of 12
ORIGINAL RESEARCH
published: 31 July 2019
doi: 10.3389/fmats.2019.00169
Frontiers in Materials | www.frontiersin.org
1
July 2019 | Volume 6 | Article 169
Edited by:
Seunghwa Ryu,
Korea Advanced Institute of Science
and Technology (KAIST), South Korea
Reviewed by:
Dongchan Jang,
Korea Advanced Institute of Science
and Technology (KAIST), South Korea
Anastasiia O. Krushynska,
University of Groningen, Netherlands
*Correspondence:
Diego Misseroni
diego.misseroni@unitn.it
Specialty section:
This article was submitted to
Mechanics of Materials,
a section of the journal
Frontiers in Materials
Received:
14 January 2019
Accepted:
01 July 2019
Published:
31 July 2019
Citation:
Kudo A, Misseroni D, Wei Y and
Bosi F (2019) Compressive Response
of Non-slender Octet Carbon
Microlattices. Front. Mater. 6:169.
doi: 10.3389/fmats.2019.00169
Compressive Response of
Non-slender Octet Carbon
Microlattices
Akira Kudo
1
, Diego Misseroni
2
*
, Yuchen Wei
1
and Federico Bosi
3
1
Division of Engineering and Applied Science, California Ins
titute of Technology, Pasadena, CA, United States,
2
Department
of Civil, Environmental and Mechanical Engineering, Unive
rsity of Trento, Trento, Italy,
3
Department of Mechanical
Engineering, University College London, London, United Ki
ngdom
Lattices are periodic three-dimensional architected soli
ds designed at the micro and
nano-scale to achieve unique properties not attainable by t
heir constituent materials. The
design of lightweight and strong structured solids by addit
ive manufacturing requires the
use of high-strength constituent materials and non-slende
r geometries to prevent strut
elastic instabilities. Low slenderness carbon octet micro
lattices are obtained through
pyrolysis of polymeric architectures manufactured with st
ereolithography technique. Their
compressive behavior is numerically and experimentally in
vestigated when the relative
density
N
ρ
ranges between 10 and 50
%
, with specific stiffness and strength approaching
the limit of existing micro and nanoarchitectures. It is sho
wn that additive manufacturing
can introduce imperfections such as increased nodal volume
, non-cubic unit cell, and
orientation-dependent beam slenderness, all of which deep
ly affect the mechanical
response of the lattice material. An accurate numerical mod
eling of non-slender octet
lattices with significant nodal volumes is demonstrated to o
vercome the limitations of
classical analytical methods based on beam theory for the pr
ediction of the lattice
stiffness, strength and scaling laws. The presented numeri
cal results and experimental
methods provide new insights for the design of structural ca
rbon architected materials
toward ultra-strong and lightweight solids.
Keywords: architected materials, additive manufacturing, st
ructural metamaterials, pyrolyzed lattices, mechanics
1. INTRODUCTION
Additive manufacturing has become one of the most promising t
echnique to fabricate advanced
materials and microstructures that exhibit properties unatt
ained by homogeneous solids or
conventionally manufactured architectures. The available
3D printing techniques have recently
grown and comprise fused deposition modeling (FDM), direct ink w
riting (DIW), selective laser
sintering (SLS), stereolithography (SLA), etc. Similarly,
the selection of materials compatible
with these processes has expanded and include thermoelastic pol
ymers (
Carneiro et al., 2015
),
transparent glasses (
Nguyen et al., 2017
), oxide ceramics (
Wilkes et al., 2013
), metallic alloys
(
Schwab et al., 2016
), and composites (
Spierings et al., 2015; Ni et al., 2018; Quintanilla et al., 20
18
).
The precise micro- and nano-scale topology control achievabl
e through additive manufacturing
has allowed the development of unique functionalities to cat
alysis (
Essa et al., 2017
), batteries (
Xia
et al., 2016; Li et al., 2017
), scaffolds (
Maggi et al., 2017
), biomedical implants (
Murr et al., 2010
), and
Kudo et al.
Mechanics of Octet Carbon Microlattices
metamaterials (
Hengsbach and Lantada, 2014; Misseroni et al.,
2016; Bertoldi et al., 2017; Bilal et al., 2017
). In particular, the field
of architected material has benefited from the advancement o
f
small-scale manufacturing that enables the design of multi
stable
solids for energy storage (
Shan et al., 2015
), the evolution of
phononic bandgap behavior (
Sugino et al., 2015; Amendola
et al., 2018
) and the exploration of previously inaccessible
mechanical property combinations (
Bauer et al., 2016
). Examples
include structural metamaterials designed to achieve extr
emely
lightweight and strong solids through a hierarchical desig
n
(
Meza et al., 2015
) or novel highly deformable and recoverable
nanolattices made up of brittle materials (
Meza et al., 2014
).
Structured solids can be classified as rigid or non-rigid
architectures depending on their nodal connectivity, state
s of
self stress, and mechanisms (
Pellegrino and Calladine, 1986
).
The former includes octet lattices and shows a stretching
dominated behavior, while the latter mostly presents a bendi
ng
dominated response as demonstrated by pyramidal lattices. Th
e
response of architected materials has been extensively analy
zed
through the investigation of their constituent unit cells u
sing
beam theory to obtain the lattice effective stiffness and streng
th
scaling laws (
Gibson and Ashby, 1997; Deshpande et al., 2001
).
These analytical tools have been proven to well predict the
mechanical response of several lattices when the relative de
nsity
N
ρ
is lower than 0.1 and the strut slenderness ratio
r
/
l
does not
exceed 0.06 (
Meza et al., 2017
). However, some computational
and experimental studies (
Schaedler et al., 2011; Meza et al.,
2015; Bauer et al., 2016
) have recently reported deviations
from the classical scaling laws due to non-slender struts and
the influence of the node geometry (
Portela et al., 2018
),
thus proposing different scaling laws. The difficult micro- and
nano-scale fabrication of slender structured solids that o
bey to
classical scaling laws motivates the investigation of non-
slender
architectures with pronounced nodal volume caused by an
imperfect 3D printing. Therefore, the study of their mechanic
al
properties is fundamental for the design of stronger lattices
that
do not suffer from strut elastic instabilities.
One of the most promising materials to fabricate extremely
lightweight and resistant architected solids is carbon, wh
ich
has recently become compatible with additive manufacturing
processes. Direct ink writing (DIW) with printable inks that
contain graphene, carbon nanotube, and graphene oxide (
Fu
et al., 2017
) has been employed for the realization of flexible,
conductive, and chemically stable prototypes (
Sun et al., 2013;
Zhu et al., 2015; Yao et al., 2016; Zhang et al., 2016
),
while 3D-printed carbon fiber reinforced composites have
been manufactured by means of FDM (
Lewicki et al., 2017;
Anwer and Naguib, 2018
). Carbon nano- and micro-lattices are
another form of 3D printed carbon which have demonstrated
elevated structural performances. Architected carbon mate
rials
are obtained by pyrolyzing 3D-printed precursor, especially
polymer lattices prepared by photocuring techniques. Carbon
nanolattices fabricated through two-photon lithography hav
e
shown a strength comparable with the theoretical strength of
flaw insensitive glassy carbon (
Bauer et al., 2016
). This printing
technique solidifies the polymeric precursor solution point-
by-point at a submicron scale in a prolonged process, thus
preventing the production of micro- and nano-architectures
at a large scale. Carbon microlattices produced by self-
propagating photopolymer waveguides (
Jacobsen et al., 2011
)
and stereolithography (SLA) (
Chen et al., 2017
) overcome the
scalability difficulties toward faster manufacturing of lar
ger
scale lattices. However, their mechanical performances are
still limited, and the development of enhanced architected
solids demands further understanding of the influence of the
manufacturing-induced imperfection on the mechanics of 3D-
printed carbon lattices.
The aim of this work is to manufacture stiff and strong
non-slender octet carbon microlattices through digital ligh
t
processing stereolithography (DLP-SLA), and to analyticall
y,
computationally, and experimentally investigate their
compressive mechanical properties and scaling laws. We show
that DLP-SLA 3D printing and pyrolysis techniques can affect
the designed lattice architecture introducing undesired f
eatures
as increased nodal volume, non-cubic unit cell and different st
rut
slenderness depending on the beam orientation with respect
to the printing direction. We investigate the influence of the
se
factors on the compressive stiffness and strength of non-slend
er
lattices with relative density
N
ρ
that ranges between 10 and 50%.
We prove the inappropriateness of classical analytical tools bas
ed
on beam theory and the derived expressions for non-slender
architectures with negligible effective Poisson’s ratio, du
e to
the topological features that are not accounted for in these
formulations. We develop computational models that faithfu
lly
predicts the experimental lattice response by reproducing the
manufactured geometry and we demonstrate that an accurate
numerical modeling of non-slender octet lattices with signi
ficant
nodal volumes allows to identify the deviation from classica
l
scaling laws and enables a proper design of advanced structura
l
DLP-SLA 3D printed carbon architectures.
2. MATERIALS AND METHODS
2.1. Sample Fabrication
Three sets (
A
,
B
, and
C
) of carbon octet microlattices were
manufactured by pyrolyzing polymeric lattices fabricated wit
h
a DLP-SLA Autodesk Ember 3D printer that employs a PR-48
transparent photoresist resin. The periodic polymeric 3D printe
d
specimens consisted of a 10
×
3
×
6 (length
×
width
×
height)
tassellation of 900

m octet unit cell with three different strut
radii of
r
A
=
52.8

m,
r
B
=
71.4

m, and
r
C
=
90.0

m (
Figure 1
). The microlattices presented a theoretical relative
density
N
ρ
A
=
0.16,
N
ρ
B
=
0.27,
N
ρ
C
=
0.40, and a beam slenderness
ratio of
r
/
l

A
=
0.08,
r
/
l

B
=
0.11,
r
/
l

C
=
0.14. The
DLP-SLA layers were deposited along the height direction and
the bottom-most anchoring layer was designed to be thicker i
n
order to sustain the microlattice. Prior to pyrolysis, the anch
oring
layer of each microstructure was removed using a razor blade
to prevent lattice distortion, thus resulting in a 10
×
3
×
5
tassellation. The polymeric lattices were inserted in a fused
quartz
tube set on a Lindberg tube furnace (model 54357) and pyrolyze
d
under vacuum. During pyrolysis, the furnace temperature was
first raised to 300
C and held constant for 4 h, then increased
to 400
C and maintained for 1 h, and finally elevated to 1,000
C
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July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
FIGURE 1 | (A)
CAD of the microlattice constructed through a three-dimens
ional tasselation of octet unit cells.
(B)
A cubic octet unit cell with its geometric
parameters
l
(strut length) and
r
(strut radius).
and kept constant for 4 h. This procedure, with all heating rate
s
carried out at 10
C/min, led to fully dense microlattices without
gasified components and preserved the 3D printed topology.
2.2. Microstructural Characterization
The polymeric and carbon microlattices obtained after 3D
printing and pyrolysis were investigated using a Thermo-
Fisher Versa 3D DualBeam Scanning Electron Microscopy
(SEM). Specimen sizes, unit cell radii and strut diameters
were measured to assess any imperfection and anisotropy
introduced during fabrication. The compressive tests on the
carbon microlattices were performed using an Instron 5569
electromechanical machine. The load
P
was applied by imposing
a constant displacement rate of 2.5

m/s on the 5
×
10 unit
cells sample surface and was measured with an Instron 2525-
804 load cell (R.C. 10 kN). The compressive displacement
1
y
was evaluated with an LE-01 (Electronic Instrument Research
)
laser extensometer interfaced with the electromechanical
testing
frame for data synchronization. Five samples for each set of oc
tet
density were characterized and tested along the y-directio
n. The
nominal lattice stress
σ
was obtained dividing the applied load
P
by the specimen footprint area, whereas the nominal strain
ǫ
was
calculated from the initial sample height
H
as
ǫ
=
1
y
/
H
.
2.3. Finite Element Analysis
Full three-dimensional finite element analyses were performe
d
in ABAQUS Standard 2018 in order to simulate the compressive
behavior of the microlattices. Numerical simulations were
carried
out on representative octet unit cells loaded in the y-direct
ion
and constrained with boundary conditions that reproduce the
response of the unit cell within the lattice. The three-dimen
sional
unit cells were parametrically designed in SolidWorks to
reproduce the measured geometrical parameters such as node
and beam radii, unit cell height and width, and fillet junction
radii between nodes and struts. In addition to the unit cells
representative of the fabricated microlattices, other unit
cells with
different relative density were modeled to investigate the st
iffness
and strength scaling laws. The microlattices were discretiz
ed with
linear elastic second-order tetrahedral elements (C3D10)
with
Young’s modulus
E
= 25.38 GPa (Kudo et.al., in preparation) and
Poisson coefficient
ν
= 0.21 (
Price and Kaae, 1969
). Compressive
forces were applied at the top nodes of the unit cell, with
F
acting
on the central node and
F
/
4 on the lateral nodes shared with
four adjacent unit cells. The effective lattice Young’s modul
us
E
y
=
E
x
in the y-direction was calculated as the ratio between
the average compressive stress on the unit cell
σ
avg
y
=
2
F
/
wh
and the average compressive strain
ǫ
avg
y
=
δ
y
/
w
, where
δ
y
represents the displacement in the y-direction,
w
and
h
are
the width and height of the unit cell (
Figure 2
). Similarly, the
effective Young’s modulus
E
z
in the z-direction was obtained as
the ratio between the average compressive stress
σ
avg
z
=
2
F
/
w
2
and the average compressive strain
ǫ
avg
z
=
δ
z
/
h
, where
δ
z
represents the displacement in the z-direction. Furthermore,
a
linear perturbation buckling analysis was conducted on each
unit
cell to assess the microlattices critical buckling strengt
h.
3. RESULTS AND DISCUSSION
3.1. Manufacturing
The geometrical features of the three sets of non-slender 3D
-
printed polymeric microlattices were investigated by using th
e
SEM. From the images in
Figure 3
, it was observed that the
additive manufacturing technique introduced undesirable
lattice
imperfections not present in the original computer-aided desi
gn
(CAD) input files, where the lattices were constituted of cubic
octet unit cells with strut radius
r
and length
l
. In particular, the
height
h
of the unit cell resulted smaller than the width
w
, leading
to a non-cubic unit cell. Therefore, the length
l
1
of the struts
deposited along the width direction (x–y plane) was greater th
an
the length
l
2
of the inclined struts 3D-printed along the height
direction (z-direction). Similarly, the in-plane strut rad
ius
r
1
resulted bigger than the out-of-plane beam radius
r
2
. Moreover,
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July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
FIGURE 2 |
CAD isometric
(A)
and lateral
(B)
views of a representative 3D printed and pyrolyzed non-slen
der octet unit cell as modeled for the finite element analysis.
The x-y in-plane beams present radius
r
1
and length
l
1
, while the out-of-plane struts have radius
r
2
and length
l
2
. The struts junction is characterized by the node
radius
r
N
and junction radius
r
j
.
FIGURE 3 |
SEM images of the DLP-SLA manufactured polymeric microlatt
ices
A
(A)
,
B
(B)
, and
C
(C)
observed from the x-z plane. The imperfections introduced b
y
additive manufacturing, visible in all the samples, are ori
entation dependent strut slenderness, increased nodal vol
ume, and corrugation on the beam surface.
the intersection of twelve struts in a single point produced qu
asi-
spherical nodes with radius
r
N
and the microlattices presented
unsought corrugated surfaces (
Figure 3
). The deviations from
the originally designed lattices were caused by the DLP-SLA
3D printer, whose resolution was limited to 50

m in the x-
y plane and the layer thickness was 25

m. For these reasons,
stereolithography appears inadequate to precisely manufacture
features as small as a few tens of microns, which would requir
e
other 3D printing techniques as two-photon lithography (
Meza
et al., 2014; Bauer et al., 2016
). The average strut slenderness for
the polymeric lattices was
r
1
/
l
1

p
,
xy
A
=
0.10,
r
1
/
l
1

p
,
xy
B
=
0.12,
and
r
1
/
l
1

p
,
xy
C
=
0.15 for the struts deposited in x-y plane, and
for the inclined struts printed along the z-direction
r
2
/
l
2

p
,
z
A
=
0.07,
r
2
/
l
2

p
,
z
B
=
0.09, and
r
2
/
l
2

p
,
z
C
=
0.12. By comparing
the designed struts (from the CAD input files) and the beams
of the manufactured lattices, we observed a decrease in the i
n-
plane struts slenderness and an increase in the out-of-plane b
eam
slenderness. This discrepancy is caused by the stereolithog
raphy
3D printing.
The polymeric lattices were subjected to pyrolysis that led
to a
70% linear shrinkage and produced pyrolytic carbon
microlattices (
Chen et al., 2017
) (
Figure 4
, left). The three
sets of pyrolyzed lattices were examined with SEM and their
averaged geometrical parameters were used to define the octet
unit cell CAD geometry employed in the numerical simulations
(
Table 1
).
Figure 4
shows progressive magnification images
which testify that the features present in the original polyme
ric
lattices were preserved throughout the heat treatment for al
l
lattices. It was observed that the strut slenderness after py
rolysis
slightly decreased with respect to the polymeric lattices val
ues
(maximum deviation was
10%), hence the heat treatment
did not significantly modify the lattice geometry. The avera
ge
ratio between the unit cell height and width was measured as
(
h
/
w
)
A
=
0.87 and (
h
/
w
)
B
,
C
=
0.97, thus showing a quasi-
cubic architecture for lattices
B
and
C
. The ratio between the
out-of-plane and in-plane strut radii was
r
2
/
r
1
0.8 and the
quasi-spherical nodes presented a radius
r
N
2
r
1
. The highest
magnification images report the lateral view of the octet unit
cell
and are paired with the CAD geometry, showing an excellent
agreement between the manufactured and simulated unit cells
(
Figure 4
, right). The only feature not reproduced in the CAD,
and therefore not accounted for in the finite element analysi
s, was
the strut corrugation.
3.2. Relative Density
The relative density of the polymeric lattices was measured b
efore
pyrolysis as
N
ρ
p
A
=
0.17,
N
ρ
p
B
=
0.27, and
N
ρ
p
C
=
0.39, thus proving
that 3D printing did not alter the desired relative density. Af
ter
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July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
TABLE 1 |
SEM measured geometrical parameters of the octet unit cell f
or the manufactured carbon microlattices.
Unit cell
w
h
r
1
r
2
r
N
r
j
̄
ρ
̄
ρ
CAD
̄
ρ
t
[
μ
m]
[
μ
m]
[
μ
m]
[
μ
m]
[
μ
m]
[
μ
m]
[-]
[-]
[-]
A
230.35
200.78
15.02
11.33
32.00
26.27
0.16
0.17
0.16
B
241.68
233.72
18.38
15.15
40.53
30.79
0.23
0.23
0.23
C
251.02
244.61
24.60
19.88
46.19
33.14
0.34
0.32
0.32
The experimental relative density
N
ρ
is reported together with the relative density
N
ρ
CAD
obtained from the CAD and the relative density
N
ρ
t
calculated through the analytical expression (1).
FIGURE 4 |
Optical microscopy (left), SEM characterization (center)
, and CAD model (right) of the manufactured microlattices
A
(A)
,
B
(B)
, and
C
(C)
before and after
pyrolysis. Progressive magnification images show that pyrol
ysis preserved the features of the polymeric lattices. The c
omparison between the fabricated carbon unit
cell and the CAD model employed for computational analysis h
ighlights the accuracy of the reproduced architectures.
pyrolysis, the density
ρ
of the carbon microlattices was measured
as
ρ
A
=
0.29
±
0.02 g/cm
3
,
ρ
B
=
0.43
±
0.02 g/cm
3
, and
ρ
C
=
0.62
±
0.03 g/cm
3
for the three sets of manufactured
microlattices. Considering the density of pyrolytic carbon
ρ
c
=
1.85 g/cm
3
(Kudo et.al., in preparation), their relative densities
were
N
ρ
A
=
0.16
±
0.01,
N
ρ
B
=
0.23
±
0.01, and
N
ρ
C
=
0.34
±
0.02.
The relative density of the carbon microlattices slightly d
ecreased
with respect to the polymeric lattices, as an effect of the pyrolysi
s.
The theoretical expression for the relative density of a non-
cubic
octet unit cell in which in-plane cylindrical struts of radiu
s
r
1
and length
l
1
and out-of-plane cylindrical beams of radius
r
2
and
length
l
2
converge into spherical nodes of radius
r
N
can be written
as
N
ρ
t
=
2
2
π
r
2
1
(
l
1
2
r
N
)
+
2
r
2
2
(
l
2
2
r
N
)

l
2
1
l
2
+
5
2
3
π
r
3
N
l
2
1
l
2
, (1)
where the first term accounts for the struts volume while
the second term considers the nodal volume. Although the
previous relation neglects the node-strut junctions and the b
eam
corrugations, it well approximates the experimental relative
density, with a maximum error of 4% over the three sets
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July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
FIGURE 5 |
Experimental stress-strain curves for the three sets of mic
rolattices considered:
A
(A)
,
B
(B)
, and
C
(C)
. Five samples were tested for each relative density
considered. The insets show the modeled unit cells and their
stress distribution during compression.
of microlattices (
Table 1
). The same table reports the relative
density
N
ρ
CAD
extrapolated from the unit cell drawing developed
with the measured geometrical parameters. The CAD relative
density resulted in excellent agreement with the measured
relative density even if it does not consider strut corrugat
ion.
3.3. Stiffness
The effective Young’s modulus
E
of an ideal octet-truss lattice
with beam radius
r
and length
l
was first examined by
Deshpande
et al. (2001)
as a stretching dominated solid with pin-jointed
struts. Later, several authors investigated the octet latt
ice as a
frame architecture and demonstrated that the assumption of
negligible bending effects is valid for a relative density
N
ρ
lower
than 0.1 and strut slenderness
r
/
l
lower than 0.06 (
Dong et al.,
2015; He et al., 2017
). From the analysis of the octet unit cell free
to deform laterally due to Poisson effect, the lattice stiffness
is
E
=
2
2
π
3
E

r
l

2
K
b
,
(2)
where
E
is the constituent material Young’s modulus, and
K
b
is
a coefficient that accounts for bending effects. In particular, f
or
truss architectures with pin-joined struts
K
b
=
1, while frame
lattices that show bending effects have nodal rigidity
K
b
>
1, and
K
b
can be written as
K
b
=
1
+
15

r
l

2
+
36

r
l

4
1
+
7

r
l

2
.
(3)
These expressions obtained by means of the beam theory predict
that the effective Poisson’s ratio
ν
is independent of the relative
density and equal to 0.33. However, numerical simulations h
ave
proven that both rigid (
Tancogne-Dejean et al., 2016
) and non-
rigid (
Thiyagasundaram et al., 2010
) architectures experience
a decrease of
ν
with an increase of
N
ρ
. Lattices with high
relative density behave as bending-dominated structures w
ith
irrotational nodes and show limited lateral expansion when
compressed. Similarly, the microlattices studied in this wo
rk
presented a relative density
N
ρ
j
greater than 0.1 and non-slender
struts (
r
i
/
l
i
)
j
>
0.06 for
i
=
1, 2 and
j
=
A
,
B
,
C
. These
features contribute to limit the lateral displacement of the
microlattices, which experimentally showed a negligible effec
tive
Poisson’s ratio
ν
. Therefore, the effective Young’s modulus of
non-slender cubic lattices can be obtained by considering the
bending effects of a laterally constrained frame unit cell. Thro
ugh
this approximation, equivalent to a null effective Poisson’s ra
tio,
the unit cell effective Young’s modulus
E
can be written as
E
=
2
π
E

r
l

2

1
+
3

r
l

2

.
(4)
It should be noticed that the derived formulation (4) predict
s the
effective stiffness for non-slender lattices with negligible effec
tive
Poisson’s ratio more accurately than the classical expressio
n
(2). However, it can be further improved by considering the
manufacturing-induced imperfections as non-cubic unit cell
s,
orientation-dependent beam slenderness, and increased nod
al
volume, in order to readily provide insights on the influence of
each one of these features on the lattice behavior. Therefor
e, the
effective stiffness
E
predicted through Equation (4) is expected
to underestimate the experimental microlattice effective Youn
g’s
modulus as obtained from uniaxial compression tests, testify
ing
the inaccuracy of currently available analytical techniqu
es
and justifying the use of numerical tools for the mechanical
characterization of the manufactured lattices.
Figure 5
reports the stress-strain curves of the three sets of
fabricated carbon microlattices, with five specimens compres
sed
for each relative density considered. It is observed that all
samples
showed a brittle linear elastic response after an initial toe
region.
An increase of the relative density yields an higher maximum
stress and strain and a lower variability of the results. The
effective Young’s modulus
E
y
resulted
E
y
,
A
= 877.90
±
48.57
MPa for
A
,
E
y
,
B
= 1565.43
±
66.99 MPa for
B
, and
E
y
,
C
=
2464.81
±
129.53 MPa for
C
. Although the achieved stiffnesses
do not represent the highest specific stiffness (stiffness to densit
y
ratio) ever recorded, they exceed the majority of natural ma
terials
and approach the values obtained for nanolattices (
Zhang et al.,
2019
).
Figure 6A
shows the measured compressive stiffness as a
function of the lattice relative density
N
ρ
. The same figure reports
the microlattice stiffness obtained from finite element analy
ses
performed on representative octet unit cells with the geometri
cal
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6
July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
FIGURE 6 |
Effective Young’s modulus
E
of non-slender carbon microlattices reported as a function
of the octet unit cell relative density
N
ρ
.
(A)
Comparison between
the stiffness measured from experiments (green/dot markers
) and values predicted from finite element simulations (orang
e/diamond markers). Effective Young’s
modulus
E
y
(B)
and
E
z
(C)
scaling laws obtained from numerical simulations for relat
ive density that ranges between 10 and 50
%
.
TABLE 2 |
Geometrical parameters of the octet unit cell employed in th
e
numerical simulations.
Microlattice
w
h
r
1
r
2
r
N
r
j
̄
ρ
CAD
̄
ρ
t
[
μ
m] [
μ
m] [
μ
m] [
μ
m] [
μ
m] [
μ
m] [-]
[-]
01
240.42 240.42 12.50 10.00 25.00 17.00 0.10 0.10
02
240.42 240.42 14.00 11.00 28.00 18.70 0.12 0.13
03
240.42 240.42 16.50 13.00 33.00 22.10 0.17 0.17
04
240.42 240.42 20.00 16.00 40.00 27.20 0.24 0.24
05
240.42 240.42 22.50 18.00 45.00 30.60 0.30 0.30
06
240.42 240.42 27.50 22.00 55.00 37.40 0.43 0.44
07
240.42 240.42 30.00 24.00 60.00 40.80 0.50 0.51
The relative density
N
ρ
CAD
obtained from the modeled geometry is reported together with
the density
N
ρ
t
calculated through the analytical expression (1).
parameters measured from SEM (
Table 1
). The computational
effective Young’s modulus for the three investigated geometr
y
resulted
E
y
,
A
=
1334.74 MPa,
E
y
,
B
=
1876.18 MPa, and
E
y
,
C
=
2849.78 MPa. The predicted values are in fair agreement
with experimental measurements, with 33.5, 16.6, and 13.5%
relative errors for the three sets of microlattices
A
,
B
, and
C
. The
discrepancy between numerical predictions and measurements
decreases with an increase of the sample relative density
N
ρ
. This
is primarily attributed to the manufacturing induced corrug
ation
that were not modeled in the computational analysis and were
less pronounced as the beam diameter and relative density
increase. In particular, the average corrugation amplitude wa
s
measured as 18.4% of the off-plane beam diameter for the
microlattice
A
, while it was limited to 11.2 and 8.7% for
lattices
B
and
C
, respectively. The influence of each one of
the manufacturing-induced imperfections on the lattice effec
tive
stiffness was not considered because these features are inher
ently
related, however, it is believed that the non-cubic unit cel
l had the
most pronounced effect for the microlattice
A
, while the nodal
geometry had a significant role for microlattices
B
and
C
, where
the deviation from cubic unit cell is small.
In order to investigate the effective stiffness scaling law, we
performed other numerical simulations on non-slender octet
unit cells with relative density
N
ρ
that varied between 0.10 and
0.50 (
Table 2
). Similarly to the manufactured microlattices, we
modeled cubic unit cells with node radius
r
N
=
2
r
1
, junction
radius
r
j
=
1.7
r
2
, and that contain struts with different in-
plane and out-of-plane diameters, with
r
2
=
0.8
r
1
. The obtained
stiffness in the y and z directions are reported as a function of t
he
relative density in
Figures 6B,C
. The effective Young’s modulus
of the considered non-slender octet lattices scales as
E
y
∝ N
ρ
1.43
and
E
z
∝ N
ρ
1.48
for the y and z directions respectively, whereas
classical scaling law for stretching dominated octet-trus
s lattices
predicts a scaling exponent equal to one (
Fleck et al., 2010
).
Deviations from classical theory were also observed for differ
ent
rigid architectures when the strut slenderness ratio
r
/
l
was larger
than 0.07, as a result of nodal contribution on the effective la
ttice
stiffness (
Portela et al., 2018
). The microlattices manufactured
and analyzed in this work present beam slenderness that ranges
between 0.065 and 0.17, pronounced nodal volumes and edge
effects caused by a limited number of unit cells along the height
and width (
Christodoulou, 2017
). Therefore a scaling exponent
m
E
>
1 testifies that the lattices experience significant bending
during compression.
3.4. Strength
The compressive strength of an octet-truss lattice material
was
determined considering the two possible failure mechanisms
of
elastic buckling or plastic yielding (
Deshpande et al., 2001
). The
analysis of an octet-truss unit cell constituted of cylindrica
l beams
reveals that the strut axial compressive stress
σ
a
is related to the
vertical stress applied to the unit cell
σ
v
through the relation
σ
v
=
2
2
π

r
l

2
σ
a
.
(5)
If the lattice presents a low relative density, elastic buckli
ng will
be responsible of the octet-truss structural collapse when th
e axial
stress
σ
a
reaches the Euler critical value
σ
b
=
n
2
π
2
E
4

r
l

2
,
(6)
where
n
depends on the strut boundary conditions, with
n
=
1
for a pin-joined strut and
n
=
2 for a doubly clamped beam.
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7
July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
FIGURE 7 |
Compressive strength
σ
of non-slender carbon microlattices reported as a function
of the octet unit cell relative density
N
ρ
.
(A)
Comparison between the
strength measured from experiments (green/dot markers) and
predicted failure
σ
,
f
y
(orange/diamond markers) and buckling
σ
,
b
y
(gray/square markers) strengths
from finite element simulations. Fracture and buckling stren
gths scaling laws along the y-direction
(B)
and z-direction
(C)
obtained from numerical simulations for a
relative density that ranges between 10 and 50
%
.
When the strut slenderness increases, buckling is preceded b
y
strut failure (yielding) as the stress reaches the material
failure
(yield) strength
σ
f
. Therefore, the compressive strength for an
octet-truss material can be expressed as
Deshpande et al. (2001)
σ
=
min
(
2
π
3
n
2
E
2

r
l

4
;
2
2
π

r
l

2
σ
f
)
.
(7)
The extension of the expression (7) to non-slender cubic frame
lattices was obtained through the analysis of an octet unit c
ell
constrained to deform laterally. Considering bending effect
s, the
relationship (5) between the strut axial stress
σ
a
and the vertical
stress
σ
v
applied to the unit cell becomes
σ
v
=
2
2
π

r
l

2

1
+
3

r
l

2

σ
a
,
(8)
while the dependence of
σ
v
on the maximum normal stress
σ
m
at
the edges of the strut was obtained considering a combinatio
n of
the bending moment and axial force effects
σ
z
=
2
2
π

r
l

2
1
+
3

r
l

2
1
+
6

r
l

2
σ
m
.
(9)
The octet-frame buckling strength was calculated by insert
ing
Equation (6) into Equation (8), while the lattice failure (y
ield)
strength is achieved when
σ
m
=
σ
f
. Hence, the compressive
strength for a cubic octet-frame lattice material writes as
σ
=
min
2
π
3
n
2
E
2

r
l

4

1
+
3

r
l

2

;
2
2
π

r
l

2
1
+
3

r
l

2
1
+
6

r
l

2
σ
f
.
(10)
As observed for the stiffness calculation, simplified analytica
l
expressions fail to capture the complex mechanical response
of non-slender architectures as the orientation-dependent
slenderness, non cubic unit cell and the nodal contribution
are not accounted for
Portela et al. (2018)
. Even though
Equation (10) provides an improved strength estimation of
octet lattices with negligible effective Poisson’s ratio com
pared
to Equation (7), the effects of the manufacturing-induced
imperfections are not fully reflected, and they were considered
through numerical analyses. Therefore, more comprehensive
analytical formulations need to be developed in order to asses
s
the mechanics of imperfect lattices. The experimental streng
th of
the octet microlattices
σ
y
increased with the relative density as
shown in
Figure 7A
, where
σ
y
,
A
= 24.80
±
4.26 MPa,
σ
y
,
B
= 65.68
±
4.45 MPa, and
σ
y
,
C
= 122.74
±
12.28 MPa refer to the three sets
of lattices. It should be noticed that the attained specific st
rengths
(strength to density ratio) exceed most of natural material
s and
approach the values obtained by carbon nanolattices (
Bauer
et al., 2016; Zhang et al., 2019
), which represent the strongest
architected materials ever realized. The achievement of st
rengths
comparable to those of nanolattices testifies that through th
e
current manufacturing technique is possible to realize larg
e-
scale ultra-strong materials, overcoming the current scal
ability
disadvantages of two-photon lithography nanofabrication.
Considering the limitations of the analytical formulation
s, the
complex microlattice stress distribution was assessed throug
h
numerical simulations, from which the maximum equivalent
stress was obtained and was compared with the strength of
the constituent material. The bulk strength of pyrolytic carb
on
has been shown to be dependent on the specimen length-
scale (
Bullock and Kaae, 1979
), with an increase as specimens
dimensions decrease. This behavior depends on the probabilit
y of
finding large critical flaws within the materials, which decr
eases
when the sample dimensions reduce. In brittle materials, the
failure strength
σ
f
is inversely proportional to the square root of
the pre-existing flaw size
σ
f
=
YK
Ic
π
a
c
,
(11)
where
K
Ic
=
0.91 MPa
m is the fracture toughness (
Zhao et al.,
1985; Brezny and Green, 1990
),
Y
=
1 is a non-dimensional
geometrical parameter for a semi-elliptical surface flaw load
ed
in tension or bending (
Bauer et al., 2016
), and
a
c
represents the
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8
July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
critical flaw size. The strength of glassy carbon fibers of 5

m
diameter made by carbonization of organic polymer reached 2
GPa, corresponding to a flaw size of 66 nm (
Kawamura and
Jenkins, 1970
). Since the strut diameters of the manufactured
microlattices range between 20 and 50

m, we assume that a
material strength
σ
f
=
1.5 GPa can be reached, equivalent to a
flaw size of 0.12

m. From the results of numerical simulations,
it is possible to obtain the octet compressive stress along the
y-direction that produces a critical equivalent stress with
in the
unit cell. The numerical lattice failure strengths were fou
nd
σ
,
f
y
,
A
=
39.31 MPa,
σ
,
f
y
,
B
=
73.17 MPa, and
σ
,
f
y
,
C
=
102.83 MPa
for the three sets of carbon microlattices and they are report
ed
in
Figure 7A
together with the experimental data. Considering
the uncertainties in the determination of the material fail
ure
strength
σ
f
and the lack of beam corrugation modeling in the
finite element analysis, these results are in fair agreement
with
the measured values, with a maximum relative error of 36.9%.
Furthermore, it should be noted that the numerical simulati
ons
were performed on a unit cell level, thus neglecting the edge
effects that a finite lattice experience.
Since the struts of the studied microlattices are non-slende
r,
buckling is unlikely to cause failure before the onset of fra
cture.
Numerical buckling analyses on the three octet unit cells
confirmed that the fracture strength was achieved before ela
stic
buckling of the out-of-plane beams. The compressive stresses
σ
,
b
y
that trigger elastic instabilities were numerically found
, with
σ
,
b
y
,
A
=
100.88 MPa,
σ
,
b
y
,
B
=
196.07 MPa and
σ
,
b
y
,
C
=
389.57 MPa.
It can be seen that the numerical buckling strength
σ
,
b
y
,
j
is always
greater than the fracture strength
σ
,
f
y
,
j
for the three microlattices
j
=
A
,
B
,
C
, thus excluding strut instability mechanisms.
The computational analyses on non-slender octet unit cells
performed to investigate the effective stiffness scaling laws we
re
employed to calculate the fracture and buckling strengths an
d
their scaling laws.
Figures 7B,C
shows that for the y loading
direction the fracture strength scales as
σ
,
f
y
∝ N
ρ
0.97
, and the
buckling strength scales as
σ
,
b
y
∝ N
ρ
2.49
, while in the z-direction
σ
,
f
z
∝ N
ρ
1.08
and
σ
,
b
z
∝ N
ρ
2.59
. As already noticed for the
effective stiffness behavior, the fracture and buckling streng
ths
scaling laws deviate from the classical analysis of octet-tr
uss
architectures, where the scaling exponents are
m
f
=
1 for
fracture and
m
b
=
2 for buckling (
Deshpande et al., 2001
).
The different scaling exponents obtained from the numerical
FIGURE 8 |
SEM fractography of the three sets of manufactures microlat
tices
A
(A)
,
B
(B)
, and
C
(C)
after catastrophic mechanical compression tests. Progres
sive
magnification images highlight the brittle fracture surface
s.
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9
July 2019 | Volume 6 | Article 169
Kudo et al.
Mechanics of Octet Carbon Microlattices
simulations depend on the non-slender architecture and the
pronounced nodal volume, and will help the evaluation of the
mechanical properties prediction of any lattice with the same
ratios between geometrical parameters. Lastly, it should be n
oted
that the results of computational analysis on non-slender cub
ic
octet unit cells with
r
2
=
0.8
r
1
,
r
N
=
2
r
2
, and
r
j
=
1.7
r
2
predict
that failure due to elastic instability occurs when the rela
tive
density
N
ρ
is lower than 0.12, corresponding to an out-of-plane
strut slenderness
r
2
/
l
2
=
0.065.
Similarly to the results reported by
Portela et al. (2018)
,
from finite element analyses it was also observed that strain
energy concentrates in the nodes and the stress concentrati
on
migrates toward the nodes as the relative density increases
and beam slenderness decreases.
Figure 8
, left shows the
SEM images of the three sets of manufactured microlattices
after catastrophic compression tests, highlighting the fail
ed
components and the brittle fracture surfaces. As numericall
y
predicted, the experimental fractographic examination proved
that in the lighter microlattice
A
the nodes remained undamaged
and only struts fractured (
Figure 8A
), while as density increases
in lattices
B
and
C
node failures were observed (
Figures 8B,C
).
Furthermore, it was commonly noticed among the three sets
of samples that fracture within the struts did not follow the
stereolithography induced corrugations.
4. CONCLUSION
In the present work, we have investigated the compressive
behavior of stiff and strong non-slender octet carbon
microlattices obtained by pyrolyzing 3D-printed polymer
architectures fabricated through stereolithography. We ha
ve
shown that additive manufacturing can lead to imperfect
lattices with significant nodal volumes and strut slenderne
ss
dependent on the beam inclination with respect to the 3D-
printing direction. The effects of these features on the lattic
e
structural response have been numerically and experimentall
y
studied considering carbon microlattices with a relative
density higher than 10%. The manufactured microarchitectur
es
have achieved superior relative stiffness and strength that
approach those of carbon nanolattices. Therefore, we have
demonstrated that the employed manufacturing technique
can lead a fast realization of large scale strong materials,
currently not achievable through nanofabrication. We have
extended the classical analytical tools based on beam theor
y to
include non-slender cubic architectures that show a neglig
ible
effective Poisson’s ratio. These formulations can be adopted
for octet lattices constituted of any materials with neglig
ible
nonlinearities and viscoelastic effects, independently of th
e
manufacturing process. Although more accurate than classica
l
octet-truss formulations when bending effects are pronounced
,
the developed analytical construction remains inadequate for
the prediction of the effective stiffness and strength of the
manufactured non-slender lattices, thus identifying the ne
eds
for future analytical investigations of structured solids w
ith
orientation dependent geometrical features and pronounced
nodal connections, in order to develop optimized materials a
nd
assess the influence of these parameters on the lattice mechan
ical
response. To overcome the identified limitations of the analy
tical
tools, we have developed computational models that reproduce
the unit cell geometry and capture the mechanical properties
of the tested architectures. The numerical investigation h
as
revealed the mechanics of low slenderness microlattices thr
ough
the prediction of compressive stiffness, failure and buckling
strengths, and their scaling laws, which provide a reliable
method to estimate the mechanical properties of imperfectly
manufactured octet lattices featuring the investigated to
pology.
The presented manufacturing process and numerical methods
represent tools to enhance the design of carbon architected
materials toward strong and lightweight solids.
AUTHOR CONTRIBUTIONS
AK, DM, and FB designed the research, discussed the results,
and wrote the paper. DM and FB carried out the analytical and
numerical analyses. AK manufactured the specimens. AK and
YW performed the experiments. All authors reviewed the paper
and gave final approval for publication.
FUNDING
Vannevar-Bush Faculty Fellowship of the US Department
of Defense. The Resnick Sustainability Institute Postdoctoral
Fellowship of Caltech. Financial support from National Group
of
Mathematical Physics (GNFM-INdAM).
ACKNOWLEDGMENTS
AK gratefully acknowledges the financial support from the
Resnick Sustainability Institute at the California Institut
e of
Technology and from Prof. Julia R. Greer (Caltech) through
the Vannevar-Bush Faculty Fellowship of the US Department of
Defense. The authors thank Prof. Julia R. Greer’s research gro
up
(Caltech) and Prof. Sergio Pellegrino (Caltech) for their su
pport
in conducting experiments in their laboratories.
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July 2019 | Volume 6 | Article 169