Materials
& Design
Materials and Design 28 (2007) 507–514
www.elsevier.com/locate/matdes
Lattice truss structures from expanded metal sheet
Gregory W. Kooistra *, Haydn N.G. Wadley
Department of Materials Science and Engineering, University of Virginia, 116 EngineerÕs Way, Charlottesville, Virginia, VA 22904, USA
Received 11 January 2005; accepted 24 August 2005
Available online 17 October 2005
Abstract
Metallic lattice truss structures are usually made by perforating a metal sheet with a periodic diamond pattern followed by folding at
node rows to create a plate of 3D interconnected trusses. For low relative density lattices, the initial sheet material is inefficiently utilized,
and the node area is small raising concerns about node bond robustness under shear or tensile loading. Here, we explore a simple
approach for making open cell, pyramidal lattice truss structures with robust nodes and close to 100% utilization of the sheet material.
Aluminium alloy lattices with a relative density of 5.7% were fabricated and bonded to aluminium alloy facesheets using a brazing technique. They have been tested in through thickness compression, and in both transverse and longitudinal shear. The lattice truss structures
made by this approach have a normalized compressive peak strength close to the predicted maximum. The non-dimensional transverse
and longitudinal shear strengths were also close to theoretical predictions. No node failures were observed during plastic shear straining
up to 20%.
Ó 2005 Published by Elsevier Ltd.
Keywords: Sandwich structures; Honeycomb; Brazing
1. Introduction
Lattice truss structures with open cell tetrahedral, pyramidal, or other topologies and relative densities below 10%
have attracted considerable interest as core replacements
for the closed cell honeycombs used in sandwich panel
structures [1,2]. Fig. 1 schematically shows several examples of such structures. When made from aluminium alloys,
lightweight sandwich structures can be created by bonding
the truss lattice to facesheets using brazing processes [3].
The ensuing structural performance has been shown to be
competitive with that of honeycomb core panels in panel
compression and bending at low core relative densities
[3–5]. Interest in these systems has also been stimulated
by the open cell topology which enables other functional-
*
Corresponding author. Present address: 26246 Twelve Trees Lane
NW, Poulsbo WA 98370. Tel.: +1 360 394 1200x257; fax: +1 360 394
1322.
E-mail address: [email protected] (G.W. Kooistra).
0261-3069/$ - see front matter Ó 2005 Published by Elsevier Ltd.
doi:10.1016/j.matdes.2005.08.013
ities such as cross flow heat exchange to be simultaneously
achieved [6–8]. The approach might also alleviate other
shortcomings of honeycomb, such as their susceptibility
to corrosion by entrapped moisture and potential for
delamination [9].
Because of delamination issues, the design of the core-tofacesheet bond in metallic honeycomb sandwich panels has
received much attention [9]. Similar joint robustness issues
have been encountered in sandwich panels utilizing metallic
lattice truss structures. When sandwich panels containing
either type of cellular core are subjected to loading the core
must transfer forces from the facesheets to the core members. It is therefore important to ensure adequate node
bond fracture strength. Even then, the area of the node contact with the facesheet determines the maximum force that
can be transmitted across the node. Several factors combine
to determine node bond robustness. They include the node
joint composition and microstructure (avoidance of brittle
phases), the degree of bond porosity, the degree of geometric constraint and the contact area of truss–truss or truss–
facesheet nodes.
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G.W. Kooistra, H.N.G. Wadley / Materials and Design 28 (2007) 507–514
made by perforating metal sheets to create a periodic array
of diamond shaped holes, Fig. 2. The sheet is then folded
along rows of nodes to create the lattice truss structure
as shown in Fig. 2 [2]. Unfortunately, as the lattice truss
structureÕs relative density is reduced, lattice truss structures make increasingly less efficient use of the initial sheet
material and therefore become increasingly costly to make.
The fraction, U, of the initial sheet used in a pyramidal
lattice of the type shown in Fig. 2 can be calculated (see
Appendix A) and has a square root dependence upon relative density:
!
pffiffiffi
2 6
1=2
1=2 ;
cos xsin x q
U¼
ð1Þ
3
Fig. 1. Three lattice truss topologies recently investigated: (a) the
tetrahedral lattice; (b) pyramidal lattice; (c) the 3D Kagomé lattice. The
tetrahedral and pyramidal lattices have been fabricated by the folding of
perforated sheet.
Metallic honeycombs can be fabricated by methods that
make very high utilization of the starting sheet material [9].
However, pyramidal lattice truss structures are usually
is the relative
where x is the angle defined in Fig. 2 and q
density (the ratio of truss density to that of the solid from
which it is made). For a regular pyramidal lattice x = 45°
and U ¼ 0:97
q1=2 . A pyramidal lattice truss with a relative
of 1% therefore utilizes less than 10% of the
density, q
starting sheet material.
Here we report a simple method for making pyramidal
truss lattices based on the in-plane expansion of partially
slit metal sheets. This is then followed by folding to create
a 3D pyramidal lattice truss core. The approach can utilize
almost 100% of the original metal sheet and coincidentally
enables fabrication of a lattice with larger (more mechanically robust) nodes. We illustrate the process by fabricating
¼ 0:057 from an alua pyramidal lattice structure with q
minium alloy and create sandwich panels by metallurgical
bonding these lattice truss structures to aluminium alloy
facesheets. We modify the mechanical property predictions
of Deshpande and Fleck [4] for ideal pyramidal lattices to
account for the fraction of the core mass apportioned to
the large area node contacts and compare these predictions
to mechanical measurements in through thickness compression as well as longitudinal and transverse shear.
Fig. 2. Schematic illustration for the manufacture of ideal pyramidal lattice cores. The principal operations include perforating a solid sheet followed by
folding.
G.W. Kooistra, H.N.G. Wadley / Materials and Design 28 (2007) 507–514
2. Fabrication
A sheet slitting, expansion and folding technique was
used to create a pyramidal lattice truss from an Al–
1.2%Mn–0.12%Cu (AA3003) alloy. Fig. 3 schematically
shows the slitting, expanding, flattening, and folding process. The expanded sheet had a thickness, t = 1.15 ±
0.01 mm, and truss member widths, w = 1.94 ± 0.01 mm.
The node row folding bends the sheet through two 35° angles (separated by the node length), which creates a truss
member inclination angle x = 45 ± 1° corresponding to
that of a regular pyramid, see Fig. 3. The length of the truss
members, l, was 11.90 ± 0.02 mm, and the length of nodes
was, b = 3.75 ± 0.02 mm.
509
For the slitting and expansion process used here, the
width of the node was twice the truss member width. The
area occupied by the node to that devoted to the truss
members was 24%. The expanded lattice trusses were
bonded to aluminium alloy facesheets using a furnace brazing technique identical to that described in [3,10]. Briefly,
the pyramidal lattice was placed between Al–Si braze alloy
clad facesheets (AA4343/AA6951), coated with a fluxing
agent (a mixture of alkali metal salts) and placed in a muffle furnace at 595 °C for 5–10 min. The samples were removed, air cooled, cleaned and tested in this annealed
condition. An example of an as-brazed sandwich panel is
shown in Fig. 4.
3. Mechanical property relations
Analytical expressions for predicting the stiffness and
strength of regular pyramidal lattice trusses (assuming all
of the core is placed in the trusses) have been derived by
Desphande and Fleck [4], and validated with finite element
simulations [5]. Here we extend the expressions for the elastic stiffness and lattice truss collapse strength under compressive and shear loading to the modified structure made
by the methods above.
To account for the fraction of the core utilized to create
a node, we begin by defining the relative density of a modified pyramidal lattice truss made by the sheet expansion
process. Its unit cell geometry is defined in Fig. 5. The rel, is found by calculating the volume fraction
ative density, q
of the unit cell occupied by metal. For a cell defined by the
included angle, x,
4ðl þ bÞ
wt
pffiffiffi
¼
q
;
ð2Þ
sin 2x ðl cos x þ b 2Þ l2
where l is the truss member length, b the length of a node,
and w and t are the width and thickness of the truss member cross-sections.
If we take x = 45°, Eq. (2) reduces to:
pffiffiffi ðl þ bÞ wt
¼4 2
q
.
ð3Þ
ðl þ 2bÞ l2
Note that Eq. (3) reduces to that of an ideal pyramidal
truss lattice when b = 0 (Eq. (A.2) in Appendix A).
Fig. 3. Schematic of the manufacturing process for the expanded
pyramidal lattice truss cores. The primary steps involve slitting, flattening
and folding the metal sheet.
Fig. 4. Photographs of the brazed AA3003 sandwich panel ð
q ¼ 0:057Þ
made for compression testing.
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G.W. Kooistra, H.N.G. Wadley / Materials and Design 28 (2007) 507–514
geometry, truss material properties and the mode of loading. Consider the out of plane compression of the unit cell
shown in Fig. 5.
If the lattice truss is made from a rigid-perfectly plastic
lattices), failure
material and the l/t ratio is small (high q
(peak strength) occurs by plastic yielding. The peak compressive strength of the lattice structure can then be
written:
q;
rpk ¼ ry sin2 x g
Fig. 5. Unit cell of the pyramidal lattice truss obtained from expanded
metal sheet. The truss member length, width and thickness are l, w and t,
respectively. The node length is given by b. The angle x was fixed at 45° so
as to obtain the regular pyramidal arrangement. The arrows indicate the
experimental loading orientations.
where ry is the yield strength of the alloy. In practice many
alloys (such as the as-brazed aluminium alloy of this study)
exhibit varying degrees of plastic strain-hardening behavior, and the peak strength is higher than that predicted
by Eq. (6).
For lattices with more slender trusses, the peak compressive strength is controlled by buckling of the lattice truss
members [4]. For trusses with the l/t ratio used here, failure
occurs by inelastic buckling. In this case the yield strength,
ry, in (6) is replaced by the inelastic column critical (buckling) stress, rcr, defined as [4]:
3.1. Elastic stiffness
rcr ¼
The flat nodes in the modified pyramidal truss structure,
Fig. 5, provide no contribution to the stiffness or strength
of the lattice truss core regardless of its mode of loading.
As a result, the stiffness is decreased over that of an ideal
lattice (where b = 0). We introduce a truss mass fraction,
g, using arguments developed in Appendix B, g = l/
(l + b). It leads to an expression that separates components
of the core topology into those that directly contribute to
the stiffness from those that are utilized to achieve robust
performance.
In Appendix B we show that Desphande and FleckÕs result [4] for the out of plane compressive stiffness of the lattice, Ec, can be rewritten as:
q;
Ec ¼ Es sin4 x g
ð4Þ
where Es is the YoungÕs modulus of the parent alloy, x the
truss member inclination angle to the facesheet, g the node
the relative density given by Eq. (2). We
coefficient, and q
note that the prediction of elastic stiffness by Deshpande
and Fleck is usually greater than that observed in experiments [4]. The reasons for this are unclear but have been
argued to result from the small geometric imperfections
of real structures.
The in-plane shear stiffness, Gc, of the ideal pyramidal
lattice trusses is isotropic [4] and for a modified lattice is given by:
Gc ¼
Es 2
sin 2x g
q.
8
ð5Þ
3.2. Peak strength
The collapse strength of a lattice truss core is determined
by the mechanism of truss failure which depends on the cell
ð6Þ
k 2 p2 Et I
;
Al2
ð7Þ
where k is determined by the column end conditions (k = 2
for built-in trusses or 1 for pin jointed trusses), A is the crosssectional area (wt) of the column, and I is the truss member
second moment of area. I for rectangular cross-sections is
wt3/12 (and is less than that of a square or circular crosssection of the same area), Et is the Shanley–Engesser tangent
modulus (dr/d) obtained from the stress–strain response of
the parent alloy [11], and t, w and l are the truss member
thickness, width and length, respectively. If Et is constant
then compressive peak strength scales with relative density
squared.
The longitudinal and transverse shear strength of a
pyramidal lattice truss as defined by the loading directions
shown in Fig. 5, are controlled by simultaneous buckling of
two of the four unit cell truss members. Again using simple
column theory [11], the shear strength for this failure mode
is given by:
sð/Þ ¼ rcr
1
sin 2x
g
q;
2 ðcos / þ sin /Þ
ð8Þ
where rcr is the critical (buckling) strength of an individual
truss member (Eq. (7)), x is the included angle, and the angle / is defined in Fig. 5. Because of pyramidal lattice symmetry, / = 45° corresponds to both the transverse and
longitudinal core shear directions.
4. Mechanical property measurements
Sandwich panels were constructed for compression and
shear tests in accordance with ASTM STP C-365 and C273, respectively. All samples were tested using a servo electric test machine (Model 4208, Instron Corp., Canton,
MA) at nominal strain rate of 103 s1. The measured load
G.W. Kooistra, H.N.G. Wadley / Materials and Design 28 (2007) 507–514
511
Table 1
Dimensions of the expanded pyramidal lattice truss members together
with the predicted and post-brazed measured relative density
Truss measurements
Coefficient
Relative density
w (mm)
t (mm)
l (mm)
b (mm)
g
Predicted
Measured
1.94
1.15
11.90
3.75
0.76
0.058
0.057
cell force was used to calculate the nominal stress applied
to the sandwich, and a nominal strain was obtained from
a laser extensometer (Model LE-01, Electronic Instrument
Research, Irwin, PA). The assessed gage length for the
compression and shear samples was the core thickness.
The dimensions required for predicting the performance
of the pyramidal lattice trusses were measured after bonding the lattices to facesheets and are summarized in Table
1. The computed g coefficient as well as the predicted and
measured relative densities are also given.
Tensile test coupons conforming to ASTM specification
E-8 accompanied the sandwich samples through the brazing cycle and were used to determine parent alloy properties and the value of the critical stress, rcr, for trusses
¼ 0:057.
with a slenderness ratio that corresponded to q
Fig. 6 shows the quasi-static (strain rate 103 s1) stress–
strain response in the as-brazed condition. The AA3003 alloy had a 0.2% offset yield strength (ry) of 37 MPa,
Es = 69 GPa and rcr = 56 MPa taking Et = 1800 MPa.
Non-dimensional compressive unload/reload modulus
values are presented in Fig. 7 as a function of strain. The
Þ, had
normalized compressive core stiffness, E Ec =ðEs q
a measured peak value of 0.14. The prediction for the
expanded pyramidal lattice was, E = 0.19 while that of
an ideal pyramidal lattice was 0.25. Both predictions are
indicated by the solid and dashed lines, respectively. Previous studies have failed to identify why lattice trusses are
significantly more compliant than prediction prior to the
peak load levels. It could be a result of non-parallelism
which impedes simultaneous loading of all truss members.
Fig. 6. The logarithmic tensile stress versus strain response for the asbrazed AA3003. The 0.2% offset yield strength is 37 MPa. The critical
column buckling stress for an individual truss member corresponding to
¼ 0:057 was calculated to be 56 MPa (Et = 1800 MPa).
the q
Fig. 7. The predicted out-of-plane compressive modulus for an expanded
¼ 0:057 is shown with experimental unload/
pyramidal lattice truss with q
reload moduli data. The measured relative density coefficient g = 0.76
accounts for the amounts of core material not contributing towards the
stiffness of the lattice.
Measurements of the core height around the sandwich
periphery revealed a deviation from the mean sandwich
thickness of 4% consistent with this possibility.
The non-dimensional out-of-plane compressive stress response is shown in Fig. 8. Predictions for the peak compressive strength (assuming built-in trusses where k = 2)
are indicated by the dashed lines. The normalized compresÞ had a measured value of
sive peak strength, R rpk =ðrY q
0.50.
Individual truss member buckling was first observed at a
nominal strain of 0.045 corresponding to the maximum
measured modulus value. Trusses on the slightly narrower
sides of the panels usually buckled first. At the peak compressive stress (0.07 nominal strain) all truss members had
fully developed plastic hinges at mid-truss locations. At
high strains of 0.47 the plastic hinges impinged on the facesheets and the core strength increased once more due to the
effective shortening of the truss members. No node failures
were observed.
Fig. 8. Shown are the mean values of unload/reload moduli data obtained
from shearing in both the longitudinal and transverse directions. The
predictions for the ideal pyramidal lattice and actual stiffnesses are also
given.
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G.W. Kooistra, H.N.G. Wadley / Materials and Design 28 (2007) 507–514
5. Discussion
Fig. 9. The measured non-dimensional compressive stress–strain response
of the expanded pyramidal lattice panel is shown with the predictions for
peak compressive strength.
Fig. 9 shows the non-dimensional longitudinal and
transverse shear unload/reload moduli. The maximum
Þ, had a
non-dimensional shear modulus, C Gc =ðrY q
measured value of 0.10 in the longitudinal direction and
0.085 in the transverse direction. The initial rise in modulus
with strain is consistent with small differences in the lengths
of the trusses. The decrease at larger strains results from
truss buckling.
Fig. 10 shows the shear response for the longitudinal
and transverse directions (/ = 45°). The non-dimensional
Þ, was measured to be
peak shear strength, T s/ =ðrY q
0.34 in the longitudinal direction, and 0.33 in the transverse. In each case, observations were consistent with failure governed by inelastic buckling of two truss members in
compression; as analyzed in Eq. (8). Fig. 11 shows a series
of photographs of the shear tests together with corresponding values of normalized stress (s) and strain. Table 2 summarizes the predicted and measured non-dimensional
moduli and strengths.
Fig. 10. The shear response in both the longitudinal and transverse load
directions is shown with the corresponding predictions. The lattice truss
collapse shear strength is predicted to be reduced by 45% from that of an
ideal lattice with square cross-section truss members and small nodal
areas.
A method for creating pyramidal lattice truss structures
from expanded metal sheets has been developed. It enables
nearly 100% utilization of the starting material and results
in large area nodes. The approach has been illustrated by
fabricating sandwich panel structures from an Al–Mn–Cu
alloy (AA3003). Extensions to other alloys that have sufficient ductility for stretching appear feasible. The use of
annealing to increase alloy ductility followed by solutionizing and precipitation hardening treatments suggests the
possibility of using the approach to make sandwich panels
from high strength aluminium alloys provided suitable
braze alloys/bonding approaches are available.
The measured maximum elastic stiffnesses of samples
with a relative density of 5.7% agree reasonably well with
those of the modified predictions of Deshpande and Fleck
[4]. The expanded pyramidal lattice stiffness is reduced
from that of an ideal pyramidal lattice by the truss mass
fraction (g). The increased metal area contact at the nodes
contributes little to the latticesÕ resistance to deformation.
Expressions for the stiffness coefficients of these pyramidal
lattices indicate that expanded pyramidal lattice truss
¼ 0:057 are equivalent in stiffness to an
structures with q
¼ 0:043.
ideal pyramidal lattice truss with q
The peak strength of expanded pyramidal lattices has
been shown to depend upon the three factors: (i) the
stress–strain response of the parent alloy (captured by the
buckling Eq. (7)), (ii) the truss mass fraction coefficient,
. During compressive loadg, and (iii) the relative density, q
ing of the expanded lattice trusses the stress–strain response is similar to that of other annealed lattice
structures [3,4]. The peak strength corresponds to the onset
of truss member buckling. Plastic hinges at truss member
mid-span were fully developed at stresses corresponding
to the peak load. The normalized peak strength of 0.50 is
consistent with a lattice that utilizes the high plastic strain
hardening capacity of the AA3003 alloy in the annealed
condition. The measured peak shear strengths are also
found to be in reasonable agreement to the modified predictions of Deshpande and Fleck [4]. Under shear loading
the peak strength corresponded to the onset of pairs of
trusses in each unit cell undergoing inelastic buckling.
The nodes showed no signs of de-bonding or shear failure in either test orientations even after nominal strains of
20% (the valid test limit). The enhanced mechanical robustness of the brazed expanded lattice trusses appears to have
been realized at a relative density of 5.7%.
Further improvements in collapse strength of these
modified pyramidal lattices appear possible. We note in
Eq. (7) that the strength scales linearly with the second area
moment of the truss. Forming nearly square truss member
cross-sections should therefore increase the buckling
strength. Extending the fabrication approach described
here to heat treatable alloys and/or new joining processes
promises to provide additional pathways for the fabrication of high efficiency sandwich panel structures from
G.W. Kooistra, H.N.G. Wadley / Materials and Design 28 (2007) 507–514
513
Fig. 11. (a) Photographs of the longitudinal shear test with cnom = 0.10 corresponding the peak shear load sustained by the lattice structure. The truss
members were observed to buckling in the out-of-the-paper direction. (b) Photographs of the transverse shear loading, again cnom = 0.10 corresponds to
the peak shear load.
Table 2
Summary of the predicted and measured non-dimensional stiffness and strength properties for the pyramidal lattice trusses
Non-dimensional property
Compressive modulus
Þ)
(E Ec =ðEs q
Shear modulus
Þ)
(C Gc =ðrY q
Compressive strength
Þ)
(R rpk =ðrY q
Shear strength
Þ)
(T s/ =ðry q
Measurement
Prediction
0.14
0.19
0.10L, 0.085T
0.09
0.50
0.57
0.34L, 0.33T
0.41
Subscripts L and T indicate the longitudinal and transverse testing orientations.
materials that cannot usually be used honeycomb construction (where post fabrication high rate quenching is very
difficult).
6. Conclusions
1. A new, more materials efficient method for fabricating
modified pyramidal lattice truss structures from aluminium sheets has been explored. The approach utilizes
close to 100% of the starting sheet material and coincidentally enables large area, mechanically robust nodes
to be fabricated.
2. The analytical predictions of Deshpande and Fleck [4]
for an ideal pyramidal lattice have been modified to
account explicitly for partitioning of material between
the nodes and the trusses and to introduce the role of
strain-hardening of the truss material in suppressing
inelastic buckling. The modified model predictions are
shown to be in reasonable agreement with experiments
conducted on samples with a relative density of 5.7%.
3. Mechanical testing of the expanded pyramidal lattice
truss structure made from a ductile Al–Mn–Cu alloy
indicate no node failure even after in plane shear strains
of 20%.
Acknowledgments
The authors express their thanks to Vikram Deshpande
and Hilary Bart-Smith for helpful discussions of this work.
We are grateful to the Office of Naval Research for support
of this research through grant N00014-01-1-1051 (program
manager, Dr. Steve Fishman).
Appendix A. Material utilization function
Consider the 2D unit cell (Fig. A.1(a)). The relative density, the volume fraction occupied by the truss members,
can be written:
2D ¼
q
VT
4 t
¼ pffiffiffi ;
VC
3l
ðA:1Þ
where VT and VC are the volumes of the truss members and
cell, respectively. The truss member thickness and length
are t and l. Using the unit cell (Fig. A.1(b)) we write the
3D relative density:
3D ¼
q
t 2
VT
2
¼
.
2
V C cos x sin x l
ðA:2Þ
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G.W. Kooistra, H.N.G. Wadley / Materials and Design 28 (2007) 507–514
placement [4]. The resulting compressive modulus can be
written:
Ec ¼ E s
2wt sin3 x
pffiffiffi
;
ðl2 cos2 x þ lb 2 cos xÞ
ðB:1Þ
where Es is the YoungÕs modulus. If we write this in terms
of the relative density Eq. (2) we find:
pffiffiffi
lsin4 xðl2 cos x þ 2lbÞ
pffiffiffi q
Ec ¼ E s
ðB:2Þ
ðl þ bÞ ðl2 cos x þ 2lbÞ
Fig. A.1. (a) 2D unit cell from diamond perforated sheet. (b) Resultant
3D unit pyramidal lattice truss.
The amount of material utilized, U, is related to the 3D relative density by:
2D ¼ C
q3D ;
U ¼q
ðA:3Þ
where C is a geometric constant for a give x angle. Taking
the ratio of relative densities we find:
pffiffiffi
2D 2 6 cos x sin1=2 x
q
¼
.
ðA:4Þ
1=2
3D
3
q
3D
q
Upon separating terms C becomes:
pffiffiffi
2 6
cos x sin1=2 x.
C¼
3
ðA:5Þ
For a regular pyramid (x = 45°) the value of C is 1
(C = 0.97), and the final expression for material utilization
is:
pffiffiffiffiffiffiffiffi
3D .
ðA:6Þ
U ¼ 0:97 q
Appendix B. Compressive stiffness
The compressive stiffness of the expanded pyramidal lattice is found by applying a point load in the compressive
direction (Fig. 4) and determining the work conjugate dis-
upon letting g = l/(l + b) it is shown that:
Ec ¼ Es sin4 x g
q.
ðB:3Þ
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