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Design and fabrication of compliant micromechanisms and structures with negative
Poisson's ratio
Larsen, Ulrik Darling; Sigmund, Ole; Bouwstra, Siebe
Published in:
I E E E Journal of Microelectromechanical Systems
Link to article, DOI:
10.1109/84.585787
Publication date:
1997
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Larsen, U. D., Sigmund, O., & Bouwstra, S. (1997). Design and fabrication of compliant micromechanisms and
structures with negative Poisson's ratio. I E E E Journal of Microelectromechanical Systems, 6, 99-106.
https://doi.org/10.1109/84.585787
JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 6, NO. 2, JUNE 1997 99
Design and Fabrication of Compliant
Micromechanisms and Structures
with Negative Poisson’s Ratio
Ulrik Darling Larsen, Ole Sigmund, and Siebe Bouwstra
Abstract— This paper describes a new way to design and
fabricate compliant micromechanisms and material structures
with negative Poisson’s ratio (NPR). The design of compliant
mechanisms and material structures is accomplished in an au-
tomated way using a numerical topology optimization method.
The procedure allows the user to specify the elastic properties of
materials or the mechanical advantages (MA’s) or geometrical
advantages (GA’s) of compliant mechanisms and returns the
optimal structures. The topologies obtained by the numerical
procedure require practically no interaction by the engineer
before they can be transferred to the fabrication unit. Fabrication
is carried out by patterning a sputtered silicon on a plasma-
enhanced chemical vapor deposition (PECVD) glass with a laser
micromachining setup. Subsequently, the structures are etched
into the underlying PECVD glass, and the glass is underetched,
all in one two-step reactive ion etching (RIE) process. The
components are tested using a probe placed on an x-y stage.
This fast prototyping allows newly developed topologies to be
fabricated and tested within the same day. [188]
Index Terms— Compliant micromechanism, fast prototyping,
negative Poisson’s ratio.
I. INTRODUCTION
M
ICROMECHANICAL devices have many promising
areas of application. Some of the main areas are tools
for microfabrication and nanofabrication, microsurgery, and
nanoprobing analysis systems. In nanofabrication, the manip-
ulation of small objects on a surface by microhandling mecha-
nisms, such as a microgripper [1] and positioning microrobotic
devices [2], are steps forward toward higher functionality
levels and higher flexibility. In microsurgery, where accuracy
is needed in microscopic and very sensitive operations, the
micromechanisms may provide very precise tweezers and
knives, etc. In microanalysis and nanoanalysis systems, the
micromechanisms can be used as positioning tools or for
precision probing of a surface [3], whereas microtweezers can
be used to hold the specimens [4].
While design tools for the electronic part of microelectrome-
chanical systems (MEMS) are very well developed, problems
such as methodical design of the mechanical parts, packaging,
Manuscript received December 5, 1995; revised October 1, 1996. Subject
Editor, D.-I. D. Cho. This work was supported by Denmark’s Technical
Research Council (Programme of Research on Computer-Aided Design).
U. D. Larsen and S. Bouwstra are with Mikroelektronik Centret (MIC),
Technical University of Denmark, DK-2800 Lyngby, Denmark.
O. Sigmund is with the Department of Solid Mechanics, Technical Univer-
sity of Denmark, DK-2800 Lyngby, Denmark.
Publisher Item Identifier S 1057-7157(97)02119-7.
and fabrication methods for MEMS still have to be solved.
Here, we shall concentrate on design and fabrication of the
mechanical parts. Because of the small dimensions, it is
difficult to use ideal hinges, bearings, and rigid bodies, as
seen in design of normal multibody systems (e.g., Erdman and
Sandor [5]) for the design of MEMS. Friction and wear would
cause the hinges to break down after a few operation cycles.
Therefore, micromechanisms should be designed as compliant
or flexible-link mechanisms. Compliant mechanisms also have
the advantage that they can be built in one piece, which lowers
the number of fabrication steps.
Fabrication of microstructures can be done using silicon-
surface micromachining in thin-film materials. Currently, two-
dimensional (2-D) fabrication procedures are well developed,
but effort is devoted to the development of fabrication meth-
ods for two-and-a-half [6] and fully three-dimensional (3-D)
micromechanisms [7]. This paper will only consider 2-D
structures. The micromachining process should be compati-
ble with general micromachining processes to facilitate the
implementation with interface electronics and actuators. Fur-
thermore, film thicknesses of more than 5
m are preferred to
prevent buckling and warping of the structures during testing.
These demands can be met by employing low-stress plasma-
enhanced chemical vapor deposition (PECVD) glass. In this
paper, the structures will be operated by external probes. On-
chip integrated actuation would, in principle, be possible using,
for example, thermal bimorph actuators [8]. In the case of
electrically conductive structures, electrostatic comb drives can
be applied.
II. M
ETHODICAL DESIGN OF COMPLIANT MECHANISMS
Design of compliant mechanisms is often accomplished
by trial and error methods. However, Howell and Midha
[9] suggest a method to design compliant mechanisms by
modifications of a rigid body model. Ananthasuresh et al.
[10] use topology optimization methods to find the compliant
mechanism topology, which, in an optimal way, can perform
a given manipulation task, and Sigmund [11] uses a truss
model and topology optimization techniques to design simple
grabbing mechanisms. Topology optimization is typically used
as a design tool when high performance and low weight
of a mechanical structure is required, which is often the
case in the aerospace and automotive industries. This field
has gained much popularity since Bendsøe and Kikuchi [12]
1057–7157/97$10.00 1997 IEEE
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100 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 6, NO. 2, JUNE 1997
introduced a topology optimization method using homoge-
nization methods. For an overview of the field, the reader
is referred to Bendsøe [13] and the references therein. The
method presented here is an improvement of the methods
presented in [11] and [12] and allows the user to specify the
mechanical advantages (MA’s) or the geometrical advantages
(GA’s) of the compliant mechanisms. The topologies obtained
require only little interpretation by the engineer before they can
be transferred to the fabrication unit. For design of compliant
mechanisms in this paper, the topology optimization problem
is defined as follows. Distribute a given amount of material
within a design domain for a prescribed MA or GA and such
that the mechanical efficiency is maximized. The optimization
problem is formulated as minimization of least-squares errors
in obtaining the prescribed elastic behavior and is solved using
a sequential linear programming (SLP) method.
A. Computational Method for Compliant Mechanisms
The main difference between rigid body mechanisms and
compliant mechanisms is that energy is no longer conserved
between the input and output point because of energy storage
in the flexible parts of the latter. This implies that the GA is
not equal to the inverse of the MA when we are considering
compliant mechanisms. In other words, defining GA as output
displacement divided by input displacement and MA as output
force divided by input force,
will be equal to one for
rigid body mechanisms with ideal hinges, whereas
will always be less than one for compliant mechanisms. One
of the goals in compliant mechanism design is to get
as close to one as possible—in that way, a high mechanical
efficiency of the compliant device is ensured. Typically, the
user wants a mechanism with some specific GA or MA
or There will also typically be some constraint on the
amount of material that can be used for the structure. By
choosing an upper limit
on the amount of material that can
be distributed in the design domain, it is indirectly possible to
control widths of beams in the designs.
The optimization problem can now be defined as follows.
Distribute a given amount of material within the design domain
such that the error in obtaining the prescribed values of
and are minimized. The optimization problem can be
written as
Minimize
subject to
and (1)
where
is the number of elements or design variables,
is the density of material in element (design variable
, is the lower side constraint on the design variables
(for computational reasons), and
is the volume of element
Typically, the algorithm requires several hundred iterations
to converge, each iteration step requiring a finite-element
analysis with two load cases [solving (1)], a simple analytical
sensitivity analysis, and a design update using the Simplex
algorithm. For more details on the computational algorithms
and methods, the reader is referred to the general literature on
topology optimization (e.g., Bendsøe [13]).
B. Compliant Mechanism Design Examples
Fig. 1 shows the design domains and input and output
load cases for five considered compliant mechanism design
problems. The rectangular design domains are discretized by
typically 3000 quadrilateral finite elements, where the density
of material in each element represents one design variable. The
resulting designs are shown in Fig. 2. By using penalization
of gray areas in the design algorithm, the amount of gray
areas is minimized, but, still, their appearance cannot be
fully eliminated. The five mechanisms can be characterized
as follows. Mechanism
is a force inverter that converts a
horizontal input force to a force in the opposite direction at the
output point. The MA’s or GA’s can be prescribed such that
force or displacement amplification can be achieved. Fig. 2(a1)
and (a2) shows a 1:1 and 4:1 force inverter, respectively. The
mechanism in Fig. 2(a2) can also be seen as a displacement
amplifier: an input displacement (black arrow) is converted
to a four-times-bigger output displacement in the opposite
direction. It should be noted that the design procedure only
considers linear displacements, and, therefore, the GA’s only
hold for small input displacements. Fig. 2(b1) and 2(b2)
shows the resulting designs for a cleaving mechanism (this
design example was used in [10]). There are two input
forces that should be converted into a compressive or tensile
force at the output piston [which is prescribed to be solid,
as seen in Fig. 1(b)]. Mechanisms
and represent the
design of gripping mechanisms. For grippers
and , the
output points are the outer points of the jaws, which are
specified to close and open, subject to the horizontal input
force, respectively. For grippers
and , the jaws are
specified to move in parallel. This was done by specifying
two separate output load cases, and the design problem was
therefore extended to include two prescribed GA’s and two
prescribed MA’s. Finally, mechanism
represents the design
of a micropositioning device. Given a design domain and two
input actuators, the problem is to find the mechanism topology
that can make the output point independently controllable in
the vertical and horizontal directions, respectively. Fig. 2(e1)
shows a design, where a horizontal input force (middle left
edge) results in a horizontal movement of the output point,
and a vertical input force (middle lower side) results in a
vertical movement of the output point. If we want to use
comb drives as actuators, we can only get attracting input
forces. Fortunately, the design method is able to produce the
micropositioning device seen in Fig. 2(e2); this mechanism
has the opposite output behavior compared to
All these
mechanisms can also be used in the opposite direction, where
a force or displacement at the output gives an associated force
or displacement at the input.
III. D
ESIGN OF STRUCTURES WITH NPR
The existence of materials with NPR’s or, in other words,
the existence of materials that expand transversely subject to
an applied tensile load has been questioned by researchers and
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LARSEN et al.: DESIGN AND FABRICATION OF MICROMECHANISMS AND STRUCTURES 101
Fig. 1. Design domains and input and output load cases for five compliant
mechanisms. The gray areas define the design domains, whereas white and
black areas define areas that are specified to be void or to consist of solid
material, respectively. Black arrows denote point and direction of the input
force, and white arrows denote points and directions of the prescribed output
forces or displacements. Numbers denote the magnitude of the forces.
engineers for a long time. However, thermodynamics allows
the Poisson’s ratio of an isotropic material to approach
1
Among other applications (see, e.g., Lakes [9]), materials with
NPR can advantageously be used in the design of hydrophones
[14] and other sensors. One reason is that the low-bulk
modulus of these materials makes them more sensitive to
hydrostatic pressure. As the bulk modulus of an isotropic
material is defined as
, where is
Young’s modulus and
is Poisson’s ratio, the sensitivity
to hydrostatic pressure is increased by almost one order of
magnitude for a Poisson’s ratio
1 material compared to a
material with an ordinary Poisson’s ratio 0.3. A theoretical
material microstructure with Poisson’s ratio
1 was first
reported by Almgren [15]. A practical material with NPR was
first developed by Lakes [16], who treated an open-walled
foam material with heat and pressure to obtain the NPR effect.
Lake’s foams have average cell sizes down to 0.3 mm and
Poisson’s ratios down to
0.8. In Sigmund [11], [17], [18], a
numerically based method is used to design materials with any
thermodynamically admissible elastic properties. In this paper,
we will show that it is possible to design and manufacture
microstructures with NPR’s and cell sizes down to 50 m using
topology optimization. These microstructures are referred to
as NPR materials and can be fabricated by utilizing silicon
surface micromachining techniques.
It might be objected that the NPR materials produced
in this work are structures rather than materials. However,
knowing that every material has a structure if one looks
at it at a sufficiently small scale, the distinction between
“materials” and “structures” is blurred. Defining materials as
repeated structures that cannot be seen by the naked eye,
the negative Poisson’s structures produced in this paper are
Fig. 2. Ten optimized compliant micromechanisms. White colors indicate
void areas, black colors indicate solid areas, and gray areas indicate areas of
medium density. Numbers denote the size of the forces.
materials indeed, and they are comparable to the naturally
existing material cork (zero Poisson’s ratio), which has a
microstructure on the same length scale.
For the design of NPR materials, we will consider periodic
microstructures, where the smallest repetitive unit, called the
base cell, will be the design domain, and the design goal
is to minimize the error in obtaining the prescribed elastic
properties for a fixed amount of material in the base cell. The
behavior of both mechanisms and NPR materials is analyzed
by discretizing the design domain by four-node quadrilateral
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102 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 6, NO. 2, JUNE 1997
Fig. 3. Material with Poisson’s ratio
0
0.8 obtained from a ground structure
with 40 by 40 elements and vertical symmetry enforcement.
Fig. 4. Four postprocessed micromechanisms: (a) a 1:1 inverter, (b) a 1:4
inverter, (c) a gripper, and (d) a positioner. These designs are now ready to
be transfered to the laser micromachining setup.
Fig. 5. Interpretation of one of the NPR material designs.
finite elements and solving the finite-element problems for
several loading cases. The design variables are given as the
density of material in each finite element. As in the previous
section, the optimization problems are solved using the SLP
method.
A. Computational Method
The behavior of a linear elastic material follows the gener-
alized Hooke’s law
(2)
where
is the elasticity tensor and and are the
stress and strain tensors. By distributing a prescribed amount
of material within the design domain (the base cell), we can
design a porous and periodic structure with the prescribed
elasticity tensor
, assuming plane stress conditions and
(a)
(b)
(c)
Fig. 6. Steps in the process sequence after (a) deposition of 6-
m PECVD
glass and 2-
m silicon, (b) laser micromachining of a prototype mask, and (c)
two-step RIE process: anisotropic glass etch followed by an isotropic silicon
etch.
with prescribed density Again, the optimization problem
is formulated as a least-squares problem
Minimize
subject to
and (3)
where, as before,
is the number of finite elements or design
variables,
is the density of material in element (design
variable
, is the lower side constraint on the design
variables (for computational reasons), and
is the volume
of element
denotes the homogenized or averaged
elasticity tensor for the inhomogeneous material, which can
be found using the standard homogenization method (e.g.,
Bourgat [19]). The homogenization method implies the solving
of a finite-element problem with three load cases, namely, a
horizontal pull, vertical pull, and shear pulling case, just as
one would do to test a real material.
B. NPR Material Design Examples
Design of an NPR material was done by specifying the
elastic properties of a material with Poisson’s ratio
0.8 and
solving the optimization problem (3) for a quadratic base
cell discretized by 1600 quadrilateral finite elements, each
representing one design variable. The resulting topology is
seen in Fig. 3 (left). The base cell is repeated in Fig. 3 (right),
where the mechanism is seen more clearly. When the material
is compressed horizontally, the triangles will collapse and
result in a vertical contraction, which is the characteristic
behavior of an NPR material.
IV. F
ABRICATION OF MICROSTRUCTURES
The achieved designs for compliant micromechanisms and
NPR materials were fabricated using silicon surface micro-
machining. The fabrication method employs a direct writing
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