Microstructure-enabled control of wrinkling in nematic elastomer
sheets
Paul Plucinsky and Kaushik Bhattacharya
California Institute of Technology, Pasadena, California 91125, USA
February 7, 2017
Abstract
Nematic elastomers are rubbery solids which have liquid crystals incorporated into their
polymer chains. These materials display many unusual mechanical properties, one such being
the ability to form fine-scale microstructure. In this work, we explore the response of taut
and appreciably stressed sheets made of nematic elastomer. Such sheets feature two potential
instabilities – the formation of fine-scale material microstructure and the formation of fine-scale
wrinkles. We develop a theoretical framework to study these sheets that accounts for both
instabilities, and we implement this framework numerically. Specifically, we show that these
instabilities occur for distinct mesoscale stretches, and observe that microstructure is finer than
wrinkles for physically relevant parameters. Therefore, we relax (i.e., implicitly but rigorously
account for) the microstructure while we regularize (i.e., compute the details explicitly) the
wrinkles. Using both analytical and numerical studies, we show that nematic elastomer sheets
can suppress wrinkling by modifying the expected state of stress through the formation of
microstructure.
1 Introduction
Nematic elastomers are rubbery solids made of cross-linked polymer chains that have nematic meso-
gens (rod-like molecules) either incorporated into the main chain or pendent from them. Their
structure enables a coupling between the entropic elasticity of the polymer network and the or-
dering of the liquid crystals, and this in turn results in fairly complex mechanical properties (see
Warner and Terentjev [30] for a comprehensive introduction and review). At high temperatures, the
mesogens are randomly oriented and the material is isotropic. However, on cooling below a critical
temperature, the mesogens undergo an isotropic to nematic phase transformation and develop a
local orientational order described by a director. This is accompanied by a spontaneous elongation
along the director and contraction transverse to it. However, since the material is isotropic in the
high temperature state, there is no preferential orientation for the director. Thus, the director may
rotate freely with respect to the material frame and form domains where the director varies spatially.
This manifests itself in a very rich range of phenomena.
Of particular interest is the soft elasticity and fine scale microstructure (textured deformation or
striped domains) observed in the clamped-stretch experiments Kundler and Finkelmann [15]. Some
of their key observations are reproduced in Figure 1. They begin with a thin rectangular sheet where
the director is uniformly oriented tangential to the sheet but in the short direction (top of Figure
1(a)). The fact that the director is uniform is evident from the fact that the sheet is transparent.
1
arXiv:1611.08621v2 [cond-mat.soft] 6 Feb 2017
6ABIRD’SEYEVIEWOFLIQUIDCRYSTALELASTOMERS
has confirmed the prediction of soft elasticity: that the fiel
d-induced director rotation
has no energy cost, can easily reach 90
o
rotation angles and has associated mechanical
strains that almost exactly follow the sketch in Fig. 1.6.
Practically, when dealing with rubbers, one might instead i
mpose a mechanical dis-
tortion (say an elongation,
λ
, perpendicular to the original director) and have the other
components of strain, and the director orientation, follow
it. The result is the same –
extension of a rubber costs no elastic energy and is accompan
ied by a characteristic
director rotation. The mechanical confirmation of the carto
on is shown in stress-strain
curves in Fig. 1.7(a) and the director rotation in Fig. 1.7(b
).
We have made liquid crystals into solids, albeit rather weak
solids, by crosslink-
ing them. Like all rubbers, they remain locally fluid-like in
their molecular freedom
and mobility. Paradoxically, their liquid crystallinity a
llows these solid liquid crystals
to change shape without energy cost, that is to behave for som
e deformations like a li-
quid. Non-ideality gives a response we call ‘semi-soft’. Th
ere is now a small threshold
before director rotation (seen in the electrooptical/mech
anical experiments of Urayama
(2005,2006), and to varying degrees in Fig. 1.7); thereafte
r deformation proceeds at
little additional resistance until the internal rotation i
s complete. This stress plateau, the
same singular form of the director rotation, and the relaxat
ion of the other mechanical
degrees of freedom are still qualitatively soft, in spite of
a threshold.
There is a deep symmetry reason for this apparently mysterio
us softness that Fig. 1.6
rationalises in terms of the model of an egg-shaped chain dis
tribution rotating in a solid
that adopts new shapes to accommodate it. Ideally, nematic e
lastomers are rotationally
invariant under separate rotations of both the reference st
ate and of the target state into
which it is deformed. If under some conditions, not necessar
ily the current ones, an
0.03
0.06
0.09
0
20
40
60
80
100
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
l
l
q
(deg)
(a)
(b)
1.15
1.3
1.45
s
(MPa)
e
1
F
IG
. 1.7. (a) Stress-deformation data of K ̈upfer and Finkelman
n (1994), for a series
of rubbers with the same composition and crosslinking densi
ty, but differing in pre-
paration history: some show a normal elastic response while
others are remarkably
soft. (b) The angle of director rotation on stretching nemat
ic elastomer perpendicu-
lar to the director for a variety of different materials, fro
m Finkelmann
etal.
(1997).
The solid line from, theoretical modeling, accurately repr
oduces singular points and
characteristic shape of data.
(
a
)
(
b
)
(
c
)
e
1
e
2
Figure 1: The clamped-stretch experiments of Kundler and Finkelmann [15] showing soft elasticity
and fine scale microstructure, but no wrinkling. (a) Snapshots of the sheet. The nematic director is
oriented vertically when undeformed (top), develops stripe domains of alternating rotated directors
depicted in (c) for moderate deformation (middle) and eventually is uniform and oriented horizon-
tally (bottom). (b) Stress-strain curves of Küpfer and Finkelmann [16] for elastomers of different
preparation histories – the lowest curve is akin to the clamped-stretch sheet in part (a) and describes
a sheet with significant soft elasticity. (c) Fine-scale strip domain microstructure.
They clamp the short edges and pull it along the long edge. The nominal stretch vs. nominal stress
behavior for this sheet is akin the bottom-most curve in Figure 1(b). Notice that nematic sheets
display essentially zero stress for very significant values of stretch: this is known as the
soft elasticity
.
The center and bottom images of Figure 1(a) capture stretch midway through the soft behavior
and at the end of the soft behavior respectively. During the soft stage of stretch, the entire sheet is
strongly scattering light and so visibly cloudy, an optical indication that the director is no longer
uniform. Beyond the soft plateau (i.e., the bottom image), the sheet becomes transparent again
in its central region, indicating uniform director arrangement in this region. However, it remains
cloudy near the clamped edges.
The cloudy regions indicate material microstructure in the form of strip domains of oscillating
director as shown in Figure 1(c). The heuristic is as follows: A nematic elastomer features a director
that can rotate through the material, and this rotation is accommodated by spontaneous elongation
along the director. Thus, the sheet can elongate along the direction of stretch with little stress by
having its director rotate from vertical to horizontal. Doing so uniformly, however, results in a shear
at intermediate orientations, but this shear can be avoided on average by breaking up the cross-
section into domains where one half the directors rotate one way (say through
θ
) while the other
half rotates the other way (through
−
θ
). This is exactly what happens in the clamped stretched
sheet at a very fine scale (microns), manifesting in stripe domains. Finally, when the director has
fully rotated to the horizontal, it becomes uniform again (since the material is invariant under the
change of sign of the director).
Bladon
et al.
[1] proposed a free energy based on entropic elasticity of the chains in the presence
of nematic order to describe the elasticity of nematic elastomers, and Verwey
et al.
[29] explained
2
how stripe domains can arise as a means of minimizing this free energy. DeSimone and Dolzmann
[11] noted that the free energy density proposed by Bladon
et al.
[1] is not quasiconvex, and thus
fine-scale microstructure can arise naturally in these materials. These include stripe domains, but
also more complex microstructure. They also computed explicitly the relaxation of the Bladon
et al.
free energy which implicitly but rigorously accounts for the microstructure. Conti
et al.
[5, 6] used
the planar version of free energy to study the stretching of sheets and were able to explain various
details of the experiments described above including the soft elasticity, formation and disappearance
of stripe domains, and the persistence of domains near the grips even at high stretches.
In this paper, we study another surprising observation inherent in these experiments, one that
that has thus far escaped notice and exploration. Even though these thin sheets have been stretched
significantly with clamped grips, they remain planar and do not wrinkle. In fact, similar experiments
have been conducted by a number of researchers, and none of them have reported any wrinkling
instability. This is surprising because thin sheets of purely elastic materials wrinkle readily when
subjected to either shear [31] or stretching with clamped grips [20, 28, 32].
The wrinkling of thin elastic membranes has been widely studied, motivated by various appli-
cation. Early research was motivated by the use of membranes for aircraft skins where wrinkling
altered their aerodynamic performance. More recent interest stems from the use of membranes in
light-weight deployable space structures including solar sails, telescopes and antennas [14, 18], and
renewed interest in fabric roofs of complex shape [3] (see also [32] and references therein). The
underlying mechanism is relatively simple: thin elastic membranes are unable to sustain any com-
pression; instead they accommodate imposed compressive strains by buckling out of plane. When
a sheet is pulled on clamped edges, the clamps inhibit the natural lateral contraction, leading to
compressive stresses in the lateral direction, which in turn leads to wrinkles or undulations elongated
along the direction of stretch. The wavelength of the undulations are large compared to thickness,
but small compared to the overall dimensions of the sheet.
Mathematically, any finite deformation theory of membranes is not quasiconvex, and thus suffers
from instabilities which can be interpreted as wrinkles (see for example [21, 24, 25]). Further, the
relaxation of such theories gives rise to tension-field theories like those of Mansfield [17] where
membranes can resist tension but not compression. Such theories are zero thickness idealizations
of the membrane which account for the consequences of wrinkles at a scale large compared to
the wrinkles but do not describe them explicitly. Alternatively, in recognizing that wrinkles cause
bending due to the non-zero thickness of the membrane, Koiter-type theories, which capture a sum
of bending and membranes energies, lead to an explicit description of wrinkles. Such theories form
the basis of the analysis of wrinkling described above [20, 28, 31, 32].
In general, there are two approaches in dealing with instabilities resulting from the failure of
(an appropriate notion of) convexity that results in features at a fine scale. The first is relaxation,
where one derives an
effective
or
relaxed
theory that describes the overall behavior after accounting
for the formation of fine-scale features. The relaxed theory of DeSimone and Dolzmann in the
context of liquid crystal elastomers and the tension-field theories for thin membranes are examples
of such relaxation. While these theories are extremely useful in describing overall behavior, they
are often difficult to compute explicitly and they do not resolve all fine scale details though it is at
times possible to
a posteriori
reconstruct them. Further, they are often degenerately convex and
therefore lead to extremely stiff numerical problems. The second approach is
regularization
where
one recognizes that the source of the nonconvexity is the neglect of some smaller order physics, and
adds some higher order term to the energy. The second gradient theories of plasticity, the phase
field theories of phase transformations and the theories of Verwey
et al.
[29] (where the Frank
elasticity regularizes the entropic elasticity) are examples. These resolve the fine-scale details, but
are computationally extremely expensive as they require a very fine resolution.
3
In this work, we are interested in the potential wrinkling behavior of stretched nematic elastomer
sheets. Therefore, we have to account for two sources of instability – a material instability that
results in the formation of fine-scale microstructure and a structural instability that results in fine-
scale wrinkles. We note that the scale of the microstructure (microns) is small compared to the scale
of wrinkles (millimeters). Therefore, we take a multiscale view and systematically develop a theory
that is a relaxation for microstructure but a regularization for wrinkles. The resulting theory is a
Koiter-type theory (4.1) with two terms; the first is the two-dimensional or plane stress reduction
of the relaxed energy of DeSimone and Dolzmann [11] and the second is bending.
To develop this theory, we build on the work of Cesana
et al.
[4] who started from an appropriate
three dimensional formulation and derived from it the relaxed membrane energy. They also provided
the explicit characterization of the instabilities or oscillations (Young measures) that underly the
relaxation. This had two sources – microstructure or stripe domains and wrinkles. Remarkably,
for the taut sheets of current interest, they found that the overall deformation gradients for which
microstructure occurs are distinct from the overall deformation gradients for which wrinkling occurs.
Here, we show, in addition, that the plane stress reduction of the relaxed entropic elastic energy
coincides with the relaxed membrane energy in all regions of interest for taut membranes except
the one involving tension wrinkles, where it is the plane stress reduction of the original entropic
energy. Consequently, this reduction accurately describes the role of microstructure in the in-plane
deformation of nematic elastomer sheets. It does not, however, accurately describe tension wrinkling
in these sheets; rather, regularization or relaxation is needed. Taking the regularization approach,
we use the Young measure characterization of tension wrinkling oscillations to compute the bending
energy for these oscillations systematically from the relaxed entropic elastic energy. This bending
together with the plane stress reduction of the relaxed entropic elastic energy gives the appropriate
Koiter-type theory for taut sheets of nematic elastomer.
We use this theory in numerical studies to demonstrate that the ability of the material to
form microstructure does indeed suppress wrinkling. Specifically, we study the clamped-stretch
experiments of Kundler and Finkelmann [15] and focus on sheets with lateral dimensions for which
purely elastic materials readily wrinkle under this stretch. We show that as a parameter that
describes the strength of the nematic order increases, the onset of wrinkling is delayed and the
amplitude is decreased, until it is completely suppressed for large enough nematic order. We further
show that the reason for this is that the ability to form microstructure alters the stress distribution
close to the clamps. These results open up the possibility of exploiting these materials in applications
where one seeks membranes that do not wrinkle.
This paper is organized as follows: In section 2, we comment on the notation and present for
clarity a visual summary capturing the hierarchy of theories for nematic elastomers described and
developed in sequel. We provide the background in Section 3, recalling the theory of Bladon
et al.
[1] for the entropic elasticity of nematic elastomers (Section 3.1), the relaxation of this energy by
DeSimone and Dolzmann [11] (Section 3.2) and the membrane theory of Cesana
et al.
[4] (Section
3.3). We develop the Koiter theory in Sections 4. We describe the numerical method in Section 5,
and apply it to study clamped extension of sheets in Section 6. We conclude in Section 7.
2 Notation and Overview
2.1 Some of the notation
We denote with
R
n
the
n
dimensional Euclidian space endowed with the usual scalar product
u
·
v
:=
u
T
v
and norm
|
u
|
:=
√
u
·
u
. We denote the unit sphere in
R
n
by
S
n
−
1
and it is defined as
the set of all vectors
u
∈
R
n
with
|
u
|
= 1
. We label with
R
m
×
n
the space of
m
×
n
matrices with
4
W
e
+ Frank Elasticity
W
qc
3
D
h
⌧
1
h,
p
/μ
!
0
h
!
0
h
!
0
p
/μ
!
0
,h
⌧
1
W
qc
2
D
p
/μ
!
0
(FreeEnergy)
( Koiter Theory )
(Relaxed3DTheory)
(MembraneTheory)
W
ps
+Bending
(a) Hierarchy of theories for nematic elastomers
Reg.
Micro.
Unst.
Micro.
Rel.
Micro.
Reg.
Micro.
Unst.
Micro.
Rel.
Micro.
Reg.
Wrink.
Unst.
Wrink.
Rel.
Wrink.
W
3
D
W
qc
3
D
W
2
D
W
qc
2
D
W
e
+
FrankEl.
W
2
D
+
Bending
h
!
0
h
⌧
1
p
/μ
#
0
3D Theories
2D Theories
p
/μ
⌧
h
W
ps
W
ps
+
Bending
(b) Strain energy densities and their treatment of instabilities
Figure 2: (a) Theories which properly account for the formation of microstructure (in 3D) and
both microstructure and wrinkling (in 2D). Each theory can be derived in the treatment of the
lengthscale as depicted. We focus here on deriving the Koiter theory (highlighted in red) starting
from the relaxed 3D theory. Elsewhere [22], we derive it starting form the free energy, where there
are simply more details to track. (b) The various strain energy densities of nematic elastomers and
whether they are stable (regularized or relaxed) or unstable to instabilities: microstructure in the
case of 3D densities, and both microstructure and wrinkling in the case of 2D densities. The stable
theories are highlighted. Starting from any strain energy density on this chart, the densities to the
right, lower diagonal, and directly below can be derived under the treatment of the lengthscales as
depicted. In principal, the top right of this chart can be populated, though such theories are less
physically relevant given the disparity in lengthscales.
5
real entries. For
n >
1
, we denote with
SO
(
n
)
the space of rotation matrices (i.e., each
F
∈
R
n
×
n
such that
F
T
F
=
I
and
det
F
= 1
). We take
R
+
to be the set of non-negative real numbers.
We often describe the material points of a three dimensional solid with
x
:=
x
1
e
1
+
x
2
e
2
+
x
3
e
3
for the fixed right-handed orthonormal basis
{
e
1
,e
2
,e
3
} ⊂
R
3
depicted in Figure 1. Similarly, we
often describe the material points of a two dimensional sheet with
̃
x
:=
x
1
̃
e
1
+
x
2
̃
e
2
for the analogous
two dimensional orthonormal basis
{
̃
e
1
,
̃
e
2
} ⊂
R
2
. As with the basis vectors and material points,
we use tilde as a mean to distinguish between two dimensional and three dimensional quantities
(if there is no conflict with previous notation). So we use
F
∈
R
3
×
3
to describe the deformation
gradient of a solid and
̃
F
∈
R
3
×
2
to describe the planar deformation gradient of a sheet; we denote
with
∇
the three dimensional gradient (with respect to
x
) and
̃
∇
the planar gradient (with respect
to
̃
x
);
...
; etc.
Lastly, we find it natural at points to introduce certain mathematical concepts:
W
1
,p
Sobolev
Spaces and weak convergence (i.e.,
⇀
) in these spaces, quasiconvexification as a means of relaxation,
Γ
-convergence as a means of dimension reduction and the theory of gradient Young measures for
characterizing instabilities. We refer to Evans [12], Dacorogna [9], Braides [2] and Müller [19]
respectively for introductions into these concepts.
2.2 Overview on the hierarchy of theories for nematic elastomers
In this work, we systematically develop a two dimensional Koiter theory for nematic elastomer
sheets by both (i) starting from an appropriate three dimensional description of these elastomers
[1, 11, 29, 30], and (ii) building off of the development of the effective or relaxed membrane theory
[4]. Thus, in the course of this development, we find it natural to introduce several variants of strain
energy densities modeling nematic elastomers, for which their is a well-characterized hierarchy. The
hierarchy is related to the mathematical treatment of small-length scales (i.e.,
√
κ/μ
and
h
) inherent
to nematic elastomer sheets, and how instabilities (both wrinkling and microstructure) are accounted
for in this treatment.
Briefly (all of this is expanded upon in Sections 3 and 4), a three dimensional nematic elastomer
can form fine-scale microstructure on a lengthscale
√
κ/μ
related to the competition between en-
tropic and Frank elasticity in these solids. In addition to this microstructure, a sheet of nematic
elastomer may wrinkle since the thickness
h
is small compared to the lateral extent of the sheet. In
typical sheets,
√
κ/μ
h
. To guide the reading of Sections 3 and 4, we provide a visual summary of
the theories which emerge in the competition of these two lengthscales and their hierarchy (Figure
2).
3 Background
3.1 Free energy density for nematic elastomers
A widely accepted theory for the free energy or entropic elasticity of nematic elastomers is due to
Bladon
et al.
[1] (see also Warner and Terentjev [30]). As with the classical neo-Hookean model for
rubbery solids, this formulation emerges from the statistics of polymer chain conformations, with
the caveat being that the distribution properly account for nematic anisotropy associated with the
liquid crystal rod-like molecules. The free energy has the form
W
e
(
F,n
) :=
μ
2
{
Tr(
F
T
`
−
1
n
F
)
−
3
,
if
n
∈
S
2
,
det
F
= 1
+
∞
otherwise
,
(3.1)
6