of 7
1
Supporting Information for:
1
Microstructure vs. Flaw:
2
Mechanisms of Failure and Strength in Nanostructures
3
X. Wendy Gu
,
Zhaoxuan Wu
̊
, Yong
-
Wei Zhang
̊
, David J. Srolovitz
+
, Julia R. Greer
*
4
Division of Chemistry and Chemical Engineering
&
5
Division of Engineering and Applied Science, California Institute of Technology,
6
1200 E. California Blvd., Pasadena, CA 91125, United States
7
8
̊
Institute of High Performance Computing, 1 Fusionopolis Way,
9
#16
-
16 Connexis, Singapore 138632
10
11
+
Departments of
Materials Science and Engineering & Mechanical Engineering and Applied
12
Mechanics, University of Pennsylvania, Philadelphia, PA 19104, United States
13
14
Modeling of Notch Geometry
15
Electroplated nc
-
Pt nanocylinders
were produced with surface
notches
of several
sizes
16
and shapes.
Circumferential length,
b
, and notch height,
h
, are used to describe the notch
17
geometry
and can be measured via
scanning electron microscopy (SEM) imaging (Fig. S1).
18
These dimensions
can be related to
notch
depth,
a
, and notch tip radius
,
r
.
a
and
r
are important
19
quantities
for
determining
stress concentrations
, but
are not easily (or accurately)
determined
20
through
SEM
observations
because of
limits on
imaging
resolution
and
imaging
ang
le
.
21
2
Observations from SEM
allowed t
he ex
perimental
notch geometries to be
classified into two
1
types
(both with rounded tips)
: (A
)
a
straight
notch that extends
across the width of the cylinder
2
(
Fig. S1c)
and (B)
a partial circumferential notch
(
Fig. S1b)
.
These idealized notch geometries
3
were used in molec
ular dynamics (MD) simulation and finite element modeling (FEM)
in order
4
to compare computer computations with experiment.
5
6
7
Supplementary
Figure S1.
(a)
Schematic of a cylinder with a rounded notch in the sidewall
8
similar
to those
in electroplated nc
-
Pt
samples
.
The
experimental
notches can be classified as (b)
9
straight notch
with a rounded tip
that crosses the width of the cylinder (geometry A) and (c) a
10
partial circumferential notch
with a rounded tip (geometry B).
Notches of both geometries are
11
characterized using circumferential length
b
,
crack height
h
, and notch radius
r
, where
r
is
12
assumed to be equal to
h
/2.
Notch depth,
a
, depends on
b
in geometry A, and is
~
5 nm
in
13
geometry B
based on SEM
observations
.
14
15
3
Finite
E
lement
M
odeling of
N
otched
C
ylinders
1
Finite element modeling (FEM) was performed using ABAQUS/CAE software package
2
in order to calculate the stress concentration at the notch on a
cylinder. The sample
was modeled
3
as a
linear elastic, isotropic
, ho
mogenous
three
-
dimensional
cylinder with the materials
4
properties of platinum (E = 172 GPa,
υ
= 0.4)
(Fig. S3
)
.
5
6
Supplementary
Figure S2
.
Simulated finite e
lement nanocylinder
with
a notch on the side
wall.
7
Eleven
of
the twelve
experimental samples were modeled using ABAQUS, with notch
8
dimensions
consistent
with SEM
measurements (notch dimens
ions were not available for the
9
remaining
sample).
FEM n
otches were
modeled
as geometry A
, straight notch,
or geometry B
,
10
partia
l circumfere
ntial notch (Fig. S3
)
.
11
12
Supplementary Figure S3
.
Stress contours
at the notch
for
FEM simulated nc
-
Pt nanocylinders
13
with
A)
a
p
artial circumferential notch
(geometry A) and (B) a straight notch (geometry B)
.
14
4
The FEM sample
was statically
loaded
, with a
surface traction applied to one cylindrical
1
base of th
e cylinder such that the sample
was pulled in tension
in the z
-
direction. The sample
2
was
fixed
in the
plane of this cylinder
face
(to model the
constraint imposed by the tension
grip).
3
The other cylind
rical base was
fixed in all three dimensions
to model a nc
-
Pt sample
fixed
to the
4
substrate. A mesh with
192023 to
463638
tetragonal elements was generated, with the majority
5
of elements concentrated at the notch.
The mesh was refined
such that the stress
values
6
converged
to
within
7%
.
7
The von Mises stress was calculated at the notch because of the multiaxial stress state
8
present at this
location, and used to find the stress concentration at the
n
otch.
The stress
9
concentration at the notch
for each sample
was plotted against ultimate tensile strength
(UTS)
in
10
Figure S4
.
Figure S4
shows that samples that broke at the notch had higher stress concentrations
11
at the notch than samples that broke away from the notch for all but one
sample, and the
12
distribution i
n UTS for these two types of failure.
13
14
5
Supplementary Figure S4
.
E
xperimental
ly
measured
ultimate tensile stress (UTS) versus stress
1
concentration factor for each flaw geometry.
F
racture
occurs away from the flaw only at
2
relatively small stress
concentrations.
3
Fracture Surface Morphology
4
After performing tensile tests to fracture, one portion of the tested nanocylinder remains
5
attached to the substrate. The fracture surface was examined by imaging the fractured
6
nanocylinder with scanning electro
n microscopy. Figure S5 shows a typical fracture surface; the
7
scale of the features on the fracture surface are similar to that of the grains.
8
9
Supplementary Figure S5
. SEM image of
a typical nanocylinder fracture surface
.
10
11
12
6
Crack Formation at Internal
Grain Boundary
1
2
Supplementary
F
igure S6
: (a
-
b) Cross
-
sectional view indicating the stress triaxiality
η
at an
3
applied tensile strain of 6% and (c) subsequent intergranular fracture at an internal grain
4
boundary in a Ni polycrystalline pillar (see Ref
.
(
1
)
for details)
.
This shows that internal grain
5
boundaries can open in regions of large triaxiality leading to the nucleation of cracks of lengths
6
approximately equal to the length of grain facets.
7
8
7
Molecular Dynamics Movies
1
Supplementary Movie S1
-
S5.
The
deformation, fracture process and stress
-
strain curves
of
2
notch
-
free nanocylinder (S1) and nanocylinders with different notch geometries (S2
-
S5) in this
3
study. In each movie, a slice at the middle section of the nanocylinder is shown to reveal the
4
internal
fracture process.
5
Supplementary Movie S6
-
S8.
The evolution of atomistic stress (
σ
yy
)
at the middle section of
6
three
nanocylinders with different notch geometries. These movies demonstrate dislocation and
7
grain boundary plasticity can effectively blunt not
ch roots and reduce stress concentrations.
8
in
-
situ
SEM Movies
9
Supplementary Movie S9
.
in
-
situ SEM video of tension test on nano
-
cylinder with external flaw
10
which breaks at the fl
aw.
11
Supplementary Movie S10.
in
-
situ SEM video of tension test on
nano
-
cylinder with external flaw
12
which breaks away from the flaw.
13
14
References
15
1.
Wu ZX, Zhang YW, Jhon MH, & Srolovitz DJ (2013)
Acta Mater.
In Press.
16
17
18