Acta Materialia 53 (2005) 303–313
www.actamat-journals.com
Cellular metal lattices with hollow trusses
Douglas T. Queheillalt *, Haydn N.G. Wadley
Department of Materials Science and Engineering, University of Virginia, 116 Engineers Way, P.O. Box 400745, Charlottesville,
Virginia 22904-4745, USA
Received 14 June 2004; received in revised form 17 September 2004; accepted 21 September 2004
Available online 22 October 2004
Abstract
Cellular metal lattice truss structures are being investigated for use as multifunctional load supporting structures where their
other functionalities include thermal management, dynamic load protection and acoustic damping. A simple method for making
lattice structures with either solid or hollow trusses is reported. The approach involves laying up collinear arrays of either solid wires
or hollow cylinders and then alternating the direction of successive layers. The alternating collinear assembly is metallically bonded
by a brazing process. The dimensions of the cylinders, the wall thickness of hollow truss structures and the spacing between the
trusses enable independent control of the cell size and the relative density of the structure. The process has been used to create stainless steel lattices with either square or diamond topologies with relative densities from 0.03 to 0.23. The through thickness elastic
modulus of these lattice truss structures is found to be proportional to relative density. The square topology has twice the stiffness
of the diamond oriented trusses. The peak compressive strengths of both topologies are similar and is controlled by plastic buckling.
The structural efficiency of hollow truss structures with a fixed cell size is approximately proportional to the relative density unlike
equivalent structures made from solid trusses whose peak strength has been predicted to scale with the cube of the relative density.
The experimental data for hollow trusses lie between predictions for trusses with built-in and pin-jointed nodes consistent with
experimental observations of constrained node rotation. The use of hollow trusses increases the resistance to buckling offsetting
the usually rapid drop in strength as the relative density decreases in cellular systems where truss buckling controls failure. The
low relative density hollow truss structures reported here have the highest reported specific peak strength of any cellular metal
reported to date.
2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Cellular material; Porous material; Stainless steel; Mechanical properties
1. Introduction
Numerous techniques have been proposed for making cellular metals [1,2]. Many initially exploited foaming techniques to create metal foams with
stochastically distributed cell sizes and shapes with
either or open or closed porosity [1]. The closed-cell variants of these metal foams have shown promise for impact energy absorption [3–6] and perhaps acoustic
*
Corresponding author. Tel.: +1 434 982 5678; fax: +1 434 982
5677.
E-mail address: [email protected] (D.T. Queheillalt).
damping [7–9]. Open cell foams have been used for cross
flow heat exchange [10–20]. Load supporting (structural) applications of these materials have been limited
by their low elastic moduli and strengths which have
power law dependencies upon density and are greatly
inferior to those of honeycombs of the same density
[21–23].
The low stiffness and strength of open cell metal
foams has generated significant interest in alternative
cell topologies which might offer strengths comparable
to honeycombs while simultaneously facilitating the
other functionalities of open cell metal foams [24–27].
Several open cell systems have been proposed based
1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2004.09.024
304
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
on truss lattices with tetrahedral [28–35], pyramidal [36–
38], 3D-Kagome [39,40] and woven (metal textile) [41–
44] topologies, Fig. 1.
The structural benefits of cellular materials are often
realized when configured as the cores of sandwich panel
structures. When appropriately oriented, the trusses are
subject to either tension or compression during panel
bending [28–30]. The strengths of these cellular core
sandwich structures are then governed by the stress at
which the various panel failure mechanisms (face sheet
stretching/wrinkling, core plastic yielding, and either
plastic or elastic buckling of core truss elements) are initiated [45–48].
During the compressive loading of cellular trusses
only core deformation mechanisms are active. Trusses
aligned with the loading direction are the most efficient
at supporting stress, while those that are inclined are
limited by force resolution considerations [45–48]. The
truss failure mechanism (elastic or plastic buckling or
plastic yield) depends upon the slenderness ratio of the
trusses. Since the slenderness ratio and relative density
of a lattice made from solid trusses are interdependent,
both the truss strength and responsible failure mecha (defined
nism depend upon the lattice relative density, q
as the density of the cellular structure, qc, divided by
that of the solid material from which it is made, qs),
the lattice topology and the material used to make the
trusses [27].
At high lattice truss relative densities, elastic-perfectly
plastic trusses fail by yielding of the truss columns even
when they are inclined. The compressive collapse
strength, rpk, scales linearly with the relative density [27]
;
rpk ¼ Rry q
ð1Þ
where R is a lattice topology dependent scaling factor
and ry is the yield stress of the solid material. Hexagonal
honeycomb cores have all of their webs aligned in the
loading direction and R = 1.0. For tetrahedral and
pyramidal lattices, Fig. 1(a) and (b), R = sin2 x, where
x is the angle of inclination between the truss elements
and the loading direction. For these two topologies the
optimal angle for resisting both through thickness compression and shear is about 55 [37] and in this case
R = 0.67.
For a diamond lattice truss structure, Fig. 1(d),
R = sin2 x but in this case x = 45 and R = 0.50. Zupan
et al. [44] have shown that edge effects are important
during compressive loading of low aspect ratio diamond
truss samples. The load supporting area is reduced because some trusses are not connected to both face sheets.
The reduction in load bearing area depends on the truss
angle and sample aspect ratio. They suggest that for low
aspect ratio samples R = sin2 x[1 (1/Atan x)], where A
is the length to height (aspect) ratio of the sample [44].
For square truss structures (made by rotating the core
in Fig. 1(d) by 45) half the trusses have x = 0 with
the remainder having x = 90. In this case the effective
value of R is again 0.5. The compressive strength of lattice structures can exceed these predictions when they
are made from made from high work hardening rate
materials.
Low relative density lattice structures have long slender trusses that fail by elastic buckling. In this case the
compressive strength is found by replacing ry in
Eq. (1) by the elastic bifurcation stress of a compressively loaded cylindrical column. The stress is given by
Fig. 1. Schematic illustrations of microtruss lattice structures with
tetrahedral, pyramidal, Kagome and woven textile truss topologies.
rcr ¼
p2 k 2 Es a2
;
l
4
ð2Þ
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
where Es is the elastic modulus, a the radius and l the
length of the column [49]. The factor k depends on the
rotational stiffness of the nodes; k = 1 for a pin-joint
that can freely rotate while k = 2 for fixed-joints which
cannot rotate [50].
We note that the relative density of a square or diamond truss lattice made from solid circular trusses depends upon the a/l ratio
p a
¼
q
:
ð3Þ
2 l
The elastic buckling strength obtained by substituting
Eqs. (2) and (3) into Eq. (1) therefore depends upon
3 . By substituting for a/l in Eq. (2) and equating the
q
plastic yield and elastic buckling strengths the square
and diamond lattice structures can be shown to collapse
by elastic buckling when
rffiffiffiffiffiffiffiffiffi
ry
<
q
:
ð4Þ
k 2 Es
At intermediate relative densities the compressive
strength of a truss lattice is controlled by inelastic buckling at an inelastic bifurcation stress given by Shanley–
Engesser tangent modulus theory [49]. In this case the
compressive strength is found by replacing ry in Eq.
(1) by the inelastic bifurcation stress of a compressively
loaded column. The peak compressive strength for inelastic buckling of a square or diamond lattice truss
structure is then given by
rpk ¼ k 2 Et sin2 x
q3 ;
ð5Þ
where Et is the tangent modulus. For both a square and
diamond topology the sin2 x term is again 0.5 and the
predicted through thickness compressive strength of
the two topologies is predicted to be the same. Note that
for solid circular trusses where either elastic or plastic
3 ,
buckling controls the peak strength scales with q
whereas when the strength is controlled by plastic yielding it scales linearly with the relative density.
Hollow truss structures provide a means for increasing the second moment of inertia of the trusses and thus
their resistance to elastic or plastic buckling. They also
provide a means for varying the cellular structures relative density without changing the cell size (while maintaining a constant truss slenderness ratio); a feature
that could be used in the optimization of multifunctional
systems.
Here, a method for making truss lattice structures
with either solid or hollow circular cross-section trusses
is described. The method enables fabrication of sandwich structures with controlled relative density, cell orientation and size with a cell topology analogous to that
of metal textile structures. However, unlike the metal
textile process, this approach does not require weaving
(so the trusses remain straight) and it is more amenable
to the fabrication of hollow trusses with potentially
305
superior buckling resistances to their solid counterparts.
The processing method and compressive mechanical
properties for solid and hollow trusses in both a 0/90
(i.e., square) and ±45 (i.e., diamond) orientation with
identical cell sizes are presented and explored.
2. Lattice truss structure fabrication
Cellular structures were assembled from 304 stainless
steel solid wires and hollow tubes using a tool to align
the cylinders in collinear layers, Fig. 2(a). The orientation of successive layers was alternated to create the lattice
truss
architecture.
The
cylinders
were
metallurgically bonded using a brazing technique
[51,52]. A Wall Colmonoy NICROBRAZ alloy 51 with
a nominal composition of Ni–25Cr–10P–0.03C (wt%)
was used for bonding. These brazing alloys contain
(a) Lattice truss layup in tool
tool
hollow tubes
or solid wires
(b) Lattice truss machined
(c) Face sheets attached
H
W
L
Fig. 2. (a) The stacking arrangement for the hollow truss structures,
(b) removing the lattice truss structure from the brazing assembly and
cutting to size and (c) attaching the face sheets to form a sandwich
panel with a diamond orientation. An analogous process is used for the
fabrication of square oriented samples.
306
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
melting point depressants such as boron, phosphorous
or silicon to achieve desirable liquid flow and wetting
behavior. The alloy was applied to each truss as a powder contained in a polymer binder. The alternating collinear assembly was placed in a vacuum furnace for
high-temperature brazing and heated at 10 C/min up
to 550 C, held for 20 min (to volatilize and remove
the polymer binder), then heated to 1020 C, for 60
min at 104 Torr before furnace cooling to ambient
temperature at 25 C/min. After removing the bonded
cellular structures from their tooling, they were cut to
size (length L = 50 mm, width W = 30 mm, height
H = 15 mm) using wire electro-discharge machining,
Fig. 2(b). A second brazing treatment was then used
to attach 304 stainless steel (0.1000 thick) face sheets to
the cellular lattice truss structure, Fig. 2(c).
Photographs of the lattice truss structures with solid
and hollow trusses and cells oriented in both the square
(0/90) and diamond (±45) orientations are shown in
Fig. 3. Fig. 4 shows micrographs of truss–truss nodes
and the truss–face sheet interface for diamond oriented
lattice structures. Fig. 5 shows cross-sectional micrographs of a truss–truss node and a truss–face sheet
interface for square oriented lattice truss structures. It
can be seen that during transient liquid phase bonding,
capillary action draws the molten braze material into
the region where the cylinders make contact and forms
a thick fillet at the contact point. The molten braze
material is also drawn to the truss face sheet contacts
forming a contact that is several times that of the tube
Fig. 4. Micrographs of nodes at: (a) truss–truss nodes; (b) truss–face
sheet connections for diamond lattice truss structures.
Fig. 3. Photographs showing: (a) square orientation; (b) diamond orientation solid and hollow microtruss lattice structures.
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
True tensile stress (MPa)
1000
307
(a)
304 stainless steel
800
600
400
200
Es = 203 GPa
0
0
0.1
0.2
0.3
0.4
0.5
0.4
0.5
True tensile strain
Tangent modulus, Et (MPa)
4000
Fig. 5. Cross-sectional micrographs showing: (a) truss–truss node;
(b) truss–face sheet bond. Dashed lines show the approximate edge
positions of the original tube and face sheet boundary.
(b)
3000
2000
1000
0
0
0.1
0.2
0.3
True tensile strain
cross-sectional area. It can be seen from Fig. 5 that
there is good braze metal penetration of the base metal
providing a strong bond between the trusses and face
sheet and at truss to truss connections (nodes).
The square and diamond oriented lattice structures
with both solid and hollow trusses were tested in compression at a nominal strain rate of 4 · 102 s1. The
measured load cell force was used to calculate the nominal stress applied to the structure. The nominal through
thickness strain was obtained from a laser extensometer
on the sample centerline.
Fig. 6. (a) Average uniaxial tensile response (true stress–true strain)
for 304 stainless steel and (b) Shanley–Engesser tangent modulus.
shown in Fig. 6(b). Note its rapid change with plastic
strain. The density, qs, of the steel was 8.0 gm/cm3 [53].
3. Relative density calculation
Fig. 7 shows a unit cell of the lattice structure created
by the alternating collinear cylinder array process. If the
Lattice Truss
Structure
2.1. Parent material response
In order to compare measured and predicted values
of strength, the uniaxial tensile response of 304 stainless
steel subjected to the same thermal history as the lattice
truss structures was determined. The tensile response
measured at a strain rate of 104 s1 is shown in
Fig. 6(a). The elastic modulus and 0.2% offset yield
strength for 304 stainless steel were 203 GPa and 176
MPa, respectively. Significant work hardening occurred
in the plastic region. The tangent modulus given by the
slope dr/de of the true stress–true strain response is
Unit cell
2ao
2ai
l
l
Fig. 7. Truss stacking sequence and a unit cell of the hollow
microtruss lattice structure.
308
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
Table 1
Tube parameters and corresponding relative densities (calculated and
measured)
Outside (2ao, mm)
1.481
1.473
1.473
1.473
1.473
Inside (2ai, mm)
0.000
1.067
1.219
1.270
1.372
Calculated
0.233
0.110
0.073
0.060
0.031
Measured
0.23 ± 0.005
0.11 ± 0.003
0.08 ± 0.003
0.06 ± 0.003
0.03 ± 0.003
Stress (MPa)
Relative density ð
qÞ
Tube diameter
80
(a) Square Orientation
60
0.23
40
0.11
20
0.08
0.06
0.03
0
The center-to-center cell size (l) was 5 mm for all samples.
0
0.2
0.4
0.6
0.4
0.6
Strain
2
V c ¼ 4ao l :
ð7Þ
, is the ratio of the metal volume
The relative density, q
to that of the unit cell
¼
q
V s pða2o a2i Þ
:
¼
2ao l
Vc
(b) Diamond Orientation
40
0.23
20
0.11
ð6Þ
If the cylinders have an inner diameter, 2ai, the volume
occupied by solid metal in the unit cell is
V s ¼ 2plða2o a2i Þ:
Stress (MPa)
60
added weight of the braze alloy is ignored; only the
diameter of the wires, the wall thickness and diameter
of the hollow cylinders, and the cell size determine the
relative density of the cellular structure. For a stacking
of cylinders of outer diameter, 2ao, and a center-to-center cylinder (cell) spacing of l; the unit cell volume
ð8Þ
¼ pao =2l which is identical to the relative
As ai ! 0, q
density expression for a plain woven textile structure,
Eq. (3) [41]. The relative density given by Eq. (8) is independent of the orientation of the cellular truss structure.
Table 1 compares the measure and predicted relative
densities using Eq. (8). It can be seen that the measured
relative densities are well predicted by Eq. (8).
4. Results and analysis
4.1. Compressive response
The through thickness compressive nominal stress–
strain responses for square and diamond oriented
samples are shown in Fig. 8(a) and (b). They exhibit
characteristics typical of many cellular metal structures
including a region of elastic response, plastic yielding,
a post yield peak stress followed by a plateau region
and finally hardening associated with densification that
began at a densification strain (eD) of 50–60%. Core
‘‘softening’’, especially for the square trusses, was observed once the peak compressive strength of the structure was surpassed.
¼ 0:11
Photographs of hollow truss structures with q
undergoing compressive loading are shown in Fig. 9 at
0.08
0.06
0.03
0
0
0.2
Strain
Fig. 8. Compressive stress–strain response for sandwich structures
with: (a) square oriented; (b) diamond oriented lattice truss structures.
nominal strains of 0%, 10%, 20%, 30% and 40%. It
can be clearly seen in Fig. 9(a) that as the square orientation lattice truss structure was compressed the trusses
parallel to the loading direction cooperatively buckled
near their midpoints. This buckling coincided with the
peak in strength, Fig. 8. The buckling half wavelength
corresponded to the face sheet separation distance. As
deformation progressed beyond peak load, the trusses
continued to cooperatively buckle causing the core to
extend laterally beyond the face sheets. This was accompanied by a significant decrease in flow stress. Some
node rotation can be observed in Fig. 8. The horizontal
trusses provided an efficient means of coupling the buckling behavior of the individual vertical trusses.
The diamond orientation lattice truss structure buckled differently. While the onset of truss buckling again
coincided with the peak in strength, Fig. 9(b), shows
that as the diamond lattice truss structure was compressed from 0% to 10% the truss members buckled with
a half wavelength approximately that of the cell size.
The nodes acted as plastic hinges and significant rotation again occurred. As deformation progressed, the
truss members continued to buckle and the core again
began to barrel outwards. The higher order deformation
buckling mode of the trusses was accompanied by a
smaller post peak drop in flow stress. The solid truss
structure buckling behaved similarly to that of the hollow trusses for both orientations.
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
309
sheets, x1 = 0 and x2 = 90, and Rel = 0.50. For a diamond lattice truss structure ignoring edge effects
x = 45 and Rel = 0.25 [44]. Edge effects for low aspect
ratio diamond truss samples can be important and in
this case, Rel = sin4 x[1 (1/A tan x)]. For the samples
tested here (A = 3.3) the predicted modulus drops from
0.25 to 0.18. The non-dimensional modulus coefficient
Rel ¼ E=
qEs for both the experimental data (collected
from unload/reload data prior to yielding) and the predicted moduli for these lattice truss structures is shown
in Fig. 10. Fig. 10 shows that the predictions and measurements are in very good agreement.
4.3. Compressive strength
The compressive peak strength of a hollow lattice
structure can be predicted from the inelastic buckling
strength of a truss member. The inelastic buckling stress
of a hollow cylinder is given by
2
ao þ a2i
p2 k 2 IEt
2 2
rcr ¼
¼ p k Et
;
ð10Þ
4l2
A c l2
where I is the second moment of inertia and Ac the
cross-sectional area of the column. Note that the stress
for the onset of buckling in Eq. (10) is minimized when
the columns are solid (i.e., ai = 0).
Inserting Eq. (10) for ry in Eq. (1) gives the peak
compressive strength for a lattice truss structure failing
by inelastic buckling
2
ao þ a2i
2 2
rpk ¼ p k Et
q:
ð11Þ
sin2 x
4l2
This result applies to both square and diamond oriented
lattice truss structures.
The elastic buckling behavior of the lattice truss
structures can by found by replacing Et in Eq. (11) with
0.75
Fig. 9. Photographs showing the deformation characteristics of: (a)
square orientation hollow truss structure; (b) diamond orientation
hollow truss samples at plastic strains of 0%, 10%, 20%, 30% and 40%.
4.2. Elastic moduli
0.45
0.3
Zupan and Fleck have shown the out-of-plane modulus, E, of a metal textile structure scales linearly with
the relative density [44]
;
E ¼ Rel Es q
0.6
ð9Þ
where Rel = sin4 x. For a square lattice truss structure
Rel = V1sin4 x1 + V2sin4 x2, where V1 is the volume fraction of trusses making contact with the face sheets and
V2 is the volume fraction of trusses parallel to the face
0.15
Square (measured)
Diamond (measured)
0
0
0.05
0.1
0.15
0.2
0.25
Fig. 10. Comparisons between the measured and predicted elastic
moduli coefficient, Rel, of square and diamond oriented lattice truss
structures as a function of relative density.
310
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
Es. The transition from elastic buckling to plastic yielding or inelastic buckling can then be found by equating
the resulting elastic buckling expression to the plastic
strength relation (Eq. (1)) or that for plastic buckling
(Eq. (11)). The relative densities of all samples investigated here were well above the elastic buckling–plastic
and elastic buckling–inelastic buckling transitions,
Table 2.
The predicted and measured non-dimensional peak
for
strength coefficient R ¼ rpk =
qry is plotted against q
both the square and diamond lattice truss structures in
Fig. 11(a) and (b), respectively. The measured value of
the peak strength coefficient for the square lattice truss
structure was approximately R = 1.2 and independent
. That of the diamond lattice truss structure was also
of q
but had a lower value of R = 0.85.
independent of q
However, the sample aspect ratio was low (A = 3.3)
and thus the strength was reduced by a factor of 0.7.
When this sample size effect was accounted for,
R = 1.2 for both the square and diamond lattices.
The predicted and measured responses for the square
and diamond topology lattices are shown on Fig. 11.
The predictions for the diamond lattices were reduced
by a factor of 0.7 to account for the samples low aspect
ratio. Only the plastic yield mechanism with R = 0.5,
Eq. (1) predicts a structural efficiency parameter that is
independent of relative density. Strengths significantly
exceeded this and were experimentally observed to be
coincident with buckling. The experimental data are
seen to lie between the k = 1 and k = 2 predictions for
plastic buckling consistent with the observed constrained node rotation, Fig. 8.
These results can be compared to recently reported
measurements for square oriented textile (solid truss)
structures where the strength coefficient varied from
0.50 to 0.65 for relative densities of 0.17 to 0.31 [41].
Recent measurements for diamond oriented textile
structures had through thickness peak strength coefficients of 0.30–0.70 for relative densities ranging from
0.18 to 0.23 [44]. The solid truss structures fabricated
by the alternating collinear cylinder lay-up process de-
Table 2
Relative densities for the transition of elastic buckling to plastic
yielding for square and diamond lattice truss samples with fixed and
pinned end conditions
Sample
relative density ð
qÞ
0.23
0.11
0.08
0.06
0.03
Transition relative density
Square
orientation
Diamond
orientation
k=2
k=1
k=2
k=1
0.015
0.006
0.004
0.003
0.001
0.030
0.011
0.007
0.006
0.003
0.018
0.007
0.004
0.003
0.002
0.011
0.014
0.009
0.007
0.003
2
(a) Square
Inelastic buckling
fixed nodes
(k=2)
1.6
1.2
Experimental
data
0.8
Elastic buckling
(solid trusses)
Plastic yielding
0.4
0
0
1.5
Inelastic buckling
pinned nodes
(k=1)
0.05
0.1
0.15
0.2
0.25
(b) Diamond
Inelastic buckling
fixed nodes
(k=2)
1.2
0.9
0.6
0.3
0
0
Elastic buckling
(solid trusses)
Experimental
data
Plastic yielding
Inelastic buckling
pinned nodes
(k=1)
0.05
0.1
0.15
0.2
0.25
Fig. 11. Comparisons between the measured and predicted peak
strength coefficient, R, of (a) square and (b) diamond oriented lattice
truss structures as a function of relative density. A sample aspect ratio
of 3.3 was used for the strength predictions of the diamond lattice.
scribed here are more than 1.7 times more efficient at
supporting compressive loads than their metal textile
counterparts. The normalized strength coefficient of
the square and diamond lattice truss structures is compared with competing stainless steel cores in Fig. 12
including, woven metal textile, pyramidal and square
honeycomb cores [44]. At a relative density of 0.03,
the hollow truss structures are 2.4 times stronger than
pyramidal trusses about 2.6 times stronger than square
honeycombs. The low relative density hollow lattice
truss structures made by the alternating collinear lay
up process appear to exhibit higher compressive
strengths than any other cellular metal topology reported to date.
4.4. Impact energy absorption
Preferred impact energy absorbing structures have
stress strain curves that exhibit yielding followed by a
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
Wv ¼
Z
1.000
(a)
0.100
Square orientation
Diamond orientation
Alternating
Colinear Layup
Wvol /
long plateau with no work hardening. Ideal structures
collapse at a constant (plateau) stress which establishes
a fixed load level transmitted to the supporting structure
[54]. Numerous structures have been assessed for such
behavior including circular tubes, square tubes, honeycombs, corrugated tubes, sandwich plates and metal
foams. Cellular structures are of considerable interest
for impact protection because of their high specific impact energy absorption and peak to plateau stress ratio
near unity [2].
The energy absorption per unit volume, Wv, can be
used as a merit index to compare different cellular structures. This specific energy is defined from the area under
the nominal stress nominal strain curve as
311
Metal
Textiles
0.010
Pyramidal
lattice truss
0.001
0.001
0.01
0.1
1.0
/
1.00
eD
(b)
ð12Þ
rðeÞ de;
o
where r(e) is the flow stress of the structure and eD is the
densification strain. The corresponding energy absorption per unit mass is calculated by dividing Eq. (12) by
the samples density, i.e., product of the relative density
and the parent alloyÕs density qs.
q
Fig. 13(a) and (b) shows the energy absorption per
unit volume and mass, respectively, for the square and
diamond oriented samples. It is compared with data recently reported for stainless steel woven metal textile lattice truss structure and pyramidal truss structures
[44,55]. Fig. 13(a) shows the energy absorption per unit
volume of the square and diamond hollow truss structures are comparable with woven textile lattice truss
and pyramidal truss structures. However, when the energy absorption is normalized per unit mass, Fig.
13(b), the square and diamond hollow truss structures
10
Square orientation
Diamond orientation
Diamond woven textile
Pyramidal truss
Square honeycomb
1
0.1
0.001
0.01
0.1
1
Fig. 12. Comparisons between the peak strength coefficient, R, of the
square and diamond oriented lattice truss structures and stainless steel
square honeycomb, pyramidal, and woven metal textile cores as a
function of relative density.
/
Alternating
Colinear Layup
Metal
Textiles
Pyramidal
lattice truss
0.10
Square orientation
Diamond orientation
0.01
0.001
0.01
0.1
1.0
/
Fig. 13. Energy absorption (a) per unit volume and (b) per unit mass
for the square and diamond orientation samples and various other core
topologies.
are comparable with the pyramidal lattice truss structures and significantly higher than the woven textile
lattice truss structures.
The superior strength and energy absorption characteristics of the structures fabricated by the alternating collinear lay-up process is due to an absence of
the plastic kinks created at crossing points in a woven
textile structure. These imperfections reduce the truss
buckling strength under axial loading which in turn
reduces the macroscopic peak strength of the cellular
structures. We also note that low relative density
(<10%) textile truss structures with solid trusses require very large cell sizes with high the truss aspect
ratios. Increasing the truss aspect ratio lowers the
plastic buckling strength of the truss element and is
responsible for a strength that varies as the cube of
the relative density. The use of the alternating collinear lay up process both eliminates cylinder kinking
and provides a means for using the higher second
moments of hollow structures to increase truss buckling resistance. Improvements in the performance of
sandwich panel structures utilizing these structures
may be achievable.
312
D.T. Queheillalt, H.N.G. Wadley / Acta Materialia 53 (2005) 303–313
5. Summary
An alternating collinear lay-up process has been
developed to manufacture lattice truss structures.
The truss assembly and face sheets were metallically
bonded by a vacuum brazing process.
The dimensions of the cylinders, the wall thickness of
hollow truss structures and the spacing between the
trusses enabled independent control of the cell size
and the relative density of the structure. The process
has been used to create stainless steel lattices with
square and diamond topologies with relative densities
from 0.03 to 0.23.
The through thickness elastic modulus of these lattice
truss structures was found to be proportional to relative density and was well predicted by theory. The
square topology has twice the through thickness stiffness of diamond oriented trusses.
The peak compressive strength was controlled by
plastic buckling. The structural efficiency of hollow
truss structures with a fixed cell size is approximately
proportional to the relative density. The data lie
between predictions for trusses with built-in and
pin-jointed nodes consistent with experimental observations of constrained node rotation.
The use of hollow trusses increases the resistance to
buckling offsetting the usually rapid drop in strength
as the relative density decreases in cellular systems
where truss buckling controls failure.
Low relative density hollow lattice truss structures
made by the alternating collinear lay up process
appear to exhibit higher compressive strengths
than any other cellular metal topology reported
to date.
Acknowledgements
We are grateful to Vikram Deshpande and Norman
Fleck (Cambridge University, UK) and Frank Zok
and Anthony Evans (UCSB) for helpful discussions.
The work was supported by the Office of Naval Research (Contract No. N00014-02-1-0614 and N0001403-1-0281, monitored by Dr. Steve Fishman and
Edward Johnson).
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