ArticlePDF Available

Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells

Authors:

Abstract and Figures

We present results from an experimental investigation of the indentation of nonspherical pressurized elastic shells with a positive Gauss curvature. A predictive framework is proposed that rationalizes the dependence of the local rigidity of an indented shell on the curvature in the neighborhood of the locus of indentation, the in-out pressure differential, and the material properties. In our approach, we combine classic theory for spherical shells with recent analytical developments for the pressurized case, and proceed, for the most part, by analogy, guided by our own experiments. By way of example, our results elucidate why an eggshell is significantly stiffer when compressed along its major axis, as compared to doing so along its minor axis. The prominence of geometry in this class of problems points to the relevance and applicability of our findings over a wide range of length scales.
Content may be subject to copyright.
Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells
A. Lazarus, H. C. B. Florijn, and P. M. Reis
*
EGS. Lab: Elasticity, Geometry and Statistics Laboratory, Department of Mechanical Engineering, Department of Civil &
Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 16 July 2012; published 5 October 2012)
We present results from an experimental investigation of the indentation of nonspherical pressurized
elastic shells with a positive Gauss curvature. A predictive framework is proposed that rationalizes the
dependence of the local rigidity of an indented shell on the curvature in the neighborhood of the locus of
indentation, the in-out pressure differential, and the material properties. In our approach, we combine
classic theory for spherical shells with recent analytical developments for the pressurized case, and
proceed, for the most part, by analogy, guided by our own experiments. By way of example, our results
elucidate why an eggshell is significantly stiffer when compressed along its major axis, as compared to
doing so along its minor axis. The prominence of geometry in this class of problems points to the
relevance and applicability of our findings over a wide range of length scales.
DOI: 10.1103/PhysRevLett.109.144301 PACS numbers: 46.70.De, 81.70.Bt, 87.64.Dz
Shells are ubiquitous as both natural and engineered
structures, including viral capsids [1], pollen grains, col-
loidosomes [2], pharmaceutical capsules, exoskeletons,
mammalian skulls [3], pressure vessels, and architectural
domes. In addition to their aesthetically appealing form,
shells offer outstanding structural performance. As such,
they typically have the function of enclosure, containment,
and protection, with an eggshell being the archetypal ex-
ample [Fig. 1(a)]. The mechanics of thin elastic shells [4,5]
have long been known to be rooted in the purely geometric
isometric deformations of the underlying curved surface,
since stretching is energetically more costly than bending
[6]. However, despite a vast literature on the isometry of
rigid surfaces [7,8], establishing a general direct theoretical
connection between the differential geometry of surfaces
and the mechanics of shells is a challenging endeavor. This
is due to the difficulties in systematically quantifying the
effect of a small but finite thickness on the mechanical
response of a curved surface. To circumnavigate this issue,
explicit boundary value problem calculations tend to be
performed by deriving the local equilibrium equations of
3D continuum mechanics and then taking the limit of a
small but finite thickness within the kinematics [9]. For
example, closed analytical solutions have been obtained in
this fashion for the indentation of spherical unpressurized
[10,11] and pressurized [12] shells. For more intricate
geometries and mechanical environments, numerical
methods can be used such as full scale finite element
simulations [13,14]. Although powerful, these computa-
tional approaches can sometimes come at the detriment
of physical insight and predictive understanding of the
interplay between the mechanics of the structure and the
geometry of its surface.
Here, we study the effect of the geometry of surfaces
with a positive Gauss curvature on the linear mechanical
response under the indentation of thin elastic shells, with or
without an in-out pressure differential. Our goal is to
quantify the geometry-induced rigidity (GIR) which we
define as the amount by which a nonspherical shell is
stiffened when compared to a spherical shell with the
same thickness and material properties [15]. For this pur-
pose, we perform precision desktop-scale experiments
where both the geometry of the shells and their material
properties are accurately custom controlled using rapid
prototyping and digital fabrication techniques. The rigidity
is quantified through indentation tests and the differential
pressure is set by a syringe-pump system under feedback
control. First, we study the rigidity at the pole of ellipsoidal
shells with different aspect ratios toward investigating the
0 0.5 1 1.5 2
0
100
200
300
400
500
F (mN)
δ (mm)
0 0.5 1 1.5 2
0
20
40
60
F (mN)
δ (mm)
F
F
F
F
b/a = 2 b/a = 1.5
b/a = 1 b/a = 0.5
b/a
K
(b)
K
p
(a)
Egg
(c)
b
a
δ
FIG. 1 (color online). (a) A chicken’s egg and 500 m thick
ellipsoidal shell samples with four aspect ratios b=a ¼
f2; 1:5; 1; 0:5g and four elastomers with Young’s modulus E ¼
f0:2; 0:5; 0:6; 1g MPa. (b) Load-displacement curves for inden-
tation at the pole for an in-out differential pressure p ¼ 0Pa,
E ¼ 1 MPa, and three different aspect ratios b=a ¼ 0:5, 1, and
2. (c) Load-displacement curves for indentation at the pole for
b=a ¼ 1:5, E ¼ 1 MPa, and 0 <p<10 kPa.
PRL 109, 144301 (2012)
PHYSICAL REVIEW LETTERS
week ending
5 OCTOBER 2012
0031-9007=12=109(14)=144301(5) 144301-1 Ó 2012 American Physical Society
specific case of axisymmetric convex surfaces. We then
turn to ellipsoids indented along their meridian, where the
surface has two distinct principal curvatures. Combining
our experimental results with the classic theory of elastic-
ity of spherical shells due to Reissner [10] as well as more
recent developments by Vella et al. [12], we find that the
effective rigidity is induced by (i) the local curvature in the
neighborhood of the locus of indentation, (ii) the material
properties of the shell, and (iii) the in-out differential
pressure.
Fabricating our thin shells involves the conception of a
computer assisted design model for a mold of the target
shell structure, which is then 3D printed out of ABS plastic,
and used to cast samples with Vinylpolysiloxane (a silicone-
based elastomer). In our experiments, we use elastomers
with four values of the Young’s modulus in the range
0:2 <E<1MPaand a Poisson ratio of 0:5.Inspired
by the eggshell geometry [Fig. 1(a )], we print molds wit h
four ellipsoidal shapes that set the geometric parameters to
be four polar radii, b ¼f1: 25; 2:5; 3:75; 5g cm, an equato-
rial circular radius, a ¼ 2:5cm, and a thicknes s t ¼
0:5mm(the latter two parameters are set constant for all
experiments). This value of thickness (t=a ¼ 0:02 1)en-
sures the validity of the thin shell assumption. In Fig. 1(a),we
show photographs of four representati v e sample s made of
different polymers (different colors correspond to different
values of E) with aspect ratios b=a ¼f0:5; 1; 1:5; 2g.
We start by vertically indenting the pole of unpressur-
ized shell specimens using a hemispherical cap indentor
(1.5 mm radius) at the constant speed of 1mm= min ,
ensuring point indentation under quasistatic conditions.
The compressive force F resulting from the indentation
by the imposed displacement is recorded using the load
cell of an Instron machine with a resolution of 100 N.
Typical load-displacement curves for shells with different
aspect ratios are presented in Fig. 1(b). We focus on the
first linear regime occurring for =t < 1 and define the
experimental rigidity K as the initial slope in the load-
displacement curves. For a given material and shell thick-
ness, the rigidity increases with the aspect ratio, suggesting
the possibility of enhancing the rigidity of a thin elastic
shell by simply changing its geometry.
To investigate this GIR behavior observed in Fig. 1(b),
we now measure K at the pole for 16 unpressurized shell
specimens (four aspect ratios b=a and four values of E).
We quantify the GIR as the dimensionless ratio K=K
s
where K
s
is the theoretical rigidity of unpressurized spheri-
cal shells under indentation defined by Reissner [10]as
K
s
¼ 8D=L
2
b
. Here, D ¼ Et
3
=12ð1
2
Þ is the bending
modulus of the shell, L
b
¼ðDa
2
=EtÞ
1=4
is a characteristic
length scale arising from the balance of bending and
stretching, and a is the radius chosen for reference. In
Fig. 2(b), we present the dependence of the GIR at the
pole on the shell’s aspect ratio, finding that it scales pro-
portional to b=a; i.e., for a given material, an ellipsoidal
shell that is twice as high as the spherical counterpart is
also twice as rigid.
We rationalize the above observation by considering the
equations for static equilibrium of thin elastic shells of
revolution under indentation [5] and computing the me-
chanical stresses at the pole. In its stress-free reference
configuration, the shell can be represented by the generat-
ing planar curves ¼ ðsÞ and z ¼ zðsÞ, where s is the
arc-length along the shell’s midplane [see the schematic
diagram in Fig. 2(a)]. In order to understand the linear
mechanical response, we consider the particular case of
small strains and small rotations under indentation at the
convex pole. Assuming axisymmetric and twistless defor-
mations, the linear equilibrium equations can be written in
the radial and axial directions, in term of s alone, as
d
ds
ð
s
0
Þ
¼ 0; (1)
1
d
ds
ð
s
z
0
Þ
D
d
ds
ð
0
Þ
0
¼f
z
; (2)
where ðsÞ is the rotation angle between the reference and
deformed configuration,
s
ðsÞ and
ðsÞ are the axial and
hoop stresses, respectively, f
z
ðsÞ is the force density per
unit area in the axial direction, and ðÞ
0
denotes differentia-
tion with respect to s. For our particular ellipsoidal geome-
tries, the generating planar curve is parametrized by
zðsÞ¼b
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
ðsÞ=a
2
p
. In the neighborhood of the inden-
tation point at the flat pole of the ellipsoid, where the
deformations occur, the local generating plane curve can
be described by the linear approximations ðsÞs,
10
−2
10
0
10
2
10
1
10
2
τ
K / (κ
2
EtD)
1/2
Eq. (4)
b/a = 0.5
b/a = 1
b/a = 1.5
b/a = 2
0 1000 2000 3000 4000
1
2
3
4
5
6
p (Pa)
GIR, K/Ks
Eq. (4)
b/a = 0.5
b/a = 1
b/a = 1.5
b/a = 2
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
b/a
GIR, K/Ks
Eq. (3)
E = 1 Mpa
E = 0.6 MPa
E = 0.5 MPa
E = 0.2 MPa
(a)
(b)
(c)
(d)
a
b
F
p
θ
ρ
δ
z
s
FIG. 2 (color online). Indentation at the pole. (a) Schematic of
a deformed ellipsoidal specimen. (b) Dependence of GIR K=K
s
on b=a for four different elastic polymers and p ¼ 0. (c) GIR
K=K
s
, as a function of the internal pressure p for four aspect
ratios b=a ¼f0:5; 1; 1:5; 2g and E ¼ 1 MPa. (d) Dimensionless
experimental rigidity Kl
2
b
=D against dimensionless pressure .
PRL 109, 144301 (2012)
PHYSICAL REVIEW LETTERS
week ending
5 OCTOBER 2012
144301-2
0
ðsÞ1, and z
0
ðsÞs=r, where r ¼ a
2
=b is the local
radius of curvature. For s r, the angle ðsÞþz
0
ðsÞ
[Fig. 2(a)] which measures the direction of the tangent
along a deformed meridian, must vanish since this tangent
is perpendicular to the pushing vertical indentor and
ðsÞ¼z
0
ðsÞ¼s=r. Replacing this local kinematics
into the equilibrium equations (1) and (2), we find that
the membrane stresses are given by ðsÞ Z EðsÞ and
therefore proportional to the local curvature at the pole,
¼ 1=r ¼ b=a
2
(the two principal curvatures are equal
due to axisymmetry, ¼
1
¼
2
). After substituting the
global natural bending length scale L
b
introduced earlier
by its local counterpart, l
b
¼ðD=Et
2
Þ
1=4
, into K
s
, we can
generalize Reissner’s rigidity at the pole of nonspherical
thin elastic shells of revolution:
K
0
t
¼
4Et
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3ð1
2
Þ
p
: (3)
In particular, for an ellipsoidal shell where ¼ b=a
2
,we
find that K
0
t
¼ðb=aÞK
s
, which is in excellent agreement
with the experimental GIR data as a function of aspect
ratios b=a presented in Fig. 2(b).
Having rationalized our results for unpressurized shells,
we now investigate the influence of an in-out differential
pressure on the rigidity of nonspherical axisymmetric
shells. We focus on four elastomeric ellipsoidal shells with
aspect ratios b=a ¼f0:5; 1; 1:5; 2g and E ¼ 1MPa,and
measure the rigidity K at their pole for differential pressures
in the range 0 <p<4kPa. Typical load-displacement
curves for increasing p are shown in Fig. 1(c) for a shell
with b=a ¼ 1:5. The corresponding dependence of the
GIR (K=K
s
), on p is presented in Fig. 2(c) and we find
that both pressure and geometry can enhance the rigidity.
Above, we showed for the unpressurized case that the
coupling between geometry and the mechanical response
was dictated by the local curvature. This now motivates us
to proceed by hypothesizing that the local nature of the
GIR also translates into the pressurized case. If this
hypothesis is correct, we should therefore find that the
indentation at the pole of a pressurized ellipsoidal shell is
locally identical to the indentation of a pressurized spheri-
cal shell of radius r ¼ 1 = ¼ a
2
=b. As such, we seek a
generalization of the theoretical rigidity of the pressurized
spherical shell of Vella et al. [12],
K
p
t
¼
8D
l
2
b
ð
2
1Þ
1=2
2 arctanhð1
2
Þ
1=2
; (4)
where we replace a by the local radius of the curvature
at the pole r ¼ 1=, where ¼
1
¼
2
, so that ¼
pr
2
=4ðEDtÞ
1=2
is a local dimensionless internal pressure
and l
b
is the local bending scale at the pole defined above.
The dependence of the dimensionless experimental rigid-
ity, Kl
2
b
=D, on the local dimensionless pressure is
showed in Fig. 2(d), where we find that all the experimental
data collapse onto the predicted rigidity K
p
t
(dashed line).
This excellent agreement confirms that Eq. (4) accurately
captures the experimental rigidity at the pole of pressurized
thin ellipsoidal shells, which corroborates our hypothesis.
We progress by generalizing the current framework to
nonaxisymmetric convex shells which are locally charac-
terized by two distinct principal curvatures
1
and
2
,a
Gauss curvature
G
¼
1
2
, which we restrict to be posi-
tive, and a mean curvature
M
¼ð
1
þ
2
Þ=2 [Fig. 3(a)].
For the moderately elongated ellipsoids we consider in our
study,
M
ffiffiffiffiffiffi
G
p
, as shown in Fig. 3(b), with representa-
tive schematics in the inset. Encouraged by our previous
success in characterizing the rigidity of convex axisym-
metric shells, we proceed again by analogy and hypothe-
size that the local mean curvature
M
generalizes the role
of in the GIR of nonaxisymmetric shell surfaces. Note
that, as required, this generalization reduces to
M
¼
1
¼
2
¼ at the pole. Thus, replacing ¼ 1 =r by
M
in
Eqs. (3) and (4), we propose a combined expression that
describes the rigidity of unpressurized or pressurized non-
spherical shells with a positive Gauss curvature,
K
t
¼
4Et
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3ð1
2
Þ
p
M
=2ð
2
1Þ
1=2
arctanhð1
2
Þ
1=2
; (5)
where is the local dimensionless pressure introduced
above with r ¼ 1=
M
.
1/
1
1/
2
F
10
0
10
2
10
1
10
2
τ
K / (κ
M
2
EtD)
−1/2
Eq. (5)
κ
M
=26 m
−1
κ
M
=32 m
−1
κ
M
=39 m
−1
κ
M
=57 m
−1
0 50 100
0
50
100
κ
G
(s)
1/2
(m
−1
)
κ
M
(s) (m
−1
)
κ
G
= κ
M
2
b/a = 1.5
b/a = 2
0 50 100
0
50
100
GIR, K(s)/K
s
(s)
κ
M
(s) (m
−1
)
Eq. (5), τ = 0
b/a = 1.5
b/a = 2
(a)
(c)
F
A
(d)
(b)
s
2
1
B
A
B
FIG. 3 (color online). (a) Experimental setup for shells with
two distinct principal curvatures
1
and
2
. (b) Evolution of the
Gauss and mean curvatures along the meridian of the ellipsoids
with b=a ¼ 1:5 and 2. (c) Evolution of geometry-induced rigid-
ity K=K
s
ðsÞ with mean curvature
M
ðsÞ for two unpressurized
shells with aspect ratios b=a ¼ 1:5; 2 and E ¼ 1 MPa.
(d) Evolution of dimensionless experimental rigidity
KðsÞl
2
b
ðsÞ=DðsÞ with dimensionless pressure for four points
along the meridian of a pressurized shell with b=a ¼ 2 and
E ¼ 1 MPa.
PRL 109, 144301 (2012)
PHYSICAL REVIEW LETTERS
week ending
5 OCTOBER 2012
144301-3
A modification of our experimental setup that allows for
indentation in regions with two different principal curva-
tures,
1
and
2
, is now used to test the proposed Eq. (5).
For this, two shells with Young’s modulus E ¼ 1 MPa
and aspect ratios b=a ¼ 1:5 and 2 are marked every
s
i
¼ 0:5cm along one meridian from the pole to the
base [Fig. 3(a)]. Once pressurized to a set constant p, the
shells are indented at the points Pðs
i
Þ with the compressive
force F aligned perpendicularly to the surface. For a given
differential pressure p, we measure the experimental local
mean curvature
M
ðs
i
Þ using digital image processing at
each indentation point Pðs
i
Þalong the meridian of our shell
specimen [Fig. 3(b)].
In Fig. 3(c) , we plot the dependence of the Geometry-
Induced Rigidity, Kðs
i
Þ=K
s
ðs
i
Þ, on the mean curvature
M
ðs
i
Þ for two unpressurized shells with aspect ratios
b=a ¼ 1:5 and 2 [Reissner’s result per unit radius K
s
ðs
i
Þ¼
4Etðs
i
Þ
2
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3ð1
2
Þ
p
is used as the reference rigidity
K
s
ðs
i
Þ]. As expected, the measured Kðs
i
Þ scales linearly
with Reissner’s rigidity K
s
ðs
i
Þ with a prefactor
M
ðs
i
Þ, i.e.,
Kðs
i
Þ¼
M
ðs
i
ÞK
s
ðs
i
Þ, which conforms to Eq. (5) with a
dimensionless differential pressure ¼ 0. The role of
M
in moderately nonaxisymmetric shells also extends to the
pressurized case, as shown in Fig. 3(d), where we plot the
evolution of dimensionless experimental rigidities
KðsÞl
b
ðsÞ
2
=D as a function of dimensionless pressure
ðsÞ for four indentation points Pðs
i
Þ along the shell with
b=a ¼ 2. We find excellent agreement between the experi-
mental rigidities Kðs
i
Þ and the predicted K
t
ðs
i
Þ (dashed
line) confirming the accuracy of Eq. (5).
Having learned about the geometry-induced rigidity in
convex elastic shells through Eq. (5), we turn our learnings
into an inverse problem toward developing a nondestruc-
tive method to measure the parameters of the shell
(e.g., shell thickness) upon knowing the geometry of the
underlying surface and the local mechanical response.
Fortunately, the thickness profile of our experimental
samples varied slightly along the meridian of our shells
due to limitations of our fabrication procedure; the male
and female parts of the molds were not perfectly aligned
during casting. Earlier, when we tested Eqs. (3)–(5), we
took for the thickness, tðs
i
Þ, the local average measured
over a curvilinear distance s ¼ 3 l
b
in the neighborhood
of the indentation point Pðs
i
Þ obtained by cutting the shell
in half and analyzing the profile with a digital flat scanner
[Fig. 4 (inset)]. Note that in our experiments, the local
bending scale, 1:6 <l
b
¼ðD=Et
2
M
Þ
1=4
< 2:5mm,isof
the order of the diameter of the hemispherical cap indentor
¼ 3mm. In Fig. 4, we compare this average thickness
tðs
i
Þ along the meridian of two shell specimens measured
directly from the scanned profiles of the cut shells, with the
thickness predicted from Eq. (5) with ¼ 0, through
indentation. The variation of the thickness profile is accu-
rately captured within a spatial resolution of s 2l
b
5mm, confirming the relevance of the local bending scale
l
b
as a measure of locality. More importantly, the fact that
we are able to recover the thickness profile using Eq. (5)
demonstrates that our proposed description can be used as
a precision nondestructive technique for nonspherical thin
elastic shell with a positive Gauss curvature.
In summary, we have quantified the relation between
geometry and the mechanical rigidity of positively curved
thin elastic shells under pressure, arriving to our predictive
framework primarily by analogy and an analysis of preci-
sion experimental data. Our work calls for a formal iden-
tification, in particular, for a higher degree of asphericity
¼ð
2
1
Þ=ð
2
þ
1
Þ, where our description is likely
to be challenged, which will require a theoretical effort that
we hope our work will help catalyze. Still, the range of
curvatures we have considered (0 <<0:6 for which
M
ffiffiffiffiffiffi
G
p
), is applicable and relevant across a wide range
of practical instances of natural and man-made nonspher-
ical shells. The scale invariance of geometry-induced
rigidity suggests that our framework should find uses
across length scales: from the mechanical testing of viral
capsids through atomic force microscopy [16], to ocular
tonometry procedures [17] or in the design of architectural
shells [18]. Moreover, one can now interpret the increased
stiffness of a chicken egg compressed along its poles as
compared to doing so along the equator as being attributed
to geometry-induced rigidity.
We thank C. Perdigou for help with preliminary experi-
ments and B. Audoly, D. Vella, A. Vaziri, and A. Boudaoud
for enlightening discussions.
Note added.—During the preparation of our manuscript,
we became aware of the complementary theoretical and
numerical work of D. Vella et al. [19], on a similar topic.
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
Curvilinear distance, s (cm)
Thickness, t (mm)
theory
b/a = 1.5
b/a = 2
3.75 cm
2.5 cm
t
s
FIG. 4 (color online). Comparison between measured and pre-
dicted thickness profile for two unpressurized shells with b=a ¼
1:5 and 2 and 0:9 <E<1 MPa. Solid and dashed lines repre-
sent mean value standard deviation for the predicted thick-
ness. (Inset) Scanned image of a shell with aspect ratio
b=a ¼ 1:5 cut in half.
PRL 109, 144301 (2012)
PHYSICAL REVIEW LETTERS
week ending
5 OCTOBER 2012
144301-4
*To whom correspondence should be addressed.
preis@mit.edu
[1] T. Baker, N. Olson, and S. Fuller, Microbiol. Mol. Biol.
Rev. 63, 862 (1999).
[2] A. Dinsmore, M. Hsu, M. Nikolaides, M. Marquez,
A. Bausch, and D. Weitz, Science 298, 1006 (2002).
[3] D. Hu, K. Sielert, and M. Gordon, J. Mech. of Mater. and
Struct. 6, 1197 (2011).
[4] G. Wempner and D. Talaslidis, Mechanics of Solids and
Shells (CRC press, Boca Raton, FL, 2003).
[5] B. Audoly and Y. Pomeau, Elasticity and Geometry: From
Hair Curls to the Nonlinear Response of Shells (Oxford
press, New York, 2010).
[6] J. H. Jellett, Trans. R. Irish Acad. 22, 343 (1849); J.
Rayleigh, The Theory of Sound (Dover Publications,
New York, 1976), Vol. 1.
[7] A. Pogorelov, Extrinsic Geometry of Convex Surfaces
(American Mathematical Society, Providence, RI, 1988).
[8] I. Ivanova-Karatopraklieva and I. Sabitov, J. Math. Sci.
Univ. Tokyo 74, 997 (1995).
[9] A. E. H. Love, A Treatise on the Mathematical Theory of
Elasticity (Dover Publications, New York, 1944); A.
Goldenveizer, Theory of Elastic Thin Shells (Pergamon
Press, New York, 1961); F. Niordson, Shell Theory, North-
Holland Series in Applied Mathematics and Mechanics
(North Holland, New York, 1985).
[10] E. Reissner, J. Math. Phys. (Cambridge, Mass.) 25,80
(1946); 25, 279 (1946).
[11] A. Pogorelov, Bending of Surfaces and Stability of Shells
(American Mathematical Society, Providence, RI, 1988).
[12] D. Vella, A. Adjari, A. Vaziri, and A. Boudaoud, J. R. Soc.
Interface 9, 448 (2012).
[13] D. Chapelle and K.-J. Bathe, The Finite Element Analysis
of Shells: Fundamentals (Springer, New York,
2010).
[14] A. Vaziri and L. Mahadevan, Proc. Natl. Acad. Sci. U.S.A.
105, 7913 (2008); A. Vaziri, Thin-Walled Struct. 47, 692
(2009).
[15] Note that our notion of GIR in thin shells is not to be
confused with the established concept of geometric stiff-
ening which is a stiffening (or weakening), typically also
for a thin structure, due to a varying stress state as
explained in R. Levy and W. R. Spillers, Analysis of
Geometrically Nonlinear Structures (Kluwer Academic
Publishers, Dordrecht, 2003).
[16] M. Arnoldi, M. Fritz, E. Ba
¨
uerlein, M. Radmacher, E.
Sackmann, and A. Boulbitch, Phys. Rev. E 62
, 1034
(2000); I. Ivanovska, G. Wuite, B. Jo
¨
nsson, and A.
Evilevitch, Proc. Natl. Acad. Sci. U.S.A. 104, 9603
(2007); W. Roos and G. J. L. Wuite, Adv. Mater. 21,
1187 (2009); W. Roos, R. Bruinsma, and G. Wuite,
Nature Phys. 6, 733 (2010).
[17] S.-Y. Woo, A. S. Kobayashi, C. Lawrence, and W. A.
Schlegel, Ann. Biomed. Eng. 1, 87 (1972); A. Elsheikh,
D. Wang, A. Kotecha, M. Brown, and D. Garway-Heath,
Ann. Biomed. Eng. 34, 1628 (2006).
[18] H.-J. Schock, Soft Shells: Design and Technology of
Tensile Architecture (Birkha
¨
user, Basel, 1997); C.
Ceccato, L. Hesselgren, M. Pauly, H. Pottmann, and J.
Wallner, Advances in Architectural Geometry 2010
(Springer Wien, New York, 2010).
[19] D. Vella, A. Adjari, A. Vaziri, and A. Boudaoud, Phys.
Rev. Lett. 109, 144302 (2012).
PRL 109, 144301 (2012)
PHYSICAL REVIEW LETTERS
week ending
5 OCTOBER 2012
144301-5
... Previous studies indicated that the stiffness of a continuous ellipsoidal shell is strongly dependent on its geometry Lazarus et al., 2012), in which the aspect ratio a/b is an important geometric parameter. As an extension to the hemisphere (with a/b = 1.0), we also create several half-ellipsoid morphing structures with aspect ratio a/b in the range 0.5 ≤ a/b ≤ 2.0. ...
... 3(c). The initial stiffness of each morphed structure is measured from the F − D curves within the linear elastic regime (D t, as suggested in (Lazarus et al., 2012)); this defines the (linear) rigidity of the morphed structure. The measured value of K is plotted as a function of a/b in fig. ...
... The rigidity of continuous ellipsoidal shells is known to depend on the aspect ratio a/b Lazarus et al., 2012). One natural question is that how stiff the morphed shell (the tessellated shell) is compared to the continuous shell of the same gross shape. ...
Preprint
Full-text available
Shape-morphing structures, which are able to change their shapes from one state to another, are important in a wide range of engineering applications. A popular scenario is morphing from an initial two-dimensional (2D) shape that is flat to a three-dimensional (3D) target shape. One of the exciting manufacturing paradigms is transforming flat 2D sheets with prescribed cuts (i.e. kirigami) into 3D structures. By employing the formalism of the 'tapered elastica' equation, we develop an inverse design framework to predict the shape of the 2D cut pattern that would generate a desired axisymmetric 3D shape. Our previous work has shown that tessellated 3D structures can be achieved by designing both the width and thickness of the cut 2D sheet to have particular tapered designs. However, the fabrication of a sample with variable thickness is quite challenging. Here we propose a new strategy -- perforating the cut sheet with tapered width but uniform thickness to introduce a distribution of porosity. We refer to this strategy as perforated kirigami and show how the porosity function can be calculated from our theoretical model. The porosity distribution can easily be realized by laser cutting and modifies the bending stiffness of the sheet to yield a desired elastic deformation upon buckling. To verify our theoretical approach, we conduct finite element simulations and physical experiments. We also examine the loading-bearing capacity of morphed structures via indentation tests in both FEM simulations and experiments. As an example, the relationship between the measured geometric rigidity of morphed half-ellipsoids and their aspect ratio is investigated in details.
... Previous investigations on snap buckling mainly focus on a single rod/ribbon-like system under different loading and boundary conditions, e.g., asymmetrical constraints [21,22], stretching [23,24], twisting [25], shearing [26], and out-of-plane compression [27,28], while the systematic investigations on the multi-stability and the complex configurations of multiple rods structure are quite limited [29,30]. Moreover, even though the buckling instability and shape shifting of 2D curved surfaces like cylinders [31,32,33,34] and spheres/hemispheres [35,36,37,38,39,31,40,41,42,43] have been largely studied, the buckling patterns in hollow gridshells would be different from the ones in continuum spherical shells [37,38]. The nonlinear responses of reticulated domes, e.g., stability, buckling, and snapping, have also drawn tremendous attention to the researchers in the structural mechanics community [44,45,46]. ...
... Previous investigations on snap buckling mainly focus on a single rod/ribbon-like system under different loading and boundary conditions, e.g., asymmetrical constraints [21,22], stretching [23,24], twisting [25], shearing [26], and out-of-plane compression [27,28], while the systematic investigations on the multi-stability and the complex configurations of multiple rods structure are quite limited [29,30]. Moreover, even though the buckling instability and shape shifting of 2D curved surfaces like cylinders [31,32,33,34] and spheres/hemispheres [35,36,37,38,39,31,40,41,42,43] have been largely studied, the buckling patterns in hollow gridshells would be different from the ones in continuum spherical shells [37,38]. The nonlinear responses of reticulated domes, e.g., stability, buckling, and snapping, have also drawn tremendous attention to the researchers in the structural mechanics community [44,45,46]. ...
... The dependence of structural rigidity and critical snapping points on the number of rods in elastic gridshells are next measured. Similar to the previous study [14], we find all rods contribute to the rigidity of an elastic gridshell, which highlights the importance of structural nonlocality in hollow grids, in contrast to the local response in continuum shells [38]. Ref. [13] mainly considered the form-finding process of an elastic gridshell, while paying only a little attention to its post-buckling behavior; Ref. [14] turned their focus to the studying of the post-buckling performance, e.g., rigidity of the whole structure, while the force-displacement curves and the mechanical responses are within a linear regime. ...
Article
Motivated by the observations of snap-through phenomena in pre-stressed strips and curved shells, we numerically investigate the snapping of a pre-buckled hemispherical gridshell under apex load indentation. Our experimentally validated numerical framework on elastic gridshell simulation combines two components: (i) Discrete Elastic Rods method, for the geometrically nonlinear description of one dimensional rods; and (ii) a naive penalty-based energy functional, to perform the non-deviation condition between two rods at joint. An initially planar grid of slender rods can be actuated into a three dimensional hemispherical shape by loading its extremities through a prescribed path, known as buckling induced assembly; next, this pre-buckled structure can suddenly change its bending direction at some threshold points when compressing its apex to the other side. We find that the hemispherical gridshell can undergo snap-through buckling through two different paths based on two different apex loading conditions. The first critical snap-through point slightly increases as the number of rods in gridshell structure becomes denser, which emphasizes the mechanically nonlocal property in hollow grids, in contrast to the local response of continuum shells. The findings may bridge the gap among rods, grids, knits, and shells, for a fundamental understanding of a group of thin elastic structures, and inspire the design of novel micro-electro-mechanical systems and functional metamaterials.
... In 1944, under supervision of F. Tölke, W. Chang (later changed to W. Zhang) completed his Dr.-Ing at TU Berlin and the second part of his thesis was published in 1949 [19] with an asymptotic solution of Tölke's toroidal equations [20] in Bessel functions. Eric Reissner [21], son of Hans Reissner, carried on his father's legacy and studied the problem of pure bending of curved tubes with orthogonal homogeneous materials and derived a general mixed-type equation of the curved tubes. [22] and [23] studied the toroidal or ring shell problems from the point of view of the small deflection theory of thin shells of revolution loaded symmetrically with respect to their axis. ...
... Many natural and man-made objects take the shape of shells to gain better rigidity. Hence [21] and [43] studied the egg-like shell geometry-induced rigidity of non-spherical pressurized elastic shells (ellipsoidal and cylindrical). The human foot evolution also takes advantage of curvature to increase its stiffness, hence [44] investigate the role of transverse curvature in stiffening the human foot. ...
Preprint
For a given material, different shapes correspond to different rigidities. In this paper, the radii of the oblique elliptic torus are formulated, a nonlinear displacement formulation is presented and numerical simulations are carried out for circular, normal elliptic, and oblique tori, respectively. Our investigation shows that both the deformation and the stress response of an elastic torus are sensitive to the radius ratio, and indicate that the analysis of a torus should be done by using the bending theory of shells rather than membrance theory. A numerical study demonstrates that the inner region of the torus is stiffer than the outer region due to the Gauss curvature. The study also shows that an elastic torus deforms in a very specific manner, as the strain and stress concentration in two very narrow regions around the top and bottom crowns. The desired rigidity can be achieved by adjusting the ratio of minor and major radii and the oblique angle.
... For shells with zero Gaussian curvature, the deformation is strongly anisotropic and influenced by the indentation location and boundary conditions [1,14,15]. The response to indentation of ellipsoidal and cylindrical shells (the latter obtained as the limit of ellipsoidal shells with one radius of curvature going to infinity) reveals the role of mean curvature in controlling the indentation stiffness, and Gaussian curvature in defining the influence of the boundary conditions on its response [16,1,17]. ...
Preprint
When poking a thin shell-like structure, like a plastic water bottle, experience shows that an initial axisymmetric dimple forms around the indentation point. The ridge of this dimple, with increasing indentation, eventually buckles into a polygonal shape. The polygon order generally continues to increase with further indentation. In the case of spherical shells, both the underlying axisymmetric deformation and the buckling evolution have been studied in detail. However, little is known about the behaviour of general geometries. In this work we describe the geometrical and mechanical features of the axisymmetric ridge that forms in indented general shells of revolution with non-negative Gaussian curvature and the conditions for circumferential buckling of this ridge. We show that, under the assumption of `mirror buckling' a single unified description of this ridge can be written if the problem is non-dimensionalised using the local slope of the undeformed shell mid-profile at the ridge radial location. In dimensional form the ridge properties evolve in quite different ways for different mid-profiles. Focusing on the indentation of shallow shells of revolution with constant Gaussian curvature, we use our theoretical framework to study the properties of the ridge at the circumferential buckling threshold and evaluate the validity of the mirror buckling assumption against a linear stability analysis on the shallow shell equations, showing very good agreement. Our results highlight that circumferential buckling in indented thin shells is controlled by a complex interplay between the geometry and the stress state in the ridge. The results of our study will provide greater insight into the mechanics of thin shells. This could enable indentation to be used to measure the mechanical properties of a wide range of shell geometries or used to design shells with specific mechanical behaviours.
... In the latter approach, a sensor is placed against the outer abdominal wall and measures the force and depth of the abdominal wall indentation. The abdominal wall acts as an elastic shell [18][19][20][21][22] , with the force of indentation proportional to the IAP 23 . Studies have been previously published that dealt with mathematical models and experimental approaches to measuring the connection between IAP and abdominal wall compliance 10,[24][25][26] . ...
Article
Full-text available
Early recognition of elevated intraabdominal pressure (IAP) in critically ill patients is essential, since it can result in abdominal compartment syndrome, which is a life-threatening condition. The measurement of intravesical pressure is currently considered the gold standard for IAP assessment. Alternative methods have been proposed, where IAP assessment is based on measuring abdominal wall tension, which reflects the pressure in the abdominal cavity. The aim of this study was to evaluate the feasibility of using patch-like transcutaneous sensors to estimate changes in IAP, which could facilitate the monitoring of IAP in clinical practice. This study was performed with 30 patients during early postoperative care. All patients still had an indwelling urinary catheter postoperatively. Four wearable sensors were attached to the outer surface of the abdominal region to detect the changes in abdominal wall tension. Additionally, surface EMG was used to monitor the activity of the abdominal muscles. The thickness of the subcutaneous tissue was measured with ultrasound. Patients performed 4 cycles of the Valsalva manoeuvre, with a resting period in between (the minimal resting period was 30 s, with a prolongation as necessary to ensure that the fluid level in the measuring system had equilibrated). The IAP was estimated with intravesical pressure measurements during all resting periods and all Valsalva manoeuvres, while the sensors continuously measured changes in abdominal wall tension. The association between the subcutaneous thickness and tension changes on the surface and the intraabdominal pressure was statistically significant, but a large part of the variability was explained by individual patient factors. As a consequence, the predictions of IAP using transcutaneous sensors were not biased, but they were quite variable. The specificity of detecting intraabdominal pressure of 20 mmHg and above is 88%, with an NPV of 96%, while its sensitivity and PPV are currently far lower. There are inherent limitations of the chosen preliminary study design that directly caused the low sensitivity of our method as well as the poor agreement with the gold standard method; in spite of that, we have shown that these sensors have the potential to be used to monitor intraabdominal pressure. We are planning a study that would more closely resemble the intended clinical use and expect it to show more consistent results with a far smaller error.
... The final force-displacement curves present the different bearing capacities between spherical and cylindrical shapes and are in good agreement with the finite element analysis (FEA) results (Fig. 4d). The stiffness (K) of the spherical shell, defined as the slope of the force-displacement curves, is higher than that of cylinder shell (Fig. S8), following the deduction as K cylinder ~ K spherical shell (t/R) 1/2 < < K spherical shell , where t is the thickness and R is the principal radius of curvature (t/R < < 1) [43][44][45]. The simulation results display the deformations like a mirror-buckling along a circular rage in spherical shell and a smooth-edge dimple formed with two d-cones linked by an inverted ridge in cylindrical one [43,46]. ...
Article
Full-text available
The processing capability is vital for the wide applications of materials to forge structures as-demand. Graphene-based macroscopic materials have shown excellent mechanical and functional properties. However, different from usual polymers and metals, graphene solids exhibit limited deformability and processibility for precise forming. Here, we present a precise thermoplastic forming of graphene materials by polymer intercalation from graphene oxide (GO) precursor. The intercalated polymer enables the thermoplasticity of GO solids by thermally activated motion of polymer chains. We detect a critical minimum containing of intercalated polymer that can expand the interlayer spacing exceeding 1.4 nm to activate thermoplasticity, which becomes the criteria for thermal plastic forming of GO solids. By thermoplastic forming, the flat GO-composite films are forged to Gaussian curved shapes and imprinted to have surface relief patterns with size precision down to 360 nm. The plastic-formed structures maintain the structural integration with outstanding electrical (3.07 × 10 ⁵ S m ⁻¹ ) and thermal conductivity (745.65 W m ⁻¹ K ⁻¹ ) after removal of polymers. The thermoplastic strategy greatly extends the forming capability of GO materials and other layered materials and promises versatile structural designs for more broad applications. Graphical abstract
... By contrast, the first-order energy term U 1 m and second-order torque energy term U q m fail at replicating the nonlinear behavior, even for small displacements where, consequently, at least a second-order reduced magnetic energy is required. Moreover, from the results in Fig. 3, we notice that the magnetic field modifies the indentation response of the shell similarly to pressure (Vella et al., 2012;Lazarus et al., 2012;Marthelot et al., 2017;Hutchinson and Thompson, 2017), since the shell can be either strengthened or weakened depending on whether the pressure is acting to inflate or deflate the shell. Figure 3: Asymmetric indentation of a spherical shell under a magnetic field in terms of the dimensionless force F R/(2πD) versus the dimensionless normal displacement at the point of indentation w/h. ...
Preprint
Full-text available
We develop a reduced model for hard-magnetic, thin, linear-elastic shells that can be actuated through an external magnetic field, with geometrically exact strain measures. Assuming a reduced kinematics based on the Kirchhoff-Love assumption, we derive a reduced two-dimensional magneto-elastic energy that can be minimized through numerical analysis. In parallel, we simplify the reduced energy by expanding it up to the second-order in the displacement field and provide a physical interpretation. Our theoretical analysis allows us to identify and interpret the two primary mechanisms dictating the magneto-elastic response: a combination of equivalent magnetic pressure and forces at the first order, and distributed magnetic torques at the second order. We contrast our reduced framework against a three-dimensional nonlinear model by investigating three test cases involving the indentation and the pressure buckling of shells under magnetic loading. We find excellent agreement between the two approaches, thereby verifying our reduced model for shells undergoing nonlinear and non-axisymmetric deformations. We believe that our model for magneto-elastic shells will serve as a valuable tool for the rational design of magnetic structures, enriching the set of reduced magnetic models.
... Our parameters place us in the high bendability regime ϵ −1 ¼ γW 2 =B > 10 3 [10]: our films buckle under minute compression. As we will show, their ability to impose their shape on a liquid is rooted in the high cost of stretching, analogous to the rigidity of a stiff mylar balloon rather than the geometric rigidity of shells that underlies the strength of architectural domes [22,23]. ...
Article
Full-text available
Thin elastic films can spontaneously attach to liquid interfaces, offering a platform for tailoring their physical, chemical, and optical properties. Current understanding of the elastocapillarity of thin films is based primarily on studies of planar sheets. We show that curved shells can be used to manipulate interfaces in qualitatively different ways. We elucidate a regime where an ultrathin shell with vanishing bending rigidity imposes its own rest shape on a liquid surface, using experiment and theory. Conceptually, the pressure across the interface “inflates” the shell into its original shape. The setup is amenable to optical applications as the shell is transparent, free of wrinkles, and may be manufactured over a range of curvatures.
Article
We develop a reduced model for hard-magnetic, thin, linear-elastic shells that can be actuated through an external magnetic field, with geometrically exact strain measures. Assuming a reduced kinematics based on the Kirchhoff–Love assumption, we derive a reduced two-dimensional magneto-elastic energy that can be minimized through numerical analysis. In parallel, we simplify the reduced energy by expanding it up to the second order in the displacement field and provide a physical interpretation. Our theoretical analysis allows us to identify and interpret the two primary mechanisms dictating the magneto-elastic response: a combination of equivalent magnetic pressure and forces at the first order, and distributed magnetic torques at the second order. We contrast our reduced framework against a three-dimensional nonlinear model by investigating three test cases involving the indentation and the pressure buckling of shells under magnetic loading. We find excellent agreement between the two approaches, thereby verifying our reduced model for shells undergoing nonlinear and non-axisymmetric deformations. We believe that our model for magneto-elastic shells will serve as a valuable tool for the rational design of magnetic structures, enriching the set of reduced magnetic models.
Article
Viruses are cellular parasites. The linkage between viral and host functions makes the study of a viral life cycle an important key to cellular functions. A deeper understanding of many aspects of viral life cycles has emerged from coordinated molecular and structural studies carried out with a wide range of viral pathogens. Structural studies of viruses by means of cryo-electron microscopy and three-dimensional image reconstruction methods have grown explosively in the last decade. Here we review the use of cryo-electron microscopy for the determination of the structures of a number of icosahedral viruses. These studies span more than 20 virus families. Representative examples illustrate the use of moderate- to low-resolution (7- to 35-Å) structural analyses to illuminate functional aspects of viral life cycles including host recognition, viral attachment, entry, genome release, viral transcription, translation, proassembly, maturation, release, and transmission, as well as mechanisms of host defense. The success of cryo-electron microscopy in combination with three-dimensional image reconstruction for icosahedral viruses provides a firm foundation for future explorations of more-complex viral pathogens, including the vast number that are nonspherical or nonsymmetrical.
Book
Geometry lies at the core of the architectural design process. It is omnipresent, from the initial determination of form to the final construction. Modern geometric computing provides a variety of tools for the efficient design, analysis, and manufacturing of complex shapes. On the one hand this opens up new horizons for architecture. On the other, the architectural context also poses new problems for geometry. The research area of architectural geometry, situated at the border of applied geometry and architecture, is emerging to address these problems. This volume, presenting the papers accepted at the 2010 Advances in Architectural Geometry conference in Vienna, reflects the substantial progress made in this field. The interdisciplinary nature of architectural geometry is reflected in the diversity of backgrounds of the contributing authors. Renowned architects, engineers, mathematicians, and computer scientists present novel research ideas and cutting-edge solutions at the interface of geometry processing and architectural design.
Book
This practical volume deals exclusively with the design and analysis of nonlinear structures, in which large deformations (displacement and other stresses) are critical. The book presents new formulations and approaches in an accessible format to aspects of structures where analysis falls short. It offers a unified approach to geometric nonlinear effects for structures from stresses to membrane shells and an innovative mathematical approach to this class of problem. The accompanying disc contains Fortran versions of source codes and examples for analysis, a valuable teaching tool extensively class tested by the authors.
Article
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
Article
We investigate the relation between the thickness and diameter of naturally occurring shells, such as the carapaces of turtles and the skulls of mammals. We hypothesize that shells used for different protective functions (for example, protection against headbutting or falling on the ground) will exhibit different power-law trends for shell thickness and diameter. To test this hypothesis, we examine over 600 shells from museum collections with diameters between 1 and 100 cm. Our measurements indicate that eggs, turtle shells, and mammalian skulls exhibit clear and distinct allometric trends. We use a theoretical scaling analysis based on elastic thin shell theory to show that the trends observed are consistent with the corresponding protective functions hypothesized. We thus provide theoretical evidence that shells can be classified by their protective function.
Article
We outline the general principles of thin plate elasticity, by emphasizing their connection with classical results of differential geometry. The relevant Föppl-von Kármán equation can be solved in some specific cases, even though they are strongly nonlinear. We present two types of solutions. The first one concerns the contact of a spherical shell with plane under increasing pressing forces, the second one describes the buckling of a thin film under pressure on a flat substrate, where we explain the observed “telephone-cord” pattern of delamination.