Multifunctional microtruss laminates: Textile synthesis
and properties
D.J. Sypeck and H.N.G. Wadley
Department of Materials Science and Engineering, 116 Engineer’s Way, School of Engineering and
Applied Science, University of Virginia, Charlottesville, Virginia 22904
(Received 1 November 2000; accepted 9 January 2001)
Open cell periodic metal truss structures can exhibit significantly higher stiffnesses and
strengths than stochastic cellular metal structures of the same relative density while
still providing high mechanical energy absorption and efficient heat exchange
opportunities. Here, a potentially inexpensive textile-based approach to the synthesis of
periodic metal microtruss laminates is reported. The process consists of selecting a
wire weave, laying up the mesh and joining using a transient liquid phase. Example
structures constructed from nichrome (Ni–24Fe–16Cr) wire cloth were made and
tested. These exhibited a linear dependence of stiffness and strength upon relative
density, absorbed large amounts of mechanical energy, and showed good potential for
efficient heat exchange.
I. INTRODUCTION
Cellular solids are made up of interconnected networks of solid struts or plates which form the edges and
faces of cells.1 Cellular solids based upon polymers are
often used for thermal insulation, protective packaging,
filtering, sandwich panel cores, or to provide buoyancy
among many other applications.1 These types of materials can also be synthesized from metals. Cellular metals
appear to possess useful combinations of thermophysical
and mechanical properties that can be tailored by adjusting the relative density, cellular architecture, or metal
type.2– 4 There is a growing interest in using cellular
metals for applications where more than one function is
required.2,3 For example, load supporting cellular metal
structures might also simultaneously provide mechanical
impact/blast absorption,5,6 thermal management,5,6 noise
attenuation,5 catalyst support,5 filtration,5 electrical energy storage,7 retardation of chemical reactions and/or
fire,5,8 or act as a host for the in-growth of biological
tissue.9
The properties of a cellular metal depend upon both
the topology of the metal and its properties. Several
classes of cellular metals are emerging. One is stochastic
in nature, based upon foaming.4 Another utilizes polymer
or wax truss patterns and investment casting to create a
periodic metal truss structure.10,11 Directional foaming
results in architectures that are predominantly closed cell,
often with wide distributions of cell size1 and many imperfections.12 Closed cell stochastic metal foams are
used for sound attenuation and impact energy absorption.4 Open cell stochastic metal foams are made using
reticulated polymer foams templates. In one approach,
890
J. Mater. Res., Vol. 16, No. 3, Mar 2001
the template is used as a pattern for an investment casting
mold which is then filled with a liquid metal and solidified.4 In other approaches, metal vapor or a fine metal
powder slurry is deposited directly on to the template.13,14 In the latter, a subsequent heat treatment removes the organic compounds and densifies the
structure. Open cell stochastic metal foams are used for
lightweight heat exchangers and as the electrodes in
nickel metal hydride batteries.4 Their Young’s moduli
and compressive yield strengths vary with relative density in a nonlinear way:1
冉冊
冉冊
E
␳ 2
=
,
Es
␳s
␳ 3Ⲑ2
␴c
= 0.3
␴ys
␳s
(1)
,
(2)
where E is Young’s modulus of the foam and Es of its
base material, ␴c is the compressive yield strength of the
foam, and ␴ys is the yield strength of its base material,
while ␳ is the density of the foam and ␳s that if its base
material. Here, a power-law dependence upon relative density indicates a rapid property loss with decreased density owing to ligament bending. Nonetheless,
these materials look to be structurally competitive when
used as the cores of sandwich panels (especially in biaxial loading).3
Recent work has sought to use rapid prototyping10 and
injection molding11 to create polymer or wax patterns
with open cell lattice architectures followed by investment casting/heat treatment. These are known as lattice
block or truss materials.15,16 Individual cells can be small
© 2001 Materials Research Society
D.J. Sypeck et al.: Multifunctional microtruss laminates: textile synthesis, and properties
(a few millimeters). By manipulating the truss architecture, you modify the properties. Young’s moduli and
compressive yield strengths for these materials vary with
relative density in a linear way (trusses are in tension/
compression with no bending).3 This is especially important at low relative density where properties far exceed
those of stochastic metal foams. These are just a few of
the benefits to be gained when good control over the
distribution of metal at the cell level is achieved. However, the casting approaches can be costly to implement
and cast metals are subject to knockdown by casting
factors. Also, many of the alloys do not favorably respond to heat treatment.
Here, a potentially inexpensive textile-based approach
for the synthesis of metal microtruss laminates is reported. It is based upon a recognition that high structural
quality porous metal objects have already been made for
many years by taking wrought metal wires and weaving
them into bundles.17 Woven wire cloth, braided cables,
and knitted metal fabrics are good examples of this. The
textile approaches used to create such articles are very
well established and relatively inexpensive. Furthermore,
a host of material choices are available (virtually all metals can be drawn into wire and then woven, braided, or
knitted) and in a variety of filament arrangements (dutch
weave, hexagonal mesh, 3-D braid, crimped mesh, etc.).
However, these metal textiles are unsuited for most multifunctional applications because the metal contacts are
not normally bonded and the articles are thin (exceptions
include 3-D braided articles). One plausible remedy for
these shortcomings involves laminating. To demonstrate
the methodology, we lay-up and join plain square woven
metal cloth at points of wire contact using a transient
liquid phase. The resulting multi-ply laminated structures
are stiff and strong in axial compression with linear
dependence on relative density. They absorb large
amounts of mechanical energy (comparable to axially
crushed thin-walled cylinders) and also show good potential for efficient heat exchange. The wide range of
wire materials, weave patterns, and lay-ups suggests
many degrees of freedom for designing structures optimized for multifunctionality at acceptable cost.
angle is reduced. From analysis of the structure as a
sequence of stacked rather than woven wires, the relative
density is
␳
␲d
=
.
␳s 4共w + d兲
When loaded parallel to a set of the wires, the wires
aligned in the loading plane support all of the load. If
they do not buckle, the relative Young’s modulus is half
of that expected by a rule of mixtures approximation:
␳
E
= 0.5
Es
␳s
(4)
.
Similarly, for the realtive compressive yield strength (assuming no wire buckling),
␳
␴c
= 0.5
␴ys
␳s
(5)
.
Similar linear relations can also be used to model other
simple geometries (e.g., rectangular weave) with adjustment of the proportionality constant to reflect the area
fraction of material in the loading plane. Designer stiffnesses and strengths that are higher (or lower) than given
by Eqs. (4) and (5) are obtained in this fashion.
To create multifunctional microtruss laminates, we
imagine stacking the mesh peak to peak (such that all
square openings are aligned) and joining together at
all points of wire contact, Fig. 2. If one neglects the joint
material, Eqs. (3)–(5) also describe the density, stiffness,
and strength of the microtruss laminate (assuming no
wire buckling).
When axially compressed, wires within the structure
can buckle before they yield. To determine which mechanism initiates failure, we treat the structure as a threedimensional space-truss constructed from many simply
supported (conservative approach) columns of length,
l ⳱ (w + d), and second moment of area, I ⳱ ␲d 4/64.18
The buckling load for an individual wire is PB 艌 ␲2
EsI/l2.18 When compressed in a direction parallel to a set
of wires, PB ⳱ 2d(w + d)␴B, where ␴B is the buckling
stress. Upon substitution, buckling of individual wires is
expected at
II. ARCHITECTURE DESIGN
We seek a metal topology that provides efficient inplane load support, excellent mechanical energy absorption, and high convective heat transfer with low pumping
requirements. Consider plain square woven cylindrical
wires of diameter d and opening width w, Fig. 1. For
mesh of low relative density, the angle that warp and
weft wires must bend as they pass over and under
one another during weaving is small and will be neglected for simplicity. The validity of this approximation
improves with decreasing relative density as the bending
(3)
␴B 艌
冉冊
Es ␳
2 ␳s
3
.
(6)
In a comparison of Eqs. (5) and (6), buckling precedes
yielding when
冉 冊
␴ys
␳
艋
␳s
Es
1Ⲑ2
,
(7)
which serves as a rough design criteria for selecting the
wire type, diameter, and spacing. Where failure by yielding dominates, a linear behavior dominates. Generally,
J. Mater. Res., Vol. 16, No. 3, Mar 2001
891
D.J. Sypeck et al.: Multifunctional microtruss laminates: textile synthesis, and properties
ally crushed thin-walled metal tubes exhibit a similar
stress–strain behavior and are efficient energy absorbers.
They plastically deform via a series of ringlike folds with
values for the absorbed energy given by4
tube
WV
= 1.26
= 1.26
W tube
m
冉冊
冉冊
␳
␳s
5Ⲑ3
␳
␳s
2Ⲑ3
␴ys⑀D ,
(8)
␴ys
⑀
,
␳s D
(9)
tube
is the absorbed energy per unit volume,
where W V
tube
W m is the absorbed energy per unit mass, and ⑀D is the
compaction strain (when the folds of the collapsed tube
lock up). A thin-walled tube is expected to be about three
times more efficient at absorbing energy than a stochastic
metal foam.4 Foam filled tubes are even better. Expectations for truss-based cellular metals are less clear.
However, their plateau stress is surmised to be higher
than that of similar density stochastic foams which leads
to a higher energy absorptive capacity. Furthermore, the
FIG. 1. (a) Cylindrical wires of diameter d are shown in a plain square
woven arrangement with an opening width w. During weaving, the
warp and weft wires must bend as they pass over and under one
another. (b) A peak to peak lay-up of the woven cloth creates a multifunctional microtruss laminate.
one prevents buckling by avoiding the use of long slender trusses or by utilizing truss cross sections of higher I
(e.g., I-beams). Tubular members (e.g., hypodermic tubing) accomplish this purpose for very low density structures with the possibility of also utilizing the space
within the tubes (e.g., cooling fluid flow, fuel storage,
etc.). In the absence of buckling, the relative stiffness and
strength of microtruss laminates is expected to be slightly
lower than predicted by Eqs. (4) and (5) due to bending
of the warp and weft wires (they are not perfectly aligned
with the direction of load) and microscopic imperfections. These imperfections cause wires (microtrusses) to
fail at reduced stresses which lowers the strength of the
laminate and alters its energy absorption behavior. This
is not necessarily a bad thing if the material is being used
to protect objects that are sensitive to excessive peak
loads and decelerations.
Good energy absorbing materials have long, flat
stress–strain curves and collapse plastically at a constant
(plateau) stress.4 Compressed open cell metal foams have
the needed open space for plastic collapse and exhibit
this type of behavior. Their ligaments bend and buckle,
absorbing work at nearly constant stress until impinging
upon one another with an accompanying stress rise. Axi892
FIG. 2. (a) A transient liquid phase joins the laminae together at all
points of wire contact to create the cellular metal core. (b) When face
sheets are added, a multifunctional sandwich panel suitable for fluid
passage and load support is made.
J. Mater. Res., Vol. 16, No. 3, Mar 2001
D.J. Sypeck et al.: Multifunctional microtruss laminates: textile synthesis, and properties
potentially large open space between trusses accommodates significant ligament plastic collapse and a lengthy
plateau.
For heat exchange, a cellular metal is envisaged as
a system that transfers heat between a structure and a
fluid.4 A dense, highly conductive open cell architecture
of large accessible surface area is preferred. A turbulent
flow of tortuous path also aids convection but requires
additional fluid pumping power. For compact applications (e.g., power electronics heat sinks, air conditioners,
etc.) a small overall size is desired. The surface area
density (surface area to volume ratio) is an important
parameter for heat exchange systems.19 For microtruss
laminates, this is approximately (using the stacked wires
model)
␲
,
(10)
␣A =
w+d
suggesting a fine mesh for compact designs (but with a
need for additional fluid pumping power). When fluids
flow through such a structure, the situation resembles a
bank of aligned cylinders in cross flow. The heat transfer
efficiency of stochastic open cell metal foams has been
evaluated using this type of approximation.19 Recent work
suggests optimal cell sizes in the millimeter range and
relative densities of about 20%.3 Materials of choice for
heat exchangers include aluminum and copper for their
high thermal conductivity whereas nichrome is often
used for electrical resistive heating.
Nickel–iron–chromium alloys are known for their excellent corrosion resistance and high strength at elevated
temperatures20 making them good experimental candidates for high-temperature multifunctionality. Nichrome
(Ni–24Fe–16Cr) is one commonly woven Ni–Fe–Cr alloy. Like other commercially available metal cloths,
wire diameters and spacings can range from a few microns to several centimeters. The Young’s modulus
and 0.2% offset yield strength for nichrome are nominally Es ⳱ 205 GPa and ␴ys ⳱ 205 MPa.20 We seek to
create nichrome test samples with aligned square openings in the millimeter size range (for efficient heat exchange) and a low relative density for structural
advantage over stochastic foams. A buckling mode of
failure is to be avoided. Equation (7) estimates the relative density range for which this occurs, ␳/␳s 艋 0.032.
This is well below the relative density of most commercially available plain square weaves. We proceed to construct samples of two different relative density for
properties comparison and evaluation.
(0.100 in.) opening width. The other was a 16 mesh with
0.508 mm (0.020 in.) diameter wires and a 1.080 mm
(0.0425 in.) opening width. To join, laminae (22 ply for
each sample) were first submersed into a polymer based
cement (Nicrobraz Cement 520 supplied by Wall Colmonoy Corp., Madison Heights, MI). They were then
coated with −140 mesh (diameter 艋 106 ␮m) Ni–25Cr–
10P braze alloy powder (Nicrobraz 51 supplied by Wall
Colmonoy Corp.). The solidus and liquidus of this braze
alloy are 880 and 950 °C whereas the solidus of nichrome is approximately 1350 °C. The coated laminae
were then stacked peak to peak (using pins to align all
square openings), and a small compressive pressure was
applied to the periphery of the lay-up. The stacked assemblies were heated in a vacuum of less than 10−2 torr
at a rate of 10 °C/min to 550 °C for 1 h to volatilize the
polymer cement. An important feature of this braze/
cement combination is that the braze alloy powders remain adhered to the wires after volatilization. The system
was then evacuated to an oxygen gettered (Ti powder)
vacuum level of less than 10−3 torr, and the temperature
was then ramped at a rate of 10 °C/min to 1120 °C and
held there for 1 h. Samples were then furnace cooled to
ambient, and their sides were machined mutually orthogonal for testing. During this final heating, the braze
alloy powders melted and were drawn by capillary action
to points of wire contact. Interdiffusion then changed the
contact composition and elevated its melting point causing solidification at the brazing temperature. The 8 mesh
sample had a density of ␳ ⳱ 1.43 g/cm3, whereas the 16
mesh sample had 2.54 g/cm3. Using ␳s ⳱ 8.30 g/cm3 for
nichrome,20 the corresponding relative densities were
␳/␳s ⳱ 17% and 31%, which are close to the (16% and
25%) predictions of Eq. (3).
The topology of the 17% dense laminate is shown in
Fig. 3. Aligned square pores were obtained in one direction for efficient cross flow with low flow resistance. In
the other two directions, the pores assumed a diamondlike shape. A large surface area to volume ratio has been
achieved. Using Eq. (10), the approximate surface area
densities are ␣A ⳱ 989 and 1979 m2/m3. For comparison, Duocel open cell aluminum foams (ERG, Inc., Oakland, CA) have 560–3320 m2/m3,21 automotive radiators
600–1000 m2/m3, cryogenic heat exchangers 1000–
2000 m 2 /m 3 , and gas turbine rotary regenerators
3000–6000 m2/m3.19 Excellent catalyst support opportunities also exist for microtruss laminates. The high surface area aids catalytic activity while the uniform pores
tend to distribute the flow evenly throughout.
III. MICROTRUSS CONSTRUCTION
Laminae of plain square woven nichrome cloth (obtained from Newark Wire Cloth Co., Newark, NJ) were
used for the construction. One was an 8 mesh with
0.635 mm (0.025 in.) diameter wires and a 2.54 mm
IV. MECHANICAL BEHAVIOR
Microtruss laminates were compression tested in ambient (25 °C) at a constant crosshead displacement rate of
0.005 mm/s using a screw driven testing instrument. A
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893
D.J. Sypeck et al.: Multifunctional microtruss laminates: textile synthesis, and properties
FIG. 3. Topology of the 17% dense nichrome microtruss laminate: (a) In one direction, the aligned square pores allow fluids to easily pass for
efficient cross flow exchange. (b) The microtruss architecture provides good stiffness and strength in the other two directions.
knife edge extensometer of 10-mm gauge and 10-mm
contact was epoxy cemented to the center of one side for
local strain measurement at low and intermediate strains.
The displacement of the crosshead determined the full
sample strain. Photographs before, after, and at 5% strain
increments during the test recorded the deformation on
the side opposite the extensometer.
The compressive stress–strain behavior for the 17%
dense laminate is shown in Fig. 4. During initial loading,
a linear elastic behavior with a Young’s modulus of E ∼
15 GPa was recorded. At a strain ⑀ ∼ 0.04%, the linear
elastic behavior ceased and a region of rapidly decreasing
slope ensued. A distinct plateau then appeared which
continued until ⑀ ∼ 7%. The compressive strength was
␴c ⳱ 18 MPa. After ⑀ ∼ 7%, the slope of the stress–strain
curve decreased (becoming negative) with a corresponding reduction in load carry capacity. At ⑀ ∼ 17%, another
plateau was formed. The stress in this second (lower) plateau region was 12 MPa.
We observed delamination of laminae near the left
sample edge as early as ⑀ ∼ 1%. This was caused by high
local stress at the joints owing to sample barreling. Some
audible acoustic emissions were noted during this period
and periodically throughout the test. This delamination,
another near the right edge of the sample, and some
bending of the debonded laminae are clearly discerned
by the time the sample had strained 5%, Fig. 5(b). Delaminations were aggravated by incomplete bonding of
laminae. We note that differences in the amount of bending experienced by warp and weft wires during weaving
resulted in at least half of all possible interlaminae contacts having a gap between them (see Fig. 3). Creating
this second set of bonds is likely to result in a triangular
(rather than diamond-like) pore structure and improved
894
resistance to delamination through the thickness. At ⑀ ∼
7%, one layer of microtrusses at the center of the sample
began to bend and collapse (in the out-of-plane direction). By ⑀ ∼ 10%, collapse of most of the remaining
cells in that cross-sectional plane had begun, Fig. 5(c).
The dark region just below the sample center was a consequence of movement by these collapsed cells in the
out-of-plane direction (toward the extensometer). As
loading continued, this region extended vertically,
Fig. 5(d). This deformation banding12 of cells led to a
decreased load carry capacity and the second (lower)
plateau dominated by microtruss bending and collapse
(rather than compressive yielding). The diamond-like
pores spread apart during the crushing. A triangulated
architecture is likely to resist the spreading and increase
the crushing stress.
For the 31% dense laminate, Young’s modulus was
E ∼ 33 GPa and its compressive strength was ␴c ⳱
41 MPa. After initial yielding, a distinct plateau was not
observed. Rather, the stress continued rising to a maximum of 61 MPa at ⑀ ∼ 5%. Numerous delaminations
accompanied a continued reduction of load carry capacity with increasing strain. Delamination frequency and
severity were much greater for this more dense laminate
owing to its reduced compliance and the higher joint
stresses (bonds now break before wires bend). Bending
and collapse of cells was not observed. Rather, entire
laminae separated from one another and bent in unison
similar to the bending seen with the outer most laminae
of the 17% dense sample. This test was stopped prematurely due to the delaminations.
Using Es ⳱ 205 GPa for nichrome, the relative
Young’s moduli for the microtruss laminates are E/Es ⳱
0.073 and 0.161 which are 14% less and 4% greater than
J. Mater. Res., Vol. 16, No. 3, Mar 2001
D.J. Sypeck et al.: Multifunctional microtruss laminates: textile synthesis, and properties
FIG. 4. Stress–strain behavior for the 17% dense nichrome microtruss
laminate at sample center: (a) low strain behavior; (b) intermediate
strain behavior. Note: Records of damage accumulation at points 5(a)–
5(f ) are found in the corresponding photographs of Fig. 5.
the (0.085 and 0.155) predictions of Eq. (2). For the 17%
dense laminate, the mean of five Vicker’s indentation
hardness tests at various points away from the bonded
joints was HV500 ⳱ 171. Ni–Fe–Cr alloys of similar
chemistry and hardness have 0.2% offset yield strengths
which are typically 200–210 MPa.20 Thus, with ␴ys ⳱
205 MPa for nichrome, the relative compressive
strengths (corresponding to the first plateau region) of
the laminates were ␴c /␴ys ⳱ 0.088 and 0.200, which are
4% and 29% greater than the (0.085 and 0.155) predictions of Eq. (3). The relative property data are plotted in
Fig. 6 along with representative values for Duocel aluminum (6101-T6) foam, the Gibson–Ashby model predictions for stochastic open cell foams, 1 and the
predictions of Eqs. (4) and (5). The anticipated linear
property dependence upon relative density is seen. The
relative Young’s moduli and compressive strengths of
nichrome microtruss laminates supersede those of stochastic open cell foams due to the reduced bending inherent in the architecture design. By comparison of
Eqs. (1) to (4) and Eqs. (2) to (5), the modeled Young’s
moduli and compressive strengths for these materials exceed those of stochastic open cell foams by factors of
0.5/(␳/␳s) and (5/3)/(␳/␳s)1/2. For example, at ␳/␳s ⳱ 5%,
the microtruss laminate is 10 times stiffer and 7.5 times
stronger than the stochastic open cell foam (provided
buckling has not preceded yielding). High transparency
metal cloth can be used to achieve these low relative
FIG. 5. Photographs of the 17% dense microtruss laminate during the test show that delaminations precede bending and collapse of cells. The
region marked “Gauge” indicates the location of the extensometer contacts on the sample side opposite to that photographed.
J. Mater. Res., Vol. 16, No. 3, Mar 2001
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D.J. Sypeck et al.: Multifunctional microtruss laminates: textile synthesis, and properties
FIG. 6. Mechanical properties of nichrome microtruss laminates supersede micromechanical models for stochastic open cell foams and
commercially available open cell aluminum foams: (a) Young’s modulus; (b) compressive strength.
densities. A less costly approach (but with a different cell
architecture) involves corrugating a more common cloth
(e.g., insect netting) and laminating. We have made
stainless steel samples with ␳/␳s 艋 4% in this fashion.
The crushing behavior and energy absorptive capacity
for the 17% dense laminate can be assessed from
Fig. 7(a). Several undulations in the stress level (due to
microtruss collapse) are seen with increasing strain until
densification (and workhardening) at 55% strain. A
larger sample size (i.e., more cells) smoothens the crushing behavior. Posttest microscopic observations of the
joints revealed slight amounts of cracking confined to
the joint areas, Fig. 7(b). A longer hold time at the
brazing temperature may be needed for further joint homogenization and toughening. Nonetheless, the bonds
were able to accommodate large amounts of microtruss bending and collapse by test end. Upon densification, the absorbed mechanical energy per unit volume
was WV ⳱ 6.9 J/cm3, which corresponds to an absorbed
energy per unit mass of Wm ⳱ 4.8 J/g. Substituting previously cited nichrome values and using the measured
densification strain (0.55) as first approximation to the
tube
⳱ 9.0 J/cm3 and
tube compaction strain, we find W V
tube
W m ⳱ 6.5 J/g from Eqs. (8) and (9). The microtruss
laminate has about three-fourths of the energy absorptive
capacity of an axially crushed thin-walled tube of the
896
FIG. 7. Stress–strain behavior and energy absorption characteristics of
the 17% dense nichrome microtruss laminate. The strain is based upon
the measured crosshead displacement. Key: (a) compression behavior;
(b) view of the laminate at its maximum strain.
same material and relative density. If one notes that delaminations near the edges of the sample had occurred
early on and these laminae were not plastically deformed
(worked) to their full potential by test end, the possibility
for additional energy absorption exists. We surmise that
at low relative density, this structural configuration has
excellent energy management potential.
V. CONCLUSIONS
A textile-based approach has been used to create low
relative density cellular metals by transient liquid
phase bonding of woven nichrome wire cloth. The resulting microtruss laminates are anisotropic. In one
direction, the topology was designed for efficient exchange by cross flow (with low flow resistance). In the
other directions, it was designed to efficiently support
loads and absorb large amounts of mechanical energy.
The relative mechanical properties in the load support
direction were higher than those of stochastic foams owing to the reduced bending inherent in the design. This is
consistent with other truss architectures and becomes
more pronounced as the relative density decreases.
J. Mater. Res., Vol. 16, No. 3, Mar 2001
D.J. Sypeck et al.: Multifunctional microtruss laminates: textile synthesis, and properties
Low-density microtruss laminates have the compliance
and necessary open space for cell collapse thereby absorbing large amounts of energy when compressed. This
synthesis technique appears to allow one to tailor the
topology of a wide variety of multifunctional cellular
materials in a simple, cost effective way. It is not necessarily limited to metals.
ACKNOWLEDGMENTS
We are grateful to Defense Advanced Research Projects
Agency/Office of Naval Research (DARPA/ONR) for the
support of this work through the University Research Initiative at Harvard (N00014-96-I-1028, program manager,
S. Fishman).
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