Materials Science and Engineering A 472 (2008) 242–250
Shear behavior of aluminum lattice truss sandwich panel structures
Gregory W. Kooistra, Douglas T. Queheillalt ∗ , Haydn N.G. Wadley
Department of Materials Science and Engineering, University of Virginia, 140 Chemistry Way, Charlottesville, VI 22904, USA
Received 13 February 2007; received in revised form 8 March 2007; accepted 9 March 2007
Abstract
Age hardenable 6061 aluminum tetrahedral lattice truss core sandwich panels have been fabricated by folding perforated sheets to form highly
flexible cellular cores. Flat or curved sandwich panels can be fabricated by furnace brazing the cores to facesheets. Flat sandwich panels with
core relative densities between 2 and 10% have been fabricated and tested in the σ ±13 shear orientation (minimum shear strength orientation for a
tetrahedral lattice) in the fully annealed (O) and aged (T6) conditions. The shear strength of the lattices increased with relative density, parent alloy
yield strength and work hardening rate. Analytical stiffness and strength predictions agree well with measured values for all relative densities and
parent alloy heat treatments investigated. The stiffness and strength of 6061-T6 aluminum tetrahedral lattice structures are shown to be comparable
to those of conventional 5052-H38 aluminum closed cell hexagonal honeycombs and more than 40% stiffer and stronger than flexible honeycombs
used for the cores of curved sandwich panels.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Lattice structures; Sandwich structures; Aluminum alloys; Brazing
1. Introduction
Millimeter cell size, aluminum alloy lattice structures with
various open cell topologies are attracting interest as lightweight
core structures for sandwich panel constructions. For bending
dominated applications of sandwich panels; the facesheets carry
the bending stresses with one facesheet in compression and one
in tension and the flexural strength of the panel is governed by the
shear response of the core and by the strength of its attachment
points (nodes) to the facesheets. The core also increases the
flexural stiffness of the panel by providing a separation between
the two facesheets.
Lattices appear to be mechanically competitive alternatives
to prismatic (corrugated) and perhaps honeycomb structures
when configured as the core of a sandwich panel. These lattice
sandwich structures are of particular current interest because
of their potential fully open interior structure which facilitates multifunctional applications [1–4]. For example, lattice
core sandwich panels appear capable of supporting significant
structural loads while also facilitating cross flow heat exchange
[5–8]. Some structures also enable high authority shape morph-
∗
Corresponding author. Tel.: +1 434 982 5678; fax: +1 434 982 5677.
E-mail address: [email protected] (D.T. Queheillalt).
0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2007.03.034
ing [9–14] and all appear to provide significant high intensity
dynamic load protection [15–21]. Lattices are also flexible and
are amenable to the creation of singly or compound curved sandwich panels. They may also alleviate some of the delamination
and corrosion concerns associated with the use of traditional
closed cell honeycomb sandwich panels [22,23].
The emergence of microscale lattice truss structures originally envisioned at the meter scale by Buckminster Fuller [24]
has been paced by the development of practical methods for their
manufacture [4,25]. Initial efforts to fabricate millimeter scale
structures employed investment casting of high fluidity casting
alloys such as copper/beryllium [26], aluminum/silicon [27–30]
and silicon brass [27]. However, the tortuosity of the lattices
and ensuing casting porosity made it difficult to fabricate high
quality structures at low relative densities (2–10%) identified
as optimal for sandwich panel constructions [31]. While some
lattice constructions appear to possess significant tolerance to
defects such as occasional weak trusses or nodes [32,33], the
low toughness of the materials used to make these as-cast lattice
materials have often lacked the mechanical robustness required
for the most demanding structural applications [34].
Efforts to exploit the inherent toughness of many wrought
engineering alloys led to the development of alternative lattice
fabrication approaches based upon perforated metal sheet folding [35]. These folded truss structures can be bonded to each
G.W. Kooistra et al. / Materials Science and Engineering A 472 (2008) 242–250
243
shown to be comparable to those of other topologies for aluminum based sandwich structures.
2. Fabrication methodology
2.1. Tetrahedral lattice truss fabrication
Fig. 1. Comparison between the normalized compressive peak strengths of the
aluminum tetrahedral lattice structures and commercially available competing
topologies that utilize aluminum alloys [42].
other or to facesheets by conventional joining techniques such
as brazing, transient liquid phase (TLP) bonding or welding
techniques to form all metallic lattice truss sandwich panels.
Panels fabricated from austenitic stainless steels with tetrahedral [35–37] and pyramidal lattice truss [38–41] topologies
have been made by node row folding of a patterned sheet
to form the core and TLP bonding to facesheets. Because of
the high temperatures normally encountered with TLP bonding, this process results in sandwich panels which remain in a
low strength, annealed condition. While these structures appear
much more robust than their investment cast counterparts, the
reduced strength of their annealed microstructure can limit their
potential uses for some structural applications.
The perforated sheet folding method has recently been
extended to age hardenable aluminium alloys such as the 6061
system, and tetrahedral lattices made from this alloy have been
shown to exhibit high specific compressive strengths (Fig. 1)
[42]. Comparisons with other cellular aluminum topologies
(Fig. 1), confirm that 6061 aluminum alloy tetrahedral lattice
structures are far superior to aluminum open cell metal foams
and prismatic corrugations. The compressive response of the
tetrahedral lattice was comparable to that of honeycomb panels
of similar specific mass and found to be sensitive to the lattices heat treatment condition. Annealed cores with high tangent
moduli were more efficient than age hardened structures and significantly exceeded elastic-ideally plastic strength predictions.
Inelastic column-buckling models robustly predict the through
thickness compressive strengths and resolved the important role
of the parent materials post-yield tangent modulus in delaying
the onset of unstable inelastic buckling.
Here, we explore the in-plane shear stiffness and strength
of these 6061 aluminum tetrahedral lattice structures described
above. The measured shear stiffness and strengths of the lattice truss structures are compared to analytical predictions and
A detailed description of the fabrication approach for making 6061 aluminum alloy lattice truss structures can be found
in Kooistra et al. [42]. Briefly, a folding process was used to
bend elongated hexagonal perforated 6061 sheet to create a single layer tetrahedral truss lattice. The folding was accomplished
using a paired punch and die tool to fold node rows into regular tetrahedrons with three trusses emanating from each node
resulting in a highly flexible core structure.
The unit cell of a tetrahedral lattice is shown in Fig. 2. The
relative density, ρ̄, of a tetrahedral lattice with 50% occupancy
of the available tetrahedral sites is given by ref. [27]:
t 2
2
1
ρ̄ = √
3 cos2 ω sinω l
(1)
where ω is the angle between the truss members and the tetrahedron base plane (ω = 54.7◦ for regular tetrahedrons) and t and
l are the sheet thickness and truss member length, respectively.
The relative density of the lattice was varied here by modification of the sheet thickness and the perforation dimensions to
maintain a square truss cross section and a constant truss length
(Table 1).
2.2. Sandwich panel fabrication
Sandwich panels were constructed from the folded lattice
structures by placing them between 6951 aluminum alloy face
sheets clad with a 4343 aluminum–silicon braze alloy. The
Fig. 2. Tetrahedral unit cell used to derive relative density and mechanical properties. The positive and negative shear directions are also shown. They result in
different stress–strain behaviors because of the different truss tensile stretching
and compressive buckling configurations.
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Table 1
Predicted and measured relative densities of tetrahedral lattice truss structures
t/l
0.063
0.079
0.099
0.125
0.178
Relative density
Model
Eq. (1)
Pre-braze
measurement
Post-braze
measurement
0.017
0.027
0.042
0.067
0.136
0.017 ± 0.003
0.025 ± 0.004
0.029 ± 0.004
0.048 ± 0.003
0.083 ± 0.002
0.020 ± 0.001
0.030 ± 0.002
0.039 ± 0.001
0.069 ± 0.002
0.106 ± 0.003
Fig. 3. Photograph of a tetrahedral lattice truss attached to facesheets by a
brazing method, ρ̄ = 0.039.
assemblies were coated with a metal-halide flux and placed in a
furnace for brazing in air at 595 ± 5 ◦ C for between 5 and 10 min
[43]. After air-cooling, the samples were solutionized at 530 ◦ C
for 60 min and either furnace cooled for the fully annealed (O)
condition or water quenched and aged at 165 ◦ C for 19 h for the
aged (T6) condition [44]. Fig. 3 shows an example of one of the
sandwich panels with a core relative density, ρ̄ = 0.039.
The nodes of the lattice truss core were ground flat prior to
brazing to ensure good contact between the lattice core and the
face sheets. This resulted in post-braze measurements of the relative densities slightly higher than those predicted by Eq. (1).
Table 1 summarizes the truss geometries, the measured and predicted relative densities. The mean post-braze relative density
(reported at a confidence level of 95%) was used to identify the
test specimens and for subsequent experimental data normalizations.
Tensile coupons of the 6061 alloy accompanied the cores
through each thermal process step and were used to determine the mechanical properties of the parent aluminum alloy in
both the O and T6 tempers. Three tensile tests were performed
according to ASTM E8 at a strain rate of 10−3 s−1 for each heat
treatment condition [45]. Table 2 shows the mechanical property
data for 6061 aluminum in the O and T6 tempers.
3. Shear experiments
The sandwich panels were tested according to ASTM C273
using a compression shear plate configuration at a nominal strain
rate 10−3 s−1 at an ambient temperature of 22 ◦ C [46]. The
applied force was used to calculate the stress imparted to the
sandwich structure, while shear displacement data of the rigid
plates was obtained by laser extensometry and the shear strain is
simply the displacement divided by the height of the sandwich
core.
Initial experiments indicated the need for a rigid bond
between the sandwich panel facesheets and shear plates, especially during evaluation of the highest relative density (strongest)
cores. The test samples were attached using four mechanisms to
prevent premature debonding and/or movement during testing.
The grit blasted surfaces were adhesively bonded using an epoxy
adhesive (Loctite Hysol® E-120HP). The facesheets were also
fastened to the shear plates using steel machine screws. Finally,
mechanical locking utilizing a leading edge stop machined into
the shear plate and a trailing edge adjustable clamping bar was
used to reduce load transfer stresses.
4. Results
Fig. 4a and *b shows representative shear stress–strain behavior for the annealed and aged samples loaded in the σ +13 direction
(see Fig. 2), which corresponds to one truss loaded in compression and the other two in tension (positive orientation). Fig. 4c
and d shows representative shear stress–strain behavior for the
annealed and aged samples loaded in the negative σ −13 direction, where one truss member is loaded in tension and the other
two in compression.
The annealed samples tested in the σ +13 direction (Fig. 4a)
sustained large macroscopic shear strains (as high as 35%)
without rupture of the tensile loaded truss members. For these
samples there was a small linear response during initial loading,
followed by yielding and increased load support. After large
deformations (which increased with sample relative density)
the tensile truss members underwent strain localization (necking) followed by rupture and a decrease in the load support.
Fig. 5 shows photograph of strain localization in a truss member (at 34.1% macroscopic strain) for an annealed sample with
ρ̄ = 0.069 sample.
The aged samples (Fig. 4b) exhibited a better defined linear
elastic region followed by observable plastic yielding of the tensile truss members. Load support continued to increase until a
peak strength was reached. Initiation of truss member rupture
then coincided with a sharp decrease in its load carrying capacity. These tensile ruptures were observed to occur at various
locations along the truss member length independent of relative density. Fig. 6 shows photographs of an aged sample with
a relative density, ρ̄ = 0.039 at various stages of deformation.
Table 2
Mechanical property data for 6061 aluminum in the O and T6 tempers from the uniaxial Cauchy (true) stress–true strain curves
Heat treatment
condition
Young’s modulus,
Es (GPa)
Yield strength,
σ ys (MPa)
Ultimate tensile
strength, σ TS (MPa)
Strain at failure,
εf (%)
Annealed (O)
Aged (T6)
68.6
69.1
70
268
208
306
21.2
15.1
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245
Fig. 4. Shear stress–strain responses for tetrahedral lattice structures. Representative responses are shown for the σ +13 orientation in the (a) annealed and (b) aged
hardened condition and for the σ −13 orientation in the (c) annealed and (d) aged hardened condition. Note the change in strain axes for the age hardened samples.
The annealed shear samples tested in the (negative) σ −13
direction (Fig. 4c), also sustained very large macroscopic strains.
For these samples there was an initial linear response, followed
by yielding and then a period of stably increasing load support.
A peak strength was eventually reached, corresponding with the
buckling of the compressively loaded truss members. The aged
samples tested in the σ −13 direction (Fig. 4d), also behaved
similarly. However, truss buckling was observed to occur at
Fig. 5. Photograph of the ρ̄ = 0.069 annealed tetrahedral lattice truss at a strain
level of 34.1%.
a significantly lower plastic shear strain consistent with the
lower tangent modulus of the heat treated condition. Some node
debonding was observed in the higher relative density samples
as indicated in Fig. 4d.
The stiffnesses for both orientations were determined from
unload/reload measurements in the nominally elastic portion of
the shear stress strain response. There was no effect of loading
direction or heat treatment. Fig. 7 shows a plot of the normalized
shear stiffness, Γ = G/(Es ρ̄), where G is the shear modulus and
Es is the Young’s modulus. The shear modulus measurements
shown in Fig. 7 are the average of two experiments conducted
for each orientation and heat treatment (eight measurements per
relative density) shown. The data indicate a linear dependence
of shear modulus upon relative density and is well fitted by a
relation of the form: G = 0.11Es ρ̄.
The normalized shear 0.2% offset yield strength for the σ ±13
orientations for both the annealed and aged samples is plotted
versus ρ̄ in Fig. 8. The experimental data for the normalized
yield strength of both the annealed and aged cores indicate a
mild dependence upon relative density for both the positive and
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negative oriented shear directions and it can be seen in Fig. 8
that the normalized yield strength increases with relative density. Note that the normalized coefficients shown in Fig. 8 are
nearly equivalent for both the annealed and aged lattice cores,
indicating a linear dependence upon σ ys of the parent alloy.
The normalized peak shear strength T = σpk /(σys ρ̄) of the
annealed and aged samples in the σ ±13 orientation is plotted
versus ρ̄ in Fig. 9. The experimental data for the normalized
peak strength of both the annealed and aged cores indicate a
mild dependence upon relative density for both the positive and
negative oriented shear samples. It can be seen in Fig. 9 that the
normalized peak strengths increase with relative density and the
rate of increase in the annealed cores is nearly double that of
the aged cores. This is indicative of the higher strain hardening
rate of 6061 in the annealed condition. For example, the ratio of
σ TS /σ ys is an indication of the strain hardening capacity. Using
the data shown in Table 2, σ TS /σ ys is 2.97 for the annealed
condition and 1.14 for the aged condition.
5. Micromechanical predictions
Fig. 6. Photographs of the ρ̄ = 0.039 T6 tetrahedral lattice truss at five values
of positive shear. Arrows indicate the loading direction. (b) The core is at the
peak stress level. Observable rupturing of truss members initiated by (c). (d)
Only the compressed truss members continued to support the shear load.
Fig. 7. The normalized shear stiffness measurements vs. relative density. Experimental data represents the initial shear stiffness for both σ 13 test orientations
and heat treatments. The error bars correspond to the 95% confidence level.
Deshpande and Fleck have developed analytical expressions
for the mechanical properties of tetrahedral lattice truss cores
assuming elastic-ideally plastic struts [27]. Table 3 summarizes
their analytical expressions for the shear stiffness, Eqs. (2) and
(3), and strength, Eqs. (4) and (5). If the lattice is constructed
from slender trusses, they can collapse by elastic buckling. In
this case, the shear strength is found by replacing σ ys in Eqs. (4)
and (5) with the elastic bifurcation stress, σ cr , of the trusses, Eq.
(6). The factor k depends on the rotational stiffness of the nodes;
for a pin-joint that can freely rotate k = 1 where k = 2 corresponds
to a built-in rigid joint. If the truss material has a non-zero
post-yield strain hardening rate, inelastic buckling defined by
Shanley–Engesser tangent modulus theory determines the lattice
strength [47]. The peak compressive strength is then obtained by
replacing σ ys in Eqs. (4) and (5) with the inelastic bifurcation
stress of a compressively loaded column. The inelastic bifurcation stress, σ cr , is given by Eq. (7) where, Et is the tangent
modulus defined as the slope dσ/dε of the uniaxial stress versus
strain curve of the parent material at a stress level, σ cr [48].
The shear stiffness in the σ ±13 directions is predicted to be the
same and its dependence upon ρ̄ is in very good agreement with
the experimental data (Fig. 7). The normalized shear strength
prediction, Eq. (5), is compared with the experimental data in
Fig. 8 using the measured yield strength of the parent aluminum
to normalize the results. The experimental results approach the
yield strength predictions as the relative density increases for
both heat treated conditions and orientations. The discrepancy
between predicted and measured yield strengths at low relative
density appears to be a manifestation of manufacturing defects
introduced during fabrication.
The peak strength of lattice structures is determined by their
mechanism of strut failure. This in turn depends on the truss
slenderness (cell geometry), strut material properties (i.e. yield
strength, Young’s and tangent moduli) and the mode of loading. Peak shear strength predictions are also given in Table 3 for
both elastic and inelastic truss buckling. A comparison between
G.W. Kooistra et al. / Materials Science and Engineering A 472 (2008) 242–250
247
Fig. 8. Normalized shear yield strength vs. relative density for tetrahedral lattice structures. Data are shown for the σ +13 orientation in the (a) annealed and (b) aged
hardened condition and for the σ −13 orientation in the (c) annealed and (d) aged hardened condition.
the measured and predicted normalized peak shear strength is
shown in Fig. 9 for both test orientations and heat treatment conditions. The peak shear strength prediction drawn on the figures
assumes inelastic buckling of the compressed truss members for
both node conditions (k = 1 and 2). Deshpande and Fleck have
shown that the in-plane shear strength is periodic for the tetrahedral lattice core and it follows from symmetry conditions that
shear strength in the σ ±13 orientations are equivalent [27]. It
is apparent that the plastic buckling model predicts the higher
relative density response of both the annealed and age hardened lattices reasonably well using a built-in node assumption.
The lower relative density data is better approximated by a pin-
jointed truss-facesheet connection condition consistent with less
end constraint of slender trusses.
6. Discussion
Numerous studies have addressed sandwich panel design for
various lattice truss sandwich structures [41,49–53]. Optimal
designs are ascertained by minimization of an objective function (normally weight) for panel bending, subject to multiple
constraints [41]. These constraints are normally defined by the
operative failure mechanisms: facesheet yielding and buckling
and truss member yielding and buckling. For most practical core
Table 3
Analytical expressions for the in-plane shear stiffness and strength of the tetrahedral lattice [27]
Mechanical property
Analytical expression
Shear stiffness
G = G13 = 18 Es sin2 2ωρ̄
1
G
= sin2 2ω = 0.11, for ω = 54.7◦
Γ =
Es ρ̄
8
σ = σ±13 = 41 σys sin2ωρ̄
1
σ
= sin2ω = 0.24, for ω = 54.7◦
T =
σys ρ̄
4
1
k2 π2 Es t 2
σ = σ±13 = σcr sin2ωρ̄, where σcr =
4
12 l 1
k2 π2 Et t 2
σ = σ±13 = σcr sin2ωρ̄, where σcr =
4
12
l
Normalized shear stiffness
Shear strength (plastic yielding)
Normalized shear strength (plastic yielding)
Shear strength (elastic buckling)
Shear strength (plastic buckling)
(2)
(3)
(4)
(5)
(6)
(7)
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G.W. Kooistra et al. / Materials Science and Engineering A 472 (2008) 242–250
Fig. 9. Normalized shear peak strength vs. relative density for tetrahedral lattice structures. Data are shown for the σ +13 orientation in the (a) annealed and (b) aged
hardened condition and for the σ −13 orientation in the (c) annealed and (d) aged hardened condition.
topologies, optimal sandwich panel designs distribute 17–34%
of the metal mass in the core, depending on topology with core
relative densities <10% and the remainder equally between the
top and bottom facesheets [52].
The panels tested here are not optimized in this formal sense,
but it is still instructive to compare their response with honeycomb cores made from aluminum alloys of similar strength.
The tetrahedral lattice is compared to an aluminum hexagonal
honeycomb, and a (reentrant) flexible honeycomb both available from Hexcel Composites (Stamford, CT, USA) under the
trade names of HEXWEB® and FLEXCORE® . These honeycombs are made from a 5052-H38 aluminum alloy for which
Es = 69 GPa, σ ys = 255 MPa and σ TS = 290 MPa [54]. The ratio
of σ TS /σ ys is an indication of the alloy strain hardening capacity. The ratio σ TS /σ y for 5052-H38 is very similar to that of the
6061-T6, 1.14 and 1.13, respectively. Both alloys also have similar tensile failure strain values (14 and 17%, respectively) in
their high strength conditions. It is therefore possible to assess
the normalized lattice truss stiffness and strengths on a purely
topological basis.
Fig. 10 shows the stiffness of the tetrahedral lattice together
with that of both the regular and flexible honeycomb. The shear
stiffness values in the figure correspond to the minimum shear
stiffness for both topologies. The tetrahedral lattice shear stiffness is nearly equivalent to the hexagonal honeycomb data and
is ∼40% stiffer than that of flexible honeycombs of the same
core relative density. The tetrahedral lattices therefore appear to
be attractive alternatives to conventional and especially flexible
honeycombs, for deflection limited applications.
Fig. 11 shows the normalized peak shear strengths for the
two honeycomb topologies loaded in the direction corresponding to the honeycombs minimum shear strength. The tetrahedral
lattice data is shown for both negative and positive orientations.
The two alloys have similar strain hardening characteristics and
so we conclude that the tetrahedral lattice topology has an efficiency approaching that of a conventional hexagonal honeycomb
Fig. 10. The shear stiffness of the 6061-T6 tetrahedral lattices compared to
commercially available 5052-H38 honeycomb cores.
G.W. Kooistra et al. / Materials Science and Engineering A 472 (2008) 242–250
249
under grant number N00014-01-1-1051. The authors are especially grateful to Vikram Deshpande (Cambridge University) for
many helpful discussions.
References
Fig. 11. The normalized shear peak strength of the 6061-T6 tetrahedral lattice
truss compared to commercially available 5052-H38 honeycomb.
and appears significantly superior to flexible honeycomb structures. Recalling the earlier study’s conclusion of comparable
compressive strengths for the two topologies, we conclude that
tetrahedral lattice structures appear to be promising alternatives
for load supporting sandwich panel structures, especially when
the structures posses significant curvature.
7. Conclusions
• Tetrahedral lattice cores have been made by folding hexagonally perforated 6061 aluminum alloy sheets. A simple open
air furnace brazing technique has then been used to metallurgically bond the tetrahedral cores to 6951 solid facesheets to
form sandwich panel structures.
• The shear stiffness and strength was experimentally measured
and their behavior was adequately captured by Deshpande
and Fleck’s analytical expressions for modulus and strength
of tetrahedral lattice truss structures.
• The structural performance of the 6061 tetrahedral lattices
lie within in the upper and lower bounds of conventional
hexagonal honeycomb shear stiffness and are competitive
with honeycomb topologies when loaded in the minimum
shear strength directions.
• The tetrahedral lattice core is highly flexible prior to bonding
to facesheets and it is more than 40% stiffer and stronger than
equivalent relative density flexible honeycombs used for the
cores of curved sandwich panels.
• The tetrahedral lattice structure offers many ancillary benefits
to traditional closed-celled honeycombs all at an equivalent
specific stiffness and strength while possessing the formability of flexible honeycomb topologies.
Acknowledgements
This work was supported by the Office of Naval Research
(ONR), monitored by Drs. Steve Fishman and David Shifler
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