of 9
1
SCIENTIFIC
RepoRts
| 6:20570 | DOI: 10.1038/srep20570
www.nature.com/scientificreports
Insensitivity to Flaws Leads to
Damage Tolerance in Brittle
Architected Meta-Materials
L. C.
Montemayor
1
, W.
H. Wong
2
, Y.-W.
Zhang
2
& J.
R.
Greer
1
Cellular solids are instrumental in creating lightweight, strong, and damage-tolerant engineering
materials. By extending feature size down to the nanoscale, we simultaneously exploit the architecture
and material size effects to substantially enhance structural integrity of architected meta-materials.
We discovered that hollow-tube alumina nanolattices with 3D kagome geometry that contained
pre-fabricated flaws always failed at the same load as the pristine specimens when the ratio of notch
length
(a)
to sample width
(w)
is no greater than 1/3, with no correlation between failure occurring at or
away from the notch. Samples with
(a/w)
>
0.3, and notch length-to-unit cell size ratios of
(a/l)
>
5.2,
failed at a lower peak loads because of the higher sample compliance when fewer unit cells span the
intact region. Finite element simulations show that the failure is governed by purely tensile loading for
(a/w)
<
0.3 for the same
(a/l)
; bending begins to play a significant role in failure as
(a/w)
increases. This
experimental and computational work demonstrates that the discrete-continuum duality of architected
structural meta-materials may give rise to their damage tolerance and insensitivity of failure to the
presence of flaws even when made entirely of intrinsically brittle materials.
Bulk ceramics are highly sensitive to flaws and fail catastrophically upon applied loads, most commonly at the
small internal flaws like cracks, voids, and inclusions
1–3
. Fiber-reinforced ceramic-matrix composites (CMCs)
have been developed to reduce their sensitivity to flaws while capitalizing on the high strength of ceramics
4–7
.
These CMCs utilize deformation of the matrix and/or fibers to delocalize strain near stress concentrators, such
as holes or notches, which leads to flaw insensitive behavior
4,5,7
. Insensitivity to notches has been reported for
silicon carbide/calcium aluminosilicate CMCs for ratios of notch to sample size of 0.2
<
(a
0
/b)
<
0.8, where
a
0
is
the notch size and
2b
is the sample width
5,7
.
It has also been postulated that reducing sample dimensions of brittle materials can give rise to flaw insensi-
tivity and to attain near-theoretical strength
8
. Gao
et al.
’s theoretical work demonstrated that a thin plate with a
penny shaped notch exhibits the fracture strength of a perfect crystal when the plate thickness falls below a critical
length scale, which is a function of surface energy, elastic modulus, and ideal material strength
8
. Nano-fracture
experiments and computations on brittle nanocrystalline platinum nanopillars, with diameters of 100
nm and a
grain size of 6
nm, revealed that their failure strength remained equivalent to the ultimate tensile strength even in
the presence of pre-fabricated flaws and that failure location was uncorrelated to the presence of the flaws
9
. This
emergence of flaw insensitivity in nano-structures was attributed to their failure at the “weakest-link,” be it at an
internal, microstructural stress concentration like a grain boundary triple junction or at an external flaw, with fail-
ure mechanism representing the intrinsic material strength. In the absence of a discrete material microstructure,
it has been reported that 75
nm-diameter metallic glass nanopillars containing external notches always failed at
those locations at lower peak loads than their un-notched counterparts
10
.
These examples demonstrate that failure tolerance of some materials to flaws cannot be solely attributed to
the length scale; it stems from the complex interplay between the internal microstructural energy landscape
within the material and the external sample dimensions and geometry
10
. Several studies have demonstrated that
incorporating architecture in material design enables proliferating lucrative material size effects that emerge at
the nano-scale onto macro-scale architected meta-materials; for example
smaller is stronger/weaker
for metals,
smaller is ductile
for brittle metallic glasses, and
smaller is tougher
for ceramics
11
. The periodic arrangements of
small-scale ordered cellular solids, such as nano- or meso-lattices, span length scales ranging from hundreds of
1
California Institute of Technology, Pasadena, CA, USA.
2
Institute of High Performance Computing, (A*STAR),
1 Fusionopolis Way, #16-16 Connexis, Singapore 138632. Correspondence and requests for materials should be
addressed to L.M. (email: lauren.c.montemayor@jpl.nasa.gov)
R
eceived: 04 September 2015
A
ccepted: 06 January 2016
P
ublished: 03 February 2016
OPEN
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2
SCIENTIFIC
RepoRts
| 6:20570 | DOI: 10.1038/srep20570
microns to tens of nanometers and facilitate the attainment of novel mechanical properties under compression,
like recoverability and enhanced specific strength compared to bulk, and these properties arise as a result of
structural and material size effects
12–14
. Existing cellular solids theories predict that mechanical behavior is deter
-
mined by the deformation mechanism of the lattice, which is either by bending or stretching and is a function
of the nodal connectivity, and the constituent material properties
15–17
. A bending dominated structure is pre-
dicted to have lower strength and stiffness when compared to a stretching-dominated structure
15–18
. In the case
of architected bulk materials, another aspect of microstructure arises in the dimensions of not only the grains of
the constituent material but also in the size of the unit cells. Fracture experiments on macro-scale cellular solids
have been explored in literature; tensile properties of nanolattices – with or without pre-fabricated defects - are
currently unknown
16,19–26
.
We explore tensile failure of 3-dimensional hollow alumina kagome nanolattices and demonstrate that they
exhibit flaw tolerance in terms of strength and failure location, which we attribute to the presence of a discrete
structure at the micron and sub-micron lengths scales within a continuum-like material.
Kagome Tension Sample Fabrication
We performed uniaxial tension experiments on hollow alumina (Al
2
O
3
) nanolattices with and without
through-thickness notches. Figure 1 shows the CAD design and SEM images of an as-fabricated dog-bone-shaped
hollow alumina kagome nanolattice thin plate embedded in an octet-truss lattice head; the kagome lattice had a
unit cell size of
l
=
3.85
±
0.16
μ
m, the octet-truss head had a unit cell size of
l
~ 4.5
μ
m, shown in Fig. 1(B,F). The
kagome lattice was created by tessellating pairs of stacked tetrahedra in three dimensions in a hexagonal tessella-
tion pattern, which gives rise to an A-B-C stacking pattern. The kagome unit cell shown in Fig. 1(C,G) contains
a single complete set of A-B-C stacked tetrahedra. The tension samples were fabricated on a silicon wafer using a
polymer scaffold created via two-photon lithography direct laser writing process and were subsequently coated
with 50
nm of Al
2
O
3
using atomic layer deposition (ALD) at 150
°C to make a composite Al
2
O
3
/polymer struc
-
ture. The octet-truss head had a width of
w
=
139.8
±
1.1
μ
m, a height of
h
=
19.2
±
0.2
μ
m, and a thickness of
t
=
24.5
±
0.3
μ
m, with the averages representing 5 samples. The Al
2
O
3
/polymer kagome test section initially had
a width of 12
unit cells, a height of 6
unit cells, and a thickness of 2
unit cells but the width was reduced at a later
step in the fabrication process to accommodate the experimental set-up. The samples had an octet-truss lattice
at both the top and bottom of the kagome test section, shown in Fig. 1(A,E); a kagome A-B-C unit cell layer was
embedded in octet-truss head and ~5
μ
m of the kagome test section was embedded in the bottom octet-truss lat-
tice at the interface between the sample and the silicon wafer. The kagome lattice was embedded in the octet-truss
Figure 1.
Representative notched nanolattice tensile specimens, designed in Solidworks (
A
) and as-fabricated
samples (
E
). The fabricated and designed unit cells, whose size denoted as
l
, are shown in (
C
,
G
), respectively.
The grip used to apply uniaxial tension to the samples is an octet-truss lattice, shown in (
B
,
F
). The notch is
shown in (
D
,
H
) and is denoted by the variable
a
while the sample width is denoted by
w
. Scale bar (
E
) denotes
25
μ
m.
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RepoRts
| 6:20570 | DOI: 10.1038/srep20570
lattices at the top and bottom of the test sections to avoid delamination at the interfaces between the kagome test
section and the octet-truss head/substrate under tensile loading.
Samples were shaped into dog-bone tensile geometries using a focused ion beam (FIB), at a current of
7 nA, to have a curved edge at the kagome/octet-truss interfaces that served to reduce stress concentrations
(Fig. 1(B,F)). The plate-like kagome Al
2
O
3
/polymer test section had a final width of
w
=
66.3
±
5.2
μ
m, a height
of
h
=
81.9
±
0.3
μ
m, and a thickness of
t
=
15.4
±
0.5
μ
m; the width and height averages were calculated using all
samples presented in this work (n
=
18) and the average thickness was determined by averaging the thickness of
10 representative samples after the deformation. The FIB was also used to pattern the notches (Fig. 1(D,H)). We
tested samples with notch lengths,
a,
ranging from 0 to 0
34.94
μ
m, and
(a/w)
varying from 0–0.54, where
w
is
the width of the test section; the unit cell size was kept constant for all samples and notch length-to-unit cell size
ratio,
(a/l)
, was varied from 0–9.1. Once the samples were shaped into their final geometry, the internal polymer
was removed using O
2
plasma (16–18
hours at 100
W) and the samples were visually inspected in the scanning
electron microscope (SEM) to non-destructively determine the amount of polymer remaining in the sample; the
amount of polymer removed was verified to be approximately constant across all samples using this technique
(see Supplementary Info). When the Al
2
O
3
kagome nanolattices were fully hollowed out, the octet-truss head
remained a composite, containing the Al
2
O
3
layer and the internal polymer. Additional details on the fabrication
process can be found in refs 12–14, 27 and 28.
We chose the 3D kagome geometry because of its high predicted fracture toughness in 2D, which stems from
elastic blunting near the crack tip
16,24
. The hollow kagome nanolattices in this work are designed to have a stiffness
of
E
=
45
MPa and a relative density of
ρ
=.
00
2
, calculated using a Solidworks model of the kagome unit cell,
shown in Fig. 1, and the measured sample dimensions from SEM images. We chose the denser Al
2
O
3
/polymer
octet-truss lattice as the sample head to stiffen this section relative to the hollow kagome test section
16,18
. The
composite Al
2
O
3
/polymer head has a 760x higher stiffness of
E
=
3.90
GPa and a relative density of
ρ
=.
03
8
(see
Supplementary Info for details). We find that this difference in stiffnesses was sufficient to perform the tensile
loading with minimized deformation of the octet-truss head.
In-Situ
Uniaxial Experiments and Simulations
The as-fabricated kagome dog-bone specimens were subjected to displacement rate-controlled uniaxial straining
in an
in-situ
nanomechanical instrument, InSEM (Nanomechanics, Inc.), at a quasi-static strain rate of
ε
=
−−
s
10
31
. Contact with the samples was made via a tension grip at the bottom faces of the octet-truss head on
either side of the kagome lattice; the tension grip was milled in the head of a 0.8
mm stainless steel screw using
electrical discharge machining (EDM), as shown in Fig. 1E. Load-displacement data, as well as real-time video of
the deformation, was captured during each uniaxial tension experiment. The displacement of the gauge section
was calculated using the observed length change,
Δ
l,
in the deformation video; uniaxial strain was defined as
ε
=
Δ
l/l
, with the original length
l
measured in SEM prior to the experiment. The stress at failure was defined as
σ
=
F/A
, where
A
is the overall cross-sectional area of the sample and
F
is the measured force at failure. The slope
of the unloading curve is not a reliable measure of the energy release-rate of the Al
2
O
3
kagome nanolattice since
the slope is an artifact of the InSEM controller.
Finite Element (FE) simulations of the as-designed notched and un-notched hollow kagome lattices were
performed to assess the ability of continuum-based models to predict deformation of architected meta-materials.
The samples in FE models were created from the SolidWorks-constructed geometries and accounted for the
interface between the octet-truss sample head and the InSEM grips. Three-dimensional 3-noded triangular shell
elements with reduced-integration were employed. The material properties of Al
2
O
3
for the FE analyses were
obtained from bulge experiments of equivalently deposited thin films of ALD Al
2
O
3
, with the modulus, E, rang
-
ing from 164–165
GPa and the ultimate tensile strength,
σ
UTS
, in the range of 1.57–2.56
GPa; for the simula-
tions in this work, the input modulus and UTS were taken to be 165
GPa and 1.57
GPa, respectively
29
. A linear
post-cracking stress-strain relationship was incorporated in the simulations to represent the brittle behavior of
Al
2
O
3
. Displacement-controlled boundary conditions were applied to the octet-truss sample head in locations
closely resembling the experimental setup. For numerical efficiency, explicit dynamics procedure was adopted
to model the quasi-static applied uniaxial tension loading. To ensure a quasi-static response, the energy bal-
ance of the modeling system was constantly monitored such that the kinetic energy of the system was negligible
compared to its internal energy and external work. Computations were performed within the finite strain set-
ting using the general-purpose finite element program ABAQUS/Explicit Version 6.13.2. To model and reflect
a quasi-static solution, the kinetic energy of the deforming kagome lattice was monitored and enforced to not
exceed 1% of its internal energy throughout the majority of the quasi-static analysis. The studied
(a/w)
ratios
were: 0, 0.11, 0.23, 0.35, 0.47 and 0.54. A constant wall thickness of 50
nm across all elements was assumed for all
simulations. The global stress and strain of the FE kagome test section was calculated using the same methodol-
ogy as in experiments.
Results
Figure 2 shows SEM images and representative load vs. strain-to-failure data for the as-fabricated and
notched geometries and reveals that all samples failed instantaneously and catastrophically, as expected for a
brittle ceramic. Three un-notched samples (
a/w
=
0) had a
F
peak
=
2.00
±
0.19
mN and a strain at failure of
ε
failure
=
0.006
±
0.001; the results for various notch sizes are tabulated in the Supplementary Information. Figure 3
shows peak-load-at-failure data for samples with notch-to-width ratios, (
a/w
), spanning from 0 to 0.54. Samples
with
(a/w)
between 0 and 0.32 had a relatively constant peak-load-at-failure of
~ 2 mN, which is equivalent to the
peak load for the un-notched samples. The peak load decreased by 32% as
(a/w)
increased from 0.32–0.54, likely
caused by the 4.2x higher compliance in the widest-notched samples when compared to un-notched samples, also
shown in Fig. 3. The experimentally obtained peak load remained nearly constant over the
(a/w)
range of 0–0.32;
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4
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RepoRts
| 6:20570 | DOI: 10.1038/srep20570
Figure 2.
Representative load-displacement curve for un-notched (
A
) and notched (
B
) kagome nanolattices
(4
μ
m unit cell, 50
nm Al2O3) in uniaxial tension. The dimensions of the notched and un-notched samples
shown are identical with the exception a notch, which spans 1/3 of the sample width and is denoted by a red
dashed red line. The red “X” denotes brittle, catastrophic failure and all data collected after this point is an
artifact of the testing instrument and not representative of the measured load on the sample Scale bar denotes
25
μ
m.
Figure 3.
Comparison of finite element and experimental data for 4
μ
m kagome lattices in uniaxial
tension.
Scale bar denotes 50
μ
m in all images. The error bars in the experimental data were calculated using the
standard deviation of the measured load and notch/sample dimensions; substantial errors were likely caused by
the variations in the compliance of the sample head.
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| 6:20570 | DOI: 10.1038/srep20570
the simulations show a monotonic reduction in the peak load with increasing
(a/w)
, decreasing by 59% as
(a/w)
widens from 0–0.35. The simulations also show a 1.5x increase in compliance as
(a/w)
increases from 0–0.54,
slightly lower than that observed experimentally.
The sample compliance was calculated as C
=
ε
/
σ
, where
σ
is the applied load divided by the full
cross-sectional area of the gauge section,
A
, for both experiments and simulations. The experimentally meas
-
ured compliance is the combined compliance of the sample/nanoindenter system and may not serve as a reliable
measure of the absolute compliance of the nanolattices; the calculated compliance serves to compare the relative
changes in sample compliance with increasing ratios of
(a/w)
.
Discussion
All samples in this work failed catastrophically for all
(a/w),
with failure always occurring along a plane of nodes
between the tetrahedral pairs forming the kagome lattice. FE simulations revealed that the highest local Von
Mises stresses occur at the nodes, along the “planes” where the tetrahedra connect; Fig. 4 shows these calcu-
lations for representative un-notched and notched (
a/w
=
0.3) samples and reveals that the nodes serve as the
weakest links when the nanolattice is tensed. This is not surprising, as it has been previously demonstrated that
the nodes in similarly-made Al
2
O
3
nanolattices were the weakest links and served as failure initiation locations
in compression
12,
14
. We discovered that the as-fabricated, un-notched samples have equivalent local stresses in
the nodes located in sample-interior and at the edge of the sample and that the notched samples have the highest
local stress concentrations at the notch roots, with minimal local stress at the sample edge immediately prior to
failure. Failure in each sample initiated at the node(s) that had a missing neighbor, whose detached side was not
constrained by the neighboring unit cells in the lattice. Failure in the notched samples always initiated at the notch
root where the adjacent unit cell had one constrained and one un-constrained boundary. All samples in this work
had one edge containing unit cells that were fully disconnected from their vertical neighbors while the other edge
had unit cells with minimal intact vertical connections along the sample edge; this is a result of the lattice unit
cell size and the sample width required to fit the tension grips. Failure in all un-notched samples initiated at the
edge with completely disconnected unit cells, which is similar to local geometry at the notch edge where the unit
cells have are disconnected from the rest of the lattice on one side. After incipient failure in a single edge node, the
crack propagates instantaneously and causes catastrophic failure of the entire structure; the applied force required
to fail the first node in uniaxial tension corresponds to the peak load at failure for a nanolattice.
While continuum-based classical mechanics theory predicts that the peak load will decrease at higher
(a/w)
;
the nanolattices here exhibit a nearly constant peak load of ~2
mN for
(a/w)
<
0.38, a value equivalent to that of
the un-notched material
1,2,30,31
. In monolithic ceramic materials with the same geometry, the stress concentration
at a notch is highest because the external notch is significantly larger than the size of the internal microstructural
Figure 4.
Finite element simulations show that the highest local Von Mises stress concentrations within the
nanolattice occur at the nodes for both the notched and un-notched specimens.
Red boxes denote the plane
where failure occurs in the samples (
B
,
E
) and red arrows denote the highest visible surface stress concentrations
immediately before failure (
A
,
D
).