Materials Sciences and Applications, 2013, 4, 471-477 Published Online August 2013 (
Copyright © 2013 SciRes. MSA
A Comparison of Bending Properties for Cellular Core
Sandwich Panels
Li Yang1, Ola Harrysson2, Harvey West2, Denis Cormier3
1Department of Industrial Engineering, University of Louisville, Louisville, USA; 2Department of Industrial & Systems Engineering,
North Carolina State University, Raleigh, USA; 3Department of Industrial & Systems Engineering, Rochester Institute of Technology,
Rochester, USA.
Received May 15th, 2013; revised June 21st, 2013; accepted July 2nd, 2013
Copyright © 2013 Li Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, various sandwich panel structures with different reticulate lattice core geometries were designed and then
fabricated in titanium via the electron beam melting (EBM) process. Bending tests were performed on the titanium sam-
ples, and mechanical properties such as modulus, bending strength, and energy absorption were evaluated. Different
failure mechanisms were observed, and it was found that sandwich structures with auxetic cores exhibited more homo-
geneous deflection and bending compliance compared with other structures. It was also demonstrated that properties of
auxetic sandwich structures can be tailored using different cell structure geometries to suit the needs of a given design
application. Furthermore, it was found that other 3D cellular sandwich structures can also exhibit high stiffness and
strength, which is desirable in potential applications.
Keywords: Auxetic Structure; Cellular Structure; Sandwich Panels; Electron Beam Melting; Bending Properties
1. Introduction
3D cellular structures possess unique advantages with
respect to specific strength, specific modulus, and energy
absorption at low densities. They are therefore promising
for applications that require light weight (e.g. aerospace,
automotive, etc.). Two-dimensional cellular structures
and foams are frequently used as sandwich structure
cores in order to provide increased bending and shearing
stiffness as well as energy absorption ability [1].
Among the range of possible cellular geometries, aux-
etic structures are of specific interest to researchers. Aux-
etic structures exhibit negative Poisson’s ratio in one or
more directions. These structures have been shown to
exhibit significantly improved shear performance com-
pared with regular structures [2-4]. According to the the-
ory of elasticity, the shear modulus of negative Poisson’s
ratio structures could become even larger than the bulk
moduli, making the structures ideal for use in sandwich
panel cores [5-7]. Furthermore, auxetic structures exhibit
synclastic bending [8-9], which also favors their potential
applications in curved sandwich panels and sandwich
skins for various applications.
Traditionally, auxetic structures have been fabricated
through multi-step processes in which specific control
over the cell geometry (i.e. strut sizes and angles) is quite
difficult. The relatively recent emergence of polymer and
metal additive manufacturing processes has given engi-
neers the ability to fabricate parts with precise cellular
geometries directly from the CAD models. It therefore
has become possible to fabricate these structures and to
compare their experimentally determined material prop-
erties with predicted values.
In the current study, sandwich structures with re-en-
trant auxetic cores as well as several other 3D reticulate
cellular core geometries were designed in CAD and then
fabricated via Arcam’s electron beam melting (EBM)
process. Mechanical properties pertinent to sandwich
structure performance were measured, including stiffness,
strength and energy absorption. A comparison of the
bending properties between different auxetic core de-
signs is provided to evaluate their potential in structural
2. Structural Designs
The unit cell of the 3D re-entrant lattice auxetic structure
used in this study is shown in Figure 1(a) [10,11]. This
auxetic structure is an orthotropic structure with direc-
tions x and y exhibiting identical properties due to the
A Comparison of Bending Properties for Cellular Core Sandwich Panels
Copyright © 2013 SciRes. MSA
(a) (b)
Figure 1. Design of the 3D re-entrant lattice structure.
symmetry. Therefore, the design of the structure could be
represented by the simplified 2D geometry shown in
Figure 1(b), which includes the length of the vertical (H)
and re-entrant (L) struts, the re-entrant angle θ, and the
thickness (t) of the strut (not shown in the figure). Li et al.
[12] showed that compressive properties of this auxetic
structure could be estimated as a function of design pa-
rameters H, L, θ, and t. The relationship between the
Poisson’s ratio values and the design parameters could be
written as:
22 2
12 34
6cos cos
sin 6sin4
Gt t
Et t
Gt tEtt
Et t
cos cos
 (2)
where νz and νx are Poisson’s ratios of the structure under
compression in the z and x directions respectively. Pois-
son’s ratios along the x and y axes are identical due to
symmetry of the structure (i.e. νx = νy). E and G are
Young’s modulus and shear modulus, α = H/L, t1 and t2
are dimensions of the re-entrant struts, and t3 and t4 are
dimensions of the vertical struts. The cross section of the
struts is taken as rectangular for this analysis, although
similar analysis’ can be performed for other cross sections.
The Poisson’s ratio values are expected to have a sig-
nificant effect on the mechanical properties of the re-en-
trant auxetic structures. With greater negative Poisson’s
ratio values, the modulus and strength of the structures
will also become greater [11,12]. In the current study,
two configurations were designed for the re-entrant au-
xetic structure with significantly different Poisson’s ra-
tios, as shown in Table 1. In Table 1, tV and tR stand for
the thickness of the square shaped vertical struts and re-
entrant struts, respectively.
The sandwich panels for bending tests were designed
as illustrated in Figure 2 using the two different auxetic-
core designs (A1 and A2). The dimensions of the sand-
wich panel cores were kept at approximately 17 mm × 20
mm × 150 mm while maintaining structural symmetry in
each direction. In addition, the thickness of the sandwich
skins was fixed at 0.75 mm. The resulting structures had 1 ×
2 × 14 auxetic unit cell repetitions for design A1, and 1.5 ×
2 × 20 auxetic unit cell repetitions for design A2.
From the unit cell orientation as shown in Figure 2, it
was known that under a bending load, the auxetic struc-
tures would be subject to compressive stress normal to
the bending load direction. According to Equation (2),
design A2 will have a higher νx value and would therefore
be expected to exhibit higher strength and higher mo-
dulus than design A1.
In order to compare the auxetic geometries with other
reticulate cellular geometries, octahedral, rhombic, and
hexagonal cellular structures were designed in CAD.
These geometries have been demonstrated elsewhere and
fabricated by electron beam melting [13-15]. The unit
cells of these structures are shown in Figure 3. Parame-
ter values for each structure are shown in Table 2. The
resulting sandwich panels had unit cell array counts of 2 ×
2 × 19 for the octahedral panel, 2 × 2 × 18 for the rhom-
bic panel, and 3 × 3 × 28 for the hexagonal panel. The
sandwich skin thickness for each design was fixed at 1
All of the cellular structure designs except for A1 pos-
sess similar relative densities as shown in Ta ble s 1 and 2.
According to the cellular theory, the relative density of a
structure has a significant influence on its mechanical
properties [16]. The selected geometries therefore permit
a high level comparison of material properties between
the different cellular geometries.
3. Experimental Procedures
Three titanium (Ti-6Al-4V) samples for each sandwich
panel design were fabricated using an Arcam A2 EBM
system using +325/100 spherical powder. The powder
was made via the plasma rotating electrode (PREP)
process. Identical default process settings for electron
Table 1. Design parameter values for the re-entrant auxetic structure.
Design H (mm) L (mm) θ (Deg.) tV (mm) tR (mm) νx νz Relative Density (%)
A1 15 7.5 45 1 0.707 1.704 0.547 6.3
A2 7.595 4 70 1 0.940 0.445 1.658 11.6
A Comparison of Bending Properties for Cellular Core Sandwich Panels
Copyright © 2013 SciRes. MSA
Figure 2. The sandwich panel with auxetic core.
(a) (b) (c)
Figure 3. Unit cell designs for (a) octahedral; (b) rhombic;
and (c) hexagonal structures.
Table 2. Designs of various cellular structures.
Design L (mm) θ (Deg.) t (mm) Relative
Density (%)
O (Octahedral) 8 - 1 11.98
R (Rhombic) 4.65 60 1 11.00
H (Hexagonal) 3.19 120 1 11.72
beam melting of lattice geometries were used to produce
all samples. All samples were oriented in the build
chamber such that the two face skins were normal to the
build direction. Due to build chamber size limitations,
the 18 samples (total) were fabricated in two batches.
After the samples were cleaned, their dimensions were
measured using digital calipers, and their masses were
measured using a digital balance having a resolution of
0.0001 g.
Bending tests were carried out using an Applied Test
System 1620 C at a constant strain rate of 1.27 mm/min.
Three point bending tests were carried out with a support
span of L = 114.3 mm as shown in Figure 4. The support
rollers had a diameter of 12.7 mm, and the load roller had
a diameter of 25.4 mm. The displacement and load (F) of
the loading roller were recorded through the crosshead.
The test was automatically stopped when the loading
level of the roller dropped below 70% of the maximum
recorded loading level. The actual experimental setup is
further illustrated in Figure 5. An FEA analysis for each
model was also performed in order to compare predicted
material properties with actual measured properties. As
shown in Figure 6, each model was fixed at two strips on
Figure 4. Experimental setup for the three point bending
Figure 5. Actual setup for the bending test.
Figure 6. FEA of the bending of the sandwich panel.
the bottom face, and then loaded at the strip on the center
of the top face. The width of each strip was 1 mm. Dur-
ing the FEA studies, loadings ranging from 1000 - 5000
N were simulated for each structure. The strength values
were obtained by determining the minimum loading level
for which the stress across the entire cross section of any
strut exceeded the yield strength of Ti-6Al-4V (1050
4. Results and Discussion
The measured dimensions of fabricated titanium sam-
ples are listed in Tab le 3. In Table 3, D1 and D2 are the
A Comparison of Bending Properties for Cellular Core Sandwich Panels
Copyright © 2013 SciRes. MSA
Table 3. Actual parameters of the samples made by EBM.
Design D1 (mm) D2 (mm) L (mm) Mass (g) Relative Density (%)
A1 20.405 ± 0.259 22.538 ± 0.125 148.675 ± 0.155 45.063 ± 2.688 14.87 ± 0.81
A2 19.829 ± 0.153 16.222 ± 0.029 150.334 ± 0.140 45.852 ± 1.504 21.40 ± 0.60
O 23.292 ± 0.116 17.052 ± 0.029 151.875 ± 0.125 48.646 ± 0.616 18.20 ± 0.22
R 19.668 ± 0.205 17.230 ± 0.039 144.848 ± 0.073 41.647 ± 1.965 19.15 ± 0.73
H 20.616 ± 0.106 17.729 ± 0.000 160.274 ± 0.076 52.112 ± 1.242 20.08 ± 0.38
overall thickness and width of the sandwich panels, and
L is the total length. It could be seen that the dimensions
of the samples were quite consistent, indicating stable
process quality. The relative densities of the samples
were significantly larger than 0.1 due to the existence of
the surface skins. All the designs, except for A1, had
measured relative densities close to 0.2.
The thickness of the sandwich skins for auxetic de-
signs A1 and A2 were around 0.75 mm, while the thick-
ness of the skins for the other sandwich structures were
very close to 1 mm. According to the classic theory of
sandwich panels, skin thickness is the dominant factor
for the performance of the structure. However, prelimi-
nary FEA results with the designed structures showed
that the skin thickness hada minimal effect on the bend-
ing modulus of the sandwich panels. It is hypothesized
that this is due to the very low relative density of the core
structure which leads to large compliance that accom-
modates local deflections of the two skin panels. Fur-
thermore, preliminary study showed that for auxetic sand-
wich structures, the dominant failure mode during bending
was core shear, which is in turn determined by the geome-
tries of individual struts rather than the surface skin.
During bending, the upper region of the sandwich
panel is subject to compressive stress, while the lower
region is subject to tensile stress. For regular core struc-
tures, compressive stresses result in lateral expansion. As
a consequence, localized stress concentrations and poten-
tial wrinkling on the face skins can be expected. Con-
versely, auxetic structures with negative Poisson’s ratios
exhibit lateral shrinkage (or inward movement) under
compression and will therefore better accommodate ma-
croscopic structural deformation.
Figure 7(a) shows the auxetic sandwich panel defor-
mation during testing, whereas Figure 7(b) shows the
corresponding FEA simulation result. Figures 7(a) and
(b) both show very little localized deformation during
bending. Instead, the deflection is distributed along the
length of the sandwich panel homogeneously. Whereas
sandwich panels with foam cores typically fail via face
yield, core shear, indentation, delamination and face
wrinkle [5], the auxetic sandwich panels failed by frac-
ture of vertical struts under the combination of bending
and tension. Failure occurred at the vertical struts located
roughly at the middle section between the loading roller
and the support roller where the local deformation was at
its maximum as seen in Figure 7(a). Because of the
highly homogeneous distribution of deflection and stress,
this auxetic sandwich structure could potentially be use-
ful for applications such as structural beams. Since the
vertical struts are subject to critical failure, new designs
involving non-uniform strut sizes could be applied to fur-
ther improve the structural performance.
Figure 8 shows the deflection of the other sandwich
structures in FEA simulation. Again, the tested titanium
structures exhibited similar behavior as that predicted
from the FEA studies. It is apparent that for octahedral,
rhombic and hexagonal sandwiches, the stress distribu-
tions, and therefore the deformation of the structures,
was largely concentrated at the surface area where the
structure was loaded. Upon failure, face yield was observ-
ed for all three of these geometries.
Table 4 shows the bending strength and modulus re-
sults, as well as the total energy absorption for each type
of structure. The results for rhombic and octahedral sam-
ples showed an unusually high variation in maximum
force and therefore, the strength and the total energy ab-
sorption. It is noted that in order to produce comparable
relative densities between sample geometries, the rhom-
bic and octahedral samples had fewer unit cell repetitions
through the thickness of the samples. It is surmised that
the small number of unit cell repetitions through the
thickness of the samples contributed to this elevated de-
gree of variance. All the other type of structures showed
relatively consistent properties.
Comparing A1 and A2, it is apparent that A2 exhibited
significantly higher strength and modulus, as predicted.
With higher νz value, the modulus of A2 is about 7 times
that of A1, while the strength of A2 had an approximately
200% increase compared with A1. On the other hand,
design A1 exhibited significantly higher resilience com-
pared with A2. The maximum deflection for A1 was al-
most twice as much as that of A2. Although design A1
could withstand much greater deflection than A2, design
A2absorbed a considerable amount of energy during the
bending, as also shown in Table 4.
The octahedral, rhombic, and hexagonal sandwich
panels showed relatively high strength and modulus val-
A Comparison of Bending Properties for Cellular Core Sandwich Panels
Copyright © 2013 SciRes. MSA
Figure 7. The shape of (a) titanium auxetic sandwich and (b)
FEA model under bending.
ues compared with the auxetic sandwiches. However, the
maximum deflections of these structures were signifi-
cantly lower, indicated that these structures had rather
low overall structural ductility.
From the comparison, it was apparent that the auxetic
structures showed significantly superior performance in
terms of maximum deflection that can be tolerated. Al-
though design A1 exhibited lower strength and modulus
compared to the other designs, it absorbed a significant
amount of energy. Design A2 showed higher modulus
and strength compared to A1. Furthermore, the total en-
ergy absorption of A2 was about 100% higher than that
of A1. The octahedral, rhombic and hexagonal structures
showed significantly lower ductility compared to the au-
xetic sandwich structures, while possessing higher modu-
lus and potentially strength values. As a result, these
structures could exhibit energy absorption abilities com-
parable to the auxetic sandwich with larger negative
Poisson’s ratio values in the thickness direction.
It is known that for many applications, the energy ab-
sorption of the sandwich panels during bending is of
great interest. For energy absorption purposes, it is de-
sired that the structure exhibit low peak response force,
and high total energy absorption. Comparing the struc-
tures in Table 4, it is apparent that the auxetic structures
possess significant advantages over the other unit cell
geometries. At a similar peak response force level
(around 3000 N), the auxetic sandwich designs absorbed
about 100% more energy than the other designs. At the
same energy absorption level (about 9 J), the auxetic
structures had a response force of about 1200 N, which
Figure 8. FEA of the bending of various sandwich struc-
tures. (a) Octahedral; (b) Rhombic; (c) Hexagonal.
was significantly lower than the others. The other struc-
tures do not seem to be ideal candidates for energy ab-
sorption applications due to their significantly higher
response force level. However, their higher modulus and
strength properties make them potential candidates for
applications where high specific stiffness and static
strength of the sandwich panels are required.
A comparison between the FEA and the average ex-
perimental results is shown in Table 5 . The strength val-
ues of the FEA agree quite well with the experimental
results, while the modulus values from the FEA study are
uniformly higher than experimental results, especially for
the octahedral and hexagonal structures. Given that the
crosshead displacement was used to monitor the deflec-
tion of the beam, the compliance of the load cell, when
testing stiffer structures would reduce the measured
It is worth noting that the ability of additive manufac-
turing processes such as EBM to fabricate engineered
cellular structures with any desired strut sizes and angles
opens up tremendous possibilities for further design op-
A Comparison of Bending Properties for Cellular Core Sandwich Panels
Copyright © 2013 SciRes. MSA
Table 4. Bending properties of various struc tures.
Design Max. Force (N) Max. Deflection (mm) Strength (MPa) Modulus (GPa) Energy Abs. (J)
A1 1206.43 ± 37.35 9.20 ± 0.77 22.04 ± 0.21 0.39 ± 0.02 8.73 ± 0.50
A2 3432.23 ± 58.58 5.96 ± 0.27 92.26 ± 1.62 2.98 ± 0.04 16.54 ± 1.61
O 5714.50 ± 977.40 2.61 ± 0.48 105.99 ± 18.83 5.23 ± 0.78 9.08 ± 0.17
R 3163.86 ± 966.96 2.92 ± 0.76 81.06 ± 23.39 4.65 ± 0.51 6.21 ± 4.39
H 5261.75 ± 199.04 3.13 ± 0.26 119.70 ± 3.48 6.20 ± 0.45 16.78 ± 2.68
Table 5. Comparison of FEA and experiments.
Design Measured Strength (MPa) FEA Strength (MPa) Measured Modulus (GPa) FEA Modulus (GPa)
A1 22.04 26.56 0.39 0.47
A2 92.26 75.00 2.98 3.26
O 105.99 103.12 5.23 10.85
R 81.06 82.50 4.65 6.46
H 119.70 111.94 6.20 9.06
timization of all of the cellular geometries examined in
this paper. For instance, critically loaded struts can be
fabricated with slightly larger diameters than other struts.
5. Conclusions
In the current work, sandwich panels with different cellu-
lar cores were designed and compared. Although the 3D
reticulate cellular core structures were not optimized for
bending, their overall performance showed promising
potential as future sandwich cores. During bending, the
auxetic sandwich panels exhibited homogeneous distri-
bution of stress and deformation. Failure by fracture of
the vertical struts located roughly at the middle section
between the loading and support rollers was seen in all
cases. Future studies should therefore focus on optimiza-
tion of the cellular structure based on the expected load-
ing patterns. Thickening of the critical vertical struts
would potentially lead to significant enhancements in
material properties with relatively little increase in mass.
The other sandwich structures showed significant stress
concentration at the loading area, and failed by face
The bending tests revealed that with different Pois-
son’s ratio values, the mechanical properties of the aux-
etic sandwich panels could be tailored over a wide range.
In addition, the auxetic sandwich panels also exhibited
extraordinary resilience under bending compared with all
the other structures. As a result, the auxetic sandwich
panels generally possess considerable potential specifi-
cally for energy absorption applications.
While the non-auxetic cellular geometries studied here
had relatively low ductility and arbitrarily chosen geo-
metrical parameters, they exhibited relatively high strength
and modulus. In addition, some of these designs also
exhibited high energy absorption during bending due to
the high modulus. In conclusion, these structures have
potential in applications where stiff and strong structural
beams are employed, and where high toughness is needed.
[1] H. G. Allen, “Analysis and Design of Structural Sand-
wich Panels,” Pergamon Press, Oxford, 1969.
[2] R. Lakes, “Foam Structures with a Negative Poisson’s
Ratio,” Science, Vol. 235, No. 4792, 1987, pp. 1038-1040.
[3] F. Scarpa and P. J. Tomlin, “On the Transverse Shear Mo-
dulus of Negative Poisson’s Ratio Lattice Structures,”
Fatigue & Fracture of Engineering Materials & Struc-
tures, Vol. 23, No. 8, 2000, pp. 717-720.
[4] F. Scarpa and G. Tomlinson, “Theoretical Characteristics
of the Vibration of Sandwich Plates with In-Plane Nega-
tive Poisson’s Ratio Values,” Journal of Sound and Vi-
bration, Vol. 230, No. 1, 2000, pp. 45-67.
[5] T. M. McCormack, R. Miller, O. Kesler and L. J. Gibson,
“Failure of Sandwich Beams with Metallic Foam Cores,”
International Journal of Solids and Structures, Vol. 38,
No. 28, 2001, pp. 4901-4920.
[6] O. Kesler and L. J. Gibson, “Size Effects in Metallic
Foam Core Sandwich Beams,” Materials Science and
Engineering: A, Vol. 326, No. 2, pp. 228-234.
[7] C. Chen, A.-M. Harte and N. A. Fleck, “The Plastic Col-
lapse of Sandwich Beams with a Metallic Foam Core,”
International Journal of Mechanical Sciences, Vol. 43,
No. 6, 2001, pp. 1483-1506.
[8] K. E. Evans, “The Design of Doubly Curved Sandwich
Panels with Honeycomb Cores,” Composite Structures,
Vol. 17, No. 2, 1991, pp. 95-111.
A Comparison of Bending Properties for Cellular Core Sandwich Panels
Copyright © 2013 SciRes. MSA
[9] R. Lakes, “Advances in Negative Poisson’s Ratio Materi-
als,” Advanced Materials, Vol. 5, No. 4, 1993, pp. 293-
296. doi:10.1002/adma.19930050416
[10] L. Yang, D. Cormier, H. West and K. Knowlson, “Non-
Stochastic Ti-6Al-4V Foam Structure That Shows Nega-
tive Poisson’s Ratios,” Materials Science and Engineer-
ing: A, Vol. 558, 2012, pp. 579-585.
[11] L. Yang, O. Harrysson, D. Cormier and H. West, “Com-
pressive Properties of Ti-6Al-4V Auxetic Mesh Struc-
tures Made by EBM Process,” Acta Materialia, Vol. 60,
No. 8, 2012, pp. 3370-3379.
[12] L. Yang, O. Harrysson, D. Cormier and H. West, “Mod-
eling of the Uniaxial Compression of a 3D Periodic Re-
Entrant Honeycomb Structure,” Journal of Materials Sci-
ence, Vol. 48, No. 4, 2012, pp. 1413-1422.
[13] O. L. A. Harrysson, O. Cansizoglu, D. J. Marcellin-Little,
D. Cormier and H. West II, “Direct Metal Fabrication of
Titanium Implants with Tailored Materials and Mechani-
cal Properties Using Electron Beam Melting Technol-
ogy,” Materials Science and Engineering: C, Vol. 28, No.
3, 2008, pp. 366-373. doi:10.1016/j.msec.2007.04.022
[14] O. Cansizoglu, O, Harrysson, D. Cormier, H. West and T.
Mahale, “Properties of Ti-6Al-4V Non-Stochastic Lattice
Structures Fabricated via Electron Beam Melting,” Mate-
rials Science and Engineering: A, Vol. 492, No. 1-2, 2008,
pp. 468-474.
[15] O. Cansizoglu, “Mesh Structures with Tailored Properties
and Applications in Hip Stems,” Ph.D. Dissertation,
North Carolina State University, Raleigh, 2008.
[16] L. J. Gibson, “Cellular Solids: Structure and Properties,”
2nd Edition, Cambridge University Press, New York, 1997.