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K. Finnegan et al.: The compressive response of carbon fiber composite pyramidal truss sandwich cores
K. Finnegana, G. Kooistraa, H.N.G. Wadleya, V.S. Deshpandeb
a
b
Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA, USA
Department of Mechanical Engineering, University of California, Santa Barbara, CA, USA
The compressive response of carbon fiber
composite pyramidal truss sandwich cores
Dedicated to Professor A. G. Evans on the occasion of his 65th birthday
Pyramidal truss sandwich cores with relative densities 4q in
the range 1 – 10 % have been made from carbon fiber reinforced polymer laminates using a snap-fitting method. The
measured quasi-static uniaxial compressive strength increased with increasing 4
q from 1 to 11 MPa over the relative density range investigated here. A robust face-sheet/
truss joint design was developed to suppress truss – face
sheet node fracture. Core failure then occurred by either (i)
Euler buckling (4
q < 2%) or (ii) delamination failure
(4
q > 2%) of the struts. Micro-buckling failure of the struts
was not observed in the experiments reported here. Analytical models for the collapse of the composite cores by Euler
bucking, delamination failure and micro-buckling of the
struts have been developed. Good agreement between the
measurements and predictions based on the Euler buckling
and delamination failure of the struts is obtained. The micro-buckling analysis indicates this mechanism of failure
is not activated until delamination is suppressed. The measurements and predictions reported here indicate that composite cellular materials with a pyramidal micro-structure
reside in a gap in the strength versus density material property space, providing new opportunities for lightweight,
high strength structural design.
Keywords: Lattice materials; Composites; Micro-buckling
1. Introduction
Rigid foams made from polymeric materials have been
widely used for the cores of sandwich panel structures
[1]. Over the last 15 years, metallic foams have attracted
significant interest because of their potentially higher
strength [2]. These metal structures can be produced by the
introduction of gas bubbles into the melt. The random bubble nucleation, expansion and subsequent melt drainage
processes lead to stochastic, closed cell structures. Open
cell structures with inter-connected struts can be made by
investment casting using reticulated polymer foam templates. However, minimization of surface energy during
the polymer precursor foaming process leads to a low nodal
connectivity, with typically three to four struts per joint.
The mechanical properties of these metal and polymer
foams are far from optimal because the cell walls deform
by local bending [2, 3]. This has led to a search for opencell microstructures with high nodal connectivities that de-
form by the stretching of constituent cell members, giving
a much higher stiffness and strength per unit mass.
Cellular solids known as lattice materials1 have recently
emerged as candidate stretch-dominated structures. They
have a stiffness and strength which scales linearly with relative density 4q (in contrast, the Young’s modulus and yield
strength of polymer and metallic foams scale with 4q2 and
4q3=2 respectively). At low relative densities, the lattice materials can therefore be more than an order of magnitude
stiffer and stronger than equivalent mass per unit volume
foams made from the same parent material, Fig. 1. Examples of lattice materials include the Octet-truss structure
with a face-centered cubic microstructure [4] and the Kagome lattice [5]. The joint connectivity of the Octet truss
is 12, and this spatially periodic material has the feature that
the cell members deform by local stretching for all macroscopic loading states [3]. Consequently, the specific
mechanical properties (stiffness, strength, toughness and
energy absorption) of the Octet-truss far exceed those of
open-cell foams. Such materials are also expected to find
applications in lightweight, compact structural heat exchangers [6]. Numerous variants of these lattice materials have
been developed over the past few years by several research
groups (see [7] for a review of this literature) including the
2D and 3D Kagome structure, the simpler to fabricate
pyramidal and tetrahedral lattices and various prismatic
topologies (diamond core, square honeycomb).
Examination of the modified Ashby material property
chart [8] shown in Fig. 1, indicates that aluminum foams
and lattices occupy the low density region of material
strength – density space. It also reveals a gap between the
strength of existing lattice materials and the unattainable
materials limit. Lattices fabricated from aluminum alloys
have begun to extend the range of cellular materials into
this gap in the material property space but it is clear that
there remains much room for further improvements.
Figure 1 also illustrates how the combination of optimized lattice topology and parent material properties can
be combined to expand material property space by creating
new engineering materials. For instance, suppose composites containing fibers configured to provide high uniaxial
specific strengths were used for the trusses or webs of a lattice structure. If buckling does not occur, the resulting lat1
We define lattice materials as periodic micro-architectured cellular
solids. They can be either closed cell or open cell.
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K. Finnegan et al.: The compressive response of carbon fiber composite pyramidal truss sandwich cores
strated in Fig. 2a. Compression tests on specimens manufactured by this methods indicated that failure occurred by
truss push-out through the face sheet (Fig. 3a) as the core
strength increased. This failure mode could only be suppressed by using very thick face sheets making this route
unfeasible for sandwich designs.
(b) The second approach utilized a water-jet cutting process to fabricate single struts of the pyramidal trusses from
a unidirectional fiber reinforced laminate and adhesively
bonding the struts to the face sheets (Fig. 2b). The nodes
of the truss in this case failed by a shear mechanism
(Fig. 3b) associated with the weak transverse strength of
the composite laminate.
The third design suppressed these two weak failure
modes by manufacturing the pyramidal truss cores from
Fig. 1. An Ashby material strength versus density map for engineering
materials [8]. The map contains gaps between existing and unattainable
materials. The maximum theoretical strength of composite lattice
structures is shown by a dashed line which falls into the high specific
strength gap at low densities. The measured properties of the composite pyramidal lattice materials investigated here are also shown.
tice structures have the anticipated compressive strengths
illustrated by the dashed line in Fig. 1. It is clear that topologically structuring composite materials show promise for
filling gaps in the strength versus density map of all known
materials. The aim of the present study is to begin an investigation into the expansion of the strength – density material
space at low densities by using carbon fiber composites to
build lattice materials.
The outline of the paper is as follows. First the composite
laminates used to manufacture the truss cores are described
along with the route for fabricating the pyramidal truss
cores. This included an assessment of node failure mechanisms and development of a node design to suppress this
fracture mode. Second, the measured compressive response
of the cores is then described along with the observed failure modes. Next, analytical models are developed for the
elastic stiffness and collapse strengths of the composite
truss cores and these are compared with measured
strengths. Finally, the measured strengths of the composite
trusses are plotted on a map of density versus strength of
all known materials in order to gauge the performance of
these materials in terms of their strength to weight ratio.
2. Sandwich panel fabrication
Composite pyramidal truss sandwich cores were manufactured from pre-fabricated composite sheet materials. Three
manufacturing approaches were investigated in an attempt
to identify a compromise core design that exploited the
strength of the composite lay-up, ease of fabrication and
avoidance of nodal failure. The first two fabrication routes
were eventually rejected due to premature node failures
(see [9] for details). They are summarized in Fig. 2:
(a) The first approach sought to exploit the high compressive axial strength and low cost of pultruded, unidirectional, fiber reinforced composite rods. These were inserted
in pre-drilled face sheets and adhesively bonded as illu2
Fig. 2. Methods for manufacturing CFRP pyramidal cores with uni-directional fiber-reinforced trusses. (a) Pultruded rod truss members are
adhesively bonded to face sheets containing pre-drilled holes and excess material removed. (b) Truss patterns are cut from unidirectional
laminates and adhesively bonded into milled slots in the face sheets.
Fig. 3. Node failure modes for the uni-directional fiber reinforced
trusses. (a) The pultruded rod core failed at the nodes by truss pushout. (b) The unidirectional laminate cut-out truss core failed by shearing of the node material in a region where the forces were transverse
to the fiber direction.
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0 – 908 laminate sheets and increasing the volume (and surface area) of the joint between the faces-sheets and the core.
We note that in this design only half the fibers are aligned
with the load and thus the design does not exploit the intrinsic strength of fiber reinforced composites fully. However,
this design proved effective at suppressing node failure.
The pyramidal truss sandwich panels tested in this study
were manufactured from 0 – 908 laminate sheets of thickness t ¼ 3 mm in three steps. First, truss patterns as shown
in Fig. 4a were water jet cut from the laminate sheets. Second these patterns were then snap-fitted into each other
(Fig. 4b) to produce a pyramidal truss. Finally, the pyramidal truss was bonded to 3 mm thick composite face-sheets
using an epoxy adhesive (Hysol EP-120). These composite
face-sheets had cruciform shaped slots of depth htab milled
into them at appropriate locations such that the nodes of
the pyramidal truss could be counter-sunk into the facesheets (Fig. 4d) providing both mechanical constraint and
adhesive contributions to the node strength.
The critical parameters describing the geometry of the
pyramidal core are sketched in Fig. 4c 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 h, respectively. The struts made an angle x with the horizontal plane
Fig. 4. Illustration of the manufacturing route for making the composite pyramidal lattice core sandwich panels studied here. (a) Semi-continuous truss patterns are water jet cut from 0/908 laminate sheets. The
fiber directions are shown in this sketch and indicate that a half of the
fibers are in the truss axial direction. (b) The pyramidal lattice is assembled by snap-fitting the truss patterns. (c) The geometry of the truss
pattern with relevant core design variables identified. (d) A schematic
illustration of a pyramidal lattice core sandwich panel. The composite
face-sheets utilized cruciform shaped slots into which the pyramidal
trusses were fitted and adhesively bonded.
Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 12
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(Fig. 4c). The unit cell of the pyramidal core is sketched in
Fig. 5a and 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
4
2ðlt þ hbÞt
2ðl4þ h4bÞ
4q ¼
0
ð1aÞ
2
42
l sin xðl cos x þ bÞ
l4sin xðl4cos x þ bÞ
where the non-dimensional lengths l4 0 l=t, b4 0 b=t and
h4 0 h=t. In the limit of vanishing node volumes (b ¼
h ! 0), this expression reduces to
4
4q 9 42
l sin 2x cos x
ð1bÞ
2.1. Pyramidal core designs
All the pyramidal cores tested and manufactured in this
study had a strut angle x ¼ 45- . Thus, the angle included
between the struts was 908 and the patterns were cut from
the laminate sheets such that half the fibers of the 0 – 908
laminate were aligned along the axis of the struts of the pyramidal core (Fig. 4a). To explore the trade-off between
node surface area and core mass efficiency, two core designs were investigated in this study (Fig. 6). These designs
differed in the design of their nodes with design 1 comprising significantly smaller nodes compared to design 2: the
node dimensions for both designs 1 and 2 are listed in Table 1. For each design four relative densities were investi-
Fig. 5. (a) Sketch of the unit cell of the pyramidal core. (b) Photograph
of the as-manufactured design 1 core (4
q = 3.4 %) in a sandwich panel
structure.
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K. Finnegan et al.: The compressive response of carbon fiber composite pyramidal truss sandwich cores
Fig. 6. Two node designs were investigated in the study. (a) Design 1
was intended to minimize node surface area. (b) Design 2 had a larger
node surface and volume which reduced the core mass efficiency.
Table 1. Node and strut dimensions in mm for both designs 1 and
2 of the composite pyramidal sandwich cores manufactured and
tested in this study. The strut angle x ¼ 45- for both core designs.
Design 1
Design 2
htab
h
w
t
c
b
1.50
1.50
3.05
1.59
3.00
3.00
3.00
3.00
3.82
9.59
6.35
15.33
gated the relative density of the core was controlled by
varying the strut length l while keeping all other geometric
parameters fixed for each core design. A photograph of the
as-manufactured design 1 core (4
q ¼ 3:4%) is included in
Fig. 5b.
2.2. Composite laminate materials
Carbon fiber laminate sheets sourced from two suppliers,
Graphitestore2 and McMaster-Carr3, were used to manufacture the pyramidal sandwich core designs 1 and 2, respectively and are subsequently referred to as laminates 1 and
2, respectively. In both cases, the laminate sheets had a
thickness t ¼ 0:3 mm and comprised 65 % by volume of
228 GPa (33 Msi) carbon fibers in a vinylester matrix. Plies
comprising unidirectional fibers were laid-up alternating at
08 and 908 to build an orthotropic laminate. Laminates 1
and 2 comprised 24 and 14 plies, respectively. The density
of both laminate materials was qs ¼ 1440 kg m%3 .
Both the laminate materials were tested in uniaxial compression along one of the fiber directions in order to deter2
GraphiteStore.com Inc., 1348 Busch Parkway Buffalo Grove,
IL 60089 USA.
3
McMaster-Carr, 6100 Fulton Industrial Blvd. Atlanta, GA 303362852, USA.
mine the relevant Young’s modulus and compressive
strength of the parent material used to manufacture the pyramidal cores. 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
in the pyramidal core. By contrast, Celanese compression
tests provide the micro-buckling failure strength of laminates: it will be subsequently seen that the micro-buckling
failure mode was not observed in the pyramidal core compression experiments reported here and a majority of the
specimens failed by strut delamination. The tests were conducted on rectangular specimens of thickness 0.3 mm,
width 20 mm and gauge length 12 mm: the specimens were
sufficiently stocky to prevent Euler buckling of the specimens. The applied load was measured via the load cell of
the test machine while a laser extensometer was used to
measure the nominal axial strain in the specimens. A nominal applied strain rate of 10%3 s%1 was employed in these
tests. The measured nominal stress versus strain curves of
laminates 1 and 2 are plotted in Fig. 7a: five repeat tests in
each case confirmed the reproducibility of the results. After
some initial “bedding-in”, the laminates display a linear
elastic response followed by delamination failure; see
Fig. 7b. The unloading Young’s modulus was measured as
Es 9 25 GPa while the delamination failure strength
rdl 9 400 MPa, for both laminates.
3. Measurements of the compressive response of the
pyramidal core
Pyramidal sandwich core specimens comprising at least
four pyramidal unit cells were employed in the compression
tests. The face-sheets of the sandwich specimens were
bonded to the platens of the test machine in order to prevent
the relative sliding of the two face-sheets. The compression
tests were conducted using a screw-driven test machine
with the measured load cell force used to calculate the nominal stress and a laser extensometer used to measure the relative approach of the two face-sheets from which the nominal applied strain was inferred. The compression tests were
conducted at a nominal applied strain rate of 10%3 s%1 and
typically unloading-reloading cycles were conducted prior
to the onset of failure in order to determine the elastic modulus of the specimens. Four relative densities of each design
were tested and five repeat tests were conducted in each
case to gauge the variability in the test measurements.
The measured compressive nominal stress versus strain
curves of designs 1 and 2 of the pyramidal sandwich cores
Fig. 7. (a) The measured compressive stress
versus strain response of the laminate materials used to manufacture cores with the two designs. The compressive response was measured along one of the fiber directions of these
0/908 unidirectional ply laminates. (b) Photographs of the specimens after delamination
failure.
4
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K. Finnegan et al.: The compressive response of carbon fiber composite pyramidal truss sandwich cores
are plotted in Fig. 8a and b, respectively for four values of 4q
in each case. For the sake of clarity the unloading – reloading cycles performed during the measurements are removed
from these figures. In all cases an initial linear response is
observed followed by a regime where the stress versus
strain response is nonlinear. Typically the peak stress occurs at the point when failure of the trusses is first observed
as marked by the arrows in Fig. 8. Subsequently, the stresses decrease with increasing strain with the serrations in
the stress versus strain curve associated with a series of failure events in the pyramidal core specimens. The significant
nonlinear behavior prior to attainment of the peak stress
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suggest that strut delamination rather than micro-buckling
is the failure mode in most of the tests.
The measured unloading moduli E of both the designs of
the pyramidal cores are plotted in Fig. 9 while the measured
peak strengths rp of designs 1 and 2 of the pyramidal cores
are included in Fig. 10a and b, respectively. The error bars
in these plots indicate the maximum and minimum values
of the measurements obtained from the five repeat tests
conducted in each case. We observe that both the initial
modulus and peak strength increase with increasing 4q.
Within the variability of the experimental data, the modulus
increases approximately linearly with 4q though rp does not
seem to exhibit a linear scaling with 4q.
3.1. Modes of failure of the pyramidal truss cores
Two failure modes were observed in the compression tests
reported above: (i) Euler buckling and (ii) delamination of
the struts. Photographs of these observed failure modes are
included in Fig. 11. The peak stress for the 4q ¼ 0:01 pyramidal core (design 1) occurs before any visible failure is observed (Fig. 8a) and is associated with the large bending deformations of the struts that precedes the onset of buckling.
Fig. 8. The measured compressive stress versus strain curves of (a) design 1 and (b) design 2 of the composite pyramidal cores. Four relative
densities of each design were tested. The instant of the first detectable
failure are indicated by arrows on each of the plots.
Fig. 9. The measured modulus of core designs 1 and 2. The error bars
indicate the maximum and minimum values obtained from five separate tests. The micromechanical predictions of the modulus for the
two designs are also included.
Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 12
Fig. 10. The measured peak strength of (a) design 1 and (b) design 2 of
the composite pyramidal cores. The error bars indicate the maximum
and minimum values obtained from the five tests. The predictions of
the strength for the two designs are also included based on the Euler
buckling, inter-ply delamination and micro-buckling failure modes of
the struts.
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K. Finnegan et al.: The compressive response of carbon fiber composite pyramidal truss sandwich cores
Fig. 11. Photographs of the compressive failure modes for the composite pyramidal cores.
(a) Euler buckling of the 4
q = 1 % design 1 lattice, (b) delamination failure of the 4q = 3.5 %
design 1 lattice and (c) delamination failure of
the 4
q = 5 % design 2. Delamination failures typically initiated near the truss “ankle”.
These bending deformations cause the struts to fail at approximately mid-span (Fig. 11a). Photographs of delamination failure of designs 1 and 2 of the pyramidal cores are
shown in Fig. 11b and c, respectively. Delamination of the
composite struts was the failure mode for design 2 for all
values of 4q tested. Typically the delamination initiates in
the vicinity of the nodes and propagated along the struts by
a combination of delamination and micro-cracking.
tions dictate that the top end of the strut is only free to move
along the x3 -direction. For an imposed displacement d in the
x3 -direction the axial and shear forces in the strut are given
by elementary beam theory as
FA ¼ Es t 2
d sin x
l
ð2aÞ
4. Analytical predictions of the composite pyramidal
truss core response
We proceed to derive analytical expressions for the “effective” transverse compressive stiffness and strength of the
composite pyramidal cores, sandwiched between two rigid
face-sheets. The pyramidal trusses are made from 0 – 908
laminates such that one set of fibers are aligned with the axial direction of the struts of the pyramidal truss (Fig. 4a).
We define a local Cartesian co-ordinate system ðe1 % e2 Þ
aligned with the orthogonal set of fibers (Fig. 4a). 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 compressive elastic modulus
E of the pyramidal core are 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 an edge clamped strut of length l and square
cross-section of side t as shown in Fig. 12. This represents
a single strut of the pyramidal core. Symmetry considera6
Fig. 12. (a) Sketch of the deformation of a single strut of the pyramidal
core under uniaxial compression and (b) the free-body diagram of a
strut loaded in a combination of compression and shear.
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K. Finnegan et al.: The compressive response of carbon fiber composite pyramidal truss sandwich cores
and
12Es Id cos x
FS ¼
ð2bÞ
l3
respectively, where I 0 t4 =12 is the second moment of area
of the strut cross-section; see Fig. 12b. The total applied
force F in the x3 -direction then follows as
>
?
9 t :2
Es t 2 d
F ¼ FA sin x þ FS cos x ¼
sin2 x þ
cos2 x ð3Þ
l
l
Now consider the unit cell of the pyramidal core sketched in
Fig. 5a. The through-thickness nominal stress r and strain e
applied to the pyramidal core are related to the force F and
displacement d via
r0
8F
ð2l cos x þ 2bÞ2
ð4aÞ
and
d
ðbÞ
l sin x
respectively. The effective Young’s modulus E 0 r=e of
the pyramidal core then follows from Eqs. (3) and (4) as
>
?
E
2l4sin x
cos2 x
2
¼
sin x þ 42
ð5Þ
4 l4cos x þ bÞ
42
Es lð
l
e0
where l4 0 l=t and b4 0 b=t are non-dimensional geometric
parameters of the core. In the limit of negligible node volumes (i. e. b4 ¼ h4 ! 0), the modulus E is related to the relative density 4q of the core via
>
?
E
2 sin x
cos2 x
2
sin
x
þ
9 42
Es l cos2 x
l42
¼ 4q sin4 x þ
4q2 3
sin x cos4 x
2
ð6Þ
The first and second terms in Eq. (6) represent the contributions to the stiffness of the pyramidal core due to the
stretching and bending of the struts, respectively.
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 will
be the one associated with the lowest value of the collapse
strength. Typically polymer matrices of fiber composites
display non-linear behavior [10] and thus elastic microbuckling is not an operative failure mode and not considered in the collapse calculations presented here.
(i) 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 [11] argued that the compressive strength rmax is given by
sY
ð7Þ
rmax ¼
4
u
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for a composite comprising inextensional fibers and a
rigid – ideally plastic matrix of shear strength sY . Kinking
initiates from a local region of fiber misalignment of angle
4. It is assumed that the micro-buckle band is transverse to
u
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, Budiansky and Fleck
[12] have shown that the micro-buckling stress is given by
rc ¼
sY % s1
4
u
ð8Þ
Prior to the micro-buckling of the struts, the struts are elastic and the analysis of Section 4.1 applies. Thus, from
Eqs. (2) and (8) it follows that the axial stress rc required
to initiate micro-buckling in the inclined strut sketched in
Fig. 12 is given by
sY
rmax
ð9Þ
rc ¼ >
9 t :2 ? ¼ >
9 t :2 ?
cot
x
4 þ cot x
u
1 þ u4
l
l
where rmax is the micro-buckling strength of the laminate
for loading in the e1 -direction in the absence of remote
shear. Recall that the normal force F per strut follows from
Eq. (2) and (3) as
>
9 t :2 ?
2
2
F ¼ rc t sin x 1 þ cot x
ð10aÞ
l
Then, taking into account that the unit cell of the pyramidal
core comprises four such struts, the nominal through thickness compressive strength of the pyramidal core follows
from Eq. (4a) as
2rc sin x½l42 þ cot2 x,
ð10bÞ
rp 0
42
l42 ðl4cos x þ bÞ
Combining Eqs. (9) and (10b), the strength of the pyramidal
core in terms of the micro-buckling strength rmax of the
laminate is given by
rp
2 sin x½l42 þ cot2 x,
>
?
ð11Þ
¼
rmax
4 2 l42 þ cot x
ðl4cos x þ bÞ
4
u
In the limit of vanishing node volume, the above expression
reduces to
rp
2 sin x½l42 þ cot2 x,
>
?
9
cot x
rmax 42
2
2
4
l cos x l þ
4
u
>
?
4q cos4 x
sin2 x4q
? 1þ
¼>
2
4q cos3 x
1þ
24
u
ð12Þ
(ii) Delamination failure of the struts
In this mode we neglect the shear stresses in the struts and
consider them as pin-jointed at their ends. An upper bound
work calculation then gives the strength of the pyramidal
core in terms of the delamination failure stress rdl of the
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composite struts as
rp
2 sin x
¼
42
rdl ðl4cos x þ bÞ
ð13Þ
In the limit of vanishing node volume, the above expression
reduces to
4q
rp
2 sin x
9
¼ sin2 x
rdl l42 cos2 x 2
ð14Þ
(iii) 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 load of an endclamped strut subjected to an axial load is given by
PE ¼
4p2 Es I
l2
ð15Þ
Thus, the nominal compressive collapse strength of the pyramidal core due to the elastic buckling of the constituent
struts is given by
rp
2p2 E4s sin x½l42 þ cot2 x,
¼
42
rmax
3l44 ðl4cos x þ bÞ
ð16Þ
where E4s 0 Es =rmax . Note that here we have assumed that
the buckling load (13) is unaffected by the transverse shear
loading of the strut. In the limit of vanishing node volume,
the above expression reduces to
rp
rmax
2p sin x½l42 þ cot2 x,
9 E4s
3l44 cos2 x
2
>
?
4
p2 cos2 x sin3 x4
q2
q cos4 x
4
¼ Es
1þ
6
2
ð17Þ
In order to illustrate the optimal performance of the composite pyramidal cores, the predicted normalized peak
strength of the composite pyramidal core rp =ð4
qrmax Þ is
plotted in Fig. 13 as a function of relative density 4q considering only the micro-buckling and Euler buckling failure
mechanisms of the struts. Predictions are shown in Fig. 13
for three selected values of E4s representative of unidirectional (E4s ¼ 167), laminated (E4s ¼ 116) and woven
(E4s ¼ 50) carbon fiber composites. For the purposes of
illustration, in Fig. 13 we have neglected the volume of the
nodes and thus employed Eqs. (12) and (17) for the microbuckling and Euler buckling collapse strengths, respec4 ¼ 2- : most experitively with the choices x ¼ 45- and u
mental evidence [10] suggests that the imperfection angle
cannot be reduced below 2- in practical designs. The normalized strength rp =ð4
qrmax Þ is a measure of the efficiency
of the topology in terms of its structural strength with
rp =ð4qrmax Þ 4 1: rp =ð4
qrmax Þ ¼ 1 corresponds to a cellular
material that attains the Voigt upper bound. We note:
(a) The normalized strength rp =ð4
qrmax Þ peaks at a 4q value at which the failure modes transition from Euler buckling to micro-buckling. Designs at this transition value of 4q
are most efficient in terms of their strength to weight ratio.
8
Fig. 13. Predictions of the variation of the normalized peak strength
rp =ð4
qrmax Þ with relative density 4
q for three selected values of the normalized laminate modulus E4s . The predictions assume that the node
volume is negligible.
This is rationalized by noting that in the Euler buckling regime the structural efficiency increases with increasing 4q
as the struts become more stocky resulting in an increase
in their Euler buckling loads. By contrast, in the microbuckling regime, with increasing 4q the shear forces on the
struts increase resulting in a decrease in their micro-buckling stress as per Eq. (8).
(b) The maximum value of rp =ð4qrmax Þ for the pyramidal
cores increases with increasing E4s with the transition from
Euler buckling failure to micro-buckling then occurring at
lower values of 4q.
4.3. Comparison with measurements
The analytical predictions (Eq. 5) of the modulus E for both
designs of the pyramidal core are included in Fig. 6 along
with the measurements. In the analytical predictions we
take x ¼ 45- and Es ¼ 25 GPa for both core designs. The
model in general under-predicts the measured values of the
modulus by about 20 % especially for high values of 4q. We
attribute this to (i) the simple beam theory employed in the
analytical predictions becoming less accurate for stubby
beams (i. e. for l=t < 8) resulting in the model under-predicting the measurements at large 4q and (ii) errors in the
measurement of the Young’s modulus of the laminate in
the column compression tests: small misalignments in these
tests are expected to give large reductions in the measured
modulus.
Comparisons between the peak strength measurements
and predictions (Eqs. (13) and (16) for the collapse
strengths by the delamination and Euler buckling modes,
respectively) are included in Fig. 10a and b for core designs 1 and 2, respectively. In line with measurements, the
Young’s modulus and delamination strengths for both laminates are taken as Es ¼ 25 GPa and rdl ¼ 400 MPa, respectively (Fig. 7a). Good agreement is observed between the
measurements and predictions based on the Euler buckling
and delamination failure modes. For comparison purposes,
the predictions of the collapse strength due to the microbuckling failure of the struts are also included in Fig. 10.
Here we have assumed that the micro-buckling strength
4 ¼ 2- , consistent with a wide body of
rmax ¼ 2 GPa and u
experimental data for polymer matrix fiber composites [10].
Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 12
MK_mk101594 – 23.10.07/druckhaus köthen
K. Finnegan et al.: The compressive response of carbon fiber composite pyramidal truss sandwich cores
The collapse strengths based on the micro-buckling failure of
the struts significantly overpredict the measurements over
most of the relative density range investigated here.
It is worth noting that the analytical model predicts that
the peak strength of the design 2 core decreases with increasing relative density for 4
q > 0:05. This is rationalized
by noting that with increasing 4
q an increasing fraction of
the composite material is present in the nodes of the pyramidal core and thus not contributing to the overall load carrying capacity of the core. Moreover, the shear stresses in the
struts also increase with increasing 4
q. These two factors together result in the peak strength decreasing with increasing
4
q above a critical value of 4
q. Note that since the nodes of the
design 1 are about half the size of the nodes of the design 2
cores, the critical density above which rp decreases with increasing 4q is significantly higher and outside the scale of
Fig. 10a.
5. Filling of material property space
The measured peak strengths of designs 1 and 2 of the composite pyramidal cores are included in Fig. 1. The density of
the cores is given as q ¼ 4
qqs with qs ¼ 1440 kg m%3 .
Clearly, the composites cores investigated here begin to fill
a gap in the known material property space in that they have
a strength greater than most known materials with densities
less 100 kg m%3 . However, the current designs of these
composite pyramidal cores do not achieve the full potential
of composite lattice materials primarily due to the inefficient utilization of material in the nodes: the node volume
in the current designs is excessively large. Also included
in Fig. 1 is the theoretical prediction of the maximum
strength of the pyramidal core based on the micro-buckling
failure mode of the struts Eq. (17) with an assumed compo4 ¼ 2- and solsite micro-buckling strength rmax ¼ 2 GPa, u
%3
id material density qs ¼ 1440 kg m (i. e. the prediction
included in Fig. 13 but with elastic buckling of the struts neglected). These theoretical predictions clearly reveal the
potential of composite lattice materials in filling gaps in
the material property space. This potential can be achieved
by appropriately designing composite truss cores such that
(i) the delamination failure mode is eliminated via suitable
node designs; (ii) the Euler buckling strength is increased
by enhancing the second moment of area of the struts of
the pyramidal core by say employing composite tubes
rather than solid members as the core struts and (iii) optimizing the volume of material in the nodes so as to prevent
node failure at a minimum node volume.
Applied
general good agreement between the measurements and
the predictions is obtained. We have demonstrated that
composite cellular materials with a pyramidal micro-structure begin to fill a gap in the strength versus density material property space. However, current designs of the pyramidal cores have not optimized the node design and thus use
material rather inefficiently. Moreover, the current designs
undergo delamination failure of the struts and thus do not
achieve the full potential of composite cores as predicted
by the micro-buckling analysis presented here.
We are grateful to the Office of Naval Research (ONR) for funding of
this project under grant number N00014-01-1-1051 (Program manager,
Dr. David Shifler).
References
[1] L.J. Gibson, M.F. Ashby: “Cellular Solids: Structure and Properties”, 2nd ed. Cambridge University Press, Cambridge (1997).
[2] M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.G. Wadley (Eds.): Butterworth Heinemann, Metal
foams: A design guide (2000).
[3] V.S. Deshpande, M.F. Ashby, N.A. Fleck: Acta. Mater. 49 (2001)
1035 – 1040.
[4] V.S. Deshpande, N.A. Fleck, M.F. Ashby: J. Mech. Phys. Solids.
49 (2001) 1747 – 1769.
[5] R.G. Hutchinson, N.A. Fleck: J. Mech. Phys. Solids. 54 (2006)
756 – 782.
[6] J. Tian, T.J. Lu, H.P. Hodson, D.T. Queheillalt, H.N.G. Wadley:
International Journal of Heat and Mass Transfer. 50 (2007)
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[7] H.N.G. Wadley, N.A. Fleck, A.G. Evans: Compos. Sci. Technol.
63 (2003) 2331 – 2343.
[8] M.F. Ashby, Y.J.M. Bréchet: Acta Mater. 51 (2003) 5801 – 5821.
[9] K. Finnegan: “Carbon fiber composite pyramidal lattice structures” Masters Thesis, Department of Engineering Physics. University of Virginia, 2007.
[10] N.A. Fleck (Ed.): Compressive Failure of Fiber Composites. Advances in Applied Mechanics, Vol 33. Cambridge University
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[11] A.S. Argon (Ed.): “Fracture of composites” Treatise of Material
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(Received June 22, 2007; accepted October 2, 2007)
Bibliography
DOI 10.3139/146.101594
Int. J. Mat. Res. (formerly Z. Metallkd.)
98 (2007) 12; page & – &
# Carl Hanser Verlag GmbH & Co. KG
ISSN 1862-5282
6. Concluding remarks
Correspondence address
A preliminary investigation of the use of pyramidal lattice
core sandwich structures fabricated from carbon fiber reinforced polymers has been conducted. Pyramidal truss sandwich cores with relative densities 4
q in the range 1 – 10 %
have been manufactured from carbon fiber reinforced polymer laminates by employing a snap-fitting method. The
measured quasi-static uniaxial compressive strength varied
in the range 1 – 11 MPa; increasing with increasing 4q. Two
failure modes were observed: (i) Euler buckling of the
struts and (ii) delamination failure of the struts.
Analytical models have been developed for the elastic response and collapse strengths of the composite cores. In
H. N. G. Wadley
Department of Materials Science and Engineering
395 McCormick Road
Wilsdorf Hall, P.O. Box 400745
Charlottesville, VVA 22904
Tel.: +1 434/982 5671
Fax: +1 434/982 5677
E-mail: [email protected]
Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 12
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