions y1 (0) = k, y1 (0) = 0, where
k is iterated so that the solution satisfies the decay condition
y1 (x) + u (rcr )r1 y1 (x) → 0 as x → ∞.
We vary r1 , find the associated y1 (0) and then compute r∞ and r0 − r∞ . With
h(0) = −1, the small parameter e can be taken as the square root of maximum
relative wall thinning (see (3.1)). It is found that the amplitude diagram is similar
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1908
Y. B. Fu and Y. X. Xie
to the solid parabolic curve in figure 3a, but with the turning point occurring at
r1 = −0.8683. The corresponding reduced critical stretch is then given by
rcr − 0.8683e = rcr (1 − 0.5146e),
which may be compared with (3.13).
4. Conclusion
In this paper, we have used the simple problem of tube inflation to quantify the
effect of material and geometrical imperfections on the critical load for the onset
of localization. For imperfections in the form of localized wall thinning, it is found
that they can reduce the critical circumferential stretch by an amount that is of
the order of the square root of the imperfection amplitude. The proportionality
coefficient is dependent on the form of the imperfection, but this square root
rule is universal. This result complements the well-known Koiter’s two-thirds
power rule for subcritical bifurcations into periodic patterns and demonstrates
that bifurcation into localized patterns is much more susceptible to imperfections
than bifurcation into periodic patterns.
We note that effects of imperfections have also previously been studied
numerically by Kyriakides & Chang [12] on a tube whose radial movement is
restricted at the two ends, and they came to the conclusion that imperfections
have little effect on the initiation pressure. This result does not contradict our
conclusion as restricted ends can be viewed as an imperfection (in the sense that
bulging occurs as soon as the tube is inflated) and so essentially Kyriakides &
Chang [12] were comparing one imperfection against another.
Our study is also motivated by possible application in the mathematical
modelling of aneurysm formation in human arteries. All existing studies on
aneurysm modelling have focused on the evolution of aneurysms after they have
already formed, with a view to guide the clinician as to whether the risk of
bursting outweighs the risk of repairs (see Watton et al. [45], Watton & Hill
[46] and references therein). We believe that our present study might be relevant
to understanding the process leading to the initial formation of aneurysms. One
scenario is that pathological changes in arteries change the constitutive behaviour
and cause localized thinning, which in turn makes localized bulging possible.
Some preliminary studies based on simple models for arterial walls have shown
that the onset pressure for localized bulging is unrealistically high in the absence
of imperfections. If localized wall thinning can reduce the critical circumferential
stretch significantly, it can then bring the onset pressure to more realistic values.
Another aspect of our results that is relevant to aneurysm modelling is that
localized wall thickening is not necessarily beneficial. On the one hand, it can
equally reduce the onset pressure, and on the other hand, a twin-hump aneurysm
can be excited at much lower pressure values; see the turning point and the middle
branch in figure 4a, respectively. Of course, the actual arterial wall constitution is
much more complicated than the single-walled isotropic model used in the present
study [47], but our present result is at least indicative of what might be expected
when more elaborate models are used.
Since this paper was submitted, it has subsequently been demonstrated by Fu
et al. [48], using a realistic constitutive model, that the initial onset of aortic
Phil. Trans. R. Soc. A (2012)
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Elastic localization
1909
abdominal aneurysms can indeed be modelled as a bifurcation phenomenon, and
that in the bifurcation interpretation of aneurysm formation, the imperfection
sensitivity determined in this paper plays a pivotal role.
This work is supported by an EPSRC grant (EP/G051704/1, awarded to Y.B.F.) and a Leverhulme
Research Fellowship (awarded to Y.X.X.).
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