Abstract— Helical structures are the basic building blocks in

biological systems, and have inspired the design and

manufacturing of helical devices with applications in

nanoelectromechanical systems (NEMS) , bio-chemical sensing,

magnetic field detection, optoelectronics, micro-robotics and

drug delivery devices. Meanwhile, multi-stable structures,

represented by the Venus flytrap and spring tape, have

attracted increasing attention due to their applications in

making artificial muscles, bio-inspired robots, deployable

aerospace components and energy harvesting devices. Here we

address the mechanical principles of self-assembly in

spontaneous bending and twisting structures, which can be

employed to manufacture self-assembling robotics at various

scales. The established theoretical framework provides a means

of guiding the on-demand design of self-assembling systems

with potential actuating mechanisms. Experimental designs of

such structures at both macroscopic and microscopic scales,

supported by finite element modeling results, demonstrate the

feasibility of creating self-assembled micro-robots with

desirable mechano-sensing and actuating capability.

I. INTRODUCTION

Mechanical self-assembly of spontaneous bending and twisting structures has been among the most desirable tasks for robotic engineers. Spontaneous helical structures are almost everywhere in natural and engineered systems [1] and have fostered tremendous research for their potential applications in nanoelecromechanical systems (NEMS), drug delivery and biological/chemical sensing, optoelectronics, and microrobotics [2]. With recent advancements in nanotechnology, physicists and engineers can now grow helical nanoribbons [3] through a ``bottom-up" approach and have also begun exploring ways to fabricate helical nanoribbons of controllable morphology [4] in a ``top-down" manner [5, 6,14].

Typically, helical ribbon shapes are achieved due to the balancing of surface stress or internal residual stress with elastic restoring forces of bending and stretching. The sophisticated interactions between elastic restoring forces and the molecular orientations and chirality often lead to the selection of different shapes [7], such as cylindrical helical ribbons and tubules with vanishing Gauss curvature, and twisted ribbons or straight helicoids with non-zero Gauss

* Resarch supported by American Academy of Mechanics Founder's

Award from the Robert M. and Mary Haythornthwaite Foundation, and

Society in Science, The Branco Weiss Fellowship, administered by ETH

Zurich (Z.C.).

Z. Chen, was with Princeton University, Princeton, NJ 08544 USA. He

is now a Society in Science – Branco Weiss Fellow with Department of

Biomedical Engineering, Washington University, St. Louis, MO 63130,

USA (e-mail: chen.z@seas.wustl.edu).

curvature. For example, charged gemini surfactants with chiral counterions exhibit a transition between helical ribbons with cylindrical curvature and twisted ribbons with Gauss curvature as a function of molecular chain length [8]. Similarly, mixed bilayers of saturated and diacetylenic phospholipids show transitions from cylindrical tubules to twisted ribbons, and then to nanometer-scale tubules, as a function of temperature [9].

Driven by mechanical anisotropy (such as in surface stress or elastic modulus) and geometric mis-orientation, helical shapes form spontaneously, and the transitions between helicoids, cylindrical helical ribbons, general helical ribbons and tubules can be achieved by tuning a few geometric parameters such as the principal curvatures and

mis-orientation angle [10, 11，13]. In twist-nematic-

elastomer films, for example, the chiral molecular arrangement of liquid crystal mesogens drives the shape selection of helicoids and spiral ribbons due to the coupling between the order of liquid crystalline and elasticity [7]. Nevertheless, the role of geometric nonlinearity in shape selections of helical geometries remains incompletely understood. In particular, geometric nonlinearity has recently been shown to be key in multi-stable structures [12] featuring more than one stable shape that arise in a variety of natural and engineered systems . Such structures have inspired design principles of deployable or smart actuation devices with multiple stable shapes each functioning in their own regimes.

In this work, we illustrate the principle of mechanical self-assembly of spontaneously bent and twisted thin structures, driven by mechanical anisotropy and geometric mis-orientation. Moreover, bistable behaviors of thin shell structures, inspired by the Venus flytrap [15,16] and slap bracelets [12,17], are revealed in structures from macro-scale to nano-scale structures, and interpreted by our theoretically model, complemented with finite element simulations. It is promising to design and manufacture nano-helical structures in a programmable manner that can be used as micro-robots [2, 6, 18, 19].

.

II. THEORY

In this theoretical framework, the strip is modeled as an

elastic sheet with length L , width w , and thickness

wH and Lw [7]. The cross-section is rectangular and the principle geometric axes are along its length, width, and thickness directions. The originally flat ribbon lies along

sincos 211 eer in the global Cartesian coordinate

Engineering shapes and instability in thin structures: towards self-

assembling micro-robots*

Zi Chen

Proceedings of the 13thIEEE International Conference on NanotechnologyBeijing, China, August 5-8, 2013

978-1-4799-0676-5/13/$31.00 ©2013 IEEE 342

system. If the ribbon only bends along one principal axes

(either 1e or

2e ) due to either surface or residual stresses,

then a cylindrical helical ribbon with zero Gauss curvature forms. A general helical ribbon (with moderately small

width), however, can bear principle curvatures 1 and

2 are

along the axes 1r and

2r with a mis-orientation angle

within the ribbon plane [11]. Then a point P

(321 )()()()( esZesYesXsP ) on the centerline of the

deformed ribbon can be parameterized by the arclength s :

)1)(cos/()( 2 ssX

cos)]sin)(/([)( 2

1 ssssY

sin)]sin)(/([)( 2

1 ssssZ ,

where 22

2

22

1 sincos and

2

2

2

1 sincos .

A variety of shapes including helical cylindrical shapes,

rings, purely twisted ribbons [11], can be achieved just by

tuning the values of 1 ,

2 (or equivalently, the mean

curvature and Gauss curvature) and the mis-orientation angle

. Although geometrically distinct, each shape represents an

element in a subset of a complete class of two-dimensional

manifolds controlled by these three independent geometric

variables [10, 11].

Fig. 1 Tunable helical ribbons manufactured by bonding two or three

layers of strips with different anisotropic pre-strains [11] (with permission).

To obtain the values of the principal curvatures, we

employ a theoretical model based on linear elasticity theory,

differential geometry and stationarity principles [12], which

takes into consideration both the non-uniform bending and

mid-plane stretching due to geometric nonlinearity. We

consider the conformation of a small piece of the ribbon onto

the surface of a torus to account for the geometric

nonlinearity. The total potential energy density per unit area

of the ribbon is

dzCff H

HHzHz )::2

1(|:|: 2/

2/2/2/

,

where C is the fourth-order elastic constant tensor. At

equilibrium, must be stationary with respect to the

unknown parameters, 1 and

2 , i.e., 0/ i (i = 1,2).

III. EXPERIMENTS

Fig. 2 One stable configuration of a bistable structure at the macroscopic scale. Scale bar: 12.0mm.

Fig. 3 The other stable configuration of the same piece as in Fig. 1.

Scale bar: 12.0mm.

A piece of a slap bracelet, cut into a triangular shape on

one end, demonstrates bistable behaviors. The slap bracelet

consists of layered stainless steel wrapped by a fabric cover.

Fig. 2 shows one stable configuration of the as-shaped

bracelet, which can transform quickly to the other stable

shape as shown in Fig. 3 upon certain mechanical stimulus.

Similar behaviors have been observed in the micro-hands

[18].

IV. FINITE ELEMENT SIMULATIONS

We employ finite element simulations (Comsol multiphysics V4.0) to gain insights on the multistable behaviors of these spontaneously curling structures. To illustrate the concept, we study the bistable behavior of a

bilayer triangular plate of total thickness mH 04.0 , and

width mw 1 with the following prescribed misfit strains.

The misfit strain tensor in the bottom layer is

110 eeb , where 04.00 , while that in the top layer

is 220 eet . Here, 1e and 2e denote the unit vectors

along the x, y axis respectively. The composite plate can be stable in two nearly cylindrical configurations. It can be stable in a configuration as shown in Fig. 3 by bending

343

downwards along y axis, where it also exhibit some characteristics of a saddle shape, due to the misfit strain in the bottom layer. On the other hand, the other possible stable shape of the composite plate features bending upwards along the x axis. This is consistent with both the macroscopic bistable plates shown in Fig. 2 and 3.

Fig. 4 One stable configuration of a bistable structure simulated

by finite element method.

.

Fig. 5 The other stable configuration of a bistable structure calculated

by finite element simulation.

Moreover, helical structures can also be simulated using

finite element methods. For example, a bi-layer ribbon

with of total thickness mH 04.0 , and width mw 1.0 ,

and length mL 1 , can deform into a helical shape given

the following misfit strains. The misfit strain tensor in the

bottom layer is

)( 122122110

' eeeeeeeeb ,

where 5.00 . The bilayer ribbon deforms into a helical

configuration as shown in Fig. 6 due to the misfit

strians.

Fig. 6 The stable helical configuration driven by misfit strains in a

bilayer ribbon simulated by finite element method.

V. CONCLUSION

We study the principles of mechanical self-assembly of

spontaneous bending and twisting structures, which can be

employed to manufacture self-assembling robotics at various

scales. The established theoretical framework provides a

means of guiding the on-demand design of self-assembling

systems with potential actuating mechanisms. Experimental

designs of such structures at both macroscopic and

microscopic scales, supported by finite element modeling

results, demonstrate the feasibility of creating self-assembled

micro-robots with desirable mechano-sensing and actuating

capability.

ACKNOWLEDGMENT

Z. C. thanks Drs. Mikko Haataja, David J. Srolovitz,

Carmel Majidi and Qiaohang Guo for helpful discussions.

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