Composites: Part A 47 (2013) 31–40
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Composites: Part A
journal homepage: www.elsevier.com/locate/compositesa
Mechanical response of carbon fiber composite sandwich panels
with pyramidal truss cores
Tochukwu George a,⇑, Vikram S. Deshpande b, Haydn N.G. Wadley a
a
b
Department of Materials Science and Engineering, 395 McCormick Road, PO Box 400745, University of Virginia, Charlottesville, VA 22904-4745, United States
Engineering Department, University of Cambridge, Cambridge, UK
a r t i c l e
i n f o
Article history:
Received 19 October 2012
Received in revised form 15 November 2012
Accepted 17 November 2012
Available online 7 December 2012
Keywords:
A. Laminates
A. Carbon fiber
a b s t r a c t
The combination of light carbon fiber reinforced polymer (CFRP) composite materials with structurally
efficient sandwich panel designs offers novel opportunities for ultralight structures. Here, pyramidal
in the range 1–10% have been manufactured from carbon
truss sandwich cores with relative densities q
fiber reinforced polymer laminates by employing a snap-fitting method. The measured quasi-static shear
strength varied between 0.8 and 7.5 MPa. Two failure modes were observed: (i) Euler buckling of the
struts and (ii) delamination failure of the laminates. Micro-buckling failure of the struts was not observed
in the experiments reported here while Euler buckling and delamination failures occurred for the low
> 1%Þ relative density cores, respectively. Analytical models for the collapse of
6 1%Þ and high ðq
ðq
the composite cores by these failure modes are presented. Good agreement between the measurements
and predictions based on the Euler buckling and delamination failure of the struts is observed while the
micro-buckling analysis over-predicts the measurements. The CFRP pyramidal cores investigated here
have a similar mechanical performance to CFRP honeycombs. Thus, for a range of multi-functional applications that require an ‘‘open-celled’’ architecture (e.g. so that cooling fluid can pass through a sandwich
core), the CFRP pyramidal cores offer an attractive alternative to honeycombs.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Sandwich panel structures with stiff, strong face sheets and low
density cellular cores are widely used to support bending loads [1].
The cores are typically made from polymeric materials such as Nomex [2], aluminum honeycombs [3] or sometimes polymer or metal foams [4,5], and are usually adhesively bonded to light, stiff face
sheet materials [6]. The recent emergence of simple methods for
making metallic sandwich panels with periodic lattice truss cores
has attracted significant interest in alternative core topologies
[7,8]. When these sandwich panels are loaded in bending, the
trusses of these lattice core structures are subjected to either uniaxial compression or tension, whereas struts of equivalent density
foams deform by bending [9]. Since the truss members more efficiently support axial loads, the modulus and strength of the
‘‘stretch – dominated’’ lattice structures significantly exceeds those
of equivalent (the same material and density) foams [10,11]. Honeycomb core sandwich structures are also able to efficiently support bending loads. However, as the core density decreases, the
slender webs begin to elastically buckle. At this point, the strength
⇑ Corresponding author. Tel.: +1 (603) 369 0689.
E-mail address: [email protected] (T. George).
1359-835X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.compositesa.2012.11.011
of lattice structures begins to exceed that of equivalent honeycombs [3].
Lattice truss core structures offer a number of other practical
advantages. If the relative density is low, they can be readily flexed
into multi-ply curved panels and then attached to curved face
sheets. The resulting panels suffer less loss of stiffness and strength
than ‘‘flex’’ honeycomb structures usually utilized for these applications [3,18]. The interconnected spaces within lattice truss core
structures can also facilitate multifunctional applications such as
cross flow heat exchange [8,10–12]. Lattice truss structures have
also received significant interest for use in shock mitigation [13],
ballistic impact protection [14], acoustic damping [15] and for
shape morphing structures [16]. In several of these applications,
the structures are loaded to their peak strength. The phenomena
governing both the stiffness and strength of these lattices are
therefore of considerable interest.
The stiffness and strength of lattice structures depends upon
the topology, relative density and the mechanical properties of
the materials used to construct them [9]. Lattices made from titanium [17–19] exhibit high compressive strengths and stiffnesses,
and offer significant potential for use in elevated temperature
aerospace applications. However, for ambient temperature weight
driven applications, sandwich structures with square honeycomb
cores made from carbon fiber reinforced polymer (CFRP) composite
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T. George et al. / Composites: Part A 47 (2013) 31–40
laminates have recently been shown to offer a high specific
strength in through thickness compression [20]. CFRP sandwich
panels with open cell pyramidal lattice cores have also been recently fabricated from 0/90 crossply laminates using a snap fit process [21]. The specific compressive strength of this lattice core was
shown to be exceeded by only that of CFRP honeycombs for core
densities around 100 kgm3 (Fig. 1). The high specific compressive
strength of this lattice resulted from the truss resistance to elastic
buckling and compressive delamination. Sandwich panel structures are usually used in situations where they are subjected to significant bending load [22]. In this case, the shear response of the
core governs the panel’s stiffness and strength. Here, we investigate the in-plane shear stiffness and strength of CFRP pyramidal
lattice truss structures made by the snap fit assembly method
[21], and compare its performance with other lightweight core
concepts.
2. Materials and manufacture
The pyramidal truss sandwich panels were manufactured from
0/90° laminate sheets of thickness t = 3.175 mm in a three step
process. First, patterns as shown in Fig. 2a were water jet cut from
the laminate sheets. Second these patterns were then snap-fitted
into each other (Fig. 2b) to produce a pyramidal truss. Finally,
the pyramidal truss was bonded to 3.175 mm thick composite face
sheets using an epoxy adhesive. (The adhesive comprised 100 parts
by weight of Dow Epoxy Resin with 13 parts of a triethylene tetramine curing agent. The cure cycle consisted of 12 h at room temperature, followed by heating to 100 °C for 2 h) These composite
face sheets had cruciform shaped slots of depth t0 milled into them
at appropriate locations such that the pyramidal nodes of the pyramidal truss could be counter-sunk into the face sheets (Fig. 2d).
The critical parameters describing the geometry of the pyramidal core are sketched in Fig. 2c and include, the strut length l, the
strut width t (which is equal to the laminate sheet thickness and
thus the struts have a square cross-section) and the node width
and thickness b and t0, respectively. The struts make an angle x
with the horizontal plane of the face sheet (Fig. 2c). The unit cell
of the pyramidal core is sketched in Fig. 3a. Simple geometric considerations dictate that the relative density of the core (defined as
the density of the ‘‘smeared-out’’ core to the density of the solid
material from which it is made) is given by
q ¼
2ðlt þ t 0 bÞt
l sin xðl cos x þ bÞ
2
2ðl þ hbÞ
;
l sin xðl cos x þ bÞ
2
ð1aÞ
b=t and h
t0 =t. In
where the non-dimensional lengths l l=t, b
the limit of vanishing node volumes ðb ¼ h ! 0Þ, this expression reduces to
q 2
4
l sin 2x cos x
:
ð1bÞ
¼ 7% specimen is inA photograph of the as-manufactured q
cluded in Fig. 3c.
2.1. Composite laminate material
The 0/90 CFRP laminate sheets were obtained from McMaster–
Carr1. They had a thickness t = 3.175 mm and comprised 65% by volume 33 Msi carbon fibers in a vinylester matrix. Plies comprising
unidirectional fibers were laid-up alternating at 0° and 90° orientation to build an orthotropic laminate comprising 14 plies. The density of the laminate material was qs = 1440 kgm3.
The laminate was tested in uniaxial compression along one of
the fiber directions in order to determine the relevant Young’s
modulus and compressive strength of the parent material used to
manufacture the pyramidal cores. Two types of compression tests
were conducted:
(i) Column compression tests were conducted in which the
specimens were compressed between two flat, parallel and
rigid platens with no end-clamping of the laminates. This
provided the delamination strength of the laminates and
best simulated the loading conditions of the struts of the
pyramidal core.
(ii) Combined loading compression (CLC) test in accordance
with ASTM D6641 [23] to determine the micro-buckling
strength of the composite.
Fig. 1. (a) Ashby material strength versus density chart of hybrid materials
including the compressive strength of CFRP pyramidal lattices [21]. (b) The Young’s
modulus versus density chart. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
In each case tests were conducted on rectangular specimens of
thickness 3.175 mm, width 20 mm and gauge length 12 mm: the
1
McMaster–Carr, 6100 Fulton Industrial Blvd. Atlanta, GA 30336-2852, USA.
T. George et al. / Composites: Part A 47 (2013) 31–40
33
Fig. 2. Illustration of the manufacturing route for making composite pyramidal lattice core sandwich panels. (a) Patterns water jet cut from 0/90 the laminate sheets. The
fiber directions are shown in this sketch. (b) Snap-fitting of the patterns to produce a pyramidal lattice. (c) Geometry of the truss pattern with relevant core design variables.
(d) Pyramidal lattice core sandwich panel. The composite face-sheets utilized cruciform shaped slots into which the pyramidal trusses were fitted and adhesively bonded. (For
interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
specimens were sufficiently stocky so as to prevent the Euler buckling of the specimens. The applied load was measured via a load
cell while a laser extensometer was used to measure the nominal
axial strain in the specimens. A nominal applied strain rate of
103 s1 was employed in these tests. The measured compressive
nominal stress versus strain curves from both types of tests are
plotted in Fig. 4. We deduce: (i) the unloading Young’s modulus Es 28 GPa, (ii) the delamination failure strength rdl 380 MPa
while and (iii) the micro-buckling strength of the composite is
rmax = 615 MPa. These laminate properties are listed in Table 1.
2.2. Pyramidal core designs
All the pyramidal cores tested and manufactured in this study
had a strut angle x = 45°. Thus, the included angle between the
struts was 90° and the patterns were cut from the laminate sheets
such that half the fibers of the 0–90° laminate were aligned along
the axis of the struts of the pyramidal core (Fig. 2a). Panels with a
ranging between 0.9% and 10%
pyramidal core of relative density q
were manufactured and tested in this study. All the cores had a
strut thickness t = 3.175 mm and node width b = 15.3 mm and
the strut length l was varied between 9.4 mm and 63.5 mm to enable manufacture of cores of different densities. The main part of
the study was carried out on cores with node thickness t0 = 0.8t:
a parametric study (reported subsequently in Section 3.1) confirmed that node failure was not an operative failure mechanism
during shear loading of the core for nodes with t 0 P 0:8t. The strut
lengths l of the six core relative densities investigated in this study
are listed in Table 2.
3. Measurements of the shear response of the pyramidal core
Single lap shear tests were conducted on the pyramidal core
panels in accordance with the standard test for sandwich core
materials, viz. ASTM C-273 [24] using a compression plate setup
as sketched in Fig. 5. The ASTM standard specifies that the length
to thickness ratio of the specimen L=H P 12, while the width
W P 2H. This was achieved by employing specimens that had 2
pyramidal cells along the width of the specimen and 5–7 cells
along the length depending upon the specimen relative density
6 10%, 6 cells for
(5 cells were used for specimens with 5% 6 q
the 3% and 1.6% specimens while the 0.9% specimen required 7
cells). In order to attach the specimen to the test fixture, holes were
drilled into the composite face sheets of the pyramidal core panels
and the panels attached to the text fixture (Fig. 5). The applied load
was measured from the load cell of the test machine and used to
infer the shear stress while the shear strain was measured using
a laser extensometer. All tests were conducted at an applied shear
strain rate of 103 s1.
In general, the shear response of the pyramidal core is anisotropic and hence dependent on the direction of shearing. We define
the direction of shearing as follows. Consider the unit cell sketched
in Fig. 3a: the shearing direction is specified via the angle a
(Fig. 3b) such that a = 0° and 90° for applied shear stresses s13
and s23, respectively. Unless otherwise specified, all measurements
reported subsequently are for a = 45° and for the sake of brevity we
shall refer to this shear stress via the symbol s and the corresponding engineering shear strain via c.
3.1. Effect of node design
¼ 5% pyramidal core
The measured s versus c curves for the q
are plotted in Fig. 6a for node thicknesses over the range
0:5 6 t 0 =t 6 0:9. The peak shear strength is seen to increase with
increasing t0/t for t0/t < 0.8 with the shear response of the t0/
t = 0.8 and 0.9 specimens identical for all practical purposes. The
lower strength of the specimens with the smaller node thicknesses
is due to tensile failure of the struts at the nodes as seen from the
photograph in Fig. 7a. A sketch illustrating this failure mode is
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T. George et al. / Composites: Part A 47 (2013) 31–40
Fig. 4. The measured uniaxial compressive stress versus strain response of the
composite material along of the fiber directions. Results from both the column
compression and CLC tests are included.
ear elastic phase followed by a plastic hardening phase, which typically precedes the peak stress and failure. Failure first occurs in the
strut under compression (i.e. the strut making an acute angle with
the loading direction) and is followed by failure of the remaining
struts and a consequent fall in the load carrying capacity of the
specimen. Two failure modes were observed:
Fig. 3. (a) Sketch of the unit cell of the pyramidal core with co-ordinate system
marked. (b) Top-view of the unit cell indicating the direction of shear. (c)
Þ ¼ 7%Þ in a sandwich panel
Photograph of the as-manufactured design 1 core ðq
structure. (For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this article.)
shown in Fig. 7b–d: the strut making an obtuse angle with the
loading direction is under tensile stresses and breaks off at the
nodes. This failure mode is prevented by employing sufficiently
stubby nodes whereby the failure mode switches to failure of the
struts of the pyramidal core and independent of the node thickness
(see Fig. 6b). Subsequently, all results are shown for t0/t = 0.8 such
that node failure is never an operative mechanism.
(i) Delamination failure of the struts: for all specimens expect
¼ 0:9% specimen, the first failure event was delamfor the q
ination of the strut making an acute angle with the loading
direction and thus under compression. A photograph of this
¼ 5% specimen just after the peak
failure event in the q
stress had been attained is included in Fig. 9a. Subsequently,
the other struts also fail by a combination of delamination
and tensile fracture.
(ii) Euler buckling of the struts: the first failure event in the
q ¼ 0:9% specimen was elastic Euler buckling of the strut
under compressive loading (i.e. the strut making an acute
angle with the loading direction). This failure event is shown
in the photograph in Fig. 9b taken immediately after the
peak stress had been attained. The buckling event results
in delamination of the strut followed by failure of the
remaining three struts as well.
The measured unloading modulus and peak strengths are summarized in Fig. 10a and b, respectively as a function of the relative
.
density q
3.2. Effect of core relative density q
3.3. Effect of shearing direction a
The measured shear stress s versus strain c response of the
pyramidal cores is plotted in Fig. 8 for the six relative densities
investigated here. In all cases, the responses display an initial lin-
All the results presented above were for loading in the a = 45°
direction. In general, we anticipate anisotropy in the shear re-
T. George et al. / Composites: Part A 47 (2013) 31–40
Table 1
The measured properties of the laminate from both the column compression and CLC
tests.
Density
Compressive
modulus
3
q (Mg/m ) E (GPa)
1.4
28
Microbuckling
strength
rmb (MPa)
615
Delamination
strength
rdl (MPa)
380
Tensile
strength
rt (MPa)
450
Tensile
modulus
Y (GPa)
77
Laminate properties.
Table 2
of specimens investigated here.
The strut length l for the six relative densities q
Þ (%)
Density ðq
Core height h (mm)
Truss length l (mm)
0.9
1.6
3.0
5.0
7.0
10.0
45.2
31.4
20.0
13.2
9.7
6.6
63.5
44.1
28.3
18.7
13.7
9.4
Sandwich core parameters.
35
4. Analytical model for the shear response
We proceed to derive analytical expressions for the ‘‘effective’’
shear stiffness and strength of the composite pyramidal cores,
sandwiched between two rigid face-sheets. The pyramidal trusses
are made from 0/90 CFRP laminates such that one set of fibers are
aligned with the axial direction of the struts of the pyramidal truss
(Fig. 2a). We define a local Cartesian co-ordinate system (e1 e2)
aligned with the orthogonal set of fibers (Fig. 2a). The Young’s
modulus and compressive plastic micro-buckling strengths of the
laminate in either the e1 or e2 directions are Es and rc, respectively
while sY is the longitudinal shear strength of the matrix material of
the laminate. The delamination strength of the composites along
the e1 or e2 directions is denoted by rdl.
4.1. Elastic properties
Analytical expressions for the shear modulus G of the pyramidal
core is obtained in terms of the core geometry and the elastic properties of the solid material by first analyzing the elastic deformations of a single strut of the pyramidal core and then extending
the results to evaluate the effective properties of the core.
Consider the unit cell sketched in Fig. 3a with an applied shear
displacement d at an angle a with the x1 axis as shown in Fig. 3a.
We resolve this applied displacement into two perpendicular
components
d1 ¼ d cos a;
ð2aÞ
Fig. 5. Sketch of the single lap compression plate setup used to measure the shear
response of the pyramidal cores. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
sponse of the pyramidal core. To investigate this anisotropy, we
¼ 5% core for loading in
conducted two additional tests on the q
the a = 22.5° and 0° directions. A comparison between the measured shear stress versus shear strain responses for the three loading directions is shown in Fig. 11: while the shear modulus seems
to be relatively insensitive to a, the peak shear strength decreases
with decreasing a from approximately 4.5 MPa in the a = 45° case
to 2.7 MPa in the a = 0° case.
Fig. 6. (a) The measured shear stress s versus strain c response (a = 45°) of the
¼ 5% core for selected values of the node thickness t0. (b) The measured peak
q
¼ 5% core.
strength as a function of node thickness for the q
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T. George et al. / Composites: Part A 47 (2013) 31–40
and
d2 ¼ d sin a:
ð2bÞ
We label the struts in the unit cell via the symbols A through D
as shown in Fig. 3a in order to refer to them in the remainder of the
analysis. The axial and shear displacements applied to struts A and
C are
da ¼ d1 cos x;
ð3aÞ
and
ds ¼ d1 sin x;
ð3bÞ
Fig. 8. The measured shear stress s versus strain c response (a = 45°) of the
and node thickness t0/
pyramidal cores for selected values of the relative density q
t = 0.8.
respectively with strut A under a compression while strut C is subjected to a tensile displacement. Elementary beam theory gives the
axial and shear forces in strut A (or C) as
¼ 5% core with t0/
Fig. 7. (a) Photograph of the node failure mechanism in the q
t = 0.5. (b) Sketch illustrating the node failure mechanism. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of
this article.)
Fig. 9. Photographs of specimens just after attainment of the peak load to illustrate
¼ 5%
the observed failure mechanisms. (a) Delamination observed in the q
¼ 0:9% specimen.
specimen. (b) Euler buckling of the struts observed in the q
(For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
37
T. George et al. / Composites: Part A 47 (2013) 31–40
FS ¼
12Es Ids
l
3
ð4bÞ
;
respectively, where I t4/12 is the second moment of area of the
strut cross-section. Equivalent expressions are valid for struts B
and D by replacing d1 with d2 in Eqs. (3). The total applied force
along the x1 direction then follow as
F 1 ¼ 2ðF A cos x þ F S sin xÞ ¼
"
#
2
2Es t 2 d1
t
2
sin x ;
cos2 x þ
l
l
ð5Þ
while the force F2 in the x2 direction is obtained by replacing d1 by
d2 in Eq. (5). The applied shear stress is then given by
s
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 F 21 þ F 22
ð2l cos x þ 2bÞ2
ð6Þ
;
and the engineering shear strain is
c
d
:
l sin x
ð7Þ
The effective shear modulus G s/c of the pyramidal core then
follows from Eqs. (6) and (7) as
"
#
2
G
sin x
sin x
2
;
¼
cos x þ 2
Es ðl cos x þ bÞ
l2
ð8Þ
in terms of
l l=t and
¼h
! 0),
b
the non-dimensional geometric parameters of the core
b=t. In the limit of negligible node volumes (i.e.
b
the dimensionless modulus G/Es is related to the rela of the core via
tive density q
"
#
2
G
sin x
sin x
2
cos x þ Es l2 cos2 x
l2
¼
q
8
2
sin 2x þ
q 2
4
sin x cos2 x:
16
ð9Þ
Eq. (9) has been written so that the first and second terms represent
the contributions to the stiffness of the pyramidal core due to the
stretching and bending of the struts respectively.
Fig. 10. Summary of the measured. (a) Unloading modulus. (b) Peak strength as a
for shear loading at a = 45° (node thickness t0/t = 0.8).
function of relative density q
The predictions of the analytical models are also included.
4.2. Collapse strength
We consider the three critical collapse mechanisms for the
pyramidal core: (i) plastic micro-buckling of the composite struts;
(ii) delamination failure of the struts and (iii) elastic Euler bucking
of the struts. The operative failure mode is the one associated with
the lowest value of the collapse strength. Typically polymer matrices of fiber composites display non-linear behavior [25] and thus
elastic micro-buckling is not an operative failure mode and not
considered in the collapse calculations presented here.
Before considering each of these failure modes, in turn, we derive expressions relating the failure strength of the pyramidal core
to the compressive failure strength rc of a single strut. The ratio of
the shear and axial forces in the struts is given from Eqs. (3) and (4)
as
FS
¼
FA
¼ 5%
Fig. 11. The measured shear stress s versus strain c response of the q
pyramidal (node thickness t0/t = 0.8) for three selected loading directions a. The
analytical model predictions of the strength are also included.
da
F A ¼ Es t2 ;
l
and
ð4aÞ
2
t
tan x:
l
ð10Þ
Substituting Eq. (10) in Eqs. (5) and (6), the peak shear stress sp
is given in terms of rc as
h
i
2
x
cos x þ tl sin
cos x
sp
¼
;
2
rc cos aðl cos x þ bÞ
2
ð11aÞ
for a 6 p=4: the peak strength in this case is set by the compressive
failure of strut A (Fig. 3) at a stress rc. For p=4 < a 6 p=2, compressive failure first occurs in strut B and the failure stress follows as
38
T. George et al. / Composites: Part A 47 (2013) 31–40
h
i
2
x
cos x þ tl sin
cos x
sp
¼
:
2
rc sin aðl cos x þ bÞ
2
ð11bÞ
We now proceed to derive expressions for rc for each of the failure modes considered here.
4.2.1. Plastic micro-buckling of the composite struts
It is generally accepted that fiber micro-buckling of composites
is an imperfection-sensitive, plastic buckling event involving the
non-linear longitudinal shear of the composite within a narrow
kink band. Argon [26] argued that the compressive strength rmax
is given by
s
rmax ¼ Y ;
/
ð12Þ
for a composite comprising straight fibers and a rigid-ideally plastic
matrix of shear strength sY. Kinking initiates from a local region of
It is assumed that the micro-buckle
fiber misalignment of angle /.
band is transverse to the axial fiber direction e1, such that the angle
b between the normal to the band and the fiber direction vanishes.
Now consider the case where the remote stress state consists of an
in-plane shear stress s1 in addition to a compressive stress parallel
to the fibers. Then, Fleck and Budiansky [27] have shown that the
micro-buckling stress is given by
rc ¼
sY s1
/
ð13Þ
:
Prior to the micro-buckling of the struts, the struts are elastic and
the analysis detailed above applies. Thus, from Eqs. (10) and (13)
it follows that the axial stress rc required to initiate micro-buckling
in the inclined strut is given by
rc ¼ h
sY
rmax
i¼h
2 i ;
þ tan x t 2
/
1 þ tan/ x tl
l
ð14Þ
where rmax is the micro-buckling strength of the laminate for loading in the e1-direction in the absence of remote shear. Substituting
Eq. (14) in Eq. (11a) gives the shear strength as
h
t2 sin2 xi
cos x þ l cos x
sp
h
i
¼
;
x 2
rmax cos a 1 þ tan
2 ðl cos x þ bÞ
ð15Þ
4.2.3. Euler buckling of the struts
Under through-thickness compression the pyramidal core may
collapse by the elastic buckling of the constituent struts. Recall that
the Euler buckling stress of an end-clamped strut subjected to an
axial load is given by
rE ¼
p2 Es t2
3l
2
;
ð18Þ
and thus the shear strength of the pyramidal core due to the elastic
buckling of the constituent struts is given by replacing rc by rE in
Eq. (11) and replacing rdl by rE in Eq. (17) for the case of vanishing
node volume.
Delamination failure of the struts can be prevented in appropriately designed composite pyramidal cores. In order to illustrate the
optimal performance of the composite pyramidal cores, the predicted normalized peak strength of the composite pyramidal core
sp =ðq rmax Þ is plotted in Fig. 12 as a function of relative density q
only considering the micro-buckling and Euler buckling failure
mechanisms of the struts. Predictions are shown in Fig. 12 for three
s Es =rmax representative of unidirectional
selected values of E
s ¼ 116Þ and woven ðE
s ¼ 50Þ carbon fiber
ðEs ¼ 167Þ, laminated ðE
composites. For the purposes of illustration, in Fig. 12 we have neglected the volume of the nodes with the choices a = 45°, x = 45°
¼ 2o : most experimental evidence [28] suggests that the
and /
imperfection angle cannot be reduced below 2° in practical de rmax Þ is a measure of the effisigns. The normalized strength sp =ðq
ciency of the topology in terms of its structural strength with
sp =ðq rmax Þ 6 1: sp =ðq rmax Þ ¼ 1 corresponds to a cellular material
that attains the Voigt upper bound. We note:
rmax Þ peaks at a q
value at
(a) The normalized strength sp =ðq
which the failure modes transition from Euler buckling to
are
micro-buckling. Designs at this transition value of q
most efficient in terms of their strength to weight ratio. This
is rationalized by noting that in the Euler buckling regime
as the
the structural efficiency increases with increasing q
struts become more stocky resulting in an increase in their
Euler buckling loads. By contrast in the micro-buckling
, the shear forces on the struts
regime with increasing q
increase resulting in a decrease in their micro-buckling
stress as per Eq. (8).
l /
for a 6 p=4. In the limit of vanishing node volume, the above
expression reduces to
h
3
1 þ q sin2
sin 2x
sp
q
i
¼ h
rmax 4 1 þ q sin xsin 2x
cos a
x
i
ð16Þ
:
4/
The shear strength for the case of
replacing cos a by sin a in Eq. (16).
p=4 < a 6 p=2 is given by
4.2.2. Delamination failure of the struts
With insufficient end-clamping, the composite struts can fail by
compressive delamination. The shear strength with compressive
delamination as the failure mode is given by Eqs. (11) with rc replaced by the delamination strength rdl of the composite struts.
In the case of vanishing node volume, the modified Eq. (11a) reduces to
"
#
sp q sin3 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ q sin 2x 1 þ
1 þ tan2 a;
rdl
2
With an analogous expression for
replacing cos a by sin a in Eq. (17).
ð17Þ
p=4 < a 6 p=2 given by
rmax Þ
Fig. 12. Predictions of the variation of the normalized peak strength sp =ðq
for three selected value of the normalized laminate modulus
with relative density q
s . The predictions assume that the node volume is negligible.
E
T. George et al. / Composites: Part A 47 (2013) 31–40
39
each relative density of the pyramidal cores are listed in Table 2
required
– these values are used to determine the values of l and b
in the analytical expressions detailed in Section 4.
5.1. Shear loading at a = 45°
The analytical predictions of the shear modulus are included in
Fig. 10a and agree well with the measurements over the range of
relative densities investigated here. The predictions of peak
strength are included in Fig. 10b with the predicted failure modes
indicated (recall that the operative failure mode for a given relative
density and set of material properties is the mode that gives the
lowest peak strength). The measured and predicted values of the
peak strengths are in good agreement. Moreover, in line with the
observations discussed in Section 3, the predicted failure mode is
¼ 0:9% specimen that lies at the
delamination expect for the q
boundary between the Euler buckling and delamination failure
modes.
5.2. 2Effect of shearing angle a
The analytical predictions indicate that the shear modulus is
independent of the shearing angle a. This is borne out by the measured s versus c responses included in Fig. 11 wherein the elastic
regime slopes are essentially independent of a. The predictions of
the peak strengths are also included in Fig. 11. The predictions
are generally in line with the measurements, suggesting that the
shear strength decreases with decreasing a.
6. Comparison with competing materials
The measured shear modulus and peak shear strengths of the
CFRP pyramidal cores are included in Fig. 13a and b, respectively
along with a number of competing materials and topologies. In this
Ashby [29] plot the density of the pyramidal cores is taken to be
q ¼ q qs with qs = 1440 kg m3. The CFRP pyramidal cores have a
performance similar to CFRP honeycombs in terms of both shear
strength and modulus. The investigation of Finnegan et al. [21]
has indicated that these CFRP pyramidal cores also have a similar
compressive performance to CFRP honeycombs. Thus, for a range
of multi-functional applications that require an ‘‘open-celled’’
architecture (e.g. so that cooling fluid can pass through a sandwich
core), the CFRP pyramidal cores offer an attractive alternative to
honeycombs.
Fig. 13. An Ashby [29] chart of (a) Shear modulus. (b) Shear strength as a function
of density. In addition to a range of competing cellular materials, properties of the
CFRP pyramidal cores measured in this study are included. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of
this article.)
rmax Þ for the pyramidal cores
(b) The maximum value of sp =ðq
s with the transition from Euler
increases with increasing E
buckling failure to micro-buckling then occurring at lower
.
values of q
5. Comparison with measurements
We proceed to compare the measurements with the analytical
predictions detailed above. In making these predictions we employ
the following material properties for the composite struts consistent with the measurements discussed in Section 3 and tabulated
in Table 1: (i) Young’s modulus Es = 28 GPa, (ii) the delamination
failure strength rdl = 380 MPa, (iii) the micro-buckling strength
¼ 2o . The strut anrmax = 615 MPa and (iv) misalignment angle /
gle is taken to x = 45° in all cases. The geometric dimensions of
7. Concluding remarks
in the
Pyramidal truss sandwich cores with relative densities q
range 1–10% have been manufactured from 0/90 crossply carbon
fiber reinforced polymer laminates by employing a snap-fitting
method. The measured quasi-static shear strength varied between
. Two failure modes
1 and 7.5 MPa; increasing with increasing q
were observed: (i) Euler buckling of the struts and (ii) delamination failure of the struts.
Analytical models are developed for the elastic response and
collapse strengths of the composite cores. In general good agreement between the measurements predictions is obtained. Along
with the companion paper [21], we have demonstrated that composite cellular materials with a pyramidal micro-structure fill a gap
in the strength versus density material property space and compete favorably with honeycomb designs. However, current designs
of the pyramidal cores have not optimized the node designs and
thus use material rather inefficiently. Moreover, the current designs undergo delamination failure of the struts and thus do not
40
T. George et al. / Composites: Part A 47 (2013) 31–40
achieve the full potential of composite cores as predicted by the
micro-buckling analysis presented here.
Acknowledgements
We are grateful to the office of Naval Research for support of
this research under Grant Number N00014-07-1-0114. The program manager was Dr David Shifler.
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