Acta Materialia 54 (2006) 5509–5518
www.actamat-journals.com
Structural performance of metallic sandwich beams
with hollow truss cores
H.J. Rathbun a, F.W. Zok a,*, S.A. Waltner a, C. Mercer a, A.G. Evans a,
D.T. Queheillalt b, H.N.G. Wadley b
b
a
Materials Department, University of California, Santa Barbara, CA 93106, USA
Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA
Received 25 April 2006; accepted 21 July 2006
Available online 5 October 2006
Abstract
The article focuses on the structural performance of sandwich beams with hollow truss lattice cores made from a ductile stainless steel.
The trusses are arranged in an orthogonal (cross-ply) configuration, in either ±45 (diamond) or 0/90 (square) orientations with respect
to the face sheets. The responses in shear, tension and compression, as well as simply supported and fully clamped bending, are measured
for specimens with both core orientations. While the two cores perform equally well in compression, the diamond orientation exhibits
higher shear strength but lower stretch resistance. For bend-dominated loadings of the sandwich beams, the core in the diamond orientation is preferred because of its higher shear strength. For stretch-dominated loadings encountered in large-displacement, fully clamped
bending, the square orientation is superior. Models of core and beam yielding are used to rationalize these observations. Optimizations
are then performed to identify strong lightweight designs and to enable performance comparisons with other sandwich structures.
2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Sandwich beams; Lightweight structures; Hollow tube core
1. Introduction
Metallic sandwich panels with various honeycomb, lattice truss and prismatic cores are being developed for structures that require high strength [1–5] and blast resistance
[6–9]. Some topological configurations are amenable to
additional functionality. For instance, shape morphing of
the structure can be achieved via actuation of core members
[10–12]. Others can be designed for thermal management,
through forced flow of fluids through the core. In a typical
application, high heat fluxes are deposited onto one of the
face sheets of a sandwich panel; the heat is conducted away
from the faces via the core elements and subsequently
removed by a coolant flowing through the core [13,14].
Any open cell metallic structure that allows coolant flow
can be used as a heat exchange medium. Through appropri*
Corresponding author. Tel.: +1 805 893 8699; fax: +1 805 893 8486.
E-mail address: [email protected] (F.W. Zok).
ate selection of material and core topology, these structures
can be designed to possess high thermal conductivity and
efficiently transfer heat to the cooling fluid [15,16].
One approach envisioned to further increase the thermal
performance of sandwich structures involves the use of
heat pipes as the truss members within the core [14,17].
Heat pipes increase the performance by homogenizing the
temperature distribution and increasing the average temperature difference at the interface between the external
pipe surface and the coolant. Additionally, because of their
high second moment of area, such pipes offer superior
buckling resistance relative to solid members of equivalent
mass [17,18]. Consequently, the cores can be designed with
enhanced structural efficiency, especially in the domain of
low core relative density.
The present study explores the structural characteristics
of metallic sandwich beams with hollow truss lattice cores.
Using recently proposed methods for the manufacture of
hollow truss cores and sandwich panels that utilize them
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2006.07.016
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H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518
Nomenclature
B
E
F
Fs
Hc
Heq
It
‘
‘0
Lc
M
Mt
r
R
S
tf
V
a
b
beam width
elastic modulus
tensile force
shear force
core thickness
thickness of a solid plate of equivalent mass
second moment of area of hollow tube
ratio of bending moment to shear force
(‘ ” M/V)
characteristic length
center-to-center tube spacing
bending moment applied to sandwich structure
bending moment in hollow tube
inner tube radius
outer tube radius
loading span
face sheet thickness
shear force applied to sandwich structure
fractional change in flow stress following
brazing
non-dimensional core strength
[17], the article focuses on plastic deformation of simply
supported and fully clamped bending beams with orthogonal (‘‘cross-ply’’) core architectures, in ±45 (diamond) and
±0/90 (square) loading orientations. Mechanical performance differences are rationalized on the basis of the core
properties in the two orientations, as well as the prevailing
core stresses in the beams. Models are used in parallel with
the experiments to glean insights into the connections
between core topology, constituent material properties
and beam performance. Finally, optimizations are performed to identify strong, lightweight sandwich panel
designs with hollow truss cores. Their performance is
shown to compare favorably with optimized panels with
square honeycomb cores.
2. Experimental procedures
2.1. Sandwich fabrication
Hollow truss cores were made from medical grade 304
stainless steel tubes (Vita Needle, Needham, MA), manufactured by a tungsten inert gas welding and plug drawing
process. The inner and outer tube radii were r = 0.61 mm
and R = 0.74 mm, respectively. In preparation for brazing,
each tube was sprayed with a mixture of a polymer-based
cement (Nicrobraz Cement 520) and a Ni–25%Cr–10%P–
0.03%C braze powder (Nicrobraz 51), both supplied by
Wall Colmonoy (Madison Heights, MI). The phosphorous
in this alloy acts as a melting point depressant and
enhances the flow and wetting characteristics of the molten
d
D
ey
c
P
P0
qc
R
rc
core
r
f
r
s
r
rt
ry
s
sp
sp
sy
W
W0
bending displacement
normalized displacement
yield strain
shear strain
non-dimensional load
load capacity at ‘ = ‘0
core relative density
normalized load
core compressive strength
tensile flow strength of core
tensile flow strength of heat-treated (not brazed)
parent alloy
panel tensile strength
maximum bending stress in hollow tube
yield strength
Shear stress
fully plastic shear strength
non-dimensional fully plastic shear strength
non-dimensional shear yield stress
non-dimensional weight index
weight of sandwich beam at ‘ = ‘0
braze. The braze alloy has a solidus of 880 C and a
liquidus of 950 C, with a suggested brazing range of
980–1095 C. Once coated, the tubes were stacked in an
orthogonal (‘‘cross-ply’’) pattern to the desired height
within an alignment tool (Fig. 1a). Alignment was achieved
by a set of uniformly spaced stainless steel dowel pins
inserted into holes in a stainless steel base plate. The dowel
spacing was selected to produce a center-to-center tube
spacing, Lc, of 5 mm. The alignment fixture was spray
coated first with Nicrobraz Green Stop-Off (Wall Colmonoy, Madison Heights, MI) and then boron nitride (GE
Advanced Ceramics, Lakewood, OH) [19].
The tube assembly was placed in a vacuum furnace
(Super VII, Centorr Vacuum Industries, Nashua, NH)
for brazing. The furnace was heated at 10 C/min to
550 C and held at temperature for 20 min to volatilize
and remove the polymer binder. Thereafter, it was heated
to 1020 C at 104 torr, held for 60 min and cooled to
ambient temperature at 25 C/min. After brazing, the
cores were removed from the tooling and cut to size (nominally 18 mm thick) using wire electro-discharge machining
(Fig. 1(b)). The measured core relative density, qc, was
7.5%. By comparison, the calculated value (neglecting the
added weight of the braze alloy) was 7.3% [15].
A second brazing operation was used to attach the
lattice cores to 304 stainless steel face sheets to produce
sandwich beams (Fig. 1(c)). For most cases, the face
sheet thickness, tf = 0.7 ± 0.1 mm. For measurement of
the core properties in compression, the face sheets were
thicker (tf = 2.5 ± 0.2 mm), to ensure that they remained
H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518
5511
2.2. Mechanical testing
The mechanical properties of the cores and the sandwich
beams were measured using standard compression, tension,
shear and bending tests [4,5,8,17,18]. Pertinent details are
presented below. In addition, for comparison with the measured strengths of the sandwich beams and assessment of
mechanical models, the uniaxial tensile properties of the
304 stainless steel face sheet material were measured following a heating cycle identical to that used for brazing.
Tests were performed in accordance with ASTM E8-01,
at a nominal strain rate of 103 s1. The elastic modulus,
E, and yield strength, ry, were 200 GPa and 200 MPa,
respectively, with a yield strain ey = ry/E = 0.001.
Despite the measurements on the appropriately heattreated parent alloy, some uncertainty in the in situ material properties in the sandwich structures remains because
of the opportunity for chemical interaction between the
alloy and the braze materials during fabrication. While previous studies have revealed minimal changes in the initial
yield strength, they indicate significant elevations in the
hardening rate and flow stress of 304 stainless steel after
exposure to similar braze alloys [5]. Some insights into
the magnitude of such effects in the present sandwich specimens are gleaned from an analysis of the tensile test
results, presented in Section 3.
3. Core properties
Effects of tube architecture (±45 vs. 0/90) on the core
properties were ascertained for three loading modes: inplane tension, transverse (out-of-plane) compression and
transverse shear.
3.1. Tension
Fig. 1. Schematic of the procedure used to fabricate the ±45 hollow truss
core sandwich beams. An analogous process was used for fabricating cores
in the 0/90 orientation.
undeformed during transverse compressive testing. Even
thicker faces (tf = 12.7 ± 0.2 mm) were attached to lap
shear specimens that were used for measuring the shear
response of the cores. To enable gripping in the clamped
bending and uniaxial tension tests, solid steel inserts were
placed between and brazed to the face sheets at the two
ends of the beams. The assembly (core, face sheets and
inserts) was placed in a clamping fixture. To prevent
bonding to the fixture, the fixture surfaces were sprayed
with Stop-Off and boron nitride. In addition, a 3 mm
thick layer of high-purity alumina–silica fiber paper was
placed between the face sheets and the adjacent fixture
surfaces. Moderate clamping pressure was used to maintain intimate contact between the faces and the core during brazing. The heating cycle was the same as that used
to produce the cores. Examples of representative test
specimens in both diamond and square orientations are
shown in Fig. 2.
Experimental results from uniaxial tension tests are plotted in Fig. 3 for sandwich specimens with both core orientations, as well as the heat-treated 304 alloy itself. The
relevant tensile stress is the load, F, divided by the areal
density, BHeq, where B is the width and Heq is the thickness
of a solid plate of equivalent mass. On this basis, the 304
alloy has the greatest stretch resistance, followed by the
sandwich beam in the 0/90 orientation; the weakest
response occurs in the ±45 orientation.
The tensile flow stress of a sandwich panel can be partitioned between the core and its face sheets, each weighted
by its respective effective thickness [4]. Making a correction
for the change in the flow stress of the steel following brazing, the resulting panel tensile strength becomes:
F
2tf
s ¼
r
¼ ð1 þ aÞ
rf ð1 bÞ
þb
ð1aÞ
BH eq
H eq
where b is a non-dimensional parameter characterizing
core strength, defined by
b
core
r
f
qc r
ð1bÞ
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Fig. 2. Representative test specimens with cores in (a) ±45 and (b) 0/90 orientations.
that after an equivalent heat treatment), the work hardening rate at small strains is elevated and the flow stress at
larger strains is about 10–30% higher than that of the pristine alloy. The inference from these comparisons is that the
in situ yield strength of the alloy is only slightly altered by
brazing. This result is utilized in subsequent comparisons
between experiment and theory. A second key conclusion
is that the stretch resistance of the 0/90 core (b = 0.5) is
indeed much higher than that of the ±45 core (b 0).
3.2. Compression
f represent the tensile flow strengths of the core
core and r
r
and the heat-treated (but not brazed) parent alloy, respectively, at a prescribed strain, and a is the fractional change
in the flow stress of the steel following brazing. Established
mechanical models for the two core types yield core
strength values b = 0.5 and b = 0 for the 0/90 and ±45
cores, respectively [2].
Predictions of Eq. (1) are plotted in Fig. 3. When the
material is assumed to be unaffected by the braze (a = 0),
the predicted yield strengths agree well with the measurements (using the appropriate value of b). However, at small
plastic strains (up to about 0.5%) the work hardening rates
are underestimated. Thereafter, at larger strains, the hardening rates are essentially the same. Conversely, upon
selecting a = 0.15, the predicted and measured curves at
large strains (>0.5%) agree well with one another, but the
yield strengths are overestimated. These observations are
the same as the braze effects reported in earlier studies
[5]. That is, following brazing, the yield strength of 304
stainless steel remains essentially unchanged (relative to
1.0
Compressive stress, σc/ρcσY
Fig. 3. Tensile response of both sandwich specimens and monolithic 304
stainless steel.
The compression test results are presented on Fig. 4. For
both orientations, yielding initiates at a normalized stress
rc/qc 0.5, the same as that predicted from stress analyses
of both cores [2,17]. Following yield, both cores strain
harden in the small plastic strain domain, ultimately reaching a peak when the tubes begin to plastically buckle. The
flow strength achieves a plateau shortly thereafter, at
rc/qcry 0.4 0.5, during which the core members
0.8
0Þ/90Þ
±45Þ
0.6
0.4
Predicted yield stress
0.2
0.0
0
10
20
30
40
50
Compressive plastic strain (%)
Fig. 4. Compressive response of the hollow truss cores.
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H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518
5513
progressively crush. Significant core densification begins at
40–50% compressive strain, marked by a rapid increase in
hardening. In contrast to the core properties under tensile
loading, the compressive properties are essentially independent of orientation.
3.3. Shear
Representative stress-strain curves from the lap shear
specimens are shown in Fig. 5. In the ±45 orientation,
yield initiates at a shear stress, sy/qcry 0.5: again consistent with the analytic prediction [3]. Following subsequent
strain hardening, the stress drops rapidly as the tubes begin
to separate from the face sheets. The low failure strain of
the ±45 orientation (c 2%) is associated with node failure at the tube-face sheet interface. Alternate brazes [4,5]
and improved node joint designs are expected to produce
more robust nodes. In the 0/90 orientation, the response
is much softer, with a yield strength, sy/qcry 0.1. The
subsequent plastic response is characterized by moderate
hardening. The nodes remain intact over the strain range
probed by these experiments (c 20%), because the low
strength of the core ensures that a correspondingly low
stress is transmitted to the face/tube interfaces.
To model the shear strength of the 0/90 core, the tubes
perpendicular to the loading direction are assumed to act
as beams, rigidly affixed to both the faces and the longitudinal tubes. An analysis of a unit cell of the tube array
(Fig. 6) yields the shear stress, s, carried by a single tube:
s¼
Fs
4Lc R
Fig. 6. Schematic of the 0/90 tube array and the stress analysis of an
individual tube member under shear loading.
Here, Lc 2R represents the edge-to-edge spacing of the
tubes, as illustrated in Fig. 6. Yielding initiates when the
maximum bending stress, rt, in the tube reaches the material yield strength, ry, That is,
rt ¼
F s ðLc 2RÞ
Mt ¼
2
ð3Þ
sy ¼
Normalized shear stress,τ/σyρc
ð5Þ
This result can be re-expressed in a non-dimensional form
as:
sy sY
R2 þ r 2
¼
qc rY 4ðLc 2RÞR
ð6Þ
where the core relative density is given by [17]:
qc ¼
s ¼
0.8
±45˚
0.6
Predicted yield
0.4
0°/90°
0.2
Predicted yield
0.02
rY pðR4 r4 Þ
8Lc ðLc 2RÞR2
pðR2 r2 Þ
2Lc R
ð7Þ
Following similar procedures, the shear stress sp needed for
the formation of a fully plastic hinge at each node is also
obtained:
1.0
0.0
0.00
ð4Þ
where It is the second moment of area of the hollow tube
about the neutral axis. Combining Eqs. (2)–(4) gives the
applied stress at yield initiation:
ð2Þ
where Fs is the applied shear force. The resulting bending
moment, Mt, acting on each tube is obtained from static
equilibrium, whereupon
M tR
4M t R
¼ ry
¼
It
pðR4 r4 Þ
0.04
Predicted limit
0.06
0.08
0.10
Shear strain, γ
Fig. 5. Shear response of hollow truss cores.
0.12
sp
2R
¼
qc rY pðLc 2RÞ
ð8Þ
Using dimensions pertinent to the present cores (R = 0.74
mm, r = 0.61 mm, Lc = 5 mm), the normalized strengths
are sY ¼ 0:09 and sY ¼ 0:13 for the 0/90 core. These results compare favorably with the experimental measurements (Fig. 5). The slight discrepancy between the
predicted fully plastic shear strength and the measured flow
strength at large strains can be attributed to strain hardening: a feature neglected in the analytical model.
To summarize, while the compressive properties of the
two cores are essentially the same, the core in the 0/90
orientation exhibits lower shear strength (by a factor of
5) but significant stretch resistance (b = 0.5). These differences manifest themselves in the properties of the sandwich
beams, as detailed below.
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H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518
4. Sandwich properties: simply supported bending
4.1. Experiments
To assess the mechanical performance of the sandwich
beams, measurements have been made using three-point
simply supported bend tests. The outer loading points consisted of 25.4 mm diameter hardened steel pins on a flat
support base. To inhibit local indentation, the inner loading platen comprised a flat central region, 12.7 mm wide,
with adjacent filets 6 mm radius. The loading span was
S = 184 mm. For comparison, tests were also performed
on monolithic beams of equivalent areal density.
The results are plotted in Fig. 7. The loads and displacements have been normalized by those needed to initiate
yielding in the equivalent-weight solid beam; the ensuing
normalized load and displacement (defined in the figure)
are denoted R and D, respectively. On this basis, it is evident that the strengths of the sandwich beams are much
greater than that of the solid beams by about an order of
magnitude. Additionally, in the plastic domain, beams with
the ±45 core are also almost twice as strong as those with
the 0/90 core. The latter differences arise from the operative failure modes, elaborated below. With the ±45 core,
large deflections are obtained without deformation localization (Fig. 8a). In this case, the test was terminated once
the outer loading pins had experienced significant lateral
displacement. The beams with the 0/90 cores exhibited
similarly large plastic deflection, but ultimately failed by
shear of the core beneath the inner loading platen (Fig. 8b).
4.2. Failure model: ±45 core
Analytic models for failure initiation of sandwich beams
with lattice cores have been developed previously [2,3],
using stress analyses of the faces and the core elements subject to a combination of transverse shear, V, and bending
Normalized load, Σ = 3FS/2BHeq 2σy
30
±45˚
moment, M (both per unit width). These models have been
adapted here for hollow truss cores. In the current implementation, the core is assumed to carry all of the shear
and the faces all of the moment. The key results are summarized below.
Failure in the face sheets can occur by either yielding or
elastic buckling. The critical loads can be described by a
non-dimensional parameter, defined as P ” V2/EM. For
beams with ±45 cores [3]:
ey t f H c
ðface sheet yieldingÞ
‘2
p2 t3f H c
P¼
ðface sheet bucklingÞ
24ð1 m2 Þ‘2 L2c
P¼
ð9Þ
ð10Þ
where Hc is the core thickness and ‘ ” M/V. The latter
quantity scales with loading span: ‘ = S/2 for simply supported three-point bending and ‘ = S/4 for clamped ends.
Similarly, the tubes in the core can fail by either yielding
or buckling. Here thes nodes at which the tube members
intersect are assumed to be pinned such that the adjoining
tube segments are free to rotate about these nodes. The
pertinent failure loads are:
P¼
pey RH c ½1 ðr=RÞ2 4Lc ‘
P¼
p3 H c R3 ½1 ðr=RÞ 16L3c ‘
ðcore yieldingÞ
ð11Þ
ðcore bucklingÞ
ð12Þ
4
Beam failure is dictated by the mode with the lowest value
of P.
For three-point bending, wherein ‘ = S/2, the load
parameter P is related to the non-dimensional quantity R
presented in Fig. 7 via the relation:
!
3S 2
R¼
P
ð13Þ
2H 2eq ey
For the present experimental configuration, the preceding models predict failure by face yielding, at a normalized
load of P = 1.3 · 106 or, equivalently, R = 10. This result
correlates well with the onset of non-linearity in the experimental measurements, shown in Fig. 7.
Onset of
roller movement
20
0˚/90˚
10
4.3. Failure model: 0/90 core
Onset of core
indentation
The most significant modification to the preceding
results (Eqs. (9)–(12)) for beams with 0/90 cores pertains
to the stress needed for core yielding. Utilizing the model
developed in Section 3 for the core shear strength, the critical load becomes:
Predicted yield points
Solid sheet
P¼
0
0
5
10
15
2
Normalized displacement, Δ = 6δHeq/S εy
Fig. 7. Comparison of simply supported bending response of the
sandwich beams with that of the equivalent weight monolithic material.
pey H c ðR4 r4 Þ
8‘Lc ðLc 2RÞR2
ð14Þ
This mode dominates in the present experiments, the loads
for face failure and core buckling being significantly higher.
The resulting critical load is R = 8. This result also agrees
favorably with the onset of non-linearity (Fig. 7).
H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518
5515
Fig. 8. Deformation of sandwich beams in simply supported bending.
5. Sandwich properties: clamped bending
Additional bend tests have been performed using fully
clamped end conditions. Apart from the boundary conditions, all testing parameters were the same. The resulting
load/displacement curves are summarized in Fig. 9 and
images of the deformed specimens are in Fig. 10. At small
displacements (d/S < 0.05), wherein the response is benddominated, the curves mimic those obtained under simply
supported conditions (Fig. 7). That is, the beam with the
±45 core outperforms that with the 0/90 core. Conversely, at larger displacements, membrane stresses arise
and the response becomes stretch-dominated. In this
domain, the strength rankings reverse; the 0/90 core
exhibits a higher flow stress than the ±45 core because
of its superior stretch resistance (Fig. 4). Moreover, when
stretch-dominated, the monolithic sheet becomes the strongest, also consistent with the results in Fig. 4.
Also shown in Fig. 9 are rudimentary predictions of the
response of the sandwich beams in the stretch-dominated
domain. These are obtained by scaling the clamped bending results of the monolithic sheet in accordance with Eq.
(1). The core strengths are again taken to b = 0 and
b = 0.5 for the ±45 and 0/90 cores, respectively, and
a = 0.15. The resulting curves agree reasonably well with
the measurements in the domain d/S > 0.1. This correlation
reaffirms that the change in strength ranking of the two
cores is a result of their respective stretch resistance.
6. Optimal designs for bending
Further assessment of the hollow truss core sandwich
beams has been made through comparison of their bending
strength with those that have been optimally designed.
Among the two hollow truss cores, only that in the ±45
orientation is considered, because of its high shear
strength. Comparisons are also made with optimal honeycomb core sandwich panels [20].
The objective of the optimization is to find the geometric
parameters of sandwich beams that can support a prescribed bending load, P, at minimum weight. The pertinent
non-dimensional weight index is [1]:
W¼
Fig. 9. Results of clamped bending experiments on both sandwich beams
and a solid sheet of equivalent mass.
W
q‘
ð15Þ
where W is the weight per unit area and q the mass density
of the solid material. From geometry, this index can be expressed as:
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Fig. 10. Deformation of sandwich beams under clamped bending. Arrows indicate locations of core separation from face sheets.
2
2tf pRH c ð1 ðr=RÞ Þ
þ
ð16Þ
2Lc ‘
‘
Each of the critical loads for the four possible failure
modes (Eqs. (9)–(12)) represents an optimization constraint. Additionally, to ensure that the designs reside in
the domain of thin beams, the core thickness is restricted
to remain below a critical value: Hc/‘ 6 0.2.
Solutions for the optimal designs have been obtained
using two complementary methods. In the first, the design
objective and the constraints were coded in an IMSL subroutine that performs the optimization numerically [1,2].
Results from the calculations show that the optima occur
at the confluence of either two or three constraints. With
knowledge of the active constraints, analytic solutions have
been derived using standard mathematical procedures.
These procedures and the resulting solutions for the hollow
truss core sandwich beams are presented in the Appendix.
Post-yield and post-buckling behaviors are not considered
here.
Variations in the non-dimensional weight W with load P
are plotted in Fig. 11. Results include those for optimized
hollow truss core and honeycomb core beams [20], as well
as a solid beam of the same material. The weight of the latter is given by [3]:
1=2
6P
W¼
ð17Þ
ey
W¼
Additionally, results for the present sandwich beams with
the ±45 hollow tube core are shown. Since failure of these
beams occurs by face yielding, Eqs. (9) and (16) yield scalings of the forms:
‘0
W ¼ W0
ð18Þ
‘
Fig. 11. Comparisons of the bend strengths of the present sandwich
beams with ±45 cores with both solid and optimized sandwich beams.
2
‘0
ð19Þ
P ¼ P0
‘
where W0 and P0 are the weight and the load capacity of
the sandwich beam at a characteristic length, ‘ = ‘0. The
latter is selected to correspond to the simply supported
three-point bend test conditions (with ‘0 = S/2 = 92 mm),
whereupon W0 = 0.029 and P0 = 1.34 · 106. Combining
Eqs. (18) and (19) (to eliminate ‘) yields:
1=2
P
W ¼ W0
ð20Þ
P0
Numerical values have been calculated for 0.009 6 W 6
0.03. This weight range was obtained by varying the characteristic length over 90 mm 6‘6 300 mm. Since the absolute core thickness remains fixed (Hc = 17 mm), its
normalized value falls in the range 0.05 6 Hc/‘ 6 0.2. Face
yielding dominates throughout.
H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518
The comparisons in Fig. 11 reveal that the present sandwich beams are far superior to the solid beams: their
strengths differing by an order of magnitude at prescribed
weight. Additionally, the beam strengths are within a factor of 2 of those with the optimal hollow truss core designs
at high loads, with the difference increasing as the load
decreases. The geometric parameters needed to achieve
higher strengths can be obtained from the solutions in
the Appendix.
The results also demonstrate that the hollow truss
core sandwiches can be designed to be stronger than
those with square honeycomb cores. However, the present truss designs only support load effectively in one
plane whereas the honeycomb cores are almost isotropic.
This deficiency could be mediated by using alternate
truss configurations, e.g. pyramidal or tetrahedral [1,2].
Indeed, methods for fabricating such cores have recently
been demonstrated [18]. More importantly, the hollow
truss structures can be used as both heat pipes and heat
exchange media within a sandwich panel. The closed cell
structure of the honeycomb core precludes its use in
these applications.
7. Summary
Metallic sandwich beams with hollow tube cores have
been fabricated in a two-step brazing process. The
responses in shear, tension, compression, as well as simply
supported and fully clamped bending, have been measured
for both diamond (±45) and square (0/90) orientations.
Among the two cores, the diamond orientation exhibits the
higher shear strength but lower stretch resistance. The two
cores perform equally well in compression. For bend-dominated loadings, wherein substantial core shear arises, the
diamond orientation is preferred. For stretch-dominated
behavior occurring at large displacements in fully clamped
tests, the square orientation is superior.
Optimizations have been performed to identify strong
lightweight sandwich designs with both hollow truss and
square honeycomb cores. Although somewhat suboptimal, the present hollow truss sandwich beams are far
superior to solid beams of equivalent weight. With some
modifications in truss topology and dimensions, exceptional strengths could be achieved. Combined with their
adaptability for efficient heat exchange, the hollow truss
core sandwiches offer outstanding potential for use as lightweight thermostructural components.
Acknowledgements
This work was supported by the Office of Naval Research through a contract to the University of Virginia
(N0014-01-1-1051) and a sub-contract to the University
of California, Santa Barbara (GG10376-114969), monitored by Dr Steve Fishman. The authors are grateful to
Professors Vikram Deshpande and Norman Fleck (Cambridge University, UK) for helpful discussions.
5517
Appendix A. Optimization of hollow truss core sandwich
beams
A.1. Preliminaries
Numerical results from the IMSL subroutine reveal
three solution domains, distinguished by the load parameter, P. Henceforth, the load domains are denoted low,
intermediate and high. In each, either two or three design
constraints are active. At low loads, the active constraints
are face yielding (FY) and core buckling (CB). The transition to the intermediate load domain occurs when the
core thickness reaches its limit: Hc/‘ = 0.2. In this
domain, face yielding and core buckling remain active.
At high loads, core yielding replaces core buckling,
whereas face yielding and the core thickness limit remain.
With knowledge of the active constraints, analytic solutions are obtained by combining the constraint functions,
as described below.
A.2. Low load domain
Upon combining the analytic solutions for the active
constraints in this domain (Eqs. (9) and (12)) with that
for the weight (Eq. (16)), tf/‘ and Hc/‘ are eliminated,
allowing W to be written in terms of two independent geometric variables, notably Lc/R and r/R:
4
W¼
p3 ð1 ðr=RÞ Þ
8ey ðLc =RÞ
3
þ
8ðLc =RÞ
2
2
p2 ð1 þ ðr=RÞ Þ
P
ðA1Þ
Over the entire physically accessible range 0 6 r/R < 1, oW/
o(r/R) < 0 and hence the optimum occurs as r/R ! 1.
However, in this limit, the walls become susceptible to
short wavelength buckling. To ensure stability against this
mode whilst retaining high resistance to long wavelength
buckling, the optimal value of r/R is taken to be 0.9 [21].
Furthermore, setting oW/o(Lc/R) = 0 yields:
"
#1=5
4
2
Lc
3p5 ð1 ðr=RÞ Þð1 þ ðr=RÞ Þ
¼
P1=5
ðA2Þ
128ey
R
With Lc/R and r/R known, Hc/‘, tf/‘ and W are readily
obtained:
"
#1=5
Hc
27ð1 þ ðr=RÞ2 Þ3
¼
P2=5
ðA3Þ
4 2
‘
2e3y ð1 ðr=RÞ Þ
!1=5
tf
2ð1 ðr=RÞ4 Þ2
¼
P3=5
ðA4Þ
2 3
‘
27e2y ð1 þ ðr=RÞ Þ
!1=5
463ð1 ðr=RÞ2 Þ2
P3=5
ðA5Þ
W¼
2
2e2y ð1 þ ðr=RÞ Þ
Although W does not depend on Lc/‘, the latter cannot exceed a critical value, dictated by the face buckling constraint. The limiting value is:
5518
Lc
6
‘
H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518
p2
24ð1 m2 Þ
4 4
4ð1 ðr=RÞ Þ
2 6
729e9y ð1 þ ðr=RÞ Þ
!1=5
From Eq. (9), the weight is:
P6=5
ðA6Þ
A.3. Intermediate load domain
The transition to the intermediate load domain occurs
when the core thickness reaches its maximum allowable
value, Hc/‘ = 0.2. The corresponding load is given by:
"
#
2 2 1=2
3
5=2 2ey ð1 ðr=RÞ Þ
ðA7Þ
Ptr ¼ ð0:2Þ
27ð1 þ ðr=RÞ2 Þ
With the core thickness fixed, the face sheet thickness is
immediately obtained from Eq. (9):
tf
5
¼
P
ðA8Þ
ey
‘
As in the low load domain, r/R is taken to be 0.9. Then,
from Eq. (12) (the core buckling constraint) and Eq. (A1):
!1=3
4
Lc
p3 ð1 ðr=RÞ Þ
¼
P1=3
ðA9Þ
80
R
"
#
2 2 1=3
10
2ð1 ðr=RÞ Þ
P1=3
ðA10Þ
W¼ Pþ
2
ey
25ð1 þ ðr=RÞ Þ
An upper allowable limit on Lc/‘ is again obtained (to prevent face buckling), now given by:
!1=2
Lc
p2
65
P
ðA11Þ
‘
24ð1 m2 Þe3y
A.4. High load domain
The transition to the high load domain occurs when core
yielding replaces core buckling. Throughout, the face yielding constraint and the limiting core thickness Hc/‘ = 0.2
remain active. The transition load is:
!1=2
!
e3y
1 ðr=RÞ2
Ptr ¼
ðA12Þ
2
10
ð1 þ ðr=RÞ Þ
W¼
12
P
ey
ðA13Þ
In this domain, W is independent of Lc/‘, Lc/R and r/R,
although the upper limit established on Lc/‘ for the intermediate load domain (Eq. (11)) remains valid.
References
[1] Wicks N, Hutchinson JW. Int J Solids Struct 2001;38:5165.
[2] Rathbun HJ, Zok FW, Evans AG. Int J Solids Struct 2005;42:
6643.
[3] Zok FW, Rathbun HJ, Wei Z, Evans AG. Int J Solids Struct
2003;40:5707.
[4] Zok FW, Waltner SA, Wei Z, Rathbun HJ, McMeeking RM, Evans
AG. Int J Solids Struct 2004;41:6249.
[5] Zok FW, Rathbun HJ, He MY, Ferri E, Mercer C, McMeeking RM,
et al. Phil Mag 2005;85:3207.
[6] Qiu X, Deshpande VS, Fleck NA. Eur J Mech – A/Solids
2003;22:801.
[7] Fleck NA, Deshpande VS. J Appl Mech 2004;71:386.
[8] Rathbun HJ, Wei Z, He MY, Zok FW, Evans AG, Sypeck DJ, et al.
J Appl Mech 2004;71:368.
[9] Xue Z, Hutchinson JW. Int J Impact Eng 2004;30:1283.
[10] Lu TJ, Hutchinson JW, Evans AG. J Mech Phys Solids 2001;49:
2071.
[11] Hutchinson RG, Wicks N, Evans AG, Fleck NA, Hutchinson JW. Int
J Solids Struct 2003;40:6969.
[12] Lucato SLDSE, Wang J, Maxwell P, McMeeking RM, Evans AG.
Int J Solids Struct 2004;41:3521.
[13] Evans AG, Hutchinson JW, Fleck NA, Ashby MF, Wadley HNG.
Prog Mater Sci 2001;46:309.
[14] Tian J, Kim T, Lu TJ, Hodson HP, Queheillalt DT, Sypeck DJ, et al.
Int J Heat Mass Transfer 2004;47:3171.
[15] Boomsa K, Poulikakos D. Int J Heat and Mass Transfer 2001;
44:827.
[16] Tian J, Kim T, Lu TJ, Hodson HP, Queheillalt DT, Sypeck DJ, et al.
Int J Heat Mass Transfer 2004;47:3171.
[17] Queheillalt DT, Wadley HNG. Acta Mater 2005;53:303.
[18] Queheillalt DT, Wadley HNG. Mater Sci Eng 2005;A397:
132.
[19] NICROBRAZ technical data sheet, Wall Colmonoy Corporation,
Madison Heights, MI, 1995.
[20] Rathbun HJ, Zok FW, Evans AG, in press.
[21] Roark RJ, Young WC. Formulas for stress and strain. Fifth ed.
McGraw-Hill Book Company; 1975.