Patterning nonisometric origami in nematic elastomer sheets
Paul Plucinsky
a
, Benjamin A. Kowalski
b
, Timothy J. White
b
, and Kaushik Bhattacharya
c
a
Aerospace and Engineering Mechanics, University of Minnesota
b
Materials and Manufacturing Directorate, Air Force Research Lab
c
Engineering and Applied Sciences, California Institute of Technology
December 14, 2017
Abstract
Nematic elastomers dramatically change their shape in response to diverse stimuli includ-
ing light and heat. In this paper, we provide a systematic framework for the design of complex
three dimensional shapes through the actuation of heterogeneously patterned nematic elastomer
sheets. These sheets are composed of
nonisometric origami
building blocks which, when appro-
priately linked together, can actuate into a diverse array of three dimensional faceted shapes.
We demonstrate both theoretically and experimentally that: 1) the nonisometric origami build-
ing blocks actuate in the predicted manner, 2) the integration of multiple building blocks leads
to complex multi-stable, yet predictable, shapes, 3) we can bias the actuation experimentally
to obtain a desired complex shape amongst the multi-stable shapes. We then show that this
experimentally realized functionality enables a rich possible design landscape for actuation us-
ing nematic elastomers. We highlight this landscape through theoretical examples, which utilize
large arrays of these building blocks to realize a desired three dimensional origami shape. In
combination, these results amount to an engineering design principle, which we hope will provide
a template for the application of nematic elastomers to emerging technologies.
Introduction
The seamless integration of function and form promises to spur innovation in technologies ranging
from MEMS and NEMS devices (e.g., with novel electrical, electromagnetic and energy function-
ality), reconfigurable and soft robotics, wearable electronics, and compliant bio-medical devices
[6, 9, 10, 14, 24, 33]. This integration can be facilitated by incorporating soft active materials
into thin or slender structures to program complex three dimensional shapes not easily achieved by
conventional means of manufacturing. This is not without its challenges: The coupling of nonlin-
earities—at the material level and at the structural level—makes a salient and general theory of
design with these systems a trying task. Even more, bridging the gap between an idealized theory
and what is capable (and practical) experimentally offers a different, but equally important, set of
challenges. In this work, we address many of these challenges in the context of active patterned
nematic elastomer sheets by developing a systematic framework for actuating complex shapes in
these systems that is grounded in both first principles theory and experimental capability.
Nematic elastomers combine the elasticity of a soft, highly deformable polymer network with
the orientational ordering of liquid crystalline monomer units. This results in a solid with dramatic
shape-changing response to temperature change and other stimuli [31, 32]: At low temperatures, the
liquid crystals prefer being aligned (in some average sense), with the orientation of this alignment
1
arXiv:1712.04585v1 [cond-mat.soft] 13 Dec 2017
T
Figure 1: A “humanoid” soft robot.
described locally by a unit vector
n
0
called the director. Upon heating, however, the molecular
order is disrupted, and this gives rise to a strongly anisotropic macroscopic deformation since the
polymer network is intrinsically coupled to this order. Typically, the elastomer contracts along the
director and expands transversely.
Building on key ideas in the study of non-Euclidean plates [4, 11, 26], Modes et al. [15, 17]
recognized that by programming the director
heterogeneously
in the plane of a thin sheet, stimula-
tion results in inhomogeneous shape-change that in turn drives complex three dimensional shapes.
Indeed, they showed theoretically how patterning azimuthal and radial director profiles about a
defect enabled conical and saddle-like shapes upon actuation. These were later realized experimen-
tally—first by de Haan et al. [3] in nematic glass sheets and later by Ware et al. [27] in nematic
elastomers. Even more, the latter work made possible the prescription of an arbitrary planar direc-
tor profile in a thin sheet (i.e., through
voxelated
LCEs), bringing questions of designable actuation
to the forefront. Further, since the entire sheet participates in the actuation, it is extremely robust
and capable large actuation force and energy. This has motivated the study of other more complex
patterns [1, 18, 20, 21, 22, 23]. Of particular interest to this work is the class of
nonisometric
origami
patterns, in which the director is piecewise uniform in the plane. These enable the design
of complex faceted (origami) shapes from simple building blocks [22, 23].
However, all the complex patterns described in these works [1, 3, 15, 17, 18, 20, 21, 22, 23, 27]
suffer from having multiple, energetically degenerate actuated configurations. For example, the
cone can actuate either up or down. This is compounded by the fact that, as patterns are made
more complex, the number of multi-stable configurations increases. We illustrate this in Figure
2, showing a nonisometric origami pattern comprising a joined pair of actuating pyramids. (The
pyramid design is discussed in more detail below). This sample exhibits two non-trivially distinct
configurations which are mechanically bi-stable. Thus, in order to deterministically actuate the
sheet into a
single
desired configuration, this degeneracy must be broken, ideally in a way that
allows us to arbitrarily program individual folds to actuate up or down as prescribed.
One obvious strategy for breaking degeneracies is to exploit the twisted nematic profile, where
the planar director varies through the thickness of the sheet. These profiles are known to generate
a spontaneous curvature upon actuation which results in complex bending deformations both in
isolated strips [19, 25, 29] and when integrated into larger patterned sheets [7, 8, 16]. However,
twisted nematics induce a spontaneous Gauss curvature, which can lead to undesirable anti-clastic
bending. Even further, Gauss’ Theorema Egregium states that a change in Gauss curvature is neces-
sarily accompanied by a stretch. Consequently, this mode of actuation always results in mechanical
frustrations in the sheet that influence the robustness of actuation [30]. So, this strategy has proven
2